We review state-of-the-art theory and experiment of the motion of cold and ultracold atoms coupled to the radiation field within a high-finesse optical resonator in the dispersive regime of the atom-field interaction with small internal excitation. The optical dipole force on the atoms together with the back-action of atomic motion onto the light field gives rise to a complex nonlinear coupled dynamics. As the resonator constitutes an open driven and damped system, the dynamics is non-conservative and in general enables cooling and confining the motion of polarizable particles. In addition, the emitted cavity field allows for real-time monitoring of the particle's position with minimal perturbation up to sub-wavelength accuracy. For many-body systems, the resonator field mediates controllable long-range atom-atom interactions, which set the stage for collective phenomena. Besides correlated motion of distant particles, one finds critical behavior and non-equilibrium phase transitions between states of different atomic order in conjunction with superradiant light scattering. Quantum degenerate gases inside optical resonators can be used to emulate opto-mechanics as well as novel quantum phases like supersolids and spin glasses. Non-equilibrium quantum phase transitions, as predicted by e.g. the Dicke Hamiltonian, can be controlled and explored in real-time via monitoring the cavity field. In combination with optical lattices, the cavity field can be utilized for non-destructive probing Hubbard physics and tailoring long-range interactions for ultracold quantum systems.Comment: 55 page review pape

We theoretically investigate the correlated dynamics of N coherently driven atoms coupled to a standing-wave cavity mode. For red detuning between the driving field and the cavity as well as the atomic resonance frequencies, we predict a light force induced self-organization of the atoms into one of two possible regular patterns, which maximize the cooperative scattering of light into the cavity field. Kinetic energy is extracted from the atoms by superradiant light scattering to reach a final kinetic energy related to the cavity linewidth. The self-organization starts only above a threshold of the pump strength and atom number. We find a quadratic dependence of the cavity mode intensity on the atom number, which demonstrates the cooperative effect.

We show that the motion of a laser-driven Bose-Einstein condensate in a high-finesse optical cavity realizes the spin-boson Dicke-model. The quantum phase transition of the Dicke-model from the normal to the superradiant phase corresponds to the self-organization of atoms from the homogeneous into a periodically patterned distribution above a critical driving strength. The fragility of the ground state due to photon measurement induced back action is calculated.PACS numbers: 05.30. Rt,37.30.+i,42.50.Nn A thermal cloud of cold atoms interacting with a single mode of a high-finesse optical cavity can undergo a phase transition when tuning the power of a laser field which illuminates the atoms from a direction perpendicular to the cavity axis [1,2,3,4]. Below a threshold power, the thermal fluctuations stabilize the homogeneous distribution of the cloud, and photons scattered by the atoms into the cavity destructively interfere, rendering the mean optical field to be zero. Above threshold, the atoms self-organize into a wavelength-periodic crystalline order bound by the radiation field which, in this case, is composed of the constructive interference of photons scattered off the atoms from the laser into the cavity. The same phase transition can happen for Bose-Einstein condensed ultra-cold atoms, that is exempt from thermal fluctuations. For low pump power at zero temperature, the homogeneous phase is stabilized by the kinetic energy and the atom-atom collisions, a sharp transition threshold is thus expected [5,6]. In both examples the selforganization is a non-equilibrium phase transition with the distinct phases being stationary states of the drivendamped dynamics.In this paper we show that the Hamiltonian underlying the spatial self-organization is analogous to the Dicke-type Hamiltonian [7] and the transition to the selforganized phase can thus be identified with the superradiant quantum phase transition [8]. Hence, the quantum motion of ultracold atoms in a cavity effectively realizes the Dicke model and may lead to the first experimental studies on this paradigmatic system. The accessibility of such a Hamiltonian dynamics is limited by the coupling to the environment. We explore how quantum noise infiltrates and depletes the ground state [9], imposing thereby a condition on the time duration allowed for the adiabatic variation of the macroscopically populated ground state by means of tuning an external parameter.We consider a zero-temperature Bose-Einstein condensate of a number of N atoms of mass m which is inside a high-Q optical cavity with a single quasi-resonant mode of frequency ω C . Such a system has been realized and manipulated in several recent experiments [10,11,12,13,14,15]. The atoms are coherently driven from the side by a pump laser field. The pump laser frequency ω is detuned far below the atomic resonance frequency ω A , so that the atom-pump (red) detuning ∆ A = ω − ω A far exceeds the rate of spontaneous emission. One can then adiabatically eliminate the excited atomic level and the at...

The spatial self-organization of a Bose-Einstein condensate (BEC) in a high-finesse linear optical cavity is discussed. The condensate atoms are laser-driven from the side and scatter photons into the cavity. Above a critical pump intensity the homogeneous condensate evolves into a stable pattern bound by the cavity field. The transition point is determined analytically from a mean-field theory. We calculate the lowest lying Bogoliubov excitations of the coupled BEC-cavity system and the quantum depletion due to the atom-field coupling.PACS. 03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow -37.10.Vz Mechanical effects of light on atoms, molecules, and ions ModelWe consider a pure Bose-Einstein condensate (BEC) interacting with a single-mode of a high-Q optical cavity. The condensate atoms are coherently driven from the side arXiv:0801.4771v2 [quant-ph]

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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. in 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 laboratory timescale, a hysteresis masks the mean-field behaviour. 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.

The quantum phase transition of the Dicke-model has been observed recently in a system formed by motional excitations of a laser-driven Bose-Einstein condensate coupled to an optical cavity [1]. The cavity-based system is intrinsically open: photons can leak out of the cavity where they are detected. Even at zero temperature, the continuous weak measurement of the photon number leads to an irreversible dynamics towards a steady-state which exhibits a dynamical quantum phase transition. However, whereas the critical point and the mean field is only slightly modified with respect to the phase transition in the ground state, the entanglement and the critical exponents of the singular quantum correlations are significantly different in the two cases.

We present a time-dependent quantum calculation of the scattering of a few-photon pulse on a single atom. The photon wave packet is assumed to propagate in a transversely strongly confined geometry, which ensures strong atom-light coupling and allows a quasi-one-dimensional treatment. The amplitude and phase of the transmitted, reflected, and transversely scattered part of the wave packet strongly depend on the pulse length ͑bandwidth͒ and energy. For a transverse mode size of the order of 2 , we find nonlinear behavior for a few photons already, or even for a single photon. In a second step we study the collision of two such wave packets at the atomic site and find striking differences between the Fock state and coherent state wave packets of the same photon number.