Multiphoton Transitions in a Macroscopic Quantum Two-State System

We have observed multiphoton transitions between two macroscopic quantum-mechanical superposition states formed by two opposite circulating currents in a superconducting loop with three Josephson junctions. Resonant peaks and dips of up to three-photon transitions were observed in spectroscopic measurements when the system was irradiated with a strong rf-photon field. The widths of the multiphoton absorption dips are shown to scale with the Bessel functions in agreement with theoretical predictions derived from the Bloch equation or from a spin-boson model.

Figure 1

(a) Scanning-electron-microscope picture of a superconducting flux qubit system. (b) Signal-to-noise ratio $\Delta I_{\mathrm{q}\mathrm{b}}/I_{\mathrm{s}\mathrm{t}\mathrm{d}}$ (solid curve) and average switching current $I_{\mathrm{s}\mathrm{w}}$ (dashed curve) as a function of applied flux through the qubit loop. The two arrows indicate the signal-to-noise ratios at ${\Phi }_{\mathrm{q}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{t}}/{\Phi }_0=0.5$ and 1.5. (c) Spectroscopic data of the qubit. Resonant frequency is plotted as a function of the half distance between the resonant peak and dip in ${\Phi }_{\mathrm{q}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{t}}/{\Phi }_0$. The solid curve represents a numerical fit to the data. The dashed line is the energy difference between localized states $|\downarrow \rangle$ and $|\uparrow \rangle$. From this fit, we obtained $E_{\mathrm{J}}/h=380$ [GHz] and $\Delta /2\pi =0.56$ [GHz].

Figure 2

Applied magnetic flux dependence of the qubit signal $\delta I_{\mathrm{s}\mathrm{w}}$. We subtracted a sinusoidal background signal from the averaged switching current of the dc SQUID $I_{\mathrm{s}\mathrm{w}}$. (a) Without microwave irradiation. The data were obtained after averaging 2000 measurements. (b) With microwave irradiation. The data were obtained after averaging 500 measurements. The microwave power and frequency were 5 dBm at the generator and 9.1 GHz, respectively. Resonant peaks of up to two-photon processes and dips of up to three-photon processes were observed.

Figure 3

Half width at half maximum (a) and normalized amplitude (b) of the resonant dips as functions of the microwave amplitude $I_{\mathrm{r}\mathrm{f}}$. The microwave frequency is ${\omega }_x/2\pi =3.8$ [GHz]. Solid (dashed) curves represent theoretical fits obtained by the Bloch equations combined with the dressed-atom approach (by the real-time path-integral method). We obtained $T_{\mathrm{r}\mathrm{1}}=31\left(27\right)$ [ns], $T_{\mathrm{r}\mathrm{2}}=84\left(66\right)$ [ns], $T_{\mathrm{r}\mathrm{3}}=430\left(240\right)$ [ns], $T_{\phi }=4.0\left(4.2\right)$ [ns], $c=0.55\left(0.55\right)$. We have obtained similar results for ${\omega }_x/2\pi =9.1$ [GHz] and $11.4$ [GHz] (not shown).