Postselected Entangled States by Photon Detection

Abstract

Postselection is a non-deterministic mechanism to entangle subsystems, often used in weakly-excited systems. We here show how highly-excited ensembles of two-level emitters can be entangled by photon detection. A collective spin is formed, characterized by a squeezing parameter detected by far-field measurements. While decoherence is detrimental to this conditional entanglement, successive photon detections act as a purification process and restores the spin squeezing. Our work opens up new avenues for the generation of postselected entanglement in open quantum systems.

Figures (4)

• Figure 1: An ensemble of atoms initially in a separable state is becomes entangled after several photon detections in direction $\mathbf{k}_{d}$. The drive $\mathbf{k}_{L}$ is used to create the (separable) initial state. The directions in which the photons are detected, $\mathbf{k}_{d}$, and the electric field measured, $\mathbf{k}_{w}$, are correlated with the direction of the created collective spin.
• Figure 2: Squeezing parameter $\xi^2$ produced in a regular chain along $\hat{z}$-axis and with a lattice step $d=2\pi/k$ for a system initially (a-b) in a CSS \ref{['eq:pure']}, with the drive, the photon and squeezing detection all along the $x$-axis; (c-d) in a steady state \ref{['eq:driven_state']} reached with a laser at an angle $\theta_{L}=\pi/3$ from the $\hat{z}$-axis, with the photon and squeezing detection along the same direction. In (a) and (c), $\xi^2$ is computed after detecting a single photon ($\nu=1$) from a chain of $N=50$ to $800$ emitters; for (b) and (d), $\nu=1$ to $8$ photons detected from a chain of $N=10$ emitters. The inset in (d) illustrates the purification process occurring with the detection events.
• Figure 3: Squeezing parameter for a three-dimensional cloud of $N=100$ two-level emitters randomly distributed inside a sphere of radius $d=100(2\pi/k)$ initially prepared in state \ref{['eq:no_coherence']}. Entanglement is detected in all the region $0<\bar{\theta}\leq \pi$ if an enough number of photons are detected (here $\nu=10, 20,\hdots, 90, 99$, with optimal case corresponding to $\nu=N/2=50$). Note that as we get close to the ground state ($\bar{\theta}\xrightarrow{}0$), more photons are necessary to detect entanglement.
• Figure 4: Squeezing parameter for a 3D random configuration of $N=100$ two-level emitters inside a sphere of radius $d=100(2\pi/k)$: (a) for the population state \ref{['eq:no_coherence']} with $\bar{\theta}=\pi/3$ and $\nu=50$ detected photons along three different directions $\theta_{d}$ (marked by the vertical dashed lines). The optimal direction $\theta_{w}$ to measure the squeezing is thus along the same direction of the photon detection, $\theta_{d}$. (b)-(c) For $N=10$ atoms initially in CSS \ref{['eq:pure']} with $\theta=3\pi/4$ and $\mathbf{k}_{L}=\pi/4$. (b) 1D linear configuration along $z$-axis with $d=2\pi/k$ with all five photons detected along the drive direction $\mathbf{k}_{L}$. (c) 2D circular configuration with radius $d=2\pi/k$, the photons are detected either in the same direction as the drive $\mathbf{k}_{L}$ (solid curve), or in 5 different angles (dash-dotted curves, with $\theta_1=0$, $\theta_2=\pi/3$, $\theta_3=\pi/2$, $\theta_4=3\pi/2$, $\theta_5=\pi$, always in the $xz$-plane). In (b-c), the dashed curve corresponds to the radiated intensity after a single photon detection (right axis).