Download Atomic and Molecular Physics for Physicists Ben-Gurion University of the Negev

Ben-Gurion University of the Negev
Atomic
Atomic and
and Molecular
Molecular Physics
Physics
for
for Physicists
Physicists
Ron Folman
Chapter 8: Zeeman effect and Stark effect. (magnetic, electric and
optical atom trapping) .
But before that, we learn what an atom data sheet is. As an
example we use the data sheet of Rb87 and from now on this will
be our example atom.
Main Reference:
Exercises:
Ran Salem.
www.bgu.ac.il/atomchip
www.bgu.ac.il/nanofabrication
www.bgu.ac.il/nanocenter
Hyperfine splitting: spin-spin interaction
F=3
F=2
5P3/2
F=1
F=0
780nm
-2
-1
0
1
2
5S1/2
-1
0
1
F=2
F=1
How a modern atom data sheet looks like:
Homework:
Prove that the
2-photon transition
From F=1 mF=-1
To F=2 mF=+1
Cannot work in the D2
Transition for large
Detunings.
Analysis of the problem
•
There are two possible Raman transitions (see on the right):
A → C → B and A → D → B
• The two photon Rabi frequency can be expressed by the dipole
matrix elements:
ΩR =
•
A |d ⋅ E | C C d ⋅ E B
2∆ AC
+
A |d ⋅ E | D D d ⋅ E B
2∆ AD
As ∆ AC ≅ ∆ AD ≡ ∆ (both are ∼ 10 GHz, mach larger then the hyperfine
split of the excited state ~150MHz), and using the Clebsch-Gordon
coefficients, we get:
1
1
3  5
1
1 1
ΩR =
J = er J ′ = 
⋅
−
=0
2∆
2
2  24 40
24 8 
Meaning the amplitude for the process is zero!
•
Applying circular polarization on
0, 1
leads to:
σ + | S,1,−1 = α 0 P,0,0 + α1 P,1,0 + α 2 P,2,0 + α 3 P,3,0
σ − | S,2,1 = β 0 P,0,0 + β1 P,1,0 + β 2 P,2,0 + β 3 P,3,0
• These states are orthogonal, thus a Raman transition is not possible
unless there is enough time:
As these state are not eigen states of the Hamiltonian they vary in time at
ω = ∆E ≈ 200MHz
However, typical Raman detunings are much larger, (>10GHz), so the
Rabi frequency will diminish.
Another way to look at it:
|F,MF>=∑|IZ,JZ>=∑ |IZ,SZ>
F=3
F=2
5P3/2
F=1
F=0
|D>
|C>
∆ AD
∆ AC
+
σ
σ−
780nm
-2
-1
0
1
2
|B>
5S1/2
|A>
-1
0
1
F=2
F=1
Zeeman
Splitting
Homework:
Why is 3.2G a magical
Number for the |1,-1>
to |2,1> transition?
Food for thought:
why are the m_f state signs reversed between the F=1 and F=2 levels?
Stark shift:
E=-1/2 α E2 only strong field seekers
There are also magical numbers for stark (light) shifts:
Zeeman Splitting in more detail:
The total magnetic moment of the atom is:
µ= [e/(2m) lz + e/(m) sz + e/(2M) iz] =
[eħ/(2m) ml + eħ/(m) ms + eħ/(2M) mi]=
µB [ml + 2ms] + µN [mi]
Stark shift in more detail:
In the first lesson we learned that:
where:
χe is the electrical susceptibility of the material,
χm is the magnetic susceptibility of the material,
ε is the electrical permittivity of the material, and
µ is the magnetic permeability of the material
The electric susceptibility χe of a dielectric material is a measure of how easily it
polarizes in response to an electric field. This, in turn, determines the electric permittivity
of the material and thus influences many other phenomena in that medium, from the
capacitance of capacitors to the speed of light.
χe = P / ε0E = Re(χe)+Im(χe)
Applying force and trapping with the 2 interactions: magnetic and optical
Optical trapping
Good
Bad
Think: why do people use a CO2 laser for optical traps of alkali atoms?
Rudolf Grimm and Matthias Weidemuller 1999
Blue detuned light can also become an atom mirror!
Electric trapping?
Re(χe) far away from the resonance is called
Polarizability (=α) and has an interaction
V= -1/2 α |E|2
Typical values are (Hal Metcalf): 100kHz x |E|2 where E is given in kV/cm.
Lets consider the case of an atom in a static electric field:
Note: The optical resonance wave length will only change in a static electric
field if there is a differential stark shift.
Think: Can atoms be trapped by a static electric field alone?
Note that an electric field cannot have a maximum in the middle of empty space.
How do ion traps solve this problem?
magnetic trapping:
In the hyperfine representation, the magnetic interaction is:
V=-µ*B
Here also you need a maximum or minimum of the magnetic field in order
to have a trapping potential, but the Ernshaw theorem forbids this!
In what conditions is the potential equal to:
V = gF mF µB h |B| where µB = 1.4 MHz/G.
Those trapped in the minimum are called weak field seekers.
Homework:
1. A macroscopic magnet can never be a weak field seeker (why?).
How does the levitron game turn it into a weak field seeker?
2. Calculate the potential (in K units) above a straight current carrying wire
(with 1A current) positioned in an homogeneous magnetic field of 10Gauss.
Atoms trapped in a static magnetic field above a current carrying wire on an AtomChip
The AtomChip laboratory is just next door in building 95!
Majorana spin flips due to adiabaticity criteria
Spin flips due to noise in transverse
Decoherence due to noise in parallel direction.