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This Demonstration shows the solution of the Maxwell–Bloch equation for a two-level system. This simplified treatment models the passage of a laser through a gas tube containing rubidium vapor, commonly studied in quantum optics experiments. Here, the "atoms" are ideal two-level quantum mechanical systems, with a ground state and an excited state. The atoms are assumed to be in their ground state at the start of the experiment. A laser beam enters the cell at and exits at m. Although 32 meters is much longer than in a typical experiment, the parameters have been chosen to follow the example in Fig. 2 of the tutorial article by Siddons . Siddons uses a completely different numerical method, so this gives us something to check our results against.[more]
The plots shown begin at the position of the laser entering the tube, namely . As shown in panel (a), the pulse builds up from zero to some fixed value over about 15 ns (nanoseconds). The atoms very close to the entrance face evolve under the influence of the laser. The population of the excited state, shown in panel (b), rises over the course of 100 ns or so toward a maximum value of about 0.0005, meaning that at most only about 0.05% of the atoms will be excited at any given time. Although the laser keeps exciting atoms, these decay through spontaneous emission back to the ground state, leading to a relatively low fraction of atoms in the excited state, even after the laser has been on for a long time.
The coherence is shown in panel (c). When the coherence is nonzero, the atoms on average have an electric dipole oscillating at the frequency of the laser. Such dipoles are themselves sources of light. In this geometry, the light generated by the atoms combines with the incident laser beam and modifies its downstream intensity. At small values of , i.e. near the entrance to the tube, relatively few atoms have a chance to modify the beam, so these atoms see nearly the same exciting light as the ones at the entrance face. Farther downstream, more of the beam is modified. Of course, that modified beam affects the downstream atoms. Since the beam is weaker, both the excited-state population and the coherence become smaller as increases.
The midpoint of the beam at the entrance is marked by a dashed black vertical line. A second such line represents that midpoint delayed by the time it takes for light to travel down the tube to the position displayed. Although the time evolution of the intensity, population and coherence changes as we vary , the vertical lines help show corresponding points on these evolving functions. You can move the slider to see what happens at various points down the tube as the laser beam passes through.[less]
Contributed by: Zachary H. Levine (May 8)
(Quantum Optics Group, National Institute of Standards and Technology)
After work by P. Siddons 
Open content licensed under CC BY-NC-SA
Siddons  gives examples of the Maxwell–Bloch equations, some of which are replicated here. The Maxwell–Bloch equations for a two-level system are given in (S9), (S26), (S43) and (S53)–(S55), where S is for Siddons. Here are the equations in slightly simplified notation:
The following symbols are used in these equations:
, the speed of light,
, the vacuum permittivity,
, the number density of atoms,
, the carrier frequency,
, the dipole matrix element,
, the population decay rate,
, the coherence decay rate,
, the detuning.
In some cases,  gives the general form for indices , which is copied here, but these are specialized to 21 in the equation. Although the equations are given with vector symbols, we assume that and are aligned with the axis and treat these as scalars.
In solving the equations, we can eliminate in favor of , leading to
The units are balanced in this equation. The left side has dimensions . The right side has dimensions
which agree with those on the left side. The density matrix is dimensionless. The other two equations have units of .
Before solving these equations, we divide them by or to get a dimensionless time and Rabi frequency. Figure 2 in  uses this time unit. The spatial unit will also be converted to a dimensionless unit.
Here, is a dimensionless distance and is a dimensionless retarded time. The population and coherence are dimensionless. Also, is the dimensionless rate of decoherence, is the dimensionless detuning and is a dimensionless constant whose physical meaning is how much electric field is generated by the coherence:
There are three constants in these equations, namely , and . If the decoherence is due to spontaneous emission, then [1, p. 12].
Siddons only solves for , leaving one dimensionless parameter . Siddons only solves for this in a single case. In this case, is a real number and is purely imaginary. Of course, is real, which is true for any . Hence, we may rewrite the equations as a set of three differential equations in :
Physically, is the Rabi frequency and is proportional to the electric field, is the upper state population of a two-level system and is the coherence.
The equations above are solved subject to , and there is an incident electric field that obeys with . (See initialization code for parameters and [1, p. 13] for details on how these are chosen.)
 P. Siddons, "Light Propagation through Atomic Vapours," Journal of Physics B: Atomic, Molecular and Optical Physics, 47(9), 2014 093001. doi:10.1088/0953-4075/47/9/093001.