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, \(\dot N_i=\sum_j(W_{ji}N_j-W_{ij}N_i)-N_i/\tau_i\), 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 \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

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 \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

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 \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

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 \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

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 \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).