Wrubel, J. P. and Schwettmann, A. and Fahey, D. P. and Glassman, Z. and

Pechkis, H. K. and Griffin, P. F. and Barnett, R. and Tiesinga, E. and Lett,

P. D. (2018) Spinor Bose-Einstein-condensate phase-sensitive amplifier

for SU(1,1) interferometry. Physical Review A - Atomic, Molecular, and

Optical Physics, 98. ISSN 1094-1622 ,

http://dx.doi.org/10.1103/PhysRevA.98.023620

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A spinor Bose-Einstein condensate phase-sensitive amplifier for SU(1,1)interferometry

J. P. Wrubel,1, 2 A. Schwettmann,3, 2 D. P. Fahey,2 Z. Glassman,2 H. K.

Pechkis,4, 2 P. F. Griffin,5, 2 R. Barnett,6, 7 E. Tiesinga,2 and P. D. Lett2

1Department of Physics, Creighton University, 2500 California Plaza, Omaha, Nebraska 68178, USA2Quantum Measurement Division, National Institute of Standards and Technology,

and Joint Quantum Institute, NIST and University of Maryland,

100 Bureau Drive, Gaithersburg, Maryland 20899-8424, USA3Homer L. Dodge Department of Physics and Astronomy,

The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA4Department of Physics, California State University, Chico, CA, 95973, USA

5Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK6Department of Mathematics, Imperial College London, London, United Kingdom, SW7 2AZ

7Department of Physics, Joint Quantum Institute and Condensed Matter Theory Center,

University of Maryland, College Park, Maryland 20742, USA

(Dated: July 19, 2018)

The SU(1,1) interferometer was originally conceived as a Mach-Zehnder interferometer with thebeam-splitters replaced by parametric amplifiers. The parametric amplifiers produce states withcorrelations that result in enhanced phase sensitivity. F = 1 spinor Bose-Einstein condensates(BECs) can serve as the parametric amplifiers for an atomic version of such an interferometer bycollisionally producing entangled pairs of |F = 1,m = ±1〉 atoms. We simulate the effect of singleand double-sided seeding of the inputs to the amplifier using the truncated-Wigner approximation.We find that single-sided seeding degrades the performance of the interferometer exactly at thephase the unseeded interferometer should operate the best. Double-sided seeding results in a phase-sensitive amplifier, where the maximal sensitivity is a function of the phase relationship betweenthe input states of the amplifier. In both single and double-sided seeding we find there exists anoptimal phase shift that achieves sensitivity beyond the standard quantum limit. Experimentally,we demonstrate a spinor phase-sensitive amplifier using a BEC of 23Na in an optical dipole trap.This configuration could be used as an input to such an interferometer. We are able to controlthe initial phase of the double-seeded amplifier, and demonstrate sensitivity to initial populationfractions as small as 0.1%.

I. INTRODUCTION

Quantum coherent states of photons or matter arethe workhorses for interferometric measurements becausethey can be made from large numbers of particles withwell-defined phases. Although classical plane waves haveno phase uncertainty, quantum coherent states of Nparticles have an inherent phase uncertainty that limitsthe interferometer’s sensitivity to the standard quantumlimit (SQL) ∆φSQL ∝ 1/

√N . Caves et al. [1, 2] realized

that an interferometer exploiting quantum correlationsbetween the input modes could surpass the SQL, reach-ing phase-sensitivities scaling with the Heisenberg limit∆φ ∝ 1/N , where N is the number of correlated par-ticles. The challenge in achieving such high sensitivityis producing and maintaining highly non-classical corre-lated states of photons or atoms.

One solution to the problem of producing quantum cor-relations for interferometry is to replace the beamsplit-ters in a Mach-Zehnder interferometer with parametricamplifiers (Fig. 1). Parametric amplifiers have a bright“pump” mode containing the vast majority of the par-ticles, and amplify the seeded or unseeded “probe” and“conjugate” states. Such a parametric amplifier may beconstructed from a single-mode Bose-Einstein condensate

within the Zeeman-split ground-state hyperfine manifold|F,m〉, which is called a spinor parametric amplifier [3–5].We consider the hyperfine states |F = 1,m〉 having

complex amplitudes represented by the spinor ξ =(ξ+1,, ξ0, ξ−1)

T , with ξm =√ρmeiθm , mean fractional

populations ρm = Nm/N and phases θm. The initialstate of the BEC of N atoms is represented by a productof the coherent states |αm〉m = |

√Nξm〉m. In the spinor

parametric amplifier |α0〉0 serves as the pump and |α±〉±provide the probe and conjugate modes.Parametric amplifiers can be modeled analytically us-

ing a linear “undepleted pump” approximation, result-ing in an interferometer design with SU(1,1) symme-try [6]. When the input parametric amplifier producesan average of N = (N+ + N−) ≫ 1 particles in theprobe and conjugate states, then the minimum phase-sensitivity of the interferometer ∆φ = 1/

√

N (N − 2)approaches the Heisenberg limit [6]. This scaling withthe number of atoms in the probe and conjugate statesN is not identical to scaling with the total number ofparticles N = Npump + N . This SU(1,1) interferometerhas been explored in various systems theoretically [6–11], and the unseeded interferometer has been exploredexperimentally [12–18].The atomic spinor SU(1,1) interferometer is sketched

in Fig. 1, where initially 0− 1% of the atoms are in the

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FIG. 1. (Color online) Configuration of the seeded spinor SU(1,1) interferometer. The initial spinor amplifier (first green box)mixes the pump coherent state |α〉0 (dashed black line) with the probe and conjugate coherent seeds |α±〉± (blue solid lines)depending on the relative spinor phase θ. For a given set of input states, the amplifier gain is determined by the parameters forspinor dynamics c, q, and tSMD (defined in section II). A shift in the spinor phase within the interferometer φ is sensed when asecond parametric amplifier (second green box) reverses the dynamics by negating the values of c and q, and evolving the statefor the same time tSMD. The output of the interferometer is the number of atoms N+,out +N−,out.

probe and conjugate states m = ±1, with the remainderin the pump state m = 0. After the input amplifier,a phase shift φ is applied to the relative spinor phaseθ = θ+1 + θ−1 − 2θ0, and the amplification is reversed ina second parametric amplifier. The measured output ofthe interferometer is the sum of the atoms in the m = ±1levels Nout = N+,out +N−,out.

Optical parametric amplifiers typically operate wellwithin the undepleted pump approximation due to theweakness of four-wave mixing and down conversion pro-cesses. Study of the seeding of the initial probe and con-jugate states of the optical SU(1,1) interferometer showedthat sensitivity beyond the standard quantum limit is lostat φ = 0, which would otherwise be the phase yieldingmaximum sensitivity in the unseeded case. By operatingthe interferometer away from φ = 0, the interferometercould still surpass the SQL [7, 9, 19].

In contrast to optical amplifiers, spinor parametricamplifiers in a single spatial mode rapidly deplete thefinite pump resource, transferring a majority of theatoms to the probe and conjugate states, and breakingthe linear approximation. Simulations of this nonlinearspinor amplifier as an input to the SU(1,1) interferometershowed that Heisenberg-limited sensitivity was retained,although the effects of initial seeding of the probe andconjugate states was not considered [10]. Extending theseresults, we simulate the nonlinear spinor amplifier as aninput to the SU(1,1) interferometer, including the casesof small initial coherent seeds in one or both of the probeand conjugate states, which has not previously been con-sidered.

Demonstrating a spinor SU(1,1) interferometer capableof surpassing the standard quantum limit is a challeng-ing task. The best results with a vacuum-seeded spinorSU(1,1) interferometer demonstrated sensitivity some-what better than the SQL: ∆φ ≈ ∆φSQL/(1.6±0.5) [15].Even in this vacuum-seeded case, it was observed that

the sensitivity of the interferometer was suppressed nearφ = 0, where the ideal interferometer (without losses)would have been maximally sensitive.

The difficulty presented by the full SU(1,1) interferom-eter of Ref. [6] is that at the operating point with maxi-mal theoretical sensitivity, the mean and variance of thedetected particles at the output, Nout, go to zero leav-ing the measurement susceptible to noise from losses [9],or imperfect reversibility [15]. We propose that a thirdreason for loss in sensitivity of the spinor SU(1,1) interfer-ometer is imperfect state preparation. If even very smallfractions of atoms are present in m = ±1, then the initialprobe and conjugate states are not exact vacuum-statesand the effects of these imperfections must be considered.

Here, we simulate the nonlinear spinor SU(1,1) in-terferometer using the truncated-Wigner approximation[20, 21] in a single spatial mode for the case of coherentsingle and double-sided seeding of the input states. Forsingle sided seeding, we find a decrease in sensitivity atφ = 0 analogous to the optical interferometer [7, 9], whilesurpassing the standard quantum limit at other phases.

We also simulate the case of double-sided seeding of theinput amplifier, resulting in finite populations in bothoutput modes, which are therefore less susceptible tomeasurement noise. Our simulations show that a nonlin-ear SU(1,1) interferometer initiated using double-sidedseeding of the m = ±1 states can significantly surpassthe standard quantum limit. For particular values of theinitial spinor phase θ of the phase-sensitive amplifier withfixed evolution time, we find a maximum sensitivity notonly surpassing the SQL, but also surpassing the expec-tation for vacuum-seeding.

Experimentally, we demonstrate the phase sensitivityof a spinor parametric amplifier that could be used toconstruct such an SU(1,1) interferometer using a 23NaBEC in a crossed optical dipole trap [22]. The BECof N ≈ 3.5 × 104 atoms meets the single spatial-mode

3

criterion so that the spatial wave-function can be ignoredin the dynamics. We show a three sigma change in theamplifier output when one input to the phase-sensitiveamplifier is seeded with a fraction of just 10−3.

II. ENTANGLEMENT IN SPINOR BECS

Entanglement in spinors has been produced both bytransferring the atoms into an unstable m = 0 level thatspontaneously decays into entangled pairs of m = ±1atoms [3, 23, 24], as well as by adiabatic passage throughquantum phase transitions [25]. The pairs of atoms pro-duced from an unstable spinor state in the undepletedpump regime are analogous to the pairs of photons pro-duced in downconversion or four-wave mixing [3]. Theatomic state that results from this process in the linearapproximation is a two-mode vacuum-squeezed state.Spontaneous emission of correlatedm = ±1 atom pairs

can be induced in a sodium BEC when the |1, 0〉 levelhas a slightly higher energy than the average energy ofthe |1,±1〉 levels. The gain of the parametric amplifiercan be controlled by varying the time tSMD allowed forthis spin-mixing dynamics (SMD) to proceed, from veryshort times for which a linear Bogoliubov approxima-tion is valid, to very long times where this approxima-tion breaks down and the depletion of the pump statemust be taken into account. During spin-mixing dynam-ics magnetization-conserving spin-flip collisions converta pair of m = 0 atoms into a correlated pair of m = ±1atoms and vice-versa.The most significant difference between entangled pho-

tons and entangled atoms is the interaction energy c =c2n between the atoms that introduces nonlinearity intothe evolution, where c2 = 2π~2(a2 − a0)/3µ, the aF arethe scattering lengths for two atoms with total spin F ,µ is the reduced mass of two atoms, and n is the meandensity of the BEC [26]. In our crossed optical dipoletrap, the atoms remain in a single spatial wavefunctionfor the duration of the experiment, so that every pair ofatoms created further increases the rate of pair produc-tion, exponentially increasing the number of atom pairsfor short times.When the spin-components of the BEC all share a sin-

gle spatial wavefunction we can make the single-mode ap-proximation (SMA). The SMA leads to a simplification ofthe dynamics of the mean-field spinor ξ = (ξ+1, ξ0, ξ−1)

T ,where |ξm〉 = |√ρmeiθm〉, down to two dynamical vari-ables: ρ0, which is the fractional population in m = 0,and the spinor phase θ = θ+1 + θ−1 − 2θ0 [26]. Thedynamics is a function of the spin-dependent interac-tion energy c, and the quadratic Zeeman energy q =(E− +E+ − 2E0)/2, where Em is the energy of the eachhyperfine state |1,m〉. The magnetization M ≡ ρ+ − ρ−is conserved in the SMA.Spinor parametric amplifiers typically avoid the non-

linear regime by restricting the dynamics to short timesso that only a very small fraction of atoms produce corre-

lated pairs [15, 27]. In the linear or “undepleted pump”regime, where the initial population in m = 0 has notchanged significantly, the quantum spin-mixing dynam-ics is modeled by making a linear Bogoliubov approx-imation. The Bogoliubov approximation remains validwhen the fraction of entangled atoms produced fromspin-mixing dynamics (SMD) is ρN = ρ+ + ρ− . 0.01.Although an SU(1,1) interferometer may have phase sen-sitivity that scales with the Heisenberg limit ∆φ ∝ 1/N ,N is small and the absolute sensitivity is limited. Nev-ertheless, an enhancement of 2.05 dB over the SQL wasachieved in a measurement of the hyperfine clock transi-tion (|F,m〉 = |1, 0〉 → |2, 0〉) in a spinor BEC using anaverage of just 0.75 entangled atom pairs [27].Here we consider the case of producing large fractions

of entangled m = ±1 atoms such that the pump is signif-icantly depleted. This nonlinear SU(1,1) interferometeris in principle capable of significantly enhanced absolutesensitivity since almost all the pump atoms participate inthe measurement [10]. The challenge introduced by thenonlinear SU(1,1) interferometer is negating the sign ofthe interaction energy c and quadratic Zeeman shift q toachieve perfect reversal of the spin dynamics. Althoughan a.c. Stark shift may be used to reverse the sign of q, itis extremely difficult to reverse the sign of the spinor in-teraction energy c. Feshbach resonances [28] can be usedto modify atomic interactions, but have limited applica-bility to the spinor system because of the need for largemagnetic fields or short experiment times. Within theBogoliubov approximation the reversal may be achievedwithout negation of c, q using an applied phase shift φinside the interferometer [15].

III. THEORY AND SIMULATIONS OF THESEEDED NONLINEAR SU(1,1) SPINOR

INTERFEROMETER

An SU(1,1) interferometer is structurally similar to theMach-Zehnder interferometer. In an unseeded SU(1,1)spinor interferometer, the input is a coherent state withall atoms in |α0〉0. Spin-mixing dynamics (SMD) ofthe atoms in the first amplifier produces pairs of entan-gled m = ±1 atoms in a time tSMD. Unlike in a pho-ton amplifier, the atoms are always within the “gain”medium, and nearly all atoms can be transferred outof m = 0. The number of atoms in m = ±1 withinthe interferometer after the spin-mixing dynamics is N .At this point a phase shift φ is applied and the spin-mixing dynamics is reversed using a second spinor am-plifier. Following Ref. [10] we treat the optimal case ofexactly reversing the Hamiltonian by mapping c → −cand q → −q. The spinor is allowed to evolve back forthe same amount of time tSMD, and the output of theinterferometer is the summed atom number in m = ±1,Nout = N+,out +N−,out.The quantum Hamiltonian of a spin-one Bose-Einstein

condensate occupying a single spatial mode [26, 29, 30] is

4

H = c/(2N)~F · ~F−qa†0a0 , where~F =

∑

mm′ a†m ~Fmm′am′

is the total spin operator, ~Fmm′ = (F xmm′ , F

ymm′ , F z

mm′),and F x,y,z

mm′ are matrix elements of the x, y, and z spin-onematrices. Finally, a†m (am) are the creation (annihilation)operators for an atom in spin state m = +1, 0,−1 andin the relevant single spatial mode. The mean field lan-guage implicitly used in the first two sections follows froma unitary transformation of H under the displacementoperator D =

∏1m=−1 exp(αma†m − α∗

mam). The part ofthe transformed Hamiltonian that is independent of thecreation and annihilation operators is minimized with re-spect to the αm, and only terms that are up to quadraticin the creation and annihilation operators are kept. Theoperator-independent terms of the transformed Hamil-tonian can be recognized as the classical or mean-fieldspinor Hamiltonian for the αm and α∗

m or equivalentlyρm and θm. The quadratic terms correspond to the Bo-goliubov Hamiltonian. Its role is discussed in the nextsection.For spin-1 alkali-metal atoms in the presence of a small

homogeneous magnetic field the parameter q correspondsto the quadratic Zeeman shift and is non-negative. Forsodium atoms c > 0 and the ground state of the meanfield Hamiltonian at zero magnetization (M = 0) is acoherent state with all atoms in m = 0, i.e. ρ0 = 1.The application of a microwave field can change the signof q and for −2c < q < 0 [31] this causes a dynamicalinstability as the classical Hamiltonian has a saddle pointat ρ0 = 1. Because of this instability pairs of m = 0atoms will spontaneously scatter into pairs of correlatedatoms in the m = ±1 states. The amplification processis called phase-insensitive because the spinor phases θ+1,θ−1 and θ0 are ill- or un-defined. Even if only one ofthe m = ±1 states is seeded so that one of ρ±1 > 0and the corresponding phase is defined, the amplificationremains phase-insensitive, since the initial spinor phaseθ = θ+1 − θ−1 − 2θ0 is still undefined.In a double-seeded SU(1,1) interferometer, the input

has most of the atoms in the m = 0 state, but also smallcoherent populations inm = ±1. Hence, the initial phaseθ of the spinor is well-defined. This initial phase controlsthe subsequent amplification or de-amplification of pairsof atoms and characterizes the phase-sensitive amplifier.The figure of merit for an interferometer is its phase

sensitivity ∆φ. We estimate ∆φ from simulations of themean Nout and variance (∆Nout)

2 of the total number ofatoms in m = ±1 at the output of the interferometer asa function of the phase shift φ within the interferometer.Using error propagation to estimate the phase sensitivitygives

∆φ = ∆Nout/

(

dNout

dφ

)

. (1)

The Fisher information F (φ) can be used to quantifythe presence of useful entanglement in a system. F (φ) isdefined in terms of the conditional probability P (Nout|φ)of obtaining Nout atoms at the output given an interfer-

ometer shift φ [10]

F (φ) =

∞∑

Nout=0

1

P (Nout|φ)

(

dP (Nout|φ)dφ

)2

. (2)

The maximum of the Fisher information Fmax ≡max[F (φ)] is achieved at an operating phase shift φopt. Ifthe measurement scheme chosen is not the optimal one,then Fmax underestimates the quantum Fisher informa-tion FQ. Whereas Fmax is the best case for the SU(1,1)measurement scheme, FQ is maximized over the set of allpossible quantum measurements [32, 33]. The inequalityFQ > N provides a condition “sufficient for entanglementand necessary and sufficient for entanglement useful forquantum metrology” [33].The Fisher information can be measured experimen-

tally using the squared Hellinger distance [24]; how-ever, this technique does not work for simulations us-ing the truncated-Wigner approximation as the distri-butions are broadened by quantum noise. We insteadcalculate 1/(∆φ)2, which provides a lower-bound for theFisher information F (φ) ≥ F<(φ) = 1/(∆φ)2 [10]. Theultimate phase-sensitivity produced by n repeated mea-surements is calculated by the Cramer-Rao lower bound∆φCR = 1/

√

nF (φ) ≤ ∆φ/√n.

The standard quantum limit for the Fisher informa-tion is FSQL = N , while the Heisenberg limit for largeN is FH = N 2. We define the scaled Fisher informa-tion f(φ) ≡ F (φ)/FSQL = F (φ)/N . With this defini-tion the SQL corresponds to a scaled Fisher informationfSQL = 1, and the Heisenberg limit is fH = N . Scal-ing the Fisher information in this way by the numberof atoms in m = ±1 after time tSMD gives an appropri-ate measure of entanglement in the system. If, however,we ask whether the interferometer performs better than aMach-Zehnder interferometer with coherent state inputs,then we should scale the Fisher information by the totalnumber of atoms in the BEC, N , instead of N .

A. Bogoliubov approximation for thephase-sensitive spinor amplifier

In this subsection we expand on the linear Bogoliubovtheory to include the cases of single and double-sided ini-tial seeding. These situations correspond to the phase-insensitive amplifier (PIA), and phase-sensitive amplifier(PSA) respectively. We use the Bogoliubov approxima-tion to derive an expression for the early-time evolutionof the spinor that is analogous to the phase-sensitive am-plification of photons in a nonlinear medium. The Bogoli-ubov approximation simplifies the Hamiltonian by con-sidering only small deviations from the initial state.In the photon case, a strong pump laser beam inter-

acts with two weak beams, the probe and conjugate. Allstates are initially coherent states, and the total numberof photons in the amplified probe and conjugate beamshas four terms. One term is the initial populations, a sec-ond term is due to spontaneous emission, a third term is

5

due to phase-insensitive gain, and a fourth term is dueto phase-sensitive gain [9].For the single-mode spinor BEC, we derive a similar

equation by using a Bogoliubov expansion of the m = ±1states, while assuming that the much larger number ofatoms inm = 0 is a constant classical field. This has beendone previously for the phase-insensitive case [10, 34],

but here we give the result for a phase-sensitive spinoramplifier, beginning with coherent states in |α±〉± with

average initial number of atoms N±, and initial phase θ(the bar indicates initial values).

The total number of atoms in m = ±1 during amplifi-cation, N (t), in the Bogoliubov approximation is

N (t) =(N+ +N−) + 2(c2n

~ω

)2

sinh2(ωt) + 2(N+ +N−)(c2n

~ω

)2

sinh2(ωt)

+4

√

N−N+c2n

~ωsinh(ωt)×

(

cosh(ωt) sin θ +c2n+ q

~ωsinh(ωt) cos θ

)

. (3)

The instability rate ω(q < 0) ≡√

|q|(2c2n− |q|)/~ isa maximum when the effective quadratic Zeeman shiftis q = −c2n. As for the optical case, we identify fourterms in the amplification process: initial population,spontaneous emission, phase-insensitive gain, and phase-sensitive gain. When q = −c2n the term proportionalto cos θ in Eq. (3) is zero, and the atom and photonamplifiers behave in closely analogous ways. The mostdistinct difference between the optical and spinor para-metric amplifiers is the ability to nearly completely de-plete the pump in the spinor amplifier.

B. Fisher information

The Fisher information of the unseeded SU(1,1) inter-ferometer in the Bogoliubov approximation [10] is

FBog(φ) =N (N + 2)

N (N + 2) sin2(φ/2) + 1cos2(φ/2), (4)

where N = N+(tSMD) + N−(tSMD) is the number ofatoms in m = ±1 after the initial amplification dueto spin-mixing dynamics. Within this approximation,FBog,max = FBog(φopt = 0) = N (N + 2), or a scaledFisher information of fBog,max = (N + 2), with N > 0.

C. Truncated-Wigner approximation

We numerically estimate the Fisher information bothwithin the Bogoliubov approximation (short tSMD) andbeyond to large depletion of m = 0 (long tSMD) by calcu-lating ∆φ (Eq. (1)) using the truncated-Wigner approx-imation to simulate the spinor wavefunction.The truncated-Wigner approximation (TWA) im-

proves upon the Gross-Pitaevskii Equation (GPE)method by allowing for the inclusion of quantum fluc-tuations [20, 21]. Briefly, the TWA simulations are aver-ages over many GPE simulations with initial noise cor-responding to the Wigner transform of the initial state.This noise has a form such that the short time dynamics

agrees with that of Bogoliubov theory. We simulate 104

trajectories to achieve good statistics on both the aver-age values and standard deviations. Using the TWA weobtain the average number of atoms at the output of theinterferometer Nout(φ) and its uncertainty (∆N )out(φ)as a function of interferometer phase. We calculate thederivative dNout/dφ for each particular instance of therandom initial conditions using a symmetric phase stepof 10−6 rad. In order to avoid the limits of the precisionof floating point numbers for the small changes in atomnumber caused by the 10−6 rad change in phase, we takethe derivative of each iteration (for instance, in Eq. (1))before averaging instead of taking the derivative of theaveraged quantities. With infinite precision these twomethods would yield the same result. Note that whensimulating using the TWA it is important to compensatethe raw averages and variances for Weyl ordering of theoperators [35].

D. Simulation results

The parameters used for the simulation are typical ofa sodium spinor BEC with N = 3 × 104 atoms, c/h =25 Hz, and q/h = −1 Hz. The results do not dependsignificantly on c or q so long as comparisons are madewith the same N . The number of atoms is increased byincreasing the time for spin-mixing dynamics tSMD up toabout 100 ms or until ρ0 is nearly depleted.We estimate the Fisher information with F<(φ) =

1/(∆φ)2 ≤ F (φ), and the scaled Fisher information withf<(φ) = F<(φ)/N . Figure 2(a) shows a plot of f<(φ) forvarious fractions of single-side seeding in m = −1 withtSMD = 20 ms. For vacuum seeding the lower bound fromthe TWA simulation agrees well with the Bogoliubov the-ory, Eq. (4). The TWA simulation is most sensitive tonoise due to random quantum fluctuations near φ = 0where ∆Nout and dNout/dφ both approach zero.The effect of even very small seed fractions is quite

pronounced. For a fraction as small as 10−6, f<(φ) issignificantly suppressed at φ = 0. As the fractional

6

35

30

25

20

15

10

5

0-0.2 -0.1 0.0 0.1 0.2

0.001

0.01

0.1

1

10-6 10

-3

ρ

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f <(φ)=

F<(φ)/N

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φ (rad)<latexit sh a1_base64="0adUtnaj2n rbnqNVOKQESvZekVo=">AA ACEHicbZDLSsNAFIYnXmu 9pbp0M1iEuimJCLosuHFZ wV6gKWUymbRDJ5Mwc6KWk Jdw71ZfwZ249Q18Ax/DaZ uFbf3hwM9/zuEcPj8RXIPj fFtr6xubW9ulnfLu3v7Bo V05aus4VZS1aCxi1fWJZo JL1gIOgnUTxUjkC9bxxzf TfueBKc1jeQ+ThPUjMpQ8 5JSAiQZ2xUtG3AP2BBmuKR Kc5wO76tSdmfCqcQtTRYW aA/vHC2KaRkwCFUTrnusk 0M+IAk4Fy8teqllC6JgMW c9YSSKm+9ns9RyfmSTAYa xMScCz9O9GRiKtJ5FvJiMC I73cm4b/9XophNf9jMskB Sbp/FCYCgwxnnLAAVeMgp gYQ6ji5ldMR0QRCobWwhU Vg4Elh3nZoHGXQaya9kXdd eru3WW1USsgldAJOkU15K Ir1EC3qIlaiKJH9IJe0Zv 1bL1bH9bnfHTNKnaO0YKs r1/tQJz8</latexit><latexit sh a1_base64="0adUtnaj2n rbnqNVOKQESvZekVo=">AA ACEHicbZDLSsNAFIYnXmu 9pbp0M1iEuimJCLosuHFZ wV6gKWUymbRDJ5Mwc6KWk Jdw71ZfwZ249Q18Ax/DaZ uFbf3hwM9/zuEcPj8RXIPj fFtr6xubW9ulnfLu3v7Bo V05aus4VZS1aCxi1fWJZo JL1gIOgnUTxUjkC9bxxzf TfueBKc1jeQ+ThPUjMpQ8 5JSAiQZ2xUtG3AP2BBmuKR Kc5wO76tSdmfCqcQtTRYW aA/vHC2KaRkwCFUTrnusk 0M+IAk4Fy8teqllC6JgMW c9YSSKm+9ns9RyfmSTAYa xMScCz9O9GRiKtJ5FvJiMC I73cm4b/9XophNf9jMskB Sbp/FCYCgwxnnLAAVeMgp gYQ6ji5ldMR0QRCobWwhU Vg4Elh3nZoHGXQaya9kXdd eru3WW1USsgldAJOkU15K Ir1EC3qIlaiKJH9IJe0Zv 1bL1bH9bnfHTNKnaO0YKs r1/tQJz8</latexit><latexit sh a1_base64="0adUtnaj2n rbnqNVOKQESvZekVo=">AA ACEHicbZDLSsNAFIYnXmu 9pbp0M1iEuimJCLosuHFZ wV6gKWUymbRDJ5Mwc6KWk Jdw71ZfwZ249Q18Ax/DaZ uFbf3hwM9/zuEcPj8RXIPj fFtr6xubW9ulnfLu3v7Bo V05aus4VZS1aCxi1fWJZo JL1gIOgnUTxUjkC9bxxzf TfueBKc1jeQ+ThPUjMpQ8 5JSAiQZ2xUtG3AP2BBmuKR Kc5wO76tSdmfCqcQtTRYW aA/vHC2KaRkwCFUTrnusk 0M+IAk4Fy8teqllC6JgMW c9YSSKm+9ns9RyfmSTAYa xMScCz9O9GRiKtJ5FvJiMC I73cm4b/9XophNf9jMskB Sbp/FCYCgwxnnLAAVeMgp gYQ6ji5ldMR0QRCobWwhU Vg4Elh3nZoHGXQaya9kXdd eru3WW1USsgldAJOkU15K Ir1EC3qIlaiKJH9IJe0Zv 1bL1bH9bnfHTNKnaO0YKs r1/tQJz8</latexit><latexit sh a1_base64="0adUtnaj2n rbnqNVOKQESvZekVo=">AA ACEHicbZDLSsNAFIYnXmu 9pbp0M1iEuimJCLosuHFZ wV6gKWUymbRDJ5Mwc6KWk Jdw71ZfwZ249Q18Ax/DaZ uFbf3hwM9/zuEcPj8RXIPj fFtr6xubW9ulnfLu3v7Bo V05aus4VZS1aCxi1fWJZo JL1gIOgnUTxUjkC9bxxzf TfueBKc1jeQ+ThPUjMpQ8 5JSAiQZ2xUtG3AP2BBmuKR Kc5wO76tSdmfCqcQtTRYW aA/vHC2KaRkwCFUTrnusk 0M+IAk4Fy8teqllC6JgMW c9YSSKm+9ns9RyfmSTAYa xMScCz9O9GRiKtJ5FvJiMC I73cm4b/9XophNf9jMskB Sbp/FCYCgwxnnLAAVeMgp gYQ6ji5ldMR0QRCobWwhU Vg4Elh3nZoHGXQaya9kXdd eru3WW1USsgldAJOkU15K Ir1EC3qIlaiKJH9IJe0Zv 1bL1bH9bnfHTNKnaO0YKs r1/tQJz8</latexit>

Unseeded 0.0001% 0.001% 0.01% 0.1% 1.0% 2.0% 5.0%

(a)50

40

30

20

10

0-0.2 -0.1 0.0 0.1 0.2

50

40

30

20

10

0-3-2-1 0 1 2 3

f <(φ)=

F<(φ)/N

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θ (rad)<latexit sha1_base64="OYaH++F3TqXUxAIIrqp/c x/ImZk=">AAACEnicbZBLSgNBEIZ74ivGVzRLN41BiJswI4IuA25cRjAPSELo6VSSJj09Q3eNGIa5hXu3egV3 4tYLeAOPYSeZhUn8oeDnryqq+PxICoOu++3kNja3tnfyu4W9/YPDo+LxSdOEsebQ4KEMddtnBqRQ0ECBEtqRB hb4Elr+5HbWbz2CNiJUDziNoBewkRJDwRnaqF8sdXEMyLoIT5jQimaDi7RfLLtVdy66brzMlEmmer/40x2EP A5AIZfMmI7nRthLmEbBJaSFbmwgYnzCRtCxVrEATC+ZP5/Sc5sM6DDUthTSefp3I2GBMdPAt5MBw7FZ7c3C/3 qdGIc3vUSoKEZQfHFoGEuKIZ2RoAOhgaOcWsO4FvZXysdMM46W19IVHaLFpUZpwaLxVkGsm+Zl1XOr3v1VuVb JIOXJKTkjFeKRa1Ijd6ROGoSTKXkhr+TNeXbenQ/nczGac7KdElmS8/ULm3ud5Q==</latexit><latexit sha1_base64="OYaH++F3TqXUxAIIrqp/c x/ImZk=">AAACEnicbZBLSgNBEIZ74ivGVzRLN41BiJswI4IuA25cRjAPSELo6VSSJj09Q3eNGIa5hXu3egV3 4tYLeAOPYSeZhUn8oeDnryqq+PxICoOu++3kNja3tnfyu4W9/YPDo+LxSdOEsebQ4KEMddtnBqRQ0ECBEtqRB hb4Elr+5HbWbz2CNiJUDziNoBewkRJDwRnaqF8sdXEMyLoIT5jQimaDi7RfLLtVdy66brzMlEmmer/40x2EP A5AIZfMmI7nRthLmEbBJaSFbmwgYnzCRtCxVrEATC+ZP5/Sc5sM6DDUthTSefp3I2GBMdPAt5MBw7FZ7c3C/3 qdGIc3vUSoKEZQfHFoGEuKIZ2RoAOhgaOcWsO4FvZXysdMM46W19IVHaLFpUZpwaLxVkGsm+Zl1XOr3v1VuVb JIOXJKTkjFeKRa1Ijd6ROGoSTKXkhr+TNeXbenQ/nczGac7KdElmS8/ULm3ud5Q==</latexit><latexit sha1_base64="OYaH++F3TqXUxAIIrqp/c x/ImZk=">AAACEnicbZBLSgNBEIZ74ivGVzRLN41BiJswI4IuA25cRjAPSELo6VSSJj09Q3eNGIa5hXu3egV3 4tYLeAOPYSeZhUn8oeDnryqq+PxICoOu++3kNja3tnfyu4W9/YPDo+LxSdOEsebQ4KEMddtnBqRQ0ECBEtqRB hb4Elr+5HbWbz2CNiJUDziNoBewkRJDwRnaqF8sdXEMyLoIT5jQimaDi7RfLLtVdy66brzMlEmmer/40x2EP A5AIZfMmI7nRthLmEbBJaSFbmwgYnzCRtCxVrEATC+ZP5/Sc5sM6DDUthTSefp3I2GBMdPAt5MBw7FZ7c3C/3 qdGIc3vUSoKEZQfHFoGEuKIZ2RoAOhgaOcWsO4FvZXysdMM46W19IVHaLFpUZpwaLxVkGsm+Zl1XOr3v1VuVb JIOXJKTkjFeKRa1Ijd6ROGoSTKXkhr+TNeXbenQ/nczGac7KdElmS8/ULm3ud5Q==</latexit><latexit sha1_base64="OYaH++F3TqXUxAIIrqp/c x/ImZk=">AAACEnicbZBLSgNBEIZ74ivGVzRLN41BiJswI4IuA25cRjAPSELo6VSSJj09Q3eNGIa5hXu3egV3 4tYLeAOPYSeZhUn8oeDnryqq+PxICoOu++3kNja3tnfyu4W9/YPDo+LxSdOEsebQ4KEMddtnBqRQ0ECBEtqRB hb4Elr+5HbWbz2CNiJUDziNoBewkRJDwRnaqF8sdXEMyLoIT5jQimaDi7RfLLtVdy66brzMlEmmer/40x2EP A5AIZfMmI7nRthLmEbBJaSFbmwgYnzCRtCxVrEATC+ZP5/Sc5sM6DDUthTSefp3I2GBMdPAt5MBw7FZ7c3C/3 qdGIc3vUSoKEZQfHFoGEuKIZ2RoAOhgaOcWsO4FvZXysdMM46W19IVHaLFpUZpwaLxVkGsm+Zl1XOr3v1VuVb JIOXJKTkjFeKRa1Ijd6ROGoSTKXkhr+TNeXbenQ/nczGac7KdElmS8/ULm3ud5Q==</latexit>

φ (rad)<latexit sha1_base64="0adUtnaj2nrbnqNVOKQES vZekVo=">AAACEHicbZDLSsNAFIYnXmu9pbp0M1iEuimJCLosuHFZwV6gKWUymbRDJ5Mwc6KWkJdw71ZfwZ24 9Q18Ax/DaZuFbf3hwM9/zuEcPj8RXIPjfFtr6xubW9ulnfLu3v7BoV05aus4VZS1aCxi1fWJZoJL1gIOgnUTx UjkC9bxxzfTfueBKc1jeQ+ThPUjMpQ85JSAiQZ2xUtG3AP2BBmuKRKc5wO76tSdmfCqcQtTRYWaA/vHC2KaR kwCFUTrnusk0M+IAk4Fy8teqllC6JgMWc9YSSKm+9ns9RyfmSTAYaxMScCz9O9GRiKtJ5FvJiMCI73cm4b/9X ophNf9jMskBSbp/FCYCgwxnnLAAVeMgpgYQ6ji5ldMR0QRCobWwhUVg4Elh3nZoHGXQaya9kXdderu3WW1USs gldAJOkU15KIr1EC3qIlaiKJH9IJe0Zv1bL1bH9bnfHTNKnaO0YKsr1/tQJz8</latexit><latexit sha1_base64="0adUtnaj2nrbnqNVOKQES vZekVo=">AAACEHicbZDLSsNAFIYnXmu9pbp0M1iEuimJCLosuHFZwV6gKWUymbRDJ5Mwc6KWkJdw71ZfwZ24 9Q18Ax/DaZuFbf3hwM9/zuEcPj8RXIPjfFtr6xubW9ulnfLu3v7BoV05aus4VZS1aCxi1fWJZoJL1gIOgnUTx UjkC9bxxzfTfueBKc1jeQ+ThPUjMpQ85JSAiQZ2xUtG3AP2BBmuKRKc5wO76tSdmfCqcQtTRYWaA/vHC2KaR kwCFUTrnusk0M+IAk4Fy8teqllC6JgMWc9YSSKm+9ns9RyfmSTAYaxMScCz9O9GRiKtJ5FvJiMCI73cm4b/9X ophNf9jMskBSbp/FCYCgwxnnLAAVeMgpgYQ6ji5ldMR0QRCobWwhUVg4Elh3nZoHGXQaya9kXdderu3WW1USs gldAJOkU15KIr1EC3qIlaiKJH9IJe0Zv1bL1bH9bnfHTNKnaO0YKsr1/tQJz8</latexit><latexit sha1_base64="0adUtnaj2nrbnqNVOKQES vZekVo=">AAACEHicbZDLSsNAFIYnXmu9pbp0M1iEuimJCLosuHFZwV6gKWUymbRDJ5Mwc6KWkJdw71ZfwZ24 9Q18Ax/DaZuFbf3hwM9/zuEcPj8RXIPjfFtr6xubW9ulnfLu3v7BoV05aus4VZS1aCxi1fWJZoJL1gIOgnUTx UjkC9bxxzfTfueBKc1jeQ+ThPUjMpQ85JSAiQZ2xUtG3AP2BBmuKRKc5wO76tSdmfCqcQtTRYWaA/vHC2KaR kwCFUTrnusk0M+IAk4Fy8teqllC6JgMWc9YSSKm+9ns9RyfmSTAYaxMScCz9O9GRiKtJ5FvJiMCI73cm4b/9X ophNf9jMskBSbp/FCYCgwxnnLAAVeMgpgYQ6ji5ldMR0QRCobWwhUVg4Elh3nZoHGXQaya9kXdderu3WW1USs gldAJOkU15KIr1EC3qIlaiKJH9IJe0Zv1bL1bH9bnfHTNKnaO0YKsr1/tQJz8</latexit><latexit sha1_base64="0adUtnaj2nrbnqNVOKQES vZekVo=">AAACEHicbZDLSsNAFIYnXmu9pbp0M1iEuimJCLosuHFZwV6gKWUymbRDJ5Mwc6KWkJdw71ZfwZ24 9Q18Ax/DaZuFbf3hwM9/zuEcPj8RXIPjfFtr6xubW9ulnfLu3v7BoV05aus4VZS1aCxi1fWJZoJL1gIOgnUTx UjkC9bxxzfTfueBKc1jeQ+ThPUjMpQ85JSAiQZ2xUtG3AP2BBmuKRKc5wO76tSdmfCqcQtTRYWaA/vHC2KaR kwCFUTrnusk0M+IAk4Fy8teqllC6JgMWc9YSSKm+9ns9RyfmSTAYaxMScCz9O9GRiKtJ5FvJiMCI73cm4b/9X ophNf9jMskBSbp/FCYCgwxnnLAAVeMgpgYQ6ji5ldMR0QRCobWwhUVg4Elh3nZoHGXQaya9kXdderu3WW1USs gldAJOkU15KIr1EC3qIlaiKJH9IJe0Zv1bL1bH9bnfHTNKnaO0YKsr1/tQJz8</latexit>

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Unseeded -2.86 rad (sep) 0 rad 2.14 rad 2.86 rad (sep) 3.02 rad

(b)

FIG. 2. Lower bound on the scaled Fisher information obtained from TWA simulations versus interferometer phase φ fortSMD = 20 ms with (a) increasing percentages of single-sided seeding into m = −1, and with (b) increasing spinor phase for0.1% double-sided seeding. The dashed line shows f(φ) calculated from Eq. (4) for the unseeded case. The inset to (a) isthe amplified fraction after spin-mixing dynamics versus the initial seed fraction ρseed. The inset to (b) is f<,max versus initialspinor phase, with colored dots corresponding to the traces shown in the main figure. Vertical dotted lines indicate the phaseof the separatrix (sep) θsep = ±2.86 rad that divides oscillating phase and running phase spinor dynamics [26].

seed increases, the narrow dip gets wider, and the maxi-mum of f<(φ) decreases until saturating near 0.1% seed-ing. For 2.0% seeding the amplified fraction of atoms isρ = N/N ≈ 0.5, and for larger seed fractions the peakdecreases as we enter the heavily depleted regime.An important consequence of the increasing seed frac-

tion ρseed = Nseed/N is the increased gain of the amplifierfor fixed tSMD, resulting in significantly larger numbersof atoms N after the initial amplification. This is demon-strated in the inset to Fig. 2(a) that shows the amplifiedfraction ρ versus ρseed for tSMD = 20 ms. Increasingthe seed fraction also increases the fraction of atoms pro-duced by the amplifier, effectively increasing the gain ofthe amplifier.Bogoliubov theory suggests that the width of f(φ) ver-

sus φ should decrease as N increases, but this is not whatis observed. Instead the full-width at half-maximum(FWHM) of f<(φ) in Fig. 2(a) is nearly constant andsimilar to the FWHM in the unseeded case, whereas thewidth of the dip at φ = 0 does increase with N . Thedeep dip at φ = 0 demonstrates that even small initialcoherent populations suppress the Fisher information, asthe mean and variance no longer approach zero. Despitethe loss of sensitivity at φ = 0, there exists instead anoptimal value of φ such that f<,max = f<(φopt).Another way of viewing the effects of single-sided seed-

ing is to plot the lower bound on the scaled Fisher in-formation f<,max = max[f<(φ)] versus ρ = N/N (seeFig. 3(a)). For a fixed seed fraction we increase the spin-mixing time tSMD until ρ ≈ 1 where the pump is nearlydepleted. It is observed in Fig. 3(a) that increasing theseed fraction causes f<,max to decrease. Even with just0.02% seed, f<,max has decreased by about a factor of30 compared to the unseeded interferometer, suggestingthat imperfect state preparation would have a significant

impact on the sensitivity of the interferometer. As ρ in-creases, the scaled Fisher information peaks at a valueorders of magnitude above the standard quantum limitbefore decreasing when ρ → 1. The dashed line in Fig.3(a) indicates F = N , or f = 1/ρ. For f<,max above thisline, the interferometer is more sensitive than a Mach-Zehnder interferometer with coherent inputs using all Natoms in the BEC. This is the limit where the nonlinearspinor SU(1,1) interferometer has truly improved mea-surement precision by exploiting quantum correlations.Seeding the initial state has the advantage that the

output of the interferometer has approximately the samenumber of atoms as in the initial seed, Nout ≈ Nseed. Fora BEC with N = 3 × 104, 2% seeding corresponds to ameasurement of 600 atoms, which is experimentally lesssusceptible to noise than an average near zero. Anotheradvantage is that the time required to perform an exper-iment is significantly decreased, since the dynamics pro-ceed more quickly. The unseeded case takes tSMD = 96ms to reach ρ = 0.50, whereas the 0.2% seeded case takesjust tSMD = 35 ms to reach the same fraction.We extend these simulations to the case of double-sided

seeding, which is sensitive to the initial spinor phase. InFig. 2(b) we simulate the SU(1,1) interferometer with0.1% coherent seeds in both m = ±1 states and plotf<(φ) for several representative initial spinor phases θwith tSMD = 20 ms. Because the initial spinor phase θis well-defined for double-sided seeding, the initial stageof the interferometer becomes a phase-sensitive amplifier,and the performance of the interferometer changes signif-icantly with θ. There are two special values of the spinorphase that for a given seed fraction separate the mean-field spinor dynamics into regions of oscillating phase andrunning phase [26]. The phases which define this sepa-ratrix can be calculated for c2n> 0 and −2c2n < q < 0

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Unseeded 0.001% 0.02% 0.2% 2%

beats MZinterferometer

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Unseeded -2.86 rad (sep) 0.14 rad 2.14 rad

beats MZinterferometer

(b)

FIG. 3. The lower bound on the scaled Fisher information versus ρ with (a) increasing percentages of single-sided seeding intom = −1, and with (b) double-sided 0.1% seeding at various initial phases. The solid black line is fBog,max = (N + 2), whichis the prediction for the unseeded case (open circles). The grey shading indicates the region of scaled Fisher information thatdoes not reach standard quantum limit fSQL = 1 (based on N atoms in the interferometer arms). The dashed line correspondsto F = N (also f = 1/ρ), which is the limit achievable using coherent states in a Mach-Zehnder interferometer having a totalof N particles. The phases of the separatrix are θsep = ±2.86 rad.

using

θsep = ± cos−1

[

(ρ+ + ρ−)(1 +q

c2n(1−ρ+−ρ−) )√4ρ+ρ−

]

. (5)

For the 0.1% seeds in Fig. 2(b) these values are θsep =±2.86 rad.One of the consequences of double-sided phase-

sensitive seeding is the loss of symmetry about φ = 0in the Fisher information in Fig. 2(b). For an averageinitial spinor phase θ = θsep = −2.86 rad, the Fisherinformation peaks at f<(φ = −0.02) = 46, which is sig-nificantly larger than f(0) = 28 for the unseeded case.We conclude that for a fixed spin-mixing time, the phase-sensitive input amplifier can yield a spinor interferometerwith greater sensitivity than the unseeded case, depend-ing on initial spinor phase θ.The inset to Fig. 2(b) plots f<,max versus θ with the

symbols indicating the phases plotted in the main fig-ure, and two dotted vertical lines indicating the phasesof the separatrix. There are two broad peaks centeredon θ = −2.8, 2.0 rad that extend significantly above theunseeded scaled Fisher information f<,max = 28. In be-tween those peaks is a broad region with decreased sen-sitivity and a minimum at θ = −0.25 rad. The phases ofthe separatrix produce initial conditions correspondingto near maximum (θ = −2.86 rad) and near minimum(θ = +2.86 rad) Fisher information.The effect of the spinor phase can also be illustrated

for double-sided 0.1% seeding by plotting f<,max versusρ (see Fig. 3(b)). We follow the performance of the in-terferometer by increasing tSMD, which again increases ρ.As in the single-side seeded interferometer, we see thatseeding decreases f<,max for a given value of ρ, but that

for sufficient tSMD such that ρ → 1, we have a scaledFisher information much larger than the standard quan-tum limit. Again we find that the optimal value of θ isnear to the one that defines the separatrix.

The peak scaled Fisher information for 0.1% double-sided seeding is f<,max ≈ 1200 and occurs for ρ ≈ 0.47.This value of f<,max is significantly larger than at thestandard quantum limit and, as for the case of single-sided seeding, occurs at much earlier times than in theunseeded case. For double-sided seeding and sufficientlyhigh gain, the sensitivity is better than that attainableusing coherent state inputs in a Mach-Zehnder interfer-ometer (dashed line in Fig. 3(b)).

IV. EXPERIMENT

To experimentally demonstrate control of seed num-ber and phase, we begin with a greater-than-90%-purecondensate of approximately 1.4×105 23Na atoms in the|1, 0〉 hyperfine ground state. The atoms are held in acrossed optical dipole trap with final trap frequencies ofω(x,y,z) = 2π(137.7(8), 140.0(8), 210(6)) Hz. The indi-cated measurement uncertainties in this paper are onestandard deviation of the mean statistical uncertainties,unless otherwise noted. The initial atomic state is puri-fied by using microwave adiabatic rapid passage to trans-fer the atoms in m = ±1 to the F = 2 manifold, andsubsequent removal from the trap with a 100 µs resonantoptical pulse. In a similar way, the total atom numberis reduced to 3.5 × 104 atoms to ensure single-mode dy-namics.

8

A. Single spatial-mode approximation

An estimate for whether the BEC stays within the sin-gle mode approximation while undergoing spinor oscil-lations can be made by considering a BEC of N atomswith a Thomas-Fermi radius RTF = aho(15Nac/aho)

1/5,

where aho =√

~/(MNaω), MNa is the atomic mass,ω ≈ 2π× 160 Hz is the geometric mean angular trap fre-quency, and ac = (a0 + 2a2)/3 = 2.79(2) nm [36]. Spin-waves can be ignored when the half-wavelength of thelowest-energy combined spatial and spin-wave is largerthan 2RTF, or λs/2 > 2RTF. The wavelength λs = 2πξs,

where ξs = 1/ks = ~/√

2MNa|c2|n is the spin-healinglength of a spin-wave with wavevector ks [37]. Within theThomas-Fermi approximation, the single-mode conditionreduces to NSMA < (π2aho/(8(a2−a0)))

5/4(15ac/aho)1/4.

The scattering length difference for sodium is (a2−a0) ≈0.29 nm [22], giving an estimate of NSMA . 2.6 × 104

atoms for the maximum number of atoms for the valid-ity of the SMA. Conversely, using N = 3 × 104, andc2n/h ≈ 23 Hz, we find a geometric mean Thomas-Fermidiameter of 2RTF = 12.6 µm, and λs/2 ≈ 9.5 µm, closeto the Thomas-Fermi single-mode criterion. In orderto achieve single-mode dynamics in the experiment, wecarefully minimize magnetic field gradients. After thishas been done, we find that additional spatial modesare not populated even for large negative q, where theBEC should be unstable and populate higher-order spa-tial modes [4, 5, 38, 39].

B. Initial phase control

Unlike in optical four-wave mixing experiments, thesingle-mode atomic BEC does not have spatially dis-tinguishable states, and we work instead with spin su-perposition states, where the fractions in m = +1 andm = −1 serve as the probe and conjugate beams respec-tively. First we prepare a probe state by transferring asmall fraction of the m = 0 state into m = −1 using twosequential microwave pulses: |1, 0〉 → |2,−1〉, and then|2,−1〉 → |1,−1〉. We can vary the probe populationbetween zero and 1%. Next, the conjugate fraction isestablished for m = +1 using similar microwave pulses.For our experiments we fix the fraction in m = +1 at1%.

The initial spinor phase θ is controlled by slightly de-tuning one of the population transfer microwave pulsesfrom resonance or by using a far-detuned phase-shiftingmicrowave pulse. Figure 4 shows our ability to set theinitial phase θ with a 500 µs long microwave pulse by ver-ifying the mean-field prediction of the spinor evolution.With single-sided seeding the final atom fraction is inde-pendent of initial phase, demonstrating phase-insensitiveamplification. For seeding both m = ±1 the amplifica-tion is phase-sensitive.

0.5

0.4

0.3

0.2

0.1

0.02.52.01.51.00.50.0

ρ=

N/N

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FIG. 4. The atom fraction in m = ±1 after 14 ms of evolutiontime versus the phase shift achieved by a microwave pulse.The amplifier is seeded with 1.0% of the atoms in m = +1(blue squares) or with 1% of the atoms in each of m = ±1(black circles) out of N = 3.6(5) × 104 atoms. Solid linesare predictions from the single-mode spinor theory. Single-sided seeding represents multiple experiments and we quotethe mean and standard deviation of the mean. Double-sidedseeding represents one experiment per phase.

C. Procedure

The spinor amplifier is switched on with a continuous-wave off-resonant microwave field that shifts the relativeenergy of the levels due to the AC stark effect [22, 31, 40,41]. In particular, for a bias magnetic field of B = 50 µT(equivalent to a 350 kHz linear Zeeman shift), we blue-detune 150 kHz from the clock transition |1, 0〉 → |2, 0〉.The effective quadratic Zeeman shift q is then

q = γB2 −∆E0 + 0.5 (∆E−1 +∆E+1)

∆Em = −∑

n

Ω2m,n

4∆m,m+n,

(6)

where the sum is over all microwave photon angu-lar momentum quantum numbers n = −1, 0, 1 withRabi frequencies Ωm,n = ωnCm,n that couple betweenatomic states with Clebsch-Gordan coefficients Cm,n =〈1,m; 1, n|2,m+ n〉, each with detuning ∆m,m+n and

γ/h = 27.7 kHz/(mT)2[22]. The measured microwave

frequencies for σ−, π, and σ+ coupling are respectivelyω−1 = 2π × 8.22(8) kHz, ω0 = 2π × 23.7(3) kHz, andω+1 = 2π × 11.76(3) kHz. This mixed microwave cou-pling complicates the calculation of q, but we intention-ally operate in this way so that we are able to makerapid transfers to all possible magnetic sublevels. Theexperimental value of q/h determined by the calibratedmicrowave fields is −1.3(4) Hz.While the value of q is stable for the duration of the

experiment, c2n varies from one experimental realizationto the next due to fluctuations in N . In order to capturethese variations within the TWA simulations, we measureN for each experimental realization and use it to scale the

9

interaction energy c2n= c2n0(N/[3×104])2/5, where c2n0

is the interaction energy for 3× 104 atoms. This scalingis expected from the Thomas-Fermi approximation forwhich n ∝ N2/5. We also compensate for slow variationsin trap frequencies by fitting each series of 35 data pointsto find the best value of c2n0. The mean and standarddeviation of the distribution from the fits for the entiredata set are c2n0/h = 22.8± 0.7 Hz.After the microwave dressing field is applied, the atoms

are allowed to evolve for a fixed time tSMD = 27 ms. Atthat time the dipole trap is switched off, and the totalnumber of atoms N and the fractional populations ρ±1

are measured by Stern-Gerlach separation and absorp-tion imaging after a short time-of-flight. By monitoringthe fluctuations in the magnetization (M ≡ ρ+ − ρ−) weestimate that our atom measurement uncertainty is 190atoms for each spin state.

D. Experimental results

In Fig. 5 we present experimental data on the initialphase-sensitive amplifier stage of the SU(1,1) interferom-eter for constant seeding into m = +1 of ρ+,seed = 0.01versus the fraction seeded into m = −1 (ρ−,seed). For alldata, tSMD is long enough to that the system has evolvedinto the depleted pump regime for which the TWA sim-ulations are necessary. The system transitions from aphase-insensitive amplifier with single-sided seeding forρ−,seed = 10−6 to a balanced double-sided phase-sensitiveamplifier when ρ−,seed = 0.01. The two sets of data dif-fer in their initial phase with θ ≈ 0 (black circles), andθ = 2.46 rad (red diamonds). The initial phases becomeincreasingly important as ρ−,seed → ρ+,seed. The differ-ence in the fractions at small ρ−,seed for the two initialphases is caused by slightly different gains produced bythe somewhat larger BEC and more negative q for thedeamplification data.For double-sided seeding with θ = 0.016 rad (black cir-

cles), the seeded atoms stimulate the production of morepairs of atoms, resulting in a larger amplified fraction af-ter a fixed amount of time. For θ = 2.45 (red diamonds),the seeded m = ±1 atoms first recombine to form pairsof m = 0 atoms before spontaneous and stimulated emis-sion produces new pairs of m = ±1 atoms. For a fixedtSMD the time required for the initial deamplification re-sults in a smaller measured fraction of atoms.The transition from phase-insensitive to phase-

sensitive amplifier can be understood by considering thestandard deviation in the initial phase of the conden-sate calculated from TWA simulations (green dashedline). ∆θ decreases as ρ−,seed increases, and for ρ−,seed >10−4 agrees well with the analytical expectation ∆θ =√

1/ρ+ + 1/ρ− + 4/ρ0/(2√N), derived from the coher-

ent state standard deviation of the components ∆θm =1/(2

√Nm). At the point that ρ−,seed = 10−3, θ is suf-

ficiently well-defined that the atom-fractions differ bythree full standard deviations from the fraction for single-

1.0

0.8

0.6

0.4

0.2

0.0

10-6

10-5

10-4

10-3

10-2

0.5

0.4

0.3

0.2

0.1

0.0

ρ=

N/N

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∆θ(π

rad)

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amplification deamplification magnetization TWA simulation

FIG. 5. Experimental atom fraction after amplification forthe seeded nonlinear amplifier versus seeded atom fractionin m = −1, with m = +1 initially populated with a frac-tion ρ+,seed = 0.01 of the atoms. Solid black circles (N =3.4(3) × 104, q/h = −1 Hz, θ = 0.016 rad, tSMD = 27 ms)have an initial spinor phase θ causing amplification, while reddiamonds (N = 7.8(9) × 104, q/h = −1.4 Hz, θ = 2.45 rad,tSMD = 25 ms) have an initial phase causing deamplification.The shading indicates the standard deviation of the mean,and the solid black lines are results based truncated-Wignerapproximation simulations. The green dashed line shows thedecrease in the standard deviation (∆θ) of the distribution ofinitial phases from the TWA simulations.

sided seeding, corresponding to less than 35 atoms seededinto m = −1. Due to spinor amplification, we achieve asensitivity to initial numbers of atoms far below the 190atoms per spin state measurement noise determined fromthe magnetization data (open circles).

V. CONCLUSION

We have used the truncated-Wigner approximationto simulate a spinor nonlinear-SU(1,1) interferometerseeded with coherent populations in the “probe” and“conjugate” states. In the case of single-sided seed-ing this realizes a phase-insensitive amplifier, while fordouble-sided seeding the input is a phase-sensitive ampli-fier. We quantify the performance of the interferometerby calculating the phase-sensitivity ∆φ, which we use toestimate the scaled Fisher information, f(φ). We haveshown that coherent seeds suppress f(φ) around φ = 0,while retaining scaled Fisher information significantly be-yond the standard quantum limit for the optimal phaseshift. An advantage of seeding the amplifier is that theoutputs have a larger mean and smaller variance, givingbetter signal-to-noise ratio for the experimental measure-ments.Experimentally, we have demonstrated that seeding

the m = ±1 input states of a 23Na spinor BEC leads to

10

an amplifier that is sensitive to probe and conjugate co-herent states with less than 35 atoms or 0.1% of the totalnumber of atoms. Using microwave pulses we can controlthe initial phase of the spinor condensate and thereforethe subsequent dynamics. Nonlinear spinor SU(1,1) in-terferometers constructed from these phase-sensitive am-plifiers are potential systems for achieving sensitivity be-yond the standard quantum limit.

ACKNOWLEDGMENTS

JPW acknowledges support from the Research Cor-poration for Science Advancement. ZG acknowledgessupport from the NSF PFC at JQI. RB received sup-port from the European Union’s Seventh Framework Pro-gramme under Grant No. PCIG-GA-2013-631002. ETacknowledges support by the US National Science Foun-dation, Grant No. PHY-1506343.

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