Thermal Equilibration between Two Quantum Systems

Two identical finite quantum systems prepared initially at different temperatures, isolated from the environment, and subsequently brought into contact are demonstrated to relax towards Gibbs-like quasiequilibrium states with a common temperature and small fluctuations around the time-averaged expectation values of generic observables. The temporal thermalization process proceeds via a chain of intermediate Gibbs-like states. We specify the conditions under which this scenario occurs and corroborate the quantum equilibration with two different models.

Figure 1

(a) A system of bosons confined into two overlapping confinements is analyzed with the Bose-Hubbard model. (b) Energy spectrum of a single system. The (red) line displays the dependence of the system mean energy, i.e., $E^S=\sum _k{ϵ}_k{e}^{-{ϵ}_k/k_{\mathcal{B}}T}/Z_S$, on temperature $T$. The initial temperatures of the “hot” system, $k_{\mathcal{B}}{T}_A/\overline{s}=94.91$, and the “cold” system, $k_{\mathcal{B}}{T}_B/\overline{s}=18.98$, are indicated by the (blue) dots. The equilibrium temperature $k_{\mathcal{B}}{T}_F=33.92\overline{s}$, calculated by using the total energy conservation [Eq. (7)], is indicated by the (red) star. (c) Instantaneous equilibrium energy level populations for systems $A$ (left column) and $B$ (right column), in the regime of arithmetic-mean (top) and thermal (bottom) equilibrations. The arithmetic-mean populations are depicted by the top (blue) solid lines, and the canonical populations for the temperature $T_F$ by the bottom (red) lines. The natural energy unit $\overline{s}$ is given by the mean energy level spacing of the single system: $\overline{s}=({ϵ}_{{\mathcal{N}}_S}-{ϵ}_1)/({\mathcal{N}}_S-1)$. The similar behavior is demonstrated by the second model; see supplementary material [18] for further model details.

Figure 2

Relaxation pathways for the model depicted in Fig. 1a. Both systems are initially prepared in canonical states (solid lines) and in pure states randomly sampled from the corresponding ensembles of typical states [Eq. (8)] (dashed lines). (a) The evolution of the mean energies and (b) the corresponding temperatures $T(t)$ of the hot system $A$ and the cold system $B$ are shown by the upper (red) and lower (blue) lines, respectively. (c) The energy level populations of both systems are displayed at different moments of time (dots), marked by the corresponding symbols in (a) and (b). The lines correspond to the canonical populations [Eq. (5)] at the temperatures evaluated from the temporal values of mean system energies [see Fig. 1b].

Figure 3

(a) Von Neumann entropy of a single system vs time for the model shown in Fig. 1a. Both systems are initially prepared in pure states randomly sampled from the ensembles of typical states [Eq. (8)]. The dashed line indicates the entropy of the Gibbs state at the temperature $T_F$. (b) The population dynamics $n_5(t)$ of the fifth site (i.e., the site making the thermal contact) for subsystem $A$ is compared with the corresponding canonical value at the temperature $T_F$. Note that the time average of $n_5(t)$, 0.9673, differs from its canonical value, 0.9651, by 0.3% only. (c) The absolute values of reduced density matrix elements $⟨{\varphi }_m|{\varrho }^A(t)|{\varphi }_n⟩$ at $t=0$ and (d) after equilibration.