L A T E X Tik Zposter Control of spatial correlations: Experimental implementation and its role on open quantum system dynamics and two-photon imaging Omar Calder´ on-Losada, Paul Diaz, Juan A. Urrea, Daniel F. Urrego and Alejandra Valencia Laboratorio de ´ optica cu´ antica, Universidad de los Andes, Bogot´ a, Colombia Control of spatial correlations: Experimental implementation and its role on open quantum system dynamics and two-photon imaging Omar Calder´ on-Losada, Paul Diaz, Juan A. Urrea, Daniel F. Urrego and Alejandra Valencia Laboratorio de ´ optica cu´ antica, Universidad de los Andes, Bogot´ a, Colombia Introduction The control of correlations of paired photons in different degrees of freedom has been of interest for its role in practical applications. In this work, we report the experimental control of the spatial correlations of paired photons and exploit this capability to study the effects of having different spatial correlations on open quantum systems and imaging procedures: • Experimentally, we control the spatial correlations of paired photons produced by spon- taneous parametric down conversion (SPDC) by using the waist of the pump beam as the tuning parameter. • Two-photon imaging: we take advantage of the experimental capability of controlling the spatial correlations in a unique setup and observe the effects of different types of spatial correlations on ghost images. • Open quantum system dynamics: we propose an all-optical setup in which a pair of entangled polarization photons is coupled with their transverse momentum degree of freedom, that plays the role of a bosonic environment, to simulate an open quantum system dynamic. We have found that controlling the initial transverse correlations, the evolution of the central system can exhibit a Markovian or non-Markovian behaviour. The Source: Type II Non-colinear SPDC APD CC APD • The state at the output face of the crystal |ψ i∝ X σ6=σ 0 Z dq s dq i Z dΩ s dΩ i × Φ σσ 0 (q s , Ω s , q i , Ω i ) |q s , Ω s ; σ 0 i|q i , Ω i ; σ 0 i with polarizations σ, σ 0 = {H, V }. • Mode function Φ σσ 0 (q s , Ω s , q i , Ω i )= N β (Ω s , Ω i )× exp " - w 2 p (Δ 2 0 +Δ 2 1 ) 4 - γ Δ k L 2 2 + i Δ k L 2 # • Phase matching conditions Δ 0 = q x s + q x i , Δ 1 = q y s cos φ s + q y i cos φ i - N s Ω s sin φ s + N i Ω i sin φ i - ρ s q x s sin φ s , Δ k = N p (Ω s +Ω s ) - N s Ω s cos φ s - N i Ω i cos φ i - q y s sin φ s + q y i sin φ i + ρ p Δ 0 - ρ s q x s cos φ e . • Spatial-Type-II-SPDC quantum state |Ψi∝ X σ6=σ 0 Z dq s Z dq i ˜ Φ σσ 0 (q s , q i ) |q s ; σ i|q i ; σ 0 i , ˜ Φ σσ 0 (q s , q i )= Z dΩ s dΩ i f s (Ω s )f i (Ω i )Φ σσ 0 (q s , Ω s , q i , Ω i ) = N exp - 1 4 q | Aq + ib | · q Mode Function and Correlation control • Degree of correlation: κ % = C % si p C % ss C % ii , C % uv = hq % u q % v i-hq % u ihq % v i , (% = x, y ) 0 50 100 150 200 - 1.0 - 0.5 0.0 0.5 1.0 w p [μm] y -MF Correlation, DOC 0 50 100 150 200 - 1.0 - 0.9 - 0.8 - 0.7 - 0.6 - 0.5 w p [μm] x-MF Correlation, DOC • Mode function orientation: The angle between the major axis of the biphoton ellipse and the corresponding horizontal axis. o/e e/o 0 50 100 150 200 - 45 0 45 w p [μm] y -MF Orientation Angle [°] o/e e/o 0 50 100 150 200 - 80 - 60 - 40 - 20 0 w p [μm] x-MF Orientation Angle [°] Lens-less Ghost Imaging • Propagated MF ˜ Φ σσ 0 (ρ A , ρ B )= Z d 2 q s Z d 2 q i g s (q s , ρ A ,z A )g i (q i , ρ B ,z B ) ˜ Φ σσ 0 (q s , q i ), with the Green function for a 2f -system g ν (q μ , ρ j , 2f )= Z d 2 ρ ‘ Z d 2 ρ c h ω (ρ j - ρ ‘ ,f )L f (ρ ‘ )h ω (ρ ‘ - ρ c ,f )e iq μ ·ρ c = C e iπ λf ρ 2 j e - iλf 4π q 2 μ δ q μ - 2π λf ρ j , that propagates the SPDC photons with a transverse momentum q μ from the source to a plane located at a distance 2f . • The Ghost Image GI σσ 0 (ρ A ) ∝ Z d 2 ρ B T (ρ B ) ˜ Φ σσ 0 2π λf ρ A , 2π λf ρ B 2 . where T (ρ) denotes the transfer function for the object to be imaged. Quantum Dynamic - Markovian vs non-Markovian • Horizontal Biphoton as an initial system-environment state: Considering ˜ Φ σσ 0 (q y s ,q y i )= ˜ Φ σσ 0 (q x s =0,q y s ,q x i =0,q y i ), the SPDC state reduces to |Ψi∝ X σ6=σ 0 Z dq y s Z dq y i ˜ Φ σσ 0 (q y s ,q y i ) |q y s ; σ i|q y i ; σ 0 i ≡|Ψi S +E = |Ψ in i , where ˜ Φ VH (q y s ,q y i )= N exp [-(q y ) | Bq y + iv | · q y ] and ˜ Φ HV (q y s ,q y i )= ˜ Φ VH (q y i ,q y s ). • Local Evolution operator: ˆ U (ξ s ,ξ i )= ˆ U s (ξ s ) ⊗ ˆ U i (ξ i ), with ˆ U ‘ (ξ ‘ ) |q y ‘ ; H i = e -iξ ‘ q y ‘ |q y ‘ ; H i ˆ U ‘ (ξ ‘ ) |q y ‘ ; V i = e i(ξ ‘ q y ‘ +ϕ) |q y ‘ ; V i where ‘ = {s, i}, ξ ‘ is the evolution parameter for each mode, and ϕ is a constant polarization-dependent phase-shift. • Reduced polarization density matrix: ˆ ρ S = tr E n ˆ U (ξ s ,ξ i ) |Ψ in ihΨ in | ˆ U † (ξ s ,ξ i ) o = |α| 2 |V,H ihV,H | + Λ(ξ s ,ξ i ) |V,H ihH, V | +Λ * (ξ s ,ξ i ) |H, V ihV,H | + |β | 2 |H, V ihH, V | with |α| 2 + |β | 2 = 1, and Λ(ξ s ,ξ i ;w p )= Z dq y s dq y i ˜ Φ VH (q y s ,q y i ;w p ) ˜ Φ * HV (q y s ,q y i ;w p )e -2i ( ξ i q y i -ξ s q y s ) being the decoherence factor. • Purity evolution: P (ξ s ,ξ i ;w p )= 1 2 +2|Λ(ξ s ,ξ i ;w p )| 2 • Trace distance: For two initial states ˆ ρ S ± the trace distance evolves as D(ξ s ,ξ i ;w p )=2|Λ(ξ s ,ξ i ;w p )| cos (arg Λ(ξ s ,ξ i ;w p )) • Purity and Purification • Trace distance evolution: Final Remarks • We have experimentally showed how DOC and orientation of MF exhibit different behaviors depending on the pump beam waist and the direction in which the correlation is been studied. 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