Floquet analysis 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.

Figure 1

Localized to delocalized transition in a 1D ring trap. (a) The Floquet ground state density at $t=0$ for the 1D ring trap as a function of driving period. At $T=4.19$ the ground state transitions from being homogeneous to the localized ground state of the attractive potential. (b) The time-averaged energy spectrum [Eq. (8)] as a function of driving period. Points are the results of Floquet simulations, and solid lines for the bound states labeled by quantum number $n$ are the result of the analytic theory given by Eq. (14). The solid line for the energy of the homogeneous state with winding number $w=0$ is from Eq. (15) with the phase given by Eq. (17). The black dashed line is the predicted transition point from equating Eqs. (14) and (17). The insets illustrate the Floquet state density corresponding to each spectral curve.

Figure 2

Properties of the Floquet ground state vs scanning period $T$ in the time-averaged limit $T\to 0$. (a) Depth of the density defect $\mathrm{\Delta }$. (b) Height of the phase defect $\delta$. (c) Fidelity of the Floquet ground state with the $k=0$ plane-wave ground state of the time-averaged limit. Circles show numerical results. Solid lines show the predictions given by Eq. (15). The blue dashed line in (c) shows the analytical approximation (24).

Figure 3

Floquet analysis of the driven harmonic oscillator. (a) A schematic of the time-averaged (left) and adiabatic (right) driven harmonic oscillators, with the lowest four energies indicated. (b) The Floquet ground state density for the driven harmonic oscillator with ${\omega }_0=2\pi /1.5$. In a small region around $T_{\text{res}}=1.5$ the strong resonance means that the numerical results for the density have not converged. Outside of this range, the numerics are stable and we obtain the expected harmonic oscillator ground state density. (c) The time-averaged energy spectrum as a function of driving period for the same parameters (markers). The analytic result Eq. (31) is shown as dashed lines. The solid lines are the result of an inverse frequency expansion to second order in $T$. The black vertical line at $T=1.5$ indicates the location of the resonance.

Figure 4

Illustration of the relative motion of the Floquet ground state density and center of the trapping potential as the driving period $T$ is increased. The red dashed line shows the analytic center-of-mass coordinate given by Eq. (29). (a) $T=0.1$, near the time-averaged limit. (b) $T=1.2$, just before resonance; the Floquet state undergoes large amplitude center-of-mass oscillations and is out of phase with the trapping potential. (c) $T=1.9$, after the resonance; the Floquet state exhibits large amplitude center-of-mass oscillations and is in phase with the trapping potential. (d) $T=9.9$, where the Floquet state is far from resonance and follows the motion of the trapping potential.

Figure 5

Large collective resonance and emergence of “photon” resonances for the driven $\left|x\right|$ potential. (a) A schematic of the time-averaged (left) and adiabatic (right) driven $\left|x\right|$ potential, with the lowest four energies indicated. (b) The Floquet ground state density for the driven $\left|x\right|$ potential as a function of driving period $T$. (c) The time-averaged energy spectrum for the driven $\left|x\right|$ potential as a function of driving period $T$. The states undergo a collective resonance, the position of which can be predicted by the energy spacings for low $T$ (vertical black line). The numerics do not converge in small regions around the collective and photon resonance points where the Floquet state reaches the edge of the $x$ grid. Solid lines are the result of an inverse frequency expansion, where the third order term has been fitted to the Floquet simulation.

Figure 6

The quartic double well displays a quasicollective resonance as well as photon resonances. (a) The time-averaged (left) and adiabatic (right) potentials. (b) The density of the Floquet ground state for the driven double well potential as a function of $T$. At large $T$ we recover the two-peaked ground state of the double-well potential. (c) The time-averaged energy spectrum for the driven double-well potential as a function of $T$. The numerics do not converge in regions where the Floquet state reaches the edge of the $x$ grid. Solid lines are the result of an inverse frequency expansion to third order.

Figure 7

Comparison of the fidelity of the Floquet ground state with the time-averaged limit for as a function of scaled driving period for the three potentials considered in this section. Solid lines are a Gaussian fit to the data, with the exception of the harmonic oscillator case, which uses the analytic fidelity given by Eq. (32). Dashed lines have been added between the quartic double-well data points to aid the eye.