We investigate the occurrence of genuine quantum effects and beyond mean-field physics in the balanced and unbalanced open Dicke models with a large yet finite number of atoms N. Such driven and dissipative quantum many-body systems have recently been realized in experiments involving ultracold gases inside optical cavities and are known to obey mean-field predictions in the thermodynamic limit N→∞. Here we show quantum effects that survive for large but finite N, by employing a novel open-system dynamics method that allows us to obtain numerically exact quantum dynamical results for atom numbers up to a mesoscopic N≈1000. While we find that beyond-mean-field effects vanish quickly with increasing N in the balanced Dicke model, we are able to identify parameter regimes in the unbalanced Dicke model that allow genuine quantum effects to persist even for mesoscopic N. They manifest themselves in a strong squeezing of the steady state and a modification of the steady-state phase diagram that cannot be seen in a mean-field description. This is due to the fact that the steady-state limit t→∞ and thermodynamic limit N→∞ do not commute.
We investigate the occurrence of genuine quantum effects and beyond mean-field physics in the balanced and unbalanced open Dicke models with a large yet finite number of atoms N. Such driven and dissipative quantum many-body systems have recently been realized in experiments involving ultracold gases inside optical cavities and are known to obey mean-field predictions in the thermodynamic limit N→∞. Here we show quantum effects that survive for large but finite N, by employing a novel open-system dynamics method that allows us to obtain numerically exact quantum dynamical results for atom numbers up to a mesoscopic N≈1000. While we find that beyond-mean-field effects vanish quickly with increasing N in the balanced Dicke model, we are able to identify parameter regimes in the unbalanced Dicke model that allow genuine quantum effects to persist even for mesoscopic N. They manifest themselves in a strong squeezing of the steady state and a modification of the steady-state phase diagram that cannot be seen in a mean-field description. This is due to the fact that the steady-state limit t→∞ and thermodynamic limit N→∞ do not commute.
Ultracold atomic gases loaded into optical cavities constitute one of the most versatile platforms for the study of out-of-equilibrium dynamics and quantum simulation of many- body systems. In this thesis, we explore these two possibilities, motivated by current state-of-the-art atom-cavity experiments, uncovering novel phenomena and opening up new theoretical and experimental avenues into the intriguing world of quantum many- body physics. First, we study a gas of ultracold atoms coupled to three optical resonators. We show that the inherent Z2 symmetries, associated with each atom-cavity coupling, can be combined into a global SO(3) rotational symmetry which can be spontaneously broken. We determine the phase diagram of the system, which shows the emergence and breaking of the continuous symmetries and displays first- and second-order phase transitions. We argue that by coupling the atoms equally to n different modes, it will in general be possible to generate a global SO(n) symmetry. Second, we investigate the dynamics of a spinor BEC inside an optical cavity. We focus on a two-component Dicke model with complex light-matter couplings, accounting for photon losses. We compute the steady-state phase diagram and find dynamical in- stabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes that generate nonreciprocal couplings between the two collective spins. Lastly, we explore the dynamics of a BEC inside an optical cavity in the presence of an optical lattice. We derive an effective master equation by adiabatically eliminating the cavity field. In the bad-cavity regime, we find an infinite-temperature steady state, but with relaxation dynamics that can be highly nontrivial. For small hopping, the interplay between dissipation and strong interactions leads to an algebraic relaxation of the system, whereas for large hopping, the approach to the steady state is exponential. In the good- cavity regime, we show that the system allows for optical pumping between momentum modes, enabling cavity cooling.
In 1973 Klaus Hepp and Elliott Lieb showed that a tightly confined ensemble of atoms placed in a quantised single-mode optical cavity will have their emission of light enhanced by constructive interference of emission events if the photons emitted are indistinguishable. This effect, known as superradiance, has proven difficult to realise experimentally due to the strong light-matter coupling required to produce such behaviour. However, over the last decade there have been several successful realisations which engineer superradiance by driving Raman transitions between atomic ground states of multilevel atoms. In this case, superradiance emerges via a quantum phase transition mediated by the laser driving power. In this work, we consider a generalisation of this model referred to as the unbalanced Dicke model. It has recently been implemented experimentally and is characterised by two lasers driving two distinct Raman transitions. We show that, in the semiclassical limit of a large number of atoms, the unbalanced Dicke model exhibits a wealth of nonlinear dynamics which describe novel dynamical phases of matter. Examples of nonlinear behaviour observed are pitchfork, saddle-node, and Hopf bifurcations, as well as two types of chaos. We explore the quantum mechanical manifestations of these bifurcations and additional physical effects which appear in the presence of quantum fluctuations. In this way, the model creates a playground in which to explore the onset of classical nonlinear dynamics from an open quantum system.
In recent years there have been significant advances in the study of many-body interactions between atoms and light confined to optical cavities.One model that has received widespread attention of late is the Dicke model, which under certain conditions exhibits a quantum phase transition to a state in which the atoms collectively emit light into the cavity mode, known as superradiance.We consider a generalization of this model that features independently controllable strengths of the corotating and counter-rotating terms of the interaction Hamiltonian.We study this system in the semiclassical (mean field) limit, i.e., neglecting the role of quantum fluctuations.Under this approximation, the model is described by a set of nonlinear differential equations, which determine the system's semiclassical evolution.By taking a dynamical systems approach, we perform a comprehensive analysis of these equations to reveal an abundance of complex dynamics.Examples of the phenomena that we observe are the emergence of superradiant oscillations arising due to Hopf bifurcations, and the appearance of a pair of chaotic attractors arising from period-doubling cascades, followed by their collision to form a single, larger chaotic attractor via a sequence of infinitely many homoclinic bifurcations.Moreover, we find that a flip of the collective spin can result in the sudden emergence of chaotic dynamics.Overall, we provide a comprehensive road map of the possible dynamics that arise in the unbalanced, open Dicke model in the form of a phase diagram in the plane of the two interaction strengths.Hence, we lay out the foundations to make further advances in the study of the fingerprint of semiclassical chaos when considering the master equation of the unbalanced Dicke model, thus, providing the possibility of studying a manifestation of quantum chaos in a specific, experimentally realizable system.
A laser-driven Bose-Einstein condensate of ultracold atoms loaded into a lossy high-finesse optical resonator exhibits critical behavior and, in the thermodynamic limit, a phase transition between stationary states of different symmetries. The system realizes an open-system variant of the celebrated Dicke model. We study the transition for a finite number of atoms by means of a Hartree-Fock-Bogoliubov method adapted to a damped-driven open system. The finite-size scaling exponents are determined and a clear distinction between the nonequilibrium and equilibrium quantum criticality is found.
The exact steady-state solution for the reduced atomic density operator for a model system of N identical two-level atoms confined to a single site (the Dicke model) and driven by a totally incoherent broad-band (chaotic) field is derived through the equivalent Fokker-Planck equation of the system. Steady-state values of the atomic observables, higher moments and atomic fluctuations are calculated for an arbitrary N and an arbitrary field strength. It is shown that when the chaotic field is a thermal (black-body) field the atomic system is driven into a steady relative occupation number that obeys the Boltzmann distribution law only for N=1. (The same result for N>1 atoms can be reached only if the cooperative interactions between the atoms are ignored.) In the thermodynamic limit N to infinity there is no critical behaviour in contrast to the resonant coherently driven case. It is also shown that semiclassical (direct) factorisation of the equations of motion does not lead to the correct results for the N to infinity limit of the exact quantum case. Within the exact theory numerical results for finite N (<or=100) are presented for the time-dependent expectation values of the atomic operators, the intensity and the intensity autocorrelation function. Superradiant pulses are emitted by N initially inverted atoms but as the incident intensity is increased this superradiant behaviour is obscured. The steady-state fluorescence spectrum is a single Lorentzian of width proportional to N in the weak-field case. In the strong-field case the spectrum is insensitive to the number of atoms and the result is similar to the one-atom situation. Photon bunching and antibunching effects are investigated for the weak- and strong-field limits. When the chaotic driving field is simply a single-mode thermal field the approach to atomic saturation is delayed in comparison with the case of the broad-band chaotic field.
The exact steady state solution for the reduced atomic density operator for a model of N identical two-level atoms occupying the same site (the Dicke model) and driven by a CW off-resonant laser field is presented. Exact expressions are given for the atomic observables and atomic correlation functions of arbitrary order. Exact thermodynamic analysis (N to infinity ) shows that the critical phase transition behaviour found in the resonant case is destroyed due to non-resonant effects. For large N and increasing detuning parameter, quantum fluctuations tend to vanish and hence the atomic observables tend to assume their semiclassical values. Also it is found that for a fixed arbitrary detuning parameter and increasing N these fluctuations disappear for the whole range of the driving field strength. The authors explain why it is not possible in the limit N to infinity to recover the correct asymptotic limit for the resonant case from the results of the dispersive case.