Atom-field entanglement is shown to play a crucial role for the onset of spatial self-organization of ultracold atoms in an optical lattice within a high-Q cavity. Like particles on a seesaw, the atoms feel a different potential depending on their spatial distribution. The system possesses two stable configurations, where all atoms occupy either only even or only odd lattice sites. While for a classical cavity field description a distribution balanced between even and odd sites is a stationary equilibrium state at zero temperature, the possibility of atom-field entanglement in a quantum field description yields an instant simultaneous decay of the homogeneous atomic cloud into an entangled superposition of the two stable atomic patterns correlated with different cavity fields. This effect could be generic for a wide class of quantum phase transitions, whenever the quantum state can act back on the control parameter.
Atom-field entanglement is shown to play a crucial role for the onset of spatial self-organization of ultracold atoms in an optical lattice within a high-Q cavity. Like particles on a seesaw, the atoms feel a different potential depending on their spatial distribution. The system possesses two stable configurations, where all atoms occupy either only even or only odd lattice sites. While for a classical cavity field description a distribution balanced between even and odd sites is a stationary equilibrium state at zero temperature, the possibility of atom-field entanglement in a quantum field description yields an instant simultaneous decay of the homogeneous atomic cloud into an entangled superposition of the two stable atomic patterns correlated with different cavity fields. This effect could be generic for a wide class of quantum phase transitions, whenever the quantum state can act back on the control parameter.
We study quantum particles at zero temperature in an optical lattice coupled to a resonant cavity mode. The cavity field substantially modifies the particle dynamics in the lattice, and for strong particle-field coupling leads to self-organization of the particles, a configuration with only every second site occupied. We study the growth of this order out of a homogeneous initial distribution for few particles. Simulations reveal that the growth dynamics crucially depends on the initial quantum many-body state of the particles and can be monitored via the cavity fluorescence. Studying the relaxation time of the ordering reveals inhibited tunnelling due to the interaction with the cavity field. However, the relaxation becomes very quick for strong coupling.
The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity.In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms.The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity.We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory.At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition.We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry.In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.
Pattern formation of atoms in high-finesse optical resonators results from the mechanical forces of light associated with superradiant scattering into the cavity mode. It occurs when the laser intensity exceeds a threshold value such that the pumping processes counteract the losses. We consider atoms driven by a laser and coupling with a mode of a standing-wave cavity and describe their dynamics with a Fokker-Planck equation, in which the atomic motion is semiclassical but the cavity field is a full quantum variable. The asymptotic state of the atoms is a thermal state, whose temperature is solely controlled by the detuning between the laser and the cavity frequency and by the cavity loss rate. From this result we derive the free energy and show that in the thermodynamic limit self-organization is a second-order phase transition. The order parameter is the field inside the resonator to which one can associate a magnetization in analogy to ferromagnetism, the control field is the laser intensity, but the steady state is intrinsically out of equilibrium. In the symmetry-broken phase, quantum noise induces jumps of the spatial density between two ordered patterns: We characterize the statistical properties of this temporal behavior at steady state and show that the thermodynamic properties of the system can be extracted by detecting the light at the cavity output. The results of our analysis are in full agreement with previous studies; we extend them by deriving a self-consistent theory which is valid also when the cavity field is in the shot-noise limit and elucidate the nature of the self-organization transition.
The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity. In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms. The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity. We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory. At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition. We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry. In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.
We present a detailed study of the spatial self-organization of laser-driven atoms in an optical cavity, an effect predicted on the basis of numerical simulations [P. Domokos and H. Ritsch, Phys. Rev. Lett. 89, 253003 (2002)] and observed experimentally [A. T. Black et al., Phys. Rev. Lett. 91, 203001 (2003)]. Above a threshold in the driving laser intensity, from a uniform distribution the atoms evolve into one of two stable patterns that produce superradiant scattering into the cavity. We derive analytic formulas for the threshold and critical exponent of this phase transition from a mean-field approach. Numerical simulations of the microscopic dynamics reveal that, on a laboratory time scale, a hysteresis masks the mean-field behavior. Simple physical arguments explain this phenomenon and provide analytical expressions for the observable threshold. Above a certain density of the atoms a limited number of ``defects'' appear in the organized phase and influence the statistical properties of the system. The scaling of the cavity cooling mechanism and the phase-space density with the atom number is also studied.
This thesis summarizes the research work we did in the past years, which focuses on the theoretical study of quantum effects of ordered atomic structures interacting with the quantum electric field of an optical resonator. Within this work we investigated the quantum properties of the light at the cavity output, emitted by this kind of system, and the quantum state of the atoms confined by the light fields of the cavity mode. Following one possible partition, we have organized the thesis in two parts. The first part of the thesis is devoted to the quantum properties of the light at the output of optical cavities, in which atoms are confined. After a brief introduction on the basics of atom-photon interactions, we show theoretically, that two atomic dipoles in a resonator can behave as a quantum non-linear medium, whose response can be controlled through the interactomic distance of the atoms inside the resonator. We identify the parameter regime, for which the system operates as a parametric amplifier, based on the atoms' collective dipole cascade emission. Moreover we also determine the corresponding field squeezing spectrum at the cavity output. These dynamics could be observed as a result of self-organization processes of laser-cooled atoms in resonators. The second part of the thesis studies the quantum ground state of an ultracold atomic gas trapped inside an optical resonator by the mechanical effects of atom-photon interactions. Due to the strong coupling regime, the ultracold atomic gas feels a quantum potential, where the fluctuations at the single photon level may be relevant in determining the trapping conditions. This potential depends on the atomic distribution, which at the same time determines the intracavity field amplitude. Here, after a brief introduction on ultracold atoms in optical lattices, we study the low temperature physics of an ultracold atomic gas in the quantum potential formed inside a pumped cavity, by mapping the dynamics to an effective Bose-Hubbard model. We predict the existence of non-compressible states of the atomic gas, and determine a phase diagram as a function of the cavity parameters. Novel effects are encountered, such as the existence of overlapping stability regions corresponding to competing insulator-like states when the laser pumps the cavity, to provide one example out of many. At the end of the thesis a concluding chapter presents some general outlooks to this thesis work.