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ISSN 0021-3640, JETP Letters, 2008, Vol. 87, No. 1, pp. 60­65. © Pleiades Publishing, Ltd., 2008. Original Russian Text © D.A. Kalashnikov, V.P. Karasev. S.P. Kulik, A.A. Solov'ev, G.O. Rytikov, 2008, published in Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, 2008, Vol. 87, No. 1, pp. 66­71.

Generation of Entangled States in Polydomain Potassium-Dihydrophosphate Crystals
D. A. Kalashnikova, V. P. Karasevb, S. P. Kulikc, A. A. Solov'evc, and G. O. Rytikovd
b

Zavoiskii Physicotechnical Institute, Kazan Scientific Center, Russian Academy of Sciences, Sibirskii trakt 10/7, Kazan 29, 420029 Russia Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 119992 Russia c Faculty of Physics, Moscow State University, Moscow, 119992 Russia d Moscow State University of Press, Moscow, 127550 Russia
Received November 27, 2007

a

It is shown experimentally that polarized entangled two-photon states in a pair of frequency modes can appear in the process of the spontaneous parametric down conversion in ferroelectric crystals with a quasi-regular domain structure (such as potassium dihydrophosphate). Entanglement measures and polarization characteristics of these states are determined. PACS numbers: 03.65.Wj, 03.67.Mn, 77.80.Dj DOI: 10.1134/S0021364008010141

1. INTRODUCTION Entangled states play important role in the theory of quantum information [1]. If a composite system is in a pure state and its wavefunction is not expressed in terms of the product of the wavefunctions of the subsystems, this state is called entangled.1 Entangled states underlie several protocols of quantum information [2, 3] and quantum key distribution [4, 5] and they are considered as tools for quantum calculations [6]. The polarization states of two qubits are the most studied. Such states are obtained in quantum optical experiments due primarily to the spontaneous parametric down conversion [7]. The aim of this work is to investigate the polarization states of two photons obtained in polydomain crystals with a certain configuration of the domain structure. The positive answer to the main question of whether the state at the output from a sample can be entangled even if a factorized state is produced in each section of the sample is obtained. To analyze the polarization state at the output from the sample, we consider two types of possible experiments. (i) The input to the polydomain sample is irradiated with laser radiation in a certain frequency­angular mode and with a certain polarization. Biphotons with known spectral, angular, and polarization properties governed by the symmetry of individual domains are produced in the bulk of the sample due to the secondorder susceptibility ((2) 0). The two-photon field in a certain frequency­angular mode is a linear superposi1

tion of the polarization states of biphotons that are produced in each domain layer and are transformed in subsequent domains. (ii) The sample operates as a linear polarization transformer. In this case, a biphoton field with given spectral, angular, and polarization properties is fed to the input. The polarization state of biphotons changes in the process of the passage through the polydomain sample with change in the configuration of the domain structure and in the input parameters of biphotons (spectral content, angular structure, and polarization state). In what follows, we consider the so-called collinear regime of spontaneous parametric down conversion when pairs of photons propagate predominantly in the same direction as a classic pump wave. Note that the frequencies of the photons in a pair can be noticeably different, because an electromagnetic field belonging to one spatial mode is not monochromatic. In this case, the usual two-mode polarization representation of the electromagnetic field should generally be refined. This refinement is appropriately performed in the framework of the P-quasispin concept [8, 9]. In the Fock representation, an arbitrary polarization two-photon state involved in spontaneous parametric down conversion is represented as a decomposition into four basis states |Hs, i and |Vs, i : 4 = c 1 |H s |H i + c 2 |H s |V i + c 3 |V s |H i + c 4 |V s |V i (1a) = ( c 1 a s a i + c 2 a s b i + c 3 b s a i + c 4 b s b i ) |0, 0 . Symbols |H and |V denote the single-photon states in the horizontal and vertical polarization modes, respec60

In this work, we consider only pure entangled states.


GENERATION OF ENTANGLED STATES

61

tively; subscripts s and i refer to the signal and idler fre quency modes, respectively; and a s, i and b s, i are the photon creation operators in the horizontal and vertical polarization modes, respectively. The terms corresponding to the presence of two photons in one frequency mode are omitted in Eq. (1a), because such states cannot be realized at the stationary regime of spontaneous parametric down conversion owing to the energy conservation law, p = s + i, s i. It was shown in [8, 9] that the wavefunction in the P-quasispin parameterization has the form bs ai + as bi ++ ++ | P = d 1 a s a i + d 2 b s b i + d 3 --------------------------2
++ ++

which is the fraction of the content of singlet (P scalar) states |P = 0; m = 0 in representation (1b); and (ii) partial (i.e., defined separately for the signal and idler frequency modes) SU(2)p-invariant depolarization degrees 1 ­ P, where P=2

j = 1, 2, 3



^2^ P j / n .

(5)

^ Here, n = 1 for states (1a) and (1b) and averages P j are expressed in terms of the square combination of the coefficients in states (1a) and (1b). From these expressions, we obtain the following simple relation between the measures C and P: C= 1 ­ 0.5 ( P s + P i ) =
2 2 2 2

b ­a + d 4 --------------------------- |0, 0 2
++ s ai ++ s bi

1 ­ Ps =

2

1 ­ P i , (6)

2

(1b)

= d 1 |P = 1; m = +1 + d 2 |P = 1; m = ­1 + d 3 |P = 1; m = 0 + d 4 |P = 0; m = 0 . Here, the basis states |P; m are defined as { P |P ; m = P ( P + 1 ) |P ; m ; P 1 |P ; m = m |P ; m , (2)
2

where P s and P i are defined by Eq. (5) separately for the signal and idler modes. Relation (6) clarifies the physical meaning of C squared as the half-sum of the partial depolarization orders in the signal and idler modes or each separate partial depolarization order, because the Ps and Pi values for states (1a) and (1b) are always equal to each other. However, the first relation in Eqs. (6) is conceptually favorable, because it includes both modes equivalently. 3. DOMAIN STRUCTURE OF POTASSIUM DIHYDROPHOSPHATE In the experiment, photon pairs are generated in a crystal of potassium dihydrophosphate KH2PO4 (KDP). Below Tc = 123 K, this crystal undergoes the ferroelectric second-order phase transition. Owing to the condition of the maximum free energy along with the requirement of the conservation of the macroscopic symmetry of the sample in the phase transition, the crystal is separated into domains, i.e., sections with the same spontaneous-polarization direction. Each domain belongs to the orthorhombic symmetry class mm2, whereas the entire sample (according to the Curie principle) belongs to the tetragonal class 4 2m (see Fig. 1). The domain boundaries can pass along the (010) and (100) planes containing the 4 and 2 axes, respectively. Therefore, two systems of domains or blocks are equiprobably formed [12]. The formation of the block domain structure in the polarization transformation is a random process, although the characteristic sizes of both blocks and domains depend on the size of a sample and the rate of its cooling near Tc. The domain thicknesses measured by different methods for samples with a typical size of about 10 mm are on the order of 10 µm [13, 14]. The distributions of the domain thicknesses measured in [15] exhibit a sharp peak near d 9 µm. Problems associated with the propagation of polarized light through polydomain crystals of the KDP group were discussed in [16, 17], where change in the polar-

and the collective P quasispin operators P2 and P1 are defined in [8, 9]. Note that operators (1/ 2 )( b s a i ± a s b i ) in representation (1b) generate two types of unpolarized twophoton states, which are maximally entangled by definition.


2. CHARACTERISTICS OF THE ENTANGLEMENT OF THE POLARIZATION STATES To analyze entanglement quantitatively, we use the measure C (concurrence) introduced by Hill and Wooters [10, 11] as C = d3 ­ d4 ­ 2d1d
2 2 2

= 2 c 2 c 3 ­ c 1 c 4 , 0 C 1, (3)

which is functionally related to linear entropy S. According to definition, C = 0 for factorized states |P = 1; m = +1 and |P = 1; m = ­1 and the maximally entangled states |P = 1; m = 0 and |P = 0; m = 0 are characterized by the maximum value C = 1. The operational characteristics of the polarization states of light fields, which are based on the P-quasispin concept and can be treated as specific measures of their entanglement, were also defined in [8, 9]. Among them are (i) the quantity X
m

2P = 1 ­ ------ , N

1 1 2 P = ­ -- + -- + P , 2 4
No. 1 2008

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KALASHNIKOV et al.

Fig. 1. Domain structure of the potassium dihydrophosphate crystal and experimental geometry. Here, X, Y, and Z are the tetragonal axes; a', b', and c' are the orthorhombic axes, Pump denotes the wave vector direction of the laser pump exciting the spontaneous parametric down conversion, and Biphoton denotes the wave vector direction for the signal and idler photons of a biphoton.

ization of normal waves at the domain boundaries was considered. Another group of works was devoted to the investigation of the spontaneous Raman scattering [18] and spontaneous parametric down conversion [19] of light in polydomain samples. The main conclusion made in both groups of works is that the domain structure serves as an efficient polarization converter. Each domain is effectively a phase plate; hence, a set of domains can be treated as a phase diffraction grating efficiently redistributing energy between the polarization and angular modes. In the general case, when describing the propagation of the optical radiation through such structures, it is necessary to take into account the redistribution of the energy between frequency modes owing to the contribution from higher order susceptibilities. 4. EXPERIMENTAL SETUP AND MEASUREMENT PROCEDURE In order to reconstruct the state vector, we used the root method of the statistical reconstruction of quasispins, which was proposed in [20] and tested for model states of a pair of polarized entangled qubits in [21]. In this method, an unknown state outlet from the sample is subjected to a set of unitary polarization transformations and projection measurements. The layout of the experimental setup is shown in Fig. 2. The given polarization transformations were performed by means of a pair of crystal-quartz plates WP1 and WP2 3716- and 437-µm thick, respectively. Then, the transformed state is projected on a fixed state with the vertical polarization with the use of the polarization prisms mounted in each channel of the Brown­ Twiss scheme. The detectors operate in the photon counting mode so that the average intensity is proportional to the number of pulses. The measurement scheme fixes both of the average intensities of radiation with the vertical polarization, R1 I1 and R2 I2, in each channel in fixed time T = 100 ms and the intensity correlation function proportional to the number of coin-

Fig. 2. Experimental setup: Ar is the 351-nm argon laser; P is the rotating prism; D is the diaphragm; K is the cryostat; KDP is the sample; LiIO3 is the lithium iodate crystal; UVM is the ultraviolet mirror; WP1 and WP2 are crystalquartz plates 3716- and 437-µm-thick, respectively; FM is the folding semitransparent mirror; PF1 and PF2 are the polarization prisms; ISP-51 is the spectrograph; F is the red cut-off filter; FD1 and FD2 are the avalanche photodiodes; and CC is the coincidence scheme.

cidences of the photocounts, Rc I1I2. The transformations of the two-photon state that are performed by the polarization elements can be represented in the form of the matrix t r t r G = s s i i , ­r* t* ­r* t* s s i i (7)

where the coefficients tj and rj are determined by the birefringence and transformer rotation angle and the subscript j = s or i refers to photons with the wavelengths s and i, respectively [22]. Thus, the polarization transformations of a given state |is are performed by the plate rotation at fixed angles and | is
kl out ^ ^ = G ( i, k ) G ( s, l ) | is .

(8)

The number of coincidences of the photocounts Rkl detected in time T is specified by the projector R kl V 1 V 2|
2 kl

.

(9)

To reconstruct the initial state, Rkl is measured for various plate positions. In this case, the number of states |iskl specified by the orientations of WP1 and WP2 plates should be at least 2D ­ 2, where D = 4 is the dimension of the Hilbert space for the system of two polarization qubits forming a biphoton. In order to increase the accuracy of the statistical reconstruction of
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GENERATION OF ENTANGLED STATES Table c1 First group Second group 0.3776 + 0.1192i ­ 0.1095 ­ 0.0299i c2 0.4074 ­ 0.0497i 0.0797 ­ 0.1295i c3 ­ 0.3179 ­ 0.0596i 0.0598 ­ 0.5975i c4 0.7551 0.7768 Xm 0.1821 0.1251 Ps 0.5135 0.9976 C
(3)

63

C

(6)

0.8581 0.0690

0.8581 0.0690

the states, it is preferable to use an excess number of measurements [21]. For the WP1 plate, the rotation angles are = 0°, 15°, 30°, and 45°, whereas the WP2 plate undergoes the complete rotation 0° 360° in every 10°. Thus, the set of measurements includes 144 various combinations of orientations of both plates. All measurements were performed at a stable sample temperature in a range of 100­115 K. An ISP-51 spectrometer with a resolution of 0.6 nm was used to separate the signal wavelength s. Contribution to the coincidences came only from the photons of the conjugate frequency mode i. However, both photons of a ( N i + N s ) pair contribute to single counts of the FD2 detector of the second channel. An argon laser operating in a continuous mode with a radiation wavelength of 351 nm and a power of 60 mW is used for pumping. Horizontally polarized pump radiation irradiated a KDP sample in which spontaneous parametric down conversion was observed in both paraelectric and ferroelectric phases. A sample 10 mm in length was cut for the type-I phase matching. We used the nondegenerate collinear regime of spontaneous parametric down conversion for which the angle between the wave vector of the pump radiation and the optical axis was approximately equal to 51°. For measurements in the ferroelectric phase, the crystal was placed into a cryostat K. After the cryostat, pump radiation was cut by the UVM mirror. For the second set of experiments, a lithium iodate (LiIO3) crystal 10 mm in length was placed in front of the cryostat. In this case, pump radiation was cut by the UVM mirror immediately after the LiIO3 crystal so that pump radiation does not enter the KDP sample. Thus, the sample was irradiated by only the biphoton field produced in the LiIO3 crystal and the sample operated as a polarization transformer. 5. RESULTS OF STATE RECONSTRUCTION AND THE ESTIMATE OF THE ENTANGLEMENT MEASURES (i) The statistical reconstruction of the states was first performed for the case where biphotons were produced in the polydomain sample for the wavelengths s = 676 nm and i = 730 nm. The factorized states are generated in each domain layer k: | = |O 676, O 730 ,
JETP LETTERS Vol. 87 No. 1 2008
(k) (k) (k)

which are subjected to SU(2) polarization transformations during the process of propagation through the sample. A superposition of the states produced in each domain appears at the output. (ii) Similar measurements were performed at the next stage for the case where the state |V1V2 appearing in spontaneous parametric down conversion in the LiIO3 crystal was transmitted through the same sample. The state at the input to the polydomain sample was factorized: | in = |V
676

,V

730

.

(11)

(iii) Entanglement measures for states at the output from the sample were calculated by Eqs. (3)­(6) with the use of the resulting data. The table presents the results of the statistical reconstruction of the polarization states of the biphoton field at the output from the samples for both groups of experiments (the first four columns) and the entanglement measures calculated by Eqs. (3)­(6). 6. DISCUSSION OF THE EXPERIMENTAL RESULTS The results presented above have a simple qualitative explanation. Indeed, if the biphoton polarization state is transmitted through the polydomain sample (the second group) and is initially factorized, it remains factorized, because the entanglement measure is not changed by local transformations, which are specified by the product of the matrices describing SU(2) polarization transformations in each domain at the frequencies s and i. In the first group of experiments, the biphoton in the polarization state, |O1O2(m), can be produced in each mth domain layer randomly over the length of the inhoN d , where d is the mogeneous sample, L = i=1 i domain thickness and N is the number of domains. The state |O1O2(m) |O1(m) |O2(m) is the state of two photons polarized along the direction O in the mth domain (the subscripts refer to the wavelengths of photons in a pair). Then, the produced biphoton propagates through the remaining part of the sample and its polarization state is changed by SU(2) rotations, which are specified by the matrices Gm, where m is the domain number



(10)


64 ° 3

KALASHNIKOV et al. H V

0

­3 630 550 500 s, nm

Fig. 3. Photograph of the frequency­angular spectrum of the KDP polydomain sample. The arrows show the predominant polarizations of the corresponding branches of the spectrum.

(1 m N). In this case, the output state is a superposition of the states produced in each domain: GNG
N­1

... G 2 |O s O i

(1)

+ GNG

N­1

... G 3 |O s O i
(N)

(2)

+ G N |O s O i

( N ­ 1)

+ ... + |O s O i

(12)

.

In particular, the first term in expression (12) describes the contribution of a biphoton that is produced in the first domain and is transformed in all subsequent domains during the process of propagation through the sample. The state specified by expression (12) can be transformed to form (1a), which is a superposition of four basis states |Hs, i and |Vs, i in the laboratory basis: it is generally entangled. The representation of the state vector in terms of symbols |Os, i means that the polarization basis in the mth domain is determined by the arrangement of its crystallographic axes and does not generally coincide with the laboratory polarization basis |Hs, i and |Vs, i . Note the complete coincidence of C values calculated by Eqs. (3) and (6) (last two columns in the table). The smallness of the Xm values in both groups of experiments indicates that the output states are far from the singlet states |P = 0; m = 0. The representation of the biphoton states at the output from the sample in form (12) implies the existence of two physical processes. The first process is the production of factorized states of biphotons in each domain and the transformation of the polarization state of biphotons during the process of propagation through the layered structure. The presence of both orthogonal polarization components of the biphoton field at the output from the crystal is an obvious consequence of these two processes. To independently verify this statement, the frequency­angular spectra of spontaneous parametric down conversion in the KDP polydomain crystal were obtained (see Fig. 3) by placing a cassette with a film at the output from the spectrograph. The polarization of the detected (signal) radiation was analyzed by rotating the polarization prism PF1. The spectrum consists of three characteristic curves, one of

which is symmetric around the zero-angle axis and two other curves are shifted by approximately 2° toward the positive and negative scattering angles. The central curve corresponds to the spontaneous parametric down conversion regime at which the signal and idler photons have a vertical polarization. Radiation corresponding to the branches shifted in angles has a horizontal polarization. It is important that the shifted branches are observed only in the polydomain samples. The appearance of the spectral sections with an orthogonal polarization can be explained by at least two mechanisms, the anisotropic linear diffraction of the signal radiation and the quasi-phase matching of spontaneous parametric down conversion. In this work, we only state the appearance of the frequency­angular components with both polarizations in the spectrum of the spontaneous parametric down conversion. This appearance confirms the statistical reconstruction of the states: if the biphoton state is polarization entangled, the state of the signal (or idler) photons is a mixture of the contributions from the orthogonal polarization components. 7. CONCLUSIONS In this work, it has been shown experimentally that entangled two-photon states can appear during the process of the spontaneous parametric down conversion in ferroelectric crystals with a quasi-regular domain structure (such as potassium dihydrophosphate). The polarization and frequency degrees of freedom are entangled due to the specific polarization transformations. Under these transformations, first, the factorized states of the photon pairs with fixed polarizations are produced in two frequency modes in the sample bulk and, second, each two-photon polarization state undergoes SU(2) rotations in each frequency mode during the process of the subsequent propagation through the sample. As a result, a two-photon polarization state, which is an information ququart, i.e., a superposition of four basis states, appears at the output from the sample. The quantitative entanglement characteristics have been measured. A comparison of the experimental data with the
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preliminary results of the simulation of various configurations of the domain structure of the crystal [23] indicates the block arrangement of the domains. Only this configuration is responsible for a high entanglement measure of the two-photon state observed in the experiment. In view of this circumstance, the search for the features in the configuration of the domain structure (the distributions of the domain-block sizes and domain thicknesses in each block) that lead to the entanglement is the next interesting problem. We are grateful to L.A. Krivitskii for assistance in obtaining the spontaneous parametric down conversion spectra and discussion of the results. This work was supported by the Russian Foundation for Basic Research (project nos. 07-02-91581-ASP.a and 06-0216769-a) and the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project no. NSh4586.2006.2). REFERENCES
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Translated by R. Tyapaev

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