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Phase Transition, Entanglement and Squeezing in a Triple-Well Condensate

We provide an in-depth characterization of a three modes Bose-Einstein condensate trapped in a symmetric circular triple well potential. We analyze how a subsystem independent measure of entanglement, the purity related to the su(3) algebra, scales for increasing number of atoms and signals correctly the quantum phase transition between two dynamical regimes in a specific arrangement. Moreover, this measure, which is intrinsically related to particle entanglement, also depicts if some squeezing is occurring when we consider the system's ground state.

Entanglement has played an important role for the understanding of quantum many body aspects [1] that traditionally belonged to statistical mechanics and quantum field theory. Several investigations in quantum critical models at T = 0 have shown that complex entangled ground state contains all the important correlations that give rise to different phases known to exist in several systems [2]. Thus, it is a fact that entanglement study in many body systems allows a deeper characterization of the ground state of the system undergoing a quantum phase transition (QPT), particularly its order. Characterization of a QPT via pairwise and collective multipartite entanglement has been given in a very conclusive way in Refs. [3,4], but are dependent on the specific partition employed, i.e., they are subsystem dependent. Another way of investigating subsystem independent entanglement in many-particle systems is through the generalized purity associated to the pertinent algebra [5]. Beyond being a measure for the quality of semiclassical approximation [6], this measure is also related to squeezing of moments of the generators of the pertinent algebra. Recently both spin squeezing and entanglement have been demonstrated for a 87 Rb condensate trapped in double and multiple wells of an optical lattice [7], and subsystem independent entanglement theoretically investigated in Ref. [6] for a double-well trapped condensate. In fact although those results were developed independently they are profoundly complementary since they relate QPT, entanglement and squeezing, for a system which is a particular realization of the Lipkin-Meshkov-Glick model [8,9]. The interplay between entanglement and squeezing has been investigated previously in many instances [10], and now seems to play an important role in QPT involving many bosons as well.
In this Letter we investigate in detail a BEC of attractively interacting neutral atoms trapped in a symmetric triple well potential in a three mode approximation and show that the ground state of the model undergoes a QPT. A time dependent variational principle using SU (3) coherent state allows for a system of semiclassical equations that enables one to find the fixed points of the model and to investigate how the lowest energy fixed points change as the collision parameters of the model are varied. Since the lowest energy state in this system corresponds to a twin condensate fixed point, where effectively the system behaves as if composed of two wells (although coherences between the twin modes are still present) we can show that the SU (3) reduces to SU (2) while in this regime. The non-linear components of the Hamiltonian provided by the interactions of our model lead to spin squeezing for the ground state as we vary the scattering parameter, which on its turn is associated to variance squeezing of some of the generators of the algebra, and shows non-trivial phenomena such as particle entanglement related to QPT and to distinct dynamical regimes. This special arrangement is promising for experimental investigation with existent trapping technology.
The model of a Bose-Einstein condensate trapped in a triple well potential has already been considered in literature in many distinct configurations [11,12]. Considering that the coupling between the potential wells is sufficiently weak, we shall use the usual local modes approximation [13] associated to the states |u j , j = 1, 2, 3, representing the ground states of harmonic approximations around each minimum of the trap. Considering this approximation to be valid even when the total number of trapped bosons N is large, the overlap ε ≡ u i |u j , (i = j) must be very small. Keeping terms up to O(ε 3 2 ), and using the fact that the total number of trapped particles N is conserved, we can write the Hamiltonian: where Ω = Ω + 2Λ(N − 1), and the operator a j (a † j ), j = 1, 2, 3, annihilates (creates) a boson in the state |u j . In the last equation we also defined the parameters of the system: Ω is the tunneling rate, Ω is the effective tunneling rate, κ is the self-collision parameter and Λ = κε 3 2 κ is the cross-collision rate, which is proportional to the interaction frequency between bosons in different sites. Although Λ is a lower order parameter of the model, we see that its presence in the effective tunneling rate Ω ≡ Ω + 2Λ(N − 1) is relevant for N 1. Since the Hamiltonian (1) preserves N , we can take advantage of the homomorphism between the commu-arXiv:1002.0557v1 [quant-ph] 2 Feb 2010 tation relations of SU (3) group generators and bilinear combinations of creation and annihilation operators in the three local modes, making use of an extension of Schwinger's pseudo-spin method [14]: for k = 1, 2, 3 and j = (k + 1) mod 3 + 1. Using these eight generators we can rewrite the Hamiltonian: showing that SU (3) is the pertinent dynamical group. Note that the tunneling term is linear in the generators, while the collision terms are quadratic.
A semiclassical dynamics of the condensate can be developed by using the time dependent variational principle [15] with coherent states of the group SU (3) [16] as test functions, allowing us to express the semiclassical Hamiltonian as H( w * , w) ≡ N ; w|H|N ; w , with the coherent states expanded on the Fock space basis for the three local modes as where C = 1/(|w 1 | 2 + |w 2 | 2 + 1) N 2 . The vector w = (w 1 , w 2 ) ∈ C 2 parametrizes the non-linear subspace of Hilbert space consisting only of coherent states and also represents a point in phase space associated with the classical analogue system [17]. The semiclassical equations of motion are then brought to the Hamilton's canonical form by the transformation of variables w j = I j /(N − I 1 − I 2 )e −iφj ; j = 1, 2. While I j is the average occupancy in the j-th local mode, φ j represents the difference in phase between the macroscopic condensates located in the j-th and the third wells. The semiclassical approximation becomes exact if the Hamiltonian is linear in the generators of the group or in the classical limit of the system, which coincides with the macroscopic limit N → ∞ [18]. The phase space position and the stability of the equilibrium points of the equations of motion depend only on the model parameters χ ≡ κ(N −1) [19]). As previously noticed in Ref. [12] there are fixed point solutions corresponding to configurations where two localized condensates are in phase and have the same occupational average, which are described by the three equivalent conditions w 1 = w 2 , w 1 = 1 or w 2 = 1, known as restrictions of twin condensates. Many of the fixed points of the semiclassical dynamics are contained in this subregime under the additional condition of φ 1 , φ 2 = 0, π, which implies that w ∈ R 2 . Without loss of generality, we choose only the condition w = w 1 = w 2 ∈ R, so that the SU (3) coherent states reduce to coherent states of SU (2), which thus brings many features observed for a two mode condensate, such as the Rabi Oscillation (RO) of population and the macroscopic self-trapping (MST) of population [20]. The phase space associated with the integrable sub-regime of twin condensates is isomorphic to S 2 , a space that parametrizes the set of SU (2) coherent states An interesting quantity for the dynamical analysis in the twin condensates subregime is the population balance I z ≡ − cos θ = 4I1−N N . The point w 1 = w 2 = 1 is a solution of the fixed point equations, independent of the values of χ and µ, which we call 1 + . Considering always µ χ, we have another solution if χ < χ + (µ), which we call 2 + ; whereas if χ > χ + (µ) two further solutions of the fixed point equations exists. In other words, a saddle-node bifurcation with critical parameter χ + (µ) is responsible for the appearance of the fixed points 3 + and 4 + . Fig. 1.a shows the semiclassical dynamics of twin condensates in the absence of the fixed points 3 + and 4 + . We see that all trajectories are around the fixed point 1 + and therefore vary around the value I z = 1 3 , which represents same average occupancy in the three local modes. This is exactly the RO dynamical regime, where the system does not show any preferential mode occupation. In figure 1.b, where χ exceeds the critical value of bifurcation, we observe the emergence of new types of orbits in the system, especially around the new stable equilibrium point 4 + . These new trajectories oscillate around negative values of I z and show a suppression of tunneling between the twin condensates and the solitary one, resulting in the preferential occupation of the third mode. Such a dynamical effect is known as macroscopic self-trapping (MST), similar to the case of two modes [13,21]. As the value of χ increases, the region of phase space occupied by orbits associated with the MST regime also grows larger, as we see in figure 1.c.
A quantitative error in the semiclassical approximation is originated when we force the state to evolve preserving minimal uncertainty on the space phase, i.e., obligating the system to remain as a coherent state. If the quantum dynamics of the system leads to an increase of the state uncertainty, we cannot expect a high quantitative accuracy in the semiclassical treatment. Including the situation where strong squeezing or spreading over the phase space occurs, leading thus to a large deviation of the coherent state. An effective measure of the quality of the semiclassical approximation is the generalized purity associated with the su(3) algebra [5], defined by This measure is derived from the total uncertainty of the algebra [22] and has maximum value P su(3) (|ψ ) = 1 only if |ψ is a coherent state of SU (3), corresponding to the state of minimum total uncertainty. In contrast, the purity presents a minimum value P su(3) (|ψ ) = 0 for states of largest uncertainty. Therefore, P su(3) can be used to measure the "distance" of a particular state to the subspace formed only by the coherent states. That is, the larger the uncertainty of a state during its evolution, the lower is P su(3) and less accurate is the semiclassical approximation using only coherent states [6].
In addition the generalized purity is also a genuine measure of separability in systems of many identical particles. Under a basis transformation of the single-particle Hilbert space, every coherent state of the type (4) can be rewritten as a separable state in each boson: where |ψ w ≡ w1|u1 +w2|u2 +|u3 (|w1| 2 +|w2| 2 +1) 1 2 is an arbitrary state of the single-particle Hilbert space [25] with associated annihilation operator a w . Due to the symmetrization principle, we see that (6) is the only possibility of a completely separable state in each particle, considering systems of many identical bosons. Therefore, the states with P su(3) (|ψ ) = 1 represent the separable states of N bosons, while the decrease of purity from this maximum value indicates the entanglement among the particles of the system. In this sense when the state of the system is less localized in the phase space, P su(3) (|ψ ) < 1, it is more entangled. Several dynamical processes can be responsible for the loss of purity of an initially coherent state. One of such processes is the squeezing of the state, and it is related to regular dynamical regimes of the semiclassical model [20]. In the semiclassical approximation we also observe chaos [19] in the three mode model, accompanied by other processes of purity loss. The generalized purity can also be written as the trace of a squared reduced density operator, however this reduction is done with respect to the algebra. Considering ρ = |ψ ψ|, we have P su(3) (|ψ ) = KTr(ρ 2 su(3) ), where K is a normalization factor dependent on N , ρ su(3) = j Tr(ρA j )A j is the density operator reduced in su(3) and {A j } is a basis of su(3) orthonormalized in relation to the trace in the Fock space of N particles. Therefore, the entanglement can be described as the decoherence of the system in its dynamical algebra. Summarizing, pure (mixed) states in the algebra are equivalent to separable (entangled) states in each boson. The semiclassical approximation is accurate only if the state remains separable in each boson, when we can completely describe the system by only one trajectory in the phase space that extremizes an action functional. Then, the entanglement is the fundamental quantum feature responsible for breaking-off the classicality of the system, and it signals QPT as we discuss now.
QPT results from non-analyticities in the ground state energy as a function of a real parameter of the Hamiltonian, characterized only in the macroscopic limit N → ∞ and at zero temperature. However, as stressed earlier, the macroscopic limit of the model is equivalent to its classical limit, so that the minimum of the semiclassical energy per particle H N , which is independent of N , can be used exactly to study the QPT. The extremes of H are the equilibrium points of the model. In Fig. 2 we show H calculated in each of its extremes as a function of χ, with N = 30, Ω = −1 and µ = 0. The occurrence of a first order QPT is evident at the critical value χ c = 2, since there is a discontinuity of the first derivative of the ground state energy, resulting from the energy level crossing of the fixed points 1 + and 4 + . As already shown, the fixed point 1 + is the center of the orbits related to the RO dynamical regime, charac-terized by no preferential occupation of the local modes. The three equivalent equilibrium points 4 + , one in each sub-regime of twin condensates, are responsible for the emergence of semiclassical trajectories associated with the MST regime in the solitary mode. Therefore, we expect the ground state to display characteristics of the RO (MST) regime for χ < χ c (χ > χ c ). Furthermore, the behavior of the low energy equilibrium points suggests a non-degenerate (triply degenerate) ground level when the self-collision parameter is lower (higher) than χ c in the limit N → ∞. Phase e −in1φ1−in2φ2 |n 1 , n 2 , n 3 , represents the probability distribution of collective phase differences [23] between the local condensates. Considering χ = µ = 0 in Fig. 3.a, the exact ground state is the coherent state |N ; w 1 = w 2 = 1 , centered at the equilibrium point 1 + , showing the non-preferential occupation of the local modes. For increasing value of χ in Fig. 3.b, we observe the state expansion in the population subspace, accompanied by squeezing in the conjugate subspace, related thus to phase squeezing. Note that the coherent states are the most localized states in the phase space, but not necessarily in a subspace. When χ = χ c there is no abrupt change in the representations of the ground state, since such changes should happen in a continuous way for finite N . When the self-collision parameter takes values slightly larger than χ c we notice a profound change in the ground state, shown in Fig. 3.c as a trifurcation of the occupational distribution, each of the components is more squeezed than the original coherent state. This is a accompanied by the emergence of an interference pattern in the conjugate subspace. This fragmentation of Φ represents the superposition in states of preferential occupancy in each local mode, also visible in the behavior of Q I . Notice that the ground state representations in phase space display a smooth transition from RO to MST regime, in contrast to the abrupt transition shown in the macroscopic limit. Now we return to the discussion of particle entanglement, as given by the generalized purity, and the signaling of the QPT [3]. In figure 4 we show dP su (3) dχ calculated at the ground state as a function of χ, considering several values of N . In the absence of bosonic collisions the ground state is a coherent state and consequently its purity value is equal to one. For χ < χ c , the region related to the expansion-compression of the ground state representations, we see that P su(3) decreases slowly with increasing χ. However, for values of χ slightly above χ c we observe a rapid decay of P su(3) caused by the ground state fragmentation associated with the tunneling suppression. The loss of purity is faster for larger number of trapped bosons, suggesting a scaling property between dP su (3) dχ and N . We define the scalable quantum critical parameter χ q c (N ) as the self-collision parameter value that minimizes dP su (3) dχ for a specific number of particles. We observe that the minimum value of  Although the critical bifurcation parameter χ + is distinct from the critical transition parameter χ c , the emergence of new fixed points in the semiclassical dynamics shows a clear relation with the occurrence of the QPT. This same characteristic has been observed in the two local modes model [24], but with some important differences. The two-mode model presents a second order QPT, accompanied by a pitchfork bifurcation with critical parameter equal to the critical parameter [20], in the absence of cross-collisions. However, in both models the equilibrium point that minimizes the semiclassical energy H can be described by a single angle θ on S 2 , considered as a function of χ. For example, the restriction w 1 = w 2 ∈ R determines the unidimensional manifold of the phase space that contains the minimum energy fixed point. Therefore, we generically consider H restricted to only one dimension in the analysis of the QPT: H(θ; χ) = H 0 (θ) + χH 1 (θ); where H 0 (H 1 ) is the portion of the semiclassical Hamiltonian obtained from the linear (quadratic) terms of H in the generators of the dynamical group. The posi- tion of the semiclassical ground state θ min ≡ θ min (χ), which satisfies the necessary conditions ∂H ∂θ θ=θmin = 0 and ∂ 2 H ∂θ 2 θ=θmin > 0, and the minimum energy curve H min (χ) ≡ H(θ min (χ), χ) are shown in figure 5 for the two and three-mode models [27]. The function θ min (χ) possesses discontinuous first derivative at χ T M c for the two-mode model, while for the three-mode model we observe a discontinuity in the curve of θ min (χ) itself at the critical value χ c . The different types of bifurcation are responsible for the distinct behavior of the curve θ min (χ) in both models, resulting also in different orders of transition. The first derivative of H min with respect to the parameter χ is given by: Therefore, the continuity of θ min implies that dHmin dχ is also continuous, since H 1 (θ min ) is a continuous function of θ min . In general, the reciprocal is also true, because H 1 (θ min ) is discontinuous in the case of θ min also discontinuous, resulting in a first order QPT. The discontinuity in the derivative of θ min can be identified only in the second derivative of H min : So, in general, the discontinuity of dθmin dχ implies discontinuity only for the second derivative of H min , resulting in a second-order QPT.
In conclusion we have described the dynamics of a BEC trapped in a symmetric triple well potential in a three mode approximation and show in details the way in that the ground state of the model undergoes a QPT. A time dependent variational principle using SU (3) coherent state allows for a system of semiclassical equations that enables one to find the fixed points of the model and to investigate how the lowest energy fixed points change as the collision parameters of the model are varied. Thus particle entanglement signals correctly the presence of QPT as one crosses distinct dynamical regimes. We have shown that the ground state of the model is related to the subdynamics of the twin condensate, a regime whose dynamics falls into the SU (2) group. We also show that the increase in the entanglement of the ground state is associated to phase compression (Rabi oscillation) and to number compression (self-trapping). In this way we relate the loss of coherence in the fundamental state to "spin squeezing". Remark that the squeezing is not the only effect that leads to the decrease of the generalized purity. Whenever there is a departure from a coherent state the purity decreases, but this behavior unambigu-ously describe particle entanglement in the system. The more entangled are the particles the smaller is the purity associated to the pertinent algebra. We remark that this three symmetric well model allows for a better characterization of QPT than the case for double well, providing not only the dynamical transitions but signaling the presence of squeezing and thus could arouse experimental investigation with existent trapping technology for this phenomenon. This amount of squeezing and entanglement, if controllable, could provide a new scenario for testing quantum physics fundamental questions.
We acknowledge Markus Oberthaler for many clarifying discussions and valuable comments on our results. This work was partially supported by FAPESP and CNPq through the Brazilian National Institute of Science and Technology on Quantum Information.