Energy backflow in strongly coupled non-Markovian continuous-variable systems

By employing the full counting statistics formalism, we characterize the first moment of energy that is exchanged during a generally non-Markovian evolution in nondriven continuous-variable systems. In particular, we focus on the evaluation of the energy flowing back from the environment into the open quantum system. We apply these results to the quantum Brownian motion, where these quantities are calculated both analytically, under the weak-coupling assumption, and numerically also in the strong-coupling regime. Finally, we characterize the non-Markovianity of the reduced dynamics through a recently introduced witness based on the so-called Gaussian interferometric power and we discuss its relationship with the energy backflow measure.

Figure 1

Consider a system of interest $S$ interacting with its environment $E$, to which it is assumed to be decoupled at time $t\le 0$. At time $t=0$ a generic observable $A$ of the environment is measured through a projective measurement $\mathrm{\Pi }$, thus obtaining a certain outcome $a_0$ belonging to its spectrum. Immediately after the interaction is switched on and the overall system evolves up to a time $t_f$, we switch off the interaction and perform another measurement of the observable $A$, this time obtaining another outcome $a_t$. In our case $A$ will be the energy of the environment ${\mathcal{H}}_E$.

Figure 2

Time evolution of (a) $\theta \left(t\right)$ and (b) $\varphi \left(t\right)$, in units of ${\omega }_0$, for $\mathrm{\Omega }=0.25{\omega }_0,\lambda =0.01$, and $T_E={\omega }_0$ and for different values of the initial system's temperature $T_S/{\omega }_0=1,2,3$ (respectively red solid line, green dot-dashed, and blue dashed).

Figure 3

Time evolution of (a) ${\langle \mathrm{\Delta }q\rangle }_t$ and (b) ${\langle \mathrm{\Delta }{E}_S\rangle }_t$, in units of ${\omega }_0$, for $\mathrm{\Omega }=0.25{\omega }_0,\lambda =0.01$, and $T_E=T_S={\omega }_0$. Note that the final value of the internal energy of the environment is lower than its initial value, meaning that the environment has cooled down.

Figure 4

Energy backflow measure as a function of the coupling strength $\lambda$ for different values of the parameters $\mathrm{\Omega }$ and $T_E$ which characterize the dynamical map.

Figure 5

Time behavior of the energy flow per unit of time $\theta \left(t\right)$ in units of ${\omega }_0$ for $\mathrm{\Omega }=0.25{\omega }_0,T_E=T_S={\omega }_0$, and coupling strength $\lambda =0.01$ (a) and $\lambda =1$ (b). The solid lines refer to the solution obtained with the numerical method, while the dashed lines are the curves predicted by the analytical approach relying on the FCS methods. Plot of the energy backflow measure in units of ${\omega }_0$ as a function of the coupling strength $\lambda$ for $\mathrm{\Omega }=0.25{\omega }_0$ (c) and as a function of the coupling strength $\mathrm{\Omega }$ for $\lambda =1$ (d), for three different values of the initial temperatures: $T_E=T_S=0.25{\omega }_0$ (green or dot-dashed line), $T_E=T_S=0.5{\omega }_0$ (red or dashed line), and $T_E=T_S={\omega }_0$ (blue or solid line). These curves were produced by means of the numerical simulation with $N=150$ environmental bosonic modes.

Figure 6

Separate contributions to the time behavior of the mean total energy in units of ${\omega }_0$, for $\mathrm{\Omega }=0.25{\omega }_0,T_E=T_S={\omega }_0$. The top three panels (a)–(c) refer to the weak-coupling case $\lambda =0.01$, the middle three (d)–(f) to $\lambda =0.8$, and finally the bottom three (g)–(i) to strong coupling $\lambda =1.8>{\lambda }^{*}$. (a), (d), (g) Mean values of the change in the environmental Hamiltonian Eq. (4). (b), (e), (h) Mean values of the change in the system Hamiltonian Eq. (19). (c), (f), (i) Mean values of the change in the interaction Hamiltonian.

Figure 7

Plots of the non-Markovianity measure ${\mathcal{N}}_{\mathcal{Q}}^{\mathbf{\sigma }}$ for the class of STSs (blue or dark grey lines) and of MTSs (red or light grey lines), with $k_{1,2}=1,r_1=r_2=0.658$ (otherwise stated), as function of $\lambda$ [panels (a) and (b)] and of the cutoff frequency [panels (c) and (d)] for fixes values of the remaining parameters. In particular, (a) $\mathrm{\Omega }=0.25{\omega }_0$ and $T_E=0.25{\omega }_0$; (b) $\mathrm{\Omega }=0.25{\omega }_0$ and $T_E={\omega }_0$; (c) $\lambda =0.01,T_E=0.25{\omega }_0$, and $r_1={10}^{-2}$; (d) $\lambda =0.01,T_E={\omega }_0$, and $r_1=r_2=0.22$.