Bose-Einstein condensation in low-dimensional traps

Photons think inside the box

Light confined to a sheet offers a glimpse into low-dimensional quantum gases.

Charged Bosons Made of Fermions in Bilayer Structures with Strong Metallic Screening

Two-dimensional monolayer structures of transition metal dichalogenides (TMDs) have been shown to allow many higher-order excitonic bound states, including trions (charged excitons), biexcitons (excitonic molecules), and charged biexcitons. We report here experimental evidence and the theoretical basis for a new bound excitonic complex, consisting two free carriers bound to an exciton in a bilayer structure. Our experimental measurements on structures made using two different materials show a new spectral line at the predicted energy with two different TMD materials (MoSe₂ and WSe₂) with both n- and p-doping if and only if all the required theoretical conditions for this complex are fulfilled, in particular, only in the presence of a parallel metal layer that significantly screens the repulsive interaction between the like-charge carriers. Because these four-carrier bound states are charged bosons, they could eventually be the basis for a new path to superconductivity without Cooper pairing.

Bose-Einstein condensation of photons in a long fiber cavity

We demonstrate photon Bose-Einstein condensation (photon-BEC) at a broad temperature range that is valid also in the long 1D fiber cavity limit. It is done with an erbium-ytterbium co-doped fiber (EYDF) cavity by overcoming the challenging requirement of sublinear light dispersion for BEC in 1D using a chirped-gratings Fabry-Perot. We experimentally show with a square-root mode-dispersion, a quadratic temperature dependence of the critical power for condensation (compared to a linear dependence in finite regular fiber-cavities) between 90 K and 382 K, as the theory predicts.

Observation of a non-Hermitian phase transition in an optical quantum gas

Quantum gases of light, such as photon or polariton condensates in optical microcavities, are collective quantum systems enabling a tailoring of dissipation from, for example, cavity loss. This characteristic makes them a tool to study dissipative phases, an emerging subject in quantum many-body physics. We experimentally demonstrate a non-Hermitian phase transition of a photon Bose-Einstein condensate to a dissipative phase characterized by a biexponential decay of the condensate's second-order coherence. The phase transition occurs because of the emergence of an exceptional point in the quantum gas. Although Bose-Einstein condensation is usually connected to lasing by a smooth crossover, the observed phase transition separates the biexponential phase from both lasing and an intermediate, oscillatory condensate regime. Our approach can be used to study a wide class of dissipative quantum phases in topological or lattice systems.

Double-layer Bose-Einstein condensates: A quantum phase transition in the transverse direction, and reduction to two dimensions

We revisit the problem of the reduction of the three-dimensional (3D) dynamics of Bose-Einstein condensates, under the action of strong confinement in one direction (z), to a 2D mean-field equation. We address this problem for the confining potential with a singular term, viz., V_{z}(z)=2z^{2}+ζ^{2}/z^{2}, with constant ζ. A quantum phase transition is induced by the latter term, between the ground state (GS) of the harmonic oscillator and the 3D condensate split in two parallel noninteracting layers, which is a manifestation of the "superselection" effect. A realization of the respective physical setting is proposed, making use of resonant coupling to an optical field, with the resonance detuning modulated along z. The reduction of the full 3D Gross-Pitaevskii equation (GPE) to the 2D nonpolynomial Schrödinger equation (NPSE) is based on the factorized ansatz, with the z -dependent multiplier represented by an exact GS solution of the 1D Schrödinger equation with potential V_{z}(z). For both repulsive and attractive signs of the nonlinearity, the 2D NPSE produces GS and vortex states, that are virtually indistinguishable from the respective numerical solutions provided by full 3D GPE. In the case of the self-attraction, the threshold for the onset of the collapse, predicted by the 2D NPSE, is also virtually identical to its counterpart obtained from the 3D equation. In the same case, stability and instability of vortices with topological charge S=1, 2, and 3 are considered in detail. Thus, the procedure of the spatial-dimension reduction, 3D → 2D, produces very accurate results, and it may be used in other settings.

Nonlinear light mode dispersion and nonuniform mode comb by a Fabry-Perot with chirped fiber gratings

We demonstrate a nonlinear light mode dispersion and a nonuniform frequency mode comb by a chirped fiber Bragg gratings (CFBG) Fabry-Perot (FP) at the 1550 nm wavelength regime. We give analytical expressions for the general chirp case, and an experimental demonstration with a linear chirp, showing a square-root dependence of the dispersion as a function of the FP mode number. Such sublinear dispersion is required, for example, for photon Bose-Einstein condensation (BEC) in a one-dimensional (1D) system like fiber cavities.

Thermally condensing photons into a coherently split state of light

The quantum state of light plays a crucial role in a wide range of fields, from quantum information science to precision measurements. Whereas complex quantum states can be created for electrons in solid-state materials through mere cooling, optical manipulation and control builds on nonthermodynamic methods. Using an optical dye microcavity, we show that photon wave packets can be split through thermalization within a potential with two minima subject to tunnel coupling. At room temperature, photons condense into a quantum-coherent bifurcated ground state. Fringe signals upon recombination show the relative coherence between the two wells, demonstrating a working interferometer with the nonunitary thermodynamic beam splitter. Our energetically driven optical-state preparation method provides a route for exploring correlated and entangled optical many-body states.

Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions

We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross-Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either byW β ( x ) = N - 1 + 2 β W ( N β x ) , for any β > 0 , or to be given byV N ( x ) = e 2 N V ( e N x ) , for some spherical symmetric, nonnegative and compactly supported W , V ∈ L ∞ ( R 2 , R ) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential V N we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

Unconventional Bloch-Grüneisen Scattering in Hybrid Bose-Fermi Systems

We report on the novel mechanism of electron scattering in hybrid Bose-Fermi systems consisting of a two-dimensional electron gas in the vicinity of an exciton condensate: We show that in certain ranges of temperatures, the bogolon-pair-mediated scattering proves to be dominating over the conventional acoustic phonon channel, over the single-bogolon scattering, and over the scattering on impurities. We develop a microscopic theory of this effect, focusing on GaAs and MoS_{2} materials, and we find the principal temperature dependence of resistivity, distinct from the conventional phonon-mediated processes. Further, we scrutinize parameters and suggest a way to design composite samples with predefined electron mobilities, and we propose a mechanism of electron pairing for superconductivity.

Bose-Einstein condensation of photons in an erbium-ytterbium 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.

Detector for positronium temperature measurements by two-photon angular correlation

We report on the design and characterization of a modular γ-ray detector assembly developed for accurate and efficient detection of coincident 511 keV back-to-back γ-rays following electron-positron annihilation. Each modular detector consists of 16 narrow lutetium yttrium oxyorthosilicate scintillators coupled to a multi-anode Hamamatsu H12700B photomultiplier tube. We discuss the operation and optimization of 511 keV γ-ray detection resulting from testing various scintillators and detector arrangements concluding with an estimate of the coincident 511 keV detection efficiency for the intended experiment and a preliminary test representing one-quarter of the completed array.

First-order spatial coherence measurements in a thermalized two-dimensional photonic quantum gas

Phase transitions between different states of matter can profoundly modify the order in physical systems, with the emergence of ferromagnetic or topological order constituting important examples. Correlations allow the quantification of the degree of order and the classification of different phases. Here we report measurements of first-order spatial correlations in a harmonically trapped two-dimensional photon gas below, at and above the critical particle number for Bose–Einstein condensation, using interferometric measurements of the emission of a dye-filled optical microcavity. For the uncondensed gas, the transverse coherence decays on a length scale determined by the thermal de Broglie wavelength of the photons, which shows the expected scaling with temperature. At the onset of Bose–Einstein condensation, true long-range order emerges, and we observe quantum statistical effects as the thermal wave packets overlap. The excellent agreement with equilibrium Bose gas theory prompts microcavity photons as promising candidates for studies of critical scaling and universality in optical quantum gases.

Phase transitions in quantum matter are related to correlation effects and they can change the ordering of material. Here the authors measure the first-order spatial correlation and the de Broglie wavelength for both thermal and condensed form of a photonic Bose gas in a dye-filled optical microcavity.

Bose-Einstein condensation in chains with power-law hoppings: Exact mapping on the critical behavior in d-dimensional regular lattices

Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.

Vortices and vortex lattices in quantum ferrofluids

The experimental realization of quantum-degenerate Bose gases made of atoms with sizeable magnetic dipole moments has created a new type of fluid, known as a quantum ferrofluid, which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to rotate through vortices with quantized circulation. In quantum ferrofluids the long-range dipolar interactions add new ingredients by inducing magnetostriction and instabilities, and also affect the structural properties of vortices and vortex lattices. Here we give a review of the theory of vortices in dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. We cover single vortex solutions, including structure, energy and stability, vortex pairs, including interactions and dynamics, and also vortex lattices. Our discussion is founded on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, ranging from analytic treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches to full numerical simulations. Routes for generating vortices in dipolar condensates are discussed, with particular attention paid to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. We also present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless transition.

CW laser light condensation

We present a first experimental demonstration of classical CW laser light condensation (LC) in the frequency (mode) domain that verifies its prediction (Fischer and Weill, Opt. Express20, 26704 (2012)). LC is based on weighting the modes in a noisy environment in a loss-gain measure compared to an energy (frequency) scale in Bose-Einstein condensation (BEC). It is characterized by a sharp transition from multi- to single-mode oscillation, occurring when the spectral-filtering (loss-trap) has near the lowest-loss mode ("ground-state") a power-law dependence with an exponent smaller than 1. An important meaning of the many-mode LC system stems from its relation to lasing and photon-BEC.

Kosterlitz-Thouless physics: a review of key issues

This article reviews, from a very personal point of view, the origins and the early work on transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in (4)Helium films and dislocations and disclinations in 2D crystals. I cover the early papers with David Thouless and describe the important insights but also the errors and oversights since corrected by other workers. I then describe some of the experimental verifications of the theory and some numerical simulations. Finally applications to superconducting arrays of Josephson junctions and to recent cold atom experiments are described.

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.

Thermodynamics of the noninteracting Bose gas in a two-dimensional box

Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.

Comparative analysis of electric field influence on the quantum wells with different boundary conditions: II. Thermodynamic properties

Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary conditions (BCs) at its edges in the perpendicular to the surfaces electric field [Formula: see text] are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity [Formula: see text] for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature T for the pure Neumann QW and their absence for other BCs are predicted and explained by the structure of the corresponding energy spectrum. Applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential. A remarkable feature of the Fermi grand canonical ensemble is, at any BC combination in zero fields, a salient maximum of [Formula: see text] observed on the T axis for one particle and its absence for any other number N of corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different N. It is proved that critical temperature [Formula: see text] of the Bose-Einstein (BE) condensation increases with the applied voltage for any number of particles and for any BC permutation except the ND case at small intensities [Formula: see text] what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than [Formula: see text] the total dipole moment [Formula: see text] may become negative for the quite moderate [Formula: see text]. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed with N growing. Different asymptotic cases of, e.g., the small and large temperatures and low and high voltages are derived analytically and explained physically. Parallels are drawn to the similar properties of the 1D harmonic oscillator, and similarities and differences between them are discussed.

Nonequilibrium model of photon condensation

We develop a nonequilibrium model of condensation and lasing of photons in a dye filled microcavity. We examine in detail the nature of the thermalization process induced by absorption and emission of photons by the dye molecules, and investigate when the photons are able to reach a thermal equilibrium Bose-Einstein distribution. At low temperatures, or large cavity losses, the absorption and emission rates are too small to allow the photons to reach thermal equilibrium and the behavior becomes more like that of a conventional laser.

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.

Observation of thermally activated vortex pairs in a quasi-2D Bose gas

We measure the in-plane distribution of thermally activated vortices in a trapped quasi-2D Bose gas, where we enhance the visibility of density-depleted vortex cores by radially compressing the sample before releasing the trap. The pairing of vortices is revealed by the two-vortex spatial correlation function obtained from the vortex distribution. The vortex density decreases gradually as temperature is lowered, and below a certain temperature, a vortex-free region emerges in the center of the sample. This shows the crossover from a Berezinskii-Kosterlitz-Thouless phase containing vortex-pair excitations to a vortex-free Bose-Einstein condensate in a finite-size 2D system.

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.

Statistical physics of Bose-Einstein-condensed light in a dye microcavity

We theoretically analyze the temperature behavior of paraxial light in thermal equilibrium with a dye-filled optical microcavity. At low temperatures the photon gas undergoes Bose-Einstein condensation, and the photon number in the cavity ground state becomes macroscopic with respect to the total photon number. Owing to a grand-canonical excitation exchange between the photon gas and the dye molecule reservoir, a regime with unusually large fluctuations of the condensate number is predicted for this system that is not observed in present atomic physics Bose-Einstein condensation experiments.

Estimating the conditions for polariton condensation in organic thin-film microcavities

We examine the possibility of observing Bose condensation of a confined two-dimensional polariton gas in an organic quantum well. We deduce a suitable parameterization of a model polynomial Hamiltonian based upon the cavity geometry, the biexciton binding energy, and similar spectroscopic and structural data. By converting the sum-over-states to a semiclassical integration over D-dimensional phase space, we arrive at a principle of correspondence between ideal and non-ideal Bose gases that share a common critical exponent. Using our results, we can calculate the properties for a model cavity containing an anthracene thin film.

Bose-Einstein condensation and superfluidity of trapped polaritons in graphene and quantum wells embedded in a microcavity

The theory for spontaneous coherence of short-lived quasiparticles in two-dimensional excitonic systems is reviewed, in particular, quantum wells (QWs) and graphene layers (GLs) embedded in microcavities. Experiments with polaritons in an optical microcavity have already shown evidence of Bose-Einstein condensation (BEC) in the lowest quantum state in a harmonic trap. The theory of BEC and superfluidity of the microcavity excitonic polaritons in a harmonic potential trap is presented. Along the way, we determine a general method for defining the superfluid fraction in a two-dimensional trap, within the angular momentum representation. We discuss BEC of magnetoexcitonic polaritons (magnetopolaritons) in a QW and GL embedded in an optical microcavity in high magnetic field. It is shown that Rabi splitting in graphene is tunable by the external magnetic field B, while in a QW the Rabi splitting does not depend on the magnetic field in the strong B limit.

Observation of the presuperfluid regime in a two-dimensional Bose gas

In complementary images of coordinate-space and momentum-space density in a trapped 2D Bose gas, we observe the emergence of presuperfluid behavior. As phase-space density ρ increases toward degenerate values, we observe a gradual divergence of the compressibility κ from the value predicted by a bare-atom model, κ(ba). κ/κ(ba) grows to 1.7 before ρ reaches the value for which we observe the sudden emergence of a spike at p = 0 in momentum space. Momentum-space images are acquired by means of a 2D focusing technique. Our data represent the first observation of non-mean-field physics in the presuperfluid but degenerate 2D Bose gas.

Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser

We have recently predicted (R. Weill, B. Fischer and O. Gat, Phys. Rev. Lett.104, 173901, 2010) condensation of light in actively mode locked lasers when the laser power increases, or the noise, that takes the role of temperature, decreases. The condensate is characterized by strong light pulses due to the dominance of the lowest eigenmode ("ground state") power. Here, we experimentally demonstrate, for the first time, light mode condensation transition in an actively mode-locked fiber laser. Following the theoretical prediction, the condensation is obtained for modulations that have a power law dependence on time with exponents smaller than 2. The laser light system is strictly one dimensional, a special opportunity in experimental physics. We also discuss experimental schemes for condensation in two- and three-dimensional laser systems.

Light-mode condensation in actively-mode-locked lasers

We show that the formation of pulses in actively mode-locked lasers exhibits in certain conditions a transition of the laser mode system to a light pulse state that is similar to Bose-Einstein condensation (BEC). The study is done in the framework of statistical light-mode dynamics with a mapping between the distribution of the laser eigenmodes to the equilibrium statistical physics of noninteracting bosons in an external potential. The light-mode BEC transition occurs for a mode-locking modulation that has a power law dependence on time with an exponent smaller than 2.

Bose-Einstein condensation of quasiparticles in graphene

The collective properties of different quasiparticles in various graphene-based structures in a high magnetic field have been studied. We predict Bose-Einstein condensation (BEC) and the superfluidity of 2D spatially indirect magnetoexcitons in a two-layer graphene. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition are shown to be increasing functions of the excitonic density but decreasing functions of a magnetic field and the interlayer separation. The instability of the ground state of the interacting 2D indirect magnetoexcitons in a slab of superlattice with alternating electron and hole graphene layers (GLs) is established. The stable system of indirect 2D magnetobiexcitons, consisting of a pair of indirect excitons with antiparallel dipole moments, is considered in a graphene superlattice. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition for magnetobiexcitons in a graphene superlattice are obtained. Moreover, the BEC of excitonic polaritons in a GL embedded in a semiconductor microcavity in a high magnetic field is predicted. While the superfluid phase in this magnetoexciton polariton system is absent due to a vanishing magnetoexciton-magnetoexciton interaction in a single layer in the limit of a high magnetic field, the critical temperature of the BEC formation is calculated. The observation of the BEC and superfluidity of 2D quasiparticles in graphene in a high magnetic field would be interesting confirmation of the phenomena we have described.

Optical lattice on an atom chip

Optical dipole traps and atom chips are two very powerful tools for the quantum manipulation of neutral atoms. We demonstrate that both methods can be combined by creating an optical lattice potential on an atom chip. A red-detuned laser beam is retroreflected using the atom chip surface as a high-quality mirror, generating a vertical array of purely optical oblate traps. We transfer thermal atoms from the chip into the lattice and observe cooling into the two-dimensional regime. Using a chip-generated Bose-Einstein condensate, we demonstrate coherent Bloch oscillations in the lattice.

Observation of a 2D Bose gas: from thermal to quasicondensate to superfluid

We present experimental results on a Bose gas in a quasi-2D geometry near the Berezinskii, Kosterlitz, and Thouless (BKT) transition temperature. By measuring the density profile after time of flight and the coherence length, we identify different states of the gas. We observe that the gas develops a bimodal distribution without long range order. In this regime, the gas presents a longer coherence length than the thermal cloud; it is quasicondensed but is not superfluid. Experimental evidence indicates that we also observe the superfluid transition (BKT transition). For a sufficiently long time of flight, we observe a trimodal distribution when the gas has developed a superfluid component.

Berry phase effect on the exciton transport and on the exciton Bose-Einstein condensate

With the exciton lifetime much extended in semiconductor quantum-well structures, the exciton transport and Bose-Einstein condensation have become a focus of research in recent years. We reveal a momentum-space gauge field in the exciton center-of-mass dynamics due to Berry phase effects. We predict a spin-dependent transport of the excitons analogous to the anomalous Hall and Nernst effects for electrons. We also predict spin-dependent circulation of a trapped exciton gas and instability in an exciton condensate in favor of vortex formation.

Kosterlitz-Thouless transition of the quasi-two-dimensional trapped Bose gas

We use quantum Monte Carlo methods to compute the density profile, the nonclassical moment of inertia, and the condensate fraction of an interacting quasi-two-dimensional trapped Bose gas with up to N ~ 5 x 10(5) atoms and parameters closely related to recent experiments. We locate the Kosterlitz-Thouless temperature T(KT) and discuss intrinsic signatures of the onset of superfluidity in the density profile. Below T(KT), the condensate fraction is macroscopic even for our largest systems and decays only slowly with system size. We show that the thermal population of excited states in the transverse direction changes the two-dimensional density profile noticeably in both the normal and the superfluid phase.

Spatial coherence of a polariton condensate

We perform Young's double-slit experiment to study the spatial coherence properties of a two-dimensional dynamic condensate of semiconductor microcavity polaritons. The coherence length of the system is measured as a function of the pump rate, which confirms a spontaneous buildup of macroscopic coherence in the condensed phase. An independent measurement reveals that the position and momentum uncertainty product of the condensate is close to the Heisenberg limit. An experimental realization of such a minimum uncertainty wave packet of the polariton condensate opens a door to coherent matter-wave phenomena such as Josephson oscillation, superfluidity, and solitons in solid state condensate systems.

Critical point of an interacting two-dimensional atomic Bose gas

We have measured the critical atom number in an array of harmonically trapped two-dimensional (2D) Bose gases of rubidium atoms at different temperatures. We found this number to be about 5 times higher than predicted by the semiclassical theory of Bose-Einstein condensation (BEC) in the ideal gas. This demonstrates that the conventional BEC picture is inapplicable in an interacting 2D atomic gas, in sharp contrast to the three-dimensional case. A simple heuristic model based on the Berezinskii-Kosterlitz-Thouless theory of 2D superfluidity and the local density approximation accounts well for our experimental results.

Effect of interface disorder on quantum well excitons and microcavity polaritons

The theory of the linear optical response of excitons in quantum wells and polaritons in planar semiconductor microcavities is reviewed, in the light of the existing experiments. For quantum well excitons, it is shown that disorder mainly affects the exciton centre-of-mass motion and is modelled by an effective Schrödinger equation in two dimensions. For polaritons, a unified model accounting for quantum well roughness and fluctuations of the microcavity thickness is developed. Numerical results confirm that polaritons are mostly affected by disorder acting on the photon component, thus confirming existing studies on the influence of exciton disorder. The polariton localization length is estimated to be in the few-micrometres range, depending on the amplitude of disorder, in agreement with recent experimental findings.

Thermodynamics of ideal quantum gas with fractional statistics in D dimensions

We present exact and explicit results for the thermodynamic properties (isochores, isotherms, isobars, response functions, velocity of sound) of a quantum gas in dimensions D > or = 1 and with fractional exclusion statistics 0 < or = g < or =1 connecting bosons (g=0) and fermions (g=1) . In D=1 the results are equivalent to those of the Calogero-Sutherland model. Emphasis is given to the crossover between bosonlike and fermionlike features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. A phase transition along the isobar occurs at a nonzero temperature in all dimensions. The T dependence of the velocity of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential.

Bose glass and superfluid phases of cavity polaritons

We report the calculation of cavity exciton-polariton phase diagram including realistic structural disorder. With increasing density polaritons first undergo a quasiphase transition toward a Bose glass: the condensate is localized in at least one minimum of the disorder potential. A further increase of the density leads to a percolation process of the polariton fluid giving rise to a Kosterlitz-Thouless phase transition toward superfluidity. The spatial representation of the condensate wave function as well as the spectrum of elementary excitations are obtained from the Gross-Pitaevskii equation for all the phases.

Evaluation of the permutational structure of quantum gases at finite temperature

Proposed is an alternative method for permutational sampling in quantum gases using the path integral formulation of statistical mechanics. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.

Superfluid transition of homogeneous and trapped two-dimensional Bose gases

Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.

Experimental evidence for the breakdown of a Hartree-Fock approach in a weakly interacting Bose gas

We investigate the physics underlying the presence of a quasicondensate in a nearly one dimensional, weakly interacting trapped atomic Bose gas. We show that a Hartree-Fock (mean-field) approach fails to predict the existence of the quasicondensate in the center of the cloud: the quasicondensate is generated by interaction-induced correlations between atoms and not by a saturation of the excited states. Numerical calculations based on Bogoliubov theory give an estimate of the crossover density in agreement with experimental results.

Trapping excitons in a two-dimensional in-plane harmonic potential: experimental evidence for equilibration of indirect excitons

We have trapped a gas of long-lifetime, high-mobility excitons in an in-plane harmonic potential. Trapping is an important step toward the goal of a controlled Bose-Einstein condensate of excitons. We show that the repulsive interaction between the excitons plays a dominant role in the behavior of the excitons, in contrast with the weak interactions in atomic gases. We show that under proper conditions the excitons thermalize in the trap to a well-defined equilibrium spatial distribution.