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Physical Review

Vortices in an Imperfect Bose Gas. III. Scattering of Single-Particle Excitations

Alexander L. Fetter

Published Oct 18, 1965• 15 citations
Publisher

Abstract

The scattering of single-particle excitatins by a vortex in the condensate of an imperfect Bose gas is studied using the Bogoliubov approximation. The $S$ matrix is calculated as in conventional scattering theory, and the phase shifts are determined with the Schwinger variational principle. The differential cross section in the long-wavelength limit is found to be $\frac{d\ensuremath{\sigma}}{d\ensuremath{\chi}}=\frac{1}{2}\ensuremath{\pi}{(\frac{\ensuremath{\hbar}}{\mathrm{mc}})}^{2}k{cot}^{2}(\frac{1}{2}\ensuremath{\chi})$, where $m$ is the mass of one atom and $c$ is the speed of sound. This cross section agrees precisely with that found previously for scattering of sound by a classical vortex with circulation $\frac{h}{m}$.

DOI
10.1103/physrev.140.a452

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The scattering of single-particle excitatins by a vortex in the condensate of an imperfect Bose gas is studied using the Bogoliubov approximation. The SSS matrix is calculated as in conventional scattering theory, and the phase shifts are determined with the Schwinger variational principle. The differential cross section in the long-wavelength limit is found to be d\ensuremathσd\ensuremathχ=12\ensuremathπ(\ensuremathℏmc)2kcot2(12\ensuremathχ)\frac{d\ensuremath{\sigma}}{d\ensuremath{\chi}}=\frac{1}{2}\ensuremath{\pi}{(\frac{\ensuremath{\hbar}}{\mathrm{mc}})}^{2}k{cot}^{2}(\frac{1}{2}\ensuremath{\chi})d\ensuremathχd\ensuremathσ​=21​\ensuremathπ(mc\ensuremathℏ​)2kcot2(21​\ensuremathχ), where mmm is the mass of one atom and ccc is the speed of sound. This cross section agrees precisely with that found previously for scattering of sound by a classical vortex with circulation hm\frac{h}{m}mh​.
The single-particle excitations of an imperfect Bose gas with a vortex in the condensate are studied using the Bogoliubov approximation. The problem is reduced to an eigenvalue equation, whose solution yields both the single-particle energies and the eigenfunction expansion of the single-particle Green's function. The spectrum of the eigenvalues contains a discrete portion (bound states) and a continuous portion (scattering states). The lowest eigenvalue corresponding to angular momentum l=0l=0l=0 and $l=\ifmmode\pm\else\textpm\fi{}1$ is determined variationally. In the long-wavelength limit, the eigenvalue for $l=\ifmmode\pm\else\textpm\fi{}1$ agrees with the frequency of normal modes calculated by Kelvin and Pitaevskii. The density of noncondensed particles is expressed in terms of the single-particle Green's function and is shown to be finite at the center of the vortex.
The translational velocity of a vortex with circulation hm\frac{h}{m}mh​ in an imperfect Bose gas is calculated by considering the time dependence of the condensate wave function corresponding to a given initial configuration of vortices. Each vortex in a system of rectilinear vortices is shown to move with the local fluid velocity at its core; a vortex ring of radius RRR is shown to move with velocity $(\frac{h}{4\ensuremath{\pi}\mathrm{mR}})\mathrm{ln}(\frac{8R}{\ensuremath{\alpha}})$, where $\ensuremath{\alpha}$ is a length the order of the core size. Both results agree with the predictions of classical hydrodynamics.
A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity vc{v}_{c}vc​ of liquid He II in a tube of radius RRR that includes the effect of the walls ${v}_{c}=\frac{C\ensuremath{\hbar}}{2mR}$, where CCC is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area $\ensuremath{\nu}$ agrees with Feyman's result $\ensuremath{\nu}=\frac{2m\ensuremath{\omega}}{h}$. In the classical limit ($h\ensuremath{\rightarrow}0$), the angular momentum and energy approach the values for solid-body rotation.
Abstract Understanding quantum dynamics in a two-dimensional Bose–Einstein condensate relies on understanding how vortices interact with each other microscopically and with local imperfections of the potential which confines the condensate. Within a system consisting of many vortices, the trajectory of a vortex–antivortex pair is often scattered by a third vortex, an effect previously characterised. However, the natural question remains as to how much of this effect is due to the velocity induced by this third vortex and how much is due to the density inhomogeneity which it introduces. In this work, we describe the various qualitative scenarios which occur when a vortex–antivortex pair interacts with a smooth density impurity whose profile is identical to that of a vortex but lacks the circulation around it.
We develop an analytical approach for the description of an atomic Bose gas at unitarity. By focusing in first instance on the evaluation of the single-particle density matrix, we derive several universal properties of the unitary Bose gas, such as the chemical potential, the contact, the speed of sound, the condensate density and the effective interatomic interaction. The theory is also generalized to describe Bose gases with a finite scattering length and then reduces to the Bogoliubov theory in the weak-coupling limit.
We consider scattering atoms from a fully Bose-Einstein condensed gas. If we take these atoms to be identical to those in the Bose-Einstein condensate, this scattering process is to a large extent analogous to Andreev reflection from the interface between a superconducting and a normal metal. We determine the scattering wave function in both the absence and the presence of a vortex. Our results show a qualitative difference between these two cases that can be understood as due to an Aharonov-Bohm effect. It leads to the possibility to experimentally study vortices in a nondestructive way.