Quantum Zeno Subspaces

Abstract

The quantum Zeno effect is recast in terms of an adiabatic theorem when the measurement is described as the dynamical coupling to another quantum system that plays the role of apparatus. A few significant examples are proposed and their practical relevance discussed. We also focus on decoherence-free subspaces.PACS numbers: 03.65. Xp, 03.67.Lx If very frequent measurements are performed on a quantum system, in order to ascertain whether it is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place [1,2]. This phenomenon stems from general features of the Schrödinger equation that yield quadratic behavior of the survival probability at short times [3,4]. The first realistic test of the quantum Zeno effect (QZE) for oscillating (two-level) systems was proposed about 15 years ago [5]. This led to experiments, discussions and new proposals [6]. A few years ago, the presence of a short-time quadratic region was experimentally confirmed also for a bona fide unstable system [7]. The same experimental setup has been used very recently [8] in order to prove the existence of the Zeno effect (as well as its inverse [9,10]) for an unstable quantum mechanical system, leading to new ideas [11,12].It is important to stress that the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multi-dimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. This continuing time evolution within the projected subspace ("quantum Zeno dynamics") has been recently investigated [13]. It has peculiar physical and mathematical features and sheds light on some subtle mathematical issues [2,14,15].All the above-mentioned investigations deal with what can be called "pulsed" measurements, according to von Neumann's projection postulate [16]. However, from a physical point of view, a "measurement" is nothing but an interaction with an external system (another quantum object, or a field, or simply another degree of freedom of the very system investigated), playing the role of apparatus. In this respect, if one is not too demanding in philosophical terms, von Neumann's postulate can be regarded as a useful shorthand notation, summarizing the final effect of the quantum measurement. This simple observation enables one to reformulate the QZE in terms of a (strong) coupling to an external agent. We emphasize that in such a case the QZE is a consequence of the dynamical features (i.e. the form factors) of the coupling between the system investigated and the external system, and no use is made of projection operators (and non-unitary dynamics). The idea of "continuous" measurement in a QZE context has been proposed several times during the last two decades [18,19], although the first quantitative comparison with the "pulsed" situation is rather recent [20].The purpose of the present article is to cast the quantum Zeno evolution in terms of an adiabatic theorem and stud...

Non-exponential Decay in Quantum Field Theory and in Quantum Mechanics: The Case of Two (or More) Decay Channels

“…the probability p(t) that an unstable state S formed at the time t = 0 did not decay at the instant t > 0, does not follow an exponential decay law for short times is now theoretically [1,2,3,4,5,6] and experimentally [7,8,9] established in the framework of quantum mechanics. Moreover, oscillations on top of the exponential function can also occur in the short time regime and could eventually last long enough to be detected [10,11]. Also at very large times deviations take place: a power law, and not an exponential law, is realized [1].…”

Thwarted dynamics by partial projective measurements

“…That is, the particle P is frozen in the initial Gaussian state |Φ 0 and never evolves (the quantum Zeno effect [5]). However, the field mode F can evolve in the "Zeno subspace" specified by the projection operatorP Z = |Φ 0 Φ 0 | ⊗ 1 1 F [5,7]. The exact formula (22) actually shows that in the Zeno limit the field mode F evolves asρ F (t) = |α 0 e −iωt α 0 e −iωt |.…”

Three Different Manifestations of the Quantum Zeno Effect

“…Let us outline the extension [10] of Misra and Sudarshan's theorem [1] on the QZE to the case of incomplete and nonselective measurements. Let Q be a quantum system, whose states belong to the Hilbert space H and whose evolution is described by the superoperator…”

“…However, as explained in the introduction, the basic features of the QZE can be obtained by making use of a continuous coupling, when the external system takes a sort of steady "gaze" at the system of interest. The mathematical formulation of this idea is contained in a theorem [18,10] on the (large-K) dynamical evolution governed by a generic Hamiltonian of the type…”

“…3 the notion of QZE to the case of unitary kicks [9] and finally discuss in Sect. 4 (unitary) continuous interactions [10]. We show in Sect.…”