A dynamic scheme for generating number squeezing in Bose-Einstein condensates through nonlinear interactions
AA dynamic scheme for generating number squeezing in Bose-Einstein condensatesthrough nonlinear interactions
Simon A. Haine
1, 2 and Mattias T. Johnsson
1, 3 Australian Research Council Centre of Excellence for Quantum-Atom Optics School of Physical Sciences, University of Queensland, Brisbane, 4072, Australia ∗ Department of Quantum Science, The Australian National University, Canberra, 0200, Australia
We develop a scheme to generate number squeezing in a Bose-Einstein condensate by utilizinginterference between two hyperfine levels and nonlinear atomic interactions. We describe the schemeusing a multimode quantum field model and find agreement with a simple analytic model in certainregimes. We demonstrate that the scheme gives strong squeezing for realistic choices of parametersand atomic species. The number squeezing can result in noise well below the quantum limit, evenif the initial noise on the system is classical and much greater than that of a Poissonian (shot noiselimit) distribution.
I. INTRODUCTION
The experimental realization of Bose-Einstein conden-sates (BECs) has allowed the creation of macroscopicquantum systems that are highly controllable, and henceprovide an excellent system to test predictions of manybody quantum dynamics. The generation of nonclassicalstates in BECs, such as number squeezed states, wouldallow for measurements of particle number statistics thatdiffer from classical predictions [1]. However, while themotional state of the atoms is reasonably simple to ma-nipulate, the quantum statistics governing the numberdistribution of the particles is difficult to control, withBECs typically produced with a large (5%) shot-to-shotvariation in the number. The generation of nonclassicalstates in BECs is currently of great interest [2], as it couldpotentially enhance the sensitivity of atomic interferom-eters [3] used to measure electric, magnetic, and gravita-tional fields, accelerations, and atomic interactions. Thegeneration of nonclassical states also provides a methodfor testing the fidelity of recent quantum state transferschemes [4], and an atom laser produced from a numbersqueezed BEC will have a reduced linewidth [5].The generation of nonclassical states via self-interaction in samples of cold atoms has been consid-ered before [6–11]. The schemes proposed in [6] and [7]describe the generation of quadrature squeezing, whichrequires the use of a well-defined phase reference in orderto be observed. A well-defined phase reference is difficultto obtain in atom optical systems, especially in the pres-ence of strong nonlinearities, which are required to pro-duce the quadrature squeezing, thus making the schemessomewhat unrealistic. In addition, both of these schemesassume the BEC is initially in a coherent state, ratherthan a more realistic statistical mixture of coherent stateswith random phases. [8–10] have demonstrated that theatomic nonlinearity can be used to generate number dif-ference squeezing, creating a state with angular momen- ∗ Electronic address: [email protected] tum projection below the standard quantum limit (spinsqueezing). These schemes assume that the total numberof particles is initially well defined. Recently, Esteve etal. [2] have directly observed squeezing in the numberdifference between two adjacent lattice sites.Chuu et al. [12] have demonstrated the ability to pro-duce small condensates which exhibit number squeez-ing. By producing a very stable trapping potential, theyfound that they were able to produce condensates witha very well specified chemical potential. As the chemicalpotential is related to the condensate number throughthe nonlinear interaction, this leads to number squeezedcondensates.In this article we describe a scheme that allows thecreation of absolute number squeezing in a BEC (asopposed to number difference squeezing), which is ex-perimentally realistic, utilizes only the relatively simpleexperimental technique of Ramsey interferometery, anddoes not require manipulation of the scattering lengthvia a Feshbach resonance or a coherent phase referencefor the atoms. We show that our scheme achieves num-ber squeezing below the quantum limit even if there isinitially considerable classical noise on the number statis-tics, and that this result holds even when we realisticallyassume that the initial state of the BEC is a statisticalmixture of states with random phase.We consider a BEC with two internal states confinedto an optical trap, with all the atoms initially in onestate. A short state-dependent coupling is applied, trans-ferring a small fraction of the population to another state.The system is then left to evolve for some time, allow-ing nonlinear interactions and interference between thetwo states, before the coupling is applied for a secondtime, transferring some of the population back to theinitial state. Provided the s -wave scattering lengths ofthe atoms in the different internal states are not all iden-tical, and by choosing appropriate coupling strengths,hold times and trap geometry, it is possible to gener-ate number squeezing. An important difference betweenour scheme and the schemes demonstrated in refs. [2]and [12] is that our scheme is based on dynamic inter-ference between the two modes to obtain absolute num- a r X i v : . [ qu a n t - ph ] O c t time |Ω| Ω t t t t Δ t Δ t t hold First coupling pulse Second coupling pulse + phase shift φ FIG. 1: Timing for the coupling pulses in the proposed ex-periment. The coupling field is turned on at t , and thenoff again at t . After a duration t hold , the coupling pulse isturned back on again at t and finally turned off at t . Afterthis, the population of state | (cid:105) atoms is measured. ber squeezing in one of the modes, where as the schemesdemonstrated in refs. [2] and [12] obtain their squeez-ing by adiabatically changing the potential to one wherethe ground state of the system exhibits number differ-ence squeezing (in the case of [2]) or absolute numbersqueezing (in the case of [12]). II. SCHEME.
Our proposed scheme is based on a Ramsey interfer-ence experiment between two hyperfine states of sodium,namely | F = 2 , m F = 0 (cid:105) ≡ | (cid:105) and | F = 1 , m F = +1 (cid:105) ≡| (cid:105) . The timing for our scheme is outlined in Figure 1. Webegin with all the atoms in a BEC in the | (cid:105) state in anoptical trap. At t = t , the microwave coupling is turnedon for a brief duration of time ∆ t = t − t . Duringthis time, a fraction of the atoms are then transferred tostate | (cid:105) . The coupling is switched off, and the system isleft to evolve for an amount of time t hold = t − t , beforewe interfere the two modes with a second microwave cou-pling pulse for a duration ∆ t = t − t . We assume thatthis pulse is phase locked to the first, with an adjustablephase shift φ . This phase shift is an important ingredi-ent, as it is what allows us to ensure that the two modesinterfere in such a way as to produce number squeezing,independently of t hold , which controls the depth of squeez-ing. Nandi et al. [11] have recently published a schemebased on a Ramsey-Bord´e interferometer, but withoutincluding this adjustable phase shift φ at the second cou-pling pulse. As a result, their squeezing is only observedat particular values of t hold , when the nonlinear phaseshift acquired during the hold time is appropriate. Theinclusion of this phase shift gives us an additional degreeof freedom in our system, as we can choose t hold inde-pendently of the phase shift required to produce numbersqueezing, and thus optimize our squeezing depth. Af-ter t , we separate the two modes with a magnetic field,and count the number of atoms in mode | (cid:105) to determinethe number statistics. The Hamiltonian for the system is ˆ H = ˆ H + ˆ H c ( t ), withˆ H = (cid:88) j =1 , (cid:90) ˆ ψ † i ( r ) H j ˆ ψ i ( r ) d r + (cid:88) i,j =1 , U ij (cid:90) ˆ ψ † i ( r ) ˆ ψ † j ( r ) ˆ ψ i ( r ) ˆ ψ j ( r ) d r , (1)and ˆ H c = (cid:90) (cid:16) (cid:126) Ω( t ) e iφ ˆ ψ † ( r ) ˆ ψ ( r ) + h . c . (cid:17) d r , (2)where ˆ ψ i ( r ) represents the annihilation operator for state | i (cid:105) , H j = − (cid:126) m ∇ + V opt ( r )+( j − (cid:126) δ is the single particleHamiltonian, and V opt ( r ) is the optical dipole potential.Ω( t ) represents the microwave coupling field, which isswitched on and off to control the coupling between thetwo hyperfine levels. The phase of this RF field can alsobe tuned between each pulse. We will make the rotatingwave approximation [1] and assume that the couplingis on resonance, such that Ω( t ) = Ω e − iδt , with (cid:126) δ thehyperfine splitting between | (cid:105) and | (cid:105) , and Ω is theRabi frequency. III. ANALYTIC MODEL.
We first consider a two mode model, which demon-strates how atomic nonlinearities can be used to gener-ate number squeezing. A two-mode model can be derivedfrom Eq. (1) and Eq. (2) by assuming that the atoms re-main in the ground motional state of the optical trap.With this assumption, the modified Hamiltonian for thesystem is ˜ H = ˜ H + ˜ H c , with˜ H = (cid:126) δ ˆ a † ˆ a + (cid:88) i,j =1 , (cid:126) χ ij ˆ a † i ˆ a i ˆ a † j ˆ a j (3)˜ H c = (cid:126) (cid:16) Ω( t ) e iφ ˆ a † ˆ a + h . c . (cid:17) (4)where ˆ a (ˆ a ) annihilates an atom from state | (cid:105) ( | (cid:105) ),and χ ij = U ij (cid:82) | ψ ( r ) | d r , where ψ ( r ) is the groundstate wavefunction of the optical potential.Beginning at t , with the coupling initially switchedoff (Ω = 0), and assuming that our initial state is aPoissonian mixture of number states for mode | (cid:105) , andvacuum for mode | (cid:105) , the density matrix for the systemis ρ ( t ) = e −| α | ∞ (cid:88) n =0 ( | α | ) n n ! | n , (cid:105)(cid:104) n , | (5)where the state | n , n (cid:105) denotes n atoms in mode | (cid:105) and n atoms in mode | (cid:105) and | α | ≡ N is the meannumber of atoms. We note that ρ ( t ) is mathematicallyequivalent to a mixture of coherent states with randomphases: ρ ( t ) = 12 π (cid:90) π | α e iθ (cid:105)(cid:104) α e iθ | dθ ⊗ | (cid:105)(cid:104) | (6)where | α (cid:105) ≡ e −| α | / (cid:80) n α n √ n ! | n (cid:105) denotes the Glauber co-herent state [1]. At times t < t , the evolution is trivialas ρ commutes with ˜ H .At t , the coupling is turned on for a duration ∆ t coupling a fraction of the atoms (sin (Ω ∆ t )) into mode | (cid:105) . If ∆ t is sufficiently short that we can ignore thecontribution to the evolution due to the nonlinear partof ˜ H , the density matrix for the system becomes ρ ( t ) = (cid:90) π | α ( t ) e iθ (cid:105)(cid:104) α ( t ) e iθ | ⊗ | β ( t ) e iθ (cid:105)(cid:104) β ( t ) e iθ | dθ π = e −| α | −| β | π (cid:88) n ,n ,m ,m (cid:90) π e iφ ( m + m − n − n ) dφα ( t ) n α ∗ ( t ) m β ( t ) n β ∗ ( t ) m √ n ! m ! n ! m ! | n , n (cid:105)(cid:104) m , m | = (cid:88) n ,m ,n A n ,m ,n | n , n (cid:105)(cid:104) m , n + n − m | , (7)with A n ,m ,n = e −| α ( t ) | −| β ( t ) | (8) × α ( t ) n α ∗ ( t ) m β ( t ) n β ∗ ( t ) n + n − m (cid:112) n ! m ! n !( n + n − m )! , and α ( t ) = α cos θ , β ( t ) = − iα sin θ , where θ =Ω ∆ t . We note that although the condensate initiallyhad no global phase, a relative phase between the twomodes has been created by this first coupling pulse.The coupling is now switched off, and the system is leftto evolve under ˜ H for a period of time t hold . At t = t ,the density matrix for the system is now ρ ( t ) = ∞ (cid:88) n ,n ,m =0 A n ,n ,m e − i Φ n ,n ,m t hold × | n , n (cid:105)(cid:104) m , n + n − m | , (9)withΦ n ,n ,m = χ ( n ( n − − m ( m − χ (( m − n n + n − m − χ ( n n − m ( n + n − m ))+ δ ( m − n ) . (10)At this point, both modes still contain a Poissonian num-ber distribution, but the relative phase created in theprevious step has been ‘sheared’ due to the nonlinear in-teraction. We note that if χ = χ = χ , there is nophase shearing due to this effect, and our scheme doesnot work. We chose sodium as our atomic species, as ithas a relatively large difference between the scatteringlengths of | F = 1 , m F = +1 (cid:105) and | F = 2 , m F = 0 (cid:105) .Finally, we describe the dynamics caused by the sec-ond microwave pulse in the Heisenberg picture, by not-ing that the Heisenberg operators after the second pulseare ˆ a H = ˆ a (0) cos θ − i ˆ a (0) sin θ e iφ , and ˆ a H = FIG. 2: (color online) log ( v ( ˆ N )) as a function of t hold and φ . For some values of t hold and φ , v ( ˆ N ) dips below 0 . χ = χ =0 .
018 s − , χ = 0 .
019 s − . The strength of the couplingpulses was θ = 0 . θ = 0 .
025 rad for the first andsecond coupling pulses respectively. The initial occupation ofthe BEC was chosen to be a Poisson distribution with (cid:104) ˆ N (cid:105) =10 . These parameters correspond to the scattering propertiesof sodium in a 500 Hz spherical harmonic trap. ˆ a (0) cos θ − i ˆ a (0) sin θ e − iφ , with θ = Ω ∆ t , where∆ t is the duration of the second microwave pulse, and φ is the phase of the microwave field relative to the firstpulse. Again, we have assumed that the duration of thepulse is sufficiently short that we can ignore the evolutiondue to the nonlinear component of ˜ H . As we are onlyinterested in the number statistics, we can neglect therest of the evolution after the second microwave pulse,as the number operators for both modes commute with˜ H . Assuming we can distinguish state | (cid:105) atoms fromstate | (cid:105) atoms, we define the normalized number vari-ance for state | (cid:105) atoms as v ( ˆ N ) ≡ ( (cid:104) ˆ N (cid:105)−(cid:104) ˆ N (cid:105) ) / (cid:104) ˆ N (cid:105) ,ˆ N ≡ ˆ a † H ˆ a H . Figure 2 shows the v ( ˆ N ) as a functionof t hold and φ , using the scattering properties of sodium( a = a = 2 . a = 3 . t hold and φ , v ( ˆ N ) dips below 0 .
01, as compared to the quantumlimit v ( ˆ N ) = 1 associated with a coherent state, indicat-ing significant number squeezing. The parameter spacefor this model is quite large, as we can adjust the lengthof the first and second coupling pulses, the hold time,and the phase of the second coupling pulse. If we did nothave the ability to adjust the phase of the second couplingpulse, we would be constrained to a vertical line in Fig-ure 2, and not necessarily be able to access the optimumvalue of the squeezing. We found that the best num-ber squeezing was obtained when the first coupling pulsewas quite weak (approximately 8% of the atoms trans-ferred). We also found we could still get a good level ofsqueezing when we began with an initial state which hadnumber fluctuations 150 times larger than a Poissoniandistribution (about 5% shot to shot fluctuations in thenumber) (see Figure 4). When starting with a such aninitial condition, the best squeezing was found when thefirst beam splitter was relatively weak. This is due to thefact that the addition of vacuum to a super-Poissoniannumber distribution drives it towards a Poissonian num-ber distribution. IV. MULTIMODE MODEL.
To investigate if the approximations we made in theprevious section to obtain an analytic solution were valid,we performed a 1D multimode simulation of the systemusing a stochastic phase space method. Specifically, weutilize a Truncated Wigner (TW) approach [14]. We re-duce Eqs. (1) and (2) to one dimension by integrating outthe dynamics in the y and z dimensions. A Fokker Plankequation (FPE) is then found from the master equationfor the system using the Wigner representation. Thisequation can then be converted into a set of stochas-tic partial differential equations (SPDEs), which can besolved numerically. By averaging over many trajecto-ries with different noises, expectation values of quanti-ties corresponding to operators in the full quantum fieldtheory can be extracted. When converting our FPE toa SPDEs, we ignore third and higher order derivativesin the FPE, as these terms do not have a simple map-ping to the stochastic PDEs, and can be assumed to benegligible when the field has a high occupation number[14]. This truncated Wigner approximation will eventu-ally fail, as it can not describe negative components ofthe Wigner function, which eventually occur when evolv-ing under a Hamiltonian such as Eq. (1). However, wehave checked our simulations in limits where the multi-mode dynamics can be neglected and they agree with theresults obtained from our two-mode analytical model. Inaddition, over our simulation times no anomalous resultssuch as signficant negative densities were seen, indicatingthat the truncation of third order derivatives was a validapproximation. The SPDEs describing the one dimen-sional system are i (cid:126) ˙ ψ ( x ) = L ψ ( x ) + (cid:126) Ω( t ) ψ ( x ) (11) i (cid:126) ˙ ψ ( x ) = L ψ ( x ) + (cid:126) Ω ∗ ( t ) ψ ( x ) , (12)with L j = H j + U jj ( | ψ j | − /dx )+ U ij ( | ψ i | − / (2 dx )) , (13)where dx is the grid spacing of the numerical simulations.The terms inversely proportional to dx compensate forthe mean field of the vacuum, which is nonzero in the −20 −10 10 200 x ( μ m) −20 −10 10 200 x ( μ m) (case I)(case II) (a)(b) FIG. 3: (color online) Normalized density profile (cid:104) ˆ ψ † ( x ) ˆ ψ ( x ) (cid:105) / (cid:104) ˆ N (cid:105) at t (blue solid line), compared to thenormalized density profile at t (black dashed line), as calcu-lated by the TW model for two different parameter regimes.In case II, more atoms are transferred in the first couplingpulse, which creates significant multimode dynamics. In caseI, the dynamics are much less pronounced and the densityprofile at t differs only slightly from that at t . Parameters:Case I: ∆ t = 1 µ s, ∆ t = 0 . µ s, t hold = 16 ms. Case II:∆ t = 6 µ s, ∆ t = 0 . µ s, t hold = 3 . = 50 rad s − , and a 500 Hz spherical harmonictrap. We assumed an initial number distribution which waspoissonian, with a mean number of atoms N = 10 . Wigner approach. The noise on the initial conditions foreach trajectory of the evolution of these equations waschosen such that they corresponded to the specific initialstate of interest.Figure (3) shows the normalized density profile of state | (cid:105) atoms, (cid:104) ˆ ψ † ( x ) ˆ ψ ( x ) (cid:105) / (cid:104) ˆ N (cid:105) , at t , compared to thenormalized density profile at t , as calculated by the TWmodel for two different parameter regimes (case I andcase II). Case II shows an example where the multimodedynamics is significant, and the density profile at t hasa pronounced difference from the ground state densityprofile. The multimode dynamics is a consequence of theunequal scattering lengths, meaning that when atoms arecreated in state | (cid:105) , they are no longer in a motionaleigenstate of the system. These dynamics are relativelyinsignificant in case I, when only 0 .
25% of the atoms aretransferred in the first coupling pulse. However, in thecase II, ∼
9% of the atoms are transferred in the first cou-pling pulse, and the perturbation to the dynamics duringthe hold time is significant, even though the system isleft to evolve for a much shorter time. Figure (4) showsthe number of atoms in state | (cid:105) and the variance in thenumber, after the second coupling pulse, for case I andcase II. In case I there is excellent agreement between themultimode model TW and the two mode analytic model.However, in case II the comparison between the two mod-els is poor, due to significant multimode dynamics (whichcan be seen in Figure (3(b)) preventing the system acting −2 −1 −1 φ/π φ/π v(N ) 〈 N 〉 (a) (b) φ/π φ/π (c) (d) >> Case I Case II
FIG. 4: (color online) Results from the multimode TW model(red dots) compared to the analytic two mode model (blacktrace), for two different parameter regimes (case I (left col-umn) and case II (right column) respectively). (a) and (b)show number variance v ( ˆ N ), while (c) and (d) show (cid:104) ˆ N (cid:105) ,at t . In both cases we assumed an initial number distri-bution which was Poissonian, with a mean number of atoms N = 10 . In case I ((a) and (c)), there is excellent agreementbetween the multimode TW model and the two mode model.In case II ((b) and (d)), there is significant disagreement be-tween the two models because a larger fraction of atoms istransferred during the first coupling pulse, which creates sig-nificant multimode dynamics, as can be seen in Figure (3(a)).The blue dashed trace indicated results from the two modemodel, when a superpoissonian distribution was used at theinitial state, with 5% number uncertainty (approximately 150times noisier than a Poisson distribution). (a) shows that evenwith large amounts of classical noise, it is possible to num-ber squeeze below the quantum limit. Parameters: Case I:∆ t = 1 µ s, ∆ t = 0 . µ s, t hold = 16 ms. Case II: ∆ t = 6 µ s, ∆ t = 0 . µ s, t hold = 3 . = 50 rad s − , and a 500 Hz spherical harmonic trap. as a two mode system. V. EXPERIMENTAL CONSIDERATIONS
This scheme relies on having good control of microwavefields in order to implement the precise timing and reso-nance conditions. Precise control of microwave intensityand pulse duration is routinely achievable in atom opticslaboratories (see, for example [15]), and control of mi-crowave frequencies with sub-Hertz stability (much lessthan the Fourier width of the pulses in our scheme) isroutinely achievable with off the shelf equipment. Asthe parameter space for the experiment is large, a goodknowledge of the parameters such as the trapping fre-quency, the Rabi frequency, and the scattering lengthswill be required such that theoretical modeling can pre-dict roughly were to seach for the squeezing. The valuesof the scattering lengths will probably be the least wellknown of these quantities. To simulate the effect of animprecise knowledge of the scattering lengths, we have −1 a / a
22 0
FIG. 5: (color online) v ( ˆ N ) as a is varied. All other param-eters (Ω , ∆ t , ∆ t , and t hold ) are the same as in Figure (4)case I, and the phase of the second coupling pulse was fixedat φ = 1 . π , which was the optimum phase for Figure (4) caseI. The squeezing is rapidly degraded as a moves away from a . Black trace: results of analytic two mode model. Reddots: Results from TW simulation. investigated the effect of varying one of the scatteringlengths ( a ), while keeping all other parameters fixed tothe values used in Figure (4) case I.Figure (5) shows v ( ˆ N ) as a is varied ( a = a corre-sponds to the value of a used in Figure (4), ie a ≡ . φ fixed at the optimum phase for squeezing ( φ = 1 . π )as found in case I. The squeezing is completely degradedas the scatting length changes by about 3%. However,this is not the because the degree of phase shearing hasbeen significantly altered. The different scattering lengthcauses a slight shift in the mean phase between the twomodes, such that it is shifted away from the optimumphase for number squeezing. If we were to re-scan thephase of the second coupling pulse to search for numbersqueezing (this would mean performing more shots of theexperiment in order to find the optimum phase), we mayfind that the squeezing is still present for a large rangeof scattering lengths. Figure (6) shows v ( ˆ N ) vs. a forthe same parameters as case I, but this time optimizingthe phase of the second coupling pulse φ for each value of a . We see that significant number squeezing can still beachieved as we alter a by 30% in either direction. Theexception is as a approaches a and a , the squeezingvanishes.When modeling the system with the multimode TWmodel, we found that squeezing could still be obtainedas we varied a by about 10%, giving decent agreementwith Figure (6) in this range. However, we found that as | a − a | became larger, the results differed significantlyfrom the two mode model. As the TW model requires sig-nificantly more computational resources, we did not op-timize the phase directly in this model. Rather, we usedthe optimum phase as found by the two mode model (Fig-ure 6). For example, with a /a = 0 .
7, the two modemodel predicts v ( ˆ N ) = 0 . φ = 0 . π ), whereas for the same value of φ the multimode TW predicts v ( ˆ N ) = 14 .
5. It is possible that the multimode TW −2 −1 a / a
22 0
FIG. 6: v ( ˆ N ) as a is varied, as calculated by the two-mode model. All other parameters (Ω , ∆ t , ∆ t , and t hold )are the same as in Figure (4) case I. The phase of the secondcoupling pulse, φ , has been optimized for maximum numbersqueezing for each value of a . Significant number squeez-ing can still be obtained as a is altered by 30% in eitherdirection. The exception is as a approaches a and a thesqueezing vanishes. model predicts squeezing for some parameters for thisvalue of the scattering length. However, as the parameterspace is large (∆ t , ∆ t , t hold , and φ can all be varied tofind the optimum parameter regime), we found it almostimpossible to find squeezing by searching using the multi-mode TW model alone. However, by utilizing the Gross-Pitaeveskii equation we were able to search for a regimewhere the system behaves approximately as a two-modesystem, and then use the two-mode model to investigatethe squeezing properties. We can then confirm the re-sults by using the multimode TW model. We found thatsignificant squeezing was achievable, with v ( ˆ N ) = 0 . t = 0 . µ s, ∆ t = 0 . µ s, t hold = 16 ms, and φ = 0 . π .Detection of atoms with high quantum efficiency willbe required in order to observe the squeezing. This isexperimentally challenging, but has been demonstratedbefore [2, 16]. An addition effect that may degrade thesqueezing is atomic loss due to inelastic collisions. Usingthe three-body recombination rates recently measured in[17], we estimate that roughly 10% of the atoms fromstate | (cid:105) (the state that we are not looking for squeez-ing) are lost during the 16 ms hold time. However, amore important concern is the two-body inelastic colli-sion rate, as it scales as (cid:82) | ψ ( r ) | d r , that is, the same way as χ ij , the nonlinear interaction parameter, so re-ducing the atomic density will not help, as the lifetimedue to collisions and the time taken to achieve squeezingscale identically. It was observed in [17] that with mix-tures of different hyperfine states decayed on timescalesof order several milliseconds. However, if the maximalstretched combination of states was used [18] (for ex-ample | F = 1 , m F = 1 (cid:105) , | F = 2 , m F = 2 (cid:105) ), lifetimesof several seconds were observed. Using this particularcombination of states would allow for squeezing underour scheme, as it has a (cid:54) = a , and as these scatteringlengths are similar one to the ones used in this paper,one would expect a similar amount of squeezing.It may be possible to create an intensity squeezed atomlaser via a similar technique to that discussed in this Let-ter. By outcoupling two co-propagating hyperfine states,and interfering them at particular distance from the con-densate, it may be possible to create intensity squeez-ing in one of the modes. However, this would require aspecies of atom with two magnetic field insensitive states,with scattering lengths a + a (cid:54) = 2 a . This require-ment could be avoided by using a separated beam pathinterferometer, as the effective χ goes to zero. However,it may be difficult to achieve the required mode matchingin this case.Finally, we wish to note that the specific states usedhere are illustrative, not optimal. We have presenteda scheme that allows the the generation of significantamounts of number squeezing in a BEC, and demon-strated its effectiveness for plausible states of an atomthat can be Bose-condensed. However scattering lengthsare not well-known for various states of many atomicspecies, and better candidates for generating numbersqueezed BECs almost certainly exist. VI. ACKNOWLEDGMENTS
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