Quantum Transducer Using a Parametric Driven-Dissipative Phase Transition

We study a dissipative Kerr resonator subject to both single- and two-photon detuned drives. Beyond a critical detuning threshold, the Kerr resonator exhibits a semiclassical first-order dissipative phase transition between two different steady states that are characterized by a $\pi$ phase switch of the cavity field. This transition is shown to persist deep into the quantum limit of low photon numbers. Remarkably, the detuning frequency at which this transition occurs depends almost linearly on the amplitude of the single-photon drive. Based on this phase-switching feature, we devise a sensitive quantum transducer that translates the observed frequency of the parametric quantum phase transition to the detected single-photon amplitude signal. The effects of noise and temperature on the corresponding sensing protocol are addressed, and a realistic circuit-QED implementation is discussed.

Figure 1

(a) Schematic representation of a KPO with nonlinearity $U$ and loss rates $\gamma$ and $\eta$ subject to single- and two-photon drives $F$ and $G$, respectively. The emitted photons by the cavity with rate $\kappa$ are collected by a heterodyne detector. (b) Steady-state Husimi $Q$ functions at points ① and ②; cf. (c) and (d). Dashed lines mark the contour of the $F=0$ $Q$ function. (c),(d) Photon density $⟨n⟩$ and phase $\mathrm{\Phi }$ of the cavity field, as a function of detuning $\mathrm{\Delta }/U$ in the steady state (dashed blue line) and for upsweeps (green) and downsweeps (red) of $\mathrm{\Delta }/U$, obtained from Eq. (2). At large positive $\mathrm{\Delta }/U$, the KPO crosses over from high $⟨n⟩$ to low $⟨n⟩$. At $\mathrm{\Delta }\approx 0$ [marked by $Ⓑ$] in (c), we see a kink in the steady state $⟨n⟩$ concomitant with a $\pi$ switch in the phase (d). The kink and phase switch are also seen for downsweeps. (e) The Liouvillian gap as a function of $\mathrm{\Delta }/U$ and phase $\theta$ of $G$. The gap vanishes at the phase-switching transition, marking the onset of a quantum DPT. System parameters are $F/U=4$, $|G|/U=6$, $\gamma /U=0.5$, and $\eta /U=0.5$. $\theta =-(\pi /2)$ in (c) and (d). $\mathrm{\Delta }/U$ is swept linearly from ${\mathrm{\Delta }}_1/U=-10$ to ${\mathrm{\Delta }}_2/U=15$ and vice versa in a total sweep time $t_s=50/U$.

Figure 2

(a) Measurement protocol: $\mathrm{\Phi }$ is measured via heterodyne detection (gray) and ${\mathrm{\Delta }}_{*}$ is extracted from an arctan fit (blue), which is then used to determine $F_{\mathrm{meas}}$ (for details, see the main text). (b) ${\mathrm{\Delta }}_{*}/U$ as a function of $F$ obtained from the master equation (2) (orange line) and its associated probability density function (PDF, blue) obtained from repeated heterodyne simulations (3). (c) PDF of $F_{\mathrm{meas}}$ for $F/U=4$, i.e., $F/(\kappa +\gamma )=2.67$; histogram from the simulated heterodyne detection (blue) and a fit to a Gaussian (blue line), with mean ${\overline{F}}_{\mathrm{meas}}/(\kappa +\gamma )=2.79\approx F/(\kappa +\gamma )$ and standard deviation ${\mathrm{\Delta }}_{F_{\mathrm{meas}}}/(\kappa +\gamma )=1.1$. The dashed line (orange) is the prediction based on the stochastic switching in the Husimi $Q$ function; cf. Eq. (5). $\kappa /U=1$ and the other parameters are the same as in Fig. 1. (d) Quantum Fisher information of estimating $F$ in the steady state (6) as a function of $\mathrm{\Delta }/U$ (solid lines) [32]. Note the prominent peak at $\mathrm{\Delta }={\mathrm{\Delta }}_{*}\approx 0$ and a smaller one at $\mathrm{\Delta }/U\approx 10$, corresponding to the crossovers $Ⓐ$ and $Ⓑ$ in Figs. 1 and 1. Both features vanish at sufficiently large temperatures $T$. For comparison, the results for $G=U=0$ (harmonic oscillator) at $T=0$ are plotted (dashed). The main peak is modulated by the resonances in the system [19, 35]. Parameters are chosen as $F/U=4.5$, $|G|/U=3$, $\gamma /U=3$, and $\eta =0$.