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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. 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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. 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