Lindblad- and non-Lindblad-type dynamics of a quantum Brownian particle

The dynamics of a typical open quantum system, namely a quantum Brownian particle in a harmonic potential, is studied focusing on its non-Markovian regime. Both an analytic approach and a stochastic wave-function approach are used to describe the exact time evolution of the system. The border between two very different dynamical regimes, the Lindblad and non-Lindblad regimes, is identified and the relevant physical variables governing the passage from one regime to the other are singled out. The non-Markovian short-time dynamics is studied in detail by looking at the mean energy, the squeezing, the Mandel parameter, and the Wigner function of the system.

Figure 1

Dynamics of the heating function $⟨n\left(t\right)⟩$ in the short-time non-Markovian regime. For the high $T$, graphics (c),(f),(i), we have used $r_0={\omega }_0∕KT=0.1$; for the intermediate $T$, graphics (b),(e),(h), we have used $r_0={\omega }_0∕KT=1$; for low $T$, graphics (a),(d),(g), we have used $r_0={\omega }_0∕KT=100$. We indicate with a solid line the analytic solution and with circles the simulations performed using the NMWF method. In the lower right corner of all graphics, we indicate whether the asymptotic long-time value of the heating function is null (zero $T$ reservoir) or not.

Figure 2

Contour plots of $\overline{\Delta \left(r,t\right)}=\Delta \left(r,t\right)∕2{\alpha }^{2}KT$ for high $T$ (a), and of the coefficient $\overline{\Delta \left(r,t\right)-\gamma \left(r,t\right)}=[\Delta \left(r,t\right)-\gamma \left(r,t\right)]∕{\alpha }^2$ for low $T$ (b). In (b) we have chosen $2\pi r_c=\hslash {\omega }_c∕KT=10$. In both (a) and (b) the contour line corresponding to $\overline{\Delta \left(r,t\right)}=0$ (a) and $\overline{\Delta \left(r,t\right)-\gamma \left(r,t\right)}=0$ (b) is indicated by a thick solid line.

Figure 3

(a) Time evolution of the position variance for an initial squeezed state having squeezing parameter $s=0.4$. The reservoir parameters are $\alpha ={10}^{-2}$, $r=0.05$, and $r_c={\omega }_c∕2\pi KT=0.2\times {10}^{-6}$. The dotted line shows the variance of the free harmonic oscillator in the absence of coupling to the environment. The system exhibits squeezing whenever ${\left(\Delta X\right)}^2\left(t\right)<0.5$. (b) Time evolution of the Mandel parameter for an initial Fock state $\mid n=3⟩$. The reservoir parameters are the same as in (a).

Figure 4

Contour plots of the Wigner function at different time instants. At $\tau =0$, the initial state is the coherent state corresponding to $\mathrm{Re}\left[{\alpha }_0\right]=1$, $\mathrm{Im}\left[{\alpha }_0\right]=0$. The reservoir parameters are $\alpha ={10}^{-2}$, $r=0.05$, and $r_c={\omega }_c∕2\pi KT={10}^{-7}$.