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arXiv:2109.03136 · Journal of Physics Condensed Matter

The quantum dynamics of two-component Bose–Einstein condensate: an S p ( 4 , R ) symmetry approach

Chang-Yan Wang, Yan He

Published Sep 2, 2022• 4 citations
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Abstract

) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.

DOI
10.1088/1361-648x/ac8f0c

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) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.
An Sp(4,R)Sp(4, R)Sp(4,R) geometrization on the quantum dynamics of two-component Bose-Einstein condensate
The method of geometrization arises as an important tool in understanding the entanglement of quantum fields and the behavior of the many-body system. The symplectic structure of the boson operators provide a natural way to geometrize the quantum dynamics of the bosonic systems of quadratic Hamiltonians, by recognizing that the time evolution operator corresponds to a real symplectic matrix in Sp(4,R)Sp(4,R)Sp(4,R) group. We apply this geometrization scheme to study the quantum dynamics of the spinor Bose-Einstein condensate systems, demonstrating that the quantum dynamics of this system can be represented by trajectories in a six dimensional manifold. It is found that the trajectory is quasi-periodic for coupled bosons. The expectation value of the observables can also be naturally calculated through this approach.
Motivated by recent advances in quantum dynamics, we investigate the dynamics of the system with SU(1,1)\text{SU}(1,1)SU(1,1) symmetry. Instead of performing the time-ordered integral for the evolution operator of the time-dependent Hamiltonian, we show that the time evolution operator can be expressed as an SU(1,1)\text{SU}(1,1)SU(1,1) group element. Since the SU(1,1)\text{SU}(1,1)SU(1,1) group describes the ``rotation'' on a hyperbolic surface, the dynamics can be visualized on a Poincar\'e disk, a stereographic projection of the upper hyperboloid. As an example, we present the trajectory of the revival of Bose-Einstein condensation and that of the scale-invariant Fermi gas on the Poincar\'e disk. Further considering quantum gases in an oscillating lattice, we also study the dynamics of the system with time-dependent single-particle dispersion.
Summary form only given. We present a quantum and semiclassical analysis of the dynamics of three coupled atomic Bose-Einstein condensates (BEC). The quantum description is most conveniently analyzed in terms of the operator algebra of SU(3). The space of pure states corresponds to a real manifold of five dimensions corresponding to the parameters of the coset SU(3)/U(2). This is the higher dimensional analogue of the Bloch sphere which has been used to describe the dynamics of two coupled BECs. The equations of motion in the Heisenberg picture can be mapped onto equations in this real manifold. The semiclassical description also involves five real parameters, and corresponds to the mean values of the Heisenberg equations with a factorisation assumption for higher order moments. The symmetries of the problem may be used to construct the ground states of the quantum system for both positive and negative scattering lengths. The structure of the ground states change as the elastic scattering strength changes with respect to the linear tunnelling coupling between the condensate components.
The semiclassical evolution of the generalized SU (2) Wigner function for quantum systems with a variable number of excitations is discussed in detail. It is shown that in the framework of the Liouvillian approach quantum dynamics can be approximately described in terms of classical trajectories on an appropriate four-dimensional symplectic manifold. The analysis of this manifold allows the introduction of a phase-like observable that can be interpreted as a relative phase ‘between’ invariant subspaces.
The dynamics of two-component Bose-Einstein condensates with coupling drive is studied using a pair of bosonic operators. We demonstrate that the effect of the coupling drive on the evolution of the relative phase of the two-component Bose-Einstein condensates depends on the initial relative phase difference between the two condensates. In some special initial phase differences $(\ensuremath{\varphi}=0$ and $\ensuremath{\pi})$ the condensate density are not sinusoidal functions of time, and the interference pattern is not steady either. The coupling drive also reduces the amplitude and the period of the oscillation. Our theoretical result confirms the experimental observations reported in Phys. Rev. Lett. 81, 1543 (1998).
A real-valued symplectic group underlies the dynamics of quantum-mechanical systems. Exploring the embedding of unitary groups clarifies the relation of non-Hermitian - and -symmetric quantum theories of recent interest. Symmetries of the full dynamical framework are quite rich and reveal new viewpoints on many topics in quantum theory. Transformations mixing 'upper' and 'lower' components of certain symplectic multiplets are indistinguishable from coupling antimatter degrees of freedom. Quantities long identified with physical observables are valid canonical coordinates of the theory, without needing support from measurement doctrine. Dirac's canonical quantization is derived from consistency, and would be redundant as a new postulate. A second-order dynamical framework exists in which observables are just the same as the underlying degrees of freedom.