Photon condensation: A new paradigm for Bose–Einstein condensation

Bose–Einstein condensation is a state of matter known to be responsible for peculiar properties exhibited by superfluid Helium-4 and superconductors. Bose–Einstein condensate (BEC) in its pure form is realizable with alkali atoms under ultra-cold temperatures. In this paper, we review the experimental scheme that demonstrates the atomic Bose–Einstein condensate. We also elaborate on the theoretical framework for atomic Bose–Einstein condensation, which includes statistical mechanics and the Gross–Pitaevskii equation. As an extension, we discuss Bose–Einstein condensation of photons realized in a fluorescent dye filled optical microcavity. We analyze this phenomenon based on the generalized Planck’s law in statistical mechanics. Further, a comparison is made between photon condensate and laser. We describe how photon condensate may be a possible alternative for lasers since it does not require an energy consuming population inversion process.     

Stationary states and rotational properties of spin–orbit-coupled Bose–Einstein condensates held under a toroidal trap

We consider a pseudospin-1/2 Bose–Einstein condensate with Rashba spin–orbit coupling in a two-dimensional toroidal trap. By solving the damped Gross–Pitaevskii equations for this system, we show that the system exhibits a rich variety of stationary states, such as vehicle wheel and flower-petal stripe patterns. These stationary states are stable against perturbation with thermal energy and can survive for a long time. In the presence of rotation, our results show that the rotating systems have exotic vortex configurations. These phenomenon originates from the interplay among spin–orbit coupling, trap geometry, and rotation.     

Interacting bosons in an optical lattice

A strongly interacting Bose gas in an optical lattice is studied using a hard‐core interaction. Two different approaches are introduced, one is based on a spin‐1/2 Fermi gas with attractive interaction, the other one on a functional integral with an additional constraint (slave‐boson approach). The relation between fermions and hard‐core bosons is briefly discussed for the case of a one‐dimensional Bose gas. For a three‐dimensional gas we identify the order parameter of the Bose‐Einstein condensate through a Hubbard‐Stratonovich transformation and treat the corresponding theories within a mean‐field approximation and with Gaussian fluctuations. This allows us to evaluate the phase diagram, including the Bose‐Einstein condensate and the Mott insulator, the density‐density correlation function, the static structure factor, and the quasiparticle excitation spectrum. The role of quantum and thermal fluctuations are studied in detail for both approaches, where we find good agreement with the Gross‐Pitaevskii equation and with the Bogoliubov approach in the dilute regime. In the dense regime, which is characterized by the phase transition between the Bose‐Einstein condensate and the Mott insulator, we discuss a renormalized Gross‐Pitaevskii equation. This equation can describe the macroscopic wave function of the Bose‐Einstein condensate in the dilute regime as well as close to the transition to the Mott insulator. Finally, we compare the results of the attractive spin‐1/2 Fermi gas and those of the slave‐boson approach and find good agreement for all physical quantities.     

Bose–Einstein condensate wave function and nonlinear Schrödinger equation

The nonlinear Schrödinger equation for the ground-state wave function of an inhomogeneous boson system is derived in the self-consistent Hartree–Fock approximation without the use of the formalism of anomalous averages. The results obtained correspond to the Gross–Pitaevskii equation for the Bose–Einstein condensate wave function when using the delta-shaped boson interaction potential.     

玻色爱因斯坦凝聚体一维轴向干涉中的非线性现象

利用含时的Ｇｒｏｓｓ－Ｐｉｔａｅｖｓｋｉｉ方程,研究了轴对称的高密度玻色爱因斯坦凝聚体在干涉过程中因原子间相互人用而产生的非线性现象。发现玻色爱因斯坦凝聚体的一维轴向干涉条纹的密度分布是一种驻波状结构。通过原子波之间非线性耦合相互作用,这种结构可以表现为物质波光栅,对其周期的原子波产生衍射现象。     

Dynamics and formation of vortices collapsed from ring dark solitons in a two-dimensional spin–orbit coupled Bose–Einstein condensate

We consider the dynamics and formation of vortices from ring dark solitons in a two-dimensional Bose–Einstein condensate with the Rashba spin–orbit coupling based on the time-dependent coupled Gross–Pitaevskii equation. Compared with previous results, the system exhibits complex dynamical behaviors in the presence of the spin–orbit coupling. With the modulation of the spin–orbit coupling, not only the lifetime of ring dark solitons is greatly prolonged, but also their attenuation kinetics is significantly affected. For two shallow ring dark solitons with the equal strength of the spin–orbit coupling, the radius of ring dark solitons increases to a maximum value over time and then shrinks into a minimum value. Due to the effect of the snake instability, ring dark solitons split into a series of ring-like clusters of vortex pairs, which perform complex oscillations. This indicates that the system is strongly dependent on the presence of the spin–orbit coupling. Furthermore, the effect of different initial modulation depths on the dynamics of ring dark solitons is investigated.     

Two-Channel Skyrme–Hartree–Fock Model for Bose–Einstein Condensate Near Feshbach Resonance

We apply a two-channel Skyrme–Hartree–Fock model to describe an atomic Bose–Einstein condensate near a Feshbach resonance. In this model the single-atom wave-function has two components corresponding to the two intrinsic states of the atom related to the Feshbach resonance. From the variational principle we derive the corresponding system of two coupled equations for the single-atom wave-function—a generalization of the Gross–Pitaevskii equation. We carry out an exploratory gaussian variational calculation and show that the two-component model can successfully describe the collapse of the condensate near a Feshbach resonance.     

Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.     

球对称非谐势阱中玻色-爱因斯坦凝聚Gross-Pitaevskii方程的精确解

考虑了描述玻色 爱因斯坦凝聚的Gross-Pitaevskii(GP)方程, 得到了在球对称非谐势阱中玻色-爱因斯坦凝聚GP方程的精确亮孤子解。In this paper, we analyze Gross Pitaevskii equation which describes the dynamics of a bright soliton in trapped atomic Bose Einstein condensates, and obtain the exact bright soliton solution of Gross Pitaevskii equation in spherically symmetric non harmonic trap.     

Internal Structure of Vortices in a Two-Component Exciton-Polariton Condensate

Vortex solutions of coupled Gross–Pitaevskii equations for a two-component Bose–Einstein condensate of exciton polaritons have been described theoretically with the inclusion of the dependence of the Rabi splitting energy on the density of the exciton component. It has been shown that the inclusion of blueshift leads to a considerable decrease in the densities of both components of the condensate. The spatial profiles of excitons and photons in the polariton system, as well as the energy of vortex excitation formation, have been calculated taking into account nonlinear corrections.     

Levitating soliton of the Bose–Einstein condensate

We have proposed a mechanical model that corresponds to the Newton equation for describing the dynamics of an oscillon, viz., a soliton-like cluster of the Bose–Einstein condensate (with atomic attraction) placed above an oscillating atomic mirror in a uniform gravitational field. The model describes the stochastic Fermi acceleration and periodic, quasi-periodic, and chaotic motion of the oscillon center, as well as hysteresis phenomena in the case of a slow variation of mirror oscillation frequency, which are in good agreement with the results obtained using the Gross–Pitaevskii equation.     

Are galaxies extending?

It is suggested that the recently observed size evolution of very massive compact galaxies in the early universe can be explained, if dark matter is in Bose–Einstein condensate. In this model the size of the dark matter halos and galaxies depends on the correlation length of dark matter and, hence, on the expansion of the universe. This theory predicts that the size of the galaxies increases as the Hubble radius of the universe even without merging, which agrees well with the recent observational data.     

Non-autonomous matter-wave solitons in hybrid atomic–molecular Bose–Einstein condensates with tunable interactions and harmonic potential

In this paper, we investigate matter-wave solitons in hybrid atomic–molecular Bose–Einstein condensates with tunable interactions and external potentials. Three types of time-modulated harmonic potentials are considered and, for each of them, two groups of exact non-autonomous matter-wave soliton solutions of the coupled Gross–Pitaevskii equation are presented. Novel nonlinear structures of these non-autonomous matter-wave solitons are analyzed by displaying their density distributions. It is shown that the time-modulated nonlinearities and external potentials can support exact non-autonomous atomic–molecular matter-wave solitons.     

Bogolubov–Hartree–Fock Theory for Strongly Interacting Fermions in the Low Density Limit

We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional.     

Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram,self-similar solutions,and dispersive shock waves

We investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.     

Three-Dimensional Numerical Simulation of Long-Lived Quantum Vortex Knots and Links in a Trapped Bose Condensate

The dynamics of the simplest vortex knots, “unknots,” and torus links in an atomic Bose condensate at zero temperature in an anisotropic harmonic trap has been simulated numerically within the three-dimensional Gross–Pitaevskii equation. It has been found that such quasistationary rotating vortex structures exist for a very long time in wide ranges of the parameters of the system. This new result is qualitatively consistent with a previous prediction based on a simplified one-dimensional model approximately describing the motion of knotted vortex filaments.     

Effective equation for quasi-one dimensional tube-shaped Bose–Einstein condensates

In this letter we derive an effective 1D equation that describes the axial dynamics of a tube-shaped Bose–Einstein condensate. The dimensional reduction is achieved by using a variational approach starting from the 3D Gross–Pitaevskii equation (GPE) in presence of a nonharmonic external potential in the transverse direction and generic in the axial one. The resulting equation is a time-dependent 1D nonpolynomial Schrödinger equation (NPSE). In view to check the accuracy of our 1D-NPSE, we numerically investigated the ground state properties of such a system that are in perfect agreement with the results produced by the 3D-GPE. We also compare the results with those from an 1D cubic nonlinear Schrödinger equation and the Thomas–Fermi approximation. Finally, the dynamics of ground states obtained from our 1D-NPSE is verified numerically by considering a small change in the strength of the axial confining potential.     

Chaotic matter shock wave of an open system

We investigate a one-dimensional open Bose-Einstein condensate with attractive interaction,by considering the effect of feeding from nonequilibrium thermal cloud and applying the time-periodic inverted-harmonic potential.Using the direct perturbation method and the exact shock wave solution of the stationary Gross-Pitaevskii equation,we obtain the chaotic perturbed solution and the Melnikov chaotic regions.Based on the analytical and the numerical methods,the influence of the feeding strength on the chaotic motion is revealed.It is shown that the chaotic regions could be enlarged by reducing the feeding strength and the increase of feeding strength plays a role in suppressing chaos.In the case of "nonpropagated" shock wave with fixed boundary,the number of condensed atoms increases faster as the feeding strength increases.However,for the free boundary the metastable shock wave with fixed front density oscillates its front position and atomic number aperiodically,and their amplitudes decay with the increase of the feeding strength.     

Anisotropic Superfluidity in a Weakly Interacting Condensate of Quasi‐2D Photons

The anisotropic superfluidity in a weakly interacting two‐dimensional Bose gas of photons in a dye‐filled optical microcavity is investigated, taking into account the dependence of the photon effective mass on the in‐plane coordinate. With the use of the generalized Gross–Pitaevskii equation and the Bogoliubov approach, it is shown that the modulation of the microcavity width leads to an effective periodic potential and the periodicity of the condensate wave function, and both the condensate energy and the spectrum of elementary excitations depend on the direction of motion. The anisotropic character of the dynamical and superfluid properties, such as helicity modulus, superfluid density, and sound velocity, as well as experimentally observable manifestations of their anisotropy are described.