Brownian motion of a matter-wave bright soliton moving through a thermal cloud of distinct atoms

Taking an open quantum system approach, we derive a collective equation of motion for the dynamics of a matter-wave bright soliton moving through a thermal cloud of a distinct atomic species. The reservoir interaction involves energy transfer without particle transfer between the soliton and thermal cloud, thus damping the soliton motion without altering its stability against collapse. We derive a Langevin equation for the soliton center-of-mass velocity in the form of an Ornstein-Uhlenbeck process with analytical drift and diffusion coefficients. This collective motion is confirmed by simulations of the full stochastic projected Gross-Pitaevskii equation for the matter-wave field. The system offers a pathway for experimentally observing the elusive energy-damping reservoir interaction and a clear realization of collective Brownian motion for a mesoscopic superfluid droplet.

Figure 1

Schematic. (a) In classical Brownian motion a heavy mass is randomly struck by lighter masses, causing stochastic motion. A bright soliton jostled by thermal atoms of a distinct atomic species provides a matter-wave analog of Brownian motion: a “quantum pollen grain.” (b) Representation of the energy-damping reservoir interaction between $|1\rangle$ atoms (coherent $\mathit{C}$ region) and $|2\rangle$ atoms (incoherent $\mathit{I}$ region), in stochastic projected Gross-Pitaevskii theory [15, 44, 45]. The interactions are number conserving and the thermal cloud of $|2\rangle$ acts as an energy reservoir for the bright soliton of $|1\rangle$. (c) Bright soliton with center-of-mass velocity $\nu \left(t\right)$, immersed in a distinct thermal cloud. Both species are confined to ring geometries, but with distinct transverse length scales: $a_{\perp ,1}\ll a_{T,2}$ (see the text).

Figure 2

Coherent $C$-field particle density scaled by interaction strength $g_{1}n\left(x\right)=g_1{\left|\psi \left(x\right)\right|}^2$ for the bright soliton analytic solution (4) (gray dashed curve) and after evolving the initial state with $\nu \left(0\right)=0.9a_{\perp ,1}{\omega }_{\perp ,1}$ numerically for $t=300{\omega }_{\perp ,1}^{-1}$ (blue curve, recentered), according to the energy-damping SPGPE (1). The green curve shows the scattering potential $V_{\varepsilon }(x,0)$ acting on the initial soliton state.

Figure 3

Variance of the soliton velocity over time for an initial position $x=0$ and initial velocity ${\nu }_0=0.9a_{\perp ,1}{\omega }_{\perp ,1}$. The blue line is the analytic expression derived from (11) and (12), while the cyan, green, and red lines give the numerically obtained values from integrating (1) for ensembles containing 50, 500, and 5000 trajectories, respectively. The inset shows the average soliton velocity over time for 5000 trajectories with an initial position $x=0$ and initial velocity ${\nu }_0=0.9a_{\perp ,1}{\omega }_{\perp ,1}$. The solid blue line is the analytic expression derived from (11), while the dashed red line gives the values obtained from the SPGPE ensemble.

Figure 4

Steady-state (a) two-time correlation function and (b) spectrum of the soliton velocity constructed using 500 trajectories of an initially stationary soliton at $x=0$. The solid blue lines are the analytic expressions (13) and (14), respectively, while the red points show the values obtained numerically from the SPGPE.