Abstract

The technique of dc-balanced, pulsed homodyne detection for the purpose of determining optical-field statistics on short time scales is analyzed theoretically. Such measurements provide photon-number and phase distributions associated with a repetitive signal light field in a short time window. Time- and space-varying signal and local-oscillator pulses are treated, thus generalizing earlier treatments of photoelectron difference statistics in homodyne detection. Experimental issues, such as the effects of imperfect detector balancing on (time-integrated) dc detection and the consequences of background noise caused by non-mode-matched parts of the multimode signal field, are analyzed. The Wigner, or joint, distribution for the two field-quadrature amplitudes during the sampling time window can be directly determined by tomographic inversion of the measured photoelectron distributions. It is pointed out that homodyne detection provides a new method for the simultaneous measurement of temporal and spectral information. Although the theory is generally formulated, with both signal and local-oscillator fields being quantized, emphasis is placed on the limit of a strong, coherent local-oscillator field, making semiclassical interpretation possible.

© 1995 Optical Society of America

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1995 (2)

For review see U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quantum Electron. 19, 89 (1995).
[CrossRef]

M. E. Anderson, M. Beck, M. G. Raymer, and J. D. Bierlein, Opt. Lett. 20, 620 (1995).
[CrossRef] [PubMed]

1994 (5)

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, Phys. Lett. A 195, 31 (1994).
[CrossRef]

K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, Opt. Lett. 19, 2152 (1994).
[CrossRef] [PubMed]

G. M. D’Ariano, C. Machiavello, and M. G. A. Paris, Phys. Rev. A 50, 4298 (1994); H. Paul, U. Leonhardt, and G. M. D’Ariano, Acta Phys. Slovaka45, 261; U. Leonhardt, H. Paul, and G. M. D’Ariano, “Tomographic detection of the density matrix,” Phys. Rev. A (to be published).
[CrossRef]

H. Kuhn, D.-G. Welsch, and W. Vogel, J. Mod. Opt. 41, 1607 (1994).
[CrossRef]

U. Leonhardt and I. Jex, J. Phys. A 49, R1555 (1994).

1993 (8)

See the special issue on Quantum phase: W. P. Schleich and S. M. Barnett, eds., Phys. Scr. T48, 5–42 (1993).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, and A. Faridani, Phys. Scr. T48, 35 (1993).
[CrossRef]

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388 (1993).
[CrossRef] [PubMed]

W. Vogel and J. Grabow, Phys. Rev. A 47, 4227 (1993).
[CrossRef] [PubMed]

U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993).
[CrossRef] [PubMed]

For review and references see U. Leonhardt, Phys. Rev. A 48, 3265 (1993).
[CrossRef] [PubMed]

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993); M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890, (1993); D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

1992 (4)

M. D. Reid and B. Yurke, Phys. Rev. A 46, 4131 (1992), Eq. (17).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. Belsley, and M. G. Raymer, Phys. Rev. Lett. 69, 2650 (1992).
[CrossRef] [PubMed]

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

Single-detector counting distributions have been expressed in terms of Wigner functions in T. Marshall and E. Santos, J. Mod. Opt. 38, 1463 (1992).
[CrossRef]

1991 (1)

W. Vogel and W. Schleich, Rev. A 44, 7642 (1991).
[CrossRef]

1990 (2)

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepard, Phys. Rev. A 42, 4102 (1990).
[CrossRef] [PubMed]

S. L. Braunstein, Phys. Rev. A 42, 474 (1990).
[CrossRef] [PubMed]

1989 (3)

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[CrossRef] [PubMed]

For a tutorial derivation see Z. Y. Ou and L. Mandel, Am. J. Phys. 57, 66 (1989).
[CrossRef]

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

1987 (6)

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987).
[CrossRef] [PubMed]

M. J. Collett, R. Loudon, and C. W. Gardiner, J. Mod. Opt. 34, 881 (1987).
[CrossRef]

P. D. Drummond, Phys. Rev. A 35, 4253 (1987).
[CrossRef] [PubMed]

For review see R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

1985 (2)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
[CrossRef] [PubMed]

1984 (1)

For discussion see J. H. Shapiro and S. S. Wagner, IEEE J. Quantum Electron. QE-20, 803 (1984), App.
[CrossRef]

1983 (2)

1982 (1)

K.-H. Brenner and K. Wodkiewicz, Opt. Commun. 43, 103 (1982).
[CrossRef]

1980 (2)

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

P. D. Drummond and C. Gardiner, J. Phys. A 13, 2353 (1980).
[CrossRef]

1979 (1)

J. H. Shapiro, H. P. Yuen, and J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

1978 (1)

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978).
[CrossRef]

1977 (1)

1973 (1)

Generally time ordering would also be required, but because the fields are treated as free fields this is irrelevant. See, for example, M. Lax and M. Zwanziger, Phys. Rev. A 7, 750 (1973).
[CrossRef]

1969 (1)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857, 1882 (1969).
[CrossRef]

1966 (1)

U. M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041 (1966).
[CrossRef]

1964 (1)

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

1958 (1)

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

1932 (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932); M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Abbas, G. L.

Anderson, M.

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

Anderson, M. E.

M. E. Anderson, M. Beck, M. G. Raymer, and J. D. Bierlein, Opt. Lett. 20, 620 (1995).
[CrossRef] [PubMed]

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A (to be published); M. Munroe, D. Boggavarapu, M. E. Anderson, U. Leonhardt, and M. G. Raymer, “High-effiency, ultrafast photon-number statistics from phase-averaged homodyne detection,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, to be published).

M. Beck, M. E. Anderson, and M. G. Raymer, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

Barnett, S. M.

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

Baumberg, J. J.

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

Beck, M.

M. E. Anderson, M. Beck, M. G. Raymer, and J. D. Bierlein, Opt. Lett. 20, 620 (1995).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, and A. Faridani, Phys. Scr. T48, 35 (1993).
[CrossRef]

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993); M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890, (1993); D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. Belsley, and M. G. Raymer, Phys. Rev. Lett. 69, 2650 (1992).
[CrossRef] [PubMed]

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

M. Beck, M. E. Anderson, and M. G. Raymer, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

Belsley, M.

D. T. Smithey, M. Beck, M. Belsley, and M. G. Raymer, Phys. Rev. Lett. 69, 2650 (1992).
[CrossRef] [PubMed]

Bierlein, J. D.

Blow, K. J.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepard, Phys. Rev. A 42, 4102 (1990).
[CrossRef] [PubMed]

Boggavarapu, D.

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A (to be published); M. Munroe, D. Boggavarapu, M. E. Anderson, U. Leonhardt, and M. G. Raymer, “High-effiency, ultrafast photon-number statistics from phase-averaged homodyne detection,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, to be published).

Bouchiat, M. A.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein, Phys. Rev. A 42, 474 (1990).
[CrossRef] [PubMed]

Brenner, K.-H.

K.-H. Brenner and K. Wodkiewicz, Opt. Commun. 43, 103 (1982).
[CrossRef]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857, 1882 (1969).
[CrossRef]

Chan, V. W. S.

Cho, Z. H.

Z. H. Cho, J. P. Jones, and M. Singh, Foundations of Medical Imaging (Wiley, New York, 1993). There is a typographical error in the Abel formula in this source.)

Collett, M. J.

M. J. Collett, R. Loudon, and C. W. Gardiner, J. Mod. Opt. 34, 881 (1987).
[CrossRef]

Cooper, J.

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. I, Chap. III.

D’Ariano, G. M.

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, Phys. Lett. A 195, 31 (1994).
[CrossRef]

G. M. D’Ariano, C. Machiavello, and M. G. A. Paris, Phys. Rev. A 50, 4298 (1994); H. Paul, U. Leonhardt, and G. M. D’Ariano, Acta Phys. Slovaka45, 261; U. Leonhardt, H. Paul, and G. M. D’Ariano, “Tomographic detection of the density matrix,” Phys. Rev. A (to be published).
[CrossRef]

DeLong, K. W.

Drummond, P. D.

P. D. Drummond, Phys. Rev. A 35, 4253 (1987).
[CrossRef] [PubMed]

P. D. Drummond and C. Gardiner, J. Phys. A 13, 2353 (1980).
[CrossRef]

P. D. Drummond, in Quantum Optics V, J. D. Harvey and D. F. Walls, eds. (Springer-Verlag, Heidelberg, 1989).

Dunn, T. J.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388 (1993).
[CrossRef] [PubMed]

Eberly, J. H.

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, and A. Faridani, Phys. Scr. T48, 35 (1993).
[CrossRef]

Fittinghoff, D. N.

Gardiner, C.

P. D. Drummond and C. Gardiner, J. Phys. A 13, 2353 (1980).
[CrossRef]

Gardiner, C. W.

M. J. Collett, R. Loudon, and C. W. Gardiner, J. Mod. Opt. 34, 881 (1987).
[CrossRef]

Glauber, R. J.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857, 1882 (1969).
[CrossRef]

U. M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041 (1966).
[CrossRef]

R. J. Glauber, in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965), pp. 331–381.

Grabow, J.

W. Vogel and J. Grabow, Phys. Rev. A 47, 4227 (1993).
[CrossRef] [PubMed]

Grangier, P.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Guena, J.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. I, Chap. III.

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Hrisoho, A.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Huttner, B.

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

Jacquier, Ph.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Jex, I.

U. Leonhardt and I. Jex, J. Phys. A 49, R1555 (1994).

Jones, J. P.

Z. H. Cho, J. P. Jones, and M. Singh, Foundations of Medical Imaging (Wiley, New York, 1993). There is a typographical error in the Abel formula in this source.)

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

Knight, P. L.

For review see R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

Kohler, B.

Kuhn, H.

H. Kuhn, D.-G. Welsch, and W. Vogel, J. Mod. Opt. 41, 1607 (1994).
[CrossRef]

LaPorta, A.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Lax, M.

Generally time ordering would also be required, but because the fields are treated as free fields this is irrelevant. See, for example, M. Lax and M. Zwanziger, Phys. Rev. A 7, 750 (1973).
[CrossRef]

Leonhardt, U.

For review see U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quantum Electron. 19, 89 (1995).
[CrossRef]

U. Leonhardt and I. Jex, J. Phys. A 49, R1555 (1994).

U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993).
[CrossRef] [PubMed]

For review and references see U. Leonhardt, Phys. Rev. A 48, 3265 (1993).
[CrossRef] [PubMed]

Lintz, M.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Loudon, R.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepard, Phys. Rev. A 42, 4102 (1990).
[CrossRef] [PubMed]

M. J. Collett, R. Loudon, and C. W. Gardiner, J. Mod. Opt. 34, 881 (1987).
[CrossRef]

For review see R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983), p. 232.

R. Loudon, in Coherence, Cooperation and Fluctuations, F. Haake, L. M. Narducci, and D. F. Walls, eds. (Cambridge U. Press, Cambridge, 1986).

Macchiavello, C.

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, Phys. Lett. A 195, 31 (1994).
[CrossRef]

Machado Mata, J. A.

J. H. Shapiro, H. P. Yuen, and J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

Machiavello, C.

G. M. D’Ariano, C. Machiavello, and M. G. A. Paris, Phys. Rev. A 50, 4298 (1994); H. Paul, U. Leonhardt, and G. M. D’Ariano, Acta Phys. Slovaka45, 261; U. Leonhardt, H. Paul, and G. M. D’Ariano, “Tomographic detection of the density matrix,” Phys. Rev. A (to be published).
[CrossRef]

Mandel, L.

For a tutorial derivation see Z. Y. Ou and L. Mandel, Am. J. Phys. 57, 66 (1989).
[CrossRef]

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Marshall, T.

Single-detector counting distributions have been expressed in terms of Wigner functions in T. Marshall and E. Santos, J. Mod. Opt. 38, 1463 (1992).
[CrossRef]

McAlister, D. F.

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), p. 149.

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[CrossRef]

Mostowski, J.

J. Mostowski and M. G. Raymer, in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, Boston, Mass., 1992), p. 187–234.
[CrossRef]

Munroe, M.

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A (to be published); M. Munroe, D. Boggavarapu, M. E. Anderson, U. Leonhardt, and M. G. Raymer, “High-effiency, ultrafast photon-number statistics from phase-averaged homodyne detection,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, to be published).

Nussenvieg, H. M.

H. M. Nussenvieg, Introduction to Quantum Optics (Gordon & Breach, New York, 1974), Chap. 3.

Ou, Z. Y.

For a tutorial derivation see Z. Y. Ou and L. Mandel, Am. J. Phys. 57, 66 (1989).
[CrossRef]

Paris, M. G. A.

G. M. D’Ariano, C. Machiavello, and M. G. A. Paris, Phys. Rev. A 50, 4298 (1994); H. Paul, U. Leonhardt, and G. M. D’Ariano, Acta Phys. Slovaka45, 261; U. Leonhardt, H. Paul, and G. M. D’Ariano, “Tomographic detection of the density matrix,” Phys. Rev. A (to be published).
[CrossRef]

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, Phys. Lett. A 195, 31 (1994).
[CrossRef]

Paul, H.

For review see U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quantum Electron. 19, 89 (1995).
[CrossRef]

U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993).
[CrossRef] [PubMed]

Phoenix, S. J. D.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepard, Phys. Rev. A 42, 4102 (1990).
[CrossRef] [PubMed]

Potasek, M. J.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Pottier, L.

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Radzewicz, C.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388 (1993).
[CrossRef] [PubMed]

Raymer, M. G.

M. E. Anderson, M. Beck, M. G. Raymer, and J. D. Bierlein, Opt. Lett. 20, 620 (1995).
[CrossRef] [PubMed]

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993); M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890, (1993); D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, and A. Faridani, Phys. Scr. T48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, M. Belsley, and M. G. Raymer, Phys. Rev. Lett. 69, 2650 (1992).
[CrossRef] [PubMed]

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

J. Mostowski and M. G. Raymer, in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, Boston, Mass., 1992), p. 187–234.
[CrossRef]

M. G. Raymer and I. A. Walmsley, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII, p. 181.
[CrossRef]

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A (to be published); M. Munroe, D. Boggavarapu, M. E. Anderson, U. Leonhardt, and M. G. Raymer, “High-effiency, ultrafast photon-number statistics from phase-averaged homodyne detection,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, to be published).

M. Beck, M. E. Anderson, and M. G. Raymer, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

Reid, M. D.

M. D. Reid and B. Yurke, Phys. Rev. A 46, 4131 (1992), Eq. (17).
[CrossRef] [PubMed]

Risken, H.

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[CrossRef] [PubMed]

Ryan, J. F.

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

Santos, E.

Single-detector counting distributions have been expressed in terms of Wigner functions in T. Marshall and E. Santos, J. Mod. Opt. 38, 1463 (1992).
[CrossRef]

Sargent, M.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[CrossRef]

Schleich, W.

W. Vogel and W. Schleich, Rev. A 44, 7642 (1991).
[CrossRef]

Shapiro, J. H.

For discussion see J. H. Shapiro and S. S. Wagner, IEEE J. Quantum Electron. QE-20, 803 (1984), App.
[CrossRef]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978).
[CrossRef]

Shepard, T. J.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepard, Phys. Rev. A 42, 4102 (1990).
[CrossRef] [PubMed]

Singh, M.

Z. H. Cho, J. P. Jones, and M. Singh, Foundations of Medical Imaging (Wiley, New York, 1993). There is a typographical error in the Abel formula in this source.)

Slusher, R. E.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Smithey, D. T.

D. T. Smithey, M. Beck, J. Cooper, M. G. Raymer, and A. Faridani, Phys. Scr. T48, 35 (1993).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993).
[CrossRef] [PubMed]

M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993); M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890, (1993); D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A 48, 3159 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. Belsley, and M. G. Raymer, Phys. Rev. Lett. 69, 2650 (1992).
[CrossRef] [PubMed]

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

Stoler, D.

B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987).
[CrossRef] [PubMed]

Sweetser, J. N.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388 (1993).
[CrossRef] [PubMed]

Titulaer, U. M.

U. M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041 (1966).
[CrossRef]

Trebino, R.

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Vogel, K.

K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[CrossRef] [PubMed]

Vogel, W.

H. Kuhn, D.-G. Welsch, and W. Vogel, J. Mod. Opt. 41, 1607 (1994).
[CrossRef]

W. Vogel and J. Grabow, Phys. Rev. A 47, 4227 (1993).
[CrossRef] [PubMed]

W. Vogel and W. Schleich, Rev. A 44, 7642 (1991).
[CrossRef]

These relations are most appropriate in the context of pulsed fields with a local description, as considered here, rather than for monochromatic plane waves, which strictly speaking extend throughout space. See W. Vogel and D.-G. Welsch, Lectures on Quantum Optics (Akademie Verlag, location, 1994), Sec. 6.3.1; L. Knoll, W. Vogel, and D.-G. Welsch, Phys. Rev. A 36, 3803 (1987).
[CrossRef]

Wagner, S. S.

For discussion see J. H. Shapiro and S. S. Wagner, IEEE J. Quantum Electron. QE-20, 803 (1984), App.
[CrossRef]

Walmsley, I. A.

T. J. Dunn, J. N. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388 (1993).
[CrossRef] [PubMed]

M. G. Raymer and I. A. Walmsley, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII, p. 181.
[CrossRef]

Welsch, D.-G.

H. Kuhn, D.-G. Welsch, and W. Vogel, J. Mod. Opt. 41, 1607 (1994).
[CrossRef]

These relations are most appropriate in the context of pulsed fields with a local description, as considered here, rather than for monochromatic plane waves, which strictly speaking extend throughout space. See W. Vogel and D.-G. Welsch, Lectures on Quantum Optics (Akademie Verlag, location, 1994), Sec. 6.3.1; L. Knoll, W. Vogel, and D.-G. Welsch, Phys. Rev. A 36, 3803 (1987).
[CrossRef]

Wigner, E. P.

E. P. Wigner, Phys. Rev. 40, 749 (1932); M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Wilson, K.

Wodkiewicz, K.

Yee, T. K.

Yuen, H. P.

H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983).
[CrossRef] [PubMed]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978).
[CrossRef]

Yurke, B.

M. D. Reid and B. Yurke, Phys. Rev. A 46, 4131 (1992), Eq. (17).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A 32, 300, 311 (1985).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Zwanziger, M.

Generally time ordering would also be required, but because the fields are treated as free fields this is irrelevant. See, for example, M. Lax and M. Zwanziger, Phys. Rev. A 7, 750 (1973).
[CrossRef]

Am. J. Phys. (1)

For a tutorial derivation see Z. Y. Ou and L. Mandel, Am. J. Phys. 57, 66 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

For discussion see J. H. Shapiro and S. S. Wagner, IEEE J. Quantum Electron. QE-20, 803 (1984), App.
[CrossRef]

IEEE Trans. Inf. Theory (3)

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

J. Mod. Opt. (4)

M. J. Collett, R. Loudon, and C. W. Gardiner, J. Mod. Opt. 34, 881 (1987).
[CrossRef]

For review see R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

Single-detector counting distributions have been expressed in terms of Wigner functions in T. Marshall and E. Santos, J. Mod. Opt. 38, 1463 (1992).
[CrossRef]

H. Kuhn, D.-G. Welsch, and W. Vogel, J. Mod. Opt. 41, 1607 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (2)

U. Leonhardt and I. Jex, J. Phys. A 49, R1555 (1994).

P. D. Drummond and C. Gardiner, J. Phys. A 13, 2353 (1980).
[CrossRef]

Opt. Commun. (3)

K.-H. Brenner and K. Wodkiewicz, Opt. Commun. 43, 103 (1982).
[CrossRef]

B. Huttner, J. J. Baumberg, J. F. Ryan, and S. M. Barnett, Opt. Commun. 90, 128 (1992).
[CrossRef]

J. Guena, Ph. Jacquier, M. Lintz, L. Pottier, M. A. Bouchiat, and A. Hrisoho, Opt. Commun. 71, 6 (1989).
[CrossRef]

Opt. Lett. (5)

Phys. Lett. A (1)

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, Phys. Lett. A 195, 31 (1994).
[CrossRef]

Phys. Rev. (4)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857, 1882 (1969).
[CrossRef]

E. P. Wigner, Phys. Rev. 40, 749 (1932); M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[CrossRef]

U. M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041 (1966).
[CrossRef]

Phys. Rev. A (13)

P. D. Drummond, Phys. Rev. A 35, 4253 (1987).
[CrossRef] [PubMed]

For review and references see U. Leonhardt, Phys. Rev. A 48, 3265 (1993).
[CrossRef] [PubMed]

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Generally time ordering would also be required, but because the fields are treated as free fields this is irrelevant. See, for example, M. Lax and M. Zwanziger, Phys. Rev. A 7, 750 (1973).
[CrossRef]

M. D. Reid and B. Yurke, Phys. Rev. A 46, 4131 (1992), Eq. (17).
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Other (24)

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A (to be published); M. Munroe, D. Boggavarapu, M. E. Anderson, U. Leonhardt, and M. G. Raymer, “High-effiency, ultrafast photon-number statistics from phase-averaged homodyne detection,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, to be published).

R. Loudon, in Coherence, Cooperation and Fluctuations, F. Haake, L. M. Narducci, and D. F. Walls, eds. (Cambridge U. Press, Cambridge, 1986).

M. Beck, M. E. Anderson, and M. G. Raymer, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 257–260.

P. D. Drummond, in Quantum Optics V, J. D. Harvey and D. F. Walls, eds. (Springer-Verlag, Heidelberg, 1989).

J. Mostowski and M. G. Raymer, in Contemporary Nonlinear Optics, G. P. Agrawal and R. W. Boyd, eds. (Academic, Boston, Mass., 1992), p. 187–234.
[CrossRef]

M. G. Raymer and I. A. Walmsley, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII, p. 181.
[CrossRef]

Strictly speaking, Eq. (2.10) should contain a sum over all modes, even those that are unexcited. However, because we use Eq. (2.10) only in evaluating normally ordered operator averages, these unexcited modes do not contribute.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), p. 149.

For a review see M. G. Raymer, D. T. Smithey, M. Beck, M. Anderson, and D. F. McAlister, “Measurement of the Wigner function in quantum optics,” in Proceedings of the Third International Wigner Symposium, Int. J. Mod. Phys. B (to be published);M. G. Raymer, D. T. Smithey, M. Beck, and J. Cooper, Acta Phys. Polon. 86, 71 (1994).

These relations are most appropriate in the context of pulsed fields with a local description, as considered here, rather than for monochromatic plane waves, which strictly speaking extend throughout space. See W. Vogel and D.-G. Welsch, Lectures on Quantum Optics (Akademie Verlag, location, 1994), Sec. 6.3.1; L. Knoll, W. Vogel, and D.-G. Welsch, Phys. Rev. A 36, 3803 (1987).
[CrossRef]

R. J. Glauber, in Quantum Optics and Electronics, C. De Witt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965), pp. 331–381.

The symbol : : indicates normal ordering of the flux operators at the detectors, i.e., all annihilation operators go to the right of creation operators. Because the transformation Eq. (2.1) does not mix positive- and negative-frequency fields, the normal ordering may be equivalently enforced at the input to the beam splitter.

One can prove this by using the relation ρ^S=(1/2) ∫dxdyPS(x,y)∣(x+iy)/2〉 〈(x+iy)/2∣, a diagonal sum over coherent states. See Ref. 49.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983), p. 232.

Note that one should not refer to sub-Poisson statistics here because even in the coherent-state case the distribution Eq. (3.3) is not a Poisson.

H. M. Nussenvieg, Introduction to Quantum Optics (Gordon & Breach, New York, 1974), Chap. 3.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[CrossRef]

Going from Eq. (3.3) to Eq. (4.1) is an example of the optical equivalence theorem by which any normally ordered expectation value can be replaced by a classical integral if the field state can be described as a mixture of coherent states. See Refs. 49 and 59 below.

An alternative, and easier, derivation of Eq. (4.3) is to expand Eq. (3.1) before it is substituted into Eq. (3.2).

It has been pointed out that for Eq. (5.1) to be valid it is not clear a priori for arbitrary states by how much |αL|2 needs to exceed the mean signal photon number. See Ref. 22.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

A computer program for filtered backprojection inverse Radon reconstruction, written by A. Faridani of Oregon State University, is available from M. Raymer.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. I, Chap. III.

Z. H. Cho, J. P. Jones, and M. Singh, Foundations of Medical Imaging (Wiley, New York, 1993). There is a typographical error in the Abel formula in this source.)

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Figures (2)

Fig. 1
Fig. 1

A signal pulse field ÊS interferes with a shorter LO pulse ÊL at a 50/50 beam splitter. The LO phase ϕ determines which quadrature amplitude of the signal is detected. The superposed fields are detected with high-efficiency photodiodes having response times much longer than the pulse durations. The photocurrents are integrated and sampled by analog-to-digital converters (ADC’s), to yield pulse photoelectron numbers n1 and n2.

Fig. 2
Fig. 2

Generalized amplitudes xϕ and yϕ are related to quadrature amplitudes x and y by a rotation in phase space. Projection integrals along dashed lines through the Wigner distribution W(x, y) yield the measured distributions Pϕ(xϕ) for generalized quadrature amplitude xϕ. One can reconstruct the Wigner distribution tomographically by measuring Pϕ(xϕ) for many different rotation angles ϕ, which is equal to the LO phase.

Equations (81)

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E ^ 1 ( + ) = t 1 E ^ S ( + ) + r 2 E ^ L ( + ) , E ^ 2 ( + ) = r 1 E ^ S ( + ) + t 2 E ^ L ( + ) ,
E ^ S ( + ) ( r , t ) = i j , l d k z ω j / 2 π D 2 a ^ j l ( k z ) × exp [ - i ω j l ( k z ) t + i k j l · r ] .
[ a ^ j l ( k z ) , a ^ n m ( k z ) ] = δ j n δ l m 2 π δ ( k z - k z ) .
ω j l ( k z ) c [ k z + ( n y l 2 + n x j 2 ) ( 2 π / D ) 2 / 2 K ] ,
Φ ^ S ( + ) ( r , t ) = i c 1 / 2 2 π D j , l 0 d k z a ^ j l ( k z ) × exp [ - i ω j l ( k z ) t + i k j l · r ] ,
I ^ i ( t ) = Det d 2 s Φ ^ i ( - ) ( s , t ) Φ ^ i ( + ) ( s , t ) ,
N ^ 12 = N ^ 1 - N ^ 2 = 0 T ( I ^ 1 - I ^ 2 ) d t ,
N ^ T = N ^ 1 + N ^ 2 = 0 T ( I ^ 1 + I ^ 2 ) d t .
N ^ 12 = 0 T d t Det d 2 s ( Φ ^ L ( - ) Φ ^ S ( + ) + Φ ^ S ( - ) Φ ^ L ( + ) ) .
Φ ^ L ( + ) = i a ^ L f L ( t ) u L ( s ) ,
0 T f L ( t ) 2 d t = 1 ,             Det u L ( s ) 2 d 2 s = 1 ,
N ^ T = a ^ L a ^ L + N ^ S ,
N ^ S = 0 T d t Det d 2 s Φ ^ S ( - ) ( s , t ) Φ ^ S ( + ) ( s , t ) .
N ^ 12 = a ^ a ^ L + a ^ a ^ L ,
a ^ = - i 0 T d t Det d 2 s f L * ( t ) u L * ( s ) Φ ^ S ( + ) ( s , t ) .
N ^ 12 = α L [ a ^ exp ( - i ϕ ) + a ^ exp ( i ϕ ) ] .
x ^ ϕ = N ^ 12 / ( α L 2 1 / 2 ) = [ a ^ exp ( - i ϕ ) + a ^ exp ( i ϕ ) ] / 2 1 / 2 .
y ^ ϕ = [ a ^ exp ( - i ϕ ) - a ^ exp ( i ϕ ) ] / i 2 1 / 2 .
( x ^ ϕ y ^ ϕ ) = ( cos ϕ sin ϕ - sin ϕ cos ϕ ) ( x ^ y ^ ) .
Δ x ^ ϕ Δ y ^ ϕ 1 / 2.
P 12 ( n 1 , n 2 ) = : exp ( - η N ^ 1 ) ( η N ^ 1 ) n 1 n 1 ! exp ( - η N ^ 2 ) ( η N ^ 2 ) n 2 n 2 ! : S , L ,
P ( n 12 ) = n 2 P 12 ( n 2 + n 12 , n 2 ) .
P ( n 12 ) = : exp [ - η ( N ^ 1 + N ^ 2 ) ] ( N ^ 2 / N ^ 1 ) n 12 / 2 × I n 12 [ 2 η ( N ^ 1 N ^ 2 ) 1 / 2 ] : S , L ,
Φ ^ B ( + ) ( s , t ) = Φ ^ S ( + ) ( s , t ) - i a ^ f L ( t ) u L ( s ) = 0 T d t Det d 2 s P B ( t , t , s , s ) Φ ^ S ( + ) ( s , t ) ,
P B ( t , t , s , s ) = δ ( t - t ) δ 2 ( s - s ) - f L ( t ) u L ( s ) f L * ( t ) u L * ( s ) .
N ^ B = 0 T d t Det d 2 s Φ ^ B ( - ) ( s , t ) Φ ^ B ( + ) ( s , t ) .
N ^ S = a ^ a ^ + N ^ B ,
N ^ 1 = ( a ^ L + a ^ 2 ) ( a ^ L + a ^ 2 ) + N ^ B 2 , N ^ 2 = ( a ^ L - a ^ 2 ) ( a ^ L - a ^ 2 ) + N ^ B 2 .
P ( n 12 ) = P ( n 12 ; α L ) P L ( α L ) d 2 α L ,
P ( n 12 ; α L ) = : exp [ - η ( N ^ 1 + N ^ 2 ) ] ( N ^ 2 / N ^ 1 ) n 12 / 2 × I n 12 [ 2 η ( N ^ 1 N ^ 2 ) 1 / 2 ] : S .
P ( n 12 ; α L ) = : exp [ - ( n 12 - η N ^ 12 ) 2 / 2 η N ^ T ] [ π 2 η N ^ T ] 1 / 2 : S ,
N ^ T = α L 2 + α ^ a ^ + N ^ B .
Φ S ( + ) ( r , t ) = i c 1 / 2 2 π D j , l 0 d k z α j l ( k z ) × exp [ - i ω j l ( k z ) t + i k j · r ] .
α = - i 0 T d t Det d 2 s f L * ( t ) u L * ( s ) Φ S ( + ) ( s , t ) .
N ¯ S = N ^ S = 0 T d t Det d 2 s Φ S ( - ) ( s , t ) Φ S ( + ) ( s , t ) = α 2 + N ¯ B ,
N ¯ B = 0 T d t Det d 2 s Φ B ( - ) ( s , t ) Φ B ( + ) ( s , t )
P ( n 12 ; α L , { α j l } ) = exp [ - ( n 12 - η { α L * α + α L α * } ) 2 / 2 η ( α L 2 + α 2 + N ¯ B ) ] [ π 2 η ( α L 2 + α 2 + N ¯ B ) ] 1 / 2 .
P ( n 12 ) = P ( n 12 ; α L , { α j l } ) P L ( α L ) P S ( { α j l } ) × d 2 α L d 2 { α j l } ,
P ( n 12 ; α L ) = : exp [ - ( n 12 - η { α L * a ^ + α L a ^ } ) 2 / 2 η ( α L 2 + N ¯ B ) ] [ π 2 η ( α L 2 + N ¯ B ) ] 1 / 2 : MMS ,
P ϕ ( x ϕ ; α L ) = : exp [ - ( x ϕ - x ^ ϕ ) 2 / 2 σ 2 ] [ π 2 σ 2 ] 1 / 2 : MMS ,
η ˜ = η ( 1 + N ¯ B α L 2 ) - 1 .
P ϕ ( x ϕ ; α L ) = exp [ - ( x ϕ - x ˜ ϕ ) 2 / 2 σ 2 ] [ π 2 σ 2 ] 1 / 2 P S ( x , y ) d x d y ,
P ϕ ( x ϕ ; α L ) = exp [ - ( x ϕ - x c cos ϕ - y c sin ϕ ) 2 / 2 σ 2 ] [ π 2 σ 2 ] 1 / 2 .
W S ( x , y ) = 1 π exp [ - ( x - x ) 2 - ( y - y ) 2 ] P S ( x , y ) d x d y .
P ϕ ( x ϕ ; α L ) = exp [ - { x ϕ - x ˜ ϕ ( x , y ) } 2 / 2 2 ] π 2 2 × W S ( x , y ) d x d y ,
P ϕ ( x ϕ ) = δ [ x ϕ - x ˜ ϕ ( x , y ) ] W S ( x , y ) d x d y ,
P ϕ ( x ϕ ) = - W S ( x ϕ cos ϕ - y ϕ sin ϕ , x ϕ sin ϕ + y ϕ cos ϕ ) d y ϕ .
W S ( x , y ) = 1 4 π 2 - d x ϕ - d ξ ξ 0 π d ϕ P ϕ ( x ϕ ) × exp [ i ξ ( x ϕ - x cos ϕ - y sin ϕ ) ] .
W S ( x , y ) = 1 π - x + x ρ ^ S x - x exp ( - 2 i y x ) d x ,
( x ^ m y ^ n ) Weyl = x m y n W S ( x , y ) d x d y .
P ϕ ( x ϕ ) = δ [ x ϕ - x ˜ ϕ ( x , y ) ] W Det ( x , y ) d x d y ,
W Det ( x , y ) = 1 π 2 2 exp [ - ( x - x ) 2 / 2 2 - ( y - y ) 2 / 2 2 ] W S ( x , y ) d x d y .
Φ S ( + ) ( s , t ) = i α S f S ( t ) u S ( s ) ,
α = η L S α S ,
η L S exp ( i β ) = 0 T d t Det d 2 s f L * ( t ) u L * ( s ) f S ( t ) u S ( s ) ,
Φ ^ B ( + ) ( s , t ) = i α B U B ( t , s ) ,
U B ( t , s ) = [ 1 - η L S 2 ] 1 / 2 [ f S ( t ) u S ( s ) - η L S exp ( i β ) f L ( t ) u L ( s ) ]
α B = [ 1 - η L S 2 ] 1 / 2 α S .
N ¯ B = [ 1 - η L S 2 ] α S 2 .
f L ( t ) = exp ( - i ω L t ) h L ( t - t L ) ,
N ¯ MMS ( ω L , t L ) = | 0 T d t exp ( i ω L t ) h L ( t - t L ) ϕ ( + ) ( t ) | 2 ,
ϕ ( + ) ( t ) = Det d 2 s u L * ( s ) Φ S ( + ) ( s , t ) ,
h L ( t - t L ) = { 0 t > t L exp [ γ ( t - t L ) ] t < t L .
N ^ 12 = δ ( N ^ B - α L 2 + a ^ a ^ ) + 2 r t ( a ^ a ^ L + a ^ a ^ L ) δ ( N ^ B - I L ) + x ϕ 2 I L ,
Δ n 12 2 η N ¯ T + η 2 Δ N 12 2 ,
η Δ N 12 2 10 - 2 N ¯ T 10 - 2 ( I ¯ L + N ¯ B ) .
Δ N 12 2 δ 2 ( Δ N B 2 + Δ I L 2 ) + x ϕ Δ I L 2 I ¯ L ( 1 2 x ϕ - δ 2 I ¯ L ) ,
δ Δ I L I ¯ L = δ f ( 10 - 2 / I ¯ L ) 1 / 2 .
f 2 ( 1 2 x ϕ 2 + x ϕ δ 2 I ¯ L ) 10 - 2 .
f 2 N ¯ MMS 10 - 2
2 f 2 δ N ¯ MMS I ¯ L 10 - 2 .
N ¯ MMS = Δ x ϕ 2 - 1 / 2 = Δ N 12 2 / 2 I L - 1 / 2 ,
[ Φ ^ S ( + ) ( s , t ) , Φ ^ S ( - ) ( s , t ) ] = c 2 π D 2 j , l 0 d k z × exp [ - i ω j l ( k z ) ( t - t ) ] × exp [ i k j l · ( s ¯ - s ¯ ) ] ,
[ Φ ^ S ( + ) ( s , t ) , Φ ^ S ( - ) ( s , t ) ] = δ ( t - t ) δ 2 ( s - s ) .
P ϕ ( x ϕ ) = Tr [ ρ ^ O ^ ] ,
O ^ = : exp [ - ( x ϕ - x ^ ϕ ) 2 / 2 σ 2 ] π 2 σ 2 : .
P ϕ ( x ϕ ) = - d x ϕ d y ϕ W ϕ ( x ϕ , y ϕ ) f O ^ ( s ) ( x ϕ , y ϕ ) ,
f O ^ ( n ) ( x ϕ , y ϕ ) = exp [ - ( x ϕ - x ϕ ) 2 / 2 σ 2 ] π 2 σ 2
f ˜ O ^ ( s ) ( λ ϕ , μ ϕ ) = exp [ ( λ ϕ 2 + μ ϕ 2 ) / 4 ] f ˜ O ^ ( n ) ( λ ϕ , μ ϕ ) ,
f ˜ O ^ ( n , s ) ( λ ϕ , μ ϕ ) = - d x ϕ d y ϕ f O ^ ( n , s ) ( x ϕ , y ϕ ) × exp [ i ( λ ϕ x ϕ + μ ϕ y ϕ ) ] .
f O ^ ( s ) ( x ϕ , y ϕ ) = exp [ - ( x ϕ - x ϕ ) 2 / ( 2 σ 2 - 1 ) ] π ( 2 σ 2 - 1 ) .

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