Figure 1

(a) Single well anharmonic potential shown with four energy levels. Arrows illustrate the dominant driven transitions and decoherence channels in our three-level $\Lambda$ system. (b) Rabi oscillation of the $f_{01}$ (blue) and $f_{02}$ (red) transitions are shown for a potential with $f_{01}=6.7\mathrm{GHz}$. The two curves are offset for clarity. (c) A micrograph of the measured qubit. The Josephson junctions in the circuit are epitaxially grown rhenium with crystalline ${\mathrm{Al}}_2{\mathrm{O}}_3$ tunneling barriers. (d) Circuit diagram of qubit, SQUID, and flux bias line. The qubit is controlled via the flux bias line and the capacitively coupled microwave line. The qubit state is probed by current biasing the SQUID and measuring the resulting voltage.Reuse & Permissions

Figure 2

Demonstration of CPT in a phase qubit. (a) Population of state $|2⟩$ ($P_2$) measured after a two-tone microwave pulse with a duration of $t_0=30\mathrm{ns}$ and frequencies $f_p$ and $f_c$. Each population measurement was performed by counting the number of SQUID switching events out of 4000 trials and accounting for measurement fidelity and background [25]. The power levels for the probe and coupling fields were $-6.5\mathrm{dBm}$ and $-15\mathrm{dBm}$ at the source, respectively. The white dot-dashed line shows the CPT resonance $2f_p-f_c=f_{01}$ (b) Theoretical data obtained by numerically solving the four level master equation (see text). We obtained the best match with experimental data by setting the resonant Rabi frequencies for these powers to be ${\Omega }_p^{(01)}=(2\pi )48\mathrm{MHz}$, ${\Omega }_c^{(12)}=(2\pi )32\mathrm{MHz}$. These values yield $A_p=|{\Omega }_p^{(01)}{|}^2/2\Delta =6.8\mathrm{MHz}$, $A_c={\Omega }_c^{(12)}=32\mathrm{MHz}$. (c)–(d) Cross sections of experimental data (with statistical error bars) (c) and theoretical data (d) taken at the values of $f_c$ indicated by the solid and dashed lines in (a) and (b). The cuts show the one-photon resonance when $f_c$ is off resonance and show the suppression of $P_2$ when $f_c=5.86\mathrm{GHz}$.Reuse & Permissions

Figure 3

Evolution of CPT as a function of microwave pulse duration, $t_0$. In this data set $f_{01}=6.19\mathrm{GHz}$, $f_{12}=5.85\mathrm{GHz}$, and $f_{02}/2=6.02\mathrm{GHz}$. $f_p$ was held fixed at 6.0158 GHz, essentially on resonance with the $|0⟩\leftrightarrow |2⟩$ two-photon transition. Each population measurement averaged 5000 trials. (a) Experimental measurement of $P_2$, and (b) theoretical calculation of $P_2$. The theoretical resonant Rabi frequencies were ${\Omega }_p^{(01)}=(2\pi )48\mathrm{MHz}$ and ${\Omega }_c^{(12)}=(2\pi )35\mathrm{MHz}$. (c)–(e) Cross sections of experimental data (symbols) and theory data (line) indicated by black dashed lines on (a) and (b). Statistical error bars are approximately equal to the symbol size. White dot-dashed lines indicate cross sections shown in Fig. 4.Reuse & Permissions

Figure 4

Cuts in time of the experimental data presented in Fig. 3 (indicated by white dot-dashed lines in a,b) at two different frequencies $f_c$ corresponding to (a) off CPT resonance $f_{12}-f_c=80\mathrm{MHz}$ and (b) on CPT resonance $f_{12}-f_c=0\mathrm{MHz}$. Also shown are theoretical simulations of the evolution of $P_2$ shown for three different values of $T_{\varphi }^{(01)}$ with other parameters the same as in Fig. 3. (inset) $P_2$ for $f_c=f_{12}$ at $t_0=40\mathrm{ns}$ plotted as a function of $T_{\varphi }^{(01)}$. The black dashed line shows the off resonant case ($f_{12}-f_c=80\mathrm{MHz}$).Reuse & Permissions