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JETP Letters, Vol. 82, No. 3, 2005, pp. 164­168. Translated from Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 82, No. 3, 2005, pp. 180­184. Original Russian Text Copyright © 2005 by Bogdanov, Galeev, Kulik, Maslennikov, Moreva.

Reconstruction of the Polarization States of a Biphoton Field
Yu. I. Bogdanova, R. F. Galeevb, S. P. Kulikb, G. A. Maslennikovb, and E. V. Morevac
a

Institute of Physics and Technology, Russian Academy of Sciences, Nakhimovskioe pr. 34, Moscow, 117218 Russia b Faculty of Physics, Moscow State University, Vorob'evy gory, Moscow, 119899 Russia c Moscow Engineering Physics Institute (State University), Kashirskoe sh. 31, Moscow, 115409 Russia
Received June 7, 2005

A method for reconstructing an arbitrary quantum state of an optical system in Hilbert space with dimension d = 4 is discussed. Such states can be realized using a collinear frequency-nondegenerate regime of generating spontaneous parametric down-conversion. The method has been tested for a number of polarization states of a biphoton field. The high accuracy of the reconstruction of the states (above 99%) indicates that the procedures proposed for reconstructing the quantum state of the system are adequate. © 2005 Pleiades Publishing, Inc. PACS numbers: 03.67.Hk, 42.25.Ja, 42.50.Dv

Methods for reconstructing quantum states have been extensively discussed in recent years. First, the properties of such states are studied in the theory of quantum information and quantum calculations. Second, this interest is stimulated by the growing requirements of experimental physics, which is closely approaching manipulation of single quantum objects. We have in mind attempts to control three stages of the evolution of quantum systems: preparation, transformation, and measurement. Quantum optical multilevel systems present a rich class of states that is very attractive for the transmission and storage of quantum information. At present, nearly complete control has apparently been achieved over two-level optical systems-- polarization, spatial, and frequency states of single photons. This statement is corroborated by operating devices for quantum key distribution that are based on certain procedures for encoding information using single photons. At the same time, multilevel states of the field are also of considerable interest [1]. In addition to fundamental aspects, an increase in the dimension is associated with an increase in the security of existing quantum key distribution systems (against certain classes of attacks on quantum key distribution protocols) [2]. A method for complete control over the preparation [3] and measurement [4, 5] of three-level polarization optical systems was recently proposed and realized. Procedures for generating and measuring other three-level optical states are being actively investigated in realization on spatial field modes [6], three-arm interferometers [7], and entangled states of biphotons [8]. Unfactorized states of two qubits based on biphotons were analyzed in detail in [9], where the complete statistical reconstruction (density matrix) of the initial state was realized. The preparation of unfactorized triplet states in four-dimensional Hilbert space on the basis of bipho-

ton field was probably reported for the first time in [10]. The removal of frequency degeneracy was used in [11] to generate a Bell singlet state in a single-beam (wavevector-degenerate) mode. This paper is devoted to discussing a method for reconstructing an arbitrary state of a four-level quantum system realized on polarization states of a frequency-degenerate biphoton field. 1. BIPHOTONS AS FOUR-LEVEL SYSTEMS As a result of spontaneous parametric down-conversion in the presence of the action of laser pumping on a crystal without an inversion center, pairs of correlated photons (biphotons) are formed. The sum of the frequencies of photons produced under stationary conditions is equal to the pumping frequency, and the scattering directions satisfy the phase-synchronism conditions [12]: 1 + 2 = p and k1 + k2 = kp. The state vector of the biphoton field in an arbitrary pure polarization state has 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 , where c i = c i exp { i i } and
4

(1)


i=1

c

2 i

=1

and, for example, |H1, H2 means that two photons are in the horizontal polarization mode and |H1, V2 implies that the photons in modes 1 and 2 have the horizontal and vertical polarizations, respectively.1 If unpolarized
1

Modes 1 and 2 are traditionally called signal and idler modes, respectively.

0021-3640/05/8203-0164$26.00 © 2005 Pleiades Publishing, Inc.


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modes are degenerate, i.e., |H1 = |H2 = |H, |V1 = |V2 = |V, then state (1) is transformed to the form | = c 1 |H, H + c 2 |H, V + c 4 |V , V , (2)

which corresponds to a three-level optical system or a qutrit [3­5]. Such a regime of generation of biphotons is called collinear and frequency-degenerate. The degeneracy can be removed by various methods. The first method is based on the choice of a crystal orientation such that spontaneous parametric scattering leads to the production of two photons with different frequencies in one spatial mode. The second method is based on the determination of the arrival times of photons with different polarizations, which can be realized by means of delay lines with polarization anisotropy. Since state (1) is represented as the expansion in terms of four basis states, it was called ququart by analogy with qubit and qutrit, which are the states of two- and three-level systems, respectively.2 The polarization properties of a two-mode biphoton field are completely determined by the coherence matrix introduced by Klyshko [13]. This matrix consisting of 16 fourth-order moments of the field can be obtained as the product of the coherence matrices of both photons [14] and is written as K4 = A E* F* G* E B I* K* F I C L* G K L D .

Fig. 1. Layout of the experimental setup for the complete tomography of ququarts: (1) horizontally oriented polarizer; (2) filter reflecting ultraviolet pumping radiation, (3QP) driving quartz plate, (1QP) and (2QP) quartz plates, (PF) polarization filter, (D) photodetector, and (CC) coincidence scheme. The preparation and measurement blocks are enclosed by the dashed lines.

modes, respectively. For a pure ququart state, the reduced polarization density matrix coincides with coherence matrix (3). Three real moments (4) and six complex moments (5) completely determine an arbitrary ququart state. Thus, if all moments are measured, the initial state of the density matrix can be reconstructed. This procedure can be performed by projecting an unknown state on certain states, as is done in medical tomography.3 The optimum tomography procedure is reduced to finding the minimum set of projectors from which the initial density matrix or state vector can be reconstructed. In this work, frequency-nondegenerate collinear states of biphotons are investigated. For this reason, the ququart is represented 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 . 2. EXPERIMENT An experimental setup for studying an arbitrary ququart (Fig. 1) consists of two blocks, "preparation" and "measurement." 2.1. The preparation block includes a lithium iodate crystal with length L = 1.5 cm and a driving quartz plate 987.7 µm in thickness. Pumping is the horizontally polarized 325-nm radiation of a He­Cd laser. The angle between the pumping wavevector and optical axis of the crystal is equal to 58°. Under these conditions, the state |0 = | V 1 , V 2 (type-1 phase matching) is generated in the crystal. The driving phase plate
3

(3)

The diagonal matrix elements are real quantities characterizing the strength of correlations between photons in two frequency­angular modes with parallel [(K4)11, (K4)44] or orthogonal [(K4)22, (K4)33] polarizations: A = a 1 a 2 a 1 a 2 = c
++ 2 1 2 3

(6)

, ,

B = a 1 b 2 a 1 b 2 = c
++ ++

2 2 4

, .

C = b 1 a 2 b 1 a 2 = c
++

D = b 1 b 2 b 1 b 2 = c

2

(4)

The off-diagonal elements are complex quantities and are determine relative phases between basis states
++ E = a 1 a 2 a 1 b 2 = c * c 2 , 1 ++ F = a 1 a 2 b 1 a 2 = c * c 3 , 1

++ ++ G = a 1 a 2 b 1 b 2 = c * c 4 , I = a 1 b 2 b 1 a 2 = c * c 3 , (5) 1 2 ++ * K = a 1 b 2 b 1 b 2 = c 2 c 4 , ++ * L = b 1 a 2 b 1 b 2 = c 3 c 4 .

Here, a+ (a) and b+ (b) are the operators of creation (annihilation) of a photon in the horizontal and vertical
2

These terms relating to energy states are incompletely correct, because no real "levels" exist in these systems. They imply the expansion of states in orthonormalized bases with two (qubit), three (qutrit), and four (ququart) basis vectors. We also use this accepted terminology. JETP LETTERS Vol. 82 No. 3 2005

The corresponding procedure is often called quantum tomography, and the complete set of measurements that provides the reconstruction of the initial state is called the quantum tomography protocol.


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transforms the vector |0 according to the rule |in = G|0, where the matrix G= r1 r2 ­t1 r* t1 t* ­r1 r* r1 t* 2 2 2 2 * t2 ­r* r2 t* t2 t* r2 ­r1 1 1 1 r* r* ­r* t* ­t* r* t* t* 12 12 12 12 t1t
2

t1r

2

r1 t

2

(7)

is the direct product of the matrices describing the polarization transformations for each of photons composing a biphoton. It depends on the complex reflection and transmission coefficients rj = i sin j sin 2 and tj = cos j + i sin j cos 2, respectively, where j is the optical thickness of the plate, is the angle between the optical axis of the plate and vertical, and subscripts j = 1 and 2 refer to the photons with frequencies 1 and 2, respectively. Since the frequencies are nondegenerate, the optical densities of the plate j = (nej ­ n0j)h/j, where h is the geometric thickness of the plate, for the signal and blank photons differ from each other. By varying the angle , one can modify the matrix G and, therefore, obtain various states of ququart. The photon frequencies are chosen arbitrarily such that they satisfy the relation 1 + 2 = p. In our scheme, in order to choose the frequency of one of the photons, a narrowband filter with the maximum transmission at a wavelength of 635 nm was used. In this case, the wavelength of the conjugate photon was equal to 667.4 nm. The spectral width of each of the photons was determined by the length of the crystal and was equal to 2 nm. 2.2. The measurement block consists of two quartz plates placed in front of the Brown­Twiss scheme with a polarization-insensitive light splitter and polarization filters in both arms. The polarization filters separate the vertical (V) polarization. Counts in each arm are detected by photodetectors pulses from which are directed to a coincidence scheme. The trigger window of the coincidence scheme was T = 1.7 ns. A feature of the experimental setup is the absence of filters in front of the detectors. Owing to this feature, each detector is recorded photons with both frequencies. A recorded event is a coincidence-scheme count that occurs when one detector records a photon with the frequency 1 and the other detector records a photon with the frequency 2. Since the vertical polarization component is separated, projection occurs on the state |VV | V 1 , V 2 . In the framework of the proposed measurement method, the thickness of plates may be chosen arbitrarily. In our experiment, plates with thicknesses h1 = 821.5 µm and h2 = 712 µm were used. 3. MEASUREMENT PROCEDURE The joint action of two plates and polarization filters provides the basis for the procedure of projecting the

initial (generally unknown) state. The quantum tomography of an arbitrary ququart state is performed by means of the detection of pair coincidences of photocounts of detectors for various positions of the quartz plates. In order to reconstruct the initial state, it is necessary to carry out no less than d2 = 16 measurements of the projections (where d = 4 is the dimension of Hil-bert space). This procedure is mathematically expressed as a system of independent linear equations relating combinations of imaginary and real parts of moments (4) and (5) to the number of detected events. Such a system can be obtained by choosing four different positions of the first plate (1QP). Then, for each fixed rotation angle, it is sufficient to conduct four measurements for various angles of the second plate (2QP). However, measurements in the experiment were carried out for a larger number of the second plate. The extra measurements make it possible to increase in the set of statistical data for more accurate reconstruction of the density matrix of the initial state [5]. The transformations made by the plates are expressed in the form | out
kl

= G ( 1, k ) G ( 2, l ) | in ,

(8)

where l and k are the rotation angles of the first and second plates, respectively. Disregarding the normalization, the number of events detected in the experiment (pair coincidences of the photocounts Rkl) is the projection of the transformed state |out onto the state |VV determined by the orientation of the polarization filters. This projection is given by the expression Rkl |VV|outkl |2. Thus, the coincidence rate for each run is determined by the rotation angles of the plates 1QP and 2QP. The experiment was aimed at reconstructing ququart states obtained for several angles of the driving plate. Specifically, the states corresponding to the plate orientation = 0 and 20° are analyzed. For each input state, four measurement runs were conducted. Each run is specified by the plate orientation angle of the 1QP plate = 90°, 105°, 120°, and 135° for the complete rotation of the 2QP plate by 360° with a step of 10°; i.e., it includes 148 measurements. Figure 2 shows the normalized coincidence number for 30 s as a function of the orientation angle of the 2QP plate for = 20°. When processing experimental data, the random coincidence rate Nran, which is expressed in terms of the rate of averaged single counts from each photodetector and the coincidence-window width of the scheme as Nran = N1N2T, is extracted from the coincidence number. To compare the reconstructed quantum state with the theoretical one, the fidelity concept is used. For pure states, this quantity is defined as F = exp|
theor 2

.

(9)

The F value lies in the range from 0 to 1. If the reconstructed state is identical to the theoretical one, F = 1. If
JETP LETTERS Vol. 82 No. 3 2005


RECONSTRUCTION OF THE POLARIZATION STATES

167

Fig. 2. Normalized coincidence rate vs. the rotation angle of the second quartz plate 2QP for the rotation angles of the first plate (1QP) = (a) 90°, (b) 105°, (c) 120°, and (d) 135°. The solid lines are the theoretical calculations.

the states are orthogonal, F = 0. To reconstruct the initial state, we used the root approach that was developed in [15, 16] and was successively applied to reconstruct states of optical qutrits [3­5]. When processing the experimental data, we separate two types of errors-- instrumental and statistical--that lead to the difference of the reconstructed state from the ideal state. The former errors are caused by the quite rough inclusion of the effect of random coincidences, errors in the specification of the plate rotation angles, error of determination of the optical thickness of the plates, finite width of the biphoton-field spectrum, etc. The latter errors are associated with statistical fluctuations caused by the Poisson character of the detected-event number. As was
Results of the statistical processing of experimental data. The left column presents the angle of the quartz plate 3QP specifying the input (reconstructed) state. The second column contains the theoretical (known) components of the prepared vector of state. The third column presents the result of the reconstruction of the state. The fourth column shows the fidelity of the experimental and theoretical values State vector 0° theory experiment |theor = (c1, c2, c3, c4) |exp = (c1, c2, c3, c4) 0 0 0 1 0.8097 ­ 0.4568 ­ 0.3527i ­ 0.0103 ­ 0.0859i ­ 0.0316 + 0.0529i
Vol. 82

shown in [5], the effect of statistical fluctuations is negligibly small compared to the effect of instrumental errors if the total number of detected events is much larger than the so-called coherence volume. Numerical calculations show that the coherence volume in a given experimental run is estimated as 2140 on average and does not exceed 4340 with a C.L. of 95%. At the same time, the total number of detected counts in the experiment was 280 000. Thus, errors of the statistical reconstruction of ququart states in the experiment under consideration were associated with instrumental errors. The table presents the results of the measurement of states |in. The high values of fidelity F corroborate the applicability of the proposed tomography procedure to the reconstruction of ququart states. We emphasize that the states |in prepared in the experiment do not cover the entire set of states (1), because they were obtained under separate phase-plateinduced SU(2) transformations (7) of each of the photons composing the biphoton. They present a class of so-called factorized states when | = | 1 | 2 . However, to verify the applicability of the general reconstruction method, the choice of this class of states is sufficient. 4. CONCLUSIONS A procedure has been proposed for reconstructing an arbitrary state of a quantum optical system in Hilbert space with the dimension d = 4. The procedure was tested in application to pure factorized ququart states, which are obtained as a result of the phase-plateinduced unitary transformation of the polarized state of an initial biphoton. The main feature of this reconstruction method is that a quantum state is subjected to unitary transformation as a whole indivisible object, which can be transmitted both through open space and fiber optic lines. This work was supported in part by the Russian Foundation for Basic Research (project no. 03-0216444a) and the Council of the President of the Russian Federation for Support of Young Russian Scientists and Leading Scientific Schools (project no. NSh166.2003.02). REFERENCES
1. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Phys. Rev. A 66, 012 303 (2002). 2. H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062 308 (2000). 3. Yu. I. Bogdanov, M. V. Chekhova, S. P. Kulik, et al., Phys. Rev. Lett. 93, 230 503 (2004). 4. Yu. I. Bogdanov, L. A. Krivitsky, and S. P. Kulik, Pis'ma Zh. èksp. Teor. Fiz. 78, 804 (2003) [JETP Lett. 78, 352 (2003)]. 5. Yu. I. Bogdanov, M. V. Chekhova, L. A. Krivitsky, et al., Phys. Rev. A 70, 042 303 (2004).

Fidelity F

20°

­ 0.0555 ­ 0.0204i ­ 0.0059 + 0.005i ­ 0.0425 + 0.0052i 0.9973 0.8067 ­ 0.4847 ­ 0.3304i 0.023 ­ 0.0554i ­ 0.0174 + 0.0413i
No. 3 2005

0.996

0.998

JETP LETTERS


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BOGDANOV et al. 12. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980) [in Russian]. 13. D. N. Klyshko, Zh. èksp. Teor. Fiz. 111, 1955 (1997) [JETP 84, 1065 (1997)]. 14. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986; Nauka, Moscow, 1973). 15. Yu. I. Bogdanov, Fundamental Problem of Statistical Data Analysis: Root Approach (Mosk. Inst. èlektron. Tekh., Moscow, 2002) [in Russian]; physics/0211109. 16. Yu. I. Bogdanov, Opt. Spektrosk. 96, 735 (2004) [Opt. Spectrosc. 96, 668 (2004)].

6. N. Langford, R. B. Dalton, M. D. Harvey, and J. L. O'Brien, Phys. Rev. Lett. 93, 053 601 (2004). 7. R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, Quantum Inform. Comput. 4, 93 (2004). 8. J. C. Howell, A. Lamas-Linares, and D. Bouwmeester, Phys. Rev. Lett. 88, 030 401 (2002). 9. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052 312 (2001). 10. Y. H. Kim, S. P. Kulik, and Y. Shih, Phys. Rev. A 63, 060 301 (2001). 11. A. V. Burlakov, S. P. Kulik, Yu. I. Rytikov, and M. V. Chekhova, Zh. èksp. Teor. Fiz. 122, 738 (2002) [JETP 95, 639 (2002)].

Translated by R. Tyapaev

JETP LETTERS

Vol. 82

No. 3

2005