Abstract

Photon number resolving detectors can be highly useful for studying the statistics of multiphoton quantum states of light. In this work we study the counts statistics of different states of light measured on multiplexed on–off detectors. We put special emphasis on artificial nonclassical features of the statistics obtained. We show new ways to derive analytical formulas for counts statistics and their moments. Using our approach we are the first, to the best of our knowledge, to derive statistics moments for multimode thermal states measured on multiplexed on–off detectors. We use them to determine empirical Mandel parameters and recently proposed sub-binomial parameters suitable for tests of nonclassicality of the measured states. Additionally, we investigate sub-Poissonian and superbunching properties of the two-mode squeezed state measured on a pair of multiplexed detectors, and we present results of the Fano factor and second-order correlation function for these states.

© 2014 Optical Society of America

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2014 (1)

2013 (4)

A. Allevi, M. Lamperti, M. Bondani, J. Perina, V. Michalek, O. Haderka, and R. Machulka, “Characterizing the nonclassicality of mesoscopic optical twin-beam states,” Phys. Rev. A 88, 063807 (2013).
[CrossRef]

T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, “Direct observation of sub-binomial light,” Phys. Rev. Lett. 110, 173602 (2013).
[CrossRef]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544, 2013.
[CrossRef]

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[CrossRef]

2012 (10)

R. Chrapkiewicz and W. Wasilewski, “Generation and delayed retrieval of spatially multimode Raman scattering in warm rubidium vapors,” Opt. Express 20, 29540 (2012).
[CrossRef]

D. A. Kalashnikov, S.-H. Tan, and L. A. Krivitsky, “Crosstalk calibration of multi-pixel photon counters using coherent states,” Opt. Express 20, 5044–5051 (2012).
[CrossRef]

D. A. Kalashnikov, S.-H. Tan, T. S. Iskhakov, M. V. Chekhova, and L. A. Krivitsky, “Measurement of two-mode squeezing with photon number resolving multipixel detectors,” Opt. Lett. 37, 2829–2831 (2012).
[CrossRef]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[CrossRef]

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109, 093601 (2012).
[CrossRef]

T. Kiesel and W. Vogel, “Complete nonclassicality test with a photon-number resolving detector,” Phys. Rev. A 86, 032119 (2012).
[CrossRef]

T. S. Iskhakov, A. M. Pérez, K. Y. Spasibko, M. V. Chekhova, and G. Leuchs, “Superbunched bright squeezed vacuum state,” Opt. Lett. 37, 1919–1921 (2012).
[CrossRef]

P. Hong, J. Liu, and G. Zhang, “Two-photon superbunching of thermal light via multiple two-photon path interference,” Phys. Rev. A 86, 013807 (2012).
[CrossRef]

R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[CrossRef]

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14, 085001 (2012).
[CrossRef]

2011 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[CrossRef]

The LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[CrossRef]

2010 (3)

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[CrossRef]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

2009 (2)

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys. 11, 093038 (2009).
[CrossRef]

2008 (4)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321, 544–547 (2008).
[CrossRef]

V. Boyer, A. Marino, and P. Lett, “Generation of spatially broadband twin beams for quantum imaging,” Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef]

J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101, 233604 (2008).
[CrossRef]

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
[CrossRef]

2007 (1)

2005 (1)

O. Haderka, J. Perina, and M. Hamar, “Direct measurement and reconstruction of nonclassical features of twin beams generated in spontaneous parametric down-conversion,” Phys. Rev. A 71, 033815 (2005).
[CrossRef]

2004 (4)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).
[CrossRef]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[CrossRef]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[CrossRef]

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).
[CrossRef]

2003 (2)

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector,” Opt. Lett. 28, 52–54 (2003).
[CrossRef]

J. Rehacek, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector,” Phys. Rev. A 67, 061801 (2003).
[CrossRef]

1996 (1)

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon chopping: new way to measure the quantum state of light,” Phys. Rev. Lett. 76, 2464–2467 (1996).
[CrossRef]

1983 (1)

D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983).
[CrossRef]

Achilles, D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).
[CrossRef]

Adami, C.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).
[CrossRef]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

Agarwal, G. S.

J. Sperling, W. Vogel, and G. S. Agarwal, “Correlation measurements with on-off detectors,” Phys. Rev. A 88, 043821 (2013).
[CrossRef]

J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
[CrossRef]

J. Sperling, W. Vogel, and G. S. Agarwal, “Sub-binomial light,” Phys. Rev. Lett. 109, 093601 (2012).
[CrossRef]

Allevi, A.

M. Lamperti, A. Allevi, M. Bondani, and R. Machulka, “Optimal sub-Poissonian light generation from twin beams by photon-number resolving detectors,” J. Opt. Soc. Am. B 31, 20–25 (2014).
[CrossRef]

A. Allevi, M. Lamperti, M. Bondani, J. Perina, V. Michalek, O. Haderka, and R. Machulka, “Characterizing the nonclassicality of mesoscopic optical twin-beam states,” Phys. Rev. A 88, 063807 (2013).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

R. Machulka, O. Haderka, J. Perina, M. Lamperti, A. Allevi, and M. Bondani, “Spatial properties of twin-beam correlations at low- to high-intensity transition,” arXiv:1405.1190 (2014).

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

I. Afek, A. Natan, O. Ambar, and Y. Silberberg, “Quantum state measurements using multipixel photon detectors,” Phys. Rev. A 79, 043830 (2009).
[CrossRef]

Andreoni, A.

Arimondo, E.

Banaszek, K.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51, 1499–1515 (2004).
[CrossRef]

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector,” Opt. Lett. 28, 52–54 (2003).
[CrossRef]

R. Chrapkiewicz, W. Wasilewski, and K. Banaszek, “High-fidelity spatially resolved multiphoton counting for quantum imaging applications,” arXiv:1405.4400 (2014).

Barbieri, M.

T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, “Direct observation of sub-binomial light,” Phys. Rev. Lett. 110, 173602 (2013).
[CrossRef]

Bartley, T. J.

T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, “Direct observation of sub-binomial light,” Phys. Rev. Lett. 110, 173602 (2013).
[CrossRef]

Berchera, I. R.

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[CrossRef]

Blanchet, J.-L.

J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101, 233604 (2008).
[CrossRef]

Bondani, M.

M. Lamperti, A. Allevi, M. Bondani, and R. Machulka, “Optimal sub-Poissonian light generation from twin beams by photon-number resolving detectors,” J. Opt. Soc. Am. B 31, 20–25 (2014).
[CrossRef]

A. Allevi, M. Lamperti, M. Bondani, J. Perina, V. Michalek, O. Haderka, and R. Machulka, “Characterizing the nonclassicality of mesoscopic optical twin-beam states,” Phys. Rev. A 88, 063807 (2013).
[CrossRef]

A. Allevi, M. Bondani, and A. Andreoni, “Photon-number correlations by photon-number resolving detectors,” Opt. Lett. 35, 1707–1709 (2010).
[CrossRef]

R. Machulka, O. Haderka, J. Perina, M. Lamperti, A. Allevi, and M. Bondani, “Spatial properties of twin-beam correlations at low- to high-intensity transition,” arXiv:1405.1190 (2014).

Boyer, V.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321, 544–547 (2008).
[CrossRef]

V. Boyer, A. Marino, and P. Lett, “Generation of spatially broadband twin beams for quantum imaging,” Phys. Rev. Lett. 100, 143601 (2008).
[CrossRef]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178–180 (2007).
[CrossRef]

Braunstein, S. L.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).
[CrossRef]

Brida, G.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14, 085001 (2012).
[CrossRef]

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4, 227–230 (2010).
[CrossRef]

Chandra, A.

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544, 2013.
[CrossRef]

Chekhova, M. V.

Chrapkiewicz, R.

R. Chrapkiewicz and W. Wasilewski, “Generation and delayed retrieval of spatially multimode Raman scattering in warm rubidium vapors,” Opt. Express 20, 29540 (2012).
[CrossRef]

M. Dabrowski, R. Chrapkiewicz, and W. Wasilewski, “Hamiltonian design in readout from room-temperature Raman quantum memory,” arXiv:1406.6489 (2014).

R. Chrapkiewicz, W. Wasilewski, and K. Banaszek, “High-fidelity spatially resolved multiphoton counting for quantum imaging applications,” arXiv:1405.4400 (2014).

Ciavarella, L.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14, 085001 (2012).
[CrossRef]

Coldenstrodt-Ronge, H.

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys. 11, 093038 (2009).
[CrossRef]

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).
[CrossRef]

Dabrowski, M.

M. Dabrowski, R. Chrapkiewicz, and W. Wasilewski, “Hamiltonian design in readout from room-temperature Raman quantum memory,” arXiv:1406.6489 (2014).

Datta, A.

T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, “Direct observation of sub-binomial light,” Phys. Rev. Lett. 110, 173602 (2013).
[CrossRef]

Degiovanni, I. P.

G. Brida, L. Ciavarella, I. P. Degiovanni, M. Genovese, L. Lolli, M. G. Mingolla, F. Piacentini, M. Rajteri, E. Taralli, and M. G. A. Paris, “Quantum characterization of superconducting photon counters,” New J. Phys. 14, 085001 (2012).
[CrossRef]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzanski, J. Kolodynski, and M. Guta, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[CrossRef]

Devaux, F.

J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101, 233604 (2008).
[CrossRef]

Donati, G.

T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, “Direct observation of sub-binomial light,” Phys. Rev. Lett. 110, 173602 (2013).
[CrossRef]

Dowling, J. P.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photostatistics from photon-number discriminating detectors,” J. Mod. Opt. 51, 1517–1528 (2004).
[CrossRef]

Eisert, J.

A. Feito, J. S. Lundeen, H. Coldenstrodt-Ronge, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Measuring measurement: theory and practice,” New J. Phys. 11, 093038 (2009).
[CrossRef]

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Nat. Commun. (1)

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V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
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Nature (2)

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New J. Phys. (2)

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J. Sperling, W. Vogel, and G. S. Agarwal, “True photocounting statistics of multiple on-off detectors,” Phys. Rev. A 85, 023820 (2012).
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A. Allevi, M. Lamperti, M. Bondani, J. Perina, V. Michalek, O. Haderka, and R. Machulka, “Characterizing the nonclassicality of mesoscopic optical twin-beam states,” Phys. Rev. A 88, 063807 (2013).
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Figures (7)

Fig. 1.
Fig. 1.

Values of conditional probabilities pN(k|n) of getting k counts when a detector is illuminated by exactly n photons, for a detector consisting of N multiplexed component on–off detectors with quantum efficiency η. (a) N=2, η=1; (b) N=4, η=1; (c) N=4, η=0.8; (d) N=4, η=0.2.

Fig. 2.
Fig. 2.

(a) Mean and variance of counts for coherent state on a detector with N=4. (b) Measured QF parameter is artificially nonclassical contrary to QB=QM=0.

Fig. 3.
Fig. 3.

(a) Mean and variance of counts and (b) input QM, measured QF, and sub-binomial QB parameters versus mean number of photons in two-mode (M=2) thermal state on a detector with N=8.

Fig. 4.
Fig. 4.

Theoretical values for empirical QF parameters for (a) single-mode thermal state and (b) multimode thermal state M=4 plotted versus mean number of photons n for different N. Depending on the parameters of the state and the detector counts statistics, it may seem classical or artificially nonclassical.

Fig. 5.
Fig. 5.

Joint count statistics ck1,k2 for two-mode squeezed state with ζ=0.8 measured by two multiplexed detectors. (a) N=, η=1; (b) N=, η=0.8; (c) N=4, η=1; (d) N=4, η=0.5.

Fig. 6.
Fig. 6.

Fano factors versus squeezing parameter ζ for different numbers of N detectors with (a) η=1 and (b) η=0.5. A lower number of detectors N can artificially increase the correlation and consequently decrease the Fano factor.

Fig. 7.
Fig. 7.

Second-order correlation function g(2) versus different numbers N of detectors for η=1 for (a) |ζ|=0.2 and (b) |ζ|=0.8 for two-mode squeezed states. A finite number N of component on–off detectors decreases g(2), which for the two-mode squeezed state is always higher than two, g(2)>2. g(2) is decreased by no more than one. For |ζ|2>1/2 and sufficiently low N, the observed counts statistics do not preserve their initial superbunching properties as in (b).

Equations (22)

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pN(k|n)=1Nn(Nk)k!S(n,k),
pN(1|n)=N(1η)n((ηηN+NNηN)n1).
χ(k)=k=0Nχ(k)(Nk)k!n=k1NnS(n,k)fn.
n=kS(n,k)xnn!=1k!(ex1)k,
n=kS(n,k)xn=(1)k(11x)k,
k=N(1en/N),
(Δk)2=N(1en/N)en/N.
QB=N(Δk)2k(Nk)1,
fn=abn,
χ(k)f=ak=0Nχ(k)(Nk)k!(1)k(1N/b)k.
kf=abN(b1)(b(N1)N),
kf2=ab(b+1)N2(b1)(b(N2)N)(b+NbN).
gn=a(n+m)!n!bn+m,
gn=ab,mbn+m=b,mbmfn.
χ(k)g=b,mbmχ(k)f.
kTh,1=nNn+N,(Δk)2Th,1=nN2(nN+n+N)(n+N)2(2n+N).
kTh,M=2=nN(n+4N)(n+2N)2.
k2Th,M=2=nN2(n3+6n2N+3nN(2N+1)+4N2)(n+N)2(n+2N)2.
ck1,k2=n1=k1n2=k2pN(k1|n1)pN(k2|n2)fn1,n2.
R=(Δ(k1k2))2k1+k2,
g(2)=k1k2k1k2.
|ψ=1|ζ|2n=0ζn|nn,

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