Spin in a Magnetic Field: Precession and Resonance

The Stern–Gerlach lesson built the static structure of spin-½: a two-dimensional space, the Pauli matrices, and measurement statistics. A magnetic field turns that structure into dynamics. The Zeeman Hamiltonian is proportional to a single Pauli operator for a static field, and to a time-dependent combination when a drive is added. Both cases solve in closed form, and together they cover Larmor precession, magnetic resonance, and the driven qubit.

The magnetic moment and the Zeeman Hamiltonian

A particle with spin $\vec{S}$ carries a magnetic moment proportional to it,

where $\gamma$ is the gyromagnetic ratio and $g$ the dimensionless $g$-factor. For the electron $q=-e$ and $g_s\approx 2$, so ${\gamma }_e=-g_{s}e/2m_e<0$; the moment points opposite the spin.¹ The energy of the moment in a field is $H=-\vec{\mu }\cdot \vec{B}=-\gamma \vec{B}\cdot \vec{S}$.

Take a uniform static field along $z$, $\vec{B}=B_0\hat{z}$. Then

The eigenstates are the $S_z$ eigenspinors $|\pm ⟩$ with energies

The splitting is linear in the field. For an electron $\Delta E=g_s{\mu }_B{B}_0$ with the Bohr magneton ${\mu }_B=e\hslash /2m_e$; the corresponding frequency is ${\gamma }_e{B}_0/2\pi \approx 28.0GHz/T$, in the microwave band (electron spin resonance). A proton has ${\gamma }_p/2\pi \approx 42.58MHz/T$, in the radiofrequency band (nuclear magnetic resonance).²

Larmor precession

The dynamics of a general spin state follow from the time-evolution operator $U(t)=e^{-iHt/\hslash }$. Since $H=-\frac{\hslash {\omega }_0}{2}{\sigma }_z$ is diagonal,

Start with a spin tilted by a polar angle $\alpha$ from $z$, $\chi (0)=(\cos \frac{\alpha }{2},\sin \frac{\alpha }{2}{)}^{\mathsf{T}}$. Evolving,

The expectation values follow from the Pauli matrices. With $a=\cos \frac{\alpha }{2}{e}^{i{\omega }_{0}t/2}$ and $b=\sin \frac{\alpha }{2}{e}^{-i{\omega }_{0}t/2}$,

The polar angle stays fixed at $\alpha$ and the vector sweeps around $z$ at angular frequency ${\omega }_0$. This is Larmor precession: the spin expectation value traces a cone, exactly like a classical magnetic moment in a field. The azimuthal sense is set by the sign of $\gamma$.

The same result comes from the Heisenberg picture without solving for the state. Ehrenfest's theorem gives the operator equation of motion, which for $⟨\vec{S}⟩$ reads

The precession frequency ${\omega }_0=\gamma B_0$ is independent of the tilt angle and of $\hslash$ — a purely classical-looking rate, even though the underlying observable is quantized to $\pm \hslash /2$.

Driving the spin: the rotating frame

Add a weak oscillating field transverse to $B_0$,

The static part precesses the spin at ${\omega }_0$; the transverse part can flip it when $\omega$ is tuned near ${\omega }_0$. To see the resonance cleanly, pass to a frame rotating about $z$ at the drive frequency $\omega$, using the unitary $R(t)=e^{i\omega t{S}_z/\hslash }$. The linear oscillation $\cos (\omega t)\hat{x}$ splits into two counter-rotating circular fields; in the rotating frame one becomes static and the other spins at $2\omega$. Dropping the fast $2\omega$ term (the rotating-wave approximation, valid for $B_1\ll B_0$) leaves a time-independent effective Hamiltonian

This is a static spin problem again, but about an effective field with a $z$-component set by the detuning ${\omega }_0-\omega$ and an $x$-component set by the drive strength ${\omega }_1$. Its magnitude is the generalized Rabi frequency

Rabi oscillations

Prepare the spin in $|+⟩$ and ask for the probability it is found in $|-⟩$ after time $t$. In the rotating frame the spin precesses about the effective field at rate $\Omega$; converting back gives the Rabi formula

The prefactor ${\omega }_1^2/{\Omega }^2$ is the maximum reachable flip probability at a given detuning. It equals $1$ only at exact resonance and falls off as the drive is mistuned. The full flopping on resonance is the coherent inversion used to manipulate qubits and to invert nuclear-spin populations in NMR.

The resonance lineshape

Fix the pulse so that on resonance it is a $\pi$-pulse, or simply read the maximum of the Rabi oscillation, and scan the drive frequency. The peak flip probability

is a Lorentzian centered at $\omega ={\omega }_0$ with full width at half maximum $2{\omega }_1$. The response peaks sharply when the drive matches the level splitting. Locating the center measures ${\omega }_0=\gamma B_0$, hence the field $B_0$ (as in magnetometry and MRI) or the moment $\gamma$ (as in precision measurements of $g$-factors). The width is set by the drive strength here; in practice it is broadened further by relaxation, and the observed linewidth reports on the spin's coupling to its environment.

The Bloch equations and relaxation

Precession plus relaxation gives the phenomenological description of a real spin ensemble. Writing $\vec{M}=N⟨\vec{\mu }⟩$ for the magnetization of $N$ spins, the precession law $\mathbf{d}\vec{M}/\mathbf{d}t=\gamma \vec{M}\times \vec{B}$ acquires two damping terms,

the Bloch equations. $T_1$ is the longitudinal (energy) relaxation time toward the equilibrium magnetization $M_0$, and $T_2$ is the transverse (phase) coherence time. Their measurement is the basis of magnetic-resonance contrast: different tissues have different $T_1$ and $T_2$, and the imaging sequence turns those into image intensity.³ The isolated two-level dynamics — precession, Rabi flopping, and the Lorentzian resonance — is the coherent core beneath that applied machinery, and the same mathematics governs the driven qubit taken up in the next lesson.