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Spatial and spectral mode selection of heralded single photons from pulsed parametric down-conversion

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Abstract

We describe an experiment in which photon pairs from a pulsed parametric down-conversion (PDC) source were coupled into single-mode fibers with heralding efficiencies as high as 70%. Heralding efficiency or mode preparation efficiency is defined as the probability of finding a photon in a fiber in a definite state, given the detection of its twin. Heralding efficiencies were obtained for a range of down-conversion beam-size configurations. Analysis of spatial and spectral mode selection, and their mutual correlation, provides a practical guide for engineering PDC-produced single photons in a definite mode and spectral emission band. The spectrum of the heralded photons were measured for each beam configuration, to determine the interplay between transverse momentum and spectral entanglement on the preparation efficiency.

©2005 Optical Society of America

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Figures (6)

Fig. 1.
Fig. 1. Setup used to herald single-photons from pulsed parametric down-conversion (PDC). Filters F1,2, either cut-off or interference and various lenses L1,2 are used, according to Table I. Distances d1,2 were chosen for practical reasons, and the direction 1 was the heralding channel except for lens configuration (b). A monochromator was inserted in the heralding channel fiber link to measure coincidence spectral distribution
Fig. 2.
Fig. 2. (a) Calculated external angular PDC emission of LiIO3 at a phase-matching angle of 44.15°, including the longitudinal mismatch and the pump transverse shape (gaussian with a 260 μm pump waist, crystal length L=5 mm). The black line is the central emission angle of PDC for the experimental condition, for a 395 nm CW pump beam, is superimposed. This calculation was done according to ref. [23], and the normalized phasematch function (perfect phasematch =1) is used. (b) Vertical slice of (a) at a wavelength of 789 nm.
Fig. 3.
Fig. 3. Spectral scan of the normalized coincidence rates performed with a monochromator in the heralding fiber path. Solid lines are theoretical curves of Eq. (19). Lens configurations (a), (b) and (c). The narrower scans were taken with a 5 nm FWHM interference filter on the heralding arm.
Fig. 4.
Fig. 4. Spectral scan coincidence for various iris diameters on the CUT arm. Data (squares for the full aperture, diamonds at 1.5 mm iris diameter, and down triangles for 1 mm iris diameter) are compared to the theoretical prediction.
Fig. 5.
Fig. 5. Theoretical prediction of the mode preparation efficiency versus the trigger bandwidth for various lens magnifications. Lens magnification is given by M 1,2 = d 1,2 f 1,2 f 1,2 . Calculations are done for w o,1 = w o,2 = w o = Mw f and a crystal 5 mm long. Solid lines are for for fixed w p = 260 μm and dotted line for w p = 3 w o .
Fig. 6.
Fig. 6. Transverse intensity profile of the heralded PDC photon by a fixed heralding single direction and wavelength, calculated exactly (a) and with the present approximation (b).

Tables (2)

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Table 1. Details of lens configurations for coupling fiber mode diameter of 4.2 μm, and M are magnifications.

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Table 2. Angular spread collection and resulting χ P’s for each lens configuration

Equations (36)

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Δ 1,2 = Δ θ 1,2 [ θ o ( λ ) λ λ o ] 1 ,
ψ = χ ( 2 ) d t V d x d y d z E p ( x , y , z , t ) E ̂ s ( ) ( x , y , z , t ) E ̂ i ( ) ( x , y , z , t ) 0 k s , ω s 0 k i , ω i .
E p ( x , y , z , t ) = d 3 k p d ω p E ̃ q p ( q p ) E ̃ ν p ( ω p ) e i k p · r i ω p t ,
ψ = d 3 k s d ω s d 3 k i d ω i A ̃ 12 ( k s , k i , ω i , ω s ) 1 k s , ω s 1 k i , ω i ,
A ̃ 12 ( k s , k i , ω i , ω s ) d 3 k p d ω p S d x d y L 0 d z E ̃ q p ( q p ) E ̃ ν p ( ω p ) e i ( Δ k x x + Δ k y y + Δ k z z )
× δ ( ω s + ω i ω p )
× δ [ k p z ( n ( ω p ) ω p c ) 2 q p 2 ]
× δ [ k s z ( n ( ω s ) ω s c ) 2 q s 2 ]
× δ [ k i z ( n ( ω i ) ω i c ) 2 q i 2 ] ,
k p z = ( n ( ω p ) ω p c ) 2 q p 2 ,
Δ k x = q p x q s x q i x
Δ k y = q p y q s y q i y θ i K i θ s K s
Δ k z = D pi v p + D is ν s + ( N p N s ) q p y K p + θ sq s y θ iq i y ,
A 12 ( ρ 1 , ρ 2 , t , τ ) E ν p ( τ t + 2 τ D pi D is 2 ) E q p ( x 1 , y 1 N p , s τ K p D is )
δ ( x 1 x 2 ) × δ ( y 1 y 2 + α s τ D is ) D is L ( τ ) ,
φ j * ( x j , y j ) = 2 π 1 w o , j exp [ x j 2 + y j 2 w o , j 2 ] ,
A 12 ( t , τ ) = d ρ 1 d ρ 2 A 12 ( ρ 1 , ρ 2 , t , τ ) φ 1 * ( ρ 1 ) φ 2 * ( ρ 2 ) .
C 12 = d t d t d τ d τ A 12 ( t , τ ) A 12 * ( t , τ ) φ 1 * I ν s ( t t + τ τ ) I ν i ( t t τ + τ ) .
A 12 ( t , τ , ρ 2 ) = d ρ 1 A 12 ( ρ 1 , ρ 2 , t , τ ) φ 1 * ( ρ 1 )
C 1 = d ρ 2 d t d t d τ d τ A 12 ( t , τ , ρ 2 ) A 12 * ( t , τ , ρ 2 ) φ 1 * I ν s ( t t + τ τ ) .
χ P = C 12 C 1 .
χ P = 4 T p w o , 1 2 w o , 2 2 w p 2 ( w o , 1 2 + w p 2 ) Δ 2 ( w o , 2 2 w p 2 + w o . 1 2 ( w o , 2 2 + w p 2 ) ) 2 8 a 6 + T p 2 ( Δ 1 2 + Δ 2 2 ) f ( c 1 , c 2 ) f ( s 1 , s 2 ) ,
f ( p , q ) = 0 1 d x e p x 2 + q 2 x 2 4 p ( Erf [ qx 2 p ] Erf [ 2 p + qx 2 p ] ) p .
c 1 = L 2 [ w o , 2 2 α 2 + w o , 1 2 ( α 2 + K p 2 D pi 2 w o , 2 2 ( Δ 1 2 + Δ 2 2 ) ) ] K p 2 [ w o , 2 2 w p 2 + w o . 1 2 ( w o , 2 2 + w p 2 ) ] [ a 2 + a 2 T p 2 ( Δ 1 2 + Δ 2 2 ) ]
α = a 2 N s , p 2 + [ a 2 N s , p 2 T p 2 + K p 2 D pi 2 w p 2 ( Δ 1 2 + Δ 2 2 ) ]
c 2 = 2 L 2 D pi 2 ( Δ 1 2 + Δ 2 2 ) a 2 + a 2 T p 2 ( Δ 1 2 + Δ 2 2 )
s 1 = L 2 [ 2 a 2 N s , p 2 T p 2 ( w o , 1 2 + 2 w p 2 ) + K p 2 D pi 2 w p 2 ( w o , 1 2 + w o , 2 2 ) ] 2 a 2 K p 2 T p 2 w p 2 ( w o , 1 2 + w p 2 )
s 2 = L 2 [ 2 a 2 N s , p 2 T p 2 w o , 1 2 + K p 2 D pi 2 w p 2 ( w o , 1 2 + w o , 2 2 ) ] 2 a 2 K p 2 T p 2 w p 2 ( w o , 1 2 + w p 2 ) .
A 12 ( ρ 1 , ρ 2 , ν s , ν i ) E ̃ ν p ( ν s + ν i ) E q p ( x 1 , y 1 + ( y 1 y 2 ) N s , p K p α s ) ) ×
exp [ I ( y 1 y 2 ) ( ν s D is + ( ν i + ν s ) D pi ) α s ] δ ( x 1 x 2 ) L ( y 1 y 2 α s ) ,
A 12 ( ν s , ν i ) = d ρ 1 d ρ 2 A 12 ( ρ 1 , ρ 2 , ν s , ν i ) φ 1 * ( ρ 1 ) φ 2 * ( ρ 2 ) .
C 12 ( ν i ) = d ν s A 12 ( ν s , ν i ) 2 I ̃ ν i ( ν i ) I ̃ ν s ( ν s ) = d ν s C 12 ( ν i , ν s ) .
C 12 ( ν i , ν s ) E ̃ ν p [ ( ν s + ν i ) ] ( 1 + 8 a 2 ζ 2 T p 2 ) 1 2 ] 2 I ̃ ν i ( ν i ) I ̃ ν s ( ν s ) ×
Erf [ γ + ζ ( ν s + ν i ) ] Erf [ ζ ( ν s + ν i ) ] 2 ,
ζ = K p D pi w o , 2 2 w p 2 + w o , 1 2 ( w o , 2 2 + w p 2 ) 2 N s , p ( w o , 1 2 + w o , 2 2 ) ,
γ = L w p 2 + w o , 2 2 w p 2 w o , 2 2 + w o , 1 2 ( w p 2 + w o , 2 2 ) N s , p K p .
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