A wave packet becomes a complicated interference pattern in a Sinai cavity; complex conjugation reverses all momenta and makes the full 2D dynamics reconstruct the initial state.
This page describes the phenomenon using the current QPE preset and numerical modules as its primary source.
This experiment separates microscopic reversibility from dynamical sensitivity. Under ideal reversal, phase information can reconstruct the state; small perturbations in the return evolution make the fidelity decay.
Press Play. A localized packet enters a rectangular cavity with a central circular scatterer. Within a short time it splits, reflects and turns into a complex interference network.
At t=T=4 a special operation is applied. For a spinless particle in a real time-independent potential, time reversal reduces to complex conjugation: psi(x,y) -> psi*(x,y). Density is unchanged at that instant, while probability current and all momenta reverse.
Watch the viewport after T. The apparently scrambled field starts to reconstruct as wavefronts retrace their previous history. Near t=2T the packet returns almost exactly to the time-reversed initial state.
Analysis -> Loschmidt. The upper panel shows the mean-position path, cyan outbound and gold on return. Below, cyan is a phase-insensitive density fidelity and gold is coherent |<psi(0)*|psi(t)>|^2.
Essential test. With Delta px=Delta py=0 both fidelities should recover nearly 1 at 2T. Then try Delta px=0.12 and Delta py=-0.07, Reset and Play again. The reversal error is small compared with the initial momentum, yet the final echo degrades strongly after the complicated interference history.
History. Loschmidt's nineteenth-century reversibility objection highlighted the tension between reversible microscopic laws and macroscopic irreversibility. In quantum mechanics Asher Peres proposed fidelity in 1984 as a probe of dynamical sensitivity; Loschmidt echo became a major quantitative tool in quantum-chaos and decoherence studies around the turn of the twenty-first century.
Model boundary. psi->psi* is the correct time-reversal operation here only for a spinless scalar wavefunction with a real Hamiltonian and no magnetic vector potential. Absorption or dissipation would destroy exact reversibility. Delta p models a controlled instantaneous reversal error, not environmental decoherence.
The term Loschmidt echo is used to quantify imperfect reversibility and sensitivity to perturbations. Jalabert and Pastawski gave an influential formulation for classically chaotic systems in 2001.
| Representative platform | GaAs-like ballistic 2DEG in a mesoscopic cavity |
|---|---|
| Mapping basis | effective-mass mapping (m*=0.067 m_e) |
| 1 QPE length unit | 50 nm |
| 1 QPE time unit | 1.45 ps |
| Energy scale E0=ħ/T0 | 0.455 meV |
| Equivalent frequency E0/h | 0.11 THz |
| Note | Representative ballistic-cavity mapping. The 40.96-unit domain is about 2.05 um across; microwave and acoustic billiards remain useful wave analogues but have different physical time conversions. |
Open Model & Constraints to see exactly what QPE computes and what is left outside the model. For current live values use Help → Current Experiment → Current Parameters.