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Physical Review A

Quantum theory of a free particle interacting with a linearly dissipative environment

Vincent Hakim, Vinay Ambegaokar

Published Jul 1, 1985• 307 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.32.423

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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 present the quantum theory of a Stern-Gerlach system in contact with a linearly dissipative environment at an arbitrary temperature. The dynamics of the reduced density matrix of the system is calculated using the path integral technique, under a general condition when the system and its environment may not be decoupled initially. We analyze the behavior of the density matrix in the long-time limit and compute the time scales of decay of the elements off-diagonal in the coordinate and the momentum space, for the cases of high temperature, zero temperature, and intermediate temperature baths.
From a quantized many-body system a wave equation for the motion of a particle linearly coupled to a heat bath is derived. The effective Hamiltonian describing the motion of the single particle is explicitly time dependent, and for a quadratic potential, has a simple dependence on the initial position and momentum of the particle. For the case of dissipative harmonic motion, a time-dependent wave equation is derived and the ground-state wave function is determined. It is also shown that if the equations of motion for the many-body system is Galilean invariant, the reduced form of equation of motion for the single particle is not. However a generalized form of transformation for the position and momentum operators, to a coordinate system moving with constant velocity, is obtained, which reduces to the Galilean transformation when the coupling to the dissipative system is turned off.
The motion of a quantum particle in a tight-binding lattice coupled to an Ohmic thermal reservoir is analyzed in the path-integral formalism. We derive exact formal expressions for moments of the probability distribution both for noncorrelated and correlated thermal initial states of the particle-plus-reservoir complex. Based on these expressions we discuss the role of initial conditions on the subsequent motion and calculate the differences in the transient behavior. For a particular value of the damping strength we find the exact solution of the dynamics in analytic form at any time, temperature, and bias for the two kinds of initial states.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson's boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann's entropy and quantum purity. In addition, we have studied Wigner's function. The time-dependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.
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.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson’s boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann’s entropy and quantum purity. In addition, we have studied Wigner’s function. The timedependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.