Optical Bloch Equations

A single atom sitting in a laser beam is the simplest laboratory in physics, and the optical Bloch equations are its complete equations of motion. They take three things the atom cannot supply on its own. From quantum mechanics they borrow the density operator $\rho$ — because a pure-state wavefunction can only rotate, never decay, yet a real atom coupled to the electromagnetic vacuum must eventually fall back to the ground state by spontaneous emission. From the electric-dipole picture they borrow the coupling $-\mathbf{d}\cdot\mathbf{E}$ , the handle by which a classical field grabs the atom's charge cloud. And from the theory of spontaneous emission they borrow the Einstein $A$ coefficient $\Gamma$ , the irreversible rate at which the excited state leaks photons into the surrounding modes. Stitch these together and you get a $2\times2$ matrix whose diagonal entries are the level populations and whose off-diagonal entries — the coherences — remember the phase of the atomic dipole relative to the driving light. That phase memory is the whole point. Rate equations keep only populations, so they can never describe an atom that absorbs a photon, holds its dipole ringing, and re-emits coherently. The Bloch equations keep the phase, and out of them fall Rabi flopping, saturation, power broadening, the natural-linewidth Lorentzian, radiation pressure, and the Mollow triplet — the whole vocabulary of laser spectroscopy and laser cooling. Two clocks govern everything: population relaxes at $\Gamma$ , coherence at $\gamma=\Gamma/2$ , exactly the $T_1$ and $T_2$ of magnetic resonance. This article builds the equations from the Hamiltonian, solves the two regimes that matter, and reads off the numbers that run a magneto-optical trap.

Key takeaways

The optical Bloch equations govern the $2\times2$ density matrix $\rho$ of a two-level atom driven by a near-resonant classical field, adding relaxation to the coherent Liouville–von Neumann dynamics.
In the rotating frame and rotating-wave approximation the drive is set by the Rabi frequency $\Omega=-\mathbf{d}\cdot\mathbf{E}_0/\hbar$ and the detuning $\delta=\omega_L-\omega_0$ .
Relaxation splits in two: the excited population decays at the spontaneous rate $\Gamma$ , but the coherence decays at $\gamma=\Gamma/2$ (collisions or laser noise add pure dephasing $\gamma_\mathrm{ph}$ ).
In steady state the excited fraction saturates, $\rho_{ee}=\tfrac12\,s/(1+s)$ , giving a power-broadened Lorentzian and a maximum photon-scattering rate $\Gamma/2$ .
The equations map exactly onto a real 3-vector — the Bloch vector $\mathbf R=(u,v,w)$ precessing about an effective field — so the atom behaves like a spin, with $T_1=1/\Gamma$ and $T_2=1/\gamma$ .
Key idea: write the two-level atom as a density matrix, evolve it coherently with the rotating-wave Hamiltonian $H=-\tfrac{\hbar}{2}(\Omega\sigma_++\Omega^*\sigma_-)-\hbar\delta\,|e\rangle\langle e|$ , then subtract population decay $\Gamma$ on $\rho_{ee}$ and coherence decay $\gamma=\Gamma/2+\gamma_\mathrm{ph}$ on $\rho_{eg}$ — everything about single-atom spectroscopy follows.

Where this lives today

Open any cold-atom laboratory and the optical Bloch equations are running silently underneath the hardware. A magneto-optical trap holds rubidium or cesium at tens of microkelvin by exploiting exactly equation (6): six red-detuned beams scatter photons at a velocity- and position-dependent rate, and the steep Lorentzian flank from Worked Example 2 turns that scattering into a restoring force. The Doppler limit $k_BT_D=\hbar\Gamma/2$ — 146 μK for rubidium — is the temperature the equations predict and the traps deliver. In optical atomic clocks, where a strontium or ytterbium transition is probed by a laser locked to a hertz-wide line, the coherence time $T_2$ sets how long the atomic dipole stays phase-coherent with the local oscillator, and hence the clock's stability. In trapped-ion and neutral-atom quantum computers, single-qubit gates are Rabi pulses — a $\pi$ -pulse flips the qubit, a $\pi/2$ -pulse builds a superposition — and the fidelity budget is written in $T_2$ . Drive a single ion hard and it emits the Mollow triplet, a central line flanked by sidebands at $\pm\Omega'$ : the fingerprint of an atom dressed by the field, a coherent effect no rate equation could produce.

The physical picture

Forget probabilities for a moment and picture the two-level atom as a tiny gyroscope. Collect its state into a real three-component arrow, the Bloch vector $\mathbf R=(u,v,w)$ : the components $u$ and $v$ are the in-phase and in-quadrature parts of the atomic dipole (the real and imaginary parts of $\rho_{eg}$ ), and $w=\rho_{ee}-\rho_{gg}$ is the population inversion. The south pole ( $w=-1$ ) is the ground state, the north pole ( $w=+1$ ) the fully excited state, and the equator a maximally coherent, radiating dipole. Drive and detuning together form an effective "magnetic field" $\boldsymbol\Omega_B=(\Omega,0,-\delta)$ , and the coherent dynamics is nothing but the arrow precessing about it — exactly a spin in an NMR magnet.

Figure

The two-level atom as a precessing Bloch vector. The drive and detuning form an effective field Ω_B = (Ω, 0, −δ) about which the arrow R precesses at the generalised Rabi rate √(Ω² + δ²); on resonance the field lies on the equator and sweeps the atom all the way from ground to excited. Relaxation then drags R back toward the ground state.

Coherent precession is only half the story. Left alone, the vacuum shortens the arrow: population leaks from north to south at the rate $\Gamma$ (the $T_1$ process), and the equatorial components — the dipole itself — dephase at $\gamma$ (the $T_2$ process). Because a coherence carries only one factor of the excited amplitude while a population carries two, radiative decay damps the coherence at exactly half the population rate, $\gamma=\Gamma/2$ ; collisions or laser phase noise add pure dephasing, so $\gamma=\Gamma/2+\gamma_\mathrm{ph}$ . The tug-of-war between precession and relaxation is the whole content of the equations: when the drive wins the arrow whirls (Rabi oscillations), and when relaxation wins it settles to a fixed tilt — steady-state saturation, capped at the equator because a two-level atom is at most half-excited on average.

Figure

Damped Rabi flopping of a two-level atom, Rb-87 D2 line (Γ = 2π × 6.07 MHz) driven at Ω = 5Γ = 2π × 30 MHz on resonance. The excited population oscillates at the ≈33 ns Rabi period, the first maximum reaching ρ_ee ≈ 0.80 (a damped π-pulse), and the envelope collapses at 3Γ/4 onto the steady-state value ρ_ee → 0.49.

The core equations

Any irreversible process — a photon lost to the vacuum, a collision that scrambles the dipole — is forbidden to a wavefunction, which can only evolve unitarily. The remedy is the density operator, whose equation of motion is a coherent commutator plus a dissipator. This master equation in Lindblad form is the trunk from which all the optical Bloch equations branch.

\frac{\mathrm d\rho}{\mathrm dt}=-\frac{i}{\hbar}[H,\rho]+\mathcal L[\rho] \qquad (1)

where: $\rho$ — the $2\times2$ density operator (diagonal = populations, off-diagonal = coherences); $H$ — the atom-plus-interaction Hamiltonian; $\mathcal L[\rho]$ — the dissipator injecting spontaneous decay and dephasing. The commutator preserves purity; the dissipator lets the atom forget. The Hamiltonian carries the bare atomic splitting and the electric-dipole coupling, taken in the dipole approximation. Written with Pauli operators and a real field of amplitude $\mathbf E_0$ at the laser frequency $\omega_L$ , it is time-dependent and inconveniently fast.

H=\frac{\hbar\omega_0}{2}\,\sigma_z-\mathbf d\cdot\mathbf E_0\cos(\omega_L t)\,(\sigma_++\sigma_-) \qquad (2)

where: $\omega_0$ — transition frequency; $\omega_L$ — laser frequency; $\mathbf d=\langle e|\mathbf d|g\rangle$ — transition dipole matrix element; $\mathbf E_0$ — real field amplitude; $\sigma_z,\sigma_\pm$ — inversion and raising/lowering operators. The first term stores the gap; the second is the lever the light uses to move population. Moving to a frame rotating at $\omega_L$ and discarding the terms oscillating at $2\omega_L$ (the rotating-wave approximation) removes the fast optical clock entirely, leaving a time-independent Hamiltonian set by just two numbers: how hard you drive, and how far off resonance you sit.

H_\mathrm{RWA}=-\hbar\delta\,|e\rangle\langle e|-\frac{\hbar}{2}\left(\Omega\,\sigma_++\Omega^*\sigma_-\right) \qquad (3)

where: $\delta=\omega_L-\omega_0$ — detuning from resonance; $\Omega=-\mathbf d\cdot\mathbf E_0/\hbar$ — the Rabi frequency, the on-resonance flopping rate. These two parameters are the effective-field components $\boldsymbol\Omega_B=(\Omega,0,-\delta)$ of the physical picture. Feeding $H_\mathrm{RWA}$ and the spontaneous-emission dissipator into equation (1) and reading off matrix elements gives the optical Bloch equations proper — one for the excited population, one for the coherence. Note the two distinct relaxation rates: $\Gamma$ on the population, $\gamma$ on the coherence.

\dot\rho_{ee}=\frac{i}{2}\left(\Omega^*\rho_{eg}-\Omega\rho_{ge}\right)-\Gamma\rho_{ee},\qquad \dot\rho_{eg}=-(\gamma+i\delta)\rho_{eg}+\frac{i}{2}\Omega\left(\rho_{ee}-\rho_{gg}\right) \qquad (4)

where: $\rho_{ee}$ — excited population, $\rho_{gg}=1-\rho_{ee}$ ; $\rho_{eg}=\rho_{ge}^*$ — slowly-varying coherence; $\Gamma=A_{21}=1/\tau$ — spontaneous decay rate; $\gamma=\Gamma/2+\gamma_\mathrm{ph}$ — coherence decay rate. The drive shuffles population through the coherence — kill the coherence and the populations freeze. Setting both derivatives to zero gives the long-time behaviour. The excited fraction climbs with drive strength but saturates: however hard you push, a two-level atom cannot exceed half-excitation on average, because stimulated emission catches up with absorption.

\rho_{ee}^{ss}=\frac12\,\frac{s}{1+s},\qquad s=\frac{\Omega^2/2}{\delta^2+\Gamma^2/4} \qquad (5)

where: $s$ — detuning-dependent saturation parameter; $\Omega^2/2$ — coherent driving strength; $\delta^2+\Gamma^2/4$ — Lorentzian denominator with half-width $\Gamma/2$ from $\gamma=\Gamma/2$ . As $s\to\infty$ , $\rho_{ee}^{ss}\to1/2$ . Each decaying atom emits one photon, so the steady scattering rate is $\Gamma$ times the excited population. Recast with the on-resonance saturation parameter $s_0=I/I_\mathrm{sat}$ , this is the master formula of laser cooling and saturation spectroscopy.

R_\mathrm{sc}=\Gamma\rho_{ee}^{ss}=\frac{\Gamma}{2}\,\frac{s_0}{1+s_0+(2\delta/\Gamma)^2},\qquad s_0=\frac{I}{I_\mathrm{sat}}=\frac{2\Omega^2}{\Gamma^2} \qquad (6)

where: $R_\mathrm{sc}$ — photons scattered per second; $s_0$ — on-resonance saturation parameter; $I_\mathrm{sat}=\pi hc\Gamma/3\lambda^3$ — saturation intensity. At high power on resonance, $R_\mathrm{sc}\to\Gamma/2$ — the maximum scattering rate of any two-level atom. Finally the system collapses into the real language of magnetic resonance. Rewriting the complex $\rho$ as the Bloch vector $\mathbf R=(u,v,w)$ turns equations (4) into a precession-plus-damping law identical, line for line, to the Bloch equation of NMR.

\dot{\mathbf R}=\boldsymbol\Omega_B\times\mathbf R-\begin{pmatrix}u/T_2\\ v/T_2\\ (w-w_\mathrm{eq})/T_1\end{pmatrix} \qquad (7)

where: $\mathbf R=(u,v,w)$ — dipole quadratures and inversion; $\boldsymbol\Omega_B=(\Omega,0,-\delta)$ — torque vector; $T_1=1/\Gamma$ , $T_2=1/\gamma$ — longitudinal and transverse relaxation times. The cross product is pure precession; the column vector is the vacuum shortening the arrow toward the ground state.

Derivation AFrom the dipole Hamiltonian to the rotating-wave Bloch equations

The goal is to turn the fast, time-dependent lab-frame Hamiltonian (2) into the time-independent form (3), then read the optical Bloch equations (4) straight off the master equation. The trick is a frame co-rotating with the laser, after which only the slowly-varying near-resonant coupling survives while the counter-rotating part averages away. Three moves, then attach the dissipator that carries spontaneous decay.

Step 1. Apply $U=e^{i\omega_L t\,|e\rangle\langle e|}$ and transform via $\tilde H=UHU^\dagger-i\hbar U\dot U^\dagger$ . The extra $-i\hbar U\dot U^\dagger$ piece subtracts $\hbar\omega_L$ from the excited energy, so the bare splitting $\hbar\omega_0$ becomes the small detuning term $-\hbar\delta\,|e\rangle\langle e|$ . Why: co-rotating with the drive freezes out the huge optical frequency and keeps only the tiny difference the physics depends on.

Step 2. Split the drive with $\cos(\omega_L t)=\tfrac12(e^{i\omega_L t}+e^{-i\omega_L t})$ . In the rotating frame $\sigma_\pm$ pick up $e^{\pm i\omega_L t}$ , so the coupling contains one stationary term and one oscillating at $2\omega_L$ . Why: only the stationary co-rotating pairing conserves quasi-energy near resonance.

Step 3. Drop the $2\omega_L$ terms — the rotating-wave approximation, valid whenever $\Omega,\delta\ll\omega_0$ — leaving the static coupling $-\tfrac{\hbar}{2}(\Omega\sigma_++\Omega^*\sigma_-)$ that completes equation (3). Why: over one optical cycle the fast terms wash out; keeping them adds only the negligible high-frequency Bloch–Siegert ripple.

Step 4. Attach the Lindblad dissipator $\mathcal L[\rho]=\Gamma(\sigma_-\rho\sigma_+-\tfrac12\{\sigma_+\sigma_-,\rho\})$ and evaluate matrix elements: the jump term feeds the ground state, and the anticommutator removes $\Gamma$ from $\rho_{ee}$ but exactly $\Gamma/2$ from $\rho_{eg}$ , yielding equations (4). Why: the anticommutator acts once on each index of the coherence, giving half the population rate — the microscopic origin of $\gamma=\Gamma/2$ .

Go deeper · Derivation B · 1 minSteady state, saturation, and the power-broadened line

Derivation BSteady state, saturation, and the power-broadened line

The aim is to solve the Bloch equations in the long-time limit and extract the three results that dominate laser spectroscopy: the saturating excited fraction (5), the ceiling scattering rate $\Gamma/2$ , and the power-broadened linewidth $\Gamma\sqrt{1+s_0}$ . The method is to set every derivative to zero, solve for the coherence, and substitute back — and because the coherence equation is linear, this is exact, with no perturbation theory needed.

Step 1. Set $\dot\rho_{eg}=0$ : $\rho_{eg}^{ss}=\dfrac{i\Omega/2}{\gamma+i\delta}(\rho_{ee}-\rho_{gg})$ . Why: the coherence adiabatically follows the populations once transients die, and the complex denominator $\gamma+i\delta$ will produce both absorptive and dispersive line shapes.

Step 2. Insert into $\dot\rho_{ee}=0$ , use $\rho_{gg}=1-\rho_{ee}$ and $\gamma=\Gamma/2$ , and solve the single linear equation to get $\rho_{ee}^{ss}=\tfrac12\,s/(1+s)$ with $s=(\Omega^2/2)/(\delta^2+\Gamma^2/4)$ . Why: eliminating the coherence reduces two coupled equations to one, and the Lorentzian falls out of $|\gamma+i\delta|^2$ .

Step 3. As $s\to\infty$ , $\rho_{ee}^{ss}\to1/2$ , so $R_\mathrm{sc}=\Gamma\rho_{ee}^{ss}\to\Gamma/2$ . Why: at saturation stimulated emission exactly cancels absorption, pinning the inversion at zero and the excited fraction at one half.

Step 4. On resonance the scattering profile has full width $\Gamma\sqrt{1+s_0}$ , not $\Gamma$ : the $s_0$ term inside the denominator of (6) broadens the line as power rises. Why: harder driving shortens the effective coherence time, and by time-frequency reciprocity a shorter coherence means a broader line — power broadening, distinct from the fixed natural width.

Go deeper · Derivation C · 1 minOn-resonance Rabi oscillations with damping

Derivation COn-resonance Rabi oscillations with damping

The purpose is to show what rate equations can never reproduce: on resonance the excited population does not merely relax, it rings at the Rabi frequency inside a decaying envelope. We set $\delta=0$ , note that the inversion couples only to the imaginary part of the coherence, and fold the two first-order equations into one damped-driven oscillator whose solution is a decaying cosine settling onto the saturation value.

Step 1. Set $\delta=0$ with real $\Omega$ : the population equation becomes $\dot\rho_{ee}=-\Omega\,\mathrm{Im}\,\rho_{eg}-\Gamma\rho_{ee}$ , so only $\mathrm{Im}\,\rho_{eg}$ drives the population. Why: on resonance the dispersive part decouples, leaving a closed pair — inversion and one quadrature.

Step 2. With $\dot{(\mathrm{Im}\,\rho_{eg})}=-\gamma\,\mathrm{Im}\,\rho_{eg}+\tfrac12\Omega(\rho_{ee}-\rho_{gg})$ and $\gamma=\Gamma/2$ , differentiate and substitute to eliminate the coherence: $\ddot\rho_{ee}+\tfrac{3\Gamma}{2}\dot\rho_{ee}+(\Omega^2+\tfrac{\Gamma^2}{2})\rho_{ee}=\tfrac{\Omega^2}{2}$ . Why: two damped first-order equations always fold into one second-order equation — here literally a damped driven oscillator.

Step 3. The particular solution is $\rho_{ee}(\infty)=\dfrac{\Omega^2/2}{\Omega^2+\Gamma^2/2}=\dfrac{\Omega^2}{2\Omega^2+\Gamma^2}$ , exactly $\tfrac12 s/(1+s)$ at $\delta=0$ . Why: the long-time limit of the oscillator must match the algebraic steady state, and it does.

Step 4. The homogeneous part oscillates at $\Omega'=\sqrt{\Omega^2-(\Gamma/4)^2}$ inside an envelope $e^{-3\Gamma t/4}$ :

\rho_{ee}(t)=\frac{\Omega^2}{2\Omega^2+\Gamma^2}\left[1-\Big(\cos\Omega' t+\tfrac{3\Gamma}{4\Omega'}\sin\Omega' t\Big)e^{-3\Gamma t/4}\right].

For $\Omega\gg\Gamma$ the atom flops at nearly $\Omega$ ; for $\Omega\lesssim\Gamma/4$ it is overdamped and merely relaxes. Why: the average of the damping rates sets the envelope $3\Gamma/4$ , and $\Omega'$ falls below $\Omega$ only when damping rivals the drive.

Limiting cases

The steady state interpolates between every regime of atom-light interaction as drive strength and detuning vary.

Case	Result	Meaning
Weak field (linear response)	$\rho_{ee}\approx\dfrac{\Omega^2/4}{\delta^2+\Gamma^2/4}$	For $\Omega\ll\Gamma$ the excited fraction is a bare Lorentzian of FWHM $\Gamma$ — the natural-linewidth profile, linear in intensity.
Strong field (saturation)	$\rho_{ee}\to\tfrac12,\ R_\mathrm{sc}\to\Gamma/2$	For $\Omega\gg\Gamma$ populations equalise; scattering caps at $\Gamma/2$ , the maximum for a two-level atom.
Power broadening	$\Delta\omega_\mathrm{FWHM}=\Gamma\sqrt{1+s_0}$	The linewidth grows with $s_0=I/I_\mathrm{sat}$ even as the peak scattering rate saturates.
Fast dephasing (rate-equation limit)	$\dot\rho_{ee}=B\rho(\omega_0)(\rho_{gg}-\rho_{ee})-\Gamma\rho_{ee}$	When $\gamma\gg\Omega$ , adiabatic elimination of $\rho_{eg}$ recovers the Einstein rate equations — coherent effects vanish.
Lossless drive ( $\Gamma\to0$ )	$\rho_{ee}(t)=\dfrac{\Omega^2}{\Omega_R^2}\sin^2\tfrac{\Omega_R t}{2}$	Undamped generalised Rabi flopping, $\Omega_R=\sqrt{\Omega^2+\delta^2}$ — the coherent-control limit behind $\pi$ and $\pi/2$ pulses.

Common trappopulation and coherence both decay at Γ

The single most common error is to give the coherence the same decay rate as the population. They differ: the excited population relaxes at the full spontaneous rate $\Gamma$ , but the optical coherence relaxes at $\gamma=\Gamma/2$ for a purely radiatively broadened transition — because a coherence carries only one factor of the excited amplitude, and amplitude decays at half the rate of probability. Only collisions or laser phase noise add pure dephasing $\gamma_\mathrm{ph}$ on top, so $\gamma=\Gamma/2+\gamma_\mathrm{ph}\ge\Gamma/2$ . Two related traps: dropping the coherences and using rate equations silently discards Rabi oscillations, dispersion, and every coherent effect; and factor-of-two confusion in the Rabi frequency — always check that $\rho_{ee}$ flops at $\Omega$ , not $\Omega/2$ , on resonance.

Worked examples

1 · How hard can light push an atom? (Rb-87 laser cooling)

The rubidium-87 D2 cooling transition has $\Gamma=2\pi\times6.07\ \mathrm{MHz}$ , $\lambda=780\ \mathrm{nm}$ , $m=87\,u$ . Using only the Bloch steady state, find the saturation intensity, the maximum scattering rate, the maximum force and its deceleration, and the recoil velocity per photon.

Step 1. Saturation intensity: $I_\mathrm{sat}=\pi hc\Gamma/3\lambda^3$ . With $\Gamma=3.81\times10^7\ \mathrm{s^{-1}}$ , $I_\mathrm{sat}=\pi(6.63\times10^{-34})(3.00\times10^8)(3.81\times10^7)/[3(7.80\times10^{-7})^3]=16.7\ \mathrm{W/m^2}=\mathbf{1.67\ mW/cm^2}$ — the standard Rb value.

Step 2. Maximum scattering rate: equation (6) caps at $R_\mathrm{sc}^\mathrm{max}=\Gamma/2=1.91\times10^7\ \mathrm{s^{-1}}$ — about 19 million photons per second, one every $2\tau$ . No extra power raises this ceiling.

Step 3. Radiation force: each photon delivers $\hbar k$ , with $k=2\pi/\lambda=8.05\times10^6\ \mathrm{m^{-1}}$ so $\hbar k=8.49\times10^{-28}\ \mathrm{kg\,m/s}$ . Then $F_\mathrm{max}=\hbar k\Gamma/2=\mathbf{1.62\times10^{-20}\ N}$ ; dividing by $m=1.44\times10^{-25}\ \mathrm{kg}$ gives $a_\mathrm{max}=\mathbf{1.1\times10^{5}\ m/s^2}\approx\mathbf{1.1\times10^{4}\,g}$ .

Step 4. What that buys: an atom from a $300\ \mathrm{m/s}$ beam stops in $2.7\ \mathrm{ms}$ over $v^2/2a=\mathbf{0.40\ m}$ — the length of a real Zeeman slower. Each photon changes the speed by only $v_\mathrm{rec}=\hbar k/m=\mathbf{5.9\ mm/s}$ , so stopping takes $\approx5\times10^{4}$ scattering events, and the Doppler limit $k_BT_D=\hbar\Gamma/2$ gives $T_D=\mathbf{146\ \mu K}$ .

2 · Saturation and power broadening

The same transition is driven at $I=3I_\mathrm{sat}$ ( $s_0=3$ ). Find the on-resonance excited population and scattering rate, the power-broadened linewidth, and the scattering rate at detuning $\delta=2\pi\times10\ \mathrm{MHz}$ .

Step 1. On resonance $s=s_0=3$ , so equation (5) gives $\rho_{ee}=\tfrac12\,s_0/(1+s_0)=\tfrac12\cdot\tfrac34=\mathbf{0.375}$ — three-quarters of the way to the ceiling. The scattering rate is $R_\mathrm{sc}=\Gamma\rho_{ee}=\mathbf{1.43\times10^{7}\ s^{-1}}$ .

Step 2. Power-broadened width: $\Gamma\sqrt{1+s_0}=\Gamma\sqrt4=2\Gamma=\mathbf{12.1\ MHz}$ , double the $6.07\ \mathrm{MHz}$ natural width — even though the peak scattering rate is capped.

Step 3. Off resonance, $2\delta/\Gamma=2(10)/6.07=3.30$ so its square is $10.9$ ; equation (6) gives $R_\mathrm{sc}=(1.91\times10^7)(3)/(1+3+10.9)=\mathbf{3.8\times10^{6}\ s^{-1}}$ and $\rho_{ee}=\mathbf{0.10}$ . Detuning by under two linewidths cut the scattering fourfold — the steep Lorentzian flank is exactly what makes a magneto-optical trap position-selective.

Physics of the portrait

The optical Bloch equations carry a borrowed name. Their algebra was written not for atoms and light but for nuclear spins in a magnet — and the transplant was so exact that the optical version simply inherited the equations, the geometry, and the two relaxation times whole.

Origins · 1 minFelix Bloch and how this was discovered

Felix Bloch

1905–1983 · magnetic resonance and the equations that became optical

Felix Bloch, a Swiss-American physicist and Stanford's first Nobel laureate, gave his name to these equations through work that had nothing to do with lasers, which did not yet exist. In 1946, describing the newly discovered nuclear magnetic resonance, Bloch wrote a phenomenological equation of motion for the bulk nuclear magnetisation: a precession about the applied field plus two independent relaxation terms — a longitudinal time $T_1$ for the component along the field to return to equilibrium, and a transverse time $T_2$ for the perpendicular components to dephase. That $T_1$ / $T_2$ split, and the picture of a vector precessing and relaxing, is the entire skeleton of the modern optical theory. When the maser and laser arrived, physicists saw that a driven two-level atom is formally a spin-½ in an effective field; Feynman, Vernon, and Hellwarth made the mapping explicit in 1957, and the resulting density-matrix equations were christened the optical Bloch equations. The lesson in the name is that a two-level quantum system is a two-level quantum system, whether its splitting is a nuclear Zeeman gap of kilohertz or an optical transition of hundreds of terahertz: the same arrow precesses, and the same two clocks tick.

History

1937

— Rabi derives the flopping formula for a spin in a rotating magnetic field

1946

— Bloch introduces the relaxation times T₁ and T₂ in nuclear magnetic resonance

1957

— Feynman, Vernon, and Hellwarth cast any two-level system as a precessing Bloch vector

1969

— Mollow predicts the three-peaked fluorescence spectrum of a strongly driven atom

1974

— Schuda, Stroud, and Hercher observe the Mollow triplet in atomic sodium

Going further

Texts. Foot, Atomic Physics (2005), Ch. 7 (the cleanest undergraduate derivation) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 and 8 · Cohen-Tannoudji, Dupont-Roc and Grynberg, Atom-Photon Interactions (the master-equation treatment) · Metcalf and van der Straten, Laser Cooling and Trapping (the scattering-force applications) · Allen and Eberly, Optical Resonance and Two-Level Atoms (the Bloch-vector geometry) · Demtröder, Laser Spectroscopy, Vol. 1.

Provenance. The equations follow the standard treatment in Foot (Ch. 7), Loudon (Ch. 2, 8), and Cohen-Tannoudji et al. A verifier should confirm: (i) the RWA Hamiltonian and Rabi frequency $\Omega=-\mathbf d\cdot\mathbf E_0/\hbar$ ; (ii) the steady state $\rho_{ee}=\tfrac12 s/(1+s)$ with $s=(\Omega^2/2)/(\delta^2+\Gamma^2/4)$ ; (iii) $\gamma=\Gamma/2$ from the Lindblad anticommutator; (iv) power-broadened FWHM $=\Gamma\sqrt{1+s_0}$ ; (v) maximum scattering rate $\Gamma/2$ and Doppler limit $k_BT_D=\hbar\Gamma/2$ ; (vi) $I_\mathrm{sat}=\pi hc\Gamma/3\lambda^3=1.67\ \mathrm{mW/cm^2}$ for Rb D2, with $R_\mathrm{sc}^\mathrm{max}=\Gamma/2=1.9\times10^7\ \mathrm{s^{-1}}$ , $a_\mathrm{max}=1.1\times10^5\ \mathrm{m/s^2}$ , $v_\mathrm{rec}=5.9\ \mathrm{mm/s}$ , $T_D=146\ \mu\mathrm K$ (standard Rb-87 values). The on-resonance damped-Rabi solution has steady value $\Omega^2/(2\Omega^2+\Gamma^2)$ , frequency $\Omega'=\sqrt{\Omega^2-(\Gamma/4)^2}$ , and envelope $e^{-3\Gamma t/4}$ , matching Loudon.

Builds on: Electric Dipole Approximation · Transition Dipole Moment · Spontaneous Emission · Einstein A and B Coefficients · Radiative Lifetime and Natural Linewidth. Leads to: Lorentzian Line Shape · Resonant Absorption · Stimulated Emission.

Ask the tutor: ask me to solve the optical Bloch equations for a moving atom and watch the Doppler cooling force emerge from the same $R_\mathrm{sc}(\delta)$ you just used.

Why do we need a density matrix instead of a wavefunction?

Because a wavefunction can only evolve unitarily — it rotates, it never decays. A real atom coupled to the vacuum does decay by spontaneous emission, an irreversible process no Schrödinger equation for the atom alone can produce. Promoting the state to a density operator $\rho$ fixes this: its diagonal elements are populations, its off-diagonal elements coherences, and the Lindblad dissipator can drain the excited state and dephase the coherence while keeping $\mathrm{Tr}\,\rho=1$ . In effect it describes an open system — atom plus vacuum bath — without tracking the infinitely many photon modes it might emit into. The moment you need decay, dephasing, or a mixed state, the wavefunction is not enough.

What physically is the difference between Γ and γ?

They measure two kinds of forgetting. $\Gamma$ is the population decay rate — how fast the atom falls to the ground state, setting $T_1=1/\Gamma$ . $\gamma$ is the coherence decay rate — how fast the dipole loses phase relative to the drive, setting $T_2=1/\gamma$ . For a purely radiatively broadened transition $\gamma=\Gamma/2$ , so $T_2=2T_1$ , because a coherence carries only one excited amplitude and amplitude decays at half the rate of probability. Collisions and laser phase noise add pure dephasing $\gamma_\mathrm{ph}$ that scrambles the phase without moving population, giving $\gamma=\Gamma/2+\gamma_\mathrm{ph}$ and $T_2<2T_1$ . It is $T_2$ , not $T_1$ , that sets the measured linewidth.

Why can't a two-level atom be more than half-excited in steady state?

In steady state the upward rate (absorption, proportional to $\rho_{gg}$ ) must balance the downward stimulated rate (proportional to $\rho_{ee}$ ). Since absorption and stimulated emission share the same cross-section, they balance precisely when $\rho_{ee}=\rho_{gg}=1/2$ . Push harder and both rates rise together, but their ratio cannot invert — you can only approach equal populations, never exceed them. This is why a two-level system can never sustain inversion and never lase: a working laser needs a third or fourth level so the pumped and lasing transitions differ, letting one population build up while the other drains.

What exactly is the rotating-wave approximation and when does it fail?

The drive $\cos\omega_L t$ splits into a co-rotating part that nearly matches the atomic phase and a counter-rotating part oscillating at roughly $2\omega_L$ . The RWA keeps the first and drops the second, since a term at twice the optical frequency averages to nothing over the slow atomic dynamics. This is superb whenever $\Omega,|\delta|\ll\omega_0$ — true by many orders of magnitude for ordinary lasers on optical transitions. It fails only under extreme driving, where the dropped term produces a small energy shift (the Bloch–Siegert shift), and in the ultrastrong-coupling regimes of circuit QED where $\Omega$ approaches $\omega_0$ .

How is the Rabi frequency related to the laser intensity?

Through the saturation parameter: $s_0=2\Omega^2/\Gamma^2=I/I_\mathrm{sat}$ , so the Rabi frequency scales as the square root of intensity, $\Omega=\Gamma\sqrt{I/2I_\mathrm{sat}}$ . This is intuitive — $\Omega\propto E_0$ , the field amplitude, and intensity $I\propto E_0^2$ . A practical consequence: to reach $\Omega=\Gamma$ you need only $I=I_\mathrm{sat}/2$ , under a milliwatt per square centimetre for rubidium. Doubling the intensity does not double the Rabi frequency; it multiplies it by $\sqrt2$ . This square-root scaling is why very high intensities are needed for the deep saturation and fast flopping of coherent-control experiments.

Why do the Bloch equations show Rabi oscillations but rate equations don't?

Because rate equations track only populations and discard the coherences — the phase memory of the dipole. Rabi flopping is fundamentally coherent: the atom absorbs a photon, its dipole keeps ringing in a definite phase relationship with the drive, and that ringing coherently drives the population back down, then up again. That cyclic exchange lives entirely in the off-diagonal $\rho_{eg}$ , which rate equations do not carry. When the coherence decays much faster than it is driven ( $\gamma\gg\Omega$ ), eliminating $\rho_{eg}$ adiabatically collapses the Bloch equations to rate equations — and the oscillations vanish, because you have discarded the very quantity that produced them.