Mesoscopic Physics meets Quantum Engineering

The following sections are included:

• Abstract
• Contents

The following sections are included:

• Mesoscopics: why it is important
• About the structure of this lecture course
• Fundamental and applied aspects of mesoscopics

The last three decades bore witness to the emergence of the new research field, Quantum Computations (also known as Quantum Information). Quantum Computations assumes the study of problems related to the processing and transfer of information using the laws and objects of quantum mechanics. The theory of quantum information appeared at the intersection of earlier fields such as information theory, computer sciences, and quantum mechanics. The formalism and methodology of quantum optics, condensed matter theory, and cryptography are also incorporated…

The problem of dynamical behaviour in a two-level system deserves detailed discussion. First, this gives results for fundamental problems. Second, it is very topical for mesoscopic systems, which have parameters tunable in a wide range, and where the regimes of control and interferometry are important. Third, it is a good example of the accurate solution of a realistic problem. Fourth, this will allow us to introduce useful formulas and approaches.

In accordance with the aims of our course, it is convenient to speak about a specific system. And here, appropriate examples are the qubits. Indeed, as discussed above, we note that (a) the qubits present the basic system of quantum mechanics, considering which, we can expound many basic problems, (b) they are interesting for possible applications in the field of quantum engineering, and (c) the possibility of these applications, in turn, is the locomotive of fundamental research in mesoscopic, superconducting and semiconducting systems…

Many of the quantum properties, inherent to superconducting circuits, can be realized in normal-metal coherent systems. This relates to systems of size much smaller than the coherence length L_φ. The phase coherence means that the particles can be described by a wave function, which evolves according to the Schrödinger equation. In the present chapter we will describe such systems, paying particular attention to specific properties such as conductance quantization and Coulomb blockade, which are important for quantum engineering. For further reading we recommend the textbooks [Moskalets 2010], [Zagoskin 2011] and the references in Footnotes 32 and 33 below.

As discussed previously, it is interesting and important to study not only single mesoscopic systems, but also their connection with other subsystems (see Footnote 9 on page 15). Below, we will consider two illustrative examples, one on the system of a quantum dot coupled to a classical nanomechanical oscillator and another for the system composed of a superconducting qubit and a quantum resonator on the base of a transmission line. In the former problem, we will consider the case of a slow resonator, of which the frequency ω₀ is much smaller than all other characteristic frequencies in the problem. In particular, if the respective distance between the quantum levels is much smaller than the thermal broadening, ћω₀ ≪ k_BT, then such a resonator should be treated as a classical one. And the system “a quantum dot — a classical resonator” will be considered in the framework of the so-called semi-classical theory. On the latter example we will consider the inverse situation, when ћω₀ > k_BT. In this case, the resonator should be considered as a quantum-mechanical object and the whole system “qubit-resonator” is described in full analogy with an atom interacting with a quantum field. Making such an analogy, we will familiarize ourselves with the formalism to describe such systems…

The following sections are included:

• CONCLUSION
• BIBLIOGRAPHY
• INDEX