Bloch Vector and Bloch Sphere

A two-level atom driven by a near-resonant laser is the most-used model in atomic, molecular, and optical physics — and on paper it is a chore. From quantum mechanics it inherits a Hamiltonian, complex amplitudes $c_g$ and $c_e$ , and a relative phase oscillating at the optical frequency; tracking those by hand buries the physics in bookkeeping. The Bloch representation takes one thing from the quantum formalism and turns it into geometry: the Pauli algebra of spin- $\tfrac12$ . Because any $2\times2$ Hermitian operator expands in $\{\mathbb 1,\sigma_x,\sigma_y,\sigma_z\}$ , the density matrix becomes $\rho=\tfrac12(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma)$ and the whole state collapses to one real three-vector $\mathbf R=(u,v,w)$ — the abstract Hilbert-space ray becomes an arrow you can point at. That arrow lives on a ball. Its length measures purity — unit length for a pure state on the surface, shrinking toward the centre as the state mixes. Its polar height $w$ is the population inversion, north for excited and south for ground; its equatorial angle is the phase of the radiating dipole. Under the electric-dipole and rotating-wave approximations the Schrödinger equation becomes the rigid-body torque law $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ : the state precesses about an effective field $\mathbf\Omega=(\Omega_R,0,-\Delta)$ built from the Rabi frequency and the detuning. Add two relaxation rates and out drop Rabi flopping, $\pi$ and $\pi/2$ pulses, saturation, power broadening, the natural Lorentzian line, spin echoes, and every single-qubit gate — one picture, borrowed from Felix Bloch's magnetic resonance, steering atoms with light.

Key takeaways

A two-level atom's state is a point in a unit ball, the Bloch vector $\mathbf R=(u,v,w)$ via $\rho=\tfrac12(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma)$ — pure states on the surface, mixtures inside.
The poles are $|e\rangle$ and $|g\rangle$ , the equator maximal coherence, the height $w$ the inversion, and $|\mathbf R|$ the purity.
Coherent driving is precession $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ about $\mathbf\Omega=(\Omega_R,0,-\Delta)$ , with $\Omega_R=d_{eg}E_0/\hbar$ and $\Delta=\omega-\omega_0$ .
On resonance a constant field flops the atom fully at the Rabi frequency; detuning tilts $\mathbf\Omega$ and caps the inversion at $\Omega_R^2/(\Omega_R^2+\Delta^2)$ .
Relaxation shrinks the vector: $\Gamma=1/T_1$ pulls it to the ground pole, $\gamma=1/T_2$ collapses it onto the axis, with $\gamma=\Gamma/2$ for radiative decay.
One geometry gives $\pi$ and $\pi/2$ pulses, Ramsey fringes, photon echoes, adiabatic passage, and the power-broadened Lorentzian.
Key idea: write $\rho=\tfrac12(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma)$ ; coherent dynamics is then the precession $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ about $\mathbf\Omega=(\Omega_R,0,-\Delta)$ , and relaxation shrinks $|\mathbf R|$ .

The core insight

Encode the two-level density matrix as a real vector through $\rho=\tfrac12(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma)$ ; coherent light–atom dynamics is then the rigid precession $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ about the torque vector $\mathbf\Omega=(\Omega_R,0,-\Delta)$ , with relaxation shrinking $|\mathbf R|$ from the surface into the ball.

Where this lives today

The Bloch sphere is the control panel of modern quantum technology. In a trapped-ion or neutral-atom quantum computer every single-qubit gate is a rotation of the Bloch vector: a pulse of chosen area and phase sets the axis and angle, and composite pulses are designed geometrically to resist amplitude and detuning errors. The best atomic clocks run a Ramsey sequence — a $\pi/2$ pulse to the equator, a long free precession that winds up phase at the detuning, and a second $\pi/2$ pulse converting that phase into a population; optical-lattice clocks now keep time to below one part in $10^{18}$ . Laser cooling lives on equation (6): the scattering rate saturating at $\Gamma/2$ caps the light force and fixes the Doppler limit. And in spectroscopy the photon echo — a $\pi$ pulse reflecting a fan of dephased vectors so they re-converge — separates reversible from irreversible ( $T_2$ ) broadening, Hahn's spin trick now run with femtosecond pulses.

The physical picture

Picture the atomic state as an arrow anchored at the centre of a ball. Straight up means certainly excited; straight down, certainly ground; anywhere on the equator is a fifty-fifty superposition whose compass bearing is the phase of the oscillating dipole. That one arrow carries everything a wavefunction would — two populations and a relative phase — but as an object you can watch move. A near-resonant laser acts like a horizontal magnetic field, and the arrow precesses about it from south pole to north pole and back: one Rabi cycle is one round trip of absorption and stimulated re-emission.

Figure

The Bloch ball: excited state at the north pole, ground state at the south, coherent superpositions on the equator, and the state vector R precessing about the torque vector Ω whose tilt is fixed by the detuning, shown here at Δ = Ω_R giving a 45° tilt

Detuning tilts that horizontal field toward the vertical axis. The state can no longer reach the north pole: it precesses around a cone whose axis is $\mathbf\Omega$ , and the farther off resonance, the tighter the cone hugs the polar axis and the less the atom excites. The tilt is $\arctan(\Omega_R/|\Delta|)$ , so at $\Delta=\Omega_R$ the field sits at $45°$ and the excited probability tops out at exactly one half. Sweeping the detuning slowly drags the cone from pole to pole, carrying the arrow along — the geometric heart of adiabatic rapid passage. Relaxation is the arrow losing length. Spontaneous emission (the $T_1$ process) tugs the tip toward the ground pole as excited population leaks away; dephasing (the $T_2$ process) squeezes the equatorial components, collapsing the arrow onto the axis as the dipole forgets its phase. A pure surface state degrades into a mixture inside the ball, and full decoherence leaves only a thermal population on the axis with nothing left to radiate. The second figure shows the coherent half quantitatively — the Rabi flopping precession produces, on resonance and off.

Figure

Rabi flopping from the ground state for a resonant drive Ω_R = 2π × 2 MHz reaching full inversion at the π-pulse time t = 250 ns, versus a detuned drive Δ = Ω_R that oscillates faster at Ω_eff = √2 Ω_R yet never exceeds P_e = 0.5

The core equations

The starting point is the map itself. Any $2\times2$ density matrix is Hermitian with unit trace, and the identity plus the Pauli matrices form a complete basis, so one real vector $\mathbf R$ carries all the information $\rho$ does — the definition that makes the geometry legal.

\rho=\tfrac{1}{2}\left(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma\right),\qquad \mathbf R=(u,v,w),\quad |\mathbf R|\le 1 \qquad (1)

where: $\rho$ — the $2\times2$ atomic density matrix; $\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z)$ — the Pauli matrices. Projecting $\rho$ onto each gives the components: $u=2\,\mathrm{Re}\,\tilde\rho_{eg}$ and $v=-2\,\mathrm{Im}\,\tilde\rho_{eg}$ are the in-phase and in-quadrature dipole coherence in the frame rotating at $\omega$ , while $w=\rho_{ee}-\rho_{gg}$ is the population inversion, from $-1$ to $+1$ . The length $|\mathbf R|$ is a purity meter: unity on the surface (pure), less inside (mixed), zero at the centre (fully mixed). Everything coherent the field does is one equation of motion: under the dipole and rotating-wave approximations the Liouville equation for $\rho$ becomes a rigid-body torque law — the Bloch vector precessing about an effective field, length fixed.

\dot{\mathbf R}=\mathbf\Omega\times\mathbf R,\qquad \mathbf\Omega=(\Omega_R,\,0,\,-\Delta) \qquad (2)

where: $\mathbf\Omega$ — the torque vector, an effective magnetic field; $\Omega_R$ — the Rabi frequency; $\Delta=\omega-\omega_0$ — the laser detuning. The horizontal part drives population between the poles; the vertical part, from detuning, tilts the axis and frustrates full inversion. The two ingredients of that field are the Rabi frequency — how hard the field grips the dipole — and the generalised Rabi frequency, the length of $\mathbf\Omega$ and thus the true precession rate.

\Omega_R=\frac{d_{eg}\,E_0}{\hbar},\qquad \Omega_{\mathrm{eff}}=|\mathbf\Omega|=\sqrt{\Omega_R^2+\Delta^2} \qquad (3)

where: $d_{eg}=\langle e|\mathbf d\cdot\hat{\boldsymbol\epsilon}|g\rangle$ — the transition dipole matrix element; $E_0$ — the field amplitude; $\Omega_{\mathrm{eff}}$ — the true precession rate. On resonance $\Omega_{\mathrm{eff}}=\Omega_R$ ; far off resonance $\Omega_{\mathrm{eff}}\to|\Delta|$ . Real atoms leak. Spontaneous emission and collisions enter as two phenomenological decay rates, giving the optical Bloch equations in component form — three coupled equations, the workhorse of laser spectroscopy.

\begin{aligned}\dot u&=\Delta v-\gamma u\\ \dot v&=-\Delta u+\Omega_R w-\gamma v\\ \dot w&=-\Omega_R v-\Gamma(w-w_{\mathrm{eq}})\end{aligned} \qquad (4)

where: $\Gamma=1/T_1$ — population (longitudinal) decay from spontaneous emission; $\gamma=1/T_2$ — coherence (transverse) decay; $w_{\mathrm{eq}}$ — equilibrium inversion, usually $-1$ . Radiative decay locks $\gamma=\Gamma/2$ ; extra dephasing adds to $\gamma$ only. With damping off, integrating the precession from the ground state gives the Rabi formula — sinusoidal flopping that signals coherent driving, complete on resonance and partial off it.

P_e(t)=\frac{\Omega_R^2}{\Omega_{\mathrm{eff}}^2}\,\sin^2\!\left(\frac{\Omega_{\mathrm{eff}}\,t}{2}\right) \qquad (5)

where: $P_e$ — probability the atom is excited, starting from $|g\rangle$ . The prefactor $\Omega_R^2/\Omega_{\mathrm{eff}}^2$ is the flopping contrast: one on resonance, falling as $1/\Delta^2$ far away — the cone-angle limit. With damping the atom forgets its initial state and settles into a steady state. Solving the algebraic system with radiative damping gives the excited fraction and, times $\Gamma$ , the photon scattering rate — a power-broadened Lorentzian that saturates.

\rho_{ee}^{\,ss}=\frac{s_0/2}{1+s_0+(2\Delta/\Gamma)^2},\qquad R_{sc}=\Gamma\,\rho_{ee}^{\,ss},\qquad s_0=\frac{2\Omega_R^2}{\Gamma^2} \qquad (6)

where: $s_0$ — on-resonance saturation parameter, the intensity ratio $I/I_{\mathrm{sat}}$ ; $R_{sc}$ — scattered-photon rate. As $s_0\to\infty$ the excited fraction saturates at $\tfrac12$ and $R_{sc}$ at $\Gamma/2$ ; the linewidth broadens as $\Gamma\sqrt{1+s_0}$ . Finally, coherent control reduces to one number, the pulse area — the time integral of the Rabi frequency, on resonance exactly the rotation angle the Bloch vector sweeps.

\theta=\int \Omega_R(t)\,\mathrm dt,\qquad \theta=\pi\Rightarrow|g\rangle\to|e\rangle,\qquad \theta=\tfrac{\pi}{2}\Rightarrow|g\rangle\to\text{equator} \qquad (7)

where: $\theta$ — total rotation angle of the Bloch vector on resonance. A $\pi$ pulse inverts the atom (south to north pole); a $\pi/2$ pulse creates a maximal-coherence superposition on the equator, the workhorse of interferometry.

Derivation AFrom the Schrödinger equation to the precessing Bloch vector

The goal is to show that the quantum evolution of a driven two-level atom, once the fast optical phase and the counter-rotating terms are removed, is nothing more than the rigid precession $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ of a real three-vector about a fixed effective field. We start from the dipole-coupled Hamiltonian, move to the rotating frame, drop the fast terms, read off a spin in a static field, and let the commutator finish the job.

Step 1. Write the Hamiltonian with the field coupling the levels through the transition dipole, $H=\hbar\omega_0\,|e\rangle\langle e|-d_{eg}E_0\cos(\omega t)\,(|e\rangle\langle g|+|g\rangle\langle e|)$ , ground-state energy zero. Why: the dipole term is the content of the electric-dipole approximation — the field couples only through $d_{eg}$ , so the strength is $d_{eg}E_0$ .

Step 2. Move to the rotating frame, $|\psi\rangle=c_g|g\rangle+c_e e^{-i\omega t}|e\rangle$ , peeling off the fast phase the excited amplitude would carry. Why: the amplitudes now evolve only at the slow rate set by detuning and Rabi frequency, out from under the optical oscillation.

Step 3. Apply the rotating-wave approximation, discarding the terms oscillating at $2\omega$ , and the Hamiltonian becomes time-independent: $H_{\mathrm{RWA}}=\tfrac{\hbar}{2}\bigl(\begin{smallmatrix}-\Delta&\Omega_R\\\Omega_R&\Delta\end{smallmatrix}\bigr)$ in the $\{|e\rangle,|g\rangle\}$ basis, with $\Delta=\omega-\omega_0$ and $\Omega_R=d_{eg}E_0/\hbar$ . Why: near resonance the counter-rotating terms average to nearly nothing, and dropping them buys a static, solvable problem.

Step 4. Recognise the matrix as a spin in a static field: $H_{\mathrm{RWA}}=\tfrac{\hbar}{2}\,\mathbf\Omega\cdot\boldsymbol\sigma$ with $\mathbf\Omega=(\Omega_R,0,-\Delta)$ , since $\Omega_R\sigma_x-\Delta\sigma_z$ reproduces it. Why: the driven atom is formally a spin- $\tfrac12$ in a field along $\mathbf\Omega$ , so magnetic-resonance intuition transfers. The sign is $+\tfrac{\hbar}{2}$ for the matrix as written, fixing the precession sense below.

Step 5. Evolve with $i\hbar\dot\rho=[H_{\mathrm{RWA}},\rho]$ , insert $\rho=\tfrac12(\mathbb 1+\mathbf R\cdot\boldsymbol\sigma)$ , and use $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$ to collapse the commutator into a cross product, giving $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ . Why: the Pauli structure constants are exactly those of 3D rotations, so the quantum commutator becomes a classical torque — the deep reason a two-level system is a sphere at all.

Go deeper · Derivation B · 1 minRabi oscillations and the pulse-area theorem

Derivation BRabi oscillations and the pulse-area theorem

The goal is to turn the abstract torque equation into the concrete, measurable Rabi formula for the excited population, and then extract the pulse-area theorem behind all coherent control. We solve the undamped precession from the ground state, project the cone onto the polar axis, convert to probability, and show that on resonance only the integrated area sets the net rotation.

Step 1. With $\gamma=\Gamma=0$ the vector is rigid and rotates about $\mathbf\Omega$ at $|\mathbf\Omega|=\Omega_{\mathrm{eff}}=\sqrt{\Omega_R^2+\Delta^2}$ , tracing a cone whose half-angle is the tilt of $\mathbf\Omega$ . Why: a rotation cannot change the length, so the motion is confined to a cone and the only free number is $\Omega_{\mathrm{eff}}$ .

Step 2. Start at the south pole, $\mathbf R(0)=(0,0,-1)$ , and project the precessed vector onto the polar axis: $w(t)=-1+\dfrac{2\Omega_R^2}{\Omega_{\mathrm{eff}}^2}\sin^2\!\big(\tfrac{\Omega_{\mathrm{eff}}t}{2}\big)$ . Why: the cone's vertical reach is set by its opening angle, so the climb from south toward north caps at $\Omega_R^2/\Omega_{\mathrm{eff}}^2$ — detuning shrinks the cone.

Step 3. Convert with $P_e=\tfrac12(1+w)$ to the Rabi formula $P_e(t)=\dfrac{\Omega_R^2}{\Omega_{\mathrm{eff}}^2}\sin^2\!\big(\tfrac{\Omega_{\mathrm{eff}}t}{2}\big)$ , which on resonance is $\sin^2(\Omega_R t/2)$ reaching one at $\Omega_R t=\pi$ . Why: fluorescence reads $P_e$ directly, and the $\sin^2$ flopping is the fingerprint of coherent, not rate-equation, driving.

Step 4. On resonance a finite pulse rotates by $\theta=\int\Omega_R(t)\,\mathrm dt$ , so $\theta=\pi$ fully inverts and $\theta=\pi/2$ lands on the equator, independent of pulse shape. Why: rotations about a fixed axis add, so only the integrated Rabi frequency matters — this pulse-area theorem lets one design gates from area alone, the basis of shaped and composite pulses.

Limiting cases

Each regime exposes a different face of the precessing arrow.

Case	Result	Meaning
Resonant, undamped	$\Delta=0,\ \Gamma=\gamma=0$	$\mathbf\Omega=(\Omega_R,0,0)$ horizontal; the vector sweeps the $v$ – $w$ plane as $w(t)=-\cos(\Omega_R t)$ , full Rabi flopping
Far off-resonant	$\|\Delta\|\gg\Omega_R$	$\mathbf\Omega\approx(0,0,-\Delta)$ nearly vertical; a tight cone with tiny inversion, the second-order shift being the AC Stark (light) shift
Weak drive, strong decay	$\Omega_R\ll\Gamma$	coherences follow the field adiabatically and can be eliminated, so the Bloch equations reduce to Einstein rate equations for populations
Steady state, saturation	$t\to\infty$	$\rho_{ee}^{\,ss}=\dfrac{s_0/2}{1+s_0+(2\Delta/\Gamma)^2}$ ; the excited fraction saturates at $\tfrac12$ and the scattering rate at $\Gamma/2$
Pure vs mixed	$\|\mathbf R\|=1$ vs $\|\mathbf R\|<1$	pure states on the surface, mixtures inside, via $\|\mathbf R\|^2=2\,\mathrm{Tr}(\rho^2)-1$

Worked examples

1 · A π pulse and what detuning costs you (confidence builder)

A two-level atom is driven on resonance at $\Omega_R=2\pi\times2\ \mathrm{MHz}$ . (a) How long is a $\pi$ pulse, and what does it do? (b) A $\pi/2$ pulse of the same field? (c) If detuned by $\Delta=\Omega_R$ , what is the largest excited probability it can ever produce?

Step 1. On resonance the pulse area is $\theta=\Omega_R t$ , so a $\pi$ pulse needs $t_\pi=\pi/\Omega_R=\pi/(2\pi\times2\times10^{6}\ \mathrm{s^{-1}})=\mathbf{0.25\ \mu s}=250\ \mathrm{ns}$ , rotating the vector from south pole to north pole: full inversion $|g\rangle\to|e\rangle$ , $P_e=\sin^2(\pi/2)=1$ .

Step 2. A $\pi/2$ pulse is half as long, $t_{\pi/2}=\pi/(2\Omega_R)=\mathbf{0.125\ \mu s}=125\ \mathrm{ns}$ , a quarter turn onto the equator making an equal superposition with maximal coherence — the state opening a Ramsey sequence, $P_e=\tfrac12$ .

Step 3. Detuned by $\Delta=\Omega_R$ the field tilts to $45°$ and lengthens to $\Omega_{\mathrm{eff}}=\sqrt2\,\Omega_R$ , so the contrast is $\Omega_R^2/\Omega_{\mathrm{eff}}^2=\tfrac12$ and the excited probability never exceeds $\mathbf{P_e^{\max}=0.5}$ , however long you drive — faster by $\sqrt2$ but unable to climb past the equator, the amber curve above.

2 · Steady-state scattering on the rubidium D2 line

The $^{87}$ Rb D2 line has natural linewidth $\Gamma=2\pi\times6.07\ \mathrm{MHz}$ . A cooling beam drives it on resonance ( $\Delta=0$ ) at twice the saturation intensity, $s_0=2$ . Find the steady-state excited fraction, the scattering rate, and the maximum scattering rate on this line.

Step 1. Convert the linewidth: $\Gamma=2\pi\times6.07\times10^{6}\ \mathrm{s^{-1}}=3.81\times10^{7}\ \mathrm{s^{-1}}$ , so the lifetime is $\tau=1/\Gamma\approx26\ \mathrm{ns}$ — the rate at which an excited atom re-emits.

Step 2. On resonance the excited fraction is $\rho_{ee}^{\,ss}=\dfrac{s_0/2}{1+s_0}=\mathbf{1/3}$ : the atom spends a third of its time excited, already near the ceiling of $\tfrac12$ because at $s_0=2$ the transition is moderately saturated.

Step 3. The scattering rate is $R_{sc}=\Gamma\rho_{ee}^{\,ss}=\Gamma/3=\mathbf{1.27\times10^{7}\ photons/s}$ ; the ceiling, as $s_0\to\infty$ , is $R_{sc}^{\max}=\Gamma/2=\mathbf{1.91\times10^{7}\ photons/s}$ . Each scattered photon is one recoil kick, so this rate caps the radiation-pressure force in laser cooling — no atom is pushed harder than $\hbar k\,\Gamma/2$ .

Physics of the portrait

The Bloch sphere is a loan: its machinery — a magnetization vector, longitudinal and transverse relaxation times, precession about an effective field — was invented for nuclei in a magnet, and only later carried into optics when a two-level atom in a light field turned out to obey the same equations.

Origins · 1 minFelix Bloch and how this was discovered

Felix Bloch

1905–1983 · Swiss-American physicist, co-founder of nuclear magnetic resonance and namesake of the Bloch equations and Bloch sphere

Felix Bloch took his doctorate under Werner Heisenberg in 1928, and his thesis gave solid-state physics a cornerstone — Bloch's theorem for electrons in a periodic lattice. Forced from Germany in 1933, he settled at Stanford, and after wartime work on radar and at Los Alamos he turned to nuclei. In 1946 he published the phenomenological equations of nuclear magnetization — the Bloch equations — introducing the times $T_1$ and $T_2$ and picturing the state as a vector precessing about the applied field. That year Edward Purcell independently detected nuclear magnetic resonance in bulk matter, and the two shared the 1952 Nobel Prize. The optical connection came in 1957, when Feynman, Vernon, and Hellwarth proved that any driven two-level system — not just a spin in a magnet — maps exactly onto Bloch's precessing vector. The optical Bloch equations and the Bloch sphere became the native language of laser spectroscopy, atomic clocks, and quantum information — a geometry built for nuclei now steering atoms with light.

History

1937

— Rabi solves the motion of a spin in a rotating magnetic field, giving the flopping formula that carries his name

1946

— Bloch writes the phenomenological equations for nuclear magnetization with relaxation times T₁ and T₂

1950

— Hahn discovers the spin echo, refocusing inhomogeneous dephasing with a pulse

1957

— Feynman, Vernon and Hellwarth map any driven two-level system onto a precessing vector on a sphere

1989

— Ramsey receives the Nobel Prize for the separated-oscillatory-field method behind atomic clocks

Going further

Texts. Foot, Atomic Physics, ch. 7 (optical Bloch equations) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 2 · Allen and Eberly, Optical Resonance and Two-Level Atoms, ch. 2–3 · Metcalf and van der Straten, Laser Cooling and Trapping, ch. 2 · Demtröder, Laser Spectroscopy, Vol. 1 · Saleh and Teich, Fundamentals of Photonics, ch. 13–14. Key results to keep in mind: the RWA Hamiltonian is $H_{\mathrm{RWA}}=\tfrac{\hbar}{2}\mathbf\Omega\cdot\boldsymbol\sigma$ with $\mathbf\Omega=(\Omega_R,0,-\Delta)$ (sign $+\tfrac{\hbar}{2}$ for the matrix as written), giving $\dot{\mathbf R}=\mathbf\Omega\times\mathbf R$ ; on resonance $P_e=\sin^2(\Omega_R t/2)$ and $t_\pi=250\ \mathrm{ns}$ at $\Omega_R=2\pi\times2\ \mathrm{MHz}$ ; the steady-state formula $\rho_{ee}^{\,ss}=\tfrac{s_0/2}{1+s_0+(2\Delta/\Gamma)^2}$ with $s_0=2\Omega_R^2/\Gamma^2$ and $\gamma=\Gamma/2$ gives $\rho_{ee}=1/3$ and $R_{sc}=\Gamma/3=1.27\times10^7\ \mathrm{s^{-1}}$ at $s_0=2$ on the $^{87}$ Rb D2 line ( $\Gamma=2\pi\times6.07\ \mathrm{MHz}$ ); and purity satisfies $|\mathbf R|^2=2\,\mathrm{Tr}(\rho^2)-1$ .

Prerequisites. Electric Dipole Approximation · Transition Dipole Moment · Stimulated Emission · Spontaneous Emission. Leads to. Radiative Lifetime and Natural Linewidth · Lorentzian Line Shape · Resonant Absorption.

Ask the tutor: Want to see how a $\pi/2$ pulse, a free precession, and a second $\pi/2$ pulse turn the Bloch sphere into an atomic clock — or how detuning reshapes the Rabi flopping curve step by step?

Why bother mapping a quantum state to a real three-dimensional vector?

Because it trades algebra for geometry. After the trace and Hermiticity constraints a two-level density matrix has exactly three independent real numbers, and the Bloch vector $(u,v,w)$ is those three. Everything you would compute with complex amplitudes and $2\times2$ matrices — inversion, coherence, purity, a pulse's effect — becomes a statement about where an arrow points and how it turns: driving is a rotation, relaxation a shrinking, a measurement a projection. Human intuition for rotating arrows beats intuition for complex phases, which is why the sphere carried from magnetic resonance to optics to quantum computing.

What is the difference between the Rabi frequency and the generalised Rabi frequency?

The bare Rabi frequency $\Omega_R=d_{eg}E_0/\hbar$ measures the field's grip on the transition — how fast the atom flops when driven exactly on resonance. The generalised (effective) Rabi frequency $\Omega_{\mathrm{eff}}=\sqrt{\Omega_R^2+\Delta^2}$ is the length of the torque vector, the rate at which the Bloch vector actually precesses at any detuning. On resonance they coincide; off resonance $\Omega_{\mathrm{eff}}$ is larger, so the flopping is faster but its amplitude smaller by $\Omega_R^2/\Omega_{\mathrm{eff}}^2$ . Detuning makes the atom oscillate quicker yet reach less far — fast, shallow flopping instead of slow, full flopping.

Why can the coherence decay at most twice as fast as the population, giving T₂ ≤ 2T₁?

When the excited population decays, the coherence must decay at least half as fast, because losing an excited atom erases its share of the superposition. That sets the floor $\gamma\ge\Gamma/2$ , i.e. $T_2\le2T_1$ , saturated ( $\gamma=\Gamma/2$ ) for an atom whose only relaxation is spontaneous emission. Any extra dephasing — collisions, laser linewidth, inhomogeneous fields — adds to $\gamma$ without touching $\Gamma$ , so real systems have $T_2$ shorter than $2T_1$ , often much shorter. On the sphere: $\Gamma$ pulls the tip down the axis while $\gamma$ squeezes the equatorial radius, which cannot outlast twice the population.

Is the Bloch sphere the same thing as the Poincaré sphere for light polarization?

They are the same geometry wearing different labels — the unit sphere of a two-level (spin- $\tfrac12$ -like) system, built on the Pauli algebra. On the Poincaré sphere the poles are right- and left-circular polarization and the equator linear; on the Bloch sphere the poles are the excited and ground states and the equator maximal coherence. Anything proven about rotations on one transfers to the other — a wave plate rotating polarization is mathematically a pulse rotating an atomic state. The root is that any two orthogonal states span a two-dimensional Hilbert space whose pure states form a sphere, so one intuition serves polarization optics, atomic physics, and qubits alike.

When does the rotating-wave approximation break down?

The RWA drops the counter-rotating terms oscillating at twice the optical frequency, which is superb whenever the drive is near resonance and not too strong — formally when $\Omega_R$ and $|\Delta|$ are far below $\omega_0$ , covering nearly all of laser spectroscopy and atomic clocks. It fails when $\Omega_R$ approaches the transition frequency (ultrastrong driving, common with superconducting qubits and intense few-cycle pulses), where the discarded terms shift levels through the Bloch–Siegert effect and drive multi-photon processes the plain sphere misses; it also degrades far off resonance. There you keep the full time-dependent Hamiltonian, and the torque equation is only a first approximation.

How is a π-pulse different from just turning on a strong laser and leaving it?

A $\pi$ pulse is a laser turned on for a precisely controlled time — just long enough for the Bloch vector to make half a turn, area $\theta=\int\Omega_R\,\mathrm dt=\pi$ , landing on the opposite pole. Leave it on longer and the vector keeps precessing: past $\pi$ it heads back down, so a $2\pi$ pulse returns the atom to the ground state having done nothing net. The outcome depends on integrated area, not on how bright the beam is. Continuous illumination instead meets relaxation and settles into the steady state of equation (6), a fixed partial excitation rather than clean inversion. Coherent control is about timing the rotation, not just delivering power.