Quantum network of superconducting qubits through an optomechanical interface

We propose a scheme to realize quantum networking of superconducting qubits based on the optomechanical interface. The superconducting qubits interact with the microwave photons, which then couple to the optical photons through the optomechanical interface. The interface generates a quantum link between superconducting qubits and optical flying qubits with tunable pulse shapes and carrier frequencies, enabling transmission of quantum information to other superconducting or atomic qubits. We show that the scheme works under realistic experimental conditions and it also provides a way for fast initialization of the superconducting qubits under 1 K instead of an operation temperature of 20 mK.

Figure 1

(a) Schematic diagram of the optomechanical quantum interface. The SQ couples with the microwave mode $a_2$ in a superconducting resonator (SR). The mechanical oscillator (MO) mode $a_m$ for vibration of the interface couples simultaneously to the mode $a_2$ of the SR and the mode $a_1$ of the optical cavity (OC). Both modes $a_2$ and $a_1$ are driven by coherent classical fields on the red sideband. (b) Energy levels of the superconducting junction, where $|g\rangle$ is the ground state, $|e\rangle$ is the first excited state, and $|s\rangle$ is the second excited state. The transition $|g\rangle$ to $|e\rangle$ couples to the mode $a_2$ with coupling rate $g_c$, while transition $|e\rangle$ to $|s\rangle$ is driven by a microwave field with Rabi frequency $\Omega \left(t\right)$.

Figure 2

(a) Shape of the output single-photon pulse $\left|f\right(t\left)\right|$. We take $g=G=3\kappa$ and the pulse duration $T_D=20/\kappa$. The driving pulse $\Omega \left(t\right)=g{e}^{-{(t-T_D/2)}^2/2t_w^2}$ is assumed to be a Gaussian shape with the peak at $t=T_D/2$ and a width $t_w=T_D/5$. The solid, dashed, and dash-dotted curves represent the analytic pulse shape in Eq. (4) derived in the adiabatic limit, the numerical result that includes the contribution of the bright state $|B\rangle$, and the exact result that includes contributions of all the modes $b,b_{\pm }$, respectively. The shape function is normalized according to ${\int \left|f\left(t\right)\right|}^{2}dt=1$ for the convenience of comparison. The overlap between the exact shape (dash-dotted curve) and the adiabatic shape (solid curve) is about $99\%$. (b) Same as (a) but with the pulse duration $T_D=5/\kappa$. The adiabatic approximation is not well satisfied in this case and the shape overlap is reduced to $80\%$. (c) Same as (a) but with the driving Rabi frequency $\Omega \left(t\right)=(g_c/\sqrt{2})e^{\kappa (t-T_D/2)/2}$, which gives a symmetric output pulse shape [34]. In the adiabatic limit, the shape (solid curve) is given by the analytic form $f\left(t\right)=\sqrt{\kappa /4}\text{sech}[\kappa (t-T_D/2)/2]$, which has an overlap of $99.7\%$ with the exact shape. (d) Same as (c) but with the pulse duration $T_D=5/\kappa$.

Figure 3

(a) Schematic to generate entanglement between remote SQs. Two SQs are located in distant cavities $A$ and $B$. The SQs with dashed boxes represent the same structure as the orange part (SQ) in Fig. 1, capacitively coupled to the SRs. The SQs couple to the output photons through optomechanical interfaces. The output photons, after propagation, interfere at a beam splitter and then are detected by single-photon counters. Registration of a photon count generates entanglement between the remote SQs. (b) The same setup can be used to entangle SQs with other kinds of matter qubits, such as trapped ions. The carrier frequency and shape of the photon from the SQ is tuned by the optomechanical interface to match with the photon pulse from other matter qubits.

Figure 4

(a) Temperature dependence of the fidelity $F$ (solid curve) of quantum interface and the fidelity $P_g$ (dashed curve) for state initialization. The dash-dotted curve shows the probability in the ground state without optomechanical sideband cooling. The parameters are taken as ${\omega }_1/2\pi =200$ THz, ${\omega }_2/2\pi =10$ GHz, ${\omega }_m/2\pi =10$ MHz [24, 25, 42], ${\kappa }_1/2\pi =10$ MHz, ${\kappa }_2/2\pi =1$ kHz, ${\kappa }_m/2\pi =10$ Hz [43], $\gamma /2\pi =5$ kHz [44, 45], $G/2\pi =1$ MHz, and $g_c/2\pi =1$ MHz. (b) Dependence of the fidelity $F$ (solid curve) and $P_g$ (dashed curve) on the optical cavity decay rate ${\kappa }_1$ at a temperature of 1 K. The other parameters are the same as in (a).