Continuous variable entanglement in a cold-atoms mirrorless optical parametric oscillator

TL;DR

The work tackles generating and characterizing multipartite continuous-variable entanglement in a mirrorless optical parametric oscillator (MOPO) implemented with four-wave mixing in a cold Cesium atomic cloud. The authors use two counter-propagating pump beams with orthogonal polarizations and heterodyne detection to measure both quadratures of selected outputs, enabling reconstruction of the covariance matrix $\mathbb{V}$ and extraction of the submatrices $\mathbb{A}_i$ and $\mathbb{C}_{ij}$. Entanglement is demonstrated for two mode pairs, $(S1,S2)$ and $(S1,S3)$, via the PPT criterion with the witness $W_{\mathrm{PPT}}=1+\operatorname{Det}[\mathbb{V}]+2\operatorname{Det}[\mathbb{C}_{ij}] - \operatorname{Det}[\mathbb{A}_i]-\operatorname{Det}[\mathbb{A}_j]<0$ and a physicality condition $W=1+\operatorname{Det}[\mathbb{V}] - 2\operatorname{Det}[\mathbb{C}_{ij}] - \operatorname{Det}[\mathbb{A}_i]-\operatorname{Det}[\mathbb{A}_j] \ge 0$, with $\operatorname{Det}[\mathbb{C}_{ij}]<0$ across the measured range. The results align with a Bloch-Messiah-based interpretation that accounts for mode-mismatch and gain-dependent behavior, and indicate the system’s multipartite entanglement structure beyond the measured pairs. The MOPO thus emerges as a scalable, memory-capable platform for quantum networks, with demonstrated potential to store and transfer orbital angular momentum in atomic degrees of freedom and to realize higher-order, multiplexed CV entanglement.

Abstract

In this work, we explore both the internal and external atomic degrees of freedom to observe quantum entanglement between the modes produced by a mirrorless optical parametric oscillator operating below the oscillation threshold in a sample of free-space cold cesium atoms. Using a new heterodyne technique, we recover the covariance matrix that reveals the quantum entanglement for two different pairs of modes, thus demonstrating the generation of four entangled modes in this system. Applications to quantum networks, and the possibilities of studying higher orders of entanglement are a direct consequence of the present study.

Figures (3)

• Figure 1: (a) Simplified experimental setup. The counter-propagating pumping beams $\left(P_1\right.$ and $\left.P_2\right)$ with linear orthogonal polarizations interact with the atoms and generate the four modes specified by $S_1, S_2, S_3$, and $S_4$. Modes $S_1, S_2$ and $S_3$ are sent to independent heterodyne detection systems. (b) Time sequence of the involved fields and detection process.
• Figure 2: Measured values for the determinants of the covariance matrices of fields $i$ ($\operatorname{Det}[\mathbb{A}_{i}]$) and j ($\operatorname{Det}[\mathbb{A}_j]$), the cross-correlation $\operatorname{Det}[\mathbb{C}_{ij}]$ and the covariance matrix $\operatorname{Det}[V]$, for modes 1 and 2 (a) and modes 1 and 3 (b). The lines are adjustment of the model, as described in the text.
• Figure 3: Physicality test of the measured covariance matrix ($W$) and entanglement witness ($W_{PPT}$), evaluated from the data in Fig. \ref{['fig:determinants']}, and their respective adjustement (for the lines)