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

Two-row figure: (a) the double-ring trapping potential, its cross-section, and the winding-number evolution of each ring over time; (b) ten density and phase snapshots showing two rings merging and equilibrating over time. — **Figure 1.** Condensate growth in the double-ring geometry, with ring radius $R = 25\ \mu$ m, width $w = 6\ \mu$ m, and inter-ring shift $\delta = 0$ , see Eq. (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, masked by the Heaviside function $\Theta(0.9V_0 - V_{\mathrm{dr}}(x,y))$ , which filters out the contributions falling within the gray shaded area in (a)(ii).

Legend followed by five phase plots: azimuthal and radial phase profiles for the left and right rings at three times, and a 2D phase map with the extraction lines overlaid. — **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 the 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 Fig. 1.

2D histogram of winding-number pairs with four highlighted cells; marginal distributions comparing noise and thermal quenches; and three columns of phase plots, velocity fields, and winding-number time traces for the four highlighted cells. — **Figure 3.** The distribution of persistent currents in the double-ring geometry. (a) The 2D histogram of winding numbers, produced by 5000 runs of the SPGPE. Colored 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 in (a). Parameters are the same as in Fig. 1.

Figure 4. Formation of persistent currents in the single-ring trap. (a) The trapping potential from Eq. (7), with

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, with a fitted Gaussian distribution (red curve) of standard deviation

\sigma = 1.5338

; insets show example 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 Fig. 3.

2D histogram of winding-number pairs for the lemniscate trap with two highlighted cells; the corresponding phase plots; and four velocity-field plots for selected cells. — **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. Colored 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 Fig. 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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