Entanglement of Two Superconducting Qubits in a Waveguide Cavity via Monochromatic Two-Photon Excitation

We report a system where fixed interactions between noncomputational levels make bright the otherwise forbidden two-photon $|00⟩\to |11⟩$ transition. The system is formed by hand selection and assembly of two discrete component transmon-style superconducting qubits inside a rectangular microwave cavity. The application of a monochromatic drive tuned to this transition induces two-photon Rabi-like oscillations between the ground and doubly excited states via the Bell basis. The system therefore allows all-microwave two-qubit universal control with the same techniques and hardware required for single qubit control. We report Ramsey-like and spin echo sequences with the generated Bell states, and measure a two-qubit gate fidelity of $F_g=90\%$ (unconstrained) and 86% (maximum likelihood estimator).

Figure 1

(a) Picture of half a cavity with two independent transmons mounted. The inset is a magnified image of one of the transmons. (b) Energy diagram of the system. Single and two-photon transitions are depicted, respectively, with one and two arrows. (c) Spectrum of the system. The activation of the transition $|00⟩\to |11⟩$ requires more power than any other transitions here measured because it is a second order transition further reduced by a factor of $J/\Delta$. The dressed cavity resonance frequency is at 11.7781 GHz.

Figure 2

(a) Two-photon coherent oscillation between the states $|00⟩$ and $|11⟩$. The oscillation saturates at approximately $1/4$ of its visibility because in the steady state all four states $|00⟩$ through $|11⟩$ are equally populated. (b) Oscillation frequency versus driving amplitude. Black dots are experimental data, the solid blue line is a multilevel numerical simulation, the dotted red line is derived from perturbation theory of coupled multilevel systems, and the dashed green line is from perturbation theory applied to coupled two-level systems. We ascribe the discrepancy observed at large amplitudes to the higher levels of the system, an effect that is captured by the numerical simulation. (c) Evaluation of the dephasing time of the Bell state with a spin-echo experiment. The refocusing bSWAP is symmetrically placed between two $\sqrt{\mathrm{bSWAP}}$ gates (inset), and the oscillation is experimentally induced by linearly ramping the phase of the last pulse in time.

Figure 3

(a)–(b) Tomographic reconstruction of the two Bell states $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$ and $\frac{1}{\sqrt{2}}(|00⟩+i|11⟩)$. (c) Expectation value of the two-qubit Pauli operators versus the relative phase between microwave sources used to generate the $\sqrt{\mathrm{bSWAP}}$ gate and the single qubit pulses.

Figure 4

Experimental [(a) and (c)] and ideal [(b) and (d)] Pauli transfer matrices for the $\sqrt{\mathrm{bSWAP}}$ [(a) and (b)] and the bSWAP [(c) and (d)] gates.