Emergence of Floquet behavior for lattice fermions driven by light pulses

As many-body Floquet theory becomes more popular, it is important to find ways to connect theory with experiment. Theoretical calculations can have a periodic driving field that is always on, but experiment cannot. Hence, we need to know how long a driving field is needed before the system starts to look like the periodically driven Floquet system. We answer this question here for noninteracting band electrons in the infinite-dimensional limit by studying the properties of the system under pulsed driving fields and illustrating how they approach the Floquet limit. Our focus is on determining the minimal pulse lengths needed to recover the qualitative and semiquantitative Floquet theory results.

Figure 1

Schematic display of the two times for the Green's function and the integration directions for the horizontal time and average time DOS. For retarded quantities, the Fourier transform runs only over the darker parts of the rectangles. Retarded quantities are nonzero only below and to the right of the diagonal labeled $t_{\mathrm{ave}}$, defined by $t_{\mathrm{rel}}=0$.

Figure 2

Upper panel: negative imaginary part of the local retarded Green's function as a function of the relative time $t_{\mathrm{rel}}$ in units of the inverse rescaled hopping $\hslash /t^{*}$. Note that the real part vanishes because $\mu =0$. Lower panel: diagonal DOS as a function of the frequency $\omega$ in units of the rescaled hopping. Because the DOS is a set of delta functions at $J_0\left(E_0/\gamma \right)=0$ the coefficients $g_{\mathrm{n}}=-1/\left(2\pi \right){\int }_0^{2\pi }d{t}_{\mathrm{rel}}\mathrm{Im}\left[g_{\mathrm{loc}}^{\mathrm{R}}\left(t_{\mathrm{rel}},0\right)\right]cos\left(n{t}_{\mathrm{rel}}\right)$ of the Fourier series are displayed for $E_0=2.404\gamma$ and $E_0=5.52\gamma$. Other parameters: average time $t_{\mathrm{ave}}=0$, driving frequency $\gamma =t^{*}/\hslash$.

Figure 3

Negative imaginary part of the local Green's function at $t_{\mathrm{ave}}=10\hslash /t^{*}$, as a function of $t_2$ in units of inverse rescaled hopping $\hslash /t^{*}$ for different electric fields. Other parameters: $\gamma =t^{*}/\hslash$. Note that the real part vanishes because $\mu =0$.

Figure 4

The diagonal DOS corresponding to the steplike pulse that starts at $t=0$ and is turned off after $n$ oscillations at a cutoff time $t_{\mathrm{c}}=2\pi n/\gamma$ and corresponding to an infinite sinusoidal drive for $\gamma =t^{*}/\hslash , t_{\mathrm{ave}}=30\hslash /t^{*}$, and two different amplitudes $E_0=5.22\gamma$ (upper panel) and $E_0=\pi \gamma /4$ (lower panel).

Figure 5

Negative imaginary part of the local Green's function at a constant average time $t_{\mathrm{ave}}=0$, as a function of $t_{\mathrm{rel}}$ in units of inverse rescaled hopping $\hslash /t^{*}$ at $E_0=3.83\gamma$ for an infinite sinusoidal drive (orange) and for two Gaussians where the product of the width of the Gaussian and the frequency of the electric field is $\gamma t_{\mathrm{E}}=10$ (blue) and $\gamma t_{\mathrm{E}}=20$ (green), respectively, at three different driving frequencies $\gamma =0.5t^{*}/\hslash$ (upper panel), $\gamma =t^{*}/\hslash$ (middle panel), and $\gamma =1.5t^{*}/\hslash$ (lower panel).

Figure 6

The diagonal DOS corresponding to an infinite sinusoidal drive (blue) and a pulsed system (green), both averaged over the Floquet period from $t_{\mathrm{ave}}=-\pi /\gamma$ to $\pi /\gamma$ and the running average in frequency $\omega$ (orange) over one period of the oscillations in the diagonal DOS of the pulsed system at different amplitudes and frequencies of the electric field and for a Gaussian pulse of varying width.

Figure 7

The upper panel: Green's function of both the pulsed system (labeled “Gauss”) and the system that is driven for an infinitely long time (labeled “Infinite”) for a constant $t_2=0$ (${\mathcal{F}}_{\mathrm{H}}$) and a constant $t_{\mathrm{ave}}=0$ (${\mathcal{F}}_{\mathrm{D}}$). Lower panel: horizontal and diagonal DOS of both systems, averaged over one Fourier period in $t_2$ and $t_{\mathrm{ave}}$, respectively. Other parameters: $E_0=3.83\gamma , \gamma =t^{*}/\hslash$, and $t_{\mathrm{E}}=30\hslash /t^{*}$.