Q014 · Prepare, couple and read out

Coherent Double Well Transfer

How can a particle get into the other well without climbing over the barrier? By tunneling. The two lowest states are split by a small tunneling energy, and the probability swings between the wells at the matching rate. A higher barrier slows the swing, and tilting the wells keeps the transfer from being complete. This two-state picture holds only while no higher levels are occupied.

Raise the central barrier or tilt the wells, and watch probability move between the two sides. Try closing the barrier at different times, reconstruct the relative coherence and find where a two-state model breaks down.

Model initial energy gap $\text{—}$

Model right-side probability $\text{—}$

First, make a prediction

Will a higher barrier stop the transfer?

Drag the top of the barrier and play time. Keep the same observation duration. Then compare a bias, a closing pulse and a narrow initial state.

Make a prediction, then gather evidence.

Measurement time — a fresh preparation for every event

Central barrier height

B

3

Central barrier width

b

0.60

Right-minus-left bias

\epsilon

0

PreparationBarrier timingMeasurement

Changing the time or measurement basis keeps the records; changing the preparation clears them. The phase analyzers use separate preparations and keep outcomes that fall outside the two-state subspace.

State details, numerics & sampling

Initial right-basis weight

r

0

Initial relative phase

\phi

0^{\circ}

Raise barrier at time

\tau_c

20

Observation duration

T

80

Sine basis size

N_b

64

Samples per acquisition

N

400

Reproducible seed

s

2614

All quantities are dimensionless; the lesson defines physical units. Raising the basis size tests truncation, not detector accuracy. The record limit is 30,000.

The model is an isolated particle in a one-dimensional box with hard walls. The spatial Hamiltonian evolves in a finite sine basis, and a separately projected two-state model serves as a comparison. Every measurement starts from a fresh preparation. Closing the barrier is a sudden quench and can add energy.

Recorded evidence & model checks

Simulated independent preparations

Does the measured transfer follow two states?

Blue points are measured right-side fractions with 95% Wilson intervals. Gray is the spatial model; dashed orange is the unnormalized two-state projection. The lower bars count both phase analyzers, including the subspace complement.

Measured coherence and phase

\text{—}

Model norm / two-state leakage

\text{—}

Initial spatial gap / projected gap

\text{—}

Basis and boundary checks

\text{—}

The outer walls at positions minus three and plus three are physical. The reference localized states come from the unbiased lowest doublet and retain small opposite-side tails. Their relative phase is a basis coherence, not the phase at a point. A sudden closing pulse can populate higher modes; the two-state projection discards that amplitude without renormalizing it.

Sources: MIT 8.04 · 11 · Feynman III · 8

Start with the essentials

Focus question: When does the two-state picture fail?
One-sentence intuition: Computed states, model references and measured outcomes play different roles, so keep them apart.

Core mathematical model

Dimensionless units

X=x/\ell,\quad \mu=m/m_*,\quad \mathcal E=Em_*\ell^2/\hbar^2,\quad \tau=\hbar t/(m_*\ell^2)

Pick a reference mass and length. The controls, axes and CSV all use these units. In the double well, the mass ratio is fixed at one.

Sine-basis approximation

u_j(X)=\sqrt{2/L}\sin\!\left[\frac{j\pi(X+L/2)}{L}\right],\quad H_{jk}=\frac{j^2\pi^2}{2\mu L^2}\delta_{jk}+\int u_jVu_k\,dX

Every retained sine mode takes part in the evolution, and the potential integrals are done analytically. The displayed matrix residual checks the eigensolver; it does not measure basis or boundary error.

Double-well potential

V(X,\tau)=B(\tau)\mathbf1_{|X|<b/2}+\frac{\epsilon}{2}\operatorname{sgn}X,\quad -3<X<3

The outer walls are infinitely high, and a positive bias raises the right side. The closing protocol suddenly sets the barrier to twelve at the chosen time. Changing a control starts a new preparation.

Left and right reference states

|L\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2},\quad |R\rangle=\frac{|0\rangle-|1\rangle}{\sqrt2},\quad |\psi(0)\rangle=\sqrt{1-r}|L\rangle+e^{i\phi}\sqrt r|R\rangle

The two lowest spatial eigenstates without bias are combined, with signs chosen so the left state sits mostly on the left. Each keeps a small tail on the other side. The narrow-packet challenge instead takes a Gaussian centered at minus 1.7 with a position standard deviation of 0.28, projects it onto the sine basis and normalizes it.

Coherent time evolution

|\psi(\tau)\rangle=e^{-iH\tau}|\psi(0)\rangle,\qquad |\psi(\tau>\tau_c)\rangle=e^{-iH_c(\tau-\tau_c)}e^{-iH\tau_c}|\psi(0)\rangle

Within the retained basis, each eigenphase advances exactly, with no time stepping. The state does not jump at a quench, but its energy can change by the work done in changing the barrier. There is no environment and no continuous measurement.

Comparison with a two-state model

H_2=P H P,\quad P=|L\rangle\langle L|+|R\rangle\langle R|,\qquad P_R(\tau)=\tfrac12+[P_R(0)-\tfrac12]\cos[(\mathcal E_1-\mathcal E_0)\tau]

The cosine law is exact for the left-localized reference state in a symmetric, static well. The projected model measures the same physical left-or-right observable and does not renormalize the amplitude it discards. Bias, narrow states and sudden barrier changes can reveal its limits.

Phase-sensitive measurements

|\pm_\theta\rangle=\frac{|L\rangle\pm e^{i\theta}|R\rangle}{\sqrt2},\quad p_+-p_-=2\operatorname{Re}(e^{-i\theta}a_L^*a_R)

The real and imaginary analyzers use angles of zero and ninety degrees, each on its own fresh preparation. Any part of the state outside the reference pair is recorded as a third outcome. Left and right counts alone cannot reconstruct the coherence.

Common difficulties

A cloud is not a path

Typical misconceptionAn animated dot crosses the wall like a classical particle.

Better mental modelThe animation shows a coherent probability density. Each recorded event comes from a freshly prepared state.

More events do not fix the model

Typical misconceptionSampling noise is the only source of error.

Better mental modelA finite basis, artificial boundaries and missing detector physics each need their own check.

Run the experiment

01
Watch the transfer
Play the default state and sample at nine times.
What to observe: Compare the gray reference with the observed fractions.
02
Hold the probability on one side
Try a tall, wide barrier, then tune the closing time.
What to observe: Closing the barrier does not always freeze the quenched state perfectly.
03
Read the phase
Sample both coherence analyzers at the same time.
What to observe: You can estimate a phase only when the coherence is large enough to measure.
04
Break the two-state approximation
Choose the narrow packet and compare the two model curves.
What to observe: Higher modes can carry a large share of the probability.