Lindblad model of quantum Brownian motion

The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.

Figure 1

Plot of the angle $\theta /\pi$ at $\gamma /\mathrm{\Omega }=0.8$. This angle is represented in the ellipse at the bottom of the picture. Here, the orange-solid (green-dashed) line represents the minor (major) axis of the Wigner function, i.e., that related to ${\delta }_l \left({\delta }_L\right)$. The axes $X$ and $P$ are those of the phase space.

Figure 2

Eccentricity of the Wigner function introduced in Eq. (28), at $\gamma /\mathrm{\Omega }=0.8$. The red dashed line represents the values of $T$ and $\mathrm{\Lambda }$ yielding ${\delta }_l^2=1$, and we have genuine squeezing below it.

Figure 3

Minimum value of ${\delta }_l^2$ over all temperatures, as a function of the cut-off frequency, at several values of the damping constant.

Figure 4

Cooling parameter $\chi$ introduced in Eq. (36), plotted for $\gamma /\mathrm{\Omega }=0.8$. The system exhibits cooling to the right of the solid line, and heating to its left. For comparison, the dashed line represents the boundary between cooling and heating obtained with the BMME (3), which is independent of $\gamma$.

Figure 5

Minimum value of the cooling parameter $\chi$ over all temperatures, as a function of the cut-off frequency, at several values of the damping constant.

Figure 6

Eccentricity $\eta$ of the Wigner function at $\gamma /\mathrm{\Omega }=0.1$, for quadratic coupling.

Figure 7

Cooling parameter $\chi$ for quadratic coupling, at $\gamma /\mathrm{\Omega }=0.1$.

Figure 8

Angle $\theta /\pi$ between the major axis of the Wigner function, and the $X$ axis of the phase space at $\gamma /\mathrm{\Omega }=0.1$, for quadratic coupling.

Figure 9

Time dependence of ${\delta }_x^2$ for several values of $\gamma$, at $\mathrm{\Lambda }/\mathrm{\Omega }=16$ and $k_{B}T/\hslash \mathrm{\Omega }=4$. The thin solid lines represent the stationary value of ${\delta }_x^2$ in the state, namely the stationary solution of Eqs. (46, 47, 48) for such a quantity.

Figure 10

Plot of the product ${\delta }_x{\delta }_p$ at $\gamma /\mathrm{\Omega }=0.1$, for quadratic coupling. This quantity is always larger than 1, in accord with the HUP.