Abstract

Photon anti-bunching, measured via the Hanbury-Brown–Twiss experiment, is one of the key signatures of quantum light and is tied to sub-Poissonian photon number statistics. Recently, it has been reported that photon anti-bunching or conditional sub-Poissonian photon number statistics can be obtained via second-order interference of mutually incoherent weak lasers and heralding based on photon counting [Phys. Rev. A 92, 033855 (2015) [CrossRef]  ; Opt. Express 24, 19574 (2016) [CrossRef]  ; https://arxiv.org/abs/1601.08161]. Here, we report theoretical analysis on the limits of manipulating conditional photon statistics via interference of weak lasers. It is shown that conditional photon number statistics can become super-Poissonian in such a scheme. We, however, demonstrate explicitly that it cannot become sub-Poissonian, i.e., photon anti-bunching cannot be obtained in such a scheme. We point out that incorrect results can be obtained if one does not properly account for seemingly negligible higher-order photon number expansions of the coherent state.

© 2017 Optical Society of America

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References

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2017 (2)

2016 (5)

2015 (3)

T. F. da Silva, G. C. Amaral, G. P. Temporaô, and J. P. von der Weid, “Linear-optic heralded photon source,” Phys. Rev. A 92, 033855 (2015).
[Crossref]

Y. Zhang, R. Okubo, M. Hirano, Y. Eto, and T. Hirano, “Experimental realization of spatially separated entanglement with continuous variables using laser pulse trains,” Sci. Rep. 5, 13029 (2015).
[Crossref] [PubMed]

Y.-S. Ra, H.-T. Lim, J.-E. Oh, and Y.-H. Kim, “Phase and amplitude controlled heralding of N00N states,” Opt. Express 23, 30807–30814 (2015).
[Crossref] [PubMed]

2014 (3)

Y.-S. Kim, O. Slattery, P. S. Kuo, and X. Tang, “Two-photon interference with continuous-wave multi-mode coherent light,” Opt. Express 22, 3611–3620 (2014).
[Crossref] [PubMed]

J.-C. Lee, H.-T. Lim, K.-H. Hong, Y.-C. Jeong, M. S. Kim, and Y.-H. Kim, “Experimental demonstration of delayed-choice decoherence suppression,” Nat. Commun. 5, 4522 (2014).
[Crossref] [PubMed]

D. Pandey, N. Satapathy, B. Suryabrahmam, J. S. Ivan, and H. Ramachandran, “Classical light sources with tunable temporal coherence and tailored photon number distributions,” Eur. Phys. J. Plus 129, 115 (2014).
[Crossref]

2013 (2)

Y.-S. Kim, O. Slattery, P. S. Kuo, and X. Tang, “Conditions for two-photon interference with coherent pulses,” Phys. Rev. A 87, 063843 (2013).
[Crossref]

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. S. Kim, and Y.-H. Kim, “Experimental realization of a delayed-choice quantum walk,” Nat. Commun. 4, 2471 (2013).
[Crossref] [PubMed]

2012 (2)

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nat. Photon. 6, 225–228 (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]

2011 (3)

R. Okamoto, J. L. O’Brien, H. F. Hofmann, and S. Takeuchi, “Realization of a Knill-Laflamme-Milburn controlled-NOT photonic quantum circuit combining effective optical nonlinearities,” Proc. Natl. Acad. Sci. USA 108, 10067–10071 (2011).
[Crossref] [PubMed]

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

Y.-S. Kim, O. Kwon, S. M. Lee, J.-C. Lee, H. Kim, S.-K. Choi, H. S. Park, and Y.-H. Kim, “Observation of Young’s double-slit interference with the three-photon N00N state,” Opt. Express 19, 24957–24966 (2011).
[Crossref]

2010 (5)

S. Barz, G. Cronenberg, A. Zeilinger, and P. Walther, “Heralded generation of entangled photon pairs,” Nat. Photon. 4, 553–556 (2010).
[Crossref]

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photon. 4, 316–319 (2010).
[Crossref]

M. Bellini and A. Zavatta, “Manipulating light states by single-photon addition and subtraction,” Prog. Opt. 55, 41–83 (2010).
[Crossref]

Y.-W. Cho, H.-T. Lim, Y.-S. Ra, and Y.-H. Kim, “Weak value measurement with an incoherent measuring device,” New J. Phys. 12, 023036 (2010).
[Crossref]

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

2009 (1)

H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuleti, “Heralded single-magnon quantum memory for photon polarization states,” Phys. Rev. Lett. 103, 043601 (2009).
[Crossref] [PubMed]

2008 (2)

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008).
[Crossref]

S.-Y. Baek, O. Kwon, and Y.-H. Kim, “Temporal shaping of a heralded single-photon wave packet,” Phys. Rev. A 77, 013829 (2008).
[Crossref]

2007 (3)

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

N. Gisin and R. Thew, “Quantum communication,” Nat. Photon. 1, 165–171 (2007).
[Crossref]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

2004 (1)

2003 (1)

T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis,” Phys. Rev. A 68, 032316 (2003).
[Crossref]

1997 (1)

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

1992 (1)

B. C. Sanders, “Entangled coherent state,” Phys. Rev. A 45, 6811–6815 (1992).
[Crossref] [PubMed]

1988 (1)

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921 (1988).
[Crossref] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[Crossref] [PubMed]

1986 (1)

C.-K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[Crossref] [PubMed]

1980 (1)

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43, 913–949 (1980).
[Crossref]

1963 (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev.  131, 2766–2788 (1963).
[Crossref]

1956 (2)

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

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921 (1988).
[Crossref] [PubMed]

Amaral, G. C.

Baek, S.-Y.

S.-Y. Baek, O. Kwon, and Y.-H. Kim, “Temporal shaping of a heralded single-photon wave packet,” Phys. Rev. A 77, 013829 (2008).
[Crossref]

Bao, X.-H.

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nat. Photon. 6, 225–228 (2012).
[Crossref]

Barz, S.

S. Barz, G. Cronenberg, A. Zeilinger, and P. Walther, “Heralded generation of entangled photon pairs,” Nat. Photon. 4, 553–556 (2010).
[Crossref]

Bellini, M.

M. Bellini and A. Zavatta, “Manipulating light states by single-photon addition and subtraction,” Prog. Opt. 55, 41–83 (2010).
[Crossref]

Bloom, B.

H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuleti, “Heralded single-magnon quantum memory for photon polarization states,” Phys. Rev. Lett. 103, 043601 (2009).
[Crossref] [PubMed]

Breitenbach, G.

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

Cao, D.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008).
[Crossref]

Chen, C.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Chen, L.-K.

L.-K. Chen, Z.-D. Li, X.-C. Yao, M. Huang, W. Li, H. Lu, X. Yuan, Y.-B. Zhang, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, X. Ma, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of ten-photon entanglement using thin BiB3O6 crystals,” Optica 4, 77–83 (2017).
[Crossref]

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

Chen, Y.-A.

L.-K. Chen, Z.-D. Li, X.-C. Yao, M. Huang, W. Li, H. Lu, X. Yuan, Y.-B. Zhang, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, X. Ma, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of ten-photon entanglement using thin BiB3O6 crystals,” Optica 4, 77–83 (2017).
[Crossref]

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref] [PubMed]

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nat. Photon. 6, 225–228 (2012).
[Crossref]

Cho, Y.-W.

Y.-W. Cho, H.-T. Lim, Y.-S. Ra, and Y.-H. Kim, “Weak value measurement with an incoherent measuring device,” New J. Phys. 12, 023036 (2010).
[Crossref]

Choi, S.-K.

Choi, Y.

Cronenberg, G.

S. Barz, G. Cronenberg, A. Zeilinger, and P. Walther, “Heralded generation of entangled photon pairs,” Nat. Photon. 4, 553–556 (2010).
[Crossref]

da Silva, T. F.

Di Franco, C.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. S. Kim, and Y.-H. Kim, “Experimental realization of a delayed-choice quantum walk,” Nat. Commun. 4, 2471 (2013).
[Crossref] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Eto, Y.

Y. Zhang, R. Okubo, M. Hirano, Y. Eto, and T. Hirano, “Experimental realization of spatially separated entanglement with continuous variables using laser pulse trains,” Sci. Rep. 5, 13029 (2015).
[Crossref] [PubMed]

Fitch, M. J.

T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis,” Phys. Rev. A 68, 032316 (2003).
[Crossref]

Franson, J. D.

T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis,” Phys. Rev. A 68, 032316 (2003).
[Crossref]

Gao, L.

D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92, 201102 (2008).
[Crossref]

Ghosh, S.

H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuleti, “Heralded single-magnon quantum memory for photon polarization states,” Phys. Rev. Lett. 103, 043601 (2009).
[Crossref] [PubMed]

Giovannetti, V.

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

Gisin, N.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photon. 1, 165–171 (2007).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev.  131, 2766–2788 (1963).
[Crossref]

Grangier, P.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

Guo, G.-C.

Han, S.-W.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

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

Hirano, M.

Y. Zhang, R. Okubo, M. Hirano, Y. Eto, and T. Hirano, “Experimental realization of spatially separated entanglement with continuous variables using laser pulse trains,” Sci. Rep. 5, 13029 (2015).
[Crossref] [PubMed]

Hirano, T.

Y. Zhang, R. Okubo, M. Hirano, Y. Eto, and T. Hirano, “Experimental realization of spatially separated entanglement with continuous variables using laser pulse trains,” Sci. Rep. 5, 13029 (2015).
[Crossref] [PubMed]

Hofmann, H. F.

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Figures (4)

Fig. 1
Fig. 1 The proposed experimental setup for manipulating photon statistics via second-order interference between two mutually incoherent weak lasers. AOM: acousto-optic modulator, BS: beam splitter, Pol.: polarizer, D : single-photon detector, HBT: Hanbury-Brown–Twiss interferometer. (a) This setup measures the second-order intensity cross-correlation Rcd(τ) between modes c and d. (b) This setup measures the conditional second-order intensity autocorrelation g C ( 2 ) ( τ , τ c ). Electronic delays τ and τc are used to explore various interference conditions.
Fig. 2
Fig. 2 The intensity cross-correlation Rcd(τ) and the conditional second-order correlation g C ( 2 ) ( τ , 0 ) for |α| = 0.1 and |α| = 1.2. Both Rcd(τ) and g C ( 2 ) ( τ , 0 ) are truncated at p = 10. The coherence time τcoh ≈ 260 ns. The blue solid lines and red dashed lines correspond to projection measurement bases {|Dc, |Dd}, and {|Ac, |Dd}, respectively. Note that g C ( 2 ) ( τ , 0 ) is never below 1, meaning that the heralded photon states always remain classical.
Fig. 3
Fig. 3 The conditional second order correlation g C ( 2 ) ( 0 , 0 ) for |α| = 0.1 and |α| = 1.2 under different p-photon Fock state truncation. For polarization projection (a) {|Dc, |Dd} and (b) {|Ac, |Dd}. (c) g C ( 2 ) ( τ , 0 ) with τ = 500 ns, i.e., τ > τcoh. Even for weak coherent state at the single-photon regime, |α| = 0.1, asymptotic behaviors are observed at relatively large p = 4. For |α| = 1.2, asymptotic behaviors are not reached until p = 9, meaning that truncation below p = 9 would result incorrect results. If Fock state truncation is made before reaching the asymptotic value corresponding to a particular α, it looks as though conditional photon anti-bunching were possible.
Fig. 4
Fig. 4 The conditional second order correlation g C ( 2 ) ( τ , τ c ), truncated at the p = 10 photon Fock state, as functions of both τ and τc. The black solid lines correspond to the case of τc = 0, presented in Fig. 2. It is clear that photon antibunching cannot be achieved by heralding if the input light is classical.

Equations (20)

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| Ψ in = e α 2 λ = 0 α λ 2 λ ! | λ ,
| Ψ = e α 4 m = 0 ( α 2 ) m 2 m ! | m a e α 4 n = 0 ( α 2 ) n 2 n ! | n b = e α 2 m = 0 n = 0 ( α 2 ) n + m 2 m ! n ! ( a ^ ) m ( b ^ ) n | 0
| Ψ ( ω a , ω b ) = e α 2 m , n = 0 ( α 2 ) n + m 2 m ! n ! { e i γ ω a a ^ ( ω a ) } m { e i γ ω b b ^ ( ω b ) } n | 0 ,
ρ = d ω a d ω b ( ω a ) ( ω b ) | Ψ ( ω a , ω b ) Ψ ( ω a , ω b ) | ,
ρ = e α d ω a ( ω a ) m = 0 ( | α | 2 ) m m ! | m a m | d ω b ( ω b ) n = 0 ( | α | 2 ) n n ! | n b n | .
a ^ H ( ω a ) 1 2 c ^ H ( ω a ) + i 2 d ^ H ( ω a ) , b ^ V ( ω b ) i 2 c ^ V ( ω b ) + 1 2 d ^ V ( ω b ) ,
ρ c d = e α d ω a d ω b ( ω a ) ( ω b ) m , n ( | α | 2 ) n + m ( m ! n ! ) 2 ( 1 2 ) m + n × { c ^ H ( ω a ) + i d ^ H ( ω a ) } m { i c ^ V ( ω b ) + d ^ V ( ω b ) } n | 0 0 | { C . C . } ,
R c d ( τ ) = Tr [ ρ c d E c ( ) ( τ ) E d ( ) ( 0 ) E d ( + ) ( 0 ) E c ( + ) ( τ ) ] Tr [ ρ c d E c ( ) ( τ ) E c ( + ) ( τ ) ] Tr [ ρ c d E d ( ) ( 0 ) E d ( + ) ( 0 ) ] ,
g C ( 2 ) ( τ , τ c ) = p = 1 I d ( p ) p = 1 I def ( p ) ( τ , τ c ) p = 1 I de ( p ) ( τ ) p = 1 I df ( p ) ( τ + τ c ) .
I d ( p ) = 1 p ! Tr [ ρ c d { E d ( ) ( 0 ) } p { E d ( + ) ( 0 ) } p ] .
I de ( p ) ( τ ) = 1 p ! Tr [ ρ c d { E d ( ) ( 0 ) } p E e ( ) ( τ ) E e ( + ) ( τ ) { E d ( + ) ( 0 ) } p ] , I df ( p ) ( τ ) = 1 p ! Tr [ ρ c d { E d ( ) ( 0 ) } p E f ( ) ( τ ) E f ( + ) ( τ ) { E d ( + ) ( 0 ) } p ] .
I def ( p ) ( τ , τ c ) = 1 p ! Tr [ ρ c d { E d ( ) ( 0 ) } p E e ( ) ( τ ) E f ( ) ( τ + τ c ) E f ( + ) ( τ + τ c ) E e ( + ) ( τ ) { E d ( + ) ( 0 ) } p ] .
g C , p = 3 ( 2 ) ( τ , 0 ) = ( 265 α 2 + 456 α + 384 ) e σ 2 τ 2 ( 3 e σ 2 τ 2 2 ) 3 ( ( 19 α + 16 ) e σ 2 τ 2 2 ( 5 α + 4 ) ) 2 ,
g C , p = 4 ( 2 ) ( τ , 0 ) = ( 9679 α 3 + 25440 α 2 + 43776 α + 36864 ) e 2 σ 2 τ 2 [ 16 ( 5 α + 4 ) e 3 σ 2 τ 2 + 6 ( 19 α + 16 ) e 4 σ 2 τ 2 + α ] / 16 [ ( 265 α 2 + 456 α + 384 ) e σ 2 τ 2 3 ( 49 α 2 + 80 α + 64 ) ] 2 .
g C , p = 5 ( 2 ) ( τ , 0 ) = 9 40 ( 1741603 α 4 + 6194560 α 3 + 16281600 α 2 + 28016640 α + 23592960 ) e 2 σ 2 τ 2 × [ 4 ( 49 α 2 + 80 α + 64 ) e 3 σ 2 τ 2 + ( 265 α 2 + 456 α + 384 ) e 4 σ 2 τ 2 + α ( 5 α + 4 ) ] / [ ( 9679 α 3 + 25440 α 2 + 43776 α + 36864 ) e σ 2 τ 2 4 ( 1415 α 3 + 3528 α 2 + 5760 α + 4608 ) ] 2 .
g C , p = 6 ( 2 ) ( τ = 0 ) = 16 3 ( 92917951 α 5 + 417984720 α 4 + 1486694400 α 3 + 3907584000 α 2 + 6723993600 α + 5662310400 ) × e 2 σ 2 τ 2 [ 3 α ( 293 α 2 + 480 α + 384 ) 16 ( 1415 α 3 + 3528 α 2 + 5760 α + 4608 ) e 3 σ 2 τ 2 + 3 ( 9679 α 3 + 25440 α 2 + 43776 α + 36864 ) e 4 σ 2 τ 2 ] / [ 5 ( 429031 α 4 + 1448960 α 3 + 3612672 α 2 + 5898240 α + 4718592 ) 2 ( 1741603 α 4 + 6194560 α 3 + 16281600 α 2 + 28016640 α + 23592960 ) e σ 2 τ 2 ] 2 .
g C , p = 7 ( 2 ) ( τ , 0 ) = 5 14 ( 5731580419 α 6 + 31220431536 α 5 + 140442865920 α 4 + 499529318400 α 3 + 1312948224000 α 2 + 2259261849600 α + 1902536294400 ) e 2 σ 2 τ 2 [ 640 α ( 350 α 3 + 879 α 2 + 1440 α + 1152 ) 10 ( 429031 α 4 + 1448960 α 3 + 3612672 α 2 + 5898240 α + 4718592 ) e 3 σ 2 τ 2 + 3 ( 1741603 α 4 + 6194560 α 3 + 16281600 α 2 + 28016640 α + 23592960 ) e 4 σ 2 τ 2 ] / [ ( 92917951 α 5 + 417984720 α 4 + 1486694400 α 3 + 3907584000 α 2 + 6723993600 α + 5662310400 ) e σ 2 τ 2 6 ( 10022813 α 5 + 42903100 α 4 + 144896000 α 3 + 361267200 α 2 + 589824000 α + 471859200 ) ] 2 .
g C , p = 8 ( 2 ) ( τ , 0 ) = 18 ( 50159199575 α 7 + 320968503464 α 6 + 1748344166016 α 5 + 7864800491520 α 4 + 27973641830400 α 3 + 73525100544000 α 2 + 126518663577600 α + 106542032486400 ) e 2 σ 2 τ 2 × [ 25 α ( 419671 α 4 + 1433600 α 3 + 3600384 α 2 + 5898240 α + 4718592 ) 16 ( 10022813 α 5 + 42903100 α 4 + 144896000 α 3 + 361267200 α 2 + 589824000 α + 471859200 ) e 3 σ 2 τ 2 + 2 ( 92917951 α 5 + 417984720 α 4 + 1486694400 α 3 + 3907584000 α 2 + 6723993600 α + 5662310400 ) e 4 σ 2 τ 2 ] / [ 7 ( 1110257215 α 6 + 5773140288 α 5 + 24712185600 α 4 + 83460096000 α 3 + 208089907200 α 2 + 33973862400 α + 0271790899200 ) 2 ( 5731580419 α 6 + 31220431536 α 5 + 140442865920 α 4 499529318400 α 3 + 1312948224000 α 2 + 2259261849600 α + 1902536294400 ) e σ 2 τ 2 ] 2 .
g C , p = 9 ( 2 ) ( τ , 0 ) = 7 6144 ( 167649791255501 α 8 + 1232712488755200 α 7 + 7888121941131264 α 6 + 42967306224009216 α 5 + 193285336879595520 α 4 + 687480221623910400 α 3 + 1806952870969344000 α 2 + 3109322676083097600 α + 2618376990385766400 ) e 2 σ 2 τ 2 [ 42 α ( 28954439 α 5 + 125901300 α 5 + 430080000 α 3 + 1080115200 α 2 + 1769472000 α + 1415577600 ) 14 ( 1110257215 α 6 + 5773140288 α 5 + 24712185600 α 4 + 83460096000 α 3 + 208089907200 α 2 + 339738624000 α + 271790899200 ) e 3 σ 2 τ 2 + 3 ( 5731580419 α 6 + 31220431536 α 5 + 140442865920 α 4 + 499529318400 α 3 + 1312948224000 α 2 + 2259261849600 α + 1902536294400 ) e 4 σ 2 τ 2 ] / [ ( 50159199575 α 7 + 320968503464 α 6 + 1748344166016 α 5 + 7864800491520 α 4 + 27973641830400 α 3 + 73525100544000 α 2 + 126518663577600 α + 106542032486400 ) e σ 2 τ 2 4 ( 8870590892 α 7 + 54402603535 α 6 + 282883874112 α 5 + 1210897094400 α 4 + 4089544704000 α 3 + 10196405452800 α 2 + 16647192576000 α + 13317754060800 ) ] 2 .
g C , p = 10 ( 2 ) ( τ , 0 ) = 256 45 ( 43577990017810513 α 9 + 362123549111882160 α 8 + 2662658975711232000 α 7 + 17038343392843530240 α 6 + 92809381443859906560 α 5 + 417496327659926323200 α 4 + 1484957278707646464000 α 3 + 3903018201293783040000 α 2 + 6716136980339490816000 α + 5655694299233255424000 ) e 2 σ 2 τ 2 × [ 49 α ( 1048007845 α 6 + 5559252288 α 5 + 24173049600 α 4 + 82575360000 α 3 + 207382118400 α 2 + 339738624000 α + 271790899200 ) 64 ( 8870590892 α 7 + 54402603535 α 6 + 282883874112 α 5 + 1210897094400 α 4 + 4089544704000 α 3 + 10196405452800 α 2 + 16647192576000 α + 13317754060800 ) e 3 σ 2 τ 2 + 12 ( 50159199575 α 7 + 320968503464 α 6 + 1748344166016 α 5 + 7864800491520 α 4 + 27973641830400 α 3 + 73525100544000 α 2 + 126518663577600 α + 106542032486400 ) e 4 σ 2 τ 2 ] / [ 3 ( 82126151511535 α 8 + 581343044698112 α 7 + 3565329025269760 α 6 + 18539077573804032 α 5 + 79357351978598400 α 4 + 268012401721344000 α 3 + 668231627754700800 α 2 + 1090990412660736000 α + 872792330128588800 ) 2 ( 16764979125550 α 8 + 11232712488755200 α 7 + 7888121941131264 α 6 + 42967306224009216 α 5 + 193285336879595520 α 4 + 687480221623910400 α 3 + 1806952870969344000 α 2 + 3109322676083097600 α + 2618376990385766400 ) e σ 2 τ 2 ] 2 .

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