The inverse parametric problem

TL;DR

The paper tackles how to determine the pump waveform needed to achieve a desired multimode scattering in a parametric oscillator. It introduces the Pump Projection Method (PPM), which uses a Frobenius inner product to project the target off-diagonal coupling onto an orthogonal pump-basis, yielding exact pump amplitudes. Numerical and experimental validations demonstrate robust reconstruction and the ability to realize complex scattering, including nonreciprocal circulation across many modes. The work enables precise design of multimode Gaussian-state generation and rapid, programmable mode routing, with broad implications for continuous-variable quantum information processing.

Abstract

We present a method to calculate the frequency components of a pump waveform driving a parametric oscillator, which realizes a desired frequency mixing or scattering between modes. The method is validated by numerical analysis and we study its sensitivity to added Gaussian noise. A series of experiments apply the method and demonstrate its ability to realize complex scattering processes involving many modes at microwave frequencies, including non-reciprocal mode circulation. We also present an approximate method to dynamically control mode scattering, capable of rapidly routing signals between modes in a prescribed manner. These methods are useful tools for encoding and manipulating continuous variable quantum information with multi-modal Gaussian states.

Figures (6)

• Figure 1: (a) General I/O relation of a parametric oscillator where the output is determined by the scattering matrix $\bm{S}$, controlled by the pump $p_L(t)$. (b) Example of a parametric oscillator, the pump port modulates the inductance of an LC oscillator. (c) We define direct problem when the knowledge of $p_L(T)$ allows to compute the scattering matrix $\bm{S}$, inverse problem when from a target $\bm{S}$ one recover the pump signal that realizes it.
• Figure 2: Robustness of the PPM against Gaussian noise. Relative reconstruction error $\| \bm{S}_\odot - \bm{S}_\text{rec} \|_2 / \| \bm{S}_\odot \|_2$ as a function of the noise-to-target ratio $\| \bm{S}_N \|_2 / \| \bm{S}_{\odot} \|_2$ for different matrix sizes $2N$. The horizontal dotted line indicates the numerical-precision for the maximum number of modes tested $N_\text{max}$. The vertical dotted line marks unity noise-to-target ratio, beyond which the PPM becomes unreliable.
• Figure 3: Experimental validation of the Pump Projection Method (PPM). (a) Target scattering matrix $\bm{S}_{\odot}$ constructed from the target pumps (blue) shown in (d). (b) Experimentally measured scattering matrix $\bm{S}_{\mathrm{exp}}$ obtained from the reconstructed pumps (cyan) shown in (d). (c) Difference between the magnitudes of the target and measured scattering matrices. (d) Pump amplitude (upper) and phase (lower). Amplitudes are normalized by the single-frequency pump amplitude at $2\omega_0$, which achieves 3dB gain in the JPA.
• Figure 4: Design and implementation of a 13-mode circulator using the PPM. (a) Target scattering matrix $|\bm{S}_\odot|$ designed in the $xp$ basis. (b) Low frequency pump amplitudes (top) and phases (bottom) of the waveform returned by the PPM. (c) Experimentally measured scattering matrix $|\bm{S}_{\text{exp}}|$, obtained by applying the pump waveform from panel (b). (d) Graphical representation of the 13-mode circulator. Black arrow shows the circulation direction.
• Figure 5: (a) Measured output mode amplitudes as a function of time $t_\ell/T$. The pump waveform is dynamically updated every measurement window $T$, encoding successive columns of the target image. The scattered output evolves in time to reproduce the desired pattern, with amplitudes expressed in decibels and normalized to the pump-off reference. (b) Example of target scattering matrix $\bm{S}_\ell$ used to generate the output at $t_\ell/T=40$ (white dashed line in panel (a)).