# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## 1 Introduction  
Many-body systems of interacting photons have come under intense investigation over the last few years, with both circuit QED- and semiconductor heterostructure-based systems [1; 2; 3; 4; 5; 6]. The main difference with traditional many-body systems, both in hard and synthetic condensed matter, is the fact that the photon lifetime is typically shorter than the time at which the system dynamics develop. To compensate for the losses, photonic systems have to be continuously driven, yielding a steady state that is a balance between driving and dissipation. Even though the basic theoretical framework for the description of open systems is well understood [7; 8], the exponential complexity of the many-body problem requires the development of practical approximate techniques to tackle systems that consist of many excitations of multiple modes, stirring the need for methods that unite the approaches of these different fields [9]. In order to study the time-evolution of an open quantum system, there are two distinct approaches [8]. The first one is the study of the ensemble as a whole. For a Markovian system, this translates into the Lindblad master equation for the density matrix \(\hat{\rho}\), where \(\hat{\rho}\) contains \(D^{2}\) elements for Hilbert-space dimension \(D\). The second approach is the trajectory approach: here, only a pure wave-function \(\ket{\psi}\) of \(D\) elements is evolved at a time. Through stochastic decoherence, the environment effectively introduces additional classical noise that complements a non-Hermitian Schrodinger evolution of the system, and the ensemble evolution is obtained by averaging over many trajectory evolutions. In addition to the reduction of complexity from \(D^{2}\) to \(D\), another advantage of the trajectory approach is that the full statistics of detector clicks are obtained. After preliminary work by [PERSON] (1984), this quantum trajectory formalism was developed towards its current shape ([PERSON], 1984; 1985) by a number of different groups ([PERSON], 1985; 1986; 1987; 1988). It is also known under a variety of other names: Monte Carlo wave function method, quantum jump method or stochastic Schrodinger equation. The act of averaging over many such trajectories to obtain an ensemble description is often called the stochastic simulation method. The formalism behind quantum-trajectories is tightly bound to quantum metrology (1985). In this context, is important to note that quantum trajectories are not a single method, but rather a class of methods, depending on the so-called unraveling, which corresponds to some (hypothetical) measurement protocol taking place on the environment. For quantum optical systems, the most common protocols are photon-counting, resulting in piecewise-deterministic processes, on the one hand, and homodyne/(far-detuned-)heterodyne detection resulting in Wiener processes, on the other hand ([PERSON], 1985). The heterodyne case has proven to be equivalent to the stochastic collapse model of quantum state diffusion ([PERSON], 1985). A summary of these common measurements and their relation with quantum trajectories is given in Appendix A. Although analytical effort can sometimes provide a serious reduction ([PERSON], 1985; 1988), computational complexity typically remains a limiting factor in exact numerical study of open systems. The aforementioned quantum trajectory method, which allows for a description in the Hilbert space of pure states \(\mathcal{H}\) instead of the superspace of density matrices \(\mathcal{H}^{2}\), can reduce the memory-cost for individual simulations to the level of closed-system problems, at the expense of the need to average over many individual Monte Carlo realizations. Still, as in the closed system case, many-body systems are usually too complex for exact computation, and one needs to use approximate methods instead. These can either occur at the level of the master equation or at the level of individual trajectories ([PERSON], 1985).