Quantum interface between a transmon qubit and spins of nitrogen-vacancy centers

Hybrid quantum circuits combining advantages of each individual system have provided a promising platform for quantum information processing. Here we propose an experimental scheme to directly couple a transmon qubit to an individual spin in the nitrogen-vacancy (NV) center, with a coupling strength three orders of magnitude larger than that for a single spin coupled to a coplanar waveguide microwave cavity. This direct coupling between the transmon and the NV center could be utilized to make a transmon bus, leading to a coherently virtual exchange among different single spins. Furthermore, we demonstrate that, by coupling a transmon to a low-density NV ensemble, a swap operation between the transmon and NV ensemble is feasible and a quantum nondemolition measurement on the state of the NV ensemble can be realized on the cavity-transmon-NV-ensemble hybrid system. Moreover, in this system, a virtual coupling can be achieved between the cavity and NV ensemble, which is much larger in magnitude than the direct coupling between the cavity and the NV ensemble. The photon state in the cavity can thus be stored into NV spins more efficiently through this virtual coupling.

Figure 1

A cavity-transmon-spin-ensemble hybrid system. A transmon is located in the place with a maximum electric field inside a coplanar waveguide (CPW) superconducting cavity. The transmon is covered by a diamond chip. The pink part is the substrate and blue one represents the superconductor film. The purple line is the electric field in the cavity and the red part in the close-up illustrates the insulator barrier of a Josephson junction. The diamond chip is denoted by the brown cuboid and the spins of the NV centers are indicated by black arrows. The size of the NV center is $L_N$ and the distance between two Josephson junctions in the transmon is $L$. The cross section of the transmon junction is $h\times h$.

Figure 2

Coupling strength $g_{\mathrm{ts}}$ as a function of the location of a single spin with $x=0$ for the case (a) a single-spin coupling to a single-JJ transmon and (b) a single-spin coupling to a double-JJ transmon. Each inset shows the schematic diagram of the coupling scenario. The quantization axis of the NV center is assumed to be along the $x$ direction. The critical current of the transmon, $I_c=500\mathrm{nA}$, is used for calculation. The white region in (a) represents a coupling strength larger than 8 kHz. The origin of the coordinate in the inset in (a) is located at the center of the insulator cuboid, so the top surface of the transmon is at $z=h/2=0.05\mu \mathrm{m}$. For (b), two junctions are identical and the distance between two junctions is $L=3\mu \mathrm{m}$. The origin in the inset in (b) is set at the center of the transmon loop and the top surface of the transmon is also at $z=h/2=0.05\mu \mathrm{m}$.

Figure 3

The coupling strength $g_{t-\mathrm{ens}}$ between a double-JJ transmon and NV spin ensemble as a function of diamond crystal size $L_N$ with different densities $n$ in (a) and as a function of density $n$ with different diamond crystal dimensions $L_N$ in (b). The diamond crystal has a volume of $L_N^3$. The coupling strength is calculated by summing over all the spins in the diamond cube. The dimension of transmon $L$ (the distance between two junctions) is $3\mu \mathrm{m}$. The purple dashed line represents $g_{t-\mathrm{ens}}=1\mathrm{MHz}$, indicating a strong coupling regime for the transmon-spin ensemble system.

Figure 4

(a) A dispersive readout procedure for the state of the NV ensemble. The whole measurement process consists of steps for preparation, excitation, swap operation, and pump-probe measurement to determine the state of the spin ensemble (b) An energy-level diagram of the NV-ensemble-transmon system. The transition frequency of the transmon depends on the state of the NV ensemble due to the dispersive interaction between the transmon and the NV ensemble. The dashed line represents the energy level of the transmon-NV-ensemble system without the interaction between them. $|\downarrow \rangle ,|\uparrow \rangle$ represents the ground and excited states of the transmon and $|G\rangle ,s^{†}|G\rangle ,{\left(s^{†}\right)}^2|G\rangle$ is the ground state and excited states of the NV ensemble, respectively.

Figure 5

(a) A schematic diagram showing the coupling of two spins to a single-JJ transmon. The blue part represents the superconductor and red part shows the insulator. Spins are shown as brown spheres with black arrows and purple circles show the magnetic field generated by the transmon. (b) Energy-level diagram of spin-transmon-spin system. Two energy levels indicated by the arrows interact with each other via the virtual exchange. Both spins are detuned from the transmon but are in resonance with each other to turn on the virtual exchange. (c) Energy-level diagram of cavity-transmon-spin ensemble system. The energy levels indicated by the arrows interact with each other to swap an excitation between them. The cavity and spin ensemble are detuned from the transmon to prohibit the real exchange between them.