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Here we describe a weakly interacting Bose gas on a curved smooth manifold, which is embedded in the three-dimensional Euclidean space. To this end we start by considering a harmonic trap in the normal direction of the manifold, which confines the three-dimensional Bose gas in the vicinity of its surface. Following the notion of dimensional reduction as outlined in [L Salasnich et al, Phys. Rev. A 65, 043614 (2002)], we assume a large enough trap frequency so that the normal degree of freedom of the condensate wave function can be approximately integrated out. In this way we obtain an effective condensate wave function on the quasi-two-dimensional surface of the curved manifold, where the thickness of the cloud is determined self-consistently. For the particular case when the manifold is a sphere, our equilibrium results show how the chemical potential and the thickness of the cloud increase with the interaction strength. Furthermore, we determine within a linear stability analysis the low-lying collective excitations together with their eigenfrequencies, which turn out to reveal an instability for attractive interactions.
We investigate, both experimentally and theoretically, the static geometric properties of a harmonically trapped Bose–Einstein condensate of 6Li2 molecules in laser speckle potentials. Experimentally, we measure the in situ column density profiles and the corresponding transverse cloud widths over many laser speckle realizations. We compare the measured widths with a theory that is non-perturbative with respect to the disorder and includes quantum fluctuations. Importantly, for small disorder strengths we find quantitative agreement with the perturbative approach of Huang and Meng, which is based on Bogoliubov theory. For strong disorder our theory perfectly reproduces the geometric mean of the measured transverse widths. However, we also observe a systematic deviation of the individual measured widths from the theoretically predicted ones. In fact, the measured cloud aspect ratio monotonously decreases with increasing disorder strength, while the theory yields a constant ratio. We attribute this discrepancy to the utilized local density approximation, whose possible failure for strong disorder suggests a potential future improvement.
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.
Photon Bose–Einstein condensates are characterised by a quite weak interaction, so they behave nearly as an ideal Bose gas. Moreover, since the current experiments are conducted in a microcavity, the longitudinal motion is frozen out and the photon gas represents effectively a two-dimensional trapped gas of massive bosons. In this paper we focus on a harmonically confined ideal Bose gas in two dimensions, where the anisotropy of the confinement allows for a dimensional crossover. If the confinement in one direction is strong enough so that this squeezed direction is frozen out, then only one degree of freedom survives and the system can be considered to be quasi-one dimensional. In view of an experimental set-up we work out analytically the thermodynamic properties for such a system with a finite number of photons. In particular, we focus on examining the dimensional information which is contained in the respective thermodynamic quantities.
Since the advent of experiments with photon Bose–Einstein condensates (phBECs) in dye-filled microcavities in 2010, many investigations have focussed upon the emerging effective photon–photon interaction. Despite its smallness, it can be identified to stem from two physically distinct mechanisms. On the one hand, a Kerr nonlinearity of the dye medium yields a photon–photon contact interaction. On the other hand, a heating of the dye medium leads to an additional thermo-optic interaction, which is both delayed and non-local. The latter turns out to represent the leading contribution to the effective interaction for the current 2D experiments. Here we analyse theoretically how the effective photon–photon interaction increases when the system dimension is reduced from 2D to 1D. To this end, we consider an anisotropic harmonic trapping potential and determine via a variational approach how the properties of the phBEC in general, and both aforementioned interaction mechanisms in particular, change with increasing anisotropy. We find that the thermo-optic interaction strength increases at first linearly with the trap aspect ratio and later on saturates at a certain value of the trap aspect ratio. Furthermore, in the strong 1D limit the roles of both interactions get reversed as the thermo-optic interaction remains saturated and the contact Kerr interaction becomes the leading interaction mechanism. Finally, we discuss how the predicted effects can be measured experimentally.
Thermo-optic interaction significantly differs from the usual particle-particle interactions in physics, as it is retarded in time. A prominent platform for realising this kind of interaction are photon Bose–Einstein condensates, which are created in dye-filled microcavities. The dye solution continually absorbs and re-emits these photons, causing the photon gas to thermalize and to form a Bose–Einstein condensate. Because of a non-ideal quantum efficiency, these cycles heat the dye solution, creating a medium that provides an effective thermo-optic photon–photon interaction. So far, only a mean-field description of this process exists. This paper goes beyond by working out a quantum mechanical description of the effective thermo-optic photon–photon interaction. To this end, the self-consistent modelling of the temperature diffusion builds the backbone of the modelling. Furthermore, the manyfold experimental timescales allow for deriving an approximate Hamiltonian. The resulting quantum theory is applied in the perturbative regime to both a harmonic and a box potential for investigating its prospect for precise measurements of the effective photon–photon interaction strength.
Although photon Bose–Einstein condensates have already been used for studying many interesting effects, the precise role of the photon–photon interaction is not fully clarified up to now. In view of this, it is advantageous that these systems allow measuring both the intensity of the light leaking out of the cavity and its spectrum at the same time. Therefore, the photon–photon interaction strength can be determined once via analysing the condensate broadening and once via examining the interaction-induced modifications of the cavity modes. As the former method depends crucially on the concrete shape of the trapping potential and the spatial resolution of the used camera, interferometric methods promise more precise measurements. To this end, the present paper works out the impact of the photon–photon interaction upon the cavity modes. A quantum mechanical description of the photon–photon interaction, including the thermal cloud, builds the theoretical backbone of the method. An exact diagonalisation approach introduced here exposes how the effective photon–photon interaction modifies both the spectrum and the width of the photon gas. A comparison with a variational approach based on the Gross–Pitaevskii equation quantifies the contribution of the thermal cloud in the respective applications.
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.