photon correlations of a sub-threshold optical parametric oscillator

Photon correlations of a sub-threshold opticalparametric oscillator

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

Republic of Trinidad and Tobago, [email protected]

E. R. Pike and Sarben SarkarDepartment of Physics, King’s College London, Strand, London WC2R 2LS, UK.

[email protected]@kcl.ac.uk

Abstract: A microscopic multimode theory of collinear type-I spontaneousparametric downconversion in a cavity is presented. Single-mode andmultimode correlation functions have been derived using fully quantizedatom and electromagnetic field variables. From a first principles calculationthe FWHM of the single-mode correlation function and the cavityenhancement factor have been obtained in terms of mirror reflectivities andthe first-order crystal dispersion coefficient. The values obtained are ingood agreement with recent experimental results [Phys. Rev. A 62 , 033804(2000)].2002 Optical Society of AmericaOCIS codes: (190.0190) Nonlinear optics; (190.4410) Parametric processes

References and links1. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon

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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

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872 (1998).8. L. Vaidman, “Teleportation of quantum states,” 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

17. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced inparametric amplification,” Phys. Rev. A 30, 1386-1391 (1984).

18. R. Andrews, E. R. Pike and S. Sarkar, ‘‘The role of second-order nonlinearities in the generation of localizedphotons,’’ Pure Appl. Opt. 7, 293-299 (1998).

19. F. De Martini, M. Marrocco , P. Mataloni, L. Crescentini and R. Loudon, ‘‘Spontaneous emission in the opticalmicroscopic cavity,’’ Phys. Rev. A 43, 2480-2497 (1991).

20. B. Zysset, I. Biaggio, and P. Gunter, “Refractive indices of orthorhombic KNbO3 : Dispersion and temperaturedependence,” J. Opt. Soc. Am. B 9, 380-386 (1992).

21. R. Andrews, E. R. Pike, and S. Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum Semiclass. Opt. 1, 588-597 (1999).

1. IntroductionThe optical process of spontaneous parametric down-conversion (SPDC) involves the virtualabsorption and spontaneous splitting of an incident (pump) photon in a transparent nonlinearcrystal producing two lower-frequency (signal and idler) photons [1-3]. The pairs of photonscan be entangled in frequency, momentum and polarization. In type-I SPDC the photons arefrequency-entangled and the signal and idler photons have parallel polarizations orthogonal tothe pump polarization.

Entangled photons have been used to demonstrate quantum nonlocality [4,5], quantumteleportation [6-8] and more recently quantum information processing [9-11]. Photon pairshave also been used to demonstrate quantum interference phenomena [12-14]. The photonpairs produced in such experiments are separated by less than a picosecond and theircorrelation properties could only be investigated indirectly, for example by fourth-orderinterference [2,12]. Ou and Lu [15,16] have recently measured the time separation of photonsproduced from a nonlinear crystal placed inside a high-Q cavity. Because of the reducedbandwidth of the photons and consequent broadening of the photon correlation functions theywere able to measure pair-photon correlations directly. Measurements of single-mode andmultimode correlation functions were made for photons with frequencies close to thedegenerate frequency for type-I collinear SPDC. Experimental results were modelled usingthe theory of Collett and Gardiner [17].

In this paper we propose an alternative microscopic multimode theory to describe thedetection 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 opticalparametric oscillator (OPO), we believe our approach will broaden the understanding of thesub-threshold operation of the OPO. We describe the situation in which collinear photonpairs, which experience multiple reflections in the cavity, are produced by pump photonswhich pass through the crystal once; this is the single-pass case and corresponds to an OPOoperating 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. Ourapproach yields similar results to the theory used by Ou and Lu, but in addition, we havebeen able to obtain exact expressions for the hitherto phenomological coupling constantsintroduced in the Ou and Lu analysis. Such constants are important in determining the FWHMof the correlation functions. We consider a high-Q cavity which corresponds to theexperimental situation with detectors positioned outside the cavity. Our theoreticalsimulations compare well with the experimental results of Ou et. al.

2. Spontaneous down-conversion amplitude for a crystal in a cavityIf we first consider a crystal atom at position 3r

r

in free space and for one-atom detectors

located at 1rr

and 2rr

, the generalized amplitude for detecting pairs of down-converted photonsfrom a single crystal atom is given by [18].


eaE ,..., 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 λα k ,

consists of a coherent state with wave-vector 0k and polarization index 0λ (the

monochromatic pump beam) with other modes in the vacuum state 0 . >321 ,,| ggg is

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

321 ,, gaa describes the detector atoms in excited states 21 , aa and a source atom

finally in the ground state 3g after the two-photon emission process. )()(,,..., tjeaµ denotes

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

atom at position jr .

Figure 1: Schematic showing a cavity of length d bounded by an input mirror M1 and anexit mirror M2. The shaded area represents the nonlinear crystal which fills the cavity.

To obtain the amplitude, )2(cavG , for pair-photon detection in the case of a crystal of length d

placed inside a cavity of length d , all embedded in a linear medium of the same refractiveindex [3] (see Figure 1), we employ the plane-wave mode functions for the quantized electricfield [19] for the cavity and external reservoir system. Since we are using the completeelectromagnetic field which describes modes both inside and outside the cavity, dampingeffects are already included in the theory and therefore need not be includedphenomenologically. For a one-sided cavity in which the amplitude transmission coefficientfor photons exiting through mirror M1 from inside the cavity is zero, the quantized electricfield is given as

)exp()()(16

dit),r(E

2/1

2,1 03

3 tiarUkkc

kjkjkj

j

ωεεπ

−

= ∫ ∑

=

rr

rrhrr(2)

)1()(

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

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

),,,(

0

0

0

0

4321

54321

3215)3(,

4)3(,

3)3(,

2)2(,

1)1(,

321

54321

5)2(

21

33321

321

rr

rrrrr

rrr

rr

rrrrr

rrr

↔+

><×

><×

−= ∫∫∫∫∫ ∞−∞−∞−∞−∞−

λλ αµµµµµ

kedcba

k

edcba

ttttt

tEtEtEtEtEa

gggtttttsaa

dtdtdtdtdth

itA

d

M1 M2


The destruction operators for the modes with spatial function )(rUjk

r

r are denotedjk

a r where

)(kj

r

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

r

r for

dzd2

1

2

1 <<− , i.e, inside the cavity is denoted by )(,

rUjkin

r

r and is defined by the

following

k

o

k

ojkin D

ikdrkit

D

rkitrU

)exp()exp()(

)(2

)(2

,

+⋅−

⋅=

+− r

r

r

r

r

r (3)

with

)2exp(1 2 ikdrD ok += (4)

ot2 and or2 being the amplitude transmission and reflection coefficients of M2 respectively

for the signal and idler photons; we have taken the reflection coefficient of M1 as 11 −=or

consistent with a perfectly reflecting and infinitely thin mirror. The wave vectors )(−kr

and)(+k

r

describe backward and forward propagating photons and are defined in terms of polar(θ ) and azimuthal (φ ) angles by

)cos,sinsin,cos(sin)( θφθφθ ±=± kkr

(5)

The angle θ is measured from the direction of the incident pump beam and therefore 0=θcorresponds to collinear propagation. In (2) the k integral is defined as

∫ ∫ ∫ ∫∞

=0

2/

0

2

0

23 sinπ π

φθθ dddkkkd (6)

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

∞<< zd2

1, are described by the mode function

)exp(')exp()( )()(,

rkiRrkirUjkjkout

r

r

r

r

r

rr ⋅+⋅= +− (7)

where

k

j

jk D

ikdrikdR

)exp()exp(' 2 −+

≈r (8)

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

3rrrr = and (7) when 2,1rr

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

simple integration over the irradiated volume of the crystal


( ) ( )( ) 2

22

2

2

2122

22211

)2(

2exp1

1

2

2sin

exp)(,;, 0

0 dvxirxvd

xvd

xidxttdtztzGo

pocav

k

k +

′

′

′= ∫−⊥ τπεω

ω

(9)

where pt1 is the amplitude transmission coefficient of M1 at the pump frequency. Crystal

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

( ) ( ) ...2

1 2

2

2*

*

*

*

*

+−∂∂

+−∂∂

+===

ii

ikiki

ii

ikiki

kk

k

ikk

k

iii

kkkk ωω

ωωω

ωωωωω

(10)

where *ikrω , *

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 kkk

rrr

=+ (11)

The correlation time ( ) ( )1221 zzvtt −+−=′τ and the integration variable

iikk

x ωω −= * ;

*ikik

ik

ikv

ωωω

=∂∂

= , is the first-order dispersion coefficient and

*

2

2

ikikik

ikv

ωωω

=∂∂

=′ , 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

beam-waist radius of2⊥ε

. 21 , zz are the positions of the detectors along the axis of the

cavity. For simplicity, we can assume that the detectors are situated at equal distances fromthe cavity. For a high-Q cavity we use the following approximation [19] to the Airy functiondenominator of the integrand in (9)

( ) ( )[ ] )12(1

1

)1(

1

2exp1

122

12

22

2

∑=

−= +−+≈

+

Nl

Nloolvdxnrdvxir π


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

wherevd

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

degenerate mode corresponds to the situation in the above equation when 0=N . We

therefore obtain the single-mode amplitude SMA as

−

+=

12

21

122

~exp

)1(

)(

nrn

ttA

o

poSM

τ(15)

3. Multimode pair-photon count rate

The multimode detection probability is equal to the single-mode probability multiplied by aprefactor which is oscillatory. It is instructive to re-express the oscillatory prefactor a

]))~2[2cos(2)(12())~2cos(2(2)12(

2

~sin

~2

12sin

2

τπτπτπ

τπ−+++=

+

NNN

N

])~2[2cos(2... τπN++ (16)

)13(1

2

2

2

1o

o

r

rn

+=

)14(~

exp

2

~sin

~2

12sin

)1(

)(

12

21

122)2(

−

+

+=

n

N

rn

ttG

o

pocav

ττπ

τπ


The term with the smallest frequency,vd

πω 2= , has a period ∼ 10-12 s. Since detectors

cannot measure such rapid oscillations in time, only an average is recorded, and therefore thecosine terms make a vanishing contribution to the count rate. Hence the multimode amplitudeis given by

)2(MMA ∼

−

++

12

21

122

~exp

)1(

)(12

nrn

ttN

o

po τ(17)

The count rate in the multimode case is therefore larger by a factor of (2N + 1) compared tothe 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 countrate per unit frequency with the crystal in free space. We first of all need to obtain thespectrum of the down-converted light with and without the cavity. In calculating the

bandwidth of the light with the cavity we use Eq. (9) for )2(cavG and integrate )*2()2(

cavcavGG

(using the approximation in Eq. (12) with 0=l ) with respect to the correlation time 'τ . The

FWHM, cav)( ω∆ , of this function is a reasonable estimate of the bandwidth and is

approximatelyvdn1

cav

3.1)( =∆ω . The numerator in Eq. (18) then works out as

42

21

42

cavity )1(2.1idth)rate/bandw(count

o

po

r

ttk

+= 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 bandwidthdv'

1.3)( cavno =∆ω . The denominator in

Eq. (18) to a good approximation is then k=cavnoidth)rate/bandw(count . The

enhancement factor, γ , is then given by,4

2

21

42

)1(2.1

o

po

r

tt

+=γ .


4. Results

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

ofo

o

r

rvd

2

2

1

2ln2

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

sm108 -19−×=v [20] the width is approximately 6 ns. This is in good qualitativeagreement with the results of Ou and Lu [15,16] where a 4.00 mm crystal was used and a3.35 mm external filter cavity was used to filter out the nondegenerate modes so that themeasurements correspond to single-mode correlations. Also the enhancement factor iscalculated to be 9.3×104 which 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 tothe fact that we assumed that M1 was a perfectly reflecting mirror.

5. Conclusion

A multimode microscopic model of the optical parametric oscillator operating well belowthreshold has been presented. Effects such as cavity damping and crystal dispersion are bothtaken into account. We have derived expressions for both the single-mode and multimodeamplitudes. 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 wellwith the experimental results of Ou et. el. In a future publication we intend to generalize thetheory to describe an oscillator operating close to threshold in which correlated squeezedstates can be generated. These sources are have been attracting much attention recentlybecause of their use in quantum information processing systems.


photon correlations of a sub-threshold optical parametric oscillator

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