Bose-Einstein Gas Research Articles

Lev Petrovich Pitaevskii

Lev Petrovich Pitaevskii

Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: Gaseous density profiles and onset of condensation.

We calculate density profiles for self-gravitating clusters of an ideal Bose-Einstein gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed to prevent escape. The length scale and energy scale in use for the gaseous phase render density profiles for gaseous macrostates independent of total mass. Density profiles for mixed-phase macrostates have a condensed core surrounded by a gaseous halo. The spatial extension of the core is negligibly small on the length scale tailored for the halo. The mechanical stability conditions as evident in caloric curves permit multiple macrostates to coexist. Their status regarding thermal equilibrium is examined by a comparison of free energies. The onset of condensation takes place at a nonzero temperature in all cases. The critical singularities and the nature of the phase transition vary with the symmetry of the cluster and the dimensionality of the space.

THE IMPACT OF INTERACTION ON BOSE-EINSTEIN CONDENSATION OF ULTRACOLD ATOMIC GASES IN TRAPPED POTENTIALS

In this paper, we discuss theoretically the behavior of ideal Bose-Einstein gases in a 3D box and in a harmonic trap. We make use of the mean filed theory based on the time dependent Gross-Pitaevskii equation which describes the main features of Bose-Einstein condensation. We investigate and discuss the density profiles and the ground state properties of a trapped condensate confined in harmonic trap. We found out that in an attractive interaction, the gas tends to increase its central density in order to lower the interaction energy.

Primordial gravitational waves spectrum in the interacting Bose–Einstein gas model

We study the evolution and power spectrum of primordial gravitational waves in the interactive Bose-Einstein gas model for dark energy, relevant, as it addresses the coincidence problem. The model is applied in the radiation, matter and dark-energy domination stages. The model introduces a scale factor associated to the radiation-matter transition which influences the gravitational spectrum. We focus on the impact of the free parameters on both the gravitational waves amplitude and its power-spectrum slope. For sets of parameters fitting Hubble's law, we show that the model's parameter for today's dark-matter energy density has a noticeable impact on such waves, while the others produce an indistinguishable effect. The feasibility of detecting such waves under present and future measurements is discussed.

Branch-counting in the Everett interpretation of quantum mechanics

A defence is offered of a version of the branch-counting rule in the Everett interpretation (otherwise known as many worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule and deliver a notion of objective probability similar to naive frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc : it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose–Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.

Generation and decoherence of soliton spatial superposition states

Due to their coherence properties, dilute atomic gas Bose-Einstein condensates seem a versatile platform for controlled creation of mesoscopically entangled states with a large number of particles and also allow controlled studies of their decoherence. However, the creation of such a state intrinsically involves many-body quantum dynamics that cannot be captured by mean-field theory, and thus invalidates the most widespread methods for the description of condensates. We follow up on a proposal, in which a condensate cloud as a whole is brought into a superposition of two different spatial locations, by mapping entanglement from a strongly interacting Rydberg atomic system onto the condensate using off-resonant laser dressing [R. Mukherjee et al., Phys. Rev. Lett. 115 040401 (2015)]. A variational many-body Ansatz akin to recently developed multi-configurational methods allows us to model this entanglement mapping step explicitly, while still preserving the simplicity of mean-field physics for the description of each branch of the superposition. In the second part of the article, we model the decoherence process due to atom losses in detail. Altogether we confirm earlier estimates, that tightly localized clouds of 400 atoms can be brought into a quantum superposition of two locations about 3 {\mu}m apart and remain coherent for about 1 ms.

Stability in turbulence: The interplay between shocks and vorticity in a superfluid with higher-order dispersion

Higher-order dispersion can lead to intriguing dynamics that are becoming a focus of modern hydrodynamics research. Such systems occur naturally, for example in shallow water waves and nonlinear optics, for which several types of novel dispersive shocks structures have been identified. Here we introduce ultracold atoms as a tunable quantum simulations platform for higher-order systems. Degenerate quantum gases are well controlled model systems for the experimental study of dispersive hydrodynamics in superfluids and have been used to investigate phenomena such as vortices, solitons, dispersive shock waves and quantum turbulence. With the advent of Raman-induced spin-orbit coupling, the dispersion of a dilute gas Bose-Einstein condensate can be modified in a flexible way, allowing for detailed investigations of higher-order dispersion dynamics. Here we present a combined experimental and theoretical study of shock structures generated in such a system. The breaking of Galilean invariance by the spin-orbit coupling allows two different types of shock structures to emerge simultaneously in a single system. Numerical simulations suggest that the behavior of these shock structures is affected by interactions with vortices in a manner reminiscent of emerging viscous hydrodynamics due to an underlying quantum turbulence in the system. This result suggests that spin-orbit coupling can be used as a powerful means to tun the effective viscosity in cold-atom experiments serving as quantum simulators of turbulent hydrodynamics, with applications from condensed matter and optics to quantum simulations of neutron stars.

A null model for Dunbar’s circles

An individual’s social group may be represented by their ego-network, formed by the links between the individual and their acquaintances. Ego-networks present an internal structure of increasingly large nested layers (or circles) of decreasing relationship intensity, whose size exhibits a precise scaling ratio. Starting from the notion of limited social bandwidth, and assuming fixed costs for the links in each layer, we propose a null model built on a grand-canonical ensemble that generates the observed hierarchical social structure. The observed internal structure of ego-networks becomes a natural outcome to expect when we assume the existence of layers demanding different amounts of resources. In the thermodynamic limit, reached when the number of ego-network copies is large, the specific layer degrees follow a Poisson distribution. We also find that, under certain conditions, equispaced layer costs are necessary to obtain a constant group size scaling. Our model presents interesting analogies to a Bose–Einstein gas, that we briefly discuss. Finally, we fit and compare the model with an empirical social network.

Out-Of-Equilibrium Transverse Momentum Spectra of Pions at LHC Energies

In order to characterize the transverse momentum spectra (pT) of positive pions measured in the ALICE experiment, two thermal approaches are utilized; one is based on degeneracy of nonperfect Bose-Einstein gas and the other imposes an ad hoc finite pion chemical potential. The inclusion of missing hadron states and the out-of-equilibrium contribute greatly to the excellent characterization of pion production. An excellent reproduction of these pT-spectra is achieved at μπ=0.12 GeV and this covers the entire range of pT. The excellent agreement with the experimental results can be understood as a manifestation of not-yet-regarded anomalous pion production, which likely contributes to the long-standing debate on “anomalous” proton-to-pion ratios at top RHIC and LHC energies.

Observational constraints on the free parameters of an interacting Bose–Einstein gas as a dark-energy model

Dark energy is modelled by a Bose-Einstein gas of particles with an attractive interaction. It is coupled to cold dark matter, within a flat universe, for the late-expansion description, producing variations in particle-number densities. The model's parameters, and physical association, are: $\Omega_{G0}$, $\Omega_{m0}$, the dark-energy rest-mass energy density and the dark-matter term scaling as a mass term, respectively; $\Omega_{i0}$, the self-interaction intensity; $x$, the energy exchange rate. Energy conservation relates such parameters. The Hubble equation omits $\Omega_{G0}$, but also contains $h$, the present-day expansion rate of the flat Friedman--Lem\^aitre--Robertson--Walker metric, and $\Omega_{b0}$, the baryon energy density, used as a prior. This results in the four effective chosen parameters $\Omega_{b0}$, $h$, $\Omega_{m0}$, $\Omega_{i0}$, fit with the Hubble expansion rate $H(z)$, and data from its value today, near distance, and supernovas. We derive wide $1\sigma$ and $2\sigma$ likelihood regions compatible with definite positive total CDM and IBEG mass terms. Additionally, the best-fit value of parameter $x$ relieves the coincidence problem, and a second potential coincidence problem related to the choice of $\Omega_{G0}$.

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose-Einstein Gas in a Box.

From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of -rds in one dimension, and -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.

Bulk viscosity of strongly interacting matter in the relaxation time approximation

This paper presents how thermal mean field effects are incorporated consistently in the hydrodynamical modelling of heavy-ion collisions. The nonequilibrium correction to the distribution function resulting from a temperature-dependent mass is obtained in a procedure which automatically satisfies the Landau matching condition and is thermodynamically consistent. The physics of the bulk viscosity is studied here for Boltzmann and Bose-Einstein gases within the Chapman-Enskog and 14-moment approaches in the relaxation time approximation. Constant and temperature-dependent masses are considered in turn. It is shown that, in the small mass limit, both methods lead to the same value of the ratio of the bulk viscosity over its relaxation time. The inclusion of a temperature-dependent mass leads to the emergence of the $\beta_\lambda$-function in that ratio, and it is of the expected parametric form for the Boltzmann gas, while for the Bose-Einstein case it is affected by the infrared cut-off. This suggests that the relaxation time approximation may be too crude to obtain a reliable form of $\zeta/\tau_R$ for gases obeying Bose-Einstein statistics.

Coincidence problem within dark energy as a coupled self-interacting Bose–Einstein gas

A late accelerated expansion of the Universe is obtained from non-relativistic Bose–Einstein particles with a short-range attractive interaction, with a temperature of the same order as the critical one, and an energy interchange with a dark-matter source. We show that for a sizable range of parameters, these particles evolve as a coupled dark-energy source, solving the coincidence problem, and in agreement with the luminosity distance versus redshift derived from supernova data.

Bose-Einstein condensate strings

We consider the possible existence of gravitationally bound general relativistic strings consisting of Bose-Einstein condensate (BEC) matter which is described, in the Newtonian limit, by the zero temperature time-dependent nonlinear Schr\"odinger equation (the Gross-Pitaevskii equation), with repulsive interparticle interactions. In the Madelung representation of the wave function, the quantum dynamics of the condensate can be formulated in terms of the classical continuity equation and the hydrodynamic Euler equations. In the case of a condensate with quartic nonlinearity, the condensates can be described as a gas with two pressure terms, the interaction pressure, which is proportional to the square of the matter density, and the quantum pressure, which is without any classical analogue, though, when the number of particles in the system is high enough, the latter may be neglected. Assuming cylindrical symmetry, we analyze the physical properties of the BEC strings in both the interaction pressure and quantum pressure dominated limits, by numerically integrating the gravitational field equations. In this way we obtain a large class of stable stringlike astrophysical objects, whose basic parameters (mass density and radius) depend sensitively on the mass and scattering length of the condensate particle, as well as on the quantum pressure of the Bose-Einstein gas.

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

We show local and global well-posedness results for the Hartree equation $$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2(\R^d)$, $\rho_\gamma(x)=\gamma(x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma(0)$ is assumed to be a perturbation of a translation-invariant state $\gamma_f=f(-\Delta)$ which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state $\gamma(t)$, counted relatively to the stationary state $\gamma_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\Delta)$. Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

Vortices and turbulence in trapped atomic condensates.

After more than a decade of experiments generating and studying the physics of quantized vortices in atomic gas Bose-Einstein condensates, research is beginning to focus on the roles of vortices in quantum turbulence, as well as other measures of quantum turbulence in atomic condensates. Such research directions have the potential to uncover new insights into quantum turbulence, vortices, and superfluidity and also explore the similarities and differences between quantum and classical turbulence in entirely new settings. Here we present a critical assessment of theoretical and experimental studies in this emerging field of quantum turbulence in atomic condensates.

Experimental Realization of Josephson Junctions for an Atom SQUID

We report the creation of a pair of Josephson junctions on a toroidal dilute gas Bose-Einstein condensate (BEC), a configuration that is the cold atom analog of the well-known dc superconducting quantum interference device (SQUID). We observe Josephson effects, measure the critical current of the junctions, and find dynamic behavior that is in good agreement with the simple Josephson equations for a tunnel junction with the ideal sinusoidal current-phase relation expected for the parameters of the experiment. The junctions and toroidal trap are created with the painted potential, a time-averaged optical dipole potential technique which will allow scaling to more complex BEC circuit geometries than the single atom-SQUID case reported here. Since rotation plays the same role in the atom SQUID as magnetic field does in the dc SQUID magnetometer, the device has potential as a compact rotation sensor.

Two polaron flavors of the Bose-Einstein condensate impurity

We show that repulsive neutral-atom impurities in a dilute gas Bose-Einstein condensate (BEC) can self-localize in bubble polaron states formally analogous to electron bubbles in helium. The BEC is then the first impurity host medium known to exhibit both Landau-Pekar polaron states akin to that of self-localized electrons in a dielectric lattice and self-localized bubble polaron states. We find that the neutral BEC-impurity system is fully characterized by only two dimensionless coupling constants and that a single BEC impurity can be steered adiabatically from the Landau-Pekar to the bubble region. The adiabatic change is that of a crossover, not a transition.

Ultrarelativistic Bose–Einstein gas on Lorentz symmetry violation

In this paper, we study the effects of Lorentz Symmetry Breaking on the thermodynamic properties of ideal gases. Inspired by the dispersion relation coming from the Carroll–Field–Jackiw model for Electrodynamics with Lorentz and CPT violation term, we compute the thermodynamics quantities for a Boltzmann, Fermi–Dirac and Bose–Einstein distributions. Two regimes are analyzed: the large and the small Lorentz violation. In the first case, we show that the topological mass induced by the Chern–Simons term behaves as a chemical potential. For Bose–Einstein gases, a condensation in both regimes can be found.

Light-induced torque for the generation of persistent current flow in atomic gas Bose-Einstein condensates

We show that a persistent current flow in an atomic gas Bose-Einstein condensate could be realized when the system is subject to two counterpropagating Laguerre-Gaussian so-called doughnut beams, creating a toroidal trap. The theory is developed involving a two-photon process within three atomic levels leading to a quantized light-induced torque which rotates the atoms, generating an atomic current flow in the ring. We also show that it is possible for the torque to be controlled and even switched on and off by varying the frequencies of the incident light, thereby allowing a mechanism for the control of the current flow.