Colloquium: Atomic quantum gases in periodically driven optical lattices

Time-periodic forcing in the form of coherent radiation is a standard tool for the coherent manipulation of small quantum systems like single atoms. In the last years, periodic driving has more and more also been considered as a means for the coherent control of many-body systems. In particular, experiments with ultracold quantum gases in optical lattices subjected to periodic driving in the lower kilohertz regime have attracted much attention. Milestones include the observation of dynamic localization, the dynamic control of the quantum phase transition between a bosonic superfluid and a Mott insulator, as well as the dynamic creation of strong artificial magnetic fields and topological band structures. This Colloquium reviews these recent experiments and their theoretical description. Moreover, fundamental properties of periodically driven many-body systems are discussed within the framework of Floquet theory, including heating, relaxation dynamics, anomalous topological edge states, and the response to slow parameter variations.

Figure 1

Effective tunneling matrix element $|J_{\text{eff}}|/J$ vs $K_0=K/\hslash \omega$. Extracted from the expansion dynamics of a condensate of about $5\times {10}^4$ ${}^{87}\mathrm{Rb}$ atoms in a shaken optical lattice of $E_R\approx 2\pi \hslash 3.16\mathrm{kHz}$ (squares: $V_0/E_R=6$, $\omega /2\pi =1\mathrm{kHz}$, circles: $V_0/E_R=6$, $\omega /2\pi =0.5\mathrm{kHz}$, triangles: $V_0/E_R=4$, $\omega /2\pi =1\mathrm{kHz}$, dashed line: theoretical prediction). Inset: $J_{\text{eff}}/J$ for $K/\hslash \omega =2$ and $V_0/E_R=9$ vs $\hslash \omega /J$ indicates a breakdown of high-frequency prediction (dashed line) for $\hslash \omega /J<2$. From [156].

Figure 2

Dynamically induced superfluid-to-Mott-insulator transition in a shaken cubic optical lattice. Two-dimensional projection of the momentum distribution obtained from time-of-flight absorption imaging at three different times during the experimental protocol: before ramping up the driving strength $K_0=K/\hslash \omega$, after $K_0$ has been ramped up linearly (middle), and after $K_0$ has been ramped down again (right). The loss and reappearance of sharp peaks indicates that the system approximately followed a many-body Floquet state undergoing a quantum phase transition from a superfluid to a Mott insulator and back. From [248].

Figure 3

(a) Anisotropic triangular lattice with effective tunneling parameters $J_{\text{eff}}$ and $J_{\text{eff}}^{\prime }$. (b) Momentum distribution averaged over many measurements (shots) and corresponding pattern of the condensate phase (indicated by the direction of the arrow) for different $J_{\text{eff}}$ and $J_{\text{eff}}^{\prime }$ (dashed and solid lines indicate positive and negative tunneling parameters). (c) Spontaneous time-reversal symmetry breaking for $J_{\text{eff}}=J_{\text{eff}}^{\prime }<0$. Each of the two spatial configurations of the condensate phase shown in A breaks time-reversal symmetry. In absorption images the two are distinguished by the position of the measured peaks, indicated by dotted or solid circles in B. The contrast between both configurations $\chi$ varies from shot to shot (C) giving a bimodal distribution (D), so that typically only one of the two configurations appears spontaneously. (b), (c) From [217].

Figure 4

(a) The dimensionless magnetic flux $\mathrm{\Phi }$ piercing a lattice plaquette equals the sum of the Peierls phases ${\theta }_{{\ell }^{\prime }\ell }$ picked up when moving around it in positive direction once. (b) Moving-secondary-lattice scheme for creating a homogeneous flux configuration. (c) Asymmetric lattice shaking (left) gives rise to complex effective tunneling matrix elements (right), plotted vs $K=F_{0}d{T}_1/T$ (in units of $\hslash \omega$) for $T_1/T_2=2.1$. (b) From [5]. (c) From [218].

Figure 5

(a) By resonantly coupling the two lowest bands of a cosine lattice a frustrated ladder is created with plaquette fluxes of $\pi$. (b) Hexagonal lattice. The effective Hamiltonian of the driven system including next-nearest-neighbor tunneling with Peierls phase $\theta$ (along the dashed lines).

Figure 6

(a) Quasienergy spectrum of a small Bose-Hubbard chain (five particles on five sites), subjected to a sinusoidal force of frequency $\hslash \omega /J=20$ and amplitude $K/\hslash \omega =2$. (b) Zoom into (a). From [71].

Figure 7

In periodically driven optical lattices, heating occurs due to the resonant creation of high-energy excitations, either interband excitations (left) or collective intraband excitations (right, illustrated using the example of a large bosonic site occupation). The time scales for these processes should be large compared to the duration of the experiment.

Figure 8

(a)–(c) Floquet spectra of periodically driven square lattice defined in Fig. 9 for the parameters specified in (d). (d) Phase diagram vs tunneling parameter $J$ and sublattice offset ${\delta }_{AB}$ (in units of $\hslash \omega /2$). Phases are characterized by the winding numbers $W_{ϵ}$ for the bulk gaps at quasienergy $ϵ$; the Chern number of the upper (lower) band is given by $C$ ($-C$). From [198].

Figure 9

(a) Simple model system on a square lattice with two sublattice states (open and filled circles). Each driving period is divided into five stages of duration $T/5$. During each stage either tunneling matrix elements $J$ are present on the highlighted bonds or an energy offset ${\delta }_{AB}$ between both sublattices. (b) Dynamics of an initially site-localized particle during one driving cycle for fine-tuned parameters $J=(5/4)\hslash \omega$ and ${\delta }_{AB}=0$ in the bulk (blue) and at the edges (green, red). (c) Quasienergy spectrum for parameters of (b). The bulk levels (thick line, blue) are dispersionless, whereas the levels for the chiral edge modes (thin lines, red, green) wrap around the Brillouin zone with constant slope. From [198].