Floquet-Network Theory of Nonreciprocal Transport

We develop a theoretical framework that lays out the fundamental rules under which a periodic (Floquet) driving scheme can induce nonreciprocal transport. Our approach utilizes an extended Hilbert space where a Floquet network with an extra (frequency) dimension naturally arises. The properties of this network (its on-site potential and the intersite couplings) are in one-to-one correspondence with the initial driving scheme. Its proper design allows for a control of the multipath scattering processes and the associated interferences. We harness this degree of freedom to realize driving schemes with narrow-band or broadband nonreciprocal transport.

Figure 1

(a) A system of two single-mode coupled cavities. The cavities can be passive or active, i.e., consist of a loss or/and gain elements. A periodic modulation of the cavity (volume or constituent parameter) leads to a periodic modulation of the associated resonant frequencies. Their position can be also periodically modulated resulting in a modulation of their coupling strength. (b) The corresponding Floquet lattice.

Figure 2

Filled (open) circles correspond to the left (right) sites of the Floquet lattice. The index $s$ indicates the sites along the Floquet lattice, while the index $n$ indicates the distance between ladder steps. The magnitude of the bond strengths (Fourier components of the driving) are indicated as $f_n$ (for coupling between only $L$ or only $R$ sites of the Floquet lattice) and $h_n$ (for cross-coupling between $L$ and $R$ sites of the Floquet lattice). (First row) Grammar associated with imaginary driving amplitudes of Fourier components that correspond to directed bonds. (Second row) Grammar associated with real driving amplitudes of Fourier components that correspond to directionless bonds. In all cases, (upwards) arrows indicate transition to higher Fourier components. The associated driving schemes are also given in terms of matrices $H(t)$, as shown below each graph. The opposite transitions associated with complex conjugate amplitudes (downward arrows—not shown) are described by the Hermitian conjugate $H^{†}(t)$.

Figure 3

Example of a Floquet lattice with typical couplings due to the driving $h(t)=h_0+2h_1\sin (\omega t)+2h_2\cos (2\omega t)$ and $\delta {\epsilon }_L=2f_1\sin (\omega t)$.

Figure 4

Interference paths that are considered in the evaluation of ${\mathcal{N}}_{0\to 0}$, associated with (a) example 1 [see Eq. (8)] and (b) example 2 [see Eq. (8)]. Numerical and analytical evaluation of ${\mathcal{N}}_{0\to 0}$ versus the incident frequency $E_0$ for ${ϵ}_R^0=0$, $f_1=h_1=0.5$ and (c) example 1 with $h_0=-0.5$ and (d) example 2 with $h_0=-1$. The simulations and calculations are done for two sets of parameters, for $\omega =0.5$ and ${ϵ}_L^0=-1$ and for $\omega =4.2$ and ${ϵ}_L^0=-1-𝚤$. The Floquet ladder in (a) and (b), is identical to Fig. 1. The numerical convergence is tested by increasing the Floquet lattice.

Figure 5

(a) The interference paths used to obtain the theoretical expression equation (9) for the nonreciprocity strength ${\mathcal{N}}_{0\to 0}$. (b) Numerical and analytical evaluation [see Eq. (9)] of ${\mathcal{N}}_{0\to 0}$ versus the incident frequency $E_0$ for $\omega =0.5$, ${ϵ}_L^0=-1$, ${ϵ}_R^0=1$, $h_0=-1$, and $h_1=h_2=0.2$ (the red line and symbols). The numerical convergence is tested by increasing the Floquet lattice.

Figure 6

Numerical (symbols) and renormalized theoretical [Eq. (10)] (lines) results of ${\mathcal{N}}_{0\to 0}$ versus the incident frequency $E_0$. The driving frequencies are $\omega =0.2$, 0.5, 0.8 associated with example 3. The other parameters are ${ϵ}_L^0=-1$, ${ϵ}_R^0=1$, $h_0=-1$, and $h_1=h_2=0.2$.

Figure 7

(a) Transport-direction independent path with directed bonds and (b) transport-direction dependent path. (c) The direction-dependent interference between two paths, which is responsible for the nonreciprocity transport.

Figure 8

Numerical evaluation of ${\mathcal{N}}_{0\to 0}$ for example 3 with $h_0=-1$ and three different values of $h_1=h_2$, 0.2, 0.4, and 1, which demonstrate a control of the NRT bandwidth. The other parameters are $\omega =0.5$, ${\epsilon }_L^0=-1$, and ${\epsilon }_R^0=1$.

Figure 9

Numerical (symbols) and theoretical (lines) results of ${\mathcal{N}}_{0\to 0}$ versus incident frequency $E_0$ when the driving frequency $\omega =0.2$, 0.5, 0.8 for (a) example 1, with the other parameters being ${ϵ}_L^0=-1$, ${ϵ}_R^0=0$, $h_0=-0.5$, and $f_1=h_1=0.5$; (b) example 2, with the other parameters being ${ϵ}_L^0=-1$, ${ϵ}_R^0=0$, $h_0=-1$, and $f_1=h_1=0.5$; and (c) example 3, with the other parameters being ${ϵ}_L^0=-1$, ${ϵ}_R^0=1$, $h_0=-1$, and $h_1=h_2=0.2$.

Figure 10

Numerical (symbols) and theoretical (lines) results of ${\mathcal{N}}_{-1\to 0}$ versus the incident energy $E_0$ for driving frequency $\omega =0.2$, 0.5, 0.8 and (a) example 1, (b) example 2, and (c) example 3. All other parameters are the same as the corresponding ones in Fig. 9.

Figure 11

The interference paths used to obtain approximated nonreciprocity strength ${\mathcal{N}}_{-1\to 0}$ for (a),(b) Eq. (13) of examples 1 and 2, and (c) Eq. (14) of example 3.