Motivated by current interest in the dynamics of trapped quantum gases, we study the microcanonical dynamics of a trapped one-dimensional gas of classical particles interacting via a finite-range repulsive force of tunable strength. We examine two questions which have been of interest in quantum dynamics: (1) the breathing mode (size oscillation) dynamics of the trapped gas and the dependence of the breathing frequency on the interaction strength, and (2) the long-time relaxation and possible thermalization of the finite isolated gas. We show that the breathing mode frequency has non-monotonic dependence on the magnitude of the mutual repulsion, decreasing for small interactions and increasing for larger interactions. We explain these dependences in terms of slowing-down or speeding-up effects of two-body collision processes. We find that the gas thermalizes within a reasonable finite timescale in the sense of single-particle energies acquiring a Boltzmann distribution, only when the interaction strength is large compared to the energy per particle.
Using a field-theoretic approach, we systematically generalize the usual semiclassical approximation for a harmonically trapped ideal Bose gas in such a way that its range of applicability is essentially extended. With this we can analytically calculate thermodynamic properties even for small particle numbers. In particular, it now becomes possible to determine the critical temperature as well as the temperature dependence of both heat capacity and condensate fraction in low-dimensional traps, where the standard semiclassical approximation is not even applicable.
We present a family of generalized unitary transformations that simplifies the Hamiltonian for a harmonically trapped two level atom (or ion) interacting with a plane wave laser field. Novel near resonant single as well as double vibrational phonon dynamical regimes are found. The validity condition of the often used motional rotating wave approximation (MRWA) is examined both numerically and analytically. Large errors are found within typical regimes of MRWA with respect to the motional degrees of freedom. The effects of MRWA in trapped ion systems are shown to be opposite to that of the rota…
An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. In contrast to previous results available in the literature, our expressions are exact for any temperature and magnetic field strength. We also present a novel factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on …
In this paper the thermodynamical parameters of a condensed Boson gas are calculated from the partial derivative of the grand potential. In particular, the analytical expressions for some important parameters, such as the condensed fraction, specific heat, critical temperature, effective size, and release energy are investigated. The mean effects which can be altered the ideal Bose gas, such as finite size, highly anisotropic of the external potential and interatomic interaction effects are considered simultaneously. Some new characteristics of the trapped interacting Bose gases in a highly an…
We discuss the scaling of the interaction energy with particle numbers for a harmonically trapped two-species mixture at thermal equilibrium experiencing interactions of arbitrary strength and range. In the limit of long-range interactions and weak coupling, we recover known results for the integrable Caldeira-Leggett model in the classical limit. In the case of short-range interactions and for a balanced mixture, numerical simulations show scaling laws with exponents that depend on the interaction strength, its attractive or repulsive nature, and the dimensionality of the system. Simple analy…
We study experimentally and theoretically a cold trapped Bose gas under critical rotation, i.e. with a rotation frequency close to the frequency of the radial confinement. We identify two regimes: the regime of explosion where the cloud expands to infinity in one direction, and the regime where the condensate spirals out of the trap as a rigid body. The former is realized for a dilute cloud, and the latter for a Bose-Einstein condensate with the interparticle interaction exceeding a critical value. This constitutes a novel system in which repulsive interactions help in maintaining particles to…
We report on the dynamics of thermalization by extending a generalization of the Caldeira-Leggett model, developed in the context of cold atomic gases confined in a harmonic trap, to higher dimensions. Universal characteristics en route to thermalization which appear to be independent of dimensionality are highlighted, which additionally suggest a scaling analogous to turbulent mixing in fluid dynamics. We then focus on features dependent on dimensionality, with particular regard to the role of angular momentum of the two atomic clouds, having in mind the goal of efficient thermalization betwe…
We derive the elastic collision rate for a harmonically-trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is $σ(k) =4π/k^2$, with $\hbar k$ the relative momentum. The collision rate is given in the form $Γ=γI(T/T_F)$--the product of a universal collision rate $γ=k_B T_F/(6 π\hbar)$ and a dimensionless function of the ratio of the temperature $T$ to the Fermi temperature $T_F$. We find $I$ has a peak value of $\simeq4.6$ at $T/T_F\simeq0.4$, $I\simeq 82 (T/T_F)^2$ for $T/T_F\leq 0.15$, and $I\simeq 2(T_F/T)^2$ for $T/T_F>1.5$. We estimate the collision r…
We theoretically study the collective excitations of an ideal gas confined in an isotropic harmonic trap. We give an exact solution to the Boltzmann-Vlasov equation; as expected for a single-component system, the associated mode frequencies are integer multiples of the trapping frequency. We show that the expressions found by the scaling ansatz method are a special case of our solution. Our findings, however, are most useful in case the trap contains more than one phase: we demonstrate how to obtain the oscillation frequencies in case an interface is present between the ideal gas and a differe…
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive analytically a differential equation connecting these densities, which so far have been proven only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is identified uniquely. We derive …
We solve the problem of a Bose or Fermi gas in $d$-dimensions trapped by $% δ\leq d$ mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature $T_{c}$ if and only if $d+δ>2$, and a jump in the specific heat at $T_{c}$ if and only if $d+δ>4$. Specific heats for both gas types precisely coincide as functions of temperature when $d+δ=2$. The trapped system behaves like an …
We compute single-particle energy spectra for a one-body hamiltonian consisting of a two-dimensional deformed harmonic oscillator potential, the Rashba spin-orbit coupling and the Zeeman term. To investigate the statistical properties of the obtained spectra as functions of deformation, spin-orbit and Zeeman strengths we examine the distributions of the nearest neighbor spacings. We find that the shapes of these distributions depend strongly on the three potential parameters. We show that the obtained shapes in some cases can be well approximated with the standard Poisson, Brody and Wigner dis…
The thermodynamical properties are calculated for a three-dimensional model of $N$ harmonically interacting spin-polarized fermions in a parabolic potential well. The obtained dependences of the chemical potential and of the internal energy on the complete range of the temperature and of the number of particles turn out to obey a scaling law, similar to the scaling from the continuum approximation for the density of states. The calculation technique is based on our path integral approach of symmetrized density matrices for identical particles in a parabolic confining well.
We use quantum Monte Carlo simulations to obtain zero-temperature state diagrams for strongly correlated lattice bosons in one and two dimensions under the influence of a harmonic confining potential. Since harmonic traps generate a coexistence of superfluid and Mott insulating domains, we use local quantities such as the quantum fluctuations of the density and a local compressibility to identify the phases present in the inhomogeneous density profiles. We emphasize the use of the "characteristic density" to produce a state diagram that is relevant to experimental optical lattice systems, rega…
For a fermion gas with equally spaced energy levels, the density and the pair correlation function are obtained. The derivation is based on the path integral approach for identical particles and the inversion of the generating functions for both static responses. The density and the pair correlation function are evaluated explicitly in the ground state of a confined fermion system with a number of particles ranging from 1 to 220 and filling the Fermi level completely.
We investigate the combined influence of a magnetic field and a harmonic interparticle interaction on the thermodynamic properties of a finite number of spin polarized fermions in a confiment potential. This study is an extension using our path integral approach of symmetrized density matrices for identical particles. The thermodynamical properties are calculated for a three dimensional model of N harmonically interacting spin polarized fermions in a parabolic potential well in the presence of a magnetic field. The free energy and the internal energy are obtained for a limited number of partic…
We have analytically explored finite size and interparticle interaction corrections to the average energy of a harmonically trapped Fermi gas below and above the Fermi temperature, and have obtained a better fitting for the excess energy reported by DeMarco and Jin [Science $\textbf{285}$, 1703 (1999)]. We have presented a perturbative calculation within a mean field approximation.
We study the Bose-Einstein condensation for a 3-d system of ideal Bose gas which is harmonically trapped along two perpendicular directions and is confined in between two slabs along the other perpendicular direction. We calculate the Casimir force between the two slabs for this system of trapped Bose gas. At finite temperatures this force for thermalized photons in between two plates has a classical expression which is independent of $\hbar$. At finite temperatures the Casimir force for our system depends on $\hbar$. For the calculation of Casimir force we consider only the Dirichlet boundary…
We consider a harmonically trapped few-Boson system under rotation and investigate the ground state properties beyond the usual ``lowest Landau level'' approximation by using exact diagonalizations in a restricted Hilbert subspace. We find that both the effective interaction energy and density distribution are strongly affected by the two-body interaction strength.
We perform high-precision measurements of the condensation temperature of a harmonically-trapped atomic Bose gas with widely-tuneable interactions. For weak interactions we observe a negative shift of the critical temperature in excellent agreement with mean-field theory. However for sufficiently strong interactions we clearly observe an additional positive shift, characteristic of beyond-mean-field critical correlations. We also discuss non-equilibrium effects on the apparent critical temperature for both very weak and very strong interactions.
Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a $d$-dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile dist…
The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: large deviations
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础科学研究所国际理论科学中心）；LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay（巴黎南大学萨克雷大学法国国家科学研究中心理论与物质物理实验室）；Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589（索邦大学法国国家科学研究中心高能理论与物理实验室）
We study the critical point for the emergence of coherence in a harmonically trapped two-dimensional Bose gas with tuneable interactions. Over a wide range of interaction strengths we find excellent agreement with the classical-field predictions for the critical point of the Berezinskii-Kosterlitz-Thouless (BKT) superfluid transition. This allows us to quantitatively show, without any free parameters, that the interaction-driven BKT transition smoothly converges onto the purely quantum-statistical Bose-Einstein condensation (BEC) transition in the limit of vanishing interactions.
We investigate the width of the Ginzburg critical region and experimental signatures of critical behavior in strongly interacting trapped Fermi gases close to unitarity, where the s-wave scattering length diverges. Despite the fact that the width of the critical region is of the order unity, evidence of critical behavior in the bulk thermodynamics of trapped gases is strongly suppressed by their inhomogeneity. The specific heat of a harmonically confined gas, for instance, is \textit{linear} in the reduced temperature $t = (T-T_{\mathrm{c}})/T_{\mathrm{c}}$ above $T_{\mathrm{c}}$. We also disc…
We discuss the transition from a fully decoherent to a (quasi-)condensate regime in a harmonically trapped weakly interacting 1D Bose gas. By using analytic approaches and verifying them against exact numerical solutions, we find a characteristic crossover temperature and crossover atom number that depend on the interaction strength and the trap frequency. We then identify the conditions for observing either an interaction-induced crossover scenario or else a finite-size Bose-Einstein condensation phenomenon characteristic of an \textit{ideal} trapped 1D gas.
In this article, we investigate mean field effects for a bosonic gas harmonically trapped above the transition temperature in the collisionless regime. We point out that those effects can play also a role in low dimensional system. Our treatment relies on the Boltzmann equation with the inclusion of the mean field term. The equilibrium state is first discussed. The dispersion relation for collective oscillations (monopole, quadrupole, dipole modes) is then derived. In particular, our treatment gives the frequency of the monopole mode in an isotropic and harmonic trap in the presence of mean fi…
We present an exact closed form expression for the {\em finite temperature} first-order density matrix of a harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where exact expressions have been limited to quantities derived from the {\em diagonal} first-order density matrix. We compare our exact results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-N limit. As an interesting applicat…
We present theoretical predictions for the equation of state of a harmonically-trapped Fermi gas in the unitary limit. Our calculations rest on a study of the equation of state of a uniform gas using three distinct perturbation schemes, compared to Monte-Carlo results. We show that in experiments the temperature can be usefully calibrated by making use of the entropy, which is invariant during an adiabatic conversion into the weakly-interacting limit of molecular BEC. We predict the entropy dependence of the equation of state.
The ground state and its structure for a rotating, harmonically trapped N-Boson system with a weak repulsive contact interaction are studied as the angular momentum L increases up to 3N. We show that the ground state is generally a fragmented condensate due to angular momentum conservation. In response to an (arbitrarily weak) asymmetric perturbation of the trap, however, the fragmented ground state can be transformed into a single condensate state. We manifest this intrinsic instability by calculating the conditional probability distributions, which show patterns analogous to the boson densit…
We compute structures of vortex configurations in a harmonically trapped Bose-Einstein condensed atom gas within three different gapless self-consistent mean-field theories. Outside the vortex core region, the density profiles for the condensate and the thermal gas are found to differ only by a few percent between the Hartree-Fock-Bogoliubov-Popov theory and two of its recently proposed gapless extensions. In the core region, however, the differences in the density profiles are substantial. The structural differences are reflected in the energies of the quasiparticle states localized near the …
In this paper, we compute exactly the average density of a harmonically confined Riesz gas of $N$ particles for large $N$ in the presence of a hard wall. In this Riesz gas, the particles repel each other via a pairwise interaction that behaves as $|x_i - x_j|^{-k}$ for $k>-2$, with $x_i$ denoting the position of the $i^{\rm th}$ particle. This density can be classified into three different regimes of $k$. For $k \geq 1$, where the interactions are effectively short-ranged, the appropriately scaled density has a finite support over $[-l_k(w),w]$ where $w$ is the scaled position of the wall. Whi…
