The quantum field theory (QFT) is a universal common language of the condensed matter community. As any live language it keeps evolving and changing. The change comes as a response to new problems and developments, trends from other branches of physics and from internal pressure to optimize its own vocabulary to make it more flexible and powerful. There are a number of excellent books which document this evolution and give snapshots of “modern” QFT in condensed matter theory for almost half a century. In the beginning QFT was developed in the second quantization operator language. It brought such monumental books as Kadanoff and Baym [1], Abrikosov, Gor'kov and Dzyaloshinski (AGD) [2], Fetter and Walecka [3] and Mahan [4]. The advent of renormalization group and Grassmann integrals stimulated development of functional methods of QFT. They were reflected in the next generation of books such as Itzykson and Zuber [5], Negele and Orland [6] and Fradkin [7]. The latest generation, e.g. Tsvelik [8], Altland and Simons [9], and Nagaosa [10], is not only fully based on functional methods, but also deeply incorporates ideas of symmetry based on universality, geometry and topology. (I do not mention here some excellent specialized texts devoted to applications of QFT in superconductivity, magnetism, phase transitions, mesoscopics, one-dimensional physics, etc.)

This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples.

Classical dissipative action

In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate Xq(t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical. Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects.

To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ-1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as Xq → ħXq. Indeed, when this is done all terms linear in Xq do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T/ħ.

Physical implementations of qubit coupling

Coupling by linear passive elements. Capacitive coupling

To have good basic elements is not enough – it is necessary to be able to connect them in a controllable way, without losing quantum coherence. Any simple effective coupling Hamiltonian (like in Eqs. (3.81, 3.82)) must be somehow implemented “in metal”. Here superconducting circuits provide a wide variety of coupling schemes to choose from (see, e.g., Wendin and Shumeiko, 2005). We will begin with the simplest case, when the interaction between the qubits is realized using linear elements (conventional capacitances and inductances), the coupling circuit stays in its ground state and adiabatically follows the evolution of the qubits (Averin and Bruder, 2003) – that is, it remains “passive”. For this to happen, the excitation energy of the coupler, ħωres, must be much higher than the interlevel spacing in the qubits (where ωres is the resonance frequency of the coupler). In other words, the evolution of the coupler is much faster than that of the qubits, and the coupler can indeed adjust to changes in the state of the latter. In the case of a purely capacitive or purely inductive coupling this condition is automatically satisfied, as then ωres → ∞.

Quantum transport in two dimensions

Formation of two-dimensional electron gas in heterojunction devices

Given all the advantages of superconducting structures, with their tunability, intrinsic protection against decoherence, well-understood physics and well-developed fabrication and experimental techniques, it would seem superfluous even to consider other possibilities for quantum engineering. Nevertheless, it would be short-sighted to neglect other possibilities, especially such rich ones as provided by devices based on a two-dimensional electron gas (2DEG). Here one has a normal electron system, which, nevertheless, maintains quantum coherence over comparatively large distances, and can literally be shaped into the desirable form (two- one- or zero-dimensional) during an experiment by a simple turn of a knob. It can use both charge and spin degrees of freedom, and serve as a basis for qubits, sensitive quantum detectors, quantum interferometers or other interesting devices. It is edifying, showing that one does not necessarily need a macroscopic quantum state (like superconductivity) to observe macroscopic quantum coherence.

As a bonus, there are interesting and useful effects that can be realized in hybrid, superconductor-2DEG structures. Probably, if we had discussed these devices first, we would have asked, who needs superconductors?

Quantum metamaterials

A qubit in a transmission line

Now we can finally discuss what kind of structures can be formed using qubits as basic units, and what would be their properties. Consider, for example, the optical properties of such a structure. For wavelengths much larger than the size of a single element, or the distance between elements, the structure will appear to be a continuous material with, possibly, strange properties. Such materials built out of artificial unit blocks are known as metamaterials (e.g., Saleh and Teich, 2007, 5.7) and, indeed, demonstrate such exotic behaviour as having a negative refractive index. Compared to them, quantum metamaterials (Rakhmanov et al., 2008), where the building blocks are qubits (or more complex artificial atoms), which maintain quantum coherence during the characteristic time of the signal propagation through them, and whose quantum state can be, in principle, selectively controlled, promise a wider and even more interesting range of behaviour. Such a device could be called an extended quantum coherent system, and our investigations will take us from the field of already fabricated structures, already conducted experiments and already verified theoretical models and into the even more interesting here-be-dragons territory of suppositions, suggestions and open questions.