Quantum properties of superpositions of oppositely squeezed states

TL;DR

The paper tackles the generation and characterization of non-Gaussian Schrödinger-cat-like states formed from oppositely squeezed states $|r;\pm\rangle$, and assesses their suitability as continuous-variable resources. It analyzes the Wigner function, revealing negativity from interference fringes, and studies entanglement generation when these states are fed into a 50:50 beam splitter. Because the ideal cross-Kerr generation is impractical, the authors propose a linear-optical heralding scheme that uses a two-mode squeezed vacuum, ancillas, beam splitters, displacements, and photon counting to approximate $|r;\pm\rangle$. They find that for small squeezing, the non-Gaussian state $|r;-\rangle$ can yield more entanglement than a pure two-mode squeezed vacuum, highlighting potential advantages in low-squeezing regimes and motivating further experimental and application-oriented work such as bosonic error correction and quantum networks.

Abstract

We investigate the quantum properties of superpositions of oppositely squeezed states, which can be regarded as Schrodinger cat states. Compared with conventional coherent-state cat states, these states exhibit distinct photon-number structures and enhanced nonclassical features. We analyze their Wigner function and quantify the entanglement generated when they are injected into a 50:50 beam splitter. For small squeezing parameters, the resulting two-mode states possess higher entanglement than pure two-mode squeezed vacuum states. We also propose a linear-optical heralding scheme that approximates this superposition of oppositely squeezed states without requiring strong Kerr nonlinearities. Our results indicate that such states are promising resources for continuous-variable quantum information processing, particularly in regimes where high non-Gaussianity and strong entanglement are desirable.

Figures (5)

• Figure 1: Schematic illustration of implementation for generating $|r;+\rangle$ approximately. The source emits the two-mode squeezed vacuum state for modes $4$ and A. Next, preparing modes $1$, $2$, and $3$ as $|0\rangle_{1}|0\rangle_{2}|0\rangle_{3}$, we apply the beam splitters $\hat{B}_{14}$, $\hat{B}_{24}$, and $\hat{B}_{34}$ to modes $1$, $2$, $3$, and $4$. Then, we apply the displacement operators $\hat{D}_{j}(\alpha_{j})$ for $j=1, 2, 3$, and $4$ to the modes. Finally, detecting single photons of modes $1$, $2$, $3$, and $4$, we obtain $|r;+\rangle$ approximately for mode A.
• Figure 2: (a) 3D plot of the success probability $P$ that all of the four unity-efficiency photon counters make click detection with the experimental setup shown in Fig. \ref{['figure01']} as a function of $r$ and $q$. (b) 3D plot of the fidelity $F$ between the projected state of mode A in the experimental setup of Fig. \ref{['figure01']} and $|r;+\rangle$ as a function of $r$ and $q$. In both (a) and (b), we set $0.26\leq r\leq 0.92$.
• Figure 3: Phase space plots of the Wigner function $W(\alpha)$ as a function of $\hbox{Re}[\alpha]$ and $\hbox{Im}[\alpha]$. (a) $|r\rangle$, $r=0.5$, (b) $|r\rangle$, $r=1.0$, (c) $|r\rangle$, $r=1.5$, (d) $|r;-\rangle$, $r=0.5$, (e) $|r;-\rangle$, $r=1.0$, (f) $|r;-\rangle$, $r=1.5$, (g) $|r;+\rangle$, $r=0.5$, (h) $|r;+\rangle$, $r=1.0$, (i) $|r;+\rangle$, $r=1.5$.
• Figure 4: Phase space plot of $W(\alpha)$ for the conventional coherent-state cat state $\{2[1-\exp(-2a^{2})]\}^{-1/2}(|a\rangle-|-a\rangle)$ with $a=1$. The graph has a negative value at the origin, $W(0)=-0.6366$.
• Figure 5: Plots of entropies as functions of the squeezing parameter $r$. The solid red, dashed blue, and dotted purple curves represent the entropies, $S^{(-)}_{\hbox{\scriptsize BS}}(r)$, $S^{(+)}_{\hbox{\scriptsize BS}}(r)$, and $S_{\hbox{\scriptsize TMSV}}$, respectively.