Charge-qubit–atom hybrid

We investigate a hybrid system of a superconducting charge qubit interacting directly with a single neutral atom via electric dipole coupling. Interfacing of the macroscopic superconducting circuit with the microscopic atomic system is accomplished by varying the gate capacitance of the charge qubit. To achieve a strong interaction, we employ two Rydberg states with an electric-dipole-allowed transition, which alters the polarizability of the dielectric medium of the gate capacitor. Sweeping the gate voltage with different rates leads to a precise control of hybrid quantum states. Furthermore, we show a possible implementation of a universal two-qubit gate.

Figure 1

Schematic diagram of a CPB interacting with an atomic qubit. A single ${}^{87}\mathrm{Rb}$ atom is placed inside the gate capacitor $C_g$ and couples to the electric field $\mathcal{E}$ between two parallel identical plates with an area $S=4\mu \text{m}\times 4\mu \text{m}$ and a separation $l=0.5\mu \text{m}$. The gate capacitance of the empty capacitor (without the extra atom) is $C_{g0}=283.3$ aF and the corresponding empty charging energy is $E_{C0}=2\pi \hslash \times 247.4$ GHz. The self-capacitance of the Josephson junction is chosen to be $C_j=30$ aF and the Josephson energy is set to $E_J=2\pi \hslash \times 100$ GHz with the corresponding critical current $I_c=200$ nA. The transition frequency between two Rydberg $|g\rangle =31s$ and $|e\rangle =31p$ states is ${\omega }_a=2\pi \times 141.1$ GHz and the corresponding electric dipole moment is $d_{eg}=565.8e{a}_0$.

Figure 2

Eigenenergy bands ${\mathcal{E}}_k$ vs $N_{g0}$. The solid curves depict the numerical results from diagonalizing $H$. The dashed lines give the asymptotic behavior ($d_{eg}=0$). ${\alpha }_{1,2}$ and ${\beta }_{1,2}$ mark two anticrossings caused by the Josephson tunneling at $N_{g0}=0.5$ and two anticrossings induced by the interqubit coupling at $N_{g0}\approx 0.3$ and $0.7$, respectively. The system eigenstates at $N_{g0}=0$, 0.5, and 1 are labeled, where $|u,\pm \rangle =\frac{1}{\sqrt{2}}\left(|u,0\rangle \pm |u,1\rangle \right)$ with $u=e,g$.

Figure 3

The expectation values ${\mathcal{N}}_k^{\left(c\right)}$ of the excess number of Cooper pairs in the box and the atomic populations ${\mathcal{N}}_k^{\left(u\right)}$ in $|u=g,e\rangle$ are plotted in (a), (b), and (c), respectively. (d) The concurrences ${\mathcal{C}}_k$ for the two-qubit system in different ${\mathrm{\Psi }}_k$. In the bands of $k=1$ and $k=2$ the charge-atom entanglement maximizes at two anticrossings ${\beta }_{1,2}$ while two qubits are weakly coupled with each other in the bands of $k=0$ and $k=3$.

Figure 4

Single-passage sweep of the hybrid system with an initial state ${\psi }_i=|e,0\rangle$. (a) The probabilities $P_{u,n}$ of the system at $N_{g0}=1$ in different basis states $|u,n\rangle$ as a function of the sweep rate $v$. Three regimes of the sweep rate $v$ can be identified: In the limit of $v\ll {\mathrm{\Delta }}_2/\left(2\pi \right)$, the system at $N_{g0}=1$ mainly remains in $|e,1\rangle$. In the intermediate regime $v\sim {\mathrm{\Delta }}_2/\left(2\pi \right), P_{g,0}$ and $P_{e,1}$ at $N_{g0}=1$ oscillate strongly. For $v\gg {\mathrm{\Delta }}_2/\left(2\pi \right)$, the system is transferred to $|e,0\rangle$. Two examples of $P_{u,n}$ vs $N_{g0}$ with $v=1{\mathrm{ns}}^{-1}$ and $v=10{\mathrm{ns}}^{-1}$ are shown in (b) and (c), respectively. (d) Three steps of the cnot gate. The atom in $|g\rangle$ is transferred in an auxiliary state; for example, $|\text{aux}\rangle =5{}^2{S}_{1/2}$, in step 1. In step 2, the gate charge bias is swept adiabatically from 0 to 0.5 and the transition between bands $k=1$ and $k=3$ is accomplished via the Raman transition. The charge bias is swept back to 0 and the atom in $|\text{aux}\rangle$ is brought back to $|g\rangle$ in step 3.