01 The Bloch sphere, refreshed
State, drive, and what the axes mean
Two levels $|g\rangle,|e\rangle$, driven by a laser with Rabi frequency $\Omega$ and detuning $\Delta=\omega-\omega_0$. Package the state as a Bloch vector $\langle\boldsymbol\sigma\rangle=(\langle\sigma_x\rangle,\langle\sigma_y\rangle,\langle\sigma_z\rangle)$: z is the population inversion $\rho_{ee}-\rho_{gg}$ (−1 = all $|g\rangle$, +1 = all $|e\rangle$); x, y are the two quadratures of the atomic dipole, only nonzero for a superposition. The whole vector precesses like a gyroscope, $\partial_t\langle\boldsymbol\sigma\rangle=\boldsymbol\wp\times\langle\boldsymbol\sigma\rangle$, $\boldsymbol\wp=\Omega\hat x-\Delta\hat z$ (Steck, Quantum and Atom Optics, §5.4).
Steck §5.5, "the optical Bloch equations." Every dual-view panel below runs this exact system, integrated numerically (RK4) in real time with different $(\Gamma,\gamma_\perp,\Delta)$ — nothing is a pre-rendered animation. $\Gamma=1/T_1$ (longitudinal/energy decay), $\gamma_\perp=1/T_2$ (transverse/coherence decay), $\gamma_\perp=\Gamma/2+\gamma_c$ with $\gamma_c$ the "extra" dephasing.
02 Coherent driving — the baseline
No environment at all: $\Gamma=0$, $\gamma_\perp=0$, on resonance. Left: the vector orbits a closed great circle forever. Right: the population inversion it produces — an undamped sine that never loses contrast.
Steck §5.2. Full-contrast Rabi flopping, forever — the trajectory every regime below decays away from.
03 Dissipation — T₁
Turn on spontaneous emission: $\Gamma>0$, and with no extra dephasing $\gamma_\perp=\Gamma/2$ exactly (natural/homogeneous broadening). Real energy now leaves into the vacuum field every time the atom sits in $|e\rangle$. Watch several full oscillations decay before the sweep resets.
Torrey's exact resonance solution, Steck §5.5.2 (starting in $|g\rangle$). Same oscillation as §02, now spiraling into the ball and settling on the pole. $\Omega_\Gamma$ and the $e^{-3\Gamma t/4}$ envelope are exactly what you fit to Rabi-flop data to extract T₁.
What actually happens physically
The atom couples to the continuum of vacuum electromagnetic modes. Each time it's in $|e\rangle$ it can emit a real photon into a mode you're not tracking — the energy genuinely leaves the two-level system. That's why z moves: population flows one way, $|e\rangle\to|g\rangle$, irreversibly.
04 Dephasing — T₂* / Tφ
Set $\Gamma=0$ again — no energy loss anywhere — but now five atoms (or five repeated shots) each see a slightly different detuning $\Delta_i$: laser-linewidth jitter, a field gradient, spatial intensity variation across a tweezer array. Watch the five thin traces: none of them decay. What decays is only their average.
The subtlety worth being precise about
Free precession by itself — one atom rotating around $\hat z$ at its own detuning rate — is not decoherence. It's exactly reversible; a single trajectory's phase is perfectly well-defined the whole time (Steck §5.4: "free precession... about $\hat z$ at the detuning rate"). Dephasing only becomes a real decoherence mechanism once you average over a spread of rates you can't track shot-to-shot (Steck §5.4.2).
That's also why it's partly fixable: a π-pulse (spin/photon echo) reverses each atom's phase and lets the faster ones catch up to the slower ones, rephasing at time $2T$. This works for the static spread shown here — it defines T₂* — but not for dephasing that's genuinely random in time (collisions, laser phase noise), which defines the intrinsic $\gamma_c$ folded into T₂.
Steck §5.5 (definitions), §5.4.2 (echo). Each member above: exact undamped generalized-Rabi solution $\tilde\Omega_i=\sqrt{\Omega^2+\Delta_i^2}$, Steck §5.2.2 — no approximation.
05 Decoherence — T₂, both at once
Real experiments have both: $\Gamma>0$ and a $\Delta_i$ spread. The same five atoms individually relax toward $|g\rangle$ while also drifting out of phase with each other — the ensemble average spirals inward and down at once, over several visible oscillations before settling.
Decoherence isn't a third mechanism — it's the name for whatever combination of dissipation and dephasing is acting on $|\rho_{eg}|$. The trace color blends rust→blue as a visual reminder of that sum, not a claim about which one "wins" first.
Same physics, a different textbook's letters
Fox, Quantum Optics: An Introduction (Ch. 9, Fig. 9.10), draws this exact pair of Bloch-sphere pictures under the labels $T_2'$ (transverse) and $T_1$ (longitudinal) instead of $T_\phi$ and $T_1$. $T_2'$ is Fox's name for the same pure/intrinsic dephasing time called $\gamma_c$/$T_\phi$ here, and his relation $1/T_2 = 1/(2T_1) + 1/T_2'$ is the identical equation boxed above — just a prime instead of a $\phi$ subscript. Don't let the different letters imply different physics across books.
Fox's caption states two things worth making explicit here too. First: "the Bloch vector of the relaxed state is only meaningful for the entire ensemble rather than for individual atoms" — true of every spiral and fan on this page. A single atom never smoothly shrinks; it stays a pure state on the Bloch sphere's surface and jumps discontinuously to $|g\rangle$ at one random instant. The smooth curves you're watching are what averaging over many such jumps — or many repeated shots — looks like; a single-trajectory ("quantum jump") picture of §03's decay would look completely different from its smooth panel here. Second: "longitudinal decay inevitably causes transverse relaxation as well" — exactly what $\gamma_\perp=\Gamma/2+\gamma_c\ge\Gamma/2$ already says in §01: T₁ decay always costs you some coherence too, never only population.
06 How you'd actually measure each one
| Protocol | Isolates | Signature |
|---|---|---|
| Rabi flopping, drive left on | a mix of both | damped oscillation: $\Omega_\Gamma$ and $e^{-3\Gamma t/4}$ envelope (§03) |
| Population decay, drive off | T₁ only | $\sigma_z(t)\to-1$ exponentially at rate Γ |
| Ramsey ($\pi/2$ – wait $T$ – $\pi/2$) | T₂* | fringe contrast decays with $T$ (§5.4.1); the atomic-clock protocol |
| Hahn echo ($\pi/2$ – $T$ – $\pi$ – $T$ – detect) | T₂ (removes the static T₂* spread) | echo amplitude recovers signal lost to inhomogeneous dephasing; extended by CPMG / DD sequences on the Dynamical Decoupling playground |
07 Cheat sheet
| Mechanism | Moves z? | Bloch picture | Echo-recoverable? | Typical cause |
|---|---|---|---|---|
| Dissipation (T₁) | Yes | spiral to pole | No | spontaneous emission, off-resonant scattering |
| Dephasing, intrinsic ($\gamma_c$) | No | shrink toward axis, fixed latitude | No | laser phase noise, collisions |
| Dephasing, inhomogeneous (T₂*) | No | fan-out, fixed latitude | Yes (spin echo) | detuning / light-shift spread across ensemble or shots |
| Decoherence (T₂, net) | whatever T₁ does | combination | partially | whatever mix of the above is actually present |
All four panels integrate Steck's boxed optical Bloch equations (§01) numerically (RK4) in real time — the "combined" panel in §05 is not a separately-derived closed form, it's the same equations with both Γ>0 and Δ≠0 plugged in at once. Reduced-motion viewers see one representative frame per panel instead of the live sweep.