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Chaos An Interdisciplinary Journal of Nonlinear Science

On the phase-space catastrophes in dynamics of the quantum particle in an optical lattice potential

M. Ćosić, Srdjan Petrović, Stefano Bellucci A

Published Oct 1, 2020• 8 citations
Publisher

Abstract

We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrödinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.

DOI
10.1063/1.5140528

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We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrödinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.
We compare the dynamics of quantum wave packets with the dynamics of classical trajectory ensembles. The wave packets are Gaussian with expectation values of position and momenta which centers them in phase space. The classical trajectory ensembles are generated directly from the quantum wave packets via the Wigner transform. Quantum and classical dynamics are then compared using several quantum measures and the analogous classical ones derived from the Wigner equivalent formalism. Comparisons are made for several model potentials and it is found that there is generally excellent classical–quantum correspondence except for certain specific cases of tunneling and interference. In general, this correspondence is also very good in regions of phase space where there is classical chaos.
Abstract As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following a quench, focusing on two-mode systems that include the two-site Bose–Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit ( N → ∞ ) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex–antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench is a δ -kick the problem can be solved analytically and we obtain scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetry-breaking dynamical phase transition.
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.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
In this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon–Heiles and Barbanis systems. To explore classical–quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincaré maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity ‖dE/dε‖ of that energy to the strength ε of the nonlinear coupling. States with extreme values of ‖dE/dε‖, corresponding to both high and to low values of ‖dE/dε‖, show regular regions in the quantum phase space, while states with intermediate values of ‖dE/dε‖ span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy Ec for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics. Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.
The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.