Abstract

We measured the time-domain quantum statistics of a pulsed, high-repetition-rate optical field by balanced homodyne detection. The measuring apparatus discriminates the time scales on which intrinsic quantum fluctuations prevail from those scales for which technical noise is overwhelming. A tomographic reconstruction of weak coherent states with various average photon numbers demonstrates the potential ability of the system to measure high-repetition-rate, time-resolved signals. Possible extensions to other physical situations are discussed.

© 2002 Optical Society of America

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References

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  1. S. Reynaud, A. Heidmann, E. Giacobono, and C. Fabre, “Quantum fluctuations in optical systems,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1992), Vol. 30, pp. 1–85.
  2. U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, Cambridge, 1997).
  3. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
    [CrossRef]
  4. M. Vasilyev, S.-K. Choi, P. Kumar, and G. M. D’Ariano, “Investigation of the photon statistics of parametric fluorescence in a traveling-wave parametric amplifier by means of self-homodyne tomography,” Opt. Lett. 23, 1393–1395 (1998).
    [CrossRef]
  5. D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]
  6. M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
    [CrossRef]
  7. D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 221–230 (1966).
    [CrossRef]
  8. 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]
  9. K. Vogel and H. Risken, “Determination of quasiprobability distribution in terms of probability distributions for rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef] [PubMed]
  10. U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of photon statistics and density matrix using homodyne detection,” Opt. Commun. 127, 144–160 (1996).
    [CrossRef]
  11. A. Montina and F. T. Arecchi, “Toward an optical evidence of quantum interference between macroscopically different states,” Phys. Rev. A 58, 3472–3476 (1998).
    [CrossRef]
  12. After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
    [CrossRef]

2001 (1)

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

2000 (1)

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

1998 (2)

1997 (1)

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

1996 (1)

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

1994 (1)

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]

1993 (1)

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

1989 (1)

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

1966 (1)

D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Aichele, T.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Arecchi, F. T.

A. Montina and F. T. Arecchi, “Toward an optical evidence of quantum interference between macroscopically different states,” Phys. Rev. A 58, 3472–3476 (1998).
[CrossRef]

Beck, M.

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

Benson, O.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

Breitenbach, G.

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

Choi, S.-K.

Crispino, M.

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

D’Ariano, G. M.

M. Vasilyev, S.-K. Choi, P. Kumar, and G. M. D’Ariano, “Investigation of the photon statistics of parametric fluorescence in a traveling-wave parametric amplifier by means of self-homodyne tomography,” Opt. Lett. 23, 1393–1395 (1998).
[CrossRef]

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]

De Martini, F.

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

Di Giuseppe, G.

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

Faridani, A.

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

Hansen, H.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

Kanatsoulis, H.

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

Kiss, T.

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

Kumar, P.

Leonhardt, U.

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

Lvovsky, A. I.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[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]

Mataloni, P.

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

Mlynek, J.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

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

Montina, A.

A. Montina and F. T. Arecchi, “Toward an optical evidence of quantum interference between macroscopically different states,” Phys. Rev. A 58, 3472–3476 (1998).
[CrossRef]

Munroe, M.

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

Raymer, M. G.

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

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

Richter, Th.

U. Leonhardt, M. Munroe, T. Kiss, Th. Richter, and M. G. Raymer, “Sampling of 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 distribution in terms of probability distributions for rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Schiller, S.

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

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

Smithey, D.

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

Vasilyev, M.

Vogel, K.

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

Fortschr. Phys. (1)

M. Crispino, G. Di Giuseppe, F. De Martini, P. Mataloni, and H. Kanatsoulis, “Towards a Fock-states tomographic reconstruction,” Fortschr. Phys. 48, 589–598 (2000).
[CrossRef]

Nature (1)

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

Opt. Commun. (1)

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

Opt. Lett. (1)

Phys. Rev. A (3)

A. Montina and F. T. Arecchi, “Toward an optical evidence of quantum interference between macroscopically different states,” Phys. Rev. A 58, 3472–3476 (1998).
[CrossRef]

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 distribution in terms of probability distributions for rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

After the submission of this manuscript, A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, reported the quantum reconstruction of a single photon Fock state, using a laser repetition rate of 800 kHz, in “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402–1–4 (2001).
[CrossRef]

D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner function 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]

Proc. IEEE (1)

D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Other (2)

S. Reynaud, A. Heidmann, E. Giacobono, and C. Fabre, “Quantum fluctuations in optical systems,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1992), Vol. 30, pp. 1–85.

U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, Cambridge, 1997).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup: HWPs, half-wave plates; NPBS, nonpolarizing beam splitter; A, amplifier; PZT, piezoelectric translation stage; ND, neutral-density filter.

Fig. 2
Fig. 2

Thicker curve, spectral power of the detected signal when one photodiode is illuminated by thermal light, giving a photocurrent of 1.16 mA. The resolution bandwidth is 300 kHz. Thinner curve, electronic noise. The narrow peaks are due to pickup from environmental noise. Inset, time-domain output signal with the laser impinging on only one photodiode (the upper and the lower traces correspond to the signals from the two detectors) and on both photodiodes (middle trace).

Fig. 3
Fig. 3

Filled circles, integral of the power spectrum; solid curve, quadratic fit; dashed curve, shot-noise contribution; dashed–dotted curve, electronic noise.

Fig. 4
Fig. 4

Filled circles, directly measured σPE2; open circles, variance deduced from the spectrum according to Eq. (1).

Fig. 5
Fig. 5

Allan variance of the pulse energy. Open circles, laser power of 3 mW; filled circles, electronic noise; solid curves, Allan variances calculated from the spectrum according to Eq. (4). The abscissa is in units of pulse separation T.

Fig. 6
Fig. 6

Reconstructed Wigner functions: (a) vacuum state, (b) weak coherent state with n1.

Fig. 7
Fig. 7

Reconstruction of the photon statistics of three coherent states. The filled circles with error bars are the reconstructed values. Rectangles, best-fit Poissonian distribution values with (a) n=0.16 photons/pulse, (b) n=0.64 photons/pulse, and (c) n=1.5 photons/pulse.

Equations (6)

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

σPE2=20 S(ν)F(ν)dν,
F(ν)=T2sin2(πνT)(πνT)2.
σA2(τ)=1/2(f2,τ-f1,τ)2,
σA2(τ)=4T2 0 S(ν)sin4(πντ)(πντ)2dν.
Pθ(x)=-+W(x cos θ-y sin θ, x sin θ+y cos θ)dy.
ρnm=n|ρˆ|m=1π0π dθ -+dxPθ(x)fnm(x)[exp i(n-m)θ],

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