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

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 letteraand the lower by the letterb. 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 DampingFor an oscillatory applied field

[ III-5 ] we see, in thethatrotating-wave approximation,

[ III-6a ]

[ III-6b ] where defines the so calledRabi flopping frequencywhich 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 theof the field is expressed as and the system's wave function -frequency detuningi.e.Equation [ III-1 ] - is written in terms of slightly modified time varying coefficients by transforming to the-rotating frame of referenceviz.

[ 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 thegeneralized Rabi flopping frequency

[ III-9a ] and the generalnon-dampedtime 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 describedphenomenologicallyby 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 theaverage decay rate constantand as thegeneralized 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 mostdamping terms (a useful attribute of density matrix formulations) --intuitively satisfyingviz.

[ III-15a ]

[ III-15b ] Transform these equations toarotating frameby taking where is assumed to be a slowly varying function of time which satisfies the equations of motion

[ III-16a ]

[ III-16b ] With andignoring terms proportionalto--i.e.the rotating wave approximation- we find that

[ III-17a ]

[ III-17b ] If we take all time derivatives in these equations equal to zero, we obtainSteady state behavior:

[ III-18a ]

[ III-18b ] or

[ III-18c ] An exceedingly valuable expression for themacroscopic polarizationis 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 issaturatedat high electromagnetic powers. In general, themacroscopic polarizationis then given by [2]

[ III-19a ]

[ III-19b ]

[ III-19c ]

The Vector Model of the Two-Level Density Matrix:There is a set ofarguments by analogywhich 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 thefamous 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 theis given byeffective Rabi precession field

[ III-23 ]

[1] The use of this form of interaction needs considerable elaboration, but we defer that discussion until later.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