UCM 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 UCM Outline • Classical description of polarization. • Quantum description of polarization. • Classical degree of polarization. • Quantum assessment of the degree of polarization. UCM Classical description of polarization • Monochromatic plane wave in a linear, homogeneous, isotropic medium E( z , t ) E0 exp[i ( t kz )] E0 is a complex vector that characterizes the state of polarization E0 aH e H aV eV linear-polarization basis: (eH, eV) circular-polarization basis: (e+, e-) 1 e e H i eV 2 UCM Stokes parameters • Stokes parameters S1 aH aV aH aV S 2 i (aH aV aH aV ) S3 aH aV 2 2 S0 aH aV 2 2 • Operational interpretation S1 I 45 I 45 S2 I I S 3 I H IV S 0 I H IV UCM The Poincaré sphere • Coherence vector S1 S2 S3 s1 , s2 , s3 S0 S0 S0 • Poincaré sphere s12 s2 s3 1 2 2 UCM Transformations on the Poincaré sphere • Polarization transformations aH aH U U SU(2) aV aV corresponding transformations in the Poincaré sphere s1 s1 s2 R (U) s2 R (U) SO(3) s s 3 3 UCM Transformations on the Poincaré sphere • Examples i H cos(V H ) sin(V H ) 0 e 0 sin(V H ) cos(V H ) 0 0 eiV 0 0 1 A differential phase shift induces a rotation about Z cos 0 sin cos( / 2) sin( / 2) 0 1 0 sin( / 2) cos( / 2) sin 0 cos A geometrical rotation of angle q/2 induces a rotation about Y of angle q UCM Quantum fields • One goes to the quantum version by replacing classical amplitudes by bosonic operators [ aH , aH ] I , ˆ† ˆ [aV , aV ] I , ˆ† ˆ [aH , aV ] 0 ˆ ˆ • Stokes parameters appear as average values of Stokes operators 1 ˆ I s σ s Tr ( σ) / 2 ˆ 2 s is the polarization (Bloch) vector ˆ Δsx Δs y Δsz2 2 S0 ˆ2 ˆ2 ˆ The electric field vector never describes a definite ellipse! UCM Classical degree of polarization • Classical definition of the degree of polarization S x2 S y S z2 2 P | s | S0 • 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). UCM A new proposal of degree of polarization A. Luis, Phys. Rev. A 66, 013806 (2002). • SU(2) coherent states 1/ 2 N N N ,q , sin N m q 2 cos m q 2 eim m H N m V m 0 m associated Q function N 1 Q q , N ,q , N ,q , ˆ N 0 4 • Q function for unpolarized light 1 Qunpol q , 4 UCM A new proposal of degree of polarization A. Luis, Phys. Rev. A 66, 013806 (2002). Distance to the unpolarized state 2 1 D 4 d Q(q , ) S 2 4 Definition D 1 P= 1 D 1 4 d [Q(q , )]2 S2 • 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) UCM Examples: SU(2) coherent states N 1 N ,q 0, N 0 Q(q , ) [cos(q / 2)]2 N H V 4 2 N P= N 1 Remarks: P= for all N. =1 The case N=0 is the two-mode vacuum with P= 0. = In the limit of high intensity tend to be fully polarized UCM Examples: number states N 1 N m H N m V Q(q , ) [sin(q / 2)] 2( N m ) [cos(q / 2)]2m 4 m 2N 2 N 1 2m P= 1 1 N 1 N 2 2 N 1 N m Remarks: For m H m V classically they would be unpolarized! The number states tend to be polarized when their intensity increases. UCM Examples: phase states N 1 N , 0 eim m N 1 m H N m V N 2, 0 N 2 1 1 Q(q , ) cos sin q sin cos q 2 4 m 2 11 P= 26 UCM Drawbacks • P= intrinsically semiclassical. is • 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. UCM Unpolarized light: classical vs. quantum • Classically, unpolarized light is the origin of the Poincaré sphere: sx s y s z 0 • Physical requirements: Rotational invariance Left-right symmetry Retardation invariance rN I N ˆ N 0 ( N 1) r N 0 N 1 The vacuum is the only pure state that is unpolarized! UCM Alternative degrees of polarization • Idea: Distance of the density matrix to the unpolarized density matrix 1 D( , unpol ) ˆ ˆ unpol ˆ ˆ 2 HS Hilbert-Schmidt distance ˆ ˆ ˆ ˆ A B Tr( A† B) • Advantages The quantum definition closest to the classical one. Invariant under polarization transformations. Feasible Related to the fidelity respect the fully unpolarized state. UCM A new degree of polarization (I) Any state Y nH , nV H V cnH cnV nH H nV V can be expressed as N Y cN m cm N m H V H m V N 0 m0 Main hypothesis: The depolarized state corresponding to Y is fN N unpol ˆY 0 N m m m N m N 0 N 1 m H V V H N fN | c H V 2 c | N m m f N 1 m 0 N 0 UCM Properties of the depolarized state • The depolarized state depends on the initial state. • The depolarized state in each su(2) invariant subspace is random 1 N ˆ N unpol 0 N m N 1 m H m V V m H N m Tr( unpol ) 1 ˆN • The extension to entangled or mixed states is trivial. UCM Example • States cos 0 sin 1 H H V cos 0 V sin 1 V then unpol cos 2 cos 2 0, 0 0, 0 ˆY 1 (cos 2 sin 2 sin 2 cos 2 )( 1, 0 1, 0 1,1 1,1 ) 2 1 2 sin sin 2 ( 2, 0 2, 0 2,1 2,1 2, 2 2, 2 ) 3 UCM A new degree of polarization (II) Definition: PQ D( unpol ) ˆ ˆ • Pure states f N2 PQ 1 N 0 N 1 UCM Examples • For any pure state in the N+1 invariant subspace N PQ N 1 • Quadrature coherent states in both polarization modes I1 (2 N ) 2 N 1 PQ 1 e 1 2 N 3/ 2 N 1 N UCM 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.