Quantum statistical ensemble for emissive correlated systems

Relaxation dynamics of complex quantum systems with strong interactions towards the steady state is a fundamental problem in statistical mechanics. The steady state of subsystems weakly interacting with their environment is described by the canonical ensemble which assumes the probability distribution for energy to be of the Boltzmann form. The emergence of this probability distribution is ensured by the detailed balance of the transitions induced by the interaction with the environment. Here we consider relaxation of an open correlated quantum system brought into contact with a reservoir in the vacuum state. We refer to such a system as emissive since particles irreversibly evaporate into the vacuum. The steady state of the system is a statistical mixture of the stable eigenstates. We found that, despite the absence of the detailed balance, the stationary probability distribution over these eigenstates is of the Boltzmann form in each $N$-particle sector. A quantum statistical ensemble corresponding to the steady state is characterized by different temperatures in the different sectors, in contrast to the Gibbs ensemble. We investigate the transition rates between the eigenstates to understand the emergence of the Boltzmann distribution and find their exponential dependence on the transition energy. We argue that this property of transition rates is generic for a wide class of emissive quantum many-body systems.

Figure 1

The stationary probability distribution of the emissive system of hard-core bosons on the 14-site lattice (geometry is shown in the inset). The initial state of the system is the maximally occupied state with $N_0=14$. The distribution includes stable states which exist for $N\le 7$. In sectors with at least three stable states the data are fitted with the Boltzmann distribution $P_n^N\propto exp(-E_n/T_N)$. The temperatures $T_N$ are presented in the table.

Figure 2

The probability distribution of the intermediate states in each $N$-particle sector. The initial state of the system is the maximally occupied state with $N_0=14$. To avoid the overlap of data in different sectors, the distribution functions are multiplied by the factor ${10}^N$. The stable states are indicated with empty symbols. In each sector the points accumulating $80\%$ of the total probability are fitted with the exponential functions $P_N\left(E\right)\propto exp(-E/T_N)$. The temperatures $T_N$ are presented in the table.

Figure 3

The transition rates $R_{nm}^N$ from the states in the $N$-particle sector to the states in the $(N-1)$-particle sector as a function of the transition energy $\varepsilon =E_n^N-E_m^{N-1}$ for $N=3,...,12$. The allowed transitions correspond to $\varepsilon >{\varepsilon }_0=0$. For illustrative purposes 900 points are shown for each $N$, and the rates are multiplied by the factor ${10}^N$.

Figure 4

The plot shows the dependence of the inverse temperatures $1/T_N$ on the number of particles for different initial conditions which are defined by the set of initial inverse temperatures $1/T_0$ (shown in the inset table). The plot demonstrates that system may undergo both heating and cooling process during the evaporation depending on the initial temperature of the system and the filling of the system.

Figure 5

Graphical representation of the Hilbert space. The system of hard-core bosons on a lattice of four sites is considered. Bars representing many-body eigenstates $|N,n\rangle$ are placed into the coordinate system “number of particles–energy.” Initial maximally occupied state is the rightmost one. Allowed transitions and some forbidden transitions are shown with solid and dashed arrows correspondingly. The achievable eigenstates are indicated with blue (gray). The stable eigenstates are represented with empty bars. After the loss of particles the systems in the ensemble can be in either the lowest energy two-particle eigenstate or the lowest energy three-particle eigenstate.

Figure 6

Number of achievable stable eigenstates. The system of hard-core bosons on a 14-site lattice is prepared in the maximally occupied state. The plot shows the number of achievable stable eigenstates as the function of the binding energy.

Figure 7

Stationary distributions. The system is prepared in eigenstates with $N_0=12$ particles and different energies: (a) $E_0=-3.00$, (b) $E_0=-0.06$, (c) $E_0=0.06$, (d) $E_0=3.00$. The stationary distributions are of the Boltzmann form in each $N$-particle sector for all these initial conditions (temperatures are presented in tables).

Figure 8

Scaling analysis. The plot shows the standard deviation of ${\log }_{10}R$ in the interval of transition energies $[-0.1,0.1]$ as the function of the number of sites in the lattice. Geometries of the lattices of different sizes are shown in the inset.