Oscillations and decay of superfluid currents in a one-dimensional Bose gas on a ring

TL;DR

This paper investigates the complex dynamics of superfluid currents in a one-dimensional Bose gas on a ring, revealing oscillations, decay, and self-trapping behaviors influenced by interactions and temperature, modeled via a dual Josephson framework.

Contribution

It introduces a dual Josephson model to describe supercurrent dynamics in a 1D Bose gas with a defect, highlighting the roles of thermal and quantum phase slips.

Findings

Identification of regimes with oscillating, decaying, or self-trapped currents

Demonstration of phase slips as key mechanisms in current decay

Validation of the dual Josephson model for describing the dynamics

Abstract

We study the time evolution of a supercurrent imprinted on a one-dimensional ring of interacting bosons in the presence of a defect created by a localized barrier. Depending on interaction strength and temperature, we identify various dynamical regimes where the current oscillates, is self-trapped or decays with time. We show that the dynamics are captured by a dual Josephson model and involve phase slips of thermal or quantum nature.

Equations

$$\hat{\mathcal{H}}={\int }_0^{L}dx{\hat{\Psi }}^{†}\left(-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x)+\frac{g}{2}{\hat{\Psi }}^{†}\hat{\Psi }\right)\hat{\Psi },$$

$$J(t)=-i\frac{\hslash }{2m}\frac{1}{N}{\int }_0^L\frac{dx}{L}⟨{\hat{\Psi }}^{†}{\partial }_x\hat{\Psi }-\left({\partial }_x{\hat{\Psi }}^{†}\right)\hat{\Psi }⟩.$$

Full text

Superfluidity is a fascinating phenomenon emerging in interacting quantum systems and governing their low temperature transport properties. Supercurrents, named in analogy with superconductivity, are characterized, among others, by frictionless flow and quantized vortices, and are most easily evidenced in ring geometries. Ultra-cold atoms confined in ring traps have proven to be a great tool to study superfluid transport properties Ramanathan et al. (2011); Moulder et al. (2012); Husmann et al. (2015). Due to their tunability and their high degree of control, they are an ideal system for studying the effect of interactions and dimensionality in the superfluid transport dynamics. As superconducting SQUIDs have provided a wealth of applications, the realization of their atomic analogs – the AQUID Mathey and Mathey (2016) – is an important step in the field of atomtronics Seaman et al. (2007); Amico et al. (2014, 2017); Gauthier et al. (2019).

From a fundamental point of view, an open question is the stability of supercurrents. This is related, but complementary to the study of setting the superfluid in rotation, also related to vortex nucleation Madison et al. (2001); Penckwitt et al. (2002); Lobo et al. (2004). For a three-dimensional (3D) ring geometry the stochastic decay of the quantized current has been studied, evidencing the role of the critical velocity Moulder et al. (2012); Dubessy et al. (2012). In the presence of a repulsive barrier crossing the ring, resulting in a weak link, hysteresis in the phase slips dynamics has been investigated Wright et al. (2013a, b); Eckel et al. (2014a); Yakimenko et al. (2015); Muñoz Mateo et al. (2015) and the role of thermal activation evidenced Kumar et al. (2017). A scenario for the phase slips dynamics induced by a weak link based on the role of vortices can be used to explain qualitatively the experimental observations Piazza et al. (2009) but fails to account quantitatively for the thermal activation Mathey et al. (2014); Kunimi and Danshita (2017). Also in a 3D fermionic double-well Josephson junction phase-slips play a role in the dynamics Burchianti et al. (2018); Xhani et al. (2019).

In this context one question naturally arises: if the phase slips dynamics are driven in 3D by vortices crossing the weak link, what happens in lower dimensions? While in two-dimensional (2D) systems vortices still play a crucial role in the superfluid dynamics Piazza et al. (2009); Mathey and Mathey (2016), they cannot exist in one-dimension (1D). Therefore the phase slips phenomenon should be of a different nature in 1D.

Previous works have shown the role of phase-slips Danshita and Polkovnikov (2012); D’Errico et al. (2017) in the decay of 1D transport in the presence of periodic perturbation Tanzi et al. (2016). For a microscopic impurity the decay rate has been estimated by computing the drag force Cherny et al. (2009). For sufficiently small obstacles stationary circulating states may exist Carr et al. (2000); Cominotti et al. (2014); Shamriz and Malomed (2018), while a forced flow past a larger obstacle results in soliton emission Hakim (1997); Freire et al. (1997); Katsimiga et al. (2018). Most of the previous studies were performed in a rotating frame, thus imposing a flow onto the ring, allowing to estimate the nucleation rate of phase-slips Khlebnikov (2005). For intermediate to strong interactions and small barriers it has been shown that the decay of persistent currents is related to the low-energy excitations in the ring Polo et al. (2018).

In this work, we investigate how a free current flows in 1D: as illustrated in Fig. 1, starting from a system initially prepared in a well-defined current state in a ring trap with a barrier, we follow the current dynamics with the aim of elucidating the dissipation mechanisms. Our study concerns both zero- and finite temperature gases, both at weak and strong interactions. We show that the dynamical behavior can be interpreted as a dual of the Josephson effect, occurring among angular momentum states. Depending on the barrier strength and the temperature regime we observe current oscillations, self-trapping or decay. In the weakly interacting regime, we show that the observed dynamics correspond to self-trapping among angular momentum states at zero temperature, and that the decay of the currents at finite temperature involves dark solitons. For strong interactions, we show that coherent quantum phase slips dominate the current dynamics at zero temperature, and incoherent ones take over at finite temperature.

Model

We consider $N$ bosons of mass $m$ with repulsive contact interactions on a ring of circumference $L$ with periodic boundary conditions, i.e. the Lieb-Liniger model, generalized to include the presence of an external barrier potential $V(x)$. The Hamiltonian reads:

[TABLE]

where $\hat{\Psi}$ is the bosonic field operator, $n=N/L$ the average density, with total number of particles $N=\int_{0}^{L}dx\,\braket{\hat{\Psi}^{\dagger}\hat{\Psi}}$. This model describes e.g. ultra-cold atoms confined in a tight ring trap. In this case $g=2\hbar\omega_{\perp}a_{s}$ is the 1D interaction strength, $\omega_{\perp}$ the radial confinement frequency and $a_{s}$ the 3D $s$-wave scattering length. In the following we consider either a delta potential $V(x)=\alpha\delta(x)$, for which analytical results can be obtained, or a Gaussian potential $V(x)=V_{0}\exp{\left(-\frac{x^{2}}{2\sigma^{2}}\right)}$, realistic from the experimental point of view. For homogeneous 1D gases the equilibrium properties at finite temperature are captured by two dimensionless parameters Kheruntsyan et al. (2003): $\gamma=\frac{mg}{\hbar^{2}n}$ quantifying the interaction regime from weak ($\gamma\ll 1$) to strong ($\gamma\gg 1$), and the reduced temperature $\tau=\frac{T}{T_{d}\gamma^{2}}$, where $T_{d}=\hbar^{2}n^{2}/2mk_{B}$ is the quantum degeneracy temperature.

Quench protocol

Our goal is to study the dynamics of the particle current in the presence of a barrier. We first prepare the system in an equilibrium state $\Psi_{0}$ in the presence of the static barrier potential. This results in a state with no current. Specific details on the implementation depend on the interaction regime and are given later. We then quench the current by phase imprinting a specific circulation onto the many-body wavefunction: $\Psi_{0}(x_{1},...x_{N})\to\Psi_{1}(x_{1},...x_{N})=\Psi_{0}\times e^{i2\pi\ell\sum_{j}x_{j}/L}$. Note that this process can be implemented in experiments using specific light potentials according to various available schemes Moulder et al. (2012); Kumar et al. (2018). We then monitor the current by computing the average of the current operator per particle:

[TABLE]

The time evolution following the quench is described by different approaches depending on the interaction and temperature regimes: (i) at $T=0$ and for a weakly interacting gas ($\gamma\ll 1$) we rely on the Gross-Pitaevskii equation (GPE) numerical solution and on an analytical two-mode model adapted from Smerzi et al. (1997); (ii) at $T>0$ and $\gamma\ll 1$ we use the Projected Gross-Pitaevskii equation (PGPE) formalism Davis et al. (2001); Blakie et al. (2008); Berloff et al. (2014) and (iii) at $\gamma\gg 1$ we use an exact time-dependent Bose-Fermi mapping describing the infinitely strong interaction Tonks-Girardeau (TG) limit for the whole temperature range Girardeau (1960); Girardeau and Wright (2000); Yukalov and Girardeau (2005), focusing on a quench with circulation $\ell=1$ 111The calculation for $\ell=2$ yields no qualitative difference in the current dynamics, just a faster decay of the oscillations..

In the weakly interacting limit we scale the Gaussian barrier strength relative to the chemical potential, i.e. we define $\lambda_{\rm GP}=V_{0}/\mu_{0}$ with $\mu_{0}=gn$ the chemical potential of the homogeneous annular gas. Figure 2 illustrates our simulation results in the weakly interacting regime as a function of $\lambda_{\rm GP}$ for a relatively narrow barrier of width $\sigma=L/50$, yet larger than the healing length $\xi=\hbar/\sqrt{2mgn}\simeq\sigma/4$. At zero temperature we observe in Fig. 2(a) that the current remains very close to the initial quenched circulating state for weak to moderate barriers, up to $\lambda_{\rm GP}\sim 1$. Above this critical value, we observe a fast decay of the current, followed by oscillations around the 0 value. This is very similar to what has been obtained in 2D simulations Mathey et al. (2014). The new feature of the 1D mean-field regime is the emergence of current oscillations at large barriers. As we discuss here below, this behavior can be interpreted as the transition from self-trapping to Josephson oscillations of the currents, in analogy to the well known Josephson effect for particle imbalance predicted in Smerzi et al. (1997) and experimentally observed using ultra-cold atoms confined in a double well trap Albiez et al. (2005). In essence sup , we derive a fully analytical two-mode model for two current states and show that this accurately captures the Gross-Pitaevskii dynamics at zero temperature and very weak interactions (see Fig. 2(c)). This model predicts a transition from self-trapping to Josephson oscillations for a critical value $\lambda_{\rm GP}^{c}$ that depends on the interaction strength as in Smerzi et al. (1997). Interestingly, a two-mode model based on current states in the linear regime also accurately describes the dynamics of vortex nucleation in stirred condensates Caradoc-Davies et al. (1999). Although the two-mode model breaks down for large barrier or higher (but still weak) interactions due to the spread of the mean-field wavefunction onto many single particle orbitals, we observe the same qualitative behavior in the simulations. Indeed, surprisingly, the current always oscillates regularly at large barriers (bottom curve of Fig. 2(a)), with a non-sinusoidal (piecewise linear) shape and very small damping rate. These oscillations can be understood by casting the GPE into the superfluid hydrodynamic form: transport of matter occurs via a density fluctuation corresponding to a shock wave Hakim (1997), propagating at the speed of sound on top of a moving fluid.

For temperature $T=\mu_{0}/k_{B}$, corresponding to the quasi-condensate regime Kheruntsyan et al. (2003), the dynamics of the current are quite different from the zero-temperature case, see Fig. 2(b). At low barriers, i.e. $\lambda_{\rm GP}\leq 0.5$, we observe an exponential decay of the current with a decay rate increasing with the barrier strength. For larger barriers we observe damped oscillations of the current. In this regime thermal phase slips occur deterministically at the position of the barrier, where the density vanishes. The transition from exponential to damped oscillation decay is observed for all our temperatures in the range $0.5\leq k_{B}T/\mu_{0}\leq 2.5$. Fig. 2(d) displays the value of the damping rate $\Gamma$ given by the fit Fit for increasing temperatures, in the range $0.5\leq k_{B}T/\mu_{0}\leq 1.75$. The damping rate increases with temperature, displaying a non-monotonous dependence on the barrier strength, with a maximum at the crossover between the two decay regimes. The crossover occurs at lower barrier strength for larger temperatures, consistent with the thermal activation of solitons, as we discuss below.

In order to elucidate the mechanisms for the current decay, Fig. 2(e) shows a single classical field trajectory, showing many spontaneous thermal gray solitons Karpiuk et al. (2012). While most of the solitons present a small density dip, hence are fast and are transmitted through the barrier Bilas and Pavloff (2005), we notice that the current undergoes discrete jumps each time a soliton is reflected on the barrier: in this case, when the soliton reaches zero velocity the density profile vanishes, allowing for a phase slip to occur. This corresponds to the adiabatic process indicated by the dashed blue line on Fig. 1(b). As the temperature increases, the probability to find slow solitons increases and the jumps occur more and more frequently, resulting in an increase of the decay rate, as seen in Fig. 2(d). Finally, as the barrier couples the soliton dynamics to the long wavelength sound excitations Bilas and Pavloff (2005) we expect this process to be intrinsically stochastic, thus resulting in an exponential decay of the average current as observed.

The description of current dynamics as dual of the Josephson effect persists at strong interactions. In this regime, the classical picture does not apply, rather, we show below that the dynamics correspond to quantum coherent oscillations among angular momentum states (see Astafiev et al. (2012) for the analog phenomenon in superconductors). We describe the dynamics of the current in the strongly interacting limit $\gamma\gg 1$ using the exact Tonks-Girardeau solution, which maps the interacting bosons onto a Fermi gas. In the TG regime the relevant dimensionless barrier strength is $\lambda_{\rm TG}=V_{b}/E_{F}$, with $V_{b}=\alpha n$ being the barrier associated energy and $E_{F}=\hbar^{2}n^{2}\pi^{2}/2m$ being the Fermi energy, corresponding to the zero-temperature chemical potential for systems displaying fermionization sup . At zero temperature, Fig. 3(a), we note that for weak barriers, $\lambda_{\rm TG}\ll 1$, in contrast to the weakly interacting regime, there is no self-trapping, rather, the current undergoes Rabi-like oscillations. These oscillations correspond to coherent quantum phase slips due to backscattering induced by the barrier, which breaks rotation symmetry thus coupling different angular momentum states Mooij and Nazarov (2006); Cominotti et al. (2014). Microscopically, it corresponds to dynamical processes involving the whole Fermi sphere, i.e. multiple-particle hole excitations where each particle coherently undergoes oscillations of angular momentum from $L_{z}=\hbar$ to $L_{z}=-\hbar$. At increasing barrier strength, an envelope appears on top of the current oscillations, degrading the Rabi oscillations. This envelope originates from the population of higher-energy modes, each transition being characterized by a different frequency (see Fig. 3(e) and sup ), leading to a mode-mode coupling and dephasing, and correspondingly more complex current oscillations.

At finite temperatures the quench dynamics of the current involve high-energy excitations with amplitude weighted by the Fermi distribution sup . The resulting dynamics correspond to an effective damping of the current oscillations with a exponential decay, see Fig. 3(b), corresponding to the effect of incoherent phase slips. The revivals observed for large barrier at zero temperature are highly suppressed due to the thermal excitations. In Fig. 3(d) we show the decay rate $\Gamma$ of the persistent currents as a function of the barrier strength Fit . We find that the decay of persistent currents grows monotonically with the barrier strength, since more and more excitations are involved in the dynamics as the barrier strength increases. In Fig. 3(c) we show the oscillation frequency as a function of $\lambda_{\rm TG}$ and observe that at increasing barrier strength the frequency crosses over from a Rabi-like regime with $\omega=\pi^{2}N\lambda_{\rm TG}$ to a Josephson-like regime with $\omega\propto\sqrt{\lambda_{\rm TG}}$, in agreement with the predictions of the low-energy Luttinger liquid theory Polo et al. (2018). Quite generally, while our results have been derived for infinite interaction strength, the predictions of the TG model, including quantum fluctuations in an exact way, are expected to closely describe a Bose gas at strong interactions.

In conclusion, we have shown that the dynamical evolution following a phase imprinting induces oscillations of the current in a 1D ring, associated to a rich excitation pattern, which can be described by a dual Josephson dynamics. At weak interactions and finite temperature we observe the formation of both sound waves and of thermally activated dark solitons. We find that phase-slippage occurs incoherently when the solitons are reflected by the barrier. In the strongly interacting regime at zero temperature we find coherent Rabi oscillations indicating quantum coherent phase slips, which are degraded by mode dephasing at large barrier strength or by thermal fluctuations at finite temperature. In the weakly-interacting limit we find self-trapping of current states, while no self-trapping is found at infinitely strong interactions, where quantum fluctuations dominate.

The dual Josephson picture is a new paradigm for dynamics of atomtronics circuits in which a current state encodes quantum information. Our work evidences the importance of the dynamics of the current in a 1D system, which can be accurately measured using existing experimental tools: an interferometric measurement accessing the local currents Corman et al. (2014); Eckel et al. (2014b) or long wavelength excitations Marti et al. (2015); Kumar et al. (2016). The stochastic decay of the current in 1D via phase slips is reminiscent of the stochastic decay due to vortex/anti-vortex recombination in 2D or 3D systems Piazza et al. (2009), where, however, oscillations are strongly damped by vortex creation Mathey et al. (2014). The main difference between 1D and the higher-dimensional counterparts is that in the former case the current dynamics are more robust: at weak interactions, the solitons properties are gradually degraded by the several interactions with the barrier, mainly by sound wave radiation Bilas and Pavloff (2005), and at strong interactions we observe the coherent dynamics of all particles. In outlook, it would be very interesting to investigate how the self-trapping disappears for large but finite interactions as well as to study the crossover to a quasi-1D geometry to explore the role of radial modes in the decay dynamics.

Acknowledgements.

We thank Maxim Olshanii and Jook Walraven for stimulating discussions. We acknowledge financial support from the ANR project SuperRing (Grant No. ANR-15-CE30-0012). LPL is a member DIM SIRTEQ (Science et Ingénierie en Région Île-de-France pour les Technologies Quantiques).