Amplification and attenuation of a probe signal by doubly dressed states

We analyze a system composed of a qubit coupled to the electromagnetic fields in two high quality quantum oscillators. A particular realization of such a system is the superconducting qubit coupled to a transmission-line resonator driven by two signals with frequencies close to the resonator's harmonics. This doubly driven system can be described in terms of the doubly dressed qubit states. Our calculations demonstrate the possibility to change the number of photons in the resonator and the transmission of the fundamental-mode signal over a wide parameter range exploiting resonances with the dressed qubit. Experiments show that in the case of high quality resonators the dressed energy levels and corresponding resonance conditions can be probed, even for high driving amplitudes. The interaction of the qubit with photons of two harmonics can be used for the creation of quantum amplifiers or attenuators.

Figure 1

Bare, dressed, and doubly dressed energy levels. The bare qubit's energy levels, $\pm \Delta E/2$, are shown in (a); when they are matched by the driving frequency ${\omega }_{\mathrm{d}}$, the qubit is resonantly excited. (At higher values of the bias ${\varepsilon }_0$, the bare-qubit multiphoton excitation should be studied.) The position of the resonance, $\hslash {\omega }_{\mathrm{d}}=\Delta E$, is described by the avoided crossing of the dressed-state levels; the dressed and averaged energy levels, $\pm \widetilde{\Delta E}/2=\pm \hslash {\Omega }_{\mathrm{R}}/2$, are plotted in panel (b). When the dressed energy levels are matched by the second (probe) signal, $\hslash {\omega }_{\mathrm{p}}=\widetilde{\Delta E}$, a resonance interaction of the coalesced system is expected, Eq. (9). Also a resonance condition is given by the two-photon process (10). This is visualized as the avoided crossings of the doubly dressed states, plotted in panel (c).

Figure 2

(a) Measured normalized transmission amplitude $\left|t\right|$ for the first qubit (see text) while a strong driving signal is applied. The results are plotted in dependence on the detuning between qubit frequency and driving signal (controlled by the qubit bias ${\varepsilon }_0$), and the probing frequency detuning $\delta {\omega }_{\mathrm{r}}={\omega }_{\mathrm{p}}-{\omega }_{\mathrm{r}}$. The amplification and the attenuation of the transmitted signal is found in agreement with the resonance conditions, $\hslash {\omega }_{\mathrm{p}}=\widetilde{\Delta E}$, Eq. (9). (b) Normalized transmission calculated for the same parameters as in (a) following Eq. (31) together with Eq. (39).

Figure 3

Normalized transmission amplitude through the resonator at a probing frequency ${\omega }_{\mathrm{p}}={\omega }_{\mathrm{r}}$ for different driving amplitudes ranging from about $-131$ dBm in (a) to $-117$ dBm in (g) in 2-dBm steps at the input of the resonator. The transmission is plotted as a function of the energy bias ${\varepsilon }_0$ of the second qubit and the driving frequency detuning $\delta {\omega }_{\mathrm{d}}={\omega }_{\mathrm{d}}-3{\omega }_{\mathrm{r}}$. The driving frequency is changed only around the third harmonic in the order of its linewidth (about $6\kappa$). The presented results correspond to a symmetric power dependence around the center frequency of the third harmonic, since the resonator acts as bandpass filter. Each plot is split into an experimental (positive detuning) and a theoretical (negative detuning) part. For the calculations the figures were split into regions for which Eq. (31) was used with a certain index $k$ in Eq. (3). Several features were highlighted with black rectangles to interrelate theory and experiment (see text).