Optical Visibility and Core Structure of Vortex Filaments in a Bosonic Superfluid
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ontribution for the JETP special issue in honor of L.P. Pitaevskii’s 85th birthday
Optical Visibility and Core Structure of Vortex Filaments in a Bosonic Superfluid1 F. Dalfovoa,*, R. N. Bisseta, C. Mordinia,b, G. Lamporesia,b, and G. Ferraria,b a
INO-CNR BEC Center and Dipartimento di Fisica, Universitä di Trento, Trento, 38123 Italy b Trento Institute for Fundamental Physics and Applications, INFN, Trento, 38123 Italy * e-mail: [email protected] Received April 6, 2018
Abstract—We use optical images of a superfluid consisting of a weakly interacting Bose–Einstein condensate of sodium atoms to investigate the structure of quantized three-dimensional vortex filaments. We find that the measured optical contrast and the width of the vortex core quantitatively agree with the predictions of the Gross–Pitaevskii equation. DOI: 10.1134/S1063776118110018
1. INTRODUCTION The Gross–Pitaevskii (GP) equation was independently derived by L.P. Pitaevskii [1] and E.P. Gross [2] in 1961. It describes a superfluid gas of weakly interacting bosons at zero temperature. The solution of the equation is a complex function Ψ = |Ψ|expiϕ, whose modulus squared represents the particle density, n = |Ψ|2, and the gradient of the phase gives the local velocity of the fluid, v = ( /m)∇ϕ, where m is the particle mass. In the derivation by L.P. Pitaevskii, the GP equation emerges as a generalization of Bogoliubov’s theory [3] to a spatially inhomogeneous superfluid2 [4]. A quantized vortex can exist as a stationary solution of the GP equation where all particles circulate with the same angular momentum around a line where the density vanishes; the solution has the form n(r ) expiϕ, where now ϕ is the angle around the vortex axis and r is the distance from the axis in cylindrical coordinates. The density n(r) is a smooth function which increases from 0 to a constant asymptotic value n0 over a length scale characterized by ξ, known as the healing length, determined by n0 and the strength of the interaction. Quantized vortices have been extensively studied in superfluid 4He [5], which is a strongly correlated liq1 The article is published in the original. 2 Notice that an equation of the same form
was derived in [4], within a phenomenological theory for superfluids close to the normal-superfluid phase transition; the meaning of the coefficients is however entirely different.
uid. The core of the vortex in 4He is only qualitatively captured by the GP equation and more refined theories are needed to account for the atom-atom interactions and many-body effects [6–10]. A direct comparison between theory and experiment for the structure of the vortex core is not available, and is likely unrealistic, the main reason being that the core size in 4He is expected to be of the same order as the atom size. The only way to observe such a vortex thus consists of looking at its effects on the motion of impurities that may be attached to it. Electrons [11–14], solid hydrogen particles [15–19], and 4 He*2 excimer molecules [20] have been used for this p
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