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East European Journal of Physics

Effective Semiclassical Evolution of Bose Einstein Condensates

Héctor H. Hernández

Published Mar 5, 2024• 1 citation
Publisher

Abstract

In this work we analyze the effective evolution of a one dimensional Bose-Einstein Condensate (BEC) within a semiclassical description of quantum systems based on expectation values of quantum dispersions and physical observables, known as momentous quantum mechanics. We show that the most prominent features and physical parameters of the system can be determined from the dynamics of the corresponding semiclassical system, consisting of an extended phase space including original classical observables and quantum dispersions, and we also show that particle trajectories for expectation values of observables are a particular characteristic in this framework. We also demonstrate that interactions with several potentials can be implemented in an intuitive way.

DOI
10.26565/2312-4334-2024-1-11

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In this work we analyze the effective evolution of a one dimensional Bose-Einstein Condensate (BEC) within a semiclassical description of quantum systems based on expectation values of quantum dispersions and physical observables, known as momentous quantum mechanics. We show that the most prominent features and physical parameters of the system can be determined from the dynamics of the corresponding semiclassical system, consisting of an extended phase space including original classical observables and quantum dispersions, and we also show that particle trajectories for expectation values of observables are a particular characteristic in this framework. We also demonstrate that interactions with several potentials can be implemented in an intuitive way.
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We analyze the problem of one dimensional quantum particle falling in a constant gravitational field, also known as the {\it bouncing ball}, employing a semiclassical approach known as momentous effective quantum mechanics. In this formalism the quantum evolution is described through a dynamical system of infinite dimension for the position, the momentum and all dispersions. Usually the system is truncated to have a finite dimensional one, however, in this case equations of motion decouple and the system can be solved. For a specific set of initial conditions we find that the time dependent dispersion in position is always around the classical trajectory.
We study the Hydrogen atom as a quantum mechanical system with a Coulomb like potential, with a semiclassical approach based on an effective description of quantum mechanics. This treatment allows us to describe the quantum state of the system as a system of infinite many classical equations for expectation values of configuration variables, their moments and quantum dispersions. It also provides a semiclassical description of the orbits and the evolution of observables and spreadings and their back-reaction on the evolution.
After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.
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.
We present an approach for the analysis of Bose-Einstein condensates in a few mode approximation. This method has already been used to successfully analyze the vibrational modes in various molecular systems and offers a perspective on the dynamics in many particle bosonic systems. We discuss a system consisting of a Bose-Einstein condensate in a triple well potential. Such systems correspond to classical Hamiltonian systems with three degrees of freedom. The semiclassical approach allows a simple visualization of the eigenstates of the quantum system referring to the underlying classical dynamics. From this classification we can read off the dynamical properties of the eigenstates such as particle exchange between the wells and entanglement without further calculations. In addition, this approach offers insights into the validity of the mean-field description of the many particle system by the Gross-Pitaevskii equation, since we make use of exactly this correspondence in our semiclassical analysis. We choose a three mode system in order to visualize it easily and, moreover, to have a sufficiently interesting structure, although the method can also be extended to higher dimensional systems.