Quantum Heating of a Nonlinear Resonator Probed by a Superconducting Qubit

We measure the quantum fluctuations of a pumped nonlinear resonator using a superconducting artificial atom as an in situ probe. The qubit excitation spectrum gives access to the frequency and amount of excitation of the intracavity field fluctuations, from which we infer its effective temperature. These quantities are found to be in agreement with theoretical predictions; in particular, we experimentally observe the phenomenon of quantum heating.

Figure 1

(a) Optical analogue of the experiment: A Fabry-Pérot cavity with resonance frequency ${\omega }_c$, quality factor $Q$, and Kerr nonlinearity $K$ is pumped at frequency ${\omega }_p$. Photons are produced at the cavity dressed mode frequency ${\tilde{\omega }}_c$ and at the idler frequency ${\omega }_i$ by SPDC of pump photon pairs. An in situ atomic probe is used as an absolute spectrometer and thermometer to characterize this quantum heating of the dressed mode. (b) Actual experiment: The cavity is a microwave superconducting $\lambda /2$ coplanar waveguide resonator, the Kerr medium a Josephson junction, and the probe a transmon artificial atom. (c) Stability diagram of the resonator in the pump power $P_p$ and frequency ${\omega }_p$ plane (in reduced units, see the text): $L$, $H$, and $B$ denote regions where the amplitude of the driven oscillations is low, high, and bistable, respectively. $P_{+}$ is the pump power at which the field switches from $L$ to $H$ when $P_p$ is increased. (d) Dressed mode frequency ${\tilde{\omega }}_c$ when increasing $P_p$ at $\Omega =3$ [along the arrows of (c)].

Figure 2

Measurement of the field radiated by the KNR pumped at $\Omega =2.8$ (${\omega }_p/2\pi =6.4395\mathrm{GHz}$). (a) Experimental setup: The pump tone, sent through a line with attenuators ($A$), is switched on and off every 2.5 ms; the reflected and radiated signals are amplified and mixed down at ${\omega }_p$, yielding two quadratures $I(t)$ and $Q(t)$ that are digitized by an acquisition card (ACQ) and Fourier transformed (FFT); the total voltage spectral densities $S_{\mathrm{ON},\mathrm{OFF}}(\Delta \omega =|\omega -{\omega }_p|)$ for the two pump states are then computed. (b) Radiated noise spectrum $S_{\mathrm{ON}}-S_{\mathrm{OFF}}$ for $P_p=-110\mathrm{dBm}$. (c) Radiated noise spectrum as a function of $P_p$. The dashed curve is the calculated ${\tilde{\Delta }}_p={\tilde{\omega }}_p-{\omega }_p(P_p)$. The vertical dotted line indicates the cut shown in (b).

Figure 3

Sideband spectroscopy of the qubit while pumping the resonator with varying power $P_p$ at $\Omega =3.9$ (${\omega }_p/2\pi =6.434\mathrm{GHz}$). (a) Experimental setup: A spectrocopic pulse is applied at ${\omega }_q$ with power $P_q$ during the pumping; the pump and spectroscopy tones are then switched off and the qubit is measured using a third pulse at ${\omega }_r$ with a power close to the bifurcation threshold $P_{+}$. The measured phase of the reflected pulse yields the qubit state, and repeating the sequence yields the qubit excited state probability $p_{|e⟩}$. (b) Measured (top) and calculated (bottom) 2D plots showing the $p_{|e⟩}({\omega }_q)$ spectra as a function of $P_p$ (bottom axis), also expressed in units of the bifurcation threshold $P_{+}$ (top axis). (c) Experimental (dots) and analytical (lines) spectra at a power $P_p=-96\mathrm{dBm}$ [dashed lines in (b)] for powers $P_q=-131$, $-120$, and $-115\mathrm{dBm}$ (from left to right). (d) Energy diagrams showing the transitions involved in the red sideband peak (Stokes), the central peak, and the blue sideband (anti-Stokes).

Figure 4

Quantum heating of the pumped resonator mode ${\tilde{\omega }}_{\mathrm{c}}$ above bifurcation: Quantity $r/(1-r)$ with $r$ being the ratio of blue to red sideband peak heights deduced from the experimental (squares) and theoretical (dots) qubit spectra of Fig. 3b, as a function of the pump power $P_{\mathrm{p}}$ (bottom axis). The calculated average number $⟨n_H⟩$ of photons in the pump mode is indicated on the top axis. In a simple theoretical picture, $r/(1-r)$ is the average photon number $⟨\tilde{n}⟩$ in the ${\tilde{\omega }}_{\mathrm{c}}$ mode, shown here as a dashed and a solid line for the $L$ and $H$ resonator states, respectively. The corresponding effective temperature $T_{\mathrm{eff}}$ is indicated on the right axis.