Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators

We propose a low-noise, triply resonant, electro-optic (EO) scheme for quantum microwave-to-optical conversion based on coupled nanophotonics resonators integrated with a superconducting qubit. Our optical system features a split resonance—a doublet—with a tunable frequency splitting that matches the microwave resonance frequency of the superconducting qubit. This is in contrast to conventional approaches, where large optical resonators with free-spectral range comparable to the qubit microwave frequency are used. In our system, EO mixing between the optical pump coupled into the low-frequency doublet mode and a resonance microwave photon results in an up-converted optical photon on resonance with high-frequency doublet mode. Importantly, the down-conversion process, which is the source of noise, is suppressed in our scheme as the coupled-resonator system does not support modes at that frequency. Our device has at least an order of magnitude smaller footprint than conventional devices, resulting in large overlap between optical and microwave fields and a large photon conversion rate ($g/2\pi$) in the range of $\sim 5$–15 kHz. Owing to a large $g$ factor and doubly resonant nature of our device, microwave-to-optical frequency conversion can be achieved with optical pump powers in the range of tens of microwatts, even with moderate values for optical $\mathit{Q}$ ($\sim {10}^6$) and microwave $Q$ ($\sim {10}^4$). The performance metrics of our device, with substantial improvement over the previous EO-based approaches, promise a scalable quantum microwave-to-optical conversion and networking of superconducting processors via optical fiber communication.

Figure 1

Electro-optic–based microwave-to-optical quantum conversion. (a) The conventional scheme wherein an electro-optic ring resonator with FSR close to but slightly different than the microwave resonance (${\omega }_M$) of a SC qubit is integrated with the microwave cavity electrode for the electro-optic conversion. The pump laser slightly blue detuned from one of the resonances results in up-conversion on resonance with the cavity mode, while down-conversion is off-resonance, and thereby suppressed. (b) Our proposed scheme, based on strongly coupled resonators, features frequency doublets corresponding to symmetric and antisymmetric supermodes, with splitting equal to the microwave resonance. The coupling between the resonators ($\mu$) can be made tunable. In this scheme the up-conversion is strongly enhanced, since both pump and the up-converted photon are resonant with the doublet. The down-conversion is strongly suppressed, since it is far detuned from the resonance of the system.

Figure 2

(a) Schematic of a coupled resonator with a tunable coupling strength that can be used to dynamically control the resonance splitting. The resonators are coupled at two points using directional couplers with $100\%$ coupling (with the coupling matrix as shown in the figure), and each has an integrated phase-shifter that imparts the phase with the opposite sign. (b) The resonance splitting can expand or contract by controlling the phase $\varphi$.

Figure 3

Calculated resonance splitting vs electrostatic voltage applied to the phase shifter for the LN coupled resonator with the structure shown in Fig. 2 and for three phase-shifter lengths of 100, 200, and 300 microns, as specified in the figure. For this calculation we assume a phase shift of $\pi$ is obtained with a 2 V cm, and use Eq. (6). We also assume the resonator has an $\text{FSR}$ of 100 GHz, $n_g=2.39,n_{\mathrm{eff}}=2$.

Figure 4

The cross section of an LN optical resonator buried in ${\mathrm{SiO}}_2$ and integrated with microwave electrodes. In (a) electrodes are on top and the bottom of the LN resonator, while in (b) one of the electrodes is on top of the LN resonator and two ground electrodes are on the sides. In both (a) and (b) the SC qubit (details not shown) is implemented on the Si substrate at some distance from the resonator. The SC qubit is capacitively connected (not shown) to the top electrode on the LN resonator via a long meander microstrip which is also an inductor. This inductor and the capacitive electrode on top of the LN resonator forms the microwave cavity. (c, d) The normalized electric potential and electric field lines in the devices in (a) and (b), respectively, assuming the microwave frequency of ${\omega }_M=2\pi \times 6$ GHz. (e) The cross section of the normalized transverse electric optical mode profile of the LN resonator at a wavelength of 1550 nm. The simulation parameters in (c)–(e) are $W=1.2\mu \text{m},H=0.75\mu \text{m},S_1=S_2=2\mu \text{m},G=3\mu \text{m},\text{and}L=1.5\mu \text{m}$. The color bar corresponds to the figures in (c)–(e).

Figure 5

Calculated $g$ factor for the Z-cut LN coupled resonator for different FSRs and for $\alpha =1$. (a) The resonator has a cross section shown in Fig. 4 with parameters of $W=1.2\mu \text{m},S_1=S_2=2\mu \text{m},G=3\mu \text{m}$, and different $H$ values as specified in the figure. (b) The resonator has a cross section shown in Fig. 4 with the parameters of $W=1.2\mu \text{m},S_1=S_2=2\mu \text{m},L=1.5\mu \text{m}$, and different $H$ values as specified in the figure. In both figures in (a) and (b) the resonator FSR values (i.e., resonator perimeters) that provide a microwave electrode capacitance $\ge 40fF$ have been specified.

Figure 6

Calculated $g$ factor for the Z-cut LN coupled resonator with tunable coupling as shown in Fig. 2 versus the FSR and for phase-shifter lengths of $200\mu \text{m}$, and $\alpha =0.55$. (a) The resonator has a cross section shown in Fig. 4 with the parameters of $W=1.2\mu \text{m},S_1=S_2=2\mu \text{m},G=3\mu \text{m}$, and different $H$ values as specified in the figure. (b) The resonator has a cross section shown in Fig. 4 with the parameters of $W=1.2\mu \text{m},S_1=S_2=2\mu \text{m},L=1.5\mu \text{m}$, and different $H$ values as specified in the figure. In both figures in (a) and (b) the resonator FSR values (i.e., resonator perimeters) that provide a microwave electrode capacitance $>40fF$ have been specified.

Figure 7

Schematic of the coupled resonator microwave-to-optical conversion with input and output parameters specified. The optical pump at a frequency of ${\omega }_s$ and the vacuum mode at frequency ${\omega }_{\mathrm{as}}$ are coupled in from a waveguide to the symmetric and antisymmetric modes of the coupled resonator, respectively.

Figure 8

Plots of (a) the optical pump power in the waveguide [Eq. (38)] and (b) the pump photon number inside the resonator [Eq. (33)] at different $g$ rates. The solid plot corresponds to $Q_{\mathrm{opt}}=Q_{\mathrm{ex},\mathrm{opt}}/2=Q_{i,\mathrm{opt}}/2={10}^6$ and $Q_M=Q_{\mathrm{ex},M}/2=Q_{i,M}/2=5000$. The dashed plot corresponds to $Q_{\mathrm{opt}}=Q_{\mathrm{ex},\mathrm{opt}}/2=Q_{i,\mathrm{opt}}/2={10}^5$ and the same $Q_M$ as in (a). Both the results in (a) and (b) are for the optimal cooperativity ($C=1$).

Figure 9

Plots of (a) the optical pump power in the waveguide [Eq. (38)] and the scattered (lost) power from the coupled resonator [Eq. (39)], and (b) the microwave-to-optical conversion factor [Eq. (35)] for different coupling $Q$ rates between the optical resonator and the waveguide characterized by $Q_{\mathrm{ex},\mathrm{opt}}$. These plots and their corresponding equations are for the optimal cooperativity ($C=1$). The other device parameters are shown in the insets of the figures.

Figure 10

Variation of the microwave cavity $Q$ due to quasiparticles generated by optical photons scattered out from the optical resonator and reaching the microwave cavity. This simulation considers a worst case scenario where all the scattered optical photons are absorbed by the microwave cavity.