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arXiv:quant-ph/0410119 · Physical Review A

Entanglement of large atomic samples: A Gaussian-state analysis

Jacob Friis Sherson, Klaus Mølmer

Published Mar 23, 2005• 20 citations
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Abstract

We present a Gaussian-state analysis of the entanglement generation between two macroscopic atomic ensembles due the continuous probing of collective spin variables by optical Faraday rotation. The evolution of the mean values and the variances of the atomic variables is determined, and the entanglement is characterized by the Gaussian entanglement of formation and the logarithmic negativity. The effects of induced opposite Larmor rotation of the samples and of light absorption and atomic decay are analyzed in detail.

DOI
10.1103/physreva.71.033813

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We present a Gaussian-state analysis of the entanglement generation between two macroscopic atomic ensembles due the continuous probing of collective spin variables by optical Faraday rotation. The evolution of the mean values and the variances of the atomic variables is determined, and the entanglement is characterized by the Gaussian entanglement of formation and the logarithmic negativity. The effects of induced opposite Larmor rotation of the samples and of light absorption and atomic decay are analyzed in detail.
Quantum interfaces between polarized atomic ensembles and coherent states of light, applied recently to manipulate bipartite and multipartite entanglement, are revisited by means of a continuous-variable formalism. The explicit use of the continuous-variable formalism facilitates significantly the analysis of entanglement between different modes, reducing it to the study of the properties of a final covariance matrix which can be found analytically. Furthermore, it allows to study matter-light interfaces for mixed states, adapting the formalism to the experimental situations in which the initial prepared Gaussian states are, unavoidably, affected by a certain amount of noise. A multipartite scenario, leading to the generation of macroscopic cluster states is presented and analyzed in detail within this formalism.
We study the effects of losses on the entanglement created between two separate atomic gases by optical probing and homodyne detection of the transmitted light. The system is well-described in the Gaussian state formulation. Analytical results quantifying the degree of entanglement between the two gases are derived and compared with the entanglement in a pair of light pulses generated by an Einstein-Podolsky-Rosen (EPR) source. For low (high) transmission losses, the highest degree of entanglement is obtained by probing with squeezed (antisqueezed) light. In an asymmetric setup where light is only sent one way through the atomic samples, we find that the logarithmic negativity of entanglement attains a constant value $\ensuremath{-}{\mathrm{log}}_{2}(N)$ with N=1∕3N=1∕3N=1∕3, irrespective of the loss along the transmission line.
We consider an ensemble of trapped atoms interacting with a continuous-wave laser field. For sufficiently polarized atoms and for a polarized light field, we may approximate the nonclassical components of the collective spin angular momentum operator for the atoms and the Stokes vectors of the field by effective position and momentum variables for which we assume a Gaussian state. Within this approximation, we present a theory for the squeezing of the atomic spin by polarization rotation measurements on the probe light. We derive analytical expressions for the squeezing with and without inclusion of the noise effects introduced by atomic decay and by photon absorption. The theory is readily adapted to the case of inhomogeneous light-atom coupling [A. Kuzmich and T.A.B. Kennedy, Phys. Rev. Lett. 92, 030407 (2004)]. As a special case, we show how to formulate the theory for an optically thick sample by slicing the gas into pieces, each having only small photon absorption probability. Our analysis of a realistic probing and measurement scheme shows that it is the maximally squeezed component of the atomic gas that determines the accuracy of the measurement.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.
We consider the optical generation and characterization of entanglement in atomic ensembles under nonuniform interaction between the ensemble and an optical mode. We show that for a wide range of parameters a system of nonuniformly coupled atomic spins can be described as an ensemble of uniformly coupled spins with a reduced effective atom-light coupling and a reduced effective atom number, with a reduction factor of order unity given by the ensemble-mode geometry. This description is valid even for complex entangled states with arbitrary phase-space distribution functions as long as the average total spin remains large, and the detection does not resolve single spins. Furthermore, we derive an analytic formula for determining the observable entanglement in the case, of relevance in practice, where the ensemble-mode coupling differs between state generation and measurement.
Summary form only given. A new approach to the problem of the quantum interface between light and atoms has been recently developed. The method utilizes free space dispersive interaction between pulses of light and spin polarized atomic ensembles. Entanglement between the polarization state of light and the collective spin state of atoms is established by measurement, more precisely by detection of light in a certain polarization basis. In the first demonstration of this approach we have generated a long-lived entangled state of two separate macroscopic atomic samples by a polarization measurement on light transmitted through the samples. We then have shown that this approach also works for mapping of a quantum state of light onto long-lived atomic spin state paving the road towards realization of the quantum memory for light. Recently we have also extended the free space light-atoms interface beyond the continuous variables in the proposal, where a coherent superposition "cat" state of an atomic ensemble is generated by a measurement sequence. Progress with other communication protocols such as atomic state teleportation and multiparty networks will be presented.