Environment-dependent dissipation in quantum Brownian motion

The dissipative dynamics of a quantum Brownian particle is studied for different types of environment. We derive analytic results for the time evolution of the mean energy of the system for Ohmic, sub-Ohmic, and super-Ohmic environments, without performing the Markovian approximation. Our results allow one to establish a direct link between the form of the environmental spectrum and the thermalization dynamics. This in turn leads to a natural explanation of the microscopic physical processes ruling the system time evolution both in the short-time non-Markovian region and in the long-time Markovian one. Our comparative study of thermalization for different environments sheds light on the physical contexts in which non-Markovian dissipation effects are dominant.

Figure 1

(Color online) Spectral distributions of different reservoirs at high temperatures. Here $\overline{I}=I/\left({\alpha }^2{k}_{B}T\right)$ and $\overline{\omega }=\omega /{\omega }_0$. For each spectral curve the location of the cutoff frequency is given by ${\overline{\omega }}_c={\omega }_c/{\omega }_0=r$. The location of the oscillator frequency has been marked with a solid vertical line.

Figure 2

(Color online) Spectral distributions at zero temperature grouped according to the resonance parameter $r$. Here $\overline{I}=2I/\left({\alpha }^2{\omega }_0\right)$ and $\overline{\omega }=\omega /{\omega }_0$. The vertical dotted-dashed line is the location of the cutoff frequency ${\overline{\omega }}_c={\omega }_c/{\omega }_0$.

Figure 3

Decay rates at high temperatures for different types of reservoir in the non-Markovian time scales. Here $\overline{\Delta }=\Delta /\left(2{\alpha }^2{k}_{B}T\right)$ and $r=0.1$

Figure 4

The decay rates at zero temperature for $r=10$. The figure on the left depicts the transition up rates, while the figure on the right corresponds to the transition down rates.

Figure 5

Thermalization times for different reservoirs. Here ${\tilde{t}}_{th}=\pi {\alpha }^2{t}_{th}$.

Figure 6

Short-time dynamics of the heating function for different high-temperature reservoirs. In the plots $\alpha =0.01$ and we have set $\frac{k_{B}T}{\hslash {\omega }_0}=100$.