Generation of Squeezed Fock States by Particle-Number Measurements on Multimode Gaussian States

TL;DR

The paper develops a universal method to generate squeezed Fock states (SFSs) by performing particle-number measurements on modes of multimode Gaussian states. It identifies a universal class of $N$-mode Gaussian states, implementable with a beam-splitter network and just two squeezed vacua, such that measuring $N-1$ modes yields an SFS whose order is determined solely by the total detected number $n$. The authors derive exact expressions for the generation probability $P(n)$ in terms of a single universal parameter $X$, show that the maximal probability matches the two-mode optimum $P(n)=\frac{n^n}{(n+1)^{n+1}}$, and provide fidelity under detector inefficiency as $F_n(\eta)=\left(\frac{(X-1)\eta^{2}+2}{X+1}\right)^{n+1}$. A detailed comparison with a non-universal cascade scheme highlights a trade-off between generation probability and squeezing resources. The work offers a practical framework for generating non-Gaussian states in scalable multimode optics, with implications for quantum information processing and metrology.

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.

Figures (8)

• Figure 1: Scheme for generating squeezed Fock states. In the figure, $S_1$ and $S_2$ denote oscillators prepared in squeezed vacuum states, $BS$ is a beam splitter with transmission coefficient $t$, PNRD is a particle-number-resolving detector, and $\Psi_{out}$ is the generated state.
• Figure 2: Scheme for generating squeezed Fock states using an $N$-mode Gaussian state and $N-1$ detectors. In the figure, $S_i$ are squeezed vacuum states, PNRDs are particle-number-resolving detectors, and $\Psi_{out}$ is the generated state.
• Figure 3: Scheme for generating a squeezed Fock state using a three-mode Gaussian state. In the figure, $S_2$ and $S_3$ are quantum oscillators in squeezed vacuum states, $V_1$ is a quantum oscillator in the vacuum state, BS denotes beam splitters, and $\Psi_{out}$ is the output state.
• Figure 4: Scheme for generating a squeezed Fock state using non-ideal detectors. In the figure, $S_i$ denotes the squeezed vacuum state, PNRDs are particle-number–resolving detectors, BS denotes a beam splitter with transmittance $t=\eta$, and $\Psi_{out}$ is the generated state.
• Figure 5: Cascade scheme for generating squeezed Fock states. In the figure, $S_1$, $S_2$, and $S_3$ denote squeezed vacuum states, $\Psi_1$ is the state at the output of the first stage of the scheme, and $\Psi_{out}$ is the final output state.