The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincar\'e recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet.
The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincar'e recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet.
We formulate continuous time quantum walks (CTQW) in a discrete quantum mechanical phase space. We define and calculate the Wigner function (WF) and its marginal distributions for CTQWs on circles of arbitrary length N. The WF of the CTQW shows characteristic features in phase space. Revivals of the probability distributions found for continuous and for discrete quantum carpets do manifest themselves as characteristic patterns in phase space.
An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincare recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.
The Poincaré recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.
Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.
The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.
This work concerns the finding of the semi-classical form of the coherent state representation for the class of quantum baker’s maps defined by Schack and Caves. It begins by introducing the finite-dimensional Hilbert space on which the quantum baker’s map is defined. Its pertinent features including the all important symmetry operators are introduced and given a full explanation. We also introduce the finite-dimensional phase space which will give the semi-classical limit a geometrical interpretation. For a D dimensional Hilbert space, the finite-dimensional phase space is found to be a grid with D2 points. Each point corresponds to a particular pair of position and momentum displacement operator eigenphases. We then detail the derivation of the finite-dimension version of the Wigner function, a quasi-distribution for the finite-dimensional phase space. We show that its most “irregular” feature, mainly its property of having more values than was thought necessary, can be explained by its correct behavior under the symmetry operations, a feature lacking in other Wigner candidates. However, even this special choice for the Wigner function proves unusable in the semi-classical limit as it is found to have a non-convergent limit. We then turn to another possible phase space function: the Q-function. It being necessary to find a suitable coherent state for the finite-dimensional Hilbert space, we begin by studying the properties of the periodically continued Gaussian states. These are the typical Weyl coherent states made periodic in both position and momentum such as to make them legitimate finite-dimensional states. Developing certain mathematical techniques allows us to show that they have compatible position and momentum representations, that a subset of them are complete and can be used to define a Q-function, and that this function obeys all of the symmetry properties. Finally, we use these coherent states to find a representation for the propagator of the quantum baker’s map. In the semi-classical limit, i.e. the large dimension limit, this representation is found, for most of the maps, to take a form of the exponentiation of the classical map’s generating function. This form was predicted long ago by Van Vleck as indicator of an operator’s classical limit. Therefore, we assert that these maps limit to the classically chaotic baker’s map. In certain limiting schemes, however, the Schack-Caves maps do not reach this form and must be given a different interpretation.