Abstract

We study theoretically how multiple scattering of light in a disordered medium can spontaneously generate quantum correlations. In particular we focus on the case where the input state is Gaussian and characterize the correlations between two arbitrary output modes. As there is not a single all-inclusive measure of correlation, we characterise the output correlations with three measures: intensity fluctuations, entanglement, and quantum discord. We find that, while a coherent input state can not produce quantum correlations, any other Gaussian input will produce them in one form or another. This includes input states that are usually regarded as more classical than coherent ones, such as thermal states, which will produce a non-zero quantum discord.

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References

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2016 (1)

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

2015 (1)

S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, “Programmable multiport optical circuits in opaque scattering materials,” Opt. Expr. 23, 229054 (2015).
[Crossref]

2014 (3)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

S. Hosseini, S. Rahimi-Keshari, J. Y. Haw, S. M. Assad, H. M. Chrzanowski, J. Janousek, T. Symul, T. C. Ralph, and P. K. Lam, “Experimental verification of quantum discord in continuous-variable states,” J. Phys. B At. Mol. Opt. Phys. 47, 025503 (2014).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39, 6090 (2014).
[Crossref] [PubMed]

2013 (1)

S. Rahimi-Keshari, C. M. Caves, and T. C. Ralph, “Measurement-based method for verifying quantum discord,” Phys. Rev. A 87, 012119 (2013).
[Crossref]

2012 (4)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref] [PubMed]

A. Ferraro and M. G. A. Paris, “Nonclassicality criteria from phase-space representations and information-theoretical constraints are maximally inequivalent,” Phys. Rev. Lett. 108, 260403 (2012).
[Crossref] [PubMed]

P. Giorda, M. Allegra, and M. G. A. Paris, “Quantum discord for Gaussian states with non-Gaussian measurements,” Phys. Rev. A 86, 052328 (2012).
[Crossref]

C. A. Prez-Delgado, M. E. Pearce, and P. Kok, “Fundamental limits of classical and quantum imaging,” Phys. Rev. Lett. 109, 123601 (2012).
[Crossref]

2010 (4)

P. Giorda and M. Paris, “Gaussian quantum discord,” Phys. Rev. Lett. 105, 020503 (2010).
[Crossref] [PubMed]

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref] [PubMed]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105, 163905 (2010).
[Crossref]

2009 (1)

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett.,  102, 193901 (2009).
[Crossref] [PubMed]

2008 (2)

T. Wellens and B. Gremaud, “Nonlinear coherent transport of waves in disordered media,” Phys. Rev. Lett. 100, 033902 (2008).
[Crossref] [PubMed]

J. W. Goodman, “Speckle with a finite number of steps,” Appl. Opt. 47, A111–A118 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (4)

P. Lodahl and A. Lagendijk, “Transport of quantum noise through random media,” Phys. Rev. Lett. 94, 153905 (2005).
[Crossref] [PubMed]

A. Datta, S. T. Flammia, and C. M. Caves, “Entanglement and the power of one qubit,” Phys. Rev. A 72, 042316 (2005).
[Crossref]

P. Lodahl, A. P. Mosk, and A. Lagendijk, “Spatial quantum correlations in multiple scattered light,” Phys. Rev. Lett. 95, 173901 (2005).
[Crossref] [PubMed]

J. Laurat, G. Keller, J.A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, “Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation,” J. Opt. B 7, S577–S587 (2005).
[Crossref]

2004 (2)

G. Adesso and A. Datta, “Quantum versus classical correlations in gaussian states,” Phys. Rev. Lett. 105, 030501 (2004).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: Comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref] [PubMed]

2001 (2)

H. Ollivier and W. H. Zurek, “Quantum Discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels,” Phys. Rev. A 63, 032312 (2001).
[Crossref]

2000 (2)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2752 (2000).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

1999 (3)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

M. Patra and C. W. J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A,  61, 063805 (1999).
[Crossref]

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726 (1999).
[Crossref]

1998 (1)

M. Paris, “Entanglement and visibility at the output of a Mach-Zehnder interferometer,” Phys. Rev. Lett. 59, 1615–1621 (1998).

1997 (1)

C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

1996 (2)

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[Crossref] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
[Crossref]

1994 (2)

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref] [PubMed]

R. Berkovits and S. Fen, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

1993 (1)

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
[Crossref] [PubMed]

1989 (1)

M. C. Teich and B. E. Saleh, “Squeezed states of light,” Quantum Opt. 1, 153–191 (1989).
[Crossref]

1988 (1)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref] [PubMed]

Abouraddy, A. F.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Adesso, G.

J. Laurat, G. Keller, J.A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, “Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation,” J. Opt. B 7, S577–S587 (2005).
[Crossref]

G. Adesso and A. Datta, “Quantum versus classical correlations in gaussian states,” Phys. Rev. Lett. 105, 030501 (2004).
[Crossref]

Aharonov, Y.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2011).

Allegra, M.

P. Giorda, M. Allegra, and M. G. A. Paris, “Quantum discord for Gaussian states with non-Gaussian measurements,” Phys. Rev. A 86, 052328 (2012).
[Crossref]

Anders, J.

J. Anders, “Estimating the degree of entanglement of unknown Gaussian states,” Diploma thesis, University of Potsdam (2003), arXiv:quant-ph/0610263.

Andersen, U. L.

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett.,  102, 193901 (2009).
[Crossref] [PubMed]

Assad, S. M.

S. Hosseini, S. Rahimi-Keshari, J. Y. Haw, S. M. Assad, H. M. Chrzanowski, J. Janousek, T. Symul, T. C. Ralph, and P. K. Lam, “Experimental verification of quantum discord in continuous-variable states,” J. Phys. B At. Mol. Opt. Phys. 47, 025503 (2014).
[Crossref]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: Comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref] [PubMed]

Barbieri, M.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39, 6090 (2014).
[Crossref] [PubMed]

Beenakker, C. W.

C. W. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

Beenakker, C. W. J.

M. Patra and C. W. J. Beenakker, “Propagation of squeezed radiation through amplifying or absorbing random media,” Phys. Rev. A,  61, 063805 (1999).
[Crossref]

Berkovits, R.

R. Berkovits and S. Fen, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Berne, B. J.

B. J. Berne, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover, 2003).

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref] [PubMed]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref] [PubMed]

Boas, D. A.

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: Comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref] [PubMed]

Bromberg, Y.

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105, 163905 (2010).
[Crossref]

Caves, C. M.

S. Rahimi-Keshari, C. M. Caves, and T. C. Ralph, “Measurement-based method for verifying quantum discord,” Phys. Rev. A 87, 012119 (2013).
[Crossref]

A. Datta, S. T. Flammia, and C. M. Caves, “Entanglement and the power of one qubit,” Phys. Rev. A 72, 042316 (2005).
[Crossref]

Chalopin, B.

Chatel, B.

Christodoulides, D. N.

Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. 105, 163905 (2010).
[Crossref]

Chrzanowski, H. M.

S. Hosseini, S. Rahimi-Keshari, J. Y. Haw, S. M. Assad, H. M. Chrzanowski, J. Janousek, T. Symul, T. C. Ralph, and P. K. Lam, “Experimental verification of quantum discord in continuous-variable states,” J. Phys. B At. Mol. Opt. Phys. 47, 025503 (2014).
[Crossref]

Cirac, J. I.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2752 (2000).
[Crossref] [PubMed]

Coudreau, T.

J. Laurat, G. Keller, J.A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, “Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation,” J. Opt. B 7, S577–S587 (2005).
[Crossref]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer, 1984).

Datta, A.

A. Datta, S. T. Flammia, and C. M. Caves, “Entanglement and the power of one qubit,” Phys. Rev. A 72, 042316 (2005).
[Crossref]

G. Adesso and A. Datta, “Quantum versus classical correlations in gaussian states,” Phys. Rev. Lett. 105, 030501 (2004).
[Crossref]

Davidovich, L.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
[Crossref] [PubMed]

Defienne, H.

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

H. Defienne, M. Barbieri, B. Chalopin, B. Chatel, I. A. Walmsley, B. J. Smith, and S. Gigan, “Nonclassical light manipulation in a multiple-scattering medium,” Opt. Lett. 39, 6090 (2014).
[Crossref] [PubMed]

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2752 (2000).
[Crossref] [PubMed]

Dunn, A. K.

D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. 15, 011109 (2010).
[Crossref] [PubMed]

Dutta, B.

R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[Crossref] [PubMed]

Fabre, C.

J. Laurat, G. Keller, J.A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, “Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation,” J. Opt. B 7, S577–S587 (2005).
[Crossref]

Fen, S.

R. Berkovits and S. Fen, “Correlations in coherent multiple scattering,” Phys. Rep. 238, 135–172 (1994).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref] [PubMed]

Ferraro, A.

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H. Ollivier and W. H. Zurek, “Quantum Discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
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P. Giorda and M. Paris, “Gaussian quantum discord,” Phys. Rev. Lett. 105, 020503 (2010).
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R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726 (1999).
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Figures (4)

Fig. 1
Fig. 1

The disordered medium is scattering N input modes âk into N output modes l. The scattering process is described by a scattering matrix S. The only non-empty input mode is âk′, the others are assumed to be in a vacuum state.

Fig. 2
Fig. 2

Map of all possible two-mode Gaussian states and their correlations, as a function of the off-diagonal elements γx and γp of the covariance matrix in Eq. (7), with diagonal values α and β: (a) α = β = 0.75, (b) α = β = 5. All physically allowed states lie within the region enclosed by the thick solid line. The region of separable states can be found as the intersection of the allowed state region and its mirror image with respect of γp (dotted line) [31]. The remaining states (in the gray area) are not separable, i.e. they are entangled. For a thermal input state the output states, Eq. (15), lie on the γx = γp line, and thus are not entangled. The dashed circle corresponds to the condition �� = 2. Inside this circle �� is less than 2, with �� = 1 at the origin. Increasing the number of photons in the input state, i.e. increasing α and β, the fraction of entangled output states decreases and some entangled states cross into the �� = 2 circle, as shown in panel (b).

Fig. 3
Fig. 3

The dependence of the Gaussian discord DG between modes l and m on the absolute values of the elements of the scattering matrix |Sl,k′| and |Sm,k′|. The input mode, k′, is in a thermal state with different average photon numbers (a) = 1, (b) = 103. The measurement is performed on the mode l.

Fig. 4
Fig. 4

Dependence of the average Gaussian discord, 〈DG〉, on the number of output modes, N, and the number of photons, , in the thermal input state.

Equations (16)

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𝒞 l . m = I l I m I l I m ,
ς μ , ν = 1 2 R ^ μ R ^ ν + R ^ ν R ^ μ R ^ μ R ^ ν , μ , ν = 1 , , 2 N ,
ς + i Ω N 0 with Ω = ( 0 1 1 0 ) ,
E k in ( r ) = e k 2 ω k 2 ε 0 V [ a ^ k e i q k r + a ^ k e i q k r ] ,
b ^ l = k N S l , k a ^ k , k , l , m = 1 , , N .
σ = ( A Γ Γ T B ) ,
σ = ( α 0 γ x 0 0 α 0 γ p γ x 0 β 0 0 γ p 0 β )
𝒞 = 1 + ( 2 γ x 2 + 2 γ p 2 ) ( 2 α 1 ) ( 2 β 1 ) .
η ± : = [ 1 2 ( Δ ± Δ 2 4 det ( σ ) ) ] 1 / 2 1 / 2 ,
I q ( ρ A B ) = S ( ρ A ) + S ( ρ B ) S ( ρ A B ) and J Π j q ( ρ A B ) = S ( ρ A ) j Tr [ ρ A B Π j ] S ( ρ A | Π j ) .
D ( ρ A B ) = inf Π j ( I q ( ρ A B ) J Π j q ( ρ A B ) ) = S ( ρ B ) = S ( ρ A B ) + inf Π j j Tr [ ρ A B Π j ] S ( ρ A | Π j ) .
D G ( σ ) = κ ( det ( B ) ) κ ( η ) κ ( η + ) + κ ( det ( A ) + 2 det ( A ) det ( B ) + 2 det ( Γ ) 1 + 2 det ( B ) ) ,
σ 2 l 1 , 2 m 1 = δ l , m 2 + W l , m Δ n ^ k + Y l , m Δ a ^ k a ^ k + Y l , m * Δ a ^ k a ^ k , σ 2 l , 2 m = δ l , m 2 + W l , m Δ n ^ k Y l , m Δ a ^ k a ^ k Y l , m * Δ a ^ k a ^ k , σ 2 l 1 , 2 m = σ 2 l , 2 m 1 = 1 2 i [ Z l , m Δ n ^ k + Y l , m Δ a ^ k a ^ k Y l , m * Δ a k a ^ k ] ,
Δ n ^ k = a ^ k a ^ k a ^ k a ^ k , Δ a ^ k a ^ k = a ^ k a ^ k a ^ k a ^ k W l , m = ( S l , k * S m , k + S m , k * S l , k ) , Z l , m = ( S l , k * S m , k S m , k * S l , k ) , Y l , m = S l , k S m , k ,
𝒞 l , m sq = 2 + | S l , k | 2 + | S m , k | 2 2 n ¯ | S l , k | | S m , k | ,
σ l , m th = ( σ α th 0 σ γ th 0 0 σ α th 0 σ γ th σ γ th 0 σ β th 0 0 σ γ th 0 σ β th )

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