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Journal of Optical Technology

Squeezed Fock states: generation and applications [Review]

Elizaveta N. Bashmakova, Sergey B. Korolev, Eduard R. Zinatullin, Yuri M. Golubev, Tatiana Yu. Golubeva

Published Mar 1, 2025• 2 citations

Abstract

Subject of study. Schemes for generating non-Gaussian quantum states and the states required for quantum error correction protocols are investigated. Aim of study. The primary objectives were to generalize methods and approaches for describing the generation of non-Gaussian states in measurement schemes to determine the most effective model for describing the generation process and to assess a model for generating squeezed Fock states and a quantum error correction code based on such states. Method. A theoretical analysis of the wavefunction evolution of various non-Gaussian states in particle-number measurement schemes was performed; in this analysis, squeezed states were used as inputs. Main results. A scheme for generating squeezed Fock states was theoretically examined. An explicit analytical expression for the output wavefunction was constructed to completely analyze the output states depending on the parameters of the scheme under consideration. A set of conditions imposed on the parameters of a two-mode entangled Gaussian state, i.e., the “universal solution regime,” was outlined. This regime guaranteed the generation of squeezed Fock states with a high probability. Practical significance. The results demonstrated the ability of the universal solution regime to generate squeezed Fock states of arbitrary order for a given squeezing parameter. The feasibility of using squeezed Fock states in quantum error correction codes was analyzed. A comparative study of squeezed Fock states and squeezed Schrödinger cat states as codewords confirmed squeezed Fock states as robust solutions to ensure information protection in channels subjected to particle loss and dephasing.

DOI: 10.1364/jot.92.000195

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Subject of study. Schemes for generating non-Gaussian quantum states and the states required for quantum error correction protocols are investigated. Aim of study. The primary objectives were to generalize methods and approaches for describing the generation of non-Gaussian states in measurement schemes to determine the most effective model for describing the generation process and to assess a model for generating squeezed Fock states and a quantum error correction code based on such states. Method. A theoretical analysis of the wavefunction evolution of various non-Gaussian states in particle-number measurement schemes was performed; in this analysis, squeezed states were used as inputs. Main results. A scheme for generating squeezed Fock states was theoretically examined. An explicit analytical expression for the output wavefunction was constructed to completely analyze the output states depending on the parameters of the scheme under consideration. A set of conditions imposed on the parameters of a two-mode entangled Gaussian state, i.e., the “universal solution regime,” was outlined. This regime guaranteed the generation of squeezed Fock states with a high probability. Practical significance. The results demonstrated the ability of the universal solution regime to generate squeezed Fock states of arbitrary order for a given squeezing parameter. The feasibility of using squeezed Fock states in quantum error correction codes was analyzed. A comparative study of squeezed Fock states and squeezed Schrödinger cat states as codewords confirmed squeezed Fock states as robust solutions to ensure information protection in channels subjected to particle loss and dephasing.

The generation of squeezed Fock states by the one or more photon subtraction from a two-mode entangled Gaussian (TMEG) state is theoretically addressed. We showed that an arbitrary order Fock state can be generated this way and we obtained a condition that should be imposed on the parameters of the TMEG state to guarantee such a generation. We called the regime, in which this condition is satisfied, the universal solution regime. We showed that, for the first squeezed Fock state, the above condition is redundant such that the generation of the first squeezed Fock state is still possible by a one-photon subtraction from an arbitrary TMEG state. At the same time, the maximum generation probability of the first squeezed Fock state generation corresponds to the universal solution regime. We applied the above results to the description of generation of the squeezed Fock states using a beam splitter and a controlled-Z operation. We have estimated the parameters of such setups and input squeezed states, which are necessary to obtain squeezed Fock states with the maximum probability.

We investigate the generation of squeezed Fock states (SFSs) via particle-number measurements in the modes of multimode Gaussian states. We identify a universal class of

N

-mode Gaussian states for which measuring

N-1

modes results in the generation of SFSs. The key feature of these states is that the generated SFSs depend only on the total number of detected particles and are independent of their distribution among the detectors. Based on the general form of the wave functions of multimode Gaussian states, we propose a universal scheme for SFS generation. For this scheme, we evaluate the probability of SFS generation and analyze the robustness of the process against imperfections in particle-number-resolving detectors. In addition, we compare the universal scheme with a nonuniversal scheme, in which the generation of SFSs depends on a specific distribution of particle numbers across the detectors. We demonstrate that the universal scheme provides a higher probability of SFS generation, at the cost of increased experimental resources.

Non-Gaussian quantum states and operations constitute essential resources for achieving quantum computational advantage and enabling quantum error correction in bosonic platforms. However, their generation in optical settings remains a challenging experimental task, often relying on probabilistic heralded protocols. Here, we present an in-depth analysis of the suitability of photon catalysis between low number Fock states and squeezed states for the generation of squeezed coherent state superpositions. We employ the stellar rank formalism to characterize the non-Gaussian complexity of input resources (including both states and measurements) and the generated states. This enables a systematic comparison of the fidelity between the catalyzed output and the target states to the maximum fidelity achievable by any protocol with the same non-Gaussian input resources. In this sense, we identify instances where the catalysis protocols considered here are provably optimal. We identify parameter regimes in which high-fidelity approximations of the target states can be achieved with minimal resources. Furthermore, we benchmark the performance of photon catalysis against Gaussian boson sampling-inspired protocols in terms of success probability and state quality, highlighting the advantages of deterministic Fock state sources. We also investigate the generation of related non-Gaussian resources including squeezed Fock states, relevant for quantum error correction. To account for experimental imperfections, we model losses across all optical modes using a Hilbert space truncation approach in the Fock basis and analyze the robustness of the generated states under realistic conditions. Our results quantify the trade-offs between non-Gaussian resource complexity, achievable fidelity, and losses in photon catalysis protocols, providing practical guidelines for near-term photonic implementations.

Abstract We investigate the generation of squeezed Fock states (SFSs) via particle-number measurements in the modes of multimode Gaussian states. We identify a universal class of N -mode Gaussian states for which measuring N − 1 modes results in the generation of SFSs. The key feature of these states is that the generated SFSs depend only on the total number of detected particles and are independent of their distribution among the detectors. Based on the general form of the wave functions of multimode Gaussian states, we propose a universal scheme for SFS generation. For this scheme, we evaluate the probability of SFS generation and analyze the robustness of the process against imperfections in particle-number-resolving detectors. In addition, we compare the universal scheme with a nonuniversal scheme, in which the generation of SFSs depends on a specific distribution of particle numbers across the detectors. We demonstrate that the universal scheme provides a higher probability of SFS generation, at the cost of increased experimental resources.

We propose an efficient and robust protocol for the generation of entanglement between a superconducting qubit and a squeezed cavity. By applying a parametric drive to the cavity coupled to the qubit, the dynamical evolution of the system is precisely described by an anisotropic Rabi model within a squeezed reference frame. Utilizing high-order time-averaging methods, we analytically derive the resonance conditions and the effective Rabi frequency for the high-order three-photon process. By implementing an adiabatic passage, slowly tuning the cavity frequency across the resonance, the system is steered into a maximally entangled state, e.g., between the three-photon state

\ket{g,3}

and the qubit excited state

\ket{e,0}

in the squeezed picture. Numerical simulation results confirm the high fidelity and robustness of the proposed protocol. Our method provides a practical pathway for generating complex non-Gaussian entangled states, which are of significant value for fault-tolerant quantum computation and quantum metrology beyond the standard quantum limit.