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ISSN 1063 7761, Journal of Experimental and Theoretical Physics, 2010, Vol. 110, No. 2, pp. 185­192. © Pleiades Publishing, Inc., 2010. Original Russian Text © S.S. Straupe, S.P. Kulik, 2010, published in Zhurnal èksperimental'no i Teoretichesko Fiziki, 2010, Vol. 137, No. 2, pp. 211­219.

ATOMS, MOLECULES, OPTICS

The Problem of Preparing Entangled Pairs of Polarization Qubits in the Frequency Nondegenerate Regime
S. S. Straupe* and S. P. Kulik
Moscow State University, Moscow, 119992 Russia *e mail: straups@yandex.ru
Received July 2, 2009

Abstract--The problems associated with the practical implementation of the scheme proposed for prepara tion of arbitrary states of polarization ququarts based on biphotons have been discussed. The influence of effects of frequency dispersion are considered, and the necessity of compensating the dispersion of group velocities in the frequency nondegenerate case even for continuous pumping is demonstrated. A method for this compensation is proposed and implemented experimentally. Physical restrictions on the quality of the prepared two photon states are revealed. DOI: 10.1134/S1063776110020019

1. INTRODUCTION The so called high dimensional quantum systems or systems with a dimension of the Hilbert space of states D 3 have attracted increasing interest. Prima rily, this interest stems from the possibility of using such systems in various quantum information proto cols. The application of these systems, which are fre quently referred to as qudits (quantum dits), as infor mation carriers offers a number of advantages over encoding with qubits [1­4]. The second important field of application of qudits is the verification of the basic principles of the quantum theory, in particular, the check of Bell type inequalities [5]. Incidentally, it appears that the use of high dimensional systems in a number of cases has made it possible to increase the quantitative gap that arises in the description of com posite systems using quantum or classical correla tions. In turn, this allows one to reduce the require ments imposed on the quantum efficiency of photo detectors and other experimental equipment [6]. Thus, the development of methods used for preparing and characterizing different states of qudits is an important problem and underlies a new direction in quantum information, i.e., so called quantum state engineering. Polarization degrees of freedom of photons are par ticularly convenient tools for experimentally prepar ing qudit states, primarily owing to the simplicity of performing polarization transformations. Specifically, in order to control the polarization state, it is sufficient to use linear optical elements of the phase plate type. Moreover, the developed statistical methods of polar ization tomography have made it possible to com pletely reconstruct the vector of the state of the initial quantum system as a result of the relatively simple pro cedure [7, 8].

A polarization ququart is considered to mean a pure two photon state that can be written in the form | = c 1 |H 1 |H 2 + c 2 |H 1 |V 2 + c 3 |V 1 |H 2 + c 4 |V 1 |V 2 , (1)

where the ket vectors |H and |V stand for the basis polarization single photon states of the field mode, and indices 1 and 2 can refer to both the frequency (longitudinal) and spatial (transverse) modes. It is common practice to distinguish polarization spatial ququarts, for which the field state is quasi degenerate in frequency but there are two spatial modes (see, for example, [7]), and polarization­frequency ququarts, when the spatial modes are degenerate but the fre quency spectrum of the field contains two components [9]. The first experiments on the preparation of entan gled pairs of polarization qubits or entangled states of ququarts (D = 4) were performed in the mid 1990s. In [10, 11], pairs of photons generated during collinear frequency degenerate spontaneous parametric down conversion (SPDC) of light with type II matching were transformed into polarization­momentum entangled states with a beam splitter and subsequent postselection. Shortly thereafter, Kwiat et al. [12] pro posed a method that does not require postselection and is based on the type II noncollinear phase match ing. There are well known schemes employed for preparation of entangled states with the use of two crystals with type I matching in noncollinear fre quency degenerate [13] and collinear but frequency nondegenerate [14, 15] regimes. The case of pulse pumping is discussed in [16, 17]. One of the advan tages of these schemes is the possibility of preparing biphotons in nonmaximally entangled states [18]. Of special interest, however, are the methods providing the preparation of polarization ququarts in an arbitrary

185


186
QWP WP QP Type I BBO HWP 1

STRAUPE, KULIK
|

pure entangled state of a qubit pair or a ququart can be represented in the form (cf. expression (1)) | = |A 1 |A 2 + 1 ­ |B 1 |B 2 , (2) where |A j and |B j are the basis vectors of the space of states of each individual qubit. This expression is known as the Schmidt decomposition. The coeffi cients and 1 ­ are eigenvalues of the single particle density matrices of each qubit (which, as is known, coincide with each other), and the vectors Aj and Bj form an orthogonal basis in which they are diagonal. Therefore, in order to prepare an arbitrary ququart state, one should be able to experimentally control the coefficients in the Schmidt decomposition and to per form switching between the bases. Note that the former operation changes the degree of entanglement of the state and affects the degrees of freedom of both qubits. In this sense, the above operation can be termed "nonlocal." However, the transformation of the basis vectors requires only "local" operations, i.e., operations that act separately on each qubit. It should also be noted that the degree of entanglement of a qubit pair can be defined in any reasonable manner; for example, it is convenient to use the measure C (concurrence) [22]. The idea of the possible implementation of the scheme used for the preparation of arbitrary states of polarization ququarts is illustrated in Fig. 1. This scheme employs two nonlinear crystals oriented so as to provide noncollinear frequency nondegenerate type I matching when both (signal and idler) photons are equally polarized. The pumping is performed by radiation from a continuous wave laser, the direction of linear polarization of the laser radiation is specified using a half wave plate (WP), and the relative phase between the horizontal and vertical components is controlled by tilting a pair of quartz plates (QPs). The state of biphotons generated in the course of spontane ous parametric down conversion under these condi tions has the form (3) | = |H 1 |H 2 + 1 ­ |V 1 |V 2 , where indices 1 and 2 correspond to different fre quency modes. The state of each individual photon in the pair is determined by the density matrix (4) j = |H j H i| + ( 1 ­ ) |V j V j| , where j = 1 and 2 is the index numbering the sub systems. In this way, we can prepare a state with the required values of the coefficients in the Schmidt decomposi tion but in the fixed (H­V) basis. The transformation to the arbitrary basis (2), however, presents no special problems. Actually, an arbitrary polarization state of
1

DBS 2 QWP HWP

Fig. 1. Schematic diagram illustrating the preparation of an arbitrary pure polarization ququart state with the use of two crystals with type I matching.

(i.e., predefined) state by means of unitary (without a loss) transformations, when all components of the vector of the polarization state of a biphoton pair can be completely controlled. The design of this universal source of ququarts is of both fundamental and practi cal interest for solving problems of quantum informa tion and quantum communication. The experimental scheme that ensures complete control of a polariza tion ququart was proposed in our recent paper [19]. In essence, it is an interferometric scheme that provides the stability of the parameters, which is not inherent in conventional interferometric schemes but is well known in classical polarization optics. To the best of our knowledge, this scheme has so far not been imple mented in full. However, specific families of states of frequency nondegenerate ququarts, of course, have already been prepared experimentally (see, for exam ple, [8, 15, 20]). In this work, we have investigated the problem of physical restrictions imposed on the quality of states prepared according to the aforementioned scheme and discussed the methods proposed for their elimina tion. It has turned out that the use of the frequency nondegenerate regime and, correspondingly, the polarization and frequency entanglement in the scheme with two crystals placed in tandem involves specific difficulties encountered in performing the experiments due to the frequency dispersion of pho tons in crystals. However, this scheme is very conve nient from the viewpoint of applications in quantum information protocols and has been frequently employed in their implementation. In this paper, we demonstrate that, in order to achieve a high quality of the prepared states, it is necessary to use compensators for group velocity dispersion, which was earlier treated as an attribute of schemes with type II phase matching [10, 11, 21] or pulse pumping [16, 17]. 2. SCHEME FOR PREPARATION OF ARBITRARY STATES OF POLARIZATION QUQUARTS Let us consider the scheme proposed in [19] for the preparation of arbitrary states of polarization ququarts. We start with the formal mathematical aspect of the problem. It is known that an arbitrary

1

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the qubit, i.e., the polarization state of a photon in one field mode, can be obtained from a specified state by means of transformations performed with the use of a quarter wave plate (QWP) and a half wave plate (HWP) placed in series. Let this transformation be Uj.
j j Then, we have |H j |A j and |V j ei |B j . The phase can be eliminated with the use of plates QP. As a result, we obtain the transformation

in each crystal of length L can be written in the form [23] ( ) L | 1, 2 = d d 1 d 2 exp i z 2



U

U

( ) L â sinc z 2


(6)

|

U1 U2

|A 1 |A 2 + 1 ­ |B 1 |B 2 ,

(5)

which ends the procedure of preparing an arbitrary state of the ququart. For experimental convenience, the modes 1 and 2 are separated in space due to the noncollinear matching. After transformations in each spatial­frequency mode, they are brought into coinci dence using a dichroic beam splitter (DBS) for the purpose of obtaining one spatial mode. An essential feature of this scheme is the use of the frequency nondegenerate regime, which makes it possible to form a collinear beam of biphotons without loss. This is undeniably convenient for practical appli cations in various quantum information protocols. However, in practice, the preparation of entangled states of ququarts in the scheme with crystals placed in tandem involves certain problems. These problems are associated with frequency dispersion effects, which are responsible for the distinguishability of the photon pairs generated in the first and second crystals and, correspondingly, for the deterioration in the purity of the prepared state. 3. EFFECTS ASSOCIATED WITH FREQUENCY DISPERSION AND THEIR COMPENSATION In expression (3), only two frequency modes are singled out. Experimentally, this corresponds to the filtration of the initial spectrum with ultimately nar row filters. When using filters with a finite bandwidth, it is necessary to take into account the structure of the frequency spectrum of SPDC. This leads to a more complex relationship for the vector of the state of the biphoton pair. For definiteness, the axes of the first and second crystals are assumed to be oriented verti cally and horizontally, respectively. Let us consider the case of continuous pumping that is linearly polarized at an angle of 45°. In the plane wave pump approxima tion, the individual states of biphoton pairs generated
2

2

â a V, H ( 1, 1 + ) a V, H ( 2, 2 ­ ) |vac , where z ( ) = k o ( 1 + , 1 ) + k o ( 2 ­ , 2 ) + k e ( p ) is the longitudinal detuning of the phase matching (the pump is an extraordinary wave, and the signal and idle photons are ordinary waves in the crystal). We will analyze the case in which the angular modes are fixed; i.e., they are separated with the use of narrow aper tures. Under these conditions, integration over angles is not performed; hence, we can restrict our analysis to the examination of the frequency spectrum. If the pump coherence length exceeds the distance between the centers of the crystals, the corresponding ampli tudes are coherently added and the total state is the superposition | = 1 2
3

p = { H, V }



d F ( , p ) a p ( 1 + )



(7)

â a p ( 2 ­ ) |vac , where the index p numbers the polarization states, and L (8) F ( , H ) F ( ) = sinc z 2 and (9) F ( , V ) = F ( ) e are the amplitudes of the biphotons in the first and sec ond crystals, respectively. The presence of the additional relative phase () is determined by two factors: the phase incursion between the polarization components of the pump in the first crystal and the appearance of an additional phase for the photons generated in the first crystal dur ing their passage through the second crystal [24]: ( ) = ( ke ( p ) ­ ko ( p ) ) L (10) + ( ke ( 1 + ) + ke ( 2 ­ ) ) L . The first term on the right hand side of expression (10) does not depend on the frequency detuning of the matching of and can be easily compensated by introducing the corresponding phase shift between the polarization components of the pump. The second term is related to the frequency dispersion and deter mines the appearance of different phase shifts for dif ferent frequencies within the spectral line of SPDC. Physically, this means that, in the separation of a fairly
3
i()

In the dichroic beam splitter, the coefficients of reflection and transmission are maximum for light beams with the frequencies 1 and 2, respectively, regardless of the polarizations.

In practice, this requirement is always satisfied. Vol. 110 No. 2 2010

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188 G
(2)

STRAUPE, KULIK 2

1

k ( ) z ( ) = o



1

k ( ) ­ o




2

1 1 ­ gr = Co , = gr vo ( 1 ) vo ( 2 )


(13)

k ( ) () = e



1

k ( ) ­ e



L
2

1 1 ­ gr L = Ce L . = gr ve ( 1 ) ve ( 2 )
gr

(14)


Fig. 2. Second order correlation functions for spontane ous parametric down conversion in two orthogonally ori ented crystals. Hatched regions correspond to the time ranges where the correlation functions overlap, thus giving rise to interference.

Here, v o, e () = /ko, e is the group velocity of wave packets with the corresponding central frequencies. Expression (13) determines the shape of the spectral line of SPDC: F()
2

= sinc ( C o L / 2 ) .

2

wide portion of the SPDC spectrum, its individual components will be added with different phases, which is equivalent to loss of coherence and to a changeover to a mixed state in the ququart representa tion. The pure polarization­frequency state will exist only in a Hilbert space with a higher dimension D 4. This dimension can be estimated, for example, by using the Fedorov parameter [25] adapted to polariza tion­frequency distributions; however, this problem is beyond the scope of our present work. Now, we analyze the influence of dispersion effects on the properties of a generated polarization state. The polarization density matrix, which corresponds to the state |, can be obtained by taking the partial trace over the frequency variables, pol = Tr | | = 1 2

In our case, it is more convenient to consider the time representation and to change over from spectral char acteristics to time correlations. The second order cor relation function and the spectrum are related by the following expression [26]: G () =
(2)



d F ( ) cos ( ) .

2

(15)

p = { H, V }



d F ( , p ) F * ( , p ) |pp pp| ,

(11)

with the obvious designation |pp a p ( 1 + ) a p ( 2 ­ ) |vac . In explicit form, we derive the following relationship for the polarization density matrix: 1 pol = ( |HH HH| + |VV VV| 2 + d F ( ) e




2 ­i ( )

|HH VV| |VV HH| ) ,

(12)

+ d F ( ) e

which, generally speaking, corresponds to a mixed polarization state. The pure state can be obtained only in the case where () = const. To the first order in the small detuning of the fre quency , we have



2 i()

When the function F() is symmetric, we obtain a conventional Fourier transform. Therefore, in our case, the correlation function of the emission from each crystal has the form of a rectangle with a width of 1 = CoL. The presence of the phase shift () in terms of time implies a relative shift of the correlation func tions by 2 = CeL (see Fig. 2). The relative shift in the correlation functions of the emission from the first and second crystals leads to the distinguishability of pairs of photons generated in these crystals at the instant of their detection and, hence, to a decrease in the visibility of the polarization interference. Let us consider this problem in detail from the experimental point of view. The scheme of the experi mental setup is depicted in Fig. 3. The pumping was radiation from a helium­cadmium laser with a wave length of 325 nm. The vertical polarization was sepa rated from the initially unpolarized radiation with the use of a Glan prism (V), after which it was rotated with a half wave plate. The relative phase between the polarization components of the pump radiation was introduced by two quartz plates with vertically ori ented optical axes. The effective thickness of these plates was determined by the relative tilt angle . The parametric down conversion occurred in two BBO crystals (each 2 mm thick) cut out for the collinear fre quency degenerate matching. The directions corre sponding to the operating wavelengths 1 = 600 nm and 2 = 710 nm were separated by the apertures (A). Then, the beams were brought into coincidence on a
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M V /2 He-Cd laser BBO 1 2 A DBS QP DBS IF D2 CC +45° +45°IF D1 A

189

Fig. 3. Experimental setup for the preparation of entangled states.

dichroic beam splitter (DBS), which transmitted radi ation at a wavelength of 1 = 600 nm and completely reflected radiation at 2 = 710 nm. In the measuring part of the scheme, the pairs were separated according to frequency with the use of a similar beam splitter. In each channel of the measuring scheme, there were film polarizers oriented at an angle of 45° with respect to the vertical and interference filters with central wavelengths of 600 and 710 nm and a bandwidth of 10 nm (IF). The coincidences of photocounts of detectors D1 and D2 were recorded using a coinci dence circuit (CC) with a window of 2 ns. In the ideal case, when the polarization state of a biphoton has the form | = 1 , i 2 ( |H 1 H 2 + e |V 1 V 2 ) (16)

the following relationship for the coincidence count ing rate: R ( ) 1 + d F ( ) cos ( ( ) ­ ) = 1 + V cos ,



2

(19)

where the visibility of the interference pattern (to the first order in detuning) is written in the form V=



2 C L d sinc o cos ( C e L ) . 2

(20)

The integral is calculated simply and leads to the fol lowing relationship for the visibility: V = 1­ Ce = 1 ­ 2, Co 1 (21)

the coincidence counting rate in this scheme is given by the formula R C +45 1, +45 2| ( |H 1 H 2 + e |V 1 V 2 ) 1 + cos ,
i 2

(17)

where the phase changes with variation in the tilt of the quartz plates in the pump beam. In actual practice, since the pairs generated in the first and second crys tals are partially distinguishable due to the incomplete overlap of the corresponding correlation functions (see Fig. 2), the visibility of the interference pattern, which is defined as V= R R
C C
max

which is consistent with the graphical representations illustrated in Fig. 2. For a fixed frequency spectrum, which, in our case, is determined by the bandwidth of the interference fil ter (it appears to be somewhat smaller than the width of the SPDC spectrum, which, in the regime used, is approximately 12 nm), the visibility can be increased by introducing an additional birefringent compensator QP into the beam with a shorter wavelength. The quartz compensator with the vertically oriented opti cal axis delays the vertically polarized photon of the pair by the time represented by the expression
comp

1 ­ gr 1 L q , = gr v eq ( 1 ) v oq ( 1 )
gr

(22)

max

­R +R

C

min min

,

(18)

C

differs substantially from unity. By using expression (12) for the polarization density matrix, we can derive

where v oq and v eq are the group velocities of the ordi nary and extraordinary waves in quartz, respectively, and Lq is the length of the compensator. By choosing the compensator length so that the condition comp = 2 will be satisfied, it is possible to achieve complete over lap of the correlation functions and, consequently, maximum visibility of the interference pattern.
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gr

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STRAUPE, KULIK

taken into account, the relationship for the biphoton amplitude additionally contains the factor corre sponding to the transmission amplitude of the inter ference filters. By choosing this factor in Gaussian form, we obtain ( ­ ) 2 F ( ) exp ­ 1 2 ( ­ â exp ­ 2 2
-4 0 4 8 Lq, mm

(23)

) ( ) L , sinc z 2
2

Fig. 4. Dependences of the visibility of polarization inter ference on the thickness of the compensator of the group velocity dispersion. The dotted line is the theoretical dependence without inclusion of the filtration. The solid line is the dependence with inclusion of the frequency fil tration with the use of 10 nm interference filters in each channel.

Figure 4 shows the experimental dependence of the visibility of the interference pattern on the length of the quartz compensator used. The dotted curve bounding the triangle corresponds to the convolution of two rectangular correlation functions, i.e., the func tions determined only by the shape of the spectral line of SPDC, without the inclusion of filtration. However, the experiment was performed using interference fil ters characterized by a Gaussian shape of the depen dence of the transmission coefficient on the wave length with the width comparable to that of the SPDC spectrum. Therefore, in actual practice, the correla tion functions should have a shape somewhat different from rectangular. When the frequency filtration is
Coincidence counting rate, Hz 0.5 V = 81% 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 , deg

where is the bandwidth of the interference filters. The solid curve in Fig. 4 represents the numerically calculated dependence of the visibility on the thick ness of the compensator of the group velocity disper sion with allowance made for the 10 nm interference filters. The calculation of the width and shift of the corre lation functions in the case under consideration leads to the following values: 1 = 96 fs and 2 = 87 fs, which corresponds to a compensator length of 2.6 mm, in good agreement with experimental data. The small difference between the experimental position of the maximum in the visibility (2.3 mm) and the calculated value is most likely explained by the errors in the values of group velocities used. The characteristic interfer ence patterns obtained with and without a compensa tor are shown in Fig. 5. The deviation of visibility from 100% even with the use of a compensator of the required length is explained by two factors. The first factor is "technical" and lies in the inaccuracy in separating the necessary spatial modes. The second factor is the influence of the terms quadratic in the detuning of the frequency in the expansion of (), which lead to different broad enings of the correlation functions and, consequently, to the deviation of visibility from 100% even in the case of ideal compensation of the part that is linear in the
Coincidence counting rate, Hz

(a)

0.5 V = 55% 0.4 0.3 0.2 0.1 0 0 5 10 15 20

(b)

25 , deg

Fig. 5. Interference patterns obtained (a) with a 2.33 mm thick compensator and (b) without a compensator. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 110 No. 2 2010


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detuning [24]. Indeed, to the second order in the detuning of the degenerate frequency, we have () =
2 1 k e 2 2

V 1.00



1

2 k e 2 2 + 2 L = B e L . (24) 2

0.98

By introducing the designations = CoL and D = Be/ C o L for the coincidence counting rate in the case of a complete compensation of the first order effects, we obtain (25) R ( ) 1 + 1 ( V c cos + V s sin ) , 2 where 2 2 2 2 V c = d sinc cos D , V s = d sinc sin D . 2 2 For the crystals used in the experiment, we have D = 0.0193 and the numerical calculation gives the follow ing value of the visibility of the interference pattern:
2

0.96

0.94 0 20 40 60 80 100 , nm





Fig. 6. Dependence of the visibility of polarization inter ference on the bandwidth of the interference filter in the case of a complete compensation of the group velocity delays.

V=

Vc + Vs = 0.89 . 2
4

2

2

(26)

4. CONCLUSIONS In this paper, we have considered a method for pre paring arbitrary entangled states of polarization ququarts in the noncollinear frequency nondegener ate type I SPDC regime according to the scheme with two crystals placed in tandem. We have elucidated the role of effects that are related to the frequency disper sion and which lead to a deterioration in the purity of the prepared states and, as a consequence, to a decrease in the visibility of the two photon polariza tion interference. It has been shown that, in the fre quency nondegenerate regime, the compensation of the group velocity dispersion, which was previously used only in schemes with pulse pumping, is necessary even in the case of continuous pumping with a large coherence length. Methods providing this compensa tion have been proposed and demonstrated experi mentally. Moreover, in the case of a complete com pensation, the maximum value of the visibility limited by dispersion effects of higher orders in the frequency detuning has been estimated. It should be emphasized that the application of fre quency nondegenerate biphotons, unlike the majority of the earlier proposed and currently known schemes (using frequency degenerate biphotons and separa tion of spatial modes), offers a number of advantages. The main advantage is the possibility of forming states, including entangled ones, which are localized in one spatial mode. Undeniably, this property is desirable from the viewpoint of possible practical applications, when two particle states must be transmitted through one communication channel. Thus, the performed investigation has revealed physical restrictions on the quality of the prepared states and demonstrated the methods for its improvement using a specific experi mental scheme.
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It should be noted that this low value of visibility is the limiting case, which corresponds to the complete absence of frequency filtration. The results from the numerical calculation of the dependence of the visibil ity on the bandwidth of the interference filters used in the case of a complete compensation are presented in Fig. 6. It can be seen from this figure that, even when sufficiently broad band interference filters are used (with a bandwidth of less than 30 nm), the values of visibility can reach 0.95 or higher. The low visibility of the polarization interference corresponds to poor quality of the preparation of entangled states. The currently obtained results of the statistical reconstruction of the state, + 1 (27) | = ( |H 1 H 2 + |V 1, V 2 ) , 2 which corresponds to the maximum of the interfer ence pattern shown in Fig. 5, lead to a fidelity of (28) F = theory| exp , which does not exceed 75%. Nonetheless, we do not see, in principle, any restrictions from above on this quantity. Higher values of the fidelity F can be reached by means of careful separation of the spatial modes and, possibly, with the use of narrower band frequency filters. However, the mere fact that we observe the two photon interference with a nonzero fidelity suggests that the preparation of entangled states according to the proposed scheme is realistic. As regards improve ment in the quality of the prepared states, it is not a crucial technical problem, and it is being solved right now.
4
2

The record high values of the visibility in experiments with nar rowband filters in the frequency degenerate regime reach 99%.

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STRAUPE, KULIK

ACKNOWLEDGMENTS We would like to thank D. Ivanov for checking the calculations. This study was supported by the Russian Foundation for Basic Research (project nos. 08 02 00741 a and 08 02 00559 a), NATO (grant no. CBP.NR.NRCL 983251), and the Council on Grants from the Presi dent of the Russian Federation for the Support of Leading Scientific Schools (grant no. NSh 796.2008.2). REFERENCES
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Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 110

No. 2

2010