Coherence Times of Dressed States of a Superconducting Qubit under Extreme Driving

We measure longitudinal dressed states of a superconducting qubit, the single Cooper-pair box, and an intense microwave field. The dressed states represent the hybridization of the qubit and photon degrees of freedom and appear as avoided level crossings in the combined energy diagram. By embedding the circuit in an rf oscillator, we directly probe the dressed states. We measure their dressed gap as a function of photon number and microwave amplitude, finding good agreement with theory. In addition, we extract the relaxation and dephasing rates of these states.

Figure 1

rf response of the strongly driven SCB coupled to an oscillator. The oscillator is formed from an inductor, with stray capacitance, in series with the pad capacitance of the sample (see inset). It has a resonance frequency $f_c=647\mathrm{MHz}$ and a quality factor $Q\approx 40$. The color images represent the rf reflection coefficient of the coupled system $\Gamma$ as a function of dc gate charge $n_g$ and microwave amplitude $A_{\mu }$. (Higher $A_{\mu }$ is in the foreground.) (a) is the magnitude response $|\Gamma |-1$, and (b) is the phase $\phi =\arg (\Gamma )$. The response is plotted as a plane in a cube along with traces that show the predicted value of the dressed gap ${\Delta }_m(A_{\mu })$, normalized to $E_J$, for different photon numbers up to $m=4$. The $z$ value at which the response plane is plotted corresponds to the normalized rf probe frequency $h{f}_c/E_J$. The data are clearly explained by formally treating the dressed states as tunable two-level systems. In (a), we clearly see that there is resonant absorption of the rf probe (red spots) at each point that ${\Delta }_m(A_{\mu })=h{f}_c$. The phase shows the dispersive shift of the resonator frequency when the dressed states are off-resonant. There are a total of only 2 free parameters, $E_J$ and ${\gamma }_{\mu }$, for all traces in both panels. The microwave frequency was 7 GHz, and we extract $E_J/h=2.6\mathrm{GHz}$.

Figure 2

Dressed state energy diagram inferred from the magnitude response. (a) Energy diagram of the combined SCB-photon system. The thin black lines are the energy bands for $E_J=0$, showing the degeneracies of states differing by $m=0,1,2,\dots$ photons regularly spaced in gate charge. The thick red lines show the degeneracies lifted for finite $E_J$. The dressed gaps ${\Delta }_m$ are functions of the normalized microwave amplitude $\alpha$. (b) Position in $n_g$. The assumed transition frequency $m\hslash {\omega }_{\mu }$ vs the splitting in $n_g$ of symmetric resonances. The data are taken from Fig. 1a and similar data for different ${\omega }_{\mu }$. All of the points fall on one line with a slope of $E_Q=62\mathrm{GHz}$. (c) The dependence of the normalized ${\Delta }_m/E_J$ on $\alpha$. We plot data for different values of $E_J/h$, ranging from 2.1 to 4.2 GHz. The data points are the positions of the resonances in $\alpha$, plotted at the normalized probe frequencies $h{f}_c/E_J$.

Figure 3

Bloch response of the 1-photon dressed state. The (color) scale is the same as Fig. 1. We model the rf response of the driven SCB by solving the Bloch equations [20] for a tunable two-level system, the dressed state described by (2), driven by the rf probe. From left to right, we have the theoretical magnitude and phase responses and the corresponding experimental data. The important fitting parameters are the relaxation rate ${\Gamma }_1^{\mathrm{deg}}$ and the dephasing rate ${\Gamma }_{\phi }^{\mathrm{deg}}$ at degeneracy. We note that it is not possible to extract unique values for ${\Gamma }_1^{\mathrm{deg}}$ and ${\Gamma }_{\phi }^{\mathrm{deg}}$ unless the magnitude and phase response are fit simultaneously. The bottom panel shows the extracted values of ${\Gamma }_1^{\mathrm{deg}}$ as a function of the detuning $\delta \omega /2\pi$ between the dressed state and oscillator, for positive detuning, giving values of $1/{\Gamma }_1^{\mathrm{deg}}\approx 0.7–3.2\mu \mathrm{s}$. The fit is explained in the text. We also extract $1/{\Gamma }_{\phi }^{\mathrm{deg}}\approx 48\mathrm{ns}$.