Quantum Theory of Collective Atomic Recoil in Ring Cavities Università degli Studi di Milano Dipartimento di Fisica via Celoria 16, 20133 Milano, Italy PhD School in Physics, Astrophysics and Applied Physics Quantum Theory of Collective Atomic Recoil in Ring Cavities Marina Samoylova Thesis advisor: Dr. Nicola Piovella 16th October 2012 “Mini Workshop - 2012”, Milano

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

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

Introduction SRyS in free space CARL in a ring cavity Scattered photons can be recycled many times Coherence time is enhanced BEC + thermal clouds (100μK) Scattered photons rapidly leave the interaction region Bose-Einstein Condensate (BEC) only

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

Experimental setup Φ 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, 203601 (2011). [2] S.Bux, H.Tomczyk, D.Schmidt, C.Zimmermann, N.Piovella, Ph.Courteille, New J. Phys., submitted (2012).

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

The semi-classical model We are interested in a 4-level system 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.

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. t t The general state of the system: , where and

Numerical Results

Numerical Results - atom-number squeezing parameter

Summary we consider CARL-type dynamics 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.

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

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