In rapidly rotating condensed Bose systems we show that there is a regime of anomalous hydrodynamics which coincides with the mean field quantum Hall regime. A consequence is the absence of a normal fluid in any conventional sense. However, even the superfluid hydrodynamics is not described by conventional Bernoulli and continuity equations. We show that there are constraints which connect spatial variations of density and phase and that the vortex positions are not the simplest description of the dynamics. We demonstrate, inter alia, a simple relation between vortices and surface waves. We show that the surface waves can emulate a "normal fluid," allowing dissipation by energy and angular momentum absorbtion from vortex motion in the trap. The time scale is sensitive to the initial configuration, which can lead to long-lived vortex patches--perhaps related to those observed at JILA.
In rapidly rotating condensed Bose systems we show that there is a regime of anomalous hydrodynamics which coincides with the mean field quantum Hall regime. A consequence is the absence of a normal fluid in any conventional sense. However, even the superfluid hydrodynamics is not described by conventional Bernoulli and continuity equations. We show that there are constraints which connect spatial variations of density and phase and that the vortex positions are not the simplest description of the dynamics. We demonstrate, inter alia, a simple relation between vortices and surface waves. We show that the surface waves can emulate a "normal fluid," allowing dissipation by energy and angular momentum absorbtion from vortex motion in the trap. The time scale is sensitive to the initial configuration, which can lead to long-lived vortex patches--perhaps related to those observed at JILA.
The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
Spontaneous pattern formation out of homogeneous media is one of the well-understood examples of hydrodynamic instabilities in classical systems, which naturally leads to the question of its manifestation in quantum fluids. Bose-Einstein condensates (BECs) of atomic gases have been an ideal platform for studying many-body quantum phenomena, such as superfluidity, and simultaneously providing an opportunity to broaden our understanding of classical hydrodynamics into quantum systems. In this review, we introduce a range of experimental studies on the pattern formation in quantum fluids of atomic gases under external driving, including Faraday waves in one and two dimensions, surface patterns, and counterflow instabilities in a mixture of superfluids. The pattern formation in the quantum system can be understood through the parametric amplification process, where an unstable dynamical mode can be exponentially amplified, similar to classical systems. Remarkably, the governing equations for surface excitations of trapped BECs can be mathematically equivalent to those of shallow water, indicating a universal description of the hydrodynamic instability across classical and quantum domains. However, the condensates, as superfluids, also possess fundamental quantum characteristics, such as quantized vorticity and a distinct dissipation channel. These unique features showcase many-body fragmentation under strong modulation and the generation of vortices in the nonlinear regime, which could offer a pathway to the study of quantum turbulence. Furthermore, the coexistence of long-range phase coherence and density modulation in driven condensates could provide unexplored features, such as those seen in supersolid-like sound modes, within nonequilibrium settings.
We study the rotational properties of a Bose-Einstein condensate confined in a rotating harmonic trap for different trap anisotropies. With simple arguments, we obtain the velocity field of the quantum fluid for condensates with or without a centered vortex. While the bosons describe open spiraling trajectories, on the frame of reference of the rotating trap, the fluid moves against the trap's rotation. We also find expressions for the angular momentum of linear and Thomas-Fermi solutions for a vortexless state. In these two limits we find the same analytic relation between the shape of the cloud and the rotation speed. Our predictions are supported by numerical simulations of the mean-field Gross-Pitaevskii model and by current experiments with rotating Bose-Einstein condensates (K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard, e-print cond-mat/0101051).
Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…
We study the surface modes of some of the vortex liquids recently found by means of exact diagonalizations in systems of rapidly rotating bosons. In contrast to the surface modes of Bose condensates, we find that the surface waves have a frequency linear in the excitation angular momentum, $\ensuremath{\Elzxh}l>0.$ Furthermore, in analogy with the edge waves of electronic quantum Hall states, these excitations are chiral, that is, they can be excited only for values of l that increase the total angular momentum of the vortex liquid. However, differently from the quantum Hall phenomena for electrons, we also find other excitations that are approximately degenerate in the laboratory frame with the surface modes, and which decrease the total angular momentum by l quanta. The surface modes of the Laughlin as well as other scalar and vector boson states are analyzed and their observable properties characterized. We argue that measurement of the response of a vortex liquid to a weak time-dependent potential that imparts angular momentum to the system should provide valuable information for characterizing the vortex liquid. In particular, the intensity of the signal of the surface waves in the dynamic structure factor has been studied and found to depend on the type of vortex liquid. We point out that the existence of surface modes has observable consequences on the density profile of the Laughlin state. These features are due to the strongly correlated behavior of atoms in the vortex liquids. We point out that these correlations should be responsible for a remarkable stability of some vortex liquids with respect to three-body losses.