Nonclassicality of multi-photon-added cat states

TL;DR

This work analyzes multi-photon-added cat states with arbitrary $m$ and relative phase $\phi$, deriving exact expressions for the photon-number distribution $P(n)$, Mandel $Q$, quadrature and amplitude-squared squeezing, and the Wigner function to map their nonclassical structure. A key finding is that photon addition induces a $\pi$ parity shift for odd $m$ and renders all states sub-Poissonian regardless of $\phi$, while erasing first-order quadrature squeezing but giving rise to amplitude-squared squeezing and second-order squeezing that grows with $m$. Wigner-function negativity reflects quantum interference and parity changes, with odd $m$ flipping the origin's sign; increasing $|\alpha|$ sharpens phase-space features. The authors propose two feasible experimental routes—atom–light interfaces and SPDC-based heralding—to realize these states with current technology, highlighting their potential for quantum imaging and information processing as higher-order nonclassical resources.

Abstract

Multi-photon-added cat states are constructed by repeatedly applying the creation operator to a cat state. We study in detail their photon-number distribution, $Q$ parameter, squeezing properties, and Wigner function. We show that photon addition induces a $π$ phase shift in the original parity configuration whenever an odd number of photons is added, reflected as swapped vanishing probabilities and phase space displacements at the origin. Remarkably, the same process drives these states into a sub-Poissonian regime regardless of the relative phase between their coherent state components, making them valuable resources for quantum imaging, at the cost of losing quadrature squeezing, but gaining amplitude-squared one. We also discuss how these states can be generated using existing hardware.

Figures (6)

• Figure 1: Photon-number distributions of the (a) cat state with $\alpha=1$ and relative phase $\phi$ (different color), (b) after its single-photon addition, and (c) after two-photon addition, restoring the original parity.
• Figure 2: The $Q$ parameter as a function of cat state strength $|\alpha|$ and different $\phi$ (color encoded) evaluated for (a) the cat state, (b) the single-photon-added cat state, and (c) the two-photon-added cat state.
• Figure 3: Variance $(\Delta x_\theta)^2$ as a function of $\theta$, for weak cat state with $\alpha=0.25$ and relative phase $\phi$ (color encoded), for the (a) non-excited, (b) single-excited and (c) double-excited cat state.
• Figure 4: Variance $(\Delta x_\theta)^2$ as a function of $|\alpha|$, with fixed phase $\theta+\varphi=\pi/2$, for (a) the cat state, (b) a single-photon-added cat, and (c) a two-photon-added cat.
• Figure 5: Second-order squeezing $Y(\theta)$ as a function of $|\alpha|$, with fixed $\theta+\varphi=0$, for the cat state (a) before photon-addition, (b) after single-photon addition, and (c) after another photon addition.