Disorder and synchronization in nonequilibrium Bose-Einstein condensates

Abstract

In this thesis, we investigate the impact of spatial disorder on driven-dissipative Bose-Einstein condensates (BECs). Bose-Einstein condensation is a collective phenomenon where many particles spontaneously occupy a single energy level and behave as a single quantum state. In recent years, new types of BECs have been made from part-light, part-matter particles called polaritons, which occur in semiconductor microcavities. Such condensates are nonequilibrium in nature, as polaritons have finite lifetimes, and so a population must be maintained through pumping. The nonequilibrium nature of these new condensates gives rise to new and interesting behaviour, distinct from traditional equilibrium BECs. We first consider a double well configuration where two condensates with different energies are localized on either side of a potential barrier. Depending on a number of factors including the energy detuning between wells, the density of particles in each condensate, and their ability to tunnel between wells, the frequencies of the condensates may be either desynchronized or synchronized. We extend previous work, which characterizes the synchronized and desynchronized regimes of this configuration, to include the effect of pumping of the condensates in each well with different strengths. This is done in the framework of a non-equilibrium extension of the Gross-Pitaevskii equation, where the phase difference between condensates is shown to behave like an overdamped pendulum. We find that this pump asymmetry acts as an effective detuning, shifting the position of the phase boundary between synchronized and desynchronized states. We then generalize the analysis of the double well to the case of lattices of many localized condensates. We derive a description for this system in terms of coupled equations for the phase of the condensate at each site. We demonstrate the similarities between this model and the Kuramoto model of coupled oscillators. Unlike the Kuramoto model, however, this lattice model permits a synchronized solution in the thermodynamic limit, and thus exhibits a phase transition. We demonstrate this through mapping to a continuum description, and outlining the connection of the model to the quantum description of a particle moving in a random potential. We produce a phase diagram characterizing the synchronized and desynchronized regimes of this system, and demonstrate the agreement between our theory and numerical simulations. Following this, we consider whether such a synchronized lattice of condensates may exhibit superfluidity. This may be tested by applying a phase twist across the boundaries of the system and measuring its energy response. By way of numerical simulations of our lattice oscillator model of driven-dissipative condensates, we confirm previous results which find that disorder inhibits superfluidity. While a uniform lattice with no detunings between sites displays a non-zero superfluid stiffness, this disappears when disorder is present in the on-site energies. We present a further perspective on this result by discussing it in the context of the connection to the wavefunction of a particle localized in a random potential. We also analyse correlation functions of the synchronized, disordered lattice in one dimension. By plotting the spatial correlation function of the condensate order parameter, we show that this system exhibits long-range phase order. This is contrasted with correlation functions of the same system, but with the static disorder replaced by spatio-temporal noise. These noisy correlation functions are seen to decay exponentially with distance. Finally, we consider the impact of both types of disorder simultaneously ? time-independent random on-site energies, and spatio-temporal noise. We find that while the phase profiles of such systems have the same long-range structure that is seen for synchronized lattices with static disorder, the additional fluctuations cause the correlation functions to decay. These numerical results enable us to draw a phase diagram of the regimes of phase and frequency ordering in 1D as a function of the strengths of both the static disorder and the noise.