In this work we study the model of flux lattice, specifically a two-leg Bose–Hubbard ladder lattice subject to effective magnetic flux, which can be implemented using spin–orbit coupled cold atoms trapped in an optical lattice. This model has attracted much recent research interest due to that it provides a platform to study the interplay between artificial gauge fields and interactions. We consider that collective dipole oscillations are activated in this model via quenching an external harmonic trapping potential and show that this system can exhibit unique dynamical behaviors in terms that oscillation amplitude and period critically dependent on the spin–orbit coupling. A dynamical slowing down in collective dipole oscillations will take place near the transition point between the Meissner superfuild and vortex superfluid phases, which agrees quantitatively with a variational analysis. We show that such a quantum phase transition can also be probed via the damping of the collective dipole oscillation. When the quenching displacement is large, the collective center-of-mass motion behaves as damped oscillation and the damping reaches its maximum near the phase transition point. The specific dynamical phenomena are explained with eigenspectra and level populations after the quenching. In the presence of interactions, the time-dependent density–matrix renormalization group (tDMRG) method is hired to study the collective dynamics. The case with weak interactions is similar to that without interactions, except that interactions tend to enhance damping. In addition to that, Bragg reflection can be observed when the quenching displacement exceeds a critical value. The case with strong interactions is also studied, in which case the phase transition points will be dramatically changed and the system can even move into the Mott insulator phase, which will greatly affect the corresponding collective dynamics. The phenomena predicted in this work can be readily observed in current available experiments on atom flux lattices.
In this work we study the model of flux lattice, specifically a two-leg Bose–Hubbard ladder lattice subject to effective magnetic flux, which can be implemented using spin–orbit coupled cold atoms trapped in an optical lattice. This model has attracted much recent research interest due to that it provides a platform to study the interplay between artificial gauge fields and interactions. We consider that collective dipole oscillations are activated in this model via quenching an external harmonic trapping potential and show that this system can exhibit unique dynamical behaviors in terms that oscillation amplitude and period critically dependent on the spin–orbit coupling. A dynamical slowing down in collective dipole oscillations will take place near the transition point between the Meissner superfuild and vortex superfluid phases, which agrees quantitatively with a variational analysis. We show that such a quantum phase transition can also be probed via the damping of the collective dipole oscillation. When the quenching displacement is large, the collective center-of-mass motion behaves as damped oscillation and the damping reaches its maximum near the phase transition point. The specific dynamical phenomena are explained with eigenspectra and level populations after the quenching. In the presence of interactions, the time-dependent density–matrix renormalization group (tDMRG) method is hired to study the collective dynamics. The case with weak interactions is similar to that without interactions, except that interactions tend to enhance damping. In addition to that, Bragg reflection can be observed when the quenching displacement exceeds a critical value. The case with strong interactions is also studied, in which case the phase transition points will be dramatically changed and the system can even move into the Mott insulator phase, which will greatly affect the corresponding collective dynamics. The phenomena predicted in this work can be readily observed in current available experiments on atom flux lattices.
The bosonic magnetic ladder, a universal and the simplest model to reproduce the coupled effects of orbital magnetic field and optical lattices, provides an ideal platform for studying spin-orbit coupling, quantum phase transition and chiral topological states of matter. Here, considering both the on-site and rungwise interactions, the ground-state and chiral nonequilibrium dynamics of the bosonic magnetic ladder under an external harmonic trap are studied analytically and numerically. The threshold for the ground-state transition from the zero momentum state to the plane wave state is obtained. We find that the external harmonic potential has an important impact on the phase transition, which strongly depends on the magnetic field and an effective atomic interaction introduced by the interplay of on-site and rungwise interactions. The ground-state phase transitions can be well manipulated by changing the strengths of the external harmonic trap and rungwise interaction. Furthermore, by quenching the harmonic trap, rich nonlinear modes coupling and chiral nonequilibrium dynamics are revealed, which have distinct characteristics in different phases. Strong spin-momentum locking, softening of the collective modes and anharmonic collective dynamics are observed near the phase transition region. Particularly, novel nonlinear chiral dynamics induced by the interplay of strong spin-momentum locking and external trapping near the phase transition region are obtained, including the chiral out-of-phase oscillation of particles in the two legs (i.e., chiral opposite oscillation of particles in the two legs), chiral alternate oscillation of particles between the two legs (i.e., particles primarily populate the right (left) leg when moving away (returning to) the initial position), and chiral focusing-expanding transition of atomic wavepacket on the two legs (i.e., the wave packet exhibits a focused (expanded) state when moving toward (away from) the equilibrium point). Chiral dephasing and phase revival occur near the phase transition region, while coherent harmonic collective dynamics persists far away from the phase transition point. The observed chiralities can be well manipulated by the magnetic field.
Motivated by a recent experiment with optical lattices that has realized a ladder geometry with an effective magnetic field [Atala et al., Nat. Phys. 10, 588 (2014)], we study the dynamics of bosons on a tight-binding two-leg ladder with complex hopping amplitudes. This system displays a quantum phase transition even without interactions. We study the nonequilibrium dynamics without and with interactions, in the presence of a harmonic trapping potential. In particular we consider dynamics induced by quenches of the trapping potential and of the magnitude of the rung hopping. We present a striking ``slowing down'' effect in the collective mode dynamics near the phase transition. This manifestation of a slowing down phenomenon near a quantum phase transition can be visualized unusually directly: The collective mode dynamics can be followed experimentally in real time and real space by imaging the atomic cloud.
We study the stability of superflow of Bose gases in optical lattices by analyzing the Bose-Hubbard model within the Gutzwiller mean-field approximation. We calculate the excitation spectra of the homogeneous Bose-Hubbard model at unit filling to determine the critical momenta for the Landau and dynamical instabilities. These two critical momenta are shown to approach each other when the on-site interaction increases towards the Mott transition point. In order to make a direct connection with realistic experiments, we next take into account a parabolic trapping potential and compute the real-time dynamics of dipole oscillations induced by suddenly displacing the trap center. We consider the following two cases: standard soft-core bosons, whose interparticle interactions include the on-site one only; and hard-core bosons with long-range dipole-dipole interactions. For both cases, we show that the dipole oscillation is significantly damped when the maximum local momentum exceeds a certain threshold, which quantitatively agrees with the critical momentum for the dynamical instability in the homogeneous system. In the case of dipolar hard-core bosons, the dynamical instability of dipole oscillations leads to the formation of checkerboard density waves in the superfluid phase near the boundary to the supersolid phase.
We consider theoretically ultracold interacting bosonic atoms confined to quasi-one-dimensional ladder structures formed by optical lattices and coupled to the field of an optical cavity. The atoms can collect a spatial phase imprint during a cavity-assisted tunneling along a rung via Raman transitions employing a cavity mode and a transverse running wave pump beam. By adiabatic elimination of the cavity field we obtain an effective Hamiltonian for the bosonic atoms, with a self-consistency condition. Using the numerical density-matrix renormalization-group method, we obtain a rich steady-state diagram of self-organized steady states. Transitions between superfluid to Mott-insulating states occur, on top of which we can have Meissner, vortex liquid, and vortex lattice phases. Also a state that explicitly breaks the symmetry between the two legs of the ladder, namely, the biased-ladder phase, is dynamically stabilized. We investigate the influence that a trapping potential has on the stability of the self-organized phases.
Flux ladders constitute the minimal setup enabling a systematic understanding of the rich physics of interacting particles subjected simultaneously to strong magnetic fields and a lattice potential. In this paper, the ground-state phase diagram of a flux-ladder model is mapped out using extensive density-matrix renormalization-group simulations. The emphasis is put on parameters which can be experimentally realized exploiting the internal states of potassium atoms as a synthetic dimension. The focus is on accessible observables such as the chiral current and the leg-population imbalance. Considering a particle filling of one boson per rung, we report the existence of a Mott-insulating Meissner phase as well as biased-ladder phases on top of superfluids and Mott insulators. Furthermore, we demonstrate that quantum quenches from suitably chosen initial states can be used to probe the equilibrium properties in the transient dynamics. Concretely, we consider the instantaneous turning on of hopping matrix elements along the rungs or legs in the synthetic flux-ladder model, with different initial particle distributions. We show that clear signatures of the biased-ladder phase can be observed in the transient dynamics. Moreover, the behavior of the chiral current in the transient dynamics is discussed. The results presented in this paper provide guidelines for future implementations of flux ladders in experimental setups exploiting a synthetic dimension.
We investigate the localization of a Bose-Einstein condensate trapped in a two-leg ladder in the presence of an artificial magnetic field via Bose-Hubbard model and discuss its abundant localized phenomena using variational approximation and direct numerical simulation. A two-leg bosonic ladder provides an ideal platform to study the complex dynamics of Bose-Einstein condensates in an optical lattice subject to an artificial magnetic field. As a main result, the system displays rich interesting localized states including self-trapping, soliton and breather. The coupling of the repulsive atomic interaction, artificial magnetic field, the rung-to-leg coupling ratio of the ladder, and the initial population difference of atoms between the two legs of the ladder results in the transitions between the localized states. The critical conditions for the transitions between the localized states are given analytically and are intuitively demonstrated in the phase diagram. Particularly, when the rung-to-leg coupling radio of the ladder satisfies a critical condition, we have the localized states with no local atomic currents between the two legs, otherwise, we observe the localized states with local atomic currents between the two legs. Finally, the results obtained via variational method are confirmed by the numerical simulations of the full discrete nonlinear Schrödinger equation describing the system.