Floquet Topological Frequency-Converting Amplifier

TL;DR

We address how a single periodically driven, lossy bosonic mode can realize a non-Hermitian Floquet lattice with a synthetic electric-field in frequency space, enabling directional amplification and frequency conversion with an effective field $E_{\rm syn}$. The method uses Floquet-Green's functions and a doubled Hermitian Hamiltonian to identify a local winding number that marks a topological phase, with edge-like Jackiw–Rebbi solitons describing the mode structure in synthetic frequency. The work shows a minimal mechanism—driven frequency modulation and driven dissipation—sufficient for topological amplification without multimode drives, and provides a concrete circuit-QED implementation via adiabatic elimination of fast auxiliary modes. These results offer a practical route to non-Hermitian topological photonics with potential applications in robust signal routing and quantum sensing.

Abstract

We introduce a driven-dissipative Floquet model in which a single harmonic oscillator with modulated frequency and decay realizes a non-Hermitian synthetic lattice with an effective electric field gradient in frequency space. Using the Floquet-Green's function and its doubled-space representation, we identify a topological regime that supports directional amplification and frequency conversion, accurately captured by a local winding number. The underlying mode structure is well described by a Jackiw-Rebbi-like continuum theory with Dirac cones and solitonic zero modes in synthetic frequency. Our results establish a simple and experimentally feasible route to non-Hermitian topological amplification, naturally implementable in current quantum technologies such as superconducting circuits.

Figures (6)

• Figure 1: Sketch of the effective model: (a) A confined particle in a periodically driven harmonic potential $\omega_0(t)$, coupled to a static decay channel $\gamma$ (also serving as a coherent drive port), a time-modulated decay channel $\kappa(t)$, and an incoherent pump $P$. (b) These periodic modulations implement a synthetic lattice with an effective constant electric field $E_{\rm syn}$ in frequency space, giving rise to directional amplification and frequency conversion due to time-reversal-symmetry breaking via asymmetric couplings.
• Figure 2: (a–d) The local winding number $\nu_n({\bar{\omega}})$ accurately predicts the topological region where $|\mathsf{G}_{nm}({\bar{\omega}})|$ reaches its off-diagonal maximum, signaling strong frequency conversion and, as $\beta\to1$, enhanced amplification ($E_0\to0$). Parameters are $\eta_P = s \times (1,9.5,11.4,19.5)$ with $\eta_\omega = \eta_\kappa/s = \eta_\gamma/s = 10$, $\phi=\pi/2$, ${\bar{\omega}}=0$, $s=3$, and $\Omega = 2\pi$. (e,f) Normalized singular vectors for the parameter sets of panels (c) and (d), corresponding to $\beta = 0.14$ and $\beta = 0.95$ (topological regime). Curves for scaling parameters $s=1$ (orange, blue) and $s=3$ (red, black) are shown. Solid lines come from numerical diagonalization of Eq. \ref{['eq:Hnm_Sambe']}, while dashed lines indicate the JR prediction (whose accuracy improves as $s\to\infty$). (e) For an input signal at harmonic $n_d$ from the $\gamma$-port, weighted by $(\boldsymbol{u}_0)_{n_d}$, the system’s steady state (and emitted spectrum $\bar{d}_{\rm out,n}({\bar{\omega}})$) contains harmonics given by $(\boldsymbol{v}_0)_n$, centered at $n_0$ (with width $\sigma_{\rm r}\to\sqrt{\eta_\kappa/2}$ as $\beta\to1$), achieving maximum conversion when $n_d = -n_0$.
• Figure 3: Maximum of the time-dependent SNR computed in the steady state as a function of $\beta(\eta_P)$SM. The blue (dashed) region identifies the topological (unstable) regime. Parameters are $\eta_\omega = \eta_\kappa = \eta_{\gamma} = 10$, $\phi = \pi/2$, and $\eta_P\in (9,30)$.
• Figure 4: Schematic of a proposed cQED setup. The slow mode $a$ couples to two fast-decaying auxiliary modes via flux-pumped Josephson elements. A dc SQUID mediates a time-modulated coupling $g_b(t)$ to the lossy mode $b$ (with decay $\kappa_b$), while a SNAIL provides a parametric coupling $g_c$ to the pumped mode $c$ ($\kappa_c$). Independent flux lines drive the SQUID at frequency $\Omega$ and pump the SNAIL at a high-frequency $\omega_p\simeq\omega_a+\omega_c$.
• Figure 5: Eigenvalues of the doubled matrix $\tilde{{\mathcal{H}}}_k(0,n)$ for different values of $n\in[-50,50]$ and $\beta=1$. The plot shows the presence of two Dirac cones at the values $(+k_0,-n_0)$ and $(+k_0,-n_0)$. The above figure is computed for the following parameters $\eta_P =20 s$, $\eta_\omega=\eta_{\gamma}/s=\eta_\kappa/s=10$, $\phi=\pi/2$, ${\bar{\omega}}=0$, and $s=3$.