Stroboscopic versus nonstroboscopic dynamics in the Floquet realization of the Harper-Hofstadter Hamiltonian

We study the stroboscopic and nonstroboscopic dynamics in the Floquet realization of the Harper-Hofstadter Hamiltonian. We show that the former produces the evolution expected in the high-frequency limit only for observables, which commute with the operator to which the driving protocol couples. On the contrary, nonstroboscopic dynamics is capable of capturing the evolution governed by the Floquet Hamiltonian of any observable associated with the effective high-frequency model. We provide exact numerical simulations for the dynamics of the number operator following a quantum cyclotron orbit on a $2\times 2$ plaquette, as well as the chiral current operator flowing along the legs of a $2\times 20$ ladder. The exact evolution is compared with its stroboscopic and nonstroboscopic counterparts, including finite-frequency corrections.

Figure 1

The Floquet realization of the Harper-Hofstadter (HH) model [32, 33]. A magnetic field gradient is applied to inhibit the hopping along the $x$ direction. At the same time, two resonant Raman-Bragg lasers result in a periodic driving protocol with site-dependent phase, which couples to the atomic number operator. As a result, in the infinite-frequency limit, the Floquet Hamiltonian coincides with the HH model of lattice bosons in a magnetic field.

Figure 2

Schematic representation of the (a) HH plaquette with the cyclotron orbit, and (b) the HH ladder with the chiral currents.

Figure 3

Floquet stroboscopic (FS) evolution. Figure taken from Ref. [18].

Figure 4

FS vs exact dynamics of the quantum cyclotron orbits on a $2\times 2$ plaquette for different values of the driving frequency. The yellow curve is the one expected from the HH model; the gray dotted curve follows the exact evolution with respect to $H^{\mathrm{lab}}\left(t\right)$, while the black curve follows the exact FS evolution. The green curve describes the system evolving with respect to the HH Hamiltonian plus the leading ${\Omega }^{-1}$ correction. The model parameters are $\left(\hslash =1\right)$ $J_x=0.85$ kHz, $J_y=0.5$ kHz, $U=0$ kHz, and ${\Phi }_{\square }=-\pi /2$, which yields $K=-0.23$ kHz, $J=0.46$ kHz, and $\zeta =-0.57$. The lattice constant is set to unity.

Figure 5

FS vs exact dynamics of the probability (particle) current expectation $\langle j_L^{10,11}\rangle$ on a $2\times 20$ ladder at unit filling: the yellow curve denotes the evolution with respect to the HH Hamiltonian starting from the ground state (GS) of the HH model. The gray dotted curve is the exact evolution of the laboratory-frame current evolved with $H^{\mathrm{lab}}\left(t\right)$ starting from the GS of the exact Floquet Hamiltonian, while its stroboscopics at times $nT$ is shown by the black line. The blue dashed-dotted line is the exact evolution of the laboratory-frame current evolved with $H^{\mathrm{lab}}\left(t\right)$ starting from the GS of the HH Hamiltonian. The model parameters are $\left(\hslash =1\right)$ $J_x=1.2$ kHz, $J_y=0.16$ kHz, $U=0$ kHz, and ${\Phi }_{\square }=-\pi /2$, which yields $K=-0.3$ kHz, $J=0.15$ kHz, and $\zeta =-0.53$; the system is in the Meissner phase [34].

Figure 6

Floquet nonstroboscopic (FNS) evolution. Figure taken from Ref. [18].

Figure 7

FNS vs exact dynamics of the quantum cyclotron orbits on a $2\times 2$ plaquette for different values of the driving frequency. The yellow and gray dotted curves are the same as in Fig. 4. The red curve shows the FNS evolution of the cyclotron orbit. The model parameters are $\left(\hslash =1\right)$ $J_x=0.85$ kHz, $J_y=0.5$ kHz, $U=0$ kHz, and ${\Phi }_{\square }=-\pi /2$, which yields $K=-0.23$ kHz, $J=0.46$ kHz, and $\zeta =-0.57$. The lattice constant is set to unity.

Figure 8

${\mathrm{FNS}}_2$ vs exact dynamics of the probability (particle) current expectation $\langle j_L^{10,11}\rangle$ on a $2\times 20$ ladder at unit filling: the yellow and blue dashed-dotted curves are the same as in Fig. 5. The gray line is the ${\mathrm{FNS}}_2$ evolution of the laboratory-frame current starting from the GS of the Floquet Hamiltonian, while the solid blue line is the ${\mathrm{FNS}}_2$ evolution of the laboratory-frame current starting from the GS of the HH Hamiltonian. The ${\mathrm{FNS}}_2$ evolution is always with respect to the Floquet Hamiltonian. The model parameters are $\left(\hslash =1\right)$ $J_x=1.2$ kHz, $J_y=0.16$ kHz, $U=0$ kHz, and ${\Phi }_{\square }=-\pi /2$, which yields $K=-0.3$ kHz, $J=0.15$ kHz, and $\zeta =-0.53$; the system is in the Meissner phase [34].