Modeling integrated frequency shifters and beam splitters

TL;DR

The paper develops a quantum input-output framework to model frequency-domain photonic processing using modulated microring resonators. By deriving effective transfer matrices within rotating-wave and SLH formalisms, it enables native N-mode frequency-beam splitters and phase shifters, and demonstrates constructions of a frequency-domain phase shifter and Mach-Zehnder interferometer from cascaded two-resonator devices. A detailed analysis of two- and four-resonator platforms reveals occupancy, loss, and cooperativity trade-offs, and a no-go theorem shows limitations for N>4 native equal-splitters due to Hadamard-diagonalizability constraints on penny graphs. The results provide a principled toolkit for rapid prototyping of frequency-encoded PQC primitives and illuminate fundamental geometric limits for scalable frequency-domain quantum optics.

Abstract

Photonic quantum computing is a strong contender in the race to fault-tolerance. Recent proposals using qubits encoded in frequency modes promise a large reduction in hardware footprint, and have garnered much attention. In this encoding, linear optics, i.e., beam splitters and phase shifters, is necessarily not energy-conserving, and is costly to implement. In this work, we present designs of frequency-mode beam splitters based on modulated arrays of coupled resonators. We develop a methodology to construct their effective transfer matrices based on the SLH formalism for quantum input-output networks. Our methodology is flexible and highly composable, allowing us to define $N$-mode beam splitters either natively based on arrays of $N$-resonators of arbitrary connectivity or as networks of interconnected $l$-mode beam splitters, with $l<N$. We apply our methodology to analyze a two-resonator device, a frequency-domain phase shifter and a Mach-Zehnder interferometer obtained from composing these devices, a four-resonator device, and present a formal no-go theorem on the possibility of natively generating certain $N$-mode frequency-domain beam splitters with arrays of $N$-resonators.

Figures (9)

• Figure 1: Schematics of the frequency-domain beam splitters based on modulated coupled ring resonators. (a) Two-resonator device which implements $2$-mode frequency beam splitters. (b) Two-resonator device in the two waveguide configuration. The output of interest is the one on the second waveguide, $b_{\rm out}^{\rm (R)}$, which we wish to split by some predetermined ratio. (c) Four-resonator device used to natively implement $4$-mode frequency beam splitters. The second waveguide might or might not be included. In all sketches the resonators have the same central frequency $\omega_0$. In (b) and (c), the superscript (L) and (R) indicate first and second waveguide, respectively. We always assume $b_{\rm in}^{(R)} = 0$.
• Figure 2: Norm squared entries of the transfer matrix for the two resonator device coupled to a single waveguide as a function of the modulation amplitude $\epsilon$, the input is at $\omega_1$. The dashed black line indicates the $0$-$100$ frequency beam splitter point, $\epsilon_{\rm GCC}$, while the two gray dashed lines indicate the $50$-$50$ frequency beam splitter points, $\epsilon_{\rm bs}^{\pm}$. The extended axis shows the asymptotic behvior of the transfer matrix entries as $\epsilon\to\infty$. Parameters: $\Delta_{12}/2\pi = 28.2~{\rm GHz}$, $\Gamma_{\rm L}/2\pi = 5.31~{\rm GHz}$, $\kappa_{\rm int}/2\pi = 0.17~{\rm GHz}$, leading to $\alpha_{\rm L}\sim 30$, which correspond to those of device (c) in Ref. Hu2021.
• Figure 3: Norm squared entries of the transfer matrix for the two resonator device in the two waveguide configuration as function of the modulation amplitude $\epsilon$. The input is taken at $\omega_1$. The dashed black line indicates the $50$-$50$ frequency beam splitter working point, $\epsilon_{\rm bs}^{(\rm R)}$, for the output of the second waveguide. The extended axis shows the asymptotic behavior of the transfer matrix entries as $\epsilon\to\infty$. Parameters are as in in Fig. \ref{['fig:fig_2']} and $\Gamma_{\rm R} = \Gamma_{\rm L}$.
• Figure 4: Frequency beam splitters for an under-coupled, $\Gamma_{\rm L}<\kappa_{\rm int}$, two-resonators-waveguide system. (a) Normalized output intensity on the second mode, $c_2$, assuming the input was resonant with $c_1$, as a function of modulation amplitude $\epsilon$. The squares indicate the best splitting ratio achievable at a given $\alpha_{\rm L}$, which occurs at the modulation amplitude $\epsilon_{\rm uc}$ in Eq. (\ref{['eqn:epsilon_uc']}). (b) Total loss of the under-coupled system as function of the modulation amplitude. The colored square corresponds to the total loss at $\epsilon_{\rm uc}$. In both panels the color map, dark-to-light, indicates values of $\alpha_{\rm L}\in[0,1]$.
• Figure 5: Schematic of two cascaded frequency beam splitters, each implemented with a pair of modulated coupled ring resonators. If the individual beam splitters are operated with $0$-$100$ splitting ratio, the cascaded device implements a frequency-domain phase shifter (shown). If the individual beam splitters are operated with $50$-$50$ splitting ratio, the cascaded device behaves as a frequency-domain Mach--Zehnder interferometer (not shown). In both cases, the desired behavior is obtained by adjusting the phases, $\phi_{1,2}$, of the individual modulations, $\epsilon_{1,2}(t)$, see text for details.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Remark 4
• Theorem 1: theorem 2.2 of Ref. breen2020hadamard
• Remark 5
• Theorem 2
• proof
• Remark 6