In this thesis, we study phase-space approximations to quantum dynamics for both Hermitian and non-Hermitian quantum systems, motivated by the need to efficiently describe quantum effects in models that are analytically intractable but admit classical analogues offering insight into their complex behaviour. Such systems arise in diverse areas, include but are not limited to quantum optics, condensed matter, open quantum systems, and quantum technologies. We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and
PT-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques. To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach. We then apply this framework to both the
su2 algebra and non-Lie algebraic bosonic conversion systems. For
su2, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of
PT-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds. Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.
The localization transition in the Hermitian Aubry-André model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-André case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.
In this thesis, we study semiclassical phase-space methods for quantum evolution in Hermitian and non-Hermitian systems. We first present the dynamics of Gaussian wave packets under non-Hermitian Hamiltonians and interpret them as a classical dynamics for complex Hamiltonians. We use these to derive exact dynamics for wave packets in the quadratic Swanson oscillator. We show that in the case of unbroken PT-symmetry there can be periodic divergences in this system and relate this to the fact that any operator mapping the system to a Hermitian counterpart is unbounded. We apply the semiclassical wave-packet dynamics to two further anharmonic example systems: a PT-symmetric wave guide, a version of which we propose as a filtering device for optical beams, and a non-Hermitian single-band tight-binding model, for which we use classical equations of motion to model both narrowly and widely distributed initial states. We further develop an exact quantum propagator for non-Hermitian dynamics using lattices of wave packets, whose evolution is governed by the semiclassical equations of motion. We demonstrate that this accurately reproduces quantum dynamics compared to the split operator method. Finally we study a Hermitian two-mode many-particle model for bosonic atom-molecule conversion, for which the classical phase-space structure is an orbifold. We show that standard semiclassical tools may be applied to recover features such as the dynamics, spectrum and density of states of the many-particle system.
In this note we consider, as in (1), finding an oscillator quantum state |z> (called the coherent state) such that <z| exp(-Hamiltonian/T |z> yields a classical-like phase space probability distribution in p and x. In (1) the approach is to use the known weights of <z|n> where Hamiltonian |n> = en |n>. Here we use instead the following ideas. First, we argue that for exp(-Hamiltonian/T) to be used at all, P(e1)P(e2)=P(e1+e2), but also P(z1*z1)P(z2*z2) = P(z1*z1+z2*z2) where z*z = pp/2+xx/2 with k=m=1. This yields a solution of exp(- b z*z) where b is unknown. In the limit of hbar w/T <<1, i.e. the classical limit, P(z*z) should become the Maxwell-Boltzmann factor exp(- (pp/2T + xx/2T). Writing |z> as a linear superposition of the energy eigenfunctions |n> one may find a solution for |z>, the coherent state which has an average of (pp/2+xx/2)/ hbar w phonons/photons. P(z*z) is called the Husimi distribution.
We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach.The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems.We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position.Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space.We present analytical solutions in L 2 (R) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.