Parametric Control of a Superconducting Flux Qubit

Parametric control of a superconducting flux qubit has been achieved by using two-frequency microwave pulses. We have observed Rabi oscillations stemming from parametric transitions between the qubit states when the sum of the two microwave frequencies or the difference between them matches the qubit Larmor frequency. We have also observed multiphoton Rabi oscillations corresponding to one- to four-photon resonances by applying single-frequency microwave pulses. The parametric control demonstrated in this work widens the frequency range of microwaves for controlling the qubit and offers a high quality testing ground for exploring nonlinear quantum phenomena of macroscopically distinct states.

Figure 1

(a) Scanning electron micrograph of a flux qubit (inner loop) and a dc-SQUID (outer loop). The loop sizes of the qubit and SQUID are $10.2\times 10.4\mu {\mathrm{m}}^2$ and $12.6\times 13.5\mu {\mathrm{m}}^2$, respectively. They are magnetically coupled by the mutual inductance $M\approx 13\mathrm{pH}$. (b) A circuit diagram of the flux qubit measurement system. On-chip components are shown in the dashed box. $L\approx 140\mathrm{pH}$, $C\approx 9.7\mathrm{pF}$, $R_{\mathrm{I}1}=0.9\mathrm{k}\Omega$, $R_{\mathrm{V}1}=5\mathrm{k}\Omega$. Surface mount resistors $R_{\mathrm{I}2}=1\mathrm{k}\Omega$ and $R_{\mathrm{V}2}=3\mathrm{k}\Omega$ are set in the sample holder. We put adequate copper powder filters CP and $LC$ filters F and attenuators A for each line.

Figure 2

Experimental results with single-frequency microwave pulses. (a) Spectroscopic data of the qubit. Each set of the dots represents the resonant frequencies $f_{\mathrm{res}}$ caused by the one- to four-photon absorption processes. The solid curves are numerical fits. The dashed line shows a microwave frequency $f_{\mathrm{MW}1}$ of 10.25 GHz. (b) One-photon Rabi oscillations of $P_{\mathrm{sw}}$ with exponentially damped oscillation fits. Both the qubit Larmor frequency $f_{\mathrm{qb}}$ and the microwave frequency $f_{\mathrm{MW}1}$ are 10.25 GHz. The external flux is ${\Phi }_{\mathrm{qb}}/{\Phi }_0=1.4944$. (c) Four-photon Rabi oscillations when $f_{\mathrm{qb}}=41.0\mathrm{GHz}$, $f_{\mathrm{MW}1}=10.25\mathrm{GHz}$, and ${\Phi }_{\mathrm{qb}}/{\Phi }_0=1.4769$. (d) The microwave amplitude dependence of the Rabi frequencies ${\Omega }_{\mathrm{Rabi}}/2\pi$ up to four-photon Rabi oscillations. The solid curves represent theoretical fits.

Figure 3

Experimental results with two-frequency microwave pulses. (a) [(b)] Two-photon Rabi oscillations due to a parametric process when $f_{\mathrm{qb}}=f_{\mathrm{MW}2}+(-)f_{\mathrm{MW}1}$. The solid curves are fits by exponentially damped oscillations. (c) [(d)] Rabi frequencies as a function of $V_{\mathrm{MW}1}$, which are obtained from the data in Figs. 3a, 3b. The dots represent experimental data when $V_{\mathrm{MW}2}=16.9$, 23.5, 33.0, and 52.0 (50.1, 62.9, 79.1, and 124.7) mV from the bottom set of dots to the top one. The solid curves represent Eq. (3). The inset is a schematic of the parametric process that causes two-photon Rabi oscillation when $f_{\mathrm{qb}}=f_{\mathrm{MW}2}+(-)f_{\mathrm{MW}1}$.