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Journal of Experimental and Theoretical Physics, Vol. 100, No. 3, 2005, pp. 521­527. Translated from Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 127, No. 3, 2005, pp. 589­596. Original Russian Text Copyright © 2005 by Krivitskioe, Kulik, Maslennikov, Chekhova.

ATOMS, MOLECULES, OPTICS

Preparation of Biphotons in Arbitrary Polarization States
L. A. Krivitskioe, S. P. Kulik*, G. A. Maslennikov, and M. V. Chekhova
Moscow State University, Moscow, 119992 Russia *e-mail: postmast@qopt.phys.msu.su
Received October 6, 2004

Abstract--An experiment on preparation of entangled photon pairs (biphotons) in an arbitrary polarization state is described. The biphotons are qutrits (three-state quantum systems). They can be used in ternary quantum cryptography protocols. A theoretically derived orthogonality criterion for the prepared biphotons is validated experimentally. The criterion can be used to identify orthogonal biphoton states. © 2005 Pleiades Publishing, Inc.

1. BIPHOTONS AS QUTRITS Most quantum cryptography protocols are based on binary encoding (with qubits) [1, 2]. Qubits can be prepared as polarization states of a single-photon wave packet, states of a spin 1/2 particle, states of a singlephoton wave packet in a two-arm interferometer, and in various other ways. In recent studies, it was proposed to use ternary logic (qutrits) in quantum cryptography instead of qubits [3­5]. These studies are generally motivated by the higher efficiency [3] and higher security [6] of quantum channels of higher dimension. A qutrit is a three-state quantum system, as a three-level atom or a spin 1 particle. However, photons are known as the best means of data transmission. There exist several methods for making photonic qutrits. In particular, ternary encoding can use photon states obtained in a three-arm interferometer [7], single-photon wave packets with helical wavefronts [8], and four-photon states created by parametric down-conversion [9]. The ternary quantum cryptography scheme proposed in [4] made use of polarization-entangled states of photon pairs (biphotons) obtained as a result of spontaneous parametric down-conversion (SPDC). It is the simplest method for preparing an arbitrary polarization state of a qutrit [10, 11], i.e., a state of the form | = c 1 |2, 0 + c 2 |1, 1 + c 3 |0, 2 (1)

frequency­angular spectrum. By virtue of the normalization c
2 1

+c

2 2

+c

2 3

=1

and the unimportance of the overall phase of state (1), a biphoton is defined by four real numbers, e.g., the two amplitudes d 1 c1 , and the two phases 13 arg ( c 1 c * ) , 3 23 arg ( c 2 c * ) . 3 d2 c
2

State (1) is also conveniently represented as a pair of photons in arbitrary pure polarization states [12]: a ( , ) a ( ', ' ) |v ac | = ----------------------------------------------------------- , a ( , ) a ( ', ' ) |v ac


(2)

with arbitrary amplitudes c1 , c2 , and c3 . The ket notation |m, n in (1) means m vertically polarized photons and n horizontally polarized ones. Theoretically, the resulting biphotons belong to the same spatiotemporal mode. Even though the biphotonic field created in an experiment always spans a frequency­angle spectrum of finite width, representation (1) is valid if the optical detector employed in the scheme does not resolve the

where a(, ) and a(', ') are the operators of creation of photons in arbitrary polarization modes characterized by azimuthal (, ') and polar (, ') angles on the PoincarÈ sphere. Representation (2) can be used to depict a photon pair on the PoincarÈ sphere. Furthermore, it can be shown [12] that important polarization characteristics of state (1), such as the Stokes vector and degree of polarization, are readily calculated by using representation (2). For example, the degree of polarization of state (1) is uniquely determined by the angular distance between the points representing the states , and ', ' on the PoincarÈ sphere: 2 cos ( /2 ) P = --------------------------------- . 2 1 + cos ( /2 ) (3)

It should be emphasized here that this quantity is interpreted as the conventional (classical) polarization

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degree defined in terms of the Stokes parameters S0 , S1 , S2 , and S3 [13]: S1 + S2 + S3 P ------------------------------- . S0
2 2 2

2. EXPERIMENT ON PREPARATION OF A BIPHOTON WITH ARBITRARY DEGREE OF POLARIZATION An arbitrary state having the form of (1) can be prepared by using an interferometric scheme in which SPDC is implemented by means of common pumping [17]. However, this scheme is impracticable because of its instability. In this study, arbitrary biphotons (qutrits) were prepared by using a different scheme, which does not require the use of an interferometer. First, we prepared and detected biphotons having an arbitrary polarization degree (4), i.e., biphotons based on photons in arbitrary polarization states (varying from similar to orthogonal ones). The experimental setup is schematized in Fig. 1. Collinear, frequency-degenerate Type I SPDC was implemented in two beta-barium borate (BBO) crystals pumped by a 325 nm He-Cd laser. The optical axes of the crystals were oriented so that vertically and horizontally polarized photon pairs (states |0, 2 and |2, 0) were generated in the first and second crystals, respectively. The ratio of the corresponding absolute amplitudes were varied by rotating a half-wave plate placed in the pump beam. In addition, the phase difference between the states |2, 0 and |0, 2 was varied by tilting two quartz plates with vertical optical axes. The light generated by the two crystals was the coherent superposition | 1 = sin ( 2 ) |2, 0 + e cos ( 2 ) |0, 2 ,
i

(4)

Degree of light polarization (4) can be measured in experiment as the maximum visibility of the modulation observed in polarization-dependent intensity [14]. Since this quantity is completely determined by second-order moments of the field, it is not an optimal characteristic of biphotonic light (whose most interesting properties manifest themselves in the behavior of fourth-order moments). An alternative definition of degree of polarization, the photon­photon polarization degree formulated in terms of fourth-order moments, has been proposed to describe the polarization state of a biphoton [14­16]. However, since the "photon­photon" polarization degree of a pure state having the form of (1) is always unity [14], it provides no information about the relative location of the two points representing a biphoton on the PoincarÈ sphere. Only the quantity defined by (4) provides information of this kind. We should also mention here the biphotons created as mixed states instead of pure state (1). In this case, the components of state (1) are multiplied by phase factors exhibiting classical fluctuations; i.e., the biphoton is a statistical mixture of several basis states. A biphoton state of this kind can be prepared by SPDC implemented by means of incoherent pumping in two or three crystals. However, it cannot be represented as (2), because (2) is a pure state.

(5)

where is the half-wave plate rotation angle relative to the vertical axis. Thus, two of the four real parameters that determine state (1) ( and ) could be varied so that

M1

325 nm

He-Cd

/2

QP

.

PC

M3

/4 /2

A2 IF P

BS

D2

M2 BBO P D1 CC
Fig. 1. Experimental setup: He-Cd = helium-cadmium laser; M1, M2, M3 = mirrors reflecting the pump beam; QP = quartz plates; BBO = beta-barium borate crystals; P = pinhole; IF = interference filter; BS = nonpolarizing beamsplitter; A1, A2 = polarizers; D1, D2 = detectors (avalanche photodiodes); CC = coincidence circuit; PC = Pockels cell. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 100 No. 3 2005

A1


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the polarization degree of the generated state would take any value between 0 and 1. Initially, the phase difference was set equal to . By varying from 0 to 45°, the generated state was transformed from |2, 0 into |0, 2, so that the two points corresponding to a biphoton state on the PoincarÈ sphere traversed its equator (see Fig. 2), remaining symmetric relative to the axis HV. This state can be represented as (2) with = ' = 2 arctan [ cot ( 2 ) ] , = 0, ' = . (6)
2

4 b'

a

V

1 b 2

In particular, we used = 22.5° to obtain the state |+45°, ­ 45°, i.e., a pair of photons polarized linearly at angles ±45° relative to the vertical axis. The generated light was totally unpolarized (with P = 0). Overall, the degree of polarization varied from 1 to 0; the angular distance between the points representing the biphoton on the PoincarÈ sphere varied from zero to . Note that the degree of polarization P of state (5) depends only on the rotation angle of the half-wave plate placed in the pump beam: P = |cos(4)|. Thus, when the plate was fixed in a certain position, the two points on the PoincarÈ sphere corresponding to biphoton (5) were located symmetrically with respect to the axis HV and separated by a constant angular distance. As the phase difference was varied, the points moved simultaneously about the axis HV. The pump beam that passed through the two crystals was eliminated by a mirror, and the biphoton light was selected both spatially and spectrally (by using a pinhole and a 10 nm bandwidth interference filter with transmittance peak at 650 nm, respectively) and directed into a Hanbury-Brown­Twiss interferometer in order to detect fourth-order moments of the field. The interferometer consisted of a 50% nonpolarizing beamsplitter (a plane-parallel plate set at a small angle of 15°, relative to the beam so that both reflected and transmitted light polarizations were similar to that of the incident light), two photodetectors (EG&G avalanche photodiodes), and a coincidence circuit with a resolution of 1.5 ns. Thin-film polarizers were inserted into the interferometer arms and used as linear polarization filters. 3. DEMONSTRATION OF ORTHOGONALITY OF BIPHOTONS (QUTRITS) The prepared biphoton states were used to validate the operational orthogonality criterion formulated in [17]. It was shown in [17] that the orthogonality of biphotons |1 and |2 is equivalent to zero counting rate in the Hanbury-Brown­Twiss interferometer output when its input is the biphoton state |1 and the polarization filters inserted into its arms select the photon polarization states that constitute |2. The experimental setup schematized in Fig. 1 always prepared the state |2 as a pair of linearly polarized photons, while

3

H

a' 4
Fig. 2. Prepared states on the PoincarÈ sphere: (1) state |2, 0 generated at = 0; (2) state |+45°, ­ 45°, = 22.5°; (3) state |0, 2, = 45°; (4) state |R, L (pair of left- and right-polarized photons), = 67.5°. The states |a, b and |a', b' prepared with = 15° and 75°, respectively, were studied in the present experiment.

different states |1 were used as input (both pairs of linearly polarized photons and pairs of elliptically polarized photons) and the biphoton polarization degree varied from 1 to 0 (see above). The state |1 with a polarization degree of 0.5 was selected as input. In this case, the biphoton |1 |a, b is represented by a pair of points located on the equator of the PoincarÈ sphere at angles of ±74.5° relative to the HV axis. According to (5), this state corresponds to = 15°. Whereas there exist an infinite number of biphoton states |2 orthogonal to the input state, the orthogonality criterion uniquely determines the polarization state of one of the photons that make up a biphoton if the state of the other is preset. In our experiment, it was convenient to set one polarizer in the Hanbury-Brown­ Twiss interferometer at an angle of 45° relative to the vertical. A calculation showed that the other polarizer must then be set at 60° to the vertical. Figure 3a shows the coincidence rate measured versus for = when the polarizers were held at 45 and 60°. According to our calculations, the minimum coincidence rate corresponds to = 15°. When the halfwave plate is fixed at = 15° and polarizer A1 is rotated while polarizer A2 is held at 45°, then the minimum coincidence rate corresponds to an angle of 60° (Fig. 3b). As the half-wave plate was rotated, the biphoton state transformed into |0, 2 (at = 45°). With further increase in the rotation angle, the two points representing the biphoton components moved in opposite direcVol. 100 No. 3 2005

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KRIVITSKIOE et al. 70 Number of coincidences per 100 s 20° 40° 60° 80° /2 plate rotation angle 100° (a) 60 50 40 30 20 10 0 0 20° 40° 60° 80° 100°

70 60 50 40 30 20 10 0 0

80 Number of coincidences per 100 s 70 60 50 40 30 20 10 0 0 40° 80° 120° 160° 200° (b)

/2 plate rotation angle
Fig. 4. Orthogonality of biphotons in the case when both input photons are elliptically polarized (state |a', b' in Fig. 2), = , and polarizers held at 45 and ­60°: coincidence rate versus half-wave plate rotation angle . The orthogonality condition is satisfied at = 75°.

imum coincidence rate was observed in the vicinity of = . The curves plotted in Figs. 3­5 were calculated for particular biphoton states and polarizer positions (e.g., see [18]). The only fitted parameter was the vertical scale.
Polarizer orientation angle

Fig. 3. Orthogonality of biphotons in the case when both input photons are linearly polarized (state |a, b in Fig. 2), = , and polarizers held at 45 and 60°: (a) coincidence rate versus half-wave plate rotation angle , the orthogonality condition is satisfied at = 15°; (b) coincidence rate versus the polarizer rotation angle relative to the vertical, the orthogonality condition is satisfied at an angle of 45°, the other polarizer is held at 60° relative to the vertical.

tions along a meridian of the PoincarÈ sphere (see Fig. 2). At = 67.5°, the output was the state |R, L consisting of circularly polarized photons. At = 75°, the degree of polarization of the output was again 0.5 (as in |a, b), but the corresponding biphoton (|a', b' in Fig. 2) consisted of elliptically polarized photons. Then, it can be verified by calculation that the orthogonality condition is satisfied when the polarizers are oriented at 45 and ­60°. Accordingly, the coincidence rate measured for these polarizer positions versus the rotation angle of the half-wave plate placed in the pump beam reaches a minimum at approximately = 75° (Fig. 4). Figure 5 shows the coincidence rate measured as a function of for = 15° (i.e., input state |a, b in Fig. 2) while the polarizers were held at 45 and 60°. The min-

4. TRANSITION TO AN ARBITRARY BIPHOTON STATE To change from a biphoton with an arbitrary degree of polarization (represented by two points separated by an arbitrary angular distance on the PoincarÈ sphere) to a biphoton in an arbitrary polarization state (represented by two arbitrary points on the PoincarÈ sphere), one must be able to perform any required transformation of state (5). This can be done by varying , , and two additional parameters characterizing state (5). These parameters can be the retardation and orientation of a retarding plate placed in the prepared biphoton beam. The feasibility of transition from state (5) to state (1) with a retarding plate of arbitrary thickness and orientation added to the setup shown in Fig. 1 can be illustrated by performing a simple geometric construction on the PoincarÈ sphere. An arbitrary biphoton state |a, b (see Fig. 6) can be obtained by using a retarding plate with certain and to transform the state |a', b' represented by two points on the PoincarÈ sphere located symmetrically relative to the axis HV. The required plate parameters are determined by the condition that the transformation on the PoincarÈ sphere maps the symmetry axis of the pair
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PREPARATION OF BIPHOTONS IN ARBITRARY POLARIZATION STATES 70 a Number of coincidences per 100 s 60 C 50 40 30 20 a' 10 0 0 1 2 3 , rad 4 5 6 H b' O b

525

Fig. 5. Coincidence rate versus phase for half-wave plate set = 15° and polarizers set at 45 and 30°. The orthogonality condition is satisfied at = .

Fig. 6. Preparation of an arbitrary polarized biphoton |a, b by using the setup schematized in Fig. 1 and a Pockels cell acting as a retarding plate with variable retardation and orientation. The Pockels cell transforms the state |a', b' created by using the setup schematized in Fig. 1 into the desired state |a, b.

|a, b (axis OC in Fig. 6) to the HV axis. Note that the state |a', b' can be prepared by using the setup shown in Fig. 1 so that the angular distance between points a and b on the sphere and the angle of their rotation relative to its equator are determined by the orientation of half-wave plate placed in the pump beam and the phase difference , respectively. In other words, the retarding plate placed after the two crystals will map state (5) to state (1) by rotating the points representing the biphoton in Fig. 1 as a whole. The resulting state (1) is characterized by the four parameters , , , and . One practical difficulty in this method is that the parameter can be varied only by changing plates with different retardations. To vary by an arbitrary amount, one should use a Pockels cell instead of a set of wave plates. Varying the voltage applied to the cell, one can use it as a retarding plate with variable retardation. The parameter can be varied gradually by revolving the cell about the biphoton beam. A state described by (1) was prepared by placing a Pockels cell after the two crystals in the optical arrangement schematized in Fig. 1. The element used in the Pockels cell was a 3 cm long crystal of lithium niobate cut along the optical z axis. When a dc voltage was applied along the x axis, the crystal became weakly birefringent, and the plane of its optical axes made an angle of 45° with the xz plane. As the dc voltage applied to the crystal was varied from zero to 2.8 kV, the phase difference between the ordinary wave (polarized in the plane of the optical axes, i.e., at an angle of 45° relative to the xz plane) and the extraordinary wave (polarized at an angle of ­ 45° relative to the xz plane) increased from zero to 2. Note that the cell acted as a zeroth-

order retarding plate with varying between 0 and . Thus, the electrically induced transformation of polarization in the cell was similar at any frequency within the SPDC bandwidth (about 40 nm) and, therefore, within the bandwidth of the interference filter. Note also that the orientation of the optical axis of a Pockels cell treated as a retarding plate is determined by the plane of the induced optical axes, i.e., makes an angle of 45° with the direction of the electric field applied to the cell.
70 Number of coincidences per 80 s 60 50 40 30 20 10 0 10° 20° 30° 40° 50°



Fig. 7. Coincidence rate versus Pockels cell orientation angle in the case of half-wave voltage applied to the Pockels cell (retardation = /2), half-wave plate orientation = 22.5°, phase = , and polarizers set at 45 and ­ 45°. Vol. 100 No. 3 2005

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3.0

3.5

cryptography. The arbitrarily polarized biphotons produced were utilized to validate a method for identifying orthogonal biphoton states based on zero counting rate in the Hanbury-Brown­Twiss scheme. The feasibility of experimental identification of orthogonal biphotons (qutrits) means that they can be used in quantum data transmission protocols, including quantum cryptography protocols. The 12 biphoton states required to implement the ternary analog of the BB84 protocol were calculated and represented on the PoincarÈ sphere in [19]. We should also note that the present demonstration of orthogonality of biphoton states generalizes the well-known experiment on anticorrelation dip reported in [20]. Previously, this effect was observed only for similarly polarized [20] or orthogonally polarized [21] photon pairs. The present study is the first demonstration of this effect for arbitrarily polarized photon pairs. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project nos. 02-02-16664 and 03-02-16444; by INTAS, project no. 2122-01; and under the State Program for Support of Leading Science Schools, grant 166.2003.02. One of us (L.A.K.) gratefully acknowledges the support provided by INTAS, YS fellowship grant no. 03-55-1971. REFERENCES
1. The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, Ed. by D. Bouwmeester, A. K. Ekert, and A. Zeilinger (Springer, Berlin, 2000; Postmarket, Moscow, 2002). 2. C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), p. 175. 3. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Phys. Rev. Lett. 76, 4656 (1996). 4. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, et al., Phys. Rev. A 60, R4209 (1999). 5. H. Bechmann-Pascuinucci and A. Peres, Phys. Rev. Lett. 85, 3313 (2000). 6. D. Bruss and C. Machiavello, Phys. Rev. Lett. 88, 127 901 (2002). 7. R. Thew, A. Acin, H. Zbinden, and N. Gisin, quantph/0307122. 8. A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240 401 (2002). 9. J. C. Howell, A. Lamas-Linares, and D. Bouwmeester, Phys. Rev. Lett. 88, 030 401 (2002). 10. Yu. Bogdanov, M. Chekhova, S. Kulik, et al., Phys. Rev. A 70, 042 303 (2004). 11. A. V. Burlakov and D. N. Klyshko, Pis'ma Zh. èksp. Teor. Fiz. 69, 795 (1999) [JETP Lett. 69, 839 (1999)].
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Fig. 8. Coincidence rate versus retardation for Pockels cell set at = 22.5°, half-wave plate orientation = 22.5°, phase = , and polarizers set at 45 and ­45°.

To begin our measurements, we switched off the voltage applied to the Pockels cell, placed the halfwave plate at the angle = 22.5° in the pump beam, and set the phase difference equal to (see (5)). This combination of parameters corresponded to the highest coincidence rate when the positions of the polarization filters were set at 45 and ­ 45° so that the input state was |+45°, ­ 45°. Next, we applied the half-wave voltage to the Pockels cell. Since the optical axis of the cell was initially aligned with the vertical direction, the biphoton remained in the same state |+45°, ­ 45° when the half-wave voltage was applied: the cell executed a rotation by relative to the axis HV on the PoincarÈ sphere. The state of the output biphoton varied as the Pockels cell was revolved, and the coincidence rate varied accordingly. Figure 7 shows the coincidence rate versus the angle between the optical axis of the Pockels cell and the vertical axis. It is clear that coincidences virtually vanished at = 22.5° because the Pockels cell transformed the selected state |+45°, ­ 45° into the state |H, V, which is orthogonal to |+45°, ­ 45°. Finally, we set = 22.5° and varied the voltage applied to the cell from zero to 2.8 kV. Figure 8 shows the corresponding coincidence rate as a function of . The minimum coincidence rate was reached at = /2, as in the preceding case, because the states |H, V and |+45°, ­ 45° are mutually orthogonal. 5. CONCLUSIONS We have experimentally demonstrated the preparation of arbitrarily polarized two-photon states (qutrits). The states were generated by using a scheme that did not include any interferometer. This highly stable preparation scheme can be employed in practical quantum

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PREPARATION OF BIPHOTONS IN ARBITRARY POLARIZATION STATES 12. A. V. Burlakov and M. V. Chekhova, Pis'ma Zh. èksp. Teor. Fiz. 75, 505 (2002) [JETP Lett. 75, 432 (2002)]. 13. W. A. Shurcliff, Polarized Light: Production and Use (Harvard Univ. Press, Cambridge, Mass., 1962; Mir, Moscow, 1965). 14. D. N. Klyshko, Zh. èksp. Teor. Fiz. 111, 1955 (1997) [JETP 84, 1065 (1997)]. 15. A. S. Chirkin, A. A. Orlov, and D. Yu. Parashchuk, Kvantovaya èlektron. (Moscow) 20, 999 (1993). 16. G. Bjork, J. Soderholm, A. Trifonov, et al., in Proceedings of SPIE on Quantum and Atomic Optics, High-Precision Measurements in Optics, and Optical Information Processing, Transmission, and Storage, ICONO 2001, Ed. by S. N. Bagayev, S. S. Chesnokov, A. S. Chirkin, and V. N. Zadkov (2002), Proc. SPIE, Vol. 4750.

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17. A. A. Zhukov, G. A. Maslennikov, and M. V. Chekhova, Pis'ma Zh. èksp. Teor. Fiz. 76, 696 (2002) [JETP Lett. 76, 596 (2002)]. 18. L. A. Krivitskioe, S. P. Kulik, A. N. Penin, and M. V. Chekhova, Zh. èksp. Teor. Fiz. 124, 943 (2003) [JETP 97, 846 (2003)]. 19. G. A. Maslennikov, M. V. Chekhova, S. P. Kulik, and A. A. Zhukov, J. Opt. B: Quantum Semiclassic. Opt. 5, 530 (2003). 20. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 21. Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29 (1994).

Translated by A. Betev

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2005