Dynamics of sound waves in an interacting Bose gas

Physical Review A, 2000

We calculate the frequencies and damping rates of the low-lying collective modes of a Bose-Einstein condensed gas at nonzero temperature. We use a complex nonlinear Schrödinger equation to determine the dynamics of the condensate atoms, and couple it to a Boltzmann equation for the noncondensate atoms. In this manner we take into account both collisions between noncondensate-noncondensate and condensate-noncondensate atoms. We solve the linear response of these equations, using a time-dependent gaussian trial function for the condensate wave function and a truncated power expansion for the deviation function of the thermal cloud. As a result, our calculation turns out to be characterized by two dimensionless parameters proportional to the noncondensate-noncondensate and condensatenoncondensate mean collision times. We find in general quite good agreement with experiment, both for the frequencies and damping of the collective modes.

Physical Review A, 1997

Physical Review A, 2000

We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finitetemperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due both to quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes. 03.75.Fi, 67.40.Db, 0530.Jp

2021

We provide the analysis of a physically motivated model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, Happ, that is quadratic in the Boson field operators for noncondensate atoms. This Happ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105–123, 1961), we apply a non-unitary transformation to Happ. Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67–101, 1972). These functions solve an eigenvalue problem for ...

Physical Review A, 1997

We study the collective and single particle excitations of trapped Bose gas with very large particle number (N ∼ 10 7 − 10 8 ) in the external harmonic trap. We utilize the correlated two-body basis function for expansion of the full many-body wave function. We calculate the ground state energy and one-body density and compare our results with those obtained using mean-field approaches. We observe that at truely large particle limit the system behaves more classically. We also present an extensive study of the excitation spectrum in a wide range of energy. We observe that the low-lying excitations are of collective nature where the effect of interaction is really important and for large particle limit those are well described by the hydrodynamic (HD) model. The asymptotic values (N → ∞) are well compared with the HD prediction. We observe that the high-lying excitations (l = 0) are of truely single particle nature and correctly reproduces the non-interacting limit at asymptotic regime (l → ∞).

Nature, 2007

We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a non-monotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures T , we demonstrate that the coefficient in T 2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weaklyinteracting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in details. At T = 0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.

In this work the propagation of sound waves in a degenerate quantum gas is considered. We modify the Wang Chang and Uhlenbeck method for description of the sounds in a Maxwell gas for the case of a degenerate quantum gas. Using this approach, we constructed a dispersion relation for sound waves in a condensed Bose gas at finite temperatures, and calculate the velocities of first and second sounds in the first approximation. The possibility of the theoretical investigation for sound damping is discussed. PACS: 03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow; 67.40.Pm Transport processes, second and other sounds, and thermal counterflow; Kapitza resistance; 67.57.Jj Collective modes.

Physical Review A, 2002

We present a convenient technique describing the condensate in dynamical equilibrium with the thermal cloud, at temperatures close to the critical one. We show that the whole isolated system may be viewed as a single classical field undergoing nonlinear dynamics leading to a steady state. In our procedure it is the observation process and the finite detection time that allow for splitting the system into the condensate and the thermal cloud. PACS numbers: 03.75.Fi Successful experimental realization of Bose-Einstein condensation in a dilute gas of alkali atoms [1] offers a new and fascinating tool to probe the border between quantum and classical worlds. The theory of the dynamical behavior of such a many body system is very difficult and cannot be solved exactly. It is particularly hard to extract quantitative predictions in the vicinity of the critical temperature. At the other extreme, at zero temperature, the weakly interacting Bose gas is well described by the mean-field approach. All particles occupy the same quantum state whose wave function is the lowest energy solution of the Gross-Pitaevskii (GP) equation . At low temperatures the Bogoliubov approximation comes handy [3] with its quasi-particles that have just been observed in a direct experiment . The theory gets much more complex at higher temperatures. There is a considerable effort to develop a working theoretical and numerical tool valid there. One group of papers from the very beginning describes the system as consisting of two (interacting) fractions: the condensate and the thermal cloud. This ambitious program is very demanding numerically. The other group interprets the high energy solutions of the time-dependent GP equation as describing the full condensate plus thermal cloud system . As a rule, these authors were able to identify the condensate as a definite part of the system only in the somewhat academic case of the gas in a rectangular box with periodic boundary conditions. In this case the condensate is just the zero momentum component of the wave function. The aim of this Rapid Communication is to show that much more may be achieved with the readily available high-energy solutions of the GP equation in harmonic traps. It is also possible to split this solution into a sum of the condensed and uncondensed parts, define the condensate wave function and approximately estimate the temperature of the resulting system.

Physical Review A, 2019

We experimentally and theoretically investigate the lowest-lying axial excitation of an atomic Bose-Einstein condensate in a cylindrical box trap. By tuning the atomic density, we observe how the nature of the mode changes from a single-particle excitation (in the low-density limit) to a sound wave (in the high-density limit). Throughout this crossover the measured mode frequency agrees with Bogoliubov theory. Using approximate low-energy models we show that the evolution of the mode frequency is directly related to the interactioninduced shape changes of the condensate and the excitation. Finally, if we create a large-amplitude excitation, and then let the system evolve freely, we observe that the mode amplitude decays non-exponentially in time; this nonlinear behaviour is indicative of interactions between the elementary excitations, but remains to be quantitatively understood.