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arXiv:math-ph/0207013 · Journal of Physics A Mathematical and General

Zeno dynamics in quantum statistical mechanics

Andreas U Schmidt

Published Jan 15, 2003• 12 citations
arXivHTMLPDF

Abstract

We study the quantum Zeno effect in quantum statistical mechanics within the operator algebraic framework. We formulate a condition for the appearance of the effect in W *-dynamical systems, in terms of the short-time behaviour of the dynamics. Examples of quantum spin systems show that this condition can be effectively applied to quantum statistical mechanical models. Furthermore, we derive an explicit form of the Zeno generator, and use it to construct Gibbs equilibrium states for the Zeno dynamics. As a concrete example, we consider the X – Y model, for which we show that a frequent measurement at a microscopic level, e.g. a single lattice site, can produce a macroscopic effect in changing the global equilibrium.

DOI
10.1088/0305-4470/36/4/319

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We study the quantum Zeno effect in quantum statistical mechanics within the operator algebraic framework. We formulate a condition for the appearance of the effect in W *-dynamical systems, in terms of the short-time behaviour of the dynamics. Examples of quantum spin systems show that this condition can be effectively applied to quantum statistical mechanical models. Furthermore, we derive an explicit form of the Zeno generator, and use it to construct Gibbs equilibrium states for the Zeno dynamics. As a concrete example, we consider the X – Y model, for which we show that a frequent measurement at a microscopic level, e.g. a single lattice site, can produce a macroscopic effect in changing the global equilibrium.
If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected "quantum Zeno subspace" is called "quantum Zeno dynamics:" for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
Abstract. If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected “quantum Zeno subspace ” is called “quantum Zeno dynamics: ” for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
We show that the quantum Zeno effect and the adiabatic change have a close connection even though the former means a halt of the dynamics by frequent measurements and the latter is the dynamics by a slowly changing Hamiltonian. We investigate the motion of a spin under stepping magnetic fields in terms of the (inverse) quantum Zeno effect. The relation between this model and the adiabatic change is studied. Another model is used to integrate the two effects into one formulation which is an extension of the proof of the adiabatic theorem. Despite their good parallelism, the asymptotic behaviors of the transition probability with respect to the total period TTT have different forms; the former effect shows a ${T}^{\ensuremath{-}1}$ behavior while the latter scales ${T}^{\ensuremath{-}2}$.
Frequent observation of a quantum system leads to quantum Zeno physics, where the system evolution is constrained to states commensurate with the measurement outcome. We show that, more generally, the system can evolve between such states through higher-order virtual processes that pass through states outside the measurement subspace. We derive effective Hamiltonians to describe this evolution, and the dependence on the time between measurements. We demonstrate application of this phenomena to prototypical quantum many-body system examples, spin chains and atoms in optical lattices, where it facilitates correlated dynamical effects.
Quantum Zeno effect can be applied to quantum information processing，and can reveal the nature of quantum measurement. In addition, it has also many potential applications. This suggests that studying the quantum Zeno effect has great theoretical and experimental significance. In this work, the system of a two-level atom interacting with a single mode field is considered and the dynamics of the system subjected to successive projection measurements is studied, and the quantum Zeno effect is presented. Moreover, the influence of the quantum Zeno effect on atomic population inversion is discussed. Based on Schrödinger equation, the survival probability of the initial state of the two-level atom subjected to frequently repeated measurements can be obtained. The survival probability depends on the time interval between measurements. It is seen that the exponential decay of the system under slowly frequent measurements is presented instead of the naturally oscillatory process. For slowly repeated measurements the atomic population inversion and the survival probability of initial state decline rapidly at the early time and then both of them become unchanged. As the time intervals of the measurements are sufficiently short, the quantum Zeno effect occurs. These results have also shown how the measurement can inhibit the atomic population inversion.
Abstract In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g., a measurement, used to restrict the time evolution (due, for example, to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a C0C_0C0​ C 0 -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate O(1n)\mathcal {O}(\tfrac{1}{n})O(n1​) O ( 1 n ) of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type n\sqrt{n}n​ n -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator, and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.