 |
[ III-1 ] |
where we know from Equation [ I-12b ] that the time varying coefficients
satisfy, in general, the following equations:
 |
[ III-2a ] |
 |
[ III-2b ] |
If we take the interaction to be the electric dipole interaction with
an applied electric field we may write
 |
[ III-3a ] |
where 
denotes the position of the center of the two-level system or atom.[1]
Thus we write
 |
[ III-3b ] |
In all but the most bizarre circumstances we may use persuasive symmetry
arguments to reason that
Thus Equations [ III-2 ] reduce to
 |
[ III-4a ] |
 |
[ III-4b ] |
where 
.
Rabi Flopping -- Without Damping
For an oscillatory applied field
 |
[ III-5 ] |
we see, in the rotating-wave approximation,that
 |
[ III-6a ] |
 |
[ III-6b ] |
where 
defines the so called Rabi flopping frequency which is, of course,
a measure of the strength of the electromagnetic interaction. Clearly,
the coupling terms have maximum effect when the frequency of the applied
field is resonant with the level splitting. In most treatments the frequency
detuning of the field is expressed as 
and the system's wave function - i.e. Equation [ III-1 ] - is written
in terms of slightly modified time varying coefficients by transforming
to the
rotating frame of reference - viz.
 |
[ III-7a ] |
 |
[ III-7b ] |
 |
[ III-7c ] |
The coupling terms in this rotating frame of reference become
 |
[ III-8a ] |
 |
[ III-8b ] |
or in matrix form
 |
[ III-8c ] |
We look for a solution in the form 
where 
is the generalization of the Rabi flopping frequency. Therefore, the condition
yields the generalized Rabi flopping
frequency
 |
[ III-9a ] |
and the general non-damped time evolving wave function
 |
[ III-9b ] |
Rabi Flopping -- With Damping:
Neglected interactions (e.g. spontaneous emission, collisions,
thermal fluctuations) limit lifetime of a state of a two-level system.
One class of lifetime limiting interactions may be described
phenomenologically
by adding decay terms to the equations of motion -
i.e. to Equations
[ III-8 ] - as follows:
 |
[ III-10a ] |
 |
[ III-10b ] |
or
 |
[ III-10c ] |
Again we look for a solution in the form 
so that
or
.
Therefore
 |
[ III-11a ] |
-- where the we refer to 
as the average decay rate constant and 
as the generalized complex Rabi flopping frequency -- and the general
time evolving wave function -- in the rotation frame -- may be written
 |
[ III-11b ] |
Density Matrix Treatment of a Two-Level Systems:
Recall from Equation [ II-30 ]
Using 
we may write
 |
[ III-12a ] |
If 
, then
 |
[ III-12b ] |
Writing matrix elements -- i.e. representatives -- of
the density operator
 |
[ III-13a ] |
 |
[ III-13b ] |
 |
[ III-13c ] |
We may again assume, by symmetry arguments, that 
so that the equations of motion for the elements of the density matrix
reduce to
 |
[ III-14a ] |
 |
[ III-14b ] |
To introduce an element of reality, we add to these equations
a pair of the most intuitively satisfyingdamping
terms (a useful attribute of density matrix formulations) -- viz.
 |
[ III-15a ] |
 |
[ III-15b ] |
Transform these equations to a rotating frame
by taking 
where 
is assumed to be a slowly varying function of time which satisfies the
equations of motion
 |
[ III-16a ] |
 |
[ III-16b ] |
With 
and ignoring terms proportionalto
-- i.e. the rotating wave approximation - we find that
 |
[ III-17a ] |
 |
[ III-17b ] |
Steady state behavior:
If
we take all time derivatives in these equations equal to zero, we obtain
 |
[ III-18a ] |
 |
[ III-18b ] |
or
 |
[ III-18c ] |
An exceedingly valuable expression for the macroscopic polarization
is then given by
 |
[ III-19a ] |
 |
[ III-19b ] |
 |
[ III-19c ] |
We have graphed Equation [ III-18b ] at resonance - i.e. 
- and, from the following expression, we see how the oscillatory polarization
is saturated at high electromagnetic powers. In general, the
macroscopic
polarization is then given by [2]
|
[ III-19a ] |
|
[ III-19b ] |
|
[ III-19c ] |
The Vector Model of the Two-Level Density Matrix:
There is a set of arguments by analogy which is exceedingly
valuable in treating transient excitation problems in optics. The basis
for the analogy lies in the fact that Equations [ III-16] are identical
in form to the
famous Bloch equations of magnetic resonance.[3]
If we make the transformation to the rotating frame -- i.e.,
-- the Bloch equations take on the form
 |
[ III-20a ] |
 |
[ III-20b ] |
where 
.
By comparing these Bloch equations with Equations [ III-16 ] and [ III-17
] we see that we have identical problems if we make the identifications
 |
[ III-21a ] |
 |
[ III-21b ] |
 |
[ III-21c ] |
 |
[ III-21d ] |
 |
[ III-21e ] |
 |
[ III-21f ] |
In other words we have the equivalent equation of motion
 |
[ III-22 ] |
where the effective Rabi precession field 
is given by
 |
[ III-23 ] |
[1] The use of this form of interaction needs considerable
elaboration, but we defer that discussion until later.
[2] That is to say
since 
by symmetry.
[3] Recall from the theory of magnetic resonance
where 
is the
gyromagnetic ratio. This equation of motion is greatly simplified
if it is written in terms of the circular polarizations 
and 
-- viz.
and
To these Bloch added the phenomenological longitudinal (thermal) relaxation
time 
and transverse dephasing time 
so that
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This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated March 9, 2000
