Detector of microwave photon pairs based on a Josephson photomultiplier

TL;DR

This work addresses fast, high-fidelity detection of microwave photon pairs in circuit QED by transforming a flux-biased Josephson photomultiplier (JPM) into a two-photon threshold detector. The authors engineer a non-perturbative two-photon coupling between storage and buffer resonators through a nonlinear inductive coupler (ASQUID or BiSQUID), enabling coherent conversion of a storage-mode photon pair into a single buffer-mode photon that the JPM can absorb. A comprehensive master-equation model—including an engineered dissipative bath—predicts detection fidelities exceeding 99% in under 50 ns under realistic circuit parameters, with tunable on/off switching of the two-photon process. The scheme offers a scalable building block for photon-number-resolving detectors in circuit QED and can be extended to higher photon numbers or detector arrays.

Abstract

We propose a viable design of a microwave two-photon threshold detector. In essence, the considered scheme is an extension of the existing single-photon detector - a Josephson photomultiplier (JPM) - an absorbing microwave detector based on a capacitively-shunted rf SQUID. To implement a two-photon threshold detector, we utilize a dimer of resonators - two lumped-element resonators interacting via an asymmetric dc SQUID, with one of the resonators capacitively coupled to the JPM. By specific tuning of the resonator frequencies and the external flux through the dc SQUID coupler, we engineer a non-perturbative two-photon coupling between the resonators. This coupling results in the coherent conversion of a photon pair from one resonator into a single photon in another resonator, enabling selective response to quantum states with at least two photons. We also consider an extended coupler design that allows on-demand \textit{in situ} switching of two-photon coupling. In addition, we propose the modified JPM design to improve its performance. Our calculations demonstrate that, for realistic circuit parameters, we can achieve more than $99\%$ fidelity of photon pair detection in less than 50 ns. The considered scheme may serve as a building block for the implementation of efficient photon-number-resolving detectors in circuit QED architecture.

Figures (10)

• Figure 1: (a) Generic scheme for detection of photon pairs using a single SPD. The shaded square represents a nonlinear element that mediates two-photon coupling. (b) Schematics of the potential circuit QED setup implementing the considered approach to detection of microwave photon pairs.
• Figure 2: (a) Dependence of the two-photon coupling strength $|g_{21}|$ on the external flux $\Phi_\mathrm{c}$ via the coupler and the asymmetry parameter $\alpha$. Solid lines correspond to combinations of $\Phi_\mathrm{c}$ and $\alpha$ providing the the odd-parity coupling regime. Circles correspond to vanishing Josephson coupling $E^\mathrm{c}_{\mathrm{J}\,\mathrm{eff}} = 0$. Here we set $I^\mathrm{c}_0 = 50\,\mathrm{n A}$ ($E^\mathrm{c}_{\mathrm{J}}/h = 24.8\,\mathrm{GHz}$). (b) Dependence of the two-photon coupling strength $|g_{21}|$ in the the odd-parity coupling regime on the critical current $I^\mathrm{c}_0$ for different values of $\alpha$. Parameters of the resonators used for calculations are as follows: $C_1 = 1\,\mathrm{pF}$ ($E_{C1}/h = 19.4\,\mathrm{MHz}$), $L_1 = 1\,\mathrm{nH}$ ($E_{L1}/h = 163.2\,\mathrm{GHz}$), $C_2 = 0.5\,\mathrm{pF}$ ($E_{C2}/h = 38.7\,\mathrm{MHz}$), and $L_2 = 0.5\,\mathrm{nH}$ ($E_{L2}/h = 326.4\,\mathrm{GHz}$).
• Figure 3: (a) Circuit diagram of the BiSQUID coupler. (b) Effective Josephson energy $E^\mathrm{c}_{\mathrm{J}\,\mathrm{eff}}$ of the BiSQUID as a function of the external fluxes $\Phi_\mathrm{c}$ and $\Phi^\prime_\mathrm{c}$ threading through its loops for $\alpha = 0.5$. Empty hexagons mark $E^\mathrm{c}_{\mathrm{J},\mathrm{eff}} = 0$. (c) Dependence of the two-photon coupling strength $|g_{21}|$ on the external fluxes $\Phi_\mathrm{c}$ and $\Phi^\prime_\mathrm{c}$ via the BiSQUID loops for the different values of $\alpha$ indicated above the corresponding plot. Solid lines indicate the values of the external fluxes providing the the odd-parity coupling. Stars mark the external fluxes providing the maximum two-photon coupling in the the odd-parity coupling regime. Circles correspond to the fluxes for which the Josephson coupling vanishes $E^\mathrm{c}_{\mathrm{J}\,\mathrm{eff}} = 0$. Here, we set $I^\mathrm{c}_0 = 60\,\mathrm{nA}$ ($E^\mathrm{c}_\mathrm{J}/h = 29.8\,\mathrm{GHz}$). (d) Dependence of the maximum two-photon coupling strength on the parameter $\alpha$ and the critical current $I_0^\mathrm{c}$. Stars correspond to the coupler parameters used in Fig. \ref{['fig:BiSQUIDResults']}(c). Parameters of the resonators are the same as used in Fig. \ref{['fig:ASQUIDResults']}.
• Figure 4: (a) Lumped-element circuit diagram of the JPM represented by the flux-biased capacitively-shunted rf-SQUID. (b) Configurations of the JPM potential for the different values of the bias flux $\Phi_\mathrm{b}$. (c) Zoomed view of the region of the JPM potential [indicated by the dashed square in Fig. \ref{['fig:JPMSchemePotential']}(b)] for $\Phi_\mathrm{b} = 0.6316\Phi_0$ in the vicinity of the shallow well for the highly asymmetric configuration with only two eigenlevels localized in the left (shallow) well, while there are 94 eigenlevels localized in the right (deep) well. We demonstrate the JPM eigenlevels and the square moduli of the corresponding wavefunctions. Note that in what follows, c.f., Sec. \ref{['sec:JPMModel']}, we relabel the pair of eigenstates $|93\rangle$ and $|95\rangle$ localized in the shallow well as $|g\rangle$ and $|e\rangle$, and the eigenlevel $|96\rangle$ localized in the deep well as $|f\rangle$. The JPM parameters used for calculations are shown in Table \ref{['tab:JPMParams']}.
• Figure 5: Modified JPM design. The JPM couples to the engineered bath represented by a waveguide of impedance $Z_\mathrm{w}$ terminated by the resistive element $R_\mathrm{w}$. The waveguide also couples to the filter resonator of frequency $\omega_\mathrm{f}$. The JPM is driven by the coherent tone of frequency $\omega_\mathrm{dr}$.