Threshold coupling strength for equilibration between small systems

In this paper we study the thermal equilibration of small bipartite Bose-Hubbard systems, both quantum mechanically and in mean-field approximation. In particular we consider small systems composed of a single-mode “thermometer” coupled to a three-mode “bath,” with no additional environment acting on the four-mode system, and test the hypothesis that the thermometer will thermalize if and only if the bath is chaotic. We find that chaos in the bath alone is neither necessary nor sufficient for equilibration in these isolated four-mode systems. The two subsystems can thermalize if the combined system is chaotic even when neither subsystem is chaotic in isolation, and under full quantum dynamics there is a minimum coupling strength between the thermometer and the bath below which the system does not thermalize even if the bath itself is chaotic. We show that the quantum coupling threshold scales like $1/N$ (where $N$ is the total particle number), so that the classical results are obtained in the limit $N\to \infty$.

Figure 1

Chaos map that shows the value of $r$ averaged over small energy windows for the possible initial states for $UN/\mathrm{\Omega }=10$. Red dots show the initial states used in Fig. 2.

Figure 2

Evolution of the probability of trimer population $x$ (upper panels) and the entropy (lower panels) for $N=70$, $UN/\mathrm{\Omega }=10$, and initial $x=0.6$. The red horizontal line in the lower panels indicates the microcanonical thermal entropy. (a) $\varepsilon =0.3$, $\omega /\mathrm{\Omega }=0.1$; (b) $\varepsilon =0.3$, $\omega /\mathrm{\Omega }=0.01$; (c) $\varepsilon =0.4$, $\omega /\mathrm{\Omega }=0.1$; (d) $\varepsilon =0.4$, $\omega /\mathrm{\Omega }=0.4$.

Figure 3

Classical evolution of $P\left(x\right)$ and $S$ for the same parameters as in Fig. 2 and corresponding initial conditions. Panels (e) and (f) show Poincaré sections corresponding to the initial states in (a)/(b) and (c)/(d). (a) $\varepsilon =0.3$, $\omega /\mathrm{\Omega }=0.1$; (b) $\varepsilon =0.3$, $\omega /\mathrm{\Omega }=0.01$; (c) $\varepsilon =0.4$, $\omega /\mathrm{\Omega }=0.1$; (d) $\varepsilon =0.4$, $\omega /\mathrm{\Omega }=0.4$; (e) $\varepsilon =0.3$; (f) $\varepsilon =0.4$.

Figure 4

Example showing how the thermalization threshold ${\omega }_T$ is determined: we simulate the evolution for different values of $\omega$ and fit or interpolate the results around $\mathrm{\Delta }{\rho }^M=0.1$ to estimate the value of ${\omega }_T$. The parameters in this example are $N=90$, $UN/\mathrm{\Omega }=10$, $\varepsilon =0.25$, and $x=0.6$.

Figure 5

Dependence of ${\omega }_T$ on $\varepsilon$ for $N=70$, $UN/\mathrm{\Omega }=10$, and $x=0.6$ (a) and on $UN/\mathrm{\Omega }$ for $N=70$, $\varepsilon =0.25$, and $x=0.6$ (b). The blue curves show the quantum results, whereas the red curves show the corresponding classical results. The smallest coupling we tested in the classical simulations was $\omega /\mathrm{\Omega }=0.01$. In cases where even this small coupling lead to thermalization, ${\omega }_T/\mathrm{\Omega }=0.01$ is only an upper bound (marked by open circles). Panels (c)–(f)/(g)–(j) show Poincaré sections corresponding to representative points in (a)/(b). (c) $\varepsilon =0.1$; (d) $\varepsilon =0.2$; (e) $\varepsilon =0.3$; (f) $\varepsilon =0.4$; (g) $UN/\mathrm{\Omega }=5$; (h) $UN/\mathrm{\Omega }=10$; (i) $UN/\mathrm{\Omega }=15$; (j) $UN/\mathrm{\Omega }=20$.

Figure 6

Dependence of the global and local mean level spacing on the particle number $N$ for $UN/\mathrm{\Omega }=10$. The global mean spacing (blue) is the level spacing averaged over all eigenstates of the uncoupled Hamiltonian. For the local level spacing (red) we average only over eigenstates with $0.2<\varepsilon <0.3$, i.e., states that are local in energy. Both curves are well described by fits of the form $a/N+b/N^2$, demonstrating the $1/N$ scaling behavior for large $N$.

Figure 7

Dependence of $\mathrm{\Delta }{\rho }^M$ on $N$ for $UN/\mathrm{\Omega }=10$, $\varepsilon =0.25$ and $x=0.6$ (left). Plotting $\mathrm{\Delta }{\rho }^M$ against $\omega N/\mathrm{\Omega }$ (right) shows that it depends only on the product $\omega N/\mathrm{\Omega }$. The data is well approximated by the curve $a/(\omega N/\mathrm{\Omega }+b)$, where $a$ and $b$ are fit parameters.