Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields

Driving a quantum system periodically in time can profoundly alter its long-time dynamics and trigger topological order. Such schemes are particularly promising for generating nontrivial energy bands and gauge structures in quantum-matter systems. Here, we develop a general formalism that captures the essential features ruling the dynamics: the effective Hamiltonian, but also the effects related to the initial phase of the modulation and the micromotion. This framework allows for the identification of driving schemes, based on general $N$-step modulations, which lead to configurations relevant for quantum simulation. In particular, we explore methods to generate synthetic spin-orbit couplings and magnetic fields in cold-atom setups.

Popular Summary

Topological states of matter constitute a fundamental aspect of quantum physics, demonstrating a remarkable fusion between elegant mathematical theories and promising technological applications. Topology guarantees the robustness of unique properties and can be intrinsic to a material. Alternatively, topology can be induced externally by subjecting the system to well-designed electromagnetic fields or mechanical deformations; these driven systems can be characterized using calculations of effective Hamiltonians. We develop a general formalism, offering new tools to identify the essential features of periodically driven quantum systems, and propose realistic schemes relevant for quantum simulations.

We illustrate the formalism by considering the generation of two families of gauge fields, particularly relevant to cold-atom experiments. First, we consider artificial magnetism, both for a uniform system and a lattice-confined gas. Then, we investigate diverse schemes leading to synthetic spin-orbit coupling in two-dimensional spin-1/2 gases. In both schemes, the driving is provided by a repeated pulse sequence whose frequency is large and off resonant with respect to transitions inherent to the system. We characterize additional effects of the driving, such as the emergence of confining or deconfining potentials. We also highlight the micromotion, i.e., the unavoidable oscillation due to the fast driving, and note that its effects are large in momentum and spin space, which might affect detection measurements that rely on these parameters.

Our formalism classifies the various driving schemes and allows one to identify simple configurations that generate the basic ingredients necessary for the emergence of nontrivial topological properties. The great flexibility of the driving schemes stemming from this work makes it possible to explore topological states of matter that remain unreachable in common static systems.

Figure 1

Sensitivity to the initial phase of the driving. The brown curve represents the classical dynamics of a particle driven by a repeating pulse sequence $\{+F,-F\}$, where $F>0$ is a uniform force; the period is $T=20$, and the particle is initially at rest $v(t_i)=0$ at $x(t_i)=0$. The red curve is the same but considers the sequence $\{-F,+F\}$, i.e., shifts the starting time $t_i\to t_i+T/2$. The blue curve is also the same but starts the sequence with $-F$ during a time $T/4$ and then repeats the sequence $\{+F,-F\}$, i.e., shifts the starting time $t_i\to t_i-T/4$. Note that the initial kick $\widehat{K}(t_i)$ is inhibited in the latter case, while it is opposite in the two former cases [Eq. (26)]. In all the plots, the particle depicts a small micromotion, captured by $\widehat{K}(t)$ in Eq. (26). The brown and red curves highlight the great sensitivity to the initial phase of the modulation.

Figure 2

Schematic representation of the general $N$-step driving sequence ${\gamma }_N$ introduced in Eqs. (27) and (28), where $\widehat{H}(t)={\widehat{H}}_0+\widehat{V}(t)$.

Figure 3

Comparison between the dynamics of the driven system following the protocol in Eq. (36) (solid red and blue curves) and the dynamics predicted by the effective Hamiltonian (42) (dotted purple curve). Shown is the center-of-mass trajectory in the $x\text{−}y$ plane for time $t\in [0,100](1/J)$. For all simulations, a Gaussian wave packet is initially prepared around $\mathit{x}(0)=0$ with a nonzero group velocity along the $+y$ direction. The blue and red curves correspond to different initial phases of the driving sequence: The blue (respectively, red) curve is obtained by starting the sequence (36) with ${\widehat{H}}_0+\widehat{A}={\overline{p}}_x^2/m^{*}$ (respectively, ${\widehat{H}}_0-\widehat{A}={\overline{p}}_y^2/m^{*}$). We set the values $\kappa =10$ and $T=\pi /80(1/J)$, such as to fulfill the low-flux condition $\mathrm{\Phi }=1/128\ll 1$.

Figure 4

The helical-Rashba scheme: dynamics of the driven system associated with the driving sequence (47). Here, $\kappa =10J/a$ and $T=0.1/J$. (a) Spin populations as a function of time. The dynamics of the pulsed system (blue and red dots) is compared with the evolution dictated by the evolution operator in Eq. (15) (blue and red lines), where ${\widehat{H}}_{\text{eff}}$ and $\widehat{K}(t)$ are given by Eqs. (17), (48), and (g4). The effective and real dynamics are sampled using the time step $\mathrm{\Delta }t=T/2$. (b) Same as above, but using a time step $\mathrm{\Delta }t=T/16$ to highlight the micromotion captured by the kick operator $\widehat{K}(t)$ in Eq. (17). (c) Center-of-mass displacement in the $x\text{−}y$ plane after a time $t=8/J$. Here, the real trajectory of the driven system (blue dots) is compared with the evolution predicted by the effective Hamiltonian formalism [Eq. (15)], considering ${\widehat{H}}_{\text{eff}}$ at first, second, and third order in $1/\omega$. The trajectories are sampled using the time step $\mathrm{\Delta }t=T/2$.

Figure 5

The dynamics associated with the driving sequence (55) on a lattice. (a) The Gaussian wave packet is initially prepared around $\mathit{x}(0)=0$ and momentum $\mathit{k}=0$, so as to probe the Dirac dispersion relation of the effective Rashba Hamiltonian in Eq. (61) ($\mathrm{\Delta }k\ll m^{*}{\lambda }_R$). (b),(c) Shown is the time-evolved particle density $\rho (x,y)$ in the $x\text{−}y$ plane at time $t=7.3(1/J)$, for $\kappa =40J/a$, and $T=7\pi /100(1/J)$, i.e., ${\lambda }_R\approx (\pi /3)Ja$. (b) The driving follows the sequence (55) starting with the pulse $+\widehat{A}$. (c) Same but starting with the pulse $-\widehat{B}$, i.e., changing the initial phase of the modulation.

Figure 6

The $xy$ scheme: dynamics of the driven system associated with the driving sequence (51). Here, $\kappa =80J/a$ and $T=\pi /20J^{-1}$, which corresponds to the special regime where ${\lambda }_R=(\pi /2)aJ$. (a) Spin populations as a function of time. Bold (respectively, thin) lines correspond to the predictions of the effective Hamiltonian in Eq. (66) (respectively, to the real dynamics of the pulsed system). (b) Center-of-mass displacement in the $x\text{−}y$ plane after a time $t=8/J$. The blue dots (respectively, purple line) correspond to the real dynamics of the pulsed system [respectively, to the effective Hamiltonian in Eq. (66)]. In both figures, the real dynamics are sampled using the time step $\mathrm{\Delta }t=T/8$ to highlight the micromotion, which is not captured by the effective Hamiltonian.