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arXiv:2604.21692

Bipartite entanglement under frequency comb pumping in parametric Josephson circuits

Mikael Vartiainen, Ilari Lilja, Ekaterina Mukhanova, Kirill Petrovnin, Gheorghe Sorin Paraoanu, Pertti Hakonen

quant-phSubmitted Apr 23, 2026• Announced Apr 24, 2026• 0 citations
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Abstract

The creation of high-quality cluster states in superconducting microwave circuits is a relevant ingredient in continuous-variable quantum computing. Although large-scale cluster states have been established in optical systems, dissipation prevents their direct applicability to the microwave realm. Recent improvements in superconducting parametric circuits, in particular Josephson parametric amplifiers (JPA) and traveling wave parametric amplifiers (TWPA), have permitted substantial progress in producing entangled states using microwave photons. In this paper, we examine experimentally and theoretically the effects of numerous parametric pump tones on the degree of two-mode squeezing in a quantum circuit and apply it to the JPA. We find that additional pumps diminish the initial two-mode correlations achieved with a single pump by redistributing it among a larger network of modes and by introducing entanglement with additional idler frequencies. Taking into account the actual heterodyne measurement conditions, the experimental results are consistent with theoretical expectations.

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The creation of high-quality cluster states in superconducting microwave circuits is a relevant ingredient in continuous-variable quantum computing. Although large-scale cluster states have been established in optical systems, dissipation prevents their direct applicability to the microwave realm. Recent improvements in superconducting parametric circuits, in particular Josephson parametric amplifiers (JPA) and traveling wave parametric amplifiers (TWPA), have permitted substantial progress in producing entangled states using microwave photons. In this paper, we examine experimentally and theoretically the effects of numerous parametric pump tones on the degree of two-mode squeezing in a quantum circuit and apply it to the JPA. We find that additional pumps diminish the initial two-mode correlations achieved with a single pump by redistributing it among a larger network of modes and by introducing entanglement with additional idler frequencies. Taking into account the actual heterodyne measurement conditions, the experimental results are consistent with theoretical expectations.
We describe an experiment demonstrating the generation of three independent square-ladder continuous-variable cluster states with up to 94 qumodes of a microwave frequency comb. This entanglement structure at a large scale is realized by injecting vacuum fluctuations into a Josephson Parametric Amplifier pumped by three coherent signals around twice its resonance frequency, each having a particular well-defined phase relation. We reach up to 1.4 dB of squeezing of the nullifier that verifies the cluster state on the square ladder graph. Our results are consistent with a more familiar measure of two-mode squeezing, where we find up to 5.42 dB for one pump, and up to 1 dB for three pumps.
Cluster states are a fundamental resource for continuous-variable quantum computing, enabling measurement-based protocols that can scale beyond the limitations of qubit-based architectures. Here, we demonstrate on-demand generation of multimode entangled microwave cluster states using a programmable Josephson Traveling-Wave Parametric Amplifier (JTWPA) operated in the three-wave mixing regime. By injecting a tailored, non-equidistant set of pump tones via an arbitrary waveform generator, we engineer frequency-specific nonlinear couplings between multiple frequency modes. The entanglement structure is verified via frequency-resolved heterodyne detection of quadrature nullifiers, confirming the target graph topology of the cluster state. Our approach allows reconfigurability through the pumps spectrum and supports scalability by leveraging the wide bandwidth and spatial homogeneity of the JTWPA. This platform opens new avenues for scalable measurement-based quantum information processing in the microwave domain, compatible with superconducting circuit architectures.
We demonstrate the experimental realization of two-dimensional, continuous variable (CV) cluster states between 191 microwave frequency modes. This result is obtained by exposing vacuum fluctuations to the input of a Josephson Parametric Amplifier, parametrically pumped by a sum of coherent tones around twice its resonant frequency. By carefully tuning pump frequencies, amplitudes, and phases we engineer the interference between mixing products and realize honeycomb and square lattice CV cluster states with three and four pump tones respectively. We prove the presence of the cluster states with a suitable nullifier test, reaching up to −1.2-1.2−1.2 dB of squeezing of the cluster state's nullifiers. We study hidden entanglement (HE) and show no hidden entanglement up to ∼−1\sim -1∼−1 dB of squeezing and negligible HE at optimal squeezing.
Traveling wave parametric amplifiers (TWPAs) have recently emerged as essential tools for broadband near quantum-limited amplification. However, their use to generate microwave quantum states still misses an experimental demonstration. In this Letter, we report operation of a TWPA as a source of two-mode squeezed microwave radiation. We demonstrate broadband entanglement generation between two modes separated by up to 400 MHz by measuring logarithmic negativity between 0.27 and 0.51 and collective quadrature squeezing below the vacuum limit between 1.5 and 2.1 dB. This work opens interesting perspectives for the exploration of novel microwave photonics experiments with possible applications in quantum sensing and continuous variable quantum computing.
Josephson traveling-wave parametric amplifiers (TWPAs) enable the amplification of weak microwave signals close to the quantum limit with large bandwidth, which has a broad range of applications in superconducting quantum computing and in the operation of single-photon detectors. While the large bandwidth allows for their use in frequency-multiplexed detection architectures, an increased number of readout tones per amplifier puts more stringent requirements on the dynamic range to avoid saturation. Here, we characterize the undesired mixing processes between the different frequency-multiplexed tones applied to a Josephson TWPA, a phenomenon also known as intermodulation distortion. The effect becomes particularly significant when the amplifier is operated close to its saturation power. Furthermore, we demonstrate that intermodulation distortion can lead to significant crosstalk and reduction of fidelity for multiplexed readout of superconducting qubits. We suggest using large detunings between the pump and signal frequencies to mitigate crosstalk. Our work provides insights into the limitations of current Josephson TWPAs and highlights the importance of performing further research on these devices.
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.