Complex quantum gases: spinor Bose–Einstein condensates of trapped atomic vapors

We discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor Bose–Einstein condensate for a dilute atomic vapor confined in an optical trap. Our results go beyond the mean field picture and are developed within a fully quantized framework.     

Bose–Einstein condensation in dilute atomic gases: atomic physics meets condensed matter physics

Bose–Einstein condensed atomic gases are a new class of quantum fluids. They are produced by cooling a dilute atomic gas to nanokelvin temperatures using laser and evaporative cooling techniques. The study of these quantum gases has become an interdisciplinary field of atomic and condensed matter physics. Topics of many-body physics can now be studied with the methods of atomic physics. Many long-standing predictions of the theory of the weakly interacting Bose gas have been verified, including thermodynamic properties of the phase transition and dynamic properties such as shape oscillations and sound propagation. Stimulated light scattering was used to determine the dynamic structure factor both in the phonon and free-particle regime. Atomic Bose condensates show a variety of novel phenomena which include multi-component spinor condensates, magnetic domain formation, miscibility and immiscibility of quantum fluids, and finite-size effects.     

Coherent state approach to the cross-collisional effects in the population dynamics of a two-mode Bose–Einstein condensate

We reanalyze the non-linear population dynamics of a Bose–Einstein condensate (BEC) in a double well trap considering a semiclassical approach based on a time dependent variational principle applied to coherent states associated to SU(2) group. Employing a two-mode local approximation and hard sphere type interaction, we show in the Schwinger’s pseudo-spin language the occurrence of a fixed point bifurcation that originates a separatrix of motion on a sphere. This separatrix corresponds to the borderline between two dynamical regimes of Josephson oscillations and mesoscopic self-trapping. We also consider the effects of interaction between particles in different wells, known as cross-collisions. Such terms are usually neglected for traps sufficiently far apart, but recently it has been shown that they contribute to the effective tunneling constant with a factor growing linearly with the particle number. This effect changes considerably the effective tunneling of the system for sufficiently large number of trapped atoms, in perfect accord with experimental data. Finally, we identify analytically the transition parameter associated to the bifurcation in the generalized phase space of the model with cross-collision terms, and show how the dynamical regime depends on the initial conditions of the system and the collisional parameters values.     

Quantum-phase dynamics of two-component Bose–Einstein condensates: Collapse–revival of macroscopic superposition states

We investigate the long-time dynamics of two-component dilute gas Bose–Einstein condensates with relatively different two-body interactions and Josephson couplings between the two components. Although in certain parameter regimes the quantum state of the system is known to evolve into macroscopic superposition, i.e., Schrödinger cat state, of two states with relative atom number differences between the two components, the Schrödinger cat state is also found to repeat the collapse and revival behavior in the long-time region. The dynamical behavior of the Pegg–Barnett phase difference between the two components is shown to be closely connected with the dynamics of the relative atom number difference for different parameters. The variation in the relative magnitude between the Josephson coupling and intra- and inter-component two-body interaction difference turns out to significantly change not only the size of the Schrödinger cat state but also its collapse–revival period, i.e., the lifetime of the Schrödinger cat state.     

A continuation BSOR-Lanczos–Galerkin method for positive bound states of a multi-component Bose–Einstein condensate

We develop a continuation block successive over-relaxation (BSOR)-Lanczos–Galerkin method for the computation of positive bound states of time-independent, coupled Gross–Pitaevskii equations (CGPEs) which describe a multi-component Bose–Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos–Galerkin method.     

Bose–Einstein condensation in atomic hydrogen

The addition of atomic hydrogen to the set of gases in which Bose–Einstein condensation can be observed expands the range of parameters over which this remarkable phenomenon can be studied. Hydrogen, with the lowest atomic mass, has the highest transition temperature, 50 μK in our experiments. The very weak interaction between the atoms results in a high ratio of the condensate to normal gas densities, even at modest condensate fractions. Using cryogenic rather than laser precooling generates large condensates. Finally, two-photon spectroscopy is introduced as a versatile probe of the phase transition: condensation in real space is manifested by the appearance of a high-density component in the gas, condensation in momentum space is readily apparent in the momentum distribution, and the phase transition line can be delineated by following the evolution of the density of the normal component.     

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ, as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ, it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ.     

Approximate solutions for half-dark solitons in spinor non-equilibrium Polariton condensates

In this work I generalize and apply an analytical approximation to analyze 1D states of non-equilibrium spinor polariton Bose–Einstein condensates (BEC). Solutions for the condensate wave functions carrying black solitons and half-dark solitons are presented. The derivation is based on the non-conservative Lagrangian formalism for complex Ginzburg–Landau type equations (cGLE), which provides ordinary differential equations for the parameters of the dark soliton solutions in their dynamic environment. Explicit expressions for the stationary dark soliton solution are stated. Subsequently the method is extended to spin sensitive polariton condensates, which yields ordinary differential equations for the parameters of half-dark solitons. Finally a stationary case with explicit expressions for half-dark solitons is presented.     

Wannier–Stark chaos of a Bose–Einstein condensate in 1D optical lattices

Using the idea of the macroscopic quantum wave function and the definition of the Melnikov chaos, we investigate the spatially chaotic features of a Bose–Einstein condensate (BEC) in a Wannier–Stark potential for the trivial phase and the non-trivial phase cases. The perturbed chaotic solutions are constructed, and the chaotic and unstable regions on the parameter space are illustrated. Numerical calculations to the spatial evolutions of the atomic number density and the energy density demonstrate the analytical results and exhibit the chaotic spatial distribution and energy distribution of the BEC atoms.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

Bose–Einstein condensation of metastable helium in a bi-planar quadrupole Ioffe configuration trap

Using a novel magnetic trapping geometry we have evaporatively cooled metastable helium atoms to form a Bose–Einstein condensate containing approximately one million atoms. This is only the fourth demonstration of a metastable condensate and the first realisation of a BEC in a bi-planar quadrupole Ioffe configuration magnetic trap.     

Bose–Einstein condensation in binary mixture of Bose gases

The Bose–Einstein condensation (BEC) in a binary mixture of Bose gases is studied by means of the Cornwall–Jackiw–Tomboulis (CJT) effective action approach. The equations of state (EoS) and various scenarios of phase transitions of the system are considered in detail, in particular, the numerical computations are carried out for symmetry restoration (SR), symmetry nonrestoration (SNR) and inverse symmetry breaking (ISB) for getting an insight into their physical nature. It is shown that due to the cross interaction between distinct components of mixture there occur two interesting phenomena: the high temperature BEC and the inverse BEC, which could be tested in experiments.     

Black Holes in Bose–Einstein Condensates

It is shown that there exist both dynamically stable and unstable dilute-gas Bose–Einstein condensates that, in the hydrodynamic limit, exhibit a behavior completely analogous to that of gravitational black holes. The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy states. We illustrate these features in two qualitatively different one-dimensional models. We have also simulated the creation of a stable sonic black hole by solving the Gross–Pitaevskii equation numerically for a condensate subject to a trapping potential that is adiabatically deformed. A sonic black hole could in this way be created experimentally with state-of-the-art or planned technology.     

Renormalized dispersion of elementary excitations in spinor polariton condensates

We analyse the polarization of spinor polariton condensates and corresponding dispersions of elementary excitations. We have considered the effects of magnetic field induced splitting in circular polarizations and residual splitting in linear polarizations in the ground state provided by the cavity asymmetry. We show that anisotropic polariton–polariton interactions fully compensate the Zeeman splitting in circular polarizations below the critical magnetic field, thus leading to the spin-Meissner effect for the polariton condensates. We also analyzed the effect of polariton–polariton interactions on the stability of the gap in linear polarizations characteristic for anisotropic microcavities. It was shown that in realistic systems this gap increases with concentration of the particles, thus contributing to the stability of the pinning of linear polarization of photoemission in semiconductor microcavities for pump intensities above the stimulation threshold.     

Quasi-two-dimensional Bose–Einstein condensates with spatially modulated cubic–quintic nonlinearities

We investigate exact nonlinear matter wave functions with odd and even parities in the framework of quasi-two-dimensional Bose–Einstein condensates (BECs) with spatially modulated cubic–quintic nonlinearities and harmonic potential. The existence condition for these exact solutions requires that the minimum energy eigenvalue of the corresponding linear Schrödinger equation with harmonic potential is the cutoff value of the chemical potential λ. The competition between two-body and three-body interactions influences the energy of the localized state. For attractive two-body and three-body interactions, the larger the matter wave order number n, the larger the energy of the corresponding localized state. A linear stability analysis and direct simulations with initial white noise demonstrate that, for the same state (fixed n), increasing the number of atoms can add stability. A quasi-stable ground-state matter wave is also found for repulsive two-body and three-body interactions. We also discuss the experimental realization of these results in future experiments. These results are of particular significance to matter wave management in higher-dimensional BECs.     

Bose–Einstein Condensation for Homogeneous Interacting Systems with a One-Particle Spectral Gap

We prove rigorously the occurrence of zero-mode Bose–Einstein condensation for a class of continuous homogeneous systems of boson particles with superstable interactions. This is the first example of a translation invariant continuous Bose-system, where the existence of the Bose–Einstein condensation is proved rigorously for the case of non-trivial two-body particle interactions, provided there is a large enough one-particle excitations spectral gap. The idea of proof consists of comparing the system with specially tuned soluble models.     

Langevin Equation of Collective Modes of Bose–Einstein Condensates in Traps

A quantum Langevin equation for the amplitudes of the collective modes in Bose–Einstein condensate is derived. The collective modes are coupled to a thermal reservoir of quasi-particles, whose elimination leads to the quantum Langevin equation. The dissipation rates are determined via the correlation function of the fluctuating force and are evaluated in the local-density approximation for the spectrum of quasi-particles and the Thomas–Fermi approximation for the condensate.I take great pleasure in dedicating this paper to Gregoire Nicolis on the occasion of his sixtieth birthday.     

Matter wave interference pattern in the collision of bright solitons

We investigate the dynamics of Bose–Einstein condensates in a quasi one-dimensional regime in a time-dependent trap and show analytically that it is possible to observe matter wave interference patterns in the intra-trap collision of two bright solitons by selectively tuning the trap frequency and scattering length.     

Harmonic Interaction Model and Its Applications in Bose–Einstein Condensation

We exactly solve the model of N harmonic interacting Bosons in a harmonic trap in any dimension. The exact ground state wavefunction, free energy, spectrum, and low excitation states are calculated. The finite particle number effect is addressed when the exact solution is compared with a mean field solution. Then we compare the harmonic interaction system with a pseudo-potential interaction system. In spite of the seemingly quite different nature of interaction, several similarities are found between the two systems.