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Characterization of a resonator using entangled two-photon states Jan Perina Jr. * Joint Laboratory of Optics of Palack y University and Institute of Physics of Academy of Sciences of the Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic Department of Optics, Faculty of Natural Sciences, Palack y University, Czech Republic Received 7 November 2002; received in revised form 11 March 2003; accepted 10 April 2003

Abstract Propagation of a photon from entangled photon pair through a Fabry–Perot resonator is utilized to obtain characteristics of the resonator. The photon exiting the resonator is mixed with its entangled twin in Hong–Ou–Mandel interferometer. Response of the resonator to an incident optical ﬁeld with the resolution up to tens of femtoseconds can be obtained from the measured coincidence-count interference proﬁle. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.65.Ky; 42.50.Dv Keywords: Spontaneous parametric downconversion; Entangled two-photon state; Fabry–Perot resonator

1. Introduction The occurrence of photons comprising an entangled photon pair generated in the process of spontaneous parametric downconversion [1] is strongly correlated in time. This property has been used for measurements with subfemtosecond time resolution [2] and determination of tunneling time of a photon through a photonic band-gap structure [3]. The newly proposed method of quantum optical coherence tomography [4] is based on strong correlations of entangled photons in a pair. Also absolute quantum eﬃciencies of detectors can be obtained using correlated photon pairs [5]. Properties of entangled photon pairs enabled to test the rules of quantum mechanics against local hidden-variables theories in many physical situations [6]. Photon pairs entangled also in polarization have been applied in quantum cryptography [7], quantum teleportation [8] or generation of GHZ states [9] recently. The ﬁeld of quantum information is perspective for their application in near future [10].

*

Tel.: +420585631509; fax: +420585224047. E-mail address: [email protected] (J. Perina Jr.).

0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01492-5

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A detailed structure of time (or frequency) entanglement of photons in a pair is revealed in the coincidence-count interference rate measured in Hong–Ou–Mandel interferometer [1,11–14]. The coincidencecount rate typically forms a dip with the width given by entanglement time (usually tens or hundreds of femtoseconds). We note that pulsed-train pumping generates entangled ﬁelds with a more complex time structure. Such ﬁelds can be described as bipartite entangled systems in Hilbert spaces with higher dimensions [15]. We show that ultrashort time correlations of photons in a pair can be successfully applied in determination of intensity response function of a resonator [16,17] to an incident optical ﬁeld. If the incident-ﬁeld spectrum covers just one peak of the resonator spectrum (one resonator mode) then the intensity response function is a damped exponential function characterized by photon lifetime in the resonator mode. Provided that the incident-ﬁeld spectrum covers spectral peaks of N resonator modes, the intensity response function of the resonator is composed of N 2 damped exponential functions characterized by photon lifetimes in these resonator modes. We let one (signal) photon from the entangled photon pair propagate through the resonator. This propagation leads to modiﬁcation of time structure of the signal photon ﬁeld that depends upon properties of the resonator. The output photon exiting the resonator is mixed with the second (idler) photon from the pair in the Hong–Ou–Mandel interferometer. This conﬁguration together with entanglement of photons in a pair then enables to obtain intensity response function of the resonator to a given incident ﬁeld from the measured coincidence-count rates. If a resonator is illuminated by a coherent ultrashort laser pulse, then the output-ﬁeld intensity as a function of time consists of N 2 damped exponential functions provided that N resonator modes were excited by the incident ﬁeld. Time resolution in hundreds of femtoseconds can also be reached in this case, but the method of optical gating based on nonlinear interaction in a medium has to be applied. Nonlinear interaction requires intense optical ﬁelds exiting the resonator. On the other hand, the method utilizing entangled photon pairs works at single-photon level. We assume a Fabry–Perot resonator in further considerations for simplicity. However, this method can be generalized to the case of a general resonator straightforwardly.

2. Experimental setup We consider the experimental setup shown in Fig. 1. Nonlinear crystal NLC generates entangled photon pairs in type-II spontaneous parametric downconversion. The emerging photons thus have diﬀerent group velocities. We also assume that generated photon pairs are not entangled in polarization. Polarization of a signal photon (denoted as 1) is rotated by 90° in a k=2-plate and the photon then propagates through delay line DL and is ﬁnally mixed with an idler photon (denoted as 2) at beamsplitter BS. The idler photon

Fig. 1. Sketch of the setup. Entangled photon pairs are generated in nonlinear crystal NLC. Polarization of the signal ﬁeld at the frequency x1 is rotated by 90° using k=2-plate and then the ﬁeld propagates through delay line DL. The structure of the idler ﬁeld at the frequency x2 is modiﬁed in Fabry–Perot resonator FPR and then the idler ﬁeld is mixed with the signal ﬁeld at beamsplitter BS. Photons are detected at detectors DA and DB ; C denotes a coincidence-count device.

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propagates through Fabry–Perot resonator (FPR) before it is mixed with its signal twin. Propagation of the idler photon through the Fabry–Perot resonator substantially modiﬁes time structure of the pair. The modiﬁed structure of the photon pair is then visible in the fourth-order interference proﬁle obtained as a coincidence-count rate of two detectors DA and DB in dependence on time delay introduced by delay line DL.

3. Entangled two-photon state Spontaneous parametric downconversion is described by the following interaction Hamiltonian [1]: Z 0 ðÞ ðÞ ^ Hint ðtÞ ¼ dzvð2Þ EpðþÞ ðz; tÞE^1 ðz; tÞE^2 ðz; tÞ þ h:c:; ð1Þ L

ð2Þ

where v is second-order susceptibility, EpðþÞ denotes the positive-frequency part of the electric-ﬁeld ðÞ ðÞ amplitude of the pump ﬁeld, and E^1 (E^2 ) is the negative-frequency part of the electric-ﬁeld operator of the downconverted signal (idler) ﬁeld. The symbol L denotes the length of the crystal and h.c. means Hermitian conjugate. The wave function jwð2Þ ð0; tÞi describing an entangled two-photon state in the output plane of the crystal (z ¼ 0) is obtained in the form (for details, see [13]): Z 1 Z 1 Z 0 Z 1 ð2Þ 0 0 ðþÞ jw ð0; tÞi ¼ Cw exp½iðx1 þ x2 Þt dz dmp dm1 dm2 Ep;env ð0; mp Þ^ ay1 ðm1 Þ^ ay2 ðm2 Þ L 1 1 ð2Þ 1 mp m1 m2 exp i z dðmp m1 m2 Þ exp ½iðm1 þ m2 Þt jvaci; vp v1 v2 ðþÞ where mj ¼ xj x0j for j ¼ p; 1; 2. The symbol Ep;env ð0; mp Þ denotes the spectrum of the positive-frequency part of the envelope of the pump-ﬁeld amplitude in the output plane of the crystal; a^y1 ðm1 Þ [^ ay2 ðm2 Þ] means the creation operator of the mode with wave vector k1 (k2 ) and frequency x01 þ m1 (x02 þ m2 ) in downconverted ﬁeld 1 (2). The symbol x0j stands for the central frequency of ﬁeld j (j ¼ 1; 2; p) and 1=vj denotes the inverse of group velocity of ﬁeld j [1=vj ¼ dkj =ðdxkj Þjxk ¼x0 ]. The j j susceptibility vð2Þ is included in the constant Cw . Frequency- and wave-vector phase matching for central frequencies (x0p ¼ x01 þ x02 ) and central wave vectors (kp0 ¼ k10 þ k20 ) are assumed to be fulﬁlled when deriving Eq. (2). An entangled two-photon state is conveniently described by a two-photon amplitude A12 ðt1 ; t2 Þ deﬁned as [13] ðþÞ

ðþÞ

A12 ðt1 ; t2 Þ ¼ hvacjE^1 ð0; t0 þ t1 ÞE^2 ð0; t0 þ t2 Þjwð2Þ ð0; t0 Þi;

ð3Þ ð2Þ

where jvaci denotes the vacuum state and the two-photon state jw ð0; t0 Þi is given in Eq. (2). The twophoton amplitude A12 in front of beamsplitter BS (i.e., after propagation of the signal photon through the delay line and modiﬁcation of the idler-photon structure in the Fabry–Perot resonator) can be obtained in the form h 2pCw si L0 A12 ðT0 ; sÞ ¼ exp iðx01 x02 Þ exp iðx01 þ x02 ÞT0 exp ix01 2 jDj g1 1 X s þ ð2l=g2 Þn ðL0 =g1 Þ qn expðin/l Þrect q1 LD n¼0 K Dp2 L0 Dp1 2l t1 þ t2 ðþÞ Ep;env ; s ¼ t1 t2 : 0; s þ T0 þ n ; T0 ¼ ð4Þ D D g1 D g2 2

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The symbols D, Dp1 , Dp2 , and K are given as D¼

1 1 ; v1 v 2

1 1 ; j ¼ 1; 2; vp vj 1 1 1 1 K¼ þ : vp 2 v1 v2

Dpj ¼

ð5Þ

The symbol g1 stands for the group velocity of the signal ﬁeld in the delay line of the length L0 . The group velocity of the idler ﬁeld in the Fabry–Perot resonator of the length l is denoted as g2 ; q1 denotes the probability amplitude that a photon propagates through the resonator without being reﬂected at the output mirror of the resonator and q expði/l Þ is determined as the ratio of the probability amplitudes that a photon exits the resonator after k and ðk þ 1Þ reﬂections at the output mirror. The rectangular function rectðxÞ ðþÞ equals 1 for 0 6 x 6 1, otherwise it equals 0. The symbol Ep;env ð0; tÞ means the positive-frequency part of the envelope of the pump-ﬁeld amplitude in the output plane of the crystal. The term proportional to qn in the expression for the two-photon amplitude A12 in Eq. (4) describes the probability amplitude appropriate for the case when the idler photon is just n times reﬂected at the output mirror of the resonator and then leaves the resonator. This means that time behaviour of an idler photon inside the resonator is written in the structure of the two-photon entangled state and can be revealed, e.g., in the coincidence-count interference proﬁle of the Hong–Ou–Mandel interferometer.

4. Normalized coincidence-count rates The normalized coincidence-count rate R for the setup shown in Fig. 1 in dependence on the length L0 of the delay line is determined as follows [13]: RðL0 Þ ¼ 1 rðL0 Þ; where

Z

ð6Þ Z

n t þ t t þt o A B A B ; tA tB A12 ; tB tA ; dtB Re A12 2 2 1 1 Z 1

2 1 tA þ tB ; tA tB : dtA dtB A12 R0 ¼ 2 1 2 1 1 2R0 Z 1

rðL0 Þ ¼

1

dtA

1

ð7Þ ð8Þ

The symbol Re denotes the real part of an argument. The use of the two-photon amplitude A12 given in Eq. (4) provides the quantities r and R0 in the form (equal central frequencies of the downconverted ﬁelds are assumed, i.e., x01 ¼ x02 ): Z 0 2 1 X 1 2p2 jCw j L 2 X 2l 2L0 n n0 0 0 q1 ðn þ n Þ q q Re exp½i/l ðn n Þ dy rect y þ rðL0 Þ ¼ R0 jDj g2 DL g1 DL y¼1 n¼0 n0 ¼0 4Kl 0 2KL0 2Dp1 l n þ ðn n0 Þ ; Cp 2KLy ð9Þ g2 D g2 D g1 D 2 1 X 1 2p2 jCw j L 2 X 2l 2Dp1 l n n0 0 0 0 q1 R0 ¼ ðn n Þ Cp ðn n Þ : q q exp½i/l ðn n Þ triang ð10Þ jDj g2 DL g2 D n¼0 n0 ¼0

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The triangular function triangðxÞ equals 1 jxj for 1 6 x 6 1, otherwise it equals 0. The autocorrelation function Cp of the pump-ﬁeld envelope is determined according to the relation Z 1 ðþÞ ðþÞ dTEp;env ð0; T ÞEp;env ð0; T þ xÞ: ð11Þ Cp ðxÞ ¼ 1

The quantity r given in Eq. (9) and determining the coincidence-count rate consists of two diﬀerent kinds of terms. Those fulﬁlling n ¼ n0 in the double sum in Eq. (9) describe propagation of the idler photon that was reﬂected just n times at the output mirror of the resonator. Intensity response function of the resonator to an incident optical ﬁeld can be derived from magnitudes of these terms. The measured distance of adjacent peaks determines optical length of the resonator. The remaining terms (n 6¼ n0 ) stem from quantum interference of probability amplitudes characterizing propagation of the idler photon through the resonator just n and n0 times. The expressions for the quantities r and R0 given in Eqs. (9) and (10) simplify for cw pumping (np is the pump-ﬁeld amplitude): 2 2 1 X 1 2p2 jCw j jnp j L 2 X 2l 2L0 n n0 0 0 rðL0 Þ ¼ q1 ðn þ n Þ 1 ; ð12Þ q q cos½i/l ðn n Þ triang g2 DL g1 DL R0 jDj n¼0 n0 ¼0 1 X 1 2p2 jCw j2 jnp j2 L 2 X 2l 0 ðn n0 Þ : R0 ¼ q1 qn qn exp½i/l ðn n0 Þ triang ð13Þ g2 DL jDj n¼0 n0 ¼0

5. Coincidence-count interference proﬁles We ﬁrst discuss the interference proﬁle of the normalized coincidence-count rate R obtained for cw pumping. The normalized coincidence-count rate R as given by Eqs. (6) and (12) consists of diﬀerent terms distinguished by ﬁxed values of n and n0 . These terms are nonzero only in localized regions of L0 . The term with given n and n0 reaches its maximum value for L0 ¼ ðg1 =g2 Þðn þ n0 Þl þ g1 DL=2 and is nonzero for ðg1 =g2 Þðn þ n0 Þl 6 L0 6 ðg1 =g2 Þðn þ n0 Þl þ g1 DL [D > 0 is assumed]. A typical behaviour of the normalized coincidence-count rate R in case when diﬀerent terms interfere constructively is shown in Fig. 2(a). The jth dip in Fig. 2(a) is determined by terms with n þ n0 ¼ ðj 1Þ (j ¼ 1; 2; . . .). We can see in Fig. 2(a) that the greater the value ðn þ n0 Þ the smaller the overall interference contribution of the corresponding terms. This

Fig. 2. Normalized coincidence-count rates R as functions of path delay L0 for (a) l ¼ 0:999873 mm (/l ¼ 2pm, m 2 Z) and (b) l ¼ 0:999976 mm (/l ¼ 2pm þ p=2, m 2 Z); L ¼ 3 mm, g2 ¼ g1 ¼ c, c ¼ 2:9979 108 m s1 , q ¼ 0:4; cw pumping is assumed. Values of the inverse group velocities appropriate for BBO crystal [18] with type-II interaction at pump wavelength kp ¼ 413 nm and downconversion wavelengths k1 ¼ k2 ¼ 826 nm apply in Figs. 2–6: 1=vp ¼ 56:85 1010 s m1 , 1=v1 ¼ 56:14 1010 s m1 , and 1=v2 ¼ 54:30 1010 s m1 .

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Fig. 3. Normalized coincidence-count rate R as a function of path delay L0 for q ¼ 0:7; values of the other parameters are the same as in Fig. 2(a).

is valid for smaller values of q. The overall interference contribution constituting the jth dip increases with the increasing value of j for the ﬁrst couple values of j for suﬃciently great values of q (see Fig. 3). The reason is that the number of interference terms contributing to the jth dip increases linearly with increasing values of j. In case when the terms interfere completely destructively, only odd dips (or peaks) survive (see Fig. 2(b)). Even dips are completely smoothed out by interference. A detailed analysis of the expression for r in Eq. (12) shows that absolute values of rðL0 Þ at the positions of local minima and maxima (positions of dips and peaks) map the intensity response function of the resonator to a given incident optical ﬁeld. Positions of interference contributions on the L0 axis in case of pulsed pumping remain the same as in cw case. However, values of interference terms depend strongly on pump-pulse duration. In general, absolute values of the interference terms are smaller for pulsed pumping in comparison with those for cw pumping. This behaviour is similar to that of the coincidence-count interference proﬁle in the Hong–Ou–Mandel interferometer and reﬂects an increase of distinguishability of the signal and idler photons with decreasing pump-pulse duration [13]. Terms with equal n and n0 are nonzero in general. However, if scor 6 KL (scor characterizes the width of the pump-pulse autocorrelation function Cp ) absolute values of these terms decrease with decreasing pump-pulse duration (see Fig. 4, compare it with Fig. 2(a)). Visibility of the coincidence-count interference proﬁle given by R then decreases. Terms with nonequal n and n0 are nonzero only if j½ðn n0 Þl =½g2 KL j 6 1 or j½ðn n0 Þl =g2 j 6 scor . These terms do not contribute to a coincidence-count interference proﬁle for pump-pulse durations in hundreds of femtoseconds, length of the Fabry–Perot

Fig. 4. Normalized coincidence-count rate R as a function of path delay L0 for pulsed pumping; L ¼ 3 mm, l ¼ 0:999873 mm (/l ¼ 2pm, m 2 Z), g2 ¼ g1 ¼ c, c ¼ 2:9979 108 m s1 , q ¼ 0:4; sp ¼ 1 1013 s, a ¼ 0.

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Fig. 5. Normalized coincidence-count rate R as a function of path delay L0 for cw pumping; l ¼ 0:99946 101 mm (/l ¼ 2pm, m 2 Z) (full curve) and l ¼ 1:004925 101 mm (/l ¼ 2pm þ p=2, m 2 Z) (dashed curve), L ¼ 3 mm, g2 ¼ g1 ¼ c, c ¼ 2:9979 108 m s1 , q ¼ 0:4.

resonator in millimeters, and values of other parameters typical for current experiments with downconverted photon pairs (see Fig. 4). The coincidence-count interference proﬁle is thus built up from terms with n ¼ n0 for femtosecond pumping. Values of rðL0 Þ at the positions of local minima (bottoms of dips) then map the intensity response function of the resonator to a given incident optical ﬁeld. We brieﬂy discuss the case when interference contributions characterized by diﬀerent values of ðn þ n0 Þ begin to overlap in RðL0 Þ; i.e., if g2 jDjL=l P 1. Proﬁle of the normalized coincidence-count rate R then depends strongly on the phase /l of the Fabry–Perot resonator (see Fig. 5). If jDjL l=g2 we can distinguish contributions characterized by diﬀerent values of ðn þ n0 Þ in the normalized coincidence-count rate R (see Fig. 5). On the other hand, if jDjL l=g2 the normalized coincidence-count rate R forms a single dip, which may be asymmetric under certain conditions (see Fig. 6). Asymmetric coincidence-count interference dips have been observed under certain experimental situations (see, e.g. [19]). Interference in a ‘‘thin resonator’’ might provide an explanation of experimental data in some cases. We note that this regime is not suitable for obtaining resonator characteristics. Frequency ﬁlters are often placed in the paths of the signal and idler photons with the aim to increase visibilities of the coincidence-count interference proﬁles. They also broaden interference contributions (dips and peaks) in the coincidence-count interference proﬁles. However, the use of ﬁlters would only make the

Fig. 6. Normalized coincidence-count rate R as a function of path delay L0 for pulsed pumping; l ¼ 1:001525 102 mm (/l ¼ 2pm þ p=2, m 2 Z) (full curve) and l ¼ 0:9912 102 mm (/l ¼ 2pm, m 2 Z) (dashed curve), L ¼ 3 mm, g2 ¼ g1 ¼ c, c ¼ 2:9979 108 m s1 , q ¼ 0:4; sp ¼ 1 1013 s, a ¼ 0.

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relation between the intensity response function of a resonator and the coincidence-count interference proﬁle more complex in our case. We note that the method of quantum optical coherence tomography developed in [4] utilizes interference of entangled two-photon states for the determination of frequency transfer function of an object in a similar setup.

6. Conclusions We have shown that propagation of a photon from an entangled two-photon state inside a resonator modiﬁes time structure of this state. This structure revealed, e.g., in the coincidence-count interference proﬁle of the Hong–Ou–Mandel interferometer then provides intensity response function of a resonator to a given incident optical ﬁeld straightforwardly under certain conditions. Optical length of the resonator and photon lifetimes of resonator modes can be determined from the intensity response function. The use of cw pumping requires maximally destructive interference of interfering terms. Pulsed-pumping case is not sensitive to changes of the ﬁeld phase inside the resonator but visibilities are lower and so the measurement precision is lower too. Resolution in tens of femtoseconds can be reached owing to ultrashort correlations of photons in an entangled photon pair. The described method is promising for determination of characteristics of unstable resonators and resonators with a complex structure.

Acknowledgements The author thanks O. Haderka, J. Soubusta and Z. Bouchal for discussions. This research was supported by the Ministry of Education of the Czech Republic under the Projects LN00A015, CEZ 314/98 and RN19982003012.

Appendix A. Coincidence-count rates for a gaussian pump-pulse proﬁle ðþÞ This appendix contains results valid for a gaussian pump-pulse proﬁle, i.e., the envelope Ep;env ð0; tÞ of the positive-frequency part of the pump-pulse amplitude in the output plane of the crystal is given as ! 1 þ ia ðþÞ Ep;env ð0; tÞ ¼ np exp 2 t2 ; ðA:1Þ sp

np means the pump-pulse amplitude, sp is the pump-pulse duration, and a denotes the chirp parameter. The pump-pulse autocorrelation function Cp given in Eq. (11) then has the form ! pﬃﬃﬃ 2 psp jnp j 1 þ a2 2 pﬃﬃﬃ Cp ðtÞ ¼ t ; ðA:2Þ exp 2s2p 2 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i.e., the correlation time scor is determined according to the relation scor ¼ 2sp = 1 þ a2 . Filtering of the downconverted ﬁelds generated in the nonlinear crystal occurs in many experimental setups. For this reason we consider gaussian frequency ﬁlters (their width is denoted as r) placed in the paths of the downconverted photons. Inclusion of frequency ﬁlters into the theory is formally done by replacing the operators a^yj ðmj Þ in Eq. (2) for jwð2Þ ð0; tÞi by the expressions a^yj ðmj Þ expðm2j =r2 Þ. The quantities r and R0 then get the form (x01 ¼ x02 is assumed):

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Z 0 Z 0 2 2 1 X 1 X r2 s2p p2 jCw j jnp j q21 n n0 0 p ﬃﬃ ﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q rðL0 Þ ¼ q q cos½i/l ðn n Þ dz dz0 2 2 R0 L L s2p r2 þ 2ð1 þ a2 Þ n¼0 n0 ¼0 2 ! r2 ð1 þ a2 Þ l 0 0 Kðz z Þ þ ðn n Þ exp 2½r2 s2p þ 2ð1 þ a2 Þ g2 ! 2 0 r2 D lðn þ n Þ L 0 ðz þ z0 Þ þ exp ; ðA:3Þ g2 4 2 g1 Z 0 Z 0 2 2 1 X 1 X r2 s2p p2 jCw j jnp j q21 0 pﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qn qn cos½i/l ðn n0 Þ dz dz0 R0 ¼ 2 2 2 2 2 R0 L L sp r þ 2ð1 þ a Þ n¼0 n0 ¼0 ! 2 2 ! r2 ð1 þ a2 Þ l r2 D lðn n0 Þ 0 0 0 ðz z Þ þ Kðz z Þ þ ðn n Þ exp exp : 2½r2 s2p þ 2ð1 þ a2 Þ g2 g2 4 2 ðA:4Þ

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