The Interaction of Radiation and Matter: Semiclassical Theory (cont.)

III. Review of Basic Quantum Mechanics: Two-Level Quantum Systems (pdf copy)

The literature of quantum optics and laser spectroscopy abounds with discussions of the two-level (two-state) system. This emphasis comes about because the interaction of such systems with the electromagnetic field may be treated in great detail to obtain valuable analytic results and, hopefully, the analysis of two-level systems generates insights that may be extended to more realistic situations. Fortunately, there are several important instances in which the application of the two-level model provides a very good approximation to a more complete theory. In the following, we label the upper level of the system by the letter a and the lower by the letter b. From Equation [ I-12a ] we write, specifically, the wave function of the two level system as
 
 
     [ 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