Figure 1

Schematic circuit for generating entanglement between two superconducting resonators. Resonator $A$ (blue) has a fundamental frequency ${\omega }_a/2\pi$, while resonator $B$ (red) has frequency ${\omega }_b/2\pi$. These are each capacitively coupled to a tunable qubit (gray) with frequency ${\omega }_q/2\pi$, with coupling strengths $g_a$ and $g_b$. The qubit is controlled by an external circuit (black). The theoretical results described in the text require ${\omega }_a<{\omega }_q<{\omega }_b$.Reuse & Permissions

Figure 2

Schematic set of operations to generate an arbitrary state of two resonators. In this Fock-state diagram, the state $|q,n_a,n_b⟩$ is represented by the node at location ($n_a,n_b$). (a) Interactions lead to couplings between these states, indicated by the arrows. Three key interactions are used: $A$ transfers quanta between the qubit and resonator $a$ (solid lines), $B$ transfers quanta between the qubit and resonator $b$ (dashed lines), and $R$ (curved arrows) rotates the qubit for states with $n_a-n_b=n$ (here with $n=0$, see text). (b) Numerical simulations of Stark-shifted Rabi oscillations for ${\omega }_a/(2\pi )=6.3\mathrm{GHz}$, ${\omega }_b/(2\pi )=7.7\mathrm{GHz}$, ${\omega }_q/(2\pi )=7\mathrm{GHz}$, $g_a/(2\pi )=g_b/(2\pi )=70\mathrm{MHz}$. The Rabi oscillations are driven at ${\omega }_d/(2\pi )=7.025\mathrm{GHz}$ and $\Omega /(2\pi )=7\mathrm{MHz}$ (with $n_a-n_b=2$). Each block corresponds to the maximum probability of the transition $|0,n_a,n_b⟩\to |1,n_a,n_b⟩$ (see text).Reuse & Permissions

Figure 3

Algorithm to generate the state $|\Psi ⟩=|3,0⟩+|0,3⟩$ of two coupled resonators. The sequence of operations is detailed in Table . The horizontal, vertical, and curved transitions are the interactions $A$, $B$, and $R$ (see text).Reuse & Permissions