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PHYSICAL REVIEW A, VOLUME 63, 060301 R

Bell-state preparation using pulsed nondegenerate two-photon entanglement
Yoon-Ho Kim,* Sergei P. Kulik, and Yanhua Shih
Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250 Received 12 July 2000; published 16 May 2001 We report a Bell-state preparation experiment. High-purity Bell states are prepared by using femtosecond pulse pumped nondegenerate collinear spontaneous parametric down-conversion. The use of a femtosecond pump pulse does not result in a reduction of quantum interference visibility in our scheme in which the postselection of amplitudes and other traditional mechanisms, such as using thin nonlinear crystals or narrowband spectral filters are not used. Another distinct feature of this scheme is that the pump, the signal, and the idler wavelengths are all distinguishable, which is very useful for quantum communications. DOI: 10.1103/PhysRevA.63.060301 PACS number s : 03.67.Hk, 03.65.Ta, 42.50.Dv

The preparation and measurement of the Bell states are two important issues in modern quantum optics, especially for quantum communications, quantum teleportation, etc. 1 . For photons, such states can be realized by using the entangled photon pairs generated in spontaneous parametric down-conversion SPDC . By making appropriate local operations on the SPDC photon pairs, one can prepare all four Bell states. The polarization Bell states, for photons, can be written as X1 ,X X1 ,Y
2

Y 1 ,Y 2 , 1 Y 1 ,X2 ,

2

where the subscripts 1 and 2 refer to two different photons, photon 1 and photon 2, respectively, and they can be arbitrarily far apart from each other. X and Y form the orthogonal basis for the polarization states of a photon; for example, it can be a horizontal ( H ) and vertical ( V ) po45° , respectively. larization state, as well as 45° and This means that the quantum interference should be independent of the choice of the bases. Such an experiment was first performed by Shih and Alley, in which noncollinear type-I SPDC and a beam splitter were used to prepare a Bell state 2 , but it is very difficult to align such a system. Collinear type-II SPDC is thus developed 3 . There is, however, a common problem: the entangled photon pairs have a 50% chance of leaving at the same output ports of the beam splitter. Therefore, the state prepared after the beam splitter may not be considered as a Bell state without amplitude postselection 4 . Only when one considers the coincidence contributing terms by throwing away two out of four amplitudes postselection of 50% of the amplitudes , is the state then said to be a Bell state. This problem is later solved by using noncollinear type-II SPDC or using two noncollinear type-I SPDCs 5 . In the cw pumped SPDC, entangled photon pairs occur randomly since the process is ``spontaneous,'' so the whereabouts of the photon pair is completely uncertain within the

*Email address: yokim@umbc.edu
Permanent address: Department of Physics, Moscow State University, Moscow, 119899, Russia.

coherence length of the pump laser beam. This huge time uncertainty makes it difficult for applications such as generation of the multiphoton entangled state, quantum teleportation, etc., as interactions between entangled photon pairs generated from different sources are required. This difficulty was thought to be solved by using a femtosecond pulse laser as a pump. Unfortunately, femtosecond pulse pumped type-II SPDC shows poor quantum interference visibility due to the very different compared to the cw case behavior of the two-photon effective wave function 6 . One has to utilize special experimental schemes to achieve complete overlap of the two-photon amplitudes. Traditionally, the following methods have been used to restore the quantum interference visibility in femtosecond pulse pumped type-II SPDC: i using a thin nonlinear crystal ( 100 m) 8 or ii using narrow-band spectral filters in front of detectors 6,7 . Both methods, however, reduce the available flux of the entangled photon pair significantly 9 and cannot achieve complete overlap of the wave functions in principle 6 . The first attempt to achieve high-visibility quantum interference in femtosecond pulse pumped type-II SPDC without using narrow-band filters and a thin crystal was reported in Ref. 10 . The observed visibility, however, was rather low, and keeping the phase coherence over a long term would be very difficult since a Michelson interferometer is used. Also, such a scheme cannot be used to prepare a Bell-state. Recently, we reported a high-visibility quantum interference experiment in which photon pairs are entangled both in polarization and space-time using femtosecond pulse pumped type-I SPDC 11 . However, it cannot be considered as a true Bell-state preparation since postselecting 50% of the amplitudes was still necessary. In this Rapid Communication, we report a Bell-state preparation experiment in which we effectively eliminate any postselection in femtosecond pulse pumped SPDC. Other features in our scheme include: i collinear SPDC makes the alignment much easier, ii Alice and Bob share photons of different frequencies entangled in both space-time and polarization, iii phase coherence is automatically kept and the visibility as high as 92% is observed, iv thick crystals can be used to increase the intensity without losing the visibility , and v the spectral bandwidth is reduced significantly compared to type-I degenerate SPDC by the use of nondegenerate SPDC. These features make our scheme a good source of Bell states for quantum information experiments.
©2001 The American Physical Society

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where t 12 t 1 t 2 and t ( t 1 t 2 ) /2. t i T i l i / c , where T i is the time at which detector i fires and l i is the optical path length from the surface of the crystal to the detector i. A( t , t 12) is the amplitude of the biphoton as explicitly calculated in Ref. 6 . For the scheme shown in Fig. 1 a , A( t , t 12) is the sum of the two amplitudes originated from the crystal in each arm of the MZI: A t ,t
12

Aa t , t

12

Ab t , t

12

,

4

FIG. 1. a Principle schematic of the experiment. The pump pulse is polarized at 45° . Nondegenerate collinear type-I SPDC occurs at the nonlinear crystal placed in each arm of the MZI. b Schematic of the experimental setup. Note that three different phases can be observed. Interference filters F 1 and F 2 are used to cut the pump noise.

where the subscripts a and b refer to the crystal from which the amplitudes are created. The delay T introduced in one T , t 12) and arm modifies the amplitude Ab ( t , t 12) Ab ( t determines the additional phase shift for the biphoton amplitudes p T K p x , where K p 2 / p , p ( p ) the central frequency wavelength of the pump, and x the spatial delay. Due to the energy conservation and negligibly small dispersion of the air, the phase shift depends only on Ks x Ki x Kp x the pump wavelength p, although the delay is introduced to the SPDC field 11,14 . If the crystals are the same and the pump fields in different arms of the MZI are identical, Aa t , t
12

Ab t , t

12

.

5

The coincidence counting rate is then calculated to be R
c

The basic idea of the experiment is illustrated in Fig. 1 a . A 45° polarized femtosecond laser pulse central wavelength 80 fsec. enters the p 400 nm and pulse duration Mach-Zehnder interferometer MZI , which contains a type-I nonlinear crystal in each arm. One crystal has its optic axis oriented vertically ( ) and another horizontally ( ) . The polarizing beam splitter PBS splits the 45° polarized pump pulse into the vertical and horizontal polarized pulses propagating along different arms of the MZI. Then nondegenerate collinear type-I SPDC occurs, with equal probability, at each crystal signal wavelength 730 nm and idler wavelength 885 nm and they are mixed at the dichroic beam splitter, which directs 730 nm photons to detector D 1 and 885 nm to detector D 2 . In the simplified single-mode approximation, the quantum state generated from the vertically oriented H 730 H 885 and that from the horizoncrystal ( ) is 1 V 730 V 885 . H and V reptally oriented one ( ) is 2 resent the horizontal and vertical polarization state of a single photon, respectively. Subscripts 730 and 885 refer to the wavelengths 730 and 885 nm, respectively. When the MZI is balanced, the quantum state after the MZI is without throwing away any amplitudes V
730 1

1 V cos

p

T,

6

where V 1 in this experiment 15,16 . Note that the angles of the analyzers A 1 and A 2 are assumed to be 45° . From Eq. 6 , we expect that the coincidence counting rate will be modulated in the pump central wavelength when T is varied. There are also two more ways to vary the phases of interference by introducing relative delays using a piece of birefringent material, such as a quartz plate after the output beam splitter, i.e., in the signal ( s ) and/or in the idler ) channels. Therefore we obtain ( i R
c

1 V cos

p

i

s

,

7

V

885 2

e

i

H

730 1

H

885 2

,

2

where is the relative phase between the two amplitudes and can easily be varied by scanning one of the mirror of the MZI. The coincidence counting rate ( R c ) is calculated as 12,13 R
c

dt dt

12

A t ,t

12

2

,

3

where p, i , and s refer to the relative phases introduced into the pump, the idler, and the signal, respectively. As we have shown so far, one can eliminate the possibility of the entangled photon pairs leaving at the same output ports of the beam splitter by employing nondegenerate twophoton entanglement. In this scheme, high-visibility quantum interference can be achieved independent of the crystal thickness and the spectral filter bandwidths, even with a femtosecond pulse pump. In practice, however, one would not like to use a MZI in the experimental setup due to stability related issues. Therefore, we use a collinear scheme where two type-I BBO crystals are placed collinearly in the pump beam path; see Fig. 1 b . Two type-I BBO crystals with a thickness of 3.4 mm each the first one is oriented horizontally and the second one is oriented vertically are then pumped by a 45° polarized pump pulse. As described before, the quantum state resulting from the first BBO is V 730 V 885 and that from the second BBO is H 730 H 885 . Since both crystals are pumped

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FIG. 2. Experimental data. a Space-time interference by varying the pump phase p when 1 2 45° . The transition from to is clearly demonstrated. b The polarization interference at . A 1 is fixed at 1 45° and A 2 is rotated, i.e., 2 is varied. The solid lines are the theoretical curves.

equally, the two amplitudes are equally probable. Due to the dispersion, however, V 730 V 885 from the first BBO ( ) and H 730 H 885 from the second BBO ( ) are distinguishable in time. To make V 730 V 885 and H 730 H 885 indistinguishable in time, one needs to compensate the delay experienced by the SPDC photon pairs at each crystal. This compensation can be made by using a properly oriented quartz rod. If the compensation is made properly, either before or after the down-conversion nonlinear crystals, one will observe high-visibility quantum interference without any spectral postselection. In the collinear scheme, having a perfect temporal compensation is difficult when the signal wavelength differs very much from the idler wavelength. This is because the signalidler photon pairs created from the first BBO ( ) experience different dispersion when they pass through the second BBO ( ). The MZI scheme does not have this disadvantage. In this experiment, for the wavelengths we are interested in, the temporal separation is rather small and does not affect the interference visibility. To prevent further dispersion effects, the compensation is made before the BBO crystals. The compensator consists of a quartz rod and two quartz plates whose optic axes are oriented vertically, see Ref. 11 , and it imposes roughly a 1.5-psec required delay between the Hand V-polarized 400-nm pump pulse, which is mainly determined by the thickness of the BBO crystals. By tilting the two quartz plates in opposite directions, the phase delay p can be varied to prepare a Bell state. After the two BBO crystals, the remaining UV radiation is blocked by a UV reflecting mirror and the collinear SPDC is selected by a diaphragm. Then a dichroic beam splitter is used to reflect

FIG. 3. Experimental data. a Space-time interference by varying s 730-nm modulation . b Space-time interference by varying i 885-nm modulation . c Two phases 730 nm and 885 nm are varied at the same time. The solid line is the theory curve based on 400-nm modulation and agrees well with the data. This confirms s i p . Note that 1 2 45° for all cases.

the signal 730 nm to D 1 and to transmit the idler 885 nm to D 2 . Two quartz plates are inserted in each beam path to vary the relative phase of the signal or the idler independently. The detector package consists of a single-photon counting module, an interference filter that is used to cut the pump noise 17 , and a polarization analyzer. To demonstrate the effectiveness of this scheme, we first study the space-time interference as a function of p by setting 1 2 45° , where 1 and 2 are the angles of the analyzers A 1 and A 2 measured from the vertical direction . According to Eq. 7 , one should observe a pump wavelength modulation in the coincidence counting rate. Note that s and i are fixed. The observed modulation period is 400 nm, see Fig. 2 a , which agrees with the theory. ( ) state, identified by constructive To prepare the destructive interference, one just needs to set p s ,3 ,5 , . . . ) , i 0,2 ,4 , ...( p s i which can be done by tilting the quartz plates so that the space-time interference fringe is at the maximum miniand Bell states can also be easily mum . Note that prepared by introducing a /2 plate in one output port of the dichroic beam splitter. We have also experimentally demonstrated the polarizaand . For , the coincidence tion interference for counting rate is calculated to be

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PHYSICAL REVIEW A 63 060301 R

R

c

2

,

1

2

cos2

1

2

.

8

This means that one should observe high-visibility modulation in polarization correlation measurement for arbitrary values of 1 and 2 . To confirm this experimentally, we first set 1 45° and varied 2 . High-visibility polarization correlation is observed, see Fig. 2 b . We then repeated this measurement for many other values of 1 and observed that the visibility remained the same. This confirms Eq. 8 . In other words, we have successfully prepared polarization Bell states from femtosecond pulse-pumped SPDC without amplitude and spectral postselection. Unlike the usual degenerate two-photon sources, this source has one distinctive feature: one can vary three different phases independently, which is very useful for quantum communications. To demonstrate this interesting feature, we observe the space-time interference by varying the relative phases of the signal ( s ) and the idler ( i ) independently. In these measurements, 1 and 2 are set at 45° . The effect of p is already demonstrated in Fig. 2 a . When the signal phase s is varied, see Fig. 3 a , the signal wavelength 730 nm modulation is observed in coincidence rate, while varying the idler phase i , see Fig. 3 b , the idler wavelength 885nm modulation is observed. Figures 2 a , 3 a , and 3 b clearly demonstrate Eq. 7 .

Finally, we tested the condition p s i by varying the signal phase s and the idler phase i at the same time. The quartz plates in both the signal and the idler paths are tilted at the same time with equal angles. To see whether the data agree with the theory, total phases accumulated in both beam path are calculated from the tilt angle, i.e., s i . As evidenced from Fig. 3 c , the data agree well with Eq. 7 . In summary, we have demonstrated a scheme to prepare pulsed entangled photon pairs from which all four Bell states can easily be obtained. Amplitude and spectral postselection are not necessary. Note also that nonmaximally entangled states can be prepared by changing the relative intensities of the pump beams. The visibility and the photon flux are greatly enhanced by this method, although a femtosecond pulse laser is used as a pump. The signal, the idler, and the pump phases can be varied independently with different modulation frequency, which is very useful for quantum communications. We would like to thank M.H. Rubin for helpful discussions. This research was supported, in part, by the Office of Naval Research, ARDA, and the National Security Agency. S.P.K. also thanks the Russian Fund for Fundamental Research, Grant No. 99-02-16419, for partially supporting his visit to UMBC.

1 S. L. Braunstein, A. Mann, and M. Revzen, Phys. Rev. Lett. 68, 3259 1992 ; C. H. Bennett et al., ibid. 70, 1895 1993 ; A. K. Ekert, ibid. 67, 661 1991 . 2 C. O. Alley and Y. H. Shih, in Proceedings of the Second International Symposium on the Foundations of Quantum Mechanics, edited by M. Namiki Physical Society of Japan, Tokyo, 1986 ; Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921 1988 . 3 T. E. Kiess et al., Phys. Rev. Lett. 71, 3893 1993 . 4 L. De Caro and A. Garuccio, Phys. Rev. A 50, R2803 1994 . 5 P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 1995 ; P. G. Kwiat et al., Phys. Rev. A 60, R773 1999 . 6 T. E. Keller and M. H. Rubin, Phys. Rev. A 56, 1627 1997 ; Y.-H. Kim, V. Berardi, M. V. Checkhova, and Y. H. Shih to be published . 7 W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 1997 ; G. Di Guiseppe et al., ibid. 56, R21 1997 ; W. P. Grice et al., ibid. 57, R2289 1998 . 8 A. V. Sergienko et al., Phys. Rev. A 60, R2622 1999 . Ё 9 Recently M. Atature et al. Phys. Rev. Lett. 84, 618 2000 claimed that one can recover high-visibility quantum interference in pulse pumped type-II SPDC from a thick crystal without spectral postselection by using narrow-band spectral fil-

10 11 12 13 14 15

16 17

ters . The theory as well as the interpretation of the experimental data presented their work are, however, shown to be in error; see Y.-H. Kim, S.P. Kulik, M.H. Rubin, and Y.H. Shih, Phys. Rev. Lett. to be published . D. Branning et al., Phys. Rev. Lett. 83, 955 1999 . Y.-H. Kim, S. P. Kulik, and Y. H. Shih, Phys. Rev. A 62, 011802 R 2000 . R. J. Glauber, Phys. Rev. Lett. 10, 84 1963 ; Phys. Rev. 130, 2529 1963 . M. H. Rubin et al., Phys. Rev. A 50, 5122 1994 . A. V. Burlakov et al., JETP Lett. 69, 831 1999 . The visibility factor V is in general a function of D 1/u 0 ( p /2) 1/u 0 ( p ) , where u 0 is the group velocity of the ordinary ray, the pump pulse duration, and the delay T. Note that V is not a function of bandwidths of the spectral filters. One can also prepare a Bell state using two type-II SPDC. Equation 7 still remains valid, but the function V has a different width and shape. See Ref. 16 . Y.-H. Kim, M.V. Chekhova, S.P. Kulik, M.H. Rubin, and Y.H. Shih, Phys. Rev. A 63, 062301 2001 . The bandwidth of the spectral filter is 10 nm at full width half maximum. This is comparable to the spectrum bandwidth of type-I nondegenerate SPDC, which is 13 nm for the crystals used in the experiment.

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