UC M Degree of polarization in quantum optics

UC M Degree of polarization in quantum optics 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

UC M Outline • • Classical description of polarization. Quantum description of polarization. Classical degree of polarization. Quantum assessment of the degree of polarization.

UC M Classical description of polarization • Monochromatic plane wave in a linear, homogeneous, isotropic medium E 0 is a complex vector that characterizes the state of polarization linear-polarization basis: (e. H, e. V) circular-polarization basis: (e+, e-)

UC M Stokes parameters • Operational interpretation

UC M The Poincaré sphere • Coherence vector • Poincaré sphere

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

UC M Transformations on the Poincaré sphere • 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

UC M Quantum fields • 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!

UC M Classical degree of polarization • Classical definition of the degree of polarization • Distance from the point to the origin (fully unpolarized state)! • Problems ü It is defined solely in terms of the first moment of the Stokes operators. ü There are states with P=0 that cannot be regarded as unpolarized. ü P does not reflect the lack of perfect polarization for any quantum state. ü P=1 for SU(2) coherent states (and this includes the two-mode vacuum).

UC M A new proposal of degree of polarization A. Luis, Phys. Rev. A 66, 013806 (2002). • SU(2) coherent states associated Q function • Q function for unpolarized light

UC M A new proposal of degree of polarization A. Luis, Phys. Rev. A 66, 013806 (2002). Distance to the unpolarized state Definition • Advantages ü Invariant under polarization transformations. ü The only states with P=0 are unpolarized states. ü P depends on the all the moments of the Stokes operators. ü Measures the spread of the Q function (i. e. , localizability)

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

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

UC M Examples: phase states

UC M Drawbacks • 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.

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

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

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

UC M Properties of the depolarized state • 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.

UC M Example • States then

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

UC M Examples • For any pure state in the N+1 invariant subspace • Quadrature coherent states in both polarization modes

UC M Conclusions • 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.