Modeling particle loss in open systems using Keldysh path integral and second order cumulant expansion

For open quantum systems, integration of the bath degrees of freedom using the second order cumulant expansion in the Keldysh path integral provides an alternative derivation of the effective action for systems coupled to general baths. The baths can be interacting and not necessarily Markovian. Using this method in the Markovian limit, we compute the particle-loss dynamics in various models of ultracold atomic gases, including a one-dimensional Bose-Hubbard model with two-particle losses and a multicomponent Fermi gas with interactions tuned by an optical Feshbach resonance. We explicitly demonstrate that the limit of strong two-body losses can be treated by formulating an indirect loss scheme to describe the bath-system coupling. The particle-loss dynamics thus obtained is valid at all temperatures. For the one-dimensional Bose-Hubbard model, we compare it to solutions of the phenomenological rate equations. The latter are shown to be accurate at high temperatures.

Figure 1

Scheme of the lossy 1D Bose-Hubbard model. $J$ is the nearest-neighbor hopping and $U$ is the on-site interaction. The loss parametrized by $\gamma$ is the one-body loss rate of doublons, i.e., doubly occupied sites

Figure 2

Left: Evolution of the lattice filling starting from an initial filling of $n_A(t=0)=0.25$ for different initial temperatures on a log-log scale. The initial temperature is determined by the initial state, which is assumed thermal. The phenomenological result shows large deviations from the microscopic theory, especially at low temperatures. Right: Evolution of the lattice filling starting from initial filling of $n_A(t=0)=0.75$ for different initial temperatures on a log-log scale. At larger initial fillings, the deviation from the phenomenological rate equation becomes smaller.

Figure 3

Top left panel: Decay of the lattice filling for an initially half-filled system and different initial temperatures. The dependence of the initial temperature is very weak (note the log-log scale). Bottom left panel: Change of kinetic energies with different initial temperatures. Systems with a lower initial temperature undergo a larger change in the kinetic energy. Upper right panel: Time evolution of the distribution of Jordan-Wigner fermions for an inverse temperature $k_{B}T=J$ (i.e., $\beta J=1$). Note a large depletion in the distribution near $k=\pm \pi /2$. Bottom right panel: Evolution of the distribution of Jordan-Wigner fermions for $k_{B}T=J/10$ (i.e., $\beta J=10$).

Figure 4

Scheme of an optical Feshbach resonance: The system consists of $N$ species of spin-$F$ ($N=2F+1$) alkaline-earth atoms in the ground state interacting via an $s$-wave potential. Using a laser colliding pairs of atoms in the ground state are coupled to pairs consisting of one ground state and one excited-state atom in a molecular state. This state has a one-body coupling parametrized by $g_{\alpha }$ to a bath to which it can be lost or gained.