M. Emre Ta g n Advisor: M. zgr Oktel Co-Advisor: zgr E. Mstecaplolu - PowerPoint PPT Presentation

Quantum entanglement and light propagation through Bose-Einstein condensate (BEC) M. Emre Ta ş g ı n Advisor: M. Özgür Oktel Co-Advisor: Özgür E. Müstecaplıoğlu

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Superradiance (SR) 1 I Nor ~ N • SR: Collective spontaneous emission 2 I SR ~ N • Must excite very quickly strong pump Intense Coherent • Scattered radiation Directional

Superradiance (SR) (Directionality) 2  L 10 W end-fire mode ˆ x ˆ z • Elongated sample SR is directional. • Modes along the long-direction ( z ) is occupied by more atoms. 2   2   I N W 1 x ˆ ˆ   N   , z : # of atoms on line.   x x ~   x , z     I N L 100  z z L 10 W

Superradiance (SR) (Pulse Shape) 3 Intensity  D Delay time T 1 ~ Decay time N Establishment of atomic coherence. at peak First experiment: [N. Skribanowitz et al. , PRL 30 , 309 (1973).]

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

BEC Superradiance (SR) (experiment*) 1 Absorption Images: ( in p -space ) *[S. Inouye et al. , Science 285 , 571 (1999).] • fan-shaped pattern Different pulse times:   35  s B) p   Establishment of atomic coherence. 75  s C) p   100  s D) p BEC p =0  p Many-atoms in the same p -state

BEC Superradiance (SR) 2 collective • SR emission: coherent directional (end-fire mode) collective • Atom scattering: coherent same-momentum (side-mode)

BEC Superradiance 3 (sequential SR)    • End-fire mode ( )   Atomic side-mode ( ) k k k 0 e e    • End-fire mode ( )   k Atomic side-mode ( ) k k e 0 e

BEC Superradiance 4 (sequential SR) 1 st -order side-modes 1 st -order SR    • End-fire mode ( )   Atomic side-mode ( ) k k k 0 e e    • End-fire mode ( )   k Atomic side-mode ( ) k k e 0 e

BEC Superradiance 5 (sequential SR) 2 nd -order SR 2 nd -order side-modes 1 st -order forms 1 st -order 2 nd -order side-modes side-modes side-modes highly occupied superradiates

BEC Superradiance 6 (sequential SR) Lattice of side-modes p -space

BEC Superradiance 6 (Pulse shape) 1 st -order SR Intensity 2 nd -order SR • normal SR: Single peak • sequential SR: Two peaks   75  s p

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Motivation-Purpose 1 Quantum Information Transfer Storage Media Flying Carriers Condensed Atoms Photons-Pulses

Motivation-Purpose 2 BEC Superradiant Scattering Normal Scattering (nonlinear regime) (linear regime) Entanglement of Entanglement of (single atom)-(single photon) (atomic wave)-(end-fire pulse) many atoms many photons Discrete-variable entanglement Continuous-variable entanglement i.e. (atom spin)-(photon polarization) i.e. Electric-fields of two pulses [ M.E. Taşgın, M.Ö. Oktel, L. You, and [M.G. Moore and P. Meystre, PRL 85 , Ö.E. Müstecaplıoğlu, PRA 79 , 053603 5026 (2000).] (2009).]

Motivation-Purpose 3 Interested in the Continuous-Variable (E-fields) Entanglement of cross-propagating end-fire pulses.

Motivation (entanglement-swap) 4 Interacts in the Interacts in the 1 st SR sequence 2 nd SR sequence entangled entangled

Motivation (entanglement-swap) 5 swap entangled entangled

Motivation (entanglement-swap) 6 Entangle systems that Entanglement swap: never before interacted. Both interact with at different times.

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Hamiltonian 1 Full second-quantized Hamiltonian of Laser-BEC: † ˆ k   a  : creates photon of momentum k , energy ck . k 2 2  q † ˆ k   c  : creates atom(boson) in side-mode q , energy . q 2 M   2 2 1 / 2    : laser detuning  : dipole coupling g ( k ) ckd / 0   r                2 i k q k q : structure factor of BEC. ( k , k ) d r ( r ) e  q , q 0

Hamiltonian 2 1) Move rotating frame. laser pulse single mode. 2) Assume end-fire pulse scattered atoms (side-modes) effective Hamiltonian:

Hamiltonian 2 Schematic acts of operators:

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Entanglement parameter 1 Separability and Entanglement If density-matrix is inseparable       1 2 p it cannot written as r r r r subsystems 1,2 are entangled. Aim : Define a parameter to test entanglement.

Entanglement parameter 2 Separability and Entanglement showed: [L.M. Duan et al. , PRL 84 , 2722 (2000).]   1 density-matrix subsystems      ˆ ˆ   2 2 2 u v c   2 c separable not entangled   1 1 subsystems        density-matrix ˆ ˆ   2 2 2 2 c u v c   2 2 c c entangled inseparable uncertainty separability limit limit    1     ˆ ˆ ˆ † ˆ ˆ ˆ u c x x / c x a a / 2  2 1 , 2   are EPR operators with  1  ˆ ˆ ˆ    ˆ ˆ ˆ † v c p p / c p a a / i 2  2 1 , 2

Entanglement parameter 3 Separability and Entanglement showed: [L.M. Duan et al. , PRL 84 , 2722 (2000).]   1 density-matrix subsystems      ˆ ˆ   2 2 2 u v c   2 c separable not entangled   1 1 subsystems        density-matrix ˆ ˆ   2 2 2 2 c u v c   2 2 c c entangled inseparable uncertainty separability limit limit   1        ˆ ˆ     2 2 2 ( t ) u v c ( t ) 0 entangled   2 c

Entanglement parameter 4   1        ˆ ˆ     2 2 2 ( t ) u v c entangled ( t ) 0   2 c 2    a ˆ ˆ a symmetry c 1      lowest possible is: 2 (uncertainty limit) low   E  x field 2   c 1  H  p field

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Swap Mechanism (analytical treatment) 1 but not exactly solvable. Seems innocent, Even numerical simulation is hard. (Keep lots of analytical expressions by hand.) First, investigate H approximately. (general behavior) Illustrate swap mechanism, analytically.

Approximation Initial Times Later Times couples couples couples

Swap Mechanism (analytical treatment) 3  (side-mode)-(end-fire) se  (end-fire)-(end-fire) Later Initial is photon-photon atom-photon swapped entanglement entanglement to     0 0 se

Outline • Superradiance and BEC Superradiance • Motivation: Entanglement of scattered pulses. • Our Model Hamiltonian • Entanglement parameter • Swap Mechanism • Simulations • Conclusions

Simulations 1 End-fire Intensity and Side-mode Occupations no damping experimental parameters    4 1 . 3 10 Hz decoherence:  I : Intensity of end-fire modes  MIT 1999 experiment n , n , n : Occupation of side-modes  0 2  8  6 N 10  2  8 M 10