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Laser Physics, Vol. 12, No. 5, 2002, pp. 1 10.
Original Text Copyright © 2002 by Astro, Ltd. Copyright © 2002 by MAIK "Nauka / Interperiodica" (Russia).

RUBRRRIKA RUBRIKA

Interference of Biphoton Fields
A. V. Burlakov, M. V. Chekhova, Yu. B. Mamaeva, O. A. Karabutova, D. Yu. Korystov, and S. P. Kulik
Department of Quantum Electronics, Moscow State University, 119899 Moscow, Russia
e-mail: postmast@qopt.phys.msu.su Received January 9, 2002

Abstract--Second-order and fourth-order interference effects in spontaneous parametric down conversion (SPDC) are considered. Two basic schemes are discussed, analogous to linear Young and MachZehnder interferometers. Interference effects are used for such applications as spectroscopy of nonlinear and linear media and preparation of novel two-photon polarization states.

1. INTRODUCTION Interference of nonclassical fields has been discussed in literature for more than 15 years [13]. This effect attracts attention from the viewpoint of both quantum theoretical description and such applications as spectroscopy and quantum information. One of the main directions in modern quantum optics is connected with the study of interference of biphoton fields. Biphoton radiation is usually obtained via spontaneous parametric down conversion (SPDC). Such radiation consists of pairs of photons correlated m frequencies, wave vectors, and polarization and differs much from classical light by its statistical properties. The paper is devoted to the experimental study of both second-order and fourth-order interference effects (with respect to the field) in the SPDC radiation. There is an essential distinction between these two types of interference. The former is essentially classical. It means that there is no principal necessity to introduce the concept of photons; in connection with this type of interference one usually speaks of photons for the sake of convenience. To observe such an interference one should overlap two fields in time and space and the possible delay should not exceed the second-order coherence length of the field. Quantum properties of SPDC manifest themselves only in the intensity level measured and do not influence the interference pattern. Note that such an interference was firstly observed as early as 1979 in Moscow State University [4]. However, a profound understanding of the matter was attained only recently. In the case of fourth-order effects, the situation changes dramatically. Now to interpret experimental results one should consider the biphoton field consisting of photon pairs. It turns out that to observe forthorder interference between photon pairs, there is no necessity to overlap these pairs in time and space. It might be claimed that the forth-order coherence length
1

of biphoton field is infinite (of course, within the pump coherence length). Quantum (or, in other words, nonclassical) essence of such a field results in the high visibility of the interference pattern which could achieve unity while the maximum visibility in classical optics could not exceed the classical limit 0.5. Numerous experiments have been realized to demonstrate nonclassical visibility in some remarkable linear interferometers [57]. The problem of interference between biphoton fields generated in different spatial macroscopic regions was clearly posed by D.N. Klyshko in 1993 [8] although the first experiments were realized several years earlier [9]. Usually such schemes are referred to as nonlinear interferometers. The first experimental work using different types of nonlinear interferometers and demonstrating interference effects in both second- and fourth orders was carried out in 1997 [10]. In the same work. a general theoretical approach to both interference kinds was proposed as well Since then, lots of different experiments have been carried out [1113]. Moreover, some new branches of biphoton quantum optics appeared due to these investigations: nonlinear two-photon spectroscopy and polarization biphoton optics. This paper might be considered as a brief review of experimental works carried out during recent years in the Laboratory of Spontaneous Parametric Scattering of M.V. Lomonosov Moscow State University. As a starting point for issues under discussion, we have to consider the approach developed in [10]. 2. SECOND-ORDER INTERFERENCE As it was shown in [10], it is possible to observe interference modulation in the intensity of spontaneous parametric radiation emitted from nonlinear matter with compound structure. By "compound structure", we mean a spatial modulation of the second-order nonlinearity or refractive index distribution. To investigate

2

BURLAKOV et al. x SP S D C L A V P Laser kp, p a d ki, i z
Fig. 1. The experimental setup for registration of frequencyangular spectra of SPDC: P, A are crossed polarizers, V is the nonlinear interaction volume, L is a lens, SP is a spectrograph, D is a detector, and C is an electronic pulse counter. Fig. 2. Nonlinear Young's scheme. A nontransparent screen with two slits is inserted into the pump beam before the nonlinear crystal. SPDC occurs in separated regions filled by the pump radiation.

L

ks, s

S

typical features of this interference we overlap SPDC radiation from two nonlinear crystals. It could seem that no interference should be observed in the case of biphoton field. The SPDC radiation has noise origin, since it is seeded by uncorrelated at different spatial points quantum fluctuations of electromagnetic field. In ordinary (classical) optics, stationary interference appears due to overlapping of some mutually coherent single-frequency waves, and the visibility of the interference pattern is a measure of their mutual coherence. Furthermore, in the SPDC process, there are three light waves with different frequencies: the pump with frequency p , the signal and idler waves with frequencies s and i , respectively. Hence, interference observed in the SPDC radiation is not ordinary in the common sense, but a three-frequency one: field moments depend in this case on the phases of all three waves. Below we consider two extreme cases of the spatial arrangement of nonlinear regions generating biphotons. By analogy with the classical optics, these schemes are called Young and MachZehnder nonlinear interferometers. In these two schemes, the SPDC radiation is registered taking into account a specific dependence of the scattering angle on the scattered light frequency. This dependence is defined by the properties of the nonlinear material and the phase matching conditions. In order to measure the relation between the angle and frequency, we used an experimental setup shown in Fig. 1. A laser pump passes through a polarizer P and enters the scattering volume V, which can have a compound structure (for example, it could consists of nonlinear crystals and linear spaces). The second polarizer A cuts the pump and transmits the scattered light. Then a lens L collects the scattered radiation and focuses it on the spectrograph SP input slit. Hence, the scattering angle is transformed into the transverse coordinate according to the relation r = F tan s , where s is the signal scattering angle outside the crystal. The scattering angle inside the crystal hereafter will be referred as s . At the output of the spectrograph, one can observe two-dimensional intensity distribution of the scattered light in the coordinates "wavelength sangle of scat-

tering s' . Intensity profiles at fixed wavelength could be detected by a detector D scanning along the s coordinate and connected to an electronic counter C. 2.1. Nonlinear Young Interferometer This scheme is a nonlinear analog of the classical two-slit Young's scheme, where a nontransparent screen with two slits is placed between a light source and the plane of observation. Two parallel nonlinear crystals are shifted in the direction transverse to the pump wave-vector. Each crystal plays the role of a separate slit in the classical Young's experiment, and the area between the crystals is the analog of the non-transparent part of the screen between the two slits. Three-frequency interference in such a scheme is observed only in the situation when the idler photons cross both nonlinear regions, i.e., L tan i a , (1)

where L and a are typical longitudinal and transverse sizes of the interaction region, and i is the angle between the pump and idler photons inside the crystal. This condition formally means that one can neglect longitudinal component of the three-frequency wave-vector mismatch in comparison with the transverse one. This scheme was experimentally realized using a non-transparent screen with two slits (the distance between the centers of the slits is d, their width is a, placed into the pump beam just before the nonlinear crystal (Fig. 2). Interference in nonlinear Young's scheme was studied at the idler frequency close to the frequency of crystal lattice vibrations (the so-called polariton scattering.) These frequencies relate to the infrared region where absorption of the idler wave is considerable.
LASER PHYSICS Vol. 12 No. 5 2002

INTERFERENCE OF BIPHOTON FIELDS 2950 19 18 17 16 15 14 610 630 650 S, nm 670 690 3560 i , cm1 4940 Intensity, a.u. 1.0 0.8 0.6 0.4 0.2 0 0.8 (a)

3

S, deg.

0.4

0

0.4

0.8 S, deg.

Visibility, V 0.7 0.8 0.6 0.4 0.2

(b)

Fig. 3. Photograph of the signal radiation frequency-angular spectrum in the nonlinear Young interferometer in the vicinity of a lattice vibration. The idle frequency is close to the resonance with the natural vibration of -iodic acid crystal OH = 2950 cm1.

It has been calculated [14] that in nonlinear Young's scheme, the angular distribution of the signal radiation intensity at fixed frequency s has the form I s ( s ) 2 I ( s, x )
a s

в ( 1 + exp ( x d ) cos [ x ( s ) d ] ) + I s ,
a

(2)

0

50

100

150

200

250

300

x = /2 sin i . Here s is the angle between the pump and the signal, x is the projection of the wave mismatch D = kp ks ki (kp, s, i are wave vectors of the interacting photons) on the direction orthogonal to the pump wave vector (see Fig. 2); is the absorption coefficient for the polariton wave propagating through transverse inhomogeneities. a The function I s (s , x ) describes the angular distribution of the signal radiation intensity scattered from a single layer with transverse size a (only one slit is opened), with the absorption coefficient in this layer a being . The shape of I s (s , x ) is intermediate between a Lorentzian and sinc2(s). The function sinc2(s) corresponds to the case x 0. The Lorentzian describes the situation when absorption is large: 1 x a , in other words, in this situation polariton free path is much smaller than the typical size of transverse inhomogeneities. For the experiment with two slits, a d. The last term in Eq. (2) has oscillating form and results from the distributed character of absorption. Its value is by two orders of magnitude smaller than the
1 x

350 400 , cm1

Fig. 4. (a) Angular distributions of signal intensity (arbitrary units) in the nonlinear Young's scheme at different signal wavelengths ( ) s = 673 nm (i = 4569 cm1) and ( ) s = 622 nm (i = 3357 cm1), = s s0 , s0 is the scattering angle of signal with maximum intensity. The nonlinear iodic acid crystal has the lattice vibration at i = 2950 cm1. Solid lines are the result of calculations. (b) Dependence of interference visibility V vs absorption at idler frequency .

amplitude of the first term in (2), and we can neglect it. So, we can see from (2) that distributed absorption influences both the visibility and the envelope of the interference pattern. Absorption localized between the two emitting regions influences only the visibility of interference [9]. An experimental photograph of signal frequencyangular spectrum in the nonlinear Young's interferometer is shown in Fig. 3. It was obtained for SPDC in -iodic acid (-HIO3) with the pump wavelength p = 514.5 nm. Figure 4a demonstrates two specific angular line shapes registered under the same conditions at the following wavelengths of signal radiation ( ) s = 673 nm (i = 4569 cm1) and ( ) s = 622 nm (i =

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ks

kp

the real part of the dielectric function at the idler photon frequency of the linear layer between the crystals influences the phase of the interference pattern, and absorption, which is related to the imaginary part of m(), decreases the visibility of interference. This fact enables us to apply methods of SPDC spectroscopy for studying optical properties of linear media. 2.2. Nonlinear MachZehnder Interferometer

ki L L' L

Fig. 5. The principal scheme of nonlinear MachZehnder interferometer. Two nonlinear crystals with length L placed one after another along the pump beam with a linear (or nonlinear) layer L' between them.

The principal scheme of nonlinear MachZehnder interferometer is shown in Fig. 5. Two crystals are set consecutively with respect to the pump beam (kp) so that the idler (ki ) and signal (ks ) photons from the first crystal pass through the second crystal [15]. In our experiments, frequencies of the pump and signal waves belong to the visible range while the idler wave belongs to near IR range. Experimental spectra are observed in coordinates s s. To observe two-photon interference, one needs to satisfy the following condition: L tan s a . (5)

3357 cm1). Solid lines are the result of calculation using (2). Clearly, the visibility of the interference pattern decreases as the idler frequency approaches the frequency of the crystal vibration (OH = 2950 cm1) and absorption increases. Thus, it was calculated and verified experimentally that the visibility V of the interference pattern observed for the signal radiation is directly related to the idler wave absorption as V = e [14]. The dependence of the visibility measured versus polariton absorption is shown in Fig. 4b. Nonlinear Young's interferometer can be also used for the study of optical characteristics of a linear medium. Consider the case where the space between the two transparent nonlinear crystals (transverse size of each crystal is a, distance between crystal centers is d) is filled by some matter with the dielectric function ' '' m() = m () + i m () and the absorption coefficient m(). The dielectric function of nonlinear crystals c() is considered to be known. Then the angular distribution of the signal intensity can be represented as I s ( s ) sinc ( x ( s ) a /2 )
2 x d

Here a is a typical transverse size, L L + L' + L is the length of the system of nonlinear crystals in the direction of the pump wave-vector, L, L' are longitudinal sizes of crystals and the space between the crystals, respectively. In other words, in order to register twophoton interference in a MachZehnder scheme, it is necessary to use short nonlinear crystals and/or small signal (registered) angles. In this case, one can neglect transverse inhomogeneity and consider transverse wave mismatch tending to zero. In the general case, the medium filling the space between the crystals has arbitrary dispersion properties. As a result of interference between biphoton fields emitted from different crystals, the frequency-angular spectrum acquires a complicated modulation structure with the parameters defined by the phase shifts for all interacting waves in nonlinear crystals and linear (intermediate layer) media. The line shape of the radiation emitted by such an interferometer can be described by ' 2 z L 2 z L + z L' I 0 ( s, s ) sinc -------- cos ------------------------- , 2 2 (6)

в [ 1 + exp ( m b ) cos ( x ( s ) d + ) ] , m ( i ) ' 2 ( i ) = bk i --------------- cos ( i ) sin ( i ) . c ( i ) '

(3)

(4)

Here b = d a is the transverse size of the space between the crystals; ki , i are the wave-vector and the angle of scattering of the idler wave inside the nonlinear crystals. Note that in contrast to MachZehnder's scheme discussed below, in this scheme, the intermediate layer does not have to be transparent at pump and signal frequencies. One can see from (3) and (4) that

where z and z' are projections of wave mismatches in nonlinear crystals and in the intermediate medium, ' respectively: z = (kp)z (ks)z (ki )z z' = ( k p )z ( k s' )z ( k i' )z , which include projections of all interacting wavevectors (pump, signal and idler) on the direction transverse to all layers. Hence, this scheme can be used for spectroscopic applications.
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INTERFERENCE OF BIPHOTON FIELDS 2000 5 4 S, deg. 3 2 1 540 560 580 600 3000 i , cm1 4000

5

S, nm

Fig. 6. The experimental photograph of the signal radiation frequency-angular spectrum observed at the output of a nonlinear MachZehnder interferometer consisting of two 400 µm LiNbO3 crystals and a linear paraffin oil layer having a resonance at 2950 cm1.

If the investigated medium has resonances at idler frequencies i , then the pattern observed for the signal radiation s (visible region) essentially depends on the features of these resonances [18]. Namely, the visibility of the interference pattern decreases if the absorption for the idler wave increases inside the layer between the crystals. Besides, the interference fringes have specific distortions similar to Rozhdestvenskiy "hooks" (of course, one should take into account principal distinctions between nonlinear MachZehnder interferometer scheme and Rozhdestvenskiy scheme, like three-wave interaction, spontaneous nature of biphoton fields, and spatial degeneracy of nonlinear interferometer arms.) The results obtained experimentally exhibited these features in the three-frequency interference spectra. Figure 6 demonstrates an experimental spectrum of interference in the case where the intermediate medium has a resonance in the range of idler frequencies (i = 2950 cm1). To obtain this spectrum we used paraffin oil placed between two 400-µm-thick crystals of lithium niobate (LiNbO3). If the idler frequency corresponds to the IR region (i = p s), visibility of the interference pattern decreases with the increase of the resonance strength. Near the edges of the resonance region, the interference fringes get specific shape of "hooks." Interference fringes inclination turns into zero (s /s = 0) when ( s i ) + ( s' i' ) = 0 , 1 1 1 where s = L --- , i = L --- , s' = L' ---ui us u s' delays of signal and idler waves in d -' mediate medium (uq = --------q , u q = dk q (7)

2.3. Spectroscopic Application of Nonlinear MachZehnder Interferometer Special research has been carried out to study an opportunity to determine dispersion characteristics of an arbitrary medium from the spectra of three-photon interference in the scheme described above [1618]. Two cases were studied: for intermediate medium transparent at all frequencies (p, s, i ) and for the one having resonances at the idler wave frequencies i (near IR-range). For the case of transparent media, experimental spectra of three-frequency interference were obtained [17]. Algorithms have been found to determine refractive indices of an arbitrary medium in the visible and near IR range. In order to determine the refractive indices of the intermediate medium, we do not need any information about the optical properties of such medium. The only thing we have to do is to measure the biphoton interference spectrum. Parameters of the interference pattern, such as frequencies of maximums (minimums) or inclination angles of the interference fringes, depend on the second factor in (6). So we have a condition to observe a maximum of interference pattern zL + z' L' = 2n (zL + z' L' = + 2n for a minimum.) The wave mismatch in the intermediate medium z' contains refractive indexes n s' and n i' at signal and idler frequencies. Thus, refractive indices of an arbitrary intermediate medium can be determined from the experimental spectra by resolving the system of nonlinear equations written for several experimental points (s, s) belonging to an intensity maximum (or minimum).
LASER PHYSICS Vol. 12 No. 5 2002

1 , i = L' --- are group u i' nonlinear and interd q -------- , q = s, i ). ' dk q

As a result, dispersive properties of an arbitrary medium can be determined by using a nonlinear Mach Zehnder interferometer in both cases: transparent on all frequencies s, i, p medium or when medium has a resonance in the region of idler waves i . 3. FORTH-ORDER INTERFERENCE Usually, interference of biphotons can be interpreted in terms of principal impossibility to treat a biphoton as a quantum object consisting of two independent photons. As a criterion for observing such interference, one might traditionally assume that the biphoton wave packets completely or partially overlap in space and time. It is equivalent to the requirement that the optical paths in a particular interferometer are equal within the accuracy of biphoton field coherence length (which is usually about several tens of micrometers). However, as it will be shown in the next section, this assumption is not true.

6 He-Cd BS

BURLAKOV et al.

D1

PBS2

P IF

QP

/2 |2.0 PBS1

LiIO3 F |0.2

Rc

D2 KDP

/2

Fig. 7. The experimental setup for observing interference of "independent" biphotons. BS is a nonpolarizing beamsplitter, PBS1, 2 are polarizing ones. QP is a halfwave plate rotating the polarization basis by 45 deg. IF and P are a 10/40 nm interference filter and a 100 µm pinhole selecting the degenerate collinear phasematching. Four KDP plates inside the dashed box were used in the spectral measurement of the synthesized polarization biphoton states (see Section 3.4).

lessly joined using a polarizing beamsplitter PBS1. Coincidences (with counting rate Rc) are measured using a polarizing beamsplitter PBS2, two detectors D1, D2, and a coincidence circuit. A halfwave plate QP at the output of the MachZehnder interferometer rotates polarization of both biphoton beams by 45°; as a result, both beams contribute to the coincidence counting rate. Interference filter IF placed before recording part selects degenerate phase matching with = 10 nm spectral width. It was shown that the probability to detect one photon by D1 at the time moment t and to detect the second photon by D2 at the time moment t + is determined by the expression 1 2 R c ( ) -- F ( ) [ 1 G ( L ) cos ( k p L ) ] , 2 (8)

Rc 2.0 1.5 1.0 0.5

0

0.2

0.4 0.6 0.8 Mirror displacement, µm

Fig. 8. Interference dependence of the coincidence counting rate Rc for SPDC radiation in the MachZehnder scheme versus optical path difference variation (corresponding to the displacement of the mirror M in Fig. 7). The modulation period equals the pump wavelength (0.325 µm).

3.1. Interference of "Independent" Biphotons We suggested a principally new scheme [19] for observing biphoton interference shown in Fig. 7. (In fact, the first observation of interference of "independent" biphotons was carried out with the help of a nonlinear Young interferometer [20].) This scheme does not contain any requirement for optical path difference between photon pairs in the interferometer arms. Collinear frequency-degenerate SPDC is excited in two spatial domains of LiIO3, crystal placed into a Mach Zehnder interferometer by 325 nm laser radiation from a HeCd laser. The residual pump is cut by filter F, polarization in one of the beams is rotated by 90° by means of a halfwave plate, and the two beams are loss-

where L L1 L2, is the interferometer arm difference, kp is the pump wavevector, F() has the meaning of effective biphoton amplitude, and G(L) is the firstorder correlation function of the pump. From Eq. (8) it is obvious that the only principal limitation for observing interference is that the path length difference should not exceed the pump coherence length lp , which is about a few meters for single-mode lasers and 30 cm for HeCd laser used in the experiment. At first sight, this is a very strange statement because biphotons are characterized by their coherence length which is about a few dozens of microns (lspdc 2/ 40 µm). Practically, this means that while interfering biphotons emitted from different arms never overlap in space and time, interference is present. It is this effect that could be called interference between "independent" biphotons. In our experiment, the optical path difference was varied by shifting mirror M and the fourth-order interference was observed (Fig. 8) for the optical path difference equal to L = 2 cm which exceeds much the biphoton field coherence length lspdc . The visibility detected is 88% which is nonclassical feature. Also note that the period of interference pattern is twice as large as the wavelength of signal and equal to the pump wavelength. In fact, this is a typical signature of a threewave parametric interaction such as SPDC and inherent in second-order interference effects too as was shown above. 3.2. Interference of "Independent" Biphotons in the Case of a Multimode Pump In the previous section, it was shown that it is impossible to obtain the forth-order interference in case of the optical path difference exceeding the coherence length of the pump. However, if the pump spectrum has a special shape, the "revival" of interference pattern at L > lp becomes possible. This is the case of multimode pump. Its correlation function has multiple local maxima, separated by time intervals corresponding to
LASER PHYSICS Vol. 12 No. 5 2002

INTERFERENCE OF BIPHOTON FIELDS M ''/4'' Visibility V 1.0 0.8 L1 QP2 0.6 0.4 0.2 ''/2'' PBS1 ''/4'' L2 QP4 IF P PBS2 0 40

7

LiIO3

F

QP1

QP3

0

40

80

120

160

200 240 L, cm

''/2'' He-Cd laser D1 Lcav

Fig. 10. Dependence of visibility V observed in the scheme of Fig. 9 versus optical path difference L. Open and closed circles are measurements made with 200 µm and 100 µm pinhole P, respectively. The solid and dotted lines are theoretical calculations taking into account all experimental conditions.

Rc

D2

In the case of equal amplitudes En , the visibility of interference pattern has a multi-peak structure sin [ N ( / c ) L ] V ( L ) = -----------------------------------------N sin [ ( / c ) L ] (11)

Fig. 9. The experimental setup for observation of biphoton interference with a multimode pump. F is a pump-cut filter, PBS1, 2 are polarizing beamsplitters, QP1, 4 are halfwave plates, QP2, 3 are quarterwave plates. IF and P are an interference filter and a pinhole selecting the collinear degenerate phase-matching. The length Lcav of HeCd laser cavity is 98 cm. The optical path difference in the Michelson interferometer L1 L2 is twice as large as 2Lcav.

the doubled length of the laser cavity, and the width of the central maximum defines the "ordinary" laser coherence length lp . Let us suppose for simplicity that the pump spectrum consists of N modes with amplitudes En , centered around the core frequency 0 with equal intermode distance , E p(t ) = e
i 0 t ( N 1 ) /2

n = ( 1 N ) /2

En e

i ( n t + n )

.

(9)

If the mode phases n are independent, the first-order correlation function in (8) takes the form

n G ( L ) = ------------------------------------- .

En e

2 in ( / c ) L

(10)

n

E

2 n

with maxima at L = (c/)m, m = 1, 2, ... In the case of a multimode laser, the intermode distance is determined by the laser cavity length = c/2Lcav and L = 2Lcavm. Such an interference of biphoton fields was observed experimentally [21]. The setup is shown in Fig. 9. A single LiIO3 crystal was pumped by multimode HeCd laser with the following parameters: = 150 MHz, N 20, Lcav = 98 cm. Collinear and frequency-degenerate biphoton field with type-I horizontal polarization was rotated by a halfwave plate QP1 by 45 deg and then was fed a Michelson interferometer with the optical path difference equal to the doubled length of the laser cavity (L = 196 cm). The interferometer consisted of a polarizing beamsplitter PBS1, two mirrors, and a couple of quarterwave plates QP2, 3 oriented at 45° and placed into the arms. Such a scheme has twice less losses compared with conventional (nonpolarizing) interferometer. The detection system was the same as used in the previous section experiment. It is necessary to emphasize that observation of 100%-visibility interference is possible only under additional condition: L cTR, where TR is the resolution time of a coincidence circuit. Otherwise, there would be a background coincidence level formed by biphoton pairs divided at the interferometer input which does not contribute to interference. This fact decreases the visibility to the classical limit 50% at L = 0.

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Figure 10 demonstrates the experimental dependence of visibility on the optical path difference L. While the first maximum at L = 0 corresponds to expected 50%, the second maximum at L = 196 cm is less than 100%. This is caused by imperfect spatial mode selection. The selection was realized by a pinhole P inserted into the beam just before the PBS2. The open circles correspond to the pinhole diameter being 200 µm while the two close circles correspond to the pinhole diameter being 100 µm In the second case the visibility in the second maximum achieves 96%. Solid and dashed curves are theoretical ones drawn with an account for all experimental factors. In fact, this interference could be also called as interference of "independent" biphotons generated by the pump at different time moments separated by 2Lcav /c periods. In some way, this effect is analogous to the interference of biphotons generated by different pump pulses which was observed recently [22]. 3.3. Biphoton Logic Interference of independent biphotons makes it possible to prepare biphotons belonging to a single space and frequency mode in an arbitrary polarization state. So far, this problem has been never posed experimentally. (However, polarization states and their transformations for non-collinear biphoton fields have been considered in [23, 24].) One can show that biphoton polarization states can be described in terms of a general approach based on the matrix representation of the SU(3) group. In the general case, the biphoton wave function can be considered as a superposition | c 1 |2 x, 0 y + c 2 |0 x, 2 y + c 3 |1 x, 1 y ,

transformations conserves the second-order polarization degree P2 = (c
2 1

c

22 2

* * ) + 2 c1 c3 + c3 c

2 2

.

(13)

It means that there is no possibility to transform completely polarized state of type-I |2x , 0y (when both signal and idler photons have the same polarization, c1 = 1, P2 = 1) into completely nonpolarized state of type-II |1x , 1y (when signal and idler photons have orthogonal polarizations, c3 = 1, P2 = 0). However, type-II biphoton field can be synthesized, using the setup shown in Fig. 6, from two beams of type-I biphotons emitted from two different macroscopic regions. In fact, this experiment is similar to the one presented in Section 3.1. In other words, two approaches are possible: the former utilizes interference terms and the latter utilizes polarization terms. Polarization approach seems to be more powerful in most cases but one should remember that polarization transformations are always due to quantum interference as classical polarization optics is based on the classical field interference. The MachZehnder interferometer in Fig. 7 prepares biphoton field in the coherent superposition | = (|2x , 0y + |0x , 2y)/ 2 with polarization vector e = (1/ 2 ){1, 1, 0}. The halfwave plate QP at the interferometer output transforms this state into the new one e' = (0, 0, 1). Such transformations are suggested as structural elements of the biphoton "ternary" logic [22]. In this case, instead of three basic states used in (12) it is more convinient to use another basis consisting of nonpolarized states as follows: | ( |2 x, 0 y + |0 x, 2 y ) / 2 = |+, ,
(+) () | ( |2 x, 0 y |0 x, 2 y ) / 2 = |+45°, 45 ° , (14)

c

2 i

= 1,

(12)

where |nx , my are basic states with a definite photon number (Fock states) in two polarization modes x, y and ck are complex amplitudes. In (12) the vacuum state |0x , 0y is omitted due to its insignificance in the description of polarization. Instead of the mode pair (x, y) one could use any other pair of orthogonal polarization modes (P1, P2). According to (12), polarization state of a biphoton field could be written as a three-dimensional complex vector e = {c1, c2, c3}. Taking into account the normalization rule and excluding the total wave function phase one can use four real parameters (two amplitudes and two phases) in the polarization state description. This consideration shows that one can transform given states into other ones without any losses using standard polarization elements (retardation plates, phase shifters etc.). The whole set of unitary polarization transformations belongs to SU(3) group and could be represented by 3 в 3 unitary unimodular matrixes {M}: e' = Me. Such

| |1, 1 = | x, y , where "+" and "" stand for right and left circular polarizations, and "± 45°" is for linear polarizations rotated by ± 45° with respect to the x-axes. It is obvious from Eq. (14) that our interferometric technique allows to prepare collinear biphoton field consisting of twin photons with arbitrary orthogonal polarizations |P1, P2. Recently, great interest has been attracted by twomode biphoton fields consisting of two polarization and two frequency (or spatial) modes. It was mainly caused by the possibility of entangled states generation which are appropriate in quantum information processes. In this case, the wave function has a general form ' ' | c 1 | x, x + c 2 | y, y ' ' + c 3 | x, y + c 4 | x , y , (15)

(0)

and instead of the basis (13) one could use four wellknown entangled Bell states (±), (±).
LASER PHYSICS Vol. 12 No. 5 2002

INTERFERENCE OF BIPHOTON FIELDS Visibility V 1.0 0.8 0.6 0.4 0.2 0 Rc per 15 s 1.0 0.8 0.6 0.4 0.2 0 80 60 40 20 0 20 40 60 80 , µm (b) (a)

9

sized pairs was performed using birefringent material with variable effective length (four KDP plates shown in Fig. 7 inside the dashed box were inserted in the experiment). Figure 11b demonstrates the dependence of coincidence rate versus the delay introduced. To obtain the undistorted dip a 40-nm interference filter IF was used which was twice as broad as the SPDC spectrum width 18 nm. Together with the anticorrelation "dip" shape, the absolute value of the first-order correlation function g(1)() for type-I SPDC was measured (Fig. 11b) [26]. Theoretical consideration gives a simple connection between the dip shape and the firstorder correlation function, Rc ( ) 1 g ( 2 ) ,
(1)

(16)

Fig. 11. (a) Module of the first-order correlation function g(1) and (b) the anticorrelation "dip" for "type-II" biphotons synthesized from type-I biphotons. Here Rc is the number of coincidences per 15 s, is the delay introduced between the signal and idler photons. Type-I biphotons were generated in a 15-mm crystal of LiIO3 from cw pump with wavelength 325 nm.

3.4. Measurements of the Synthesized Biphoton States The next step in our research is the measurement of the synthesized polarization states of biphoton field. To measure biphoton field means to get information about its spectrum and polarization state. For the first purpose, in our experiment we used well-known in quantum optics interference technique for anticorrelation "dip" measurement [25]. While in all experiments considered above a biphoton consists of signal and idler photons travelling without any delay between them, now one should introduce such a delay. When the delay exceeds the biphoton field coherence length determined by spectrum, an interference of "independent" biphotons vanishes. Applying this technique to synthesized type-II biphoton states, we verified our assumption that the spectral and correlation characteristics of synthesized state are defined by the spectrum of the initial type-I biphoton states. Variation of the delay between signal and idler photons of syntheLASER PHYSICS Vol. 12 No. 5 2002

i.e., the dip width is exactly twice smaller then g(1)() width. This fact clearly agrees with the results shown in Figs. 11a, 11b. Thus, the synthesized type-II biphoton states maintain the original spectrum of type-I SPDC while having polarization properties of type-II SPDC. This shows an essential difference between the spectral (and correlation) characteristics of the synthesized and natural type-II biphoton fields generated via SPDC. Full measurement of a biphoton polarization state includes measurement of both amplitudes and phases in Eq. (12) or Eq. (15). Such a measurement could be called polarization "tomography." In the classical optics, polarization tomography is related to the measurement of the Stokes parameters (in general approach to the matter was suggested by D.N. Klyshko in [27]). In the case of biphoton fields, polarization vector e contains both the second-order and forth-order properties. Hence, polarization tomography in this case is equivalent to polarization vector e measurement. Experimental realization of a "quantum tomograph" has not yet been realized and remains one of the important tasks of modern biphoton quantum optics. CONCLUSION The paper contains a brief review of the recent experiments carried out in the Laboratory of Spontaneous Parametric Scattering of M.V. Lomonosov Moscow State University. It joins the works in the area of both second-order and forth-order interference and polarization effects in biphoton fields. Possible applications in nonlinear spectroscopy were discussed and demonstrated. Also, a polarization description of single mode biphoton field is introduced. It should be mentioned that collinear degenerate or non-degenerate biphoton field seems to be most suitable in quantum communication and quantum cryptography and could find a wide application. This work was supported in part by the Russian Foundation for Basic Research, projects nos. 99-0216419 and 00-15-96541.

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REFERENCES
1. Ghosh, R. and Mandel, L., 1987, Phys. Rev. Lett., 59, 1903. 2. Ou, Z.Y, Wang, L.J., Zou, X.Y., and Mandel, L., 1990, Phys. Rev. A, 41, 566. 3. Herzog, T.J. et al., 1994, Phys. Rev. Lett., 72, 629. 4. Aktsipetrov, O.A. et al., 1979, Solid State Phys., 21, 1833. 5. Franson, J.D., 1991, Phys. Rev. A, 44, 4552. 6. Kwiat, P.G., Steinberg, A.M., and Chiao, R.Y., 1993, Phys. Rev. A, 47, 2472. 7. Brendel, J., Mohler, E., and Martienssen, W., 1991, Phys. Rev. Lett., 66, 1142. 8. Klyshko, D.N., 1993, JETP, 77, 222. 9. Zou, X.Y., Wang, L.J., and Mandel, L., 1991, Phys. Rev. Lett., 67, 318. 10. Burlakov, A.V. et al., 1997, Phys. Rev. A, 56, 3214. 11. Fonseca, E.J.S., Monken, C.H., and Padua, S., 1999, Phys. Rev. Lett., 82, 2868. 12. Kim, Y.-H. et al., 2000, Phys. Rev. Lett., 84, 1.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

White, A.G. et al., 1999, Phys. Rev. Lett., 83, 3103. Mamaeva, Yu.B. et al., 2001, JETP, 120, 67. Burlakov, A.V. et al., 1997, JETP Lett., 65, 19. Burlakov, A.V. et al., 1998, JETP, 113, 1991. Korystov, D.Yu., Kulik, S.P., and Penin, A.N., 2000, Quantum Electron., 30, 10. Korystov, D.Yu., Kulik, S.P., and Penin, A.N., 2001, JETP Lett., 73, 248. Burlakov, A.V. et al., 1999, Phys. Rev. A, 60, R4209. Burlakov, A.V. et al., 1999, JETP Lett., 69, 788. Burlakov, A.V. et al., 2001, Phys. Rev. A, 63, 053801. Kim, Y.H. et al., 1999, Phys. Rev. A, 60, R37. Mattle, K. et al., 1996, Phys. Rev. Lett., 76, 4656. Cerf, N.J., Adami, C., and Kwiat, P.G., 1998, Phys. Rev. A, 57, 1477. Hong, C.K., Ou, Z.Y., and Mandel, L., 1987, Phys. Rev. Lett., 59, 2044. Burlakov, A.V. et al., 2001, Phys. Rev. A, 64, 041803. Klyshko, D.N., 1997, JETP, 84, 1065.

SPELL: uncorrelated, polarizer, polariton, losslessly

LASER PHYSICS

Vol. 12

No. 5

2002