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Physical Review Letters

Localization and anomalous diffusion of a damped quantum particle

Hermann Grabert, Peter Schramm, Gert‐Ludwig Ingold

Published Mar 30, 1987• 64 citations
Publisher

Abstract

The time evolution of an initially localized state of a quantum particle coupled to a dissipative environment with spectral density I(\ensuremath{\omega})\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{\alpha}}}$ for low frequencies is discussed. At finite temperatures, the width of the state can grow subdiffusively or superdiffusively, depending on \ensuremath{\alpha}. For \ensuremath{\alpha}>2 damping becomes ineffective for long times and the state spreads kinematically. At zero temperature the spreading is slower and for \ensuremath{\alpha}<1 an initially localized state remains localized for all times.

DOI
10.1103/physrevlett.58.1285

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The time evolution of an initially localized state of a quantum particle coupled to a dissipative environment with spectral density I(\ensuremath{\omega})\ensuremath{\propto}\ensuremathω\ensuremathα{\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{\alpha}}}\ensuremathω\ensuremathα for low frequencies is discussed. At finite temperatures, the width of the state can grow subdiffusively or superdiffusively, depending on \ensuremath{\alpha}. For \ensuremath{\alpha}>2 damping becomes ineffective for long times and the state spreads kinematically. At zero temperature the spreading is slower and for \ensuremath{\alpha}<1 an initially localized state remains localized for all times.
We report the results related to the two-time dynamics of the coordinate of a quantum free particle, damped through its interaction with a fractal thermal bath (non-Ohmic coupling approximately omega(delta) with 0<delta <1 or 1<delta <2). When the particle is localized, its position does not age. When it undergoes anomalous diffusion, only its displacement may be defined. It is shown to be an aging variable. The finite temperature aging regime is self-similar. It is described by a scaling function of the ratio t(w)/tau of the waiting time to the observation time, as characterized by an exponent directly linked to delta.
The common perception is that strong coupling to the environment will always render the evolution of the system density matrix quasiclassical (in fact, diffusive) in the long time limit. We present here a counterexample, in which a particle makes quantum transitions between the sites of a ddd-dimensional hypercubic lattice while strongly coupled to a bath of two-level systems that ``record'' the transitions. The long-time evolution of an initial wave packet is found to be most unusual: the mean square displacement of the particle density matrix shows long-range ballistic behavior, with $⟨{n}^{2}⟩\ensuremath{\sim}{t}^{2}$, but simultaneously a kind of weakly localized behavior near the origin. This result may have important implications for the design of quantum computing algorithms, since it describes a class of quantum walks.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.
For a free particle, the coupling to its environment can be the relevant mechanism to induce quantum behavior as the temperature is lowered. We study general linear environments with a spectral density proportional to ω^s at low frequencies and consider in particular the specific heat of the free damped particle. For super-Ohmic baths with s>=2, a reentrant classical behavior is found. As the temperature is lowered, the specific heat decreases from the classical value of k_B/2, thereby indicating the appearence of quantum effects. However, the classical value of the specific heat is restored as the temperature approaches zero. This surprising behavior is due to the suppressed density of bath degrees of freedom at low frequencies. For s<2, the specific heat at zero temperature increases linearly with s from -k_B/2 to k_B/2. An Ohmic bath, s=1, is thus very special in the sense that it represents the only case where the specific heat vanishes at zero temperature.
The analysis of a strongly coupled harmonic quantum Brownian motion has been performed in [1] for a special class of spectral densities obtained as a generalization of the Drude model. In the present scenario, we extend the study of the strongly coupled harmonic quantum Brownian motion to regular spectral densities that are structured as sub-Ohmic at low frequencies and arbitrarily shaped at high frequencies. The bosonic environment is initially in the vacuum state unentangled from the coherent state of the main oscillator. As a generalization of the previous results, we obtain that the long time dynamics is determined uniquely by the initial condition and the low frequency structure of the spectral density. Also in the present framework, inverse power law regressions to the asymptotics appear. The position and the momentum tend to undamped oscillations. The number of excitations relaxes to its initial value with damped oscillations enveloped in inverse power law relaxations. For the momentum and the number of excitations the inverse power law decays become arbitrarily slow in critical configurations by approaching the upper bound of the sub-Ohmic regime. The critical frequencies of the main oscillator are determined by the first negative moment of the spectral densities. Differences with respect to the weak coupling regime arise.
Quantum diffusion (QD) is studied via dissipative Madelung hydrodynamics. Initially, the wave packet spreads ballistically, then passes for an instant through normal diffusion and later tends asymptotically to a sub-diffusive law. It is shown that the apparent QD coefficient is not a universal physical parameter because it depends on the initial wave packet preparation. The overdamped QD of an electron in the field of a periodic potential is also investigated; in this case, the wave packet spreads logarithmically in time. Thermo-QD of heavier particles such as hydrogen, deuterium and tritium atoms in periodic potentials is studied and a simple estimate of the tunneling effect is obtained in the framework of a quasi-equilibrium semiclassical approach. The effective thermo-quantum temperature is also discussed in relation to the known temperature dependence of muon diffusivity in solids.