LATEX TikZposter

Control of spatial correlations: Experimental implementation and its roleon open quantum system dynamics and two-photon imaging

Omar Calderon-Losada, Paul Diaz, Juan A. Urrea, Daniel F. Urrego and Alejandra ValenciaLaboratorio de optica cuantica, Universidad de los Andes, Bogota, Colombia

Control of spatial correlations: Experimental implementation and its roleon open quantum system dynamics and two-photon imaging

Omar Calderon-Losada, Paul Diaz, Juan A. Urrea, Daniel F. Urrego and Alejandra ValenciaLaboratorio de optica cuantica, Universidad de los Andes, Bogota, Colombia

Introduction

The control of correlations of paired photons in different degrees offreedom has been of interest for practical applications. In this work,we report the experimental control of the spatial correlations of pairedphotons and discuss how to exploit this capability to study the effectsof having different spatial correlations on open quantum systems andimaging procedures:

• Experimentally, we control the spatial correlations of pairedphotons produced by spontaneous parametric down conversion(SPDC) by using the waist of the pump beam as the tuningparameter.

• Two-photon imaging: we take advantage of the experimen-tal capability of controlling the spatial correlations in a uniquesetup to observe the effects of different types of spatial correla-tions on ghost images.

• Open quantum system dynamics: we propose an all-optical setup in which a pair of entangled polarization photonsis coupled with their transverse momentum degree of freedom,which plays the role of a bosonic environment, to simulate anopen quantum system dynamic.

The Source: Type II Non-colinear SPDC

1

APD

CC

APD

• The state at the output face of the crystal

|ψ〉 ∝∑σ 6=σ′

∫dqsdqi∫

dΩsdΩiΦσσ′(qs,Ωs,qi,Ωi) |qs,Ωs;σ′〉 |qi,Ωi;σ

′〉

with polarization σ, σ′ = H,V .

• Mode function

Φσσ′(qs,Ωs,qi,Ωi) = Nβ(Ωs,Ωi)×

exp

[−

w2p(∆

20 + ∆2

1)

4− γ

(∆kL

2

)2

+ i∆kL

2

]

• Phase matching conditions

∆0 = qxs + qxi ,

∆1 = qys cosφs + qyi cosφi−NsΩs sinφs + NiΩi sinφi − ρsqxs sinφs,

∆k = Np(Ωs + Ωs)−NsΩs cosφs −NiΩi cosφi− qys sinφs + qyi sinφi + ρp∆0 − ρsqxs cosφe.

• Spatial-Type-II-SPDC quantum state

|Ψ〉 ∝∑σ 6=σ′

∫dqs

∫dqi Φσσ′(qs,qi) |qs;σ〉 |qi;σ′〉 ,

Φσσ′(qs,qi) =

∫dΩsdΩifs(Ωs)fi(Ωi)Φσσ′(qs,Ωs,qi,Ωi)

= N exp

[−1

4qᵀAq + ibᵀ · q

]with q = (qxs , q

ys , q

xi , q

yi ), A is a 4 × 4 real-valued, sym-

metric matrix and b a 4-dimensional vector. Besides,ΦV H(qs,qi) = ΦHV (qi,qs).

• Rate of Coincidences:

Sσσ′(qs,qi) = |Φσσ′(qs,qi)|2

Mode Function and Correlation control

• Degree of correlation:

κ% =C%si√

C%ssC

%ii

, C%uv = 〈q%uq%v〉 − 〈q%u〉 〈q%v〉 , (% = x, y)

0 50 100 150 200-1.0

-0.5

0.0

0.5

1.0

wp [μm]

y-MFCorrelation,DOC

0 50 100 150 200-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

wp [μm]

x-MFCorrelation,DOC

• Mode function orientation: The angle between the major axis of the biphotonellipse and the corresponding horizontal axis.

o/e

e/o

0 50 100 150 200

-45

0

45

wp [μm]

y-MFOrientationAngle

[°]

o/e

e/o

0 50 100 150 200

-80

-60

-40

-20

0

wp [μm]

x-MFOrientationAngle

[°]

Ghost Imaging with tunable momentum correlation

• Propagated MF

Φσσ′(ρA,ρB) =

∫d2qs

∫d2qi gs(qs,ρA, zA)gi(qi,ρB, zB)Φσσ′(qs,qi),

with the Green function for a 2f -system

gν(qµ,ρj, 2f ) =

∫d2ρ`

∫d2ρchω(ρj − ρ`, f )Lf(ρ`)hω(ρ` − ρc, f )eiqµ·ρc

= Ceiπλf ρ

2je−

iλf4π q

2µδ

(qµ −

2π

λfρj

),

that propagates the SPDC photons with a transverse momentum qµ fromthe source to a plane located at a distance 2f .

• The Ghost Image

GIσσ′(ρA) ∝∫d2ρB

∣∣∣∣T (ρB)Φσσ′

(2π

λfρA,

2π

λfρB

)∣∣∣∣2 .where T (ρ) denotes the transfer function for the object to be imaged.

• wyp ≈ wx

p ≈ 35µm,

• wyp = 950µm, wx

p = 740µm

• Experimental Ghost Images

• Auto- and Cross-correlation

Quantum Dynamic - Markovian vs non-Markovian

• Horizontal MF as an initial system-environment state:With Φσσ′(q

xs = 0, qys , q

xi = 0, qyi ) = Φσσ′(q

ys , q

yi ), the SPDC state

reduces to

|Ψ〉 ∝∑σ 6=σ′

∫dqys

∫dqyi Φσσ′(q

ys , q

yi ) |q

ys ;σ〉 |q

yi ;σ

′〉 ≡ |Ψ〉S+E = |Ψin〉 ,

whereΦV H(qys , q

yi ) = N exp [−(qy)ᵀBqy + ivᵀ · qy]

and ΦHV (qys , qyi ) = ΦV H(qyi , q

ys).

• Local Evolution operator: U(ξs, ξi) = Us(ξs)⊗ Ui(ξi), with

U`(ξ`) |qy` ;H〉 = e−iξ`qy` |qy` ;H〉

U`(ξ`) |qy` ;V 〉 = ei(ξ`qy`+ϕ) |qy` ;V 〉

where ` = s, i, ξ` is the evolution parameter for each mode, andϕ is a constant polarization-dependent phase-shift.

• Reduced polarization density matrix:

ρS = trE

U(ξs, ξi) |Ψin〉 〈Ψin| U †(ξs, ξi)

= |α|2 |V,H〉 〈V,H| + Λ(ξs, ξi) |V,H〉 〈H,V |

+ Λ∗(ξs, ξi) |H,V 〉 〈V,H| + |β|2 |H,V 〉 〈H, V |

with |α|2 + |β|2 = 1, and

Λ(ξs, ξi; wp) =

∫dqysdq

yi ΦV H(qys , q

yi ; wp)Φ

∗HV (qys , q

yi ; wp)e

−2i(ξiqyi−ξsqys)

being the decoherence factor.

• Purity evolution:

P(ξs, ξi; wp) =1

2+ 2|Λ(ξs, ξi; wp)|2

• Trace distance: For two initial states ρS± the trace distanceevolves as

D(ξs, ξi; wp) = 2|Λ(ξs, ξi; wp)| cos (arg Λ(ξs, ξi; wp))

• Purity and Purification

• Trace distance evolution:

Final Remarks

• We have experimentally showed how DOC and orientation of MF exhibit different behaviorsdepending on the pump beam waist and the direction in which the correlation is been studied.The momentum distribution is also affected by the polarization post-selection.

• Our experiment allows to learn about the role of the spatial correlations of photon pairs on theresolution of ghost images.

• By controlling the initial correlation, the evolution of the central system can exhibit a Marko-vian or non-Markovian behaviour, inducing a non-local quantum memory effect that allows torecover the purity (and entanglement) in the central system when this has been degraded bylocal interactions.

References

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[2] O. Calderon-Losada, J. Florez, J. P. Villabona-Monsalve, and A. Valen-cia, Opt. Lett. 41, 1165 (2016).

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[4] M. D’Angelo, A. Valencia, M. H. Rubin, and Y. Shih, Phys. Rev. A72, 013810 (2005).

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Email: o.calderon31@uniandes.edu.co

Webpage: http://opticacuantica.uniandes.edu.co