--- title: "Quantum Time Mirror Interactive Lab | Sandbox Physics" description: "Scatter the wave. Reverse the story. 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." canonical_url: "https://sandboxphysics.com/labs/loschmidt-quantum-time-mirror/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/loschmidt-quantum-time-mirror/" last_modified: "2026-10-18" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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 $0$ 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. SPREAD PHASE IMPRINT + REVERSE REASSEMBLE $\tau=32$ 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. 0 $2\pi$ 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. 0 / 8000 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 $\text{—}$ 95% Wilson interval $\text{—}$ Model probability in gold region $\text{—}$ Effective occupied sites $\text{—}$ ### 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. $\text{—}$ $$ F(t)=|\langle\psi(0)|\psi(t)\rangle|^2,\qquad N_{\mathrm{eff}}=\frac{(\sum_jp_j)^2}{\sum_jp_j^2} $$ 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. $0$ [Doncheski et al. (2003) · Wave packets in two-dimensional quantum billiards](https://arxiv.org/abs/quant-ph/0307070) [Wisniacki et al. (2001) · Sensitivity to perturbations in a quantum chaotic billiard](https://arxiv.org/abs/nlin/0111051) Physics tutorial ## Quantum Time Mirror **Background** 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. **Why it matters** Follow one state through a complete causal experiment. 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 misconception** The particle rides along a material surface. **Better mental model** The particle is confined to a two-dimensional lattice; height encodes square-root probability with fixed display gain. #### Finite model **Typical misconception** These images prove quantum chaos or the arrow of time. **Better mental model** 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. ### Run the experiment 1. 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. 2. 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. 3. 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. Physical conclusion 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. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/) 5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/) 6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/) [View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)