Part II, Chapter 1

Optical Bloch Equations

The two-level atom driven by a classical field: Rabi oscillations, decoherence, and the Bloch sphere.

1.1 The Two-Level Atom Model

When a near-resonant laser field interacts with an atom, only two energy levels participate significantly. The ground state |g〉 and excited state |e〉 are separated by energy ℏω₀. The atom's state is:

$$|\psi(t)\rangle = c_g(t)|g\rangle + c_e(t)|e\rangle$$

The interaction with a classical monochromatic field E(t) = E₀ cos(ωt) is described by the dipole Hamiltonian:

$$\hat{H} = \frac{\hbar\omega_0}{2}\hat{\sigma}_z - \hat{\mathbf{d}} \cdot \mathbf{E}(t) = \frac{\hbar\omega_0}{2}\hat{\sigma}_z + \hbar\Omega\cos(\omega t)(\hat{\sigma}_+ + \hat{\sigma}_-)$$

where Ω = -d·E₀/ℏ is the Rabi frequency and d is the transition dipole moment.

Figure 1. The Bloch sphere representation of a two-level atom. The north pole corresponds to |0〉 (ground state), the south pole to |1〉 (excited state), and the equator to equal superpositions. Rabi oscillations correspond to rotation of the state vector around the drive axis (dashed cyan arc).

Derivation 1: Rabi Oscillations

Step 1. Transform to the rotating frame via c̃_e = c_ee^iωt/2, c̃_g = c_ge^-iωt/2. The Schrödinger equation becomes:

$$i\hbar\frac{d}{dt}\begin{pmatrix}\tilde{c}_e \\ \tilde{c}_g\end{pmatrix} = \frac{\hbar}{2}\begin{pmatrix}-\delta & \Omega \\ \Omega & \delta\end{pmatrix}\begin{pmatrix}\tilde{c}_e \\ \tilde{c}_g\end{pmatrix}$$

where δ = ω - ω₀ is the detuning and we have applied the rotating wave approximation (RWA), dropping terms oscillating at 2ω.

Step 2. For an atom starting in |g〉, the excited state probability is:

$$P_e(t) = |c_e(t)|^2 = \frac{\Omega^2}{\Omega_R^2}\sin^2\!\left(\frac{\Omega_R t}{2}\right)$$

where Ω_R = √(Ω² + δ²) is the generalized Rabi frequency. On resonance (δ = 0), the atom undergoes complete population inversion at frequency Ω.

1.2 Density Matrix and Optical Bloch Equations

To include dissipation (spontaneous emission, dephasing), we use the density matrix ρ = |ψ〉〈ψ|. The optical Bloch equations are derived from the master equation with Lindblad dissipators.

Derivation 2: Optical Bloch Equations from Lindblad Master Equation

Step 1. The Lindblad master equation for a two-level atom with spontaneous decay rate Γ is:

$$\dot{\rho} = -\frac{i}{\hbar}[\hat{H}, \rho] + \Gamma\left(\hat{\sigma}_-\rho\hat{\sigma}_+ - \frac{1}{2}\{\hat{\sigma}_+\hat{\sigma}_-, \rho\}\right)$$

Step 2. Defining the Bloch vector components u = 2Re(ρ̃_eg), v = 2Im(ρ̃_eg), w = ρ_ee - ρ_gg, in the rotating frame:

Optical Bloch Equations:

$$\dot{u} = -\frac{u}{T_2} + \delta v$$

$$\dot{v} = -\frac{v}{T_2} - \delta u + \Omega w$$

$$\dot{w} = -\frac{w + 1}{T_1} - \Omega v$$

where T&sub1; = 1/Γ (population relaxation time) and T&sub2; = 2/Γ (coherence dephasing time, in the absence of additional pure dephasing).

1.2.1 T&sub1; and T&sub2; Relaxation

The two characteristic times describe distinct physical processes:

T&sub1; (longitudinal): Population relaxation. Governs the decay of the population inversion w toward thermal equilibrium (w = -1). Set by spontaneous emission: T&sub1; = 1/Γ.
T&sub2; (transverse): Coherence dephasing. Governs the decay of the off-diagonal elements (u, v). In general: 1/T&sub2; = 1/(2T&sub1;) + 1/T&sub2;* where T&sub2;* accounts for pure dephasing (collisions, inhomogeneous broadening).

1.3 Bloch Sphere Representation

The state of a two-level system is represented as a point on or inside the Bloch sphere. Pure states lie on the surface (|R| = 1), mixed states inside (|R| < 1).

Bloch Sphere Geometry

• North pole (w = +1): Excited state |e〉
• South pole (w = -1): Ground state |g〉
• Equator: Equal superpositions (|g〉 ± |e〉)/√2
• u-axis: In-phase component of the coherence
• v-axis: Quadrature component of the coherence

Derivation 3: Steady-State Solution and Saturation

Step 1. Setting all time derivatives to zero in the OBE and solving the linear system:

$$w_{\text{ss}} = -\frac{1}{1 + s}, \qquad s = \frac{\Omega^2 T_1 T_2}{1 + \delta^2 T_2^2}$$

Step 2. The saturation parameter s determines the steady-state excited population:

$$\rho_{ee}^{\text{ss}} = \frac{1 + w_{\text{ss}}}{2} = \frac{s}{2(1+s)} = \frac{\Omega^2/(4\delta^2 + \Gamma^2)}{1 + \Omega^2/(4\delta^2 + \Gamma^2)/2}$$

Step 3. On resonance (δ = 0), defining the saturation intensity I_sat = πhcΓ/(3λ³):

$$\rho_{ee}^{\text{ss}} = \frac{1}{2}\frac{s_0}{1 + s_0}, \qquad s_0 = \frac{I}{I_{\text{sat}}} = \frac{2\Omega^2}{\Gamma^2}$$

As I → ∞, ρ_ee → 1/2: maximum population in the excited state is 50%, never fully inverted in steady state (unlike coherent Rabi oscillations).

1.4 Ramsey Interferometry

Ramsey's method of separated oscillatory fields achieves spectroscopic resolution limited by the free-evolution time T, not the interaction time. Two short π/2 pulses separated by dark time T produce interference fringes in the excitation probability.

Derivation 4: Ramsey Fringe Pattern

Step 1. A resonant π/2 pulse rotates the Bloch vector from the south pole to the equator. The rotation matrix for a pulse of area θ = Ωτ is:

$$R_{\pi/2} = \begin{pmatrix}\cos(\pi/4) & -i\sin(\pi/4) \\ -i\sin(\pi/4) & \cos(\pi/4)\end{pmatrix}$$

Step 2. During free evolution time T, the state acquires a phase δT:

$$U_{\text{free}} = \begin{pmatrix}e^{-i\delta T/2} & 0 \\ 0 & e^{i\delta T/2}\end{pmatrix}$$

Step 3. After the second π/2 pulse, the excitation probability is:

$$P_e = \cos^2\!\left(\frac{\delta T}{2}\right)$$

The fringe spacing is Δν = 1/(2T). For T = 1 s (as in atomic fountain clocks), this gives sub-Hz resolution, enabling the most precise frequency measurements in physics.

1.5 Power Broadening and Spectral Lineshape

The steady-state absorption profile of a two-level atom is a Lorentzian whose width increases with driving intensity — this is power broadening.

Derivation 5: Power-Broadened Lorentzian

Step 1. From the steady-state OBE solution, the scattering rate (proportional to ρ_ee) is:

$$R_{\text{sc}} = \Gamma \rho_{ee} = \frac{\Gamma}{2} \frac{s_0}{1 + s_0 + (2\delta/\Gamma)^2}$$

Step 2. This is a Lorentzian in detuning δ with FWHM:

$$\Delta\omega_{\text{FWHM}} = \Gamma\sqrt{1 + s_0}$$

Step 3. The peak scattering rate on resonance:

$$R_{\text{sc}}^{\text{max}} = \frac{\Gamma}{2}\frac{s_0}{1 + s_0} \xrightarrow{s_0 \gg 1} \frac{\Gamma}{2}$$

At high intensity, the linewidth broadens as Γ√(1 + I/I_sat) while the peak scattering rate saturates at Γ/2 — each atom can scatter at most Γ/2 photons per second.

1.6 Beyond the Two-Level Atom

Real atoms have many levels. When the detuning from intermediate levels is large, they can be adiabatically eliminated, reducing the problem to an effective two-level or few-level system.

1.6.1 Three-Level Systems: Λ, V, and Ξ Configurations

Λ-System

Two ground states |g&sub1;〉, |g&sub2;〉 coupled to a common excited state |e〉. Enables electromagnetically induced transparency (EIT), slow light, and stimulated Raman adiabatic passage (STIRAP).

V-System

One ground state coupled to two excited states. Used in quantum beat spectroscopy and fluorescence studies of excited-state coherences.

Ξ (Ladder) System

Sequential coupling |g〉 → |e〉 → |r〉. Enables two-photon excitation to Rydberg states, the basis for Rydberg-mediated quantum gates.

1.6.2 Electromagnetically Induced Transparency (EIT)

In a Λ-system, a strong coupling field makes the medium transparent to a weak probe field at a frequency where it would normally be absorbed. The probe experiences:

• A narrow transparency window of width Δω_EIT = Ω_c²/Γ
• Extreme slow light: group velocity v_g = cΩ_c²/(Γnσ) can reach m/s
• Storage and retrieval of light pulses by turning the coupling field on and off (quantum memory)

Hau et al. (1999) demonstrated v_g = 17 m/s in a BEC, and light has been stopped completely using EIT in warm atomic vapors and cold atoms.

1.6.3 Dark States and STIRAP

In a Λ-system, there exists a dark state that is an eigenstate of the Hamiltonian with zero excited-state component:

$$|D\rangle = \cos\theta\, |g_1\rangle - \sin\theta\, |g_2\rangle, \qquad \tan\theta = \frac{\Omega_1}{\Omega_2}$$

STIRAP (STImulated Raman Adiabatic Passage) uses a counter-intuitive pulse ordering (coupling field before pump field) to adiabatically transfer population from |g&sub1;〉 to |g&sub2;〉 through the dark state, avoiding any population in the lossy excited state. Transfer efficiencies exceed 99.9%.

1.7 Cavity Quantum Electrodynamics

When an atom is placed inside a high-finesse optical or microwave cavity, the atom-field interaction can dominate over dissipation. This is the strong coupling regimeof cavity QED, described by the Jaynes-Cummings model.

$$\hat{H}_{\text{JC}} = \hbar\omega_c \hat{a}^\dagger\hat{a} + \frac{\hbar\omega_0}{2}\hat{\sigma}_z + \hbar g(\hat{a}\hat{\sigma}_+ + \hat{a}^\dagger\hat{\sigma}_-)$$

The vacuum Rabi splitting 2g is the frequency splitting of the dressed states when a single photon is in the cavity. Strong coupling requires g > κ, γ (cavity decay rate, atomic decay rate).

Key Cavity QED Parameters

• Single-atom cooperativity: C = g²/(κγ). Strong coupling: C >> 1
• Purcell effect: For C > 1, spontaneous emission is enhanced into the cavity mode by a factor F_P = 3Qλ³/(4π²V_mode)
• Vacuum Rabi oscillations: A single excited atom in an empty cavity oscillates between |e,0〉 and |g,1〉 at frequency 2g
• Photon blockade: The nonlinear level spacing of the Jaynes-Cummings ladder prevents two photons from entering the cavity simultaneously — a single-photon transistor

Cavity QED has been realized in optical cavities (Kimble group, Caltech), microwave cavities (Haroche group, ENS Paris — Nobel Prize 2012), and superconducting circuits (circuit QED — Schoelkopf, Devoret at Yale). These systems are foundational platforms for quantum information processing and quantum networking.

1.8 Photon Echo and Coherent Transients

Inhomogeneous broadening causes apparent dephasing of the macroscopic polarization on a timescale T&sub2;*. However, the individual atomic coherences still evolve, and can be refocused using echo techniques.

1.8.1 Free Induction Decay

After a short π/2 pulse, the macroscopic polarization oscillates at the mean transition frequency and decays on a timescale T&sub2;* = 1/(πΔν_inh) due to inhomogeneous broadening. This is free induction decay — much faster than the homogeneous T&sub2;.

1.8.2 Photon Echo (π/2 - π sequence)

A π/2 pulse creates coherence. After time τ, a π pulse reverses the phase evolution. At time 2τ, all frequency components rephase, producing a photon echo:

$$P_{\text{echo}}(2\tau) \propto e^{-4\tau/T_2}$$

The echo amplitude decays with the homogeneous T&sub2;, not T&sub2;*. This technique extracts the intrinsic coherence time even in highly inhomogeneous samples. It is the optical analog of the Hahn spin echo in NMR.

1.8.3 Dynamical Decoupling

Sequences of many π pulses (CPMG, XY-N, Uhrig sequences) can protect quantum coherence from environmental noise by repeatedly refocusing the dephasing. This extends the effective coherence time by orders of magnitude and is essential for quantum computing and quantum sensing.

Applications

Atomic Clocks

Ramsey interferometry is the basis of cesium fountain clocks (NIST-F2) and optical lattice clocks. The current best clocks achieve fractional frequency uncertainties of 10&sup-18;, losing less than one second over the age of the universe.

Magnetic Resonance Imaging (MRI)

The Bloch equations for nuclear spins in magnetic fields are mathematically identical to the optical Bloch equations. T&sub1;-weighted and T&sub2;-weighted MRI sequences exploit the different relaxation times of water in various tissues.

Quantum Computing Gates

Rabi oscillations implement single-qubit gates: a π pulse is a NOT gate, a π/2 pulse creates a Hadamard-like superposition. Precise pulse calibration (duration and amplitude) is essential for high-fidelity quantum operations.

Laser Spectroscopy

Understanding saturation and power broadening is essential for precision spectroscopy. Techniques such as saturated absorption spectroscopy exploit the saturation behavior to resolve sub-Doppler features, enabling laser frequency stabilization.

1.9 Quantum Control with Pulse Sequences

Precise control of quantum states requires carefully designed pulse sequences that compensate for errors and implement desired unitary operations.

1.9.1 Composite Pulses

Composite pulse sequences (e.g., BB1, CORPSE, Knill) replace a single rotation with a sequence of rotations that achieve the target operation with reduced sensitivity to systematic errors (pulse duration, Rabi frequency calibration, detuning):

• BB1 (Broadband-1): A 4-pulse sequence that suppresses errors to 4th order in the pulse area error
• CORPSE: Compensates for off-resonance errors using three pulses with specific rotation angles
• SK1: Suppresses both amplitude and frequency errors simultaneously

1.9.2 Adiabatic Rapid Passage (ARP)

Sweeping the laser frequency through resonance while maintaining the adiabatic condition Ω >> dδ/dt inverts the population with near-unit probability, independent of the exact Rabi frequency. This is more robust than a π-pulse but slower. The Landau-Zener formula gives the non-adiabatic transition probability:

$$P_{\text{non-ad}} = e^{-\pi\Omega^2/(2|\dot{\delta}|)}$$

1.9.3 Optimal Control Theory

GRAPE (Gradient Ascent Pulse Engineering) and Krotov algorithms numerically optimize the time-dependent Rabi frequency and phase to implement a target unitary operation with maximum fidelity in minimum time, subject to experimental constraints (bandwidth, power limits). These techniques routinely achieve gate fidelities >99.99% in NMR and are increasingly used in atomic and superconducting qubit systems.

Key Equations Summary

Rabi Oscillation:

$$P_e(t) = \frac{\Omega^2}{\Omega_R^2}\sin^2\!\left(\frac{\Omega_R t}{2}\right), \qquad \Omega_R = \sqrt{\Omega^2 + \delta^2}$$

Optical Bloch Equations:

$$\dot{u} = -u/T_2 + \delta v, \quad \dot{v} = -v/T_2 - \delta u + \Omega w, \quad \dot{w} = -(w+1)/T_1 - \Omega v$$

Steady-State Population:

$$\rho_{ee}^{\text{ss}} = \frac{s_0/2}{1 + s_0 + (2\delta/\Gamma)^2}$$

Power-Broadened Linewidth:

$$\Delta\omega = \Gamma\sqrt{1 + s_0}$$

Ramsey Fringes:

$$P_e = \cos^2(\delta T/2)$$

Jaynes-Cummings Energy:

$$E_\pm(n) = (n+\tfrac{1}{2})\hbar\omega \pm \frac{\hbar}{2}\sqrt{\delta^2 + 4g^2(n+1)}$$

Historical Context

1937 — Rabi: Observed resonance transitions in molecular beams using oscillating fields, introducing the concept of Rabi oscillations. Nobel Prize 1944.

1946 — Bloch: Formulated the Bloch equations for nuclear magnetic resonance, describing the dynamics of spin precession and relaxation. Nobel Prize 1952.

1950 — Ramsey: Invented the method of separated oscillatory fields, dramatically improving spectroscopic precision. Nobel Prize 1989.

1969 — Allen & Eberly: Published a comprehensive treatment of the optical Bloch equations, establishing the theoretical framework for laser-atom interactions.

1975 — Haroche & Raimond: Pioneered cavity QED experiments observing single-atom Rabi oscillations in microwave cavities. Nobel Prize 2012.

Interactive Simulation

This simulation solves the optical Bloch equations numerically and visualizes Rabi oscillations, damped dynamics, Ramsey fringes, and the power-broadened lineshape.

Optical Bloch Equations: Rabi Oscillations & Ramsey Fringes

Python

script.py103 lines

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# Optical Bloch equations: du/dt, dv/dt, dw/dt
def bloch_eqs(y, t, Omega, delta, Gamma):
    u, v, w = y
    T1 = 1.0 / Gamma
    T2 = 2.0 / Gamma  # no pure dephasing
    dudt = -u / T2 + delta * v
    dvdt = -v / T2 - delta * u + Omega * w
    dwdt = -(w + 1) / T1 - Omega * v
    return [dudt, dvdt, dwdt]

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# Plot 1: Rabi oscillations with different damping
ax1 = axes[0, 0]
Omega = 5.0
delta = 0.0
t = np.linspace(0, 10, 1000)

for Gamma_val, label, col in [(0.0, "Gamma=0 (undamped)", "#a78bfa"),
                                (1.0, "Gamma=1.0", "#c084fc"),
                                (3.0, "Gamma=3.0", "#f97316")]:
    if Gamma_val == 0:
        Pe = (Omega**2 / (Omega**2 + delta**2)) * np.sin(np.sqrt(Omega**2 + delta**2) * t / 2)**2
    else:
        sol = odeint(bloch_eqs, [-1, 0, -1], t, args=(Omega, delta, Gamma_val))
        w_sol = sol[:, 2]
        Pe = (1 + w_sol) / 2
    ax1.plot(t, Pe, color=col, linewidth=2, label=label)

ax1.set_xlabel("Time (1/Omega)", color="white")
ax1.set_ylabel("Excited state population", color="white")
ax1.set_title("Rabi Oscillations with Damping", color="#c4b5fd", fontsize=14)
ax1.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white", fontsize=9)
ax1.tick_params(colors="white")
for spine in ax1.spines.values():
    spine.set_color("#7c3aed")

# Plot 2: Bloch vector trajectory (u,w plane)
ax2 = axes[0, 1]
for Gamma_val, col in [(0.5, "#a78bfa"), (2.0, "#c084fc"), (5.0, "#f97316")]:
    sol = odeint(bloch_eqs, [0, 0, -1], t, args=(Omega, 0.0, Gamma_val))
    ax2.plot(sol[:, 1], sol[:, 2], color=col, linewidth=1.5,
             label="Gamma=" + str(Gamma_val), alpha=0.8)
ax2.set_xlabel("v (coherence)", color="white")
ax2.set_ylabel("w (inversion)", color="white")
ax2.set_title("Bloch Vector Trajectories (v-w plane)", color="#c4b5fd", fontsize=14)
ax2.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white", fontsize=9)
ax2.tick_params(colors="white")
for spine in ax2.spines.values():
    spine.set_color("#7c3aed")

# Plot 3: Ramsey fringes
ax3 = axes[1, 0]
T_free = 5.0  # free evolution time
delta_vals = np.linspace(-5, 5, 500)
# Ideal Ramsey fringes (no decay)
P_ramsey = np.cos(delta_vals * T_free / 2)**2
# With T2 decay
T2_decay = 3.0
P_ramsey_decay = 0.5 * (1 + np.exp(-T_free / T2_decay) * np.cos(delta_vals * T_free))

ax3.plot(delta_vals, P_ramsey, color="#a78bfa", linewidth=2, label="No decay")
ax3.plot(delta_vals, P_ramsey_decay, color="#f97316", linewidth=2, label="T2 = 3.0")
ax3.fill_between(delta_vals, P_ramsey, alpha=0.1, color="#a78bfa")
ax3.set_xlabel("Detuning delta", color="white")
ax3.set_ylabel("Excitation probability", color="white")
ax3.set_title("Ramsey Fringes (T_free = 5.0)", color="#c4b5fd", fontsize=14)
ax3.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax3.tick_params(colors="white")
for spine in ax3.spines.values():
    spine.set_color("#7c3aed")

# Plot 4: Power-broadened lineshape
ax4 = axes[1, 1]
delta_scan = np.linspace(-10, 10, 500)
Gamma_line = 1.0
for s0, label, col in [(0.1, "I/Isat=0.1", "#a78bfa"),
                        (1.0, "I/Isat=1.0", "#c084fc"),
                        (10.0, "I/Isat=10", "#f97316"),
                        (100.0, "I/Isat=100", "#ef4444")]:
    R_sc = (Gamma_line / 2) * s0 / (1 + s0 + (2 * delta_scan / Gamma_line)**2)
    ax4.plot(delta_scan, R_sc, color=col, linewidth=2, label=label)

ax4.set_xlabel("Detuning delta/Gamma", color="white")
ax4.set_ylabel("Scattering rate (Gamma units)", color="white")
ax4.set_title("Power-Broadened Lineshape", color="#c4b5fd", fontsize=14)
ax4.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white", fontsize=9)
ax4.tick_params(colors="white")
for spine in ax4.spines.values():
    spine.set_color("#7c3aed")

plt.tight_layout()
plt.savefig("output.png", dpi=150, bbox_inches="tight", facecolor="#0a0a0a")
plt.show()
print("Plot saved: Optical Bloch equations - Rabi oscillations, Ramsey fringes, power broadening.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Conceptual Questions

Q1: Why does a two-level atom never reach full inversion in steady state?

In steady state, the atom reaches a balance between stimulated absorption (driving population up) and stimulated emission + spontaneous emission (driving it down). Since spontaneous emission always brings the atom back to the ground state, the maximum steady-state excited population is 50% — even with arbitrarily intense driving.

Q2: What is the physical meaning of T&sub1; vs T&sub2;?

T&sub1; governs how fast energy is exchanged with the environment (population relaxation). T&sub2; governs how fast phase information is lost (coherence dephasing). T&sub2; ≤ 2T&sub1; always, since losing energy also destroys coherence, but additional pure dephasing processes can make T&sub2; much shorter than T&sub1;.

Q3: Why are Ramsey fringes narrower than Rabi spectroscopy?

In Ramsey interferometry, the fringe width is 1/(2T) where T is the free evolution time, which can be much longer than the pulse duration. The resolution is set by how long the atom evolves freely between the two π/2 pulses, not by how long it interacts with the field.

Further Topics

Extensions and Connections

• Mollow triplet: The fluorescence spectrum of a strongly driven two-level atom splits into three peaks — the carrier and two sidebands at ±Ω_R. This is a direct manifestation of the dressed states and has been observed in single quantum dots and single molecules.
• Autler-Townes splitting: A strong coupling field splits the absorption line of a weak probe field in a three-level system. This is distinct from EIT (which involves quantum interference) though the spectra can appear similar.
• Superradiance and subradiance: N atoms within a wavelength of each other collectively interact with the radiation field. Superradiant states decay N times faster (Dicke superradiance) while subradiant states are long-lived dark states.
• Quantum regression theorem: Multi-time correlation functions of the emitted field obey the same equations as the OBE, enabling calculation of the fluorescence spectrum and higher-order correlations.