First-order sidebands in circuit QED using qubit frequency modulation

Sideband transitions have been shown to generate controllable interaction between superconducting qubits and microwave resonators. Up to now, these transitions have been implemented with voltage drives on the qubit or the resonator, with the significant disadvantage that such implementations only lead to second-order sideband transitions. Here we propose an approach to achieve first-order sideband transitions by relying on controlled oscillations of the qubit frequency using a flux-bias line. Not only can first-order transitions be significantly faster, but the same technique can be employed to implement other tunable qubit-resonator and qubit-qubit interactions. We discuss in detail how such first-order sideband transitions can be used to implement a high fidelity controlled-not operation between two transmons coupled to the same resonator.

Figure 1

Average error with respect to the perfect red sideband process $|1;0\rangle \leftrightarrow |0;1\rangle$. A Gaussian FC pulse is sent on the first qubit at the red sideband frequency assuming the second qubit is in its ground state. Solid red line, average error of the red sideband as given by Eq. (39) when the second qubit is excited; blue dashed line, population transfer error $1-P_t$, with $P_t$ given by Eq. (41); black dots, numerical results for the average error. We find the evolution operator after time $t_p$ for each eigenstate of the second qubit. The fidelity is extracted by injecting these unitaries in Eq. (34). The qubits are taken to be transmons, which are modeled as four-level Duffing oscillators (see Sec. 5a) with $E_J^{\left(1\right)}=25$ GHz, $E_J^{\left(2\right)}=35$ GHz, $E_C^{\left(1\right)}=250$ MHz, $E_C^{\left(2\right)}=300$ MHz, yielding ${\omega }_{01}^{\left(1\right)}=5.670$ GHz and ${\omega }_{01}^{\left(2\right)}=7.379$ GHz, and $g_{01}^{\left(1\right)}=100$ MHz. The resonator is modeled as a five-level truncated harmonic oscillator with frequency ${\omega }_r=7.8$ GHz. As explained in Sec. 5, the splitting between the first two levels of a transmon is modulated using a time-varying external flux $\phi$. Here, we use Gaussian pulses in that flux, as described by Eq. (71) with $\tau =2\sigma$, $\sigma =6.6873$ ns, and flux drive amplitude $\Delta \phi =0.075{\phi }_0$. The length of the pulse is chosen to maximize the population transfer.

Figure 2

Sideband transitions for a three-level system coupled to a resonator. Applying an FC drive at frequency ${\Delta }_{i,i+1}$ generates a red sideband transitions between states $|i+1;n\rangle$ and $|i;n+1\rangle$, where the numbers represent, respectively, the logical MLS and resonator states. Similarly, driving at frequency ${\Sigma }_{i,i+1}$ leads to a blue sideband transition, that is, $|i;n\rangle \leftrightarrow |i+1;n+1\rangle$. Transitions between states higher in the Fock space are not shown for reasons of readability. This picture is easily generalized to an arbitrary number of levels.

Figure 3

FC driving of a transmon with an external flux. The transmon is modeled using the first four levels of the Hamiltonian given by Eq. (56), using parameters $E_J/2\pi =25$ GHz and $E_C/2\pi =250$ MHz. We also have $g_{ge}/2\pi =100$ MHz and ${\omega }_r/2\pi =7.8$ GHz, which translates to ${\Delta }_{ge}/2\pi \simeq 2.1$ GHz. (a) Frequency of the transition to the first excited state obtained by numerical diagonalization of Eq. (56). As obtained from Eqs. (65) to (68), the major component in the spectrum of ${\omega }_{ge}\left(t\right)$ when shaking the flux away from the flux sweet spot at frequency ${\omega }_{\mathrm{FC}}$ also has frequency ${\omega }_{\mathrm{FC}}$. However, when shaking around the sweet spot, the dominant harmonic has frequency $2{\omega }_{\mathrm{FC}}$. Furthermore, the mean value of ${\omega }_{ge}$ is shifted by $G$. (b) Rabi frequency of the red sideband transition $|1;0\rangle \leftrightarrow |0;1\rangle$. The system is initially in $|1;0\rangle$ and evolves under the Hamiltonian given by Eq. (42) and a flux drive described by Eq. (61). Solid red line, analytical results from Eq. (70) with $m=1$ and ${\phi }_i=0.25$; dotted blue line, $m=2$ and ${\phi }_i=0$; black dots and triangles, exact numerical results. (c) Geometric shift for ${\phi }_i=0.25$ (solid red line) and 0 (dotted blue line). (d) Increase in the Rabi frequency for higher coupling strengths with ${\phi }_i=0.25$ and $\Delta \phi =0.075$. (e) Behavior of the resonance frequency for the flux drive. As long as the dispersive approximation holds ($g\lesssim g_{\mathrm{crit}}/2\pi =1061$ MHz), it remains well approximated by Eq. (69), as shown by the solid red line. The same conclusion holds for the Rabi frequency.

Figure 4

Amplitude of the Gaussian pulse over time. $\Delta {\phi }^{\prime }$ is such that the areas ${\mathcal{A}}_{+}$ and $2{\mathcal{A}}_{-}$ are equal. Then, driving the sideband at its resonance frequency for the geometric shift that corresponds to the flux drive amplitude $\Delta {\phi }^{\prime }$ allows population inversion.

Figure 5

Average fidelity of the pulse sequence $U_{\mathrm{ent}}$ as a function of cavity damping $\kappa$. The simulation includes realistic damping $T_1=2\mu$s of the transmons and neglects pure dephasing. Purcell decay is taking into account in the simulation. The other parameters are given in Table .