Two-Level Systems and the Bloch Sphere

The precession lesson solved a spin in a magnetic field. That problem is not special to spin. Any quantum system with two relevant states — two atomic levels near resonance, the two positions of a tunneling atom, two coupled molecular configurations — has a two-dimensional Hilbert space and a $2\times 2$ Hamiltonian, and every such Hamiltonian is a spin in an effective field. The geometry of spin-½, the Pauli matrices, and the Bloch sphere therefore describe every two-level system at once.

The generic two-level Hamiltonian

Any Hermitian $2\times 2$ matrix expands in the identity and the three Pauli matrices with real coefficients,

because $\{I,{\sigma }_x,{\sigma }_y,{\sigma }_z\}$ is a basis for the real vector space of Hermitian $2\times 2$ matrices.¹ The identity term $d_{0}I$ shifts both eigenvalues equally and never affects dynamics or measurement probabilities; discard it. The remaining traceless part $\vec{d}\cdot \vec{\sigma }$ is $|\vec{d}|$ times $\hat{d}\cdot \vec{\sigma }$, formally identical to the spin coupling $-\gamma \vec{B}\cdot \vec{S}$ with an effective field along $\hat{d}$. Using $(\hat{d}\cdot \vec{\sigma }{)}^2=I$, the eigenvalues are

and the eigenstates are the spinors pointing along $\pm \hat{d}$, the ${\chi }_{\pm }(\hat{d})$ built in the Stern–Gerlach lesson. Every two-level problem reduces to reading off the vector $\vec{d}$ and applying the spin-½ results. The gap between the levels is $2|\vec{d}|$, and it closes only if all three components of $\vec{d}$ vanish at once.

The Bloch sphere

A normalized state of a two-level system, up to an overall phase, is

with $|0⟩,|1⟩$ the eigenstates of ${\sigma }_z$. The two real angles $(\theta ,\phi )$ are the polar and azimuthal coordinates of a point on a unit sphere, the Bloch sphere. The correspondence is faithful: distinct points give physically distinct states, and the two removed real parameters (normalization and global phase) carry no observable content.

The point's Cartesian coordinates are the expectation values of the Pauli operators, the Bloch vector

The north and south poles are $|0⟩$ and $|1⟩$; the equator holds the equal superpositions, which differ only by the phase $\phi$. Orthogonal states sit at antipodal points: $|0⟩$ and $|1⟩$ are a full $\pi$ apart on the sphere though only $\pi /2$ apart as spinors, the half-angle again. The density matrix packages the same information,

with $|\vec{r}|=1$ for the pure states considered here; mixed states fill the interior $|\vec{r}|<1$ and are developed with the density matrix.

Evolution as rotation

Drop the identity term and write the Hamiltonian as $H=\frac{\hslash \Omega }{2}\hat{n}\cdot \vec{\sigma }$, where $\hslash \Omega /2=|\vec{d}|$ and $\hat{n}=\hat{d}$. The evolution operator exponentiates directly, using $(\hat{n}\cdot \vec{\sigma }{)}^2=I$ to sum the series:

Acting on the Bloch vector, this unitary rotates $\vec{r}$ by the angle $\Omega t$ about the axis $\hat{n}$, at angular rate $\Omega$. The spinor picks up half that angle, the $SU(2)$-to-$SO(3)$ two-to-one map: a $2\pi$ rotation of the Bloch vector returns it to itself while multiplying the spinor by $-1$, and only $\Omega t=4\pi$ restores the spinor. Every closed-form two-level dynamics from the previous lessons is a special case. Larmor precession is rotation about $\hat{z}$; on-resonance Rabi flopping is rotation about $\hat{x}$ in the rotating frame; general driving is rotation about the tilted effective-field axis.

Avoided crossings and level repulsion

Take a two-level system whose diagonal energies depend on a control parameter $\lambda$ and would cross at some ${\lambda }_0$, with a constant coupling $v$ off the diagonal:

with $\bar{\epsilon }=\frac{1}{2}({\epsilon }_1+{\epsilon }_2)$. Where the bare energies cross, ${\epsilon }_1={\epsilon }_2$, the exact eigenvalues are separated by $E_{+}-E_{-}=2|v|$. The coupling prevents the levels from touching: they approach, repel, and exchange character. The two eigenstates at the crossing are the symmetric and antisymmetric combinations of the bare states, not the bare states themselves. A genuine crossing of two levels requires both the diagonal difference and the coupling to vanish at once — two conditions on the parameters, so in a single-parameter family crossings are avoided generically.²

The ammonia molecule and the maser

The ammonia molecule NH${}_3$ is a nitrogen atom joined to a triangle of three hydrogens. The nitrogen has two equilibrium positions, one on each side of the hydrogen plane, separated by a potential barrier. Call the localized states $|L⟩$ and $|R⟩$; by symmetry they have the same energy $E_0$, and quantum tunneling through the barrier couples them with amplitude $A$:

The symmetric and antisymmetric combinations are the stationary states, with energies $E_0-A$ and $E_0+A$. Tunneling lifts the degeneracy and splits the level by $2A$. For ammonia the splitting corresponds to a transition frequency of about $23.87\mathrm{GHz}$ (a microwave wavelength near $1.25\mathrm{cm}$), the inversion line that drove the first maser.³ Placing molecules in the upper state $|A⟩$ and stimulating the $23.87\mathrm{GHz}$ transition gives coherent microwave amplification — the ammonia maser.

A molecule prepared in $|L⟩$ is not stationary. Writing it as $|L⟩=\frac{1}{\sqrt{2}}(|S⟩+|A⟩)$ and letting the two stationary states accumulate their phases, the probability of finding the nitrogen still on the left is

a complete oscillation between the two wells at angular frequency $2A/\hslash$. This is the two-level flopping of the previous lesson with the tunneling amplitude playing the role of the drive: the nitrogen tunnels back and forth through the barrier, and the same ${\cos }^2$ law that governed Rabi inversion governs the inversion of the molecule.

The qubit

A two-level system with a controllable Hamiltonian is a qubit. Its state is a point on the Bloch sphere; the computational basis is the pair of poles $|0⟩,|1⟩$; and a measurement in that basis returns $0$ or $1$ with probabilities ${\cos }^2(\theta /2)$ and ${\sin }^2(\theta /2)$. Any single-qubit operation is a unitary, hence a Bloch-sphere rotation, realized physically by turning on a Hamiltonian $\frac{\hslash \Omega }{2}\hat{n}\cdot \vec{\sigma }$ for a chosen duration — exactly the driven-spin control of magnetic resonance. The spin-½ algebra assembled across this module is therefore the operational core of a single qubit; entangling two of them turns the single Bloch vector into a joint state whose correlations no pair of individual vectors can reproduce.