Perfect displacement of a superconducting resonator via fast-forward scaling theory

Abstract

We investigate the fast-forward and time-scaling properties of superconducting resonators under a coherent drive. We propose a scheme for perfect displacement of a superconducting resonator by modulating the drive amplitude based on fast-forward scaling theory. Furthermore, we propose a scheme exploiting both the fast-forward and time-scaling properties that enables perfect displacement through detuning modulation. The proposed schemes are also applicable to a subsystem that can be effectively represented by a driven resonator. In particular, we apply the latter scheme to fast and high-fidelity displacement of a coupler between Kerr parametric oscillators.

Figures (6)

• Figure 1: Schematic illustration of displacement trajectories under different types of control. The triangle and star represent the initial and final displacements, $\alpha_{\mathrm{i}}$ and $\alpha_{\mathrm{f}}$, respectively. The green dashed line and the red solid curve correspond to the trajectories of the fast-forwarded dynamics and the adiabatic dynamics, respectively. The latter coincides with the trajectory obtained using the counter-diabatic protocol. The dynamics based solely on the fast-forward (FF) protocol and that based on both the FF and time-scaling (TS) protocols follow the same trajectory at different speeds.
• Figure 2: The time dependence of $\alpha_0(t)$, $\dot{\alpha}_0(t)/\Delta$, $\ddot{\alpha}_0(t)/\Delta^2$, and $\alpha_{\mathrm{FF}}(t)$ (a–c) and of $\Omega_0(t)$ and $\Omega_{\mathrm{FF}}(t)$ (d–f). The form of $\Omega_0(t)$ is given in Eq. \ref{['eq:Omegat']}. We plot $\ddot{\Omega}_0(t)/\Delta^2$ in (f). We set $\Delta/2\pi=30$ MHz, $\Omega_{\mathrm{i}}/2\pi=0$ Hz, $\Omega_{\mathrm{f}}/2\pi=120$ MHz. $T_{\mathrm{f}}=20$ ns in (a) and (d), $T_{\mathrm{f}}=40$ ns in (b) and (e), and $T_{\mathrm{f}}=80$ ns in (c) and (f).
• Figure 3: The infidelity between the target state $\Ket{\alpha_0(T_{\mathrm{f}})}$ and the final state $\Ket{\psi_{\mathrm{nonad}}(T_{\mathrm{f}})}$ under $\hat{H}_0(t)$ in Eq. \ref{['eq:H0t2']} with $\Omega_0(t)$ in Eq. \ref{['eq:Omegat']}, $1-|\Braket{\psi_{\mathrm{nonad}}(T_{\mathrm{f}})|\alpha_0(T_{\mathrm{f}})}|^2$. We set $\Delta/2\pi=30$ MHz, $\Omega_{\mathrm{i}}/2\pi=0$ Hz, and $\Omega_{\mathrm{f}}/2\pi=120$ MHz.
• Figure 4: The infidelity between the target state $|\Omega_{\mathrm{i}}/\Delta_{\mathrm{f}}\rangle$ and the final state $|\psi_{\rm lin}(t_{\mathrm{f}})\rangle$ under the Hamiltonian $\hat{H}_{\mathrm{lin}}(t)$ in Eq. \ref{['eq:H_lin1']}. We set $\Delta_{\mathrm{i}}/2\pi=200$ MHz, $\Delta_{\mathrm{f}}/2\pi=20$ MHz, and $\Omega_{\mathrm{i}}/2\pi=80$ MHz.
• Figure 5: (a) Schematic of a system consisting of a frequency-tunable resonator (coupler, subsystem c) and two Kerr parametric oscillators (KPOs, subsystems $1$ and $2$). (b) Mechanism of the $ZZ$ coupling between Kerr-cat qubits in the adiabatic regime. Top: the coupler's detuning, $\Delta_{\mathrm{c}}(t)$. Middle: the four eigenenergies of the system in the first order of perturbation in Eqs. \ref{['eq:E00E11']} and \ref{['eq:E01E10']} corresponding to the four states in Eqs. \ref{['eq:psi00t']}--\ref{['eq:psi11t']}. $\Theta$ is the rotation angle of the $R_{ZZ}$ gate. Bottom: the amplitudes of the four coherent states of the coupler, $\pm\alpha_{\mathrm{c}}^{+}(t)$ and $\pm\alpha_{\mathrm{c}}^{-}(t)$. $t_{\mathrm{g}}$ is the gate time.