Spectroscopy on a single nonlinear mode recognizes quantum states

TL;DR

This work shows that a single nonlinear optical mode can serve as a quantum reservoir for recognizing optical quantum states, by reading the input squeezing from the mode’s emission spectrum rather than performing full tomography. The authors formalize a Kerr-type nonlinear cavity under Lindblad dynamics, encode the spectrum into moments and higher-order features, and demonstrate that simple linear regression or neural-network readouts can extract squeezing parameters with percent-level accuracy. They reveal how nonlinearity strength controls expressiveness and generalization across quadratures, and extend the approach to multivariate learning with neural networks and to realistic sources using an Optical Parametric Oscillator coupled to a polariton microcavity. The results suggest a compact, robust pathway for quantum-state recognition in photonic platforms, with potential applicability to other quantum states and time-domain enhancements such as time lenses.

Abstract

Characterising optical quantum states is essential for the development of quantum technologies. While traditional approaches to perform full quantum state tomography are often experimentally demanding, neuromorphic architectures may provide an effective alternative. In this work, we demonstrate how a quantum nonlinear driven-dissipative mode is sufficient to act as a quantum reservoir. By analyzing the occupations at different frequencies in the emission spectrum, a linear regression suffices in many cases to recognize the relevant parameters of incident squeezed states. Beyond highlighting the general potential of this approach under continuous driving, we illustrate its effectiveness in an explicit nontrivial example where the source is a degenerate optical parametric oscillator (OPO), coupled to a nonlinear polariton microcavity.

Figures (8)

• Figure 1: Setup: Quantum-optical squeezed states are continuously injected in a single-mode nonlinear cavity. The cavity's output radiation is then sent to a spectrometer, and the resulting parameters are then learned from the resulting emission spectrum
• Figure 2: Learning squeezing strength (a): Spectra of a polariton microcavity coupled to a squeezed environment, for different values of squeezing strength $r$ at $\theta=0, \alpha_D=5,\Bar{n}=0$ (b): moments $M_m$ as a function of $r$ for the same training data as the left panel (circles) and 100 random testing data in the same range (stars). The moments are rescaled to their respective maximum values for display (c): individual comparison of actual values of $r$ versus the predicted on testing data . This leads to a NRMSE of $\varepsilon\approx1.7\%$. Parameters $U=12~\mu eV$Kasprzak2007Liew2010 and $\gamma_c= 67~ns^{-1}=44.1~\mu eV$.
• Figure 3: Dependence on reservoir nonlinearity (a): the prediction error on $r$ (same task as in Fig. \ref{['fig:rdepspectra']}) decreases with $U$, as is well captured by a power law decay $\varepsilon\propto U^{-2/3}$. At higher values of $U$, the performance saturates. (b): Wigner-functions in optical phase space of the steady-states in the cavity for the different values of $U$ that are indicated by the red arrows in (a). At the lowest values $U= 10^{-4}~meV$ the nonlinearity has little effect and the microcavity steady state is directly proportional to the squeezed input state. For increasing $U$, the behavior of the state becomes progressively richer while $\varepsilon$ decreases. At $U=0.1~meV$ the squeezing becomes suppressed, to which we can attribute the saturation in performance. The quadrature variables in the reservoir mode are defined as $\hat{X}=\frac{\hat{a}+\hat{a}^\dagger}{2}$ and $\hat{Y}=\frac{\hat{a}-\hat{a}^\dagger}{2i}$.
• Figure 4: Generalisability of learning: rescaled moments $M_m$ as function of $r$, when training is performed at $\theta=\pi/2$ (circles) and testing at $\theta=0$ (stars). The prediction error on $r$ is still relatively small at $\varepsilon=8.0\%$, highlighting that learning in the network generalises.
• Figure 5: Multivariate prediction with a hybrid quantum-classical model. In a dataset with variability both in $r$ and $\theta$, learning with moments $M_0-M_4$ as before works well to predict $r$ (inset of panel (a)), but less for $\theta$ (inset of (b)). Results improve significantly by training on the pre-processed (see panel c) full spectrum, as depicted on the main panels of (a,b). Further improvement is obtained using a feedforward neural network model trained on the pre-processed spectral data (panels d,e). Panel c depicts the renormalization procedure. The differences $\hat{S}_{i}(\omega)$ obtained by subtracting a reference spectrum $S_{ref}(\omega)$ from each of the spectra $S_{i}(\omega)$ in the dataset are renormalized (divided by the mean over $\omega$$\left\langle S_{i}(\omega)\right\rangle$) and used as training examples for training the neural network. Panel f depicts a polar coordinate representation of the predicted data, with colored points corresponding to predicted squeezing parameters, and black points to the target values. The color of the predicted points corresponds to the prediction error (measured as the Euclidean distance $d=\sqrt{r_p^2+r_t^2-2r_pr_t\cos{(\theta_p-\theta_t})}$ between predicted $(r_p, \theta_p)$ and target $(r_t,\theta_t)$ squeezing parameters).