Dressed-State Amplification by a Single Superconducting Qubit

We demonstrate amplification of a microwave signal by a strongly driven two-level system in a coplanar waveguide resonator. The effect, similar to the dressed-state lasing known from quantum optics, is observed with a single quantum system formed by a persistent current (flux) qubit. The transmission through the resonator is enhanced when the Rabi frequency of the driven qubit is tuned into resonance with one of the resonator modes. Amplification as well as linewidth narrowing of a weak probe signal has been observed. The stimulated emission in the resonator has been studied by measuring the emission spectrum. We analyzed our system and found an excellent agreement between the experimental results and the theoretical predictions obtained in the dressed-state model.

Figure 1

(a) Electron micrograph of the central part of the resonator: the central line of the resonator (left), qubit (middle), and the gold film resistor (right). (b) Sketch of the dressed system manifolds $N$ and $N+1$. The dashed lines correspond to the uncoupled states. The qubit relaxation with rate $\Gamma$ is sketched for the uncoupled states. When coupled the levels will split depending on the photon state $N$. The level spacing between those split states is given by the generalized Rabi frequency Eq. (4). (c) After tracing over the photon number $N$ an effective two-level system, denoted with states $|1⟩$ and $|2⟩$ is obtained. Both the sign and strength of the relaxation (excitation) in this system depend on the detuning [see Eq. (7)].

Figure 2

(a) Normalized transmission amplitude of the probe signal as a function of $({\omega }_p-{\omega }_r)/2\pi$ and qubit energy bias $ϵ/2\pi$. The inset shows two cuts of this figure, visible as the Lorentzian shaped transmission vs. $({\omega }_p-{\omega }_r)/2\pi$ in the amplification point (corresponding to $\delta =1.4\mathrm{GHz}$, dashed line) and out of the Rabi resonance (solid line). The experiments were done in the presence of a strong driving field, corresponding to 8 pW at the input of the resonator, applied at the third harmonic frequency. The driving strength defines the on-resonance Rabi frequency. (b) Numerical simulations by making use of Eq. (6) in a four photon space. The probing power is assumed to be weak creating a coherent state with a mean photon number less than one in the fundamental mode.

Figure 3

(a) Normalized transmission amplitude as a function of the detuning $\delta$ (between the strong driving ${\omega }_d$ and the qubit transition frequency ${\omega }_q$) and the driving power. The dashed line is calculated according to Eq. (4) and used to map the on-resonance Rabi frequency ${\Omega }_R^{⟨N⟩}(\delta =0)$ to the driving power. (b) Numerical simulations of the experimental results shown in (a) by use of Eq. (6).

Figure 4

Spectral emission from the resonator. The solid lines correspond to the best Lorentzian fits of the measured data and are used to reconstruct the linewidths. The power spectral density measured without driving field (dots) corresponds to the thermal occupation of the resonator at an effective temperature of 30 mK. The cold amplifier adds a background corresponding to a noise temperature of 7 K. If the lasing process is turned on by driving the resonator’s third harmonic we observe a clear increase of the emission and a lower linewidth (open circles).