1. Optically polarized atoms
Marcis Auzinsh, University of Latvia
Dmitry Budker, UC Berkeley and LBNL
Simon M. Rochester, UC Berkeley

2. surface for J=2, octupole,
Atomic density matrix
Why the density matrix?
Definition of the density operator
Density matrix elements
Density matrix evolution
Angular-momentum probability
surface for J=2, octupole,
in z-directed E-field

3. Why the density matrix? No such thing as an unpolarized atom
Spin ½ state:
only two free
parameters
Normalized:
+ arbitrary phase
relative phase
relative magnitude

4. Why the density matrix? Expectation value of spin: x component: All
components:

5. Why the density matrix? Spin “points” in the (θ,Ф) direction
An unpolarized sample has no preferred direction
state of atom i
We can't use a wave function to describe the average state of an unpolarized sample

6. Definition of the density matrix
operator:
Average over
all N atoms
Identity
operator
complete
set of basis
states
Trace of an
operator
Basis set can be truncated

7. Density matrix elements
Expansion
coefficients
Diagonal matrix
elements are real:
“population” of state n
Off-diagonal matrix elements average
to zero if atoms are uncorrelated

8. Density matrix elements
Unpolarized sample in state with angular momentum J:
Equal probability to be in any sublevel
No correlation between the atoms
Trace is 1
For J=1:
Total number
of states

9. Density matrix evolution
Schrödinger eq.:
h.c.:
Time derivative
of DM:
“Liouville equation”

10. Density matrix evolution
In practice, there are other terms not described by the (semiclassical) Hamiltonian
Repopulation matrix
Relaxation matrix
These terms describe, e.g., spontaneous decay and atom transit

11. Example: 2-level system, subject to monochromatic light field
Rabi frequency
Transit rate
Natural width

12. Rotating wave approximation
We would like to get rid of the time-dependence at the optical frequency
Use
With unitary transformation
conserves total probability
drop off-resonant
terms

13. Rotations Classical rotations Quantum rotations Visualization
Commutation relations
Quantum rotations
Finding U (R )‏
D – functions‏
Visualization
Irreducible tensors
Polarization moments

14. Classical rotations Rotations use a 3x3 matrix R:
position or other vector
Rotation by
angle θ
about z axis:
For θ=π/2:
For small
angles:
For arbitrary
axis:
Ji are “generators of infinitesimal rotations”

15. Commutation relations
Rotate green around x, blue around y
From picture:
Rotate blue
around x, green around y
For any two axes:
Using
Difference is a rotation around z

16. Quantum rotations Want to find U (R) that corresponds to R
U(R) should be unitary, and should rotate various objects as we expect
E.g., expectation
value of vector
operator:
Remember, for spin ½,
U is a 2x2 matrix
A is a 3-vector of 2x2 matrices
R is a 3x3 matrix

17. Quantum rotations Infinitesimal rotations
Like classical formula, except
i makes J Hermitian
For small θ:
 gives J units of angular momentum
minus sign is conventional
The Ji are the generators of infinitesimal rotations
They are the QM angular momentum operators.
This is the most general definition for J
We can recover arbitrary rotation:

18. Quantum rotations Determining U (R)‏ Start by demanding that U(R)‏
satisfies same commutation relations as R
The commutation relations specify J, and thus U(R)‏
That's it!
E.g., for
spin ½:

19. Quantum rotations Is it right?
We've specified U(R), but does it do what we want?
Want to check
J is an observable, so check
Do easy case: infinitesimal rotation around z
Neglect δ2 term
Same Rz matrix as before

20. D -functions Matrix elements of the rotation operator
Rotations do not change j .
D-function
z-rotations are simple:
so we use Euler angles (z-y-z):

21. Visualization Angular momentum probability surfaces
“probability to measure m=j along quantization axis”
rotate basis set to measure along arbitrary axis:
ρjj(θ,Φ) contains all the information of the DM
Can be expanded in spherical harmonics

22. Irreducible tensors rotation of basis kets:
rotation of spherical harmonics:
these are irreducible tensors:
rank κ, components -κ<q<κ
for irreducible tensor operators:
generalizes
reduced matrix element
Wigner-Eckart theorem:

23. Polarization operators
define irreducible tensor operators with
reduced matrix element
W-E theorem:
# of operators:
complete basis
expand DM:
The PM's are physically significant and have useful symmetries

24. Visualizing polarization operators
Calculate ρjj(θ,Φ) for polarization operator:
(rotation of irr. tensor)‏
(matrix elem. of pol. op.)‏
Each polarization moment corresponds to a spherical harmonic