1320 OPTICS LETTERS / Vol. 32, No. 10 / May 15, 2007

Four-photon interference with asymmetric beamsplitters

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, and G. C. GuoKey Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of

Sciences, Hefei 230026, China

Z. Y. OuDepartment of Physics, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis,

Indiana 46202, USA

Received November 28, 2006; accepted January 29, 2007;posted March 1, 2007 (Doc. ID 77528); published April 17, 2007

Two experiments of four-photon interference are performed with two pairs of photons from parametric down-conversion with the help of asymmetric beam splitters. The first experiment is a generalization of the Hong–Ou–Mandel interference effect to two pairs of photons while the second one utilizes this effect to demon-strate a four-photon de Broglie wavelength of � /4 by projection measurement. © 2007 Optical Society ofAmerica

OCIS codes: 270.0270, 270.4180, 260.3160, 190.4410.

Multiphoton quantum interference plays a pivotalrole in quantum information sciences. Although two-photon interference has been widely studied1 and isapplied to some quantum information protocols,2

quantum interference of more than two photons hasonly recently been the focus of research because ofits role in the fundamental study of quantumnonlocality3,4 and the improvement in the precisionphase measurement.5–9

The most well-known two-photon interference isthe Hong–Ou–Mandel effect,10 where two photonsenter a symmetric beam splitter (50:50) from twosides. Because of two-photon destructive interfer-ence, the probability for the two photons to exit atseparate ports is zero. However, generalization to ahigher photon number is not straightforward. For ex-ample, for an input state of �2a ,2b�, i.e., two from eachside [Fig. 1(a)] to a symmetric beam splitter, there isa nonzero probability for the �2A ,2B� state at theoutput,11 contrary to the two-photon counterpart.

Recently, however, Wang and Kobayashi12 pro-posed a generalization of the Hong–Ou–Mandel ef-fect to three photons with an asymmetric beam split-ter �T�R�. With a state of �2a ,1b� input at a beamsplitter with T=2R=2/3, the probability is zero forthe state �2A ,1B� in the output state, due to three-photon interference.12 Sanaka et al.13 demonstratedexperimentally this three-photon Hong–Ou–Mandeleffect.

Wang and Kobayashi12 went further and proposeda three-photon interferometer, which shows a three-photon de Broglie wavelength. The de Broglie wave-length of a multiphoton state is the equivalent mat-ter wavelength for all the photons as one entity.14

Thus the de Broglie wavelength for an N-photonstate is simply � /N with � as the single-photon wave-length. It will show up in an interference fringe thatis N times finer than the regular single-photonfringe. It has applications in precision phase mea-

5–9 15

surement and lithography. Liu et al. imple-

0146-9592/07/101320-3/$15.00 ©

mented the scheme by Wang and Kobayashi12 and ob-served an interference pattern with the three-photonde Broglie wavelength.

In this Letter, we will generalize the idea of Wangand Kobayashi12 to the four-photon case. We send theinput state of �2a ,2b� to an asymmetric beam splitterand observe a four-photon Hong–Ou–Mandel effectwith proper adjustment of the transmissivity of thebeam splitter. Then we form an interferometer anddemonstrate the de Broglie wavelength for four pho-tons.

When a state of �2a ,2b� enters an asymmetric beamsplitter with T�R [Fig. 1(a)], the output state is16

��4�out = �6TR��4A,0B� + �0A,4B��

+ �6TR�T − R���3A,1B� − �1A,3B��

+ ��T − R�2 − 2TR��2A,2B�. �1�

For a symmetric beam splitter with T=R=1/2, wefind a nonzero probability for the state �2A ,2B� in theoutput, as we discussed earlier. But when �T−R�2

−2TR=0 or T= �3±�3� /6, R= �3��3� /6, the �2A ,2B�term disappears from Eq. (1) and the probability ofdetecting two photons at each side is zero, i.e.,P4�2A ,2B�=0. Hence, we realize a generalized Hong–Ou–Mandel effect for two pairs of photons.

If we follow the outputs by another symmetricbeam splitter as shown in Fig. 1(b), the �3A ,1B� and�1A ,3B� states in Eq. (1) will not contribute to the

Fig. 1. (a) Hong–Ou–Mandel interferometer with asym-metric beam splitter and (b) formation of an interferometer

for the de Broglie wavelength of four photons.

2007 Optical Society of America

May 15, 2007 / Vol. 32, No. 10 / OPTICS LETTERS 1321

probability P4�2C ,2D� of detecting four photons withtwo at each side due to a two-photon Hong–Ou–Mandel effect. Since the �2A ,2B� state in Eq. (1) iscanceled out when T= �3±�3� /6, R= �3��3� /6, onlythe part of �4,0�+ �0,4� in Eq. (1) will contribute toP4�2C ,2D�, leading to a four-photon interference ef-fect with

P4�2C,2D� � 1 + cos 4�, �2�

where � is the single-photon phase difference be-tween A and B. This can be easily confirmed byevaluating the four-photon detection probabilityP4�2C ,2D�� �2a ,2b � C†2D†2D2C2 �2a ,2b� with

C = �A + ej�B�/�2, A = �Ta + �Rb,

D = �ej�B − A�/�2, B = �Tb − �Ra, �3�

where T= �3±�3� /6, R=1−T.Experimental implementation is shown in Fig. 2,

where the four-photon state of �2a ,2b� is producedfrom a type-II parametric downconversion processpumped by 150 fs frequency-doubled pulses from amode-locked Ti:sapphire laser operating at 780 nm.The 2 mm long �-barium borate (BBO) crystal is sooriented that it produces two beamlike orthogonallypolarized fields at the degenerate wavelength of780 nm.17 The horizontal (H) and the vertical (V) po-larized fields are first coupled into single-mode fibersand are recombined with a polarization beam splitter(PBS1) into one beam before passing through an in-terference filter (IF) of 3 nm bandpass. The filteredfield is then fed into the four-photon interferometer ofFig. 1. But the beam splitters of Fig. 1 are equiva-lently replaced by two polarization rotators (HWPs)and another polarization beam splitter (PBS2). Thusit is a polarization interferometer. A phase shifter(PS), made of two synchronically rotating birefrin-gent quartz plates, is inserted between the twoHWPs to introduce a variable single-photon phaseshift � between the two orthogonal polarizations. Theinput four-photon state of �2H ,2V� is generated viatwo pairs of downconverted photons.

In the first experiment, the rotation angle fromHWP1 is set to zero so that it has no effect on the Hand V polarizations except for a relative delay butthat from HWP2 is set at �=13.7° so that cos22�= �3+�3� /6 (angle of polarization rotation is 2�). HWP2and PBS2 together are equivalent to an asymmetric

2 2

Fig. 2. (Color online) Layout of the experiment. PBS, po-larization beam splitter; HWP, half-wave plate; PS, phaseshifter; IF, interference filter; D1–D4, photodetectors.

beam splitter of T=cos 2� and R=sin 2�. The fiber

coupler for the H-polarized photons is mounted on amicrotranslation stage to introduce a delay � be-tween the H and V polarizations. Four-photon coinci-dence counts are registered to measure the probabil-ity P4�2C ,2D� as a function of the delay between theH and V polarizations. The data are shown in Fig. 3after background subtraction. It shows the typicalHong–Ou–Mandel dip with a visibility of 88% and afull width at half height of 196 m, which are derivedfrom a least-squares fit to a mathematically conve-nient Gaussian shape (the solid curve). The less than100% visibility is a result of imperfect temporal modematch between the two pairs of downconvertedphotons.18–20 To verify that we indeed have the cor-rect T and R with � at 13.7°, we fix the delay � at thebottom of the dip in Fig. 3 but change �. The mea-sured four-photon coincidence counts after back-ground subtraction is plotted as a function of � in Fig.4, which shows four minima at �=13.7°, 31.3°, 58.7°,76.3°, corresponding to cos22�=T= �3±�3� /6. Again,the minimum values do not go to zero, due to imper-fect temporal mode match. The solid curve is a least-squares fit to the function,21

P4��� = C��1 − 1.5 sin24��2 + �3 sin24� − 1�

�1 − sin24���1 − E/A�/2�, �4�

where C is a scaling factor and E /A is a parameterthat characterizes the temporal mismatch.18,20 Notethat when E /A=1, the function in Eq. (4) toucheszero at the four minima, corresponding to the perfectmode match.

In the next experiment, we set the delay � at thebottom of the dip in Fig. 3 and turn HWP1 to 13.7°and HWP2 to 22.5° so that HWP1 serves as theasymmetric beam splitter (BS1) with T=cos2�213.7° �= �3+�3� /6 and HWP2 as the 50:50 symmet-ric beam splitter (BS2) in Fig. 1. We measure thefour-photon coincidence as a function of the single-photon phase difference � between the H and V po-larizations via the phase shifter (PS in Fig. 2). In thisway, we form a polarization interferometer equiva-lent to that in Fig. 1. The result of this measurementis shown in Fig. 5(a). Although it reaches minimumat the values near �=� /4, 3� /4, 5� /4, 7� /4, as pre-dicted by Eq. (2), the coincidence has very uneven

Fig. 3. (Color online) Four-photon coincidence as a func-tion of the relative, delay � between the H and V

polarizations.

1322 OPTICS LETTERS / Vol. 32, No. 10 / May 15, 2007

maxima. This is caused by the imperfect cancellationof the �2A ,2B� term in Eq. (1) due to temporal modemismatch21 as shown in the nonzero minimum inFig. 3. The existence of the �2A ,2B� term will add acos 2� term to Eq. (2) resulting from interference be-tween �4A ,0B�+ �0A ,4B� and �2A ,2B�. The data in Fig.5(a) fit very well to the function,

P4�2C,2D� = C�1 + V4 cos 4� + V2 cos 2��, �5�

with V4=0.62 and V2=0.39.Fortunately, the uneven peaks in Fig. 5(a) can be

balanced21 by slightly adjusting HWP1 away from13.7° to 13.1°, as shown in Fig. 5(b). The least-squares fit for the data in Fig. 5(b) to the function inEq. (5) gives V4=0.59 and V2=−0.03. The smallnessof V2 indicates a good cancellation of the cos 2� term

Fig. 5. (Color online) Four-photon coincidence as a func-tion of the phase difference � between H and V polariza-tions. HWP1 is set at (a) �=13.7° and (b) �=13.1°.

Fig. 4. (Color online) Four-photon coincidence as a func-tion of the rotation angle of HWP2.

in Eq. (5).

In conclusion, we demonstrated both the general-ized Hong–Ou–Mandel effect and the de Brogliewavelength of four photons with two pairs of down-converted photons in a scheme involving asymmetricbeam splitters. These two effects are a result of four-photon interference.

This work was funded by National FundamentalResearch Program of China (grant 2001CB309300),the Innovation funds from Chinese Academy of Sci-ences, and National Natural Science Foundation ofChina (grant 60121503). Z. Y. Ou is supported by theU.S. National Science Foundation under grants0245421 and 0427647. Z. Y. Ou’s e-mail address iszou@iupui.edu.

References

1. L. Mandel, Rev. Mod. Phys. 71, S274 (1999).2. A. Zeilinger, Rev. Mod. Phys. 71, S288 (1999).3. D. M. Greenberger, M. A. Home, and A. Zeilinger, in

Bell’s Theorem, Quantum Theory, and Conceptions ofthe Universe, M. Katafos, ed. (Kluwer Academic, 1989).

4. D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter,and A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999).

5. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J.Heinzen, Phys. Rev. A 54, R4649 (1996).

6. Z. Y. Ou, Phys. Rev. A 55, 2598 (1997).7. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P.

Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733(2000).

8. P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S.Gasparoni, and A. Zeilinger, Nature 429, 158 (2004).

9. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg,Nature 429, 161 (2004).

10. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett.59, 2044 (1987).

11. Z. Y. Ou, J.-K. Rhee, and L. J. Wang, Phys. Rev. Lett.83, 959 (1999).

12. H. Wang and T. Kobayashi, Phys. Rev. A 71, 021802(R)(2005).

13. K. Sanaka, K. J. Resch, and A. Zeilinger, Phys. Rev.Lett. 96, 083601 (2006).

14. J. Jacobson, G. Björk, I. Chuang and Y. Yamamoto,Phys. Rev. Lett. 74, 4835 (1995).

15. B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, Z. Y. Ou,and G. C. Guo, arxix.org e-print archives quant-ph/06/0266, October 31, 2006, http://arxiv.org/abs/quant-ph/0610266.

16. R. A. Campos, B., E. A. Saleh, and M. C. Teich, Phys.Rev. A 40, 1371 (1989).

17. S. Takeuchi, Opt. Lett. 26, 843 (2001).18. Z. Y. Ou, J.-K. Rhee, and L. J. Wang, Phys. Rev. A 60,

593 (1999).19. K. Tsujino, H. F. Hofmann, S. Takeuchi, and K. Sasaki,

Phys. Rev. Lett. 92, 153602 (2004).20. Z. Y. Ou, Phys. Rev. A 72, 053814 (2005).21. A more detailed multimode theory similar to Refs.

18 and 20 will take into account the temporalmismatch between the two pairs of downconverted

photons. This theory will be presented elsewhere.