Degree of polarization in quantum optics

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Degree of polarization in quantum optics
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Luis L. Sánchez-Soto, E. C. Yustas Universidad
Complutense. Madrid. Spain Andrei B.
Klimov Universidad de Guadalajara. Jalisco.
Mexico Gunnar Björk, Jonas Söderholm Royal
Institute of Technology. Stockholm. Sweden.
Quantum Optics II. Cozumel 2004
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Outline
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• Classical description of polarization.
• Quantum description of polarization.
• Classical degree of polarization.
• Quantum assessment of the degree of polarization.

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Classical description of polarization
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• Monochromatic plane wave in a linear,
  homogeneous, isotropic medium
• E0 is a complex vector that characterizes the
  state of polarization
• linear-polarization basis (eH, eV)
• circular-polarization basis (e, e-)

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Stokes parameters
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• Stokes parameters
• Operational interpretation

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The Poincaré sphere
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• Coherence vector
• Poincaré sphere

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Transformations on the Poincaré sphere
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• Polarization transformations
• corresponding transformations in the Poincaré
  sphere

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Transformations on the Poincaré sphere
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• Examples
• A differential phase shift induces a rotation
  about Z
• A geometrical rotation of angle q/2 induces a
  rotation about Y of angle q

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Quantum fields
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• One goes to the quantum version by replacing
  classical amplitudes by bosonic operators
• Stokes parameters appear as average values of
  Stokes operators
• s is the polarization (Bloch) vector
• The electric field vector never
• describes a definite ellipse!

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Classical degree of polarization
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• Classical definition of the degree of
  polarization
• Distance from the point to the origin (fully
  unpolarized state)!
• Problems
• It is defined solely in terms of the first moment
  of the Stokes operators.
• There are states with P0 that cannot be regarded
  as unpolarized.
• P does not reflect the lack of perfect
  polarization for any quantum state.
• P1 for SU(2) coherent states (and this includes
  the two-mode vacuum).

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A new proposal of degree of polarization
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A. Luis, Phys. Rev. A 66, 013806 (2002).

• SU(2) coherent states
• associated Q function
• Q function for unpolarized light

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A new proposal of degree of polarization
UCM
A. Luis, Phys. Rev. A 66, 013806 (2002).

• Distance to the unpolarized state
• Definition
• Advantages
• Invariant under polarization transformations.
• The only states with P0 are unpolarized states.
• P depends on the all the moments of the Stokes
  operators.
• Measures the spread of the Q function (i.e.,
  localizability)

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Examples SU(2) coherent states
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• Remarks
• 1 for all N.
• The case N0 is the two-mode
• vacuum with 0.
• In the limit of high intensity
• tend to be fully polarized

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Examples number states
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• Remarks
• For classically they
  would be unpolarized!
• The number states tend to be polarized when their
  intensity increases.

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Examples phase states
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Drawbacks
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• is intrinsically semiclassical.
• The concept of distance is not well defined.
• There is no physical prescription of unpolarized
  light.
• States in the same excitation manifold can have
  quite different polarization degrees.

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Unpolarized light classical vs. quantum
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• Classically, unpolarized light is the origin of
  the Poincaré sphere
• Physical requirements
• Rotational invariance
• Left-right symmetry
• Retardation invariance
• The vacuum is the only pure state that is
  unpolarized!

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Alternative degrees of polarization
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• Idea Distance of the density matrix to the
  unpolarized density matrix
• Hilbert-Schmidt distance
• Advantages
• The quantum definition closest to the classical
  one.
• Invariant under polarization transformations.
• Feasible
• Related to the fidelity respect the fully
  unpolarized state.

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A new degree of polarization (I)
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• Any state
• can be expressed as
• Main hypothesis The depolarized state
  corresponding to Y is

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Properties of the depolarized state
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• The depolarized state depends on the initial
  state.
• The depolarized state in each su(2) invariant
  subspace is random
• The extension to entangled or mixed states is
  trivial.

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Example
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• States
• then

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A new degree of polarization (II)
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• Definition
• Pure states

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Examples
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• For any pure state in the N1 invariant subspace
• Quadrature coherent states in both polarization
  modes

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Conclusions
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• Quantum optics entails polarization states that
  cannot be suitably described by the classical
  formalism based on the Stokes parameters.
• A quantum degree of polarization can be defined
  as the distance between the density operator and
  the density operator representing unpolarized
  light.
• Correlations and the degree of polarization can
  be seen as complementary.