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arXiv:quant-ph/0411083 · Physical Review A

Multimode entanglement of light and atomic ensembles via off-resonant coherent forward scattering

Dmitriy V. Kupriyanov, O. S. Mishina, Igor M. Sokolov, Brian Julsgaard, Eugene Simon Polzik

Published Mar 29, 2005• 55 citations
arXivHTMLPDF

Abstract

Quantum theoretical treatment of coherent forward scattering of light in a polarized atomic ensemble with an arbitrary angular momentum is developed. We consider coherent forward scattering of a weak radiation field interacting with a realistic multilevel atomic transition. Based on the concept of an effective Hamiltonian and on the Heisenberg formalism, we discuss the coupled dynamics of the quantum fluctuations of the polarization Stokes components of propagating light and of the collective spin fluctuations of the scattering atoms. We show that in the process of coherent forward scattering, this dynamics can be described in terms of a polariton-type spin wave created in the atomic sample. Our work presents a general example of an entangling process in the system of collective quantum states of light and atomic angular momenta, previously considered only for the case of spin-$\frac{1}{2}$ atoms. We use the developed general formalism to test the applicability of the spin-$\frac{1}{2}$ approximation for modeling the quantum nondemolishing measurement of atoms with a higher angular momentum.

DOI
10.1103/physreva.71.032348

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Quantum theoretical treatment of coherent forward scattering of light in a polarized atomic ensemble with an arbitrary angular momentum is developed. We consider coherent forward scattering of a weak radiation field interacting with a realistic multilevel atomic transition. Based on the concept of an effective Hamiltonian and on the Heisenberg formalism, we discuss the coupled dynamics of the quantum fluctuations of the polarization Stokes components of propagating light and of the collective spin fluctuations of the scattering atoms. We show that in the process of coherent forward scattering, this dynamics can be described in terms of a polariton-type spin wave created in the atomic sample. Our work presents a general example of an entangling process in the system of collective quantum states of light and atomic angular momenta, previously considered only for the case of spin-12\frac{1}{2}21​ atoms. We use the developed general formalism to test the applicability of the spin-12\frac{1}{2}21​ approximation for modeling the quantum nondemolishing measurement of atoms with a higher angular momentum.
We study the three-dimensional nature of the quantum interface between an ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled through a dispersive interaction. To achieve strong entanglement between the collective atomic spin and the photons, one must match the spatial mode of the collective radiation of the ensemble with the mode of the laser beam while minimizing the effects of decoherence due to optical pumping. For ensembles coupling to a probe field that varies over the extent of the cloud, the set of atoms that indistinguishably radiates into a desired mode of the field defines an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is not characterized by a single parameter, the optical density, but by a collection of different effective atom numbers that characterize the coherence and decoherence of the system. To model the dynamics of the system, we develop a full stochastic master equation, including coherent collective scattering into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and backaction due to continuous measurement of the light entangled with the spin waves. This formalism is used to study the squeezing of a spin wave via continuous quantum nondemolition (QND) measurement. We find that the greatest squeezing occurs in parameter regimes where spatial inhomogeneities are significant, far from the limit in which the interface is well approximated by a one-dimensional, homogeneous model.
We discuss general polarization-sensitive correlation measurements of light fluctuations based on the Hanbury Brown and Twiss interference detection scheme. Such measurements are considered as a basis for correlation spectroscopy, which allows us to observe the correlations of the fluctuations of all polarization components (Stokes parameters) of the probe light. The interferometer scheme is discussed in application to the study of the polarization-dependent interaction of light with an atomic ensemble with polarized angular momentum and put into an external magnetic field. We describe the general procedure for evaluating the photocurrent correlation function in the case of an optically thin atomic medium. Our calculations are based on the Keldysh diagram technique which is usually used for the study of non-equilibrium systems by means of the second quantized formalism. It allows us to express the Raman field-correlation functions and parametric scattering processes describing the output light fluctuations in terms of a different type of time-ordered electric field and atomic Green functions. We show that with definite conditions the photocurrent correlation function can be written as the sum of two terms proportional to the correlation functions of atomic density matrix fluctuations in symmetric and antisymmetric forms. This expression of the photocurrent correlation function can be used in various different applications.
Development of a theoretical treatment of multiple coherent light scattering in an ultracold atomic gas is reported. Specific application is made to coherent backscattering of a weak-radiation field from realistically modeled samples of ultracold atomic 85Rb.{}^{85}\mathrm{Rb}.85Rb. Comprehensive Monte Carlo simulations of the spatial, spectral, and polarization dependence of the backscattering line shape are made and compared with available experimental results.
This thesis contributes to the theory of a light-matter quantum interface based on the Kerr-effect arising in the dispersive interaction of pulsed laser light with a spin-polarized atomic ensemble. The scattering process can be described on the basis of a simple, yet fully quantum mechanical model involving only a small number of bosonic modes referring to transverse spin components and quadratures of forward scattered light. Based on this model we derive protocols for the creation of entangled states of light and atoms and the teleportation of quantum states of light onto atoms employing this entanglement. Furthermore we present a protocol, which allows one to exchange the state of light and atoms and thus provides a quantum memory for states of the light field. For both, the storage and the teleportation of coherent states we prove a benchmark on the average fidelity, which is achievable by purely classical protocols. Both protocols allow one to significantly surpass this threshold and to demonstrate thereby a gain in employing quantum strategies. Under common experimental conditions the initial state of light and atoms is Gaussian and the given interaction preserves this property. Motivated by this observation, we examine in this context, how a given interaction can be used to simulate others and how it can be employed to create entanglement at optimal rates. The results are applied to construct protocols based on several passes of light through an atomic ensemble.
In this thesis I explore the use of collective spin angular momentum as a platform for quantum information processing. Such systems have several nice features that make them excellent choices for such protocols, especially ones where information is also stored in photonic variables. The longer coherence times of atoms makes it possible to store information from light in atoms for future use, and it is generally easier to couple atomic variables than to create nonlinear interactions of light. Above and beyond the advantages of atomic systems, collective atomic systems have additional strengths. In the limit of a large number of atoms, the collective variables of atomic systems have a natural connection to the bosonic algebra of light (known as the Holstein-Primakoff or HP approximation) where components of the collective spin angular momentum effectively act as quadratures, making them natural systems for coupling to light. Also, collectively addressing the atoms allows one to access a large state space without individual addressing of the atoms. A natural first step towards control of these collective variables is generating large amounts of atomic spin squeezing. In the HP approximation going from rotations to squeezing corresponds to going from linear to quadratic interactions in the atomic quadratures; for extremely large squeezing one should see the breakdown of the HP approximation and the ability to generate arbitrary collective atomic states. I have sought to improve previous schemes for the spin squeezing of atomic ensembles, such as the proposal of Takeuchi et. al. based on coherent quantum feedback. In this scheme a beam of linearly polarized light passes through the atomic ensemble (prepared in a coherent state), coupling to the atoms through a state-dependent index of refraction (the Faraday effect). The light is then passed through a wave-plate and reflected back through the atoms for a second pass. This double-pass scheme leads to an effective nonlinearity as the atomic fluctuations are mapped onto the light on the first pass and then back on to the atoms in the second pass. The light acts as a bus coupling each atom to each of the others. This nonlinear interaction forms a shearing of the atomic coherent state that results in squeezing. The light is entangled to the atoms through these interactions, and remains entangled as it escapes the system. This leads to decoherence of the atoms as the light is lost to the environment, reducing the amount of spin squeezing achieved. The first step towards improving the double-pass scheme was to add a quantum eraser step in which the light is disentangled from the squeezed atoms. By first measuring one quadrature of the light, and then performing a measurement-dependent rotation on the atomic ensemble, it is possible to decouple the atoms and light so that the loss of the light does not reduce the atomic squeezing. This results in an improvement of the rate of atomic spin squeezing. The nonlinear shearing interaction that remains still falls short of the exponential squeezing seen in optical parametric amplification. The reason for this can be seen by decomposing the shearing interaction into squeezing along a given quadrature followed by a rotation. The rotation leads to a constantly changing axis along which the squeezing occurs, and thus the squeezing builds up more slowly. This can be corrected for by breaking up the light pulse into a sequence of small pulses, and performing a small phase matching rotation after each pulse. The squeezing then adds up along a constant direction, rather than along a constantly varying direction. This results in still further improvement of the rate of atomic spin squeezing. A complete model includes the effects of photon-atom scattering and other noise and loss effects on the overall rate of squeezing. Previous derivations of noise due to photon-atom scattering have started with the unjustified assumptions that the atomic and photonic decoherence channels were both Gaussian and independent of correlations between the two subsystems. They then proceed with very general statistical arguments that rely upon these simplifications. My work begins with the more fundamental master equation picture in which I find the Linblad jump operators. I find that in general the photonic and atomic loss channels are not independent, with the intensity of the light dictating the details of this dependence, and find the conditions under which the Gaussian approximation holds. Squeezing and loss is initially treated assuming ensembles of spin-1/2 atoms, but this work is further extended to higher dimensional subsystems. For a higher spin case, preparing the atoms in spin coherent states is not optimal. One can engineer a stronger interaction by preparing the atoms in an atomic "cat state", i.e., a superposition of the two stretch states along the direction of propagation of the light beam. The fluctuations of such a state are more strongly coupled to the light, resulting in a stronger nonlinearity. This leads to strong correlations between the atoms, but they are not immediately useful for squeezing; the cat state must be coherently mapped to a coherent state to achieve atomic spin squeezing. The state created in this manner is ultimately more squeezed than that achieved with the same interaction but prepared initially in a coherent state.
In this paper we consider the optical coupling of two four-level atoms via cooperative Raman scattering of correlated photons of the nondegenerate spontaneous parametric radiation. The Raman scattering of twin photons incident on the atoms is additionally stimulated by coherent classical radiation in such a way that it is slightly off resonant for each atom considered independently but resonant for the joint system. We discuss the special excitation conditions when the amplitude of the cooperative stimulated Raman scattering from quantum to classical modes becomes greater than the normal transition rates associated with independent Raman scattering. In certain conditions weak entanglement between the Zeeman sublevels in the ground states of these atoms can appear. The small admixture of entanglement in the system of two atoms leads to strong entanglement between transverse macroscopic spin fluctuations of two spatially separated atomic ensembles coupled by such quantum light. The proposed mechanism of entanglement is relevant to the ground state spin subsystem of two distant macroscopic and optically thin atomic ensembles. Such an entangled state can be stored for a long time and used as a basic Einstein-Podolsky-Rosen system in different applications pursuing the goal of spreading quantum information schemes in atomic subsystems.