Coupled Wells and Avoided Crossings

Coupled wells give a concrete wave-mechanics origin for two-level Hamiltonians. Start with two states localized in separated wells, allow a small tunneling matrix element between them, and then vary a bias that raises one well relative to the other. Without tunneling, the two localized energies can cross. With tunneling, the exact energies repel and form an avoided crossing.

This page is the static energy-level picture. Time-dependent passage through such a crossing is introduced in Landau–Zener Problem: First Encounter.

Localized Basis

Let $\lvert L\rangle$ and $\lvert R\rangle$ denote approximate low-energy states localized in the left and right wells. They are useful when the barrier is high enough that each well still looks like an identifiable subsystem.

Project the full Hamiltonian into the subspace spanned by these two states. After choosing phases so the tunneling amplitude is real, the effective Hamiltonian is

$$H_{\mathrm{eff}} = \begin{pmatrix} E_L & -K\\ -K & E_R \end{pmatrix}, \qquad K\geq0.$$

Here $E_L$ and $E_R$ are the energies of the localized states before mixing, while $K$ measures tunneling through the barrier. The minus sign is a convention chosen so that, for a symmetric double well, the symmetric combination is lower in energy.

If $\lvert L\rangle$ and $\lvert R\rangle$ are not exactly orthogonal, one should first orthonormalize the two-state subspace or solve the corresponding generalized eigenvalue problem. The simple matrix above assumes that this cleanup has already been done.

Symmetric Limit

For a symmetric double well,

$$E_L=E_R=E_0.$$

Then

$$H_{\mathrm{eff}} = E_0I-K\sigma_x.$$

The eigenstates are

$$\lvert+\rangle = \frac{1}{\sqrt2} \left( \lvert L\rangle+\lvert R\rangle \right),$$

and

$$\lvert-\rangle = \frac{1}{\sqrt2} \left( \lvert L\rangle-\lvert R\rangle \right),$$

with energies

$$E_+=E_0-K, \qquad E_-=E_0+K.$$

The splitting is

$$\Delta E=2K.$$

This is the two-state version of the even-odd splitting explained in Double-Well Potential.

Adding A Bias

A bias tilts the two wells so their localized energies are no longer equal. Define

$$\bar E=\frac{E_L+E_R}{2}, \qquad \epsilon=\frac{E_L-E_R}{2}.$$

Then

$$H_{\mathrm{eff}} = \bar E I+\epsilon\sigma_z-K\sigma_x.$$

This is exactly the Pauli-matrix form of a two-level Hamiltonian, with effective vector

$$\mathbf b=(-K,0,\epsilon).$$

The exact eigenvalues are

$$E_\pm = \bar E \pm \sqrt{\epsilon^2+K^2}.$$

When $K=0$, the energies are just $E_L$ and $E_R$, so they cross when $\epsilon=0$. When $K\ne0$, the minimum gap is

$$\Delta E_{\mathrm{min}}=2K.$$

The dashed curves are localized-state energies that would cross if the tunneling matrix element vanished. The solid curves are the exact two-level eigenvalues for $K\ne0$, with minimum gap $2K$.

Diabatic And Adiabatic Labels

The localized states $\lvert L\rangle$ and $\lvert R\rangle$ are often called diabatic states in avoided-crossing language. Their unmixed energies $E_L$ and $E_R$ are the dashed curves in the figure.

The exact instantaneous eigenstates of $H_{\mathrm{eff}}$ are called adiabatic states. They are the solid curves. Far from the crossing, where

$$\lvert\epsilon\rvert\gg K,$$

the adiabatic states are almost localized. Near the crossing, where $\lvert\epsilon\rvert\lesssim K$, the two wells strongly mix.

For the lower energy eigenstate, the probability of occupying the left localized state is

$$P_L^{(-)} = \frac12 \left( 1-\frac{\epsilon}{\sqrt{\epsilon^2+K^2}} \right).$$

Thus, as the bias passes through $\epsilon=0$, the lower adiabatic state changes character from mostly left-localized to mostly right-localized. The change is smooth when $K\ne0$ and abrupt only in the uncoupled limit $K=0$.

Wave-Mechanics Meaning Of The Coupling

The matrix element $K$ is controlled by the overlap of the localized wavefunctions in the barrier region. Schematic notation writes

$$K = -\langle L\vert H\vert R\rangle,$$

after the phase convention above has been chosen.

For a high and wide barrier, this overlap is small. Semiclassically, it is often exponentially suppressed:

$$K\sim A e^{-S/\hbar},$$

where $S$ is a barrier action and $A$ is a prefactor. This page uses only the consequence: the avoided-crossing gap is small when the barrier weakly couples the wells.

The detailed Herring, WKB, and instanton estimates for the coupling belong to Tunneling Splittings.

Physical Interpretation

Coupled-well avoided crossings appear whenever a control parameter changes which localized configuration is energetically favorable while tunneling keeps the two configurations connected. Examples include:

• a double well tilted by an electric field or mechanical force;
• molecular inversion coordinates with a symmetry-breaking bias;
• semiconductor double quantum wells with gate-controlled detuning;
• superconducting circuits whose effective potential has two relevant minima;
• cold-atom double wells with adjustable barrier height and bias.

The microscopic systems differ, but the two numbers that matter near an isolated crossing are the detuning $2\epsilon$ and the coupling $K$.

Relation To Dynamics

The avoided-crossing diagram by itself is a spectrum as a function of a parameter. If the parameter is slowly changed in time, the system may follow an adiabatic branch. If it is changed quickly, the state may remain close to a diabatic localized state.

For a linear time sweep,

$$\epsilon(t)=\frac{vt}{2},$$

the Hamiltonian becomes the standard Landau–Zener form

$$H(t) = \begin{pmatrix} vt/2 & -K\\ -K & -vt/2 \end{pmatrix}$$

up to an irrelevant average energy. The static gap $2K$ is the same minimum gap that controls the Landau–Zener transition probability.

Common Mistakes

• Treating localized well states as exact energy eigenstates when $K\ne0$.
• Confusing the tunneling matrix element $K$ with the full energy splitting away from resonance.
• Calling a level crossing avoided without identifying the coupling that prevents the crossing.
• Forgetting that a large bias can localize eigenstates even when tunneling is nonzero.
• Applying a two-level avoided-crossing model when nearby excited states are also strongly mixed.
• Confusing the static avoided-crossing spectrum with the time-dependent Landau–Zener transition probability.

Where This Is Used

• Double-Well Potential gives the symmetric wave-mechanics source of the localized basis.
• Two-State Hamiltonians derives the eigenvalues and mixing formulas in general notation.
• Pauli-Matrix Hamiltonians gives the effective-field representation.
• Bloch Sphere: Wave-Mechanics Perspective visualizes the same model as rotation around $\mathbf b=(-K,0,\epsilon)$.
• Tight-Binding Dimer recasts the same Hamiltonian as a two-site hopping model.
• Landau–Zener Problem: First Encounter treats a first time-dependent sweep through the avoided crossing.
• Landau–Zener Transition gives the advanced transition-formula treatment.
• Adiabatic Approximation as a Method develops gap diagnostics, local schedules, and leakage estimates for following an avoided-crossing branch.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• C. Zener, “Non-adiabatic crossing of energy levels,” Proceedings of the Royal Society A 137, 696-702, 1932.

Exercises

1. Diagonalize

$$H = \begin{pmatrix} \epsilon & -K\\ -K & -\epsilon \end{pmatrix}.$$

What is the minimum gap as $\epsilon$ varies?

Solution

The trace is zero, and

$$H^2 = (\epsilon^2+K^2)I.$$

Thus the eigenvalues are

$$E_\pm = \pm\sqrt{\epsilon^2+K^2}.$$

The gap is

$$E_+-E_- = 2\sqrt{\epsilon^2+K^2},$$

which is minimized at $\epsilon=0$. The minimum gap is

$$\Delta E_{\mathrm{min}}=2K.$$

2. For the lower eigenstate, evaluate $P_L^{(-)}$ when $\epsilon=0$ and when $\epsilon\gg K\gt 0$.

Solution

Use

$$P_L^{(-)} = \frac12 \left( 1-\frac{\epsilon}{\sqrt{\epsilon^2+K^2}} \right).$$

At $\epsilon=0$,

$$P_L^{(-)}=\frac12.$$

For $\epsilon\gg K\gt 0$,

$$\frac{\epsilon}{\sqrt{\epsilon^2+K^2}}\approx1,$$

so

$$P_L^{(-)}\approx0.$$

The lower eigenstate is then mostly right-localized, because the left localized energy is higher.

3. Explain why increasing the barrier height reduces the avoided-crossing gap.

Solution

The gap at resonance is $2K$. The coupling $K$ comes from the overlap of the left and right localized states through the barrier. Raising the barrier reduces that overlap, so $K$ becomes smaller. Therefore the avoided-crossing gap narrows.

4. What changes when the bias is swept in time rather than held fixed?

Solution

For a fixed bias, the avoided crossing is just a spectrum and a set of stationary eigenstates. When the bias changes in time, the state may or may not follow an instantaneous eigenstate. The transition probability then depends on the sweep rate, the gap, and the initial condition; this is the Landau–Zener problem for a linear sweep.