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arXiv:2601.04006 · Physical Review A

Anomalous minimization for critical velocity of superflow along a step potential

Akihiro Kanjo, Hiromitsu Takeuchi

Published May 7, 2026• 1 citation
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Abstract

To reveal a microscopic mechanism for the anomalous minimization and dependence of the superfluid critical velocity on a moving obstacle potential in a atomic Bose-Einstein condensate [\href{https://link.aps.org/doi/10.1103/PhysRevA.91.053615}{Phys.~Rev.~A \textbf{91}, 053615 (2015)}], we introduce a considerably simplified model of superflow along a step potential. The energy spectrum and wave functions of the lowest-energy excitations in this system are well described by the semi-classical analysis based on the Bogoliubov theory. We found that the critical velocity is minimized and becomes zero when the potential height equals the hydrostatic chemical potential, which corresponds to the critical point of the local condensation phase transition inside the step potential. In a finite-size system, the critical velocity $v_\mathrm{c}$ obeys a power-law scaling with the system size $L_x$ as $v_\mathrm{c}\propto L_x^{-0.963}$. This criticality provides an explanation of the power-law scaling of the minimum critical velocity observed in the experiment.

DOI
10.1103/h68l-lpjf

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To reveal a microscopic mechanism for the anomalous minimization and dependence of the superfluid critical velocity on a moving obstacle potential in a atomic Bose-Einstein condensate [\href{https://link.aps.org/doi/10.1103/PhysRevA.91.053615}{Phys.~Rev.~A \textbf{91}, 053615 (2015)}], we introduce a considerably simplified model of superflow along a step potential. The energy spectrum and wave functions of the lowest-energy excitations in this system are well described by the semi-classical analysis based on the Bogoliubov theory. We found that the critical velocity is minimized and becomes zero when the potential height equals the hydrostatic chemical potential, which corresponds to the critical point of the local condensation phase transition inside the step potential. In a finite-size system, the critical velocity vcv_\mathrm{c}vc​ obeys a power-law scaling with the system size LxL_xLx​ as vc∝Lx−0.963v_\mathrm{c}\propto L_x^{-0.963}vc​∝Lx−0.963​. This criticality provides an explanation of the power-law scaling of the minimum critical velocity observed in the experiment.
When a superfluid flows past an obstacle, quantized vortices can be created in the wake above a certain critical velocity. In the experiment by Kwon et al. [Phys. Rev. A 91, 053615 (2015)], the critical velocity vc{v}_{c}vc​ was measured for atomic Bose-Einstein condensates (BECs) using a moving repulsive Gaussian potential, and vc{v}_{c}vc​ was minimized when the potential height V0{V}_{0}V0​ of the obstacle was close to the condensate chemical potential $\ensuremath{\mu}$. Here we numerically investigate the evolution of the critical vortex shedding in a two-dimensional BEC with increasing V0{V}_{0}V0​ and show that the minimum vc{v}_{c}vc​ at the critical strength ${V}_{0c}\ensuremath{\approx}\ensuremath{\mu}$ results from the local density reduction and vortex-pinning effect of the repulsive obstacle. The spatial distribution of the superflow around the moving obstacle just below vc{v}_{c}vc​ is examined. The particle density at the tip of the obstacle decreases as V0{V}_{0}V0​ increases to Vc0{V}_{c0}Vc0​, and at the critical strength, a vortex dipole is suddenly formed and dragged by the moving obstacle, indicating the onset of vortex pinning. The minimum vc{v}_{c}vc​ exhibits power-law scaling with the obstacle size $\ensuremath{\sigma}$ as ${v}_{c}\ensuremath{\sim}{\ensuremath{\sigma}}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}1/2$.
We consider the setup employed in a recent experiment (Ramanathan et al 2011 Phys. Rev. Lett. 106 130401) devoted to the study of the instability of the superfluid flow of a toroidal Bose–Einstein condensate in the presence of a repulsive optical barrier. Using the Gross–Pitaevskii mean-field equation, we observe, consistently with what we found in Piazza et al (2009 Phys. Rev. A 80 021601), that the superflow with one unit of angular momentum becomes unstable at a critical strength of the barrier and decays through the mechanism of phase slippage performed by pairs of vortex–antivortex lines annihilating. While this picture qualitatively agrees with the experimental findings, the measured critical barrier height is not very well reproduced by the Gross–Pitaevskii equation, indicating that thermal fluctuations can play an important role (Mathey et al 2012 arXiv:1207.0501). As an alternative explanation of the discrepancy, we consider the effect of the finite resolution of the imaging system. At the critical point, the superfluid velocity in the vicinity of the obstacle is always of the order of the sound speed in that region, v barr = c l . In particular, in the hydrodynamic regime (not reached in the above experiment), the critical point is determined by applying the Landau criterion inside the barrier region. On the other hand, the Feynman critical velocity v f is much lower than the observed critical velocity. We argue that this is a general feature of the Gross–Pitaevskii equation, where we have v f = c l with being a small parameter of the model. Given these observations, the question still remains open about the nature of the superfluid instability.
We numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept an obstacle formed by an optical dipole potential through the long axis of the condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of v≈0.3v\approx 0.3v≈0.3 mm/s, suggestive of a superfluid critical velocity substantially less than the average speed of sound. However, our analysis shows that the that the experimental observations of Engels and Atherton are in fact consistent with a superfluid critical velocity equal to the local speed of sound. We construct a model of energy transfer to the system based on the local density approximation to explain the experimental observations, and propose and simulate experiments that sweep potentials through harmonically trapped condensates at a constant fraction of the local speed of sound. We find that this leads to a sudden onset of excitations above a critical fraction, in agreement with the Landau criterion for superfluidity.
We present measurements of the critical velocity for vortex shedding in a highly oblate Bose-Einstein condensate with a moving repulsive Gaussian laser beam. As a function of the barrier height V0{V}_{0}V0​, the critical velocity vc{v}_{c}vc​ shows a dip structure having a minimum at ${V}_{0}\ensuremath{\approx}\ensuremath{\mu}$, where $\ensuremath{\mu}$ is the chemical potential of the condensate. At fixed ${V}_{0}\ensuremath{\approx}7\ensuremath{\mu}$, we observe that the ratio of vc{v}_{c}vc​ to the speed of sound cs{c}_{s}cs​ monotonically increases for decreasing $\ensuremath{\sigma}/\ensuremath{\xi}$, where $\ensuremath{\sigma}$ is the beam width and $\ensuremath{\xi}$ is the condensate healing length. We explain our results with the density reduction effect of the soft boundary of the Gaussian obstacle, based on the local Landau criterion for superfluidity. The measured value of vc/cs{v}_{c}/{c}_{s}vc​/cs​ with our stiffest obstacle is about 0.4, which is in good agreement with theoretical predictions for a two-dimensional superflow past a circular cylinder.
This thesis examines the manifestation of the critical velocity for superfluidity in dilute gas Bose-Einstein condensates (BECs). Superfluidity is a macroscopic quantum phenomenon characterised by unique properties such as frictionless flow, high thermal conductivity and the presence of quantised vortices. Our study focuses on what is arguably the most well-known attribute: the ability for an object to move through a superfluid without experiencing a drag force when travelling below a threshold critical velocity.The Landau criterion for superfluidity predicts the critical velocity in a BEC is equal to the speed of sound, and this conclusion is supported by a large body of theoretical work. However, a number of experiments have observed lower critical velocities than theory predicts. In the first part of this thesis, we numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept a repulsive obstacle, formed by a optical dipole potential, through the long axis of a condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of v _ 0:3 mm s
We solve the Gross-Pitaevskii (GP) and Bogoliubov equations to investigate the metastability of superfluidity in a Bose-Einstein condensate in the presence of a uniformly moving defect potential in a two-dimensional torus. We calculate the total energy and momentum as functions of the driving velocity of the moving defect and find metastable states with negative effective mass near the critical velocity. We also find that the first excited energy (energy gap) in the finite-sized torus closes at the critical velocity, that it obeys one-fourth power-law scaling, and that the dynamical fluctuation of the density (amplitude of the order parameter) is strongly enhanced near the critical velocity. We confirm the validity of our results near the critical velocity by calculating the quantum depletion. We find an unconventional swallowtail structure (multiple-swallowtail structure) through calculations of the unstable stationary solutions of the GP equation.