FM3F.4.pdf

Frontiers in Optics 2013/Laser Science XXIX © OSA 2013

Superdense Coding with Vector Vortex Beams: A Classical Analogy of Entanglement Giovanni Milione1,2,7,*, T. A. Nguyen1,7, D. A. Nolan2,7, E. Karimi4, S. Slussarenko5, L. Marrucci5,6,7, and R. R. Alfano1,2,7

1 Institute for Ultrafast Spectroscopy and Lasers, Physics Department City College of the City University of New York, 160 Convent Ave., New York, NY 11010, USA 2 Graduate Center of the City University of New York, 365 Fifth Ave., New York, NY 10016, USA 3 Corning Incorporated, SullivanPark, Corning, NY 14831, USA 4 Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada 5 Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy 6 CNR-SPIN Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy 7 New York State Center for Complex Light, 160 Convent Ave., New York, NY 11010 USA [email protected]

Abstract: Superdense coding using a classical analogy of quantum entanglement is experimentally investigated. The spatially inhomogeneous polarization of vector vortex beams, as described by a higher-order Poincare sphere, play the role of maximally entangled Bell states. OCIS codes: (080.4865) Optical Vortices; (270.5565) Quantum Communications; (260.5430) Polarization

Introduction Quantum entanglement can be considered the foundation of quantum information science [1]. Entanglement is the “spooky action at a distance” that can be exhibited by two spatially separated particles; the measurement of one particle defines the other. A classical analogy of quantum entanglement can be made, referred to as classical entanglement, where a physical entity has the same mathematical relationship (non-factorizability) as two quantum entangled particles [2]. While there is no “spooky action at a distance” (non-locality) in classical entanglement, the limits and utility of the analogy in classical optics have recently been explored [3]. This includes the relationship of an equivalent Bell’s inequality to a light beam’s degree of coherence [4], and the classical entanglement between light’s spin and orbital angular momentum degrees of freedom [5]. In this presentation, a new limit and utility of the classical analogy of quantum entanglement is explored. A method of quantum communication referred to as superdense coding using classical entanglement is experimentally investigated where the spatially inhomogeneous states of polarization of vector vortex beams, as described by a higher-order Poincare sphere, play the role of maximally entangled Bell states. Superdense Coding with Vector Vortex Beams The spatially inhomogeneous states of polarization of vector vortex beams also referred to as cylindrical vector beams, such as radial and azimuthal polarization, have received recent interest because of their unique rotational symmetry [6]. A vector vortex beam, as represented by a higher-order Poincare sphere [6], can be expressed as the linear combination of right and left circular polarized optical vortices of opposite topological charge (Fig. 1b):

ψ = ψ R R ⊗ + + ψ L L ⊗ −

.

(1)

ˆ / 2 are right and left circular polarization, respectively, being Eigen states of light’s spin R , L = ( xˆ + σ iy) angular momentum

σ (σ = ±1) . ± = exp(±iφ ) are optical vortices, being Eigen states of light’s orbital ( = 0, ±1, ±2,...) . As can be seen from Eq. 1 a vector vortex beam is mathematically

angular momentum similar to a Bell state describing the quantum entanglement of two spatially separated particles [1]. In Eq. 1, referred to as an optical spin-orbit Bell states, the spin and orbital angular momentum degrees of freedom play the role of the two spatially separated particles. The method of quantum communication referred to as superdense coding relies on quantum entanglement. Apply the identity

Iˆ and three Pauli operators σˆ i (i = x, y, z) to one particle (one bit) that

is in a maximally entangled Bell state; it can be transformed into each of the four orthogonal Bell states

Φ+ , Φ− , Ψ + , Ψ − comprising two bits of information that can be communicated. This ability to

FM3F.4.pdf

Frontiers in Optics 2013/Laser Science XXIX © OSA 2013

communicate two bits of information by transforming one bit is the superdense part of superdense coding [1]. These four transformations on the optical spin-orbit Bell state of Eq. 1 (the vector vortex beam) are experimentally investigated. A vector vortex beams is generated using a tunable liquid crystal q-plate [7]. These four transformations are carried out with linear optical elements - a quarter wave plate (QWP)/ half wave plate (HWP) /quarter wave plate (QWP) arranged in series and rotated at an appropriate angle (Fig. 1(c)). If we begin with a state

ψ = Φ+

the four transformations, and the four resulting optical spin-orbit Bell states, are given by [5]:

ˆ σˆ Φ+ → Φ± = ( R ⊗ + ± L ⊗ − ) / 2 I, x

(2)

σˆ y , σˆ z Φ+ → Ψ ± = ( R ⊗ − ± L ⊗ + ) / 2

Each transformation and each optical spin-orbit Bell state of Eq. 2 corresponds to a rotation and/or “jump” between higher-order Poincare spheres and a different/orthogonal vector vortex beam, respectively (Fig. 1(a) and Fig. 1(b)).

σˆ y

σˆ x

Iˆ

σˆ z

(a)

(a) (b)

(c) +

R

R ⊗ +

R ⊗ −

−

L

R ⊗ +

R ⊗ −

q=1

QWP

HWP

QWP

(b) Figure 1: (a) Experimentally measured polarization of optical spin-orbit Bell states (vector vortex beams) (b) Transformations of Eq. 2 as represented by a higher-order Poincare sphere. (c) q-plate and wave plates to carry out transformations of Eq. 2

Conclusion Superdense coding using classical entanglement is experimentally investigated where vector vortex beams play the role of maximally entangled Bell states. In contrast to quantum entanglement, here there is no “spooky action at a distance” and therefore no quantum security [1]. Yet, the 4-dimensional state space of light’s spin and orbital angular momentum spanned by the optical spin-orbit Bell states can be accessed by only transforming the polarization degree of freedom. This is a result of the mathematical similarity of classical entanglement to quantum entanglement and may find applications in classical communication, for example, as a “vector beam shift keying” to encode four bits of information on one vector vortex beam by simply manipulating its polarization with conventional optical elements. References [1] S. Barnett. Quantum information. (Oxford, 2009). [2] R. J. C. Spreeuw, "A classical analogy of entanglement." Foundations of physics 28(3), 361-374 (1998). [3] B. N. Simon, et al. "Nonquantum entanglement resolves a basic issue in polarization optics." Physical review letters 104(2), 023901 (2010); Q. Xiao-Feng, and J. H. Eberly. "Entanglement and classical polarization states." Optics letters 36(20), 4110-4112 (2011). [4] K. Kagalwala, et al. "Bell's measure in classical optical coherence." Nature Photonics 7(1), 72-78 (2012). [5] E. Karimi, et al. "Spin-orbit hybrid entanglement of photons and quantum contextuality." Physical Review A 82(2), 022115 (2010). [6] Giovanni Milione, et al. "Higher order Pancharatnam-Berry phase and the angular momentum of light." Physical Review Letters 108(19), 190401 (2012); Giovanni Milione, et al. "Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light." Physical Review Letters 107(5), 053601 (2011). [7] Slussarenko, Sergei, et al. "Tunable liquid crystal q-plates with arbitrary topological charge." Opt. Express 19(5), 4085-4090 (2011); F. Cardano, et al. "Polarization pattern of vector vortex beams generated by q-plates with different topological charges." Applied Optics 51(10), C1C6 (2012); F. Cardano, et al. “Generation and dynamics of optical beams with polarization singularities,” Optics Express, 21(7), 8815-8820 (2013)

Frontiers in Optics 2013/Laser Science XXIX © OSA 2013

Superdense Coding with Vector Vortex Beams: A Classical Analogy of Entanglement Giovanni Milione1,2,7,*, T. A. Nguyen1,7, D. A. Nolan2,7, E. Karimi4, S. Slussarenko5, L. Marrucci5,6,7, and R. R. Alfano1,2,7

1 Institute for Ultrafast Spectroscopy and Lasers, Physics Department City College of the City University of New York, 160 Convent Ave., New York, NY 11010, USA 2 Graduate Center of the City University of New York, 365 Fifth Ave., New York, NY 10016, USA 3 Corning Incorporated, SullivanPark, Corning, NY 14831, USA 4 Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5 Canada 5 Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II,” Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy 6 CNR-SPIN Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy 7 New York State Center for Complex Light, 160 Convent Ave., New York, NY 11010 USA [email protected]

Abstract: Superdense coding using a classical analogy of quantum entanglement is experimentally investigated. The spatially inhomogeneous polarization of vector vortex beams, as described by a higher-order Poincare sphere, play the role of maximally entangled Bell states. OCIS codes: (080.4865) Optical Vortices; (270.5565) Quantum Communications; (260.5430) Polarization

Introduction Quantum entanglement can be considered the foundation of quantum information science [1]. Entanglement is the “spooky action at a distance” that can be exhibited by two spatially separated particles; the measurement of one particle defines the other. A classical analogy of quantum entanglement can be made, referred to as classical entanglement, where a physical entity has the same mathematical relationship (non-factorizability) as two quantum entangled particles [2]. While there is no “spooky action at a distance” (non-locality) in classical entanglement, the limits and utility of the analogy in classical optics have recently been explored [3]. This includes the relationship of an equivalent Bell’s inequality to a light beam’s degree of coherence [4], and the classical entanglement between light’s spin and orbital angular momentum degrees of freedom [5]. In this presentation, a new limit and utility of the classical analogy of quantum entanglement is explored. A method of quantum communication referred to as superdense coding using classical entanglement is experimentally investigated where the spatially inhomogeneous states of polarization of vector vortex beams, as described by a higher-order Poincare sphere, play the role of maximally entangled Bell states. Superdense Coding with Vector Vortex Beams The spatially inhomogeneous states of polarization of vector vortex beams also referred to as cylindrical vector beams, such as radial and azimuthal polarization, have received recent interest because of their unique rotational symmetry [6]. A vector vortex beam, as represented by a higher-order Poincare sphere [6], can be expressed as the linear combination of right and left circular polarized optical vortices of opposite topological charge (Fig. 1b):

ψ = ψ R R ⊗ + + ψ L L ⊗ −

.

(1)

ˆ / 2 are right and left circular polarization, respectively, being Eigen states of light’s spin R , L = ( xˆ + σ iy) angular momentum

σ (σ = ±1) . ± = exp(±iφ ) are optical vortices, being Eigen states of light’s orbital ( = 0, ±1, ±2,...) . As can be seen from Eq. 1 a vector vortex beam is mathematically

angular momentum similar to a Bell state describing the quantum entanglement of two spatially separated particles [1]. In Eq. 1, referred to as an optical spin-orbit Bell states, the spin and orbital angular momentum degrees of freedom play the role of the two spatially separated particles. The method of quantum communication referred to as superdense coding relies on quantum entanglement. Apply the identity

Iˆ and three Pauli operators σˆ i (i = x, y, z) to one particle (one bit) that

is in a maximally entangled Bell state; it can be transformed into each of the four orthogonal Bell states

Φ+ , Φ− , Ψ + , Ψ − comprising two bits of information that can be communicated. This ability to

FM3F.4.pdf

Frontiers in Optics 2013/Laser Science XXIX © OSA 2013

communicate two bits of information by transforming one bit is the superdense part of superdense coding [1]. These four transformations on the optical spin-orbit Bell state of Eq. 1 (the vector vortex beam) are experimentally investigated. A vector vortex beams is generated using a tunable liquid crystal q-plate [7]. These four transformations are carried out with linear optical elements - a quarter wave plate (QWP)/ half wave plate (HWP) /quarter wave plate (QWP) arranged in series and rotated at an appropriate angle (Fig. 1(c)). If we begin with a state

ψ = Φ+

the four transformations, and the four resulting optical spin-orbit Bell states, are given by [5]:

ˆ σˆ Φ+ → Φ± = ( R ⊗ + ± L ⊗ − ) / 2 I, x

(2)

σˆ y , σˆ z Φ+ → Ψ ± = ( R ⊗ − ± L ⊗ + ) / 2

Each transformation and each optical spin-orbit Bell state of Eq. 2 corresponds to a rotation and/or “jump” between higher-order Poincare spheres and a different/orthogonal vector vortex beam, respectively (Fig. 1(a) and Fig. 1(b)).

σˆ y

σˆ x

Iˆ

σˆ z

(a)

(a) (b)

(c) +

R

R ⊗ +

R ⊗ −

−

L

R ⊗ +

R ⊗ −

q=1

QWP

HWP

QWP

(b) Figure 1: (a) Experimentally measured polarization of optical spin-orbit Bell states (vector vortex beams) (b) Transformations of Eq. 2 as represented by a higher-order Poincare sphere. (c) q-plate and wave plates to carry out transformations of Eq. 2

Conclusion Superdense coding using classical entanglement is experimentally investigated where vector vortex beams play the role of maximally entangled Bell states. In contrast to quantum entanglement, here there is no “spooky action at a distance” and therefore no quantum security [1]. Yet, the 4-dimensional state space of light’s spin and orbital angular momentum spanned by the optical spin-orbit Bell states can be accessed by only transforming the polarization degree of freedom. This is a result of the mathematical similarity of classical entanglement to quantum entanglement and may find applications in classical communication, for example, as a “vector beam shift keying” to encode four bits of information on one vector vortex beam by simply manipulating its polarization with conventional optical elements. References [1] S. Barnett. Quantum information. (Oxford, 2009). [2] R. J. C. Spreeuw, "A classical analogy of entanglement." Foundations of physics 28(3), 361-374 (1998). [3] B. N. Simon, et al. "Nonquantum entanglement resolves a basic issue in polarization optics." Physical review letters 104(2), 023901 (2010); Q. Xiao-Feng, and J. H. Eberly. "Entanglement and classical polarization states." Optics letters 36(20), 4110-4112 (2011). [4] K. Kagalwala, et al. "Bell's measure in classical optical coherence." Nature Photonics 7(1), 72-78 (2012). [5] E. Karimi, et al. "Spin-orbit hybrid entanglement of photons and quantum contextuality." Physical Review A 82(2), 022115 (2010). [6] Giovanni Milione, et al. "Higher order Pancharatnam-Berry phase and the angular momentum of light." Physical Review Letters 108(19), 190401 (2012); Giovanni Milione, et al. "Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light." Physical Review Letters 107(5), 053601 (2011). [7] Slussarenko, Sergei, et al. "Tunable liquid crystal q-plates with arbitrary topological charge." Opt. Express 19(5), 4085-4090 (2011); F. Cardano, et al. "Polarization pattern of vector vortex beams generated by q-plates with different topological charges." Applied Optics 51(10), C1C6 (2012); F. Cardano, et al. “Generation and dynamics of optical beams with polarization singularities,” Optics Express, 21(7), 8815-8820 (2013)