Phase Space Consideration in Quantum Dynamical Fluctuation Expansion

We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024

Quantum dynamics of an electron moving in coupled quartic and coupled double-well oscillators under intense laser fields

Non-Gaussian correlations in a pure state are inextricably linked with certain nonclassical features, such as a non-positive-definite Wigner function. In a commonly used simulation technique in ultracold atoms and quantum optics, known as the truncated Wigner method, the quantum dynamics is mapped to stochastic trajectories in phase space, governed by a positive approximation to the true Wigner distribution. The question thus arises: How accurate is this approach in predicting truly nonclassical behavior? In this article, we benchmark the ability of the truncated Wigner phase-space method to reproduce the non-Gaussian statistics of the single-mode anharmonic oscillator. We find that the this method can reliably predict departures from Gaussian statistics over a wide range of particle numbers, whereas the positive-$P$ representation, which involves no approximations, is limited by rapidly growing statistical uncertainty. The truncated Wigner function, furthermore, is able to reproduce the non-Gaussian correlations while satisfying the condition for purity.

The classical and quantum synchronization between two nonlinear mechanical modes of Bose–Einstein condensates is investigated by different types of measures in order to reveal macroscopic and microscopic properties of synchronized behaviors in a closed quantum system. The classical measure synchronization (CMS) is studied by Pearson correlation coefficient, the orbital overlapping and covering areas in the phase space based on mean-value dynamical equations. The dynamical transitions of CMS are analyzed with phase diagrams in the parametric plane of population imbalance and phase difference between two modes in a wide range of mode coupling rate. Based on Husimi Q functions, the synchronized behaviors of quantum measure synchronization (QMS) are displayed by density overlapping and correlated probability dynamics in phase space, and further investigated by two quantum measures: Mari measure and mutual information. These results demonstrate that the “revival and collapse” of quantum fluctuations beyond mean-value dynamics discriminates QMS from CMS. The overwhelming dynamics of error fluctuations not only excludes complete CMS and perfect phase overlap in QMS, but also leads to upper bound to Mari measure and unceasing oscillations of mutual information. We reveal that the correlation between Mari measure and mutual information for QMS is derived from the similar dynamics of error fluctuations with respect to their opposite mean-value behaviors.

We construct a symplectic realisation of the twisted Poisson structure on the phase space of an electric charge in the background of an arbitrary smooth magnetic monopole density in three dimensions. We use the extended phase space variables to study the classical and quantum dynamics of charged particles in arbitrary magnetic fields by constructing a suitable Hamiltonian that reproduces the Lorentz force law for the physical degrees of freedom. In the source-free case the auxiliary variables can be eliminated via Hamiltonian reduction, while for non-zero monopole densities they are necessary for a consistent formulation and are related to the extra degrees of freedom usually required in the Hamiltonian description of dissipative systems. We obtain new perspectives on the dynamics of dyons and motion in the field of a Dirac monopole, which can be formulated without Dirac strings. We compare our associative phase space formalism with the approach based on nonassociative quantum mechanics, reproducing extended versions of the characteristic translation group three-cocycles and minimal momentum space volumes, and prove that the two approaches are formally equivalent. We also comment on the implications of our symplectic realisation in the dual framework of non-geometric string theory and double field theory.

Electromagnetic modes are instrumental in building quantum machines. In this experiment, we introduce a method to manipulate these modes by effectively controlling their phase space. Preventing access to a single energy level, corresponding to a number of photons N, confined the dynamics of the field to levels 0 to N - 1. Under a resonant drive, the level occupation was found to oscillate in time, similarly to an N-level system. Performing a direct Wigner tomography of the field revealed its nonclassical features, including a Schrödinger cat-like state at half period in the evolution. This fine control of the field in its phase space may enable applications in quantum information and metrology.

In this chapter we begin by examining simple models of electron hopping between bound states located on sites or atoms. In particular, we investigate electron hopping on classical dynamic lattices. Then we discuss models of molecular dynamics of electrons based on effective potential interactions, including models of Wigner dynamics with coordinate- and momentum-dependent interactions in phase space. Finally, we study several molecular dynamics models based on wave packet dynamics. The powerful methods of quantum Monte Carlo simulations will be treated in Chaps. 9 and 10.

The quantum Liouville equation is solved in the Wigner representation using generalized Monte Carlo techniques. For small increments of time, the solution is represented as a sequential classical evolution in phase space followed by a quantum "jump" distribution in momentum space, with the latter simulated via a stochastic method. Extending the work initiated by John and Remler [Ann Phys. (N.Y.) 180, 152 (1987)] the technique is developed and validated for higher dimensions. Also, an alternative algorithm is developed and applied to study motion of a quantum system in an anharmonic quartic potential well, with significantly improved results.

To study the kinetic properties of dense quantum plasma, a new quantum dynamics method in the Wigner representation of quantum mechanics has been developed for extreme conditions, when analytical approximations based on different kinds of perturbation theories cannot be applied. This method combines the Feynman and Wigner formulation of quantum mechanics and uses for calculation the path integral Monte-Carlo (WPIMC) in phase space and quantum generalization of the classical molecular dynamics methods (WMD) allowing to solve the quantum Wigner–Liouville-like equation. The Fermi–Dirac statistical effects are accounted for by the effective pair pseudopotential depending on coordinates and momenta and allowing to avoid the famous “sign problem” due to realization of the Pauli blocking of fermions. Significant influence of the interparticle interaction on the high energy asymptotics of the momentum distribution functions have been observed. According to the quantum Kubo formula, we also study the electron conductivity of dense plasma media.

A geometric approach to quantum control in projective hilbert spaces

A phase-space Poisson-bracket structure induced by the Schrodinger equation is addressed in order to show the existence of a phase space representation of quantum dynamics where action-angle variables can be defined. Quantum dynamics can be embedded into a classical picture of coupled harmonic oscillators in phase space, providing for an internal contextual hidden variable theory. This approach also provides for a natural scenario in order to understand the different manifestations of phase space structures in quantum mechanics as well as to study the ocurrence of chaotic phenomena in the classical limit.

A key feature of gauge theories is the presence of non-physical degrees of freedom whose time evolution is not determined by equations of motion. It seems natural to eliminate such degrees of freedom from the theory and obtain an effective dynamics only for physical variables. In this chapter this problem is studied in detail. There are two ways to achieve this goal, invariant and non-invariant. In the invariant approach, only gauge-invariant variables are used, while the non-invariant route is associated with the use of quantities that are not gauge invariant. However, it only appears non-invariant, similarly to the use of local coordinates on a manifold to compute coordinate-invariant quantities, e.g. curvature or geodesic length. Both approaches describe the very same theory and, therefore, must be equivalent. Unfortunately, there is a shortage of rigorous studies to establish the equivalence of the invariant and non-invariant approaches for general gauge systems. For that reason the discussion is limited to concrete models. The analysis of the physical phase space structure is in the focus of the discussion since this problem has been studied only recently and is not discussed in the monograph literature and textbooks on gauge theories. It appears that for a large class of gauge theories the physical phase space may have a non-trivial structure, i.e. different from the conventional symplectic plane for each physical degree of freedom. This leads to several new phenomena in classical as well as quantum dynamics [183] (see reviews [26, 195]).

We used quantum Fisher information (QFI) to distinguish quantum chaotic and regular dynamics in Dicke model. The QFI, which reflects the geometric properties of a state in parameter space, characterizes the sensitivity of the state with respect to SU(2) rotations. With numerical computation we found that, quantum dynamics in chaotic and regular cases are well distinguished by QFI, especially in the long-time evolution, where the QFI in the chaotic situation is evidently larger than that in regular one, i.e. the chaotic nature enhances the sensitivity of the system, as expected usually. We also plotted the time-averaged QFI in phase space, and found a good quantum-classical correspondence.

We identify a transition to chaos in the semiclassical spin-boson model that occurs for relatively large boson fields, as one of two periodic orbits becomes unstable. We have studied the quantum dynamics in the vicinity of this transition, as characterized by (i) phase-space trajectories followed by quantum expectation values, (ii) spectra of such trajectories, (iii) subsystem entropies for the spin and boson systems, and (iv) growth of operator variances for the boson system. We find that the transition to chaos in the classical system has no apparent effect on any of these variables, in the spin-1/2 case. This is in disagreement with some claims made in earlier studies of this system. \textcopyright{} 1996 The American Physical Society.

Microscopic description of heavy ion collisions