Abstract

The effect of boundary deformation on the classical entanglement which appears in the classical electromagnetic field is considered. A chaotic billiard geometry is used to explore the influence of the mechanical modification of the optical fiber cross-sectional geometry on the production of classical entanglement within the electromagnetic fields. For the experimental realization of our idea, we propose an optical fiber with a cross section that belongs to the family of Robnik chaotic billiards. Our results show that a modification of the fiber geometry from a regular to a chaotic regime can enhance the transverse mode classical entanglement.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
  58. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  59. K. B. Møller, T. G. Jørgensen, and J. P. Dahl, “Displaced squeezed number states: Position space representation, inner product, and some applications,” Phys. Rev. A 54, 5378–5385 (1996).
    [Crossref] [PubMed]
  60. S. K. Joseph, L. Y. Chew, and M. A. F. Sanjuán, “Effect of squeezing and planck constant dependence in short time semiclassical entanglement,” Eur. Phys. J. D 68, 238 (2014).
    [Crossref]

2015 (5)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Y. Sun, X. Song, H. Qin, X. Zhang, Z. Yang, and X. Zhang, “Non-local classical optical correlation and implementing analogy of quantum teleportation,” Sci. Rep. 5, 9175 (2015).
[Crossref] [PubMed]

B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: The Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

2014 (4)

S. K. Joseph, L. Y. Chew, and M. A. F. Sanjuán, “Effect of squeezing and planck constant dependence in short time semiclassical entanglement,” Eur. Phys. J. D 68, 238 (2014).
[Crossref]

P. Ghose and Mukhrjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

V. S. Manu and A. Kumar, “Quantum simulation using fidelity-profile optimization,” Phys. Rev. A 89, 052331 (2014).
[Crossref]

S. K. Joseph, L. Y. Chew, and M. A. F. Sanjuán, “Impact of quantum-classical correspondence on entanglement enhancement by single-mode squeezing,” Phys. Lett. A 378, 2603–2610 (2014).
[Crossref]

2012 (1)

G. B. Lemos, R. M. Gomes, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Experimental observation of quantum chaos in a beam of light,” Nat. Commun. 3, 1211 (2012).
[Crossref] [PubMed]

2011 (1)

M. Lombardi and A. Matzkin, “Entanglement and chaos in the kicked top,” Phys. Rev. E 83, 016207 (2011).
[Crossref]

2010 (2)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

A. Rai, S. Das, and G. Agarwal, “Quantum entanglement in coupled lossy waveguides”, Opt. Express 18, 6241–6254 (2010).
[Crossref] [PubMed]

2009 (3)

L. Carretero, P. Acebal, S. Blaya, C. García, A. Fimia, R. Madrigal, and A. Murciano, “Nonparaxial diffraction analysis of Airy and SAiry beams”;, Opt. Express,  25, 22432–22441(2009).
[Crossref]

N. N. Chung and L. Y. Chew, “Dependence of entanglement dynamics on the global classical dynamical regime,” Phys. Rev. E 80, 016204 (2009).
[Crossref]

S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, “Quantum signs of chaos in a kicked top,” Nature 461, 768–771 (2009).
[Crossref] [PubMed]

2008 (3)

S.-H. Zhang and Q.-L. Jie, “Quantum-classical correspondence in entanglement production: Entropy and classical tori,” Phys. Rev. A 77, 012312 (2008).
[Crossref]

M. A. Man’ko, “Analogs of time-dependent quantum phenomena in optical fibers,” J. Phys.: Conf. Series 99, 012012 (2008).

T. Betcke, “The generalized singular value decomposition and the method of particular solutions,” SIAM J. Sci. Comput. 30, 1278–1295 (2008).
[Crossref]

2007 (4)

T. W. Hijmans, T. N. Huussen, and R. J. C. Spreeuw, “Time and frequency domain solutions in an optical analogue of Grovers search algorithm,” J. Opt. Soc. Am. B 24, 214–220 (2007).
[Crossref]

S. Chávez-Cerda, J. R. Moya-Cessa, and H. M. Moya-Cessa, “Quantumlike systems in classical optics: applications of quantum optical methods,” J. Opt. Soc. Am. B 24, 404–407 (2007).
[Crossref]

V. Doya, O. Legrand, C. Michel, and F. Mortessagne, “Optical scar in a chaotic fiber,” Eur. Phys. J. Special Topics 145, 49–61 (2007).
[Crossref]

S. Chávez-Cerda, H. M. M. Cessa, and J. R. M. Cessa, “Quantum-like entanglement in classical optics,” Opt. Photon. News 18, 38 (2007).
[Crossref]

2006 (1)

A. Bogdanov, Y. Bogdanov, and K. Valiev, “Schmidt modes and entanglement in continuous-variable quantum systems,” Russ. Microelectron. 35, 7–20 (2006).
[Crossref]

2005 (2)

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005).
[Crossref]

J. Fu, Z. Si, S. Tang, and J. Deng, “Analogs of time-dependent quantum phenomena in optical fibers,” Phys. Rev. A 71, 059901 (2005).
[Crossref]

2004 (3)

S. Ghose and B. C. Sanders, “Entanglement dynamics in chaotic systems,” Phys. Rev. A 70, 062315 (2004).
[Crossref]

X. Wang, S. Ghose, B. C. Sanders, and B. Hu, “Entanglement as a signature of quantum chaos,” Phys. Rev. E 70, 016217 (2004).
[Crossref]

J. N. Bandyopadhyay and A. Lakshminarayan, “Entanglement production in coupled chaotic systems: Case of the kicked tops,” Phys. Rev. E 69, 016201 (2004).
[Crossref]

2003 (3)

H. Fujisaki, T. Miyadera, and A. Tanaka, “Dynamical aspects of quantum entanglement for weakly coupled kicked tops,” Phys. Rev. E 67, 066201 (2003).
[Crossref]

S. Hiroyuki and O. Megumi, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[Crossref]

R. Fedele and M. A. Man’ko, “Beam optics applications: quantumlike versus classical-like domains,” Eur. Phys. J. D 27, 263 (2003).
[Crossref]

2002 (3)

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[Crossref]

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref] [PubMed]

J. N. Bandyopadhyay and A. Lakshminarayan, “Testing statistical bounds on entanglement using quantum chaos,” Phys. Rev. Lett. 89, 060402 (2002).
[Crossref] [PubMed]

2001 (3)

A. Lakshminarayan, “Entangling power of quantized chaotic systems,” Phys. Rev. E 64, 036207 (2001).
[Crossref]

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

M. A. Man’ko, V. I. Man’ko, and R. V. Mendes, “Quantum computation by quantumlike systems,” Phys. Lett. A 288, 132 (2001).
[Crossref]

2000 (1)

S. Parker, S. Bose, and M. B. Plenio, “Entanglement quantification and purification in continuous-variable systems,” Phys. Rev. A 61, 032305 (2000).
[Crossref]

1999 (2)

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

P. A. Miller and S. Sarkar, “Signatures of chaos in the entanglement of two coupled quantum kicked tops,” Phys. Rev. E 60, 1542–1550 (1999).
[Crossref]

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1996 (1)

K. B. Møller, T. G. Jørgensen, and J. P. Dahl, “Displaced squeezed number states: Position space representation, inner product, and some applications,” Phys. Rev. A 54, 5378–5385 (1996).
[Crossref] [PubMed]

1992 (1)

J. Stein and H.-J. Stöckmann, “Experimental determination of billiard wave functions,” Phys. Rev. Lett. 68, 2867–2870 (1992).
[Crossref] [PubMed]

1991 (2)

F. Haake, G. Lenz, P. Seba, J. Stein, H.-J. Stöckmann, and K. Życzkowski, “Manifestation of wave chaos in pseudointegrable microwave resonators,” Phys. Rev. A 44, R6161–R6164 (1991).
[Crossref] [PubMed]

D. D. Holm and G. Kovačič, “Homoclinic chaos for ray optics in a fiber,” Physica D 51, 177–188 (1991).
[Crossref]

1990 (1)

H.-J. Stöckmann and J. Stein, “Quantum chaos in billiards studied by microwave absorption,” Phys. Rev. Lett. 64, 2215–2218 (1990).
[Crossref]

1988 (1)

J. Rai and C. L. Mehta, “Coordinate representation of squeezed states,” Phys. Rev. A 37, 4497–4499 (1988).
[Crossref] [PubMed]

1983 (1)

M. Robnik, “Classical dynamics of a family of billiards with analytic boundaries,” J. Phys. A: Math. Gen. 16, 3971 (1983).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

1979 (2)

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
[Crossref]

L. Bunimovich, “On the ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1979).
[Crossref]

Acebal, P.

L. Carretero, P. Acebal, S. Blaya, C. García, A. Fimia, R. Madrigal, and A. Murciano, “Nonparaxial diffraction analysis of Airy and SAiry beams”;, Opt. Express,  25, 22432–22441(2009).
[Crossref]

Agarwal, G.

Aiello, A.

Anderson, B. E.

S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, “Quantum signs of chaos in a kicked top,” Nature 461, 768–771 (2009).
[Crossref] [PubMed]

Bandyopadhyay, J. N.

J. N. Bandyopadhyay and A. Lakshminarayan, “Entanglement production in coupled chaotic systems: Case of the kicked tops,” Phys. Rev. E 69, 016201 (2004).
[Crossref]

J. N. Bandyopadhyay and A. Lakshminarayan, “Testing statistical bounds on entanglement using quantum chaos,” Phys. Rev. Lett. 89, 060402 (2002).
[Crossref] [PubMed]

Banzer, P.

Berg-Johansen, S.

Betcke, T.

T. Betcke, “The generalized singular value decomposition and the method of particular solutions,” SIAM J. Sci. Comput. 30, 1278–1295 (2008).
[Crossref]

T. Betcke and L. Trefethen, “Reviving the method of particular solutions,” SIAM Rev.47, 469–491 (2005).
[Crossref]

Blaya, S.

L. Carretero, P. Acebal, S. Blaya, C. García, A. Fimia, R. Madrigal, and A. Murciano, “Nonparaxial diffraction analysis of Airy and SAiry beams”;, Opt. Express,  25, 22432–22441(2009).
[Crossref]

Bogdanov, A.

A. Bogdanov, Y. Bogdanov, and K. Valiev, “Schmidt modes and entanglement in continuous-variable quantum systems,” Russ. Microelectron. 35, 7–20 (2006).
[Crossref]

Bogdanov, Y.

A. Bogdanov, Y. Bogdanov, and K. Valiev, “Schmidt modes and entanglement in continuous-variable quantum systems,” Russ. Microelectron. 35, 7–20 (2006).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Bose, S.

S. Parker, S. Bose, and M. B. Plenio, “Entanglement quantification and purification in continuous-variable systems,” Phys. Rev. A 61, 032305 (2000).
[Crossref]

Braunstein, S. L.

S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

Bunimovich, L.

L. Bunimovich, “On the ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1979).
[Crossref]

Carretero, L.

L. Carretero, P. Acebal, S. Blaya, C. García, A. Fimia, R. Madrigal, and A. Murciano, “Nonparaxial diffraction analysis of Airy and SAiry beams”;, Opt. Express,  25, 22432–22441(2009).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Cessa, H. M. M.

S. Chávez-Cerda, H. M. M. Cessa, and J. R. M. Cessa, “Quantum-like entanglement in classical optics,” Opt. Photon. News 18, 38 (2007).
[Crossref]

Cessa, J. R. M.

S. Chávez-Cerda, H. M. M. Cessa, and J. R. M. Cessa, “Quantum-like entanglement in classical optics,” Opt. Photon. News 18, 38 (2007).
[Crossref]

Chaudhury, S.

S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, “Quantum signs of chaos in a kicked top,” Nature 461, 768–771 (2009).
[Crossref] [PubMed]

Chávez-Cerda, S.

S. Chávez-Cerda, J. R. Moya-Cessa, and H. M. Moya-Cessa, “Quantumlike systems in classical optics: applications of quantum optical methods,” J. Opt. Soc. Am. B 24, 404–407 (2007).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

This figure shows the geometry at the core of a family of the Robnik optical fiber. When λ = 0, the core of the optical fiber takes the standard circular shape. By increasing the deformation parameter λ, different quantum chaotic optical fibers are obtained. A schematic of the optical ray propagating in the z-direction is also shown as well as its projection on the xy plane which contains patterns reminiscent of a quantum chaotic billiard. Note that the z axis is the analog of time in the standard Schrödinger equation.

Fig. 2
Fig. 2

The von Neumann entropy of entanglement Svn of the four lowest eigenfunctions vs the deformation parameter λ of the Robnik fiber is shown in (a), while (b) shows the four lowest state eigenfunctions with λ = 0.499. It can be seen that the von Neumann entropy of entanglement Svn in the eigenmodes saturates as the geometries of these fibers approach the completely chaotic regime. The solid black curve in (a) shows the average von Neumann entropy of entanglement of the ten lowest eigenmodes S ¯ v n for different boundary geometries of the Robnik fiber.

Fig. 3
Fig. 3

A plot of the classical entanglement of the squeezed coherent state versus the deformation parameter λ of the Robnik fiber is shown in (a). We observe that the classical entanglement is higher when the system is in the chaotic regime and the initial squeezing has enhanced the classical entanglement in the system. In (b), we have plotted the classical entanglement dynamics as the light propagates along the z-axis. It is important to note that the light enters the fiber without any classical entanglement. Entanglement increases as it propagates, and as it traverses a distance of 50 units the classical entanglement saturates. Note that the blue and green curves represent the coherent state and the squeezed coherent state respectively. We have chosen the initial state at the point (x, px,y, py) = (0.25,0.1,0.0,0.1).

Fig. 4
Fig. 4

The probability densities |ψ(r,θ)|2 of the entangled coherent states coming out of the Robnik fiber with length z = 50 units are shown in (a), (b) and (c) with λ = 0.15, λ = 0.2 and λ = 0.499 respectively. The probability densities |ψ(r,θ)|2 are shown in the cross-sectional u − v plane of the fiber. The blue color shows the regions where the probability is zero and the red color shows the regions of higher probability. The entangled state is obtained from a tensor product coherent state centered at (x, px,y, py) = (0.25,0.1,0.0,0.1) with λw = 0.01. The entangled state occurs after the light beam has propagated a distance of z = 50 units. Note that we have also indicated the von Neumann entanglement entropy Svn of the polar coordinates below these figures. In (d), (e) and (f) the corresponding classical phase space is shown for λ = 0.15,λ = 0.2 and λ = 0.499 respectively. From the figure it can be observed that the time evolved coherent state has a higher classical entanglement when the corresponding geometry possesses a larger degree of chaotic behavior.

Equations (15)

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i λ w ψ ( x , y , z ) z = λ w 2 2 n 0 ( z ) x y 2 ψ ( x , y , z ) + 1 2 n 0 ( z ) [ n 0 2 ( z ) n 2 ( x , y , z ) ] ψ ( x , y , z ) ,
H = 1 n 0 ( z ) ( p x 2 2 + p y 2 2 ) + U ( x , y , z ) ,
( ( u + B ) 2 + v 2 2 B ( u + B ) ) 2 = A 2 ( ( u + B ) 2 + v 2 ) .
ϕ m , n ( r , θ ) = J m ( j m n r ) e i m θ .
ψ ( r , θ ) = m = 0 N C m J m ( β r ) e i m θ .
ρ 1 ( r , r ) = ψ ( r , θ ) ψ * ( r , θ ) r r d θ .
S v n ( t ) = η i log η i ,
ρ 1 ( r , r ) ϕ i ( r ) d r = η i ϕ i ( r ) ,
u = A cos θ + B cos 2 θ
v = A sin θ + B sin 2 θ .
| α k , ζ k = D ^ ( α k ) S ^ ( ζ k ) | 0 ,
D ^ ( α k ) = exp ( α k a ^ k α k * a ^ k ) ,
S ^ ( ζ k ) = exp ( 1 2 ζ k a ^ k 2 1 2 ζ k * a ^ k 2 ) ,
α k = 1 2 h ¯ ( q k + i p k ) ,
ψ ( x , α k , ζ k ) = ( 1 π h ¯ ) 1 / 4 ( cosh r k + e i θ sinh r k ) 1 / 2 exp { 1 2 h ¯ ( cosh r k e i θ sinh r k cosh r k + e i θ sinh r k ) ( x q 1 ) 2 + i h ¯ p 1 ( x q 1 / 2 ) } .

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