Transcript Optically polarized atoms

Optically polarized atoms
Marcis Auzinsh, University of Latvia
Dmitry Budker, UC Berkeley and LBNL
Simon M. Rochester, UC Berkeley
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Atomic density matrix
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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
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Why the density matrix?
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No such thing as an unpolarized atom
Spin ½ state:
Normalized:
+ arbitrary phase
relative phase
only two free
parameters
relative magnitude
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Why the density matrix?
Expectation
value of spin:
x component:
All
components:
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Why the density matrix?
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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
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Definition of the density matrix
Density
operator:
Average over
all N atoms
Identity
operator
complete
set of basis
states
Trace of an
operator
Basis set can be truncated
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Density matrix elements
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
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Density matrix elements
Unpolarized sample in state with angular momentum J:
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Equal probability to be in any sublevel
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No correlation between the atoms
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Trace is 1
For J=1:
Total number
of states
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Density matrix evolution
Schrödinger eq.:
h.c.:
Time derivative
of DM:
“Liouville equation”
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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
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Example:
2-level system, subject to
monochromatic light field
Rabi frequency
Transit rate
Natural width
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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
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Rotations
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Classical rotations
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Commutation relations
Quantum rotations
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Finding U (R )
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D – functions
Visualization
Irreducible tensors
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Polarization moments
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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”
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Commutation
relations
Rotate green
around x, blue
around y
From picture:
For any two axes:
Using
Rotate blue
around x,
green around y
Difference is
a rotation
around z
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Quantum rotations
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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
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Quantum rotations
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Infinitesimal rotations
Like classical
formula, except
i makes J Hermitian
For small θ:
minus sign is
conventional
 gives J units of
angular momentum
 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:
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Quantum rotations
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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:
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Quantum rotations
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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
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D -functions
Matrix elements of the rotation operator
Rotations do
not change j .
D-function
z-rotations are
simple:
so we use
Euler angles (zy-z):
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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
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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
Wigner-Eckart
theorem:
reduced matrix
element
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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
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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
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