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Università degli Studi di Milano Dipartimento di Fisica via Celoria 16, Milano, Italy Quantum Theory of Collective Atomic Recoil in Ring Cavities 16th October 2012 Mini Workshop , Milano PhD School in Physics, Astrophysics and Applied Physics Thesis advisor: Dr. Nicola Piovella Marina Samoylova

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Outline The advantages of studying a Bose-Einstein Condensate (BEC) in a ring cavity A possible experimental realization of such a system The semi-classical and quantum models of the system The numerical analysis of the exact solution The summary of the results Future doctoral research

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Introduction Superradiant Rayleigh Scattering in free space (SRyS) [ Collective Atomic Recoil Lasing in free space (CARL) ] incident laser beam

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Introduction SRyS in free spaceCARL in a ring cavity

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2D CARL configuration System: a BEC in a high-finesse ring cavity X Z pump field Φ

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Experimental setup A Bose-Einstein condensate is prepared in an Ioffe-Pritchard type magnetic trap in a high- finesse ring cavity (F=135000). The BEC is illuminated by s-polarized pump light incident under the angle Φ=37˚. The pump beam is provided by a Ti:sapphire laser. The condensate scatters the light superradiantly into two counter-propagating cavity modes. The atomic momentum distribution is taken via absorption imaging. A single-photon counter records the photons transmitted through one of the cavity mirrors. [1] S. Bux, C. Gnahm, R. Maier, C. Zimmermann and Ph. Courteille, Phys. Rew. Lett. 106, (2011). [2] S.Bux, H.Tomczyk, D.Schmidt, C.Zimmermann, N.Piovella, Ph.Courteille, New J. Phys., submitted (2012). Φ

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Results of the experiment Results of the experiment N=80000 is the number of atoms, t= 200μs is the duration of the pump laser pulse At certain conditions only 4 momentum states can be populated individual momentum state

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The semi-classical model closed systems of equations ! In the semi-classical limit the four states configuration can be solved in terms of two independent two-level systems for the left and right cavity modes. We are interested in a 4-level system

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The Quantum Model The Hamiltonian of the system in the interaction picture: where and are constants of motion representing the sum of excitations for the systems 1 and 2, respectively. The general state of the system:, where and t t

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Numerical Results

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- atom-number squeezing parameter

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Summary we consider CARL-type dynamics we consider CARL-type dynamics we investigate 4-level system we investigate 4-level system In the semi classical limit the four states configuration can be represented in terms of two independent two-level systems. The quantum problem can be solved exactly where its full quantum properties are determined.

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Future Plans Why is it so interesting? BEC in optical lattice 2D 3D Easy to realize Large variety of optical lattices Fascinating optical effects Photonic band gaps (PBG)

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Future Plans What is PBG? We consider propagation of light through an optical lattice loaded with cold atoms The goal is to study photonic band gaps in cold atomic structures range of frequencies where no propagation modes exist in any directions a -π/a π/a Access to real time manipulations Perfect long range order Why?

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