Quantum Zeno Crossover — Bloch Phase-Space Portrait

Double-Well System

Tunnel coupling J 1.00

Measurement (dephasing) rate γ 0.00

Simulation speed 1.0×

Live State

Coherent tunneling

P(left) = 100%P(right) = 0%

Coherence |ρ_LR|

0.500

Effective decay Γ

0.00

Free period T=π/J

3.14

Elapsed time t

0.0

How it works

A particle in a symmetric double well can tunnel between the left (L) and right (R) wells. Restricted to the two lowest states this is a two-level system with tunnel coupling J. The Bloch-like vector (z, y) — population difference z = P(L) − P(R) and imaginary coherence y — obeys:

dz/dt = -2J y
dy/dt =  2J z - γ y

Instead of an animated 3D well, this page draws the (z, y) trajectory directly as a phase-space spiral on a 2D plane, and plots the crossover curve Γ_eff(γ) explicitly next to it:

γ = 0 — pure rotation on the unit circle at angular rate 2J, no inward spiral: coherent Rabi-like tunneling, period T = π/J.
0 < γ < 4J — anti-Zeno regime: the vector spirals inward while still winding around the origin; envelope decays as exp(-γt/2).
γ > 4J — quantum Zeno regime: the spiral degenerates into a straight inward slide (no winding — the eigenvalues go real), and the surviving decay rate collapses to Γ_eff ≈ 4J²/γ → 0 as γ grows. The particle freezes in its starting well.

This γ/2 → 4J²/γ crossover at γ = 4J is the Kofman–Kurizki quantum Zeno / anti-Zeno crossover — the general theory behind Itano's original 1990 trapped-ion Zeno experiment and continuous-measurement cold-atom demonstrations (Streed et al., Phys. Rev. Lett. 97, 260402, 2006).

Phase vector (z,y) Γ_eff(γ) crossover curve