Bose condensation and Bogoliubov excitation in resonator-embedded superconducting qubit network

Abstract

Superconducting qubit networks (SQNs) embedded in a low-dissipative resonator is a promising device allowing one not only to establish the collective quantum dynamics on a macroscopic scale but also to greatly enhance the sensitivity of detectors of microwave photons. A quantum ac Stark effect provided by coupling between an SQN and microwave photons of a resonator, leads to a strong nonlinear interaction between photons. Here, we present a two-tone spectroscopy experiment in which a set of 10 superconducting flux qubits is coupled to the input R- resonator and the output T- transmission line. An external microwave pump field close to the resonance frequency populates macroscopically the resonator mode as a Bose-Einstein condensate, while a second probe beam scans the resonances referred also as Bogoliubov-like excitations. The corresponding excitation frequency measured from the transmission coefficient, |S21(f)| displays an abrupt change of the resonant dip position once the power of the pump field overcomes a critical value Pcr. This sharp shift occurs in a narrow region of pump frequencies, and can be tuned by an applied magnetic field. It is a signature of bistability of the photon number inside the resonator, in agreement with theory.

Figures (12)

• Figure 1: Schematic of a proposed device: an SQN composed of $N$ flux qubits coupled to a low-dissipative resonator (bottom) and a transmission line (top).
• Figure 2: Device layout and experimental setup. (a) (Left) Layout of the device. (Right) The optical micrograph of 10 capacitively-shunted flux qubits integrated with $R$- resonator and $T$-transmission line. Port numbers are indicated. (b) Schematic of the experimental setup highlighting the microwave paths and components.
• Figure 3: The dependence of the transmission coefficient $|S_{21}|$ on the probe frequency $f_{p}$ for three different values of pump signal power: $P_{pump}<P_{cr}$(a); $P_{pump} \simeq P_{cr}$ (b); $P_{pump} \geq P_{cr}$ (c). The frequency of the pump signal was 7.74328 GHz. The externally applied magnetic field is zero. Note the weaker mirror idler resonance at about 7.742 GHz in graph (a) and (b) corresponding to the second branch of the Bogoliubov excitation.
• Figure 4: (a) The experimental dependence of the dip frequency position on the pump power $P_{pump}$ for different frequencies of the pump signal.(b) Calculated dependencies of the dip frequency position on the pump power $P_{pump}$ for different frequencies of the pump signal. (c) Deduced photon number inside the cavity. To obtain the best fit to the experimental curves we choose the following parameters as: $\omega_c=2\pi \times 7.7430 GHz$, $\gamma_c=2\pi \times 3.2 MHz$, $\Delta=2\pi \times 1.78\,GHz$ and the coupling $g=2\pi \times 25\, MHz$. The dashed line corresponds to the real line as seen in experiment. Within the bistability region, the photon number in the resonator is in the lower branch in the green and black curves while it is in the higher branch in the red curve.
• Figure 5: The dependence of the dip frequency position on the pump signal power $P_{pump}$ for different directions of power sweep. The frequency of a pump signal was 7.745 GHz