Quantum Zeno–Anti-Zeno Crossover in Double-Well Tunneling

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%

P(left well) over time

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. Starting purely in L, free evolution gives a coherent Rabi-like oscillation:

P(L,t) = cos²(Jt),   P(R,t) = sin²(Jt),   period T = π/J

A continuous position measurement — in real experiments this is fluorescence imaging of ultracold atoms tunneling between sites of an optical-lattice double well (Streed et al., Phys. Rev. Lett. 97, 260402, 2006) — dephases the L/R coherence at rate γ without changing the populations directly. The population difference z = P(L) − P(R) and the imaginary coherence y obey:

dz/dt = -2J y
dy/dt =  2J z - γ y

The two decay eigenvalues are λ = [-γ ± √(γ² - 16J²)] / 2, which produce three regimes as γ increases:

γ = 0 — coherent tunneling, no decay, period T = π/J.
0 < γ < 4J — anti-Zeno regime: weak measurement adds friction to the oscillation; its envelope decays at rate ≈ γ/2, so watching more makes the particle switch wells less predictably but decoheres faster.
γ > 4J — quantum Zeno regime: overdamped, and the surviving decay rate collapses to Γ_eff ≈ 4J²/γ → 0 as γ grows. The particle is frozen in whichever well it started in — frequent measurement suppresses the very transition it is monitoring.

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 later continuous-measurement demonstrations.

Left-well population Right-well population

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