Thermal Decay of Planar Jones-Roberts Solitons

TL;DR

This work presents a thermally driven decay analysis of planar Jones-Roberts solitons in two-dimensional Bose-Einstein condensates within the stochastic projected Gross-Pitaevskii framework. By deriving analytical expressions for two damping channels—number damping (particle exchange) and energy damping (energy transfer, number-conserving)—the authors characterize the decay across low- and high-velocity regimes and show that energy damping is typically dominant at experimentally relevant phase-space densities. Numerical simulations across the full velocity range corroborate the analytical predictions and reveal distinct damping behavior, with interaction energy emerging as a practical, density-dependent measure of JRS decay, particularly for rarefaction pulses. The results provide experimentally testable signatures and emphasize the usefulness of the SPGPE approach for finite-temperature dynamics in planar quantum fluids, including near dipole annihilation and in intermediate regimes.

Abstract

Homogeneous planar superfluids exhibit a range of low-energy excitations that also appear in highly excited states like superfluid turbulence. In dilute gas Bose-Einstein condensates, the Jones- Roberts soliton family includes vortex dipoles and rarefaction pulses in the low and high velocity regimes, respectively. These excitations carry both energy and linear momentum, making their decay characteristics crucial for understanding superfluid dynamics. In this work, we develop the theory of planar soliton decay due to thermal effects, as described by the stochastic projected Gross-Pitaevskii theory of reservoir interactions. We analyze two distinct damping terms involving transfer between the condensate and the non-condensate reservoir: particle transfer that also involves energy and usually drives condensate growth, and number-conserving energy transfer. We provide analytical treatments for both the low and high velocity regimes and identify conditions under which either mechanism dominates. Our findings indicate that energy damping prevails at high phase space density. These theoretical results are supported by numerical studies covering the entire velocity range from vortex dipole to rarefaction pulse. We use interaction energy to characterize rarefaction pulses, analogous to the distance between vortices in vortex dipoles, offering an experimentally accessible test for finite temperature theory in Bose-Einstein condensates.

Figures (15)

• Figure 1: Density and phase plot of a planar Jones-Roberts soliton with a velocity below ((a) and (c), respectively; the velocity is $v=0.05c$) and above ((b) and (d), respectively; the velocity is $v=0.85c$) the critical velocity $v_\text{c}\simeq0.636c$jones_motions_1986. Below $v_\text{c}$ it consists of two counter-rotating vortices, with a $2\pi$-winding around each of them. Above, there remains only one minimum that does not reach to zero anymore. Additionally, the phase does not take all values between $0$ and $2\pi$. With further increasing velocity the JRS flattens and widens.
• Figure 2: Interaction energy as a function of the velocity. The blue dots stem from numerical calculations (see appendix \ref{['Simulation']}), the orange dashed line is the interaction energy in the low-velocity limit kuznetsov_instability_1995 and the orange solid line is the high-velocity result (\ref{['Eintlinear']}). The grey dots signify the critical velocity $v_\text{c}\simeq 0.64c$ at which the annihilation occurs jones_motions_1986. The violet dash-dotted line is derived using Padé approximations for the density as discussed in section \ref{['Pade']}. We observe good agreement to the dispersion relation ansatz approximation found in smirnov_dynamics_2012 (grey solid line). The early decline near $v\sim 0.1c$ is due to the weak mutual distortion of independent vortex cores when $d_\textrm{v}\lesssim 14\xi$. The velocity was extracted by calculating the momentum and interaction energy and employing (\ref{['Integral-relation2']}). For high velocities $v\sim c$ the interaction energy and momentum become sensitive to small changes in velocity. Additionaly, JRS become increasingly flat making the estimation of these quantities sensitive to collisions with sound waves. These effects lead to an overestimation of the velocity in the region marked with a grey band. For smaller interaction energies $mE_\text{int}/(n_0\hbar)\lesssim 0.5$, the soliton reaches an extension comparable to the size of the grid used in the simulation.
• Figure 3: Normalized decay of the momentum under number damping (blue) compared with its decay under energy damping in the low (orange, solid) and high (orange, dashed) 2D phase space density regime. (a) shows the analytical results (\ref{['numberd']}), (\ref{['MehdiResult']}) for the damping of the vortex distance of a dipole, (b) the results (\ref{['numberdamping']}), (\ref{['energydamp']}) concerning the decay of the interaction energy of a rarefaction pulse. As in both cases the normalized decay of the momentum is plotted against the momentum (this is because the momentum is proportional to $d_\text{v}$ in the low-velocity limit and to $E_\text{int}$ in the high-velocity limit), a direct comparison of the damping strengths is possible. A cutoff is chosen so that $N_\text{cut}=1$. The number damping strength is set as $\gamma=4a_\text{s}^2/\lambda_\text{th}^2$, corresponding to $\beta\mu_{3D}\simeq0.58$ (other choices of the temperature and chemical potential can change $\gamma$ only by less than a factor of 2). The thickness of the atomic cloud is set to $l_z=\sqrt{2}\xi$. In the low density regime number damping is the dominant process for all interaction energies for which our analytical results can be expected to lie in the right order of magnitude. On the other hand, in the high density limit, energy damping is dominant for all but extremely large vortex distances or small interaction energies. Hence, it is the relevant process for JRS observable in experiment. As the normalized number damping strength in the rarefaction pulse regime is constant while the normalized energy damping strength varies strongly with interaction energy, the relevant process should be easily identifiable.
• Figure 4: Distance between two vortices against time under evolution according to the $\gamma$GPE with a damping rate $\gamma=0.003$. The blue dots stem from a numerical simulation, where the positions of the two vortices were determined as the global minima of the density in the upper and lower half plane. The orange solid line is the analytical result (\ref{['numberd']}) and the violet dashed line corresponds to antiproportional behaviour of the damping rate as suggested in tornkvist_vortex_1997mehdi_mutual_2023. The logarithmic correction shows much better agreement with the numerics than the antiproportional damping rate previously claimed. Note that using a phenomenological antiproportional damping rate capturing the early decay correctly (purple dash-dotted line) predicts to strong damping in the long term, thereby demonstrating the necessity of the logarithmic correction. On the other hand, considering only the hydrodynamic limit (corresponding to $\alpha=1/\sqrt{2}$, orange dash-dot-dotted line) predicts considerably to weak damping. Hence, the inclusion of the interaction energy and quantum pressure in the total energy is crucial to gain an accurate description.
• Figure 5: Comparison of the (normalized) analytically derived damping rates (\ref{['numberdamping']}), (\ref{['energydamp']}) (orange solid lines) with numerical results (orange dots and violet diamonds). (a) shows the number damping case while (b) presents energy damping induced decay. Our analytical result shows an improving prediction for the damping rate with decreasing interaction energy $E_\text{int}$. The poor agreement for large interaction energies is inevitable, as we work in lowest order of the reciprocal Lorentz factor $\epsilon$ and thus the interaction energy.