Periodically driven quantum matter: The case of resonant modulations

Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micromotion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micromotion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time evolution of the system.

Figure 1

Superlattices and resonant modulations. (a) The two-site superlattice $\left(\alpha =0,1\right)$, (b) the three-site superlattice $\left(\alpha =0,1,2\right)$, and (c) the Wannier-Stark ladder $\left(\alpha =0,1,2,3,\cdots \right)$. Colors refer to the different sectors $\alpha \in \mathbb{Z}$. The cases (a) and (c) involve a single-harmonic modulation, with frequency $\omega =\Delta /\hslash$; see Sec. 5. The case (b) involves a higher-order offset $2\Delta$, requiring the use of a two-harmonic modulation to restore the tunneling over the entire lattice; see Sec. 7.

Figure 2

(a) Labeling of sites $m=m^{\pm }$ according to the energy offset $\pm \Delta$; see Eqs. (39) and (40). (b) The corresponding plaquettes ${\square }^{\pm }$, with Peierls phase factors $exp(\pm i{\varphi }_{m,n})$ along the $x$ direction; the flux penetrating these two plaquettes are given by $2\pi {\Phi }^{+}={\varphi }_{m^{+},n}-{\varphi }_{m^{+},n+1}$ and $2\pi {\Phi }^{-}={\varphi }_{m^{-},n+1}-{\varphi }_{m^{-},n}$, respectively. The phases ${\varphi }_{m,n}$, defined in Eq. (43), are established by the time-periodic modulation (35).

Figure 3

(a) Momentum distribution associated with the ground state $|\text{g.s}.\rangle$ of the effective Hamiltonian ${\widehat{\mathcal{H}}}_{\text{eff}}$, corresponding to a staggered-flux lattice with flux ${\Phi }^{\pm }=\pm 1/4$. (b) Momentum distribution of the time-evolved state $\left|\psi \right(t)\rangle$, at time $t=T/8$, when initially starting the evolution with the ground state $\left|\psi \right(t_0)\rangle =|\text{g.s.}\rangle$. The amplitudes of the density peaks oscillate within a period of the driving, but the position of the peaks remain constant. Here $\kappa =8J$ and $\omega =20J$, where $J$ is the hopping amplitude. This latter figure is to be compared with the experimental data shown in Fig. 2 in Ref. [34], which corresponds to ${\Phi }^{\pm }=\pm 1/4$ and $2\kappa /\left(\hslash \omega \right)\approx 0.48$. The reduced FBZ, $k_x\in [-\pi /2a,\pi /2a[$ and $k_y\in [-\pi /a,\pi /a[$, is highlighted in all figures.