Bose-Einstein Condensation Of Photons Research Articles

Observation of nonlinear response and Onsager regression in a photon Bose-Einstein condensate

The quantum regression theorem states that the correlations of a system at two different times are governed by the same equations of motion as the single-time averages. This provides a powerful framework for the investigation of the intrinsic microscopic behaviour of physical systems by studying their macroscopic response to a controlled external perturbation. Here we experimentally demonstrate that the two-time particle number correlations in a photon Bose-Einstein condensate inside a dye-filled microcavity exhibit the same dynamics as the response of the condensate to a sudden perturbation of the dye molecule bath. This confirms the regression theorem for a quantum gas, and, moreover, demonstrates it in an unconventional form where the perturbation acts on the bath and only the condensate response is monitored. For strong perturbations, we observe nonlinear relaxation dynamics which our microscopic theory relates to the equilibrium fluctuations, thereby extending the regression theorem beyond the regime of linear response.

Quantum matter and gravitation: photons in a waveguide

The conditions required by quantum matter to modify space-time geometry are explored within the framework of the general theory of relativity. The required characteristics for space-time modification in solid state structures, are met in either (a) massive photon Bose–Einstein condensate in a waveguide, or (b) the massive photons in superconductor’s bulk, or (c) the Bose–Einstein condensate of acoustic phonons, or (d) a metal–insulator-topological insulator heterostructure.

Dicke superradiance, Bose-Einstein condensation of photons and spontaneous symmetry breaking

Dicke superradiance, Bose-Einstein condensation of photons and spontaneous symmetry breaking

Photon thermalization and a condensation phase transition in an electrically pumped semiconductor microresonator.

We report on an experimental study of photon thermalization and condensation in a semiconductor microresonator in the weak-coupling regime. We measure the dispersion relation of light and the photon mass in a single-wavelength, broad-area resonator. The observed luminescence spectrum is compatible with a room-temperature, thermal-equilibrium distribution. A phase transition, identified by a saturation of the population at high energies and a superlinear increase of the occupation at low energy, takes place when the phase-space density is of order unity. We explain our observations by Bose-Einstein condensation of photons in equilibrium with a particle reservoir and discuss the relation with laser emission.

Fluctuation dynamics of an open photon Bose-Einstein condensate

Bosonic gases coupled to a particle reservoir have proven to support a regime of operation where Bose-Einstein condensation coexists with unusually large particle-number fluctuations. Experimentally, this situation has been realized with two-dimensional photon gases in a dye-filled optical microcavity. Here, we investigate theoretically and experimentally the open-system dynamics of a grand canonical Bose-Einstein condensate of photons. We identify a regime with temporal oscillations of the second-order coherence function $g^{(2)}(\tau)$, even though the energy spectrum closely matches the predictions for an equilibrium Bose-Einstein distribution and the system is operated deeply in the regime of weak light-matter coupling. The observed temporal oscillations are attributed to the nonlinear, weakly driven-dissipative nature of the system which leads to time-reversal symmetry breaking.

Collective modes of a photon Bose–Einstein condensate with thermo-optic interaction

Although for photon Bose–Einstein condensates the main mechanism of the observed photon–photon interaction has already been identified to be of a thermo-optic nature, its influence on the condensate dynamics is still unknown. Here a mean-field description of this effect is derived, which consists of an open-dissipative Schrödinger equation for the condensate wave function coupled to a diffusion equation for the temperature of the dye solution. With this system at hand, the lowest-lying collective modes of a harmonically trapped photon Bose–Einstein condensate are calculated analytically via a linear stability analysis. As a result, the collective frequencies and, thus, the strength of the effective photon–photon interaction turn out to strongly depend on the thermal diffusion in the cavity mirrors. In particular, a breakdown of the Kohn theorem is predicted, i.e. the frequency of the centre-of-mass oscillation is reduced due to the thermo-optic photon–photon interaction.

Theory of Bose condensation of light via laser cooling of atoms

A Bose-Einstein condensate (BEC) is a quantum phase of matter achieved at low temperatures. Photons, one of the most prominent species of bosons, do not typically condense due to the lack of a particle number-conservation. We recently described a photon thermalization mechanism which gives rise to a grand canonical ensemble of light with effective photon number conservation between a subsystem and a particle reservoir. This mechanism occurs during Doppler laser cooling of atoms where the atoms serve as a temperature reservoir while the cooling laser photons serve as a particle reservoir. Here we address the question of the possibility of a BEC of photons in this laser cooling photon thermalization scenario and theoretically demonstrate that a Bose condensation of photons can be realized by cooling an ensemble of two-level atoms (realizable with alkaline earth atoms) inside a Fabry-Perot cavity.

Bose\u2013Einstein condensation of photons in an erbium\u2013ytterbium co-doped fiber cavity

Bose–Einstein condensation (BEC) is a special many-boson phenomenon that was observed in atomic particles at ultra-low temperatures. Later, BEC was also shown for non-atomic bosons, such as photons. Those experiments were usually done in micron-size cavities, where the power (particle number) was varied, and not the temperature, until condensation was reached. Here we demonstrate BEC of photons in a few-meters-long one-dimensional (1D) erbium–ytterbium co-doped fiber cavity at, below and above room temperature, between 100 K and 415 K. The experiments were done at about the 1550 nm wavelength regime having a few to tens of μW intra-cavity light power (107−108 photons). By varying the power and also the temperature, we found linear dependence of the condensation on power for various temperatures and of the critical power (for condensation) on temperature. These findings agree, functionally and quantitatively, with the theoretical BEC prediction without any adjustable parameter.

Interplay of coherent and dissipative dynamics in condensates of light**This paper is dedicated to the memory of Tobias Brandes

Based on the Lindblad master equation approach we obtain a detailed microscopic model of photons in a dye-filled cavity, which features condensation of light. To this end we generalise a recent non-equilibrium approach of Kirton and Keeling such that the dye-mediated contribution to the photon–photon interaction in the light condensate is accessible due to an interplay of coherent and dissipative dynamics. We describe the steady-state properties of the system by analysing the resulting equations of motion of both photonic and matter degrees of freedom. In particular, we discuss the existence of two limiting cases for steady states: photon Bose–Einstein condensate and laser-like. In the former case, we determine the corresponding dimensionless photon–photon interaction strength by relying on realistic experimental data and find a good agreement with previous theoretical estimates. Furthermore, we investigate how the dimensionless interaction strength depends on the respective system parameters.

Bose-Einstein condensation of photons from the thermodynamic limit to small photon numbers

Photons can come to thermal equilibrium at room temperature by scattering multiple times from a fluorescent dye. By confining the light and dye in a microcavity, a minimum energy is set and the photons can then show Bose–Einstein condensation. We present here the physical principles underlying photon thermalization and condensation, and review the literature on the subject. We then explore the ‘small’ regime where very few photons are needed for condensation. We compare thermal equilibrium results to a rate-equation model of microlasers, which includes spontaneous emission into the cavity, and we note that small systems result in ambiguity in the definition of threshold.

Polarization dynamics in a photon Bose-Einstein condensate

It has previously been shown that a dye-filled microcavity can produce a Bose-Einstein condensate of photons. Thermalization of photons is possible via repeated absorption and re-emission by the dye molecules. In this paper, we theoretically explore the behavior of the polarization of light in this system. We find that in contrast to the near complete thermalization between different spatial modes of light, thermalization of polarization states is expected to generally be incomplete. We show that the polarization degree changes significantly from below to above threshold, and explain the dependence of polarization on all relevant material parameters.

Coherent coupling between an optomechanical membrane and an interacting photon Bose–Einstein condensate

In this paper, we study theoretically the optomechanical interaction of an interacting condensate of photons with an oscillating mechanical membrane in a microcavity. We show that in the Bogoliubov approximation, due to the large number of photons in the condensate, there is a linear strong effective coupling between the Bogoliubov mode of the photonic Bose–Einstein condensate (BEC) and the mechanical motion of the membrane which depends on the photon–photon scattering potential. This coupling leads to the cooling of the mechanical motion, the normal mode splitting (NMS), the squeezing of the output field and the entanglement between the excited mode of the cavity and the mechanical mode. Since the photon condensation occurs at room temperature, this hybrid system can be potentially considered as a room temperature source of squeezed light as well as a suited candidate for exploring the quantum effects. We show that, on one hand, the non-linearity of the photon gas increases the degree of the squeezing of the output field of the microcavity and the efficiency of the cooling process at high temperatures. On the other hand, it reduces the NMS in the displacement spectrum of the oscillating membrane and the degree of the optomechanical entanglement. In addition, the temperature of the photonic BEC can be used to control the above-mentioned phenomena.

Spontaneous Symmetry Breaking and Phase Coherence of a Photon Bose-Einstein Condensate Coupled to a Reservoir.

We examine the phase evolution of a Bose-Einstein condensate of photons generated in a dye microcavity by temporal interference with a phase reference. The photoexcitable dye molecules constitute a reservoir of variable size for the condensate particles, allowing for grand canonical statistics with photon bunching, as in a lamp-type source. We directly observe phase jumps of the condensate associated with the large statistical number fluctuations and find a separation of correlation time scales. For large systems, our data reveal phase coherence and a spontaneously broken symmetry, despite the statistical fluctuations.

Coexistence of photonic and atomic Bose-Einstein condensates in ideal atomic gases

We have studied conditions of photon Bose-Einstein condensate formation that is in thermodynamic equilibrium with ideal gas of two-level Bose atoms below the degeneracy temperature. Equations describing thermodynamic equilibrium in the system were formulated; critical temperatures and densities of photonic and atomic gas subsystems were obtained analytically. Coexistence conditions of these photonic and atomic Bose-Einstein condensates were found. There was predicted the possibility of an abrupt type of photon condensation in the presence of Bose condensate of ground-state atoms: it was shown that the slightest decrease of the temperature could cause a significant gathering of photons in the condensate. This case could be treated as a simple model of the situation known as "stopped light" in cold atomic gas. We also showed how population inversion of atomic levels can be created by lowering the temperature. The latter situation looks promising for light accumulation in atomic vapor at very low temperatures.

Bose-Einstein condensation of photons with nonlocal nonlinearity in a dye-doped graded-index microcavity

We consider a microcavity made by a graded-index (GRIN) glass, doped by dye molecules, placed within two planar mirrors and study Bose-Einstein condensation (BEC) of photons. The presence of the mirrors leads to an effective photon mass, and the index grading provides an effective trapping frequency; the photon gas becomes formally equivalent to a two dimensional Bose gas trapped in an isotropic harmonic potential. The inclusion of nonlinear effects provides an effective interaction between photons. We discuss, in particular, thermal lensing effects and nonlocal nonlinearity, and quantitatively compare our results with the reported experimental data.

Observation of Grand-Canonical Number Statistics in a Photon Bose-Einstein Condensate

We report measurements of particle number correlations and fluctuations of a photon Bose-Einstein condensate in a dye microcavity using a Hanbury Brown-Twiss experiment. The photon gas is coupled to a reservoir of molecular excitations, which serve as both heat bath and particle reservoir to realize grand-canonical conditions. For large reservoirs, we observe strong number fluctuations of the order of the total particle number extending deep into the condensed phase. Our results demonstrate that Bose-Einstein condensation under grand-canonical ensemble conditions does not imply second-order coherence.

Schwinger-Keldysh theory for Bose-Einstein condensation of photons in a dye-filled optical microcavity

We consider Bose-Einstein condensation of photons in an optical cavity filled with dye molecules that are excited by laser light. By using the Schwinger-Keldysh formalism we derive a Langevin field equation that describes the dynamics of the photon gas, and in particular its equilibrium properties and relaxation towards equilibrium. Furthermore we show that the finite lifetime effects of the photons are captured in a single dimensionless damping parameter, that depends on the power of the external laser pumping the dye. Finally, as applications of our theory we determine spectral functions and collective modes of the photon gas in both the normal and the Bose-Einstein condensed phase.

Bose-Einstein condensation of light: General theory

A theory of Bose-Einstein condensation of light in a dye-filled optical microcavity is presented. The theory is based on the hierarchical maximum entropy principle and allows one to investigate the fluctuating behavior of the photon gas in the microcavity for all numbers of photons, dye molecules, and excitations at all temperatures, including the whole critical region. The master equation describing the interaction between photons and dye molecules in the microcavity is derived and the equivalence between the hierarchical maximum entropy principle and the master equation approach is shown. The cases of a fixed mean total photon number and a fixed total excitation number are considered, and a much sharper, nonparabolic onset of a macroscopic Bose-Einstein condensation of light in the latter case is demonstrated. The theory does not use the grand canonical approximation, takes into account the photon polarization degeneracy, and exactly describes the microscopic, mesoscopic, and macroscopic Bose-Einstein condensation of light. Under certain conditions, it predicts sub-Poissonian statistics of the photon condensate and the polarized photon condensate, and a universal relation takes place between the degrees of second-order coherence for these condensates. In the macroscopic case, there appear a sharp jump in the degrees of second-order coherence, a sharp jump and kink in the reduced standard deviations of the fluctuating numbers of photons in the polarized and whole condensates, and a sharp peak, a cusp, of the Mandel parameter for the whole condensate in the critical region. The possibility of nonclassical light generation in the microcavity with the photon Bose-Einstein condensate is predicted.

Theory of Bose-Einstein condensation of light in a microcavity

A theory of Bose-Einstein condensation (BEC) of light in a dye microcavity is developed. The photon polarization degeneracy and the interaction between dye molecules and photons in all of the cavity modes are taken into account. The theory goes beyond the grand canonical approximation and allows one to determine the statistical properties of the photon gas for all numbers of dye molecules and photons at all temperatures, thus describing the microscopic, mesoscopic, and macroscopic light BEC from a general perspective. A universal relation between the degrees of second-order coherence for the photon condensate and the polarized photon condensate is obtained. The photon Bose-Einstein condensate can be used as a new source of nonclassical light.

Hierarchical maximum entropy principle for generalized superstatistical systems and Bose-Einstein condensation of light

A principle of hierarchical entropy maximization is proposed for generalized superstatistical systems, which are characterized by the existence of three levels of dynamics. If a generalized superstatistical system comprises a set of superstatistical subsystems, each made up of a set of cells, then the Boltzmann-Gibbs-Shannon entropy should be maximized first for each cell, second for each subsystem, and finally for the whole system. Hierarchical entropy maximization naturally reflects the sufficient time-scale separation between different dynamical levels and allows one to find the distribution of both the intensive parameter and the control parameter for the corresponding superstatistics. The hierarchical maximum entropy principle is applied to fluctuations of the photon Bose-Einstein condensate in a dye microcavity. This principle provides an alternative to the master equation approach recently applied to this problem. The possibility of constructing generalized superstatistics based on a statistics different from the Boltzmann-Gibbs statistics is pointed out.