Superconducting resonator and Rydberg atom hybrid system in the strong coupling regime

We propose a promising hybrid quantum system, where a highly excited atom strongly interacts with a superconducting $LC$ oscillator via the electric field of capacitor. An external electrostatic field is applied to tune the energy spectrum of the atom. The atomic qubit is implemented by two eigenstates near an avoided-level crossing in the dc Stark map of a Rydberg atom. Varying the electrostatic field brings the atomic-qubit transition on or off resonance with respect to the microwave resonator, leading to a strong atom-resonator coupling with an extremely large cooperativity. Like the nonlinearity induced by Josephson junctions in superconducting circuits, the large atom-resonator interface disturbs the harmonic potential of the resonator, resulting in an artificial two-level particle. Different universal two-qubit logic gates can also be performed on our hybrid system within the space where an atomic qubit couples to a single photon with an interaction strength much larger than any relaxation rates, opening the door to the cavity-mediated state transmission.

Figure 1

(a) A ${}^{87}\mathrm{Rb}$ atom interacts with an $LC$ oscillator. The capacitor $C$ is composed of a pair of equal-sized conducting spheres with a radius of $0.3\mu \mathrm{m}$ and an intercenter distance of $3.1\mu \mathrm{m}$ along the $z$ axis, resulting in $C=20$ aF. The inductance is chosen to be $L=24.8\mu \mathrm{H}$, leading to the oscillation frequency ${\omega }_0=2\pi \times 7.1$ GHz. The ${}^{87}\mathrm{Rb}$ atom is placed at the halfway point (the origin of the coordinates) between two spheres and couples to the quantized electric field (the amplitude ${\mathcal{E}}_o$) of capacitor. A pair of parallel plates (in the $x-y$ plane) with an imposed voltage difference of $U$ is applied to generate an electrostatic-field bias ${\mathcal{E}}_s$ at the position of atom, inducing the dc Stark shifts of atomic states. Distributions of ${\mathcal{E}}_o$ and an example of ${\mathcal{E}}_s$ around the position of atom in the $x-z$ plane are shown in panels (b) and (c), respectively.

Figure 2

Atomic qubit. (a) dc Stark map in the electrostatic field ${\mathcal{E}}_s$ with the $z$ component of total angular momentum $m=\frac{5}{2}$. The detailed energy spectrum surrounded by the rectangle frame is shown in panel (b), where an avoided crossing with an energy separation $\mathrm{\Omega }=2\pi \times 3.2$ GHz occurs at ${\mathcal{E}}_s^{\left(a\right)}\equiv 550.7$ V/cm. Two eigenstates $|\downarrow \rangle$ and $|\uparrow \rangle$ at ${\mathcal{E}}_s^{\left(q\right)}\equiv 530$ V/cm near the anticrossing are chosen to form an atomic qubit. The qubit-transition frequency is ${\omega }_q=2\pi \times 27.1$ GHz. When ${\mathcal{E}}_s$ is adiabatically reduced to zero, $|\uparrow \rangle$ and $|\downarrow \rangle$ approach $22D_{5/2}$ and a superposition state ${\mathrm{\Phi }}_{n=20}$ combined by a set of $|n=20,l\ge 3,j,m=\frac{5}{2}\rangle$, respectively. (c) Vacuum Rabi oscillation of atom between $|\downarrow \rangle$ and $|\uparrow \rangle$ with ${\mathcal{E}}_s={\mathcal{E}}_s^{\left(a\right)}$ and the atom initially in $|\uparrow \rangle$. The Rabi frequency is $\mathrm{\Omega }$.

Figure 3

(a) Energy spectrum of hybrid system as a function of the electrostatic field ${\mathcal{E}}_s$. The energy curves associated with $|\downarrow \rangle$ and $|\uparrow \rangle$ are thickened. The energy spacings of avoided-level crossings occurring between $|\downarrow ,N+1\rangle$ and $|\uparrow ,N\rangle$ at ${\mathcal{E}}_s={\mathcal{E}}_s^{\left(c\right)}\equiv 543.3$ V/cm are $\sqrt{N+1}g$ with $g=2\pi \times 0.90$ GHz. (b) Vacuum Rabi oscillations of ${\mathcal{N}}_{\uparrow }$ and ${\mathcal{N}}_p=\langle a^{†}a\rangle$ by solving the time-dependent Schrödinger equation with ${\mathcal{E}}_s={\mathcal{E}}_s^{\left(c\right)}$. The atom is initially prepared in $|\uparrow \rangle$ while no microwave photons exist inside the cavity.

Figure 4

cnot operation, where the atom acts as the control qubit while the photonic qubit plays the target role. The different steps are discussed in text.