Unitary Cavity Spin Squeezing by Quantum Erasure

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by  Ian D. Leroux

Unitary Cavity Spin Squeezing by Quantum Erasure
Ian D. Leroux,∗ Monika H. Schleier-Smith,† Hao Zhang, and Vladan Vuletić
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Dated: January 9, 2012)
Deterministic light-induced spin squeezing in an atomic gas is limited by photon shot noise or,
equivalently,byatomicstateinformation escapingwiththelightfieldmediatingtheeffectiveatom-
atom interaction. We show theoretically that the performance of cavity spin squeezing [M.H.
Schleier-Smith, I.D. Leroux, and V. Vuletić, Phys. Rev. A 81, 021804(R) (2010)] can be sub-
stantially improvedbyerasingthelight-atom entanglement,andproposeseveralmethodsfordoing
so. Accounting for light scattering into free space, quantum erasure improves the scaling of cavity
squeezing from S−1/2 toS−2/3, where S is the total atomic spin.
2
1
0
I. INTRODUCTION
2

n
Squeezedspinstates[1]areamongthe simplestmany-
a
body entangledstatesto describeandcharacterize,since
J
theirquantumcorrelationsappearasanimprovedsignal-
5
to-noise ratio in certain measurements. This improve-

] ment makes squeezed spin states potentially useful for
h precision metrology beyond the standard quantum limit
p
(SQL) that is set by the quantum fluctuations of inde-
-
t pendent particles. In particular, spin squeezing might
n
increase the stability of atomic clocks, magnetometers,
a
u and other measurements based on atom interferometry
q [2–5]. Manyapproachestospinsqueezinghavebeenpro-
[ posed [2, 6–21] and a number of them have been demon-
Figure 1. (Color online) Setup for cavity squeezing. (a)
strated experimentally, including atomic absorption of
2 Atomswithtwopseudospinstates|↑i,|↓iopticallycoupledto
v squeezed light [22], entangling gates in an ion trap [23], an excited state |ei. (b) The atoms are uniformly coupled to
2 projection by quantum non-demolition (QND) measure- anopticalcavity,eithersymmetric(i)orone-sided(ii),whose
5 ment [24–26], atom-atom collisions [27–29], and light- resonancefrequencyω liesbetweenthetwoopticaltransition
c
7 mediated interaction between distant atoms in an op- frequencies. (c)Thecavityisdrivenbyapulsedetunedfrom
3 tical resonator [30, 31]. This last method, cavity squeez- its resonance frequency, so that the intracavity intensity de-
9. ing [19, 20], has generated the strongest spin squeezing pends on the atom-induced cavity frequency shift. (d) The
0 to date, with 5.6 dB observed (no noise subtracted) and resultingACStarkshiftimpartsanSz-dependentphaseshift
1 10 dB inferred (detection noise subtracted) [30]. to the atoms, shearing the uncertainty distribution of a co-
1 herent spin state on the Bloch sphere of total atomic spin
Cavity squeezing relies on the off-resonant interaction
: S.
v between an ensemble of atoms and a light field circulat-
i ing inanopticalresonatorcavity[19]. The atoms’state-
X
dependent index of refraction modifies the cavity reso-
r observer can gain information about the atomic state.
nance frequency. If the cavity is driven by a probe laser,
a
While that information can be used to reduce the vari-
the atom-induced resonance frequency shift changes the
anceofatransversespincomponent[6,32,33]andhence
optical power circulating in the cavity, modifies the AC
to perform conditional spin squeezing by QND measure-
Stark shift it imparts to the atoms, and thus affects the
ment[24–26],itisignoredinunconditionalcavitysqueez-
phase of the atomic state. This phase shift, which de-
ing and thus causes non-unitary evolution in the spin
pends on the states of all atoms in the ensemble, intro-
subspace. Equivalently, the AC Stark shift fluctuations
ducescorrelationsbetweenthemandproducesasqueezed
associatedwiththeprobephotonshotnoisecauseanun-
state of the total atomic spin. In the cavity squeez-
desirablegrowthofthe spinuncertaintyregion,reducing
ing scheme analyzed in Ref. [19], the atomic spin is en-
the achievable cavity squeezing.
tangled with the outgoing light field so that an outside
Inthispaperweproposeschemesthatsuppresstheef-
fects ofphotonshotnoise, disentanglethe outgoinglight
field and the atomic spin, and ideally result in unitary
∗ PresentAddress: DepartmentofPhysicsandAstronomy,Aarhus
evolutioninthespinsubspace. Weshowthatthephoton
University,DK-8000AarhusC,Denmark
† Present Address: Max-Planck-Institut für Quantenoptik shot noise suppression can be achieved either by creat-
and Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 ing effective photon number states using high-efficiency
München,Germany photodetectors and classical feedback or by a spin echo
2
technique for canceling quantum noise. Practically, this reflected, and nothing about the intracavity dynamics
proposal shows that the performance of unconditional can be inferred from the reflected power. If the near-
light-inducedsqueezingcanbeimprovedwellbeyondthe monochromatic input light is in a photon number state
limit set by photon shot noise or light-atom entangle- rather than a coherent state, then any phase shift im-
ment. Fundamentally, it shows that one can engineer parted to the reflected light cannot be measured either.
light-mediated interactions between distant atoms with- With no way for an outside observer to learn the state
out leaving a trace of the atomic state in the light field. of the atoms inside the cavity, their dynamical evolution
A related idea has recently been proposed [35] in a must be unitary, as we show explicitly below.
modification of the free-space scheme put forward by Briefly, we model the system with an effective Hamil-
Takeuchiet al. for spinsqueezing by two-foldlight-atom tonian that dispersively couples the atoms’ pseudospin
interaction [15]. The modified scheme of Ref. [35] can degrees of freedom to the optical resonator mode. After
produce exponential squeezing for short times, but is ul- diagonalizing this Hamiltonian, we use it to analyze the
timately limitedbylightscatteringinto unobservedfree- evolution of an arbitrary initial pseudospin state under
space modes. The near-unity single-atom cooperativity the action of an incident light pulse with definite pho-
readily available in optical resonators [36] reduces the ton number. For a sufficiently monochromatic pulse, the
rate of this scattering relative to the squeezing, and al- systemevolvesintoaproductstateofthefieldandpseu-
lows the scheme we presenthere to achieve substantially dospin degrees of freedom, where the transformation of
greaterspinsqueezingforagivenatomnumberthanany thepseudospinstateisgeneratedbytheone-axistwisting
free-space technique. Hamiltonian [1].
In Sec. II we analyze cavity squeezing in a one-sided The system we consider consists of N identical atoms
cavity where the coupling of the intracavity light field uniformly coupled to an optical resonator that, in con-
to the continuum of external light modes is treated ex- trast to the system considered in Ref. [19], is one-sided
actly in the input-output formalism [37, 38]. For an in- (Fig. 1). The atoms have two stable states separated
put lightfield preparedina near-monochromaticphoton in energy by ~ωa, such as hyperfine or magnetic sub-
number state no information about the atomic state is levels, which we label as the pseudospin states and
|↑ii
containedintheoutgoinglightfieldandthespindynam- for the i-th atom. We define total pseudospin op-
|↓ii
ics are therefore unitary. We show that the evolution of erators 2S = N ( ), the population
z i=1 |↑iih↑|i−|↓iih↓|i
the collective atomic spin is then described by the one- differencebetweenthestablestates,andS =S +iS =
+ x y
axis twisting Hamiltonian introduced by Kitagawa and N , SP = S†. For simplicity, we consider the
Ueda[1]. InSec.III weintroducethe effects ofnon-ideal i=1|↑iih↓|i − +
case where the two optical transitions e and
input light states, and in Sec. IV we use the results to P |↑i ↔ | i
e connecting the stable states to the optically
evaluateseveralpracticalimplementationstrategies: one |↓i ↔ | i
active excited state e couple to the resonator with the
using a spin-echo technique to erase the phase informa- | i
same single-photon Rabi frequency 2g. The cavity res-
tion from the outgoing light field, one using a squeezed
onance frequency ω is chosen halfway between the two
c
input light field, and one using high-efficiency photode-
opticaltransitionfrequencies,sothattheirsingle-photon
tectors to generate an effective photon number state of
detunings ∆= ω /2 are of equal magnitude and op-
a
the light field. In Sec. V we consider the unavoidable ± ±
positesign. Withintherotating-waveapproximationand
effects of light scattering into free space by the atomic
neglecting, for now, the scattering of photons into free
ensemble, deriving the associatedlimit on cavitysqueez-
space by the atoms, the Hamiltonian for the intracavity
ing with quantum erasure.
system is
N
H
cav =ω S + ω e e +ω c†c+
II. UNITARY CAVITY SQUEEZING USING AN ~ a z c| iih |i c
INPUT PHOTON FOCK STATE i=1
X
N
+ g( e + e )c†+H.c. . (1)
In Ref. [19] the authors analyze cavity squeezing for |↑iih |i |↓iih |i
atoms in a symmetric Fabry-Pérot cavity driven by Xi=1(cid:2) (cid:3)
a coherent field tuned to the slope of the resonance In this equation the first two terms describe the energy
[Fig. 1(b)(i)]. The performance of the scheme is limited ofthe atoms,thethirddescribesthe energyofthe cavity
by the entanglement of the atomic spin variables with modewithphotonannihilationoperatorc,andtheterms
the light field leaving the resonator. If the information proportional to g describe the coupling of the atoms to
inthe lightfieldis notused,then itmustbe tracedover, the light field. We are interestedin the linear,dispersive
leading to decoherence of the atomic state. This adds regime of atom-field interactions where ∆ is large com-
spin uncertainty which ultimately limits the attainable paredto g, to the cavitylinewidth κ, andto the excited-
amount of spin squeezing. state linewidth Γ. Assuming that the intracavityphoton
That this decoherence can be eliminated is most eas- numberc†cremainslowenoughtokeeptheexcited-state
ily seen in a setup with a one-sided cavity [Fig. 1(b)(ii)]. population negligible (i.e. c†c g2/∆2 1), we adia-
≪
Provided that the cavity is lossless, all incident light is batically eliminate the excited state e to replace the
(cid:10) (cid:11) | i
3
full atom-cavity interaction by an AC Stark shift of the m , a setof decoupledharmonicoscillators. The excita-
two pseudospin states. In so doing we also neglect stim- t|ioins created by the a† operators are photons with am-
ω
ulated Raman processes which, under the same set of plitudes to be either in the cavity (c†) or in the outside
assumptions, are too far off resonance to be significant. continuum (b† ), with the intracavity component reso-
ω′
Usingtheinput-outputformalism[37,38]todescribethe nantly enhanced near the atom-shifted cavity resonance
coupling with the field outside the cavity, we obtain the frequency ω ω +ΩS . A one-photon eigenstate can
c z
following effective Hamiltonian for the dynamics of the now be gener≈ated by acting with the total field raising
pseudospin and of the light field: operator a† on any of the vacuum states,
ω
H~ =ωaSz +ωcc†c+ dωωb†ωbω Ha†ω|mi⊗|0i=~(ωam+ω)a†ω|mi⊗|0i, (5)
Z
κ and repeated applications of a† can yield arbitrary n-
+Ωc†cS +i dω[b†c c†b ]. (2) ω
z 2π ω − ω photon states.
r Z We are now equipped to calculate the evolutionof the
The first three terms of the Hamiltonian account for the atomic spin state under the action of input light pulses.
energy of the atoms, of the cavity field, and of the field In particular, we consider an incident pulse containing
outside the cavity with its continuous spectrum of cre- exactlynphotonsincidentonthe cavity,asdescribedby
ation operators b†. The fourth term represents the dis- the initial state
ω
persive coupling between the cavity field and the atoms,
n
and may be interpreted either as an AC Stark shift of Ψ = ψ 1 dωeiωt0B(ω)b† 0 (6)
the atomic levels by the light field or as a modification (−t0) | ai⊗ √n! ω | i
(cid:18)Z (cid:19)
of the cavity resonance frequency by the atomic index (cid:12) (cid:11)
(cid:12)
of refraction [19]. The coefficient Ω=2g2/∆ is both the where ψa is an arbitrary state of the atoms and B(ω)
| i
cavityfrequencyshiftperatomicspinflipandtheatomic is the pulse amplitude spectrum. We have written this
transitionfrequencyshiftperintracavityphoton. The fi- initial state explicitly as a product state of the atoms
nal integral is the coupling between the cavity and the andfield, emphasizingthatthe creationoperatorb†ω acts
external field through the cavity’s partially transmissive only on the field outside the cavity. We take the initial
input mirror, leading to a damping of the energy stored time t0 to be far in the past, before the pulse arrives
−
in the cavity field at rate κ. at the resonator (specifically t max dB(ω)/dω 2/3).
0
≫ | |
The Hamiltonian H may be exactly diagonalized. Re-expressedintermsoffieldeigenstates,theinitialstate
First, note that both S and the total photon number is
z
c†c+ dωb†b are conserved. Thus, all product states
ω ω
m 0 ofanatomiceigenstate m ofS withthe elec- ( i)n
|troim⊗aRg|nieticvacuum 0 areeigens|taitesofzthefullHamil- Ψ(−t0) = −n! × (7)
| i r
tonian. The eigenstates with non-zero photon number (cid:12) (cid:11) n
require additional labels to specify the spectral distribu- (cid:12) × dωei(ωt0−Φω)B(ω)a†ω |ψai⊗|0i
tion of the photons. To find those eigenstates, we fol- (cid:18)Z (cid:19)
low Fano’s procedure for diagonalizing a discrete state where we have introduced an atom-dependent operator
(the cavity mode) coupled to a continuum (the external
field) [39], which yields an operator that annihilates a κ/2 π
Φ =arctan (8)
photon in an eigenstate of the total field: ω ω ω ΩS − 4
(cid:18) − c− z(cid:19)
1 κ which represents the phase lag of the intracavity field’s
a = i c
ω
(ω ω ΩS )2+κ2/4 2π response to an external drive at frequency ω. In this
− c− z (cid:20)r
form,allcomponentsof Ψ evolvewithknownfrequency,
p+2κπP dω′ωbω′ω′ +(ω−ωc−ΩSz)bω . (3) soitisstraightforwardt|ofiindthefinalstateat+t0,long
Z − (cid:21) afterthelightpulsehasreflectedfromthecavity. Writing
this final state in terms of operators that act separately
P is a reminder that the Cauchy principal value of the
integralmustbetakenoverthepoleatω =ω′. Thisfield onthe field(b†ω)andonthe atoms(Sz),weseethatitis,
in general, entangled:
operatorhasthe usualcommutationrelation a ,a† =
ω ω′
δ(ω′ ω) and allows us to rewrite the Hamihltoniani(2) ( i)ne−2iωat0Sz
− Ψ = − (9)
into the much simpler form (+t0) √n! ×
(cid:12) (cid:11) n
H~ =ωaSz+ dωωa†ωaω (4) (cid:12) × dωe−i(ωt0+2Φω)B(ω)b†ω |ψai⊗|0i.
Z (cid:18)Z (cid:19)
inwhichthefirsttermisjustthebareenergyoftheatoms However,foranear-monochromaticinputpulse centered
and the second describes, for a given atomic eigenstate on a frequency ω with a bandwidth much less than κ,
p
4
the atomic operator Φ is approximately Φ over the III. EFFECTS OF UNCERTAIN PHOTON
ω ωp
spectrum of the pulse and the state factorizes into NUMBER
Ψ(+t0) =e−2i(nΦωp+ωat0Sz)|ψai⊗ (10) For input pulses with uncertain photon number, we
(cid:12)(cid:12) (cid:11) ⊗ (√−in)!n dωB(ω)e−iωt0b†ω n|0i. cbaynavfienrdagtihnegoevxeprectthaetipoonssviballeuephooftaonnnautommbiecrsobosfetrhveabinle-
(cid:18)Z (cid:19) put state. In general,such an averagewill depend on all
Inthislimitofanincidentmonochromaticn-photonFock moments of the photon number distribution. However,
state the final field state is independent of the atomic animportantspecialcaseariseswhenthephaseperpho-
state (the pulse has simply been reflected by the one- ton φ is small and the photon numbers are large such
0
sided cavity), while the atomic state has been trans- thatthe distributionsforn andn approachGaussians.
1 2
formed by the unitary operator Inthecommoncasewherethefluctuationsinthephoton
number difference (n n )havea variancewhichscales
2 1
Un =e−2i(nΦωp+ωat0Sz). (11) withthetotalphotonn−umber,therotationanglevariance
can be expressedas ∆ρ2 =γ µ , with γ a proportion-
h i
Thedynamicsofinterestinthespinsubspaceareencoded alityconstantwhichis1forthecaseofindependentpho-
inthenonlinearoperatorΦωp. Whenthepulsefrequency tonshotnoisefluctua(cid:10)tions(cid:11)onn1 andn2. Thevarianceof
is tuned to the slope of the Lorentzian cavity resonance, theshearing ∆µ2 ,meanwhile,ishigher-orderinφ and
0
ωp = ωc +κ/2, and provided that NΩ κ so that the vanishes in the limit we are considering, so we will not
atoms do not shift the cavity resonance≪frequency by a distinguishb(cid:10)etwee(cid:11)nµanditsexpectationvalue µ here-
h i
largefractionofthecavitylinewidth,wecanexpandΦωp after. In this regimethe squeezingis determinedonlyby
to second order in the spin rotation angle φ0 = 2Ω/κ the total number of atoms 2S, the shearing µ set by the
imparted by a single incident photon. We find mean total photon number n +n , and the ratio γ of
1 2
h i
rotationuncertaintytoshearingsetbythevarianceofthe
Un =e−i(2ωat0Sz+nφ0Sz+n2φ20Sz2), (12) photonnumberdifferencen1−n2betweenthetwopulses.
ForlargeS,aninitialatomic pseudospinpolarizedalong
where the terms linear in S generatea precessionofthe xˆ, µ 1, and ρ = 0, the final spin expectation values
z
≪ h i
atomic pseudospin and the term quadratic in S gener- are approximated as
z
ates a shearing or S -dependent rotation.
z
To isolate the shearing term it is convenient to apply Sx =Se−21v, (14)
this transformationtwice, separated by a π pulse on the h i
S = S =0, (15)
atoms which inverts Sz. The overall effect is the two- h yi h zi
pulse transformation S2 = S 1+S 1 e−2v , (16)
y 2 −
Uρ,µ =e−i(ρSz+21µSz2) (13) (cid:10)S2(cid:11)= Sh, (cid:0) (cid:1)i (17)
z 2
bobertsainnedabnydtnhe. sHeqeureenρce=U(nn1–π–Unn)2φwitishtphheoptohnasneurmo-- hSySz+S(cid:10)zSy(cid:11)i=2 Sz2 µhSxi=S2µe−12v, (18)
1 2 2 1 0
tation angle and µ = (n +n )−φ2 is the strength of the (cid:10) (cid:11)
2 1 0 where v = γµ+µ2 S2 is the characteristic phase vari-
shearing action expressed as an atomic phase shift per z
ance resulting from both the uncertainty on the photon
unit change in S . In the ideal case where the incident
z (cid:10) (cid:11)
number and from the shearing. The mean spin length
photonnumberisidenticalinthetwopulses,n =n ,we
1 2
findtheone-axistwistingtransformationU0,µ =e−iµSz2/2 hcSerxtiaiinstryedreugcieodnaasrotuhnedpthhaeseBblorcohadsepnhienrge.vTwhreatprsatnhseveurnse-
considered by Kitagawa and Ueda [1]. Their results for
variance S2 is initially the projection noise S/2, then
the spin squeezing achievable by this transformation are y
grows as the phase variance scaled by the length of the
reviewed in Appendix A. Bloch vec(cid:10)tor(cid:11)S2v, before saturating near S2/2 when the
Thus we find that the two-fold interaction of an n-
uncertainty distributionhas completely wrappedaround
photonpulsewiththeatom-cavitysystemcanrealizethe
the Bloch sphere. The cross correlation S S +S S
S2 one-axis twisting Hamiltonian, at least when scat- h y z z yi
z is the product of the variance of S , the phase change µ
tering into free space is ignored (see Sec. V). The uni- z
per unit S , and the change in S per unit phase. Using
tary evolution in the spin subspace can be understood z y
Eq. (A7) to compute the minimum transverse spin vari-
as a consequence of the absence of photon shot noise, or
ance ∆S2 andcomparingthisvariancetothereduced
equivalently, of the absence of phase information in the min
mean spin length S gives the metrological squeezing
reflected light that would revealthe atomic state. While (cid:10) (cid:11) h xi
parameter [2, 3]
thesametransformationcanbeaccomplishedwithasin-
gle input pulse followed by a photon-number-dependent
rotationoftheatomicspin,itiseasiertoimplementusing ζ = 2S ∆Sm2in . (19)
the double-pulse sequence, as we shall see below. S 2
(cid:10) x (cid:11)
h i
5
1 n
1
n
2
0.1
ζ
0.01
n
1
0.001
0 2 4 6 8 n′
1
103µ
Figure 3. (Color online) Cavity squeezing with coher-
Figure 2. (Color online) Cavity squeezing for S = 104 ent pulses. Previous experimental demonstrations of cavity
and varying degrees of photon shot noise suppression: none squeezing [30] used two separate coherent pulses with sta-
(γ = 1, dotted red), 90% (γ = 0.1, dashed blue), 99% tistically independent photon numbers n1 and n2 (top). If,
(γ = 0.01, chain dotted green), and complete (γ = 0, solid instead, a single coherent pulse is used for both interactions
black). Inthisfigure,noiseduetophotonscatteringintofree withthecavity(bottom),thetwophotonnumbersn1 andn′1
space is ignored, corresponding to a cavity with single-atom differ only by the optical losses between the two interactions
cooperativity parameter η≫1 (see Sec. V). andtheeffectoftheircorrelatedfluctuationscanbecanceled
out by a spin-echosequence.
ζ−1 is the squared signal-to-noise ratio, relative to the
uncoupled modes and failing to interact with the cavity
standard quantum limit, of a measurement of the total
mode is 1(1 √1 2ℓ) ℓ/2, yielding a binomial dis-
pseudospin’s direction. 2 − − ≈
tribution for the cavity-coupled photons whose variance
Figure 2 shows the squeezing parameter calculated as
is γ ℓ/2 times photon shot noise. In other words, the
a function of shearing strength µ for S = 104 and resid- ≈
noise in the photon number difference n n to which
ual fractions of photon shot noise γ of 1, 0.1, 0.01, and 2− 1
the squeezingissensitiveis justtheshotnoiseofthe lost
0. In general the squeezing parameter first decreases
photons.
as ζ 2γ/(Sµ) under the action of the shearing be-
fore ri≈sing again as ζ S2µ4/24 as the curvature of the
≈
Blochspheredeformstheuncertaintyregion. Inthelimit
IV. PRACTICAL PHOTON SHOT NOISE
γ 0 we recover the ideal squeezing from perfect one-
→ SUPPRESSION SCHEMES
axis twisting as considered in Ref. [1] and Appendix A:
ζ (Sµ)−2+S2µ4/24,leadingtoanoptimumsqueezing
th≈at scales as ζ 122/3/(8S2/3). In this section we show how to achieve reduced pho-
≈ ton shot noise γ < 1 using several schemes of practical
Evenifthe inputfieldcontainsadefinite photonnum-
interest.
ber, imperfections in the resonatorcan introduce optical
loss. Sincethelossofphotonsisarandomprocess,itrein-
troduces noise on the number of photons which interact
with the atoms, and thus decoherence. Such optical loss A. Spin Echo Quantum Eraser for Coherent Input
Pulses
processes can quite generally be understood as coupling
thecavitytoasecondcontinuumofmodesintowhichthe
photons are scattered. For example, we might imagine Iftheinputconsistsoftwoindependentcoherentpulses
a second continuum of modes b′ behind the right-hand (Fig. 3, top), ∆(n n )2 = n + n and γ = 1.
ω 2− 1 h 2i h 1i
cavity mirror of Fig. 1(b)(i) and then allow this mirror In this case we recover the scaling obtained by Takeuchi
(cid:10) (cid:11)
to be partly transparent. Since the cavity field c couples et al. [15] for their polarization-feedback scheme in free
to a weighted sum of the incident field mode b and the space: ζ 2/(Sµ)+S2µ4/24, with the best squeezing,
variousindependentunobservedmodesb′ wecaωn,follow- obtained≈for a shearing parameter µ 121/5/S3/5, scal-
ω ≈
ing Fano, identify a single linear combination of all the ing as ζ 5/(3841/5S2/5). Note that these results do
≈
field modes at a given frequency that couples maximally nottakeinto accountfree-spacescattering,whichcanbe
tothecavitymodeandoneormoreorthogonalcombina- ignored only if a cavity with cooperativity η >1 is used
tions(asmanyasthereareunobservedfields)thatdonot (see Sec. V).
coupletothe cavitymode atall[39](seeAppendix Bfor However,if the same coherentpulse is reused for both
a practical consequence of this separation into coupled interactions by storing it in an optical delay line while
anduncoupledmodes). Iftheaveragefractionlostofthe the π pulse is applied to the atoms (Fig. 3, bottom),
input pulse atonehalf-linewidth detuning isℓ 1,then then ideally n = n and ρ = γ = 0. Information about
2 1
≪
theprobabilityofanincidentphotonbeinginoneofthese S is encoded in the optical phase shift of the coherent
z
6
pulse after its first reflection from the cavity, but this
is erased during the second reflection, which applies the
opposite phase shift since S has changed sign in the
z
interim. Thus the light is disentangled from the atoms
at the end of the sequence and the overall evolution of
the atoms is unitary. In a realimplementation there will
belossesinthedelaylineusedtostorethepulsebetween Figure 4. (Color online) By counting photons after their in-
teraction with the cavity, the squeezing light pulse can be
its twointeractionswiththe cavity,suchthatn willnot
2
projected onto a definite photon number state. With fast
be precisely equal to n and ρ will not exactly vanish. If
1 classical feedback it can even be steered to a predetermined
afractionℓofthephotonsislostbetweenthetwopulses,
numberstateforunconditionalphotonshotnoisesuppression.
then ∆(n n )2 = n ℓ ( n + n )ℓ/2, where
2 1 1 2 1
− h i ≈ h i h i
the second approximation holds for small losses, giving
(cid:10) (cid:11)
γ ℓ/2. Again,thesqueezingislimitedbytheshotnoise
≈ spin-squeezing setup does not change the photon num-
of the lost photons.
ber we may place it inside the feedback loop, before the
photodetector (Fig. 4), thus sidestepping this difficulty.
In the simplest scheme, a coherent laser pulse is sent
B. Squeezed Input Pulses
onto the cavity and the reflected light is collected by
a photon counter. If a perfect photon counter detects
Onedemonstratedapproachtoimprovingopticalatom n photons in the light pulse reflected from the cavity,
detection for a given number of photons is to use a then the system is projected into the state obtained for
squeezed state of the input light field [40]. We there- an incident n-photon Fock state. The photon-counting
fore consider the effect of incident pulses whose fluc- measurementhas removedthe uncertaintyonthe energy
tuations in-phase with the coherent amplitude α are of the incident light pulse and has destroyed the com-
squeezed so as to reduce intensity noise in the cav- plementary information in the phase of the light field,
ity. The photon number variance of such pulses can which would have revealed the atomic state via the cav-
be parametrized as [41, 42] ∆n2 = α2e−2s + ity frequency shift. Conditioned on the reflected pho-
| |
2sinh2scosh2s where α is the coherent amplitude of ton number, the atomic dynamics are unitary: although
(cid:10) (cid:11)
the pulse. The usual optical squeezing parameter s is the evolution of the spin state depends a priori on the
related to the shot noise suppression factor by γ = uncertain photon number in the incident pulse, a poste-
e−2s + (2sinh2scosh2s)/ n +n . Note that γ does riori the experimenter knows exactly which unitary op-
2 1
h i
not decrease monotonically with s but reaches a mini- eration was performed by the light pulse whose photon
mum which improves with photon number as γ (n + number was measured. Note that no useful informa-
2
n )−1/3, because squeezing reduces the fluctuati∼ons in a tion can be obtained from the photon arrival times at
1
quadratureofthe incidentelectricfieldratherthaninits the counter. Since the spectrum of the incident light
magnitude. Delivering such squeezed light to the cavity must be much narrower than the cavity linewidth, the
in the presence of optical losses remains experimentally arrival time of the photons has a Fourier-limited uncer-
challenging,butsourcesproviding9dB-squeezedlightto tainty much larger than the cavity lifetime and one can-
gravitationalobservatorieshave been demonstrated [43], notdeterminewhetheraphotonenteredtheresonatoror
so that suppression factors of order γ 10−1 may be merely bounced off the input mirror.
∼
attainable by this approach. Rather than contenting oneself with conditional uni-
tary evolution, one can deterministically generate the
equivalentofaninputFockstatebyapplyingdirectfeed-
C. Generation of Effective Fock States by back to the incident light. One must merely count re-
Measurement flectedphotonsandswitchthe lightsourceoffoncesome
target photon number n has been reached.
Anotherapproachtosuppressingphotonshotnoisere- Finitephotodetectorquantumefficiencywillintroduce
lies on the conservation of total photon number in this an uncertainty on the number of input photons for a
scheme: every photon sent onto the cavity must leave it given detected photon count. The residual photon shot
and, for a single-ended cavity with negligible loss, every noiseobtainedbythis techniquewillthereforebeatbest
photon leaves in the same reflected mode. This iden- γ =1 Q,whereQis thequantumefficiencyofthe pho-
−
tity between photon numbers of the input and output todetector. Note that a transmission-based QND mea-
fields allowsaneffective input Fockstate tobe produced surement of the atomic spin as used in Ref. [25] can
using high-efficiency photon counters and classical feed- squeeze initially as ζ 2/(SµQ), which is slower than
≈
back. Such feedback-generated Fock states were studied the ζ 2γ/(Sµ) of cavity squeezing for any finite quan-
≈
intheearlydaysoflightsqueezingresearch[44]buthave tum efficiency. Even a perfect photon-shot-noise-limited
notseenwideusebecause,intheabsenceofaQNDpho- phasemeasurementofthelightreflectedfromaone-sided
todetector,thephotonFockstateisdestroyedinthevery cavity could squeeze only as ζ 1/(4SµQ), so that for
≈
detection process that generates it. However, since the Q & 85% conditional squeezing by measurement still
7
1
proceeds more slowly than cavity-feedbacksqueezing us-
ingthesamephotodetectortosuppressphotonshotnoise
fluctuations.
0.1
V. EFFECTS OF SCATTERING INTO FREE ζ
SPACE
0.01
So far, we have neglected scattering of photons into
free space. Like atom loss in collisional squeezing of
0.001
Bose-Einstein condensates [45], such photon loss de- 0 2 4 6 8
grades the performance of light-induced spin-squeezing 103µ
schemes [13, 14] unless a suitable level scheme is used to
avoidits effects [18]. Inthe simple andsymmetricmodel
Figure 5. (Color online) Cavity squeezing for S = 104 with
we consider, the average number of photons scattered
input pulses of definite photon number for a perfect cavity
into free space per atom in the ensemble is given by
(solid black), and for finite single-atom cooperativities η =1
(chaindottedgreen),η=0.1(dashedblue)andη=0.01(dot-
φ Γ µ
0
2r=(n +n ) = ted red). For reference, the gray curve shows squeezing for a
1 2
2 ∆ 2η
perfectcavity(η→∞)withoutphotonshotnoisesuppression
(γ =1).
where the numbers of photons Raman- and Rayleigh-
scatteredareequaltoeachotherandtor. Thescattering
depends only on the shearing µ and the cavity cooper- is reduced to v′ γµ+µ2 S¯2 . Similarly, the factor
ativity η = 4g2/(κΓ), so that for any finite single-atom of S2 in the S≈–S correlhatizoni [Eq. (18)]—which ex-
cooperativity η scattering into free space is inescapable z y z
pressed the correlation between the S found at the end
at any detuning ∆. Scattering leads directly to atomic (cid:10) (cid:11) z
ofthesqueezingandtheatom-inducedchangetothelight
decoherence by revealing the state of certain atoms to a
shift that modifies S —becomes S S¯ (1 r)S/2,
hypothetical observeroutside the cavity and, in the case y z z ≈ −
again to leading order in r.
of Raman scattering, by randomly flipping some spins (cid:10) (cid:11)
Combiningtheseeffects,wefindanadjustedsetofspin
in the ensemble. Here we apply the treatment of these
moments
effects given in Ref. [19] to the case of unitary cavity
feedback. Sx =S e−21v′, (20)
Allphotonsscatteredintofreespacerevealtheinternal h i C
S = S =0, (21)
state of the scattering atom to a hypothetical observer. h yi h zi
For Raman scattering, the state is encoded in the fre- S2 = S, 1+S 2 1 e−2v′ , (22)
quency of the scattered light. For Rayleigh scattering, y 2 C −
it is encoded in the phase of the scattered field, because (cid:10) (cid:11) S h (cid:0) (cid:1)i
S2 = , (23)
the laser detuning from resonance has opposite sign for z 2
the twospinstates [46]. Any atomwhichscattersa pho- SySz +S(cid:10)zSy(cid:11)=S2(1 r) µe−12v′. (24)
toninto freespacethereforeacquiresanunknownphase, h i − C
entangledwith the informationlostinthe scatteredfield Note thatphotonnumber fluctuations due to atomic ab-
and uncorrelatedwith that of the other atoms in the en- sorption can be neglected because, while the fraction of
semble. The mean length of the Bloch vector is thus atomswhichscatteraphotonisfixedbyµandη,thefrac-
reducedfrom S by a factor of =e−2r correspondingto tion of photons scattered vanishes in the large-photon-
C
the fraction of atoms which have scattered no photons. numberlimit consideredhere. The resultingsqueezingis
Raman scattering, in addition, modifies the relative plottedinFig.5fordifferentcavitycooperativitiesη and
population of and , and forces us to distinguish for perfect photon shot noise suppression (γ =0).
|↑i |↓i
between the spin component Sz found at the end of For the weak- and moderate-coupling regimes where
the squeezing and its time-averaged value S¯z during the scattering is the dominant limitation on squeezing (η <
squeezingprocess. ThedistributionofSz isunalteredby 1), we find ζ (Sµ)−2 + µ/(3η). As in the ideal
the Raman scattering, since it already corresponded to case, the squeeze≈d variance is initially suppressed by the
thesumofindependentrandom 1/2contributionsfrom square of the squeezing parameter, but the noise from
±
the uncorrelated atoms in the initial state. But since scattering into free space adds a variance which scales
spins which flip partway through the squeezing pulse linearly with the shearing. This leads to an optimum
contribute less to the time average, S¯2 is reduced to squeezing ζ 61/3/2(Sη)2/3 for a shearing parameter
z
(1 2r/3)S/2,to leading orderin r. Shinceiit is S¯ which µ (6η)1/3/≈S2/3. Note that 4Sη corresponds to the
z
− ≈
sets the averageatom-induced shift of the light intensity resonant optical depth of the atomic ensemble probed
inside the cavity, and thus the phase shift imparted to through the cavity, and that the achievable squeezing
the atomsby the Starkeffect, the phase varianceinturn therefore scales as optical depth to the 2/3. This is
−
8
the same scaling reported by Trail et al. for their analo- Ueda [1]. In order to prepare a squeezed state, we begin
gous polarization-basedspin-squeezing scheme [35]. The with the atoms in a totally symmetric but unentangled
dashed curve of Fig. 5 shows the squeezing achievable coherentspinstate (CSS) alongthe xˆ axis ofpseudospin
with photon shot noise suppression in a setup otherwise
similar to that used in Ref. [30], using 2S = 2 104 + ⊗N
atoms of 87Rb (Γ/ ∆ =1.8 10−3) in a resonator×with ψa = |↑i |↓i (A1)
a single-atom coop|era|tivity ×for the relevant transitions | i (cid:18) √2 (cid:19)
of η = 0.1 so that φ = ηΓ/ ∆ = 1.8 10−4. For a S 1 2S
shearing parameter of0µ = 1.8| |10−3 co×rresponding to = s22S S+m |mi. (A2)
a photon number of 2.7 104 in×each incident pulse, the mX=−S (cid:18) (cid:19)
×
squeezing reaches20dB,a substantialimprovementover
The second form explicitly shows the CSS’s binomial
the 13dB achievable in the same system without photon
distribution of S eigenvalues. In this state S =
z x
shot noise suppression. h i
S = N/2, S = S = 0, S S +S S = 0 and
y z y z z y
beFcoomr etshesisgtnriofincga-nctoubpelfionrgersecgaimtteeriηng≫ca1ntdheecochuervreattuhree Sy2 = Sz2h =i S/2h. Aifer thehtransformatioin Uρ,µ de-
fined in Eq. (13), the S distribution is unmodified but
z
ensemble, and the ideal squeezing behavior of Sec. II is (cid:10) (cid:11) (cid:10) (cid:11)
the phases between m levels have acquired a quadratic
restored. Note that the scaling of the achievable squeez- | i
dependence on m. The expectation values become
ing with atom number (as S−2/3) is the same for finite
η as it is in this ideal limit of η . Once the effect µ
→ ∞ S =Scos(ρ)cos2S−1 , (A3)
of photon shot noise has been suppressed, the scattering h xi 2
into free space costs only a constant factor in squeezing (cid:16)µ(cid:17)
S =Ssin(ρ)cos2S−1 , (A4)
performance. h yi 2
S 1 (cid:16) (cid:17)
S2 = 1+ S , 1 cos(2ρ)cos2S−2(µ) ,
y 2 − 2 −
VI. CONCLUSION (cid:10) (cid:11) h (cid:16) (cid:17) (cid:0) (cid:1)(iA5)
and
In this paper we have shown how to improve cavity
squeezing performance by disentangling the atomic vari-
S S +S S =S(2S 1)cos(ρ)
ables fromthe light fieldwhich mediates the interatomic y z z y
h i − ×
interaction. We have suggested several ways of doing sin µ cos2S−2 µ . (A6)
this,includingaspin-echosequencethaterasesthephase × 2 2
(cid:16) (cid:17) (cid:16) (cid:17)
informationinacoherentlightpulse,andtheuseofpho-
Forρ=0themeanspinremainsalignedalongxˆandthe
todetectors and classical feedback to generate effective
minimumvariancetransversetothisdirectionisgivenby
Fockstatesoftheinputfield. Oncetheentanglementbe-
tween atoms and outgoing light field is eliminated, even
1
a moderate cavity cooperativity η 1 suffices to obtain ∆S2 = u u2 + S S +S S 2 (A7)
∼ min 2 +− − h y z z yi
squeezing performance close to the limit set by the cur- (cid:18) q (cid:19)
(cid:10) (cid:11)
vature of the Bloch sphere. This limit, in turn, could be
with u = S2 S2 . For large S and in the region
overcome by two-axis counter-twisting [1], which can be ± y ± z
realized by alternating periods of one-axis twisting with of significant squeezing S−1 µ S−1/2 the squeez-
rotationsoftheatomicspin[47]. Furthermore,sinceuni- ing paramet(cid:10)er i(cid:11)s ap(cid:10)pro(cid:11)ximate≪ly ζ ≪(Sµ)−2 +S2µ4/24,
≈
tary S2 evolution in combination with rotations suffices, decreasingunderthe actionoftheshearinguntilthecur-
z
inprinciple,toimplementanyunitarymaponthe Bloch vature of the Bloch sphere deforms the uncertainty re-
sphere [34], the techniques we have presented could en- gion. Thebestsqueezingobtainedisζ 122/3/8S2/3 for
able the production of non-Gaussian entangled states of a shearing parameter µ 121/6/S2/3. ≈
≈
ensembles comprising tens of thousands of atoms. Fur-
ther studies are needed to determine which states are
attainable given realistic experimental imperfections. Appendix B: Simulating a One-Sided Cavity
I.D.L. acknowledges support from NSERC; M.H.S.-S.
acknowledgessupportfromtheHertzFoundationandthe
Throughout this paper we have considered a single-
NSF.ThisworkwassupportedbytheNSF,DARPA,and
endedcavity. Manycavity-QEDexperimentsfinditcon-
ARO.
venient to use two-sided Fabry-Pérot resonators. Such
two-sided cavities would mix a Fock state input from
one end with vacuum fluctuations admitted through the
Appendix A: Squeezing by One-Axis Twisting other end of the cavity, restoring much of the photon
shot noise we wish to suppress. Equivalently, informa-
This appendix summarizes the results for squeezing tiononthe cavitydetuning (andhencethe atomicstate)
by one-axis twisting first obtained by Kitagawa and isavailableintheratiooftransmittedtoreflectedphoton
9
numbers, and this information leak entails atomic deco-
herence. Fortunately, it is possible to converta symmet-
ric cavity into an effective one-sided cavity using only
external optics.
Figure6illustratestheprincipleofthisconversion. For
any given frequency, transverse mode, and polarization,
the symmetric cavity couples to two spatially separated
inputfields(rightandleft). Sincethereisonlyonecavity
mode near the givenfrequency with the giventransverse
modeandpolarization,itmustcoupleonlytosomelinear
combination, labeled here as b†, of the two input fields.
The orthogonal combination d† does not couple to the
cavity mode at all. Classically, b† (d†) correspondsto si-
multaneous illumination from left and right with phases
Figure 6. (Color online) Equivalence of one-sided (top left)
chosen so as to give constructive (destructive) interfer-
andtwo-sided(topright)cavities: theleftandrightportsofa
encewithintheresonator. EnclosingtheFabry-Pérotina
symmetricFabry-Pérotresonatorcanbecombinedonabeam
Sagnacinterferometeroverlapstheleftandrightfieldsat
splittertoisolatethelinearcombinationofthetwofieldsthat
theinput,sothatwithappropriatelychosenpathlengths
couples to the intracavity field (bottom right). Conversely,
the single input of a one-sided cavity can be mixed with an the maximally-coupled superposition b† is isolated from
auxiliarymodetoyieldaneffectivetwo-sidedresonator(bot- the uncoupled mode d†. To an observer looking into the
tom left). b† portoftheinputbeamsplitter,theapparatusappears
tobeasingle-endedcavitywhichcouplestonootherfield
modes. Thed† input,uncoupledfromthecavity,remains
available for interferometer stabilization.
[1] M.KitagawaandM.Ueda,Phys.Rev.A47,5138(1993). [17] D. Meiser, J. Ye, and M. J. Holland, New J. Phys. 10,
[2] D.J.Wineland,J.J.Bollinger,W.M.Itano,F.L.Moore, 073014 (2008).
and D. J. Heinzen, Phys.Rev.A 46, R6797 (1992). [18] M.Saffman,D.Oblak,J.Appel, andE.S.Polzik,Phys.
[3] D.J. Wineland, J. J. Bollinger, W.M. Itano, and D. J. Rev. A 79, 023831 (2009).
Heinzen,Phys. Rev.A 50, R67(1994). [19] M.H.Schleier-Smith,I.D.Leroux, andV.Vuletić,Phys.
[4] A. Louchet-Chauvet, J. Appel, J. J. Renema, D. Oblak, Rev. A 81, 021804 (2010).
N.Kjærgaard, andE.S.Polzik,NewJ.Phys.12,065032 [20] M.H.Schleier-Smith,I.D.Leroux, andV.Vuletić,Phys.
(2010). Rev. A 83, 039907(E) (2011).
[5] I.D.Leroux,M.H.Schleier-Smith, andV.Vuletić,Phys. [21] M. E. Taşgın and P. Meystre, Phys. Rev. A 83, 053848
Rev.Lett. 104, 250801 (2010). (2011).
[6] A.Kuzmich,N.P.Bigelow, andL.Mandel,Europhysics [22] J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik,
Lett.42, 481 (1998). Phys. Rev.Lett. 83, 1319 (1999).
[7] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 83, 2274 [23] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett,
(1999). W. M. Itano, C. Monroe, and D. J. Wineland, Phys.
[8] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Rev. Lett.86, 5870 (2001).
Nature409, 63 (2001). [24] J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff,
[9] A. André, L.-M. Duan, and M. D. Lukin, Phys. Rev. N. Kjærgaard, and E. S. Polzik, Proceedings of theNa-
Lett.88, 243602 (2002). tional Academy of Sciences 106, 10960 (2009).
[10] I.BouchouleandK.Mølmer,Phys.Rev.A65,041803(R) [25] M.H.Schleier-Smith,I.D.Leroux, andV.Vuletić,Phys.
(2002). Rev. Lett.104, 073604 (2010).
[11] J. Zhang, K. Peng, and S. L. Braunstein, Phys. Rev. A [26] Z. Chen, J. G. Bohnet, S. R. Sankar, J. Dai, and J. K.
68, 035802 (2003). Thompson, Phys.Rev. Lett.106, 133601 (2011).
[12] J. K. Stockton, J. M. Geremia, A. C. Doherty, and [27] J.Estève,C.Gross,A.Weller,S.Giovanazzi, andM.K.
H.Mabuchi, Phys. Rev.A 69, 032109 (2004). Oberthaler, Nature455, 1216 (2008).
[13] K.Hammerer, K.Mølmer, E. S.Polzik, and J. I.Cirac, [28] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K.
Phys.Rev.A 70, 044304 (2004). Oberthaler, Nature464, 1165 (2010).
[14] L. B. Madsen and K. Mølmer, Phys. Rev. A 70, 052324 [29] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra,
(2004). and P. Treutlein, Nature464, 1170 (2010).
[15] M. Takeuchi, S. Ichihara, T. Takano, M. Kumakura, [30] I.D.Leroux,M.H.Schleier-Smith, andV.Vuletić,Phys.
T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett. 94, Rev. Lett.104, 073602 (2010).
023003 (2005). [31] I.D.Leroux,M.H.Schleier-Smith, andV.Vuletić,Phys.
[16] C. Genes and P. R. Berman, Phys. Rev. A 73, 063828 Rev. Lett.106, 129902(E) (2011).
(2006). [32] A. Kuzmich, L. Mandel, and N. P. Bigelow, Phys. Rev.
10
Lett.85, 1594 (2000). M. Koschorreck, and M. W. Mitchell, Phys. Rev. Lett.
[33] T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, 105, 053601 (2010).
Phys.Rev.Lett. 102, 033601 (2009). [41] H. P. Yuen,Phys. Rev.A 13, 2226 (1976).
[34] S. Chaudhury, S. Merkel, T. Herr, A. Silberfarb, I. H. [42] R. Loudon and P. L. Knight, Journal of Modern Optics
Deutsch, and P. S. Jessen, Phys. Rev. Lett. 99, 163002 34, 709 (1987).
(2007). [43] H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gräf,
[35] C.M.Trail, P.S.Jessen, andI.H.Deutsch,Phys.Rev. K. Danzmann, and R.Schnabel, Class. QuantumGrav.
Lett.105, 193602 (2010). 27, 084027 (2010).
[36] H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, [44] S. Machida and Y. Yamamoto, Optics Communications
M. Cetina, A. Grier, J. Simon, and V. Vuletić, in 57, 290 (1986).
Advances in Atomic, Molecular, and Optical Physics, [45] A. Sinatra, J.-C. Dornstetter, and Y. Castin, “Spin
Vol. 60, edited by E. Arimondo, P. R. Berman, and squeezing in Bose-Einstein condensates: Limits im-
C. C. Lin (Academic Press, 2011) pp.201–237. posed by decoherence and non-zero temperature,”
[37] M.J.CollettandC.W.Gardiner,Phys.Rev.A30,1386 arXiv:1109.2401v1 (2011).
(1984). [46] H.Uys,M.J.Biercuk,A.P.VanDevender,C.Ospelkaus,
[38] C.W.GardinerandM.J.Collett,Phys.Rev.A31,3761 D.Meiser,R.Ozeri, andJ.J.Bollinger,Phys.Rev.Lett.
(1985). 105, 200401 (2010).
[39] U.Fano, Phys. Rev.124, 1866 (1961). [47] Y. C. Liu, Z. F. Xu, G. R. Jin, and L. You, Phys. Rev.
[40] F. Wolfgramm, A. Cerè, F. A. Beduini, A. Predojević, Lett. 107, 013601 (2011).