arXiv subjects

Alexander L. Fetter

Publications and source records attributed to Alexander L. Fetter.

At least 19 recordsLinked to original sources

arXiv2024

Conformal maps and superfluid vortex dynamics on curved and bounded surfaces: the case of an elliptical boundary

Recent advances in cold-atom platforms have made real-time dynamics accessible, renewing interest in the motion of superfluid vortices in two-dimensional domains. Here we show that the energy and the trajectories of arbitrary vortex configurations may be computed on a complicated (curved or bounded) surface, provided that one knows a conformal map that links the latter to a simpler domain (like the full plane, or a circular boundary). We also prove that Hamilton's equations based on the vortex energy agree with the complex dynamical equations for the vortex dynamics, demonstrating that the vortex trajectories are constant-energy curves. We use these ideas to study the dynamics of vortices in a two-dimensional incompressible superfluid with an elliptical boundary, and we derive an analytical expression for the complex potential describing the hydrodynamic flow throughout the fluid. For a vortex inside an elliptical boundary, the orbits are nearly self-similar ellipses.

cond-mat.quant-gas↗

arXiv2022

Dynamics of a massive superfluid vortex in $r^k$ confining potentials

We study the motion of a superfluid vortex in condensates having different background density profiles, ranging from parabolic to uniform. The resulting effective point-vortex model for a generic power-law potential $\propto r^k$ can be experimentally realized with recent advances in optical-trapping techniques. Our analysis encompasses both empty-core and filled-core vortices. In the latter case, the vortex acquires a mass due to the presence of distinguishable atoms located in its core. The axisymmetry allows us to reduce the coupled dynamical equations of motion to a single radial equation with an effective potential $V_{\rm eff}$. In many cases, $V_{\rm eff}$ has a single minimum, where the vortex precesses uniformly. The dynamics of the vortex and the localized massive core arises from the dependence of the energy on the radial position of the vortex and from the $r^k$ trap potential. We find that a positive vortex with small mass orbits in the positive direction, but the sense of precession can reverse as the core mass increases. Early experiments and theoretical studies on two-component vortices found some qualitatively similar behavior.

cond-mat.quant-gas↗

arXiv2021

Dynamics of massive point vortices in binary mixture of Bose-Einstein condensates

We study the massive point-vortex model introduced in Ref. [Phys. Rev. A 101, 013630 (2020)], which describes two-dimensional point vortices of one species that have small cores of a different species. We derive the relevant Lagrangian itself, based on the time-dependent variational method with a two-component Gross-Pitaevskii (GP) trial function. The resulting Lagrangian resembles that of charged particles in a static electromagnetic field, where the canonical momentum includes an electromagnetic term. The simplest example is a single vortex with a rigid circular boundary, where a massless vortex can only precess uniformly. In contrast, the presence of a sufficiently large filled vortex core renders such precession unstable. A small core mass can also lead to small radial oscillations, which are, in turn, clear evidence of the associated inertial effect. Detailed numerical analysis of coupled two-component GP equations with a single vortex and small second-component core confirms the presence of such radial oscillations, implying that this more realistic GP vortex also acts as if it has a small massive core.

cond-mat.quant-gas↗

arXiv2021

Superfluid vortex dynamics on a spherical film

Motivated by ongoing experimental efforts to make closed Bose-Einstein condensate (BEC) shells in microgravity environments, this work studies the energy and dynamics of singly quantized vortices on a thin spherical superfluid shell, where the overall vortex charge must vanish (as on any compact surface). For each vortex, stereographic projection yields the corresponding complex potential on the tangent plane. The resulting stream function then provides both the total energy and the dynamics of a system of overall neutral vortices on a spherical film. Although a single vortex dipole follows a simple dynamical orbit, four vortices can present a variety of situations. We study a few symmetric initial configurations and then focus on the special case of two small vortex dipoles.

cond-mat.quant-gas↗

arXiv2019

Superfluid vortex dynamics on a torus and other toroidal surfaces of revolution

The superfluid flow velocity is proportional to the gradient of the phase of the superfluid order parameter, leading to the quantization of circulation around a vortex core. In this work, we study the dynamics of a superfluid film on the surface of a torus. Such a compact surface allows only configurations of vortices with zero net vorticity. We derive analytic expressions for the flow field, the total energy, and the time-dependent dynamics of the vortex cores. The local curvature of the torus and the presence of non-contractable loops on this multiply connected surface alter both the superfluid flow and the vortex dynamics. Finally we consider more general surfaces of revolution, called toroids.

cond-mat.quant-gas↗

arXiv2019

Superfluid Vortex Dynamics on Planar Sectors and Cones

We study the dynamics of vortices formed in a superfluid film adsorbed on the curved two-dimensional surface of a cone. To this aim, we observe that a cone can be unrolled to a sector on a plane with periodic boundary conditions on the straight sides. The sector can then be mapped conformally to the whole plane, leading to the relevant stream function. In this way, we show that a superfluid vortex on the cone precesses uniformly at fixed distance from the apex. The stream function also yields directly the interaction energy of two vortices on the cone. We then study the vortex dynamics on unbounded and bounded cones. In suitable limits, we recover the known results for dynamics on cylinders and planar annuli.

cond-mat.quant-gas↗

arXiv2017

Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli

Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being $\pm \hbar/(2MR)$, with $M$ the mass of the superfluid particles and $R$ the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function $χ({\bf r}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\bf r}_{12}$ becomes larger than the cylinder radius. An Appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second Appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.

cond-mat.quant-gas↗

arXiv2016

Vortex dynamics in coherently coupled Bose-Einstein condensates

In classical hydrodynamics with uniform density, vortices move with the local fluid velocity. This description is rewritten in terms of forces arising from the interaction with other vortices. Two such positive straight vortices experience a repulsive interaction and precess in a positive (anticlockwise) sense around their common centroid. A similar picture applies to vortices in a two-component two-dimensional uniform Bose-Einstein condensate (BEC) coherently coupled through rf Rabi fields. Unlike the classical case, however, the rf Rabi coupling induces an attractive interaction and two such vortices with positive signs now rotate in the negative (clockwise) sense. Pairs of counter-rotating vortices are instead found to translate with uniform velocity perpendicular to the line joining their cores. This picture is extended to a single vortex in a two-component trapped BEC. Although two uniform vortex-free components experience familiar Rabi oscillations of particle-number difference, such behavior is absent for a vortex in one component because of the nonuniform vortex phase. Instead the coherent Rabi coupling induces a periodic vorticity transfer between the two components.

cond-mat.quant-gas↗

arXiv2015

Vortex Dynamics in a Spin-Orbit Coupled Bose-Einstein Condensate

Vortices in a one-component dilute atomic ultracold Bose-Einstein condensate (BEC) usually arise as a response to externally driven rotation. Apart from a few special situations, these vortices are singly quantized with unit circulation. Recently, the NIST group has constructed a two-component BEC with a spin-orbit coupled Hamiltonian involving Pauli matrices, and I here study the dynamics of a two-component vortex in such a spin-orbit coupled condensate. These spin-orbit coupled BECs use an applied magnetic field to split the hyperfine levels. Hence they rely on a focused laser beam to trap the atoms. In addition, two Raman laser beams create an effective (or synthetic) gauge potential. The resulting spin-orbit Hamiltonian is discussed in some detail. The various laser beams are fixed in the laboratory, so that it is not feasible to nucleate a vortex by an applied rotation that would need to rotate all the laser beams and the magnetic field. In a one-component BEC, a vortex can also be created by a thermal quench, starting from the normal state and suddenly cooling deep into the condensed state. I propose that a similar method would work for a vortex in a spin-orbit coupled BEC. Such a vortex has two components, and each has its own circulation quantum number. If both components have the same circulation, I find that the composite vortex should execute uniform precession, like that observed in a single-component BEC. In contrast, if one component has unit circulation and the other has zero circulation, then some fraction of the dynamical vortex trajectories should eventually leave the condensate, providing clear experimental evidence for this unusual vortex structure. In the context of exciton-polariton condensates, such a vortex is known as a "half-quantum vortex".

cond-mat.quant-gas↗

arXiv2014

Vortex dynamics in spin-orbit coupled Bose-Einstein condensates

I use a time-dependent Lagrangian formalism and a variational trial function to study the dynamics of a two-component vortex in a spin-orbit coupled Bose-Einstein condensate (BEC). For a single-component BEC, various experiments have validated this theoretical approach, for example a thermal quench that yields a quantized vortex in roughly 25% of trials. To be definite, I assume the specific spin-orbit form used in recent NIST experiments, which introduces a spatial asymmetry because of the external Raman laser beams. I here generalize this formalism to include a two-component order parameter that has quantized circulation in each component but not necessarily with the same circulation. For example a singly quantized vortex in just one component yields a BEC analog of the half-quantum vortex familiar in $^3$He-A and in $p$-wave chiral superconductors. This and other unusual two-component vortices have both periodic trajectories and unbounded trajectories that leave the condensate, depending on the initial conditions. The optimized phase of the order parameter induces a term in the particle current that cancels the contribution from the vector potential, leaving pure circulating current around the vortex.

cond-mat.quant-gas↗

arXiv2012

Bose gas: Theory and Experiment

For many years, $^4$He typified Bose-Einstein superfluids, but recent advances in dilute ultra-cold alkali-metal gases have provided new neutral superfluids that are particularly tractable because the system is dilute. This chapter starts with a brief review of the physics of superfluid $^4$He, followed by the basic ideas of Bose-Einstein condensation (BEC), first for an ideal Bose gas and then considering the effect of interparticle interactions, including time-dependent phenomena. Extensions to more exotic condensates include magnetic dipolar gases, mixtures of two components, and spinor condensates that require a focused infrared laser for trapping of all the various hyperfine magnetic states in a particular hyperfine $F$ manifold of $m_F$ states. With an applied rotation, the trapped BECs nucleate quantized vortices. Recent theory and experiment have shown that laser coupling fields can mimic the effect of rotation. The resulting synthetic gauge fields have produced vortices in a nonrotating condensate.

cond-mat.quant-gas↗

arXiv2011

Self-similar expansion of the density profile in a turbulent Bose-Einstein condensate

In a recent study we demonstrated the emergence of turbulence in a trapped Bose-Einstein condensate of Rb-87 atoms. An intriguing observation in such a system is the behavior of the turbulent cloud during free expansion.The aspect ratio of the cloud size does not change in the way one would expect for an ordinary non-rotating (vortex-free) condensate. Here we show that the anomalous expansion can be understood, at least qualitatively, in terms of the presence of vorticity distributed throughout the cloud, effectively counteracting the usual reversal of the aspect ratio seen in free time-of-flight expansion of non-rotating condensates.

cond-mat.quant-gas↗

arXiv2010

Vortices and dynamics in trapped Bose-Einstein condensates

I review the basic physics of ultracold dilute trapped atomic gases, with emphasis on Bose-Einstein condensation and quantized vortices. The hydrodynamic form of the Gross-Pitaevskii equation (a nonlinear Schr{ö}dinger equation) illuminates the role of the density and the quantum-mechanical phase. One unique feature of these experimental systems is the opportunity to study the dynamics of vortices in real time, in contrast to typical experiments on superfluid $^4$He. I discuss three specific examples (precession of single vortices, motion of vortex dipoles, and Tkachenko oscillations of a vortex array). Other unusual features include the study of quantum turbulence and the behavior for rapid rotation, when the vortices form dense regular arrays. Ultimately, the system is predicted to make a quantum phase transition to various highly correlated many-body states (analogous to bosonic quantum Hall states) that are not superfluid and do not have condensate wave functions. At present, this transition remains elusive. Conceivably, laser-induced synthetic vector potentials can serve to reach this intriguing phase transition.

cond-mat.quant-gas↗

arXiv2009

Bose-Einstein condensate in a rapidly rotating non-symmetric trap

A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional harmonic trap can be described with the lowest Landau-level set of single-particle states. The condensate wave function psi(x,y) is a Gaussian exp(-r^2/2), multiplied by an analytic function f(z) of the complex variable z= x+ i y. The criterion for a quantum phase transition to a non-superfluid correlated many-body state is usually expressed in terms of the ratio of the number of particles to the number of vortices. Here, a similar description applies to a rapidly rotating non-symmetric two-dimensional trap with arbitrary quadratic anisotropy (omega_x^2 < omega_y^2). The corresponding condensate wave function psi(x,y) is a complex anisotropic Gaussian with a phase proportional to xy, multiplied by an analytic function f(z), where z = x + i β_- y is a stretched complex variable and 0< β_- <1 is a real parameter that depends on the trap anisotropy and the rotation frequency. Both in the mean-field Thomas-Fermi approximation and in the mean-field lowest Landau level approximation with many visible vortices, an anisotropic parabolic density profile minimizes the energy. An elongated condensate grows along the soft trap direction yet ultimately shrinks along the tight trap direction. The criterion for the quantum phase transition to a correlated state is generalized (1) in terms of N/L_z, which suggests that a non-symmetric trap should make it easier to observe this transition or (2) in terms of a "fragmented" correlated state, which suggests that a non-symmetric trap should make it harder to observe this transition. An alternative scenario involves a crossover to a quasi one-dimensional condensate without visible vortices, as suggested by Aftalion et al., Phys. Rev. A 79, 011603(R) (2009).

cond-mat.quant-gas↗

arXiv2008

Rotating trapped Bose-Einstein condensates

After reviewing the ideal Bose-Einstein gas in a box and in a harmonic trap, I discuss the effect of interactions on the formation of a Bose-Einstein condensate (BEC), along with the dynamics of small-amplitude perturbations (the Bogoliubov equations). When the condensate rotates with angular velocity Omega, one or several vortices nucleate, with many observable consequences. With more rapid rotation, the vortices form a dense triangular array, and the collective behavior of these vortices has additional experimental implications. For Omega near the radial trap frequency omega_perp, the lowest-Landau-level approximation becomes applicable, providing a simple picture of such rapidly rotating condensates. Eventually, as Omega approaches omega_perp, the rotating dilute gas is expected to undergo a quantum phase transition from a superfluid to various highly correlated (nonsuperfluid) states analogous to those familiar from the fractional quantum Hall effect for electrons in a strong perpendicular magnetic field.

cond-mat.stat-mech↗

arXiv2006

Lowest Landau-level description of a Bose-Einstein condensate in a rapidly rotating anisotropic trap

A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional trap can be described with the lowest Landau-level set of states. In this case, the condensate wave function psi(x,y) is a Gaussian function of r^2 = x^2 + y^2, multiplied by an analytic function P(z) of the single complex variable z= x+ i y; the zeros of P(z) denote the positions of the vortices. Here, a similar description is used for a rapidly rotating anisotropic two-dimensional trap with arbitrary anisotropy (omega_x/omega_y le 1). The corresponding condensate wave function psi(x,y) has the form of a complex anisotropic Gaussian with a phase proportional to xy, multiplied by an analytic function P(zeta), where zeta is proportional to x + i beta_- y and 0 le beta_- le 1 is a real parameter that depends on the trap anisotropy and the rotation frequency. The zeros of P(zeta) again fix the locations of the vortices. Within the set of lowest Landau-level states at zero temperature, an anisotropic parabolic density profile provides an absolute minimum for the energy, with the vortex density decreasing slowly and anisotropically away from the trap center.

cond-mat.stat-mech↗

arXiv2006

Motion of a vortex line near the boundary of a semi-infinite uniform condensate

We consider the motion of a vortex in an asymptotically homogeneous condensate bounded by a solid wall where the wave function of the condensate vanishes. For a vortex parallel to the wall, the motion is essentially equivalent to that generated by an image vortex, but the depleted surface layer induces an effective shift in the position of the image compared to the case of a vortex pair in an otherwise uniform flow. Specifically, the velocity of the vortex can be approximated by $U \approx (\hbar/2m)(y_0-\sqrt 2 ξ)^{-1}$, where $y_0$ is the distance from the center of the vortex to the wall, $ξ$ is the healing length of the condensate and $m$ is the mass of the boson.

cond-mat.soft↗

arXiv2005

Dynamics of rapidly rotating Bose-Einstein condensates in a harmonic plus quartic trap

A two-dimensional rapidly rotating Bose-Einstein condensate in a harmonic plus quartic trap is expected to have unusual vortex states that do not occur in a pure harmonic trap. At a critical rotation speed $Ω_h$, a central hole appears in the condensate, and at some faster rotation speed $Ω_g$, the system undergoes a transition to a giant vortex state with pure irrotational flow. Using a time-dependent variational analysis, we study the behavior of an annular condensate with a single concentric ring of vortices. The transition to a giant vortex state is investigated by comparing the energy of the two equilibrium states (the ring of vortices and the giant vortex) and also by studying the dynamical stability of small excitation modes of the ring of vortices.

cond-mat.other↗