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ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2009, Vol. 108, No. 1, pp. 33­42. © Pleiades Publishing, Inc., 2009. Original Russian Text © D.A. Kalashnikov, V.P. Karasev, K.G. Katamadze, S.P. Kulik, A.A. Solov'ev, 2009, published in Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, 2009, Vol. 135, No. 1, pp. 40­50.

ATOMS, MOLECULES, OPTICS

Generation of Arbitrary Frequency-Entangled States of Two-Photon Light
D. A. Kalashnikova, V. P. Karasevb, K. G. Katamadzec, S. P. Kulikc, *, and A. A. Solov'evc
a

Zavoiskii Physical Technical Institute, Russian Academy of Sciences, Kazan, 420029 Tatarstan, Russia b Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119991 Russia c Moscow State University, Moscow, 119899 Russia *e-mail: sergei.kulik@gmail.com
Received July 13, 2008

Abstract--We consider a new method for the generation of polarization-frequency entangled states of photon pairs. We use a frequency-nondegenerate regime of spontaneous parametric down conversion where the photon pairs (biphotons) are produced with identical polarizations, propagate mostly in the same direction, but differ in frequency. Entanglement is achieved by a coherent superposition of pairs emitted from two nonlinear crystals, with the polarization of the biphotons from the first crystal being changed by a transformer placed between the two crystals. We show that this scheme allows the degree of entanglement to be controlled by the choice of biphoton frequencies. PACS numbers: 03.67.Bg, 42.50.Dv, 42.50.Ex DOI: 10.1134/S1063776109010051

1. INTRODUCTION Pairs of correlated photons generated in the process of spontaneous parametric down conversion (SPDC) are currently the most convenient tool in various problems in quantum optics, quantum communication, and quantum information. Strong correlations that have no quantitative analog in the classical description of an electromagnetic field are usually termed as "entanglement" in quantum information and quantum communication. In the case of pure states, the entangled states are those for which the wave function of the system is not factorized, | 12 | 1 | 2 . (1)

As a rule, the so-called polarization­spatial entanglement, where the subsystems differ in photon propagation direction (1 and 2) and are characterized by polarization degrees of freedom, occurs in experiment. In this case, the expression for the wave function takes one of the following forms: | 12 = c |H 1 |V 2 + d |V 1 |H 2 , | 12 = c |H 1 |H 2 + d |V 1 |V 2 , (3a) (3b)

The wave function of the composite system is on the left-hand side of Eq. (1), while the direct product of the wave functions for subsystems 1 and 2 are on the righthand side. The biphoton wave function is a linear superposition of the contributions from the various spatial­ frequency field modes involved in SPDC, 1 | = |0 + -2

depending on the chosen type of phase matching and the configuration of nonlinear ((2) 0) crystals. The polarization states on the right-hand sides of Eqs. (3) are generated by a single action of the production operators in the polarization modes H1, 2 and V1, 2 on the vacuum state, for example, |H1 = a 1 |0, |V2 = b 2 |0, etc.


= H , V k s, k


i

|1

, k

s

,1

, k

i

,

(2)

and satisfies condition (1). The ket vector |1 , ks, 1 , ki on the right-hand side of Eq. (2) denotes a two-photon Fock state for which there is one photon in each of ks and ki modes, the index is responsible for the polarization, and the term |0 ( 1) points to the dominant contribution from the vacuum component.
33

There are several methods for producing states (3). When frequency-degenerate type II phase matching is used (the signal and idler photons have orthogonal polarizations), the crystal orientation is chosen in such a way that correlated photons of both polarizations are emitted in certain directions. Thus, for example, a collinear regime of SPDC followed by the separation of spatial modes by a beamsplitter was proposed in [1], and a noncollinear regime of SPDC was proposed in [2]. Type I phase matching, where the photons have identical polarizations, allows states (3b) to be generated if two orthogonally oriented identical crystals are used [3]. Burlakov et al. [4] suggested placing two crystals in different Mach­Zehnder interferometer arms,


34

KALASHNIKOV et al.

which is physically equivalent to the scheme in [3]. A scheme was considered in [5] in which a pump beam passes twice through one crystal in a frequency-degenerate noncollinear regime of SPDC. States (3b) were produced by the superposition of the spatial biphoton field modes generated on different passages using a mirror. The polarization of the biphotons generated on the first passage was rotated through 90° by a successive transformation: /4 plate­mirror­/4 plate. Subsequently, the mentioned schemes have been repeatedly modified and used in a number of quantum information and quantum communication protocols as well as for generating new classes of two-photon entangled states, such as an analog of "scalar light" [6], and cluster and "hyperentangled" states [7]. The most general wave function of a pair of photons that describes their polarization state is the so-called ququart, or vector state, that belongs to the four-dimensional Hilbert space: | = c 1 |1 + c 2 |2 + c 3 |3 + c 4 |4 = c 1 |H 1 H 2 + c 2 |H 1 V 2 + c 3 |V 1 H 2 + c 4 |V 1 V 2 . (4)

entanglement. According to [18], the latter can be defined via the coefficients c2 and c3 (c1 = c4 = 0): 1 c 2 = -------------------- , 2 1+ c 3 = -------------------- . 2 1+ (5)

The state is maximally entangled if = 1 and is factorized if = 0. In the more general (nondegenerate) case where all coefficients ci in decomposition (4) are nonzero, the quantity C introduced by Wooters [19, 20] is a convenient measure of entanglement: C = 2 c2 c3 ­ c1 c4 , 0 C 1. (6)

The indices of the polarization states are responsible for the external degrees of freedom of the field--the frequencies or directions of propagation. Obviously, states (3) are a special case of (4). The frequency-nondegenerate regime of SPDC makes it possible to produce the so-called frequencypolarization entangled states. This regime is convenient because the correlated photons propagate in the same direction. Consequently, entanglement can be transferred from one spatial point to another through one channel [8, 9], which is required in quantum communication, for example, during quantum teleportation [10]. In this case, the form of states (3) formally does not change: indices "1" and "2" now refer to different frequency modes. The polarization properties of spatial­ temporal multimode quantum light fields were consistently described by Karasev [11, 12]. Based on frequency-nondegenerate biphotons, the authors of [13­ 17] investigated ququarts and their possible applications in quantum communication. In recent years, great attention has been focused on achieving full control over the quantum state (4)1. In fact, the question has arisen of producing a universal source of photon pairs in which the complex amplitudes ci would be controlled by an experimenter, depending on the problem to be solved. For example, for the source of entangled states (3a) of photon pairs, the problem is reduced to control over the degree of
1

In this paper, we consider a new source of frequencynondegenerate entangled photon pairs. Control over the degree of entanglement (6) is exercised by controlling the biphoton field spectrum. This can be achieved by several methods, depending on the chosen parameters of the problem. One of the stimuli for the conducted experiments was a recent report on the observation of a high degree of entanglement of the photon pairs generated in polydomain potassium dihydrophosphate crystals [21]. The model considered here probably provides a basis for explaining several peculiarities of the effect, primarily the spectral dependence of the degree of entanglement. 2. THE IDEA OF THE METHOD Let a laser pump beam (kp, p) pass through two identically oriented crystals of thickness L. Let us place a plate WP of anisotropic material, for example, crystalline quartz, oriented at an angle of 45° to the plane containing the optical axes of the crystals in the gap between the crystals (Fig. 1). The crystals are oriented in such a way that SPDC is excited in each of them in a collinear frequency-nondegenerate regime and the biphoton state is, for example, |H1H2 (the subscripts refer to the central frequencies in the spectrum of the signal and idler emissions). However, the biphotons generated in the first crystal undergo a polarization SU(2) transformation in the plate: t r t r ^^ ^ G G1 G2 = 1 1 2 2 . ­r* t* ­r* t* 11 22 (7)

The coefficients t1, 2 and r1, 2 are defined by the optical thickness = cos r
1, 2 1, 2

= (n

1, 2 o

­n

1, 2 e

)l/

1, 2

and orientation of the plate: t
1, 2 1, 2

+ i sin
1, 2

1, 2

cos 2
= 45°

= 45°

cos
1, 2

1, 2

, (8a) (8b)

Such problems are usually combined under the term "quantum state engineering."

= i sin

sin 2

i sin

.

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No. 1

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GENERATION OF ARBITRARY FREQUENCY-ENTANGLED STATES WP NC1 Laser radiation NC2

35

| H 1 H 2

^ G | H 1 H 2

2 ^ 1 ­ a |H 1 H 2 + aG |H 1 H 2

Fig. 1. Scheme illustrating the idea of the method for generating polarization-frequency entangled two-photon states. NC1 and NC2 are two identical nonlinear crystals, WP is a polarization transformer.

The state remains factorized after the plate: its polarization representation |P(1)P(2). At the exit from the system of two polarization state of the two-photon field | =
2

transforming is ph = crystals, the is (9)

1 ­ a |H 1 H 2 + a ph .

The complex coefficient a results from the polarization ^ transformation G 1 that the plate makes on the classical pump field. Its inclusion in the form of a normalization factor is needed, because only the vertically polarized pump component contributes to SPDC. State (9) can be expressed in terms of nondegenerate decomposition (4) or a ququart, and it is generally entangled. Indeed, it is easy to see that for the wavelengths 1 and 2 at which the plate is a half-wave one, i.e., 1 = 2 = -- , 2 the final state is | = 1 ­ a | H 1 H 2 + a |V 1 V 2 ,
2

(10)

formed. If, however, at least one of the conditions (10) and (12a) is violated, then the two-photon state is a nonmaximally entangled one with an intermediate (between zero and one) value of the measures , C, etc.2 For a plate with a fixed thickness l, condition (12b) uniquely specifies the coefficient a in Eqs. (9) and (11). Thus, the optical thickness (1, 2), which depends on the wavelength explicitly, remains the only free parameter that defines the form of (9). In addition, a weak implicit dependence manifests itself through the bire1, 2 1, 2 fringence dispersion n o ­ n e = n(1, 2). It is convenient to consider the polarization transformation made by the plate in terms of its "polarization spectrum." Operationally, such spectra would be obtained if a broadband light beam were passed through the following system: vertical polarizer­plate oriented at 45° to the vertical line­vertical polarizer. In this case, as follows from (8), the wavelength dependence of the intensity of the transmitted light would be I
1, 2

= ( cos

1, 2

)

2

(13)

(11)

and the state becomes maximally entangled when the condition 1 a = exp { i } -----2 (12a)

with a characteristic period of P 2/nl. Consequently, two spectral parameters emerge in the problem: the biphoton spectrum width biphoton and P . biphoton in a nondegenerate regime is known [22] to depend on the dispersion of the signal and idler photon group velocities, I
biphoton

is satisfied. Condition (12a) is achieved by choosing a plate thickness of p = ( no ­ ne ) l / p = m ,
p p

sin ( DL 2 ------------------------) , DL

where L is the crystal length, p p = ----- ­ 1 = 2 ­ ----- , 2 2 k1 k2 11 D = ---- ­ ---- = -------- ­ -------- , u1 u2 1 2
2

m = 1, 2, ...

(12b)

at which it becomes a wave one at the pump wavelength. Thus, a maximally entangled state is produced in the scheme under consideration when, first, the polarization of all spectral components of the biphotons generated in the first crystal is transformed into an orthogonal one and, second, an equilibrium superposition of the contributions from |H1H2 and |V1V2 is

There exist several other measures of entanglement, such as entropy, the Schmidt number, and the Fedorov parameter, but here we will mainly consider parameter C. Vol. 108 No. 1 2009

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36 I, rel. units 1.0

KALASHNIKOV et al. I, rel. units (a) 0.8 0.6 0.4 0.4 0.2 0 6800 1.2 1.0 0.8 0.6 0.6 0.4 0.4 0.2 0 6300 6500 6700 6900 , å 7100 7300 7500 0.2 0 6300 0.2 6900 7000 7100 7200 (c) 7300 0 6300 1.0 0.8 6500 6700 6900 7100 7300 (d) 7500 1.2 1.0 0.8 0.6 (b)

6500

6700

6900 , å

7100

7300

7500

Fig. 2. Spectral dependences of the intensity illustrating the main regimes investigated here. (a) The spectrum of a frequency-quasinondegenerate biphoton field; (b­d) the polarization spectra of quartz (crystalline) plates for l = 0.820 (b), 3.725 (c), and 10.060 mm (d).

so that biphoton 22/cDL. The relationship between biphoton and P defines the observed peculiarities of the polarization effects in the scheme under consideration. (1) biphoton P . All spectral components of the biphotons undergo identical polarization transformations--the case of the so-called single-mode transformations typical of quantum optics. The degree of entanglement can be varied as the crystal orientation changes through the tuning of the central frequencies in the biphoton spectrum. P . This corresponds to the so(2) biphoton called multimode polarization transformation. In this case, at a high spectral resolution of the recording equipment, the ququart biphoton polarization state can change when the frequency is scanned within biphoton and when the crystal orientation is fixed. (3) biphoton P . This is an intermediate case. We have experimentally implemented and investigated all three regimes. The three listed cases are illus-

trated in Fig. 2, which shows the measured spectral distributions of the biphoton field and the polarization spectra of the quartz plates used. 3. EXPERIMENT The experimental setup is schematically shown in Fig. 3. We used two lithium iodate (LiIO3) crystals, each 10 mm long, cut for frequency-degenerate collinear type I phase matching. The central frequencies 1 and 2 (1 + 2 = p) in the spectrum of the signal and idler radiations were chosen by slightly tilting the crystal with respect to the pump radiation direction. The crystals were arranged in such a way that their optical axes were antiparallel (see below). The radiation from a continuous-wave argon laser at 351 nm was directed into the optical channel of the setup by a prism (P) and a pair of mirrors (M1 and M2). A Glan­Thompson prism (V) separated the vertical pump polarization component in such a way that the same polarization state |H1H2 was generated in both crystals. Quartz
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GENERATION OF ARBITRARY FREQUENCY-ENTANGLED STATES

37

FD1

P Ar D M1 V D M2 2 LiIO3 WP LiIO3 UVM LiIO3 TP1 TP2 H D 1

ISP-51

D

CC

FM

CF D
FD2

Fig. 3. Scheme of the experimental setup. Ar is an argon laser operating at 351 nm; P is a prism; D are diaphragms; M1 and M2 are mirrors; V is a Glan­Thompson prism that separates the vertical pump polarization component; LiIO3 is a lithium iodate crystal; WP is a quartz plate; UVM is an ultraviolet mirror; TP1 and TP2 are tomographic crystalline quartz plates 537 and 439 µm in thickness, respectively; H is a Glan­Thompson prism that separates the horizontal biphoton radiation component; FM is a semitransparent mirror; CF is a KS-series cutoff filter; ISP-51 is a spectrograph; FD1 and FD2 are avalanche photodiodes; and CC is a coincidence scheme.

plates (WP) of different thicknesses cut parallel to the optical axis and rotated through = 45° relative to the vertical line were placed between the crystals. The plates made a polarization transformation of the states of the biphotons generated in the first crystal. An ultraviolet mirror (UVM) that cut off the pump radiation was located after the second crystal. An additional LiIO3 crystal of the same length as the first two crystals was placed next. It served to compensate for the spatial displacement of the biphoton beam (see below). To reconstruct the density matrix, we used the root method of statistical quantum state reconstruction suggested by Bogdanov [23, 24] and tested previously on model states of ququart biphotons [13, 15]. In this case, the unknown state emerging from the sample is subjected to a set of unitary polarization transformations and projective measurements. The specified polarization transformations were carried out using a pair of plates with a known thickness cut from crystalline quartz (TP1 and TP2). The transformed state was then projected onto the horizontally polarized state using a Glan­Thompson prism (H). Subsequently, the signal and idler photons were separated by a semitransparent mirror (FM) and directed into two channels of the Brown­Twiss scheme. The recording system was assembled from photon-counting avalanche photodiodes FD1 and FD2. The measuring scheme records the number of detector counts R1,2 in a fixed time T proportional to the average intensity of the horizontally polarized light, R1 I1 and R2 I2, and the intensity correlator proportional to the number of photocount coincidences, Rc I1I2. The unknown state emerging from the sample can be reconstructed by measuring Rc for various plate positions. In this case, the number of states specified by the TP1 and TP2 orientations should be at least 2K ­ 2,

where K = 4 is the dimension of the Hilbert space for the system of two polarization qubits representing the biphoton. To increase the accuracy of statistical state reconstruction, it is more preferable to use an excessive number of measurements, i.e., a larger number of intermediate states. The crystalline quartz plates were cut in the plane containing the optical axis. The thicknesses of the first (TP1) and second (TP2) plates were 537 and 439 µm, respectively. The plate rotation angles were chosen to provide excessive statistics for a more accurate state reconstruction. Thus, for example, the rotation angles for TP1 were = 0, 15°, 30°, and 45°, while TP2 made a complete turn, 0 360°, every 10°. Thus, the series of measurements included 148 various combinations of TP1 and TP2 orientations. In channel (1), a frequency selection of the specified spectral emission mode of SPDC was made with an ISP-51 spectrograph and the emission was then recorded by the avalanche photodiode FD1. In channel (2), only a broadband KS-series filter CF was located in front of the avalanche photodiode FD2 and, thus, the detector recorded a broad frequency spectrum.3 The pulses from the detectors were fed to a discriminator. The signal recording system could perform scanning based on a delay with a 1-ns step of the pulses coming from both detectors using an electronic delay line through which the pulses were fed to the coincidence
3

The choice of this regime of measurement was justified by the fact that the frequency could be tuned only in one channel (containing the spectrograph). Only the frequency-coupled modes 1 + 2 = p const contributed to the photocount coincidences, which was ensured automatically via broadband recording in the second channel. Vol. 108 No. 1 2009

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38 LiIO3 ' S k z

KALASHNIKOV et al. WP LiIO3 UVM LiIO3

z

|HH | ph = |P ( 1 ) P ( 2 )

| =

1 ­ a | H 1 H 2 + a

2

ph

Fig. 4. Scheme illustrating the spatial displacement of the beams in SPDC and its compensation. LiIO3 are nonlinear crystals; the first two crystals serve to generate horizontally polarized biphotons and the last crystal serves to compensate for the displacement; WP is a quartz plate that makes polarization transformations on the biphotons and pump radiation; UVM is a filter that cuts off the radiation at the pump wavelength. and ' are the angles between the vector k, the Umov­Poynting vector S, and the crystal optical axis z. The dashed curves indicate the ray path during the displacement in the second crystal and its compensation using the third crystal. The sections of the spatial regions occupied by the biphotons and the pump in the plane perpendicular to the figure are shown in the lower part of the figure: the horizontal and dotted vertical lines correspond to the horizontal (initial) and vertical biphoton polarizations; the circles indicate the sections of the pump beam.

scheme CC with a time resolution of about 5 ns. The number of pulses in time T was recorded by the counters. Subsequently, the recorded events were analyzed statistically. 4. COMPENSATION FOR THE SPATIAL DISPLACEMENT OF THE BEAMS When analyzing the schemes in which entangled states are prepared, taking into account the overlap between the spatial­temporal contributions to the wave function plays an important role. The loss of control over the overlap between the components causes an effective increase in the number of degrees of freedom and we then go outside the scope of the chosen Hilbert space dimension. In an experiment, this leads to an averaging of the observed state over the "redundant" degrees of freedom and manifests itself in measurements as a mixed state. Thus, for example, the distinguishability of a photon pair by any parameter, except the polarization one, leads to the transition from a qutrit to a ququart through the removal of degeneracy in indices "1" and "2": c '1 |HH + c '2 |HV + c '2 |VV c 1 |H 1 H 2 + c 1 |H 1 V 2 + c 1 |V 1 H 2 + c 2 |V 1 V 2 . (14)

an extraordinary wave for a LiIO3 crystal, the front propagation direction kp does not coincide with the energy propagation vector Sp (Fig. 4). If the crystal optical axis makes an angle with the wave vector and an angle ' with the Poynting vector, then the relation between these angles is given by the formula [25] no tan ' = ---- tan . 2 ne
2

(15)

Another example is the necessity of introducing compensators of the group delays during the generation of a Bell state from two crystals with type I phase matching and pulse pumping [8] or compensators of the lateral displacement when a type II crystal is used [2, 9]. Let us consider the generation and propagation of biphotons through a sequence of anisotropic crystals, both linear (quartz plates) and nonlinear (LiIO3 crystals) ones. Since the vertically polarized pump beam is

Biphotons with an ordinary polarization are generated in the entire pump beam propagation path in the crystal and S1, 2 || kp in view of the phase matching conditions. Thus, the spatial region in which the biphoton field at the exit from the first crystal is localized is determined both by the cross-sectional sizes of the pump beam and by the displacement angle (15). In the transforming phase plate, the polarization of the biphoton emission changes and both vertical and horizontal components generally arise. The pump beam is deflected in the opposite direction as it propagates through the second crystal, since the second crystal is oriented in such a way that its axis is antiparallel to the axis of the first crystal. The biphoton generation process is similar to that in the first crystal. Since the biphotons generated in the first crystal also pass through the second crystal, the horizontally polarized component of the beam does not change, while the vertically polarized component is deflected in the same way as the pump beam. Thus, after the two crystals, there is a region of overlap between the biphoton fields emitted from the first and second crystals. To ensure maximum overlap between these regions, we place the third crystal of the same length as the two previous crystals and oriented in the same way as the first one after the UV mirror. In this case, the two spatial regions occupied by the biphotons
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GENERATION OF ARBITRARY FREQUENCY-ENTANGLED STATES Calculated F for various wavelengths for a plate l = 3.725 mm in thickness Wavelength 1, nm F without compensation F with compensation 711.0 0.86 0.95 714.5 0.81 0.93 715.1 0.68 0.78 723.7 0.93 0.99 724.5 0.89 0.98

39

emitted from the first and second crystals will coincide. Consequently, there is no need to separate the overlapping part by diaphragms and the entangled state will be generated most efficiently. Note that in a well-known scheme for the generation of entangled states, using two orthogonally oriented crystals with type I phase matching [3] does not solve the problem of the lateral displacement. Placing additional compensators, we can only symmetrize the configuration of the two spatial regions [9] but not bring them into coincidence! 5. DISCUSSION The criterion for comparing the experiments with and without the compensation scheme was the F (fidelity) parameter, which for pure states takes the form F
pure

= th| exp .
2

(16)

As we see from the table, the F parameter for the experiments with the third (compensational) crystal is larger, suggesting that the theoretical and experimental (reconstructed) states coincide more closely. Indeed, as was noted above, incomplete compensation for the spatial displacement of the beams manifests itself as a mixture of the sought-for state with additional states, while the reconstruction procedure was performed by assuming the states to be pure. To estimate the wavelength dependence of the degree of entanglement, the statistical state reconstruction procedure was performed in different parts of the spectrum both when the crystal orientation changed (conditions 1 and 3 in Section 2) and for a fixed orientation (condition 2). The wavelengths were chosen so as to cover one or more theoretically calculated periods of the variation in the degree of entanglement. As follows from Fig. 2, the thicker the transforming plate WP, the shorter the period of the variation in the degree of entanglement as a function of the wavelength C(). Figures 5a and 5b plot the variations in C when there are plates 820 and 3725 µm in thickness, respectively, between the crystals. For such (relatively thin) plates, the variation in C is possible only when the crystal orientation changes. Obviously, this is not very convenient from an experimental viewpoint, since the phase matching condition for the chosen spectral range is highly sensitive to the orientation and good reproducibility of the results is difficult to achieve. Using a thicker plate, we can vary the degree of entanglement in a narrower wavelength range, since the

polarization properties of the plate change rapidly with wavelength (Fig. 2d). This makes it possible to control parameter C by choosing the biphoton emission wavelength for a fixed crystal orientation (postselection). This requires choosing the optical thickness of the plate in such a way that the period of the variation in C is much smaller than the SPDC spectrum width. For this purpose, in our experiment, we used a quartz plate 10.06 mm in thickness with a characteristic period of the variation in the polarization spectrum of about 5 nm (Fig. 2d). This situation corresponds to condition 2 (Section 2). In this case, a high spectral resolution is required to accurately separate the frequency modes and it is necessary to make a spatial selection of the emission. However, a significant difficulty arises in calculating polarization transformations, since the result depends strongly on the optical thickness of the transformer: both the geometric thickness of the plate and its refractive index should be known. For example, an error in the fifth decimal place of the refractive index leads to a phase shift of /2 for a thickness l 10 mm. At the same time, the geometric thickness of the plate should be measured with an accuracy as high as 10-5 cm. This presents an experimental problem--only the interferometric methods, which, in turn, require knowledge of the refractive indices, provide such accuracy. In the long run, the accuracy of determining the optical thickness of the quartz plates was the main factor in comparing the experimental and theoretical spectral dependences. The gray-shaded regions in Fig. 5 show what the calculation gives when the transformer parameters are varied within the indicated limits and when the uncertainty in setting the plate rotation angle up to 1% is taken into account. In fact, these regions were outside the experimental possibilities of controlling the setting parameters. Their area was determined both by the uncertainty in the specified parameters (the plate orientation and optical thickness) and by the instability of the statistical state reconstruction procedure to these parameters [26]. Thus, for example, the relatively large area in Fig. 5b probably stems from the fact that the l = 3.725 mm plate at all pairs of coupled wavelengths is close to a wave one and the eigenvalue spectrum for the instrumental matrix in the state reconstruction procedure becomes degenerate. Note the discrepancy between the experimental and theoretical values for small C. We associate this fact with an inadequate correspondence of the measure C to the "amount of entanglement" near its extreme values (0 and 1). For example, suppose that in
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40 C 1.0 0.8

KALASHNIKOV et al. C = 2c (a) 1.0 0.8 0.6 0.4 0.2 0 710 1.0 0.8 0.6 0.4 0.2 0 710 1.0 0.8 0.6 0.4 0.2 0 708 0.6 0.4 0.2 0 720 730 (b) 740 750 0 0.2 0.4 0.6 0.8 1.0 c4
4

1 ­ ( c4 )

2

Fig. 6. Entanglement parameter C versus real amplitude c4 at c2 = c3 = 0. The region of a sharp change in C near large amplitudes c4 1 is hatched.

715

720 (c)

725

730

710

712

714

716 , nm

Fig. 5. Wavelength dependences of the entanglement parameter C for various optical thicknesses of the transforming plate: (a) l = 820 µm (condition 1: biphoton P), (b) l = 3725 µm (condition 3: biphoton P), and P). The (c) l = 10 060 µm (condition 2: biphoton regions uncontrollable in the experiment are highlighted in gray.

Eq. (4) c2 = c3 = 0, while c1 and c4 are real quantities. The normalization condition gives one independent amplitude, c 4 = 1 ­ c 1 , and the term c1c4 appearing in definition (6) changes sharply near zero at c4 1
2 2

(Fig. 6). Thus, if one of the amplitudes c1 and c4 is much larger than the other,4 then an error in determining this component will lead to a significant spread in C at low values (i.e., ~10-1). The instability of the laser power, which fluctuated over the exposure time, served as an additional source of errors leading to an increase in the statistical scatter of the data shown in Fig. 5. Each point in Fig. 5 represents the result of statistical state reconstruction based on 148 measurements (see above). The total time spent on complete measurement for a pair of fixed wavelengths was about 50 min, while the characteristic time of stable laser operation was about 20 min. At the same time, it took several hours of continuous operation of the setup to measure the complete spectral dependence. Over this time, all fluctuations of the pump power and the temperatures of the crystals and quartz plates were averaged, which led to an increase in statistical errors. The choice of such an exposure time was dictated by the compromise between the total optimal number of recorded events, photocount coincidences (~104) [13, 24], and the characteristic time scale of the instrumental fluctuations of the setup (20 min). Note that SPDC from two nonlinear crystals with (anti)parallel optical axes should be accompanied by second-order (in field) interference [27­29]. This effect manifests itself as spatial and frequency modulations of the biphoton field intensity R1,2. In some cases, this is regarded as an obstacle in analyzing the fourth-order moments, since it gives additional modulation of the photocount coincidences Rc. However, in our case, the entire wave function (4) was reconstructed and the
4

This corresponds to the dominant contribution from one polarization component (|V1V2) in decomposition (4) and leads to a nearly factorized state. Vol. 108 No. 1 2009

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41

interference of the second- and fourth-order moments could be analyzed independently. Note only that the second-order interference clearly showed up in the spectra obtained, in accordance with the conditions formulated earlier in [27­29]. 6. CONCLUSIONS We would like to note that the effect investigated here is probably the same in physical nature as the observed "anomalies" of the degree of entanglement in polydomain potassium dihydrophosphate crystals found in [21]--parameter C reached 0.8! It was known from symmetry considerations that biphotons with identical polarizations could be generated in each domain--the case considered here. During their subsequent propagation, the biphotons underwent polarization transformations and a coherent superposition of two-photon fields with different polarization states arose at the exit from the sample. Because of dispersion and complex spatial structure of the sample, the polarization transformations depend strongly on the wavelength and, as a result, the degree of polarization should depend on the relationship between the spectral composition of the biphotons and the characteristic "spectral scale" of the inhomogeneous medium. Such experiments have not yet been carried out--the results of [21] were limited to measuring the parameter C at a pair of fixed (coupled) wavelengths. At the same time, a strong spectral dependence of the polarization properties of polydomain ferroelectrics was noted in a number of experimental works in which both elastic [30] and inelastic [31] light scattering was investigated. It is clear from our results that the model of polarization transformations in a (strongly) dispersing medium should form the basis for theoretical calculations of the propagation or generation of a two-photon field in spatially inhomogeneous media, including natural polydomain ferroelectrics ((2) 0) [29, 31, 32] and optical fibers ((3) 0) [33, 34]. ACKNOWLEDGMENTS We thank E.V. Moreva, A.N. Penin, and P.A. Prudkovskii for discussion of the results. This work was supported by the Russian Foundation for Basic Research (project nos. 08-02-00741-a, 06-02-39015-GFEN-a, and 06-02-16769-a) and the Program for Support of Leading Scientific Schools (grant no. NSh-796.2008.2). The experimental results were obtained at Moscow State University thanks to the support to D.A.K by the Russian Foundation for Basic Research (project no. 07-0290800-mob_st).

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Translated by V. Astakhov

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2009