Abstract

We report the experimental realization of first- and second-order optical stochastic interferometry with single-photon Fock states and with a couple of photons generated by spontaneous parametric downconversion. The behavior of the constitutive element of the stochastic interferometer, the stochastic beam splitter, is theoretically described, both for first- and second-order interferometry. The theory predicts a reduction of the visibility from 1 to π/4 and to 1/2, respectively, for the two cases. These results are a direct consequence of the presence of Bose–Einstein correlations within the electromagnetic field. The visibility reduction obtained in the two experiments and their comparison with theoretical predictions are discussed in detail.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
  32. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
    [CrossRef]
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  34. G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
    [CrossRef]

1998

D. Boschi, F. De Martini, and G. Di Giuseppe, “Quantum interference and Feynman path indistinguishability for particles in entangled states,” Fortschr. Phys. 46, 6–14 (1998).

1997

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

1994

L. De Caro and A. Garuccio, “Reliability of Bell-inequality measurements using polarization correlations in parametric-down-conversion photon sources,” Phys. Rev. A 50, R2803–R2805 (1994). In virtue of expressions (33) and (50) the factorizability of the product state is not affected by the unitary transformation performed by the Sto-BS. The effective entangled state is determined at the BS output by state postselection obtained by the coincidence phodetection process.
[CrossRef] [PubMed]

1992

N. Hussain, N. Imoto, and R. Loudon, “Quantum theory of dynamic interference experiments,” Phys. Rev. A 45, 1987–1996 (1992).
[CrossRef] [PubMed]

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

F. De Martini and L. De Dominicis, “Bose–Einstein photon correlations in the stochastic interferometer,” Opt. Lett. 17, 429–431 (1992).
[CrossRef] [PubMed]

1990

1989

M. O. Scully, B. G. Englert, and J. Schwinger, “Spin coherence and Humpty-Dumpty. III. The effects of observation,” Phys. Rev. A 40, 1775–1784 (1989).
[CrossRef] [PubMed]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
[CrossRef]

F. De Martini and S. Di Fonzo, “Transition from Maxwell–Boltzmann to Bose–Einstein partition statistics by stochastic splitting of degenerate light,” Europhys. Lett. 10, 123–128 (1989).
[CrossRef]

1988

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

Y. Shih and C. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

1987

H. Rauch, J. Summhammer, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, “Measurement of amplitude prob-ability distributions for photon-number-operator eigenstates,” Phys. Rev. A 36, 1955–1958 (1987).
[CrossRef] [PubMed]

1986

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

D. Stoler and B. Yurke, “Generating antibunched light from the output of a nondegenerate frequency converter,” Phys. Rev. A 34, 3143–3147 (1986).
[CrossRef] [PubMed]

1985

F. De Martini, “The concept of photon and the paradoxes of the complementarity in optics,” in Niels Bohr Symposium Proceedings, Roma, 1985, Riv. Storia Sci. 2, 557 (1985).

1984

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44 (1984).
[CrossRef]

1983

B. De Finetti, Theory of Probability (Wiley, New York, 1975). The Fermi Dirac statistics has been obtained in the context of stochastic scattering by inserting in the integral of Eq. (11) a formal condition expressing a “particle exclusion principle,” i.e., the occupancy 0 or 1 of the output modes:J. Tersoff and D. Bayer, “Quantum statistics for distinguishable particles,” Phys. Rev. Lett. 50, 553–554 (1983).
[CrossRef]

1980

L. S. Bartell, “Complementarity in the double-slit experiment: on simple realizable systems for observing intermediate particle-wave behavior,” Phys. Rev. D 21, 1698–1699 (1980).
[CrossRef]

1979

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[CrossRef]

Alley, C.

Y. Shih and C. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

Bartell, L. S.

L. S. Bartell, “Complementarity in the double-slit experiment: on simple realizable systems for observing intermediate particle-wave behavior,” Phys. Rev. D 21, 1698–1699 (1980).
[CrossRef]

Basseras, P.

Bayer, D.

B. De Finetti, Theory of Probability (Wiley, New York, 1975). The Fermi Dirac statistics has been obtained in the context of stochastic scattering by inserting in the integral of Eq. (11) a formal condition expressing a “particle exclusion principle,” i.e., the occupancy 0 or 1 of the output modes:J. Tersoff and D. Bayer, “Quantum statistics for distinguishable particles,” Phys. Rev. Lett. 50, 553–554 (1983).
[CrossRef]

Boschi, D.

D. Boschi, F. De Martini, and G. Di Giuseppe, “Quantum interference and Feynman path indistinguishability for particles in entangled states,” Fortschr. Phys. 46, 6–14 (1998).

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
[CrossRef]

Cioccolanti, V.

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

De Caro, L.

L. De Caro and A. Garuccio, “Reliability of Bell-inequality measurements using polarization correlations in parametric-down-conversion photon sources,” Phys. Rev. A 50, R2803–R2805 (1994). In virtue of expressions (33) and (50) the factorizability of the product state is not affected by the unitary transformation performed by the Sto-BS. The effective entangled state is determined at the BS output by state postselection obtained by the coincidence phodetection process.
[CrossRef] [PubMed]

De Dominicis, L.

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

F. De Martini and L. De Dominicis, “Bose–Einstein photon correlations in the stochastic interferometer,” Opt. Lett. 17, 429–431 (1992).
[CrossRef] [PubMed]

De Martini, F.

D. Boschi, F. De Martini, and G. Di Giuseppe, “Quantum interference and Feynman path indistinguishability for particles in entangled states,” Fortschr. Phys. 46, 6–14 (1998).

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

F. De Martini and L. De Dominicis, “Bose–Einstein photon correlations in the stochastic interferometer,” Opt. Lett. 17, 429–431 (1992).
[CrossRef] [PubMed]

F. De Martini and S. Di Fonzo, “Transition from Maxwell–Boltzmann to Bose–Einstein partition statistics by stochastic splitting of degenerate light,” Europhys. Lett. 10, 123–128 (1989).
[CrossRef]

F. De Martini, “The concept of photon and the paradoxes of the complementarity in optics,” in Niels Bohr Symposium Proceedings, Roma, 1985, Riv. Storia Sci. 2, 557 (1985).

Di Fonzo, S.

F. De Martini and S. Di Fonzo, “Transition from Maxwell–Boltzmann to Bose–Einstein partition statistics by stochastic splitting of degenerate light,” Europhys. Lett. 10, 123–128 (1989).
[CrossRef]

Di Giuseppe, G.

D. Boschi, F. De Martini, and G. Di Giuseppe, “Quantum interference and Feynman path indistinguishability for particles in entangled states,” Fortschr. Phys. 46, 6–14 (1998).

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

Englert, B. G.

M. O. Scully, B. G. Englert, and J. Schwinger, “Spin coherence and Humpty-Dumpty. III. The effects of observation,” Phys. Rev. A 40, 1775–1784 (1989).
[CrossRef] [PubMed]

Garuccio, A.

L. De Caro and A. Garuccio, “Reliability of Bell-inequality measurements using polarization correlations in parametric-down-conversion photon sources,” Phys. Rev. A 50, R2803–R2805 (1994). In virtue of expressions (33) and (50) the factorizability of the product state is not affected by the unitary transformation performed by the Sto-BS. The effective entangled state is determined at the BS output by state postselection obtained by the coincidence phodetection process.
[CrossRef] [PubMed]

Heiberger, L.

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

Hong, C. K.

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

Hussain, N.

N. Hussain, N. Imoto, and R. Loudon, “Quantum theory of dynamic interference experiments,” Phys. Rev. A 45, 1987–1996 (1992).
[CrossRef] [PubMed]

Imoto, N.

N. Hussain, N. Imoto, and R. Loudon, “Quantum theory of dynamic interference experiments,” Phys. Rev. A 45, 1987–1996 (1992).
[CrossRef] [PubMed]

Loudon, R.

N. Hussain, N. Imoto, and R. Loudon, “Quantum theory of dynamic interference experiments,” Phys. Rev. A 45, 1987–1996 (1992).
[CrossRef] [PubMed]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

Milani, G.

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

Miller, R. J. D.

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

Rauch, H.

H. Rauch, J. Summhammer, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[CrossRef] [PubMed]

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44 (1984).
[CrossRef]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
[CrossRef]

Schwinger, J.

M. O. Scully, B. G. Englert, and J. Schwinger, “Spin coherence and Humpty-Dumpty. III. The effects of observation,” Phys. Rev. A 40, 1775–1784 (1989).
[CrossRef] [PubMed]

Scully, M. O.

M. O. Scully, B. G. Englert, and J. Schwinger, “Spin coherence and Humpty-Dumpty. III. The effects of observation,” Phys. Rev. A 40, 1775–1784 (1989).
[CrossRef] [PubMed]

Sergienko, A. V.

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

Shih, Y.

Y. Shih and C. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

Stoler, D.

B. Yurke and D. Stoler, “Measurement of amplitude prob-ability distributions for photon-number-operator eigenstates,” Phys. Rev. A 36, 1955–1958 (1987).
[CrossRef] [PubMed]

D. Stoler and B. Yurke, “Generating antibunched light from the output of a nondegenerate frequency converter,” Phys. Rev. A 34, 3143–3147 (1986).
[CrossRef] [PubMed]

Summhammer, J.

H. Rauch, J. Summhammer, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[CrossRef] [PubMed]

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44 (1984).
[CrossRef]

Sweetser, J.

Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
[CrossRef]

Tersoff, J.

B. De Finetti, Theory of Probability (Wiley, New York, 1975). The Fermi Dirac statistics has been obtained in the context of stochastic scattering by inserting in the integral of Eq. (11) a formal condition expressing a “particle exclusion principle,” i.e., the occupancy 0 or 1 of the output modes:J. Tersoff and D. Bayer, “Quantum statistics for distinguishable particles,” Phys. Rev. Lett. 50, 553–554 (1983).
[CrossRef]

Tuppinger, D.

H. Rauch, J. Summhammer, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[CrossRef] [PubMed]

Walmsley, I. A.

Wang, X. D.

Wootters, W. K.

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[CrossRef]

Yurke, B.

B. Yurke and D. Stoler, “Measurement of amplitude prob-ability distributions for photon-number-operator eigenstates,” Phys. Rev. A 36, 1955–1958 (1987).
[CrossRef] [PubMed]

D. Stoler and B. Yurke, “Generating antibunched light from the output of a nondegenerate frequency converter,” Phys. Rev. A 34, 3143–3147 (1986).
[CrossRef] [PubMed]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[CrossRef]

Europhys. Lett.

F. De Martini and S. Di Fonzo, “Transition from Maxwell–Boltzmann to Bose–Einstein partition statistics by stochastic splitting of degenerate light,” Europhys. Lett. 10, 123–128 (1989).
[CrossRef]

Fortschr. Phys.

D. Boschi, F. De Martini, and G. Di Giuseppe, “Quantum interference and Feynman path indistinguishability for particles in entangled states,” Fortschr. Phys. 46, 6–14 (1998).

Opt. Lett.

Phys. Lett. A

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44 (1984).
[CrossRef]

Phys. Rev. A

H. Rauch, J. Summhammer, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[CrossRef] [PubMed]

N. Hussain, N. Imoto, and R. Loudon, “Quantum theory of dynamic interference experiments,” Phys. Rev. A 45, 1987–1996 (1992).
[CrossRef] [PubMed]

F. De Martini, L. De Dominicis, V. Cioccolanti, and G. Milani, “Stochastic interferometer,” Phys. Rev. A 45, 5144–5151 (1992).
[CrossRef] [PubMed]

L. De Caro and A. Garuccio, “Reliability of Bell-inequality measurements using polarization correlations in parametric-down-conversion photon sources,” Phys. Rev. A 50, R2803–R2805 (1994). In virtue of expressions (33) and (50) the factorizability of the product state is not affected by the unitary transformation performed by the Sto-BS. The effective entangled state is determined at the BS output by state postselection obtained by the coincidence phodetection process.
[CrossRef] [PubMed]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 13711384 (1989).
[CrossRef]

G. Di Giuseppe, L. Heiberger, F. De Martini, and A. V. Sergienko, “Quantum interference and indistinguishability with femtosecond pulses,” Phys. Rev. A 56, R21–R24 (1997).
[CrossRef]

M. O. Scully, B. G. Englert, and J. Schwinger, “Spin coherence and Humpty-Dumpty. III. The effects of observation,” Phys. Rev. A 40, 1775–1784 (1989).
[CrossRef] [PubMed]

D. Stoler and B. Yurke, “Generating antibunched light from the output of a nondegenerate frequency converter,” Phys. Rev. A 34, 3143–3147 (1986).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, “Measurement of amplitude prob-ability distributions for photon-number-operator eigenstates,” Phys. Rev. A 36, 1955–1958 (1987).
[CrossRef] [PubMed]

Phys. Rev. D

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[CrossRef]

L. S. Bartell, “Complementarity in the double-slit experiment: on simple realizable systems for observing intermediate particle-wave behavior,” Phys. Rev. D 21, 1698–1699 (1980).
[CrossRef]

Phys. Rev. Lett.

B. De Finetti, Theory of Probability (Wiley, New York, 1975). The Fermi Dirac statistics has been obtained in the context of stochastic scattering by inserting in the integral of Eq. (11) a formal condition expressing a “particle exclusion principle,” i.e., the occupancy 0 or 1 of the output modes:J. Tersoff and D. Bayer, “Quantum statistics for distinguishable particles,” Phys. Rev. Lett. 50, 553–554 (1983).
[CrossRef]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

Y. Shih and C. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

Riv. Storia Sci.

F. De Martini, “The concept of photon and the paradoxes of the complementarity in optics,” in Niels Bohr Symposium Proceedings, Roma, 1985, Riv. Storia Sci. 2, 557 (1985).

Other

F. De Martini, Foundations of Quantum Mechanics, M. O. Scully and M. M. Nieto, eds. (World Scientific, New York, 1992).

M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974).

G. Di Giuseppe, Laurea thesis (Universita’ di Roma “La Sapienza,” Rome, 1995).

J. A. Wheeler, “The past and the delayed-choice double slit experiment,” Mathematical Foundations of Quantum Theory, A. R. Marlow, ed. (Academic, New York, 1978), pp. 250–253.

D. Bouwmeester, “The physics of quantum information: basic concepts,” The Physics of Quantum Information, D. Bouwmeester, A. Ekert, and A. Zeilinger, eds. (Springer, Berlin, 2000), pp.1–14.

D. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988).

J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Mass., 1994).

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J. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, 1987).

F. De Martini, S. Di Fonzo, and R. Tommasini, in Coherence and Quantum Optics VI, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1990).

F. De Martini and R. Tommasini, “Transition from classical “Maxwell–Boltzmann” to quantum Bose–Einstein partition statistics by stochastic scattering of degenerate light,” in New Frontiers of QED and Quantum Optics, A. Barut, ed. (Plenum, New York, 1990), pp. 54–58.

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Figures (7)

Fig. 1
Fig. 1

First-order stochastic interferometer (Sto-If).

Fig. 2
Fig. 2

Poincarè sphere representation of the rotation performed by the stochastic beam splitter on the input single-photon state.

Fig. 3
Fig. 3

First-order fringe visibility V(1), Eq. (22), as a function of the stochastic parameter σ : the solid curve represents the theoretical prediction; the dotted curve corresponds to the best-fit analysis of the experimental data.

Fig. 4
Fig. 4

Experimental apparatus for first-order stochastic interferometry.

Fig. 5
Fig. 5

Second-order Sto-BS: (a) noncollinear case, (b) collinear case.

Fig. 6
Fig. 6

Second-order fringe visibility V(2) as a function of the stochastic parameter σ. The dashed curve corresponds to the expression of the theoretical V(2) for the noncollinear case, Eq. (39). The solid curve represents the curve for V(2) in the collinear case, Eq. (57). It can be compared with the dotted curve, which corresponds to the best-fit representation of the experimental data.

Fig. 7
Fig. 7

Experimental apparatus for second-order stochastic interferometry.

Equations (61)

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T=1+x2,R=1-x2,
p(x|σ)12πσ erf12σ exp-x2σ2,
Uˆ(1)=exp-i π2 n·σˆ=-in·σˆ,
Uˆ(1)=-i2(σˆx+σˆz).
Uˆ(2)=exp-i δ2σˆz=cos δ2Iˆ-i sin δ2σˆz.
Uˆ(3)=expi π2σˆx=12(Iˆ+iσˆx).
Uˆ=Uˆ(3)·Uˆ(2)·Uˆ(1)=Rˆyδ+π2·Rˆzπ2,
Rˆj(θ)=exp-i θ2σˆj.
Uˆ=cosδ2+π4exp-i π4-sinδ2+π4exp+i π4sinδ2+π4exp-i π4cosδ2+π4exp+i π4.
t=cosδ2+π4exp-i π4,
r=sinδ2+π4exp+i π4.
T=|t|2=1+sin δ2,R=|r|2=1-sin δ2,
|Ψint=Uˆ|Ψin=exp-i π4cosδ2+π4|o+sinδ2+π4|e,
Pnh=2-nChn-1+1[(1+x)(n-h)(1-x)h]p(x|σ)dx.
|Ψout=Rˆz(ϕ)|Ψint=Rˆz(ϕ)Uˆ|Ψin=Uˆ(R)|Ψin.
ρˆout(p)=Uˆ(R)ρˆin(n)Uˆ(R)=Iˆ+p·σˆ2,
ρˆin(n)=|oo|=Iˆ+n·σˆ2,
Uˆ(R)n·σˆ Uˆ(R)=n·Rσˆ=RTn·σˆ.
R=Rzπ2Ryδ+π2Rz(ϕ)=-sin ϕcos ϕ0-cos ϕ sin δ-sin ϕ sin δcos δcos ϕ cos δsin ϕ cos δsin δ,
P(m,±)=Tr1±m·σˆ2ρˆout(p)=Tr1±Uˆ(R)m·σˆUˆ(R)2ρˆin(n).
P(m, ±)=12[1±m·RTn]=12[1±m·p]=12[1±Rm·n].
P(δ, φ)=12(1±cos δ cos φ).
V(1)(δ)=|cos δ|,
V(1)(x)x=-1+1p(x)V(1)(x)dx.
V(1)(σ)=π4  1F132, 2; 12σ2 1F11, 32; 12σ2.
V(1)(σ)=π2 -+P(χ) J1(2πχ)χ dχ-+P(χ) sin(2πχ)χ dχ.
Vm(1)=1ΔT ΔTV(1)(δ)dt.
1ΔT dt=p(x|σ)dx.
dtdx=ΔTp(x|σ)=ΔT2πσ erf12σ exp-sin δ2σ2.
Vm(1)=1ΔT -π/2π/2V(δ) dtdx dxdδ dδ,
Vm(1)=12πσ erf(1/2σ) -π/2π/2 expsin δ2σ2cos2 δdδ.
Vˆh(Sh)=expiδh2+π4σˆx=cosδh2+π4Iˆ+i sinδh2+π4σˆx,
|Ψin=|ψ11|ψ22,
|Ψint=Vˆ1|ψ11Vˆ2|ψ22.
|Ψout=[t|ψ˜13+r|ψ˜14][t|ψ˜24+r|ψ˜23],
|Ψout=[t2|ψ˜13|ψ˜24+r2|ψ˜14|ψ˜23]+tr[|ψ˜13|ψ˜23+|ψ˜14|Ψ˜24],
 outΨ|j,lo,ek3,3q4,4:Eˆjk(-)Eˆjk(+)Eˆlq(-)Eˆlq(+):|Ψout.
R0=Γ[|t|4+|r|4+t2r*2|ψ˜2|ψ˜1|2+c.c],
R=Γ[|t|4+|r|4].
V(2)=|R-R0|R=2|r|2|t|2|t|4+|r|4|ψ˜2|ψ˜1|2.
V(2)(δ1, δ2)=12[1+cos(δ1-δ2)],
V(2)(σ)=12{[1+[V(1)(σ)]2}.
|Ψin=12[|ψa1|ψb2+exp(iϕ)|ψb1|ψa2].
|ψint=12[|ψ˜a11|ψ˜b22+exp(iϕ)|ψ˜b11|ψ˜a22],
|Ψout=12[t2|ψ˜a13|ψ˜b24+r2|ψ˜b23|ψ˜a14+exp(iϕ)(t2|ψ˜b13|ψ˜a24+r2|ψ˜a23|ψ˜b14)]+|NoCoinc.
R0=Γ{[|t|4+|r|4][1+|ψa|ψb|2 cos ϕ]+t2r*2Re[ψ˜b2|ψ˜b1ψ˜a1|ψ˜a2exp(iϕ)]+c.c.},
R=Γ[|t|4+|r|4][1+|ψa|ψb|2 cos ϕ].
V(2)=2|r|2|t|2|t|4+|r|4|Re[ψ˜b2|ψ˜b1ψ˜a1|ψ˜a2exp(iϕ)]|,
V(2)(δ1, δ2; ϕ)=12[1+cos(δ1-δ2)]|cos(ϕ)|,
V(2)(δ1, δ2)=cos δ1 cos δ21+sin δ1 sin δ2,
V(2)(δ1, δ2; ϕ)
=cos δ1 cos δ2+(1-sin δ1 sin δ2)cos ϕ1-sin δ1 sin δ2+cos δ1 cos δ2 cos ϕ.
|Ψint=|ψ˜11|ψ˜21.
|Ψout=[co1,e2|o3|e4+ce1,o2|e4|o3]
+[co1,o2|o3|o3+ce1,o2|e4|e4],
R0=Γ|o|ψ˜1e|ψ˜2+e|ψ˜1o|ψ˜2|2|,
R=Γ[|o|ψ˜1e|ψ˜2|2+|e|ψ˜1o|ψ˜2|2].
V(2)=2 |o|ψ˜1e|ψ˜2e|ψ˜1o|ψ˜2||o|ψ˜1e|ψ˜2|2+|e|ψ˜1o|ψ˜2|2.
V(2)(δ)=cos2 δ1+sin2 δ.
V(2)(x)x=-1+1p(x)V(2)(x)dx.
Vm(2)(σ)=|Rx-R0x|/R0x.

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