Abstract

We examine the photon statistics of photon-subtracted thermal light using photon-number-resolved detection. We demonstrate experimentally that the photon number distribution transforms from a Bose–Einstein distribution to a Poisson distribution as the number of subtracted photons increases. We also show that second- and higher-order photon correlation functions can be directly determined from the photon-number-resolved detection measurements of a single optical beam.

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
    [CrossRef]
  3. S. M. Barnett and P. M. Randmore, Methods in Theoretical Quantum Optics (Oxford University, 1997).
  4. R. Hanbury Brown and R. Q. Twiss, Nature 178, 4541 (1956).
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    [CrossRef]
  6. M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, Opt. Express 18, 1430 (2010).
    [CrossRef]
  7. A. E. Lita, A. J. Miller, and S. W. Nam, Opt. Express 16, 3032 (2008).
    [CrossRef]
  8. A. J. Miller, A. E. Lita, B. Calkins, I. Vayshenker, S. M. Gruber, and S. W. Nam, Opt. Express 19, 9102 (2011).
    [CrossRef]
  9. D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa, H. Tsuchida, H. Fujino, H. Ishii, T. Itatani, S. Inoue, and T. Zama, Opt. Express 19, 870 (2011).
    [CrossRef]
  10. P. Marek and R. Filip, Phys. Rev. A 81, 022302 (2010).
    [CrossRef]
  11. M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
    [CrossRef]
  12. V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
    [CrossRef]
  13. Z. H. Levine, T. Gerritis, A. Migdall, D. V. Samarov, B. Calkins, A. E. Lita, and S. W. Nam, J. Opt. Soc. Am. B 29, 2066 (2012).
    [CrossRef]

2012 (1)

2011 (2)

2010 (4)

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

P. Marek and R. Filip, Phys. Rev. A 81, 022302 (2010).
[CrossRef]

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

M. J. Stevens, B. Baek, E. A. Dauler, A. J. Kerman, R. J. Molnar, S. A. Hamilton, K. K. Berggren, R. P. Mirin, and S. W. Nam, Opt. Express 18, 1430 (2010).
[CrossRef]

2008 (2)

A. E. Lita, A. J. Miller, and S. W. Nam, Opt. Express 16, 3032 (2008).
[CrossRef]

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

2007 (1)

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, Nature 178, 4541 (1956).

Adhikari, R.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Amemiya, K.

Andersen, U.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Avenhaus, M.

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

Baek, B.

Barnett, S. M.

S. M. Barnett and P. M. Randmore, Methods in Theoretical Quantum Optics (Oxford University, 1997).

Bellini, M.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

Berggren, K. K.

Calkins, B.

Chekhova, M. V.

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

Dauler, E. A.

Filip, R.

P. Marek and R. Filip, Phys. Rev. A 81, 022302 (2010).
[CrossRef]

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Fujii, G.

Fujino, H.

Fukuda, D.

Gerritis, T.

Goda, K.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Gruber, S. M.

Hamilton, S. A.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, Nature 178, 4541 (1956).

Inoue, S.

Ishii, H.

Itatani, T.

Kerman, A. J.

Kim, M. S.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

Laiho, K.

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

Leuchs, G.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Levine, Z. H.

Lita, A. E.

Mandel, L.

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

Marek, P.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

P. Marek and R. Filip, Phys. Rev. A 81, 022302 (2010).
[CrossRef]

Marquardt, C.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Mavalvala, N.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

McKenzie, K.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Migdall, A.

Mikhailov, E. E.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Miller, A. J.

Mirin, R. P.

Miyakawa, O.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Molnar, R. J.

Müller, C.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Nam, S. W.

Numata, T.

Parigi, V.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

Randmore, P. M.

S. M. Barnett and P. M. Randmore, Methods in Theoretical Quantum Optics (Oxford University, 1997).

Samarov, D. V.

Saraf, S.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Silberhorn, C.

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

Stevens, M. J.

Tsuchida, H.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Nature 178, 4541 (1956).

Usuga, M.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Vass, S.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Vayshenker, I.

Ward, R.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Weinstein, A. J.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

Wittmann, C.

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Wolf, E.

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

Yoshizawa, A.

Zama, T.

Zavatta, A.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Phys. (2)

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nat. Phys. 4, 472 (2008).
[CrossRef]

M. Usuga, C. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. Andersen, Nat. Phys. 6, 767 (2010).
[CrossRef]

Nature (1)

R. Hanbury Brown and R. Q. Twiss, Nature 178, 4541 (1956).

Opt. Express (4)

Phys. Rev. A (1)

P. Marek and R. Filip, Phys. Rev. A 81, 022302 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Phys. Rev. Lett. 104, 063602 (2010).
[CrossRef]

Science (1)

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science 317, 1890 (2007).
[CrossRef]

Other (2)

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

S. M. Barnett and P. M. Randmore, Methods in Theoretical Quantum Optics (Oxford University, 1997).

Supplementary Material (1)

» Media 1: JPG (54 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of experiment with photon-subtracted thermal light and photon-number-resolved detection. SMF, single-mode fiber; att., optical attenuators; HWP, half-wave plate; PBS, polarizing beam splitter; TES1(2), transition-edge-sensors. Inset: measured photon number distribution P ( n ) of pseudo-thermal light (blue bars), calculated Poisson distribution (red curve), and Bose–Einstein distribution with μ B = 1.31 (green curve).

Fig. 2.
Fig. 2.

Photon number distribution P ( n ) of photon-subtracted thermal light (arm 2 with μ B = 0.43 conditioned on arm 1 with μ A = 0.96 ). Experimental data (blue bars), calculations (red bar) using Eq. (8). Poisson (red dashed curve) and Bose–Einstein (green curve) distributions are shown as lines as a guide to the eye. Inset: calculated trace distances D TH ( P ) = Tr | ρ ^ ρ ^ TH ( P ) | / 2 .

Fig. 3.
Fig. 3.

Mean photon numbers of photon-subtracted thermal light (arm 2) versus subtracted-photon number (arm 1). Experimental data (points); predictions (lines) using Eq. (8) with experimentally determined values of ( μ A , μ B ), which are, from top to bottom, (0.96, 0.43), (0.51, 0.23), and (0.59, 0.11).

Fig. 4.
Fig. 4.

Correlations of photon-subtracted thermal light (arm 2 with μ B = 0.43 ) versus subtracted-photon numbers (arm 1 with μ A = 0.96 ), with a lower bound of 1 (dashed line). Experimental data (dots); predictions (lines) using Eq. (3) (see Media 1).

Equations (11)

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ρ ^ k = ( a ^ ) k ρ ^ ( a ^ + ) k Tr ( ( a ^ ) k ρ ^ ( a ^ + ) k ) = n = 0 ( n + k ) ! n ! k ! μ n ( 1 + μ ) k + n + 1 | n n |
μ k = ( k + 1 ) μ
g k ( n ) ( 0 ) = ( a ^ + ) n ( a ^ ) n a ^ + a ^ n = ( k + n ) ! k ! ( k + 1 ) n .
ρ ^ ph = 0 2 π d φ 2 π | | α | e i φ | α | e i φ | i = 0 | α | 2 i ! | i i |
ρ ^ = n = 0 k = 0 n j = 0 n ( n k ) ( n j ) τ k ( 1 τ ) n k τ j ( 1 τ ) n j × μ A B n ( 1 + μ A B ) n + 1 | n k | k n j | j |
Π ^ k = i = k ( i k ) η 1 ( 2 ) k ( 1 η 1 ( 2 ) ) i k | i i | ,
P ( j , k ) = Tr ( Π ^ j ρ ^ Π ^ k ) = ( j + k ) ! j ! k ! μ A k μ B j ( 1 + μ A + μ B ) j + k + 1 ,
μ 2 | k = μ 2 | 0 ( k + 1 ) ,
P ( j | k ) = P ( j , k ) j = 0 α P ( j , k ) α [ k μ B / ( 1 + μ A + μ B ) ] j j ! .
g ( n ) ( 0 ) = ( a ^ + ) n ( a ^ ) n a ^ + a ^ n = j = 0 ( j + n ) ! j ! Tr ( ρ ^ Π ^ j + n ) ( j = 0 ( j + 1 ) Tr ( ρ ^ Π ^ j + 1 ) ) n .
g ( n ) ( 0 ) = j = 0 ( j + n ) ! j ! P ( j + n | k ) ( j = 0 ( j + 1 ) P ( j + 1 | k ) ) n .

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