Persistent current formation in double-ring geometries

Bland, T; Marolleau, Q; Comaron, P; Malomed, B A; Proukakis, N P

doi:10.1088/1361-6455/ab81e9

Persistent current formation in double-ring geometries

T. Bland, Q. Marolleau, P. Comaron, B. A. Malomed, N. P. Proukakis

Journal of Physics B: Atomic, Molecular and Optical Physics 53, 115301 (2020) · 10.1088/1361-6455/ab81e9 · arXiv:1911.12802 · PDF

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Abstract

Quenching an ultracold bosonic gas in a ring across the Bose–Einstein condensation phase transition is known, and has been experimentally observed, to lead to the spontaneous emergence of persistent currents. The present work examines how these phenomena generalize to a system of two experimentally accessible explicitly two-dimensional co-planar rings with a common interface, or to the related lemniscate geometry, and demonstrates an emerging independence of winding numbers across the rings, which can exhibit flow both in the same and in opposite directions. The observed persistence of such findings in the presence of dissipative coupled evolution due to the local character of the domain formation across the phase transition and topological protection of the randomly emerging winding numbers should be within current experimental reach.

Figures5

Colour map of the double-ring potential: two overlapping rings drawn dark against a uniform high plateau, annotated with the ring radius R and the ring width w. — **Figure 1.** Condensate growth in the double-ring geometry, with ring radius $R = 25\ \mu$ m, width $w = 6\ \mu$ m, and the inter-ring shift is $\delta = 0$ , see equation (5). (a), (i): the double-ring potential, (ii) its cross section along $y = 0$ (blue solid curve) and (iii) the evolution of the winding number over scaled time $\gamma t$ for the left (blue squares) and right (red circles) rings. (b), (i)–(v): the density evolution in scaled time, with $n_{\mathrm{max}} = 73\ \mu$ m $^{-2}$ . (vi)–(x): the corresponding phase profiles. For clarity, these have been masked by the Heaviside function $\Theta(0.9V_0 - V_{\mathrm{dr}}(x,y))$ , which filter out the contributions falling within the grey shaded area in (a), (ii).

Cross section of the potential along y = 0, showing three wells reaching zero, with the region above the barrier height shaded grey and the chemical potential marked by a red dashed line. — **Figure 1.** Condensate growth in the double-ring geometry, with ring radius $R = 25\ \mu$ m, width $w = 6\ \mu$ m, and the inter-ring shift is $\delta = 0$ , see equation (5). (a), (i): the double-ring potential, (ii) its cross section along $y = 0$ (blue solid curve) and (iii) the evolution of the winding number over scaled time $\gamma t$ for the left (blue squares) and right (red circles) rings. (b), (i)–(v): the density evolution in scaled time, with $n_{\mathrm{max}} = 73\ \mu$ m $^{-2}$ . (vi)–(x): the corresponding phase profiles. For clarity, these have been masked by the Heaviside function $\Theta(0.9V_0 - V_{\mathrm{dr}}(x,y))$ , which filter out the contributions falling within the grey shaded area in (a), (ii).

Key identifying the left ring in blue and the right ring in red, and the three times by line style: dotted, dashed and solid. — **Figure 2.** Azimuthal [(i) and (ii)] and radial [(iii) and (iv)] condensate phase shown for times $\gamma t = (1.35, 3.75, 30)$ ms (dotted, dashed, solid lines respectively, with increasing opacity for later times). The azimuthal phase is extracted around the lines (i) $\rho(x + R, y) = R$ and (ii) $\rho(x - R, y) = R$ , while the radial phase profile is displayed along cross section $y = 0$ . (v) References to the azimuthal (solid) and radial (dashed) phase profiles are superimposed on the 2D phase plot. Other parameters are the same as in figure 1.

Azimuthal phase of the left ring against the polar angle, in blue, wildly oscillating at the earliest time and falling smoothly across a full turn at the latest. — **Figure 2.** Azimuthal [(i) and (ii)] and radial [(iii) and (iv)] condensate phase shown for times $\gamma t = (1.35, 3.75, 30)$ ms (dotted, dashed, solid lines respectively, with increasing opacity for later times). The azimuthal phase is extracted around the lines (i) $\rho(x + R, y) = R$ and (ii) $\rho(x - R, y) = R$ , while the radial phase profile is displayed along cross section $y = 0$ . (v) References to the azimuthal (solid) and radial (dashed) phase profiles are superimposed on the 2D phase plot. Other parameters are the same as in figure 1.

Two-dimensional histogram of the winding numbers of the two rings, brightest at the origin and falling away in all directions, with four cells outlined in pink, red, orange and green and labelled c to f. — **Figure 3.** The distribution of persistent currents in the double-ring geometry. (a): the 2D histogram of winding numbers, produced by results of 5000 runs of the SPGPE. Coloured squares correspond to the phase plots and velocity fields below. (b): marginal histogram distributions for the left ( $n_L$ ) and right ( $n_R$ ) rings, comparing growth from noise to instantaneous thermal quench simulations, error bars contain the true probability within a 95% confidence interval. (c)–(f): (i) phase plots, (ii) velocity fields, and (iii) winding number evolutions of a typical single numerical run from within the selected squares from (a). Parameters are the same as figure 1.

Marginal probability of each winding number, peaked at zero, with four overlapping series for the left and right rings under growth from noise and under a thermal quench, carrying error bars. — **Figure 3.** The distribution of persistent currents in the double-ring geometry. (a): the 2D histogram of winding numbers, produced by results of 5000 runs of the SPGPE. Coloured squares correspond to the phase plots and velocity fields below. (b): marginal histogram distributions for the left ( $n_L$ ) and right ( $n_R$ ) rings, comparing growth from noise to instantaneous thermal quench simulations, error bars contain the true probability within a 95% confidence interval. (c)–(f): (i) phase plots, (ii) velocity fields, and (iii) winding number evolutions of a typical single numerical run from within the selected squares from (a). Parameters are the same as figure 1.

Colour map of the single-ring potential, one dark annulus against a uniform high plateau. — **Figure 4.** Formation of persistent currents in the single-ring trap. (a) The trapping potential from equation (7) with parameters $R = 25\ \mu$ m and $w = 6\ \mu$ m. (b) An equilibrium density profile produced by a single run of the SPGPE. (c) The distribution of winding numbers $n_w$ after 5000 realisations of the SPGPE, shown along with a fitted Gaussian distribution (red curve), whose standard deviation is $\sigma = 1.5338$ . Insets show examples of phase profiles for $n_w = (-1, 0, 5)$ from left to right, masked by the Heaviside function $\Theta(0.9V_0 - V_{\mathrm{sr}}(x,y))$ for clarity. (d) Marginal distributions for the single-ring $n_w$ , and for the left ( $n_L$ ) and right ( $n_R$ ) rings from figure 3.

Equilibrium density of the single ring, a bright annulus on a dark background. — **Figure 4.** Formation of persistent currents in the single-ring trap. (a) The trapping potential from equation (7) with parameters $R = 25\ \mu$ m and $w = 6\ \mu$ m. (b) An equilibrium density profile produced by a single run of the SPGPE. (c) The distribution of winding numbers $n_w$ after 5000 realisations of the SPGPE, shown along with a fitted Gaussian distribution (red curve), whose standard deviation is $\sigma = 1.5338$ . Insets show examples of phase profiles for $n_w = (-1, 0, 5)$ from left to right, masked by the Heaviside function $\Theta(0.9V_0 - V_{\mathrm{sr}}(x,y))$ for clarity. (d) Marginal distributions for the single-ring $n_w$ , and for the left ( $n_L$ ) and right ( $n_R$ ) rings from figure 3.

Two-dimensional histogram of the winding numbers of the two lobes of the lemniscate, brightest at the origin, with four cells outlined in white, orange, red and green. — **Figure 5.** The quench in the 2D Bose gas confined in the lemniscate potential. (a) The 2D histogram of winding numbers produced by 5000 runs of simulations of the SPGPE. Coloured squares correspond to the surrounding phase plots and velocity fields. (b)–(c) Selected phase plots, showing winding-number sets $(2, -2)$ and $(4, 3)$ . (d) Panels (i)–(iv) display velocity fields of selected squares from (a). Other parameters are the same as in figure 1.

Two phase plots of the figure-of-eight condensate, one for the winding-number pair plus two and minus two, one for plus four and plus three, with red arrows for the direction of flow. — **Figure 5.** The quench in the 2D Bose gas confined in the lemniscate potential. (a) The 2D histogram of winding numbers produced by 5000 runs of simulations of the SPGPE. Coloured squares correspond to the surrounding phase plots and velocity fields. (b)–(c) Selected phase plots, showing winding-number sets $(2, -2)$ and $(4, 3)$ . (d) Panels (i)–(iv) display velocity fields of selected squares from (a). Other parameters are the same as in figure 1.

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Supplemental figures3

Azimuthal phase of the left ring against the polar angle, in blue, sawtoothing three times across the turn, with the region of overlap between the rings shaded grey. — **Figure 6.** Centre: 2D phase profile for the case $(-3, -1)$ . Left: azimuthal phase profile for the left ring, grey highlights the region of overlap between the rings. Right: azimuthal phase for the right ring, shifted in the $x$ -axis such that the overlap region of the two rings occurs in the grey region. Other parameters are the same as in figure 1.

Two-dimensional phase map of the double ring for the winding-number pair minus three and minus one, with a blue circle on the left ring and a red circle on the right. — **Figure 6.** Centre: 2D phase profile for the case $(-3, -1)$ . Left: azimuthal phase profile for the left ring, grey highlights the region of overlap between the rings. Right: azimuthal phase for the right ring, shifted in the $x$ -axis such that the overlap region of the two rings occurs in the grey region. Other parameters are the same as in figure 1.

Mean absolute winding number against the ratio of ring circumference to width, on logarithmic axes, blue points lying along a dashed square-root line, with the region having no central hole in the density shaded grey. — **Figure 7.** (a) A power-law dependence given by equation (B.1). Each blue point is the measurement of $\langle|n_w|\rangle$ after 5000 SPGPE simulations for varying $w = (3, 6)\ \mu$ m and $R = (12.5, 18.75, 25, 37.5, 50)\ \mu$ m. The grey region covers the range of $R < 2w$ , when there is no central hole in the density. (b) The distribution of observed winding numbers, produced by the thermal quench with ramp time $\tau_r = 0.025$ s. Red (wide) bars are experimental data from Corman *et al.* for the single ring, based on 36 runs, while blue (thin) bars represent numerical data collected from 5000 realisations of the SPGPE. Parameters are specified in the main text.

Probability of each winding number as overlaid bar charts, wide red bars for the experimental data and narrow blue bars for the simulations, both peaked at zero and carrying error bars. — **Figure 7.** (a) A power-law dependence given by equation (B.1). Each blue point is the measurement of $\langle|n_w|\rangle$ after 5000 SPGPE simulations for varying $w = (3, 6)\ \mu$ m and $R = (12.5, 18.75, 25, 37.5, 50)\ \mu$ m. The grey region covers the range of $R < 2w$ , when there is no central hole in the density. (b) The distribution of observed winding numbers, produced by the thermal quench with ramp time $\tau_r = 0.025$ s. Red (wide) bars are experimental data from Corman *et al.* for the single ring, based on 36 runs, while blue (thin) bars represent numerical data collected from 5000 realisations of the SPGPE. Parameters are specified in the main text.

Column of four panels, alternating phase plots and velocity fields, framed in orange and green and annotated with the winding number of each ring. — **Figure 8.** Distribution of persistent currents in a double-ring geometry, with $\delta = w/2$ . (a) 2D histogram of observed winding numbers after 5000 runs of the SPGPE. Coloured squares correspond to the surrounding phase plots and velocity fields. (b) Measure of the relative difference between the $\delta = 0$ and $\delta = w/2$ histograms. Note the different spatial extent of the colorbar in this case, corresponding to a maximum relative error of 6.3%. (i)–(viii) Phase plots and velocity fields of selected squares from (a). Other parameters are the same as figure 1.

Two-dimensional histogram of the winding numbers for rings offset by half a ring width, brightest at the origin, with four cells outlined in colour. — **Figure 8.** Distribution of persistent currents in a double-ring geometry, with $\delta = w/2$ . (a) 2D histogram of observed winding numbers after 5000 runs of the SPGPE. Coloured squares correspond to the surrounding phase plots and velocity fields. (b) Measure of the relative difference between the $\delta = 0$ and $\delta = w/2$ histograms. Note the different spatial extent of the colorbar in this case, corresponding to a maximum relative error of 6.3%. (i)–(viii) Phase plots and velocity fields of selected squares from (a). Other parameters are the same as figure 1.

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Conclusion

We have explored the spontaneous growth of persistent currents in a quenched 2D Bose gas in the co-planar side-by-side double-ring geometry. The emerging persistent currents are stable and long-lived, and do not transfer between the rings. The overall value of the winding number in each ring behaves independently, which can be directly attributed to the importance of the local nature of phase establishment in the azimuthal direction, an observation that remains valid even when allowing for radial random phase variations. While the distribution of winding numbers across the two rings can therefore be directly mapped onto the well-known results for the single-ring setting, it is important to note that the azimuthal phase gradient in each ring (for a given non-zero winding number) is not constant, in direct contrast to the constant phase gradient of the single-ring case. This is one manifestation of the non-homeomorphism between a single torus and a 2-torus. Remarkably, however, this is not necessarily an impediment to potential future atomtronic devices, which could utilize the observed robust nature of the superfluid current configuration. The independence of the ring winding numbers and stability of the formed supercurrents may facilitate a well-controlled current transfer protocol between rings. Such deterministic transfer of winding numbers across multiple-loop atomtronic architectures is a promising direction for future research.

Varying the separation of the rings demonstrates the ability to modify the atom transfer between the rings and generate vortices between them. However, there is no sign of angular momentum transfer between the rings. The robustness of the persistent currents after the quench holds steadfast even against the change of trap geometry, such as replacement of the double ring by the lemniscate potential, where one might naively expect uncorrelated persistent currents in individual rings to be suppressed due to atomic motion along a “figure-of-eight” path. We also varied the phenomenological damping parameter $\gamma$ in a broad range of values, and extended the simulation time, which produced no evidence of decay, confirming the efficiency of the topological protection of the spontaneously generated winding numbers.

A natural question suggested by the present results is the transfer of the winding numbers between the rings, which will be the subject of further work. The ability to control the transfer of winding numbers could be envisaged as a prototypical atomtronic switch, providing an avenue for controllable realization of coherent quantum phase slips required for the Mooij-Harmans qubit. A hot topic of current studies of toroidal BECs is the supercurrent decay mechanism, an understanding of which could help control the transfer of current states between the rings. The Gross-Pitaevskii equation and its many extensions do not capture supercurrent decay, even at finite temperature, without an external barrier. However, due to roughness of the potential, the decay time for large winding numbers in experiments is on the order of seconds, whereas $n_w \sim 1$ may be stable on times on the order of minutes. Theoretical studies of the decay mechanism so far have all relied on effects along the annulus, caused by a repulsive barrier, the decay being visualized as vortices crossing the barrier region radially. It has been shown that the temperature-induced decay does not fit the Caldeira-Leggett superconductivity model, i.e. the observed decay rate does not match simple models of quantum tunnelling and thermal activation of phase slips. Using a truncated Wigner approximation, some of the disagreements between theory and experiment have been attributed to thermal fluctuations; however, exact identification of the decay mechanism remains an open question.

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