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arXiv:2001.04033 · Physical Review A

Unlocked-relative-phase states in arrays of Bose-Einstein condensates

Dengling Zhang, Haibo Qiu, Antonio Muñoz Mateo

Published Jun 16, 2020• 1 citation
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Abstract

Phase engineering techniques are used to control the dynamics of long-bosonic Josephson-junction arrays built by linearly coupling Bose-Einstein condensates. Just at the middle point of the underlying discrete energy band of the system, unlocked-relative-phase states are shown to be stationary along with the locked-relative-phase Bloch waves. In finite, experimentally feasible systems, such states find ranges of dynamical stability that depend on the ratio of coupling to interaction energy. The same ratio determines different decay regimes, which include the recurrence of staggered-soliton trains in the condensates around Josephson loop currents at the junctions. These transient solitons are also found in their stationary configurations, which provide striped-density states by means of either dark-soliton or bright-soliton trains. The parametric connection between the latter solitonic states and unlocked-relative-phase states is shown. Additionally, the preparation of maximally out-of-phase (or splay) states is demonstrated to evolve into an oscillation of the uniform density of the condensates that keeps constant the total density of the system and is robust against noise at low coupling.

DOI
10.1103/physreva.101.063623

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Phase engineering techniques are used to control the dynamics of long-bosonic Josephson-junction arrays built by linearly coupling Bose-Einstein condensates. Just at the middle point of the underlying discrete energy band of the system, unlocked-relative-phase states are shown to be stationary along with the locked-relative-phase Bloch waves. In finite, experimentally feasible systems, such states find ranges of dynamical stability that depend on the ratio of coupling to interaction energy. The same ratio determines different decay regimes, which include the recurrence of staggered-soliton trains in the condensates around Josephson loop currents at the junctions. These transient solitons are also found in their stationary configurations, which provide striped-density states by means of either dark-soliton or bright-soliton trains. The parametric connection between the latter solitonic states and unlocked-relative-phase states is shown. Additionally, the preparation of maximally out-of-phase (or splay) states is demonstrated to evolve into an oscillation of the uniform density of the condensates that keeps constant the total density of the system and is robust against noise at low coupling.
Soon after its theoretical prediction, striped-density states in the presence of synthetic spin-orbit coupling were realized in Bose-Einstein condensates of ultracold neutral atoms [J.-R. Li et al., Nature 543, 91 (2017)]. The achievement opens avenues to explore the interplay of superfluidity and crystalline order in the search for supersolid features and materials. The system considered is essentially made of two linearly coupled Bose-Einstein condensates, that is, a pseudo-spin-1/21/21/2 system, subject to a spin-dependent gauge field ${\ensuremath{\sigma}}_{z}\ensuremath{\hbar}{k}_{\ensuremath{\ell}}$. Under these conditions, the stripe phase is achieved when the linear coupling $\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}/2$ is small against the gauge energy $m\mathrm{\ensuremath{\Omega}}/\ensuremath{\hbar}{k}_{\ensuremath{\ell}}^{2}<1$. The resulting density stripes have been interpreted as a standing-wave interference pattern with approximate wave number $2{k}_{\ensuremath{\ell}}$. Here, we show that the emergence of the stripe phase is induced by an array of Josephson vortices living in the junction defined by the linear coupling. As happens in superconducting junctions subject to external magnetic fields, a vortex array is the natural response of the superfluid system to the presence of a gauge field. Also similar to superconductors, the Josephson currents and their associated vortices can be present as a metastable state in the absence of gauge field. We provide closed-form solutions to the 1D mean-field equations that account for such vortex arrays. The underlying Josephson currents coincide with the analytical solutions to the sine-Gordon equation for the relative phase of superconducting junctions [C. Owen and D. Scalapino, Phys. Rev. 164, 538 (1967)].
We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped targetlike trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [R. Mathew, A. Kumar, S. Eckel, F. Jendrzejewski, G. K. Campbell, M. Edwards, and E. Tiesinga, Phys. Rev. A 92, 033602 (2015)]. In addition to the ground-state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability and, when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.
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Collisional dynamics of solitary matter waves of superfluid hard core bosons, consisting of dark and bright non-linear waves as well as supersonic periodic trains, reveals remarkable richness and coherence, with the phase of the condensate playing a key role. Depending upon the condensate density and their relative velocity, two distinctive collisional types emerge: the intuitively expected repulsive collisions due to the hard core boson constraint; and, also collisions in which they "pass through" one another. In addition to confirming the soliton status of both bright and dark solitary waves, our studies reveal a variety of multi-solitons including multiple families of breathers, that can be produced and precisely controlled via quantum phase engineering.
The stacks of Bose–Einstein condensates coupled by long Josephson junctions present a rich phenomenology feasible to experimental realization and specially suitable for technological applications as the nonlinear-optics and superconducting analogs have already proved. Among this, we show that transverse Bloch waves excited in arrays of one-dimensional coupled condensates can carry tunneling superflows whose dynamical stability depends on the quasimomentum. Across the stacks with periodic boundary conditions, forming closed ring-shaped systems, such Bloch states yield transverse Josephson vortices with a generic non-integer circulation in units of h / m . Additionally, the superpositions of degenerate linear Bloch waves can suppress the supercurrents and give rise to families of nonlinear standing-wave states with strong (transverse) spatial localization. Stable states of this type can also be found in finite size systems.