Chapter 31: Magnetic-Dipole Transitions
=======================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 31.
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 31.1: Basic Properties Of Magnetic-Dipole Transitions
-------------------------------------------------------------

Problem 31.1.1 — Research problem: magnetic dipole moments in real atoms
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.1.2 — Multipole expansion of a real atom
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.2: The Iodine Laser: A Magnetic-Dipole Laser Transition
------------------------------------------------------------------

Problem 31.2.1 — Iodine laser transition cross section
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Propagate irradiance with :math:`dI/dz=g(I)I`, using :math:`g(I)=g_0/(1+I/I_s)` when saturation matters; integrate before inserting boundary values.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.

Section 31.5: Transverse Response: The Ac Susceptibility
--------------------------------------------------------

Problem 31.5.1 — Radiative decay rate for a classical magnetic dipole
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.5.2 — Polarization changes for a circularly polarized wave
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.6: Longitudinal Response: Rate Equation
--------------------------------------------------

Problem 31.6.1 — Alternative approach to solving the Bloch equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.7: Large-Signal And Coherent-Transient Effects
---------------------------------------------------------

Problem 31.7.1 — 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 31.7.2 — Conversion between electric-dipole and magnetic-dipole models
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Use the magnetic Bloch equations :math:`\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1` and solve in the rotating frame.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.7.3 — Rabi flopping behavior: alternative derivation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
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 31.7.4 — Rabi frequency with both detuning and relaxation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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`.