Two-Level System

A two-level system is any quantum degree of freedom whose relevant state space is two-dimensional. It is the smallest model that supports superposition, relative phase, noncommuting observables, coherent population transfer, avoided crossings, and nontrivial unitary control.

The abstract model is universal, but its physical interpretation is not. A spin, two localized orbitals, two atomic levels, and an engineered qubit can share the same $2\times2$ Hamiltonian while assigning different meanings to the basis, Pauli operators, controls, and measured quantities.

Model at a Glance

Field	Closed two-level model
Hilbert space	$\mathbb C^2$
Pure state	$c_1\lvert1\rangle+c_2\lvert2\rangle$
Mixed state	$\rho=(I+\mathbf r\cdot\boldsymbol\sigma)/2$
General Hamiltonian	$c_0I+\mathbf b\cdot\boldsymbol\sigma$
Static eigenvalues	$E_\pm=c_0\pm\lvert\mathbf b\rvert$
Static energy gap	$2\lvert\mathbf b\rvert$
Pure-state geometry	Surface of the Bloch sphere
Mixed-state geometry	Interior of the Bloch ball
Static solvability	Exact
Time-dependent solvability	Exact only for special protocols; otherwise time ordered or numerical
Canonical home	Two-Level Systems

The Two-Level System Hamiltonian is the convention-complete operator card. This page focuses on how the matrix becomes a physical model.

State Space

Choose an ordered orthonormal basis

$$\{\lvert1\rangle,\lvert2\rangle\}.$$

A normalized pure state is

$$\lvert\psi\rangle =c_1\lvert1\rangle+c_2\lvert2\rangle, \qquad \lvert c_1\rvert^2+\lvert c_2\rvert^2=1.$$

After global phase is removed, a pure state has two real parameters and can be written

$$\lvert\psi\rangle = \cos\frac{\theta}{2}\,\lvert1\rangle +e^{i\phi} \sin\frac{\theta}{2}\,\lvert2\rangle.$$

The associated Bloch vector is

$$\mathbf r = \left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right).$$

A general density operator is

$$\rho =\frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right), \qquad \lvert\mathbf r\rvert\leq1.$$

Pure states have $\lvert\mathbf r\rvert=1$; mixed states have $\lvert\mathbf r\rvert<1$. The Bloch Sphere: Wave-Mechanics Perspective owns the pure-state geometry, while Bloch Sphere for Density Operators owns the mixed-state extension.

Hamiltonian and Parameters

In the chosen basis, the most general closed-system Hamiltonian is

$$\hat H = \begin{pmatrix} \epsilon_1 & \kappa\\ \kappa^* & \epsilon_2 \end{pmatrix}.$$

Equivalently,

$$\hat H=c_0I+\mathbf b\cdot\boldsymbol\sigma,$$

where

$$c_0=\frac{\epsilon_1+\epsilon_2}{2}, \qquad \mathbf b = \left( \operatorname{Re}\kappa, -\operatorname{Im}\kappa, \frac{\epsilon_1-\epsilon_2}{2} \right).$$

Parameter	Meaning	Units in this convention
$c_0$	Common energy offset	Energy
$b_z$	Half the bare-basis detuning	Energy
$b_x,b_y$	Coupling quadratures	Energy
$\kappa$	Complex off-diagonal coupling	Energy
$\lvert\mathbf b\rvert$	Half the exact splitting	Energy

Atomic, magnetic-resonance, and qubit literature often writes

$$\hat H =\frac{\hbar}{2} \left( \Omega_0I+\boldsymbol\Omega\cdot\boldsymbol\sigma \right).$$

Then $\mathbf b=\hbar\boldsymbol\Omega/2$, and the Bloch rotation angular frequency is $\lvert\boldsymbol\Omega\rvert$. A factor-of-two error usually signals that energy-valued $\mathbf b$ and frequency-valued $\boldsymbol\Omega$ have been mixed.

Static Solvability

For constant $\mathbf b\neq0$, define

$$\hat{\mathbf b} =\frac{\mathbf b}{\lvert\mathbf b\rvert}.$$

The eigenvalues and projectors are

$$E_\pm =c_0\pm\lvert\mathbf b\rvert,$$ $$P_\pm =\frac12 \left( I\pm\hat{\mathbf b}\cdot\boldsymbol\sigma \right).$$

The exact propagator over $\tau=t-t_0$ is

$$\begin{aligned} U(t,t_0) &=e^{-ic_0\tau/\hbar} \Bigg[ \cos\left( \frac{\lvert\mathbf b\rvert\tau}{\hbar} \right)I\\ &\qquad -i\sin\left( \frac{\lvert\mathbf b\rvert\tau}{\hbar} \right) \hat{\mathbf b}\cdot\boldsymbol\sigma \Bigg]. \end{aligned}$$

The term $c_0I$ contributes only a global phase inside one isolated fixed sector. The vector $\mathbf b$ sets the energy eigenstates, gap, and rotation axis.

For

$$\hat H = \begin{pmatrix} \epsilon_1&\kappa\\ \kappa^*&\epsilon_2 \end{pmatrix},$$

an initial $\lvert1\rangle$ has transition probability

$$P_{1\to2}(t) = \frac{\lvert\kappa\rvert^2} {\delta^2+\lvert\kappa\rvert^2} \sin^2\left[ \frac{t}{\hbar} \sqrt{\delta^2+\lvert\kappa\rvert^2} \right],$$

where $\delta=(\epsilon_1-\epsilon_2)/2$. Complete transfer is possible at zero detuning; nonzero detuning reduces the maximum.

Two-State Hamiltonians owns the diagonalization and mixing-angle derivation.

Observables and Bloch Dynamics

Every Hermitian two-level observable has the form

$$\hat A=a_0I+\mathbf a\cdot\boldsymbol\sigma.$$

For $\rho=(I+\mathbf r\cdot\boldsymbol\sigma)/2$,

$$\langle\hat A\rangle =\operatorname{Tr}(\rho\hat A) =a_0+\mathbf a\cdot\mathbf r.$$

Thus specifying the physical meaning of the Pauli axes specifies the observable dictionary. In one realization $\sigma_z$ may be spin polarization; in another it may be site-population imbalance or an internal-state population difference.

Closed Hamiltonian evolution gives

$$\frac{d\mathbf r}{dt} =\frac{2}{\hbar}\mathbf b\times\mathbf r.$$

The Bloch vector precesses about $\mathbf b$ at angular frequency $2\lvert\mathbf b\rvert/\hbar$. Its length is conserved.

Common measured quantities include:

• basis-state populations;
• relative-phase interference;
• energy-basis populations;
• Pauli expectation values and tomography data;
• transition probabilities under pulses;
• susceptibility to static or oscillatory controls;
• leakage probability when the model is embedded in a larger system.

Physical Realizations

Spin in a magnetic field

For a spin-$1/2$ magnetic moment,

$$\hat H =-\gamma\hat{\mathbf S}\cdot\mathbf B =-\frac{\gamma\hbar}{2} \mathbf B\cdot\boldsymbol\sigma.$$

The Pauli matrices represent physical spin components, and the basis may be chosen as eigenstates of $S_z$. Spatial motion may still exist; freezing it is another model assumption. See Spin in Magnetic Fields.

Coupled wells or sites

A localized basis often gives

$$\hat H =\bar E I+\epsilon\sigma_z-K\sigma_x.$$

Here $\sigma_z$ measures left-versus-right population, while $K$ is a tunneling or hopping amplitude. The energy eigenstates are delocalized mixtures unless $K=0$. See Coupled Wells and Avoided Crossings and Tight-Binding Dimer.

Selected atomic levels

A real atom has many levels. A two-level atom retains one pair and the coupling between them. Its validity depends on selection rules, detunings, drive bandwidth, polarization, and leakage to spectator levels. See Two-Level Atom.

Engineered qubit

An engineered qubit requires more than a two-dimensional subspace. It also needs preparation, calibrated controls, measurement, coherence, and an operational computational basis. A protected logical qubit can live in a much larger physical Hilbert space. Bits, Qubits, Qudits, and Modes gives the distinctions.

Avoided crossing

The standard swept model is

$$\hat H(t) =\frac{vt}{2}\sigma_z+\Delta_0\sigma_x.$$

The instantaneous gap is smallest at the crossing point. Landau–Zener Problem: First Encounter states the ideal sweep assumptions and transition result.

Validity of the Two-Level Approximation

A fundamental spin-$1/2$ internal degree of freedom is intrinsically two-dimensional, although other degrees of freedom may remain. Atomic levels, double-well states, oscillator encodings, and solid-state qubits are usually truncations of larger spaces.

Let $P$ project onto the retained pair and $Q=I-P$ onto omitted states. The naive projected Hamiltonian

$$\hat H_{\mathrm{proj}}=P\hat H P$$

is reliable only when $Q$ remains weakly populated and virtual excursions are controlled.

Check:

1. Leakage gap: omitted levels remain far in energy compared with coupling matrix elements.
2. Drive bandwidth: pulse spectra do not overlap unwanted transitions.
3. Drive strength: off-resonant amplitudes stay small over the protocol.
4. Selection rules: nominally forbidden couplings are negligible at the required accuracy.
5. Virtual shifts: eliminated states do not produce unaccounted Stark, dispersive, or Lamb-like shifts.
6. Time scale: small leakage or phase errors do not accumulate beyond tolerance.
7. State preparation: the initial state actually lies in the retained subspace.
8. Measurement: the detector’s outcomes are represented correctly within the two-state dictionary.
9. Environment: decay into omitted states is absent or included in an open-system model.

A useful perturbative warning parameter for one omitted level is

$$\eta_{\mathrm L} \sim \frac{\lvert g_{\mathrm L}\rvert} {\lvert\Delta_{\mathrm L}\rvert},$$

where $g_{\mathrm L}$ is its coupling to the retained sector and $\Delta_{\mathrm L}$ is the detuning. Small $\eta_{\mathrm L}$ suppresses direct admixture, but pulse bandwidth and accumulated virtual shifts still require separate checks.

Time-Dependent and Driven Models

For

$$\hat H(t) =c_0(t)I+\mathbf b(t)\cdot\boldsymbol\sigma,$$

the instantaneous eigenvalues are easy to write, but the evolution is generally

$$U(t,t_0) =\mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}\hat H(s)\,ds \right].$$

Two Hamiltonians at different times obey

$$[\hat H(t),\hat H(t')] =2i \left[ \mathbf b(t)\times\mathbf b(t') \right]\cdot\boldsymbol\sigma.$$

An ordinary exponential of the integral is valid when the effective-field directions commute at all relevant times. Otherwise, pulse order and time ordering matter.

Important driven variants include:

Variant	Added structure	Solution status
Resonant constant pulse	Fixed transverse field in a rotating frame	Exact within the chosen frame and approximation
Detuned Rabi model	Static effective detuning plus transverse coupling	Exact within the rotating-wave model
Landau–Zener sweep	Linearly changing detuning	Exact transition formula for the ideal infinite sweep
Arbitrary shaped pulse	Time-dependent axis and magnitude	Time ordered; usually numerical
Periodic drive	Floquet structure	Analytic only in special regimes or approximations

Rabi Oscillations: First Encounter owns the introductory closed-form dynamics. The Time-Dependent Two-Level Systems Notebook separates propagation error, rotating-wave error, pulse-shape effects, and model error.

Open-System Extension

A Hermitian $2\times2$ Hamiltonian preserves purity and Bloch-vector length. Relaxation, dephasing, thermalization, and measurement backaction require a channel or master equation.

For Markovian dynamics,

$$\frac{d\rho}{dt} =-\frac{i}{\hbar}[\hat H,\rho] +\sum_\mu \left( L_\mu\rho L_\mu^\dagger -\frac12 \left\lbrace L_\mu^\dagger L_\mu,\rho \right\rbrace \right).$$

The jump operators and rates are additional model data. Optical Bloch Equations develops driven relaxation and dephasing, while Lindblad Generator is the compact generator card.

An effective non-Hermitian Hamiltonian can describe conditional no-jump evolution, but it does not by itself define a trace-preserving ensemble dynamics.

What the Model Teaches

The two-level system is the minimal laboratory for:

• basis dependence versus spectral invariants;
• relative phase and interference;
• noncommuting observables;
• avoided crossings and level repulsion;
• coherent population transfer;
• Bloch-sphere geometry;
• unitary control as rotations;
• adiabatic and nonadiabatic following;
• effective Hamiltonians and subspace truncation;
• relaxation and dephasing once open dynamics is added;
• qubit state, gate, and tomography notation.

It is small enough for exact algebra but rich enough to expose nearly every conceptual distinction that later reappears in larger Hilbert spaces.

Related Models

Model	Added degree or approximation
Rabi Model	Couple one two-level system to one quantized oscillator mode
Jaynes–Cummings Model	Apply the rotating-wave approximation to the one-mode coupling
Dicke Model	Couple many two-level systems collectively to one mode
Spin-boson model	Couple a two-level system to a bath of bosonic modes
Qutrit or multilevel atom	Retain leakage levels explicitly
Coupled qubits	Tensor several two-dimensional factors and add interactions
Stabilizer models	Restrict operations and measurements rather than the Hamiltonian alone

A two-level atom coupled to a field is not equivalent to replacing the field by another two-level system. Bosonic modes have unbounded occupation, while the emitter saturates after one excitation.

Benchmark Targets

Test	Expected result
Hermiticity	$\hat H^\dagger=\hat H$
Static energies	$c_0\pm\lvert\mathbf b\rvert$
Unitary propagation	$U^\dagger U=I$
State normalization	$\operatorname{Tr}\rho=1$
Closed pure-state evolution	$\lvert\mathbf r\rvert=1$ remains fixed
Constant-field motion	Rotation about $\mathbf b$ at $2\lvert\mathbf b\rvert/\hbar$
Resonant pulse	Complete transfer at the appropriate pulse area
Time-step convergence	Observable changes decrease under time-step refinement

Norm conservation alone does not validate a rotating-wave approximation, a pulse model, or a two-state truncation. Numerical error and physical-model error must be reported separately.

Common Mistakes

• Assuming the chosen basis is the energy basis.
• Calling the diagonal matrix entries exact energies when the off-diagonal coupling is nonzero.
• Treating every two-level degree of freedom as a literal spin.
• Treating every two-level system as an operational qubit.
• Mixing energy and angular-frequency conventions by a factor of two.
• Ignoring basis ordering and the phase convention of $\sigma_y$.
• Dropping $c_0I$ where phases between sectors or thermodynamic energies matter.
• Replacing a time-ordered exponential by an ordinary exponential when the rotation axis changes.
• Interpreting norm conservation as evidence that the physical truncation is accurate.
• Applying a weak-drive rotating-wave result to strong or ultrafast driving.
• Ignoring leakage, virtual shifts, and spectator levels.
• Modeling relaxation or dephasing with a Hermitian Hamiltonian alone.
• Calling a non-Hermitian conditional Hamiltonian a complete open-system model.

Exercises

1. Density matrix to Bloch vector

Let

$$\rho = \begin{pmatrix} p&c\\ c^*&1-p \end{pmatrix}.$$

Find its Bloch vector and translate positivity into a condition on $p$ and $c$.

Solution

Comparing with

$$\rho =\frac12 \begin{pmatrix} 1+r_z&r_x-ir_y\\ r_x+ir_y&1-r_z \end{pmatrix}$$

gives

$$r_x=2\operatorname{Re}c, \qquad r_y=-2\operatorname{Im}c, \qquad r_z=2p-1.$$

For a unit-trace Hermitian $2\times2$ matrix, positivity is equivalent to nonnegative determinant:

$$\det\rho =p(1-p)-\lvert c\rvert^2 \geq0.$$

Thus

$$0\leq p\leq1, \qquad \lvert c\rvert^2\leq p(1-p).$$

This is equivalent to $\lvert\mathbf r\rvert\leq1$.

2. Resonant population inversion

For

$$\hat H=\frac{\hbar\Omega}{2}\sigma_x,$$

the system starts in the $+1$ eigenstate of $\sigma_z$. Find the probability of the $-1$ outcome and the shortest pulse duration that gives complete inversion.

Solution

The propagator is

$$U(t) = \cos\left(\frac{\Omega t}{2}\right)I -i\sin\left(\frac{\Omega t}{2}\right)\sigma_x.$$

Because $\sigma_x$ exchanges the two $\sigma_z$ basis states,

$$P_{+\to-}(t) =\sin^2\left(\frac{\Omega t}{2}\right).$$

The first complete inversion occurs when

$$\frac{\Omega t_\pi}{2} =\frac{\pi}{2},$$

so

$$t_\pi=\frac{\pi}{\Omega}.$$

This is a $\pi$ pulse in the stated angular-frequency convention.

3. Leakage estimate

A retained transition is driven with characteristic coupling $\Omega$, while a spectator state is detuned by $\Delta_{\mathrm L}$ and coupled with strength $g_{\mathrm L}$. What checks are needed beyond requiring $\lvert g_{\mathrm L}/\Delta_{\mathrm L}\rvert\ll1$?

Solution

The small ratio suppresses static admixture and off-resonant population at leading order, but it is not a complete validation. One must also check:

• whether the pulse bandwidth overlaps the spectator transition;
• whether long evolution accumulates a relevant ac Stark or dispersive shift of order $\lvert g_{\mathrm L}\rvert^2/\lvert\Delta_{\mathrm L}\rvert$;
• whether selection-rule-breaking terms or additional spectators are present;
• whether the drive changes the detuning during the protocol;
• whether decay through the spectator state creates irreversible error;
• whether the target observable is sensitive to small phase errors even when leakage probability is small.

The accepted error threshold belongs to the physical task, not to the two-level algebra.

Canonical Links

• Two-Level Systems is the canonical physical overview.
• Two-Level System Hamiltonian is the convention-complete Hamiltonian card.
• Two-State Hamiltonians owns static mixing and coherent-transfer derivations.
• Pauli-Matrix Hamiltonians owns the effective-field geometry.
• Bloch Sphere: Wave-Mechanics Perspective owns pure-state visualization.
• Two-Level Atom owns the multilevel-to-two-level atomic reduction.
• Time-Dependent Two-Level Systems Notebook supplies reproducible driven benchmarks.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions, Wiley, 1992.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.