Beating the 3 dB Limit for Intracavity Squeezing and Its Application to Nondemolition Qubit Readout

While the squeezing of a propagating field can, in principle, be made arbitrarily strong, the cavity-field squeezing is subject to the well-known 3 dB limit, and thus has limited applications. Here, we propose the use of a fully quantum degenerate parametric amplifier (DPA) to beat this squeezing limit. Specifically, we show that by simply applying a two-tone driving to the signal mode, the pump mode can, counterintuitively, be driven by the photon loss of the signal mode into a squeezed steady state with, in principle, an arbitrarily high degree of squeezing. Furthermore, we demonstrate that this intracavity squeezing can increase the signal-to-noise ratio of longitudinal qubit readout exponentially with the degree of squeezing. Correspondingly, an improvement of the measurement error by many orders of magnitude can be achieved even for modest parameters. In stark contrast, using intracavity squeezing of the semiclassical DPA cannot practically increase the signal-to-noise ratio and thus improve the measurement error. Our results extend the range of applications of DPAs and open up new opportunities for modern quantum technologies.

Figure 1

(a) Schematic of our proposal with a fully quantum DPA. We use two cavities to represent the pump mode ${\widehat{a}}_p$ (frequency ${\omega }_p$, loss rate ${\kappa }_p$) and the signal mode ${\widehat{a}}_s$ (frequency ${\omega }_s$, loss rate ${\kappa }_s$). The single-photon parametric coupling between them has a strength $g$. A driving tone at frequency ${\omega }_d$ is applied to the pump mode and, simultaneously, the signal mode is driven by the other two tones at frequencies ${\omega }_{\pm }^d$. (b) Time evolution of the squeezing parameter ${\xi }_p^2$ for $G_{+}/G_{-}=0.5$ and 0.7. We assumed that ${\mathrm{\Delta }}_s=100g$, ${\mathrm{\Delta }}_p=0.1{\mathrm{\Delta }}_s$, ${\mathrm{\Omega }}_{2\mathrm{pd}}=0.05{\mathrm{\Delta }}_s$, ${\kappa }_s=100{\kappa }_p=0.4g$, and $G_{-}=g_0$. Curves are the effective predictions, while symbols are the exact results. (c) Steady-state squeezing parameter ${({\xi }_p^2)}_{\mathrm{ss}}$ versus the cooperativity $\mathcal{C}$ for ${\kappa }_s=100{\kappa }_p$, and for $G_{+}/G_{-}=0.8$, 0.9, and 0.99. In (b) and (c), the gray shaded areas refer to the regime below the 3 dB limit.

Figure 2

(a) SNR improvement, i.e., ${\mathrm{SNR}}_z^{\mathrm{fq}}/{\mathrm{SNR}}_z^{\mathrm{std}}$, versus the degree $r_p$ of intracavity squeezing for different values of the DPA cooperativity: $\mathcal{C}=5$, 10, 30, and $\infty$. An exponential improvement can be obtained for $\mathcal{C}\gg \exp (2r_p)/4$. (b) SNR and (c) measurement error versus the measurement time. The solid and dashed curves correspond to the longitudinal readout using intracavity squeezing of the fully quantum ($r_p=2$, $\mathcal{C}=5$) and semiclassical ($r_{\mathrm{out}}^{\mathrm{sc}}=2$) DPAs, respectively, while the dash-dotted curves are results of the standard longitudinal readout with no squeezing. The green shaded area represents the experimentally most interesting regime. In (b) and (c), all the parameters are the same except that ${\chi }_z=\kappa$ in (c). (d) SNR (left axis) and measurement error (right axis) versus $\mathcal{C}$ in the fully quantum-DPA case for $\tau =1/\kappa$. Other parameters are the same as in (c).