Time-averaged trapping potentials have played an important role in the development of the field of ultracold atoms. Despite their widespread application, there is not yet a complete understanding of when a system can be considered time-averaged. Here we use Floquet theory to analyze the lowest energy state of time-periodic trapping potentials, and characterize the transition from a localized state in a slowly moving trap to a delocalized state in a rapidly oscillating time-averaged potential. We investigate how the driving parameters affect the density and phase of the Floquet ground state, and provide a quantitative measure of the degree to which they can be considered time-averaged. We study a number of simple representative systems, and comment on the features affecting the experimental realization of time-averaged trapping potentials.
Time-averaged trapping potentials have played an important role in the development of the field of ultracold atoms. Despite their widespread application, there is not yet a complete understanding of when a system can be considered time-averaged. Here we use Floquet theory to analyze the lowest energy state of time-periodic trapping potentials, and characterize the transition from a localized state in a slowly moving trap to a delocalized state in a rapidly oscillating time-averaged potential. We investigate how the driving parameters affect the density and phase of the Floquet ground state, and provide a quantitative measure of the degree to which they can be considered time-averaged. We study a number of simple representative systems, and comment on the features affecting the experimental realization of time-averaged trapping potentials.
Bose-Einstein condensation (BEC) and superfluidity are both remarkable manifestations of quantum mechanics on a macroscopic scale and have opened up the exploration of quantum phenomena in a qualitatively novel regime. The effort to realise BEC in dilute atomic gases, culminating in 1995, brought with it a powerful new set of ways to control and manipulate atoms and opened up a host of new scientific goals including quantum control, precision measurement, the study of phase transitions and the study of superfluidity – a peculiar phase of matter which is characterised by the absence of dissipation. In the 1940s, the Soviet physicist Lev Landau provided a qualitative theory for these dissipationless “super”fluids by providing an explanation of the critical velocity, below which the formation of fluid excitations is forbidden by the conservation of energy and momentum. This theory has stood the test of time and still forms the basis of our understanding today. Building on the work of Landau, we investigate the interplay between BEC and superfluidity using a novel Floquet analysis, which forms a natural way to characterise time-periodic systems. As a starting point to achieve this goal, we apply our Floquet analysis to another area of interest – the engineering of trapping potentials via the technique of time-averaging. We use Floquet theory to analyse the lowest energy state of time-periodic trapping potentials and characterise the transition from a localised state in a slowly moving trap to a delocalised trap in a rapidly oscillating time-averaged potential. We investigate how the driving parameters affect the density and phase of the Floquet ground state, and provide a quantitative measure of the degree to which they can be considered time-averaged. We investigate the conditions under which resonances occur in Floquet systems which can inform the implementation of experimental time-averaged traps since such resonances can lead to experiment-destroying, uncontrollable heating. We then investigate the breakdown of superfluidity in systems with moving obstacles using nonlinear conjugate gradient optimisation. We first perform a stability analysis, finding that superfluid Floquet states likely become unstable at the critical velocity and thus will not converge with optimisation techniques. We are unable to obtain convergence in a large region of parameter space and find that our optimisers converge best when the driving frequency is a multiple of the Bogoliubov frequency. We find that we are able to characterise the breakdown of superfluidity with two complementary measures of optimality, providing an operational definition of the critical velocity and transcritical regime which can replace the drag force in situations when it cannot be calculated.
Experiments with ultracold atoms in optical lattices usually involve a weak parabolic trapping potential which merely serves to confine the atoms, but otherwise remains negligible. In contrast, we suggest a different class of experiments in which the presence of a stronger trap is an essential part of the set-up. Because the trap-modified on-site energies exhibit a slowly varying level spacing, similar to that of an anharmonic oscillator, an additional time-periodic trap modulation with judiciously chosen parameters creates nonlinear resonances which enable efficient Floquet engineering. We employ a Mathieu approximation for constructing the near-resonant Floquet states in an accurate manner and demonstrate the emergence of effective ground states from the resonant trap eigenstates. Moreover, we show that the population of the Floquet states is strongly affected by the phase of a sudden turn-on of the trap modulation, which leads to significantly modified and rich dynamics. As a guideline for further studies, we argue that the deliberate population of only the resonance-induced effective ground states will allow one to realize Floquet condensates which follow classical periodic orbits, thus providing challenging future perspectives for the investigation of the quantum-classical correspondence.
Abstract Experiments with ultracold atoms in optical lattices usually involve a weak parabolic trapping potential which merely serves to confine the atoms, but otherwise remains negligible. In contrast, we suggest a different class of experiments in which the presence of a stronger trap is an essential part of the set-up. Because the trap-modified on-site energies exhibit a slowly varying level spacing, similar to that of an anharmonic oscillator, an additional time-periodic trap modulation with judiciously chosen parameters creates nonlinear resonances which enable efficient Floquet engineering. We employ a Mathieu approximation for constructing the near-resonant Floquet states in an accurate manner and demonstrate the emergence of effective ground states from the resonant trap eigenstates. Moreover, we show that the population of the Floquet states is strongly affected by the phase of a sudden turn-on of the trap modulation, which leads to significantly modified and rich dynamics. As a guideline for further studies, we argue that the deliberate population of only the resonance-induced effective ground states will allow one to realize Floquet condensates which follow classical periodic orbits, thus providing challenging future perspectives for the investigation of the quantum–classical correspondence.
Clean, interacting, periodically driven systems are believed to thermalize to featureless, infinite-temperature states that preclude the formation of stable ordered states. Here, we show stable quantum order, distinct from prethermal order, and its phase transitions in the ground state of the average-energy operator. Decomposing the time evolution into geometric and average-energy dynamics, we apply Geometric Floquet theory to identify the Floquet ground state. We derive a general variational principle to compute the Floquet ground state in large interacting periodically driven systems, and use it to develop a matrix product state algorithm. In driven nonintegrable chains of more than 100 interacting spins, our numerical simulations exhibit a gapped, area-law entangled Floquet ground state in extended parameter regimes; the ground-state manifold can support both dressed equilibrium order (e.g., Floquet antiferromagnets) and inherently nonequilibrium order (e.g., ground-state time crystals). This quantum order is protected by a finite average-energy gap, which closes at phase transition points upon changing model parameters. We decompose Floquet phases of matter into their equilibrium and nonequilibrium elements: whereas the Floquet ground state at any fixed time is a representative of an equilibrium phase of matter, the nonequilibrium content is entirely produced by the drive holonomy and is geometric. Finally, we identify quasilocal counterdiabatic driving as a mechanism for the stability against heating of gapped Floquet ground states in nonintegrable quantum systems, and argue why Floquet ground states evade the Eigenstate Thermalization Hypothesis. Our work enables the study of interacting nonequilibrium matter deep into the many-body regime, and lays the foundations for the engineering of strongly correlated ordered quantum states.
We report on the experimental realization of a Kapitza trap for ultracold atoms. Using time-periodic attractive and repulsive Gaussian potentials, we create an effective trap for ultracold neutral atoms in a regime where the time average of the potential is equal to zero. We analyze the role of experimental imperfections, the stability of the trapped atomic cloud, and the magnitude of the effective potential. We find good agreement with the high-frequency expansion of the underlying system dynamics. Our experimental approach opens up new possibilities to study Floquet systems of neutral atoms.
We report an experimental realization of ultracold atoms confined in a time-averaged, adiabatic potential (TAAP). This trapping technique involves using a slowly oscillating (kHz) bias field to time-average the instantaneous potential given by dressing a bare magnetic potential with a high-frequency (MHz) magnetic field. The resultant potentials provide a convenient route to a variety of trapping geometries with tunable parameters. We demonstrate the TAAP trap in a standard time-averaged orbiting potential trap with additional Helmholtz coils for the introduction of the radio frequency dressing field. We have evaporatively cooled $5\ifmmode\times\else\texttimes\fi{}{10}^{4}$ atoms of 87Rb to quantum degeneracy and observed condensate lifetimes of longer than 3 s.