Figure 1

Jaynes-Cummings Hamiltonian transitions. (a) Energy level diagram, with the four basic transitions ${\omega }_{n,+}$ (red solid arrows, on left), ${\omega }_{n,-}$ (blue solid arrows, on right), ${\omega }_{n,⤡}$ (blue dashed arrows, right to left), and ${\omega }_{n,⤢}$ (red dashed arrows, left to right). (b) Transition frequencies ${\omega }_n/2\pi$ as a function of $n$, for ${\omega }_r/2\pi =6\mathrm{GHz}$, $g/2\pi =180\mathrm{MHz}$, and $\Delta =0$. (c) Transition matrix elements $⟨j|{\sigma }_x|k⟩$ due to a drive on the qubit. (d) Transition matrix elements $⟨j|x|k⟩$, with $x=a+a^{†}$, due to a drive on the resonator. Dark values indicate larger matrix elements, while the corresponding transitions are indicated by the symbols.Reuse & Permissions

Figure 2

(a) Error as a function of the pulse time $T$ for the Fock state transitions $|0⟩\to |N,-⟩$, for $N=4$ and using the same parameters as Fig. 1. The circles are for a numerically optimized two-quadrature pulses with $M=1$ Fourier component, while the squares are for $M=3$ Fourier components. (b) Time-dependent probabilities $|⟨v|\Psi (t)⟩{|}^2$ as a function of time $t$ for an $M=3$ pulse with $T=50\mathrm{ns}$, where $|v⟩=|0,0⟩,|1,+⟩,|2,-⟩,|3,+⟩$, and $|4,-⟩$.Reuse & Permissions

Figure 3

Qudit operations with the Jaynes-Cummings ladder. (a) Coupling graph for the states $|0⟩$ and $|n,-⟩$, for $n=1\to 7$. The darker edges correspond to couplings through the transitions ${\omega }_{⤢,⤡}$, while the lighter edges correspond to the transitions ${\omega }_{\pm }$. The intermediate states $|n,+⟩$ have been omitted for simplicity. (b) Schematic rotation ${\mathcal{R}}_{0,4}(\theta )$, consisting of the sequence ${\mathcal{R}}_{0,3}(\pi ){\mathcal{R}}_{3,4}(\theta ){\mathcal{R}}_{0,3}(\pi )$, whose second step is indicated by the curved arrows (see text).Reuse & Permissions