Now how do we know that we really have an optical lattice consisting
of a series of potential wells containing atoms? Imagine the atoms
in the lattice as marbles sitting in the individual pockets of an egg
Suppose we suddenly shift the carton to one side - the marbles would
begin to oscillate back and forth in their "potential wells". It
is possible to observe the same motion in an optical lattice!
To form the optical lattice (see figure), we split
a single beam into two using a polarizing beam splitter (BS). The
two beams, which have orthogonal linear polarizations, intersect at the
location of the atoms (shown as a circle). They are not exactly counterpropagating,
but rather intersect at a small angle.
To shift our optical lattice, we use a phase modulator
(PM) to provide a sudden change in the optical path length of one of the
lattice beams (see figure 7). This phase delay translates the standing
wave that forms the optical lattice.
After the shift, the atoms find themselves on one
side of a well, where they feel a restoring force toward the minima of
the well (see top of figure 8). If we think of the atoms as classical
particles, then it is apparent that the atoms will start to oscillate back
and forth in the well.
Now we look at this from the point of view of quantum
mechanics. Each of the potentials can be
thought to approximate a harmonic potential. The allowable energies
in a harmonic potential are quantized into discrete levels, each one corresponding
to an eigenfunction of the potential. These eigenfunctions are stationary
states, in that they do not evolve in time, with the exception of a phase
Suddenly shifting the potential, however, creates
a coherent superposition between adjacent levels, resulting in a spatially
localized wavepacket that does evolve in time. This wavepacket approximates
what is known as a coherent state, which is a wavepacket that maintains
minimal uncertainty in both position and momentum coordinates for all times
(see Sakurai). The wavepacket proceeds to oscillate back and forth
in the wells.
Now that the atoms are oscillating in their wells,
we need a way to detect their motion. To do so, we should understand
exactly what mechanism is responsible for producing the force on the atoms.
When an atom absorbs or emits a photon, its momentum must change in
accordance with the principle of
conservation of momentum. A photon of frequency ω and wavevector
k carries a momentum
ħk and so the momentum of an atom must change by ħk when
it absorbs or emits a photon.
Let's consider the case of when the atom is on the
left side of the well, where we know that it must feel a force to the right.
Remember that the atom is surrounded by photons that are moving in both
the left and right directions because the two lattice beams are counterpropagating.
To explain the force to the right, we say that the atom first absorbs
a photon from the right-going beam (thus receiving a momentum kick of ħk
to the right). Now that the atom is in the excited state, a photon
from either the left or right going beam can cause stimulated emission.
If a right going photon does the "stimulating", then the emitted photon
imparts a momentum of ħk to the left, which cancels out momentum kick
from the absorbtion. Thus
the absorption and stimulated emission of photons from the same beam
have no net effect on the atom.
On the other hand, if the left going photon did the stimulating, the
left going emitted photon would impart a kick to the right - the same direction
as the kick received in the absorption process.
Thus, one photon has been removed from the right going beam and one
photon has been added to the left going beam - in essence, one photon
has been exchanged or redistributed from the right to the left (see bottom
on figure 8). This photon redistribution imparts to the atom a momentum
of 2ħk to the right (summing the contributions of absorbtion and emission).
Now if the rate of photon redistribution events is given by R, then
the change of momentum per unit
time is 2Rħk. We know that force is defined as change
of momentum per unit time (dp/dt), so we conclude that the force on the atom is 2Rħk.
Therefore, to know the force we need
to measure R.
We do this by measuring the power of the beams
after they interact with the atoms and exit the chamber. We shine
each beam onto a photodiode and then take the difference of the
two photodiode signals (see figure 7). The difference in power is
given by 2Rħkc, where c is the speed of light. If there is no force
on the atom (i.e. the atom is sitting in the potential minimum) then there
is no photon exchange - and the difference signal will be 0. However
if the atom is situated on the left side of the well and feels a force
to the right, then there will be a finite difference signal. If the
atom is located at the corresponding location on the right side of the
well and feels a force to the left, then the difference will have the same
magnitude, but the polarity of the signal will change. A typical
plot of this signal as a function of time is shown in figure 9 - this is
the force on the atom as it oscillates. By integrating
the signal twice, we can determine the position of the atoms as function
of time. As can be seen from the figure, the period of oscillation
in this particular lattice is about 20 microseconds. Also, the amplitude
of the oscillation decays in time. This is due both to dephasing,
resulting from the fact that the
energy levels are spaced by slightly different energies, because the
potentials are not exactly harmonic,
and decoherence which results from spontaneous emission.
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