Toroidal Bose-Einstein Condensate Research Articles

Formation of local and global currents in a toroidal Bose-Einstein condensate via an inhomogeneous artificial gauge field

We study the effects of a position-dependent artificial gauge field on an atomic Bose-Einstein condensate in quasi-one-dimensional and two-dimensional ring settings. The inhomogeneous artificial gauge field can induce global and local currents in the Bose-Einstein condensate via phase gradients along the ring and vortices, respectively. We observe two different regimes in the system depending on the radial size of the ring and the strength of the gauge field. For weak artificial gauge fields, the angular momentum increases, as expected, in a quantized manner; however, for stronger values of the fields, the angular momentum exhibits a linear (nonquantized) behavior. We also characterize the angular momentum for noncylindrically symmetric traps.

Inflationary dynamics and particle production in a toroidal Bose-Einstein condensate

We present a theoretical study of the dynamics of a Bose-Einstein condensate (BEC) trapped inside an expanding toroid that can realize an analogue inflationary universe. As the system expands, we find that phonons in the BEC undergo redshift and damping due to quantum pressure effects. We predict that rapidly expanding toroidal BEC's can exhibit spontaneous particle creation, and study this phenomenon in the context of an initial coherent state wavefunction. We show how particle creation would be revealed in the atom density and density correlations, and discuss connections to the cosmological theory of inflation.

Symmetry Breaking in Interacting Ring-Shaped Superflows of Bose–Einstein Condensates

We demonstrate that the evolution of superflows in interacting persistent currents of ultracold gases is strongly affected by symmetry breaking of the quantum vortex dynamics. We study counter-propagating superflows in a system of two parallel rings in regimes of weak (a Josephson junction with tunneling through the barrier) and strong (rings merging across a reduced barrier) interactions. For the weakly interacting toroidal Bose–Einstein condensates, formation of rotational fluxons (Josephson vortices) is associated with spontaneous breaking of the rotational symmetry of the tunneling superflows. The influence of a controllable symmetry breaking on the final state of the merging counter-propagating superflows is investigated in the framework of a weakly dissipative mean-field model. It is demonstrated that the population imbalance between the merging flows and the breaking of the underlying rotational symmetry can drive the double-ring system to final states with different angular momenta.

Collisional dynamics of multiple dark solitons in a toroidal Bose–Einstein condensate: quasiparticle picture

We study the collisional dynamics of multiple dark solitons in a Bose-Einstein condensate confined by a toroidal trap. We assume a tight enough confinement in the radial direction to prevent possible dissipative effects due to the presence of solitonic vortices. Analytical expressions for the initial order parameters with imprinted phases are utilized to generate different initial arrays of solitons, for which the time-dependent Gross-Pitaevskii equation is numerically solved. Given that the soliton velocity is conserved due to the lack of dissipation, we are able to apply a simple quasiparticle description of the soliton dynamics. In fact, the trajectory equations are written in terms of the velocities and the angular shifts produced at each collision, in analogy to the infinite one-dimensional system. To calculate the angular shifts, we directly extract them from the trajectories given by the Gross-Pitaevskii simulations and, on the other hand, we show that accurate values can be analytically obtained by adapting a formula valid for the infinite one-dimensional system that involves the healing length, which in our inhomogeneous system must be evaluated in terms of the sound velocity along the azimuthal direction. We further show that very good estimates of such a sound velocity can be directly determined by using the ground state density profile and the values of the imprinted phases. We discuss the possible implementation of the system here proposed using the current experimental techniques.

Temperature-induced decay of persistent currents in a superfluid ultracold gas

We study how temperature affects the lifetime of a quantized, persistent current state in a toroidal Bose-Einstein condensate (BEC). When the temperature is increased, we find a decrease in the persistent current lifetime. Comparing our measured decay rates to simple models of thermal activation and quantum tunneling, we do not find agreement. We also measured the size of hysteresis loops size in our superfluid ring as a function of temperature, enabling us to extract the critical velocity. The measured critical velocity is found to depend strongly on temperature, approaching the zero temperature mean-field solution as the temperature is decreased. This indicates that an appropriate definition of critical velocity must incorporate the role of thermal fluctuations, something not explicitly contained in traditional theories.

Stochastic phase slips in toroidal Bose-Einstein condensates

Motivated by recent experiments we study the influence of thermal noise on the phase slips in toroidal Bose-Einstein condensates with a rotating weak link. We derive a generalized Arrhenius-like expression for the rate of stochastic phase slips. We develop a method to estimate the energy barrier separating different superflow states. The parameters at which the energy barrier disappears agree with the critical parameters for deterministic phase slips obtained from dynamics simulations, which confirms the validity of our energetic analysis. We reveal that adding thermal noise lowers the phase-slip threshold. However, the quantitative impact of the stochastic phase slips turns out to be too small to explain the significant discrepancy between theoretical and the experimental results.

Realizing and optimizing an atomtronic SQUID

We demonstrate how a toroidal Bose–Einstein condensate with a movable barrier can be used to realize an atomtronic SQUID. The magnitude of the barrier height, which creates the analogue of an SNS junction, is of crucial importance, as well as its ramp-up and -down protocol. For too low of a barrier, the relaxation of the system is dynamically suppressed, due to the small rate of phase slips at the barrier. For a higher barrier, the phase coherence across the barrier is suppressed due to thermal fluctuations, which are included in our Truncated Wigner approach. Furthermore, we show that the ramp-up protocol of the barrier can be improved by ramping up its height first, and its velocity after that. This protocol can be further improved by optimizing the ramp-up and ramp-down time scales, which is of direct practical relevance for on-going experimental realizations.

Self-heterodyne detection of thein situphase of an atomic superconducting quantum interference device

We present theoretical and experimental analysis of an interferometric measurement of the {\it in-situ} phase drop across and current flow through a rotating barrier in a toroidal Bose-Einstein condensate (BEC). This experiment is the atomic analog of the rf-superconducting quantum interference device (SQUID). The phase drop is extracted from a spiral-shaped density profile created by the spatial interference of the expanding toroidal BEC and a reference BEC after release from all trapping potentials. We characterize the interferometer when it contains a single particle, which is initially in a coherent superposition of a torus and reference state, as well as when it contains a many-body state in the mean-field approximation. The single-particle picture is sufficient to explain the origin of the spirals, to relate the phase-drop across the barrier to the geometry of a spiral, and to bound the expansion times for which the {\it in-situ} phase can be accurately determined. Mean-field estimates and numerical simulations show that the inter-atomic interactions shorten the expansion time scales compared to the single-particle case. Finally, we compare the mean-field simulations with our experimental data and confirm that the interferometer indeed accurately measures the {\it in-situ} phase drop.

Persistent currents supported by solitary waves in toroidal Bose-Einstein condensates

We analyze the generation of persistent currents in Bose-Einstein condensates of ultracold gases confined in a ring. This phenomenon has been recently investigated in an experiment [Nature \textbf{506}, 200 (2014)], where hysteresis loops have been observed in the activation of quantized persistent currents by rotating weak links. In this work, we demonstrate the existence of 3D stationary currents with non-quantized angular momentum. They are generated by families of solitary waves that show a continuous variation in the angular momentum, and provide a bridge between different winding numbers. We show that the size of hysteresis loops is determined by the range of existence within the weak link region of solitary waves which configure the energy barrier preventing phase slips. The barrier vanishes when the critical rotation leads winding numbers and solitonic states to a matching configuration. At this point, Landau and Feynman criteria for phase slips meet: the fluid flow reaches the local speed of sound, and stationary vortex lines (which are the building blocks of solitons) can be excited inside the system.

Spin squeezing by tensor twisting and Lipkin-Meshkov-Glick dynamics in a toroidal Bose-Einstein condensate with spatially modulated nonlinearity

We propose a scheme for spin-squeezing in the orbital motion of a Bose-Einstein condensate (BEC) in a toroidal trap. A circular lattice couples two counter-rotating modes and squeezing is generated by the nonlinear interaction spatially modulated at half the lattice period. By varying the amplitude and phase of the modulation, various cases of the twisting tensor can be directly realized, leading to different squeezing regimes. These include one-axis twisting and the two-axis counter-twisting which are often discussed as the most important paradigms for spin squeezing. Our scheme naturally realizes the Lipkin-Meshkov-Glick model with the freedom to vary all its parameters simultaneously.

Vortices in a toroidal Bose-Einstein condensate with a rotating weak link

Motivated by a recent experiment [K. C. Wright et al., Phys. Rev. Lett. 110, 025302 (2013)], we investigate deterministic discontinuous jumps between quantized circulation states in a toroidally trapped Bose-Einstein condensate. These phase slips are induced by vortex excitations created by a rotating weak link. We analyze the influence of a localized condensate density depletion and atomic superflows, governed by the rotating barrier, on the energetic and dynamical stability of the vortices in the ring-shaped condensate. We simulate in a three-dimensional dissipative mean-field model the dynamics of the condensate using parameters similar to the experimental conditions. Moreover, we consider the dynamics of the stirred condensate far beyond the experimentally explored region and reveal surprising manifestations of complex vortex dynamics.

Collective excitation interferometry with a toroidal Bose-Einstein condensate

The precision of compact inertial sensing schemes using trapped- and guided-atom interferometers has been limited by uncontrolled phase errors caused by trapping potentials and interactions. Here, we propose an acoustic interferometer that uses sound waves in a toroidal Bose-Einstein condensate to measure rotation, and we demonstrate experimentally several key aspects of this type of interferometer. We use spatially patterned light beams to excite counter-propagating sound waves within the condensate and use \emph{in situ} absorption imaging to characterize their evolution. We present an analysis technique by which we extract separately the oscillation frequencies of the standing-wave acoustic modes, the frequency splitting caused by static imperfections in the trapping potential, and the characteristic precession of the standing-wave pattern due to rotation. Supported by analytic and numerical calculations, we interpret the noise in our measurements, which is dominated by atom shot noise, in terms of rotation noise. While the noise of our acoustic interferometric sensor, at the level of $\sim \mbox{rad}\, \mbox{s}^{-1}/\sqrt{\mbox{Hz}}$, is high owing to rapid acoustic damping and the small radius of the trap, the proof-of-concept device does operate at $10^4 - 10^6$ times higher density and in a volume $10^9$ times smaller than free-falling atom interferometers.

Self-interference of a toroidal Bose–Einstein condensate

We demonstrate the self-interference of a single Bose–Einstein condensate on a non-simply connected geometry, focussing on a toroidally trapped ring-shaped condensate. First, we show how the opposite parts of the ring can interfere using the Wigner function representation. Then, using analytical expressions for the time-evolution of a freely expanding ring-shaped condensate with and without a persistent current, we show that the self-interference of the ring-shaped condensate is possible only in the absence of the persistent current. We conclude by proposing an experimental protocol for the creation of ring dark solitons using the toroidal self-interference.

Creation of matter wave Bessel beams and observation of quantized circulation in a Bose–Einstein condensate

Bessel beams are plane waves with amplitude profiles described by Bessel functions. They are important because they propagate ‘diffraction-free’ and because they can carry orbital angular momentum. Here we report the creation of a Bessel beam of de Broglie matter waves. The Bessel beam is produced by the free evolution of a thin toroidal atomic Bose–Einstein condensate (BEC) which has been set into rotational motion. By attempting to stir it at different rotation rates, we show that the toroidal BEC can only be made to rotate at discrete, equally spaced frequencies, demonstrating that circulation is quantized in atomic BECs. The method used here can be viewed as a form of wavefunction engineering which might be developed to implement cold atom matter wave holography.

Threshold for creating excitations in a stirred superfluid ring

We have measured the threshold for creating long-lived excitations when a toroidal Bose-Einstein condensate is stirred by a rotating (optical) barrier of variable height. When the barrier height is on the order of or greater than half of the chemical potential, the critical barrier velocity at which we observe a change in the circulation state is much less than the speed for sound to propagate around the ring. In this regime we primarily observe discrete jumps (phase slips) from the non-circulating initial state to a simple, well-defined, persistent current state. For lower barrier heights, the critical barrier velocity at which we observe a change in the circulation state is higher, and approaches the effective sound speed for vanishing barrier height. The response of the condensate in this small-barrier regime is more complex, with vortex cores appearing in the bulk of the condensate. We find that the variation of the excitation threshold with barrier height is in qualitative agreement with the predictions of an effective 1D hydrodynamic model.

Phase diagram of the Bose-Hubbard model on a ring-shaped lattice with tunable weak links

Motivated by recent experiments on toroidal Bose-Einstein condensates in all-optical traps with tunable weak links, we study the one-dimensional Bose-Hubbard model on a ring-shaped lattice with a small region of weak hopping integrals using quantum Monte Carlo simulations. Besides the usual Mott insulating and superfluid phases, we find a phase which is compressible but non superfluid with a local Mott region. This `local Mott' phase extends in a large region of the phase diagram. These results suggest that the insulating and conducting phases can be tuned by a local parameter which may provide a new insight to the design of atomtronic devices.

Critical velocity for a toroidal Bose–Einstein condensate flowing through a barrier

We consider the setup employed in a recent experiment (Ramanathan et al 2011 Phys. Rev. Lett. 106 130401) devoted to the study of the instability of the superfluid flow of a toroidal Bose–Einstein condensate in the presence of a repulsive optical barrier. Using the Gross–Pitaevskii mean-field equation, we observe, consistently with what we found in Piazza et al (2009 Phys. Rev.A 80021601), that the superflow with one unit of angular momentum becomes unstable at a critical strength of the barrier and decays through the mechanism of phase slippage performed by pairs of vortex–antivortex lines annihilating. While this picture qualitatively agrees with the experimental findings, the measured critical barrier height is not very well reproduced by the Gross–Pitaevskii equation, indicating that thermal fluctuations can play an important role (Mathey et al 2012 arXiv:1207.0501). As an alternative explanation of the discrepancy, we consider the effect of the finite resolution of the imaging system. At the critical point, the superfluid velocity in the vicinity of the obstacle is always of the order of the sound speed in that region, vbarr = cl. In particular, in the hydrodynamic regime (not reached in the above experiment), the critical point is determined by applying the Landau criterion inside the barrier region. On the other hand, the Feynman critical velocity vf is much lower than the observed critical velocity. We argue that this is a general feature of the Gross–Pitaevskii equation, where we have vf = ϵ cl with ϵ being a small parameter of the model. Given these observations, the question still remains open about the nature of the superfluid instability.

Superflow in a Toroidal Bose-Einstein Condensate: An Atom Circuit with a Tunable Weak Link

We have created a long-lived (≈40 s) persistent current in a toroidal Bose-Einstein condensate held in an all-optical trap. A repulsive optical barrier across one side of the torus creates a tunable weak link in the condensate circuit, which can affect the current around the loop. Superflow stops abruptly at a barrier strength such that the local flow velocity at the barrier exceeds a critical velocity. The measured critical velocity is consistent with dissipation due to the creation of vortex-antivortex pairs. This system is the first realization of an elementary closed-loop atom circuit.

Time-resolved density correlations as a probe of squeezing in toroidal Bose–Einstein condensates

The author studies the evolution of mean-field and linear quantum fluctuations in a toroidal Bose–Einstein condensate, whose interaction strength is quenched from a finite (repulsive) value to zero. The azimuthal equal-time density–density correlation function is calculated and shows temporal oscillations with twice the (final) excitation frequencies after the transition. These oscillations are a direct consequence of positive and negative frequency mixing during non-adiabatic evolution. The author argues that a time-resolved measurement of the equal-time density correlator might be used to calculate the moduli of the Bogoliubov coefficients and thus the amount of squeezing imposed on a mode, i.e. the number of atoms excited out of the condensate.

Vortex-induced phase-slip dissipation in a toroidal Bose-Einstein condensate flowing through a barrier

We study superfluid dissipation due to phase slips for a BEC flowing through a repulsive barrier inside a torus. The barrier is adiabatically raised across the annulus while the condensate flows with a finite quantized angular momentum. At a critical height, a vortex moves from the inner region and reaches the barrier to eventually circulate around the annulus. At a higher critical height, an anti-vortex also enters into the torus from the outer region. Both vortex and anti-vortex decrease the total angular momentum by leaving behind a $2\pi$ phase slip. When they collide and annihilate or orbit along the same loop, the condensate suffers a global $2\pi$ phase slip, and the total angular momentum decreases by one quantum. In hydrodynamic regime, the instability sets in when the local superfluid velocity equals the sound speed inside the barrier region.