We generalize the Power-Zineau-Woolley transformation to obtain a canonical Hamiltonian of cavity quantum electrodynamics for arbitrary geometry of boundaries. This Hamiltonian is free from the A-square term and the instantaneous Coulomb interaction between distinct atoms. The single-mode models of cavity QED (Dicke, Tavis-Cummings, Jaynes-Cummings) are justified by a term by term mapping to the proposed microscopic Hamiltonian. As one straightforward consequence, the basis of no-go argumentations concerning the Dicke phase transition with atoms in electromagnetic fields dissolves.PACS numbers: 05.30. Rt,37.30.+i,42.50.Nn,42.50.Pq The fundamental description of the interaction of atomistic matter with the electromagnetic field in the Coulomb gauge is known to suffer from the presence of an awkward term containing the square of the vector potential. In most of the practical cases, in the framework of a diluteness assumption for the atoms, this term can be neglected and the observable effects are ultimately accounted for in terms of a simplified model, such as the Jaynes-Cummings one, for example. In typical quantum optical systems, such a phenomenological approach with properly adjusted parameters usually gives a satisfactory quantitative accuracy. However, there are situations in which even the qualitative behaviour of the system is questionable because of the confusion around this term. A prominent example is the Dicke model, where the very existence of the predicted superradiant phase transition depends on the validity of the adopted effective model. [1][2][3][4] Further discrepancies due to the A-square term occur in relation with novel artificial systems in which the electromagnetic field confined into a small volume is coupled to some kind of polarizable material in the so-called ultrastrong coupling regime. [5,6] In this Letter, we show that cavity quantum electrodynamics, i.e., when the field itself as well as the light-matter interaction are significantly influenced by the presence of boundaries, can be established at a fundamental level on a Hamiltonian which eliminates the problem of the A-square term. We present a canonical transformation which makes manifest that this term is compensated by a dipole-dipole interaction term, and the remaining terms are of a simple linear form.[7] From our approach it follows, for example, that there is no principle that would prevent the superradiant phase transition in the case of an ensemble of atomic dipoles in a cavity. The canonical transformation is analogous to the Power-Zienau-Woolley (PZW) transformation in free space, however, in our approach we allow for arbitrary geometry, thereby treating general cavity QED system. All our vector fields are thus defined on a generic (possibly even multiply connected) domain D in the threedimensional real space bounded by (possibly several disjunct) sufficiently smooth surfaces ∂D, which consist of a perfect conductor. Overall, D is assumed to be bounded.Consider an arbitrary number of point charges coupled to the elect...

We study spatial self-organisation and dynamical phase-space compression of a dilute cold gas of laser-illuminated polarisable particles in an optical resonator. Deriving a non-linear Fokker-Planck equation for the particles' phase-space density allows us to treat arbitrarily large ensembles in the far-detuning limit and explicitly calculate friction forces, momentum diffusion and steady-state temperatures. In addition, we calculate the self-organisation threshold in a self-consistent analytic form. For a homogeneous ensemble below threshold the cooling rate for fixed laser power is largely independent of the particle number. Cooling leads to a q-Gaussian velocity distribution with a steady-state temperature determined by the cavity linewidth. Numerical simulations using large ensembles of particles confirm the analytical threshold condition for the appearance of an ordered state, where the particles are trapped in a periodic pattern and can be cooled to temperatures close to a single vibrational excitation.

Collective off-resonant scattering of coherent light by a cold gas induces long-range interactions via interference of light scattered by different particles. In a 1D configuration, these interactions grow particularly strong by coupling the particles via an optical nanofiber. Above a threshold pump laser intensity, we predict a phase transition from a homogeneous density to a self-sustained crystalline order. In the dispersive regime, we determine the critical condition for the onset of order as well as the forms of gas density and electric field patterns above threshold. Surprisingly, there can coexist multiple ordered states with distinct appearances.

In a recent work of ours [1], we generalized the Power-Zineau-Woolley gauge to describe the electrodynamics of atoms in an arbitrary confined geometry. Here we complement the theory by proposing a tractable form of the polarization field to represent atomic material with well-defined intra-atomic potential. The direct electrostatic dipole-dipole interaction between the atoms is cancelled. This theory yields a suitable framework to determine limitations on the light-matter coupling in quantum optical models with discernible atoms. We find that the superradiant criticality is at the border of covalent molecule formation and crystallization.

We investigate the possibility of a Dicke-type superradiant phase transition of an atomic gas with an extended model which takes into account the short-range depolarizing interactions between atoms approaching each other as close as the atomic size scale, which interaction appears in a regularized electric-dipole picture of the QED of atoms. By using a mean field model, we find that a critical density does indeed exist, though the atom-atom contact interaction shifts it to a higher value than it can be obtained from the bare Dicke-model. We argue that the system, at the critical density, transitions to the condensed rather than the "superradiant" phase.

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The collective dynamics of mobile scatterers and light in optical resonators generates complex behaviour. For strong transverse illumination a phase transition from homogeneous to crystalline particle order appears. In contrast, a gas inside a single-side pumped ring cavity exhibits an instability towards bunching and collective acceleration called collective atomic recoil lasing (CARL). We demonstrate that by driving two orthogonally polarized counter propagating modes of a ring resonator one realises both cases within one system. The corresponding phase diagram depending on the two pump intensities exhibits regions in which either a generalized form of self-ordering towards a travelling density wave with constant centre of mass velocity or a CARL instability is formed. Controlling the cavity driving then allows to accelerate or slow down and trap a sufficiently dense beam of linearly polarizable particles.

We study the dynamics of a multispecies mixture of laserilluminated polarizable particles moving inside an optical resonator. Above a certain pump threshold the collective enhanced scattering of laser light into the cavity induces a phase transition from a homogeneous spatial distribution to a common crystalline order. We analytically show that adding particles of any mass and temperature always strictly lowers the minimum pump power required for self-ordering and trapping. This allows to capture and trap new species of atoms, molecules or even polarizable nanoparticles in combination with proven examples, for which a high phase-space density is readily available. Cooperative light scattering mediates effective energy exchange and thus sympathetic cooling between different species without the need for direct collisional interaction. The predicted ordering thresholds and cooling timescales are in the range of current technology for particles with a wide range of mass, polarizability and initial temperature.