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arXiv:2512.15453 · Physical Review Applied

The inverse parametric problem

Michele Cortinovis, Fabio Lingua, David B. Haviland

quant-phphysics.app-phSubmitted Dec 17, 2025• Announced Dec 18, 2025• 0 citations
v1 · Dec 17, 2025v2 · Jul 7, 2026
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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.

DOI
10.1103/1bdh-lcxq
Comments
9 pages, 6 figures

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We investigate nonreciprocal scattering within the modes of a microwave frequency comb. Adjusting the pump frequencies, amplitudes, and phases of a Josephson parametric oscillator, we control constructive interference for the m ⟶ ℓ scattering processes, while concurrently achieving destructive interference for the inverse process ℓ ⟶ m . We outline the methodology for realizing nonreciprocity in the context of two-mode isolation and a three-mode circulation, which we extend to multiple modes. We find good agreement between the experiments and a linearized theoretical model. Nonreciprocal scattering expands the toolset for parametric control, with the potential to engineer alternative quantum correlations.
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This thesis demonstrates the experimental generation of two-dimensional Continuous Variable (CV) cluster states in a superconducting microwave frequency comb using parametric pumping. We develop the Pump Projection Method (PPM) to solve the inverse parametric problem: given a target multimode correlation structure, we determine the required pump configuration by exploiting the orthogonality of elementary coupling matrices in Hilbert-Schmidt space. We experimentally generate square-lattice and honeycomb-lattice cluster states with up to several tens of modes, verified through nullifier measurements showing squeezing below the vacuum level. The states exhibit scalability with respect to mode number and robustness against frequency spacing variations. A dissipative model explains the optimal pumping regime, while hidden entanglement analysis confirms negligible unwanted correlations. These results establish a systematic framework for engineering programmable multimode entanglement in superconducting circuits, providing a versatile platform for Measurement-Based Quantum Computation (MBQC) in the microwave domain.
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