Loschmidt echo · phase memory

Quantum Time Mirror

Can a scattered wave be gathered back into exactly its starting state? Yes, in this ideal model. Flip the sign of the Hamiltonian halfway and the wave retraces its spread back to its starting state. The return depends on phase as well as density. Shift the phase of the upper half of the arena by half a cycle at the turnaround: the density is unchanged at that instant, yet the final overlap with the start can drop below 1 percent. A full-cycle shift restores the echo.

Let a wavepacket spread through a stadium, then reverse the sign of its Hamiltonian so the wave gathers back. Twist the phase over half the arena at the turning point: the density does not change at that moment, yet the final echo fails.

The model is a finite lattice with one particle, and the sign of its Hamiltonian is reversed ideally. Two perturbations test the echo: a coherent phase imprint and an extra potential during the return leg. This is not a time machine or experimental hardware; it does not model decoherence or demonstrate thermodynamic irreversibility.

COHERENT WAVEFIELD

Overlap with the initial quantum state

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Model · phase sensitive

Height shows the square root of site probability, with fixed gain. The gold circle marks the preparation region. The surface is a plot of the wavefunction, not a moving sheet of material.

SPREADPHASE IMPRINT + REVERSEREASSEMBLE

\tau=32

01 / THE FULL STORY

An echo has to bring back the phase as well.

Cyan: selected protocol. Dashed coral: perfect reversal, same initial state. Drag the gold cursor to inspect.

$\tau=Jt/\hbar$

02 / PHASE

Same brightness can hide a different state.

Hue: complex phase. Brightness: amplitude, fixed scale. The hue origin is arbitrary; relative phase matters. Top is positive vertical coordinate.

2\pi

03 / SPACE AND TIME

Here the wave history folds back.

Horizontal: lattice column, left to right. Vertical: time, 0 at top and 64 at bottom. Color shows the model position marginal, with fixed square-root scale.

04 / POSITION READOUTS

One preparation. One position.

Counts from the recorded positions only, divided by total shots. Empty until you measure. The gold region is fixed; positions alone do not measure quantum-state overlap.

Freeze the imprint: identical density, different phase

Both maps show the midpoint, with no time between them. Turn up the phase knob and only the upper half changes hue. The potential on the way back has not acted yet.

Before imprint

Immediately after

$|e^{-i\phi}\psi_j|^2=|\psi_j|^2$

Sample the wavefunction

Build the pattern from single detections.

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Each detection is a fresh independent preparation at the selected time. Ideal lattice-position measurement; all shots are kept. Fixed simulation seed, reproducible on reset.

Measured fraction in gold region

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95% Wilson interval

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Model probability in gold region

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Effective occupied sites

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Do the counts follow the wave?

Gold bars: recorded horizontal-position frequency. Cyan: model marginal. Linear probability scale. Columns run from 0 to 63.

Latest 20 records · complete CSV export

Trial	$x/a$	$y/a$	Gold region

Model, units, numerical checks & sources

H_0/J=4I-\sum_{\langle i,j\rangle}(|i\rangle\langle j|+|j\rangle\langle i|),\qquad\tau=Jt/\hbar

A finite square lattice, spacing a, uniform hopping J and no interactions. The surrounding and island sites are removed. The diagonal stays four even at a boundary: a discrete Dirichlet wall. This finite lattice model is the quantitative claim; the usual continuum interpretation would require a separate mesh-convergence study.

\psi_j(0)=C\exp\!\left[-\frac{(x_j/a+10)^2+(y_j/a+4)^2}{25}\right]e^{0.9i[(x_j/a+10)\cos\theta+(y_j/a+4)\sin\theta]}

The truncated Gaussian is normalized once, at preparation. Horizontal coordinate increases rightward, vertical upward. Angle is counterclockwise from the positive horizontal axis. The circle has radius 18; stadium semicircle centers are at horizontal coordinates minus and plus 9; the rectangle has half-width 27 and half-height 18. The island is centered at (4, 0). All lengths use lattice spacing.

|\psi(2T)\rangle=e^{i(H_0+\epsilon G)T/\hbar}e^{-i\phi P_+}e^{-iH_0T/\hbar}|\psi(0)\rangle

JT/\hbar=32,\quad P_+=\sum_{y_j>0}|j\rangle\langle j|,\quad G_{jj}=e^{-[(x_j/a-6)^2+(y_j/a)^2]/18}

Just after the midpoint, imprint the upper half and evolve under the negative of the original Hamiltonian plus its potential error. The displayed midpoint is before the imprint; the next frame includes it. Ideal sign reversal is stipulated, not engineered here. The Gaussian error is centered at (6, 0), and is absent on the outward leg.

\epsilon=0:\quad F(2T)=1-4p_+(1-p_+)\sin^2(\phi/2)

The upper-half probability just before imprint gives an independent exact check of the final overlap when the reverse potential error is zero.

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F(t)=|\langle\psi(0)|\psi(t)\rangle|^2,\qquad N_{\mathrm{eff}}=\frac{(\sum_jp_j)^2}{\sum_jp_j^2}

The gold region is a disk of radius four around the initial center. Its position count is distinct from phase-sensitive state fidelity. Colors saturate at site probability 0.025; height is 24 times its square root in scene units. Spacetime color saturates at horizontal marginal probability 0.2. These display gains never change propagation or counts.

Exponential action uses norm-limited substeps, without renormalization. No environment, loss, continuum extrapolation, chaos exponent or thermodynamic claim. Sampling uses a fixed pseudorandom seed; intervals describe shot uncertainty only.

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Doncheski et al. (2003) · Wave packets in two-dimensional quantum billiards

Wisniacki et al. (2001) · Sensitivity to perturbations in a quantum chaotic billiard

Start with the essentials

Focus question: Can a scattered wave be gathered back into exactly its starting state?
One-sentence intuition: Two states with exactly the same position density can carry different phases, and so give different echoes.

Core mathematical model

Evolve one state

|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle

Every surface, diagnostic plot and sampled position comes from this one state.

Compare the full state

F(t)=|\langle\psi(0)|\psi(t)\rangle|^2

This fidelity is computed from the model and includes phase. It is not the fraction of detections near the starting point.

Common difficulties

The raised surface is only a display

Typical misconceptionThe particle rides over a real, material surface.

Better mental modelThe particle moves only on a two-dimensional lattice. The height shows the square root of the probability, scaled by a fixed display gain.

A finite model

Typical misconceptionThese images prove quantum chaos or the arrow of time.

Better mental modelThe model is a finite lattice with one particle, and the sign of its Hamiltonian is reversed ideally. Two perturbations test the echo: a coherent phase imprint and an extra potential during the return leg. This is not a time machine or experimental hardware; it does not model decoherence or demonstrate thermodynamic irreversibility.

Run the experiment

01
Compare the presets
Select the three presets and scrub the full timeline.
What to observe: An ideal reversal brings the state back. A phase twist on half the plane spoils the final overlap.
02
Inspect phase
Toggle phase color, then inspect the phase map below the arena.
What to observe: The colors follow a phase convention. Phase differences across space are physical; the overall hue is not.
03
Collect detections
Collect positions, pin the experiment and export all trials.
What to observe: As detections pile up, their frequencies approach the model density, but they cannot reveal all of the phase.