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.PACS numbers: 03.75. Lm, 03.67.Mn, 64.70.Tg, 03.75.Kk 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-MeshkovGlick 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 effect...
Abstract. We present a complete analysis of the dynamics of a Bose-Einstein condensate trapped in a symmetric triple-well potential. Our classical analogue treatment, based on a time-dependent variational method using SU(3) coherent states, includes the parameter dependence analysis of the equilibrium points and their local stability, which is closely related to the condensate collective behaviour. We also consider the effects of off-site interactions, and how these 'cross-collisions' may become relevant for a large number of trapped bosons. Besides, we have shown analytically, by means of a simple basis transformation in the single-particle space, that an integrable sub-regime, known as twin-condensate dynamics, corresponds in the classical phase space to invariant surfaces isomorphic to the unit sphere. However, the quantum dynamics preserves the twin-condensate defining characteristics only partially, thus breaking the invariance of the associated quantum subspace. Moreover, the periodic geometry of the trapping potential allowed us to investigate the dynamics of finite angular momentum collective excitations, which can be suppressed by the emergence of chaos. Finally, using the generalized purity associated to the su(3) algebra, we were able to quantify the dynamical classicality of a quantum evolved system, as compared to the corresponding classical trajectory.
We present a detailed derivation of the semiclassical propagator in the SU(n) coherent state representation. In order to provide support for immediate physical applications, we restrict this work to the fully symmetric irreducible representations, which are suitable for the treatment of bosonic dynamics in n modes, considering systems with conservation of total particle number. The derivation described here can be easily extended to other classes of coherent states, thus representing an alternative approach to previously published methods.Comment: 16 page
We reanalyze the non-linear population dynamics of a Bose-Einstein Condensate (BEC) in a double well trap considering a semiclassical approach based on a time dependent variational principle applied to coherent states associated to SU(2) group. Employing a two-mode local approximation and hard sphere type interaction, we show in the Schwinger's pseudo-spin language the occurrence of a fixed point bifurcation that originates a separatrix of motion on a sphere. This separatrix corresponds to the borderline between two dynamical regimes of Josephson oscillations and mesoscopic self-trapping. We also consider the effects of interaction between particles in different wells, known as cross collisions. Such terms are usually neglected for traps sufficiently far apart, but recently it has been shown that they contribute to the effective tunneling constant with a factor growing linearly with the particle number. This effect changes considerably the effective tunneling of the system for sufficiently large number of trapped atoms, in perfect accord with experimental data. Finally, we identify analytically the transition parameter associated to the bifurcation in the generalized phase space of the model with cross-collision terms, and show how the dynamical regime depends on the initial conditions of the system and the collisional parameters values.
The semiclassical propagator in the representation of SU(n) coherent states is characterized by isolated classical trajectories subjected to boundary conditions in a doubled phase space. In this paper, we recast this expression in terms of an integral over a set of initial-valued trajectories. These trajectories are monitored by a filter that collects only the appropriate contributions to the semiclassical approximation. This framework is suitable for the study of bosonic dynamics in n modes with fixed total number of particles. We exemplify the method for a Bose-Einstein condensate trapped in a triple-well potential, providing a detailed discussion on the accuracy and efficiency of the procedure.
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