The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$ -number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

Physical Review AAug 1, 1971208 citations

Brownian motion of a quantum particle

Boris B. Kadomtsev, Михаил Борисович Кадомцев

Physics Letters AJun 1, 19973 citations

Brownian motion of a quantum oscillator

Salvatore Carusotto

The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.

Physical Review AApr 1, 197515 citations

Brownian motion of quantum particles

V. I. Mel'Nikov, S. V. Meshkov

Aug 1, 19831 citation

c

-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.