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					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
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                                      Outline




      •   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


      • 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
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                             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
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                             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       eiV    
                                     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
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                                    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!
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                     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  eim 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)
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                  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
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                                  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!
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                  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 m0
      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.

				
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posted:2/17/2011
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