Generation of macroscopic singlet states in atomic ensembles

We study squeezing of the spin uncertainties by quantum non-demolition (QND) measurement in non-polarized spin ensembles. Unlike the case of polarized ensembles, the QND measurements can be performed with negligible back-action, which allows, in principle, perfect spin squeezing as quantified by [G. Toth et al., Phys. Rev. Lett. 99, 250405 (2007)]. The generated spin states approach many-body singlet states, and contain a macroscopic number of entangled particles, even when individual spin is large. We introduce the Gaussian treatment of unpolarized spin states and use it to estimate the achievable spin squeezing for realistic experimental parameters. Our proposal might have applications for magnetometry with a high spatial resolution or quantum memories storing information in decoherence free subspaces.


Introduction
Realization of large coherent quantum systems exhibiting non-classical behavior is at the center of attention in many-body quantum experiments with cold atoms [1] and ions [2]. In a system of spin-1/2 particles, spin squeezing [3,4] is one of the most successful approaches for creating large scale quantum entanglement 6 . In a spin squeezed state, the collective spin of an ensemble of particles is almost completely set in one direction, while the variance of an orthogonal spin component is decreased below the standard quantum limit. In the special case of spin-1/2 atoms this implies atom-atom entanglement [7] 7 . In a quantum non-demolition (QND) scheme, the particles interact with a light field, which is subsequently measured projecting the atoms into a squeezed state [9,10]. Typically, the length of the squeezing dynamics is limited by losses, and a short-time approximation can be applied. In this regime, two of the spin components orthogonal to the large mean spin behave like the canonical variables x and p. The Gaussian formalism can then be used for modeling [11] and can include realistic effects of noise and imperfections [12]- [15], [17].
At this point the questions arise: Is it possible to realize a protocol for the creation of entanglement by squeezing the spin uncertainties that would also work for higher spins? This is important since most experiments use atoms with larger spins. The solution is not easy: Known methods for creating spin-1/2 entanglement by spin squeezing cannot straightforwardly be generalized to higher spins, without restricting dynamics or the detection to a spin-1/2 subspace [18,19]. Moreover, from the point of view of modeling spin systems, one might ask: Is it possible to extend the Gaussian formalism to unpolarized spin states? This is important if we want to depart from the usual setups with large collective spin.
In this paper, we show how to create and detect entanglement by QND measurement of collective spin in an unpolarized ensemble. We show how to create a many-body singlet state of atoms without requiring that they interact with each other and the system settles in a ground state of some antiferromagnetic Hamiltonian. Our proposal works even for particles with a large spin. With realistic experimental parameters for 87 Rb [15], the predicted squeezing dynamics is 3 robust to decoherence, and produces a many-atom singlet state. Unlike standard spin squeezing, the method creates entanglement even in the limit of very strong interaction, which might be used in experimental implementations with cavities [20,21], or in any multi-atomic system in which a von Neumann measurement of the collective spin is possible. We demonstrate the validity of the Gaussian approximation for unpolarized spin states. For the lossless case, we confirm our finding with comparison to the exact model.
The paper is organized as follows. In section 2, we present the spin squeezing parameter to detect the entanglement of many-body singlet states and also discuss the properties of the singlets we aim to prepare. In section 3, we describe the squeezing process. First, we consider the lossless case and present a model based on a Gaussian approximation. Later, we include decoherence in the model. For the lossless case, we compare our results to the results of the exact model. In the appendix, we present the details of the calculations for the exact model.

Detecting the entanglement of singlet states
In this paper, we will use the generalized spin squeezing parameter where J l are the components of the collective angular momentum, ( J l ) 2 = J 2 l − J l 2 , and for a system of N spin-j particles we define J := N j. It has already been shown in [22]- [26] that any state giving ξ s < 1 is entangled (i.e. not fully separable). For completeness, we present briefly the proof for (1). For pure product states of the form | p = ⊗ N k=1 |ψ k , we have l=x,y,z where j (k) l denotes the spin coordinates of particle (k) for l = x, y and z. Here, we used the fact that l ( j (k) l ) 2 |ψ k j. For a mixture of pure product states, i.e. for separable states, (2) remains true since the variance is concave in the state.
The states giving ξ s = 0 are called many-body singlet states [27]. In particular, an equal mixture of all pure singlets, expected to arise in permutationally invariant systems, has intriguing entanglement properties [22,24,25]. The bipartite entanglement of this state has already been determined for qubits [28]. It is very mixed, yet its entanglement is robust to noise [26]. For qubits, the singlet state studied in this paper is an equal mixture of all states composed of twoqubit singlets as can be seen in figure 1.
For an imperfect realization, N ξ 2 s gives an upper bound on the number of particles unentangled with other particles [24,25]. This can be seen as follows. Let us consider a pure state of the form ⊗ M k=1 |ψ k ⊗ |ψ M+1,...,N , which has M particles unentangled with the rest. For such a state, we have Hence, Due to the concavity of the variance, mixing pure states with M or more unentangled particles, we still have N ξ 2 s M. If N ξ 2 s for some quantum state is smaller than this bound, then the state cannot be obtained by preparing pure states having at least M unentangled spins and mixing them. If N (1 − ξ 2 s ) is a large number, then we can say that entanglement between macroscopic numbers of particles is present in the system in this sense.
Next, we will review the relevant properties of singlet states. Many-body singlet states are at the center of attention in condensed matter physics and also in other areas of quantum physics. They appear as ground states of many fundamental spin models. There have been numerous experiments and experimental proposals for studying such states [29]- [32]. Surprisingly, the permutationally invariant singlet appears even in quantum gravity calculations of black hole entropy [28]. Its realization is difficult as a T = 0 thermal state of a spin system, since in this case the system Hamiltonian is J 2 x + J 2 y + J 2 z , which is essentially the sum of two-body interactions connecting all spin pairs.
Singlets play an important role in quantum information processing. They are invariant under transformations of the form U β = exp(−iβ J n ), which describe the effect of an external homogeneous magnetic field on the spins and n is the direction of the field. ξ s can be directly related to the decrease of fidelity of a pure state | under an external magnetic field, defined as Straightforward calculations show that for small β 8 . Thus, small ξ s indicates insensitivity to external fields, which makes singlet states applicable to encoding quantum information in decoherence free subspaces [34], for sending information independent of a reference direction [35] or, possibly, for metrological applications in which insensitivity to external homogeneous magnetic fields is needed, e.g. measuring the spatial gradient or fluctuations of the fields. Note that measuring field gradients is possible with two almost completely polarized atomic ensembles in an entangled state [36,37]. A bipartite singlet can also be used for this aim [38]. However, using the singlet of a single atomic ensemble can lead to magnetometry with a better spatial resolution.

Outline of the squeezing procedure
In this paper, we consider a procedure to produce an atomic state with ξ s < 1 from a nonentangled state with ξ s ∼ 1. For the initial state and throughout the dynamics, we have J = 0. Suitable initial states include the thermal state (completely mixed state), for which ξ s = √ j + 1, and pure product states of the form | p = ⊗ k |ψ k , with | j ψ k | = j for which ξ s = 1. Spin coherent states also have ξ s = 1; however, for them J = 0. After the initial state was created, ξ s is reduced by sequentially measuring and if necessary correcting by feedback the spin components.
Next, we will examine the dynamics of the expectation values and variances of collective spin operators and we will determine how they depend on the particle number N . In particular, A ∼ N p will denote that the quantity A is proportional to N p for large N .
The first squeezing step is a QND measurement of the atomic spin component J x . The realization is similar to QND measurements in polarized atomic ensembles [9,10]. The atoms interact with light, which is subsequently measured. As discussed later, this procedure can reduce the value of ( J x ) 2 considerably but with only a minor increase in ( J y ) 2 and ( J z ) 2 . However, the procedure results in nonzero J x . Fortunately, the expectation value of the spin component J x remains small and we obtain | J x | √ N . Feedback, using the data coming from the measurement of the light, can then be applied to restore J x = 0. This feedback step is similar to the coherent rotation in polarized spin squeezing [39,40], but does not require a large average polarization. However, it is now not simply a rotation, but an incoherent process. The feedback introduces negligible noise: It can be shown that for making J x zero, ∼ √ N atoms are affected, introducing extra noise ( J x ) 2 ∼ √ N , which is negligible relative to the initial noise of ∼N in the large N limit 9 . Alternatively, instead of feedback, post-selection could be used to identify cases with low J x .
If the initial state was a pure product state of the type mentioned above, measurement of J x is sufficient to produce a squeezed state with ξ s < 1. For the thermal state, further squeezing steps are needed for J y and J z . Since J = 0, the generalized uncertainty relations ( J k ) 2 ( J l ) 2 h 2 4 J m 2 do not enforce a measurement back-action. As such, the QND measurements can produce a state with ( J x ) 2 , ( J y ) 2 and ( J z ) 2 all significantly reduced so that ξ 2 s 1.

Lossless case
We now describe the details of our calculations. We employ methods developed for treating the Gaussian dynamics of continuous variable systems [9]- [11], [14], which we adapt to the case of unpolarized ensembles of spin-j particles. First, the QND measurement is modeled in the absence of decoherence, which allows comparison with an exact calculation. Then, we incorporate decoherence due to light scattering [13,44]. 9 Scattering sets the two-body correlations j (m) l j (n) l to zero, where m is one of the spins that went through scattering, while n is one of the non-affected spins. From J 2 l 0, a lower bound on the average two-point correlations is ∼ −1/N . Since ∼ √ N atoms are affected by scattering, the number of two-point correlations set to zero by scattering is ∼N √ N . Thus, the increase of ( J l ) 2 is at most ∼ √ N .

6
For modeling the QND pulse, the atoms are described by the J l operators, while the light pulse is characterized by the Stokes operators S l [13,17]. We choose the initial state to be the completely mixed atomic state, 0 := (1/(2 j + 1) N )1, and a fully polarized optical state with S = (S 0 , 0, 0). The full system is described by the operators with a covariance matrix As shown by simple calculations, for large N the initial state is Gaussian for the R k operators. That is, symmetric moments with order higher than the second can be obtained from lower-order ones according to the theory of Gaussian distributions, knowing that cumulants with order three and higher are zero [41]. In other words, concerning the moments of R k , the state is completely characterized by , S and J . The first step of the QND measurement of J x is interaction between the atoms and light via the Hamiltonian This suggests a characteristic time scale [12] τ : The dynamical equations of mn can be obtained from the Heisenberg equation of motion for the operators R k given as withh = 1. For example, the dynamics of R 5 is obtained as where the coupling constant is defined as κ := t τ . Hence, for the dynamics of the square of R 5 , we obtain where {A, B} + is the anticommutator of A and B. Knowing that due to symmetries of the setup for all times R k = 0 for k = 1, 2, 3, 5 and 6 we obtain for the variance of R 5 where we used the notation R k := R k − R k . Let us now consider dynamics for t τ . Knowing that R 4 = √ S 0 for t = 0 and R k R l R m · · · 1, we can examine how the different terms depend on S 0 . We find that 7 on the right-hand side of (14) the second term is of order (1/S 0 ), the fourth term is of order (1/ √ S 0 ) and the rest of the terms are of order 1. Thus, assuming a large number of photons, that is, a large S 0 , several terms can be neglected and we obtain Long but straightforward calculations show that several terms can also be neglected in the dynamical equation for variances of the other R k variables, assuming large J and S 0 . Hence, for the evolution of the covariance matrix, is obtained, where M is the identity matrix, apart from The evolution of the expectation values is described as (17) Similar analysis shows that the dynamical equations for higher order moments of R k can also be simplified if t τ . When computing the dynamics of these moments, instead of the Heisenberg equation of motion, R (out) can be used. Under this dynamics, which is a linear mapping between operators, the state remains Gaussian. Note, however, that this approximation breaks down for much larger times t ∼ τ √ J . (See also the appendix.) The second step of the QND process is measuring S y of the light. The theory of Gaussian states can be used to describe the measurement, which is modeled with a projection [42] 10 M = P − P (P y P P y ) MP T P .
Here MP denotes the Moore-Penrose pseudoinverse and P y = diag(0, 0, 0, 0, 1, 0). As described above, the QND measurement minimally disturbs the unmeasured components. In the Gaussian approximation the variance of the other spin components remains unchanged. Measuring the other J k components is analogous to the J x case. Note that magnetic fields could be used to rotate the collective spin to facilitate the measurement of the different spin components. Finally, as we have already noted, after each squeezing step, feedback has to be used to restore the zero expectation value of the angular momentum coordinates. We have also mentioned that post-selection can be used in the place of feedback. In this case, a feedback scheme does not have to be realized; however, part of the experiments must be discarded. Thus, when a QND measurement determines J l , only cases with | J l | B are retained. We define  Let us now make the calculations for realistic parameters. We consider N = 10 6 87 Rb atoms with spin j = 1, and for the light field S 0 = 5 × 10 7 . Sequential squeezing of the x, y and z spin components is shown in figure 2. The horizontal axis indicates the total interaction time, with successive intervals of up to 2τ for measurement of J x , J y and J z , respectively. Results are shown for squeezing from a thermal state and also from We obtain ξ 2 s = 0.32 and ξ 2 s = 0.20 for the completely mixed initial atomic state and for (19), respectively. Remarkably, the QND interaction can be solved exactly for the initial state (19), as shown in the appendix. The results are presented in figure 2. In that calculation we find that, for large N , time t ∼ τ × J 0.25 gives ( J x ) 2 ∼ √ J and the two halves of the atoms remain almost fully polarized into the +z and −z directions, respectively. For much longer times, the two halves are not fully polarized any more. In particular, for t ∼ τ 2 := τ × √ J , we obtain ( J x ) 2 ∼ 1, and ξ 2 s = 1 2 . Thus, we have squeezing even in the long-time (von Neumann) limit.

Model including losses
We now incorporate decoherence, following ideas from [13,14,17], adapted to our use of a correlation matrix of all the three spin components. In particular, a parameter η describes the probability that an atom suffers spontaneous excitation due to the off-resonant probe and thus describes the fraction of atoms that decohere during the QND process. For simplicity, we assume that the atoms end up in the completely mixed state, which is usual for handling the effects of noise. We obtain where D = diag(1, 1, 1, 0, 0, 0) and noise = diag(1, 1, 1, 0, 0, 0) × n j j . 11 Here n j is the variance of j x for a spin-j particle in a totally mixed state. The decoherence is connected to κ through where α is the resonant optical depth of the sample and Q = 1 for the spin-1/2 case. For the spin-1 case, Q = 8 9 if the near-resonant intermediate state has j = 0 [45]. Using these techniques, we calculate the degree of squeezing as a function of the time for different, experimentally feasible, values of α [14,15].
The results, shown in figure 2, indicate that considerable squeezing is indeed possible under realistic conditions. For the completely mixed atomic initial state, we obtained ξ 2 s = 0.74, 0.61 and 0.54 for α = 50, 75 and 100, respectively. For comparison, polarized spin states have been squeezed in variance by ∼50% under similar conditions. Thus, our proposal is realistic with present-day technology. The values of α chosen reflect the state-of-the-art for single-pass optical probing [16,44]. The use of an optical cavity could boost the effective α by orders of magnitude [21].

Conclusions
In summary, spin squeezing of unpolarized atomic ensembles by QND measurement shows several intriguing differences relative to polarized samples. The absence of a significant measurement back-action allows simultaneous squeezing of all spin components and approaches a macroscopic singlet state. Spin squeezing of this type implies entanglement of a macroscopic number of particles, for arbitrary spin. To treat this problem, we have extended the Gaussian formalism to include the dynamics of all three spin components and their correlations for an unpolarized ensemble 12 . The advantage of our approach is that it is possible to determine the area of validity for our model, and we can also incorporate von Neumann measurements. In the lossless case, the results agree with an exact calculation. Realistic calculations including decoherence indicate that production of these macroscopic singlet states should be possible with existing ensemble systems. In the future, it will be interesting to look at the possibility of storing quantum information in the decoherence free subspace [47], formed by different singlet states, obtained for j > 1 2 when using different initial states for our squeezing procedure. The method of modeling large spin systems can be generalized to multiple ensembles and a series of light pulses or even for modeling many-particle quantum systems in other areas of physics. a von Neumann measurement of J x . For large systems the summation in (A.5) can be replaced by integration. Using these ideas, we obtain Due to the absolute value sign, the (−1) j 2 term in (A.5) can be neglected for this calculation, thus the integral of a smooth function must be computed numerically for J 2 x . The dynamics of the other two variances can be computed similarly, knowing the matrix elements of M := J 2 y + J 2 z . The change of M during this dynamics is negligible. Note that [J x , M] = 0; thus a von Neumann measurement of J x does not change M .
A.2. The j > 1 2 case Finally, the j > 1 2 case is analogous to the j = 1 2 case, if we note that starting from a product state with N 1 particles in state | j and N 2 particles in state |− j gives the same dynamics for J k,l as starting from a state with 2N 1 j particles in state |+ 1 2 and 2N 2 j particles in state |− 1 2 .