# Photon correlations of a sub-threshold optical parametric oscillator.

**ABSTRACT** A microscopic multimode theory of collinear type-I spontaneous parametric downconversion in a cavity is presented. Single-mode and multimode correlation functions have been derived using fully quantized atom and electromagnetic field variables. From a first principles calculation the FWHM of the single-mode correlation function and the cavity enhancement factor have been obtained in terms of mirror reflectivities and the first-order crystal dispersion coefficient. The values obtained are in good agreement with recent experimental results [Phys. Rev. A 62 , 033804 (2000)].

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**ABSTRACT:**In this paper we describe theoretically quantum control of temporal correlations of entangled photons produced by collinear type II spontaneous parametric down-conversion. We examine the effect of spectral phase modulation of the signal or idler photons arriving at a 50/50 beam splitter on the temporal shape of the entangled-photon wave packet . The coincidence count rate is calculated analytically for photon pairs in terms of the modulation depth applied to either the signal or idler beam with a spectral phase filter. It is found that the two-photon coincidence rate can be controlled by varying the modulation depth of the spectral filter.Journal of Optics B Quantum and Semiclassical Optics 09/2005; · 1.81 Impact Factor

Page 1

Photon correlations of a sub-threshold optical

parametric oscillator

R. Andrews

Department of Physics, Faculty of Agriculture and Natural Sciences, The University of the West Indies, St. Augustine,

Republic of Trinidad and Tobago, W.I..

randrews@fans.uwi.tt

E. R. Pike and Sarben Sarkar

Department of Physics, King’s College London, Strand, London WC2R 2LS, UK.

erp@maxwell.ph.kcl.ac.uk

Sarben.Sarkar@kcl.ac.uk

Abstract: A microscopic multimode theory of collinear type-I spontaneous

parametric downconversion in a cavity is presented. Single-mode and

multimode correlation functions have been derived using fully quantized

atom and electromagnetic field variables. From a first principles calculation

the FWHM of the single-mode correlation function

enhancement factor have been obtained in terms of mirror reflectivities and

the first-order crystal dispersion coefficient. The values obtained are in

good agreement with recent experimental results [Phys. Rev. A 62 , 033804

(2000)].

2002 Optical Society of America

OCIS codes: (190.0190) Nonlinear optics; (190.4410) Parametric processes

and the cavity

References and links

1.D. C. Burnhamand D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon

pairs,” Phys. Rev. Lett. 25, 84-87 (1970).

2.Z. Y. Ou, X. Y. Zou, L. J. Wang and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys.

Rev. A 42, 2957-2965 (1990).

3.C. K. Hong and L. Mandel, ‘‘Theory of parametric frequency down-conversion of light,’’ Phys. Rev. A 31,

2409-2418 (1985).

4.Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation

experiment,” Phys. Rev. Lett. 61, 50-53 (1988).

5.P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75,

4337-4341 (1995).

6.C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown

quantumstate via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895-1899 (1993).

7.S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantumvariables,” Phys. Rev. Lett. 80, 869-

872 (1998).

8. L. Vaidman, “Teleportation of quantumstates,” Phys. Rev. A 49, 1473-1476 (1994).

9.D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, “Experimental quantum

teleportation,” Nature 390, 575-579 (1997).

10.J-W Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger, “Experimental entanglement swapping : Entangling

photons that never interacted,” Phys. Rev. Lett. 80, 3891-3894 (1998).

11.D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 82, 1345-1349 (1999).

12. J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A

41, 5139-5146 (1990).

13.X. Y. Zou, L. J. Wang and L. Mandel, “Induced coherence and indistinguishability in optical interference,”

Phys. Rev. Lett 67, 318-321 (1991).

14.C. K. Hong, Z. Y. Ou and L. Mandel, “Measurement of subpicosecond time intervals between two photons by

interference,” Phys. Rev. Lett. 59, 2044-2046 (1987).

15.Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric down-conversion for the prolongation of

correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556-2559 (1999).

16. Y. J. Lu and Z. Y. Ou, “Optical parametric oscillator far below threshold: Experiment vs theory,” Phys. Rev. A

62, 033804-033804-11 (2000).

(C) 2002 OSA 3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 461

#1044 - $15.00 US Received March 28, 2002; Revised May 27, 2002

Page 2

17.M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in

parametric amplification,” Phys. Rev. A 30, 1386-1391 (1984).

R. Andrews, E. R. Pike and S. Sarkar, ‘‘The role of second-order nonlinearities in the generation of localized

photons,’’ Pure Appl. Opt. 7, 293-299 (1998).

F. De Martini, M. Marrocco , P. Mataloni, L. Crescentini and R. Loudon, ‘‘Spontaneous emission in the optical

microscopic cavity,’’ Phys. Rev. A 43, 2480-2497 (1991).

B. Zysset, I. Biaggio, and P. Gunter, “Refractive indices of orthorhombic KNbO3: Dispersion and temperature

dependence,” J. Opt. Soc. Am. B 9, 380-386 (1992).

R. Andrews, E. R. Pike, and S. Sarkar, “Photon correlations and interference in type-I optical parametric down-

conversion,” J. Opt. B: QuantumSemiclass. Opt. 1, 588-597 (1999).

18.

19.

20.

21.

1. Introduction

The optical process of spontaneous parametric down-conversion (SPDC) involves the virtual

absorption and spontaneous splitting of an incident (pump) photon in a transparent nonlinear

crystal producing two lower-frequency (signal and idler) photons [1-3]. The pairs of photons

can be entangled in frequency, momentum and polarization. In type-I SPDC the photons are

frequency-entangled and the signal and idler photons have parallel polarizations orthogonal to

the pump polarization.

Entangled photons have been used to demonstrate quantum nonlocality [4,5], quantum

teleportation [6-8] and more recently quantum information processing [9-11]. Photon pairs

have also been used to demonstrate quantum interference phenomena [12-14]. The photon

pairs produced in such experiments are separated by less than a picosecond and their

correlation properties could only be investigated indirectly, for example by fourth-order

interference [2,12]. Ou and Lu [15,16] have recently measured the time separation of photons

produced from a nonlinear crystal placed inside a high-Q cavity. Because of the reduced

bandwidth of the photons and consequent broadening of the photon correlation functions they

were able to measure pair-photon correlations directly. Measurements of single-mode and

multimode correlation functions were made for photons with frequencies close to the

degenerate frequency for type-I collinear SPDC. Experimental results were modelled using

the theory of Collett and Gardiner [17].

In this paper we propose an alternative microscopic multimode theory to describe the

detection of photon pairs produced from a nonlinear crystal placed inside a high-Q cavity.

Given the limited number of studies in the sub-threshold regime of operation of the optical

parametric oscillator (OPO), we believe our approach will broaden the understanding of the

sub-threshold operation of the OPO. We describe the situation in which collinear photon

pairs, which experience multiple reflections in the cavity, are produced by pump photons

which pass through the crystal once; this is the single-pass case and corresponds to an OPO

operating far below threshold [16]. We calculate analytic expressions for both the single-

mode and multimode correlation functions in terms of the crystal and cavity parameters. Our

approach yields similar results to the theory used by Ou and Lu, but in addition, we have

been able to obtain exact expressions for the hitherto phenomological coupling constants

introduced in the Ou and Lu analysis. Such constants are important in determining the FWHM

of the correlation functions. We consider a high-Q cavity which corresponds to the

experimental situation with detectors positioned outside the cavity. Our theoretical

simulations compare well with the experimental results of Ou et. al.

2. Spontaneous down-conversion amplitude for a crystal in a cavity

If we first consider a crystal atom at position

3rr

in free space and for one-atom detectors

located at

from a single crystal atom is given by [18].

1rrand

2rr, the generalized amplitude for detecting pairs of down-converted photons

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 462

#1044 - $15.00 US Received March 28, 2002; Revised May 27, 2002

Page 3

ea

E

,...,are components of the electric field vector and it should be noted that in (1) repeated

superscripts are being summed over. The initial state of the electromagnetic field,

0

0

, 0

λ

αk

,

consists of a coherent state with wave-vector

0 k

and polarization index

0 λ

,

g

(the

>

monochromatic pump beam) with other modes in the vacuum state 0 .

the wave-function describing the ground state of the detector atoms and the crystal atom and

,,

gaa

describes the detector atoms in excited states

321

,|

gg

is

32121,aa

and a source atom

µ

finally in the ground state

components of the interaction-picture electric dipole moment operator for the multi-level

atom at position

3 g

after the two-photon emission process.

)(

) ( ,

e

,...,

t

ja

denotes

jr .

Figure 1: Schematic showing a cavity of length d bounded by an input mirror M1and an

exit mirror M2. The shaded area represents the nonlinear crystal which fills the cavity.

To obtain the amplitude,

placed inside a cavity of length d , all embedded in a linear medium of the same refractive

index [3] (see Figure 1), we employ the plane-wave mode functions for the quantized electric

field [19] for the cavity and external reservoir system. Since we are using the complete

electromagnetic field which describes modes both inside and outside the cavity, damping

effects are already included in the theory and therefore need not be included

phenomenologically. For a one-sided cavity in which the amplitude transmission coefficient

for photons exiting through mirror M1from inside the cavity is zero, the quantized electric

field is given as

/ 1

kc

k

j

επ

=

) 2(

cav

G

, for pair-photon detection in the case of a crystal of length d

) exp()()(

16

di t), r (E

2

2 , 1

∑

0

3

3

tiarUk

j k

r

j k

r

j

ωε−

=∫

r

r

h

r

r

(2)

) 1 ()(

}, 0 | )

5

,(),(),(),(),(|, 0

,,| )

5

()()()()(|,,

),,,(

0

0

0

0

4321

4321

321

λ

k

) 3 ( ,

e

4

E

) 3 ( ,

d

3

) 3 ( ,

c

2

) 2 ( ,

b

1

) 1 ( ,

a

321

54321

5

) 2 (

21

r

r

333

r

r

21

r

r

3

r

r

21

r

r

r

r

r

r

r

r

r

r

r

r

↔+

><×

><×

a

−

=

∫∫∫∫∫

∞−∞−∞−∞−∞−

λ

k

α

µµµµµ

edcba

ttttt

ttEtEtEtEa

gggtttttsa

dtdtdtdtdt

h

i

tA

d

M1

M2

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 463

#1044 - $15.00 USReceived March 28, 2002; Revised May 27, 2002

Page 4

The destruction operators for the modes with spatial function

)(rU

j k

r

r

are denoted

j kar where

(rU

j k

r

r

)(k

1

j

r

ε

(j=1,2) denotes the mode polarization vector. The spatial function

)

for

dzd

2

1

2

<<−

, i.e, inside the cavity is denoted by

)(

,

j k

r

r

r

U

in

and is defined by the

following

k

o

k

o

j k

,

r

in

D

ikdr

r

k it

D

r

r

k it

r

r

U

) exp() exp(

)(

)(

2

)(

2

+⋅

−

⋅

=

+−

rr

(3)

with

)2 exp(1

2

ikdrD

ok

+=

(4)

o

t2

for the signal and idler photons; we have taken the reflection coefficient of M1as

and

o

r2

being the amplitude transmission and reflection coefficients of M2 respectively

1

1

r

−=

o r

k

consistent with a perfectly reflecting and infinitely thin mirror. The wave vectors

)(+

k

r

describe backward and forward propagating photons and are defined in terms of polar

(θ ) and azimuthal (φ ) angles by

)(−

and

)cos,sin sin, cos(sin

)(

θφθφθ±=

±

kk

r

(5)

The angle θ is measured from the direction of the incident pump beam and therefore

corresponds to collinear propagation. In (2) the k integral is defined as

0

=

θ

∫∫∫∫

∞

=

0

2/

0

2

0

23

sin

ππ

φ

d

θ

d

θ

dkkkd

(6)

Similarly, the parts of the modes outside the cavity in the region in front of the crystal, i.e.,

1

, are described by the mode function

∞<< zd

2

)exp(') exp()(

)()(

,

j k

r

r

r

k iRr

r

k ir

r

U

j k

r

out

rr

⋅+⋅=

+−

(7)

where

k

j

j k

'

r

D

ikd

−

rikd

R

) exp()exp(

2

+

≈

(8)

After substituting the appropriate expressions for the electric fields in (1) using (3) when

3rr

rr=

and (7) when

r

rr=

for the spatial mode function, we obtain, on performing a

simple integration over the irradiated volume of the crystal

2 , 1r

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 464

#1044 - $15.00 USReceived March 28, 2002; Revised May 27, 2002

Page 5

()()

()

2

2

2

2

2

2

1

2

2

2

⊥

2211

) 2 (

cav

2 exp1

1

2

2

sin

exp)(,;,

0

0

dvxir

x

′

vd

x

′

vd

x

′

i dxttdtztzG

o

po

k

k

+

=

∫−

τ πε

ω

ω

(9)

where

dispersion has been taken into account with the following wave-vector expansion:

p

t1is the amplitude transmission coefficient of M1at the pump frequency. Crystal

()()

...

2

1

2

2

k

2

*

i

*

i

*

*

i

*

+−

∂

∂

+−

∂

∂

ω

+=

=

=

i

i k

i k

i

i

i k

i k

i

k

k

i

k

k

k

i

i

kk

kk

ωω

ω

ωω

ωω

ωω

(10)

where

*

ik

r

ω

,

*

ik

are the perfectly phase-matched frequencies and wave-vectors respectively

which satisfy the following energy-conservation and phase-matching conditions:

*

0

*

2

*

1

kkk

ωωω

=+

;

*

0

*

2

*

1

k

r

k

r

k

r

=+

(11)

Thecorrelationtime

()()

1221

zzvtt

−+−=

′ τ

andtheintegrationvariable

i

i

k

k

x

ωω−=

*

;

*i k

i k

ik

ik

v

ω=ω

ω∂

∂

=

, isthefirst-orderdispersion coefficientand

*

2

ik

2

ik

i k

ik

v

ω=ω

ω∂

∂

=

′

, is the second-order dispersion coefficient. In obtaining (9) we have

assumed that the pump has a spatial profile which is Gaussian in the x-y plane [18] with a

ε

.

2

z

are the positions of the detectors along the axis of thebeam-waist radius of

2

⊥

1,z

cavity. For simplicity, we can assume that the detectors are situated at equal distances from

the cavity. For a high-Q cavity we use the following approximation [19] to the Airy function

denominator of the integrand in (9)

()

()

[]

)12(

1

1

−

)1 (

1

r

2exp1

1

2

2

1

2

2

2

2

∑

−=

=

+

+

≈

+

Nl

Nl

o

o

l vdxn

dvxir

π

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 465

#1044 - $15.00 US Received March 28, 2002; Revised May 27, 2002

Page 6

where

The summation in Eq. (12) describes the detection of the degenerate mode plus N non-

degenerate modes on either side of the central degenerate mode. The term in Eq. (12)

corresponding to l = 0 describes the detection of the degenerate mode. After substituting Eq.

(12) in Eq. (9) we obtain to a good approximation the following amplitude

where

vd

τ

τ =

~

. The single-mode amplitude which describes the detection of the central

degenerate mode corresponds to the situation in the above equation when

therefore obtain the single-mode amplitude

0

=

N

. We

SM

t

A

as

−

+

=

1

2

21

1

r

2

2

~

τ

exp

) 1 (

)(

n

n

t

A

o

po

SM

(15)

3. Multimode pair-photon count rate

The multimode detection probability is equal to the single-mode probability multiplied by a

prefactor which is oscillatory. It is instructive to re-express the oscillatory prefactor a

]))

~

τ

2 [ 2cos(2 )(1 2 (

+

))

~

τ

2 cos(2 (2 ) 1

+

2 (

2

~

τ

sin

~

τ

2

π

12

sin

2

ππ

π

−+=

+

NNN

N

])

~

τ

2 [

N

2cos(2...

π++

(16)

) 13(

1

2

2

2

1

o

o

r

r

n

+

=

) 14(

~

τ

exp

2

~

τ

sin

~

τ

2

π

12

sin

)1 (

1

n

)(

1

2

2

1

r

2

2

) 2(

cav

−

+

+

=

n

N

tt

G

o

po

π

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 466

#1044 - $15.00 USReceived March 28, 2002; Revised May 27, 2002

Page 7

The term with the smallest frequency,

vd

π

2

ω

=

, has a period ∼ 10-12s. Since detectors

cannot measure such rapid oscillations in time, only an average is recorded, and therefore the

cosine terms make a vanishing contribution to the count rate. Hence the multimode amplitude

is given by

) 2(

MM

A

∼

−

+

+

1

2

21

1

2

2

~

τ

exp

) 1 (

) ( 12

n

n

r

ttN

o

po

(17)

The count rate in the multimode case is therefore larger by a factor of (2N + 1) compared to

the single-mode count rate, but shows the same time dependence as the single-mode case.

5. Enhancement-factor per mode

The enhancement factor per mode,γ , is defined by the following:

cavityno

cavity

idth)rate/bandw(count

idth)rate/bandw (count

=

γ

(18)

i.e., it is the ratio of the count rate per unit frequency with the crystal in the cavity to the count

rate per unit frequency with the crystal in free space. We first of all need to obtain the

spectrum of the down-converted light with and without the cavity.

bandwidth of the light with the cavity we use Eq. (9) for

0

=

l

FWHM,

cav

)( ω∆

, of this function is a reasonable estimate of the bandwidth and is

3 . 1

)(

=∆ω

. The numerator in Eq. (18) then works out as

In calculating the

and integrate

) 2(

cav

G

)*2(

cav

) 2(

cavGG

(using the approximation in Eq. (12) with) with respect to the correlation time ' τ . The

approximately

vdn1

cav

4

2

2

1

4

2

cavity

) 1 (

2 . 1

=

idth)rate/bandw (count

o

po

r

tt

+

k

where k is a constant. In the absence of

the cavity we use the right-hand-side of Eq. (32) in [21] for the spectrum of the degenerate

photons. This gives us an approximate bandwidth

dv'

1 . 3

)(

cav no

=∆ω

. The denominator in

Eq. (18) to a good approximation is then

k

=

cav no

idth)rate/bandw(count

. The

enhancement factor, γ , is then given by,

4

2

2

1

4

2

) 1 (

2 . 1

=

o

po

r

tt

+

γ

.

(C) 2002 OSA3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 467

#1044 - $15.00 USReceived March 28, 2002; Revised May 27, 2002

Page 8

4. Results

If we consider the case of single-mode detection, our analysis predicts a theoretical FWHM

r vd

2

1

+

sm 108

×=

v

[20] the width is approximately 6 ns. This is in good qualitative

agreement with the results of Ou and Lu [15,16] where a 4.00 mm crystal was used and a

3.35 mm external filter cavity was used to filter out the nondegenerate modes so that the

measurements correspond to single-mode correlations. Also the enhancement factor is

calculated to be 9.3×104which compares well with the results of Ou et. al. who obtained 5.5

× 104. Any differences between our predictions and that of Ou et. al. is most probably due to

the fact that we assumed that M1was a perfectly reflecting mirror.

of

o

o

r

2

2 ln2

. For a 4.1 mm cavity, M2 transmission coefficient of 1.5% and

-19

−

5. Conclusion

A multimode microscopic model of the optical parametric oscillator operating well below

threshold has been presented. Effects such as cavity damping and crystal dispersion are both

taken into account. We have derived expressions for both the single-mode and multimode

amplitudes. For a cavity of a given finesse the calculated value of the FWHM of the single-

mode correlation function and the cavity enhancement factor are calculable and compare well

with the experimental results of Ou et. el. In a future publication we intend to generalize the

theory to describe an oscillator operating close to threshold in which correlated squeezed

states can be generated. These sources are have been attracting much attention recently

because of their use in quantum information processing systems.

(C) 2002 OSA 3 June 2002 / Vol. 10, No. 11 / OPTICS EXPRESS 468

#1044 - $15.00 US Received March 28, 2002; Revised May 27, 2002