Quantum trajectories approach

The second-order approach described above corresponds to a first-level approximation to the description of correlations and fluctuations in the cavity dynamics. Moving to higher-order expansions increases the number of ordinary differential equations needed to resolve the dynamics, which soon becomes cumbersome and impractical. An alternative approach to solve the master Eq. (¹) is to use the quantum trajectories (or Monte Carlo wavefunction) method^39–42. Here, the Lindblad dynamics of the density operator ρ is replaced by a wavefunction whose evolution is given by a non-Hermitian effective Hamiltonian, interspersed with stochastic quantum jumps. Subsequently, evolution of ρ is approximated by an ensemble average of wavefunctions, or trajectories, say ψi. For a large number of trajectories, z, the average of any observable is then given by

The effective non-Hermitian Hamiltonian for the non-equilibrium cavity model in Eq. (¹) is given by

where J_k are the jump operators defining the stochastic dynamics. In the non-equilibrium cavity model, coherences cannot be created by Heff. Hence, if a quantum trajectory starts in a particular number state, say ψi(0)=n0,m0, the action of Heff alone does not change the state in this number basis. The complete dynamics of the trajectory is simply governed by the stochastic jumps J_k, occuring at rates R_k:

A particular quantum trajectory is constructed by drawing a series of stochastic events, with their individual probabilities proportional to the rates R_k. The time between consecutive events is drawn from an exponential distribution, whose mean is the inverse of total rate of events. From a large ensemble of trajectories, we can calculate the non-stationary second-order correlation function, g⁽²⁾(t₁, t₂), as

where 〈⋅〉 denotes ensemble average, over the entire set of trajectories. The same approximations as discussed in “Two-time statistics” section are assumed here as well.