Two-Level Systems

A two-level system is a quantum system whose relevant Hilbert space is two-dimensional. It describes an isolated spin-$1/2$ degree of freedom exactly, or an approximately isolated pair of states within a larger system. The choice of states determines what the two amplitudes mean and which physical measurements they predict.

Helpful background. State vectors explains expansion in an orthonormal basis.

Two-dimensional state space

Choose an orthonormal basis

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

A general pure state is

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

Measurement in this basis gives probabilities $|c_1|^2$ and $|c_2|^2$. Their values do not determine the relative phase: $(|1\rangle+|2\rangle)/\sqrt2$ and $(|1\rangle+i|2\rangle)/\sqrt2$ have equal populations but different predictions in other measurement bases. The Bloch sphere displays both population and phase.

The basis states need not be energy eigenstates. A localized basis can describe preparation or detection, while the energy basis describes stationary states. Changing coordinates must transform the state, Hamiltonian, and measured observables consistently; it does not change the physical probabilities.

Basis states and coupling

In a chosen basis, a closed two-level Hamiltonian is a Hermitian two-by-two matrix:

$$H= \begin{pmatrix} E_1 & \Delta\\ \Delta^* & E_2 \end{pmatrix}.$$

The diagonal entries are always the energy expectation values in the basis states. They are energy eigenvalues when the coupling $\Delta$ vanishes. The off-diagonal entry $\Delta$ mixes the two basis states.

For a static Hamiltonian with $\Delta\ne0$, an initial basis state is a superposition of energy eigenstates. Their evolving relative phase produces population oscillations. Two-State Hamiltonians derives the spectrum, eigenvectors, energy projectors, and exact transition probability. A common energy shift contributes only a global phase within the isolated two-level system.

The coupling is not the energy gap: even when $E_1=E_2$, the gap is $2|\Delta|$. If also $\Delta=0$, the Hamiltonian is proportional to the identity and every state evolves only by a common phase.

Physical realizations and model validity

The basis must be connected to a preparation and measurement:

• In a double well, basis states may be left-localized and right-localized wavefunctions.
• For spin-$1/2$, basis states may be eigenstates of one spin component.
• For an atom, basis states may be two selected internal energy levels.
• For a qubit, basis states are computational states selected by a control architecture.
• For a dimer, basis states may be localized site orbitals.

Except for an intrinsically two-dimensional degree of freedom, retaining two levels is an approximation. Identify which states have been omitted and why their populations remain negligible over the energy range and duration of interest. A large undriven energy separation alone does not protect against a resonant drive or a sufficiently broad pulse. Couplings can also shift the retained levels through virtual transitions even when leakage remains small.

Use coupled wells and the tight-binding dimer for spatial realizations. The two-level atom explains the selected-level approximation for internal states. Driven Rabi oscillations and a Landau–Zener sweep each require additional time-dependence and validity assumptions.

An abstract two-dimensional quantum system is a qubit. Using a physical two-level subspace as a working quantum-information device additionally requires sufficiently coherent preparation, control, and measurement; see Bits, Qubits, Qudits, and Modes.

Exercises

1. Normalize the state $\lvert\psi\rangle=A(\lvert1\rangle+i\lvert2\rangle)$.

Solution

The norm is

$$\langle\psi\vert\psi\rangle =\lvert A\rvert^2 \left( \langle1\vert1\rangle+\langle2\vert2\rangle \right) =2\lvert A\rvert^2.$$

Normalization requires $2\lvert A\rvert^2=1$, so one may choose

$$A=\frac{1}{\sqrt2}.$$

2. Explain why a two-level approximation can break down under a strong drive.

Solution

A two-level approximation assumes all other states remain dynamically irrelevant. A strong drive can couple the chosen two states to additional levels, shift energies, or make off-resonant transitions significant. Once other states acquire appreciable amplitude, the two-dimensional model no longer gives a closed description.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• 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.