This paper is a short introduction to cold atom physics and Bose-Einstein condensation. Light forces on atoms are presented, together with laser cooling, and a few atom traps: the magneto-optical trap, dipole traps and magnetic traps. A brief description of Bose-Einstein condensation is given together with some important links with condensed matter physics. The reader is referred to comprehensive reviews and to other lecture notes for further details on atom cooling, trapping and Bose-Einstein condensation.
We describe Bose-Einstein condensation of strongly interacting particles into a quantum state which is an excited single-particle state, but becomes the ground state as density increases because it minimizes the interaction energy compared to other states. Mean field calculations for a graphene potential just wide enough for two closely interacting layers of molecular hydrogen show condensation at temperatures up to 60 K. In the condensed state, molecules hop between layers, increasing the first peak in the pair-correlation function just past the hard core repulsion diameter.
In the quantum metrology protocol described by Tacla et al. [Tacla et al., Phys. Rev. A 82, 053636 (2010)] where a two mode Bose-Einstein condensate (BEC) is used for parameter estimation, the measured quantity is to be obtained by doing a one parameter fit of the observed data to a theoretically expected signal. Here we look at different levels of approximation used to model the two mode BEC to see how the estimate improves when increasing level of detail is added to the theory while at the same time keeping the expected signal computable.
The critical temperature of Bose-Einstein condensation essentially depends on internal properties of the system as well as on the geometry of a trapping potential. The peculiarities of defining the phase transition temperature of Bose-Einstein condensation for different systems are reviewed, including homogenous Bose gas, trapped Bose atoms, and bosons in optical lattices. The method of self-similar approximants, convenient for calculating critical temperature, is briefly delineated.
We study the impact of spin-exchange collisions on the dynamics of Bose-Einstein condensation, by rapidly cooling a chromium multi-component Bose gas. Despite relatively strong spin-dependent interactions, the critical temperature for Bose-Einstein condensation is reached before the spin-degrees of freedom fully thermalize. The increase in density due to Bose-Einstein condensation then triggers spin dynamics, hampering the formation of condensates in spin excited states. Small metastable spinor condensates are nevertheless produced, and manifest strong spin fluctuations.
Exciton-polaritons in semiconductor microcavities have been used to demonstrate quantum effects such as Bose-Einstein condensation, superfluity, and quantized vortices. However, in these experiments, the polaritons have not reached thermal equilibrium when they undergo the transition to a coherent state. This has prevented the verification of one of the canonical predictions for condensation, namely the phase diagram. In this work, we have created a polariton gas in a semiconductor microcavity in which the quasiparticles have a lifetime much longer than their thermalization time. This allows t…
There is a wide-spread belief in the literature on Bose-Einstein condensation of interacting atoms that all variants of mean-field theory incorrectly describe the condensation phase transition, exhibiting, instead of the necessary second-order transition, a first-order transition, even for weakly interacting Bose gas. In the present paper, it is shown that a self-consistent mean-field approach is the sole mean-field theory that provides the correct second-order condensation transition for Bose systems with atomic interactions of arbitrary strength, whether weak or strong.
A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infrared bound for the suitable two-point Bogolyubov's inner product. We do not use the reflection positivity or some expansion methods.
Bose-Einstein condensation in the presence of a synthetic spin-momentum interaction is considered, focusing on the case where a Dirac or Rashba potential is generated via a tripod scheme. We found that the ground states can be either plane wave states or superpositions of them, each characterized by their unique density distributions.
The two-component mixture of Bose particles with short-range pairwise interaction at finite temperatures in three dimensions is considered. Particularly we examine, by means of the large-$N$ expansion technique, the stability of mixed state below the Bose-Einstein transition point and the temperature dependence of the condensate density for symmetric mixture of Bose gases. The presented analysis reveals the importance of finite-temperature excitations of the non-condensed particles in formation of the phase diagram of two-component Bose systems.
We report Bose-Einstein condensation of 84-Sr in an optical dipole trap. Efficient laser cooling on the narrow intercombination line and an ideal s-wave scattering length allow creation of large condensates (N0 ~ 3x10^5) even though the natural abundance of this isotope is only 0.6%. Condensation is heralded by the emergence of a low-velocity component in time-of-flight images.
Squeezed states, a special kind of entangled states, are known as a useful resource for quantum metrology. In interferometric sensors they allow to overcome the "classical" projection noise limit stemming from the independent nature of the individual photons or atoms within the interferometer. Motivated by the potential impact on metrology as wells as by fundamental questions in the context of entanglement, a lot of theoretical and experimental effort has been made to study squeezed states. The first squeezed states useful for quantum enhanced metrology have been proposed and generated in quan…
The Bose-Einstein condensation in the hard-core boson limit (HCB) of the Bose-Hubbard model with two local states and the particle hopping in the excited band only is investigated. For the purpose of considering the non-ergodicity, a single-particle spectral density is calculated in the random phase approximation by means of the temperature boson Green functions. The non-ergodic contribution to the momentum distribution function of particles (connected with the static density fluctuations) increases significantly and becomes comparable with the ergodic contribution in the superfluid phase near…
We review the properties of chaoticity and coherence in Bose-Einstein condensation and correlations, for a dense boson system in its mean-field represented approximately by a harmonic oscillator potential. The order parameter and the nature of the phase transition from the chaotic to the condensate states are studied for different fixed numbers of bosons. The two-particle correlation function in momentum space is calculated to investigate how the Bose-Einstein correlation depends on the degree of condensation and other momentum variables. We generalize the Bose-Einstein correlation analysis to…
After reviewing the interpretation of laser operation as a non-equilibrium Bose-Einstein condensation phase transition, we illustrate the novel features arising from the non-equilibrium nature of photon and polariton Bose-Einstein condensates recently observed in experiments. We then proposea quantitative criterion to experimentally assess the equilibrium vs. non-equilibrium nature of a specific condensation process, based on fluctuation-dissipation relations. The power of this criterion is illustrated on two models which shows very different behaviours.
We report on the attainment of Bose-Einstein condensation of 86Sr. This isotope has a scattering length of about +800 a0 and thus suffers from fast three-body losses. To avoid detrimental atom loss, evaporative cooling is performed at low densities around 3x10^12 cm^-3 in a large volume optical dipole trap. We obtain almost pure condensates of 5x10^3 atoms.
The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential $U=\sum_{i=1} ^d c_i |\frac{x_i}{a_i}|^{n_i}$ are studied carefully. Detailed calculation of Kim $et$ $al.$ (S. H. Kim, C. K. Kim and K. Nahm, J Phys. Condens. Matter 11 10269 (1999).) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume $C_V$ are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition…
We study the Bardeen-Cooper-Shrieffer (BCS) pairing state of a two-component Bose gas with a symmetric spin-orbit coupling. In the dilute limit at low temperatures, this system is essentially a dilute gas of diatomic molecules. We compute the effective mass of the molecule and find that it is anisotropic in momentum space. The critical temperature of the pairing state is about eight times smaller than the Bose-Einstein condensation (BEC) transition temperature of an ideal Bose gas with the same density.
We study the thermalization and the Bose-Einstein condensation of a paraxial, spectrally narrow beam of quantum light propagating in a lossless bulk Kerr medium. The spatiotemporal evolution of the quantum optical field is ruled by a Heisenberg equation analogous to the quantum nonlinear Schrödinger equation of dilute atomic Bose gases. Correspondingly, in the weak-nonlinearity regime, the phase-space density evolves according to the Boltzmann equation. Expressions for the thermalization time and for the temperature and the chemical potential of the eventual Bose-Einstein distribution are foun…
This paper introduces a quasiequilibrium one-dimensional Bose-Einstein condensation of photons trapped in a microtube. Light modes with a cut-off frequency (a photon's "mass") interact through different processes of absorption, emission, and scattering on molecules and atoms. In this paper, we study the conditions for the one-dimensional condensation of light and the role of photon-photon interactions in the system. The technique in use is the Matsubara's Green's functions formalism modified for the quasiequilibrium system under study.
We open a new discussion of generalized canonical partition function in standard statistical mechanics and apply it for the study of Bose-Einstein condensation. We discuss the possible cases for the generalized canonical partition function and arrives at a conclusion that the system of trapped bose gas will not be existing at absolute zero. We analyse the present study with an experimental result and point out the general difficulties in the analyses of experimental observations, which can possibly suppress the effect of generalized canonical partition function over standard canonical partitio…
Open quantum systems can be systematically controlled by making changes to their environment. A well-known example is the spontaneous radiative decay of an electronically excited emitter, such as an atom or a molecule, which is significantly influenced by the feedback from the emitter's environment, for example, by the presence of reflecting surfaces. A prerequisite for a deliberate control of an open quantum system is to reveal the physical mechanisms that determine the state of the system. Here, we investigate the Bose-Einstein condensation of a photonic Bose gas in an environment with contr…
Bose-Einstein condensation of rotons in helium was considered long ago. It was shown that the relative velocity of the normal motion in this state must be equal to the Landau critical velocity. We argue that the condensation can be attained at a smaller velocity if the temperature is low enough.
We review recent experiments on the Bose-Einstein condensation of photons in a dye-filled optical microresonator. The most well-known example of a photon gas, photons in blackbody radiation, does not show Bose-Einstein condensation. Instead of massively populating the cavity ground mode, photons vanish in the cavity walls when they are cooled down. The situation is different in an ultrashort optical cavity imprinting a low-frequency cutoff on the photon energy spectrum that is well above the thermal energy. The latter allows for a thermalization process in which both temperature and photon num…
We propose an experiment, where the Bell inequality is violated in a many-body system of massive particles. The source of correlated atoms is a spinor $F=1$ Bose-Einstein condensate residing in an optical lattice. We characterize the complete experimental procedure--- the local operations, the measurements and the inequality---necessary to run the Bell test. We show how the degree of violation of the Bell inequality depends on the strengths of the two-body correlations and on the number of scattered pairs. We show that the system can be used to demonstrate the Einstein-Podolsky-Rosen paradox. …
A self-consistent field model for a mixture of Bose and Fermi particles is formulated. There is explored in detail the case of a delta-like interaction, for which the thermodynamic functions are obtained, and Bose-Einstein condensation of interacting particles in the presence of the admixture of fermions is studied. It is shown that the admixture of Fermi particles leads to reducing of the temperature of Bose-Einstein condensation and smoothing of features of thermodynamic quantities at the transition temperature. As in the case of a pure Bose system, in the state of a mixture with condensate …
We review recent work on the Bose-Einstein condensation of photons in a dye microcavity environment. Other than for material particles, as e.g. cold atomic Bose gases, photons usually do not condense at low temperatures. For Planck's blackbody radiation, the most ubiquitous Bose gas, photon number and temperature are not independently tunable and at low temperatures the photons simply disappear in the system's walls, instead of massively occupying the cavity ground mode. In the here described approach, this obstacle is overcome by a fluorescence-induced thermalization mechanism in a dye-filled…
We consider the problem when there are two kinds of Bosons with an attraction between them. We find the system to consist of two Bose condensates with an additional pairing order between the Bosons. The properties of this state are discussed.
The role of geometrical frustration in strongly interacting bosonic systems is studied with a combined numerical and analytical approach. We demonstrate the existence of a novel quantum phase featuring both Bose-Einstein condensation and spin-glass behaviour. The differences between such a phase and the otherwise insulating "Bose glasses" are elucidated.
We use the Gross-Pitaevskii equation to determine the spatial structure of the condensate density of interacting bosons whose energy dispersion epsilon_k has two degenerate minima at finite wave-vectors q. We show that in general the Fourier transform of the condensate density has finite amplitudes for all integer multiples of q. If the interaction is such that many Fourier components contribute, the Bose condensate is localized at the sites of a one-dimensional lattice with spacing 2pi/q; in this case Bose-Einstein condensation resembles the transition from a liquid to a crystalline solid. We…
We propose a new theoretical formalism which describes the Bose Einstein condensation of weakly interacting bosons with finite life time interacting with a thermal bath. We show that if a quasi-thermal distribution function of particles is achieved, the elementary excitations of the condensate show a linear spectrum characteristic of a superfluid, with a renormalized sound velocity with respect to the equilibrium case.
The peculiarities in the Bose-Einstein condensation of particles and quasiparticles are discussed. The difference between the condensation of conserved and unconserved particles is analyzed. A classification of quasiparticles is given. The emphasis is made on the ability of particles and quasiparticles to condense. Illustrations include: general Bose-condensed atomic systems, such as ensembles of trapped atoms, Bose gases with conserved and unconserved number of atoms, vibrating atoms in double-well lattices, Holstein-Primakoff magnons, Schwinger bosons, slave bosons, and the condensation of s…
A well-known conjecture in mathematical physics asserts that the interacting Bose gas exhibits Bose-Einstein condensation (BEC) in the thermodynamic limit. We consider the Bose gas on certain hyperbolic spaces. In this setting, one obtains a short proof of BEC in the infinite-volume limit from the existence of a volume-independent spectral gap of the Laplacian.
The relation between the bulk correlation length and the decay length of thermodynamic Casimir forces is investigated microscopically in two three-dimensional systems undergoing Bose-Einstein condensation: the perfect Bose gas and the imperfect mean-field Bose gas. For each of these systems, both lengths diverge upon approaching the corresponding condensation point from the one-phase side, and are proportional to each other. We determine the proportionality factors and discuss their dependence on the boundary conditions. The values of the corresponding critical exponents for the decay length a…
The density of bosonic states are calculated for spinless free massive bosons in generalised d dimensions. The number of bosons are calculated in the lowest energy state. The Bose Einstein condensation was found in generalised d dimensions (on and above d = 3) and the condensation temperature is calculated which is observed to drop abruptly above three dimensions and decreases monotonically as the dimensionalities of the system increases. The rate of fall of the condensation temperature decreases as the dimensionality increases. Interestingly, in the limit of infinite dimensions, the condensat…
We examine the Bose-Einstein condensation of exciton-polaritons in a semiconductor microcavity via an electrical current. We propose that by embedding a quantum dot p-i-n junction inside the cavity, the tunneling current through the device can reveal features of condensation due to a one-to-one correspondence of the photons to the condensate polaritons. Such a device can also be used to observe the phase interference of the order parameters from two condensates.
Bose-Einstein condensation is a remarkable manifestation of quantum statistics and macroscopic quantum coherence. Superconductivity and superfluidity have their origin in Bose-Einstein condensation. Ultracold quantum gases have provided condensates close to the original ideas of Bose and Einstein, while condensation of polaritons and magnons have introduced novel concepts of non-equilibrium condensation. Here, we demonstrate a Bose-Einstein condensate (BEC) of surface plasmon polaritons in lattice modes of a metal nanoparticle array. Interaction of the nanoscale-confined surface plasmons with …
We show that in a two-component Bose gas with Rashba spin-orbit coupling (SOC) two atoms can form bound states (Rashbons) with any intra-species scattering length. At zero center-of-mass momentum there are two degenerate Rashbons due to time-reversal symmetry, but the degeneracy is lifted at finite in-plane momentum with two different effective masses. A stable Rashbon condensation can be created in a dilute system with weakly attractive intra-species and repulsive inter-species interactions. The critical temperature of Rashbon condensation is about six times smaller than the BEC transition te…
We detail the use of simple machine learning algorithms to determine the critical Bose-Einstein condensation (BEC) critical temperature $T_\text{c}$ from ensembles of paths created by path-integral Monte Carlo (PIMC) simulations. We quickly overview critical temperature analysis methods from literature, and then compare the results of simple machine learning algorithm analyses with these prior-published methods for one-component Coulomb Bose gases and liquid $^4$He, showing good agreement.
Bose-Einstein condensation is a unique phase transition in that it is not driven by inter-particle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question \emph{`How is this ideal Bose gas condensation modified in the presence of interactions between the particles?' } This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some ans…
We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.
We clarify the condition for the occurrence of magnon Bose-Einstein condensation (BEC) in spin pumping systems without using external pumping magnetic fields. The Goldstone model is generalized and the stability of the vacuum is closely investigated. By applying the generalized Goldstone model to spin pumping systems, the condition for the experimental realization of the stable magnon BEC in spin pumping systems is theoretically proposed.
We consider interacting Bose particles in an external local potential. It is shown that large class of external quasicrystal potentials cannot sustain any type of Bose-Einstein condensates. Accordingly, at spatial dimensions $D\leq 2$ in such quasicrystal potentials a supersolid is not possible via Bose-Einstein condensates at finite temperatures. The latter also hold true for the two-dimensional Fibonacci tiling. However, supersolids do arise at $D\leq 2$ via Bose-Einstein condensates from infinitely long-range, nonlocal interparticle potentials.
We consider the low density limit of a Fermi gas in the BCS approximation. We show that if the interaction potential allows for a two-particle bound state, the system at zero temperature is well approximated by the Gross-Pitaevskii functional, describing a Bose-Einstein condensate of fermion pairs.
We study Bose gases in $d$ dimensions, $d \ge 2$, with short-range repulsive pair interactions, at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose--Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the eigenstates of the underlying one-particle Hamiltonian.
