UA Math 2010

Cavity optomechanics …

• From the top down: Review of optomechanical mirror cooling

• Coupling to ultracold atoms and molecules

• From the bottom up: Bosonic & fermionic atomic mirrors

• Outlook: All-optical optomechanics

ARO NSF ONR

Mishkat Bhattacharya ( U. Rochester)Wenzhou ChenDan GoldbaumRina Kanamoto ( Ochanomizu University)Greg PhelpsSwati SinghEwan WrightKey Zhang ( Shanghai)

UA Math 2010

quantum metrology

• How can quantum resources be exploited with maximal robustness and minimal technical overhead?

• What are the inherent sensitivity-reliability tradeoffs in quantum sensors?

• What is the role of particle statistics in quantum metrology?

• How do decoherence mechanisms set fundamental limits to measurement precision?

This talk:

• From the top down: Cooled nanoscale cantilevers for sensing and quantum control of AMO systems

• From the bottom up: optomechanics in quantum-degenerate atomic systems

UA Math 2010

From the top down…

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micromechanical cantilevers –single-particle detectors

Protein detectionThree cantilevers coated with antibodies to PSA, a prostate cancer marker found in the blood. The left cantilever bends as the protein PSA binds to the antibody. Photo courtesy of Kenneth Hsu/UC Berkeley & the Protein Data Bank

Micromechanical cantilever with a single E. Coli cellA micromechanical cantilever with a single E. Coli cell attached. The cantilever is used to detect changes in mass due to selectively attached biological agents present in small quantities.Craighead Research Group, Cornell University

UA Math 2010

micro-mirrors

D. Kleckner & D. Bouwmeester, Nature 444, 75 (2006)

SEM photograph of a vertical thermal actuator with integrated micromirror. Application of a current to the actuator arm produces vertical motion of the mirror, which can either reflect an optical beam or allow it to be transmitted. (Southwest Research Institute, http://www.swri.org/3pubs)

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• Optical spring effect: - laser radiation increases - Requires cavity tuned below laser frequency

• Optical damping: - Requires cavity tuned above laser frequency

• Use two lasers!

eff

radiation pressure mirror cooling

Basic idea: change frequency and damping of a mirror mounted on a spring using radiation pressure

enveff

eff ef efff

MB Bk Tn

kT

Number of quanta of vibrational motion:

[ H. Metzger and K. Karrai, Nature 432, 1002 (2004) ]

[ T. Corbitt et al, PRL 98, 150802 (2007) ]

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optical spring effect

Goal: increasing

Trapping due to `optical spring’: rp

opt

( )dFk

x

dx

B. S. Sheard et al. PRA 69, 051801(R) (2004)

eff

Need red-detuned cavity

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optical damping

0

0

0

0

F

Goal: decreasing

Cooling due to asymmetric pumping of sidebands (resolved sidebands limit)

M. Lucamarini et al., PRA 74, 063816 (2006)A. Schliesser et al. PRL 97, 243905 (2006)

env effeff ( / )MTT

Need blue-detuned cavity

00 0

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• Adiabatic and non-adiabatic (optical delay) contributions:

• Adiabatic effect – “optical spring” (also bistability and multistability)

• Non-adiabatic effect

a simple theory

2 22

eff RP2eff eff

1 | |( )

2 / 2

( ) / 2

d z dz az F t

dt Q dt m m c c L

dai x a a i s

d

m

t

2 2

2 /

4 / 1P

eff

2 2 2

2

16 /

(4 1/ )

Q

0( )z zR

• Linear coupling: , integrate equation for a formally, keeping terms up to

dz

dt

laser cavity( ) ( )z z

eff( )

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prehistory

• V. Braginsky, “Measurement of Weak Forces in Physics Experiments” (Univ. of Chicago Press, Chicago, 1977).

• V.B. Braginsky and F.Y. Khalili, "Quantum Measurement'' (Cambridge University Press, Cambridge, 1992)

• C. M. Caves, Phys. Rev. D23, 1693 (1981). • A. Dorsel, J. D. McCullen, PM, E. Vignes and H. Walther,” Optical bistability and

mirror confinement induced by radiation pressure,” PRL 51, 1550 (1983).

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early history

18K

Cavity cooling of a microlever, C. Höhberger-Metzger & K. Karrai, Nature 432, 1002 (2004)

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S. Gigan et al. (Zeilinger group) “Self-cooling of a micromirror by radiation pressure”, Nature 444, 67 (2006)

O. Arcizet et al. (Pinard/Heidman group) “Radiation pressure cooling and optomechanical instability of a micromirror”, Nature 444, 71 (2006)

D. Kleckner et al. (Bouwmeester group) “Sub-Kelvin optical cooling of a micromechanical resonator”, Nature 444, 75 (2006)

A. Schliesser et al. (Vahala/Kippenberg groups) “Radiation pressure cooling of a micromechanical oscillator using dynamical back-action” PRL 97, 243905 (2006)

T. Corbitt et al. (LIGO) “An all-optical trap for a gram-scale mirror”, PRL 98, 150802 (2007), “Optical dilution and feedback cooling of a gram-scale oscillator”, PRL 99, 160801 (2007)

A. Vinante et al (AURIGA gravitational wave detector mirror), Meff= 1,100 Kg, PRL 101, 033601 (2008)

recent history

< 10K

10K

135 mK

11K

0.17 mK

6.9 mK

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hot off the press…

(17 March 2010)

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a broad range of systems

From T. Kippenberg and K. VahalaScience 321, 1172 (2008)

(CPW: coplanar plane wave)

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cantilevers for quantum detection and control

• Pushing quantum mechanics to truly macroscopic systems• Quantum superpositions and entanglement in macroscopic systems• Novel detectors• Coherent control

Single-domain ferromagnet with oscillatory component B(t) couples to atomic spin F

Described by Tavis-Cummings Hamiltonian[P. Treutlein et al., PRL 99, 140403 (2007)]

Recent experimental results: Treutlein et al., coupling via Casimir-Polder forcehttp://www.munichatomchip.de/microresonator.html

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cantilever coupling to dipolar molecules

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single dipolar molecule

0

4ddG

0

1

4ddG Electric dipoles

Magnetic dipoles

Used in Magnetic Force Microscopy

Frequency shift squeezing

1/22 2( )c mr R x x

1/2

2' 5

0

3m

m cm

d d

mR

ò

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motional squeezing

2 †2IV C b b

Classical cantilever motion, '2 mc

1/2

6

2

15

4 2

/

m c

o t c c

B c c

d dC N

m R m

N k

t

T

u C

ò

S. Singh, M. Bhattacharya, O. Dutta, PM, PRL101, 263603 (2008)

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… and from the bottom up…

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cavity optomechanics with atoms

• Single atom J. Kimble, G. Rempe, …

• Utracold thermal gas D. Stamper-Kurn, Nature Phys. 4, 561 (2008)

• BEC T. Esslinger, Science 322, 235 (2008)

• Fermions R. Kanamoto & PM, PRL 104, 063601(2010) (theory)

(also: Interferometers, quantum simulators,, clocks,…)

Moving mirror mechanical center-of-mass motion of atom(s)

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cantilever coupling to Bose condensates

2 2 2†

2† †† 2

2† 22 ( ) (

1c( )

2 2os )

2( )c m m

a a

ga

p da a a a a kxx xi a a m q dx

mH

mq

dx

light mirror rad. pressure condensate-light interaction

Condensate recoil: 0 2 2cos(2 ) cos(( ) 2 )x Nc kx c kx c

2 2† † 2

2† †2 †

2

1 2(

2( ))

2 4c m ma

p g Na a i a a ca m q

mH caq a a

rad. pressure rad. pressure on BEC “mirror”

T. Esslinger et al, Science 322, 235 (2008)D. Stamper-Kurn et al, Nature Physics 4, 56 (2008)

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optomechanical coupling

† † †02 2 sm0

2ˆ ˆ ˆ ˆ ˆ ˆ

4ˆˆ

N Ua a c a Qc c aN

sm

sm

2

sm 2 2

0

†

4

1ˆ ˆ ˆ( )2

c

r

c

r

L m

mL NU

Q c c

Classical dynamics, adiabatic elimination of optical field:

22

22 2sm

22 sm

22 2sm

2

244

mm

rr

Q

Q

d

dt

d

dQ

Q

qq

q

qt

Effective mirror:

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potential

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quasi-periodic orbits

K. Zhang, W. Chen, M. Bhattacharya and PM, PRA 81, 013802 (2010)

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chaotic orbits

Next:

• quantum dynamics, entanglement• quantum control• quantum chaos (?)• …

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ring resonator system

† * †

1,2

† 2 † 22 2

†2 1 1 2

† †0 1 1 2 22

ˆ ˆ ˆ ˆ ˆ( )

ˆ ˆˆ ˆ ˆ ˆ( ) ( ),ˆ ˆ2

ˆ ˆikx i

i i i

kx

i i ii

H a a i a a

ddx x U a a a a x

m dxa a e a a e

† * †

†

† † † †

† † †

† †0

2 1 1

2 1 1 2 0 0

00

0

0

2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ(

ˆ ˆ ˆ

) 4 ( )

2ˆ ˆ ˆ ˆ ˆ( )

2

)(

i i i i i r c

s

c

c s s

c

i

s

a a

H U N a a i a a c c c c

U

i U

a a c c c

a a c c c c

c

a a

0 cos(2 ) sin(2 )1 2 2

ˆ ˆ ˆ ˆ( ) c skx c c cL L

kL

x x Let

2 Counter-propagating running waves

1

2

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three-mode system†

†

0

ˆ ˆ ˆ( ) / 2

ˆ ˆ ˆ( ) / 2

ˆ

j j j

j j j

X c c

P i c c

c N

† †0 2 1

† * †0

2 2 2 2

1 2

† †0 2 1 1 2

ˆ ˆ ˆ ˆ( )

ˆ ˆ ˆ ˆ2 (

ˆ

ˆ ˆ ˆ ˆ( )

ˆ ˆ ˆ ˆ(

)

ˆ

ˆ )

i i i i ii

r c c

c

s s

sX

H

U N a a

U N a a i a a

X P X

a a

i U N

X

a a a a

P

Analog of 2 coupled optomechanical mirrors

(W. Chen, D. S. Goldbaum, M. Bhattacharya and PM, arXiv:1003.1812)

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classical dynamics

* *2 1 1 2

*

2 20

2 20

1 0

*2 1 1

2 1 1

2

2

2 0 1 2

( )

( )

(

2 (16 ) 4

2 (16 ) 4

( )

( )

)

( )

c c

c

r c r

s

s

c

s r s

s

ri

X i

X X X U

X

N

X X X U N

i

X iX

U N i i

i U N i i

Dark states! Dark states!

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steady state intensity and phase

Spontaneous symmetry breaking

Symmetric pumping

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steady state “positions”

1 2

1 2

Spontaneous Symmetry breaking

1 2

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Outlook – toward all-optical optomechanics

optical tweezers

fixed FP mirror

Main advantage:

Optical spring is coupled to reservoir at T~ 0

(S. Singh, D. Goldbaum, E. M. Wright, PM, in preparation)

UA Math 2010

effects of laser linewidth

M

laser spectrum

2v

(G. Phelps)

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UA Math 2010

nano-MRI

Cantilever force sensor at the heart of IBM's"nano-MRI" microscope. IBM scientists have used this nano-MRI to visualize structures at resolution 60,000 times better than current magnetic resonance imaging technology allows. This technique brings MRI capability to the nanoscale level for the first time and represents a major milestone in the quest to build a microscope that could "see" individual atoms in three dimensions. With further development, applications could include understanding how individual proteins interact with drugs for discovery and development, and analyzing computer circuits only a few atoms wide.

UA Math 2010

two-wavelengths approach

T. Corbitt et al, PRL 98, 150802 (2007)

Spring constant

damping

k>0 >0 Dynamically unstable

k>0 <0 Stable

k<0 >0 “anti-stable”

k<0 <0 Statically unstable(carrier)

(su

bca

rrie

r)

UA Math 2010

basic optomechanics − quantum approach

2† 2 21

( )2 2n M

pq a a m q

mH

1( ) 1

1 /n n n

n c qq

L q q L L

† †2

2 21

2 2n M a ap

a a mm

qH q

But:

So:

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three-mirror geometry

[ A. Dorsel et al., PRL 51, 1550 (1983); PM et al, JOSA B11, 1830 (1985); J. D. McCullen et al., Optics Lett. 9, 193 (1984) ]

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three-mirror cavity

See also: M. Bhattacharya and PM, PRL 99, 073601 (2007) M. Bhattacharya et al, PRA 77, 033819 (2008)

Early discussion: PM et al., JOSA B2, 1830 (1985)

…

Nature 452, 72 (2008)

UA Math 2010

three-mirror cavity, R=1

,

,

(1 / )

(1 / )n l n

n r n

q L

q L

† † †2

† 2 21( ( ))

2 2n M

pa a b b m q a a b b qH

m

L L0 q

UA Math 2010

three-mirror cavity, R<1

, 0( )n e n e L q q

, 0( )n o n o L q q

2† † 2 † †21

( ) ))2

((2n e n o M L

pa a b b m a a bH q b q

m

0 0q

W. J. Fader, IEEE J. Quant. Electron. 21, 1838 (1985)

/ 2q L0

freq

uenc

y

0q

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three-mirror cavity, R<1

20, ( )Qn e n q q

20, ( )Qn o n o q q

2† † 2 2 † † 21

( ) (( )2 2

)Qn e n o MHp

a a b b m a aqm

b b q

0 0q

allows preparation of energy eigenstates& observation of quantum jumps

/ 2q L0

freq

uenc

y

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a flavor of the full theory (R=1)

† † †2 22

† 1

2 2 Mc qp

H a a b b m a a b b qm

Quantum Langevin equations of motion, input-output formalism:

laser cavity

2 2

in

in

in† †

/ 2

/ 2

/

/ 2M M

a i Q a

b i Q b

p m

m a a b b m p

q

a

p

b

q

Noise operators:

†in in, in in in, ' ( ')sa t a a t a t a t t t

'in in in0 ' 1 coth

2 2 2i t tM

B

dt t t e

m k T

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experimental signatures

2

2eff

eff

e f

f

f

e f

( )

1 1

2 2

B

q

B

m

k

d q

k T

S

q

Tn

k

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future ring cavity work

• Small quantum fluctuations

• Feeble quantized fields

• Revisit the Fabry-Pérot case

• Damping of the matter-wave side modes Heating and cooling mechanisms

• Optical monitoring of matter-wave fluctuations via dark states

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asymmetric pumping

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spin-polarized fermions in Fabry-Pérot

Rina KanamotoRina Kanamoto

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elementary excitations – bosons vs. fermions

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bosonization of fermionic field

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cantilever state measurement

How can we measure the quantum state of the cantilever? … No phonon counter (yet)!

Idea: monitor the state of a detector atom coupled to both the field to be characterized and to an additional classical field

Needed:

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field state measurement in the microwave regime

Wigner state tomography

[Banaszek and Wodkiewicz, PRL 76, 4344 (1996)]

Atomic homodyne detection

[Wilkens and PM PRA 43, 3832 (1991)]

MW pulse

( )D

qubit

Measurement

LO

qubit

Measurement

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nonlinear atomic homodyning detection of cantilever state

40

2

3 / 4

ac B F Fx Bc c

B B

g g m Gm

G r

†2 0Raman ( )

2e

k k

gH a a

†( )ac B F Fx B c acg m G x g cH c

Need:2 0

2e

acL

gg

Zeeman interaction:

(magnetic field gradient)

(see P. Treutlein et al)

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Wigner characteristic function measurement

cantilev

†

er

2ex

*

1 1. .

2 4

( ) ( )

( )

ai

ig I

c cW

W

ig e

P e h c

C Tr e

C

• classical electromagnetic field• weak cantilever excitation• interaction time

† † †k k k kc c a a a a I

S. Singh + PM., arXiv 0908.1415v1

† †ex

1 1 1cos(2 2 1) cos(2 2 )

2 2 2

( ) / 2k

P g A A g A A

A a c

atom | | | | Initial atomic state:

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measurement back-action

, 1,

| ( ) | ;|

( )c i

c i

g

g U g

P

State of field after detection of atom in ground state:

,

, 1

1| cos( ) ( / 2) ( / 2)

2

1sin( ) ( / 2) ( / 2) |

2

c i

g

c i

e

g I D DP

i g I D DP

Cantilever initially in ground state

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Dissipation – some numbers

6Li atoms coupled to Si resonator: 65 s c1/ eg

Typical lifetime of trapped atoms: 100 ms to a few seconds

Dipole-allowed transitions rates: MHz to GHz

Raman transitions are to virtual state: population weighting factor

2/L k L

Raman decay time: a few msec

MECHANICAL COUPLING

Thermal dissipation: thermal thermal thermalc B

c

k Tn n n

Q

OPTICAL COUPLING

6bath1MHz Q 10 10mKc T 3

dissipation 4 10 sec