Measurement-induced long-distance entanglement of superconducting qubits using optomechanical transducers

Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge because they cannot be interfaced to light, the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediate such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. The approach is scalable to the generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.

Figure 1

Schematic of the setup. Each of two qubits (e.g., superconducting transmon qubits, shown as black circuits) interacts with a mechanical oscillator (nanobeam, shown in yellow) that, in turn, couples to an optical mode. The optical resonators (blue toroids) are unidirectionally coupled; the output of the first (top) cavity enters the second (bottom) one. Homodyne measurement at the output of the second cavity provides information about the joint state of the two qubits and can be used to postselect an entangled state.

Figure 2

Possible experimental realizations of the proposed scheme. Transmon qubits can interact with mechanical oscillators via position-dependent gate capacitance. The mechanical oscillator can be formed by a nanobeam that interacts with the near field of a toroidal optical resonator (a) or a membrane used as an end mirror or in the middle of a Fabry-Perot cavity (b). Mechanical oscillators can further be integrated into the circuit of a flux qubit (c). Other solid-state qubits—such as magnetic sublevels of a nitrogen-vacancy centers—can be used as well, exploiting magnetic coupling to cantilevers (d). Superconducting circuits are indicated in black, a nitrogen-vacancy center in red, mechanical oscillators in yellow, and optical modes in blue.

Figure 3

Entanglement generation with the nanobeam optomechanical transducer of Fig. 2. In panel (a) we show the time dependence of the concurrence for success probability $P_{\mathrm{succ}}=0.1$ (solid blue line) and $P_{\mathrm{succ}}=0.5$ (dashed green line). In the bottom panels, we plot the concurrence (optimized over the measurement time) versus optical transmission $\tau$ between the two nodes. In (b) we plot the concurrence for several detection efficiencies $[\eta =1$ (solid blue line), $\eta =0.8$ (dashed green line), $\eta =0.6$ (dotted red line), $\eta =0.4$ (dot-dashed magenta line), and $\eta =0.2$ (thin black line)]; in (c) the plotted curves represent different success probabilities $[P_{\mathrm{succ}}=0.1$ (solid blue line), $P_{\mathrm{succ}}=0.3$ (dashed green line), and $P_{\mathrm{succ}}=0.5$ (dotted red line)]. We consider qubit coupling $\chi =2\pi \times 5.8\mathrm{MHz}$, mechanical frequency ${\omega }_m=2\pi \times 8.7\mathrm{MHz}$ and quality $Q_{\mathrm{m}}=5\times {10}^4$, optical decay rate $\kappa =2\pi \times 39\mathrm{MHz}$, optomechanical coupling $g=2\pi \times 900\mathrm{kHz}$, and temperature $20\mathrm{mK}$ (corresponding to thermal occupation $\overline{n}=48$); we assume the intrinsic relaxation and coherence lifetimes of the qubits $T_{1,2}=20\mu \mathrm{s}$. Furthermore, we use the values $\tau =\eta =1$ for panel (a), $P_{\mathrm{succ}}=0.1$ for (b), and $\eta =1$ for (c).

Figure 4

Comparison of the analytical model (dashed red line) with numerical simulations (solid blue line). In the first row [panels (a)–(c)], we consider system parameters ${\gamma }_{-}=0.1\mathrm{\Gamma },{\gamma }_1={\gamma }_2=0.2\mathrm{\Gamma },\eta =0.6$; in the second row [panels (d)–(f)], the parameters are ${\gamma }_{-}=0.1\mathrm{\Gamma },{\gamma }_1=\mathrm{\Gamma },{\gamma }_2=0.3\mathrm{\Gamma },\eta =1$; and for the last row [panels (g)–(i)], we use the parameters ${\gamma }_{-}=0.8\mathrm{\Gamma },{\gamma }_1={\gamma }_2=0,\eta =1$. The success probability $P_{\mathrm{succ}}=0.1$ for the first column [panels (a),(d),(g)], $P_{\mathrm{succ}}=0.3$ for the second column [panels (b),(e),(h)], and $P_{\mathrm{succ}}=0.5$ for the last column [panels (c),(f),(i)].

Figure 5

Comparison of the total spin measurement (solid blue line) with the measurement of the spin difference (dashed red line). (a) Decoherence dominated by dephasing, ${\gamma }_1={\gamma }_2=0.2\mathrm{\Gamma },{\gamma }_{-}=0.05\mathrm{\Gamma },P_{\mathrm{succ}}=0.2$. (b),(c) Comparable dephasing and relaxation, ${\gamma }_1={\gamma }_2=0.1\mathrm{\Gamma },{\gamma }_{-}=0.3\mathrm{\Gamma },P_{\mathrm{succ}}=0.1$ (b), $P_{\mathrm{succ}}=0.5$ (c). For all plots, the detection efficiency $\eta =1$.