Q022 · Avoided crossing / interference

Landau–Zener Passage

Cross an avoided level crossing, inspect the energy and fixed bases, then turn back. Vary the waiting time to reveal interference from the same finite-time evolution. Measure independently prepared trials and compare the record 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 slow and fast sweeps. Follow the colored populations, then measure. A return passage lets the alternatives interfere.

Choose, compare, then collect evidence.

Preview time — every measurement runs the complete 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.

Dimensionless two-level Hamiltonian with finite endpoints and optional Markovian dephasing in the fixed basis. The numerical density-matrix evolution includes both sweeps and the hold. The infinite-sweep exponential is a separate reference, applicable only to a noiseless single passage from the lower energy state.

Recorded evidence and model limits

SIMULATED INDEPENDENT TRIALS

Count the final upper-energy outcomes

Blue dots and pointwise 95% Wilson intervals come from the recorded trials. Gray is the finite-time model. These are not simultaneous bands or an optional-stopping guarantee. Preview and replay never collect trials.

Current setting: fraction / 95% range

\text{—}

Infinite-sweep reference, when applicable

\text{—}

Evolution steps / recorded trials

\text{—}

Finite endpoints, initial state and dephasing can invalidate the infinite-sweep reference. The return protocol is evolved in full, including the hold. The detector measures 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: An adiabatically followed energy state can exchange its fixed-basis character. A second crossing reads the relative phase accumulated between 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

Displayed energies are in the chosen scale and displayed times in its inverse angular frequency. Fixed zero has positive Pauli-z eigenvalue. The coupling and detuning signs are fixed by this equation.

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 positive endpoint detuning. A return protocol holds there, then reverses 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 preview is a different observable; neither is a particle path.

Infinite-endpoint reference

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

This reference assumes a linear sweep from negative to positive infinity, initially lower energy, without dephasing. It does not generate the finite-time trajectories or the return fringes.

Specified dephasing channel

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

Markovian pure dephasing acts in the fixed basis throughout sweeps and holds. A midpoint unitary step is bracketed by exact half dephasing steps. There is no relaxation, bath spectrum or material calibration.

From a record to a fraction

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

Every recorded outcome comes from a fresh preparation and the complete pulse at that scan setting. Pointwise Wilson intervals describe Bernoulli uncertainty under the stated model; they are not a joint scan band.

Common difficulties

Basis confusion

Typical misconceptionSlow evolution always means no state change.

Better mental modelAn energy eigenstate changes its fixed-basis composition as the Hamiltonian changes.

Multiplying probabilities

Typical misconceptionTwo passages are two independent coin flips.

Better mental modelCoherent amplitudes retain their relative phase. The simulator evolves that coherence through the full protocol.

Reference versus prediction

Typical misconceptionThe exponential must exactly match every finite pulse.

Better mental modelIts endpoint, initial-state and noise assumptions differ. The gray finite-time curve uses the actual configured evolution.

Run the experiment

01
Follow slowly
Choose Slow passage and replay to the end. Inspect both population curves.
What to observe: Upper-energy population stays small while the fixed-basis character changes.
02
Rush across
Choose Fast passage, then collect a speed scan.
What to observe: The state increasingly remains in its fixed-basis character, ending in upper energy.
03
Return and interfere
Choose Go there and back. Scan waiting time, then drag the waiting control.
What to observe: The final counts oscillate because phase accumulation changes before the second crossing.
04
Check the limits
Increase dephasing or move endpoints inward and repeat. Export the raw record.
What to observe: The finite-time record may depart from the infinite-sweep reference; noise changes the actual evolution.