Populations and Coherences

Quantum mechanics hands optics a single object for an atom that is neither a definite pure state nor a purely classical one: the density operator $\hat\rho=\sum_k p_k\,|\psi_k\rangle\langle\psi_k|$ , the quantum-statistical generalisation of the wavefunction carrying both superposition and classical ignorance in one device. Pick a basis $\{|n\rangle\}$ and its matrix elements split into two kinds with completely different physics. The diagonal $\rho_{nn}$ are the populations — probabilities of finding the atom in level $n$ , the quantities the Einstein rate equations already balance. The off-diagonal $\rho_{nm}$ are the coherences — complex numbers storing the definite phase relationship between amplitudes of different levels, which no rate equation ever sees. That phase is not decoration. It makes a driven atom radiate a macroscopic polarisation — the source of absorption, dispersion and gain — and interferes to give quantum beats, photon echoes, Ramsey fringes and electromagnetically induced transparency. Keep only populations and you have the rate-equation limit, valid only when coherences dephase fast enough to follow the field instantaneously. The two relax differently: populations by spontaneous emission at the Einstein rate $\Gamma$ , coherences at the larger $\gamma=\tfrac12\Gamma+\gamma_\mathrm{coll}$ that adds phase-scrambling from collisions. Erasing coherences without touching populations is decoherence — the bridge from a pure superposition to a classical mixture, and the origin of the Lorentzian line.

Key takeaways

The density operator $\hat\rho$ describes an atom that may be a statistical mixture; diagonal $\rho_{nn}$ are populations (occupation probabilities, real and summing to one), off-diagonal $\rho_{nm}$ are coherences (complex, bounded by $|\rho_{nm}|\le\sqrt{\rho_{nn}\rho_{mm}}$ , saturated only for a pure state).
Evolution follows $\mathrm i\hbar\,\dot{\hat\rho}=[\hat H,\hat\rho]$ ; adding relaxation gives the optical Bloch equations, populations relaxing at $\Gamma$ , coherences dephasing at $\gamma=\tfrac12\Gamma+\gamma_\mathrm{coll}$ .
The radiating dipole $\langle\hat d\rangle=2\mu\,\mathrm{Re}\,\rho_{eg}$ is a coherence, so all optical response — index, absorption, gain — lives off-diagonal; populations only set how strongly.
Decoherence destroys coherences while leaving populations untouched, turning a superposition into a mixture — the origin of collisional broadening and the quantum-to-classical transition.
Key idea: populations $\rho_{nn}$ are occupation probabilities that obey rate equations; coherences $\rho_{nm}$ are complex superposition phases that generate the radiating dipole and decay by dephasing — keeping both is what separates quantum optics from the Einstein rate-equation picture.

The core insight

Populations $\rho_{nn}$ are occupation probabilities that obey rate equations; coherences $\rho_{nm}$ are complex superposition phases that generate the radiating dipole and decay by dephasing — keeping both is what distinguishes quantum optics from the Einstein rate-equation picture.

Where this lives today

Nowhere is the distinction more operational than in quantum computing, where Bloch's two relaxation times are the headline figures of merit for a qubit: $T_1$ the population (energy) relaxation time, $T_2$ the coherence time, with $T_2\le 2T_1$ just $\gamma\ge\Gamma/2$ read backwards. Pushing $T_2$ up by shielding qubits from phase noise is the central fight: superconducting transmons now reach hundreds of microseconds, trapped-ion and nuclear-spin qubits seconds to minutes. A Ramsey sequence — split a superposition, precess in the dark, read fringe contrast versus delay — directly measures coherence decay, and the same sequence on an optical clock transition is how the best clocks reach fractional stabilities near $10^{-18}$ : the narrower the coherence linewidth $2\gamma$ , the finer the clock. Coherences also do constructive work: in electromagnetically induced transparency a laser-prepared ground-state coherence (a dark state) makes an opaque vapour transparent and slows light dramatically — Lene Hau's group reached 17 metres per second in 1999 — and the same engineering stores photons in quantum memories. From a spectral lamp's linewidth to a quantum processor's fidelity, one population rate and one coherence rate set what is possible.

The physical picture

Picture a two-level atom as an arrow — the Bloch vector — inside a unit sphere. Its vertical component is the population inversion $w=\rho_{ee}-\rho_{gg}$ : how far it tilts between ground (south pole) and excited (north). Its horizontal component is the coherence, $u=2\,\mathrm{Re}\,\rho_{eg}$ , $v=2\,\mathrm{Im}\,\rho_{eg}$ : a phase clock hand saying where in its cycle the superposition sits. Rate equations track only how high the arrow points; the density matrix also tracks the clock, deciding whether many atoms radiate in step or wash out. That clock is the whole difference between a superposition and a mixture: an equal coherent superposition of $|e\rangle$ and $|g\rangle$ and an incoherent fifty-fifty mixture share populations $\rho_{ee}=\rho_{gg}=\tfrac12$ , yet the first radiates a definite dipole (vector on the equator), the second nothing (zero horizontal length). Everything optical hangs on the off-diagonal part.

Figure

The Bloch vector splits the state into a population part w = ρ_ee − ρ_gg (vertical) and a coherence part u = 2 Re ρ_eg (horizontal). Pure states reach the surface; decoherence shrinks the horizontal part toward the axis at rate γ while the vertical population is untouched, turning a pure superposition into a classical mixture

Decoherence is exactly this erosion of the horizontal component while the vertical survives. A collision that kicks the optical phase leaves the atom just as likely excited — energy untouched — but scrambles the clock; that is why elastic collisions broaden lines without shortening lifetimes. And because the observable dipole is a coherence decaying at $\gamma$ , that decay sets the line width. The next figure makes the asymmetry concrete for the sodium D2 transition.

Figure

Populations and coherences of the sodium D2 line relax differently. The excited population ρ_ee decays with lifetime τ = 16.2 ns (rate Γ = 6.17×10⁷ per second). Buffer-gas collisions make the coherence decay faster, at γ = Γ∕2 + γ_coll = 8.09×10⁷ per second, so T₂ = 12.4 ns is shorter than τ though collisions change no level occupation

The core equations

The starting object is the density operator, a classical average over pure states $|\psi_k\rangle$ each prepared with probability $p_k$ . Picking a basis $\{|n\rangle\}$ makes it a matrix whose diagonal and off-diagonal entries are the populations and coherences — the whole distinction lives in this one definition.

\hat\rho=\sum_k p_k\,|\psi_k\rangle\langle\psi_k|,\qquad \rho_{nm}=\langle n|\hat\rho|m\rangle \qquad (1)

where: $p_k$ — classical probability of preparing pure state $|\psi_k\rangle$ , with $\sum_k p_k=1$ ; $\rho_{nn}$ — population, probability of finding the system in $|n\rangle$ ; $\rho_{nm}$ — coherence between $|n\rangle$ and $|m\rangle$ (complex, $n\ne m$ ); $|n\rangle$ — orthonormal basis. The diagonal is what a level-population measurement returns; the off-diagonal exists only when the ensemble shares a definite phase. Three properties follow from $\hat\rho$ being Hermitian, positive and trace-one, making populations behave like probabilities: they are real and nonnegative, sum to one, and swapping a coherence's indices conjugates it.

\rho_{nn}\ge 0,\qquad \sum_n\rho_{nn}=\mathrm{Tr}\,\hat\rho=1,\qquad \rho_{mn}=\rho_{nm}^{*} \qquad (2)

where: $\mathrm{Tr}$ — the trace, sum of diagonal elements; $\rho_{nm}^{*}$ — complex conjugate; Hermiticity $\hat\rho=\hat\rho^\dagger$ guarantees real, positive populations. The diagonal is a genuine probability distribution; the coherences carry the non-classical information in conjugate pairs. Coherences cannot be arbitrarily large: positivity of $\hat\rho$ ties every coherence to the two populations it connects. This bound is the sharpest statement of how much phase information an ensemble can hold; its saturation defines a pure state.

|\rho_{nm}|\le\sqrt{\rho_{nn}\,\rho_{mm}} \qquad (3)

where: $|\rho_{nm}|$ — magnitude of a coherence. The ratio $|\rho_{nm}|/\sqrt{\rho_{nn}\rho_{mm}}\in[0,1]$ is a degree of coherence: one for a pure superposition, zero for an incoherent mixture. Time evolution of the closed system obeys the density-matrix form of the Schrödinger equation, the Liouville–von Neumann equation — the master clock that makes populations flow and coherences precess before relaxation is added.

\mathrm i\hbar\,\frac{\mathrm d\hat\rho}{\mathrm dt}=[\hat H,\hat\rho] \qquad (4)

where: $\hat H$ — total Hamiltonian (atom plus field); $[\,\cdot\,,\cdot\,]$ — commutator. Its structure sorts the two: a diagonal $\hat H$ leaves populations fixed and makes each coherence precess at the Bohr frequency $\omega_{nm}=(E_n-E_m)/\hbar$ — free coherences oscillate, free populations stand still. The reason optics cares which elements are which is the induced dipole: the expectation of $\hat d$ with $\hat\rho$ , using that $\hat d$ connects only opposite-parity states, is built purely from a coherence.

\langle\hat d\rangle=\mathrm{Tr}(\hat\rho\,\hat d)=\mu\,(\rho_{eg}+\rho_{ge})=2\mu\,\mathrm{Re}\,\rho_{eg} \qquad (5)

where: $\hat d$ — electric-dipole operator, $\langle e|\hat d|g\rangle=\mu$ ; $\mu$ — transition dipole matrix element (chosen real); $\rho_{eg}$ — excited–ground coherence. This is the crux: the polarisation $P=N\langle\hat d\rangle$ in Maxwell's equations is set by coherences alone; populations do not radiate, only setting the amplitude to which the coherence responds. Coupling to reservoirs — vacuum field, thermal photons, collisions — adds dissipation that acts unequally: populations decay by spontaneous emission, coherences dephase at least half as fast, plus any pure phase-scrambling.

\dot\rho_{ee}=-\Gamma\rho_{ee}+\ldots,\qquad \dot\rho_{eg}=-(\mathrm i\omega_0+\gamma)\rho_{eg}+\ldots,\qquad \gamma=\tfrac12\Gamma+\gamma_\mathrm{coll} \qquad (6)

where: $\Gamma$ — population decay rate $=A_{eg}=1/\tau=1/T_1$ ; $\gamma$ — coherence dephasing rate $=1/T_2$ ; $\gamma_\mathrm{coll}$ — pure dephasing from collisions or phase noise; $\omega_0$ — atomic (Bohr) frequency. The relation $\gamma=\tfrac12\Gamma+\gamma_\mathrm{coll}$ is the article's most important line: even collision-free, $\gamma=\Gamma/2$ , and collisions only push it higher. Driving the atom and letting it settle gives the steady-state coherence in the rotating frame — the quantity that, times $N\mu$ , becomes the polarisation, and in the weak-field limit the susceptibility $\chi(\omega)=N\mu\langle\rho_{ge}\rangle/\varepsilon_0 E_0\propto(\Delta+\mathrm i\gamma)/(\Delta^2+\gamma^2)$ .

\rho_{eg}^{\,\mathrm{ss}}=\frac{\tfrac12\Omega\,(\Delta-\mathrm i\gamma)}{\Delta^2+\gamma^2+\tfrac12\Omega^2\,\gamma/\Gamma}\;(\rho_{gg}-\rho_{ee}) \qquad (7)

where: $\Omega=\mu E_0/\hbar$ — Rabi frequency; $\Delta=\omega-\omega_0$ — detuning; $\rho_{gg}-\rho_{ee}$ — population difference driving the response. The real part gives dispersion, the imaginary part absorption (a Lorentzian of full width $2\gamma$ — the Fourier transform of a coherence decaying at $\gamma$ ); the $\tfrac12\Omega^2\gamma/\Gamma$ term is power broadening as the drive saturates.

Derivation AOptical Bloch equations for a driven two-level atom

The goal is to turn the Liouville–von Neumann equation into concrete, closed equations for the population difference and coherence of a driven two-level atom, so the distinct damping of the two — populations at $\Gamma$ , coherences at $\gamma$ — appears explicitly rather than by decree. We take $\hat H=\hat H_0+\hat V$ with $\hat H_0=\hbar\omega_0|e\rangle\langle e|$ and dipole coupling $\hat V=-\hat d\,E(t)$ , $E(t)=E_0\cos\omega t$ , in the electric-dipole and rotating-wave approximations, adding relaxation last.

Step 1. Insert $\hat H$ into $\dot{\hat\rho}=\tfrac{1}{\mathrm i\hbar}[\hat H,\hat\rho]$ and read off elements: the population feeds from the coherence, $\dot\rho_{ee}=\tfrac{\mathrm i}{\hbar}(V_{eg}\rho_{ge}-\rho_{eg}V_{ge})$ , and the coherence precesses at $\omega_0$ while driven by the population difference, $\dot\rho_{eg}=-\mathrm i\omega_0\rho_{eg}+\tfrac{\mathrm i}{\hbar}V_{eg}(\rho_{gg}-\rho_{ee})$ . Why: this couples "how much" and "in what phase" — neither evolves alone once the field is on.

Step 2. Move to the rotating frame, $\rho_{eg}=\tilde\rho_{eg}\,e^{-\mathrm i\omega t}$ , discarding the counter-rotating $2\omega$ terms (rotating-wave approximation). Define $\Omega=\mu E_0/\hbar$ and $\Delta=\omega-\omega_0$ ; the fast oscillation is gone and $\tilde\rho_{eg}$ is slowly varying. Why: only the slow envelope couples to populations on the relaxation timescale, so peeling off the carrier makes steady states meaningful.

Step 3. Add relaxation. Spontaneous emission empties $|e\rangle$ at rate $\Gamma$ and, interrupting the amplitude's phase, damps the coherence at $\Gamma/2$ ; elastic collisions add $\gamma_\mathrm{coll}$ to the coherence alone. Hence populations relax with $\Gamma$ , the coherence with $\gamma=\Gamma/2+\gamma_\mathrm{coll}$ . Why: this asymmetry is the heart of the topic — phase-scrambling can be fast without moving an occupation number.

Step 4. Collecting terms gives the optical Bloch equations, $\dot{\tilde\rho}_{eg}=-(\gamma+\mathrm i\Delta)\tilde\rho_{eg}+\tfrac{\mathrm i}{2}\Omega(\rho_{ee}-\rho_{gg})$ and $\dot\rho_{ee}=-\Gamma\rho_{ee}+\tfrac{\mathrm i}{2}\Omega(\tilde\rho_{ge}-\tilde\rho_{eg})$ . With $u=2\,\mathrm{Re}\,\tilde\rho_{eg}$ , $v=2\,\mathrm{Im}\,\tilde\rho_{eg}$ , $w=\rho_{ee}-\rho_{gg}$ these are torque equations with transverse damping $\gamma$ and longitudinal damping $\Gamma$ . Why: the two constants are the decay rates of two perpendicular parts of one arrow — population along the axis, coherence around the equator.

Go deeper · Derivation B · 1 minPure dephasing turns a superposition into a mixture

Derivation BPure dephasing turns a superposition into a mixture

The goal is to exhibit decoherence in its cleanest form: a process that annihilates coherence while leaving populations exactly fixed, driving a pure state into a maximally mixed one, with the crossover tracked by the purity. This is the mechanism behind collisional broadening and the emergence of classical statistics from quantum superposition. We take an equal superposition, $\rho_{ee}=\rho_{gg}=\tfrac12$ , $\rho_{eg}(0)=\tfrac12$ — pure — under pure dephasing at $\gamma_\mathrm{coll}$ with $\Gamma=0$ and no drive.

Step 1. With $\Gamma=0$ the populations obey $\dot\rho_{ee}=\dot\rho_{gg}=0$ , frozen at $\tfrac12$ ; the coherence obeys $\dot\rho_{eg}=-(\mathrm i\omega_0+\gamma_\mathrm{coll})\rho_{eg}$ . Why: dephasing is purely off-diagonal — the diagonal equations have nothing on their right-hand side.

Step 2. Integrate to $\rho_{eg}(t)=\tfrac12\,e^{-\gamma_\mathrm{coll}t}\,e^{-\mathrm i\omega_0 t}$ : the magnitude decays exponentially while the phase winds at $\omega_0$ , populations pinned at $\tfrac12$ . Why: the atom is just as likely excited at any later time, yet its ability to interfere bleeds away — energy conserved, phase lost.

Step 3. The purity $\mathrm{Tr}\,\hat\rho^2=\rho_{ee}^2+\rho_{gg}^2+2|\rho_{eg}|^2=\tfrac12+\tfrac12\,e^{-2\gamma_\mathrm{coll}t}$ falls from $1$ (pure) to $\tfrac12$ (maximally mixed) as the coherence vanishes. Why: purity is the basis-free scalar certifying the pure-to-mixed transition — its fall is quantumness leaking into the reservoir.

Step 4. Once $t\gg1/\gamma_\mathrm{coll}$ the state is indistinguishable from a fifty-fifty incoherent mixture — no dipole, no interference. This is decoherence, the bridge to classical behaviour and, in frequency, the origin of collisional (pressure) broadening. Why: the exponential that kills the dipole in time becomes, by Fourier transform, the extra Lorentzian width — one process, two faces.

Limiting cases

Case	Result	Meaning
Pure state	$\hat\rho^2=\hat\rho,\ \mathrm{Tr}\,\hat\rho^2=1$	coherences saturate their bound $\lvert\rho_{nm}\rvert=\sqrt{\rho_{nn}\rho_{mm}}$ ; all relative phase is definite
Fully mixed / incoherent	$\rho_{nm}=0\ (n\ne m),\ \mathrm{Tr}\,\hat\rho^2<1$	only populations remain; the Einstein rate-equation description is exact
Fast-dephasing (rate-equation) limit	$\gamma\gg\Omega,\Gamma\Rightarrow\dot{\tilde\rho}_{eg}\approx0$	coherence follows the field adiabatically; eliminating it gives rate equations with Lorentzian rate $W=\tfrac{\Omega^2}{2}\gamma/(\Delta^2+\gamma^2)$
Radiatively limited coherence	$\gamma=\tfrac12\Gamma$	no collisions; the dipole decays at half the population rate — the natural linewidth
Thermal equilibrium	$\rho_{nn}=e^{-E_n/k_BT}/Z,\ \rho_{nm}=0$	Boltzmann populations, zero coherence in the energy basis; consistent with detailed balance

Common trapCoherences must relax at the same rate as populations

They need not, and generally do not. Population and coherence decay obey two different constants, $\Gamma$ and $\gamma=\tfrac12\Gamma+\gamma_\mathrm{coll}$ , and the second can vastly exceed the first: collisions, laser phase noise and inhomogeneous fields scramble the phase without changing which level the atom occupies, so $\gamma_\mathrm{coll}$ — and the linewidth $2\gamma$ — can dwarf the natural width while the lifetime $1/\Gamma$ is untouched. Even collision-free the two are unequal: $\gamma=\Gamma/2$ . The other half of the trap is treating the split as absolute — it is basis-dependent, so a state that is pure population in the bare basis carries coherences in the dressed or rotating basis. "Population" and "coherence" are labels on a choice of axes, not on the state itself.

Worked examples

1 · Natural linewidth of the sodium D2 line (confidence builder)

The sodium 3p state producing the yellow D2 line at 589 nm has radiative lifetime $\tau=16.2\ \mathrm{ns}$ ; with no collisions the coherence is radiatively limited, $\gamma=\Gamma/2$ . Find $\Gamma$ , $\gamma$ , the coherence lifetime $T_2$ , and the natural linewidth (FWHM).

Step 1. The population decays as $\rho_{ee}\propto e^{-\Gamma t}$ with $\Gamma=1/\tau=1/(16.2\times10^{-9}\ \mathrm s)=\mathbf{6.17\times10^{7}\ s^{-1}}$ — the Einstein $A$ -coefficient.

Step 2. With no collisions, $\gamma=\Gamma/2=3.09\times10^{7}\ \mathrm s^{-1}$ , so $T_2=1/\gamma=\mathbf{32.4\ ns}=2\tau$ . The dipole outlives the excited population — with only radiative damping the phase lasts twice as long as one might guess.

Step 3. The emitted line is the Fourier transform of a coherence decaying at $\gamma$ — a Lorentzian of FWHM $2\gamma=\Gamma$ , so $\Delta\nu=\dfrac{\Gamma}{2\pi}=\dfrac{6.17\times10^{7}}{2\pi}=\mathbf{9.82\ MHz}$ , matching the tabulated sodium D2 linewidth ( $\approx9.79$ MHz). Population at $\Gamma=6.17\times10^{7}\ \mathrm s^{-1}$ , coherence at $\gamma=3.09\times10^{7}\ \mathrm s^{-1}$ ( $T_2=32.4$ ns), natural linewidth $\Delta\nu=9.82$ MHz.

2 · Collisions broaden the line but leave the lifetime alone

Now put the same atoms in a buffer gas, so elastic collisions add pure dephasing at $\gamma_\mathrm{coll}=5.0\times10^{7}\ \mathrm s^{-1}$ . Find the new $\gamma$ , $T_2$ and line FWHM, and compare with the excited-state lifetime.

Step 1. Elastic collisions leave populations untouched: the excited state still decays only radiatively, so $\Gamma=6.17\times10^{7}\ \mathrm s^{-1}$ and $\tau=\mathbf{16.2\ ns}$ , unchanged — collisions redirect phase, not energy.

Step 2. The coherence now dephases at $\gamma=\tfrac12\Gamma+\gamma_\mathrm{coll}=3.09\times10^{7}+5.0\times10^{7}=8.09\times10^{7}\ \mathrm s^{-1}$ , so $T_2=1/\gamma=\mathbf{12.4\ ns}$ — the dipole now dies faster than the population, the reverse of example 1.

Step 3. The width follows the coherence: $\Delta\nu=\dfrac{\gamma}{\pi}=\dfrac{8.09\times10^{7}}{\pi}=\mathbf{25.7\ MHz}$ , a factor $25.7/9.82=\mathbf{2.6}$ broader than the natural line — while the lifetime has not moved. Collisions raise $\gamma$ to $8.09\times10^{7}\ \mathrm s^{-1}$ , broadening the line 9.82→25.7 MHz, yet leave $\tau=16.2$ ns exactly — pure dephasing at work.

3 · How fast does a superposition become a mixture?

Take the pure superposition of Derivation 3 — $\rho_{ee}=\rho_{gg}=\tfrac12$ , $\rho_{eg}(0)=\tfrac12$ — under pure dephasing at $\gamma_\mathrm{coll}=1.0\times10^{8}\ \mathrm s^{-1}$ with $\Gamma=0$ . When is the purity halfway to fully mixed, and what is the state at 10 ns?

Step 1. The purity $\mathrm{Tr}\,\hat\rho^2=\tfrac12+\tfrac12 e^{-2\gamma_\mathrm{coll}t}$ runs from $1$ to $\tfrac12$ ; "halfway" is $0.75$ , giving $e^{-2\gamma_\mathrm{coll}t}=\tfrac12$ and $t=\dfrac{\ln 2}{2\gamma_\mathrm{coll}}=\dfrac{0.693}{2\times10^{8}}=\mathbf{3.47\ ns}$ .

Step 2. At $t=10\ \mathrm{ns}=1/\gamma_\mathrm{coll}$ the coherence is $\tfrac12 e^{-1}=\mathbf{0.184}$ and the purity $\tfrac12+\tfrac12 e^{-2}=\mathbf{0.568}$ — nearly fully mixed — while the populations stay at $\rho_{ee}=\rho_{gg}=\tfrac12$ . Purity hits 0.75 at 3.47 ns; by 10 ns coherence is 0.184 and purity 0.568, yet both populations remain $\tfrac12$ — phase erased, no atom moved between levels.

Physics of the portrait

The whole language of this article — a state as an arrow, vertical part a population, horizontal part a coherence, the two relaxing at $1/T_1$ and $1/T_2$ — was minted not in optics but in magnetic resonance, by the physicist whose name the equations still carry.

Origins · 1 minFelix Bloch and how this was discovered

Felix Bloch

1905–1983 · architect of the Bloch equations and the Bloch-vector picture

Felix Bloch was a Swiss-American physicist whose fingerprints are on solid-state, nuclear and optical physics alike. As Heisenberg's first doctoral student he proved in 1928 that electron wavefunctions in a crystal are plane waves modulated by the lattice (Bloch's theorem, the cornerstone of band theory), and later explained ferromagnetism through spin waves and domain walls. His decisive contribution here came after the war at Stanford: in 1946 he introduced the phenomenological Bloch equations for nuclear magnetisation, a vector precessing under a magnetic torque while relaxing along the field at $1/T_1$ and losing its transverse component at $1/T_2$ . That distinction — slow longitudinal relaxation of "how much", faster transverse relaxation of "in what phase" — is exactly the populations-versus-coherences split; when Feynman, Vernon and Hellwarth showed in 1957 that any two-level system maps onto such a vector, Bloch's equations became the optical Bloch equations, $T_1$ set by spontaneous emission and $T_2$ by dephasing. He shared the 1952 Nobel Prize with Purcell and became CERN's first Director-General in 1954. The vocabulary of every laser-spectroscopy and quantum-computing lab — $T_1$ , $T_2$ , the Bloch sphere — is his.

History

1927

— von Neumann and Landau independently introduce the density operator for mixed quantum states

1932

— von Neumann's Mathematical Foundations formalises the trace rule and mixed states

1946

— Bloch's phenomenological equations describe precessing, relaxing nuclear magnetisation with distinct T₁ and T₂

1957

— Fano names the diagonal populations and off-diagonal coherences of the density matrix

1957

— Feynman, Vernon and Hellwarth map any two-level system onto a Bloch vector

1975

— Allen and Eberly codify the optical Bloch equations for two-level atoms

Going further

Texts. Foot, Atomic Physics, Ch. 7 (density matrix and optical Bloch equations) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2–5 · Cohen-Tannoudji, Diu and Laloë, Quantum Mechanics, complement E-III · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 2–3 · Allen and Eberly, Optical Resonance and Two-Level Atoms.

Provenance. The definitions and coherence bound follow Cohen-Tannoudji, Diu and Laloë (complement E-III) and Loudon Ch. 2; the naming of populations and coherences is Fano's 1957 review. The optical Bloch equations, $\gamma=\Gamma/2+\gamma_\mathrm{coll}$ , and the Lorentzian line of width $2\gamma$ are standard in Foot (Ch. 7) and Demtröder (Ch. 2–3). A verifier should reproduce the adiabatic rate $W=\tfrac{\Omega^2}{2}\,\gamma/(\Delta^2+\gamma^2)$ (weak-field excited fraction $\rho_{ee}=\tfrac{\Omega^2}{4}/(\Delta^2+\gamma^2)$ , matching Steck) and the purity decay $\mathrm{Tr}\,\hat\rho^2=\tfrac12+\tfrac12 e^{-2\gamma_\mathrm{coll}t}$ . The worked numbers use $\tau=16.2$ ns, giving $\Gamma=6.17\times10^{7}\ \mathrm s^{-1}$ and $\Gamma/2\pi=9.82$ MHz.

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

Ask the tutor: Ask me to walk through the optical Bloch equations on the Bloch sphere, or to show how the $T_2 \le 2T_1$ bound sets the coherence time of a real qubit or atomic clock.

What is the actual difference between a coherent superposition and a statistical mixture?

They can share identical populations yet behave oppositely. A coherent superposition $\tfrac{1}{\sqrt2}(|e\rangle+|g\rangle)$ is a pure state with definite relative phase: full off-diagonal elements, a Bloch vector on the equator, a radiating dipole. A fifty-fifty mixture — the atom is $|e\rangle$ or $|g\rangle$ , you just do not know which — has the same diagonal but zero coherence, a vector on the axis, no dipole. Only the coherence tells them apart, and decoherence carries the first into the second.

Why do coherences decay at Γ∕2 even with no collisions at all?

Because spontaneous emission does two things: it empties the excited state at rate $\Gamma$ and, being an amplitude process, interrupts the phase. The excited amplitude falls as $e^{-\Gamma t/2}$ (its square, the population, as $e^{-\Gamma t}$ ), and the coherence, bilinear in the amplitudes, inherits that $\Gamma/2$ decay. So the irreducible collision-free rate is $\gamma=\Gamma/2$ — the natural linewidth; collisions only add to this floor.

Is the split into populations and coherences physical, or a matter of choice?

It is basis-dependent. "Population" means a diagonal element, "coherence" an off-diagonal one — but diagonal in which basis? A state that is pure population in the bare atomic basis generally carries coherences in the dressed-state or rotating basis; rotate your axes and you reshuffle occupation into phase. Only invariants like $\mathrm{Tr}\,\hat\rho=1$ and the purity $\mathrm{Tr}\,\hat\rho^2$ are basis-free, so name the basis before calling a number a population or a coherence.

T₁, T₂, T₂* — how do these relate to Γ and γ?

$T_1=1/\Gamma$ is the population (energy) relaxation time. $T_2=1/\gamma$ is the single-atom coherence time, set by $\gamma=\Gamma/2+\gamma_\mathrm{coll}$ , obeying $T_2\le2T_1$ . $T_2^*$ is the ensemble coherence time, usually shorter: different atoms precess at slightly different frequencies (Doppler shifts, field inhomogeneity) and dephase relative to one another with no energy loss. A spin or photon echo reverses this inhomogeneous spreading, recovering $T_2$ from $T_2^*$ — what echoes are for.

Is decoherence the same thing as dissipation or energy loss?

No, and the distinction is the point. Dissipation changes populations — energy leaves the atom (spontaneous emission at rate $\Gamma$ ), dragging the coherence down too. Pure decoherence changes only coherences: an elastic collision randomises the phase while conserving energy, so $\gamma_\mathrm{coll}$ adds to $\gamma$ with no change to $\Gamma$ . In worked example 3 the populations stay frozen while the purity collapses. Both broaden lines, but only dissipation shortens lifetimes.

When is it safe to ignore coherences and just use rate equations?

When coherences dephase far faster than anything else — $\gamma\gg\Omega,\Gamma$ . There the coherence locks instantly to the populations (adiabatic elimination), leaving rate equations with a Lorentzian rate $W=\tfrac{\Omega^2}{2}\gamma/(\Delta^2+\gamma^2)$ , and coherence-specific effects vanish; this covers much ordinary spectroscopy with broad, collision-dominated lines. It fails whenever coherence has time to act: strong driving (Rabi oscillations), short pulses (echoes, free-induction decay), or engineered long-lived coherences (EIT, memories, qubits). Rule of thumb: if your timescale is shorter than $T_2$ , keep the coherences.