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Journal of Experimental and Theoretical Physics, Vol. 97, No. 4, 2003, pp. 846­857. Translated from Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 124, No. 4, 2003, pp. 943­955. Original Russian Text Copyright © 2003 by Krivitskioe, Kulik, Penin, Chekhova.

MISCELLANEOUS

Biphotons as Three-Level Systems: Transformation and Measurement
L. A. Krivitskioe, S. P. Kulik*, A. N. Penin, and M.V. Chekhova
Moscow State University, Vorob'evy gory, Moscow, 119992 Russia *e-mail: skulik@qopt.phys.msu.su
Received April 28, 2003

Abstract--Two algorithms for measuring the polarization state of the biphoton field prepared in the form of a three-level system (qutrit) are considered. On the basis of the general approach developed by Klyshko [Zh. Eksp. Teor. Phys. 111, 1955 (1997)] for describing the polarization properties of single-mode electromagnetic fields in the fourth order in the field, a procedure for measuring the polarization density matrix of qutrits is proposed and implemented. © 2003 MAIK "Nauka / Interperiodica".

2

1. INTRODUCTION In a number of protocols of quantum information, use is made of states of light with definite (i.e., preset) properties. For instance, the so-called Bell states [1] are used in quantum teleportation, high-density coding, or quantum cryptography. Such states are obtained as a result of spontaneous parametric scattering of light [2- 4]. Polarization-compressed states, in which the fluctuations of one or several Stokes parameters are suppressed, are widely discussed in the literature [5, 6]. A particular case of such states is "scalar light" for which fluctuations of all Stokes parameters are suppressed [7] (light of this type was recently obtained in experiments [8]). In quantum cryptography [9], the secrecy in the key distribution increases when multilevel (in particular, threelevel) states of light are used [10­12]. The manifestation of the geometrical phase for such (three-level) optical systems is of certain interest [13, 14]. Obviously, the application of states of light with preset properties in certain experiments presumes the solution of the following three interconnected problems: (i) generation of such states; (ii) their transformation during transmission over a communication channel; (iii) application of a reliable procedure for controlling a state at a given instant. In recent years, the procedure known as quantum tomography of light (tomos stands for layer and grapho means to write). This procedure aims at reconstructing the initial state of an electromagnetic field by measuring several projections of this state in different bases. Such a state can be recorded with the help of a wave function, density matrix, or quasiprobability function. It is probably more appropriate to apply the term "quantum tomography" to the quasiprobability function since this function permits a visual representation of the state in the form of a three-dimensional image. However,

analysis of the literature shows that the term tomography is being used now in a wider sense as reconstruction of the initial state. Among a large number of publications in this field, we will mention only those in which this procedure was used directly in experiment. For example, the reconstruction of the initial quasiprobability function was carried out in [15-17] during the measurement of states characterized by continuous variables, such as quadrature- or polarization-compressed light. Tomography of the states of discrete variables, which are realized with the help of polarizationspatial cubits, was carried out in [18]. In this case, the polarization state of the biphoton field generated in the frequency-degenerate noncollinear mode can be treated as a pair of qubits. First attempts of implementation of the qutrit tomography were made in [19]. 2. BIPHOTON AS A THREE-LEVEL SYSTEM (QUTRIT) This study is devoted to an analysis of the procedure for transformation and measurements of the states of light that can be described in terms of a three-level system. We consider the polarization state of frequencydegenerate biphoton radiation in a single spatial mode. In the general case, such states can be prepared using spontaneous parametric scattering of light from three nonlinear crystals oriented in a special way. The states of light generated during spontaneous parametric scattering can be represented in the form 1 | = |vac + -2

1 2

2


k, k'

F

k, k'

|1 k, 1 k' ,

(1)

where |vac is the vacuum state, the quantity Fk, k' is called the biphoton amplitude, and |1k, 1k' is a state with one (signal) photon in mode k and one (idler) photon in mode k'. The modes are characterized by frequency,

1063-7761/03/9704-0846$24.00 © 2003 MAIK "Nauka / Interperiodica"


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direction, and polarization. We will henceforth assume that the modes differ only in the polarization. The meaning of quantity Fk, k' is that the square of its modulus gives the probability of recording two photons in the two polarization modes k and k'. Real states of the biphoton field possess fixed frequency and spatial (angular) spectra. When we speak of the single-mode approximation in frequency and angle, we assume that states (1) cannot be distinguished in respect of these parameters in a certain experiment. In other words, for different frequencies and spatial components of such radiation, no delays exceeding the corresponding coherence scales of the field can emerge during its preparation and propagation. Disregarding the vacuum component and taking into account all possible ways of distribution of two photons (that are indistinguishable in frequency and direction of propagation) between the two polarization modes, we obtain | = c 1 |H, H + c 2 |H, V + c 3 |V , V = c 1 |2, 0 + c 2 |1, 1 + c 3 |0, 2 . (2)

matrix known as the coherent matrix of the fourth order in the field, A DE K4 = D* C F E* F* B ,

(4)

with six moments as components. The diagonal components are formed by the real moments A a a ,
2 2

B b b ,
2 2

C a ab b .


(5)

These moments characterize the correlation of intensities in polarization modes H and V. In the general case, the nondiagonal components are complex-valued: D a ab ,
2

F a b b ,
2

E a b .
2

(6)

In particular, for a single-mode biphoton field, averaging is carried out over wave function (2), and moments (5) and (6) assume the form A = 2c
2 1

= 2d1, C= c
2 2

2

B = 2c = d2,
2

2 3

= 2d3,

2

Here, we have used two equivalent representations of state (1) in the Fock basis. For example, notation |H, H = |2, 0 indicates that both photons are in the horizontal polarization mode. The complex amplitudes of states (ci = di exp(ii ), i = 1, 2, 3) satisfy the normalization condition

(7)

D=

2 c* c2 , 1

F=

2 c* c3 , 2

E = 2 c* c3 . 1

(8)


i=1

3

c

2 i

= 1.

(3)

It can be seen that moments (6) and (8) contain relative phases of three basis states. It should be noted that the moments defined in relations (7) and (8) coincide except for constants with the components of the density-of-states matrix (2), | | ,
mk

32

State (2) describes a three-level system; consequently, the biphoton field can be treated as the object of investigation of the properties of this system. Since the state of a two-level system is known as qubit in the quantum theory of information, state (2) will be referred to as a quantum trit, or qutrit. A detailed description of the characteristics of biphotons (qutrits) and their visual geometrical representation can be found in [20]. We assume that polarization properties of a singlemode electromagnetic field are described by the creation ( a H a, a V b) and annihilation (aH a, aV b) operators for photons in the polarization modes H and V. These operators satisfy the conventional commutation relations [ a, a ] = [ b, b ] = 1 ,


= cm c* , k

(9)

which will be referred to as the polarization density matrix (i.e., the matrix written in the polarization basis). The condition 2 = of the purity of the state imposes additional constraints on the moments. For example, we obtain the constraint ABC = DFE *, (10)

[ a, a ] = [ b, b ] = [ a , a ] = 0 .


which makes it possible, for example, to eliminate moment E. In a mixed state, averaging over the wave function should be supplemented with averaging over the classical probability distribution function Pj , where Pj is the probability of finding the system in a pure state j , P j = 1. j



It was shown in [21] that the parameters determining the polarization properties of a single-mode field in the fourth order can be combined into a Hermitian

Since the wave function is defined to within a constant phase factor, we have only two independent phases characterizing state (2), viz., 21 = 2 ­ 1 and 31 = 3 ­ 1 .
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848 s = /4 s s

KRIVITSKIOE et al.



BS

D2

The action of nonpolarizing beam splitter executing the spatial separation of signal and idler photons is described by the relation a 1 as 1 = ------ + -----b 2 2 b a i . b (11)

i i D1

i = /4

Then the complete transformation for each channel has the form
CC



a b

2

Fig. 1. Schematic diagram of the setup for measuring an arbitrary state of the qutrit: BS is the beam splitter, and D1 and D2 are photodetectors with a quarter-wave plate and a rotating polarizer in from of each detector; photocurrent pulses from the detectors are fed to the paired coincidence circuit CC, detecting moment R.

' 1 sin2 cos sin = -----' 2 cos sin cos2

t r a . (12) ­ r * t * b

Here, a and b are the photon annihilation operators in the polarization modes H and V prior to the transformation, is the angle of orientation of the polarizer relative to the vertical axis, and r and t are the amplitude coefficients of reflection and transmission of /4 phase plates, respectively, t = cos + i sin cos 2 , r = i sin sin 2 . (13)

2

Thus, in order to define a pure state of the three-level system (2), we must define 2S ­ 2 = 4 real parameters. In order to define a mixed states, S2 ­ 1 = 8 real-valued parameters are required. Here, S = 3 is the dimension of the Hilbert space of a biphoton (qutrit). 3. QUTRIT TOMOGRAPHY The idea of measuring the parameters of state (2) can be formulated as follows. The initial unknown state is subjected to preset polarization transformations so that the final state, which is fed to the input of the recording system, corresponds to the known combination of moments (5) and (6). Since we are speaking of the measurement of fourth-order moments in field, the system of recording of biphotons (qutrits) must consist of a beam splitter and a pair of detectors with inputs connected to the photocount coincidence circuit (the Brown­Twiss circuit [22]). We can consider two methods for carrying out polarization transformations. Method 1. First, the initial biphoton beam is split into two channels by a beam splitter (Fig. 1).1 These channels will be referred to as the signal (s) and idler (i) channels. Further, in each spatial mode of the beam splitter, transformations on the signal photon and the idler photon forming the initial biphoton takes place. Transformations are carried out with the help of quarter-wave phase plates and polarization prisms (polarizers). We can write these polarization transformations in the Heisenberg representation.
1

Here we have introduced the parameter = (no ­ ne)/ of the plates (h is the plate thickness) and is the angle of their rotation relative to the vertical axis. This gives t = ( 1 + i cos 2 ) / 2 , t = i sin 2 / 2 . (14) (15)

2

Thus, we have four parameters (two for each channel) defining the polarization transformations. These are the angles and of rotation of the plates and the polarizer. In experiments with a biphoton field, the number of photocount coincidences for the detectors mounted in the signal and idler channels or the fourth-order moment of the form R
s, i ( b 's ) ( b 'i ) b 's b 'i = R ( s, s, i, i )

(16)

We will take into account only the events leading to the coincidence of photocounts, i.e., the events when the signal and idler photons fall into different input modes of the beam splitter. This takes place in half the total number of trials.

is measured. In the general case, this moment is a combination of six moments forming matrix K4 . The task of polarization tomography is to find six moments, (5) and (6), from relations of the form (16) upon a change in parameters s , s , i , and i of the phase plates and polarizers. Several remarks can be made concerning the experimental procedure of measurement of a state. 1. We assume that the source generating biphotons prepares them in state (2), which is stationary. Each act of measurement is accompanied by the destruction of such a state. However, the experimenter has a sufficiently large set of copies of the initial state, so that the destruction of some of these states does not affect the remaining ones. Such an "ensemble" approach leads to the conclusion that in each next measurement we are
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BIPHOTONS AS THREE-LEVEL SYSTEMS: TRANSFORMATION AND MEASUREMENT Table 1 2 3 4 5 6 7 8 9 s1 0 0 0 45° 45° 45° 45° ­ 45° 45° s ­ 90° ­ 90° 0 0 ­45° ­45° 0 ­22.5° ­45° i1 0 0 0 0 0 0 0 45° 45° i ­ 90° 0 0 0 0 ­ 90° ­ 90° 22.5° 45° Field moment being measured A/4 C/4 B/4 (B + C + 2Im F)/8 (B + C ­ 2Re F)/8 (A + C ­ 2Re D)/8 (A + C + 2Im D)/8

849

(A + C ­ 2Im E)/16 (A + C ­ 2Re E)/16

dealing with the same unperturbed state and, hence, the results of measurement provide information on the initial state. 2. The number of measurements required for complete reconstruction of arbitrarily defined state (2) is determined by the number of independent real numbers defining this state. However, in experiment, we are dealing with non-normalized states and, hence, the number of measurements increases. Normalization is set for measuring moments A, B, and C defined by formulas (5) and (7). Using relations (3) and (7), we obtain A + 2C + B = 2. (17)

II. Let us suppose that s = ­ 45°, s = ­ 22.5°, i = 45°, and i = 22.5°. In this case, we have 1 t s = ------ , 2 i r s = ­ ------ , 2 1 t i = ------ , 2 i r i = ------ . 2

The remaining three moments, D, E, and F, carry information about phases 21 = 2 ­ 1, 31 = 3 ­ 1, and 32 = 31 ­ 21. However, since these moments are complex-valued, their real and imaginary parts (which are associated with cosine and sines of the phases, respectively) must be measured separately. Thus, the number of measurements required for complete reconstruction of state (2) is equal to seven for a pure state and nine for a mixed state. 3. In order to reduce the effect of errors on the parameters of the state being measured, we must minimize the number of moments appearing in relation (16). Let us consider several examples. I. Let us suppose that s = 0, s = ­ 90°, i = 0, and i = 0. In this case, we have 1+i t s = ---------- , 2 r s = 0, 1+i t i = ---------- , 2 r i = 0.

Then it follows from relations (12) and (16) that Rs, i C/4.

We find that Rs, i (A + B + Im E )/16. The complete set of measurements required for determining the unknown state (2) can be represented in the form of a table. The experimental setup corresponding to this set contains a quarter-wave plate and a rotating polarizer each in the signal and idler channels. It should be noted that the number of polarization transformers appearing in each channel of the circuit (see Fig. 1) is equal to two and not to three, as proposed in [18], where an additional half-wave plate was used. This is possible since the action of a half-wave plate with parameter /2} and a polarizer with the fixed orientation 90° is equivalent to the action of a single rotatable polarizer with parameter = ­ 2/2. Thus, the first seven rows of the table contain orientations of the plates and polarizers, which make it possible to measure three real (A, B, C) and two complex (D, F) moments. These measurements are sufficient for complete reconstruction of a pure state. If the state is mixed, two more measurements must be made and the complex moment E must be determined. The verification of condition (10) can give the answer to the question concerning the purity of the state. Method 2. The polarization transformations are carried out on biphotons (qutrits) as integral objects. Then these objects are directed to the modified Brown­Twiss circuit, where their spatial separation and detection takes place (Fig. 2). The first beam splitter (PBS) is of the polarization type: it completely reflects light with the vertical (V) polarization and transmits light with the horizontal (H) polarization. The nonpolarization beam splitters BS1 and BS2 mounted behind the PBS spatially separate light of the same polarization, while detectors
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850 , ; = 45° D4 D1 PBS BS2

KRIVITSKIOE et al. D3
2 2 tr t 2 G ­ 2 tr * t ­ r 2 r *2 ­ 2 t * r *

2t*r , 2 r* r
2

(18)

so that ' = G or c c '1 1 c '2 = G c 2 . c3 c '3

BS1 D2 CC

(19)

4

Fig. 2. Schematic diagram of a setup for partial tomography of state of qutrits. The polarization transformations on biphotons as integral objects are carried out with the help of /2 and /3 phase plates. Beam splitters PBS, BS1, and BS2 are intended for the spatial separation of signal and idler photons with parallel or orthogonal polarizations. The coincidence circuit registers the number of coincidences between all possible pairs of detectors: D1­D2, D1­D3, D1­D4, D2­D3, D2­D4, and D3­D4. Dashed contour encircles a phase plate and a polarizer oriented at an angle of = 45°, which are mounted in an additional protocol.

Coefficients t and r were introduced by relations (13). In the SchrÆdinger representations, the moments A', B', and C' being measured contain averaging over the wave function ' connected with the initial wave function through transformation (19). Our aim is to establish one-to-one correspondence between the results of measurements of moments A', B', and C' and moments (5) and (6) prior to the polarization transformations. Let us consider the action of various transformers on biphotons. 1. The /2 plate: = /2, = 22.5°. In this case, we have i t = r = ------ , 2 ­ 1/2 ­ 1 / 2 ­ 1/2 G ­ 1/ 2 0 1 / 2 . ­ 1/2 1 / 2 ­ 1/2 (20)

are placed in their output modes. As a result, the coincidence count rate between photocounts of the pair of detectors D1­D2 is proportional to the fourth-order moment of the field, R
D 1­ D 2

b b B',
2 2

while that between detectors D3­D4 is proportional to the moment R
D 3­ D 4

a a A'.
2 2

The three moments being measured can be expressed with the help of relations (19) and (20) in terms of the parameters of the input state :
2 12 2 2 A' = 2 c '1 = -- [ d 1 + d 3 + 2 d 1 d 3 cos ( 1 ­ 3) ] + d 2 2 (21) 2 2 + ------ d 1 d 2 cos ( 1 ­ 2 ) + ------ d 2 d 3 cos ( 2 ­ 3 ) , 2 2 2 12 2 2 B' = 2 c '3 = -- [ d 1 + d 3 + 2 d 1 d 3 cos ( 1 ­ 3) ] + d 2 2 (22) 2 2 ­ ------ d 1 d 2 cos ( 1 ­ 2 ) ­ ------ d 2 d 3 cos ( 2 ­ 3 ) , 2 2

Coincidences between photocounts of any pairs of detectors (D1­D3, D1­D4, D2­D3, and D2­D4) give the value of the moment for fields with orthogonal polarizations: R
D 1, 2 ­ D 3, 4

a ab b ---------------------- C '. 4


Let us consider the polarization transformations carried out in the circuit depicted in Fig. 2 in the SchrÆdinger representation. For this, we write the wave function (2) in the form of a column whose elements are normalized amplitudes ci (i = 1, 2, 3). The transformations executed over these polarization states of the biphoton field by phase plates can be described by the unitary 3 â 3 matrix [23]

1 C ' = -- c '2 4

2

12 2 = -- [ d 1 + d 3 ­ 2 d 1 d 3 cos ( 1 ­ 3 ) ] . (23) 8

It should be noted that the last expression demonstrates the polarization anticorrelation effect in explicit form.
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BIPHOTONS AS THREE-LEVEL SYSTEMS: TRANSFORMATION AND MEASUREMENT

851

The number of coincidences of photocounts between the detectors recording radiation in orthogonal polarization modes decreases to zero if the number of biphotons in the vertical polarization mode is equal to the number of biphotons in the horizontal polarization mode (7) (i.e., A = B) and the phase shift between the basis states |2, 0 and |0, 2 vanishes (1 = 3). 2. The /2 plate: = /2, = 30°. In this case, we have i t = -- , 2 A' = 2 c '1
2

3 r = i ------ , 2
2 3

12 2 = -- { d 1 + 6 d 2 + 9 d 8

­2 6 d 1 d 2 cos ( 1 ­ 2 ) + 6 d 1 d 3 cos ( 1 ­ 3 ) ­6 6 d 2 d 3 cos ( 2 ­ 3 ) } . B' = 2 c '3
2

(24)

1 2 2 = -- { 9 d 1 + 6 d 2 + d 8

2 3

+6 6 d 1 d 2 cos ( 1 ­ 2 ) + 6 d 1 d 3 cos ( 1 ­ 3 ) +2 6 d 2 d 3 cos ( 2 ­ 3 ) } .

(25)

3. The /3 plate: /3, = 45°. In this case, we have 1 t = -- , 2 A' = 2 c '1
2

3 r = i ------ . 2
2 3

1 2 2 = -- { 9 d 1 + 6 d 2 + d 8

(ii) transformer = /2, = 22.5°; from relation (23), we find cos(1 ­ 3); (iii) transformer = /2, = 30°; from relations (24) and (25), we find cos(1 ­ 2) and cos(2 ­ 3); (iv) transformer = /3, = 45°; from relations (26) and (27), we find sin(1 ­ 2) and sin(2 ­ 3). Comparing two of the above protocols, we note that the second protocol is less preferable from the viewpoint of the accuracy that can be attained. This is due to the fact that expressions (24)­(27) for the cosines and sines of the relative phases of the basis states contain a large number of quantities being measured, which increases the error in their determination. In addition, the second method is applicable only to pure states (2); in this case, we can speak only of partial tomography of qutrits. On the other hand, the circuit depicted in Fig. 2 is more convenient since it permits the simultaneous measurement of three moments (A', B', and C') for a given transformer and, hence, the entire protocol can be realized in a shorter time. This factor must probably be taken into account for using the results of tomography in a more complex protocol (e.g., in quantum cryptography on the basis of qutrits [10]). It should also be noted that the choice of the polarization transformers in the second tomographic method is not limited to /2 and /3 plates. Any phase plate can ensure the relation between the observed (A', B', C') and input (A, B, C) moments. The only exception is the circuit with only one half-wave plate (t = i cos 2, r = i sin 2); in this case, only cosines of the corresponding phases can be determined. The modified version of the circuit depicted in Fig. 2 gives the relations A, B, C
f (, ; = 45°)

­6 6 d 1 d 2 sin ( 1 ­ 2 ) ­ 6 d 1 d 3 cos ( 1 ­ 3 ) ­2 6 d 2 d 3 sin ( 2 ­ 3 ) } , B' = 2 c '3
2

(26)

A', B', C '

12 2 = -- { d 1 + 6 d 2 + 9 d 8

2 3

+2 6 d 1 d 2 sin ( 1 ­ 2 ) ­ 6 d 1 d 3 cos ( 1 ­ 3 ) ­6 6 d 2 d 3 sin ( 2 ­ 3 ) } .

(27)

obtained using a single arbitrary phase plate with parameters and and a polarizer separating the linear polarization at an angle of 45°, which are mounted in front of the polarization beam splitter enclosed in the dashed contour in Fig. 2). However, the calculation of inverse transformers using such a circuit is quite cumbersome and is not included in this work; we just mention that such a method of qutrit tomography is possible in principle. 4. EXPERIMENT Below, we describe the procedure of qutrit tomography, in which polarization transformations are carried out over the signal and idler photons separately after their spatial separation. The schematic diagram of the experimental setup for single-mode biphoton tomography is shown in Fig. 3. Pumping is carried out by a CW argon laser with wavelength = 35 nm and power P = 120 mW. The pumping beam is directed to a nonlinear lithium iodate (LiIO3) crystal of length l = 1 cm; in the bulk of this
Vol. 97 No. 4 2003

2

4. Finally, in the absence of a transformer, when G I (I is the unit matrix), the moduli of the amplitudes of the basis states (2), A' A = 2 d 1 ,
2

2

B' B = 2 d 3 ,
2

4 C ' C = d 2 (28)
2

can be established in three dimensions. Thus, for the circuit considered here, the protocol of polarization tomography of qutrits is as follows: (i) transformer is absent; the real amplitudes of states d1 , d2 , and d3 are determined from relations (28);

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852 M 351 nm LiO3 351 nm 702 nm M ISP51 M D1 QWP1 PF1 IF CC UVM QP D FM D Ar

KRIVITSKIOE et al.

QWP2 PF2 IF D2 BS

Fig. 3. Experimental setup: Ar is an argon laser; M are mirrors; LiIO3 is a nonlinear crystal; UVM is a mirror reflecting the ultraviolet pumping radiation and transmitting the biphoton radiation; QP is the driving phase plate D are diaphragms; FM is the folding mirror; BS is the beam splitter, QWP1 and QWP2 are quarter-wave plates; PF1and PF2 are polarization prisms; IF are interference filters; D1 and D2 are photodetectors; and ISP51 is the spectrograph intended for controlling the biphoton field spectrum.

crystal, spontaneous parametric scattering takes place. The crystal is oriented so that biphotons are emitted in the frequency-degenerate collinear regime. The polarizations of both photons are orthogonal to the polarization of the pumping beam (type I synchronism, or e oo interaction). In this regime, the optical axis of the crystal forms an angle = 58° with the wave vector of pumping radiation. After the passage of pumping radiation through the crystal, it is extracted from the system by mirror UVM reflecting radiation at a wavelength of 351 nm and transmitting light at the double wavelength. A system of diaphragms D and an interference filter IF with a central wavelength of 700 nm and a half-amplitude width of 10 nm is used to separate one spatial and one frequency mode of the biphoton field. The spectral width of spontaneous parametric scattering for the given crystal is 20 nm. The width of the spatial correlation function of spontaneous parametric scattering was determined in our experiments by the divergence of the pumping beam and was estimated as p 3 â 10­4 rad. This quantity determined the diameter of diaphragms D. As a result of such a mode discrimination, the effect of the finite spatial frequency biphoton radiation spectrum on subsequent polarization transformations can be disregarded. Various polarization states were prepared directly using a thin quartz plate QP (driving plate). Each value of the optical axis orientation of plate QP corresponds to a certain polarization state of biphotons fed to the input of the measuring system. The plate thickness is h = 824 ± 0.5 µm; the radiation loss due to reflection

from the plate faces amounts approximately to 8% and the error in the orientation of its optical axis is equal approximately to 1°. The plate thickness was chosen so that the transformation executed by the plate depended on the wavelength insignificantly within the filter transmission band. The measuring block is formed by an intensity interferometer with a 50-% nonpolarization beam-splitting mirror BS and two FEU-79 detectors D1 and D2 operating in the photon count mode; the quantum efficiency is ~ 10­2. The beam-splitting mirror is mounted at a small angle at a small angle (about 12°) to the beam so that the polarization state of light does not change as a result of reflection and transmission. A quarter-wave plate QWP and a rotating polarization filter PF (GlanThompson prisms) is installed in each arm of the interferometer. Zero-order quartz plates for a wavelength of s = 702 nm with bleached faces are used. The loss introduced by the polarization filters amounts to 8­12% and the error in the orientation of the optical elements is approximately equal to 2.0°. After amplification and amplitude discrimination, signals from detectors are supplied to the coincidence circuit CC with a resolution time of T 5 ns. The frequency-angular spectrum of the biphoton field is monitored with the help of an ISP-51 spectrograph. Since the SU2 transformation executed by the phase plate QP is used for preparing the initial states, it is impossible to obtain an arbitrary state of a biphoton (qutrit) in the experiment discussed here.2 Nevertheless, the qutrit state behind the plate belongs to a subclass of states (2) with certain relations between parameters di and i and is described by the expression c1 0 c = G 0 3 c3 1

2 2

(29) 2 ­ 0.9725 sin 2 = 2 sin 2 ( 0.9724 cos 2 + i â 0.1636 ) 2 0.0275 ­ 0.9724 cos 2 + i â 0.3273 cos 2 where matrix G for a QP plate with parameters n = no ­ ne = 0.0089 and = 32.82 was calculated using formula (18). Here, is the angle of orientation of the plate relative to the vertical axis, which determines the state (29) of the qutrit and which is measured from the direction of polarization of the initial biphoton beam (not transformed by the quartz plate QP). In measurements 1-3 (see table), the diagonal components of matrix K4 are determined, while a combination of diag2

2

In order to obtain such a state, a transformation of the SU3 group is required; this can be done via nonlinear transformations with an extremely low efficiency.
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BIPHOTONS AS THREE-LEVEL SYSTEMS: TRANSFORMATION AND MEASUREMENT |d1| 1.0 (a) 0.8 0.6 0.4 0.2 0 0° |d
2 3| 2

853

|d2| 1.0

2

(b) 0.8 0.6 0.4 0.2 0 10° 20° 30° 40° 50° 0° 10° 2 ­ 3, 2, 0 ­1 20° 30° 40° 50°

1.0 0.8 0.6 0.4 0.2 0 0° 10° 20° 30° 40° 50° (c)

­2

(d)

­3 0° 10° 20° 30° 40° 50°

Fig. 4. The results of measurement of three amplitudes, d1 (a), d2 (b) and d (c), and two phases, 2 ( ), 2 ­ 3 ( ) (d). Different initial states of qutrits correspond to different orientations of driving plate QP. Solid and dashed curves correspond to calculated dependences.

onal and nondiagonal components is determined in measurements 4-9. It was shown in Section 3 that the first seven measurements are sufficient for a pure state. It should be noted that, proceeding to each next measurement, the orientation of only one polarization transformer has to be changed, which is convenient from the experimental point of view. The results of experiments in the form of dependences of amplitudes di and phases i of the states pre-

pared by the quartz plate QP for various values of orientation are represented in Fig. 4. Figure 5 shows the experimental dependences of the components of the coherence matrix K4 on . The results of calculation of components di and i (Fig. 4) and moments (Fig. 5) for different input states are shown for comparison. For a fixed value of the orientation 0 = 25° of the driving plate, the polarization density matrix of the corresponding state was reconstructed experimentally: i .



exp

0.271 0.345 + 0.074 i ­ 0.24 ­ 0.114 i = 0.345 ­ 0.07 i 0.508 ­ 00.316 ­ 0.075 0.221 ­ 0.24 + 0.114 i ­ 0.316 + 0.075 i

(30)

The set of eigenvalues for this matrix is as follows: 1 = 0.99, 2 = ­0.021, and 3 = 0.03. The trace of the 2 squared matrix (30) is found to be Tr( exp ) = 0.981. It can be seen from the curves shown in Figs. 4 and 5 that the maximal relative errors appear in the recon-

structed values of phases of states in the nondiagonal elements of the density matrix also. This is due to the fact that the phases are calculated using simultaneously the results of several measurements. The errors are superimposed and make a considerable contribution to the result.
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854 Re D, Im D 0.6 0.4 0.2 0 ­ 0.2 (a)

KRIVITSKIOE et al. Re F, Im F 0.2 0 ­ 0.2 ­ 0.4 ­ 0.6 ­ 0.4 0° 10° 20° 30° 40° 50° 0° 10° 20° 30° 40° 50° (b)

Fig. 5. The results of measurement of components of matrix K4 . Real ( ) and imaginary ( ) parts of moments D (a) and F (b). Different initial states of qutrits correspond to different orientations of the driving plate QP. The solid and dashed curves correspond to calculated dependences.

Several remarks concerning the properties of the reconstructed density matrix exp are appropriate here. This matrix is Hermitian and normalized; i.e.,
2 exp

will be eliminated in near future from the measuring procedure discussed here. 5. CONCLUSIONS We have considered two algorithms of measurement of the polarization state of a biphoton field prepared in the form of a three-level system (qutrit). On the basis of the general approach developed by Klyshko [21} for describing the polarization properties of single-mode electromagnetic fields in the fourth order in the field, we proposed and realized a procedure for measuring the polarization density matrix. As the initial states, we used a set of states that can be obtained with the help of spontaneous parametric scattering in a nonlinear crystal with a type I synchronism in the frequency-degenerate collinear regime. We hope that the procedure of measurement considered here will be useful in the realization of the quantum cryptography protocol based on qutrits [10] as well as in spectroscopic studies of the phase transition in ferroelectric crystals. Experimental data show that the anomalies observed in some crystals during the formation of the domain structure in the processes of elastic [25] and inelastic [25] small-angle scattering of light cannot be explained in the framework of the existing models. It would be interesting to apply the approach developed by us here in the solution of the inverse scattering problem for a phase transition. ACKNOWLEDGMENTS This study was supported financially by the Russian Foundation for Basic Research (project nos. 02-02-16664 and 03-03-16444) and INTAS (grant no. 2122-01).
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=

exp and Tr( exp ) = 1. It follows from the general properties of the density matrix that it must have positive 2 eigenvalues and satisfy the condition 0 Tr( exp ) 1. For the pure states studied by us here, there exists only one nondegenerate eigenvalue equal to unity, and 2 Tr( exp ) = 1. Due to the effect of various experimental errors, the reconstructed density matrix exp obviously does not meet the above requirements. In order to put in correspondence a "realistic" physical state satisfying the above properties to the experimentally measured matrix, we must carry out the procedure of the maximum likelihood estimation [18, 24]; this procedure and the results of its application are given in the Appendix. Let us briefly consider the main sources of errors in our experiments. First, this is a low quality of the phase plates (QWP) used for transforming the polarization state; the plate thickness does not satisfy the condition = /4. In addition, since experiments were made with a biphoton field possessing a finite spectral width (10 nm), zero-order plates ensuring the independence of the transformation of the wavelength should be used. However, the driving plate QP was a higher-order plate and different spectral components of the biphoton field were transformed in different manners. Second, we must mention the error in alignment of all polarization transformers. In our experiments, this error was about 2°, which gave rise to random errors in various realizations. The above sources of errors are of technical rather than of fundamental nature. We believe that these errors

2

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APPENDIX Procedure of Maximum Likelihood Estimation of Experimental Results One of the results obtained in this study is the reconstruction of the density matrix of a given state from the experimental results. However, the reconstructed density matrix of the system does not correspond to a real pure biphoton state in view of various kinds of experimental errors. The procedure of maximum likelihood estimation (MLE) that will be described below makes it possible to indicate the pure biphoton state corresponding to the experimental result with the highest probability. We will apply the MLE method [18] realized for a system of two polarization-spatial qubits (the dimension of space S = 4). The algorithm considered below was realized for three-level systems, or qutrits (the dimension of space S = 3). The MLE algorithm consists of the following three main parts. 1. Obtaining the expression for the density matrix of the three-level system in the general form. This matrix is a function of nine real variables (we denote this matrix as ph(q1, ..., q9) and satisfies the following three properties: it is nonnegative definite, normalized, and Hermitian. 2. The introduction of the likelihood function that shows the extent to which the obtained experimental data are close to the real "physical" state described by the density matrix ph(q1, ..., q9). The likelihood function (q1, ..., q9; R1, ..., R9) depends on nine arguments {q1, ..., q9} of the ideal density matrix and on nine experimental values {R}, where R is the coincidence count rate of photocounts (16) and = 1, 2, ..., 9 is the number of the corresponding measurement in the table. 3. Using the standard numerical optimization methods for the available experimental data {R1, ..., R9}, we obtain the set of variables { q 1 , ..., q 9 } for which function (q1, ..., q9; R1, ..., R9) assumes the minimal value. In this case, the best estimate for the experimental density matrix has the same form as the ideal density opt matrix of the optimized set of variables; i.e., ph( q 1 , ..., q 9 ). Let us consider the MLE procedure in greater detail. "Physical" Density Matrix The property of nonnegative definiteness for an arbitrary matrix can be represented in the form U 0. (A.1)
opt opt opt

into inequality (A.1), we obtain Q Q = ' ' 0,


(A.2)

where |' = Q| . The Hermiticity of the matrix is also obvious: (Q Q) = Q (Q ) = Q Q.


The matrix can be normalized as follows: u = Q Q /Tr ( Q Q ) .


(A.3)

1

A system with three degrees of freedom has a density matrix of dimension 3 â 3 with eight independent realvalued variables. For the convenience of subsequent computations, matrix Q is defined in the triangular form q1 0 0 Q ( q 1, ..., q 9 ) = q 4 + iq 5 q2 0 . q 8 + iq 9 q 6 + iq 7 0

(A.4)

Then, in accordance with Eq. (A.3), the physical density matrix can be written in the form ph ( q 1, ..., q 9 ) = Q Q /Tr ( Q Q ) .


(A.5)

It is also necessary for subsequent computations to express the elements of matrix Q in terms of the elements of matrix ph , i.e., to obtain a relation inverse to (A.5): Q M 12 M 21 31 13 11 ­ ------------------ ­ ------------- 33 M 11 33 M 12 -------------------M 11 33 31 --------- 33 0 . (A.6) 33 0

0 M 11 ------- 33 32 --------- 33

Here, Mij is the first-order minor of matrix ph , i.e., the determinant of the matrix obtained as a result of crossing out the ith row and the jth column of matrix ph . Likelihood Function In a system with the physical density matrix (A.5), the expected value of the number of coincidences being recorded is equal to R
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ph = N [ ( b 's ) ( b 'i ) b 's b 'i ] ,

The matrix written in the form U = QQ is nonnegative definite and Hermitian, Indeed, substituting this matrix
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where N is the normalization factor. We assume that the spread in the number of coincidences has a Gaussian statistics. Then the probability that the given set of values R1, ..., R9 will be obtained is given by 1 P ( R 1, ..., R 9 ) = ----------N norm
ph

set experimental values {R1, ..., R9} assumes its minimal value, we used the standard procedure with the packet Mathcad 2000-MINIMIZE. It was found that the state with the density matrix ph{ q 1 , ..., q 9 } describes the experimental results with the highest probability. The required initial estimate of parameters {q1, ..., q9} was obtained using relation (A.6) with a density matrix reconstructed from the experiment. It
opt opt

=1



9

( R ­ R ) exp ­ -------------------------- , (A.7) 2 2

ph 2

where ~ R is the standard deviation and Nnorm is the normalization constant. From the viewpoint of mathematical procedure, the problem of determining the maximum of functional (A.7) is equivalent to determining the minimum of the exponent. This function is called the likelihood function of the MLE method. This function can be written explicitly in the form ( q 1, ..., q 9 ; R 1, ..., R 9 ) =

should be noted that generally ph{ q 1 , ..., q 9 } does not necessarily correspond to a class of pure states; it must only satisfy the physical requirements imposed on the density matrix. In order to find out which pure state corresponds to the experimentally reconstructed matrix with the highest probability, we must require that
opt opt

=1



9

( R ­ R ) -------------------------- . (A.8) ph 2 R

ph

2

ph{q1, ..., q9} satisfies the condition Tr( ph (q1, ..., q9)) = 1. This reduces the number of independent variables to four and simplifies the numerical procedure significantly.
2

Numerical Optimization In order to find the set of parameters { q 1 , ..., q 9 } for which the probabilistic function (A.8) with the preopt opt

The MLE procedure considered above was carried out on the experimentally reconstructed density matrix exp in the form (30). As a result, the optimized density matrix



opt

0.33 0.329 ­ 0.207 i ­ 0.263 ­ 0.01 i = 0.329 + 0.207 i 0.459 ­ 0.257 ­ 0.176 i 0.211 ­ 0.263 + 0.01 i ­ 0.257 + 0.176 i

,

was obtained; the trace of this matrix is Tr( ph ) = 0.9994 and its eigenvalues are 1 = 0.99969, 2 = 00031, and 3 = 0.
2

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Translated by N. Wadhwa

SPELL: 1. qubits, 2. qutrit, 3. trit, 4. splitters

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