Elementary understanding on Hanle effect No'1 atomic polarization

1
Elementary understanding on Hanle effect No.1
atomic polarization

• Rev. -2 6 March 2009
• Saku Tsuneta
• (NAOJ)

2
Table of contents

• Atomic polarization (this handout)
• Hanle effect (atomic polarization with B)
• Van Vleck effect
• More formal treatment
• Density matrix approach

3
Hanle effect in general

• Hanle effect or atomic polarization should be
  understood with quantum mechanics. I do not find
  any merit to rely on the classical picture.
• It is a beautiful application of very fundamental
  concept of quantum mechanics such as quantum
  state and angular momentum, scattering, and
  conceptually should not be a difficult topic.
• This is an attempt to decode the following
  excellent text
• Javier Trujillo Bueno, Atomic Polarization and
  the Hanle Effect, AIP conference series volume
  236, 161-195, 2001.
• Please point out any incorrect description for
  better understanding!
• The outstanding textbooks for basic quantum
  mechanics are
• R. P. Fynman, Lectures on physics Quantum
  Mechanics
• J. J. Sakurai, Modern Quantum Mechanics

4
Atomic polarization is merely conservation of
angular momentum Example 1 1-0 system
J0, m0gt1,0gt
Take quantanization axis to be Direction of
Incident photons
1/2
1/2
J1, m-1gt-1gt
J1, m1gt1gt
J1, m0gt0gt
1gt, 0gt,-1gt B0 degenerated state
Unpolarized light from a star
Rgt
Lgt
A right-circularized photon carrying angular
momentum -1 Lgt causes transition to m1 state
of atom (1gt to 0gt). A left-circularized photon
carrying angular momentum 1 Rgt causes transition
to m-1 state of atom (-1gt to 0gt).
5
Atomic polarization is merely conservation of
angular momentumExample 2 0-1 system
J1, m-1gt-1gt
J1, m1gt1gt
J1, m0gt0gt
1gt, 0gt,-1gt B0 degenerated state
1/2
1/2
Take quantanization axis to be Direction of
Incident photons
J0, m0gt0,0gt
Unpolarized light from a star
Lgt
Rgt
A right-circularized photon carrying angular
momentum 1 Rgt causes transition to m1 state
of atom (0gt to 1gt). A left-circularized photon
carrying angular momentum -1 Lgt causes
transition to m-1 state of atom (0gt to -1gt).
6
If unpolarized light comes in from horizontal
direction,
Take quantanization axis to be direction of
Incident photons
J1, m-1gt-1gt
Lgt
Exactly the same atomic polarization take place
but in the different set of quantum base states
-1gt, 0gt,1gt Note that 1gt and 1gt are
different quantum states. For instance 1gt is
represented by linear superposition of 1gt,
0gt and -1gt.
J1, m0gt0gt
Un-polarized light from a side
J0, m0gt0,0gt
Rgt
J1, m1gt1gt
7
What is the relation betweenJmgt and Jmgt base
states?
Rotation matrix for spin 1 (any text book in
quantum mechanics)
Normal to stellar surface
1gt,0gt,-1gt
1gt,0gt,1gt
?
If ? is 90 degree, 1gt (1cos?)/21gt
(1-cos?)/2-1gt 1/2 (1gt-1gt )
lt11gt1/4 0gt -sin?/v2 1gt sin?/v2
-1gt 1/v2 (-1gt -1gt) ) lt00gt1/2
-1gt (1-cos?)/21gt (1cos?)/2-1gt 1/2
(1gt-1gt ) ) lt-1-1gt1/4 If ? is 90 degree,
1gt 1/ v2 lt11gt1/2 0gt 0
lt00gt0 -1gt 1/ v2 lt-1-1gt1/2 Thus,
illumination from side provides different atomic
polarization!
8
This mean that
Quantanization axis
Quantanization axis
9
Uniform radiation case
1gt (1cos?)/21gt (1-cos?)/2-1gt 0gt
-sin?/v2 1gt sin?/v2 -1gt -1gt
(1-cos?)/21gt (1cos?)/2-1gt sum over
0lt?lt p (dO2psin?/4 p) lt11gt ? (1cos?)²/8
(1-cos?)²/8 dO 1/3 lt00gt ? sin²?/4
sin²?/4 dO 1/3 lt-1-1gt ? (1-cos?)²/8
(1cos?)²/8 dO1/3 Thus, uniform irradiation
results in no atomic polarization!
10
He 10830

• Blue 10829.09A
• J(low)1 J(up)0
• Red1 10830.25A
• J(low)1 J(up)1
• Red2 10830.34A
• J(low)1 J(up)2

11
He10830 red wing
Dark filament No Stokes-V No Stokes-LP (LP
exists with horizontal B) Hanle effect! Can not
exist without B due to symmetry
J1, m-1gt-1gt
J1, m1gt1gt
J1, m0gt0gt
Incoherent states
1/3
1/2
1/2
1/3
1/3
Prominence No Stokes Stokes LP even with zero B
J0, m0gt0,0gt
Lgt
Rgt
Unpolarised light from a star
12
To understand LP from prominence with zero
horizontal B,

• 1gt state is created by absorption of an Lgt
  photon from below (photosphere).
• Consider the case of 90 degree scattering, we
  rotate the quantization axis normal to
  photosphere by 90 degree i.e. parallel to
  photosphere.
• With 1,1gt to 0,0gt transition, a photon with
  state ½Rgt ½ Lgt is emitted (90 degree
  scattering).
• This is a linearly polarized photon with state
  xgt 1/v2 (Rgt Lgt) !
• Likewise, for -1gt state, -xgt -1/v2 (Rgt
  Lgt)

13
He10830 blue wings
Dark filament No Stokes-V No Stokes-LP (LP
exists with horizontal B) Hanle effect! Can not
exist without B due to symmetry
J0, m0gt1,0gt
1/2
1/2
1/3
1/3
1/3
J1, m-1gt-1gt
J1, m1gt1gt
J1, m0gt0gt
Prominence No Stokes-V No Stokes-LP (even with B)
Rgt
Lgt
14
He 10830 with horizontal B

• prominence filament
• Blue 10829.09A no LP LP
• J(low)1 J(up)0
• Red1 10830.25A LP LP
• J(low)1 J(up)1
• Red2 10830.34A LP LP
• J(low)1 J(up)2

15
If with magnetic field, the story becomes
different, namely

• to be continued to No.2 memo