Abstract

We present a multichannel optical homodyne detection setup suitable for measurements of quantum light statistics with ultrafast time resolution. We employ the setup to measure light statistics of diode lasers driven by both a direct and a pulsed current source. In particular, we measure the time-resolved second-order correlation function g(2) and photon number distribution. Our results provide information about the dynamics of the lasing transition.

© 2013 Optical Society of America

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  18. M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, “Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution,” Opt. Express 18, 20229–20241 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
    [CrossRef]
  23. M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).
  24. D. F. McAlister and M. G. Raymer, “Correlation and joint density matrix of two spatial-temporal modes from balanced-homodyne sampling,” J. Mod. Opt. 44, 2359–2383 (1997).
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    [CrossRef]
  29. M. Blazek, S. Hartmann, A. Molitor, and W. Elsaesser, “Unifying intensity noise and second-order coherence properties of amplified spontaneous emission sources,” Opt. Lett. 36, 3455–3457 (2011).
    [CrossRef]
  30. G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. 138, B1012–B1016 (1965).
    [CrossRef]
  31. F. T. Arecchi and V. Degiorgio, “Statistical properties of laser radiation during a transient buildup,” Phys. Rev. A 3, 1108–1124 (1971).
    [CrossRef]
  32. S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
    [CrossRef]
  33. P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
    [CrossRef]
  34. P. Horowitz and W. Hill, The Art of Electronics, 2nd ed.(Cambridge University, 1989).

2013 (1)

2011 (2)

M. Blazek, S. Hartmann, A. Molitor, and W. Elsaesser, “Unifying intensity noise and second-order coherence properties of amplified spontaneous emission sources,” Opt. Lett. 36, 3455–3457 (2011).
[CrossRef]

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

2010 (1)

2009 (5)

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[CrossRef]

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

2008 (1)

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

2006 (1)

M. Kira and S. W. Koch, “Quantum-optical spectroscopy of semiconductors,” Phys. Rev. A 73, 013813 (2006).
[CrossRef]

2005 (1)

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

2001 (1)

1997 (2)

D. F. McAlister and M. G. Raymer, “Correlation and joint density matrix of two spatial-temporal modes from balanced-homodyne sampling,” J. Mod. Opt. 44, 2359–2383 (1997).
[CrossRef]

D. F. McAlister and M. G. Raymer, “Ultrafast photon-number correlations from dual-pulse, phase-averaged homodyne detection,” Phys. Rev. A 55, R1609–R1612 (1997).
[CrossRef]

1995 (1)

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]

1993 (1)

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]

1990 (1)

P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
[CrossRef]

1988 (1)

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

1986 (1)

S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
[CrossRef]

1971 (1)

F. T. Arecchi and V. Degiorgio, “Statistical properties of laser radiation during a transient buildup,” Phys. Rev. A 3, 1108–1124 (1971).
[CrossRef]

1967 (2)

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. I. general theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

M. Lax and W. Louisell, “Quantum noise IX: quantum Fokker–Planck solution for laser noise,” IEEE J. Quantum Electron. 3, 47–58 (1967).
[CrossRef]

1965 (1)

G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. 138, B1012–B1016 (1965).
[CrossRef]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

1956 (1)

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

Allerman, A.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

E. Blansett, M. Raymer, G. Khitrova, H. Gibbs, D. K. Serkland, A. Allerman, and K. Geib, “Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” Opt. Express 9, 312–318 (2001).
[CrossRef]

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]

Arecchi, F. T.

F. T. Arecchi and V. Degiorgio, “Statistical properties of laser radiation during a transient buildup,” Phys. Rev. A 3, 1108–1124 (1971).
[CrossRef]

Aßmann, M.

M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, “Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution,” Opt. Express 18, 20229–20241 (2010).
[CrossRef]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Bachor, H.-A.

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. (Wiley-VCH, 2004).

Bayer, M.

M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, “Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution,” Opt. Express 18, 20229–20241 (2010).
[CrossRef]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[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]

Berstermann, T.

M. Aßmann, F. Veit, J.-S. Tempel, T. Berstermann, H. Stolz, M. van der Poel, J. M. Hvam, and M. Bayer, “Measuring the dynamics of second-order photon correlation functions inside a pulse with picosecond time resolution,” Opt. Express 18, 20229–20241 (2010).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Blansett, E.

Blansett, E. L.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

Blazek, M.

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]

Boisen, A.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

Boyd, S.

S. Boyd, and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds. (Springer-Verlag, 2008), pp. 95–110.

Brown, R. H.

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

Cui, G.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

Cundiff, S.

M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).

Cundiff, S. T.

G. Roumpos and S. T. Cundiff, “Photon number distributions from a diode laser,” Opt. Lett. 38, 139–141 (2013).
[CrossRef]

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

Degiorgio, V.

F. T. Arecchi and V. Degiorgio, “Statistical properties of laser radiation during a transient buildup,” Phys. Rev. A 3, 1108–1124 (1971).
[CrossRef]

Elsaesser, W.

Fabre, C.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

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]

Figueroa, E.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Forchel, A.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Geib, K.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

E. Blansett, M. Raymer, G. Khitrova, H. Gibbs, D. K. Serkland, A. Allerman, and K. Geib, “Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” Opt. Express 9, 312–318 (2001).
[CrossRef]

Gibbs, H.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

E. Blansett, M. Raymer, G. Khitrova, H. Gibbs, D. K. Serkland, A. Allerman, and K. Geib, “Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” Opt. Express 9, 312–318 (2001).
[CrossRef]

Gies, C.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Godard, A.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

Grant, M.

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds. (Springer-Verlag, 2008), pp. 95–110.

Hartmann, S.

Hill, W.

P. Horowitz and W. Hill, The Art of Electronics, 2nd ed.(Cambridge University, 1989).

Höfling, S.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Hommel, D.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Horowitz, P.

P. Horowitz and W. Hill, The Art of Electronics, 2nd ed.(Cambridge University, 1989).

Hunter, A. E.

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

Hvam, J. M.

Jahnke, F.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Kalden, J.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Khitrova, G.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

E. Blansett, M. Raymer, G. Khitrova, H. Gibbs, D. K. Serkland, A. Allerman, and K. Geib, “Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” Opt. Express 9, 312–318 (2001).
[CrossRef]

Kira, M.

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

M. Kira and S. W. Koch, “Quantum-optical spectroscopy of semiconductors,” Phys. Rev. A 73, 013813 (2006).
[CrossRef]

M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).

Kistner, C.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Kjærgaard, N.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Koch, S.

M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).

Koch, S. W.

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

M. Kira and S. W. Koch, “Quantum-optical spectroscopy of semiconductors,” Phys. Rev. A 73, 013813 (2006).
[CrossRef]

Korystov, D.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Koschorreck, M.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Kruse, C.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Kubasik, M.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Kupchak, C.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Kuwata, M.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Lachs, G.

G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. 138, B1012–B1016 (1965).
[CrossRef]

Lamb, W. E.

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. I. general theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

Lax, M.

M. Lax and W. Louisell, “Quantum noise IX: quantum Fokker–Planck solution for laser noise,” IEEE J. Quantum Electron. 3, 47–58 (1967).
[CrossRef]

Lobino, M.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Louisell, W.

M. Lax and W. Louisell, “Quantum noise IX: quantum Fokker–Planck solution for laser noise,” IEEE J. Quantum Electron. 3, 47–58 (1967).
[CrossRef]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

McAlister, D. F.

D. F. McAlister and M. G. Raymer, “Ultrafast photon-number correlations from dual-pulse, phase-averaged homodyne detection,” Phys. Rev. A 55, R1609–R1612 (1997).
[CrossRef]

D. F. McAlister and M. G. Raymer, “Correlation and joint density matrix of two spatial-temporal modes from balanced-homodyne sampling,” J. Mod. Opt. 44, 2359–2383 (1997).
[CrossRef]

Mecozzi, A.

P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
[CrossRef]

Molitor, A.

Müller, J. H.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Munroe, M.

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]

Nagasawa, N.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Polzik, E. S.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Ralph, T. C.

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. (Wiley-VCH, 2004).

Raymer, M.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

E. Blansett, M. Raymer, G. Khitrova, H. Gibbs, D. K. Serkland, A. Allerman, and K. Geib, “Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” Opt. Express 9, 312–318 (2001).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

D. F. McAlister and M. G. Raymer, “Ultrafast photon-number correlations from dual-pulse, phase-averaged homodyne detection,” Phys. Rev. A 55, R1609–R1612 (1997).
[CrossRef]

D. F. McAlister and M. G. Raymer, “Correlation and joint density matrix of two spatial-temporal modes from balanced-homodyne sampling,” J. Mod. Opt. 44, 2359–2383 (1997).
[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]

Reitzenstein, S.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Rosencher, E.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

Roumpos, G.

G. Roumpos and S. T. Cundiff, “Photon number distributions from a diode laser,” Opt. Lett. 38, 139–141 (2013).
[CrossRef]

M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).

Roy, R.

S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
[CrossRef]

Sanders, B. C.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

Sapia, A.

P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
[CrossRef]

Schneider, C.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Scully, M. O.

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. I. general theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

Serkland, D.

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

Serkland, D. K.

Smith, R. P.

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

Smithey, D. T.

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]

Spano, P.

P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
[CrossRef]

Stolz, H.

Takiguchi, Y.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Tempel, J.-S.

Tsuchiya, Y.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Twiss, R. Q.

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

Ueda, M.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Urakami, T.

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

van der Poel, M.

Vandenberghe, L.

S. Boyd, and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Veit, F.

Wiersig, J.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Windpassinger, P. J.

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yu, A. W.

S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
[CrossRef]

Zhu, S.

S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
[CrossRef]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

IEEE J. Quantum Electron. (2)

M. Lax and W. Louisell, “Quantum noise IX: quantum Fokker–Planck solution for laser noise,” IEEE J. Quantum Electron. 3, 47–58 (1967).
[CrossRef]

E. L. Blansett, M. Raymer, G. Cui, G. Khitrova, H. Gibbs, D. Serkland, A. Allerman, and K. Geib, “Picosecond polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 287–301 (2005).
[CrossRef]

J. Mod. Opt. (1)

D. F. McAlister and M. G. Raymer, “Correlation and joint density matrix of two spatial-temporal modes from balanced-homodyne sampling,” J. Mod. Opt. 44, 2359–2383 (1997).
[CrossRef]

Meas. Sci. Technol. (1)

P. J. Windpassinger, M. Kubasik, M. Koschorreck, A. Boisen, N. Kjærgaard, E. S. Polzik, and J. H. Müller, “Ultra low-noise differential AC-coupled photodetector for sensitive pulse detection applications,” Meas. Sci. Technol. 20, 055301 (2009).
[CrossRef]

Nature (2)

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249(2009).
[CrossRef]

Nature Phys. (2)

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nature Phys. 7, 799–804 (2011).
[CrossRef]

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270(2009).
[CrossRef]

Opt. Commun. (1)

M. Ueda, M. Kuwata, N. Nagasawa, T. Urakami, Y. Takiguchi, and Y. Tsuchiya, “Picosecond time-resolved photoelectric correlation measurement with a photon-counting streak camera,” Opt. Commun. 65, 315–318 (1988).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (3)

G. Lachs, “Theoretical aspects of mixtures of thermal and coherent radiation,” Phys. Rev. 138, B1012–B1016 (1965).
[CrossRef]

M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. I. general theory,” Phys. Rev. 159, 208–226 (1967).
[CrossRef]

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Phys. Rev. A (5)

D. F. McAlister and M. G. Raymer, “Ultrafast photon-number correlations from dual-pulse, phase-averaged homodyne detection,” Phys. Rev. A 55, R1609–R1612 (1997).
[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]

M. Kira and S. W. Koch, “Quantum-optical spectroscopy of semiconductors,” Phys. Rev. A 73, 013813 (2006).
[CrossRef]

F. T. Arecchi and V. Degiorgio, “Statistical properties of laser radiation during a transient buildup,” Phys. Rev. A 3, 1108–1124 (1971).
[CrossRef]

S. Zhu, A. W. Yu, and R. Roy, “Statistical fluctuations in laser transients,” Phys. Rev. A 34, 4333–4347 (1986).
[CrossRef]

Phys. Rev. Lett. (2)

P. Spano, A. Mecozzi, and A. Sapia, “Statistical distribution of trajectories in the time-intensity plane during semiconductor-laser gain switching,” Phys. Rev. Lett. 64, 3003–3006 (1990).
[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]

Rev. Mod. Phys. (1)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[CrossRef]

Science (2)

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science 322, 563–566 (2008).
[CrossRef]

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009).
[CrossRef]

Other (8)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 2nd ed. (Wiley-VCH, 2004).

P. Horowitz and W. Hill, The Art of Electronics, 2nd ed.(Cambridge University, 1989).

M. Kira, G. Roumpos, S. Koch, and S. Cundiff, “Shaping quantum-optical statistics by conditioning ultrafast pulses” (manuscript in preparation, available from cundiffs@jila.colorado.edu).

S. Boyd, and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

CVX Research, Inc., “CVX: matlab software for disciplined convex programming, version 2.0 beta” (2012), http://cvxr.com/cvx .

M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds. (Springer-Verlag, 2008), pp. 95–110.

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Figures (16)

Fig. 1.
Fig. 1.

(a) Schematic of the homodyne detection technique. LO, local oscillator; S, source; PBS, polarizing beam splitter; HWP, half-wave plate. (b) Schematic of the detector electronics. For every light pulse, the photodiode box produces a current pulse, whose number of electrons is a measurement of the operator Q^ in Eq. (2). The pulse is then integrated by the charge-sensitive preamplifier and shaped to a Gaussian pulse of fixed width by the shaping amplifier. The height of the output pulse is proportional to the number of electrons in the input current pulse.

Fig. 2.
Fig. 2.

Raw data for the case of no source (source is the vacuum). (a) Trace collected on the digitizer. The height of each pulse is one measurement of the photoelectron difference operator Q^. (b) Histogram of measurements of the operator Q^.

Fig. 3.
Fig. 3.

Standard deviation of the measured photoelectron difference number per pulse as a function of the number of LO photoelectrons per pulse when only the LO is present. Blue circles show experimental data, and the red curve shows the square root behavior, as expected for the shot noise of the LO pulse. For small NLO, we are limited by the thermal noise of <1000electrons/pulse.

Fig. 4.
Fig. 4.

Schematic of the double-channel homodyne detection setup. Both LO beams originate from the same laser, and their delays can be controlled independently with two translation stages. The source light is coupled into a single-mode fiber. The output of the fiber is properly collimated to maximize spatial mode matching between the source and LO beams. The triggering circuits are discussed in Appendix D.

Fig. 5.
Fig. 5.

Schematic of the triple-channel homodyne detection setup. The third channel (Ch3) uses the same LO as Ch2, but the LO phase is rotated by π/2 using the quarter-wave plate QWP1.

Fig. 6.
Fig. 6.

Top: representation of the solutions of Eq. (10) for the same experimental data as in Fig. 7(d). Each distribution pn is represented by a point (fsmooth(pn),log(l(pn))) in the two-dimensional plane. The shaded area covers the suboptimal points. The black curve shows the Pareto optimal points, which are obtained by solving (10) for varying γ. Bottom: reconstructed photon number distribution pn for the three Pareto optimal points shown with symbols of the same color in the top panel.

Fig. 7.
Fig. 7.

Experimental results for the external-cavity diode laser driven by direct current. (a) Measured optical intensity as a function of the input current. The left axis shows the intensity measured with the homodyne setup, whereas the right axis shows the intensity measured by a calibrated power meter. (b)–(e) Reconstructed photon number distributions for four different drive currents. We compare the experimental data to the distributions of three different states of equal intensity: a coherent state, a mixed state with the same g(2) as the experimental data, and the solution of the semi-classical single-mode laser model (see text).

Fig. 8.
Fig. 8.

Experimental results for the external-cavity diode laser driven by the edge-triggered current source (see Appendix D). We plot the measured intensity Ns(t)=a^s(t)a^s(t) and second-order correlation g(2)(t,t) as a function of time for one particular peak current (120 mA). The threshold for direct current excitation is 40mA [28]. The 0 time is arbitrary.

Fig. 9.
Fig. 9.

Photon number distributions pn reconstructed from the same data as in Fig. 8 for different time delays. In the top three panels, we plot pn for three different time delays during the initial decrease of g(2) (see Fig. 8) and compare with a mixed state with the same intensity and g(2). In the bottom panel, we plot pn at a later time, when g(2)>2, namely the distribution is superthermal, and compare it with a thermal state with the same intensity.

Fig. 10.
Fig. 10.

Experimental results for the external-cavity diode laser driven by the triggered current source (see Appendix D). We plot the measured intensity Ns(t)=a^s(t)a^s(t) and second-order correlation g(2)(t,t) as a function of time for three different peak currents.

Fig. 11.
Fig. 11.

Experimental results for a 796 nm bare laser diode driven by the edge-triggered source (see Appendix D). We plot the measured intensity Ns(t)=a^s(t)a^s(t) and second-order correlation g(2)(t,t) as a function of time for three different peak currents. The 0 ns time delay is arbitrary.

Fig. 12.
Fig. 12.

Experimental results for the bare laser diode driven by the edge-triggered source (see Appendix D). We plot the measured intensity Ns(t) and second-order correlation g(2)(t,t) at time t=0ns as a function of peak current.

Fig. 13.
Fig. 13.

Dual-time correlation functions for the bare laser diode measured using the double-channel setup of Fig. 4 at three different peak currents. We excite the diode using the 8 ns edge-triggered current source, keep t1 fixed around the center of the pulse, and adjust t2 around t1.

Fig. 14.
Fig. 14.

Schematic of the triggered current source. The input is a negative square pulse. The output is a positive square current pulse of the same duration as the input and peak current determined by the variable resistor R3.

Fig. 15.
Fig. 15.

Schematic of the edge-triggered current source. The input is a negative square pulse. The circuit triggers on the negative edge of the input pulse and results in an almost Gaussian current pulse with peak current determined by the variable capacitor C1. The duration of the pulse is determined by the fall time of the input triggering pulse and the frequency response of Q1 and D1.

Fig. 16.
Fig. 16.

Current pulses achieved using the edge-triggered current source. Lines of different color correspond to different values of the tuning capacitor C1, which controls the peak current. The 0 ns time delay is arbitrary.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

a^±=12(eiϕLOa^LO±a^S),
Q^=a^LO(eiϕLOa^S+eiϕLOa^S).
inoise=4kBTΔfR.
Q^2=a^LO20|(eiϕLOa^S+eiϕLOa^S)2|0=a^LO20|a^Sa^S|0=a^LO2=NLO,
ΔQ=Q^2Q^2=NLO.
ΔQ=(NLO2)2+(NLO2)2=NLO.
Q^2NLO=S|(eiϕLOa^S+eiϕLOa^S)2|S=S|(2a^Sa^S+1)|Sa^Sa^S=12(Q^2NLO1),
Q^4NLO2=S|(6a^Sa^Sa^Sa^S+12a^Sa^S+3)|Sa^Sa^Sa^Sa^S=Q^46NLO22a^Sa^S12.
g(2)=a^Sa^Sa^Sa^Sa^Sa^S2.
minimizelog(l(pn))+γfsmooth(pn)subject topn0,n=0,1,Nmax,andn=0Nmaxpn=1.
l(pn(γ))<l(pn(γ=0)),andfsmooth(pn(γ))<fsmooth(pn(γ=0)).
fsmooth(pn)=[n=1Nmax|pnpn1|d]1/d,
exp(λ*a^+λa^)=exp(|λ|22[a^,a^])exp(λ*a^)exp(λa^)=exp(|λ|22)exp(λ*a^)exp(λa^)n=0(λ*a^+λa^)nn!=n1=0|λ|2n12n1n1!n2=0(λ*a^S)n2n2!n3=0(λa^S)n3n3!.
(eiϕLOa^+eiϕLOa^)22!=1211!+a^a^(1!)2,
(eiϕLOa^+eϕLOa^)44!=1222!+a^a^211!(1!)2+a^a^a^a^(2!)2,
(eiϕLOa^+eiϕLOa^)66!=1233!+a^a^222!(1!)2+a^a^a^a^211!(2!)2+a^a^a^a^a^a^(3!)2.
Q^12=a^LO1(eiϕLO1a^S(t1)+eiϕLO1a^S(t1))+a^LO2(eiϕLO2a^S(t2)+eiϕLO2a^S(t2))a^LO1q^1+a^LO2q^2.
Q^122=(a^LO1q^1+a^LO2q^2)2=a^LO12q^12+a^LO22q^22+2a^LO1a^LO2q^1q^2=NLO1(1+2a^S(t1)a^S(t1))+NLO2(1+2a^S(t2)a^S(t2)),
Q^124=NLO12q^14+NLO22q^24+(42)NLO1NLO2q^12q^22.
q^12q^22=(1+2a^S(t1)a^S(t1))(1+2a^S(t2)a^S(t2))=1+2a^S(t1)a^S(t1)+2a^S(t2)a^S(t2)+4a^S(t1)a^S(t2)a^S(t2)a^S(t1).
a^S1a^S2a^S2a^S1=1412(a^S(t1)a^S(t1)+a^S(t2)a^S(t2))+124NLO1NLO2(Q^124Q^14Q^24)=14+124NLO1NLO2(Q^124Q^14Q^246NLO1Q^226NLO2Q^12).
|Q^122ei(ϕLO2ϕLO1)|=NLO1NLO2a^S(t1)a^S(t2).

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