Chapter 5: The Rabi Frequency
=============================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 5.
Use each section/problem identifier with the book; the original prompts are
not reproduced here.  Each entry gives the governing model, the decisive
solution route, and a physical verification.

Section 5.1: Validity Of The Rate-Equation Model
------------------------------------------------

Problem 5.1.1 — Harmonic response of a two-level atomic system
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write one balance equation per level, :math:`\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i`, add population conservation, and solve the resulting linear steady-state system.
Check that every population is nonnegative, their sum is conserved, and the unpumped and strongly pumped limits are sensible.

Section 5.2: Strong-Signal Behavior: The Rabi Frequency
-------------------------------------------------------

Problem 5.2.1 — The slowly varying envelope approximation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write the resonant coupling as :math:`\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar`; integrate the Bloch rotation angle :math:`\Theta=\int\Omega(t)\,dt` before reading off the populations.
Verify population conservation and recover the weak-field rate-equation limit when :math:`\Omega T_2\ll1`.

Problem 5.2.2 — Analysis of off-resonance Rabi Bopping behavior
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write the resonant coupling as :math:`\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar`; integrate the Bloch rotation angle :math:`\Theta=\int\Omega(t)\,dt` before reading off the populations.
Verify population conservation and recover the weak-field rate-equation limit when :math:`\Omega T_2\ll1`.

Problem 5.2.3 — Coherent transients: The 90° pulse
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write the resonant coupling as :math:`\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar`; integrate the Bloch rotation angle :math:`\Theta=\int\Omega(t)\,dt` before reading off the populations.
Verify population conservation and recover the weak-field rate-equation limit when :math:`\Omega T_2\ll1`.

Problem 5.2.4 — Large-signal atomic response: Two-frequency mixing and intermodulation effects
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison.
Write the resonant coupling as :math:`\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar`; integrate the Bloch rotation angle :math:`\Theta=\int\Omega(t)\,dt` before reading off the populations.
Verify population conservation and recover the weak-field rate-equation limit when :math:`\Omega T_2\ll1`.

Problem 5.2.5 — Quantum transition matrix element for an electric-dipole atom
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Write the resonant coupling as :math:`\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar`; integrate the Bloch rotation angle :math:`\Theta=\int\Omega(t)\,dt` before reading off the populations.
Verify population conservation and recover the weak-field rate-equation limit when :math:`\Omega T_2\ll1`.