LOSCHMIDT ECHO / PHASE MEMORY

Quantum Time Mirror

Let a wavepacket spread through a stadium. Reverse its Hamiltonian and gather it back. Twist the phase of half the arena: the density stays unchanged at that instant, but the final echo breaks.

An ideal sign-reversal protocol in a finite one-particle lattice model. A coherent phase imprint and a reverse-leg potential test the echo. This is not a time machine, experimental hardware, decoherence or a demonstration of thermodynamic irreversibility.

COHERENT WAVEFIELD

OVERLAP WITH THE INITIAL QUANTUM STATE

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MODEL · PHASE SENSITIVE

Height: square root of site probability, fixed gain. Gold circle: the preparation region. The surface is a wavefunction diagnostic, not a moving material sheet.

SPREADPHASE IMPRINT + REVERSEREASSEMBLE

\tau=32

01 / THE FULL STORY

The echo must recover the phase, too.

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

A fold in the wave history.

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 are at the midpoint. No time elapses between them. Increase the phase knob: only the upper half changes hue. The reverse-leg potential has not acted yet.

Before imprint

Immediately after

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

SAMPLE THE WAVEFUNCTION

Make the pattern out of 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: Scatter the wave. Reverse the story.
One-sentence intuition: Identical position densities can carry different phases and produce different echoes.

Core mathematical model

Evolve one state

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

Every surface, diagnostic and sampled position is derived from this state.

Compare the full state

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

This model fidelity includes phase; it is not the fraction detected near the starting point.

Common difficulties

A raised wave is a diagnostic

Typical misconceptionThe particle rides along a material surface.

Better mental modelThe particle is confined to a two-dimensional lattice; height encodes square-root probability with fixed display gain.

Finite model

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

Better mental modelAn ideal sign-reversal protocol in a finite one-particle lattice model. A coherent phase imprint and a reverse-leg potential test the echo. This is not a time machine, experimental hardware, decoherence or a demonstration of thermodynamic irreversibility.

Run the experiment

01
See the cause
Select the three presets and scrub the full timeline.
What to observe: An ideal reversal returns the state. A half-plane phase twist spoils the final overlap.
02
Inspect phase
Toggle phase color, then inspect the phase map below the arena.
What to observe: Hue is a phase convention. Spatial differences matter; global hue does not.
03
Acquire evidence
Collect positions, pin the experiment and export all trials.
What to observe: The measured frequency converges toward the model density, but does not reveal all phases.