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Acta Physica Polonica A

Non Saturation of Energy in a Quantum Kicked Oscillator

Mark V. Daly, D.M. Heffernan

Published May 1, 1996• 5 citations
Publisher

Abstract

We present a quantum mapping for the kicked harmonic oscillator which relates the probability amplitudes of the undriven oscillator's eigenfunctions over successive kicks.We show how for various kick strengths the wave functions lave a linear energy increase up to the limit imposed b y the finite matrix size of the evolution matrix.We use this linear energy increase to define a quantum diffusion-like coefficient.We also show how this increase in energy causes the wave functions to spread out and become diifuse with fittle or no discernible structure.This model may serve as a paradigm for the study of quantum chaos.

DOI
10.12693/aphyspola.89.571

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We present a quantum mapping for the kicked harmonic oscillator which relates the probability amplitudes of the undriven oscillator's eigenfunctions over successive kicks.We show how for various kick strengths the wave functions lave a linear energy increase up to the limit imposed b y the finite matrix size of the evolution matrix.We use this linear energy increase to define a quantum diffusion-like coefficient.We also show how this increase in energy causes the wave functions to spread out and become diifuse with fittle or no discernible structure.This model may serve as a paradigm for the study of quantum chaos.
We explore the properties of the quantum kicked harmonic oscillation governed by the Hamiltonian . The classical version of this model is known to exhibit certain type of stochastic behavior. We reduce the quantum equation of the model to quantum mapping equations. Numerical results of the mappings show that the behavior of the average energy as a function of time depends on the ration of the frequency of the external and that of the unperturbed system . The energy increases first very rapidly and then the energy growth saturates for irrational . The break time is independent of the strength of the external force, but is roughly proportional to . For a fixed rational the energy is damped oscillation for smaller T, and it is seemingly recurrent with time in the sense of Hogg and Huberman. In both cases, these behaviors of energy are independent of the strength of the external force and differ from the behavior of the quantum kicked rotator.
The evolution of the density matrix is calculated in the Wigner representation for the periodically kicked particle [H=(1/2p2{\mathit{p}}^{2}p2+1/2) Kx2{\mathit{Kx}}^{2}Kx2${\mathrm{\ensuremath{\Sigma}}}_{\mathit{n}}$\ensuremath{\delta}(t-nT)] in the presence of dissipation. The dissipation is introduced via a linear coupling to a bath of harmonic oscillators. The behavior in various regions of the parameter space for the problem is analyzed in detail and compared with the Markovian approximation. The degree of agreement with the Markovian approximation is classified. In various regions in parameter space the model reduces to various examples that were studied in the past (damped particle, harmonic oscillator). Most regions of the parameter space were not explored in the past and are of potential interest. One of these regions exhibits a limit cycle.
The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.
We study the effect of many-body quantum interference on the dynamics of coupled periodically kicked systems whose classical dynamics is chaotic and shows an unbounded energy increase. We specifically focus on an N-coupled kicked rotors model: We find that the interplay of quantumness and interactions dramatically modifies the system dynamics, inducing a transition between energy saturation and unbounded energy increase. We discuss this phenomenon both numerically and analytically through a mapping onto an N-dimensional Anderson model. The thermodynamic limit N→∞, in particular, always shows unbounded energy growth. This dynamical delocalization is genuinely quantum and very different from the classical one: Using a mean-field approximation, we see that the system self-organizes so that the energy per site increases in time as a power law with exponent smaller than 1. This wealth of phenomena is a genuine effect of quantum interference: The classical system for N≥2 always behaves ergodically with an energy per site linearly increasing in time. Our results show that quantum mechanics can deeply alter the regularity or ergodicity properties of a many-body-driven system.
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.