Abstract

We develop a scheme to reconstruct the optical quantum state of a single-mode bright light field by using the dispersion characteristics of the empty cavity. The input field has a strong coherent component at frequency ω0, which serves as a local oscillator (LO) to measure its two-sideband mode at ω0±Ω. We control the relative phase of the 0–2π range between the LO and the two-sideband mode by scanning the cavity length, so the optical quantum state is tomographically reconstructed. In the proposed scheme the influence of the space-mode mismatch between the LO and measured mode on the quantum efficiency is eliminated, and this scheme can conveniently be used in some quantum optical systems in which LO field cannot be available.

© 2000 Optical Society of America

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  1. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef] [PubMed]
  2. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef] [PubMed]
  3. D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
    [CrossRef] [PubMed]
  4. G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
    [CrossRef]
  5. H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
    [CrossRef]
  6. A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).
  7. G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
    [CrossRef] [PubMed]
  8. U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
    [CrossRef]
  9. M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
    [CrossRef]
  10. S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
    [CrossRef] [PubMed]
  11. J. H. Shapiro and A. Shakeel, “Optimizing homodyne of detection of quadrature noise squeezing by local-oscillator selection,” J. Opt. Soc. Am. B 14, 232–239 (1997).
    [CrossRef]
  12. G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
    [CrossRef] [PubMed]
  13. G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
    [CrossRef]
  14. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
    [CrossRef] [PubMed]
  15. P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
    [CrossRef]
  16. Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
    [CrossRef] [PubMed]
  17. T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
    [CrossRef]
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  19. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
    [CrossRef]
  20. G. Breitenbach and S. Schiller, “Homodyne tomography of classical and non-classical light,” J. Mod. Opt. 44, 2207–2225 (1997).
    [CrossRef]

1998

G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
[CrossRef]

1997

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

G. Breitenbach and S. Schiller, “Homodyne tomography of classical and non-classical light,” J. Mod. Opt. 44, 2207–2225 (1997).
[CrossRef]

J. H. Shapiro and A. Shakeel, “Optimizing homodyne of detection of quadrature noise squeezing by local-oscillator selection,” J. Opt. Soc. Am. B 14, 232–239 (1997).
[CrossRef]

1996

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

1995

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

1994

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

1993

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

1992

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

1991

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

1989

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

1986

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Anderson, M. E.

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

Boggavarapu, D.

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

Bramati, A.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Breitenbach, G.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

G. Breitenbach and S. Schiller, “Homodyne tomography of classical and non-classical light,” J. Mod. Opt. 44, 2207–2225 (1997).
[CrossRef]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Cooper, J.

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

D’Ariano, G. M.

G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

Devoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Galatola, P.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

Giacobino, E.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Grangier, P.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Grelu, P.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Holliday, G. S.

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

Jost, V.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Kimble, H. J.

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

Kiss, T.

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Kuhn, H.

H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Kumar, P.

G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
[CrossRef]

Leonhardt, U.

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

Leuchs, G.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

Levenson, M. D.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Lugiato, L. A.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

Macchiavello, C.

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

Marin, F.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Mlynek, J.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Muller, T.

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Munroe, M.

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

Paris, M. G. A.

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

Paul, H.

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

Pereira, S. F.

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Poizat, J. P.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Poizat, J. Ph.

Porreca, M. G.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

Qu, Y.

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

Raymer, M. G.

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

Richter, T.

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Risken, H.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Roch, J. F.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Schiller, S.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

G. Breitenbach and S. Schiller, “Homodyne tomography of classical and non-classical light,” J. Mod. Opt. 44, 2207–2225 (1997).
[CrossRef]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Muller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Shakeel, A.

Shapiro, J. H.

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Singh, S.

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

Smithey, D. T.

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Tasche, M.

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

Tombesi, P.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

Vasilyev, M.

G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
[CrossRef]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Vogel, W.

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Walls, D. F.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Welsch, D. G.

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

Xiao, M.

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

Zhang, T. C.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Zuchetti, A.

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

J. Mod. Opt.

H. Kuhn, D. G. Welsch, and W. Vogel, “Determination of density matrices from field distributions and quasiprobilities,” J. Mod. Opt. 41, 1607–1613 (1994).
[CrossRef]

G. Breitenbach and S. Schiller, “Homodyne tomography of classical and non-classical light,” J. Mod. Opt. 44, 2207–2225 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Nature

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

Opt. Commun.

P. Galatola, L. A. Lugiato, M. G. Porreca, P. Tombesi, and G. Leuchs, “System control by variation of the squeezing phase,” Opt. Commun. 85, 95–103 (1991).
[CrossRef]

U. Leonhardt, M. Munroe, T. Kiss, T. Richter, and M. G. Raymer, “Sampling of the photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
[CrossRef]

Phys. Rev. A

M. Munroe, D. Boggavarapu, M. E. Anderson, and M. G. Raymer, “Photon number statistics from the phase-averaged quadrature field distribution: theory and ultrafast measurement,” Phys. Rev. A 52, R924–R927 (1995).
[CrossRef]

G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A 52, R1801–R1804 (1995).
[CrossRef] [PubMed]

G. M. D’Ariano, M. Vasilyev, and P. Kumar, “Self-homodyne tomography of a twin-beam state,” Phys. Rev. A 58, 636–648 (1998).
[CrossRef]

A. Zuchetti, W. Vogel, D. G. Welsch, and M. Tasche, “Direct sampling of density matrices in field-strength bases,” Phys. Rev. A 54, 1–4 (1996).

G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris, “Detection of the density matrix through optical homodyne tomography without filtered back projection,” Phys. Rev. A 50, 4298–4302 (1994).
[CrossRef] [PubMed]

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Y. Qu, M. Xiao, G. S. Holliday, S. Singh, and H. J. Kimble, “Enhancement of photon antibunching by passive interferometry,” Phys. Rev. A 45, 4932–4943 (1992).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, “Measurement of number-phase uncertainty relations of optical fields,” Phys. Rev. A 48, 3159–3167 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. Devoe, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

S. Schiller, G. Breitenbach, S. F. Pereira, T. Muller, and J. Mlynek, “Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation,” Phys. Rev. Lett. 77, 2933–2936 (1996).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt.

T. C. Zhang, J. P. Poizat, P. Grelu, J. F. Roch, P. Grangier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, “Quantum noise of free-running and externally-stabilized laser diodes,” Quantum Semiclassic. Opt. 7, 601–613 (1995).
[CrossRef]

Other

C. Fabre and S. Reynaud, “Quantum noise in optical systems: a semiclassical approach,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1991), pp. 1–42.

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Figures (5)

Fig. 1
Fig. 1

Schematic of optical homodyne tomography.

Fig. 2
Fig. 2

Self-homodyne tomography with an empty cavity. The input field has a strongly coherent component at center frequency ω0. The output field reflected by the empty cavity is directly detected, and a narrow band of the output photocurrents is selected, centered about frequency Ωω0. PZT, piezoelectric transducer; M2, M3, high reflectors.

Fig. 3
Fig. 3

Phase-shift angle as a function of phase detuning ϕ0. r=0.9998.

Fig. 4
Fig. 4

Variance of the amplitude-squeezed coherent state numerically calculated relative to the detuning frequency of the scanned empty cavity at sideband mode frequency 2πΩ=500 MHz. L=0.1 m,r=0.9998, and the amplitude squeezing is 90%.

Fig. 5
Fig. 5

Magnitude and phase of the reflected field relative to the input field for a nonideal empty cavity r=0.9998.

Equations (33)

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cˆ(t)=(2/2)[aˆ(t)+exp(iθ)aˆLO(t)],
dˆ(t)=(2/2)[aˆ(t)-exp(iθ)aˆLO(t)].
Iˆ-(t)=cˆ+(t)cˆ(t)-dˆ+(t)dˆ(t)=exp(iθ)aˆ+(t)aˆLO(t)+exp(-iθ)aˆLO+(t)aˆ(t),
a¯(ω0)=ass;a¯(ω0±Ωω0)=0.
Iˆ-(Ω)=a¯LO(ω0)[exp(-iθ)aˆ(ω0-Ω)+exp(iθ)aˆ+(ω0+Ω)].
Xˆ(θ, t)=[aˆ(t)exp(-iθ)+aˆ+(t)exp(iθ)]/2,
Xˆ(θ, Ω)=[aˆ(ω0-Ω)exp(-iθ)+aˆ+(ω0+Ω)exp(iθ)]/2,
[aˆ(ω0-Ω), aˆ+(ω0-Ω)]=2πδ(Ω-Ω).
Pθ(xθ)=-W(xθ cos θ-yθ sin θ,xθ sin θ+yθ cos θ)dyθ,
aˆ=raˆ+taˆin,
aˆout=taˆ-raˆin,
aˆ=r1aˆ exp(iϕ0)+1-r12aˆvac,
aˆ0out=r1-r exp(-iϕ0)exp(-iϕ0)-rr1aˆ0in(ω0)+t1-r12exp(-iϕ0)-rr1aˆvac,
aˆ+out=r1-r exp[-i(ϕ0+ΩL/c)]exp[-i(ϕ0+ΩL/c)]-rr1aˆ0in(ω0+Ω)+t1-r12exp[-i(ϕ0+ΩL/c)]-rr1aˆvac,
aˆ-out=r1-r exp[-i(ϕ0-ΩL/c)]exp[-i(ϕ0-ΩL/c)]-rr1aˆ0in(ω0-Ω)+t1-r12exp[-i(ϕ0-ΩL/c)]-rr1aˆvac,
aˆ0out=1-r exp(-iϕ0)exp(-iϕ0)-raˆ0in(ω0),
aˆ+out=1-r exp[-i(ϕ0+ΩL/c)]exp[-i(ϕ0+ΩL/c)]-raˆ0in(ω0+Ω),
aˆ-out=1-r exp[-i(ϕ0-ΩL/c)]exp[-i(ϕ0-ΩL/c)]-raˆ0in(ω0-Ω).
1-r exp[-iϕ0]exp[-iϕ0]-r=1,aˆ0out¯=aˆ0in(ω0)¯=const.
θ(ϕ0)=Arg1-r exp(-iϕ0)exp(-iϕ0)-r.
Iˆ(Ω)-+dt exp(iΩt):|ˆ(t)|2:
=-+dωˆ+(ω+Ω)ˆ(ω),
Iˆ(Ω)=aˆ0out+aˆout+aˆ+out+aˆ0out.
ıˆ(Ω)=lim|aˆ0out| TrLO[Iˆ(Ω)ρˆLO]2|aˆ0out|,
ıˆ(Ω)=12{exp[iθ(ϕ0)]aˆ-out+exp[-iθ(ϕ0)]aˆ+out+}=12 expiθ(ϕ0)-θϕ0-ΩLcaˆ-in+exp-iθ(ϕ0)-θϕ0+ΩLcaˆ+in+.
θϕ0+ΩLc-π,
θϕ0-ΩLcπ,θ(ϕ0)[-π, π],
ıˆ(Ω)=-(1/2){exp[iθ(ϕ0)]aˆ-in+exp[-iθ(ϕ0)]aˆ+in+}.
aˆ0out¯=r1-r exp(-iϕ0)exp(-iϕ0)-rr1 aˆ0in(ω0)¯.
θ(ϕ0)=Argrr1 exp(-iϕ0)-1r1 exp(-iϕ0)-r.
aˆ0out¯=r1-r exp(-iϕ0)exp(-iϕ0)-rr1 aˆ0in(ω0)¯
=r1-r exp(-iϕ0)exp(-iϕ0)-rr1exp[iθ(ϕ0)]aˆ0in(ω0)¯,θ(ϕ0)[-π, π],aˆ+out-aˆ0in(ω0+Ω),aˆ-out-aˆ0in(ω0-Ω).
ıˆ(Ω)=-12{exp[iθ(ϕ0)]aˆ-in+exp[-iθ(ϕ0)]aˆ+in+}.

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