Millimeter-wave interconnects for microwave-frequency quantum machines

Superconducting microwave circuits form a versatile platform for storing and manipulating quantum information. A major challenge to further scalability is to find approaches for connecting these systems over long distances and at high rates. One approach is to convert the quantum state of a microwave circuit to optical photons that can be transmitted over kilometers at room temperature with little loss. Many proposals for electro-optic conversion between microwave and optics use optical driving of a weak three-wave mixing nonlinearity to convert the frequency of an excitation. Residual absorption of this optical pump leads to heating, which is problematic at cryogenic temperatures. Here we propose an alternative approach where a nonlinear superconducting circuit is driven to interconvert between microwave-frequency ($7\times {10}^9$ Hz) and millimeter-wave-frequency photons ($3\times {10}^{11}$ Hz). To understand the potential for quantum state conversion between microwave and millimeter-wave photons, we consider the driven four-wave mixing quantum dynamics of nonlinear circuits. In contrast to the linear dynamics of the driven three-wave mixing converters, the proposed four-wave mixing converter has nonlinear decoherence channels that lead to a more complex parameter space of couplings and pump powers that we map out. We consider physical realizations of such converter circuits by deriving theoretically the upper bound on the maximum obtainable nonlinear coupling between any two modes in a lossless circuit, and synthesizing an optimal circuit based on realistic materials that saturates this bound. Our proposed circuit dissipates less than ${10}^{-9}$ times the energy of current electro-optic converters per qubit. Finally, we outline the quantum link budget for optical, microwave, and millimeter-wave connections, showing that our approach is viable for realizing interconnected quantum processors for intracity or quantum data center environments.

Figure 1

(a) Spectrum of the system, showing the three relevant modes, and (b) a diagram representing their mutual interaction via the ${\varphi }^4$ nonlinearity of the kinetic inductor and coupling to their respective continuum input-ouput fields. (c) The effective coupled system of modes $a$ and $b$ when $c$ is coherently displaced by a strong pump signal. (d) Model of heat dissipation due to the pump tone at the base stage of the cryostat. The terms shown in red are, from top left to bottom right, the internal dissipation in the pump mode and the pump line losses of the incoming and reflected field.

Figure 2

(a) Schematic representation of an arbitrary linear circuit and the vacuum fluctuation amplitude $\mathrm{\Delta }{\varphi }_{ij}^{\left(k\right)}$ in a normal mode $k$ of the flux difference across an inductance $L$ (highlighted in purple) connecting its two nodes $i$ and $j$. According to the sum rule expressed by Eq. (7), this vacuum fluctuation amplitude is related to the effective inductance $L_{ij}^{\left(\mathrm{eff}\right)}$ (b) we would measure by an ideal impedance meter in a circuit from which the capacitive elements have been removed (c). This inductance is in turn bounded from above by $L$. (d) An example of a circuit in a Cauer topology in which the product ${\varphi }_a{\varphi }_b{\varphi }_c^2$ of vacuum fluctuation amplitudes across the inductance $L$ is maximized for given mode frequencies ${\omega }_a/2\pi =7$ GHz, ${\omega }_b/2\pi =300$ GHz, ${\omega }_c=({\omega }_b-{\omega }_a)/2$, and for $L=1$ nH.

Figure 3

(a) Plot of the dissipated energy per qubit $E_{\text{qbit}}$ as a function of the linewidths of modes $a$ and $b$, optimized with respect to the linewidth of mode $c$, as given by Eq. (16). The dashed red lines represent different values of the converter efficiency $T$ limited by the ratio between the external coupling of the modes and their internal losses. Close to the dashed black line, corresponding to unity quality factor of the microwave mode, the discrete mode model of the system becomes invalid. The solid black line indicates a value of the mm-wave linewidth equal to the microwave frequency, implying breakdown of the rotating wave approximation. The solid blue line shows the boundary between the two regimes in Eq. (16). Below the line, the nonlinearity of mode $c$ is low enough to let us set ${\kappa }_{\mathrm{ext}}={\kappa }_{\mathrm{int}}$ while above the line, Eq. (15) forces ${\kappa }_{\mathrm{ext}}>{\kappa }_{\mathrm{int}}$. (b) and (c) Loss of photon state purity due to (b) dephasing by the pump shot noise and (c) heating due to its back-action. Evaluated using Eqs. (C2) and (D4), respectively, with the linewidth of mode $c$ chosen to maximize $E_{\text{qbit}}$, as in (a). The red dot indicates the operating point chosen in our example as a compromise between the dephasing and heating effects.