View source for Magneto-Optical Traps

The Magneto-Optical Trap (MOT) is a configuration of atoms in laser
cooling beams and coils producing static magnetic fields, which is
astonishingly effective in cooling atoms from room temperature to
microkelvin scale temperatures.  Today, the MOT is a workhorse of
atomic physics, and the starting point for many atomic physics
experiments, such as with Bose-Einstein condensates.

We begin our study of the MOT with an explanation of how it was
originally thought to be impossible, due to the optical Earnshaw
theorem.  We then present the physics of two-level atoms in a
one-dimensional MOT, from which generalization to 3D MOT's trapping
real multi-level atoms is straightforward.

=== Optical Earnshaw Theorem ===

Some history helps place the MOT in context.  In the 1970's and
1980's, many people discussed prospects for, and limitations of the
stimulated and spontaneous force.  In 1983, Gordon and Ashkin derived
the optical Earnshaw theorem.  The (standard, electromagnetic)
Earnshaw theorem states that a charged particle will never be in a
stable equilibrium in a static, 3D configuration of electric fields.
This holds because for a particle to be stable, all field lines must
point inwards, but we know that the divergence of the electric field
in vacuum is zero.

The optical Earnshaw theorem states, in a similar way, that a 3D
sponteneous light force trap is not possible.  This was not a very
positive start for atom trapping.  The proof goes as follows.  Recall
that the sponeaneous light force is related to the Poynting vector as
:<math> 
\vec{F}_S = c\vec{S}
\,,
</math>
where <math>\vec{S}</math> is the Poynting vector.  From electromagnetism,
however, we know that
:<math> 
	\frac{\partial U}{\partial t} + \vec{\nabla}\cdot \vec{S} = 0
\,,
</math>
but we know that <math>\vec{\nabla}\cdot \vec{F}_S = 0</math> if there is zero
local energy density <math>U</math>.  This contradicts the requirement that
<math>\vec{\nabla}\cdot \vec{F}_S < 0</math>.

Where is the flaw in this reasoning?  In 1986, David Pritchard
realized there were many ways to circumvent the optical Earnshaw
theorem.  What is ignored here is the {\em internal structure of the
atom}.  After all, the spontaneous light force is actually not just
proportional to the incident light flux.  The assumption that every
atom which is exposed to the same Poynting vector experiences the same
force, is just wrong.  Namely,
:<math> 
\nabla( c(\vec{r})\vec{S}(\vec{v})) \neq 0
\,.
</math>
The force can thus be spatially dependent.  First, <math>c(\vec{r})</math> can
experience saturation.  Second, optical pumping can be used, taking
advantage of internal structure of the atoms.  Third, there can be
line shifts due to external static fields, as when hyperfine states
are used which are magnetic field sensitive.

The perfect solution was demonstrated in 1987, in a collaboration
between MIT (paper by Raab et al) and Steve Chu's group at Bell Labs.

=== 1-dimensional MOT ===

Consider two counterpropagating laser beams with opposite circular
polarizations, incident on some atoms, in the presence of a static
magnetic field gradient:

::[[Image:Magneto-Optical_Traps-mot-fig1.png|thumb|306px|none|]]

The energy levels of three hyperfine levels, with <math>m=0</math> and <math>m=\pm 1</math>,
become a function of space due to the magnetic field, so that the
<math>\sigma_-</math> and <math>\sigma_+</math> polarized light beams interact with
spatially localized classes of atoms:

::[[Image:Magneto-Optical_Traps-mot-fig2.png|thumb|408px|none|]]

This scenario is similar to an optical molasses, but the frequency
shifts <math>\pm B z</math> are due to a magnetic field gradient <math>\beta = \mu
B'/\hbar</math>.  Just as in the molasses analysis, we must add the effects
of the left and right laser beams.  Assume we have low intensity, so
we find that
:<math> 
F = F_R + F_L = \frac{\hbar k \Gamma}{2} \left[
\frac{I/I_0}{1+4\left(\frac{\delta-kv-\beta z}{\Gamma}\right)^2} -
\cdots \right]
</math>
For small <math>v</math>, <math>z</math>,
:<math> 
F(v,z) = \frac{\hbar k \Gamma}{2} \left[ -kv - \beta z\right]
</math>
The equation of motion for the atom is thus that of a harmonic
oscillator, because the force is linear with displacement:
:<math> 
\ddot{z} + v\dot{z} + \omega_{\rm trap}^2 z = 0
\,,
</math>
where the trap frequency is given by the ratio of the light momentum
<math>k</math> and the field gradient strenght <math>\beta</math>,
:<math> 
\omega_{\rm trap} \propto \frac{\beta}{k}
\,.
</math>
Typically, experimental parameters lead this to be an overdamped
harmonic oscillator, with <math>\omega_{\rm trap}\sim 2\pi \times 1</math> kHz,
and a damping time of about <math>50</math> <math>\mu</math>s.  The static well depth is
usually around a few Kelvin, but the dynamic depth is even larger.
The dynamic depth refers to how much kinetic energy an atom can get
before it is recaptured by the trap, because it is cooled by optical
molasses effects even after it leaves the local potential well.

=== 3D MOT trapping with real atoms ===

Real atoms have a multilevel structure, with complicated hyperfine
structure.  If you have a complicated ground state structure, you need
to include optical pumping effects between all the levels, but the
combination of these effects and the simple Zeeman shift model
discussed above works very well.

Recall that optical molasses works better than originally anticipated,
in which it was found that the damping due to polarization gradients
was much much larger than the damping due to doppler cooling,
<math>\gamma_{pg}\gg \gamma_{\rm doppler}</math>  The same turns out to be true
for the restoring force in magneto-optical traps.

We have discussed only the 1D scheme above.  You may not appreciate
it, but a lot of spontaneous light forces which were discussed to
circumvent the optical  Earnshaw theorem worked in 1D, but were very
difficult to generalize to 3D.  The MOT scheme, on the other hand,
generalizes very well to 3D.  By using three pairs of anti-Helmholtz
coils to create the field gradients, and six laser beams, you get the
generalizeation to 3D and it works as well as in one dimension.

::[[Image:Magneto-Optical_Traps-mot-fig3.png|thumb|510px|none|]]

==== Example: vapor cell MOT ====

Consider a cesium vapor cell MOT.  The atoms, initially at 300K, can
be cooled to <math>3</math> <math>\mu</math>K.   The density starts at <math>10^8</math> cm<math>^{-3}</math>,
which is increased to <math>10^{11}</math> cm<math>^{-3}</math>.  This means that the phase
space density, which is <math>n/T^{3/2}</math>, has been increased by <math>15</math> order
of magnitude.  Initially, the phase space density before cooling is
<math>20</math> orders magnitide away from Bose-Einstein condensation; after MOT
trapping, the system is much closer - only <math>5</math> orders of magnitude
closer, but not yet there.

[[Category:Light forces]]