Typical, finite baths as a means of exact simulation of open quantum systems

There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the electromagnetic field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open systems: using a finite bath chosen to have certain key properties of thermalizing many-body systems. To explore the numerical resources required by this method to approximate an open system coupled to an infinite bath, we simulate a weakly damped system and compare to the evolution given by the relevant Markovian master equation. We obtain the Markovian evolution with reasonable accuracy by using an additional averaging procedure, and elucidate how the typicality of the bath generates the correct thermal steady state via the process of “eigenstate thermalization.”

Figure 1

Here we show the relationships between the energy ranges of the bath, the system, and the initial state of the bath. The energy range of the bath is the interval $[E_{\text{bath}}^{\text{min}},E_{\text{bath}}^{\text{max}}]$, depicted by the light gray region in diagram. The initial state of the bath is chosen to be a random superposition of all the bath energy eigenstates with energies in the interval $[E_{\psi }^{\text{min}},E_{\psi }^{\text{max}}]$, for a total energy width of $\Delta E_{\psi }=E_{\psi }^{\text{max}}-E_{\psi }^{\text{min}}$. The energy range of the initial state is denoted by the dark gray region in the diagram. The energy range of the system is denoted in the diagram by $\Delta E_{\text{sys}}$. The energy windows given by the light gray areas on either side of the dark gray region must be at least as wide as the energy range of the system. This is so that the system has the ability to dump all its energy into the bath, and extract all its energy from the bath. Without this ability the bath cannot thermalize the system to the Boltzmann state.

Figure 2

Evolution of the populations of the energy eigenstates of a nonlinear four-level system coupled to a bath with 5000 states, for two initial states of the system. The horizontal lines give the populations for the Boltzmann thermal distribution at the relevant temperature. In plot (a) the initial state is the one with lowest energy, and in (b) it is the state with the highest energy. The insets are expanded versions of the plots for early times, showing the initial relaxation to the thermal state.

Figure 3

Each of the four plots shows the populations of one of the system energy levels (of which there are four). In each plot: the dashed horizontal line (yellow) is the desired thermal (Boltzmann) value. The light-gray solid horizontal line (green) is the actual steady-state population when the initial state is state 4. The dark noisy line (dark blue) is the population given by each of the energy eigenstates of the universe, as a function of their energy. The solid light gray curve (cyan) is the moving average of the noisy line over a window of 200 adjacent eigenstates. Thermalization happens in the region where the light gray curve (cyan), light gray horizontal line (yellow), and horizontal dashed line (green) coincide. In the dashed box in the plot for level 1 we display the distribution of the initial state of the universe, over its eigenstates, when the system starts in state 1 (left, red) and 4 (right, mauve).

Figure 4

Here we plot an estimate of the mean square of the off-diagonal elements of the projector onto the ground state, $P_1=|1\rangle \langle 1|$. This estimate is calculated, for each value of $n$, by averaging the square of the off-diagonal elements for $m=n+1$ to $m=n+10$.

Figure 5

Here we plot the dynamics of the thermal relaxation of the system, averaged over 200 initial states of the bath, randomly chosen within the same energy window (dashed lines). This averages out the fluctuations due to the finite size of the bath, and we obtain a good approximation to the evolution for the macroscopic bath. We also plot the thermal relaxation given by the equivalent Markovian Redfield master equation (solid line). The main plot is for a bath with 5000 states, and the inset gives the result for a bath of 2500 states.