Non-Markovian Quantum Friction of Bright Solitons in Superfluids

We explore the quantum dynamics of a bright matter-wave soliton in a quasi-one-dimensional bosonic superfluid with attractive interactions. Specifically, we focus on the dissipative forces experienced by the soliton due to its interaction with Bogoliubov excitations. Using the collective coordinate approach and the Keldysh formalism, a Langevin equation of motion for the soliton is derived from first principles. The equation contains a stochastic Langevin force (associated with quantum noise) and a nonlocal in time dissipative force, which appears due to inelastic scattering of Bogoliubov quasiparticles off of the moving soliton. It is shown that Ohmic friction (i.e., a term proportional to the soliton’s velocity) is absent in the integrable setup. However, the Markovian approximation gives rise to the Abraham-Lorentz force (i.e., a term proportional to the derivative of the soliton’s acceleration), which is known from classical electrodynamics of a charged particle interacting with its own radiation. These Abraham-Lorentz equations famously contain a fundamental causality paradox, where the soliton (particle) interacts with excitations (radiation) originating from future events. We show, however, that the causality paradox is an artifact of the Markovian approximation, and our exact non-Markovian dissipative equations give rise to physical trajectories. We argue that the quantum friction discussed here should be observable in current quantum gas experiments.

Figure 1

(a) Spectral function of the bath formed by Bogoliubov quasiparticles $J(\omega )=J_{\mathrm{sc}}(\omega )+J_{\mathrm{ac}}(\omega )$ (red line) and the contribution of creation or annihilation processes $J_{\mathrm{ac}}(\omega )$ to it (dashed blue line). (b) The low-frequency part of (a) with the asymptotics $J_M(\omega )$ (dashed purple line) given by Eq. (16). $J_0$ is a constant defined after Eq. (16).

Figure 2

(a) Time dependence of the friction kernel $\eta (t)$, corresponding to the exact spectral function, given by Eq. (15) (red line), and to the approximate one, given by $J_M^{*}=J_0(\omega \tau {)}^3\exp [-\omega {\tau }_{*}]$ (dashed blue line), at $T=\mu$ and ${\tau }_{*}=3\tau$. (b) Poles in the complex plane $\omega ={\omega }^{\prime }+i{\omega }^{\prime \prime }$ of the response function $\chi (\omega )$ in the Markovian approximation leading to the Abraham-Lorentz friction force. In the upper half-plane there is the spurious pole corresponding to causality violation. (c) Poles of the response function $\chi (\omega )$ calculated exactly for the spectral function $J_M^{*}$ or in the Markovian approximation, if the Abraham-Lorentz force is treated as perturbation. Note that there is no unphysical pole in the upper half-plane of $\omega$.