Persistent current induced by turbulent cascade and geometric quench in superfluid Bose−Einstein condensates

Abstract

The persistent flow of superfluids is essential for understanding the fundamental characteristics of superfluidity and shows promise for applications in high-precision metrology and atomtronics. We proposed a protocol for generating persistent flows with significant winding numbers by employing a geometric quench and leveraging two-dimensional (2D) quantum turbulence. By subjecting the trap potential to sudden geometric quenches to drive the system far from equilibrium, we can reveal intriguing nonequilibrium phenomena. Our study demonstrates that transitioning from a single ring-shaped configuration to a double concentric ring-shaped configuration through a geometric quench does not induce a persistent current in Bose−Einstein condensates (BECs). The energy transfer from small to large length scales during the 2D turbulent cascade of vortices can generate persistent flow with a small winding number in toroidal BECs. Nonetheless, the interplay of geometric quench and turbulent cascade can lead to circulation flows that exhibit high stability, uniformity, and are devoid of topological excitations. We showcase the intricate nature of turbulence in our investigation, which is influenced by factors like boundaries and spatial dimensionality. This advancement holds promise for innovative atomtronic designs and provides insights into quantum tunneling and interacting quantum systems under extreme non-equilibrium conditions.

Graphical abstract

Keywords

persistent current / turbulent cascade / geometric quench / superfluid / Bose–Einstein condensates

1 Introduction

The study of persistent flow in superfluids contributes to the understanding of the fundamental characteristics of superfluidity and offers potential applications in high-precision metrology and atomtronics [1-3]. Recent advancements in technical capabilities have enabled the creation of tailored trapping potentials with arbitrary geometries, facilitating quantum transport experiments with quantum gases in Bose−Einstein condensates (BECs) characterized by weak interatomic interactions that can be effectively described by the mean-field Gross−Pitaevskii (GP) equation [4]. Two-dimensional (2D) annular BECs confined in ring-shaped traps [1, 5-8] maintain persistent currents with quantized circulation along closed paths [3, 7, 9-11], where the phase winding of the macroscopic wave function is a multiple of 2\pi. These systems have garnered increased interest as atomic analogs of superconducting quantum interference devices, serving as foundational components for atomtronic circuits [2, 12-14]. They exhibit similarities to persistent currents in superconducting rings with quantized magnetic flux.

The direct macroscopic circulation flow of superfluid BECs in annular traps can be achieved by inducing rotation through perturbing potential [15, 16]. However, fast rotation leads to inevitable excitations in the condensate flow. Alternatively, circulation states can be attained through internal atomic state manipulation [7, 9, 17], although this method is challenging in magnetically trapped condensates due to the coupling of different Zeeman substates. Phase imprinting for generating circulation states [10, 18, 19] is difficult to apply in systems with complex structures. Experimental realization of matter-wave guiding over large distances with BECs in a neutral-atom accelerator ring, based on magnetic time-averaged adiabatic potentials [20], has demonstrated acceleration of BECs to hypersonic velocities but with nonuniform density.

Adding energy transiently to a system typically results in increased disorder. However, in 2D superfluids, the suppression of vortex bending, tilting, and Kelvin wave perturbations inhibits disorder. This leads to the phenomenon of the well-known inverse energy cascade [21-25], where order emerges from turbulence due to the prevention of energy dissipation at small length scales [26-29]. Notably, Onsager vortex clusters are formed as a remarkable feature, serving as a tool for vortex thermometry [30, 31]. Furthermore, in toroidal BECs, a large-scale persistent flow of atoms can be induced by the dissipation of 2D quantum turbulence (2DQT) in the form of disordered vortex distributions, demonstrating energy transport from small to large length scales [32].

Although the decay of turbulent cascade has been extensively studied, the inherent unpredictability of turbulence is further complicated by physical properties like boundaries and spatial dimensionality. There is significant interest in exploring the dissipative dynamics of 2DQT across various geometries. Furthermore, by inducing significant deviations from equilibrium through a sudden geometric alteration of the trapping potential, novel nonequilibrium phenomena may be discovered. In this study, we investigate the dynamic effects of quenching condensates in a BEC confined within a double concentric ring-shaped trap. The focus is on quantitatively analyzing the emergence of persistent atom flows (currents) with high winding numbers in this unique geometry. These findings aim to advance research on superfluidity and quantum tunneling in diverse transport regimes.

2 Model

At zero temperature, the complex-valued mean-field order parameter of a trapped 2D condensate containing N atoms obeys the GP equation given by

(1)\mathrm{i}\hslash {\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}}=\left(-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}^2+V_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}+g_{2\mathrm{D}}N{\left|\psi \right|}^2\right)\psi ,

where the 2D coupling constant is g_{2\mathrm{D}} =2\sqrt{2\pi }{a}_s{a}_z\hslash {\omega }_z with a_s being the s-wave scattering length of the atoms and a_z=\sqrt{\hslash /(m{\omega }_z)}. The order parameter \psi is normalized to 1. To minimize density variations, our system utilizes an experimentally accessible box potential [33, 34], ensuring spatial uniformity of the condensates away from the boundaries. Specifically, the quenching dynamics of a BEC can be described as a sudden change in the potential well during the dynamical process:

(2)V_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}(t) =\{\begin{array}{rc}{V}_1,& t=0,\\ V_1+V_2,& t>0,\end{array}

with

(3)

where r=\sqrt{x^2+y^2}. V_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}(t >0) can be achieved using an optical-box trap created by two hollow tube beams and two sheet beams, as illustrated in Fig.1(a). Here, R_j(j =1,3) and R_k(k =2,4) represent the inner and outer radii of the two concentric rings, respectively, with the width of the potential barrier between the rings denoted by d=R_3-R_2. Time and length are scaled with the units t_0=1 /{\omega }_z and a_0=a_z, respectively. By default, we set V_0=1.43\hslash {\omega }_z and d=1.25a_0 unless otherwise specified. In our research, the superfluid consists of a BEC containing N=1\times {10}^5{}^{87}Rb atoms with a trapping frequency {\omega }_z=2\pi \times 700 Hz and a_s=5.4\times {10}^{-9} m. The injection of angular momentum through vortex imprinting involves embedding vortices in the condensate distribution instead of the macroscopic annular superflow, aligning with the approach used in Ref. [32]. These vortices, randomly distributed, can be introduced into the condensate cloud using either phase imprinting techniques [35-39] or optical obstacle stirring [40]. The initial states are obtained through numerical simulation of the GP equation using imaginary time evolution. It is worth mentioning that the vortices can be introduced into the BEC either initially or during the dynamic evolution, with no impact on the fundamental physics. However, the latter approach may induce more pronounced density wave oscillations in the system.

As previously mentioned, an inverse energy cascade induced by 2DQT can promote persistent flow in ring-shaped condensates. To identify the occurrence of a turbulent state, it is essential to utilize the spectral scaling approach [41, 42]. Further characterization of the quantum turbulence state through spectrum analysis is better accomplished using the Madelung transformation given by \psi (\mathit{r},t )=\sqrt{n(\mathit{r},t)}\exp (\mathrm{i}\varphi (\mathit{r},t) ), where n and \varphi represent the density and phase distribution of the condensate, respectively. By substituting this form into Eq. (1), the total energy-functional is composed by four part,

E_{tot}={\displaystyle \int \left({\displaystyle \frac{1}{2}}|\mathrm{\nabla }\sqrt{n}{\mathrm{e}}^{-\mathrm{i}\varphi }{|}^2+V_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}n+{\displaystyle \frac{1}{2}}{g}_{2D}{n}^2\right)\mathrm{d}\mathit{r}}

(4)\begin{array}{rl}& ={\displaystyle \frac{1}{2}}{\displaystyle \int |\sqrt{n}\mathit{u}{|}^2\mathrm{d}\mathit{r}+{\displaystyle \frac{1}{2}}{\displaystyle \int (\mathrm{\nabla }\sqrt{n}{)}^2\mathrm{d}\mathit{r}+{\displaystyle \int n{V}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}\mathrm{d}\mathit{r}}}}\\ & +{\displaystyle \frac{1}{2}}{\displaystyle \int g_{2D}{n}^2\mathrm{d}\mathit{r},}\end{array}

representing the superfluid kinetic energy, the quantum pressure energy, the trap energy, and the interaction energy, respectively. The spectra of kinetic energy (incompressible part) can further characterize 2DQT with a large quantity of randomly distributed vortices. Following Ref. [43], to obtain the incompressible kinetic energy spectrum, we initially decompose the superfluid kinetic energy into two components, E_{kin}=E_{tot}^i+ E_{tot}^c. This decomposition involves splitting the vector field \sqrt{n} \mathit{u} into its solenoidal and irrotational parts since (\sqrt{n}\mathit{u}{)}^i and (\sqrt{n}\mathit{u}{)}^c are orthogonal: \sqrt{n}\mathit{u} =(\sqrt{n}\mathit{u}{)}^i+(\sqrt{n}\mathit{u}{)}^c\equiv \mathrm{\nabla }\times \mathit{A}+\mathrm{\nabla }\mathrm{\Theta }, where \mathrm{\nabla } \cdot (\sqrt{n}\mathit{u}{)}^i=0 and \mathrm{\nabla } \times (\sqrt{n}\mathit{u}{)}^c=0. Here, \mathit{A} and \mathrm{\Theta } represent the vector and scalar potentials of the flow field obtained by solving the equations

(5){\mathrm{\nabla }}^2 \mathrm{\Theta }=\mathrm{\nabla }\cdot (\sqrt{n}\mathit{u})\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\nabla }\times \mathit{A}=-\sqrt{n}\mathit{u}+\mathrm{\nabla } \mathrm{\Theta }.

Physically, E_{tot}^i=\int {\left|(\sqrt{n}\mathit{u}{)}^i\right|}^2\mathrm{d}\mathit{r} and E_{tot}^c=\int {\left|(\sqrt{n}\mathit{u}{)}^c\right|}^2\mathrm{d}\mathit{r} correspond to the kinetic energies of vortices and sound waves in the superfluid, respectively. The spectrum of kinetic energy (incompressible part) is [41]

(6)E_{kin}^i(k)={\displaystyle \frac{k}{2}}{\displaystyle \int {\left|{\displaystyle \int {\mathrm{e}}^{\mathrm{i}\mathit{k}\cdot \mathit{r}}{\left(\sqrt{n(\mathit{r})}\mathit{u}\mathbf{(}\mathit{r}\mathbf{)}\right)}^i\mathrm{d}\mathit{r}}\right|}^2\mathrm{d}{\varphi }_k.}

Here, {\varphi }_k is the polar angle in k space.

The spectral analysis reveals a k^{-5/3} power law in the range k_{\tau }\equiv 2\pi {\tau }^{-1} as shown in Fig.1(b), where \tau =11 μm represents the cross-sectional radial thickness of the toroidal BEC, corresponding to the largest intervortex distance, and \xi =0.54 μm is the healing length associated with the vortex core scale. We examine two scenarios. One involves a double concentric ring-shaped condensate with N_v vortices imprinted within the inner ring, while the other entails a single ring-shaped condensate with N_v vortices released into a double ring-shaped trap. For a fixed total number of atoms, maintaining the same angular momentum per atom may necessitate varying trap parameters between these scenarios. The energy spectra of the two cases at t=14 ms with N_v=20 are depicted by the blue dash-dotted and red solid lines in Fig.1(b). The E(k)\propto k^{-3} region is attributed to the vortex core structure, while the E(k)\propto k^{-5/3} region aligns with Kolmogorov’s analysis of turbulence spectra.

3 Results and analysis

3.1 Geometric quench of the trapping potential from a single ring to a double-ring

The behavior of the system under geometry-quench deviating from equilibrium is not straightforward. To investigate the impact of quenching the trapping potential, we initially trap atoms in a single ring configuration to form an annular condensate, as illustrated in Fig.2(a). Subsequently, we adjust the trapping potential to create a double-ring geometry, depicted schematically in Fig.2(b). In the absence of interatomic interactions (g_{2D} =0), the Josephson oscillation occur during the dynamic evolution. However, when the strength of interatomic interaction exceeds a threshold value, it is completely suppressed [44, 45]. Throughout this study, we maintain the interatomic interaction strength above the threshold to prevent the adverse effects of Josephson oscillations on achieving a stable current.

Initially, the system lacks vortices and circular currents. Upon sudden release into the double-ring trap, atoms emerge in the outer ring region through tunneling across the barrier between the two rings, exhibiting a highly disordered density distribution as depicted in Fig.2(b). As the density distribution of the condensate in the outer ring trap becomes more homogeneous, vortex−antivortex pairs are excited, as shown in Fig.2(c), without altering the total angular momentum. Subsequently, the system reaches a steady state with a few symmetrically distributed vortex pairs within the outer ring, while the density distribution in the inner ring gradually becomes uniform, as illustrated in Fig.2(d). Notably, the quadrupole will maintain its structure, with all vortices remaining fixed in position over an extended period of time. In a 2D disk-shaped condensate, a quadrupole structure is generally unstable [46]. The generation of topological excitations is akin to that observed during the interference and merging of BECs in a double well potential [47, 48].

3.2 2DQT induced persistent current in a double-ring geometry

For BECs initially in a double concentric ring trap without quench, the ground state leads to a uniform density distribution in both the inner and outer rings. N_v vortices of the same charge (s=-1) are seeded in the inner ring region through phase imprinting and imaginary time evolution. The vortices are initialized with random positions, as depicted in Fig.3(a). As stated in Fig.1(b), the initial condition meets the criterion for 2DQT. Due to the narrow width of the ring-shaped condensate, these vortices remain close to the boundary (barrier potential separating the two rings). During the time evolution, ghost vortices with opposite charges emerge within the interface between the two parts of the condensate, identifiable by green points in Fig.3(b)−(d). Through time evolution, some of the ghost vortices can escape into the condensate cloud as indicated in Fig.3(c). Additionally, the originally seeded vortices encounter difficulty transiting from the inner ring-shaped condensate to the outer one, with comparably fewer vortices excited in the outer ring-shaped condensate when compared to the scenario in Fig.2.

The number of ghost vortices is approximately equal to the number of vortices, which aligns with previous research on 2DQT in an annular quantum fluid with vortices. As illustrated in Fig.3(e), a persistent atom flow spontaneously forms in the inner ring as the vortex count varies (decreasing from the initial number of vortices). A current appears in the outer ring-shaped condensate following the appearance of the current in the inner condensate, ultimately reaching stable values. The modulation of the circular current is correlated with the creation and destruction of vortices within the BECs. However, the annihilation of vortex-antivortex pairs cannot result in an inverse energy cascade [49], thus failing to induce a circular current. An inverse energy cascade, which can transfer energy from a small scale to a large scale, occurs only when incompressible energy dissipates as a result of vortices dissipating at the boundary. Reference [50] documented the decay of atomic current through a barrier in a toroidal BEC as abrupt 2\pi drops, leaving observable steps in the time evolution of winding (circulation) number. The growth or decay of the current is quantized in integer increments, as depicted by the steps in Fig.3(e), representing transitions between distinct quantized persistent current states. Vortices that escape through the boundary of the inner ring are challenging to recover, resulting in the absence of discernible steps in the S curve of the inner ring. As the count of vortices and ghost vortices decreases, the winding number of both rings stabilizes.

To further demonstrate the angular momentum transfer in this scenario, we also analyze the temporal evolution of the angular momentum, L (in units of \hslash), of the inner and outer rings, as depicted in the inset of Fig.3(e). The angular momentum operator \hat{\mathit{L}} is given by

(7)\hat{\mathit{L}}={\displaystyle \int \mathrm{d}\mathit{r}{\hat{\mathbf{\Psi }}}^{†}\mathrm{i}\hslash \left(y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}-x{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}\right)\hat{\mathbf{\Psi }}.}

In the mean field approach, the angular momentum can be calculated by

(8)L=⟨ \psi |\widehat{L_z}|\psi ⟩=\mathrm{i}\hslash {\displaystyle \int \mathrm{d}\mathit{r}{\psi }^{\ast }\left(y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}-x{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}\right)\psi .}

Due to the cylindrical symmetry of the trap, the Hamiltonian \hat{\mathit{H}}

(9)\begin{array}{rl}\hat{\mathit{H}}=& \int \mathrm{d}\mathit{r}{\hat{\mathbf{\Psi }}}^{†}\left(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}\right)\hat{\mathbf{\Psi }}\\ & +\iint \mathrm{d}\mathit{r}\mathrm{d}\mathit{r}{}^{\mathbf{\prime }}{\hat{\mathbf{\Psi }}}^{†}{\hat{\mathbf{\Psi }}}^{{}^{\prime }†}{g}_{2\mathrm{D}}\delta (r-r{}^{\prime })\hat{\mathbf{\Psi }}{}^{\prime }\hat{\mathbf{\Psi }}\end{array}

satisfies the commutation relation [\hat{\mathit{H}}, \hat{\mathit{L}}]= 0. As shown in the inset of Fig.3(e), the total angular momentum is conserved during the evolution and the angular momentum L_{in} of the inner ring-shaped condensate and L_{out} of the outer condensate gradually converge in this scenario. It is important to highlight that when the number of initially randomly seeded vortices is fewer than three, the criterion for 2DQT is not fulfilled. As a result, the persistent current does not emerge in either the inner or the outer ring-shaped condensate.

3.3 2DQT and geometric quench induced persistent current

In this case, the initial state is set as a single ring-shaped condensate with randomly distributed vortices of the same charge, as depicted in Fig.4(a). Subsequently, the trap is deformed into a double concentric ring-shaped geometry, mimicking the process applied to the condensate without the initial vortices shown in Fig.2. The atoms and vortices, initially within the inner ring, traverse the barrier and disperse into the outer ring-shaped region of the trap, as illustrated in Fig.4(b). Vortex pairs continue to form in the outer ring-shaped condensate during the tunneling, while the condensate’s density distribution becomes increasingly uniform, depicted in Fig.4(c) and (d). This behavior mirrors the evolution observed in the initial state lacking vortices in the inner ring-shaped condensate, as shown in Fig.4(c) and (d). Over time, all initially seeded and dynamically excited vortices decay, giving rise to persistent currents in both the inner and outer rings.

A comparison with the scenario devoid of geometric quench, displayed in Fig.3(e), reveals a significantly larger winding number of the current in the outer ring in Fig.4(e), indicating a more efficient transfer of angular momentum from vortices to global atom flows. Notably, the circulation flow in the outer ring-shaped condensate stabilizes much more rapidly than in the inner ring-shaped condensate. This disparity is further evidenced by the evolution of the angular momentum of the inner and outer rings, as depicted in the inset of Fig.4(e). Specifically, the angular momentum in the outer ring exceeds that of the inner ring (L_{out}\sim 1.5L_{in}). Notice that L_{in} and L_{out} gradually converge in the above scenario, displayed in Fig.3(e). The difference of the angular momentum between this case and the above scenario reveals the impact of the geometrically quenching in the dissipating dynamics of quantum turbulence. Reference [16] showed that the circulation induced by a rotating barrier remains stable only at low angular frequencies, while high angular frequencies lead to dynamic instability due to excitation of vortices within the condensate cloud.

3.4 Relations between the winding number and the system parameters

In Fig.5(a), it is demonstrated that the absolute value of the average winding number of the inner and outer ring-shaped condensate increases linearly with the initial number of seeded vortices for the case with 2DQT and geometric quench. The error bars represent the standard deviation of 64 realizations for a given initial vortex distribution. When starting with 30 initial vortices, the winding number of the circulation flow is −23, while that of the inner ring-shaped BEC is only −4. This discrepancy highlights the crucial role of geometric quench in energy transfer during the two-dimensional turbulent cascade within ring-shaped condensates. In contrast to the scenario without a quench in the concentric double ring-shaped condensate, as depicted in Fig.3, the persistent current can be induced regardless of the number of initially seeded vortices after the quench. This phenomenon stems from the quench itself introducing vortices into the system. When combined with the initial vortices, a state of 2DQT can emerge.

Furthermore, the impact of the thickness and height of the middle barrier on the winding number of the circulation flow is explored in Fig.5(b) and (c), respectively, while maintaining a constant number of 20 initial vortices. When d>5a_0 or V_0>5\hslash {\omega }_z, tunneling is significantly suppressed to prevent the occurrence of the circulation flow in the outer ring-shaped trap. The winding number of the outer ring-shaped condensate does not exhibit a monotonic variation concerning both d and V_0; a specific combination of these parameters is required to achieve the maximum persistent current of the circulation flow. It is observed that the peak occurs at the healing length, denoted as \xi, and in proximity to the chemical potential \mu =gn, where n represents the average density. The variations in the winding number with respect to these parameters follow Gaussian distributions.

4 Conclusion

In conclusion, we propose a novel approach with geometric quench and 2DQT protocol to create persistent currents of circular condensate flows with high winding numbers and establish double-concentric coherent persistent currents with varying winding numbers. For a double ring-shaped condensate with vortices of the same charge and randomly distributed in the inner ring, circular condensate flows can form in both the inner and outer ring-shaped condensates due to the 2D turbulent cascade and quantum tunneling mechanisms. The discrepancies in the winding numbers between the two flows are not significant. Conversely, if a condensate with randomly distributed vortices is quenched by transforming the trap from a single ring-shaped geometry to a double ring-shaped one, the outer ring-shaped condensate’s winding number will be significantly larger than that of the inner ring-shaped flow. Nonetheless, if the initial state lacks vortices, a geometric quench alone will not induce persistent atom flows. The persistent currents produced in our setup exhibit nearly uniform density distributions and are devoid of vortex excitations. This innovative approach proves to be more effective in creating atom circulation flows compared to the conventional technique of rotating barriers and can be employed in devising intricate systems based on ultracold atoms.

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