Onsager vortex formation in Bose-Einstein condensates in two-dimensional power-law traps

We study computationally dynamics of quantized vortices in two-dimensional superfluid Bose-Einstein condensates confined in highly oblate power-law traps. We have found that the formation of large-scale Onsager vortex clusters prevalent in steep-walled traps is suppressed in condensates confined by harmonic potentials. However, the shape of the trapping potential does not appear to adversely affect the evaporative heating efficiency of the vortex gas. Instead, the suppression of Onsager vortex formation in harmonic traps can be understood in terms of the energy of the vortex configurations. Furthermore, we find that the vortex-antivortex pair annihilation that underpins the vortex evaporative heating mechanism requires the interaction of at least three vortices. We conclude that experimental observation of Onsager vortices should be the most apparent in flat or inverted-bottom traps.

Figure 1

Comparison of the time evolution of the vortex configuration between the $\alpha =2$ harmonic trap (a)–(c) and $\alpha =100$ uniform trap (d)–(f). The gray-scale value represents the superfluid density, and the color bars are normalized to the maximum density: $4.3\times {10}^{-3}{a}_{\mathrm{osc}}^{-2}$ and $2.7\times {10}^{-3}{a}_{\mathrm{osc}}^{-2}$ for the top and bottom rows, respectively. Vortices and antivortices are denoted by blue (dark) and green (light) circles, respectively. The red line represents the effective dipole moment of the vortex distribution. Movies S1 and S2 in the Supplemental Material [50] show the dynamics of each simulation.

Figure 2

Comparison of the vortex number decay and dipole moment evolution (inset) for the harmonic (red/light) and uniform (blue/dark) traps. The black circles in the inset correspond to the time frames displayed in Fig. 1. The fluctuations in the vortex number are due to vortices crossing the counting radius of $0.9R_{\mathrm{o}}$, in addition to occasional vortex-antivortex pair creation.

Figure 3

Comparison of statistical behavior between (a) the harmonic trap and (b) the uniform trap. For each dynamical simulation, the dipole moment of the vortex configuration is shown as a function of the incompressible kinetic energy per vortex number squared. The initial state in each plot is the bottom-left corner, and the evaporative heating increases the energy per vortex number squared over time. The data appear as columns because each vortex annihilation increases the energy per vortex number squared by a discrete amount.

Figure 4

Statistical data obtained from 100 000-step Markov chain Monte Carlo simulations for a harmonic (red/light) and a uniform (blue/dark) trap for a total of 12 vortices with equal numbers of vortices and antivortices. The panels show (a) the specific heat, (b) the incompressible kinetic energy per vortex number squared, and (c) the dipole moment of the configuration, each plotted as a function of the statistical temperature. The shaded regions in (b) and (c) correspond to the standard deviation of each observable at a given temperature. The maximum in the specific heat indicates the transition to the Onsager vortex state in each trap, and is accompanied by an increase in both the energy per vortex number squared and the dipole moment. Panels (d)–(e) and (f)–(g) show typical vortex configurations at the temperature extremes in the harmonic and uniform traps, respectively, with labeling as in Fig. 1. The temperatures shown in these frames are indicated in (c) with vertical dashed lines.

Figure 5

Incompressible kinetic energies of a vortex-antivortex pair for a range of power-law traps computed using the GPE. The pair is placed symmetrically in the trap and both vortices are an equal radial distance from the center. In order of peak location from left to right, the power-law exponents are $\alpha =2$ (red), $\alpha =\{4,6,8,14,30\}$ (thin black lines), and $\alpha =100$ (blue). In addition, the dipole energy for an inverted trap (as described in the main text) is shown in light green (far right peak). The maximal separation is indicated on each curve with a circle, and is emphasized further on the two extreme power-law traps, as well as the inverted trap, with a vertical dashed line.

Figure 6

A vortexonium state (formed from a vortex-antivortex pair) highlighted in (a) with a dashed oval colliding with an antivortex and dissipating into fluid sound waves which disperse radially, indicated in (b) and (c) with dashed circles. The vortices and color bar are labeled as per Fig. 1. Supplemental movie S4 shows the dynamics of this event [50], including the wave function phase.

Figure 7

Feynman diagram depicting the entire vortex-antivortex annihilation process observed, with time flowing from left to right. The straight lines represent vortices ($v^{+}$) and antivortices ($v^{-}$), the double line represents vortexonium ($v^{*}$), and the wavy line denotes the sound waves emitted at each vertex (the magnitude of the second burst of sound is far greater than the first). The light blue lines indicate participating catalyst vortices, which are not annihilated during the process.

Figure 8

Ensemble-averaged vortex number decay curves for harmonically trapped systems at zero temperature ($\gamma =0$, solid red/dark line) and nonzero temperature ($\gamma ={10}^{-3}$, solid green/light line). The fits for each curve to Eq. (4) are shown as black dashed lines, with ${\mathrm{\Gamma }}_1=0.14{\text{s}}^{-1}, {\mathrm{\Gamma }}_2=0.044{\text{s}}^{-1}, {\mathrm{\Gamma }}_3={\mathrm{\Gamma }}_4=0$ for the nonzero-temperature case, and ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0, {\mathrm{\Gamma }}_3=1.2\times {10}^{-4}{\text{s}}^{-1}, {\mathrm{\Gamma }}_4=8.1\times {10}^{-7}{\text{s}}^{-1}$ for the zero-temperature case.

Figure 9

(a)–(c) Unbinding of a vortex pair at the boundary in the uniform trap; (d)–(f) reflection of a vortexonium state at the boundary in the uniform trap. The green arrows show the direction each excitation is traveling. The insets in (a) and (d) show the phase of the wave function in the corresponding frame, showing the two singularities in (a) and the phase step in (d). The sound wave produced by each collision event propagates outwards in (c) and (f). The color bar is normalized to the maximum condensate density, as in Fig. 1. Movies S6 and S7 in the Supplemental Material [50] show each event in full.

Figure 10

Comparison of dipole moment evolution in traps of varying steepness. Each curve is averaged over five simulations. The trap steepnesses are (from bottom to top) $\alpha =2$ (solid red line), $\alpha =8$ (dot-dashed black line), $\alpha =14$ (dotted black line), $\alpha =30$ (dashed black line), $\alpha =100$ (solid blue line), and an $\alpha =100$ inverted trap (solid green line).