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Physica Scripta

Probing the quantum–classical connection with open quantum dots

David K. Ferry, Richard Akis, Roland Brunner

Published Oct 1, 2015• 7 citations
Publisher

Abstract

Open quantum dots provide a natural system in which to study both classical and quantum features of transport. From the classical point of view these dots possess a mixed phase space which yields families of closed, regular orbits as well as an expansive sea of chaos. As a closed test bed, they provide a natural system with a very rich set of eigen-states. When coupled to the environment through a pair of quantum point contacts, each of which passes several modes, the original quantum environment evolves into a set of decoherent and coherent states, which eventually couple to the classical states discussed above. The manner of this connection is governed strongly by decoherence theory. The remaining coherent states possess all the properties of pointer states. Here, we discuss the quantum–classical connection and how it appears within the experimental world.

DOI
10.1088/0031-8949/2015/t165/014010

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Open quantum dots provide a natural system in which to study both classical and quantum features of transport. From the classical point of view these dots possess a mixed phase space which yields families of closed, regular orbits as well as an expansive sea of chaos. As a closed test bed, they provide a natural system with a very rich set of eigen-states. When coupled to the environment through a pair of quantum point contacts, each of which passes several modes, the original quantum environment evolves into a set of decoherent and coherent states, which eventually couple to the classical states discussed above. The manner of this connection is governed strongly by decoherence theory. The remaining coherent states possess all the properties of pointer states. Here, we discuss the quantum–classical connection and how it appears within the experimental world.
Recent work on the role of decoherence in quantum physics has suggested that the quantum to classical decay is governed by a discrete set of pointer states, which are quite stable and uncoupled from other states in the system. We show that the conductance oscillations exhibited by open quantum dots are governed by a discrete set of stable quantum states which have the properties of the pointer states, and which are closely related to trapped classical orbits in the open dot. These states are essentially classical in nature, as evidenced by their energy level spacing, and their decay is apparently in the environment as opposed to within the dots.
Edge states have been a backbone of our understanding of the experimental basis of the quantum Hall effect for quite some time. Interestingly, this comprises a quantum system with well defined currents and particle trajectories. The role of trajectories in quantum mechanics has been a problematic question of interpretation for quite some time, and the open quantum dot is a natural system in which to probe this question. Contrary to early speculation, a set of well defined quantum states survives in the open quantum dot. These states are the pointer states and provide a transition into the classical states that can be found in these structures. These states provide resonances, which are observable as oscillatory behavior in the magnetoconductance of the dots. But, they have well defined current directions within the dots. Consequently, one expects trajectories to be a property of these states as well. As one crosses from the low to the high field regime, quite steady trajectories and consequent wave functions can easily be identified and examined. In this talk, we review the current understanding and the support for the decoherence theory.
Three paradigms commonly used in classical, pre-quantum physics to describe particles (that is: the material point, the test-particle and the diluted particle (droplet model)) can be identified as limit-cases of a quantum regime in which pairs of particles interact without getting entangled with each other. This entanglement-free regime also provides a simplified model of what is called in the decoherence approach islands of classicality, that is, preferred bases that would be selected through evolution by a Darwinist mechanism that aims at optimising information. We show how, under very general conditions, coherent states are natural candidates for classical pointer states. This occurs essentially because, when a (supposedly bosonic) system coherently exchanges only one quantum at a time with the (supposedly bosonic) environment, coherent states of the system do not get entangled with the environment, due to the bosonic symmetry.
We consider the quasi-classical situation in which a quantum system interacts with another quantum system or with a quantum environment without getting entangled with it. We show how this regime is intimately linked to three paradigms commonly used in classical, pre-quantum physics to describe particles (that is, the material point, the test particle and the diluted particle (droplet model)). This entanglement-free regime also provides a simplified insight on what is called in the decoherence approach "islands of classicality", that is, preferred bases that would be selected through evolution by a Darwinist mechanism which aims at optimizing information. We show how, under very general conditions, coherent states are natural candidates for classical pointer states. This occurs essentially because, when a (supposedly bosonic) system coherently exchanges only one quantum at a time with its (supposedly bosonic) environment, coherent states of the system do not get entangled with the environment, due to the bosonic symmetry.
The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ``Schr\"odinger-cat states.'' The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ``collapse of the wave packet.'' The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ``classical reality'' as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ``found out'' without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ``fitness'' in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.