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Quantum Gases in Pancake and Two-dimensional Regimes

Лев П. Питаевский, S. Stringari

Published Jan 1, 2016• 0 citations
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Abstract

Abstract This chapter addresses the question of quantum gases in two dimensions. It investigates both the pancake and the deep two-dimensional regimes. In the former case the gas is locally three-dimensional along the axial direction. In contrast, in the deep two-dimensional regime the axial motion is frozen. The chapter investigates the new physical phenomena exhibited by these two-dimensional quantum gases exploring the behaviour of the equation of state and of the collective oscillations. It discusses the behaviour of a fast-rotating Bose gas in two-dimensions, and the structure of vortex lines during the transition between the two-dimensional Thomas–Fermi and the lowest Landau level regimes. In the deep two-dimensional regime quantum gases exhibit novel and important phenomena at finite temperatures, due to the absence of long-range order and the emergence of the Berezinskii–Kosterlitz–Thouless transition. The peculiar behaviour of the BCS–BEC transition in deeply two-dimensional Fermi gases is also discussed.

DOI
10.1093/acprof:oso/9780198758884.003.0023

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Abstract This chapter addresses the question of quantum gases in two dimensions. It investigates both the pancake and the deep two-dimensional regimes. In the former case the gas is locally three-dimensional along the axial direction. In contrast, in the deep two-dimensional regime the axial motion is frozen. The chapter investigates the new physical phenomena exhibited by these two-dimensional quantum gases exploring the behaviour of the equation of state and of the collective oscillations. It discusses the behaviour of a fast-rotating Bose gas in two-dimensions, and the structure of vortex lines during the transition between the two-dimensional Thomas–Fermi and the lowest Landau level regimes. In the deep two-dimensional regime quantum gases exhibit novel and important phenomena at finite temperatures, due to the absence of long-range order and the emergence of the Berezinskii–Kosterlitz–Thouless transition. The peculiar behaviour of the BCS–BEC transition in deeply two-dimensional Fermi gases is also discussed.
Abstract This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its one-dimensional-like geometrical shape, and the deep one-dimensional regime where the motion is frozen in the radial direction. The cigar configuration is well suited to investigating the novel features exhibited by quantized vortices and solitons. In the deep one-dimensional regime Bose gases exhibits a very different behaviour with respect to usual three-dimensional BECs as a consequence of the quantum and thermal fluctuations of the phase. The Lieb–Liniger transition between the one-dimensional mean-field and the Tonks–Girardeau regimes is discussed, and important features concerning the frequency of the collective oscillations and the superfluid behaviour are pointed out. Finally, the chapter discusses some key features exhibited by one-dimensional Fermi gases.
This course on two-dimensional quantum gases, prepared for the S{ã}o Paulo ICTP-SAIFR IVth School on Light and Cold Atoms, is organized into three lectures. The first introductory lecture addresses Bose-Einstein condensation in an ideal gas, the effect of interactions in the weakly interacting limit, Gross-Pitaevskii equation and its hydrodynamic formulation, as well as superfluidity. The second lecture is devoted to the main features of two-dimensional quantum gases: importance of the interactions, enhanced phase fluctuations and the emergence of a quasi long-range order, superfluidity through the Berezinskii-Kosterlitz-Thouless mechanism, scaling symmetry. The third lecture has taken the form of a seminar on fast rotating quasi two-dimensional Bose gases and thermal melting of the vortex lattice. The slides accompanying these notes can be found on the school website: https://www.ictp-saifr.org/slca2025/.
In this work we investigate the unique properties of ultracold Bose gases in one and two dimensions. In two dimensions, we present simulations of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the projected Gross-Pitaevskii equation formalism. We confirm the presence of the BKT phase by observing algebraic decay in the first-order correlation function, unbinding of vortex pairs, and behaviour of the superfluid fraction. In one dimension we use a transverse Gaussian ansatz and the Lagrangian approach to derive an effective equation of motion for the one dimensional Bose gas at zero temperature.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.