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.

Basic notions of quantum mechanics

Quantum axioms

Let us start with a brief recapitulation of quantum mechanics on the “how to” level. According to the standard lore, the instantaneous state of any quantum system (that is, everything that can be known about it at a given moment of time) is given by its wave function (state vector) – a complex-valued vector in some abstract Hilbert space; the nature of this space is determined by the system. All the observables (i.e., physical quantities defined for the system and determined by its state – e.g., the position or momentum of a free particle, the energy of an oscillator) are described by Hermitian operators defined in the same Hilbert space. All three elements – the Hilbert space, the state vector, and the set of observables – are necessary to describe the outcome of any experiment one could perform with the system. Since humans cannot directly observe the behaviour of quantum objects, these outcomes are also called measurements, being the result of using some classical apparatus in order to translate the state of a quantum system into the state of the apparatus, which can then be read out by the experimentalist. The classical (i.e., non-quantum) nature of the apparatus is essential, as we shall see in a moment.

It is always risky to combine well-known and well-tested notions in order to describe something new, since the future usage of such combinations is unpredictable. After “quantum leaps” were appropriated by the public at large, nobody, except physicists and some chemists, seems to realize that they are exceedingly small, and that breathless descriptions of quantum leaps in policy, economy, engineering and human progress in general may actually provide an accurate, if sarcastic, picture of the reality. When the notion of the “marketplace of ideas” was embraced by academia, scientists failed to recognize that among other things this means spending 95% of your resources on marketing instead of research. Nevertheless, “quantum engineering” seems a justified and necessary name for the fast-expanding field, which, in spite of their close relations and common origins, is quite distinct from both “nanotechnology” and “quantum computing” in scope, approaches and purposes. Its subject covers the theory, design, fabrication and applications of solid-statebased structures, which can maintain quantum coherence in a controlled way. In a nutshell, it is about how to build devices out of solid-state qubits, and how they can be used.

The miniaturization of electronic devices to the point where quantum effects must be taken into account produced much of the momentum behind nanotechnology, together with the need to better understand and control matter on the molecular level coming from, e.g., molecular biology and biochemistry (see, e.g., Mansoori, 2005, Chapter 1).

Josephson effect

Superconductivity: A crash course

The transition from the theoretical description of hypothetical building blocks of a quantum coherent device to something which can be actually fabricated and controlled is made much easier by the existence of superconductivity. This phenomenon, roughly speaking, allows a macroscopic quantum coherent flow of electrons in a sufficiently cold piece of an appropriate material by establishing a specific long-range order among them. Due to this one can, for example, use macroscopically different states of a superconductor as quantum states of a qubit, with the obvious advantage over “microscopically quantum” systems (like actual atoms) from the point of view of control, measurement and, last but not least, fabrication of structures with the desired parameters and on the desired scale.

The phenomenon of superconductivity – the history of its discovery, experimental manifestations, theoretical explanation, open questions, relevance to other branches of physics, and technological applications – requires a thorough treatment, which can be found in any number of books. For our purposes Tinkham (2004) will provide more than sufficient background.

We will not wander into the field of exotic/high-temperature superconductors for two simple reasons: the quantum coherent behaviour of the kind we need to realize qubits or other quantum coherent devices has not yet been properly and/or routinely achieved in these systems; and the fabrication of such devices is not yet reliable enough or even feasible.

In this chapter we discuss a theory for the freezing of an isotropic liquid into a crystalline solid state with long-range order. The transformation is a first-order phase transition with finite latent heat absorbed in the process. We focus on a first-principles orderparameter theory of freezing that originated from the pioneering work of Ramakrishnan and Yussouff (1979). The theory approaches the problem from the liquid side and views the crystal as a liquid with grossly inhomogeneous density characterized by a lower symmetry of the corresponding lattice. This is in contrast to description of the crystal in terms of phonons. The crucial quantity characterizing the physical state of the system in this non-phonon-based model is the average one-particle density function n(x). The thermodynamic description of either phase involves a corresponding extremum principle for a relevant potential. The latter, obtained as a functional of the one-particle density n(x) and the stable thermodynamic state of the system, is identified by the corresponding density required for invoking the extremum principle. This approach, which is generally referred to as the density-functional theory (DFT) of freezing (Haymet, 1987; Baus, 1987, 1990; Singh, 1991; Löwen, 1994; Ashcroft 1996), has been improved over the years and successfully applied for the study of liquid-to-crystal transitions in various simple liquids, the solid–liquid interface, two-dimensional systems, metastable glassy states, etc. For applications of density-functional methods in statistical mechanics there exist general reviews (Evans, 1979; Henderson, 1992).