We exhibit a specific implementation of the creation of geometrical phase through the state-space evolution generated by the dynamic quantum Zeno effect. That is, a system is guided through a closed loop in Hilbert space by means a sequence of closely spaced projections leading to a phase difference with respect to the original state. Our goal is the proposal of a specific experimental setup in which this phase could be created and observed. To this end we study the case of neutron spin, examine the practical aspects of realizing the “projections, ” and estimate the difference between the idealized projections and the The effect of the observer in quantum mechanics is perhaps nowhere more dramatic than in the collection of phenomena loosely (and casually) known as the “quantum Zeno effect. ” This was first formulated by von Neumann [1, 2], and is deeply rooted in fundamental features of the temporal behavior of quantum systems [3]. During
We exhibit a specific implementation of the creation of geometrical phase through the state-space evolution generated by the dynamic quantum Zeno effect. That is, a system is guided through a closed loop in Hilbert space by means a sequence of closely spaced projections leading to a phase difference with respect to the original state. Our goal is the proposal of a specific experimental setup in which this phase could be created and observed. To this end we study the case of neutron spin, examine the practical aspects of realizing the “projections, ” and estimate the difference between the idealized projections and the The effect of the observer in quantum mechanics is perhaps nowhere more dramatic than in the collection of phenomena loosely (and casually) known as the “quantum Zeno effect. ” This was first formulated by von Neumann [1, 2], and is deeply rooted in fundamental features of the temporal behavior of quantum systems [3]. During
The usual, "static" version of the quantum Zeno effect consists in the hindrance of the evolution of a quantum systems due to repeated measurements. There is however a "dynamic" version of the same phenomenon, first discussed by von Neumann in 1932 and subsequently explored by Aharonov and Anandan, in which a system is forced to follow a given trajectory. A Berry phase appears if such a trajectory is a closed loop in the projective Hilbert space. A specific example involving neutron spin is considered and a similar situation with photon polarization is investigated.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
The dynamics of a quantum system undergoing frequent ``measurements,'' leading to the so-called quantum Zeno effect, is examined on the basis of a neutron-spin experiment recently proposed for its demonstration. Unlike in all previous studies, the spatial degrees of freedom of the neutron are duly taken into account. Their inclusion in the analysis is important for two reasons: first, neutron-reflection effects are shown to be very important; second, the evolution may turn out to be totally different from the ideal case. Our results can be interpreted in terms of a rigorous theorem due to Misra and Sudarshan: indeed we clarify that, in contrast with a widespread belief, a quantum Zeno effect does not halt the evolution of a quantum system; it rather modifies it, by forcing the system to remain in a certain subspace, defined by the very measurement performed.
A purely dynamical explanation for the quantum Zeno effect is proposed. It is argued that a quantum system undergoes a quantum-Zeno-type dynamics as a consequence of a particular type of evolution involving a series of frequent spectral decompositions. The role of quantum measurements and of the ``collapse of the wave function'' is investigated and it is clarified that, provided a final observation is performed, the dynamical quantum Zeno effect can be obtained without making use of von Neumann's projection postulate. The meaning of infinitely frequent measurements is critically discussed and it is argued that it should be regarded as a mathematical idealization, impossible to realize from a physical point of view.
The influence of the geometric phase, in particular the Berry phase, on an entangled spin-21 system is studied. We discuss in detail the case, where the geometric phase is generated only by one part of the Hilbert space. We are able to cancel the effects of the dynamical phase by using the ``spin-echo'' method. We analyze how the Berry phase affects the Bell angles and the maximal violation of a Bell inequality. Furthermore we suggest an experimental realization of our setup within neutron interferometry.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.