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PHYSICAL REVIEW A 67, 010301 R

2003

Experimental entanglement concentration and universal Bell-state synthesizer
Yoon-Ho Kim,1,* Sergei P. Kulik,2, Maria V. Chekhova,2, Warren P. Grice,1 Yanhua Shih2
1

Center for Engineering Science Advanced Research, Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 2 Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250 Received 16 April 2002; revised manuscript received 16 September 2002; published 10 January 2003 We report a Bell-state synthesizer in which an interferometric entanglement concentration scheme is used. An initially mixed polarization state from type-II spontaneous parametric down-conversion becomes entangled after the interferometric entanglement concentration. This Bell-state synthesizer is universal in the sense that the output polarization state is not affected by spectral filtering, crystal thickness, and, most importantly, the choice of pump source. It is also robust against environmental disturbance and a more general state, partially mixed ­ partially entangled state, can be readily generated as well. DOI: 10.1103/PhysRevA.67.010301 PACS number s : 03.67. a, 03.65.Ud, 42.50.Dv

I. INTRODUCTION

Multiparticle quantum entanglement traditionally has been associated with fundamental issues in quantum physics, such as, the uncertainty principle and the locality, reality, and causality problems in quantum theory 1 . Recently, quantum entanglement has found its applications in metrology, communication, and information processing 2 . To successfully implement these new ideas, one must be able to generate and manipulate entangled states at will. It is, however, generally recognized that even the generation, not to mention manipulation, of multiparticle entanglement is not trivial. Nonetheless, a great deal of work has been carried out with entangled two-qubit quantum states, or Bell states. These states are important not only because of their simplicity, but also because of their utility in applications, such as, quantum cryptography. In addition, entangled two-qubit states may one day serve as building blocks for the construction of states of three or more entangled qubits 3 . The first direct generation of an entangled two-qubit state involved photon pairs produced in the process of cw-pumped type-II spontaneous parametric down-conversion SPDC 4,5 . Although this method is still widely used for the generation of polarization entangled states, it has its limitations. In particular, the photon pair emission times are completely random. This is a drawback in applications such as quantum teleportation, multiphoton state generation, practical quantum cryptography, etc., where knowledge of the approximate times of emission is required. Much of the uncertainty in emission time is eliminated when the SPDC process is pumped by an ultrafast laser. Unfortunately, differences in the spectral and temporal properties of the photon pair cause the polarization entanglement to suffer with this type of pumping scheme 6,7 . It is possible to ``concentrate'' following the definition in Ref. 8 the entanglement by passing the photons through narrow spectral filters, effectively retaining only the more highly entangled

pairs. The entanglement concentration based on a local filtering process such as this is not desirable, however, since most of the photons are simply wasted. A number of schemes involving multiple crystals have been devised to circumvent these problems 9 , although none can match the simplicity and stability of a single-crystal scheme. In this paper, we report the experimental demonstration of a general entanglement concentration scheme in a two-qubit state of type-II SPDC. Our entanglement concentration scheme does not rely on local filtering. Therefore, the degree of entanglement is not affected by the pump bandwidth, the thickness of nonlinear crystal, the bandwidth of spectral filters, etc. As a result, no photons are wasted: all qubit pairs, which are initially in a mixed state, exit the entanglement concentrator as entangled qubit pairs.
II. ENTANGLEMENT CONCENTRATION SCHEME

Consider the polarization state of the photon pair generated from a type-II BBO crystal pumped either by a cw or by an ultrafast pump laser, see Fig. 1 a . As in Ref. 4 , attention is restricted to the intersections of the cones made by the e and o rays exiting the BBO crystal. In each of these two directions, a photon of either polarization horizontal or vertical may be found, with the orthogonal polarization found in the conjugate beam. Unlike common misconcep-

*Electronic address: kimy@ornl.gov; yokim@umbc.edu
Permanent address: Department of Physics, Moscow State University, Moscow, Russia. 1050-2947/2003/67 1 /010301 4 /$20.00


FIG. 1. a Noncollinear type-II SPDC is used to prepare an initial two-qubit mixed state. b Entanglement concentration scheme. c Detectors and polarization analyzers. ©2003 The American Physical Society

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FIG. 2. Two possible quantum-mechanical paths that a photon pair may follow.

distinguishable only by polarization. Therefore, they are in quantum superposition 1/ 2( H 1 H 2 e i V 1 V 2 ) . In this expression, is the phase between the two terms and may be varied by tilting the phase plates 0 , the density matrix of QP1 and QP2. With a setting of () () . the output state can be written as ent If the delay is introduced in one arm relative to the other, then the overall overlap between the two amplitudes becomes smaller. In this case, a more general state, partially ) mix , mixed ­ partially entangled state, ent (1 where 0 1 , is generated.
III. UNIVERSAL BELL-STATE SYNTHESIZER

tion, the photon pairs found in these two directions are not polarization entangled. In fact, the polarization state of the photon pair is best represented as the mixed state mix 1 V 1 H 2 H 2 V 1 ) , where H 2 ( H1 V2 V2 H1 and V refer to the horizontal and vertical polarization state of a single photon, respectively. The reason the state is mixed has to do with timing information carried by the photon pair. Because the group velocities experienced by the different polarizations are not the same, one polarization always precedes the other. Thus, the two amplitudes H 1 V 2 and V 1 H 2 are distinguishable in principle. In cw-pumped type-II SPDC, a pair of birefringent compensators, with the effective thickness equal to half the down-conversion crystal, can remove this timing information, thus transforming the mixed state to a pure polarization entangled state 4,10 . But in ultrafast type-II SPDC, the pump pulse introduces additional timing information, which cannot be eliminated with the birefringent compensators 6 . It is therefore impossible to transform the mixed state into a pure polarization entangled state 7 . The interferometric entanglement concentration scheme shown in Fig. 1 b , nevertheless, allows us to transform the mixed state to a polarization entangled state, regardless of the pump bandwidth, crystal thickness, and spectral filters used 11 . Let us now discuss the entanglement concentration scheme in detail, see Fig. 1 b . The photon pairs exit the 0 ) to a polarizacrystal and travel equal distances delay tion beam splitter PBS . A /2 plate inserted in one arm rotates the polarization by 90° and ensures that the photon pairs have the same polarization when they reach the PBS. Thus, there are two possible outcomes: both photons are reflected ( r -r ) ; or both are transmitted ( t -t ) . These two biphoton paths are illustrated in Fig. 2. Since it is never the case that two photons exit the same port of the PBS, no postselection is required. Note also that the photon that leaves the crystal with e polarization ( o polarization is always detected by D 1 ( D 2 ) . Therefore, the intrinsic timing information present in type-II SPDC cannot be used to distinguish between the t -t and the r -r paths. As a result, the temporal and spectral differences between the photon pair have no bearing on the polarization entanglement. We, therefore, have effectively ``disentangled'' the timing information and the polarization information of the qubit pair. Note that the photon pair arrival times are still known within the coherence time of the pump pulse. The resulting amplitudes H 1 H 2 and V 1 V 2 are then

To understand why this entanglement concentration scheme works as a universal Bell-state synthesizer, it is necessary to carry out a quantum-mechanical calculation of the joint detection rate. What we would like to show here is that 0 balanced interferometer the quantum interference at does not depend on any parameters that are related to the pump source, spectral filtering, and the crystal properties. We present a brief summary of the calculation here, with a detailed calculation to be published elsewhere 12 . The coincidence count rate R c has the form R
c

dt 1 dt

2

0E

() 1

t1 ,

E

() 2

t2 ,

2

,

is the state of type-II SPDC 6 . Assuming that the where 0 , the electricquartz phase plates are adjusted so that field operators that reach the detectors in this experiment can be written as E
() 1

t1 ,

d sin
1

cos e
i

1

e a

i

(t1

)

a

Ve

t1

He

,
i t2

E

() 2

t2 ,

d sin
2

cos e
i

2

e

a

Vo

(t2

)

a

Ho

,

where, for example, a Vo ( ) is the annihilation operator for with vertical polarization which a photon of frequency was originally created as the o ray of the crystal. 1 and 2 are the angles of the polarization analyzers A 1 and A 2 , respectively. Upon carrying out the calculation, we find that the coincidence count rate has the form R dt dt sin
1

c

cos
2

1

cos t

2

t

/2, t
2

sin

/2, t

.

1

The detailed expression of ( t , t ) is given in Refs. 9,12 and it contains all the important factors such as crystal length, pump bandwidth, bandwidth of spectral filters, etc.

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EXPERIMENTAL ENTANGLEMENT CONCENTRATION AND . . .

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FIG. 3. Experimental data for cw-pumped type-II SPDC. 0 for solid circle ( ( ) ) . Inset for open circle ( ( ) ) and shows the polarization interference for ( ) state.

FIG. 4. Experimental data for ultrafast type-II SPDC. Analyzer angles were A 1 ( A 2 ) 45° ( 45° ) for the peak and 45° ( 45° ) for the dip.

Let us now look at Eq. 1 carefully. If the interferometer 0, ( t , t ) factors out so that R c is balanced, i.e., cos( 1 2)2, a clear signature of the polarization entangled 0 , all parameters state ( ) . This means that, when the that affect the temporal shape of the ( t , t ) , including the pump bandwidth, the crystal properties, the crystal thickness, and the filter bandwidth, simply do not have any effect on the quantum interference. For this reason, this scheme can be considered as a universal Bell-state synthesizer.

IV. EXPERIMENT

As described above, the initial polarization state, a mixed state, was prepared by type-II noncollinear frequencydegenerate SPDC. A 3-mm BBO crystal was pumped by a cw argon ion laser operating at 351.1 nm, producing photons with a central wavelength of 702.2 nm. The delay was introduced through the motion of one of the trombone prisms and the phase between the two alternatives shown in Fig. 2 was adjusted by slightly tilting the 600- m quartz plates QP1 in opposite directions. The optic axes of the quartz crystals were oriented vertically. The outputs from the two detectors were fed to a time-to-amplitude converter TAC and the TAC output was analyzed by a multichannel analyzer with a coincidence window set to 3 nsec. No spectral filters were used in this experiment. We first carried out typical space-time interference experiments by varying the delay with both polarization analyzers set at 45° . The phase term was adjusted by tilting the 0 and were sequartz plates QP1. The settings lected to prepare Bell states ( ) and ( ) , respectively. Figure 3 shows the experimental data. When the delay is zero, i.e., no path-length difference between the two arms, complete destructive or constructive interference is observed. Note that this is different from that of typical type-II case in which equal to half the crystal thickness should be inserted to observe complete quantum interference 4,10 . The typical triangular two-photon wave packet is clearly demonstrated and the base width of the triangular wave packet agrees well with the theoretically expected value of 742 fsec 10,12 . The inset of Fig. 3 shows the polarization interference for

state. A 1 was fixed at 45° and A 2 was rotated. The expected cos( 1 2)2 correlation is clearly demonstrated. The above space-time and polarization measurements were repeated with several different spectral filters with almost no change in the quality of quantum interference. The stability of the interferometer was also checked by repeating the polarization interference measurement at different times. We have found almost no change in the visibility of the polarization interference. This is due to the fact that a small 0 has very little effect on the quantum change around interference, as shown in Fig. 3. We have also carried out the experiment with an ultrafast laser used as the pump. A second harmonic 390 nm of a commercial mode-locked Ti:Sa laser operating at 780 nm was used as a pump. The pump pulse duration was about 130 fsec. The thickness of the type-II BBO crystal was 3 mm and the SPDC radiation was centered at 780 nm. Interference filters with 20-nm full width at half maximum FWHM were inserted in front of each detector to reduce noise counts. The effective coincidence window for this experiment was 3 nsec. Figure 4 shows a typical data set for this measurement. The observed visibility is greater than 90%, by far the highest in ultrafast type-II SPDC generated from a thick crystal for comparison, see Ref. 7 for usual ultrafast type-II SPDC data . In principle, the visibility does not depend on crystal, pump, and filter parameters, as shown in Eq. 1 . The FWHM of the interference is roughly estimated to be 140 ­ 150 fsec, which is very close to the theoretically expected value of 160 fsec assuming exactly 2 nm FWHM pump bandwidth 12 .

()

V. DISCUSSION

It is helpful to introduce the terms ``entanglement'' or ``entanglement of formation'' E and ``entropy'' or ``entropy of entanglement'' S formally, as done in Ref. 8 , to visualize the entanglement concentration process in this work. Under these definitions, our initial mixed two-qubit state has S 0.5 and E 0 13 . The output state of the balanced entanglement concentrator has E 1 and S 0 . Most importantly, the entanglement concentration has been obtained

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without discarding any subensemble of the initial two-qubit system. A more general state, partially mixed ­ partially entangled, can be readily prepared by introducing the delay in one 0 and t , where t is the two-photon coherarm. If ence length, we obtain a state which is not completely entangled and not completely mixed. Such states are called Werner states 14 and lie between two points ( S , E ) ( 0.5,0 ) and ( S , E ) ( 0,1) . Only recently, researchers have started to study these states experimentally in the cw domain 15 . We have recently finished tomographic measurements of several such Werner states in pulsed two-photon polarization states using this scheme and the results will be published elsewhere 16 . Such states are important in studying controllable decoherence in multiqubit systems for now, however, it is limited to two-qubit systems . Since our scheme readily offers controllable decoherence in the pulsed domain, we believe that it will be a useful tool to generate a multiqubit entangled state and to study its decoherence in a controlled environment.

In conclusion, we have reported the experimental realization of entanglement concentration without local filtering and demonstrated a universal Bell-state synthesizer. Although typical type-II SPDC is used as a source, our Bell-state synthesizer is not affected by the pump source, the crystal properties, and the use of spectral filters. In addition, a more general state, partially mixed ­ partially entangled state can be readily generated. We believe that this new Bell-state synthesizer will be useful in experimental studies of quantum information science and as a building block of multiparticle entanglement.
ACKNOWLEDGMENTS

We would like to thank M.H. Rubin for helpful discussions. This research was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, NSA, NSF, ONR, and the LDRD Program of the Oak Ridge National Laboratory managed for the U.S. DOE by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725.

1 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 1935 ; J.S. Bell, Speakable and Unspeakable in Quantum Mechanics Cambridge University Press, New York, 1987 . 2 D.N. Klyshko and A.N. Penin, Sov. Phys. Usp. 30, 716 1987 ; A. Steane, Rep. Prog. Phys. 61, 117 1998 ; J.P. Dowling, Phys. Rev. A 57, 4736 1998 . 3 T.E. Keller et al., Phys. Rev. A 57, 2076 1998 . 4 P.G. Kwiat et al., Phys. Rev. Lett. 75, 4337 1995 . 5 Here we only consider experiments in which amplitude postselection does not have to be assumed. 6 T.E. Keller and M.H. Rubin, Phys. Rev. A 56, 1534 1997 ; W.P. Grice and I.A. Walmsley, ibid. 56, 1627 1997 . 7 G. Di Giuseppe et al., Phys. Rev. A 56, R21 1997 ; W.P. Grice et al., ibid. 57, R2289 1998 ; Y.-H. Kim et al., ibid. 64, 011801 R 2001 ; Y.-H. Kim et al., Phys. Rev. Lett. 86, 4710 2001 . 8 R.T. Thew and W.J. Munro, Phys. Rev. A 63, 030302 R 2001 .

9 Y.-H. Kim et al., Phys. Rev. A 62, 011802 R 2000 ; 63, 060301 R 2001 ; 63, 062301 2001 . 10 Y.H. Shih and A.V. Sergienko, Phys. Lett. A 186, 29 1994 ; 191, 201 1994 ; M.H. Rubin, D.N. Klyshko, Y.H. Shih, and A.V. Sergienko, Phys. Rev. A 50, 5122 1994 . 11 Our entanglement concentration scheme does not remove the intrinsic timing information present in type-II SPDC: it is simply made irrelevant to the quantum interference effect. 12 Y.-H. Kim and W.P. Grice, J. Mod. Opt. to be published ; e-print quant-ph/0207047. 4 13 S i 1 i log4 i , where i are the eigenvalues of the density matrix . S 0.5 because the initial state is a mixture of H 1 V 2 and V 1 H 2 of all four possible states. 14 R.F. Werner, Phys. Rev. A 40, 4277 1989 . 15 P.G. Kwiat et al., Nature London 409, 1014 2001 ; A.G. White et al., Phys. Rev. A 65, 012301 2001 . 16 Y.-H. Kim and W. P. Grice unpublished .

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