Q022 · Avoided crossing / interference

Landau–Zener Passage

Does a slow sweep through an avoided crossing switch a two-level system into its other state? Yes, if the sweep is slow enough. The state stays on the lower energy branch, and across the avoided crossing that branch turns from one basis state into the other. A fast sweep leaves the state where it started. A second crossing shows interference.

Sweep a two-level system through an avoided crossing, compare its state in the energy basis and the fixed basis, then sweep back. Change the waiting time between the two passes, and the same finite-time evolution produces interference. Measure many freshly prepared trials and compare the counts with the model.

Model final upper-energy probability $\text{—}$

Total protocol time $\text{—}$

Make a prediction

If the energy levels exchange sides, will a slow state follow?

Compare a slow sweep with a fast one. Watch the colored populations, then measure. Sweeping back across the crossing lets the two paths interfere.

Predict first, then compare and collect evidence.

Preview time (each measurement runs the full protocol)

Return across the crossing after a waitInitial stateSweep speed

v

1

Minimum energy gap

\Delta

1

Wait between passages

t_w

0

Single passage scans speed; return passage scans waiting time. That scanned setting preserves records. Other evolution changes clear them. Waiting has no effect on a single passage.

Model details and sampling

Endpoint detuning magnitude

A

8

Diabatic dephasing rate

\gamma

0

Trials at each setting

N

400

Reproducible sampling seed

s

1984

Energy uses a chosen scale; time uses the corresponding inverse angular frequency. Values are dimensionless. Sampling is reproducible and capped at 30,000 trials.

The model is a dimensionless two-level Hamiltonian swept between finite endpoints, with optional Markovian dephasing in the fixed basis. The density matrix is evolved numerically through both sweeps and the hold between them. The exponential formula for an infinitely long sweep is only a separate reference, and it applies only to a single noiseless pass that starts in the lower energy state.

Recorded evidence and model limits

Simulated independent trials

Count how often the state ends in the upper level

Blue dots and their pointwise 95% Wilson intervals come from the recorded trials. The gray curve is the finite-time model. The intervals are not simultaneous bands and carry no guarantee under optional stopping. Preview and replay never add trials.

Current setting: fraction / 95% range

\text{—}

Infinite-sweep reference, when applicable

\text{—}

Evolution steps / recorded trials

\text{—}

Finite endpoints, the choice of initial state and dephasing can each make the infinite-sweep reference invalid. The return protocol is evolved in full, including the hold. The detector always measures in the final energy basis, even when the preview shows the fixed basis.

Start with the essentials

Focus question: Does remaining in the lower energy state mean remaining in the same fixed-basis state?
One-sentence intuition: A state that follows its energy level adiabatically can swap its fixed-basis character. A second crossing reads out the relative phase built up between the two crossings.

Core mathematical model

Hamiltonian and units

\frac{H}{E_0}=\frac{\Delta\sigma_x+\epsilon(\tau)\sigma_z}{2},\qquad \tau=E_0t/\hbar

Energies are shown in units of the chosen energy scale, and times in units of the matching inverse angular frequency. The fixed zero state has the positive Pauli-z eigenvalue. This equation fixes the signs of the coupling and the detuning.

Energy states and finite sweep

E_\pm/E_0=\pm\tfrac12\sqrt{\Delta^2+\epsilon^2},\qquad \epsilon(\tau)=-A+v\tau

The forward sweep ends at the positive endpoint detuning. In the return protocol the system waits there, then sweeps back at the same speed. The initial lower or upper energy state is defined at the actual finite starting point.

What the final detector measures

P_+=\frac{1+\boldsymbol r\cdot\boldsymbol n_f}{2},\qquad P_0=\frac{1+r_z}{2}

The final detector projects onto the upper energy eigenstate. The blue fixed-zero population is a different observable. Neither curve is the path of a particle.

Reference for an infinitely long sweep

P_{\mathrm{LZ}}=\exp\!\left(-\frac{\pi\Delta^2}{2v}\right)

This formula assumes a linear sweep from minus infinity to plus infinity, a start in the lower energy state and no dephasing. The finite-time trajectories and the return fringes are not computed from it.

The dephasing model

\dot\rho=-i[H/E_0,\rho]+\frac{\gamma}{2}(\sigma_z\rho\sigma_z-\rho)

Markovian pure dephasing acts in the fixed basis during the sweeps and the hold. Each numerical step applies an exact half step of dephasing, a midpoint unitary step, then another exact half step of dephasing. Relaxation, bath spectra and material calibration are not modeled.

From counts to a fraction

\widehat P_+(s)=\frac{n_+(s)}{N(s)}

Each outcome comes from a freshly prepared state that runs through the complete pulse at that scan setting. The Wilson interval at each point describes the Bernoulli uncertainty under this model; together the intervals do not form a joint band for the whole scan.

Common difficulties

Mixing up the two bases

Typical misconceptionIf the sweep is slow, the state does not change.

Better mental modelThe state stays in the same energy eigenstate, but that eigenstate changes its fixed-basis composition as the Hamiltonian changes.

Two crossings are not two coin flips

Typical misconceptionTwo passages are two independent coin flips, so their probabilities multiply.

Better mental modelThe two amplitudes stay coherent and keep their relative phase. The simulation carries that coherence through the whole protocol.

The formula is only a reference

Typical misconceptionThe exponential formula should match every finite pulse exactly.

Better mental modelThe formula assumes different endpoints, a particular initial state and no noise. The gray finite-time curve follows the evolution actually set up in the Lab.

Run the experiment

01
Follow slowly
Choose Slow passage and replay to the end. Inspect both population curves.
What to observe: The upper-energy population stays small, yet the state swaps its fixed-basis character.
02
Rush across
Choose Fast passage, then collect a speed scan.
What to observe: The faster the sweep, the more the state keeps its fixed-basis character, which leaves it in the upper energy state at the end.
03
Return and interfere
Choose Go there and back. Scan waiting time, then drag the waiting control.
What to observe: The final counts oscillate with waiting time, because the phase built up before the second crossing changes.
04
Check the limits
Increase dephasing or move the endpoints inward, and repeat. Export the raw record.
What to observe: The finite-time results can differ from the infinite-sweep formula, and noise changes the evolution itself.