Evaporative damping in open system theory of Bose-Einstein Condensates

TL;DR

This work adds evaporative damping to the stochastic projected Gross-Pitaevskii equation, deriving it from three-$C$-region interaction terms and showing it is non-negligible compared with number damping. By mapping to a multimode Wigner representation and employing a constant-energy approximation, the authors obtain a nonlocal diffusion-drift description that includes two main channels: a number-damping-like term and an energy-damping-like term, plus interdependent noise. A pseudo-local form and a dimensionally reduced version are developed, enabling practical implementation and exploration in both three-dimensional and reduced-dimensional Bose gases. The result provides a full first-principles picture of finite-temperature BEC dynamics and suggests evaporative damping can be important near the critical temperature and in high-density regimes, with potential implications for solitons and vortices in reduced geometries.

Abstract

We derive a new damping mechanism in the open quantum systems description of Bose-Einstein condensates. It stems from previously neglected terms in the derivation of the stochastic projective Gross-Pitaevskii equation (SPGPE), accounting for a nonlinear evaporation of particles from the coherent into the incoherent region. We demonstrate that the mechanism, while so far assumed to be of minor importance, is comparable in strength to the widely employed number damping. We also provide a simplified (pseudo)-local and a dimensionally reduced form of this evaporative damping. The process completes the SPGPE description of ultracold Bose gases giving a full first-principles picture of their evolution at finite temperature.

Figures (5)

• Figure 1: Schematic of collisional damping processes in the SPGPE: (a) Number damping, (b) Energy damping, and (c) Evaporative damping, the focus of the present work. The final process (d) is the Hamiltonian nonlinear interaction in the projected Gross-Pitaevskii evolution describing the coherent dynamics of the classical field. Each process describes an interaction between the coherent region and the thermal cloud involving increasing numbers of $C$-region field operators. The time reverse of each process also occurs.
• Figure 2: Number damping strength $\gamma$ vs evaporative damping strength $\bar{\sigma}$ in the high phase space density regime. Number damping and evaporative damping are of comparable size.
• Figure 3: Schematic of the evaporative damping process. The lines symoblise the single particle states with (single particle) energies ranging from the single particle ground state energy $\epsilon_0$ to the cut-off $\epsilon_\text{cut}$. Energy conservation considerations require that, under neglect of the interaction energy, the (single particle) energies of two particles (grey dots) in the coherent region combined has to be at least $\epsilon_\text{cut}$ for an evaportive scattering to occur.
• Figure 4: To estimate the effect of unphysical scattering events due to the constant energy approximation we introdue a second cut-off with momentum at the cut-off $K_\text{cut}$ for the projector. The coherent region then consists of particles that occupy states below this new cut-off, while the incoherent region encompasses still states above the original cut-off. States between these cut-offs are ignored.
• Figure 5: Estimate of the strength of evaporative damping $\bar{\sigma}_{K\text{cut}}$ for different choices of the cut-off for the coherent region $K_\text{cut}$ relative to the cut-off for the incoherent region $k_\text{cut}$. In the SPGPE the cut-offs coincide $K_\text{cut}=k_\text{cut}$, so that each state is either covered by the coherent or the incoherent region. The grey dotted line marks $K_\text{cut}=k_\text{cut}/\sqrt{2}$, so that left of it only unphysical scattering events (violating energy conservation) contribute to $\bar{\sigma}_\text{Kcut}$. For $K_\text{cut}<k_\text{cut}/2$ evaporative damping vanishes, in consistency with momentum conservation of the scattering processes. For $K_\text{cut}>k_\text{cut}$ further physical scattering events contribute to the strength of evaporative damping. Those, however, are in SPGPE treatment already accounted for by number damping.