We investigate thermalization in finite quantum systems using coherent state-based approaches to solve the time-dependent Schr\'odinger equation. Earlier, a lot of work has been done in the quantum realm, to study thermalization in spin systems, but not for the case of continuous systems. Here, we focus on continuous systems. We study the zero temperature thermalization i.e., we consider the ground states of the bath oscillators (environment).
In order to study the quantum dynamics of a system under investigation, we require numerical methods to solve the time-dependent Schr\'odinger equation. We describe different numerical methods like the split-operator fast fourier transform, coupled coherent states, static grid of coherent states, semiclassical Herman-Kluk propagator and the linearized semiclassical initial value representation to study the quantum dynamics. We also give a comprehensive comparison of the most widely used coherent state based methods. Starting from the fully variational coherent states method, after a first approximation, the coupled coherent states method can be derived, whereas an additional approximation leads to the semiclassical Herman-Kluk method. We numerically compare the different methods with another one, based on a static rectangular grid of coherent states, by applying all of them to the revival dynamics in a one-dimensional Morse oscillator, with a special focus on the number of basis states (for the coupled coherent states and Herman-Kluk methods the number of classical trajectories) needed for convergence.
We also extend the Husimi (coherent state) based version of linearized semiclassical theories for the calculation of correlation functions to the case of survival probabilities. This is a case that could be dealt with before only by use of the Wigner version of linearized semiclassical theory. Numerical comparisons of the Husimi and the Wigner case with full quantum results as well as with full semiclassical ones is given for the revival dynamics in a Morse oscillator with and without coupling to an additional harmonic degree of freedom. From this, we see the quantum to classical transition of the system dynamics due to the coupling to the environment (bath harmonic oscillator), which then can lead ultimately to our final goal of thermalization for long-time dynamics. In regard to thermalization in quantum systems, we address the following questions--- is it enough to increase the interaction strength between the different degrees of freedom in order to fully develop chaos which is the classical prerequisite for thermalization, or if, in addition, the number of those degrees of freedom has to be increased (possibly all the way to the thermodynamic limit) in order to observe thermalization.
We study the ``toppling pencil'' model, i.e., an excited initial state on top of the barrier of a symmetric quartic double well to investigate thermalization. We apply the method of coupled coherent states to study the long-time dynamics of this system. We investigate if the coupling of the central quartic double well to a finite, environmental bath of harmonic oscillators in their ground states will let the central system evolve towards its uncoupled ground state. This amounts to thermalization i.e., a cooling down to the bath ``temperature'' (strictly only defined in the thermodynamic limit) of the central system.
It is shown that thermalization can be achieved in finite quantum system with continuous variables using coherent state-based methods to solve the time-dependent Schr\'odinger equation. Also, here we witness thermalization by coupling the system to a bath of only few oscillators (less than ten), which until now has been seen for more than ten to twenty bath oscillators
