Generic map from non-Lindblad to Lindblad master equations

Many current problems of interest in quantum nonequilibrium are described by time-local master equations (TLMEs) for the density matrix that are not of the Lindblad form, that is, that are not strictly probability-conserving and/or Markovian. Here we describe a generic approach by which the system of interest that obeys the TLME is coupled to an ancilla, such that the dynamics of the combined system-plus-ancilla is Markovian and thus described by a Lindblad equation. This in turn allows us to recover the properties of the original TLME dynamics from a physical unraveling of this associated Lindblad dynamics. We discuss applications of this generic mapping in two areas of current interest. The first one is that of “thermodynamics of trajectories”, where non-Lindblad master equations encode the large-deviation properties of the dynamics, and we show that the relevant large-deviation functions (i.e., dynamical free energies) can be recovered from appropriate observables of the ancilla. The second one is that of quantum filters, where we show that tracking a quantum system undergoing a continuous homodyne measurement with another quantum system of the same size will inherently be inefficient in our framework.

Figure 1

We consider how to map a system that obeys a TLME [Eq. (2)] that is not in Lindblad form, (a), to a hybrid system-ancilla, (b), whose total evolution is Markovian, Eq. (1), which under a quantum weighted average, Eq. (3), reproduces the TLME for the original system. In this case (a) is a micromaser—a cavity pumped by excited two-level atoms—biased by a “counting” field $s$. This results in non-Lindblad evolution (see the text for details) where the probability to detect a ground-state atom leaving the cavity is biased by the factor $e^{-s}$ with respect to unbiased dynamics. Such non-Lindblad evolution is mimicked by a micromaser coupled to an ancilla. The composite system (b) evolves under a physical Markovian dynamics. We evolve a Markovian quantum jump Monte Carlo of the composite system and then extract the dynamics of the embedded non-Lindblad evolution. The decay of the coherence of the ancilla is plotted in (c). The decay rate of the coherence in the Markovian simulation is precisely the large-deviation function $\theta \left(s\right)$, which corresponds to the largest eigenvalue of the non-Lindblad TLME. $\theta \left(s\right)$ can be extracted in this manner and contains the full counting statistics of the atoms leaving the cavity. In (d) we plot the function of $\theta \left(s\right)$ and also the points determined using the decay rates of the coherence in (c). In (d) we can see the micromaser at these conditions $\theta \left(s\right)$ has a first-order singularity at $s_c\approx 3.4\times {10}^{-4}$, indicating a dynamical phase transition (for the micromaser it is actually a very sharp crossover) [63, 64].