Universität des Saarlandes | Universität des Saarlandes ......form a simple harmonic oscillator, a...

Quantum optomechanicsMarkus Aspelmeyer, Pierre Meystre, and Keith Schwab Citation: Phys. Today 65(7), 29 (2012); doi: 10.1063/PT.3.1640 View online: http://dx.doi.org/10.1063/PT.3.1640 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v65/i7 Published by the American Institute of Physics. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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www.physicstoday.org July 2012 Physics Today 29

Aided by optical cavities and superconducting circuits,researchers are coaxing ever-larger objects to wiggle, shake, and flex in ways that are distinctly quantum mechanical.

Markus Aspelmeyer, Pierre Meystre, and Keith Schwab

optomechanics

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Quantum

Give me a place to stand and with a lever I will move thewhole world. —Archimedes

Over two millennia ago, scholars fromantiquity had already come to under-stand the power of simple mechanicalelements. And from that understand-ing they formulated an enduring,

common-sense notion of the nature of reality, de-scribed thusly in Plato’s The Republic:

The same thing cannot ever act or beacted upon in two opposite ways, or betwo opposite things, at the same time,in respect of the same part of itself, andin relation to the same object.

Today researchers at the cutting edge of physics arestill exploiting simple mechanical elements as toolswith which to carefully probe our world. But unliketheir predecessors, they are preparing those ele-ments deeply in the quantum regime and, in theprocess, challenging ancient notions of reality. Iron-ically, today’s devices, though similar in many waysto those of antiquity, steer us to a completely differ-ent worldview—one in which an object, possiblyeven a macroscopic one, can indeed act in two waysat the same time.

Two key developments, born of two converg-ing perspectives on the physical world, have en-abled the advance. From the top-down perspective,nanoscience and the semiconductor industries havedeveloped advanced materials and processing tech-

niques, which in turn have given rise to ultrasensi-tive micromechanical and nanomechanical devices.Such devices can probe extremely tiny forces, oftenwith spatial resolution at atomic scales, as exempli-fied by the recent measurements of the Casimirforce (see the article by Steve Lamoreaux, PHYSICSTODAY, February 2007, page 40) and the mechanicaldetection and imaging of a single electron spin (seePHYSICS TODAY, October 2004, page 22). From the bottom-up perspective, quantum optics and atomicphysics have yielded an exquisite understanding ofthe mechanical aspects of light–matter interaction,including how quantum mechanics limits the ultimate sensitivity of measurements and howback- action—the influence a quantum measure-ment necessarily exerts on the object being meas-ured—can be harnessed to control quantum statesof mechanical systems.

Quantum optomechanics combines the twoperspectives: By pairing optical or microwave cavi-ties with mechanical resonators to form a cavityopto mechanical system, one acquires a means toachieve quantum control over mechanical motionor, conversely, mechanical control over optical ormicrowave fields. The laws of quantum physics canthen be made to reveal themselves in the motion of objects ranging in size from nanometers to

Markus Aspelmeyer is a professor of physics at the University of Vienna.Pierre Meystre is Regents Professor of Physics and Optical Sciences atthe University of Arizona in Tucson. Keith Schwab is a professor ofapplied physics at the California Institute of Technology in Pasadena.

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centi meters, from femtograms to kilograms. Cavityopto mechanical systems hold promise as a means toboth observe and control the quantum states ofmacroscopic objects and to measure feeble forcesand fields with a sensitivity, precision, and accuracyapproaching the quantum limit1 (see the article byKeith Schwab and Michael Roukes, PHYSICS TODAY,July 2005, page 36).

Early optomechanicsPut simply, a cavity optomechanical system is anoptical or microwave cavity that contains a mechan-ical element, a moving part that can support collec-tive oscillation modes whose quanta of excitation

are known as phonons. The system could be as sim-ple as an optical cavity in which one of the end mir-rors oscillates as if attached to a spring.

Among the first well-understood cavity opto-mechanical systems were the early gravitational -wave detectors developed in the late 1970s and early1980s, with major contributions by Vladimir Bragin-sky, Kip Thorne, Carlton Caves, William Unruh, andothers.2 Such detectors are essentially giant inter -ferometers, with each arm being a kilometers-longoptical cavity bounded by mirrors several kilo-grams in mass (see figure 1). In theory, a ripple inthe local curvature of spacetime due to a passinggravitational wave should alter each cavity’s opticalpath length, modulate its resonance frequency, and,in turn, alter the optical transmission to a photo -detector. The Laser Interferometer Gravitational -Wave Observatory, currently the gold standard ofgravitational-wave detectors, can achieve dis -placement sensitivities as high as 10−19 m Hz−1/2. Inother words, it can detect a displacement of about1/1000 of a proton radius based on a one-secondmeasurement.

A related approach to detecting gravitationalwaves calls for using a massive, multiton cylinderas a gravitational -wave antenna. In theory, the cylin-der should undergo bending oscillations in the pres-ence of a passing gravitational wave. Provided thecylinder is integrated into a high-quality supercon-ducting microwave cavity, that bending should de-tectably modulate the cavity’s resonance frequency.Although the interferometer and bar- antenna ap-proaches to gravitational -wave detection deployvery different technologies, both rely on the under-lying concept that mechanical motion can be har-nessed to modulate an electromagnetic resonance.

Thirty years after the first deep studies of thelimits of gravitational wave detectors, it’s evident to us that Braginsky, Caves, and their contempo-raries had two very exciting things to say: First, gravitational-wave astronomy might be possible,and second, so might the measurement and manip-ulation of macroscopic objects at their quantum lim-its.3 The second message has motivated an increas-ing number of mostly young researchers trained inareas as diverse as solid-state physics, quantum in-formation, and computation to look for and exploitthe quantum behavior of large mechanical objects intabletop experiments.

Getting to zeroQuantum effects in any system are most pro-nounced when the influence of thermal fluctuationscan be ignored. So, ideally, a mechanical quantumexperiment would start with the mechanical ele-ment in its quantum ground state of motion, inwhich all thermal quanta have been removed. Inpractice, however, one settles for cooling the ele-ment such that for a given mechanical mode thetime-averaged number of thermal phonons, the so-called occupation number N, is less than one. Putanother way, the mean thermal energy kBT shouldbe less than the quantum of mechanical energy ħωm,so that N ≈ kBT/ħωm < 1. Here kB is Boltzmann’s con-stant, ħ is the reduced Planck’s constant, T is the

30 July 2012 Physics Today www.physicstoday.org

Quantum optomechanics

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Figure 1. Chasing waves. (a) TheLaser Interferometer Gravitational -Wave Observatory in Livingston,Louisiana, and similar gravitational -wave detectors were among thefirst cavity optomechanical systems.(b) They typically consist of massivemirrors suspended to form a pair ofoptical cavities, each some kilo -meters long. The cavities make upthe arms of a Michelson inter -ferometer and together can detectchanges in distance as small as10−21 relative to the cavity length.(c) Mirrors used in the gravitational-wave detector GEO600, locatednear Sarstedt, Germany.

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temperature, and ωm is the frequency of the vibra-tional mode of the mechanical element.

Removing thermal phonons from the mechan-ical element is a key experimental challenge. Inter-estingly, the ideas and methods for doing so weretheoretically developed as early as the 1960s. Themain idea is to exploit the intracavity radiation pres-sure, the force due to momentum transfer associ-ated with photon scattering. In particular, Bragin-sky realized that the finite time delay between achange in position of the mechanical element andthe response of the intracavity field allows the radi-ation field to extract work from or perform work onthe mechanical system.

The process is best illustrated for the case of abasic Fabry–Perot resonator (see figure 2). Whenboth of the resonator’s mirrors are held fixed, theoptical transmission is sharply peaked near the cav-ity resonance frequencies ωp = pπc/L, where L is themirror separation, p is a positive integer, and c is thespeed of light. The resonances result from the con-structive interferences between the partial wavespropagating back and forth inside the cavity. Thehigher the quality of the mirror, the more roundtripslight takes before exiting the cavity, and the sharperare the resonance peaks.

If one end mirror is mounted on a spring toform a simple harmonic oscillator, a pump laser offrequency ωL will be modulated by the mechanicalfrequency and form sidebands with frequenciesωL ± ωm. From a quantum mechanical perspective,the process is analogous to the generation of Stokesand anti-Stokes sidebands in Raman scattering: Theupper sideband is a result of pump-laser photonsacquiring energy by annihilating thermal phononsin the mechanical element; the lower sideband re-sults from photons depositing phonons and shed-ding energy. The first process occurs at a rate pro-portional to the occupation factor N of themechanical mode of interest; the second, at a rateproportional to N + 1.

By carefully detuning the frequency of thepump field relative to a specific cavity resonance ωc ,one can resonantly enhance one of the processes. Inparticular, red-detuning from the cavity resonanceenhances the upper sideband and promotes extrac-tion of energy from the mechanical element. As longas the up-converted photons leave the cavity suffi-ciently fast, carrying with them their newly ac-quired energy, the process can cool the motion of themechanical element to well below the temperatureof its surroundings. Although the quantum noise ofthe optical source imposes a fundamental coolinglimit, it is nonetheless theoretically possible to coolthe mechanical mode arbitrarily close to the quan-tum ground state, N = 0. Furthermore, the coherentinteraction between photons and phonons allowsmanipulations in the quantum regime, as pointedout early on by one of us (Meystre), Peter Knight,Paolo Tombesi, and Claude Fabre.

The technique, a form of sideband cooling, wasfirst demonstrated in experiments by Braginsky andby David Blair in the microwave regime as a way toreduce noise in gravitational -wave antennas.4 Since2004, several laboratories around the world have

used the method to cool nano- and micromechanicallevers, in both the optical and microwave domains.Today, high-quality optomechanical devices pro-duce couplings strong enough to cool low-masslevers—ranging from a few picograms to hundredsof nanograms—to their ground state of motion.

The methods used in cavity opto mechanics arein many ways analogous to the conventional laser-cooling techniques developed for quantum informa-tion processing with trapped ions (see the article byIgnacio Cirac and Peter Zoller, PHYSICS TODAY, March2004, page 38). There, the collective normal-mode os-cillations of a string of ions modify the response ofthe pump laser that drives their internal state. The re-sulting optomechanical coupling between the ions’motional and internal degrees of freedom allows one


Figure 2. Quantum optomechanics: the basics. (a) If both end mirrorsof a Fabry–Perot cavity are fixed in place, pump-laser photons havingfrequency ωL tuned to a cavity resonance arrive at a detector with nofrequency modulation. (The total transmission to the detector is indi-cated in the plot at right by the solid line; the cavity transmission spec-trum is indicated by the dashed line.) (b) However, if one mirror is al-lowed to oscillate harmonically, pump photons are modulated by theoscillation frequency ωm: A pump beam tuned to a cavity resonance willyield sidebands of equal amplitude at frequencies ωL ± ωm. Each photonin the upper sideband acquires energy by extracting a phonon from theoscillator, and each photon in the lower sideband sheds energy by de-positing a phonon. (c) By red-detuning the pump laser, one can en-hance the upper sideband and thereby cool the oscillating mirror. (d) Byblue-detuning the pump laser, one enhances the lower sideband andamplifies the mirror oscillations. (Figure prepared by Jonas Schmöle.)


to cool the ions or prepare other quantum states ofinterest.5 However, cavity optomechanics differs inan important and attractive aspect: Whereas conven-tional laser cooling relies on the fixed internal reso-nances of materials to enhance light– matter interac-tions, cavity opto mechanical cooling allows one toengineer the resonance-enhancing structure. Thatstructure could be an optical cavity with a series ofnarrow resonances or a microwave cavity such as asuperconducting LC circuit.

A tale of few phononsUntil recently, the pioneering developments in opto -mechanical coupling went largely unnoticed outsideof the experimental gravitation and quantum opticscommunities. Michael Roukes, who realized nearlytwo decades ago that high-frequency nanoscale me-chanical devices could be chilled to the quantumregime, is a notable exception. Advances in materialsscience and nanofabrication—particularly the rise ofnano- and microelectromechanical systems and opti-

cal microcavities—have sinceopened the possibility of cou-pling quantum optical modeswith mechanical devices in table-top experiments.6 The originalideas of optomechanical couplingwere extended in many new di-rections and realized in widelyvaried optomechanical systems(see figure 3). In the past year,those efforts have culminated inthe cooling of at least three differ-ent micro mechanical systems towithin a fraction of a phonon oftheir ground state of vibrationalmotion. (Here and below, unlessotherwise specified, mechanicalcooling refers to cooling of thecenter-of-mass motion.)

In a NIST experiment led byJohn Teufel and Ray Simmonds,7

the mechanical resonator was acircular aluminum membrane,15 μm across and 100 nm thick,that underwent drum-like vibra-tions with a resonance frequencyof 10 MHz (see figure 3e). Themembrane was tightly coupledto a superconducting microwavecavity and chilled in a cryostat to20 mK, at which the phonon occupation N was about 40. Sideband cooling was then usedto cool the membrane to N ≈ 0.3.

At Caltech, Oskar Painterand colleagues were similarlysuccessful using a 15-μm-long,600-nm-wide, and 100-nm-thicksilicon beam as the opto -mechanical system (see fig-ure 3b).8 Clamped at both endsto a silicon wafer, the suspendedbeam acts simultaneously as amechanical resonator and an

optical cavity. The mechanical mode of interest wasa breathing mode, a periodic widening and nar-rowing that is most pronounced near the beam’smidpoint and has a remarkably high quality factorof 105. (On average, a phonon survives 105 oscilla-tions before being lost to the environment.) Andperiodic perforations patterned into the beam cre-ate a photonic crystal cavity that confines light tothe same region around the beam’s midpoint.

The co- localization of light and vibrational mo-tion in such a small volume facilitates large opto-mechanical coupling. Thus, after cryogenicallychilling the structure to 20 K, at which N ≈ 100, theresearchers could use sideband cooling to removethe remaining phonons and cool the beam toN ≈ 0.8. At that point, the group was able to observeanother genuine quantum feature: Near the groundstate, a mechanical resonator is significantly morelikely to absorb phonons than to emit them, andthat asymmetry reveals itself experimentally as apreferential sideband scattering of blue-detuned



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Figure 3. Cavity optomechanical devices range from nanometer-sized structures of as little as 107 atoms and 10−20 kg to micromechanical structures of 1014 atoms and 10−11 kg tomacroscopic, centimeter-sized mirrors comprising more than 1020 atoms and weighing sev-eral kilograms. They include (a) gases of ultracold atoms, (d) micro spheres, and (g) micro -scale membranes, all of which have mechanical resonances that can couple with the lightinside an optical cavity; (b, c) flexible, nanoscale waveguides that have both optical and mechanical resonances; (e) superconducting membranes that exhibit drum-like vibrationsand can be integrated into microwave cavities; (f) microtoroidal waveguides having bothoptical and mechanical resonances; and mechanically compliant mirrors, which can rangefrom the microscopic (h) to the macroscopic (i, j) and which introduce mechanical degreesof freedom to an optical cavity when incorporated as an end mirror. (Figure prepared byJonas Schmöle. Images courtesy of (a) Ferdinand Brennecke, ETH Zürich; (d) the ViennaCenter for Quantum Science and Technology; (i) Christopher Wipf; and (j) LIGO Laboratory.Other images adapted from (b) ref. 8, J. Chan et al.; (c) M. Li et al., Nature 456, 80, 2008; (e) ref. 7; (f ) ref. 10, E. Verhagen et al.; (g) J. D. Thompson et al., Nature 452, 72, 2008; and (h) G. D. Cole et al., Nat. Commun. 2, 231, 2011.)


light over red- detuned light. (See figure 4.) The next challenge is to control the quantum

state of the mechanical resonator. A group led by An-drew Cleland at the University of California, SantaBarbara, took an important first step in that directionby coupling an acoustic resonator to a qubit, a two-state quantum system.9 The resonator was a 300-nm-thick sheet of aluminum nitride, 40 μm long and20 μm wide, whose thickness oscillates at a fre-quency of 6 GHz. At such high oscillation frequen-cies, the phonon energy ħωm is large, and thereforea conventional dilution refrigerator—which canreach temperatures near 25 mK—sufficed to cool theresonator to an occupation factor N < 0.07.

Exploiting the piezoelectric nature of AlN, theSanta Barbara team then coupled the resonator to thequbit, a superconducting Josephson junction, whichcould in turn detect the presence of a single phonon.A null result meant the mechanical resonator was inthe ground state. The technique is analogous to thosethat have been developed over the years in cavityand circuit quantum electrodynamics, except thatthe photons are replaced by phonons.

Not only did the resonator–qubit couplingallow observation of the energy quantization andother quantum features of the resonator, it also en-abled controlled manipulation of the mechanical res-onator at the few-phonon level. The Santa Barbaragroup was able to observe a few coherent oscillationsof a single quantum exchanged between the qubitand the resonator—a first demonstration of coherentcontrol over single quantum excitations in a micro-mechanical resonator. Recent experiments at Caltechand Harvard University and in Grenoble, France,have made important further steps by coupling me-chanical devices to a variety of other qubits.

Single-quantum or few-quanta control canoccur only in a strong-coupling regime, where en-ergy is exchanged between the mechanical res-onator and the qubit or optical mode with very littledissipation; loss of photons and phonons to the en-vironment must be minimal. That regime has nowbeen reached with several micromechanical devicesin addition to the one used in the Santa Barbara ex-periment.10 Eventually, such strong optomechanicalcoupling will allow high-fidelity transfer of quan-tum states between light and mechanical systems. Itshould even be possible to generate entanglementbetween photons and phonons. Conversely, phononfields can be mapped onto an optical mode to takeadvantage of the reliable, high- efficiency detectionschemes available in optics.

Promise in the fieldThe lure of quantum optomechanics goes far beyond simply adding another class of objects—mechanical resonators—to the list of “tamed” quan-tum systems. Rather, the promise is that just aswe’ve learned to couple mechanical elements withthe photons in an optical cavity, we can functional-ize those same elements to couple with, say, thespins in a magnetic material or the charges at a con-ducting surface. That way, a mechanical elementwould serve as a universal transducer, an interme-diary between otherwise incompatible systems. Fly-

ing photons could be linked with stationary, non -optical qubits, for instance. Only recently, a groupled by Philipp Treutlein at the University of Basel,Switzerland, has demonstrated a hybrid opto -mechanical system coupling ultracold atoms to amicromechanical membrane. Such hybrid quantumsystems may be important in classical and quantuminformation processing, for which the ability to con-vert information from one form to another is crucial.In fact, several laboratories are now working to cou-ple single mechanical elements to both optical andmicrowave frequency resonators, with the goal ofconnecting superconducting microwave circuitsand qubits to optical fields.

The connection between optomechanics andatomic physics is particularly interesting. Not onlydid laser cooling of atoms inspire and enable therapid progress in quantum optomechanics, it alsoled to the discovery of other phenomena such as optomechanically induced transparency, the ana-logue to electromagnetically induced transparency.The effect, which exploits optical interference be-tween a mechanical resonator’s excitation paths tocontrol the cavity transmission, may allow storage


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Figure 4. Quantum signatures near the ground state. (a) Thenanoscale beam shown here undergoes breathing-mode oscilla-tions—successive expansions and contractions—that are strongestnear the center, as indicated in the simulated image at top. The perfo-rations along the beam’s length form a photonic cavity that confineslight to the same region, as indicated in the bottom image. (b) A laserfield at an appropriately detuned frequency ωL can be coupled to thewaveguide via a tapered optical fiber and used to cool the breathing-mode oscillations to near the ground state. (c) At that point, red- detuned photons are less likely to extract phonons and shift upward in frequency (blue arrow) than are blue-detuned photons to createphonons and shift downward (red arrow). (Here, ωc is the cavity reso-nance frequency and ωm is the breathing-mode frequency.) (d) Theasymmetry is detectable in experiments: With roughly three phononsresiding in the beam, the upper sideband (blue) generated from a red-detuned laser is significantly smaller than the lower sideband (red)generated from an equivalently blue-detuned laser. (Panels a–c courtesy of Oskar Painter and colleagues. Panel d adapted from ref. 8, H. Safavi-Naeini et al.)


of light in arrays of cavityoptomechanical devices.11

Atomic systems canalso serve as mechanical elements in cavity opto -mechanical systems.12 Aftertrapping 105 ultracold ru-bidium atoms inside an op-tical cavity, Dan Stamper-Kurn and colleagues at theUniversity of California,Berkeley, coupled the cloudof atoms with a pump laserthat was detuned by the res-onance frequency of theatoms’ collective motion,about 40 kHz. By blue-detuning the pump laser sothat it deposited phononsinto the cloud, the groupwas able to observe theback-action exerted on theatoms by the quantum fluc-tuations of the optical field.

The generation and ma-nipulation of mechanicalquantum states is also achallenging and importantgoal for quantum metrology and sensing applica-tions. To date, the force sensi tivity of atomic forcemicroscopes and other classical mechanical devicesalready exceeds 10−18 N Hz−1/2. In other words, in lessthan a second, one can measure a force as small asthe gravitational attraction between a person in LosAngeles and another in New York. Still, we haven’treached the ultimate limit. Current implementa-tions suffer from thermal noise and, eventually,from noise associated with quantum uncertainty.

Fortunately, quantum physics provides a wayaround thermal and quantum noise by way of whatare known as quantum nondemolition measure-ments. Such measurements, first posited in the 1970sby Braginsky and coworkers, typically call for gen-erating a squeezed state—that is, confining the un-wanted but unavoidable quantum noise to a variablethat is complementary to the variable of interest.

Heisenberg’s uncertainty principle states thatcertain pairs of physical properties—say, the ampli-tude and phase of an electromagnetic wave—cannotsimultaneously be known with arbitrary precision.But a measurement of the wave’s amplitude can beperformed in such a way that most of the uncertaintyis carried by the phase, or vice versa. Such squeezedstates of light have recently been shown to enhancethe sensitivity of gravitational -wave detectors13 (seePHYSICS TODAY, November 2011, page 11).

In principle, it is also possible to prepare mechan-ical squeezed states in which nearly all of the quantumuncertainty is confined to either the position or the mo-mentum. In fact, the classical squeezing of micro -mechanical oscillators below the thermal noise limitwas first demonstrated several years ago by DanielRugar and colleagues at IBM.14 However, squeezingbelow the standard quantum limit—the precision limitfor the case when quantum uncertainty is distributed

evenly among complemen-tary properties—has yet to beachieved. A number of strate-gies based on Braginsky’s orig-inal schemes, which can bereadily implemented in opto -mechanical systems, are beingactively pursued.

Macroscale quantummechanicsAlthough still speculative,micromechanical oscillatorscould offer a route to newtests of quantum theory atunprecedented size andmass scales. Since funda-mental particles behavequantum mechanically, onewould by induction expectthat large collections of par-ticles should also behavequantum mechanically. Butthat conclusion certainlyseems contrary to our every-day classical experiencewith ordinary matter. Evenlarge quantum conden-

sates—a cupful of superfluid helium, for in-stance—which do display quantum propertiessuch as frictionless, quantized flow, do not displaymacroscopic superposition states.

To explain the so-called quantum measure-ment problem, also notoriously known asSchrödinger’s cat, some theorists propose thatstandard quantum mechanics breaks down formacroscopic objects in such a way that their super-position is forbidden. In one such theory, gravita-tion, which is always unshieldable, ultimatelycauses massive objects to decohere, or transitionfrom quantum to classical behavior. In another the-ory, objects couple to a stochastic background fieldthat localizes the object at a rate that scales with the number of particles.

Quantum optomechanics offers a promisingway to produce spatial superpositions in massiveobjects such as mechanical levers or quartz nano -spheres and to directly test theories of how they de-cohere.15 Ongoing work in that direction builds onoptical-trapping and optical-cooling techniquesoriginally proposed by Arthur Ashkin16 and shouldeventually allow a single trapped particle to be pre-pared in a quantum superposition of two distinct center-of-mass states.

Approaching the problem from the opposite direction—from the bottom up—researchers in Vienna used conventional molecular-beam tech-niques to produce matter-wave interferences withlarge, 430-atom molecules.17 Ultimately, it may bepossible to conduct similar quantum experimentswith even more massive mechanical systems. Agroup led by Nergis Mavalvala of MIT recentlytook a first step in that direction by cooling a kilogram-size oscillator to within about 200phonons of the quantum ground state.18



Today’s devices,though similar in

many ways to thoseof antiquity, steer us

to a completely different worldview—

one in which an object, possibly

even a macroscopicone, can indeed act

in two ways at the same time.


Macroscale mechanical quantum experi-ments will have to overcome a number of daunt-ing technical issues. Some of those issues—identifying gravity’s role in decoherence, for example—might be resolved by conducting ex-periments in free fall, perhaps aboard a satellite.(Last year’s Caltech experiment, in which aphonon occupancy of less than one was achievedat a bath temperature of 20 K, shows that ground-state cooling is now within the range of commer-cial cryocoolers that can be flown on satellites.)We are confident that coming experiments willlead to a more profound understanding of quan-tum mechanics, establish limits to its validity, orconfirm what we, and likely many others, be-lieve—that technical issues such as environmen-tal decoherence, and not the appearance of newphysical principles, establish the transition fromthe quantum world to the classical. We have neverbeen so close to being able to truly address thoseprofound questions and to challenge Plato’s com-monsense notion of reality in the laboratory.

References 1. For recent reviews, see T. J. Kippenberg, K. J. Vahala,

Science 321, 1172 (2008); M. Aspelmeyer et al., J. Opt.Soc. Am. B 27, A189 (2010); F. Marquardt, S. M. Girvin,Physics 2, 40 (2009).

2. V. B. Braginsky, Y. I. Vorontsov, K. S. Thorne, Science209, 547 (1980); C. M. Caves et al., Rev. Mod. Phys. 52,341 (1980). For a historical perspective, see P. Meystre,M. O. Scully, eds., Quantum Optics, Experimental Grav-itation, and Measurement Theory, Plenum Press, NewYork (1983).

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Naeini et al., Phys. Rev. Lett. 108, 033602 (2012). 9. A. D. O’Connell et al., Nature 464, 697 (2010).

10. S. Gröblacher et al., Nature 460, 724 (2009); J. D. Teufelet al., Nature 471, 204 (2011); E. Verhagen et al., Nature482, 63 (2012).

11. S. Weis et al., Science 330, 1520 (2010); A. H. Safavi-Naeini et al., Nature 472, 69 (2011).

12. K. W. Murch et al., Nat. Phys. 4, 561 (2008); F. Bren-necke et al., Science 322, 235 (2008); M. Schleier-Smithet al., Phys. Rev. Lett. 107, 143005 (2011).

13. J. Abadie et al. (LIGO Scientific Collaboration), Nat.Phys. 7, 962 (2011).

14. D. Rugar, P. Grütter, Phys. Rev. Lett. 67, 699 (1991); J. Suh et al., Nano Lett. 10, 3990 (2010).

15. W. Marshall et al., Phys. Rev. Lett. 91, 130401 (2003); O. Romero-Isart et al., Phys. Rev. Lett. 107, 020405(2011); O. Romero-Isart, Phys. Rev. A 84, 052121 (2011).

16. A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); O. Romero-Isart et al., New J. Phys. 12, 033015 (2010); D. E. Changet al., Proc. Natl. Acad. Sci. USA 107, 1005 (2010); P. F.Barker, M. N. Shneider, Phys. Rev. A 81, 023826 (2010).

17. S. Gerlich et al., Nat. Commun. 2, 263 (2011). 18. B. Abbott et al., New J. Phys. 11, 073032 (2009). ■

July 2012 Physics Today 35


Gravitational-wave astronomy: new methods of measurements

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Abstract. The current status of gravitational-wave astronomy isreviewed. Advances in ground-based antennae are discussed andnew methods of measurement that promise a considerable im-provement in sensitivity are described. The promise held out bythe improved antenna technologies is discussed.

1. Historical introduction. Searching for sourcemodels and main characteristics of antennae

The purpose of this paper is to describe to the reader the stateof the art in the part of astrophysics usually called gravita-tional-wave astronomy. This part includes the programs ofstudies and developments, as well as the discussion ofunsolved problems, which are related to ground-basedantennae using free masses. During the next 4 ± 8 years,these antennae are expected to bring qualitatively newastrophysical information: the registration of bursts ofgravitational radiation from astrophysical catastrophes thatoccurred hundreds of millions of years in the past. The mainpart of the paper is devoted to the key problem, the free-massantenna sensitivity, and to the development of qualitativelynewmethods of measurements. The first introductory sectionis devoted to the reader not acquainted with this part ofphysics, and also includes a short historical description of thedevelopment of gravitational-wave astronomy. Sections 2and 3 give the state of the art in the construction of ground-based antennae as of the moment of writing of this paper andpoint to possible ways of solving some non-trivial problems of

measurements, which determine the antenna's sensitivity. InSections 4 and 5 we give a short description of possibleconsequences resulting from successive operation of anten-nae, for gravitational-wave astronomy and for other fields ofphysics.

The emission of gravitational waves by masses movingwith alternating accelerations, predicted by A Einstein in1918 (see [1] and also [2, 3]), was first met without interest byexperimentalists. The reason for this was the smallness of theeffect for any reasonable laboratory experiment. The powerof gravitational radiation for small derivatives of theacceleration is

_Egrav � G

45c5

�q3Dab

qt3

�2

; �1�

where G is the gravitational constant, c is the speed ofelectromagnetic and gravitational wave propagation, Dab isthe quadrupole moment of masses

Dab ��V

r�3xaxb ÿ dabr2� dv : �2�

The factor Gcÿ5 � 10ÿ60 (in CGS units) is due to thequadrupole character of radiation from gravitational`charges' (gravitational masses). The quadrupole character,in turn, is the consequence of the experimental fact, usuallycalled the equivalence principle, which is one of the postulatesof general relativity (GR). For two equal point masses Mseparated by a distance l and rotating around the barycenterwith a frequency o, equation (1) takes the form

_Egrav � 128

5

G

c5M2l 4o6 ; �3�

which shows that forM � 106 g, l � 102 cm,o �3� 102 sÿ1,the value of _Egrav is as small as 10ÿ23 erg sÿ1.

It is such estimates that underlay the pessimism ofexperimentalists in detecting this radiation in the laboratory.

Now it seems evident that the smallness of the factorGcÿ5

can be `compensated' by a high value of the factor M 2, ifM ' 1033 g (i.e. of order of the solar mass). In other words,one can expect a significant values of _Egrav from astrophysical

V B Braginski| MV Lomonosov Moscow State University,

Physics Department,

Vorob'evy gory, 119899 Moscow, Russian Federation

Tel. (7-095) 939-55 65

Received 12 April 2000

Uspekhi Fizicheskikh Nauk 170 (7) 743 ± 752 (2000)

Translated by K A Postnov; edited by M S Aksent'eva

PHYSICS OF OUR DAYS PACS numbers: 04.30. ± w, 04.80.Nn, 95.55.Ym, 95.85.Sz

Gravitational-wave astronomy: new methods of measurements

V B Braginski| }

DOI: 10.1070/PU2000v043n07ABEH000775

Contents

1. Historical introduction. Searching for source models and main characteristics of antennae 6912. Thermal and non-thermal noise in free-mass antennae 6933. Antenna measurement system, the sensitivity limits of quantum origin and the problem of energy

in the system 6954. Other sources of gravitational waves. Other antennae 6975. New physical information which can be obtained with gravitational-wave antennae 698

References 699

Physics ±Uspekhi 43 (7) 691 ± 699 (2000) #2000 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences

} The author is also known by the name V B Braginsky. The name used

here is a transliteration under the BSI/ANSI scheme adopted by this

journal.

sources. However, 30 years had passed after Einstein'sprediction when Fock [4] pointed out a good prospect forastrophysical sources: In 1948 he noted that for the planetJupiter orbiting around the Sun _Egrav ' 4��109 erg sÿ1 ' 400 W. A simple examination of the catalogsof known binary stars [6] made at the beginning of the 1960s,immediately revealed that there are about ten binary systemswith orbital periodso ' 10ÿ3 sÿ1 (one turn per several hours)and l ' 1011 cm, which emit _Egrav ' 1030 ± 1031 erg sÿ1 (i.e.about 1% of the electromagnetic power of the Sun).

The next logical step was to answer the question:Are thereany explosive sources (i.e. astrophysical catastrophes) with amass of the order of the solar mass but with a frequency ograv

much higher than in close binary stars? At the beginning ofthe 1960s, some papers evaluated _Egrav and ograv for simplemodels of such catastrophes. At this stage, a great contribu-tion was made by Ya B Zeldovich, N S Kardashev,I D Novikov, I S Shklovski| [6, 7]. The discovery of pulsars(rotating neutron stars with a large magnetic moment whoseaxis does not coincide with the rotational one) by J Bell andAHewish in 1967 [8] was an important argument favoring theexistence of such sources. A few years later, J Taylor andRHulse discovered a pulsar in a binary systemwith a neutronstar companion [9]. Careful measurements of the orbitalperiod decrease in this binary system due to the energy lossesby gravitational radiation allowed the validity of Eqn (1) to bechecked for the first time with an accuracy of a few percent.Clearly, at the moment of coalescence of the two componentsthe power _Egrav should bemaximal. This type of cataclysm hasbecome `candidate number one' for experimentalists, who bythat time already had some experience in constructing suchantennae (see below). A detailed analysis of the coalescenceprocess of binary neutron stars has been conducted for manyyears in several groups. Although this analysis has not yetbeen completed (see the review by K S Thorne [10] andreferences therein), one can say that a certain consensusbetween theoreticians has been achieved, which states thatthe total energy emitted should be about Egrav ' 1052 ergs andthe wave packet of the radiation should last for severalseconds with an increasing mean frequency from tens toseveral hundreds Hz. The key question for experimentalists,namely, how often such events occur, has not been definitelysolved as yet, and there exist both pessimistic and optimisticprognoses.

In the `semi-optimistic' scenario (prognosis) by H BetheandGBrown [11] the coalescence of two neutron stars shouldhappen on average once per 104 years in one galaxy. Thisimplies that in a space volume with radius R � 1026 cm (i.e.about 30 Mpc), which comprises near 105 galaxies, about 10coalescences (and, hence, bursts of gravitational radiation)should occur during one year. Therefore, the terrestrialobserver could register about one gravitational-wave burstwith intensity ~I ' 10 ± 0.1 erg sÿ1 cmÿ2 with changingfrequency at hundreds Hz during an integration time of onemonth. It is such intensities and event rates that are expectedto be detected by two large ground-based free-mass antennaeLIGO1 and VIRGO2 (see review [2] and references therein).These and other projects will be discussed below.

A gravitational wave is the wave of acceleration gradients(components of the Riemann curvature tensor) perpendicularto the direction of the wave vector alternating in time andspace, which propagates with the speed of light. For example,if a sine-like gravitational wave propagates along the z-axis, inone half period the acceleration gradient is positive along thex-axis and is negative along the y-axis. Over the next half aperiod, the direction of the gradients reverses. According to aconvenient expression by K S Thorne, the gravitational waveis a ripple over the static curvature. Thus, it is impossible todiscover a gravitational wave at one point. However, thisbecomes possible using two point masses separated by a finitedistance L or an extended body. In the first case one needs tomeasure the change in the distanceDLgrav between themasses,which for some optimal orientation of the wave and the twomasses is

DLgrav � 1

2hL ; �4�

where h is the dimensionless amplitude of the wave (metricperturbation amplitude). For further description it is impor-tant to stress that this shift in the position is due to a changingforce with an amplitude

Fgrav � 1

2hLmo2

grav ; �5�

where ograv is the gravitational wave frequency. The valueho2

grav is just the amplitude of the wave component of theRiemann tensor. Therefore, both gravitational-wave emittersand detectors are of quadrupole type. The term `free-massantennae' means that the frequency of real mass suspensionsis much smaller, and the frequency of their internal mechan-ical modes is much higher, than the gravitational-wavefrequency ograv. It is relevant to notice that a plain gravita-tional wave, i.e. far away from the source, has twoindependent polarizations turned with respect to each otherby 45� and usually denoted by h� and h�. So the totalamplitude of the wave h in equations (4) and (5) is

h ��h2� � h2�

q: �6�

It is not difficult to express h through the intensity eI:h � 1

ograv

��16pGc3

eIr: �7�

Assuming eI � 10 or 0.1 erg sÿ1 cmÿ2 and ograv �� 103 rad sÿ1, we obtain from Eqn (7) h ' 10ÿ21 orh ' 10ÿ22, respectively. If L � 4� 105 cm and m � 104 g(these are the distance between the test bodies and theirmass, respectively, in the LIGO project), Eqns (4) and (5)imply that the amplitudes DLgrav ' 2� 10ÿ16 cm or2� 10ÿ17 cm should be measured, which are due to forcesFgrav ' 2� 10ÿ6 dyne or 2� 10ÿ7 dyne, respectively. Theseestimates will be used below to evaluate the requiredparameters of antenna and measurement equipment. It isimportant to emphasize that the force Fgrav reveals itself in therelative motion of one test mass with respect to another, andhence the measuring devise should be set up between themasses.

As was already noted, the antenna should not necessa-rily use two separated masses: one can take an extendedmassive body and measure variable tensions inside it causedby Fgrav. The first to suggest such an antenna was J Weber.

1 LIGO from Laser Interferometric Gravitational-wave Observatory. The

project was initiated by California and Massachusetts Institutes of

Technology.2 VIRGO is the name of the Franco-Italian project.

692 V B Braginski| Physics ±Uspekhi 43 (7)

In 1968, he announced the discovery of gravitationalradiation in the coincidence scheme with two such anten-nae [13]. The sensitivity reached in his experiments corre-sponded to h ' 10ÿ16 (i.e. five orders of magnitude (!) worsethan what is being planned at the initial stage of the LIGOand VIRGO projects). At the beginning of the 1970s,Weber's experiments were repeated by several independentgroups (and in particular, by the MSU± IKI group [14])using similar detectors. The tests gave negative results: nobursts were discovered. Despite this fact, Weber, in myopinion, deserves recognition by experimentalists as thepioneer of a new astrophysical field. One of the reasonsfor the failure of his experiments was the absence of areliable astrophysical prognosis of the event rates, theiramplitude and frequency.

The invention of lasers in 1961 by T Meiman had animportant effect on choosing the scheme for the antenna.Immediately after this discovery, in 1961, M E Gertsenshteinand V I Pustovoit [15] suggested using two widely separatedfreely-suspended massive mirrors (as the test masses), whichform an optical Fabry ± Perot resonator. The pumping of thisresonator with powerful coherent laser radiation shouldprovide a high sensitivity to slight oscillations of suchmirrors caused by a gravitational wave. This concept under-lies the LIGO and VIRGO projects. In the middle of the1970s, R Drever and R Weiss had already gained significantexperience with small laboratory models of such an antenna.Together with K S Thorne, they organized a triumvirate,which justified the necessity of developing a large-scaleantenna based on this principle (it is this antenna that hasbeen called LIGO). K S Thorne played a special role in thismatter. Being an outstanding theoretician and specialist ingeneral relativity, he simultaneously has a deep understand-ing of experimental physics. Apparently, his intuitionsuggested that the accumulation of astrophysical informa-tion should allow soon meanwhile provide experimentalistswith a reliable prediction of the values of expected h, ograv,and event rates, and that the period could be spent in gettingexperience using a large prototype of the full-scale antenna.In 1981, the National Science Foundation (NSF) in the USAstarted to finance the design and construction of a prototypeof the large antenna (which has L � 4� 103 cm). After 15years of work, researchers from Caltech and MIT havereached a sensitivity for this prototype of h ' 10ÿ19 (withan integration time t ' 10ÿ2 s and a signal-to-noise ratio oforder one). Thus, about two decades after Weber's experi-ments, the sensitivity has been increased by three orders ofmagnitude (see [16] and references therein). In 1996, NSFbegan financing the construction of the full-scale antennae(L � 4� 105 cm). The construction has been completed inthe last year, and now the gradual adjustment of two suchantennae is under way with the aim of reaching a sensitivityof h ' 10ÿ21 by the fall of 2001 (the LIGO-I project), afterwhich a long-term recording of the distance variationsbetween the mirrors in the coincidence scheme must begin.In about 2005, significant modifications are planned, amongwhich are a change of the mirrors' suspension and theirinsulation from noise. As a result, the sensitivity is expectedto increase up to h ' 10ÿ22 (the LIGO-II project). Originally,the LIGO project was purely national. However, the NSFencouraged cooperation not only between American uni-versities and institutes, but also the participation of foreigngroups (in particular, two Russian groups: one fromM V Lomonosov Moscow State University (MSU), another

from the Institute of Applied Physics of Russian Academy ofSciences). Presently, more than 200 researchers participatedirectly in the LIGO project, and there is a large exchangewith participants form the VIRGO project (which uses oneantenna with arms L � 3� 105 cm), as well as with the moremodest (in the size of L) projects GEO-600 and TAMA. Fora detailed description of wave solutions in general relativityand many other interesting events related to gravitational-wave astronomy, two large reviews [17, 18] are recommendedto the reader. The author has restricted himself to a veryshort historical introduction above, which, however, issufficient to get acquainted with the experimental achieve-ments over the last decade and an understanding of unsolvedproblems.

2. Thermal and non-thermal noisein free-mass antennae

Through a cross-section with area

l2grav �4p2c2

o2grav

�8�

during half a period of a gravitational wave a very largenumber of gravitons pass:

Ngrav �eIc2

4p�ho4grav

' 1045eI

0:1 erg sÿ1 cmÿ2

��

ograv

103 rad sÿ1

�ÿ4: �9�

It is this ensemble of gravitons that produces the gravitationalforce Fgrav acting on the test masses of the antenna. The largevalue of Ngrav allows us to consider Fgrav as being a classicalforce. It is seen from estimate (9) that this can be done even forvalues of eImuch smaller than quoted in the previous section.The classicality of Fgrav means that there is no need toquantize the gravitational field.

According to the quantum theory of measurements, thereis no limit on detecting a small classical force acting on amass.Consequently, the gravitational antenna sensitivity isexpected to increase with improving methods of measuringsmall DLgrav (or other observables related to Fgrav). Thisimportant feature has already been used in planning the two`steps' LIGO-I and LIGO-II, which we discussed in Section 1.Apparently, after having reached the sensitivity h ' 10ÿ22,the LIGO-III stage with a smaller level of h will beconstructed. However, this does not imply that there are noconstraints at all to the antenna sensitivity, which isdependent on the Planck constant �h. In reality they do existand are determined by the specific method of measurement ofthe response of the system consisting of two masses on Fgrav.These quantum limitations will be considered in Section 3,and in this section we shall describe the most importantsensitivity constraints determined by random actions ofthermal and non-thermal origin.

The mirror in the LIGO-I (a fused-quartz cylinder 25 cmin diameter with a thickness of 10 cm) has a mass of about 104

g. It is suspended on a thin steel fiber loopwith a free length ofabout 20 cm (the fiber girds about the middle of the cylindricsurface). If one considers such a suspended mirror as aGalilean pendulum with a point-like mass and takes intoaccount that the pendulum oscillation frequency is muchsmaller than ograv ' 103 sÿ1, then the obvious condition fordetection of Fgrav (provided that the only source of random

July, 2000 Gravitational-wave astronomy: new methods of measurements 693

force action is the thermostat) will be

Fgrav >

��4kTH

t

r�

��4kTm

t�mt

s; �10�

where T is the thermostat temperature, H is the frictioncoefficient in the pendulum oscillation mode, and t�m � m=His the mode relaxation time. Formula (10) is a directconsequence of the fluctuation-dissipative theorem (FDT)for this simple model. Assuming in (10) Fgrav � 2� 10ÿ6 dyne(i.e. for the sensitivity level of LIGO-I) and t � 10ÿ2 s, weobtain that it is necessary to have t�m > 4� 104 s for a signal-to-noise ratio of the order of one. In LIGO-I, such asuspension on steel fibers gives t�m ' 105 s. Clearly, this isnot sufficient for LIGO-II, in which Fgrav must be smaller byan order of magnitude.

Adding the condition that the signal-to-noise ratio is 10,formula (10) implies that t�m must be 4� 108 s, i.e. about 12years at T � 300 K, m � 104 g, t � 10ÿ2 s. Note thatcondition (10) holds for the so-called viscous friction model,inwhichH � const. If we consider the losses in the suspensionas in the structure friction model, the requirement on t� willbe an order of magnitude smaller. My colleagues from theMSU V P Mitrofanov, O A Okhramenko, and K V Tokma-kov, starting from 1991, developed the mirror suspension forthe LIGO using fibers made of a super-pure fused quartz(whose internal losses correspond to a mechanical quality ofQm ' 3� 107 [19]). Several new technological tricks sug-gested and realized by them have allowed the achievementof t�m ' 1:7� 108 , i.e. about 5.4 years [20], which is close towhat is required in LIGO-II. Such a long relaxation time ofmechanical oscillations corresponds to a relative amplitudedecrease by 0.3% in five days. It is this value that wasmeasured in the experiment. The volume and goal of thepresent paper does not permit us to consider the technologicaldetails of these methods in more detail. However, thefollowing note is pertinent. Although FDT suggests norecommendations as to what should be done to increase t�m,nevertheless, by using not too strict semi-empirical relation-ships confirmed experimentally, one can affirm that in such asuspension under real laboratory conditions t�m ' 109 s andeven higher values can be reached. As will be clear from thenext section, the value of the dimensionless ratiot=t�m ' 5� 10ÿ11 already achieved today plays an importantrole in attaining the so-called standard quantum limit of thesensitivity.

The simplemodel considered above (a point-likemass andthe unique pendulum mode) is not full: it is also necessary totake into account the contribution to the fluctuation actionon the mirror's center of mass from thermal fluctuations overthe entire length of the fiber. In terms of the Langevinlanguage for Brownian motion, it is necessary to take intoaccount spectral components of numerous violin modes ofproper oscillations of the fiber near ograv, as well as thetransfer function of the fiber oscillations to the mirror'sbarycenter. The calculations, which we omit here, give thefollowing condition: for h ' 10ÿ22 the quality of the lowestfrequency modes of the fiber must be at the level 107. Themeasurements by Mitrofanov and Tokmakov have shownthat the quality of the low-frequency eigen modes of the samefibers made of the super-pure quartz falls within the range5� 107 ± 1� 108 [21].

There is onemore feature of the fluctuation force action ofthe fiber on the mirror. This feature is that the sources of

fluctuations are within the fiber itself and cause its motion.This motion (the change of the fiber coordinates relative to,say, a platform towhich the fiber is attached) can bemeasuredby a separate sensor. The calculations by S P Vyatchanin andYuM Levin [22] have shown that such a measurement allowsone to subtract about 99% of random shifts, caused bythermal fluctuations inside the fiber, from the mirror'smotion.

To conclude this part of the section, one can say that theproblem of `suppression' of random fluctuations of themirror's barycenter position, caused by the thermostat in thesuspension fibers, can also be solved in the case where theplanned sensitivity must be much better than h ' 10ÿ22.

When considering fluctuations in the mirror suspensionfibers, we have so far restricted ourselves to the case ofthermal equilibrium, to which FDT can be applied. How-ever, in addition to such fluctuations, there are others whichdo not obey FDT. It is easy to see that to attain larger t�m, it isadvantageous to decrease the diameter of the fiber (of steel orfused quartz), i.e. it is profitable to approach the stress closeto the fiber disruption threshold. The free energy density inpolycrystal solids and amorphous bodies is usually of theorder of 106 erg gÿ1. As established long ago, at the tensionclose to solid body disruption, an acoustic emission isobserved, which is the consequence of a partial redistributionof the free energy (vacancy group jumps, dislocation creation,etc.) At present, no theoretical models of such processes havebeen proposed.

It is important to note that the kinetic energymo2

grav DL2grav=2 ' 2� 10ÿ22 erg for LIGO-I is many orders

of magnitude smaller than the total free energy storage in thefiber. In other words, even a tiny fraction of the free energy inthe fiber transformed by tension into acoustic emission (i.e.into additional to thermal oscillations of the fiber), can mimicFgrav. Recently, A Yu Ageev and I A Bilenko (also from theMSU group), when making detailed measurements of thetemporal structure of the Brownian oscillations of the steelstring used inLIGO-I have found random short-term changesin the oscillation amplitude exceeding the ordinary values,which correspond to the Langevin model of the Brownianmotion:��

�dx2p

��kT

mo2m

s ��2tt��m

s; �11�

where m is the effective mass of a mode with frequency om

and relaxation time t��m [23]. These rare bursts exactlyrepresent excessive (non-thermal) noise. As established fromthe measurements, such bursts, which can mimic metricvariations at the level h ' 10ÿ21, are observed in the fibers,on average, once every several hours under tensions of about50% of the breaking stress. Decreasing the tension stronglydecreases the false event rate. A coincidence scheme betweentwo antennae, of course, makes it easier to reject such `signals'imitating the action of Fgrav, however it is natural that theexperimentalists consider it as the last `line of defense'.

So far, no measurements of the excess fluctuations (ofnon-thermal origin) in the fibers of super-pure quartz havebeen performed. This is a much more complicated problem,since for such fibers the value of t��m is about four orders ofmagnitude higher than for steel fibers, and, correspondingly,the sensitivity to small fiber oscillations must be at least twoorders of magnitude better. A new method of measurementsrecently suggested by I A Bilenko and M L Gorodetski| [24]will, in my opinion, allow this problem to be solved.


Not only thermal fluctuations in the suspension fibersprevent accurate measurements of small Fgrav, which causesoscillations DLgrav of the mirror's center of mass. The opticalFabry ± Perot resonator is formed by two almost flat (with acurvature radius of about 10 km) surfaces of two quartzcylinders. The mirrors are constructed from a multi-layercoating of the quartz cylinders' surface, which has a highreflectivity. It is clear that the laser interferometer measuresnot the displacement of the cylinder's barycenter, but the sumof this displacement and the proper internal oscillations of thecylinder. If they are due to internal forces (for instance, due tothe inner friction inside the material which, according toFDT, is the source of the volume forces), they do not shift thebarycenter, but can give an appreciable noise contribution tothe quantity being measured.

To calculate this contribution, AGillespi and FRaab [25],using the same Langevin model for modes, estimated thefraction of the total variance of the mean coordinate of themirror (i.e. the cylinder's surface) nearograv. Note thatograv issignificantly lower than the proper frequencies of thecylinder's modes, so the contributions from low-frequencywings of several tens of modes have been taken into account.This calculations showed the possibility of attaining a levelclose to h ' 10ÿ22 only at low frequenciesograv 4 103 rad sÿ1

at a signal-to-noise ratio close to one. The same calculationsfor a sapphire cylinder (whose internal losses are an order ofmagnitude smaller than for quartz, and the density andvelocity of sound is two times as large) show that it is possibleto gain three times in sensitivity nearograv ' 103 rad sÿ1, if thelosses inside the material are of so-called structural character.

Unfortunately, fluctuations of the cylinder's surface (themean coordinate of the surface of the mirror) in the frequencydomain of interest include not only pure Brownian fluctua-tions, which can be calculated in the linear Langevin model.There is at least one more effect of a non-linear origin thatcontributes almost as much as Brownian fluctuations. Thiseffect is the consequence of anharmonism of the lattice, whichgives rise to thermal expansion and temperature fluctuationsdT. If for some reason the temperature inside the layeradjacent to the mirror's surface l changes by some value dT,the displacement of the outer surface of the mirror will besignificant even for tiny temperature changes:

dl � aTldT � 10ÿ17 cmaT

5�10ÿ7 Kÿ1l

10ÿ2 cmdT

2�10ÿ9 K ;

�12�where aT is the coefficient of linear expansion. As seen fromestimate (12), under these conditions dl is of order ofDLgrav ' 2� 10ÿ17 cm for dT of about 1 nanokelvin. Suchtemperature fluctuations can result from ordinary equili-brium fluctuations, whose total variance is [26]

hdT 2i � kT2

CrV; �13�

where C is the specific heat capacity, r is the density, and V isthe volume. In order to find the fluctuations, correspondingto (13), of themean coordinate of themirror's surface near thefrequency ograv of interest here, it is necessary to know thefrequency dependence of the dissipation inside the mirror'smaterial. Assuming the losses inside the mirror to correspondonly to the thermoelastic model, M L Gorodetski| andS P Vyatchanin obtained analytical expressions (see [27, 28])for the spectral density of fluctuations of the mean coordinate

in the mirror's surface inside a circle of radius r0. Theintroduction of the parameter r0 is significant, because onlyfrom such a spot does the laser beam get information aboutthe mirror's motion. Calculations show that for r0 ' 1:6 cm,which was initially planned for LIGO-II, and themirrormadeof fused quartz, the sensitivity limitations due to this effect arethree times smaller than due to `purely' Brownian limitations.At present, monocrystal sapphire is being widely discussed asa possible `candidate' for the mirror material. For thismaterial, the above effect is twice as large as the Brownianone for the same r0. There are at least two possibilities toreduce the contribution of this effect to the total fluctuations,which do not relate to the mirror's barycenter motion. Thefirst possibility is to appreciably increase r0 (see [27]), thesecond suggests subtracting fluctuations of the mirror'ssurface coordinate (recall that they are due to internalforces, which do not shift the position of the mirror'sbarycenter).

It is quite possible that the analysis [27, 28] neglects otherfluctuation processes that could lead to additional randomvariations of the mirror's surface and, correspondingly, toincrease the threshold value of h (in particular, excessivenoise). Clearly, only direct measurements can give a definiteanswer.

Concluding this section, we should note that here we havedescribed the state of the art in solving two of the mostdifficult experimental problems. It is also relevant to stressthat our description had a semi-qualitative character withsimple numerical estimates. In the cited references, the readercan find a lot of details which were omitted, in particular, thespectral representation of the limiting sensitivity in thefrequency range 10 rad sÿ1 < ograv < 104 rad sÿ1. Due tolimit of space and bearing in mind the purpose of the presentpaper, we have not considered the solutions of simplerproblems, such as antiseismic insulation of the mirrors and astrategy of monitoring other possible sources of force actionon the mirrors.

Finally, we can emphasize that the problem of choice ofthemirror and its suspension proved to be, in essence, a ratherdifficult experimental task, which has required a lot of time.This can be illustrated as follows. It took in total 15 years toreach a sensitivity of h ' 10ÿ19 in the LIGO-I prototype, andabout 8 years to obtain a relaxation time of t�m ' 5:4 years forLIGO-II [20].

3. Antenna measurement system,the sensitivity limits of quantum origin andthe problem of energy in the system

The scheme of the measurement system in LIGO (with somesimplifications) is shown in Fig. 1.

Its principal elements include two optical Fabry ± Perotresonators formed by two pairs of mirrors AA0 and BB0. Thedistance between the mirrors in each pair is L � 4� 105 cm.Optical oscillations inside them are excited by laser 1 (with thewavelength l � 1064 nm) through beam-splitter 2, whichtogether with additional mirrors 3 and 4 connects theresonators. Mirrors A and B are `deaf', in other words theirreflection coefficient R is close to one. Mirrors A and B aremore transparent with a lower R. In LIGO-II, mirrors withfineness F � p=�1ÿ R� ' 3� 103 are planned. The conceptof this scheme was first suggested by Drever [29]. Later thisscheme was analyzed in detail and modified by several groups(see references cited in [12]).


If a gravitational wave propagates along the axis x, theforce Fgrav acts between the mirrors AA 0 and their separationchanges with amplitude DLgrav ' hL=2, while the distancebetween B and B0 does not change. If the wave propagatesalong the y-axis, the roles of the mirrors reverse. When thewave propagates along the normal to the plane xy and thewave polarizations are optimal with respect to the axes x andy, both pairs of mirrors will oscillate in antiphase so that thetotal response will be twice as large.

A special control scheme (with a feed-back) allowsadjustment of the mutual positions of the mirrors and thetuning of resonance mode frequencies in these schemesaccordingly. Note that in the feed-back the upper boundaryof monitoring frequencies is much lower than ograv, so themirrors react on Fgrav as essentially free masses. The accuracyof mirror position control up to this upper boundary is smallfractions of l=F .

For a certain position adjustment of all six mirrors, theamplitude of oscillations of the phase difference of the opticalfield in two arms Dj is proportional to h.

The depth of modulation of the power flux coming intodetector 5 is, in turn, proportional toDj. Note that the tuningof the optical system is such that the detector is workingalmost in the `dark field' regime.

If a burst of gravitational radiation has a wave-packetform with duration t and frequency ograv, one can obtainsimple formulas for the two cases: when p=ograv is smallerthan the relaxation time of optical oscillations t�opt, andp=ograv > t�opt in the opposite case. The time t�opt can beexpressed through R and F and in one of the LIGO-IIvariants under discussion is of order

t�opt �L

c�1ÿ R� �FLpc' 4� 10ÿ3s

F103

L

4� 105 cm: �14�

If p=ograv 5t�opt, then

Dj ' 2pLl�1ÿ R�

DLgrav

L' FL

lh ' 4� 10ÿ10 rad

F103

� L

4� 105 cm

h

10ÿ22

�l

10ÿ4 cm

�ÿ1: �15�

If p=ograv < t�opt, then

Dj ' 1

2hoopt

pograv

' 4� 10ÿ10 radh

10ÿ22oopt

2� 1015 sÿ1

��

p=ograv

4� 10ÿ3 s

�ÿ1: �16�

These numerical estimates show that the values of Dj from(16) and (15) are nearly equal at p=ograv ' t ' t�opt.

It is important to note that measurements of smallquantities Dj over a short time interval t in such aninterferometer are essentially the matching of the opticalfield oscillations in resonators AA0 and BB0. The ratio Dj=tis the deviation of one frequencywith respect to other, and theratio DLgrav=L � h=2 is the relative value of this deviationequal to Doopt=oopt. From this point of view, the sensitivityh ' 10ÿ19 already reached in the LIGO prototype [16] (inwhich L � 4� 103 cm) is, undoubtedly, an outstandingresult. For comparison, it is sufficient to mention that thesmallest relative deviations of two stable generators (Allan'svariance) measured so far are only about 10ÿ17.

At the end of this section, we shall use the above estimatesfor Dj ' 10ÿ10 rad (for LIGO-II).

As was noted in the Introduction, in gravitational-waveantennae at a certain sensitivity level there appear some kindof `barriers', sensitivity limitations of quantum origin. Theydepend on the choice of the observable and measurementprocedure. Such restrictions were predicted in experimentswith macroscopic masses in the coordinate measurementswith a finite averaging time in 1967 [30]. A few years later [31]it became clear that the same type of limitations can bedirectly derived from Heisenberg's uncertainty relations forcoordinate measurements, taking into account the finitenessof the measurement time t. Clearly, there exist a whole familyof such limits if measurements of an observable involve thedirect measurement of a coordinate (geometrical distance,field strength, etc.) They are called `standard quantum limits'.For example, in the case of a freemass we are interested in, thestandard quantum uncertainty of its coordinate is

DxSQL ��ht2m

r' 2� 10ÿ17 cm

�t

10ÿ2 s

�1=2�m

104 g

�ÿ1=2;

�17�if two point observations of the coordinate x are made at thebeginning and the end of a time interval t.

By equating the free mass coordinate uncertainty toDxSQL, one can find an analytical expression for thestandard limit of the metric perturbation amplitude hSQL

caused by a wave packet with duration t and meanfrequency ograv in the scheme of the LIGO-II interferom-eter (with four test masses)

hSQL ��

8�h

mo2gravL

2t

s' 2� 10ÿ23

�m

104 g

�ÿ1=2��

ograv

103 sÿ1

�ÿ1�L

4� 105 cm

�ÿ1� t10ÿ2 s

�ÿ1=2: �18�

Equation (18) is correct for t5 2p=ograv. As seen fromestimates (17) and (18), the planned LIGO-II sensitivityh ' 10ÿ22 is very close to hSQL. In other words, the contribu-tion of purely quantum fluctuations to the total measurementerror will be significant (see [27] for more detail).

A `recipe' to get around this obstacle has been knownfor quite a long time. As noted in Section 1, for antennae

BB0

A0

A

3

1 2

4

5

L

L

x

y

Figure 1.


there are practically no sensitivity limits of quantum origin.It is only necessary to choose other observables to detectFgrav. The `recipe' usually referred to as quantum non-demolition (QND) measurements was initially suggestedfor gravitational antennae (see review [32]). However, italso proved attractive for opticians studying quantumphenomena, and QND-measurements have been success-fully realized in optical experiments, although with a ratherlarge number of photons (see review [33]). Recently, thesame measurements were performed with microwave quantaas well: the presence of a single microwave quantum in aresonator was measured without absorption of the quantum[34]. Up to the present time, this has not yet been done withmechanical test masses (a free mass or an oscillator).However, without solving this, a much more difficult,problem, it is impossible to increase the gravitationalantenna sensitivity keeping the values of L and M.

Possible schemes for QND-measurements in antennaehave been analyzed by the MSU group over last few years.The first suggestion was to use larger fineness values F inmirrors thereby drastically increasing t�opt [35]. At thealready reached F � 2� 106 [36] the value t�opt ' 10 s forL � 4� 105 cm. Such a long relaxation time opens thepossibility of using the small parameter t=t�opt ' 10ÿ3,which determines the attainable degree of `squeezing' ofthe quantum state of the optical field inside resonator.Naturally, the meter itself should be installed inside theresonator and should not absorb optical photons. Thevariant of such a meter, suggested in [35], was rather thedescription of a thought experiment than a realisticengineering scheme.

The second attempt to develop such a meter [37] led to thescheme shown in Fig. 2.

Three free mirrors with a maximal F (and, correspond-ingly, t�opt) form an optical Fabry ± Perot resonator. Anexternal laser (not shown in the figure) excites oscillations inthe resonator. Inside it, near the tilted mirror B, an additionalfourth mirror D is inserted with reduced fineness and,correspondingly, with relatively high transmittance, whichshould be specified. In such a scheme, all modes in theresonator ABC are split into doublets due to mirror D. Atsome transmittance of mirror D, when the upper componentof the doublet is excited, the ponderomotive force producedby the optical field shifts mirror D if mirror C is displaced. Ifsuch a resonator receives some amount of energy of about

what is needed to reach hSQL, the displacement isD ' hL=2 � DLgrav.

The meter itself must be placed between mirror K, locatedoff the optical field (the ponderomotive force does not act onit), and mirror D. For an ideal QND device (for instance, forspeed measurements), the finite value of F , as was calculatedby M L Gorodetskii and F Ya Khalili, bounds the sensitivityof such an antenna with the limit

h � hSQL��ogravt�opt

p � 10ÿ2hSQL

�ograv

103 sÿ1

�ÿ1=2� t�opt10 s

�ÿ1=2:

�19�The analysis of the speedmeter recentlymade by theMSU

group in collaboration with K S Thorne using the parametersattainable in the modern cryogenic microwave electronics,showed that one can reach a sensitivity not better thanh ' 0:3hSQL [38]. Possibly, to find a much simpler technolo-gical solution, it would be convenient to turn the free massesD and K into a mechanical oscillator with eigen frequencynear ograv, by `adding' to them a stiffness of optical originwith a very low noise level [39], or to use symphoton quantumstates in a different antenna topology [40].

The last problem to discuss in this section is the amount ofadmissible energy inside resonators (or circulating powerW).For anordinary (coordinate)measurement scheme (seeFig. 1)in LIGO-II the projected circulating powerW ' 1013 erg sÿ1 ' 1 MW. Provided that the mirrors havemulti-layer reflection coatings of highest quality, the powerdissipated in the coating will be of order 107 erg sÿ1' 1 W.The absorption of each optical photon with energy�hoopt ' 2� 10ÿ12 erg will give rise to a local burst consistingof around 50 additional thermal photons, which in turn,because of a small free-path length (both in quartz andsapphire), will lead to a local shot-like heating of the mirrorsurface. Due to a non-zero thermal expansion coefficient aT,the mirror surface will fluctuate. Such a specific thermo-photon noise was analyzed in detail in paper [27]. Thecalculations showed that W ' 1013 erg sÿ1 is indeed veryclose to the limiting admissible value, if one wish to approachhSQL. Clearly, a possible solution is the use of squeezedquantum states in the main resonators, which can beprepared at t=t�opt 5 1.

4. Other sources of gravitational waves.Other antennae

The main purpose of the present article is to describeachievements in development of ground-based free-massantennae and the discussion of novel methods of measure-ments that can substantially increase the sensitivity. In thissection, we restrict ourselves to only a brief enumeration ofother directions of studies in gravitational-wave astronomy.

In previous sections we used a simple example to illustratethe conditions required to reach a given sensitivity: agravitational-wave burst has a duration of t ' 10ÿ2 s and amean frequency of ograv ' 103 rad sÿ1. The free-massantennae operate over a very broad frequency range10 rad sÿ1 4ograv 4 104 rad sÿ1. Bursts of radiation fromcoalescing neutron stars (the preferred source in the LIGOand VIRGO prognoses) must have t of the order of severalseconds and a changing frequency from tens to a few hundredHz. The a priori knowledge of waveforms of such burstswould significantly facilitate the detection and increase the

CK

B

A

D

x

y

DL DLL

L

Figure 2.


signal-to-noise ratio. This is one of the problems that is beingstudied by some theoretical astrophysics groups. Ignorance ofthe exact equation of state of neutron star material makes thisproblem even more difficult. In addition, two coalescingcomponents may have strongly different values and orienta-tions of angular momentum. Experimentalists still hope thattheoreticians hold their promise to calculate around 105 burstwaveforms from such sources when the antennae startworking in a stationary observational regime.

Coalescing binary neutron stars are not the only sourcesto be sought by gravitational wave antennae. Stronger burstscan be expected from coalescing neutron stars and black holesand coalescing binary black holes. One more unsolvedtheoretical problem are the waveforms from such sourcesand their expected event rate.

Non spherically-symmetric supernova explosions canalso generate bursts of gravitational radiation. The eventrate of such events in a galaxy is about two orders ofmagnitude higher than that of coalescing neutron stars inthe model by H Bethe and G Brown. Attempts to construct areliable model of the initial stage of the supernova explosionhave failed so far.

With increasing sensitivity, ground-based antennae willbe capable of registering gravitational wave noise back-grounds. Some of them should have the same origin as thecosmic microwave background discovered in the mid-60s.The first models of the relic gravitational radiation suggestedby L P Grishchuk [41] and A A Starobinski| [42] were furtherdeveloped by other scientists. Of these new models, in myopinion, the most interesting is the model by R Brustein andG Veneziano [43] based on the superstring theory. This isessentially the first direct prediction of the superstring theory.A notable feature of this model is that it has no divergencebetween `before' and `after' the Big Bang.

At present, two other projects are also developing free-mass antennae. Both these programs invoke satellites. Thefirst of them exploits the Doppler effect as a measure of the`response' on metric perturbations, as was suggested manyyears ago [44]. These antennae have a threshold sensitivity ofhmin ' 10ÿ14 ± 10ÿ15 for gravitational wave bursts with amean frequency 10ÿ2 rad sÿ1 4ograv 4 10ÿ4 rad sÿ1. Thesensitivity is determined, first of all, by the non-stability ofmicrowave autogenerators, which are used in the Earth-satellite channel. In such a program, it would be natural tohave two pairs of free masses (the first satellite Ð Earth andthe second satelliteÐEarth) and to use a coincidence scheme.Unfortunately, there are no such dedicated programs and theexperimentalists have to use, as a `by-product', free hours ofcommunication with single satellites, whose principal goal isto study remote planets of the Solar system (see references in[45, 46] for more detail).

Another program of free-mass antennae (the LISAproject) also uses satellites and plans to search the samefrequency range of ograv as the previous one. In manyrespects this program is similar to LIGO and VIRGO: itintends to use three (drag-free) satellites at the terrestrialdistance from the Sun and separated by L ' 5� 1011 cm(i.e. by six orders of magnitude larger than the mirrors inLIGO and VIRGO). Distance variations between thesatellites should be measured by a laser interferometer. Atthe present time, a significant number of ground-basedlaboratory tests of model units of such antennae and manycalculations have already been done, however the finaldecision about the practical realization of this program has

not yet been taken (see, e.g. [47] and a special issue ofjournal [48]).

5. New physical information which can beobtained with gravitational-wave antennae

In conclusion of this paper, it is relevant to enumerate whatnew can be expected by physics from using ground-basedgravitational-wave antennae.

1. The discovery of bursts of gravitational radiation andthe study of their rate of occurrence can give informationabout the space density of binary neutron stars in galaxies andthe contribution they give to the so-called dark matter. Burstwaveforms can possibly be used to study the preferentialequation of state of the neutron star matter. Short burstwaveforms will, perhaps, allow the construction of a model ofthe initial stage of supernova explosions.

2. More than twenty five years ago General Relativity(GR) turned into an engineering discipline for high-precisionspace navigation. This essentially means that GR has beenchecked to a high degree of accuracy, however high-precisionnavigation inside our Solar system implies at the same timethat GR is valid only in the case where gravitational potentialis much smaller than c2. The validity of GR in theultrarelativistic case (when gravitational potential is of orderof c2) has never been tested. The possibility of such a test willemerge if theoretical astrophysicists can predict the wave-forms of the signal from coalescing black holes and if suchbursts are really detected.

3. The detection of the relic gravitational wave back-groundwill, undoubtedly, provide an invaluable contributionto cosmology. It is however possible that detected chaoticfluctuations of the mirror's barycenter, which are not due tothermal (or non-thermal) fluctuations in the suspension orinside the mirror itself and the fluctuating action of thequantum detector, will not be correlated with the motion ofthe two pairs of mirrors in a way that the wave components ofthe Riemann tensor produce. In such a case the prediction ofSHawking [49] (see also [50] and [51]) about the interaction ofspace-time fluctuations on the Planckian scale with ordinarymatter may be realized. These fluctuations are sometimescalled the space-time foam. Hawking's prediction is that suchan interaction appears in the decoherence of the wavefunction of an ordinary body, i.e. in small random displace-ments of the body's barycenter.

2. It is natural to expect that methods of measurementsdeveloped for LIGO and VIRGO can be used in other fieldsof physics. It is worthwhile noting that when QND-detectorsof the relative velocity of motion of masses, which allow thedetermination of the linear momentum with an accuracybetter than the standard quantum limit

DPSQL ��hm

2t

r;

are used in antenna measurement systems, the energy,determined through the momentum, will be measured withan error DE less than �h=t [52].

To conclude, it should be said that all the expenses on theLIGO project over 20 years, most of which were spent toconstruct buildings and vacuum equipment, are less than afourth part of the cost of a nuclear submarine. Yet mankindcontinues manufacturing several such submarines each year,which, in contrast to LIGO, are incapable of making it anywiser.


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PRL 96, 043003 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending3 FEBRUARY 2006

Feedback Cooling of a Single Trapped Ion

Pavel Bushev,1,* Daniel Rotter,1 Alex Wilson,1 Francois Dubin,1 Christoph Becher,1,† Jurgen Eschner,1,‡ Rainer Blatt,1,3

Viktor Steixner,2,3 Peter Rabl,2,3 and Peter Zoller2,3

1Institute for Experimental Physics, University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria2Institute for Theoretical Physics, University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria

3Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria(Received 16 September 2005; published 3 February 2006)

0031-9007=

Based on a real-time measurement of the motion of a single ion in a Paul trap, we demonstrate itselectromechanical cooling below the Doppler limit by homodyne feedback control (cold damping). Thefeedback cooling results are well described by a model based on a quantum mechanical master equation.

DOI: 10.1103/PhysRevLett.96.043003 PACS numbers: 32.80.Lg, 32.80.Pj, 42.50.Lc, 42.50.Vk

PMT 1

Mirror

Fluorescence modulation

Ba+ ionBS

in-loop

bandpass

phase

to trap electrodes

rfsa

out-loop

Cold damping feedback

PMT 2

Gain

FIG. 1 (color online). A single 138Ba� ion in a Paul trap(parabola) is laser excited and cooled on its S1=2 to P1=2

transition at 493 nm. A retro-reflecting mirror 25 cm awayfrom the trap and a lens (not shown) focus back the fluorescenceonto the ion. The resulting interference fringes with up to 73%contrast are observed by a photomultiplier (PMT 1). The ion’soscillation in the trap creates an intensity modulation of the PMTsignal which is observed as a sideband on a spectrum analyzer(rfsa) [16]. For feedback cooling, the sideband signal is filtered,phase-shifted, and applied to the ion as a voltage on a trapelectrode.

Quantum optics, and more recently mesoscopic con-densed matter physics, have taken a leading role in realiz-ing individual quantum systems, which can be monitoredcontinuously in quantum limited measurements, and at thesame time can be controlled by external fields on timescales fast in comparison with the system evolution.Examples include cold trapped ions and atoms [1], cavityQED [2–5], and nanomechanical systems [6]. This settingopens the possibility of manipulating individual quantumsystems by feedback, a problem which is not only of afundamental interest in quantum mechanics, but alsopromises a new route to generating interesting quantumstates in the laboratory. During the last decade a quantumfeedback theory [7,8] has been developed, its basic ingre-dients are the interplay between quantum dynamics and thebackaction of the measurement on the system evolution.First experimental efforts to realize feedback on variousquantum systems [5,9–12] have been reported only re-cently. In this Letter we present an experiment on feedbackcontrol, i.e., feedback cooling, of a single trapped ion infront of a mirror. We establish a continuous measurementof the position of the laser driven ion by monitoring itsfluorescence, then feed back a damping force proportionalto the momentum thus demonstrating ‘‘cold damping’’[13,14]. We use a quantum control theory based on aquantum optical modelling of the system dynamics andcontinuous measurement theory of photodetection. Thisprovides us with a quantitative understanding of the ex-perimental results.

We study a single 138Ba� ion in a miniature Paul trapwhich is continuously laser excited and laser cooled to theDoppler limit on its S1=2 to P1=2 transition at 493 nm, asoutlined in Fig. 1. The ion is driven by a laser near theatomic resonance, and the scattered light is emitted into theradiation modes reflected by the mirror, as well as the other(background) modes of the quantized light field [15]. Lightscattered into the mirror modes can either reach the pho-todetector directly, or after reflection from the mirror. Thisleads to an interference of the emitted photons, determinedby the instantaneous position of the ion with respect to the

06=96(4)=043003(4)$23.00 04300

mirror. Therefore, the motion of the ion (its projection ontothe ion-mirror axis) modulates the photon counting signal,and is detected in the spectrum as a vibrational sideband[16], superimposed on the background shot noise. Of thethree sidebands at about (1, 1.2, 2.3) MHz, correspondingto the three axes of vibration, we observe the one at � �1 MHz. It has a width of about 400 Hz due to the Dopplercooling rate.

Our goal is to continuously read the position of the ion,and feed back a damping force proportional to the momen-tum to achieve feedback cooling. For a weakly driven atomthe emitted light is dominantly elastic scattering at the laserfrequency. The information on the motion of the ion isencoded in the sidebands of the scattered light, displacedby the trap frequency �. For an ion trapped in the Lamb-Dicke regime,�

��Np� 1 (N is the mean excitation number

3-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevLett.96.043003


of the motional oscillator), the motional sidebands are thensuppressed by the Lamb-Dicke parameter,�, relative to theelastic component at the laser frequency. In our experi-ment, � � 2�a0=�� 0:07, a0 being the rms size of thetrap ground state. The light reaching the detector will thusbe the sum of the elastic component and weak sidebands[16], a situation reminiscent of homodyne detection, wherea strong oscillator beats with the signal field to provide ahomodyne current at the detector. This physical pictureallows us to formulate the continuous readout of the posi-tion of the ion as well as the quantum feedback cooling inthe well-developed language of homodyne detection andquantum feedback.

The homodyne current at the photodetector (see Fig. 1)with the (large) signal from the elastic light scatteringsubtracted has the form

Ic�t� � ��hzic�t� ��2

r��t�: (1)

The first term is proportional to the conditioned expecta-tion value of the position of the trapped ion, hzic�t�, and thesecond term is a shot-noise contribution with Gaussianwhite noise ��t�. We have defined z � a� ay � z=a0

with a (ay) destruction (creation) operator of the harmonicoscillator, and we have assumed that the trap center isplaced at a distance L � n�=2� �=8 (n integer) fromthe mirror, corresponding to a point of maximum slopeof the standing wave intensity of the mirror mode. Thecurrent Ic�t� scales with � / �, which is the light scatteringrate into the solid angle (4��) of the mirror mode inducedby the laser. The expectation value h�ic � Trf��c�t�g isdefined with respect to a conditioned density operator�c�t�, which reflects our knowledge of the motional stateof the ion for the given history of the photocurrent.According to the theory of homodyne detection, �c�t�obeys the Ito stochastic differential equation [17]

d�c�t� � i�aya; �c�t��dt�L0�c�t�dt

��2��2

qH�c�t�dW�t�; (2)

where H�c�t� � z�c�t� � �c�t�z 2hzic�t��c�t��. Thefirst line determines the unobserved evolution of the ion,including the harmonic motion in the trap with frequency �and the dissipative dynamics, L0, due to photon scattering.The latter is given by

L 0� � ��N � 1�Da�� NDay�� (3)

where we have defined the superoperator Dc�� c�cy �cyc�� cyc�=2. The laser cooling rate � �400 Hz and the steady-state occupation number N �hayai can either be estimated from the motional sidebandsor deduced from the cooling laser parameters [1], whichyields N � 17 for the present experimental parameters.The last term of Eq. (2) is proportional to the Wienerincrement dW�t� � ��t�dt and corresponds to an updateof the observers knowledge about the system according to a

04300

certain measurement result Ic�t�. In summary, Eq. (1) dem-onstrates that observation of the sidebands of the lightscattered into the mirror mode provides us with informa-tion of the position of the ion, while the system densitymatrix is updated according to Eq. (2). This is the basis fordescribing feedback control of the ion, as shown in thefollowing.

For feedback cooling, the vibrational sideband is ex-tracted with a bandpass filter of bandwidth B � 30 kHz,shifted by ��=2�, and amplified, and the resulting outputvoltage is applied to an electrode which is close to the trapinside the vacuum. Thereby we create a driving forcewhich is proportional to (and opposed to) the instantaneousvelocity of the ion and which thus adds to the damping ofits vibration, i.e., cold damping. The overall gain of thefeedback loop depends on the interference contrast, pho-tomultiplier tube (PMT) characteristics, etc., It is variedelectronically by setting the amplification G of the finalamplifier in the loop. We can also set the phase to othervalues than the optimum of ��=2� to compare experi-ment and theory.

To analyze the result of the feedback we look at thechanges in the sideband spectrum. The modified spectrarequire careful interpretation. The spectrum observed in-side the feedback loop (‘‘in-loop’’ or PMT 1 in Fig. 1)shows not only the motion of the ion, but the sum of themotion and the shot noise. As the feedback correlates thesefluctuations, a reduction of the signal below the shot-noiselevel may occur, similar in appearance to signatures ofsqueezed light. This effect is known as ‘‘squashing’’ [18],or ‘‘anticorrelated state of light’’ in an opto-electronic loop[19], and it does not constitute a quantum mechanicalsqueezing of the fluctuations [20,21]. The effect on themotion can only be reliably detected by splitting the opticalsignal before it is measured and recording it outside thefeedback loop with a second PMT (‘‘out-loop’’ in Fig. 1),whose shot noise is not correlated with the motion.

In Fig. 2 we show spectra recorded with the spectrumanalyzer, and measured outside and inside the feedbackloop. The first curve of each row, showing the largestsideband, is the one without feedback (gain G � 0). Theother two curves are recorded with the loop closed (gainvalues G � 1:3 and 8.7). The sub shot-noise fluctuationsinside the loop, when the ion is driven to move in antiphasewith the shot noise, are clearly visible in Fig. 2(f). Themain cooling results are Figs. 2(b) and 2(c), which showthe motional sideband reduced in size and broadened,indicating a reduced energy (proportional to the area underthe curve) and a higher damping rate (the width). Fromcase (b) to (c) the area increases, as the injected andamplified shot noise overcompensates the increased damp-ing. As shown below, in our model the incorporation ofquantum feedback competing with laser cooling predictssuch behavior, i.e., the existence of an optimal gain formaximal cooling (for a detailed description cf. Ref. [22]).

We model the effect of the feedback force acting on theion by extending Eq. (2) with the feedback contribution,

3-2

c)

f)e)d)

b)a)

FIG. 2 (color online). Feedback cooling spectra. The vibra-tional sideband around � � 1 MHz is shown on top of thespectrally flat shot-noise background. The solid line representsa Lorentzian fitting function. The upper curves (a), (b), (c) aremeasured outside the feedback loop, while the lower curves (d),(e), (f) are the in-loop results. Spectra (a) and (d) are for lasercooling only, the other curves are recorded with feedback at theindicated gain values. The feedback phase is set to ��=2�. Thegain values indicate the settings of the final amplifier in thefeedback loop.


d�c�t��fb � i ~GIfb;c�t �z; �c�t��; (4)

where Ifb;c�t� denotes the measured current after the feed-back circuit. The latter is proportional to the voltage ap-plied to the trap electrodes. All conversion factors betweenthe feedback current and the actual force applied on the ionare included in the overall gain ~G / G. The time delay inthe feedback loop preserves causality and is small com-pared to the fastest time scale �1 of the motion of the ionwhich allows us to consider the Markovian limit (! 0�).

To obtain an expression for the feedback current Ifb;c�t�,we change into a frame rotating with the trap frequency �and define the density operator, c�t� � exp�i�ayat� �c�t� exp�i�ayat�, evolving on the (slow) cooling timescales. This is convenient due to the large separationbetween the time scale of the harmonic oscillations, �1

and the time scale of laser and feedback cooling in ourexperiment. For our experimental parameters, �� B��, the feedback current for a phase shift of ��=2� has theform [22]

Ifb;c�t� ��hpic�t� �

��2

r��t�

�cos��t�; (5)

where hpic�t� � Trfpc�t�g, and p � i�ay a� is the mo-mentum operator conjugated to z. The first term in thiscurrent therefore provides damping for the motion of theion. The second term of Eq. (5) describes the shot noisewhich passes through the electronic circuit and is fed backto the ion. The stochastic variable ��t� is Gaussian whitenoise on a time scale given by the inverse bandwidth B1,

04300

whereby B� � implies that it is spectrally flat on thefrequency range of the cooling dynamics.

For a full record of the photocurrent Ic�t�, Eqs. (2) and(4) determine the evolution of the ion’s motional state inthe presence of feedback. As it is impractical to keep trackof the whole photocurrent in the experiment, we derive amaster equation for the density operator averaged over allpossible realizations of Ic�t�, �t�. Along the lines of theWiseman-Milburn theory of quantum feedback [7], for aphase shift of ��=2�, we obtain the quantum feedbackmaster equation [22]

_ � L0 i~G��

4z; p�p�

~G2�16�z; z; ��:

(6)

The second and third terms are the additional contributionsdue to the feedback. The part linear in ~G induces dampingof the motion of the ion, and the term quadratic in ~Gdescribes the effect of the fed back noise leading to diffu-sion of the momentum. The competition between lasercooling, damping, and injected noise leads to the character-istic behavior of the steady-state number expectation value

hniss �N � �� ~G�2N 1�=2�� ~G2=8�

1� 2�� ~G=�: (7)

For small gain, damping dominates, and the energy of theion is decreased below the Doppler limit. For higher gain,the diffusive term describing the noise fed back into thesystem overcompensates cooling, i.e., heats the ion.Consequently, for ��=2� feedback phase and optimalgain conditions the steady-state energy is minimized. Onthe contrary, for a � phase shift �z� replaces p in thesecond term of Eq. (6), the feedback force then merelyinduces a frequency shift, �! � ~G��=2, but no damping.Increasing the gain then always enhances the steady-statenumber expectation value, i.e., the mean ion energy. Wenow compare the theoretical predictions and the experi-mental results for the two selected feedback phases��=2� and �.

The measured ion energy as a function of the feedbackelectronic gain is shown in Fig. 3. On the first side, asexpected, for a ��=2� feedback phase, cooling by morethan 30% below the Doppler limit is achieved, whilefurther increase of the gain drives the shot noise and there-fore heats the motion of the ion. On the other side, a �phase shift in the feedback loop does not yield any damp-ing; in such conditions the motion of the ion is only driven.This results in an increase of the measured sideband area(as shown in the inset of Fig. 3), as well as a shift of thesideband center frequency (graph not shown). Both casesdemonstrate good agreement between experiment and the-ory. Finally, let us stress that the optimal cooling rate isgoverned by the collection efficiency of the fluorescencegoing into the mirror mode, �. In the experiments presentedabove, � � 1% leads to a decrease of the steady-stateoccupation number N from 17 to 12. For the experimen-

3-3

2 4 6 8

0.7

0.8

0.9

1.1

1.2

1.3

1.0

aera dnabedis desilamro

N

Electronic gain

Phase (-π/2)

0.2 0.4 0.6 0.8

1.2

1.4

1.6

1.8

2

2.2

Nor

am

lised

sid

eban

d ar

ea

Electronic gain

Phase π

FIG. 3 (color online). Steady-state energy of the cooled oscil-lator: measured sideband area, normalized to the value withoutfeedback, versus gain of the feedback loop, for ��=2� (mainplot) and � (inset) feedback phase. The curves are the modelcalculations. The gain axis is scaled to the experimental valuesof the electronic gain.


tally realistic � � 15% and under optimal experimentalcooling conditions a final state occupation N � 3 can beachieved, which would allow tests even further in thequantum regime.

To summarize, we have demonstrated real-time feed-back cooling of the motion of a single trapped ion.Electromechanical backaction based on a sensitive real-time measurement of the motion of the ion in the trapallowed us to cool one motional degree of freedom by30% below the Doppler limit. Unlike with laser cooling,the presented method allows us to cool one of the ion’smotional mode without heating the two others, and ourprocedure can easily be extended to cooling all motionalmodes. The cooling process is shot-noise limited and thefraction of scattered photons recorded to observe the mo-tion of the ion limits the ultimate cooling at optimum gain.The latter can yield a steady-state occupation number N �3 for realistic experimental conditions, offering a possibleway to efficiently cool the motion of ions unsuitable forsideband cooling.

This work has been supported by the Austrian ScienceFund (FWF) in the project SFB15, by the EuropeanCommission (QUEST network, HPRNCT-2000-00121,QUBITS network, IST-1999-13021, SCALA IntegratedProject, Contract No. 015714), and by the ‘‘Institut furQuanteninformation GmbH.’’ We thank S. Mancini, A.Masalov, and D. Vitali for clarifying discussions. P. B.thanks the group of theoretical physics at U. Camerino,Italy, for hospitality.

04300

*Present address: Laboratory of Physical Chemistry, ETHZurich, Switzerland.

†Present address: Fachrichtung Technische Physik,Saarland University, Saarbrucken, Germany.

‡Present address: ICFO—Institut de Ciencies Fotoniques,08860 Castelldefels (Barcelona), Spain.

[1] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.Mod. Phys. 75, 281 (2003), and references cited.

[2] M. Keller, B. Lange, K. Hayasaka, W. Lange, andH. Walther, Nature (London) 431, 1075 (2004).

[3] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R.Buck, A. Kuzmich, and H. J. Kimble, Science 303, 1992(2004); P. Maunz, T. Puppe, I. Schuster, N. Syassen,P. W. H. Pinkse, and G. Rempe, Nature (London) 428, 50(2004).

[4] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.Phys. 73, 565 (2001).

[5] J. E. Reiner, W. P. Smith, L. A. Orozco, H. M. Wiseman,and Jay Gambetta, Phys. Rev. A 70, 023819 (2004).

[6] K. C. Schwab and M. L. Roukes, Phys. Today 58, No. 7,36 (2005).

[7] H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548(1993).

[8] S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 80,688 (1998); D. Vitali, S. Mancini, L. Ribichini, andP. Tombesi, Phys. Rev. A 65, 063803 (2002).

[9] D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya, andS. Habib, Phys. Rev. Lett. 92, 223004 (2004).

[10] B. D’Urso, B. Odom, and G. Gabrielse, Phys. Rev. Lett.90, 043001 (2003).

[11] N. V. Morrow, S. K. Dutta, and G. Reithel, Phys. Rev. Lett.88, 093003 (2002).

[12] J. M. Geremia, J. K. Stockton, and H. Mabuchi, Science304, 270 (2004); D. Oblak, P. G. Petrov, C. L. GarridoAlzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L.Sørensen, and E. S. Polzik, Phys. Rev. A 71, 043807(2005).

[13] J. M. W. Milatz and J. J. Van Zolingen, Physica (Utrecht)19, 181 (1953).

[14] P. F. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev.Lett. 83, 3174 (1999).

[15] U. Dorner and P. Zoller, Phys. Rev. A 66, 023816(2002).

[16] P. Bushev, A. Wilson, J. Eschner, C. Raab, F. Schmidt-Kaler, C. Becher, and R. Blatt, Phys. Rev. Lett. 92, 223602(2004); J. Eschner, Ch. Raab, F. Schmidt-Kaler, andR. Blatt, Nature (London) 413, 495 (2001).

[17] C. W. Gardiner and P. Zoller, Quantum Noise (Springer,Berlin, 2004), and references cited.

[18] B. C. Buchler, M. B. Gray, D. A. Shaddock, T. C. Ralph,and D. E. McClelland, Opt. Lett. 24, 259 (1999).

[19] A. V. Masalov, A. A. Putilin, and M. V. Vasilyev, J. Mod.Opt. 41, 1941 (1994).

[20] H. M. Wiseman in Quantum Squeezing (Springer, Berlin,2004), and references cited.

[21] J. H. Shapiro, G. Saplakoglu, S.-T. Ho, P. Kumar,B. E. A. Saleh, and M. C. Teich, J. Opt. Soc. Am. B 4,1604 (1987).

[22] V. Steixner, P. Rabl, and P. Zoller, Phys. Rev. A 72,043826 (2005).

3-4

Shot-Noise-Limited Monitoring and Phase Locking of the Motion of a Single Trapped Ion

P. Bushev,1 G. Hetet,2,3 L. Slodicka,2 D. Rotter,2 M.A. Wilson,2 F. Schmidt-Kaler,4 J. Eschner,5 and R. Blatt2,3

1Physikalisches Institut, Karlsruher Institut fur Technologie, D-76128 Karlsruhe, Germany2Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria

3Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria4QUANTUM, Institut fur Physik, Universitat Mainz, D-55128 Mainz, Germany

5Experimentalphysik, Universitat des Saarlandes, D-66123 Saarbrucken, Germany(Received 10 February 2012; revised manuscript received 5 December 2012; published 26 March 2013)

We perform a high-resolution real-time readout of the motion of a single trapped and laser-cooled Baþ

ion. By using an interferometric setup, we demonstrate a shot-noise-limited measurement of thermal

oscillations with a resolution of 4 times the standard quantum limit. We apply the real-time monitoring for

phase control of the ion motion through a feedback loop, suppressing the photon recoil-induced phase

diffusion. Because of the spectral narrowing in the phase-locked mode, the coherent ion oscillation is

measured with a resolution of about 0.3 times the standard quantum limit.

DOI: 10.1103/PhysRevLett.110.133602 PACS numbers: 42.50.Lc, 03.65.Ta, 37.10.Ty, 42.50.St

The control of a system often relies on gathering infor-mation about its evolution in real time and using it in afeedback loop to drive it into a desired state [1]. In classicalphysics, the accuracy of such feedback operations can intheory be controlled with infinite precision. In quantumphysics, however, there is a fundamental limit to theamount of knowledge one can gain about a system, withimportant consequences for its controllability [2–5]. If oneobserves the motion of a body, the uncertainty in a positionmeasurement inevitably arises due to the Heisenberg rela-tion. A measurement of its position disturbs its momentum,leading to an overall blur of the observed motion. Theresulting displacement uncertainty is minimized in theso-called ‘‘standard measurement procedure’’ which leadsto the standard quantum limit (SQL) [6]. In the case of aharmonic oscillator, the SQL is equal to the position vari-

ance of its ground state �xSQL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih0jr2j0ip ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@=2M!

p,

where M is the mass of the oscillator and ! its resonantfrequency.

Cooling to the quantum mechanical ground state hasbeen demonstrated for elementary harmonic oscillatorssuch as trapped ions [7] and electrons [8], as well as fornanomechanical oscillators [9,10]. The measurement reso-lution achieved in experiments with mechanical oscillatorsof very different sizes typically settles at the level of5�xSQL � 10�xSQL, from the kilogram size oscillator in

gravitational wave astronomy in the LIGO experiment[11], down to various nanomechanical systems [12–14].Only recently, resolution below the standard quantum limithas been demonstrated with nanomechanical oscillators[15,16]. In experiments measuring the trajectory of a singleatom in real time, the resolution achieved so far hasremained bounded by a few �xSQL [17,18].

In this Letter, we present a high-resolution real-timereadout of the motion of a single trapped and laser-cooledbarium ion. The relevant oscillation frequency of one of its

macromotion degrees of freedom is !0=2�� 1 MHz,corresponding to a ground state wave function with aspatial extension of�xSQL � 6 nm. Using an interferomet-

ric setup [19,20], we are able to monitor the trajectory ofthe ion motion with 4�xSQL resolution, only limited by the

resonance fluorescence noise. We furthermore demonstrateefficient suppression of the photon recoil-induced phasediffusion of the single-ion motion by phase locking it to anexternal oscillator. The motion of the locked oscillator isdetected with resolution below the SQL.Figure 1 shows a schematic of the setup. We use a single

138Baþ ion which is held in a miniature Paul trap andcontinuously laser-excited and cooled on its S1=2 to P1=2

transition at 493 nm [21]. The ion oscillates in three non-degenerate modes with frequencies of about 1, 1.2, and2 MHz. A retroreflecting mirror L ¼ 25 cm away from the

PMT

Spectrum analyzer

Mirror

Loop filterAmp

To power of the trap

Ba ion

Trap potential

LO

Quadrature detection

X (t) X (t)

M1

M2

Laser 493 nm

Laser 650 nm1 2

FIG. 1 (color online). Optical and electronic setup for thecontinuous readout of ion oscillation and feedback operation.LO denotes local oscillator, PMT stands for photomultiplier (inthe photon counting mode), M1 and M2 are radio frequencymixers, and Amp is an amplifier. The parabola represents thetrapping potential confining the ion.

PRL 110, 133602 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

29 MARCH 2013

0031-9007=13=110(13)=133602(5) 133602-1 � 2013 American Physical Society


trap and a lens (not shown) are arranged such that theyimage the ion onto itself. The 493 nm fluorescence of theion which is scattered directly towards the photomultiplier(PMT) interferes with the part of the fluorescence that isretroreflected from the mirror. Scanning the ion-mirrordistance then leads to interference fringes at the positionof the PMT with a contrast of up to 73% [19].

For a continuous readout of the ion position, the mirrorposition is fixed in such a way that the ion stays at themidpoint of the fringes [20]. Then, the motion of the ionleads to a modulation of the intensity of the scattered light,i.e., a modulation in the arrival times of the photocounts.When the signal from the PMT is measured with an rfspectrum analyzer, the motion along the trap axis at fre-quency !0 ¼ 2�� 1:039 MHz appears as a clear reso-nance line above the shot-noise background, as shown inFig. 2(a).

The instantaneous displacement of the single trapped ionis defined as xðtÞ ¼ �xSQLðaþ ayÞðtÞ, where a and ay are

the destruction and creation operators of the harmonicoscillator. Using the quantum mechanical formalism intro-duced in our previous work [22], the detected photocurrentiðtÞ is

iðtÞ / I0f1þ Vhsin½2kxðtÞ cos��icg þ �IðtÞ: (1)

Here, I0 is the mean fluorescence rate in counts= sec , V isthe visibility of the interference fringes limited by imper-fect optics and imaging as well as by the motion alongother trap axes, k is the wave vector of the 493 nm light,� ¼ 55� is the projection angle between the trap axis andthe optical axis, and �IðtÞ represents the fluorescence shotnoise. The detected motion of the ion xðtÞ � hxðtÞic ¼TrfxðtÞ�cðtÞg is calculated with the conditioned densityoperator �cðtÞ which accounts for the observation of theelastically scattered photons. In the following, we show

that the sideband signal xðtÞ yields sufficient informationabout the motion of the ion such that it can be used forfeedback.The single Baþ ion is trapped in the Lamb-Dicke

regime; therefore, Eq. (1) can be rewritten as

iðtÞ / I0½1þ 2VkxðtÞ cos�� þ �IðtÞ: (2)

The spectrum of the photocurrent detected by the rf spec-trum analyzer consists of three terms:

hi2ð!Þi / I0 þ 2�I20�ð!Þ þ 4I20V2k2Sxð!Þcos2�: (3)

The first term corresponds to the shot noise, which scaleslinearly with I0; the second one is the dc-offset or meanintensity level, which is outside our observation frequencyrange. The last term reveals the power spectral density ofthe ion motion Sxð!Þ. It is calculated using the fluctuation-dissipation theorem [11,12,23,24] for a harmonic oscillatorcoupled to a thermal bath with temperature T, which yields

Sxð!Þ ¼ 4kBT�

M

1

ð!20 �!2Þ2 þ �2!2

; (4)

whereM is the mass of the ion and � is the damping rate ofthe oscillation. In our case, the measured resonance width� ¼ 2�� 380 Hz directly gives the cooling rate of thetrapped ion [21]. The area under the resonance curve isproportional to the thermal energy of the oscillator,

hx2i ¼ 1

2�

Z 1

0Sxð!Þd! ¼

Z 1

0

~SxðfÞdf: (5)

To calibrate the power spectral density ~SxðfÞ measuredon the spectrum analyzer in terms of displacement, weapply a weak sinusoidal signal to one of the trap electrodes[22,25]. This weak excitation is detuned by 2.5 kHz from!0 and coherently excites the ion motion, as shown inFig. 2(a). The relation between the motional amplitude ofthis coherent oscillation and the spectrum analyzer signalis found by measuring the change of the (time-averaged)fringe contrast at different excitation levels. Using Eq. (1),we then obtain the calibrated spectral density for the ionmotion in nm2=Hz, as shown in Fig. 2(b).The resonance line is well fitted using Eq. (4), which

yields through Eq. (5) the mean (rms) amplitude of the ion

excursion hxðtÞ2i1=2 ¼ 51 nm. This value is higher than theone estimated from the Doppler cooling limit xD � 27 nmset by the linewidth � ¼ 2�� 20:4 MHz of the S to Pcooling transition. That is because the laser detunings andintensities chosen to maximize the signal-to-noise ratio donot provide optimal cooling conditions.From Fig. 2(b), we determine the achievable resolution

of the position measurement, i.e., the minimum attainableuncertainty of any point of the trajectory when the ion’smotion is monitored in real time. It is set by the shot-noisepedestal with spectral density SSN ¼ 1:0 nm2=Hz, which isshown by the dashed line in Fig. 2(b), and corresponds toan excursion of the ion motion of 24 nm, or 4�xSQL

1036

1

Frequency (kHz)104210401038

Spe

ctra

l den

sity

, S

(nm

/H

z)x

6

3

4

5

2

Sig

nal p

ower

(dB

)

-68

-64

-60

-72

1036Frequency (kHz)

104210401038

ba

2~

FIG. 2 (color online). Shot-noise-limited detection of the ion’soscillations. (a) Spectrum of the photocurrent showing the mo-tional sideband (wide resonance line) and the coherent excitation(sharp peak) used for calibration of the ion’s motion. Thespectrum was measured with a 30 Hz resolution bandwidth(RBW). (b) The blue curve is the calibrated spectral density ofthe ion displacement ~SxðfÞ. The solid red line is a Lorentzian fit,as described by Eq. (4). The precision of the position measure-ment is set by the shot-noise background, which is shown as adashed green line.


29 MARCH 2013

133602-2

[11,12]. If the oscillation of the ion was aligned with theoptical axis, the resolution would be improved by a factorof cos� to 2:3�xSQL.

To monitor the ion motion, we employ the homodynetechnique depicted in Fig. 1 (see also Ref. [26]). Part of thedetected PMT signal is mixed with a local oscillator (LO)on two rf mixers M1 and M2 with a �=2 phase shiftbetween them, and the two outputs are low-pass filtered.The resulting signals X1ðtÞ and X2ðtÞ are given by

XiðtÞ ¼Z t

�1Hðt� �Þxið�Þd� ði ¼ 1; 2Þ; (6)

where x1ðtÞ and x2ðtÞ are the in-phase and quadraturecomponents of the homodyne sideband signal (includingshot noise) and Hðt� �Þ is the time response function ofthe filter.

A time trace of the X1 component, with the LO set onand off resonance with the oscillation, is shown in Fig. 3.To observe the motion of the ion in real-time conditions,i.e., with resolution better than the �1 ms coherence timeof the oscillator, we used a filter that matches the oscillatorbandwidth (300 Hz), and the acquisition time resolutionwas set to 0.2 ms. The measured standard deviations are48 nm on and 25 nm off resonance, in good agreement withthe values determined from Fig. 2(b).

To verify that such a homodyne signal bears the infor-mation about the motion, despite the large fraction ofnoise, we now use it to drive the ion motion into a definedcoherent state by phase locking it to the local oscillator. Tothis end, the output of mixer M1 is low-pass filtered with a300 Hz cutoff frequency and amplified, thus providing anoutput proportional to the phase lag between the ion’svibration and the LO. This signal is used to control theintensity of the trap drive in order to change the trapfrequency (see Fig. 1), thereby counteracting the phase

diffusion due to photon recoils. This operation is reminis-cent of a classical phase-locked loop (PLL), whereby thevibration of the single ion plays the role of the voltage-controlled oscillator, synchronized to the phase of the LOreference.To demonstrate the operation of the PLL, the vibrational

mode with 1.04 MHz central frequency is first locked to asinusoidal LO. We show the relevant spectra with andwithout feedback in Fig. 4(a). Without feedback, the side-band spectrum has a Lorentzian shape with about a 500 Hzwidth, determined by the equilibrium between laser cool-ing and recoil heating. In contrast, with the feedback loopclosed, phase locking is clearly observed: The spectrumconsists of a narrow central peak and two side lobes. Thepeak contains about 1=8 of the total motional energy, andphase noise is suppressed over more than a 700 Hz band-width, indicating that the ion motion follows the LOclosely. The side lobes indicate the regime where the phasenoise is increased, around the oscillation frequency of thefeedback loop (servo bumps). The total energy remainsconstant; i.e., the PLL has no damping effect, in contrast tofeedback cooling, as in Ref. [22]. The coherent, phase-locked ion oscillation is detected with much higher preci-sion than the free thermal motion due to the narrowing ofthe resonance line and the detection bandwidth. As can beseen from Fig. 4(b), the width of the coherent line is not yetresolved with a 1 Hz resolution bandwidth. Using 1 Hz asthe upper limit, we find that the coherent motion is detectedwith better than 0:3�xSQL resolution against the residual

phase noise pedestal in the closed-loop operation: Thespectral density of the residual noise of 2:6 nm2=Hz cor-responds to motion with a 1.6 nm mean excursion.We demonstrate further the potential of the phase lock

by the response to a frequency-modulated (FM) reference,whose spectrum consists of several bands separated by themodulation frequency. The modulation frequency and theindex are chosen such that the FM harmonics do notcoincide with 50 Hz noise, and most of the spectral energyfalls into the bandwidth of the loop filter. The resultingspectrum of the motional sideband is displayed in Fig. 4(c),with the spectrum of the LO as a reference. The spectrumof ion motion clearly reproduces the three main bands ofthe FM reference. This measurement shows, in conjunctionwith Fig. 4(b), that we can encode and decode phase orfrequency information into the motion of the single trappedatom with sensitivity below the standard quantum limit.To resolve the coherent peak in the locked state even

better, we use fast Fourier tranform (FFT) analysis of the

second order correlation function gð2Þ, following Ref. [27].Figure 4(d) shows the central portion of the FFT spectrum,now using a 50 mHz resolution. The resulting linewidth ofthe locked signal is 120 mHz. This residual width isdetermined mainly by the fact that the time base used in

recording gð2Þ is not synchronized to the 10 MHz clockused for generating the local oscillator. In principle, the

Sideband Shot noiseIn-p

hase

dis

plac

emen

t, X

(nm

)

-100

0

50

-50

100

0Time (ms)

300200100 400

1

FIG. 3 (color online). Time evolution of the in-phase homo-dyne signal X1ðtÞ measured when the local oscillator is tuned inresonance with the ion’s oscillation (red curve, T ¼ 0 to 200 ms)and when it is tuned to the off-resonant detection of shot noiseonly (blue curve, T ¼ 200 to 400 ms). The data are filtered witha 300 Hz bandwidth, and the sampling rate was ð0:2 msÞ�1.


29 MARCH 2013

133602-3

narrow linewidth would allow deriving a new limit for theresolution of the ion motion, but calibration proves lessreliability for the FFT spectrum than for the one recordedwith the rf spectrum analyzer.

In conclusion, we have demonstrated a shot-noise-limited continuous measurement of the thermal oscillationof a single ion with 4�xSQL resolution. The resolution

could be moved closer to the standard quantum limit bycollecting more fluorescent light or by introducing asteeper gradient in the interferometric detection of themotion. The first method would employ the use of a lenswith a higher numerical aperture [28,29]; the second onemay be implemented by setting up the trap inside an opticalcavity [30,31]. We have also realized the phase control ofthe ion motion, suppressing the photon recoil-inducedphase diffusion through a PLL-type feedback loop.Because of the narrowing of the resonance line in thePLL mode, the coherent ion oscillation is measured witha 0:3�xSQL resolution.

The measured trajectory of the ion has three noise con-tributions arising from its ground state motion, the shotnoise of the detected fluorescence, and the recoil back-action of the scattered photons [32]. The latter also deter-mines the steady state of the motion through the lasercooling process. The PLL control diminishes the shot-noise contribution by narrowing the observed line. Theresulting improvement of the displacement resolutiondoes not affect the quantum mechanical part of the trajec-tory �x associated with the zero-point motion of theion and hence does not influence its momentum uncer-tainty �p ’ @=2�x. Also, the backaction part remainsunchanged since the PLL control stabilizes the phase butdoes not alter the temperature of the ion.The potential of the PLL control has been demonstrated

by synchronizing the ion’s oscillation to a modulated ref-erence signal. The phase locking technique may be used toattain the common-mode rejection of recoil effects inphotonic atom-atom interactions [33,34]. It may also becombined with feedback cooling [22] in order to imple-ment combined amplitude and phase control.This work has been supported by the Austrian Science

Fund (SFB15), the European Commission (QUEST,HPRNCT-2000-00121; QUBITS, IST-1999-13021), theSpanish MICINN (QOIT, CSD2006-00019; QNLP,FIS2007-66944), and the ‘‘Institut fur Quantenin-formation GmbH.’’ P. B. and J. E. thank M. Chwalla, M.Aspelmeyer, and G. Morigi for stimulating discussions.

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1037 1038 1039

-78

-76

-74

-72

-70

Sig

nal p

ower

(dB

)

Frequency (kHz)

c

-75

1038

-70

10391037

10

100

Detuning (Hz)0 50-50

b

10-10 0

~120 mHz

Detuning (Hz)

Sig

nal p

ower

den

sity

(a.

u)

10

20

d

a

Frequency (kHz)

-80Sig

nal p

ower

(dB

)

FIG. 4 (color online). Phase locking of a trapped ion’s vibra-tion. (a) Motional sideband spectra when the PLL is off (bluecurve) and when the PLL is on (red curve), measured withRBW ¼ 10 Hz. (b) Calibrated spectral density ~SxðfÞ aroundthe central peak of the phase-locked ion motion, measuredwith a 1 Hz RBW. The dashed line indicates the residual noisefloor of 2:6 nm2=Hz. (c) Sideband spectrum when the ion islocked to an FM signal with modulation frequency 56.3 Hz andmodulation index 1 (red curve). The spectrum of the FMreference is also shown (blue curve). (d) Center peak of thephase-locked ion motion measured with a 50 mHz resolution,using the FFT of the gð2Þ correlation function [27].


29 MARCH 2013

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http://dx.doi.org/10.1038/35102129







VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER1999

igh-anicaltheor a

3174

Cooling of a Mirror by Radiation Pressure

P. F. Cohadon,* A. Heidmann,† and M. Pinard‡

Laboratoire Kastler Brossel,§ Case 74, 4 place Jussieu, F75252 Paris Cedex 05, France(Received 30 March 1999; revised manuscript received 24 June 1999)

We describe an experiment in which a mirror is cooled by the radiation pressure of light. A hfinesse optical cavity with a mirror coated on a mechanical resonator is used as an optomechsensor of the Brownian motion of the mirror. A feedback mechanism controls this motion viaradiation pressure of a laser beam reflected on the mirror. We have observed either a coolingheating of the mirror, depending on the gain of the feedback loop.

PACS numbers: 42.50.Lc, 04.80.Nn, 05.40.Jc

e5].

isnnn

.ic,

eyntedr

ed

Thermal noise is a basic limit for many very sensitive optical measurements such as interferomegravitational-wave detection [1–3]. Brownian motion osuspended mirrors can be decomposed into suspenand internal thermal noises. The latter is due to thermainduced deformations of the mirror surface and constituthe major limitation of gravitational-wave detectors in thintermediate frequency domain [4,5]. Observation acontrol of this noise have thus become an important issin precision measurements [6–8]. In order to reduthermal noise effects, it is not always possible to lowthe temperature and other techniques have been proposuch as feedback control [9].

In this Letter we report the first experimental observatiof the cooling of a mirror by feedback control. The principle of the experiment is to detect the Brownian motioof the mirror with an optomechanical sensor and thenfreeze the motion by applying an electronically controlleradiation pressure on the mirror. Mechanical effectslight on macroscopic objects have already been observsuch as the dissipative effects of electromagnetic radiat[10], the optical bistability and mirror confinement incavity induced by radiation pressure [11], or the regulatiof the mechanical response of a microcantilever by feeback via the photothermal force [12]. In our experimethe radiation pressure is driven by the feedback loopsuch a way that a viscous force is applied to the mirrIt thus plays a role somewhat similar to the one in opticmolasses for atoms.

The cooling mechanism can be understood from texperimental setup shown in Fig. 1. The mirror is usedthe rear mirror of a single-ended Fabry-Pérot cavity. Tphase of the field reflected by the cavity is very sensitito changes in the cavity length [13–15]. For a resonacavity, a displacementdx of the rear mirror induces aphase shiftdwx of the reflected field on the order of

dwx �8Fl

dx , (1)

where F is the cavity finesse andl is the opticalwavelength. This signal is superimposed to the quantphase noise of the reflected beam. Provided thatcavity finesse is high enough, this quantum noise

0031-9007�99�83(16)�3174(4)$15.00

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negligible and the Brownian motion of the mirror can bdetected by measuring the phase of the reflected field [1

To cool the mirror we use an auxiliary laser beamreflected from the back on the mirror. This beamintensity modulated by an acousto-optic modulator driveby the feedback loop so that a modulated radiatiopressure is applied to the mirror. The resulting motion cabe described by its Fourier transformdx�V� at frequencyV which is proportional to the applied forces

dx�V� � x�V� �FT �V� 1 Frad�V�� , (2)

wherex�V� is the mechanical susceptibility of the mirrorIf we assume that the mechanical response is harmonthis susceptibility has a Lorentzian shape

x�V� �1

M�V2M 2 V2 2 iGV�

, (3)

characterized by a massM, a resonance frequencyVM ,and a dampingG related to the quality factorQ of themechanical resonance byG � VM�Q.

The forceFT in Eq. (2) is a Langevin force responsiblefor the Brownian motion of the mirror. At thermal equi-librium its spectrumSFT �V� is related to the mechanical

FIG. 1. Experimental setup. The Brownian motion of thmirror is detected with a high-finesse cavity. A frequenc(F. stab.) and intensity (I. stab.) stabilized light beam is seinto the cavity, and the phase of the reflected field is measurby homodyne detection. This signal is fed back to the mirrovia the radiation pressure exerted by a beam with modulatintensity (AOM).

© 1999 The American Physical Society

VOLUME 83, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 18 OCTOBER 1999

susceptibility by the fluctuation-dissipation theorem

SFT �V� � 22kBT

VIm

√1

x�V�

!� 2MGkBT , (4)

where T is the temperature. The resulting thermal noisespectrum ST

x �V� of the mirror motion has a Lorentzianshape centered at frequency VM and of width G.

The second force Frad in Eq. (2) is the radiation pres-sure exerted by the auxiliary laser beam and modulated bythe feedback loop. Neglecting the quantum phase noisein the control signal, this force is proportional to the dis-placement dx of the mirror [Eq. (1)]. We choose thefeedback gain in such a way that the radiation pressureis proportional to the speed y � iVdx of the mirror

Frad�V� � iMVgdx�V� , (5)

where g is related to the electronic gain. The radiationpressure exerted by the auxiliary laser beam thus corre-sponds to an additional viscous force for the mirror. Theresulting motion is given by

dx�V� �1

M�V2M 2 V2 2 i�G 1 g�V�

FT �V� . (6)

This equation is similar to the one obtained without feed-back [Eq. (2) with Frad � 0] except that the radiationpressure changes the damping without adding any fluc-tuations. The noise spectrum S

fbx �V� of the mirror mo-

tion still has a Lorentzian shape but with a different widthGfb � G 1 g and a different height. The variation ofheight can be characterized by the amplitude noise reduc-tion R at resonance frequency

R �

vuut STx �VM�

Sfbx �VM�

�Gfb

G�

G 1 gG

. (7)

The resulting motion is then equivalent to a thermalequilibrium at a different temperature Tfb which can beeither reduced or increased depending on the sign of thegain g

Tfb

T�

G

Gfb�

G

G 1 g. (8)

We now describe our experiment. The mirror is coatedon the plane side of a small plano-convex mechanicalresonator made of silica (Fig. 1). The coating has beenmade at the Institut de Physique Nucléaire de Lyonon a1.5 mm thick substrate with a diameter of 14 mm and acurvature radius of the convex side of 100 mm. Internalacoustic modes correspond to Gaussian modes confinedaround the central axis of the resonator [16,17]. Thefundamental mode studied in this paper is a compressionmode with a waist equal to 3.4 mm and a resonance fre-quency close to 2 MHz [15,17]. The front mirror of thecavity is a Newport high-finesse SuperMirror(curvatureradius � 1 m, transmission � 50 ppm) held at 1 mmfrom the rear mirror. We have measured the followingparameters of the cavity: free spectral range � 141 GHz,cavity bandwidth � 1.9 MHz, beam waist � 90 mm.These values correspond to a finesse F of 37 000.

The light entering the cavity is provided by a titane-sapphire laser working at 810 nm and frequency locked toa stable external cavity which is locked to a resonance ofthe high-finesse cavity by monitoring the residual lighttransmitted by the rear mirror. The beam is intensitystabilized and spatially filtered by a mode cleaner. Onegets a 100 mW incident beam on the high-finesse cavitywith a mode matching of 98%. The phase of thefield reflected by the cavity is measured by homodynedetection. The reflected field is mixed with a 10 mWlocal oscillator and a servoloop monitors the length ofthe local oscillator arm so that we measure the phasefluctuations of the reflected field. This signal is sent bothto the feedback loop and to a spectrum analyzer.

The feedback loop consists of an amplifier with variablegain and phase which drives the acousto-optic modulator.The 500 mW auxiliary beam is uncoupled from the high-finesse cavity by a frequency shift of the acousto-opticmodulator (200 MHz) and by a tilt angle of 10± withrespect to the cavity axis. We have checked that thisbeam has no spurious effect on the homodyne detection.A bandpass filter centered at the fundamental resonancefrequency of the mirror is also inserted in the feedbackloop to reduce its saturation. For large gains, the radiationpressure Frad can become of the same order as theLangevin force FT , and it must be restricted in frequencyin order to get a finite variance. The electronic filterhas a quality factor of 200 and limits the efficiency ofthe feedback loop to a bandwidth of 9 kHz around thefundamental resonance frequency.

Figure 2 shows the phase noise spectrum of the re-flected field obtained by an average of 1000 scans of thespectrum analyzer with a resolution bandwidth of 10 Hz.Curve a is obtained at room temperature without feed-back. It reproduces the thermal noise spectrum ST

x �V�of the mirror which is peaked around the fundamental

FIG. 2. Phase noise spectrum of the reflected field normalizedto the shot-noise level for a frequency span of 1 kHz aroundthe fundamental resonance frequency of the mirror. The peakreflects the Brownian motion of the mirror without feedback (a)and with feedback for increasing gains (b) to (d).

3175


resonance frequency (1858.9 kHz) with a width G�2p of45 Hz (mechanical quality factor Q � 40 000). The spec-trum is normalized to the shot-noise level, and it clearlyappears that the thermal noise is much larger than thequantum phase noise [15].

Curves b to d are obtained with feedback for increasingelectronic gains. The phase of the amplifier is adjustedto maximize the correction at resonance. From Eqs. (5)and (6) this corresponds to a global imaginary gain for theloop and to a purely viscous radiation pressure force. Thecontrol of the mirror motion is clearly visible on thosecurves. The thermal peak is strongly reduced while itswidth is increased. The amplitude noise reduction R atresonance is larger than 20 for large gains.

The effective temperature Tfb can be deduced from thevariance Dx2 of the mirror motion which is equal to theintegral of the spectrum S

fbx �V�. From Eqs. (4), (6), and

(8) one gets the usual relation for a harmonic oscillator atthermal equilibrium

12

MV2MDx2 �

12

kBTfb . (9)

The decrease of the area of the thermal peak observed inFig. 2 thus corresponds to a cooling of the mirror.

Figure 3 shows the effect of feedback for a reverse gain(g , 0). Noise spectra are obtained by an average of500 scans with a resolution bandwidth of 1 Hz. Curve bexhibits a strong increase of the thermal peak which nowcorresponds to a heating of the mirror. The feedback alsoreduces the damping from G to G 2 jgj, thus increasingthe quality factor of the resonance. We have obtaineda maximum effective quality factor of 2.2 3 106 (Gfb �G�50), limited by the saturation of the feedback loop.

It is instructive to study the efficiency of the coolingor heating mechanism with respect to the gain of thefeedback loop. Figure 4 shows the variation of thedamping Gfb�G, of the amplitude noise reduction R at

FIG. 3. Phase noise spectrum of the reflected field in dB scalenormalized to the shot-noise level for a frequency span of250 Hz, without feedback (a) and with feedback for a negativegain (b).

3176

resonance, and of the cooling factor T�Tfb , as a functionof the feedback gain. These parameters are derived fromthe experimental spectra by Lorentzian fits which givethe width and the area of the thermal peak, the latterbeing related to the effective temperature by Eq. (9). Tomeasure the feedback gain, we detect the intensity ofthe auxiliary beam after reflection on the mirror. Theratio between the modulation spectrum of this intensityat frequency VM and the noise spectrum S

fbx �VM� is

proportional to the gain g. This measurement takes intoaccount any nonlinearity of the gain due to a possiblesaturation of the acousto-optic modulator.

As expected from Eqs. (6) and (7), the damping and theamplitude noise reduction R have a linear dependencewith the gain, as well for cooling (g . 0) as for heating(g , 0). The straight line in Fig. 4 is in excellentagreement with experimental data and allows one tonormalize the gain g to the damping G, as this has beendone in the figure.

This figure also shows that large cooling factors T�Tfb

can be obtained. This cooling factor does not, however,evolve linearly with the gain as it would be expected fromEq. (8). This is due to the presence of a backgroundthermal noise visible in Fig. 2. This noise is related toall other acoustic modes of the mirror and to the thermalnoise of the coupling mirror of the cavity. The feedbackloop does not have the same effect on this noise andon the fundamental thermal peak. The solid curve inFig. 4 corresponds to a theoretical model in which thebackground noise is assumed to be unchanged by thefeedback. As a consequence, only the fundamental modeis cooled at a temperature Tfb , whereas all other modes stayin thermal equilibrium at the initial temperature T . Theresulting cooling factor T�Tfb is in excellent agreementwith experimental data.

The cooling mechanism is not limited to the mechani-cal resonance frequencies. Figure 5 shows the cool-ing obtained at frequencies well below the fundamental

FIG. 4. Variation of the damping Gfb�G (squares), of theamplitude noise reduction R at resonance (circles), and of thecooling factor T�Tfb (diamonds), as a function of the feedbackgain g normalized to G. Solid curves are theoretical results.


FIG. 5. Cooling far from resonance (800 kHz). The back-ground thermal noise (a) is reduced in the presence of feedback(b). Curves represent the phase noise spectra of the reflectedfield in dB scale normalized to the shot-noise level.

resonance frequency. The electronic filter is now centeredaround 800 kHz and the feedback loop reduces the back-ground thermal noise (curve a). The width of the noisereduction (curve b) is related to the filter bandwidth. Thephase and the gain of the feedback loop have been ad-justed since the electronic gain g has now to be comparedto the real part MV

2M of the inverse of the mechanical

susceptibility at low frequency [Eq. (6)]. Note that theamplitude of the radiation pressure exerted by the auxil-iary laser beam is, however, approximately the same as inthe resonant case. Large noise reduction is actually ob-tained when the radiation pressure Frad is on the order ofthe Langevin force FT whose amplitude is independent offrequency [Eq. (2)].

In conclusion, we have observed a thermal noise reduc-tion by a factor of 20 near the fundamental resonance fre-quency. The radiation pressure exerted by the feedbackloop corresponds to a viscous force which increases thedamping of the mirror without adding thermal fluctuations.For large gains, the thermal peak of the fundamental modebecomes of the same order as the background thermal noiseand no global thermal equilibrium is reached. As far as aneffective temperature can be defined for the fundamentalmode, we have obtained a reduction of this temperatureby a factor of 40. We have also observed a heating ofthe mirror for reverse feedback gains, and a cooling of thebackground thermal noise at low frequencies.

The cooling mechanism demonstrated in this paper maybe useful to increase the sensitivity of gravitational-waveinterferometers. The main difficulty is to freeze the ther-mal noise without changing the effect of the signal. Wepropose in Fig. 6 a possible scheme to control the ther-mal noise of one mirror of the interferometer. A cav-ity performs a local measurement of the mirror motionwhich is fed back to the mirror via the radiation pressureof a laser beam. The coupling mirror of the cavity is a

Interferometer

Mirror

Cavity

Feedback

FIG. 6. Application of the cooling mechanism to reduce thethermal noise in a gravitational-wave interferometer.

small plano-convex mirror with a high mechanical reso-nance frequency and a low background thermal noise atlow frequency. As a consequence the cavity measures thethermal noise of the mirror of the interferometer. For ashort cavity, this measurement is not sensitive to a gravi-tational wave and the cooling can reduce the backgroundthermal noise at the gravitational-wave frequencies with-out changing the response of the interferometer.

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]§Laboratoire de l’Université Pierre et Marie Curie et del’Ecole Normale Supérieure associé au Centre Nationalde la Recherche Scientifique

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Observation of a kilogram-scale oscillator near itsquantum ground state

B Abbott1, R Abbott1, R Adhikari1, P Ajith2, B Allen2,3, G Allen4,R Amin5, S B Anderson1, W G Anderson3, M A Arain6, M Araya1,H Armandula1, P Armor3, Y Aso7, S Aston8, P Aufmuth9,C Aulbert2, S Babak10, S Ballmer1, H Bantilan11, B C Barish1,C Barker12, D Barker12, B Barr13, P Barriga14, M A Barton13,M Bastarrika13, K Bayer15, J Betzwieser1, P T Beyersdorf16,I A Bilenko17, G Billingsley1, R Biswas3, E Black1, K Blackburn1,L Blackburn15, D Blair14, B Bland12, T P Bodiya15, L Bogue18,R Bork1, V Boschi1, S Bose19, P R Brady3, V B Braginsky17,J E Brau20, M Brinkmann2, A Brooks1, D A Brown21, G Brunet15,A Bullington4, A Buonanno22, O Burmeister2, R L Byer4,L Cadonati23, G Cagnoli13, J B Camp24, J Cannizzo24,K Cannon1, J Cao15, L Cardenas1, T Casebolt4, G Castaldi25,C Cepeda1, E Chalkley13, P Charlton26, S Chatterji1,S Chelkowski8, Y Chen10,27, N Christensen11, D Clark4, J Clark13,T Cokelaer28, R Conte29, D Cook12, T Corbitt15,56, D Coyne1,J D E Creighton3, A Cumming13, L Cunningham13, R M Cutler8,J Dalrymple21, S Danilishin17, K Danzmann2,9, G Davies28,D DeBra4, J Degallaix10, M Degree4, V Dergachev30, S Desai31,R DeSalvo1, S Dhurandhar32, M Díaz33, J Dickson34, A Dietz28,F Donovan15, K L Dooley6, E E Doomes35, R W P Drever36,I Duke15, J-C Dumas14, R J Dupuis1, J G Dwyer7, C Echols1,A Effler12, P Ehrens1, E Espinoza1, T Etzel1, T Evans18,S Fairhurst28, Y Fan14, D Fazi1, H Fehrmann2, M M Fejer4,L S Finn31, K Flasch3, N Fotopoulos3, A Freise8, R Frey20,T Fricke1,37, P Fritschel15, V V Frolov18, M Fyffe18, J Garofoli12,I Gholami10, J A Giaime5,18, S Giampanis37, K D Giardina18,K Goda15, E Goetz30, L Goggin1, G González5, S Gossler2,R Gouaty5, A Grant13, S Gras14, C Gray12, M Gray34,R J S Greenhalgh38, A M Gretarsson39, F Grimaldi15, R Grosso33,H Grote2, S Grunewald10, M Guenther12, E K Gustafson1,R Gustafson30, B Hage9, J M Hallam8, D Hammer3, C Hanna5,J Hanson18, J Harms2, G Harry15, E Harstad20, K Hayama33,

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T Hayler38, J Heefner1, I S Heng13, M Hennessy4, A Heptonstall13,M Hewitson2, S Hild8, E Hirose21, D Hoak18, D Hosken40,J Hough13, S H Huttner13, D Ingram12, M Ito20, A Ivanov1,B Johnson12, W W Johnson5, D I Jones41, G Jones28, R Jones13,L Ju14, P Kalmus7, V Kalogera42, S Kamat7, J Kanner22,D Kasprzyk8, E Katsavounidis15, K Kawabe12, S Kawamura43,F Kawazoe43, W Kells1, D G Keppel1, F Ya Khalili17, R Khan7,E Khazanov44, C Kim42, P King1, J S Kissel5, S Klimenko6,K Kokeyama43, V Kondrashov1, R K Kopparapu31, D Kozak1,I Kozhevatov44, B Krishnan10, P Kwee9, P K Lam34, M Landry12,M M Lang31, B Lantz4, A Lazzarini1, M Lei1, N Leindecker4,V Leonhardt43, I Leonor20, K Libbrecht1, H Lin6, P Lindquist1,N A Lockerbie45, D Lodhia8, M Lormand18, P Lu4, M Lubinski12,A Lucianetti6, H Lück2,9, B Machenschalk2, M MacInnis15,M Mageswaran1, K Mailand1, V Mandic46, S Márka7, Z Márka7,A Markosyan4, J Markowitz15, E Maros1, I Martin13, R M Martin6,J N Marx1, K Mason15, F Matichard5, L Matone7, R Matzner47,N Mavalvala15, R McCarthy12, D E McClelland34, S C McGuire35,M McHugh48, G McIntyre1, G McIvor47, D McKechan28,K McKenzie34, T Meier9, A Melissinos37, G Mendell12,R A Mercer6, S Meshkov1, C J Messenger2, D Meyers1, H Miao14,J Miller1,13, J Minelli31, S Mitra32, V P Mitrofanov17,G Mitselmakher6, R Mittleman15, O Miyakawa1, B Moe3,S Mohanty33, G Moreno12, K Mossavi2, C Mow-Lowry34,G Mueller6, S Mukherjee33, H Mukhopadhyay32,H Müller-Ebhardt2, J Munch40, P Murray13, E Myers12, J Myers12,T Nash1, J Nelson13, G Newton13, A Nishizawa43, K Numata24,J O’Dell38, G Ogin1, B O’Reilly18, R O’Shaughnessy31,D J Ottaway15, R S Ottens6, H Overmier18, B J Owen31, Y Pan22,C Pankow6, M A Papa3,10, V Parameshwaraiah12, P Patel1,M Pedraza1, S Penn49, A Perreca8, T Petrie31, I M Pinto25,M Pitkin13, H J Pletsch2, M V Plissi13, F Postiglione29,M Principe25, R Prix2, V Quetschke6, F Raab12, D S Rabeling34,H Radkins12, N Rainer2, M Rakhmanov50, M Ramsunder31,H Rehbein2, S Reid13, D H Reitze6, R Riesen18, K Riles30,B Rivera12, N A Robertson1,13, C Robinson28,E L Robinson8, S Roddy18, A Rodriguez5, A M Rogan19,J Rollins7, J D Romano33, J Romie18, R Route4,S Rowan13, A Rüdiger2, L Ruet15, P Russell1, K Ryan12,S Sakata43, M Samidi1, L Sancho de la Jordana51, V Sandberg12,

New Journal of Physics 11 (2009) 073032 (http://www.njp.org/)

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V Sannibale1, S Saraf52, P Sarin15, B S Sathyaprakash28,S Sato43, P R Saulson21, R Savage12, P Savov27, S W Schediwy14,R Schilling2, R Schnabel2, R Schofield20, B F Schutz10,28,P Schwinberg12, S M Scott34, A C Searle34, B Sears1, F Seifert2,D Sellers18, A S Sengupta1, P Shawhan22, D H Shoemaker15,A Sibley18, X Siemens3, D Sigg12, S Sinha4, A M Sintes10,51,B J J Slagmolen34, J Slutsky5, J R Smith21, M R Smith1,N D Smith15, K Somiya2,10, B Sorazu13, L C Stein15, A Stochino1,R Stone33, K A Strain13, D M Strom20, A Stuver18,T Z Summerscales53, K-X Sun4, M Sung5, P J Sutton28,H Takahashi10, D B Tanner6, R Taylor1, R Taylor13, J Thacker18,K A Thorne31, K S Thorne27, A Thüring9, K V Tokmakov13,C Torres18, C Torrie13, G Traylor18, M Trias51, W Tyler1,D Ugolini54, J Ulmen4, K Urbanek4, H Vahlbruch9, C Van DenBroeck28, M van der Sluys42, S Vass1, R Vaulin3, A Vecchio8,J Veitch8, P Veitch40, A Villar1, C Vorvick12, S P Vyatchanin17,S J Waldman1, L Wallace1, H Ward13, R Ward1, M Weinert2,A Weinstein1, R Weiss15, S Wen5, K Wette34, J T Whelan10,S E Whitcomb1, B F Whiting6, C Wilkinson12, P A Willems1,H R Williams31, L Williams6, B Willke2,9, I Wilmut38, W Winkler2,C C Wipf15, A G Wiseman3, G Woan13, R Wooley18, J Worden12,W Wu6, I Yakushin18, H Yamamoto1, Z Yan14, S Yoshida50,M Zanolin39, J Zhang30, L Zhang1, C Zhao14, N Zotov55,M Zucker15 and J Zweizig1 (LIGO Scientific Collaboration)1 LIGO—California Institute of Technology, Pasadena, CA 91125, USA2 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167Hannover, Germany3 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA4 Stanford University, Stanford, CA 94305, USA5 Louisiana State University, Baton Rouge, LA 70803, USA6 University of Florida, Gainesville, FL 32611, USA7 Columbia University, New York, NY 10027, USA8 University of Birmingham, Birmingham B15 2TT, UK9 Leibniz Universität Hannover, D-30167 Hannover, Germany10 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik,D-14476 Golm, Germany11 Carleton College, Northfield, MN 55057, USA12 LIGO Hanford Observatory, Richland, WA 99352, USA13 University of Glasgow, Glasgow G12 8QQ, UK14 University of Western Australia, Crawley, WA 6009, Australia15 LIGO—Massachusetts Institute of Technology, Cambridge, MA 02139, USA16 San Jose State University, San Jose, CA 95192, USA17 Moscow State University, Moscow, 119992, Russia


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18 LIGO Livingston Observatory, Livingston, LA 70754, USA19 Washington State University, Pullman, WA 99164, USA20 University of Oregon, Eugene, OR 97403, USA21 Syracuse University, Syracuse, NY 13244, USA22 University of Maryland, College Park, MD 20742 USA23 University of Massachusetts, Amherst, MA 01003 USA24 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA25 University of Sannio at Benevento, I-82100 Benevento, Italy26 Charles Sturt University, Wagga Wagga, NSW 2678, Australia27 Caltech-CaRT, Pasadena, CA 91125, USA28 Cardiff University, Cardiff, CF24 3AA, UK29 University of Salerno, 84084 Fisciano (Salerno), Italy30 University of Michigan, Ann Arbor, MI 48109, USA31 The Pennsylvania State University, University Park, PA 16802, USA32 Inter-University Centre for Astronomy and Astrophysics, Pune-411007, India33 The University of Texas at Brownsville and Texas Southmost College,Brownsville, TX 78520, USA34 Australian National University, Canberra, 0200, Australia35 Southern University and A& M College, Baton Rouge, LA 70813, USA36 California Institute of Technology, Pasadena, CA 91125, USA37 University of Rochester, Rochester, NY 14627, USA38 Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK39 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA40 University of Adelaide, Adelaide, SA 5005, Australia41 University of Southampton, Southampton SO17 1BJ, UK42 Northwestern University, Evanston, IL 60208, USA43 National Astronomical Observatory of Japan, Tokyo 181-8588, Japan44 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia45 University of Strathclyde, Glasgow G1 1XQ, UK46 University of Minnesota, Minneapolis, MN 55455, USA47 The University of Texas at Austin, Austin, TX 78712, USA48 Loyola University, New Orleans, LA 70118, USA49 Hobart and William Smith Colleges, Geneva, NY 14456, USA50 Southeastern Louisiana University, Hammond, LA 70402, USA51 Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain52 Sonoma State University, Rohnert Park, CA 94928, USA53 Andrews University, Berrien Springs, MI 49104 USA54 Trinity University, San Antonio, TX 78212, USA55 Louisiana Tech University, Ruston, LA 71272, USAE-mail: [email protected]

New Journal of Physics 11 (2009) 073032 (13pp)Received 20 February 2009Published 16 July 2009Online at http://www.njp.org/doi:10.1088/1367-2630/11/7/073032

56 Author to whom any correspondence should be addressed.


mailto:[email protected]

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Abstract. We introduce a novel cooling technique capable of approachingthe quantum ground state of a kilogram-scale system—an interferometricgravitational wave detector. The detectors of the Laser InterferometerGravitational-wave Observatory (LIGO) operate within a factor of 10 of thestandard quantum limit (SQL), providing a displacement sensitivity of 10−18 min a 100 Hz band centered on 150 Hz. With a new feedback strategy, wedynamically shift the resonant frequency of a 2.7 kg pendulum mode to lie withinthis optimal band, where its effective temperature falls as low as 1.4 µK, and itsoccupation number reaches about 200 quanta. This work shows how the exquisitesensitivity necessary to detect gravitational waves can be made available to probethe validity of quantum mechanics on an enormous mass scale.

Contents

1. The LIGO interferometers 62. The cooling mechanism 63. Measurement results and discussion 94. Cooling to the quantum limit 105. Future prospects with LIGO 11Acknowledgments 11References 11

Observation of quantum effects such as ground state cooling [1]–[15], quantum jumps [16],optical squeezing [17], mechanical squeezing [18]–[20] and entanglement [21]–[26] thatinvolve macroscopic mechanical systems are the subject of intense experimental effort [27].The first step toward engineering a non-classical state of a mechanical oscillator is to coolit, minimizing the thermal occupation number of the mode. Any mechanical coupling to theenvironment admits thermal noise that randomly drives the system’s motion, as dictated by thefluctuation–dissipation theorem [28], but ‘cold’ frictionless forces, such as optical or electronicfeedback, can suppress this motion, hence cooling the oscillator.

Two types of forces have recently proven valuable for cooling. The first is africtionless damping force, originating either from an electronic servo system (‘cold damping’)[4, 29, 30] or from photothermal or radiation pressure forces in a detuned cavity (‘cavitycooling’) [1]–[3], [5, 6, 8]; this force reduces the motion of the oscillator while also diminishingits quality factor. The second is an optical restoring force, which increases the resonantfrequency of the oscillator without additional friction, effectively increasing its quality factor[7, 10]. To reach the quantum regime in experiments exploiting these techniques, a low noiseoscillator’s position must be monitored by a highly sensitive readout device. By providingboth of these features, the Laser Interferometer Gravitational-wave Observatory (LIGO)interferometers present a unique opportunity to cool kilogram-scale mirrors to enticingly lowtemperatures. Although the LIGO interferometers do not have sufficiently large optical restoringforces for the second effect to be significant, their active control systems may instead be used toreproduce the effect.


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1. The LIGO interferometers

LIGO operates three kilometer-scale interferometric detectors with the goal of directly detectinggravitational waves of astrophysical origin [31, 32]. The measurements reported here wereperformed at LIGO’s Hanford Observatory. The detector shown in figure 1 comprises aMichelson interferometer with a 4 km long Fabry–Perot cavity of finesse 220 placed in eacharm to increase the sensitivity of the detector. Each mirror of the interferometer has massM = 10.8 kg, and is suspended from a vibration-isolated platform on a fine wire to form apendulum with frequency �p = 0.74 Hz, to shield it from external forces and to enable itto respond to a gravitational wave as a mechanically free mass above the natural resonantfrequency. To minimize the effects of laser shot noise, the interferometer operates with highpower levels; approximately 400 W of laser power of wavelength 1064 nm is incident on thebeam splitter, resulting in over 15 kW of laser power circulating in each arm cavity. The presentdetectors are sensitive to changes in relative mirror displacements of about 10−18 m in a 100 Hzband centered around 150 Hz (figure 2). This low noise level allows for the preparation of low-energy states for the oscillator mode considered next.

The four mirrors of the LIGO interferometer (figure 1) are each an extended object witha displacement xi (i = 1, . . . , 4) defined along the optical beam axis. The servo control systemthat keeps the interferometer mirrors at the resonant operating point is an essential componentof this study. While all longitudinal and angular degrees of freedom of the mirrors are activelycontrolled, our discussion is limited to the differential arm cavity motion, which is the degree offreedom excited by a passing gravitational wave, and hence also the most sensitive to mirrordisplacements. This mode corresponds to the differential motion of the centers of mass ofthe four mirrors, xc = (x1 − x2) − (x3 − x4), and has a reduced mass of Mr = 2.7 kg. A signalproportional to differential length changes is measured at the antisymmetric output of the beamsplitter, as shown in figure 1. This signal is filtered by a servo compensation network beforebeing applied as a force on the differential degree of freedom by voice coils that actuate magnetsaffixed to the mirrors.

2. The cooling mechanism

The degree of freedom that is of interest as a quantum particle is the differential mirror motionxc. However, optical measurements probe the location of the mirror surface (averaged over theoptical beam), which differs from center-of-mass location due to the mirror’s internal thermalnoise, and include a sensing noise due to the laser shot noise. Combining these noises into atotal displacement noise XN, the output signal is written as

xs = xc + XN. (1)

The center-of-mass motion is also subject to a noise force FN (including, for example, thethermally driven motion of the mirror suspensions and the seismic motion of the ground thatcouples through the suspensions) and a feedback force that is proportional to xs. The resultingequation of motion in the frequency domain is given by:

−Mr

[�2

− i � �p φ(�) − �2p

]xc = FN − K (�)xs. (2)

Here K (�) is the frequency-domain feedback filter kernel, and the φ(�) term accountsfor mechanical damping. For a viscously damped pendulum with quality factor Qp = �p/0p


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Figure 1. (a) Optical layout of a LIGO interferometer. Light reflected fromthe two Fabry–Perot cavities formed by input and end mirrors, M1–M4, isrecombined at the beam splitter (BS). To control the differential degree offreedom, an optical signal proportional to mirror displacement is measured onthe photodetector (PD), and fed back as a differential force on the mirrors,after appropriate filtering to form restoring and damping forces. (b) The spectraldisplacement noise density of the differential mode of motion of the LIGO 4 kminterferometer at the Hanford Observatory is shown. Also shown is the targetsensitivity and the quantum noise contribution, which consists of shot noiseabove 30 Hz and radiation pressure noise below. The standard quantum limit(SQL) is also shown, and the closest approach to the measured sensitivity isabout a factor of 10 near 150 Hz. (c) An aerial photograph of the LIGO Hanfordsite in the state of Washington is shown. (d) A photograph of a 10.8 kg mirror isshown. Photographs courtesy of the MIT/Caltech LIGO Laboratory.

(�p and 0p correspond to the real part and twice the imaginary part of the complexeigenfrequency of the pendulum), φ(�) = 1/Qp. If the damping is not viscous, but insteadcaused by internal friction, φ(�) takes on a more complex form [28]. Combining equations (1)and (2), the equation of motion for the center-of-mass is obtained:

−Mr

[�2

− i � �p φ(�) − �2p − K (�)/Mr

]xc = FN − K (�)XN. (3)

In this experiment, the control kernel is adjusted so that

K (�)/Mr ≈ �2eff + i�0eff (4)


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Figure 2. Response function of mirror displacement to an applied force, forvarious levels of damping. The points are measured data, the thin lines are azero fit parameter model of the complete feedback loop, and the thick linesspanning the resonance (shown in the shaded region) are fitted Lorentzians, fromwhich the effective resonant frequency and quality factor are derived for eachconfiguration.

with �eff and 0eff much larger than �p and 0p, respectively, such that the modified dynamics ofxc are given by a damped oscillator driven by random forces:

−Mr[�2− i�0eff − �2

eff] xc = FN − K (�)XN. (5)

An electro-optical potential well in which the mirrors oscillate is thus created.The output of our experiment measures xs, and in order to deduce true mirror motion xc,

the limiting sources of noise must be considered. If noise predominantly drives the center-of-mass motion, i.e. FN � K (�)XN, then xs ≈ xc (see equation (1)) and the measured signalcorresponds to the center-of-mass motion. However, in the case that surface or sensing noisedominates, i.e. K (�)XN � FN, then a correction factor must be applied to the measured signalto deduce the center-of-mass motion. Taking equations (1) and (5), in the limit that FN = 0, weobtain

xc =K (�)

Mr�2xs. (6)

If the levels of each noise XN and FN are not precisely known, then one can make a conservativecorrection by applying a factor max(1, |K (�)/Mr�

2|) to determine the worst possible center-

of mass motion, thereby accounting for the fact that the servo can inject noise back onto theoscillator. The effective temperature of the mode may then be obtained:

Teff =Mr�

2effδx2

rms

kB, (7)


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where

δx2rms =

∫∞

0max

(1,

K (�)

Mr�2

)2

Sxs(�)d� ≡

∫∞

0Sxdd�. (8)

Sx s is the single-sided power spectral density of the measured motion xs and Sxd includes thecorrection factor. At large feedback gains, the measured noise Sx s may be arbitrarily suppressed,however, the mirror motion will reach a finite level as limited by the detection noise XN. This‘squashing’ effect has been explored previously [9, 33], and the calculation of Sxd avoidsunderestimates of the mirror motion. It is impossible to reliably measure the mirror motion atarbitrarily high frequencies, and the integral in equation (8) will diverge in any real system.The integration must therefore be limited in its frequency band, as is later discussed. Finally,the corresponding occupation number may be determined by

Neff =kBTeff

h�eff. (9)

K (�) of equation (4) is formed by convolving the position-dependent output signal withfilter functions corresponding to the real and imaginary parts of the feedback kernel K (�).In the LIGO feedback system, there are additional filters and propagation delays that causedeviations from the ideal cold, damped spring, at high and low frequencies. Below 100 Hz,K (�) increases sharply to suppress seismically driven motion; at high frequencies (abovea few kilohertz), K (�) decreases precipitously to prevent the control system from feeding shotnoise back onto the mirrors. However, in the frequency band important for this measurement(near the electro-optical resonance), the feedback is well approximated by a spring and dampingforce, as shown in figure 2.

3. Measurement results and discussion

The servo control loops of the LIGO interferometers are optimized to minimize noise couplingto measurement of the differential mode motion of the mirrors. The modifications to the servoloops to create a nearly ideal cold spring at �eff = 140 Hz do not significantly affect the noiselimits, shown in figure 1. Figure 3 shows the amplitude spectral density of mirror displacementfor varying levels of cold damping. To infer the effective temperature of the mode, its effectivefrequency �eff and an estimation of the root-mean-square displacement fluctuation δxrms mustbe determined. First the differential mirror motion is driven and the response is measured, asshown in figure 2. These response functions are fit to a damped oscillator model; �eff and Qeff

are products of the fit. Then δxrms is computed by integrating the spectrum in the band from 100to 170 Hz, as described in equation (8). The sensitivity in this frequency band is limited by lasershot noise that enters into XN. To correct for the finite integration band, the result is scaled bysetting our measured spectrum equal to the integral over the same frequency band of a thermallydriven oscillator spectrum,

Sxth(�) =4kBTeff0eff/Mr

(�2eff − �2)2 + �202

eff

. (10)

In this way, a minimum effective temperature Teff = 1.4 ± 0.2 µK is measured, correspondingto thermal occupation number Neff = 234 ± 35. Systematic error of 15% in the calibration


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Figure 3. Amplitude spectral density of displacement in the frequency band ofintegration. The curves (from highest to lowest) were produced by applyingincreasingly strong cold damping to the oscillator, corresponding to themeasurements of figure 2. The depression in the lowest curve is due to theshape of the background noise spectrum; the effects of the servo are correctedfor according to equations (1)–(8). The narrow line features between 100 and110 Hz are mechanical resonances of auxiliary subsystems, and a 120 Hz powerline harmonic is also visible. The predominant noise is laser shot noise.

dominates statistical error in these uncertainty estimates. The limits to integration were chosenas a compromise between having a wide limit, and choosing frequencies at which mirror motionis sensed. In the limit that the width of the integration band approaches 0, the lowest temperatureachieved approaches 0.9 µK. For larger integration limits, the temperature diverges because ofthe increased uncertainty at high frequency caused by shot noise (as occurs in all experiments).The spectra in figure 3 are predominantly limited by shot noise in the measurement band. Itmay at first appear unusual to associate a temperature with a device limited by shot noise, ratherthan thermal noise. However, the above calculations are justified, since the ultimate limit toexperiments such as this is known to arise from optical noise [34].

4. Cooling to the quantum limit

An interesting question arises as to whether this technique can lead to ground state coolingof the electromechanical oscillator. To mitigate the shot noise limit, which arises due to thefluctuating number of photons detected, the laser power could be increased. However, radiationpressure noise (a fluctuating force exerted on the mirrors due to the shot noise of the laser)increases with laser power and will ultimately limit the sensitivity. The SQL is obtained when


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shot noise and radiation pressure noise contribute equally to the total quantum noise [35].Hence, the continuous displacement measurement required for servo feedback does introduce anadditional term to the uncertainty relation for the oscillator position and momentum fluctuationsdue to measurement-induced steady state decoherence. If, however, the classical noises (such asthermal) are reduced significantly below the SQL, active feedback, with the appropriate controlkernel, is capable of cooling the electro-optic oscillator to its motional ground state [36].

5. Future prospects with LIGO

In the coming years, two upgrades of the LIGO detectors are planned. The first, EnhancedLIGO, is presently underway with an expected completion date in 2009, and seeks to improvethe sensitivity of the instruments above 40 Hz. The improvement in displacement sensitivity inthe frequency band around 150 Hz, where the cold spring measurements were performed, isexpected to be about a factor of 2. Subsequently, a major upgrade, Advanced LIGO, expectedto be completed in 2014, should give a factor of 10–15 improvement in displacement sensitivityrelative to that of the detector used for this work (with a concomitant factor of 4 increasein mass). In Advanced LIGO, the laser power circulating in the Fabry–Perot cavities shouldexceed 800 kW, permitting strong restoring forces to be generated optically. Enhanced LIGOis expected to reach ∼6 times lower occupation number, approaching 40 quanta, and withAdvanced LIGO, the detectors will be operating at the SQL, allowing the ground state to beapproached.

As they approach the SQL, these devices should enable novel experimental demonstrationsof quantum theory that involve kilogram-scale test masses [25, 37, 38]. The present work,reaching microkelvin temperatures, provides evidence that interferometric gravitational wavedetectors, designed as sensitive probes of general relativity and astrophysical phenomena, canalso become sensitive probes of macroscopic quantum mechanics.

Acknowledgments

We gratefully acknowledge the support of the United States National Science Foundationfor the construction and operation of the LIGO Laboratory and the Science and TechnologyFacilities Council of the United Kingdom, the Max-Planck-Society, and the State ofNiedersachsen/Germany for support of the construction and operation of the GEO600 detector.We also gratefully acknowledge the support of the research by these agencies and by theAustralian Research Council, the Council of Scientific and Industrial Research of India, theIstituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Educacion y Ciencia,the Conselleria d’Economia Hisenda i Innovacio of the Govern de les Illes Balears, the ScottishFunding Council, the Scottish Universities Physics Alliance, The National Aeronautics andSpace Administration, the Carnegie Trust, the Leverhulme Trust, the David and Lucile PackardFoundation, the Research Corporation and the Alfred P Sloan Foundation.

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[10] Corbitt T, Wipf C, Bodiya T, Ottaway D, Sigg D, Smith N, Whitcomb S and Mavalvala N 2007 Opticaldilution and feedback cooling of a gram-scale oscillator to 6.9 mK Phys. Rev. Lett. 99 160801

[11] Mow-Lowry C M, Mullavey A J, Goßler S, Gray M B and McClelland D E 2008 Cooling of a gram-scalecantilever flexure to 70 mK with a servo-modified optical spring Phys. Rev. Lett. 100 010801

[12] Vinante A et al 2008 Feedback cooling of the normal modes of a massive electromechanical system tosubmillikelvin temperature Phys. Rev. Lett. 101 033601

[13] Regal C A, Teufel J D and Lehnert K W 2008 Measuring nanomechanical motion with a microwave cavityinterferometer Nat. Phys. 4 555

[14] Schliesser A, Arcizet O, Rivire R and Kippenberg T J 2009 Resolved-sideband cooling and measurement ofa micromechanical oscillator close to the quantum limit arXiv:0901.1456

[15] Groeblacher S, Hertzberg J B, Vanner M R, Gigan S, Schwab K and Aspelmeyer M 2009 Demonstration ofan ultracold micro-optomechanical oscillator in a cryogenic cavity arXiv:0901.1801

[16] Thompson J D, Zwickl B M, Jayich A M, Marquardt F, Girvin S M and Harris J G E 2008 Strong dispersivecoupling of a high-finesse cavity to a micromechanical membrane Nature 452 72

[17] Corbitt T, Chen Y, Khalili F, Ottaway D, Vyatchanin S, Whitcomb S and Mavalvala N 2006 Squeezed-statesource using radiation-pressure-induced rigidity Phys. Rev. A 73 023801

[18] Rabl P, Shnirman A and Zoller P 2004 Generation of squeezed states of nanomechanical resonators byreservoir engineering Phys. Rev. B 70 205304

[29] Ruskov R, Schwab K and Korotkov A N 2005 Squeezing of a nanomechanical resonator by quantumnondemolition measurement and feedback Phys. Rev. B 71 235407

[20] Clerk A A, Marquardt F and Jacobs K 2008 Back-action evasion and squeezing of a mechanical resonatorusing a cavity detector New J. Phys. 10 095010

[21] Bose S, Jacobs K and Knight P L 1997 Preparation of nonclassical states in cavities with a moving mirrorPhys. Rev. A 56 4175

[22] Armour A D, Blencowe M P and Schwab K C 2002 Entanglement and decoherence of a micromechanicalresonator via coupling to a Cooper-pair box Phys. Rev. Lett. 88 148301

[23] Marshall W, Simon C, Penrose R and Bouwmeester D 2003 Towards quantum superpositions of a mirror2003 Phys. Rev. Lett. 91 130401

[24] Vitali D, Gigan S, Ferreira A, Böhm H R, Tombesi P, Guerreiro A, Vedral V, Zeilinger A and AspelmeyerM 2007 Optomechanical entanglement between a movable mirror and a cavity field Phys. Rev. Lett.98 030405

[25] Müller-Ebhardt H, Rehbein H, Schnabel R, Danzmann K and Chen Y 2007 Entanglement of macroscopictest masses and the standard quantum limit in laser interferometry Phys. Rev. Lett. 100 013601

[26] Wipf C, Corbitt T, Chen Y and Mavalvala N 2008 Route to observing ponderomotive entanglement with anoptically trapped mirror New J. Phys. 10 095017







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13

[27] Kippenberg T J and Vahala K J 2008 Cavity optomechanics: back-action at the mesoscale Science 321 1172[28] Saulson P R 1990 Thermal noise in mechanical experiments Phys. Rev. D 42 2437[29] Mancini S, Vitali D and Tombesi P 1998 Optomechanical cooling of a macroscopic oscillator by homodyne

feedback Phys. Rev. Lett. 80 688[30] Cohadon P F, Heidmann A and Pinard M 1999 Cooling of a mirror by radiation pressure Phys. Rev. Lett.

83 3174[31] Abramovici A et al 1992 LIGO: The Laser Interferometer Gravitational-Wave Observatory Science 256 325[32] Abbott B et al 2009 LIGO: The Laser Interferometer Gravitational-Wave Observatory Rep. Prog. Phys. 72

076901[33] Pinard M, Cohadon P F, Briant T and Heidmann A 2000 Full mechanical characterization of a cold damped

mirror Phys. Rev. A 63 013808[34] Genes C, Vitali D, Tombesi P, Gigan S and Aspelmeyer M 2008 Ground-state cooling of a micromechanical

oscillator: generalized framework for cold damping and cavity-assisted cooling schemes Phys. Rev. A77 033804

[35] Braginsky V B and Khalili F Ya 1992 Quantum Measurement (Cambridge: Cambridge University Press)[36] Danilishin S et al 2008 Creation of a quantum oscillator by classical control Phys. Rev. Lett. submitted

(arXiv:0809.2024)[37] Kimble H J, Levin Y, Matsko A B, Thorne K S and Vyatchanin S P 2002 Conversion of conventional

gravitational-wave interferometers into quantum nondemolition interferometers by modifying their inputand/or output optics Phys. Rev. D 65 022002

[38] Buonanno A and Chen Y 2001 Quantum noise in second generation, signal-recycled laser interferometricgravitational-wave detectors Phys. Rev. D 64 042006


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http://dx.doi.org/10.1088/0034-4885/72/7/076901






http://www.njp.org/

Everything moves! In a world dominated by electronicdevices and instruments it is easy to forget that all

measurements involve motion, whether it be the motion ofelectrons through a transistor, Cooper pairs or quasiparti-cles through a superconducting quantum interference de-vice (SQUID), photons through an optical interferome-ter—or the simple displacement of a mechanical element.Nanoscience today is driving a resurgence of interest inmechanical devices, which have long been used as frontends for sensitive force detectors. Among prominent his-torical examples are Coulomb’s mechanical torsion bal-ance, which allowed him in 1785 to establish the inverse-square force law of electric charges, and Cavendish’smechanical instrument that allowed him in 1798 to meas-ure the gravitational force between two lead spheres.

Today, micro- and nanoelectromechanical systems(MEMS and NEMS) are widely employed in ways similarto those early force detectors, yet with vastly greater forceand mass sensitivity—now pushing into the realm ofzeptonewtons (10⊗21 N) and zeptograms (10⊗21 g). These ul-traminiature sensors also can provide spatial resolution atthe atomic scale and vibrate at frequencies in the giga-hertz range.1 Among the breadth of applications that havebecome possible are measurements of forces between in-dividual biomolecules,2 forces arising from magnetic reso-nance of single spins,3 and perturbations that arise frommass fluctuations involving single atoms and molecules.4

The patterning of mechanical structures with nanometer-scale features is now commonplace; figure 1 and the coverdisplay examples of current devices.

The technological future for small mechanical devicesclearly seems bright, yet some of the most intriguing ap-plications of NEMS remain squarely within the realm offundamental research. Although the sensors for the appli-cations mentioned above are governed by classical physics,the imprint of quantum phenomena upon them can nowbe readily seen in the laboratory. For example, the Casimireffect, arising from the zero-point fluctuations of the elec-tromagnetic vacuum, can drive certain small mechanicaldevices with a force of hundreds of piconewtons and pro-duce discernible motion in the devices.5 But it is now pos-sible, and perhaps even more intriguing, to consider theintrinsic quantum fluctuations—those that belong to themechanical device itself. The continual progress in shrink-ing devices, and the profound increases in sensitivity

achieved to read out those devices, now bring us to therealm of quantum mechanical systems.

The quantum realmWhat conditions are required to observe the quantum prop-erties of a mechanical structure, and what can we learnwhen we encounter them? Such questions have receivedconsiderable attention from the community pursuing grav-itational-wave detection: For more than 25 years, that com-munity has understood that the quantum properties of me-chanical detectors will impose ultimate limits on their forcesensitivity.6,7 Through heroic and sustained efforts, theLaser Interferometer Gravitational Wave Observatory(LIGO) with a 10-kg test mass, and cryogenic acoustic de-tectors with test masses as large as 1000 kg, currentlyachieve displacement sensitivities only a factor of about 30from the limits set by the uncertainty principle.

But the quantum-engineering considerations for me-chanical detectors are not exclusive to the realm of gravi-tational-wave physics. In the introduction to their pio-neering book on quantum measurement, VladimirBraginsky and Farid Khalili envisage an era when quan-tum considerations become central to much of commercialengineering.7 Today, we are approaching that time—advances in the sensitivity of force detection for new typesof scanning force microscopy point to an era when me-chanical engineers will have to include \ among their listof standard engineering constants.

Several laboratories worldwide are pursuing mechan-ical detection of single nuclear spins. That goal is espe-cially compelling in light of the recent success of DanRugar and colleagues at IBM in detecting a single electronspin with a MEMS device (see figure 1b and PHYSICSTODAY, October 2004, page 22).3 However, nuclear spinsgenerate mechanical forces of about 10⊗21 N, more than1000 times smaller than the forces from single electronspins. One ultimate application of this technique, struc-tural imaging of individual proteins, will involve millionsof bits of data and require measurements to be carried outnot on the current time scale of hours, but over microsec-onds. Detecting such small forces within an appropriatemeasurement time will necessitate new quantum meas-urement schemes at high frequencies, a significant chal-lenge. Yet the payoff, originally envisaged by John Sidles,will be proportionately immense: three-dimensional,chemically specific atomic imaging of individual macro-molecules.8

The direct study of quantum mechanics in micron- andsubmicron-scale mechanical structures is every bit as at-tractive as the actual applications of MEMS and NEMS.9With resonant frequencies from kilohertz to gigahertz, low

36 July 2005 Physics Today © 2005 American Institute of Physics, S-0031-9228-0507-010-9

Keith Schwab is a senior physicist at the National SecurityAgency’s Laboratory for Physical Sciences in College Park,Maryland. Michael Roukes is Director of the Kavli NanoscienceInstitute and is a professor of physics, applied physics, and bio-engineering at the California Institute of Technology in Pasadena.

Nanoelectromechanical structures are starting to approach the ultimate quantummechanical limits for detecting and exciting motion at the nanoscale. Nonclassical states ofa mechanical resonator are also on the horizon.

Keith C. Schwab and Michael L. Roukes

Putting Mechanics into Quantum Mechanics

dissipation, and small masses (10⊗15–10⊗17 kg), these devicesare well suited to such explorations. Their dimensions notonly make them susceptible to local forces, but also make itpossible to integrate and tightly couple them to a variety ofinteresting electronic structures, such as solid-state two-level systems (quantum bits, or qubits), that exhibit quan-tum mechanical coherence. In fact, the most-studied sys-tems, nanoresonators coupled to various superconductingqubits, are closely analogous to cavity quantum electrody-namics, although they are realized in a very different pa-rameter space.

Quantized nanomechanical resonatorsThe classical and quantum descriptions of a mechanical res-onator are very similar to those of the electromagnetic fieldin a dielectric cavity: The position- and time- dependent me-chanical displacement u(r,t) is the dynamical variable anal-ogous to the vector potential A(r, t). In each case, a waveequation constrained by boundary conditions gives rise to aspectrum of discrete modes. For sufficiently low excitationamplitudes, for which nonlinearities can be ignored, the en-ergy of each mode is quadratic in both the displacement andmomentum, and the system can be described as essentiallyindependent simple harmonic oscillators.

Spatially extended mechanical devices, such as thosein figure 1, possess a total of 3A modes of oscillation, where

A is the number of atoms in the structure. Knowing theamplitude and phase of all the mechanical modes is equiv-alent to having complete knowledge of the position and mo-mentum of every atom in the device. Continuum mechan-ics, with bulk parameters such as density and Young’smodulus, provides an excellent description of the modestructure and the classical dynamics, because the wave-lengths (100 nm–10 mm) of the lowest-lying vibrationalmodes are long compared to the interatomic spacing.

It is natural to make the distinction between nano-mechanical modes and phonons: The former are low-frequency, long-wavelength modes strongly affected by theboundary conditions of the nanodevice, whereas the latterare vibrational modes with wavelengths much smallerthan typical device dimensions. Phonons are relatively un-affected by the geometry of the resonator and, except indevices such as nanotubes that approach atomic dimen-sions, are essentially identical in nature to phonons in aninfinite medium.

It is an assumption that quantum mechanics shouldeven apply for such a large, distributed mechanical struc-ture. Setting that concern aside for the moment, one canfollow the standard quantum mechanical protocol to es-tablish that the energy of each mode is quantized:E ⊂ \w(N ⊕ 1/2), where N ⊂ 0, 1, 2, . . . is the occupation fac-tor of the mechanical mode of angular frequency w. The

http://www.physicstoday.org July 2005 Physics Today 37

2 mm

a

c

e

b

d

f

Source Drain

Gate

W

dz

Figure 1. Nanoelectro-mechanical devices. (a) A 20-MHz nanome-chanical resonator ca-pacitively coupled to asingle-electron transistor(Keith Schwab, Labora-tory for Physical Sci-ences).11 (b) An ultrasen-sitive magnetic forcedetector that has beenused to detect a singleelectron spin (DanRugar, IBM).3 (c) A tor-sional resonator used tostudy Casimir forces andlook for possible correc-tions to Newtonian grav-itation at short lengthscales (Ricardo Decca,Indiana University–Purdue University Indi-anapolis). (d) A paramet-ric radio-frequency me-chanical amplifier thatprovides a thousandfoldboost of signal displace-ments at 17 MHz(Michael Roukes, Cal-tech). (e) A 116-MHznanomechanical res-onator coupled to a single-electron transistor(Andrew Cleland, Uni-versity of California,Santa Barbara).10

(f) A tunable carbonnanotube resonator op-erating at 3–300 MHz (Paul McEuen, CornellUniversity).14

quantum ground state, N ⊂ 0, has a zero-point energy of\w/2 and is described by a Gaussian wavefunction of width∀x2¬1/2 ⊂ DxSQL ⊂ =\/(2mw). This quantity, known as thestandard quantum limit, is the root mean square ampli-tude of quantum fluctuations of the resonator position.7

The larger the zero-point fluctuations, the easier they areto detect. For example, a radio-frequency (10–30 MHz)nanomechanical resonator, with a typical mass around 10⊗15 kg, has DxSQL ⊂ 10⊗14 m, some 105–106 times larger thanDxSQL for the macroscopic test masses in the gravitational-wave detectors. Although 10⊗14 m represents a distanceonly a little larger than the size of an atomic nucleus, it isreadily detectable by today’s advanced methods, such asthose described below. At the other extreme, a carbonnanotube 1 mm long has DxSQL � 10⊗10 m, about the size ofa small atom. That relatively large value, although smallby absolute standards, makes nanotubes and nanowiresvery attractive for displaying and exploring quantum phe-nomena with mechanical systems.

A crucial consideration for reaching the quantum limitof a mechanical mode is the thermal occupation factor Nth,set by the mode frequency w and the device temperatureT. The average fluctuating energy of an individual me-chanical mode coupled to a thermal bath is expected to be

where Nth follows the Bose–Einstein distribution. For hightemperatures this expression reduces to the classicalequipartition of energy: Each mode carries kBT of energy.Figure 2a displays the deviation from classical behaviorthat occurs at low temperatures: When kBT � \w, Nth isless than 1 and the mode becomes “frozen out.” For a1-GHz resonator, this freeze-out occurs for T < 50 mK—aregime well within the range of standard dilution refrig-erators employed in low-temperature laboratories. Further-more, a 1-GHz device can be readily shielded from parasiticexternal driving forces. Nanomechanical resonators witha fundamental flexural resonance exceeding 1 GHz havebeen demonstrated by one of us (Roukes) at Caltech,1 al-

though position-detectionschemes with the band-width and sensitivity tomeasure the tiny ampli-tude of the zero-point fluc-tuations are still underdevelopment.

Fluctuations, boththermal and quantum, areof great practical and fun-damental importance:They set ultimate limitsfor sensitive force detec-tion and for the coherencetime of quantum states.These limits are deter-mined by the nanores-onator quality factor Q,which parameterizes thecoupling strength to thethermal bath. Currently,all atomic-force microscopesand experiments are in thehigh-temperature limit:kBT � \w and ∀E¬ � kBTper mode. For an acoustic-frequency mechanical res-onator typically found in

an atomic-force microscope setup, with a resonant fre-quency around 5 kHz, a mass of 10⊗12 kg, and a qualityfactor Q on the order of 104, one finds at room temperaturethat Nth ⊂ 109 and the rms displacement fluctuations,=∀xth

2 ¬ ⊂ 10⊗9 m, are more than 104 times larger than DxSQLfor this device. Thermal fluctuations limit the force sensi-tivity to 240 aN/Hz1/2 (where 1 aN ⊂ 10⊗18 N). By cooling avery thin and compliant low-frequency cantilever to near200 mK, the IBM group has been able to reduce the fluc-tuations to 6 × 10⊗13 m and achieve a record force sensi-tivity of 0.8 aN/Hz1/2.

Despite the close analogy between the quantization ofthe motion of an extended mechanical device and the quan-tization of the electromagnetic field in a cavity, some im-portant fundamental differences exist. The total zero-pointenergy in a mechanical system is finite because the num-ber of mechanical degrees of freedom is finite. Also, evenfor a rectangular flexural resonator, the mechanical modestructure is very complex and differs from the harmonicstructure for a rectangular electromagnetic cavity res-onator. In addition, due to the resonator’s finite stretchingand consequent tensioning, there is an intrinsic mechani-cal nonlinearity that is easily observable for modest me-chanical amplitudes. The nonlinearity provides amode–mode coupling, analogous to photon–photon cou-pling in nonlinear dielectrics, and drives instabilities. Thisnonlinearity has already been put to effective use for bothparametric mechanical amplification and square-law me-chanical detection, as described below.

The challenge of motion transductionA prerequisite for attaining the ultimate potential fromnanomechanical devices is displacement sensing—that is,reading out the NEMS motional response induced by anapplied stimulus. Most typically, this requirement distillsto transduction, or conversion, between the mechanicaland electrical domains: Ultimately one wants a voltage sig-nal that provides the time record of the NEMS response.Many displacement-sensing techniques for mechanical de-vices have been proposed and demonstrated (see figure 3);perhaps not surprisingly, displacement transducers that

∀ ¬ ⊂ ⊂E N\w \wth12

1e\w /k TB 1⊗( (⊕ ,

38 July 2005 Physics Today http://www.physicstoday.org

100

10

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0.5OC

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1 mK 10 mK 100 mK 1 KTEMPERATURE

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19.656 19.658 19.660 19.662 19.664FREQUENCY (MHz)

tv

tN

3.8 fm/Hz1/2

TN

= 73 mK= 75th

a b

Figure 2. Quantum limits. (a) The occupation factor Nth (black curves) of various mechan-ical resonator frequencies is a function of resonant frequency and temperature T. Shownin red is the lifetime tN of a given number state for a 10-MHz resonator with quality factorQ ⊂ 200 000 (recently demonstrated at the Laboratory for Physical Sciences).11 Also inred is the expected decoherence time tv for a superposition of two coherent states in thatresonator displaced by 100 fm. (b) The measured noise-power spectrum of the thermalmotion (black line, with a Lorentzian fit in red) atop the white noise (blue baseline) of theposition detector. The curve corresponds to the green point in panel a, with T ⊂ 73 mKand Nth ⊂ 75. These data show the closest approach to date to the uncertainty-principlelimit: The detector noise gives a displacement sensitivity a factor of 5.8 from the quantummechanical limit.

are important for microscale devices do not prove optimalfor nanoscale devices. For example, such optical methodsas fiber-optic interferometry and reflecting off a cantilever(the so-called optical lever) have been used extensively inMEMS, especially on commercially available scanningprobe microscopes. However, because nanomechanical de-vices are small compared to the wavelength of light, dif-fraction dramatically complicates such approaches. Near-field optical methods may play an important future role.But optical adsorption and the resulting heating of the me-chanical structure prove problematic for the most sensi-tive regimes. So it appears that optimal coupling at thequantum limit may be difficult to achieve optically. Nev-ertheless, optical techniques have recently enabled the de-tection of microcantilever motion induced by a single elec-tron spin.3

Piezoresistive displacement transduction has foundwidespread applications in larger micron-scale devices.Use of this technique for nearly quantum-limited detectionof NEMS devices, however, does not appear to be straight-forward. The dissipation involved will cause appreciableheating of tiny devices, which will likely preclude opera-tion at ultralow temperatures.

One technique that has been immensely successful ismagnetomotive detection, which utilizes the electromotiveforce generated by a metallic conductor (typically a metalli-zation layer) affixed to a mechanical device that moves ina large magnetic field—on the order of several tesla. Thistechnique is especially well suited for low-temperaturemeasurements for which the imposition of large magneticfields is commonplace; it has been employed to monitor thehighest-frequency devices measured to date, with funda-mental flexural modes exceeding 1 GHz.1 Unfortunately,magnetomotive motion detection becomes increasinglyawkward at higher frequencies. The effective electrical im-pedance, which arises from electromechanical coupling tothe motional part of the device, scales inversely with fre-quency and hence tends to be swamped by parasitic, staticcircuit impedances as the frequency increases into thegigahertz domain. But even at lower frequencies, at hun-dreds of megahertz, observing the thermal fluctuations ofa nanomechanical resonator by this technique has so farproven elusive—the intrinsic noise of state-of-the-art cryo-genically cooled amplifiers is too high to surmount thebugaboo of insufficient mechanical-to-electrical transduc-tion efficiency. The problem is generic for most motion-transduction methods at microwave frequencies.

Using parametric amplification, the Caltech grouphas demonstrated an attractive way to work around thisproblem. They employ the Euler instability—the nonlin-earity that causes a beam under compression to buckle—to realize a high-frequency amplifier that works entirelyon mechanical principles. Their device, depicted in figure1d, provides stable gain and up to a thousandfold dis-placement amplification in response to a weak stimulusforce at 17 MHz. With sufficient gain in the mechanical do-

main, the challenge of displacement transduction becomessubstantially easier and thermomechanical fluctuations ina cryogenically cooled device have indeed been observedwith such an amplifier.

Some of the most exciting recent transduction effortsare focused on coupling high-frequency nanomechanicalsystems to various nanoelectronic and mesoscopic devicesthat serve essentially as integrated amplifiers. Examples arequantum point contacts, quantum dots, or single-electrontransistors (SETs) used in configurations where the mo-tion of a nanomechanical device modulates the electrontransport properties. Among these readout strategies, theSET—shown by many research groups to be a very sensi-tive detector of charge—has now enabled nearly quantum-limited position detection for NEMS devices. Figures 1aand 1e each show a charged nanomechanical resonatorcapacitively coupled to an SET. The resonator’s motioninduces a change in the charge on the gate electrode of theSET; the resulting change in the SET’s conductance can bedirectly monitored. Careful consideration of noise sourcesindicates that the SET can provide motion detection withsensitivity down to the quantum limit.

Robert Knobel and Andrew Cleland at the Universityof California, Santa Barbara,10 were the first to demon-strate position detection using an SET. They used the de-vice as a narrowband mixer—with a bandwidth of only100 Hz—and detected the flexural resonance of a 100-MHzdevice. Their effort yielded continuous position detection


B

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c

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Figure 3. Detection techniques for nanomechanical dis-placement. (a) An optical detection scheme that is typicallyused in atomic-force microscopy but that does not workwell for NEMS. Light from an optical fiber reflects off theresonator and returns up the fiber. (b) A magnetomotivetechnique that measures the electromotive force generatedwhen the metal layer on top of the resonator moves in themagnetic field B. (c) Coupling to a mesoscopic detectorsuch as a quantum dot, a quantum point contact, or a single-electron transistor. The current IDS through the detector ismodulated by the NEMS motion.

to within a factor of about 100 of the quantum limit, buttheir detection bandwidth was very limited.

Recently a group led by one of us (Schwab) at the Lab-oratory for Physical Sciences has shown that a radio-frequency SET can simultaneously provide both largebandwidth (75 MHz) and ultrasensitive detection.11 Thegroup directly detected the motion of a 20-MHz resonator(figure 1a), which had a Q of 50 000, and demonstratednanomechanical measurements only a factor of about 6from the quantum limit, the closest approach to date forany position measurement (see figure 2b). With such posi-tion sensitivity, they observed random vibrations driven bythe thermal energy at an effective temperature as low as60 mK. The experiment essentially constituted millikelvinnoise thermometry on a single mechanical degree of free-dom. At the experiment’s lowest temperature, the thermaloccupation factor Nth for the mechanical mode was on theorder of 58; that low value indicates that the quantumground state (Nth < 1) of NEMS devices is within reach.Necessary steps toward achieving that goal include opti-mized transduction at higher resonant frequencies and im-provements in thermalization, that is, the cooling of themechanical mode.

Cooling is less straightforward than one might ini-tially assume. Precisely in the regime where device modesbecome frozen out (and quantum effects begin to emerge),thermal conductance quantization (see PHYSICS TODAY,June 2000, page 17) imposes limits on the rate at which ananoscale device can thermalize to its environment.12

Work-arounds are few. One can perhaps attempt to engi-neer the modes mediating thermal contact—to optimizeheat transfer while preserving the quantum nature of thespecific mechanical mode under study—but it is not clearhow practical this approach will be. Perhaps a more prom-ising alternative is active cooling of the mechanical modethrough controllable external interactions.13 For example,using very low-noise temperature detection to provide op-timal feedback, it should be possible to actively cool themechanical mode close to the ground state. The prospectsare reminiscent of the strategy taken with trapped atoms:Laser-cooling of samples into a low-energy state allows forvarious quantum measurements before the atoms begin towarm up from their interaction with the environment.

Ultimately, molecular mechanical systems, assembledwith atomic precision, will subsume today’s nanomechan-ical systems patterned by top-down methods. Devicesbased on single-wall carbon nanotubes offer very excitingpossibilities because nanotubes are electronically activeand naturally form nanoelectronic devices such as single-electron transistors. Exploiting the strain dependence ofelectron transport through a carbon nanotube, PaulMcEuen’s group at Cornell University has recently de-tected the vibrational mode spectrum of a suspended nano-tube (see figure 1f).14 Although the “bottom up” fabricationis today still an art rather than a technology, nanotube and nanowire NEMS offer great promise for achievingboth very high resonant frequencies and sensitive force detection.

Uncertainty-principle limits on position detectionHow precisely can one continuously measure the position ofan object? This basic question is of central importance formechanical force detection, such as in atomic-force mi-croscopy. It is clear that quantum mechanics should placelimits on the ultimate answer, since it’s possible to constructfrom continuous measurement records both the mechanicalresonator’s position and its momentum. One should not ex-pect to be able to violate the uncertainty principle.

To continuously record the position, one must couple

the mechanical device to some form of linear detector;practically, the detector would take the form of a displace-ment transducer (such as one that linearly converts posi-tion to voltage) and an amplifying device. During the de-velopment of the maser in the 1960s, it became clear thatthe performance of any linear amplifier is limited by theuncertainty principle. In the context of position detection,the ultimate sensitivity was clarified by Carleton Cavesand colleagues6: Even for optimal engineering, the meas-urement record is obscured by an equal contribution ofquantum noise from the mechanical resonator plus itstransducer and from the subsequent linear amplifier usedas a readout. With the resonator at zero temperature, theminimum possible variance of the position record is

Two distinct forms of noise arise from the readoutprocess and affect nanomechanical measurements. Thefirst, measurement noise, is independent of the resonatorand is added to the output signal. In practice, this contri-bution may take the form of electrical or photon shot noise.The second is the somewhat more subtle back-action noise,which emanates from the readout and drives the resonatorstochastically. Back-action noise might arise from fluctuat-ing potentials or currents that are, in turn, converted by thetransducer into physical forces subsequently imposed on themechanical device. Electrical engineers are familiar withthese two forms of noise from linear amplifiers and repre-sent them as voltage and current noise, respectively. Opti-mally engineered coupling between the resonator and theamplifier is achieved when the two amplifier noise contri-butions are equal. But very few linear amplifiers, even whenoptimized in this way, can approach the quantum limit. Ac-cordingly, reaching the quantum limit for position detectioninvolves choosing an amplifier that is capable of quantum-limited detection in principle, and that can be optimally cou-pled to the mechanical resonator in practice.

Coupling nanomechanics to quantum systemsA large effort in condensed matter physics today is focusedon investigating the coherent quantum mechanical be-havior of individual solid-state devices. That researchbegan with mesoscopic physics in the 1980s and continuestoday, motivated by interest in quantum information andcomputation. Beautiful and convincing quantum behav-ior—quantized energy, coherent evolution, superpositionstates, and entanglement—has been observed with vari-ous single-electron devices, SQUIDs, and quantum dots. Agrowing number of researchers now believe that we havethe necessary tools to observe similar behavior in smallmechanical structures.

To explore quantum effects in NEMS, it is essential tomove beyond continuous measurement and linear cou-pling: Continuous linear measurements cannot distin-guish between classical and quantum oscillators. Thequantum properties of the most heavily studied simpleharmonic oscillator—the modes of the electromagneticfield—have been revealed by coupling to a nonlinear de-tector such as a photon counter (for energy detection) or toa coherent two-level system, as in a single atom in an elec-tromagnetic cavity.

Analogous measurement concepts in nanomechanicsare starting to take shape. Reviewing the experimentalstatus of quantum nondemolition measurements in 1996,Braginsky and Khalili wrote that “no experimentalscheme had been proposed so far” for measuring the en-ergy of a mechanical resonator.7 The Caltech group has set

\mw ln 3

.DxQL⊂ 1.35 xSQLD= �

40 July 2005 Physics Today http://www.physicstoday.org

about to change that; recently they demonstrated how tobuild a mechanical square-law detector—whose output isproportional to the square of the position coordinate—analogous to Braginsky and Khalili’s energy detector. Ex-perimental work on this front is under way, with theoret-ical investigation led by researchers at Caltech and theUniversity of Queensland.15

The detection of a single electron spin by Rugar’s IBMgroup is the first realization of a single quantum two-levelsystem coupled to a mechanical mode.3 This success demon-strates both that the coupling between the cantilever andspin can be strong enough to produce measurable displace-ments and that the coherence of the spin is sufficient to per-form many coherent manipulations, even while the spin iscoupled to the resonator. A positive-feedback scheme ap-pears possible: The coherent back action of the cantileveronto the spins can directly stimulate the transition of morespins and give rise to amplified cantilever oscillations, a me-chanical lasing scheme dubbed the cantilaser.16 This pro-posed technique has the advantage that the dynamics of thecantilever–spin system is not limited to the mechanical re-laxation time tm ⊂ Q/w0, and it may prove useful in the fastmechanical detection of spin ensembles.

The quantum system formed by capacitively coupling aNEMS resonator to a solid-state qubit has garnered perhapsthe most attention from theorists. Andrew Armour, MilesBlencowe, and one of us (Schwab) first described one suchsystem, a NEMS coupled to a Cooper-pair box.17 The CPB isa superconducting device in which both the electrostaticcharging energy of a small conducting island and theJosephson coupling energy are important in the coherentquantum dynamics of the device’s charge states. For certainbias configurations, the CPB is well described by a simpletwo-level qubit Hamiltonian in which the island’s capaci-tance determines the spacing between the levels +0¬ and +1¬,which have zero and one extra Cooper pair, respectively; theJosephson energy determines the transition rate betweenthe two charge states. Experiments by a number of re-searchers have demonstrated CPB eigenstates with excited-state lifetimes of up to 7 ms and coherence times of super-position states as long as 0.5 ms.

The CPB and the mechani-cal resonator interact simplythrough the electrostatic forcemediated by the charge on thebox. The coupling strength canbe comparable to the resonatorquantization energy for reason-able parameters. Adding in thesingle-mode harmonic-oscillatorHamiltonian that describes themechanical resonator, the sys-tem is directly analogous to cav-ity quantum electrodynamics. Infact, for certain CPB biases, thesystem is described by a Hamil-tonian (the so-called Jaynes–Cumming Hamiltonian) widelyemployed throughout atomicphysics. Various groups explor-ing this analogy have concludedthat the NEMS–CPB systemshould allow for experimentssuch as number-state detection,quantum nondemolition meas-urements, and a resonator cool-ing process analogous to lasercooling of atoms.

These simple descriptions ofcoupled quantum systems suggest an intriguing possibility,first described by Armour and colleagues.17 The interactiondisplaces the resonator according to the charge state of theCPB: The device moves to the left if the CPB is in the +0¬charge state, and moves to the right if the CPB is in the +1¬charge state. This coupling raises the simple question,Which way does the mechanics move if one prepares theCPB into the superposition state (+0¬ ⊕ +1¬)/=2+? If the me-chanics truly is quantum, then naively one expects the res-onator to displace in both directions and ultimately form anentangled state with the qubit (see figure 4).

To design and fabricate experimental devices that allowfor such observations, at least two criteria need to be satis-fied. First, the coupling between the qubit and the resonatormust be sufficient so that the difference in displacementgenerated by the two qubit states should be greater thanDxSQL. Meeting this condition will ensure that an entangledstate is formed with two nearly orthogonal positions of theresonator. Second, the decoherence time of the resonatormust be larger than the measurement time. The decoher-ence time of a state formed by a superposition of two co-herent states displaced by a distance Dx and coupled to athermal bath modeled by simple harmonic oscillators wascalculated by Wojciech Zurek and colleagues at Los AlamosNational Laboratory in 1992. Their result,

is plotted in figure 2a. For a very small displacement of thetwo superposition states, Dx ⊂ 3 DxSQL ⊂ 100 fm, for whichthe states are still quantum mechanically distinct, tv is ex-pected to be surprisingly long—greater than 1 ms at 100 mK.For an entangled state with Dx comparable to the width ofa typical resonator, 100 nm, tv is estimated to be 10⊗18 s!Thus Schrödinger-cat states formed by a mechanical res-onator coupled to a qubit appear to be feasible, although thedistance between these states would be subatomic.

Illuminating the quantum worldNanomechanical resonators will be increasingly useful for ex-plorations of quantum mechanics, whether as ultrasensitive

\k TB

( (,tv ⊂ QDxSQL

Dx

2


P x( )

Two-levelsystem

x

TIME

Figure 4. Quantum superpositions may be possible with a quantum two-level sys-tem (TLS) coupled to a nanomechanical resonator. In this schematic, the flexure ofthe cantilever depends on whether the TLS is in the down state +A¬ (red) or the upstate +R¬ (blue). The plot shows the probability P of finding the resonator center-of-mass at a given position x for different times after coupling the resonator to the TLS.If the TLS is prepared in a superposition state, an entangled state is formed and theresonator will be in both locations.

probes of quantum and mesoscopic forces, as detectors ofsingle quantum systems, as “buses” in quantum informa-tion devices,18 or as devices to allow the study of quantumbehavior in ordinary bits of matter. The construction of ascanning microscope that can detect single nuclear spinsremains a grand challenge of nanomechanics. Success inthese areas and others will require highly engineered me-chanical structures that routinely operate near quantummechanical limits.

The generation and detection of the uniquely quan-tum states of a small mechanical device, such as energyeigenstates (so-called Fock states), superposition states, orentangled states, are particularly interesting because themechanical structures may be considered “bare systems”:There is no macroscopic quantum condensate to protectthe device from excitation or decoherence. In supercon-ductors, superfluids, or Bose–Einstein condensates, thenumber of quantum states of a macroscopic sample is dras-tically reduced to only a few degrees of freedom. For in-stance, in a CPB containing billions of electrons, super-conductivity suppresses almost all the normal-statedegrees of freedom down to just two: number and phase.

The mechanical devices shown in figure 1 are com-posed of ordinary matter, full of defects and imperfections,and help answer the question: What does it take to observethe quantum nature of an ordinary system? The motiva-tion is similar to that behind the recent beautiful work ofAnton Zeilinger and colleagues at the University of Vi-enna, who have shown matter-wave interference with mol-ecules of increasing complexity and have studied decoher-ence by directly controlled interactions. In this vein, wehope experiments with nanomechanical devices at thequantum limit may further illuminate the boundary be-

tween the microscopic realm, governed by quantum me-chanics, and our macroscopic world, governed by classicalmechanics.

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Friedsam et al., J. Phys. Condens. Matter 15, S1709 (2003);M. Benoit, H. E. Gaub, Cells Tissues Organs 172, 174 (2002).

3. D. Rugar et al., Nature 430, 329 (2004).4. K. L. Ekinci, Y. T. Yang, M. L. Roukes, J. Appl. Phys. 95, 2682

(2004).5. For a review, see M. Bordag, U. Mohideen, V. M. Mostepa-

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Rev. Mod. Phys. 52, 341 (1982). A recent treatment from acondensed matter approach has been given by A. A. Clerk,Phys. Rev. B 70, 245306 (2004).

7. V. B. Braginsky, F. Ya. Khalili, Quantum Measurement, Cam-bridge U. Press, New York (1992).

8. J. A. Sidles et al., Rev. Mod. Phys. 67, 249 (1995).9. For an excellent review, see M. Blencowe, Phys. Rep. 395, 159

(2004).10. R. Knobel, A. N. Cleland, Nature 424, 291 (2003).11. M. D. LaHaye et al., Science 304, 74 (2004).12. K. C. Schwab et al., Nature 404, 974 (2000).13. I. Wilson-Rae, P. Zoller, A. Immamoglu, Phys. Rev. Lett. 92,

075507 (2004); A. Hopkins et al., Phys. Rev. B 68, 235328(2003).

14. V. Sazonova et al., Nature 431, 284 (2004).15. D. Santamore et al., Phys. Rev. A 70, 052105 (2004).16. I. Bargatin, M. L. Roukes, Phys. Rev. Lett. 91, 138302 (2003).17. A. D. Armour, M. P. Blencowe, K. C. Schwab, Phys. Rev. Lett.

88, 148301 (2002).18. E. K. Irish, K. C. Schwab, Phys. Rev. B 68, 155311 (2003); X.

Zou, W. Mathis, Phys. Lett. A 324, 484 (2004); A. N. Cleland,M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004). �

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