**Difficulty**

## Part 1: Superconductivity saves the day

**By Christian Dickel**

In a series of blog posts, I want to introduce the bread and butter of the DiCarlo group within QuTech: Studying quantum effects in superconducting electrical circuits. In the title, I suggest that we are building artificial atoms, but that depends on the definition of “atomness”. I hope to give the reader some insight to judge for him or herself whether our work comes short of this or goes beyond it. Also, I want to convey some of the amazement I feel working on a subject that brings together electrical engineering, superconductivity, and quantum mechanics in its purest form.

This blog post is rather long, but I have marked non-essential sections with a *.

## Of atoms and circuits

Most people know what an electrical circuit is. Many people even understand roughly what the diagrams that we use to represent them contain: resistors, capacitors and inductors. Here, it is important that for scientists and engineers, some diagrams are not merely illustrations but really have a precise mathematical meaning. The circuit, together with the parameters, corresponds to a mathematical object. In the case above, the circuit is made of an inductor and a capacitor.

Many people also have some image, that they associate with atoms. The image I chose here already contains some information about a specific atom: the oxygen atom. However, it does not have a precise mathematical meaning like a circuit diagram. It shows eight electrons (small black circles), eight protons (white circles) and eight neutrons (grey circles). The negatively charged electrons orbit around the positively charged nucleus like planets around a sun, only with electromagnetism instead of gravity as the attractive force.

The notion that everything is made of atoms is mainstream today, unchallenged even by religious fanatics or rogue scientists. Atoms are tiny (around 0.000 000 000 1 m in diameter) and behave according to the rules of quantum mechanics – supposedly a hard to understand theory. How can an electrical circuit behave like an atom?

## Atoms are quantum!

What is “atomness”? Certainly, electrical circuits are big compared to atoms and they are not the elementary building blocks of our world. What circuits and atoms have in common is that they both are governed by the laws of electrodynamics. But the artificial atom analogy has to do with quantum behavior, which in case of atoms first manifested in spectral lines and the fact that atoms exist at all. Electrodynamics would predict that there is no stable way that electrons can orbit the nucleus. Moving charge has to radiate electromagnetic waves (light) and this would cause the electrons to lose energy and spiral into the nucleus. This caused great confusion and unease amongst the physicists around 1910. At this point, they had found that different chemical elements emit and absorb light at specific frequencies (or colors, for the layman). Below the diagram representing oxygen, is the spectrum of oxygen, whose lines at certain colors are frequencies that oxygen atoms interact more strongly with.

Scientists knew these spectra for many different elements and knew atoms were stable, but there was no theory explaining it. This changed in 1913, when Niels Bohr postulated that the electron could have stable orbits around the nucleus when the angular momentum of the electron was equal to multiples of – bringing in the Planck constant . Planck had introduced this constant to solve another big problem at the time: the thermal radiation of blackbodies. “Classical” theories (we use the word classical as the opposite of quantum in this context) predicted this to have infinite energy. Planck postulated that energy is not exchanged in arbitrary amounts but in packages he called “quanta”. The new constant was related to the size of these energy packets. Bohr extended the “quantization” from light to matter. He used and the known masses and charges of the electron and nucleus to calculate the spectrum of hydrogen and got it right to impressive precision. This was the start of explaining the spectra and the stability of atoms and kicked off a golden age of physics.

The spectral lines of atoms are only part of the story. At the time Planck and Bohr just postulated it, but the discrete exchange of energy is an essential feature of quantum mechanics. With it also emerge the more counterintuitive quantum effects, such as superposition and entanglement mentioned in previous blog posts. The idea of pushing electrical circuits into the quantum regime is really aiming for a new platform where quantum effects can be engineered and studied with current nanotechnology.

## Circuits are not?

Long before quantum mechanics, the theory of electrical circuits was already known – Kirchhoff had layed out the rules to calculate voltages and currents in an electrical circuit in 1845. The circuit diagram above shows the circuit of a harmonic oscillator. It is the electrical equivalent of a pendulum where potential energy (height) and kinetic energy (speed of the pendulum) are replaced by the energy in the capacitor (potential) and inductor (kinetic). The resonance frequency is given by . At that frequency, the circuit will be more prone to energy exchange, a principle we use in for example antennas. This is similar to a spectral line where the atom exchanges energy, but it is not the same: most electrical circuits behave classically; currents and voltages do not show quantization. We do not notice our antennas sending and receiving tiny packages. Pushing circuits into the quantum regime is not easy, but another stunning experimental discovery would ultimately lead to the artificial atoms I work with: superconductivity.

Why do we not see quantum mechanical effects when we measure the resistance of a wire? In Bohr’s hydrogen atom, there is a single electron spinning around the nucleus in a vacuum. In the electrical circuits described by Kirchhoff’s Laws, charge is treated more like a fluid. While the total amount of charge is conserved, charge, in this theory does not come in discrete chunks. The reason is of course that there are generally so many electrons flowing through a circuit that their discrete nature disappears. In the limit of many interacting particles or degrees of freedom, quantum mechanical features can be hidden. One of the ways to make them appear is to work with circuits that have very few electrons.

The second key parameter that can hide quantum behavior is high temperature. You might ask: High compared to what? Atoms show quantum behavior at room temperature because their spectral lines correspond to very high temperatures. Remember, the light we use to see is emitted by the sun which burns at 5500° C and most atoms around us are in their ground states. The electrical circuits in our lab have energy transitions that lie much lower. Therefore, to see the quantum behavior, we have to cool our systems down to a fraction of a degree above absolute zero (−273.15° C). This might seem ironic to people who think of quantum mechanics as random, because temperature is related to energy stored in the random movement of atoms. However, this classical randomness related to temperature is quite distinct from quantum randomness. It can simply smear out the clear spectral lines of quantum mechanics such that they will look more like a classically expected continuum.

Lastly, there is energy loss. Electrical resistivity (which generally can strongly depend on temperature), is due to the electrons who carry currents losing some of their energy to the atomic lattice of the material and heating the material in the process. The electrons do not really move through metals like fish through water but rather like a ball through a pinball machine. Loss of energy can also hide quantum effects. If an electron, an atom or a circuit is in a superposition and loses its energy, it usually falls into the ground state. The energy has to go somewhere, for example as light or heat into the environment. In fact, the three points I make here are quite interrelated and together form the big picture of the transition between classical and quantum physics.

## Superconductivity to the rescue

As if to solve all our problems at once, in 1911 Heike Kammerlingh Onnes, then a professor in Leiden, discovered that the electrical resistance of metals can drop to zero at very low temperatures. This phenomenon was puzzling, because it is not a gradual reduction in resistance but a sudden jump at a temperature that depended on the material. Some metals like aluminum, mercury, titanium, and lead show this behavior, while others such as iron, copper, gold, and silver do not. The theoretical explanation for this phenomenon took almost 50 years and again involved quantum mechanics.

The hand-wavy explanation is that there is effectively an attractive force between the electrons which conduct the electricity, instead of the repelling force one would expect because negative charges repel each other. This is due to the Coulomb repulsion being weakened by the many conducting electrons actually shielding each other from it. At the same time, the lattice vibrations in the crystal made of the nuclei can lead to an attractive force and under certain conditions the attraction wins. The attractive interaction between the electrons makes them bond together in pairs named Cooper pairs after one of the theoretical physicists who developed this particular theory of superconductivity.

The Cooper pairs behave different from the original electrons, again because of quantum mechanics. While the electrons were bumping into the crystal lattice (specifically defects in that lattice), the Cooper pairs do not scatter on these defects. That is because the two electrons of the Cooper pair are not actually spatially close to each other which they would need to be to both scatter on a given defect or lattice vibration. Quantum mechanics does not allow the Cooper pairs to scatter without breaking up, which requires a minimum energy cost being payed. As long as this energy is not available they cannot scatter. They therefore do not experience energy loss while still transferring charge from A to B, thus leading to zero electrical resistance.

As soon as the temperature, which is the mean random energy available in the degrees of freedom of a system, exceeds the energy of the Cooper pair bond – also called the superconducting gap, they can be broken up into their constituent electrons and the electrical resistance will come back. That’s why superconductivity has a threshold temperature. But the threshold is sharper than would be explained by that alone.

## Macroscopic quantum coherence *

In addition to the zero electrical resistance, superconductivity has additional consequences, which I will struggle to illustrate here. The electrons themselves are particles called fermions (protons and neutrons are also fermions by the way, all building blocks of everyday-matter are fermions), which have the weird property that two of them cannot be in exactly the same state. This realization together with postulating another internal degree of freedom – the spin of the electron – ultimately explained the periodic table of elements and as such might be held amongst the highest scientific achievements of all time. The fermionic nature of electrons is also essential to describe semiconductors and metals.

However, two electrons bound by an attractive force form something much like an atom (think hydrogen atom and substitute the proton with another electron). This atom doesn’t behave like a fermion, but like a different type of particle, the boson. While two fermions cannot be in the same state, bosons can be, in fact due to quantum statistics they prefer it. The textbook example of a boson is the photon. Photons will bunch together into the same quantum state in phenomena such as lasing. In the superconductor, all Cooper pairs collapse into one massive ground state forming a so-called Cooper pair condensate. Basically the electrical conductivity is mediated by one collective quantum state of all Cooper pairs. This solves the problem of having many degrees of freedom because the Cooper pair condensate, while consisting of many particles only has two degree of freedom. An island of superconductor is characterized by two parameters: The number of cooper pairs on the island and the superconducting phase of the condensate. Writing down the state of a normal-metal island is quite a bit more complicated. Probably this blog article is too long and confusing but this is one of the most amazing features of quantum mechanics in my opinion and I wanted to share it!

## Quantum dots: “real” artificial atoms? *

In an atom the transition from one state of the other is actually the transition of an electron from one orbital to another. Electrons are fermions as mentioned above meaning that two of them cannot occupy exactly the same state. Our Cooper pairs do not behave like that and the states of our circuits are not fermionic excitations like in atoms. This actually means they exhibit less complex behavior, because in an atomic multi level system the transitions from one state to the other depends on if this state is already occupied. In our case this hardly matters, because – while I always talked about artificial atoms here – we usually only use two-levels of a quantum system – qubits. That is because we generally build devices that resemble digital information processors. But the higher levels do play a role in a multi-qubit context.

One can also make traps for single electrons in semiconductors, which are closer to artificial atoms. Essentially this is just making a box which can be occupied by one or more electrons. In QuTech, these boxes for electrons, called quantum dots, are also studied . One can also model them as circuits. They are operated in the few-electron regime and show fermionic behavior. The electronic cloud of an atom usually extends about 0.1 nm, while quantum dots can be on the order of 100 nm, so they are already larger. In comparison, our superconducting circuits are usually well in the regime – most features of our chips can be seen with the bare eye. Superconductivity is essential for pushing these huge circuits into the quantum regime. To conclude, while the excitations in our systems show quantum behavior, they are arguably a bit different from the excitations in atoms or quantum dots. However, in superconducting circuits parameter regimes can be achieved, that can be difficult to reach with atoms or quantum dots. The different ways of engineering quantum systems are really quite complementary.

## Design constraints

Using superconducting materials, we can now make a dissipation-less electrical circuit. The degrees of freedom of this circuit will behave quantum mechanically, if we chose their energy scales to be above the energy scale associated with temperature. But superconductors require very low temperatures anyway. In our case, most of our experiments take place around 20 mK. A mind-blowing 20 milli-degrees above absolute zero. This is a factor 50 below the transition temperature of aluminum, our workhorse superconductor. We’re safely in the superconducting regime and quantum mechanics will only allow photons with energies above (twice) the superconducting gap to cause dissipation.

So the energy of our superconducting circuit atoms is bounded by temperature from below and by the superconducting gap from above. The critical temperature of aluminum is about 1.2 K, which corresponds to a gap of about 40 GHz. Transitions close to that gap would be more prone to quasiparticles and radiation close to the gap degrades the superconductors performance. We usually don’t fabricate systems with transitions far past 8 GHz, which is on the safe side. The operating temperature should also be significantly below the superconducting gap. With a Helium 3 dilution refrigerator, which itself relies on quantum effects, we can cool macroscopic systems down to about 20 mK. This corresponds to a frequency of ~400 MHz. At a factor of 10 above this, thermal excitations in the system should be largely frozen out. Usually we do not operate our artificial atoms below 4 GHz.

The 4-8 GHz band is called the C-band by electrical engineers. Coincidentally the higher frequencies of common Wi-Fi also fall in this band. It has been used for a long time in satellite communications, such that there are good coherent sources of radiation available, which we use to manipulate our atoms. Also, at these frequencies there are still very good amplifiers available. At higher frequencies, amplifiers tend to become more noisy, equipment becomes more expensive and residual dielectric losses increase.

Within these constraints, we can make our artificial atoms. There are some additional constraints due to the materials we use. Mainly it is smart to avoid small charge islands where the fluctuating charges in the environment can have a big impact. But largely we get about one octave in frequency where we can aim our atoms’ transition energies. We can also engineer coupled atoms and microwave resonators.

## A toolbox for quantum engineering

This is how capacitors and inductors become the tools of quantum engineers. Another essential circuit element is the Josephson junction, but this is another story and shall be told in my next blog post. Atomic physicist have to live with discrete numbers of electrons, protons and neutrons. They have to find a suitable element and possibly isotope for a given application which can be quite non-trivial. We have to draw a circuit and do some calculations to get the right atomic properties. We don’t only make circuits that resemble atoms but we also make circuits for specific applications: We use amplifiers based on quantum circuits that only add the minimal noise allowed by quantum mechanics, or we make microwave lasers. Below, you can see one of my chips, as a drawing and the finished chip I made in the cleanroom. It is a 7mm by 2 mm chip and serves as a teaser for the next, more concrete blog article on superconducting circuits.

This is a very simplified account. Truth be told, one cannot fully understand all these phenomena without having had years of physics education. While it is a privilege to work on problems that combine so many fascinating effects and that required several generations of ingenuity, the downside is that our work will remain obscure to most people. We usually circumvent this problem by only telling bits and pieces of our work to the general public and presenting them as a complete picture. Here, I tried to give a brute-force account of the essential effects, but I still had to leave many essential logical steps unexplained and probably, I omitted some as well. But don’t be fooled, there is a coherent logic that underlies our scientific work and that ties everything together. If you want to understand these things in more detail, look at hyperphysics for more detailed explanations of different effects that I mentioned. More specific introductions to superconducting circuits can be found here and here. As a take away-message, imagine your job was designing and making your own atoms.

Editors Note [30/06/17]: changed treatment of superconducting island degrees of freedom to be more accurate at request of the author.

### About Christian Dickel

Chris came to the Netherlands for the food and the weather but stayed for the quantum computer work. Apart from work he enjoys playing music with friends, ranting and soap-boxing.

This toolbox for quantum engineering is very important for design. Hope everyone get enough help from this. Thanks for shear it.

A toolbox for quantum engineering for nice project i know. i have learn enough from here.

its very helpful,thanks for shearing.