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arXiv:1112.3220 · Physical Review A

Three-fluid hydrodynamics of spin-1 Bose-Einstein condensates

Gergely Szirmai, P. Szépfalusy

Published May 4, 2012• 11 citations
arXivHTMLPDF

Abstract

We study excitations of the spin-1 Bose gas at finite temperatures and in the presence of a not-so-strong magnetic field or, equivalently, when the gas sample is partially polarized. Motivated by the success of two-fluid hydrodynamics of scalar superfluids we develop a three-fluid hydrodynamic description to treat the low-frequency and long-wavelength excitations of the spin-1 Bose gas. We derive coupled linear hydrodynamic equations of the three sounds and evaluate them numerically in a self-consistent mean-field approximation valid for dilute gas in the intermediate and critical temperature regions. In this latter region we identify the critical mode.

DOI
10.1103/physreva.85.053603

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We study excitations of the spin-1 Bose gas at finite temperatures and in the presence of a not-so-strong magnetic field or, equivalently, when the gas sample is partially polarized. Motivated by the success of two-fluid hydrodynamics of scalar superfluids we develop a three-fluid hydrodynamic description to treat the low-frequency and long-wavelength excitations of the spin-1 Bose gas. We derive coupled linear hydrodynamic equations of the three sounds and evaluate them numerically in a self-consistent mean-field approximation valid for dilute gas in the intermediate and critical temperature regions. In this latter region we identify the critical mode.
A finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the many-particle quantum hydrodynamic method. It is presented as the two fluid model of the Bose–Einstein condensate (BEC) and normal fluid. The continuity, Euler, spin evolution, and nematic tensor evolution equations are derived for each fluid. The linear and quadratic Zeeman effects are included. Scalar and spin–spin like short-range interactions are considered in the first order by the interaction radius. Obtained hydrodynamic equations are also represented as the set of two nonlinear Pauli equations. The spectrum of the bulk collective excitations is considered for the ferromagnetic phase in the small temperature limit. The spin wave is not affected by the presence of the small temperature in the described minimal coupling model, where the thermal part of the spin-current of the normal fluid is neglected. The two sound waves are affected by the spin evolution in the same way as the change of spectrum of the single sound wave in BEC, where speed of sound is proportional to g1+g2 with gi as the interaction constants.
The two-fluid model is fundamental for the description of superfluidity. In the nearly incompressible liquid regime, it successfully describes first and second sound, corresponding, respectively, to density and entropy waves, in both liquid helium and unitary Fermi gases. Here, we study the two sounds in the opposite regime of a highly compressible fluid, using an ultracold ^{39}K Bose gas in a three-dimensional box trap. We excite the longest-wavelength mode of our homogeneous gas, and observe two distinct resonant oscillations below the critical temperature, of which only one persists above it. In a microscopic mode-structure analysis, we find agreement with the hydrodynamic theory, where first and second sound involve density oscillations dominated by, respectively, thermal and condensed atoms. Varying the interaction strength, we explore the crossover from hydrodynamic to collisionless behavior in a normal gas.
We derive coupled equations of motion for the condensate (superfluid) and noncondensate (normal fluid) degrees of freedom in a trapped Bose gas at finite temperatures. Our results are based on the Hartree-Fock-Popov approximation for the time-dependent condensate wave function, and thermodynamic local equilibrium for the noncondensate atoms. In the special case of a uniform weakly interacting gas, our hydrodynamic equations are shown to be consistent with the two-fluid equations of Landau. The collective modes in a parabolically trapped Bose gas include the analog of the out-of-phase second-sound mode in superfluid 4He{}^{4}\mathrm{He}4He.
The field of Bose-Einstein condensation (BEC) in dilute atomic gases provides a fruitful playground to test well-developed theories of quantum fluids. Research using BECs can address open questions relating to the many-body aspects of two-component quantum liquids, namely the interaction between the hydrodynamic normal and the superfluid component at finite temperatures. After the first realization of BEC some pilot experiments have been carried out, but detailed experiments are missing. This has to be compared to the case of liquid helium II, where many experiments since the 1950s have added to our understanding of novel phenomena in quantum liquids, like collective excitations, first and second sound, and others. One of the drawbacks of liquid helium is that the interactions are so strong that a clear distinction between the two components is difficult and the interpretation of the phenomena remains ambiguous. Most experimental studies of dilute Bose-Einstein condensed gases use clouds, which are in the collisionless regime. The reason for the lack of detailed experiments in BECs to study quantum liquids and in particular the hydrodynamical aspects of it, is the limited number of atoms (typically 1--10 million) in the experiments leaving the thermal atoms virtually collisionless. Efforts to decrease the mean free path by increasing the confinement limits the lifetime of the sample, since the density is limited by three-body decay. Experiments on dilute clouds of cold atoms are generally conducted in highly asymmetric traps. In these elongated, cigar-shaped geometries the mean free path of the atoms can become much shorter than the size of the cloud in the long, axial direction, but at the same time exceeds the size in the other, radial directions. In this so-called hydrodynamic-collisionless regime the system can be considered radially collisionless and axially hydrodynamic. In our setup we have created BECs containing up to 300 million sodium atoms by evaporation of atoms in an axially strongly decompressed trap with an aspect ratio of 1:65. Hot atoms created in three-body collisions are able to leave the sample in this highly asymmetric trap, before they can heat other atoms in an avalanche. The sample is axially hydrodynamic, but due to the large aspect ratio collisionless in the radial direction. Such samples are ideal for the observation of the interaction between the superfluid and normal fluid component of the cloud. In this thesis we describe the setup which is used to reach the hydrodynamic regime and we describe a novel implementation of the well-known phase contrast imaging technique. Next, we study collective excitations in the crossover from the collisionless toward the hydrodynamic regime, we determine the heat conduction in a cold, hydrodynamic thermal cloud and observe the propagation of a thermal wave in the two-fluid regime. Furthermore, we study the friction between the superfluid and normal fluid component of the cloud. In the final chapters of this thesis we study the propagation of second sound, as well as the formation and propagation of shock waves.
Abstract We develop a novel model of the magnetized spin-1 Bose–Einstein condensate (BEC) of neutral atoms, using the method of many-particle quantum hydrodynamic (QHD) and propose an original derivation of the system of continual equations. We consider bosons with a spin-spin interaction and a short range interaction in the first order in the interaction radius, on the of basis of the self-consistent field approximation of the QHD equations. We demonstrate that the dynamics of the fluid velocity and magnetization is determined by a nontrivial modification of the Euler and Landau–Lifshitz equation, and show that a nontrivial modification of the spin density evolution equation contains the spin torque effect that arises from the self-interactions between spins of the bosons. The properties of the dispersion spectrum of collective excitations are described. We obtain the new contribution of the self-interaction of spins in the spin wave spectrum together with the influence of an external magnetic field and spin-spin interactions between polarized particles. The anisotropic spin wave instability is predicted.
We establish a complete set of hydrodynamic equations for a spin-1 Bose-Einstein condensate, which are equivalent to the multicomponent Gross-Pitaevskii equations and expressed in terms of only observable physical quantities: the spin and the nematic (or quadrupolar) tensor in addition to the density of particles and the mass current that appear in the hydrodynamic description of a scalar condensate. The obtained hydrodynamic equations involve a generalized Mermin-Ho relation that is valid regardless of the spatiotemporal dependence of the spin polarization. Low-lying collective modes for phonons and magnons are reproduced by linearizing the hydrodynamic equations. We also apply the single-mode approximation to the hydrodynamic equations and find a complete set of analytic solutions.