One can lift the restriction of the generalized Rabi frequency being constant; however, the situation becomes more complex. One of the limitations on the functions is quite basic. The integral must be finite---remember that t is an arbitrarily large amount of time. Two functions will be examined: the gaussian and the hyperbolic secant, both of which have finite areas. The detuning will be set to zero; thus, these functions will have the form of the electric field envelope.
The gaussian is a familiar function to students of physics and mathematics. It is probably the first function that one can think of that has a finite area which is easily scaled. The form of the gaussian that shall be used is:
where c is a constant that determines the width of the gaussian, and is the time at which the gaussian will peak. Enter this function into the amplitude parser. Set c=256 and =50. Make sure that Amp=0.00 and Freq=1.00. Place all the atoms in the lower state. Run BlochApp and compare the result to the pulse experiment. Notice that the result is the same because the integral of is .
Let us examine what happens when . Plot the three dimensional projection of the Bloch vector for values of Freq between 1.00 and 1.10 in steps of 0.02. The results in Figure are rather interesting. Notice that as the detuning increases the population is less inverted. The curves are fairly evenly spaced and very smooth.
Figure: The effect of detuning on a gaussian of area with Freq=1.00 ... 1.10
Perform the same experiment with a pulse. The curves in Figure are not very evenly spaced, and most of the curves have discernible cusps. What is the origin of this cusp? (Hint: Write the equation for a square pulse and take the derivative.) An interesting discussion of the gaussian and the pulse in this light can be found in a paper by Jean--Philippe Grivet.
Figure: The effect of detuning on a Pulse with Freq=1.00 ... 1.10
The hyperbolic secant is just as interesting as this gaussian and far more mysterious. This function has the equation:
where c determines the width of the function and is again the time at which the function is maximized. The integral of this function over all time is equal to . Deriving the value of this integral is a challenge in and of itself. The use of complex analysis simplifies the work.
Perform the same experiment which was done with the gaussian and the pulse. The resulting set of curves is connected, smooth, and elegant. The closure also occurs exactly as the atoms return to the lower state. This plot, in Figure , shows immediately that the hyperbolic secant is a special function.
Figure: The effect of detuning a hyperbolic secant of area with Freq=1.00 ... 1.20
To understand how odd these curves are, perform this experiment again with the pulse and the gaussian, after multiplying both of them by two in order to generate an area of . Should these curves bring one back to their initial position just as the atoms move back into the lower state? We assumed earlier that , and for this value, every single curve does the same thing. However, if , then none of these curves should bring the atoms back to their initial state precisely. There should be some error. The gaussian does not complete the reinversion and the pulse over--inverts; yet, the hyperbolic secant does neither.
The hyperbolic secant is a solution to the Maxwell--Bloch equations which model an electromagnetic field interacting spatially with atoms in a cavity. It is also a soliton which is a special kind of solution invariant under propagation. These kinds of solutions have been studied in many areas of physics including hydrodynamics and optics. The Maxwell--Bloch equations can be used to generate a semiclassical laser theory. One of the results of this theory is that an electromagnetic wave which has a hyperbolic secant envelope is never attenuated in a cavity, i.e., it is a soliton. The theory behind the derivation of this result is beyond the scope of this work. The text by Allen and Eberly provides an excellent discussion of pulse propagation inside a cavity, and the interested reader is encouraged to read it.