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Journal de physique

Quantum dynamics of a damped free particle

Claude Aslangul, No�lle Pottier, Daniel Saint-James

Published Jan 1, 1987• 14 citations
Publisher

Abstract

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.

DOI
10.1051/jphys:0198700480110187100

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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.
We consider a free particle coupled with finite strength to a bath and investigate the evaluation of its specific heat. A harmonic oscillator bath of Drude type with cutoff frequency ω D is employed to model an ohmic friction force with dissipation strength γ. Two scenarios for obtaining specific heat are presented. The first one uses the measurement of the kinetic energy of the free particle, while the second one is based on the reduced partition function. Both descriptions yield results which are consistent with the Third Law of thermodynamics. Nevertheless, the two methods produce different results that disagree even in their leading quantum corrections at high temperatures. We also consider the regime where the cutoff frequency is smaller than the friction strength, i.e. ω D < γ. There, we encounter puzzling results at low temperatures where the specific heat based on the thermodynamic prescription becomes negative. This anomaly is rooted in an ill-defined density of states of the damped free particle, which assumes unphysical negative values when γ/ω D > 1.
The quantum-mechanical motion of a free particle coupled to a linearly dissipative environment is analyzed in the Hamiltonian formalism. The total Hamiltonian is diagonalized and its eigenstates displayed. These results are used to discuss the role of initial conditions on the subsequent motion. A widely used initial condition---heat bath uncoupled from particle---is compared with another one in which the initial off-diagonal coherence of the reduced density matrix is comparable to that in the state of thermal equilibrium of the coupled system. Different transient behaviors, on time scales longer than the inverse cutoff frequency of the bath are found. The mean-square momentum of the particle in the steady state is found to depend on this highest bath frequency. An analysis of the correlated states of particle and heat reservoir shows that the latter behaves in some sense like a position-measuring apparatus.
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 Brownian motion of a quantum particle in a harmonic confining potential and coupled to harmonic quantum thermal bath is exactly solvable. Though this system presents at high temperatures a pedagogic example to explain the laws of thermodynamics, it is shown that at low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. In physical terms, this happens when the cloud of bath modes around the particle starts to play a nontrivial role, namely, when the bath temperature T is smaller than the coupling energy. Indeed, equilibrium thermodynamics of the total system, particle plus bath, does not imply standard equilibrium thermodynamics for the particle itself at low T. Various formulations of the second law are found to be invalid at low T. First, the Clausius inequality can be violated, because heat can be extracted from the zero point energy of the cloud of bath modes. Second, when the width of the confining potential is suddenly changed, there occurs a relaxation to equilibrium during which the entropy production is partly negative. In this process the energy put on the particle does not relax monotonically, but oscillates between particle and bath, even in the limit of strong damping. Third, for nonadiabatic changes of system parameters the rate of energy dissipation can be negative, and, out of equilibrium, cyclic processes are possible which extract work from the bath. Conditions are put forward under which perpetuum mobility of the second kind, having one or several work extraction cycles, enter the realm of condensed matter physics. Fourth, it follows that the equivalence between different formulations of the second law (e.g., those by Clausius and Thomson) can be violated at low temperatures. These effects are the consequence of quantum entanglement in the presence of the slightly off-equilibrium nature of the thermal bath, and become important when the characteristic quantum time scale variant Planck's over 2pi /k(B)T is larger than or comparable to other time scales of the system. They show that there is no general consensus between standard thermodynamics and quantum mechanics. The known agreements occur only due to the weak coupling limit, which does not pertain to low temperatures. Experimental setups for testing the effects are discussed.
We consider a quantum system linearly coupled to a reservoir of harmonic oscillators. For finite coupling strengths, the stationary distribution of the damped system deviates from the predictions of standard thermodynamics. With the help of the quantum Hamiltonian of mean force, we quantify this deviation exactly for a harmonic oscillator and provide approximations in the limit of high and low temperatures and weak and strong couplings. Moreover, in the semiclassical regime, we use the quantum Smoluchowski equation to obtain results valid for any potential. We finally give a physical interpretation of the deviation in terms of the initial system-reservoir coupling.