Quantum Zeno effect and dynamics

Abstract

If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its -- not necessarily bounded from below -- generator as a generalized mean value Hamiltonian

Generalized Adiabatic Theorem and Strong-Coupling Limits

“…The standard way to induce the QZD is to frequently perform projective measurements [2,22,23]. Consider a finite-dimensional quantum system with a Hamiltonian H. The unitary evolution of its density matrix ρ is given by e −itH (ρ) = e −itH ρe itH , with H = [H, q ].…”

Large-time limit of the quantum Zeno effect

“…uniformly in t on compact subsets of R, see [2][3][4] . Therefore, if one performs frequent measurements on a quantum system in a given time interval [0, t], a QZE takes place 1 : the transitions to states different from the initial one are hindered, despite the action of the Hamiltonian (in general the state ψ is not an eigenstate of the Hamiltonian H).…”

Kick and Fix: The Roots of Quantum Control

“…where P is a projection operator and H Z = PHP the "Zeno" Hamiltonian [10]. Notice that in general U N (t) is not unitary on the range of P, while the limit is, under suitable conditions and in particular cases, such as bounded H or finite-dimensional P [11]. A general proof of the above formula is still missing, together with a rigorous definition of the Zeno Hamiltonian for infinite-dimensional projections and unbounded Hamiltonian.…”