Abstract
We consider the open two-site Bose–Hubbard dimer, a well-known quantum mechanical model that has been realised recently for photons in two coupled photonic crystal nanocavities. The system is described by a Lindblad master equation which, for large numbers of photons, gives rise to a limiting semiclassical model in the form of a four-dimensional vector field. From the situation where both sites trap the same amount of photons under symmetric pumping, one encounters a transition that involves symmetry breaking, the creation of periodic oscillations and multistability as the pump strength is increased. We show that the associated one-parameter bifurcation diagram of the semiclassical model captures the essence of statistical properties of computed quantum trajectories as the pump strength is increased. Even for small numbers of photons, the fingerprint of the semiclassical bifurcations can be recognised reliably in observables of quantum trajectories.
Highlights
Phase transitions describe a fundamental change in the behaviour of a system as its parameters are changed
We investigate the predictive power of the semiclassical ordinary differential equations (ODEs) model (5) by simulating Heμff in (8) for increasing values of μ; this is achieved by performing quantum trajectory simulations with the method detailed in Appendix A
We considered the case of negative intermode coupling when the semiclassical model features a transition, as the pump strength is increased, from symmetric dynamics via symmetry breaking at a pitchfork bifurcation to oscillatory dynamics and to multistability between different types of asymmetric states
Summary
Phase transitions describe a fundamental change in the behaviour of a system as its parameters are changed. We are interested here in the relationship between dynamics and bifurcations of the semiclassical ODE model and the observable behaviour of the quantum system when the number of particles of the considered system is relatively small. In this case, the system is far from the thermodynamic limit, and quantum fluctuations may be significant and cannot be neglected. Quantum simulations of the underlying Hamiltonian, which are only feasible computationally for small numbers of particles, can be used to investigate the predictive power of semiclassical models for novel types of quantum systems that operate with very few atoms and/or photons. Visualisation and postprocessing of the data are performed with Matlab R
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