---
author:
- 'Yuta Tsuchimoto, Patrick Knüppel, Aymeric Delteil, Zhe Sun, Martin Kroner,'
- Ataç Imamoğlu
title: |
    SUPPLEMENTAL MATERIAL\
    Quantum interface between photonic and superconducting qubits
---

Quantum Monte Carlo method
--------------------------

We consider optical to microwave conversion. In the quantum trajectory formalism, the time evolution of the system is given by the Schrödinger equation: $$\frac{d}{dt}\ket{\psi(t)} = \frac{1}{i\hbar} H_{\mathrm{eff}} \ket{\psi(t)},$$ where $\psi(t)$ is a stochastic wavefunction and $H_{\mathrm{eff}}$ is the non-Hermitian Hamiltonian given by $$\label{hamiltonian}
H_{\mathrm{eff}} = H_{\mathrm{s}} + H_{\mathrm{t}} + H_{\mathrm{st}} - \frac{i \hbar}{2} \sum_k \hat{C}^\dagger_k \hat{C}_k.$$ where $H_{\mathrm{s}}$ ($H_{\mathrm{t}}$) is the source (target) Hamiltonian and $H_{\mathrm{st}}$ the interaction Hamiltonian. The collapse operators $\hat{C}_k$ of this system are $$\begin{aligned}
\hat{C}_1  &=  \sqrt{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} +  \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} \eta} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\ 
\hat{C}_2  &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})} (1-\eta)} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C}_3  &=  \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C}_4  &=  \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}},
\end{aligned}$$ where $\sigma_{ij} = \ket{i}\bra{j}$ express the projection ($i=j$) and lowering or rising operator ($i \neq j$) respectively. (s) and (t) stand for source and target (interface). $\hat{C}_1$ corresponds to a detection event where photons are emitted either in the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ or $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transitions. The latter transition is coupled to the incident light with a coupling efficiency $\eta$. The operator $\hat{C}_2$ denotes an event originating from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition which does not couple to the incident mode. $\hat{C}_3$ describes an event associated with the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition. Finally, $\hat{C}_4$ accounts for the cavity decay event. By substituting these collapse operators, equation \[hamiltonian\] becomes: $$\begin{aligned}
H_{\mathrm{eff}} = && \hbar \Omega_{\mathrm{L}} (\sigma_{\mathrm{GF}}^{({\mathrm{s}})} + \sigma_{\mathrm{FG}}^{({\mathrm{s}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ &&- i\hbar \frac{\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{s}})} - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})} - i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})}\nonumber\\ && -i\hbar \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}\eta} \sigma_{\mathrm{EF}}^{({\mathrm{s}})} \sigma_{\mathrm{FG}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_c}{2}\hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}.\end{aligned}$$ Here, we neglected the term $\sigma_{\mathrm{FE}}^{({\mathrm{s}})}\sigma_{\mathrm{GF}}^{({\mathrm{t}})}$ describing the reverse process where an emitted photon from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition drives the $\ket{E}_{\mathrm{s}} \rightarrow \ket{F}_{\mathrm{s}}$ transition because we assume unidirectional coupling realized by a Faraday rotator or a chiral waveguide. We calculated the time evolution of the stochastic wave function using the quantum Monte Carlo wave function approach. We set the initial state as the ground states for both the source and interface and assumed that the cavity does not have microwave photons, i.e. $\ket{\psi_{\mathrm{initial}}} = \ket{G}_{\mathrm{s}}\ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$. As the excitation laser pulse, we chose a Gaussian pulse with a peak Rabi frequency $\Omega_0$ which is of the same order as $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The bandwidth of the pulse was set to be smaller than $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}$. The generated single-photon pulse shape from the $\ket{F}_{\mathrm{s}} \rightarrow \ket{E}_{\mathrm{s}}$ transition was ensured to be Gaussian by keeping $\Omega_0/\Gamma_{\mathrm{FE}}^{({\mathrm{s}})}$ small. $\eta$ was assumed to be 1.0. By assuming a realistic CQD and SC cavity, we set each parameter as follows: $\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/2\pi = 300{\mskip3mu}{\mathrm{MHz}}$, $\kappa_c/2\pi = 3 {\mskip3mu}{\mathrm{MHz}}$, $g_c/2\pi = 50-400{\mskip3mu}{\mathrm{MHz}}$, $\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}/2\pi = 0-750 {\mskip3mu}{\mathrm{MHz}}$. Since the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$ transition heralds a successful optical-to-microwave photon conversion, we counted this event to estimate the conversion efficiency and rate.

Next, we consider microwave to optical conversion. Here, the collapse operators of this scheme are as follows: $$\begin{aligned}
\hat{C'}_1 &= \sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GF}}^{({\mathrm{t}})}, \\
\hat{C'}_2 &= \sqrt{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}} \sigma_{\mathrm{GE}}^{({\mathrm{t}})}, \\
\hat{C'}_3 &= \sqrt{\kappa_{\mathrm{c}}} \hat{a}_{\mathrm{c}}.
\end{aligned}$$ The effective Hamiltonian is then given by $$\begin{aligned}
H_{\mathrm{eff}} =&& \hbar \Omega_{0} (\sigma_{\mathrm{GE}}^{({\mathrm{t}})} + \sigma_{\mathrm{EG}}^{({\mathrm{t}})}) + \hbar g_{\mathrm{c}} (\hat{a}_{\mathrm{c}} \sigma_{\mathrm{FE}}^{({\mathrm{t}})} + \hat{a}_{\mathrm{c}}^\dagger \sigma_{\mathrm{EF}}^{({\mathrm{t}})})\nonumber\\ && - i\hbar \frac{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{FF}}^{({\mathrm{t}})}
- i \hbar \frac{\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{2}\sigma_{\mathrm{EE}}^{({\mathrm{t}})} -i \hbar \frac{\kappa_{\mathrm{c}}}{2} \hat{a}_{\mathrm{c}}^\dagger \hat{a}_{\mathrm{c}}\end{aligned}$$ The initial state is $\ket{\psi_{\mathrm{initial}}} = \ket{G,1_{\mathrm{c}}}_{\mathrm{t}}$. The parameters used for this simulation are the same as those of the optical to microwave conversion except for $\Omega_0 = \Gamma_{\mathrm{FG}}^{({\mathrm{t}})}/3$. We counted the decay event $\hat{C'}_1$ to estimate the conversion efficiency and rate.

Analytical formula for the optical-to-microwave conversion efficiency
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We consider a simple case where weak coherent field resonantly couples the target CQD with perfect mode matching $\eta = 1.0$ based on van Enk [@vanEnk]. Here, the coherent field satisfies $$b_{\mathrm{in}}(t)\ket{\beta} = \beta {\mathrm{exp}}(-i\omega_{\mathrm{in}}t)\ket{\beta},$$ where $b_{\mathrm{in}}$ is the input field operator, $\omega_{\mathrm{in}}$ is the incident photon frequency, and $\beta$ is the amplitude of the input field. Based on the input-output formalism, we write the mean output field $\braket{b_{\mathrm{out}}}$ generated from the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition as follows: $$\braket{b_{\mathrm{out}}} = \beta+\sqrt{\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}}\braket{\sigma_{\mathrm{FG}}^{({\mathrm{t}})}}.$$ Assuming that $\kappa_{\mathrm{c}}$ is sufficiently smaller than $g_{\mathrm{c}}$ and all the other decay rates, we find that an analytical formula of the mean field is $$\braket{b_{\mathrm{out}}} = \beta-\frac{2\beta\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}}{4g_{\mathrm{c}}^2/\Gamma_{\mathrm{FG}}^{({\mathrm{t}})}+\Gamma_{\mathrm{EG}}^{({\mathrm{t}})}-ig_{\mathrm{c}}}.$$ Here, we define a normalized mean field $\braket{b_{\mathrm{out}}^{\mathrm{n}}} = \braket{b_{\mathrm{out}}}/\beta$. The efficiency of the $\ket{F,0_{\mathrm{c}}}_{\mathrm{t}} \rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition is given by the conjugate product $\braket{b_{\mathrm{out}}^{\mathrm{n\dagger}} b_{\mathrm{out}}^{\mathrm{n}}}$. One can calculate the conversion efficiency $\zeta$ as the complement of this decay efficiency as shown in the main text.

![image](SP1.pdf){width="75.00000%"}

Erasure of time-bin information of the herald photons
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Figure \[S1\] shows a proposed optical setup which compensates for the delay between the two time-bin components of photons emitted from the $\ket{E,1_{\mathrm{c}}}_{\mathrm{t}}\rightarrow \ket{G,0_{\mathrm{c}}}_{\mathrm{t}}$ transition, which allows to remove entanglement between the transferred qubits and the herald photons. The polarization of these photons is defined by the QD selection rules. Here, we note $\ket{H}$ ($\ket{V}$) the horizontally (vertically) polarized state, and we assume that the photons are initially $H$-polarized. A Pockels-cell driven synchronously with the state transfer protocol selectively rotates the early component from $\ket{H}$ to $\ket{V}$. This allows to use a polarized beam splitter to channel the two time components into two separate optical paths, such that the early component is delayed by $t_2 - t_1$ and rotated back to $\ket{H}$. Therefore the two components arrive simultaneously and in the same polarization state onto a second (non-polarized) beam splitter. A single-photon detector placed at one of the output port heralds a successful operation. This process is inherently probabilistic but does not require additional local operation on the qubit. On the other hand, the scheme can be in principle rendered deterministic by using two single-photon detectors of high ($\sim 100\%$) efficiency at the two output ports of the beam splitter. In this case, an additional qubit rotation would have to be performed on the transmon qubit state depending on which detector has clicked, due to the $\pi$ phase difference between the final qubit states left after a photon detection in either of the two output ports of the beam splitter.

Coupling strength
-----------------

We show that the large dipole of the CQD interacting with enhanced cavity vacuum electric field leads to large $g_{\mathrm{c}}$ necessary for the fast and high-efficient conversion. The dipole moment of a CQD is given by $p \sim a \cdot e/2$ where $e$ is a charge of an electron and $a$ is the distance between the two quantum dot layers. We assume a typical distance $a \sim 10{\mskip3mu}\text{nm}$, resulting in $p \sim 8 \times 10^{-28} {\mskip3mu}\text{C$\cdot$m}$. This large dipole moment interacts with the cavity vacuum electric field given by $$E_{\mathrm{rms}} = \sqrt{\frac{\hbar\omega_{\mathrm{c}}}{2\epsilon_{\mathrm{0}}\epsilon_{\mathrm{eff}}V_{\mathrm{eff}}}}
   \label{eq:Vrms}$$ where $\omega_{\mathrm{c}}$ is the resonant frequency of the SC cavity. $\epsilon_{\mathrm{0}}$ and $\epsilon_{\mathrm{eff}}$ are the permittivity of vacuum and the effective dielectric constant of the SC cavity, respectively. This vacuum field can also be expressed as a function of the cavity impedance as follows: $$E_{\mathrm{rms}} = \sqrt{\frac{Z_{\mathrm{cav}}\hbar\omega_{\mathrm{c}}^2}{\pi d^2}}. 
   \label{eq:Evac}$$ where $d$ is the the gap between the center and ground conductor. $Z_{\mathrm{cav}}$ is the characteristic impedance of the SC cavity. For a typical SC cavity, $Z_{\mathrm{cav}}$ is about $50 {\mskip3mu}{\Omega}$. A high impedance cavity can be used to enhance the vacuum field [@Stoc17]. If we assume $Z_{\mathrm{cav}} \sim 2000 {\mskip3mu}\Omega$, $d \sim 7 {\mskip3mu}\mu\text{m}$ and $\omega_{\mathrm{c}}/2\pi \sim11{\mskip3mu}\text{GHz}$, the vacuum field is $E_\text{rms} \sim 3{\mskip3mu}\text{V/m}$. Furthermore, $E_\text{rms}$ is enhanced between the top gate and the ground plane by factor of $\sqrt{\epsilon_{\mathrm{eff}}/\epsilon_{\mathrm{GaAs}}} \cdot d/d'$ (see Fig. 1 in the main text). $\epsilon_{\mathrm{GaAs}}$ is the dielectric constant of GaAs, which is about 13. $\epsilon_{\mathrm{eff}}$ is given by $(\pi c/l\omega_{\mathrm{c}})^2$ where $c$ is the speed of light in vacuum and $l$ is the length of the cavity. We assumed $l \sim3{\mskip3mu}\text{mm}$ and $d' \sim 200 {\mskip3mu}\text{nm}$ as realistic values. The coupling strength is therefore $$g_{\mathrm{c}}/2\pi = \frac{p \cdot \sqrt{\epsilon_{\mathrm{eff}}/\epsilon_{\mathrm{GaAs}}} \cdot d/d' \cdot E_{\mathrm{rms}}}{2\pi\hbar} \sim 200{\mskip3mu}\text{MHz}.
  \label{eq:g}$$ This large coupling strength satisfies the conditions for the high conversion efficiency and rate discussed in the paper.

S. J. van Enk, Phys. Rev. A **69**, 043813 (2004).

A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Phys. Rev. X **7**, 011030 (2017).
