Degree of polarization in quantum optics UCMUCM 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

Outline Classical description of polarization. Quantum description of polarization. Classical degree of polarization. Quantum assessment of the degree of polarization. UCMUCM

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 - ) UCMUCM

Stokes parameters Operational interpretation UCMUCM

The Poincaré sphere Coherence vector Poincaré sphere UCMUCM

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

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

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! UCMUCM

Classical degree of polarization 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 P=0 that cannot be regarded as unpolarized. P does not reflect the lack of perfect polarization for any quantum state. P=1 for SU(2) coherent states (and this includes the two-mode vacuum). UCMUCM

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

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

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

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

Examples: phase states UCMUCM

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. UCMUCM

Unpolarized light: classical vs. quantum 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! UCMUCM

Alternative degrees of polarization 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. UCMUCM

A new degree of polarization (I) Any state can be expressed as Main hypothesis: The depolarized state corresponding to  is UCMUCM

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. UCMUCM

Example States then UCMUCM

A new degree of polarization (II) Definition: Pure states UCMUCM

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

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. UCMUCM