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Physical Review A

Classical and quantum models for phase imprinting

Yi Zheng, Juha Javanainen

Published Mar 17, 2003• 10 citations
Publisher

Abstract

We investigate theoretically the generation of a persistent current in a Bose-Einstein condensate in a toroidal trap by means of phase imprinting. The phase-imprinting potential must be periodic around the torus, and will have parts that accelerate atoms in the direction opposite to the intended flow. We study the evolution of the average momentum and the momentum spread under a two-slope model potential numerically by using the Gross-Pitaevskii equation, and also in a simple classical-mechanics analogy. Because of the inevitable pollution caused by the opposite slopes of the potential, it is not possible to generate a pure persistent current. The same caveat presumably applies to vortices in higher-dimensional systems.

DOI
10.1103/physreva.67.035602

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We investigate theoretically the generation of a persistent current in a Bose-Einstein condensate in a toroidal trap by means of phase imprinting. The phase-imprinting potential must be periodic around the torus, and will have parts that accelerate atoms in the direction opposite to the intended flow. We study the evolution of the average momentum and the momentum spread under a two-slope model potential numerically by using the Gross-Pitaevskii equation, and also in a simple classical-mechanics analogy. Because of the inevitable pollution caused by the opposite slopes of the potential, it is not possible to generate a pure persistent current. The same caveat presumably applies to vortices in higher-dimensional systems.
Persistent current, or "flow without friction", as well as quantum vortices are the hallmarks of superfluidity. Recently a very long-lived persistent flow of atoms has been experimentally observed in Bose-Einstein condensates trapped in a ring-shaped potential. This enables fundamental studies of superfluidity and may lead to applications in high-precision metrology and atomtronics. We overview our recent theoretical studies of the generation of the persistent current in a stirred toroidal atomic Bose-Einstein condensate, and discuss our new investigation of the hysteresis in the atomtronic circuit.
We investigate the role of vortices in the decay of persistent current states of annular atomic superfluids by solving numerically the Gross–Pitaevskii equation, and we directly compare our results with the 6Li experiment at LENS data. We theoretically model the optical phase-imprinting technique employed to experimentally excite finite-circulation states in the Bose–Einstein condensation regime, accounting for imperfections of the optical gradient imprinting profile. By comparing simulations of this realistic protocol to an ideal imprinting, we show that the introduced density excitations arising from imperfect imprinting are mainly responsible for limiting the maximum reachable winding number wmax in the superfluid ring. We also investigate the effect of a point-like obstacle with variable potential height V0 on the decay of circulating supercurrents. For a given obstacle height, a critical circulation wc exists, such that for an initial circulation w0 larger than wc the supercurrent decays through the emission of vortices, which cross the superflow and thus induce phase slippage. Higher values of the obstacle height V0 further favor the entrance of vortices, thus leading to lower values of wc. Furthermore, the stronger vortex-defect interaction at higher V0 leads to vortices that propagate closer to the center of the ring condensate. The combination of both these effects leads to an increase in the supercurrent decay rate for increasing w0, in agreement with experimental observations.
Experimental stirring of a toroidally trapped Bose-Einstein condensate at high temperature generates a disordered array of quantum vortices that decays via thermal dissipation to form a macroscopic persistent current [T. W. Neely et al., Phys. Rev. Lett. 111, 235301 (2013)]. We perform three-dimensional numerical simulations of the experimental sequence within the stochastic projected Gross-Pitaevskii equation using ab initio determined reservoir parameters. We find that both damping and noise are essential for describing the dynamics of the high-temperature Bose field. The theory gives a quantitative account of the formation of a persistent current, with no fitted parameters.
Using a numerical implementation of the truncated Wigner approximation, we simulate the experiment reported by Ramanathan et al. in Phys. Rev. Lett. 106, 130401 (2011), in which a Bose-Einstein condensate is created in a toroidal trap and set into rotation via a phase imprinting technique. A potential barrier is then placed in the trap to study the decay of the superflow. We find that the current decays via thermally activated phase slips, which can also be visualized as vortices crossing the barrier region in the radial direction. Adopting the notion of critical velocity used in the experiment, we determine it to be lower than the local speed of sound at the barrier, in contradiction to the predictions of the zero-temperature Gross-Pitaevskii equation. We map out the superfluid decay rate and critical velocity as a function of temperature and observe a strong dependence. Thermal fluctuations offer a partial explanation of the experimentally observed reduction of the critical velocity from the phonon velocity.
Several aspects of superfluidity in an atomic Bose-Einstein Condensate are studied in a torus-type quasi-1D model. The critical velocity of the superfluid in this quasi-1D model is found to differ from the conventional concept of the limit of velocity for frictionless flow. This result is one of the unique features of the quasi-1D system. For a generalized discrete quasi-1D model, the stability of Bloch oscillations is studied. The quasi-particle spectrum of small oscillations is analyzed, which in turn gives the phase diagram of the stable-unstable region. The results are in good agreement with a recent experimental observation. Last, we consider an experiment proposal where a phase-imprinting technique is used to generate superfluid flow with fixed momentum or a vortex in atomic BEC. Our study shows that because of backflow generated by part of the imprinting potential, phase imprinting is not an ideal tool to generate a single-momentum superflow or single vortex. ^
The dynamics of quantum vortices in a two-dimensional annular condensate held under a toroidal trapping potential composed of a harmonic term plus Gaussian term are considered by numerically simulating the Gross-Pitaevskii equation. Families of solitary wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometry the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previously. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with different velocities. The stability of the solitary wave sequences for the annular condensate without a persistent flow is also investigated by numerically evolving the solitary wave solutions in time. In addition, the interaction of vortex-vortex pairs and vortex-antivortex pairs is considered and it is demonstrated that the collisions are either elastic or inelastic depending on the magnitude of the angular velocity. The similarities and differences between numerically simulating the Gross-Pitaevskii equation and using a point vortex model for these collisions are elucidated.