Brownian motion of solitons in a Bose–Einstein condensate

Abstract

We observed and controlled the Brownian motion of solitons. We launched solitonic excitations in highly elongated 87 Rb BoseEinstein condensates (BECs) and showed that a dilute background of impurity atoms in a different internal state dramatically affects the soliton. With no impurities and in one dimension (1D), these solitons would have an infinite lifetime, a consequence of integrability. In our experiment, the added impurities scatter off the much larger soliton, contributing to its Brownian motion and decreasing its lifetime. We describe the soliton's diffusive behavior using a quasi-1D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent with experiment.W e studied the diffusion and decay of solitons in the highly controlled quantum environment provided by atomic Bose-Einstein condensates (BECs), where density maxima can be stabilized by attractive interactions [i.e., bright solitons (1, 2)] or, as here, where density depletions can be stabilized by repulsive interactions [i.e., dark solitons such as kink solitons (3, 4) and solitonic vortices (5)]. By contaminating these BECs with small concentrations of impurity atoms, we quantitatively studied how random processes destabilize solitons.Our BECs can be modeled by the one-dimensional (1D) Gross-Pitaevski equation (GPE): an integrable nonlinear wave equation with soliton solutions as excitations above the ground state. For a homogeneous 1D BEC of particles with mass m Rb with density ρ0, speed of sound c, and healing length ξ = / √ 2m Rb c, the dark soliton solutionsare expressed in terms of time t, axial position z , soliton velocity vs , and soliton width ξs = ξ/ 1 − (vs /c) 2 . Such dark solitons have a minimum density ρ0(vs /c) 2 and a phase jump −2 cos −1 (vs /c), both dependent on the soliton velocity vs . These behave as classical objects with a negative inertial mass ms ≈ − 4 ρ0/c, essentially the missing mass of the displaced atoms. The negative mass implies that increasing the soliton velocity reduces its kinetic energy; thus, dissipation accelerates dark solitons (6). This can be seen from the soliton equation of motionwhere −γż is the friction force and V is the confining potential due to the mean-field effects of the condensate. The random Langevin force f (t) has a white noise correlator f (t)f (t ) = 2γkBT δ(t − t ), where T is temperature and kB is Boltzmann's constant. The connection between the friction coefficient γ and f (t) derives from the same microscopic dynamics that yield the fluctuation-dissipation theorem for positive mass objects-f (t) is responsible for Brownian motion, whereas γ describes friction, but both have contributions from impurity atoms. Conventionally, the diffusion coefficient D is inversely proportional to the friction coefficient: D ∝ 1/γ. For negative mass objects, we show that the diffusion coefficient is instead proportional to the friction coefficient D ∝ γ; this reflects that friction is an antidamping force for negative mass objects. The interplay between ...

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“…Recent years have seen renewed theoretical interest in the effects of friction and dissipation on solitons, which can greatly affect their lifetime and stability [18][19][20][21][22][23][24][25]. Additionally, the diffusion coefficient of a soliton was recently measured experimentally for the first time [26].…”

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