Modeling mechanical equilibration processes of closed quantum systems: A case study

We model the dynamics of a closed quantum system brought out of mechanical equilibrium, undergoing a nondriven, spontaneous, thermodynamic transformation. In particular, we consider a quantum particle in a box with a moving and insulating wall, subjected to a constant external pressure. Under the assumption that the wall undergoes classical dynamics, we obtain a system of differential equations that describes the evolution of the quantum system and the motion of the wall. We study the dynamics of such a system and the thermodynamics of the process of compression and expansion of the box. Our approach is able to capture several properties of the thermodynamic transformations considered and goes beyond a description in terms of an ad hoc time-dependent Hamiltonian, considering instead the mutual interactions between the dynamics of the quantum system and the parameters of its Hamiltonian.

Figure 1

Particle wave function after a time $T=L_0/2V$ when the wall is moving with constant speed $V$, for different values of $V$. The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 2

Wall speed as a function of time for initial compression and expansion when there is no friction [panels (a) and (d), respectively]; under the presence of the frictional term $-\gamma V$ [panels (b) and (e)]; under friction described as in Eq. (1) [panels (c) and (f)]. The system's dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 3

(a) Population of the ground state of ${\widehat{H}}^{*}\left(t\right)$ as a function of time for different values of $\mathrm{\Gamma }$. (b) [(c)] Populations of the first excited state (coherences between the ground and of the first excited state) of $\widehat{H^{*}}\left(t\right)$ as a function of time, for different values of $\mathrm{\Gamma }$. (d) Purity of the state of the system against for various choices of $\mathrm{\Gamma }$. The system dynamics is approximated considering only the first 40 eigenstates of $\widehat{H}\left(0\right)$.

Figure 4

Relative difference in the length of the box with and without dephasing $\mathrm{\Delta }L=\left[L(\mathrm{\Gamma }=10)-L(\mathrm{\Gamma }=0)\right]/L(\mathrm{\Gamma }=0)$. The system dynamics is approximated considering only the first 40 eigenstates of $\widehat{H}\left(0\right)$.

Figure 5

(a) Final length of the box ($T=2$) as a function of $P_0/P\left(0\right)$. (b) Minimum length of the box as a function of $P_0/P\left(0\right)$. The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 6

Energy dissipated due to the frictional force and quantum entropy production in the case of initial expansion (a, b) and compression (c, d). The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 7

Change in the expectation value of the Hamiltonian of the particle and average energy change as expected from Eq. (13) (a), (c) and their difference (b), (d) as functions of time in the case of initial expansion and compression. The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 8

Left- and right-hand sides of Eq. (14) (a), (c) and their difference (b), (d) as functions of time in the case of initial expansion and compression. The system's dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 9

Length of the box in the case of initial expansion ($P_0/P\left(0\right)=0.9$) as a function of time when the system is prepared in the ground state of its Hamiltonian, for $\gamma =10$ (a), $\gamma =1$ (b), and $\gamma =0.1$ (c). The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 10

Length of the box and speed of the wall when $M=0.001$ (a), (b) and $M=1$ (c), (d) as functions of time in the case of initial expansion ($P_0/P\left(0\right)=0.9$). The system dynamics is approximated considering only the first 20 eigenstates of $\widehat{H}\left(0\right)$.

Figure 11

Infidelity between the final state reached by truncating to the first $K$ eigenstates of $H\left(0\right)$ and to the final state reached by truncating to the first $K+1$ eigenstates of $H\left(0\right)$ for different values of initial inverted temperature $\beta$ in the case of initial compression (a) and expansion (b).