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arXiv:cond-mat/0507125 · Physical Review Letters

Berry Phase Effects on the Dynamics of Quasiparticles in a Superfluid with a Vortex

Chuanwei Zhang, Artem M. Dudarev, Qian Niu

Published Jul 28, 2006• 28 citations
arXivHTMLPDF

Abstract

We study quasiparticle dynamics in a Bose-Einstein condensate with a vortex by following the center of mass motion of a Bogoliubov wave packet, and find important Berry-phase effects due to the background flow. We show that the Berry phase invalidates the usual canonical relation between the mechanical momentum and position variables, leading to important modifications of quasiparticle statistics and thermodynamic properties of the condensates. Applying these results to a vortex in an infinite uniform superfluid, we find that the total transverse force acting on the vortex is proportional to the superfluid density. We propose an experimental setup to directly observe Berry phase effects through measuring local thermal atoms' momentum distribution around a vortex.

DOI
10.1103/physrevlett.97.040401

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We study quasiparticle dynamics in a Bose-Einstein condensate with a vortex by following the center of mass motion of a Bogoliubov wave packet, and find important Berry-phase effects due to the background flow. We show that the Berry phase invalidates the usual canonical relation between the mechanical momentum and position variables, leading to important modifications of quasiparticle statistics and thermodynamic properties of the condensates. Applying these results to a vortex in an infinite uniform superfluid, we find that the total transverse force acting on the vortex is proportional to the superfluid density. We propose an experimental setup to directly observe Berry phase effects through measuring local thermal atoms' momentum distribution around a vortex.
Motivated by understanding Majorana zero modes in topological superfluids in particle-number conserving framework beyond the present framework, we study the effect of particle number conservation on the Berry phase resulting from transport of a bound quasiparticle around a superfluid vortex. We find that particle-number non-conserving calculations based on Bogoliubov-de Gennes (BdG) equations are unable to capture the correct physics when the quasiparticle is within the penetration depth of the vortex core where the superfluid velocity is non-zero. Particle number conservation is crucial for deriving the correct Berry phase in this context, and the Berry phase takes non-universal values depending on the system parameters and the external trap imposed to bind the quasiparticle. Of particular relevance to Majorana physics are the findings that superfluid condensate affects the part of the Berry phase not accounted for in the standard BdG framework, and that the superfluid many-body ground state of odd number of fermions involves superfluid condensate deformation due to the presence of the bound quasiparticle - an effect which is beyond the description of the BdG equations.
We consider the inertial mass of a vortex in a superfluid. We obtain a vortex mass that is well defined and is determined microscopically and self-consistently by the elementary excitation energy of the kelvon quasiparticle localized within the vortex core. The obtained result for the vortex mass is found to be consistent with experimental observations on superfluid quantum gases and vortex rings in water. We propose a method to measure the inertial rest mass and Berry phase of a vortex in superfluid Bose and Fermi gases.
The nondissipative transverse force acting on one moving vortex under the influence of another vortex is discussed in fermionic superfluid systems, where the relative velocity between the vortices is finite. On the basis of detailed numerical solutions of the Bogoliubov-de Gennes equation, the Berry phase for an adiabatic motion of the vortex line is examined for a two-vortex system. It is found that the detailed electronic structure of a vortex core can affect the transverse force, without abandoning the previous discussions of the robust Magnus force on a single vortex.
Motivated by understanding Majorana zero modes in topological superfluids in particle-number conserving framework beyond the present framework, we study the effect of particle number conservation on the Berry phase resulting from transport of a bound quasiparticle around a superfluid vortex. We find that particle-number non-conserving calculations based on Bogoliubov-de Gennes (BdG) equations are unable to capture the correct physics when the quasiparticle is within the penetration depth of the vortex core where the superfluid velocity is non-zero. Particle number conservation is crucial for deriving the correct Berry phase in this context, and the Berry phase takes non-universal values depending on the system parameters and the external trap imposed to bind the quasiparticle. Of particular relevance to Majorana physics are the findings that superfluid condensate affects the part of the Berry phase not accounted for in the standard BdG framework, and that the superfluid many-body ground state of odd number of fermions involves superfluid condensate deformation due to the presence of the bound quasiparticle - an effect which is beyond the description of the BdG equations.
Starting from the time-dependent version of the Feynman-Hellmann theorem, the Magnus force acting on a vortex in fermion superfluid is expressed via the adiabatic curvature over the space of vortex positions. With use of the Bogoliubov--de Gennes approximation, the Magnus force in a homogeneous superfluid at T=0 is shown to originate from virtual transitions between the lowest quasiparticle core bound states. Nonadiabatic corrections to the curvature are obtained to second order in vortex velocity. The adiabatic approximation is shown to break down at a critical velocity equal to the vortex velocity in the first Landau level. The effect of elastic scattering on the Magnus force is discussed in terms of the relaxation-time approximation. It is suggested that this approximation is appropriate only for a large-scale vortex motion. In this case, the effective Magnus force is drastically reduced when the elastic-scattering rate exceeds the core excitation frequency. We conjecture that quantum vortex tunneling is governed by a local Magnus force obtained from the Berry phase approach.
When vortices are displaced in Bose-Einstein condensates (BECs), the Magnus force gives the system a momentum transverse in the direction to the displacement. We show that BECs in long channels with vortices exhibit a quantization of the current response with respect to the spatial vortex distribution. The quantization originates from the well-known topological property of the phase around a vortex; it is an integer multiple of $2\ensuremath{\pi}$. In a way similar to that of the integer quantum Hall effect, the current along the channel is related to this topological phase and can be extracted from two experimentally measurable quantities: the total momentum of the BEC and the spatial distribution. The quantization is in units of m/2hm/2hm/2h, where mmm is the mass of the atoms and hhh is Planck's constant. We derive an exact vortex momentum-displacement relation for BECs in long channels under general circumstances. Our results present the possibility that the configuration described here can be used as a novel way of measuring the mass of the atoms in the BEC using a topological invariant of the system. If an accurate determination of the plateaus are experimentally possible, this gives the possibility of a topological quantum mass standard and precise determination of the fine structure constant.