**Difficulty**

**By Hans Mooij**

# Introduction

When did we have our first quantum bit? To answer, one needs to agree on the definition. When does a two-level system become a qubit? In my view, only when coherent quantum dynamics is demonstrated. In the summer of 2002, Rabi oscillations of a superconducting flux qubit were observed in our laboratory. They were published in Science [1]; the primary authors were Irinel Chiorescu (postdoc, now professor at Florida State University) and Yasunobu Nakamura (on sabbatical from NEC Japan, now professor at University of Tokyo). As we all know, much has happened in the years after. Here I want to describe what happened before. How did we come to this point? I concentrate on my personal story and on superconducting circuits. In our Quantum Transport group we had the parallel research line on semiconductor quantum dots by Leo Kouwenhoven and his people that led to our first spin qubit in 2006.

Developing a quantum bit was not a goal for the international communities of research on small Josephson junctions or semiconductor quantum dots until late in the nineties. We studied quantum effects in fabricated nanostructures for their own interest and for application in new devices. In 1995 very few people in our field had even heard of quantum bits and quantum computers. Quantum information processing was studied in single optical photon circuits, trapped ions and atoms, as well as in nuclear magnetic resonance on many identical molecules. Solid-state circuits were considered too messy; they had too many uncontrolled degrees of freedom. Among ourselves, some of us started to discuss whether a solid-state quantum bit was completely out of the question.

I can pinpoint my own entrance in the quantum information area. In the summer of 1996 I spent two months in Santa Barbara at the ITP (Institute of Theoretical Physics, not yet Kavli) in a program on quantum computing and quantum coherence moderated by Wojciek Zurek and David DiVincenzo. I learned the language and had to strain my brains to understand what was going on. These were exciting times for quantum information. With Grover’s and Shor’s search and factorization algorithms it had become clear that a quantum computer could do things beyond the reach of any classical computer. The very recent discovery of error correction made it possible to think that a practical quantum computer could be built. I was an outsider there and knew very few people, but I was welcomed and helped. In particular Seth Lloyd and Ike Chuang (still a PhD student at that time) took me under their wings. I gave a talk at the meeting on the possibility of using charge states in Cooper pair boxes as quantum bits. As usual, preparing a talk helped me to understand my subject but I do not think that I convinced many people. I remember that Murray Gell-Mann (co-inventor of the Standard Model) asked the question why in the world one would try to do this.

# Quantum effects in superconducting circuits

Our Quantum Transport group in Delft had been studying quantum effects in small fabricated structures for considerable time. At the end of the eighties, submicron fabrication had turned over into nanofabrication; controlled dimensions of around 50 nm were in reach for some industry and university groups. In Delft there was a strong particle optics group. It had initiated a Center for Submicron Technology with a professional electron beam pattern generator (EBPG). Once it worked, they had no clear ideas what to do with it but we jumped in with pleasure. At the same time, we bought our first dilution refrigerator (the Oxford S200 that is now on display in the B-wing hallway) and our bath temperature came down from 300 mK to 10 mK. The combination led to our first experiments on single electron devices. These were metallic tunnel junction circuits, fabricated with shadow evaporation. Aluminum was used and superconductivity was killed by a 1 tesla field. We collaborated closely with the Saclay group of Michel Devoret and Daniel Estève. The first PhD student in this area (Bart Geerligs) made a turnstile device that allowed controlled transmission of one, two or three electrons for each cycle of a microwave signal. In all these experiments, the islands contained more than a billion electrons, each occupying its own quantum state. Of course, a bit later, we switched off the magnetic field and looked at transport in the superconducting state. We expected to see Coulomb blockade at twice the voltage, but much more happened. Many sharp peaks appeared. In hindsight, we ran into resonances between the states of the Cooper pair condensate and modes in the environment.

We and other experimental groups were moving forward like persons in a dark room. In theory there were some attempts towards a quantum description of Josephson elements, but the actors did not agree and there was no feedback from experiments. Very famous theorists in solid-state physics and in superconductivity told us that what we tried was impossible. They explained that an aluminum island of a square micrometer had too many bulk and surface states, and that the difference between confined core electrons and conduction electrons was arbitrary. We would never see single electron states. In superconductivity things should be even worse. The BCS concept of Cooper pairs that occupied electron states in a vaguely defined range around the Fermi surface was maybe adequate to explain certain macroscopic properties, but not applicable on the single Cooper pair level. A loop as we used in the flux qubit was far too big for quantum mechanics to apply. Absolutely impossible. I myself worked from a naive, intuitive, experimental approach. Every new result in Delft or elsewhere encouraged us to try something else to see how far we could go.

The very first experimental indication for quantum behavior of a Josephson junction came from the group of John Clarke at Berkeley in 1985 [2]. Michel Devoret and John Martinis measured the escape from the zero-voltage state of a current-biased Josephson junction while microwaves were applied. It was seen to be in accordance with excitation from a quantum state rather than from a classical resonance. Tony Leggett [3] developed the theoretical framework to understand and describe the interaction with a messy multi-mode environment, which in turn helped to engineer that environment with cooled filters. In all the groups that worked on superconducting quantum objects a very large fraction of the effort was devoted to the systematic suppression of decoherence mechanisms. Filtering of the connecting lines, shielding of radiation and in general the systematic design and construction of all circuits was essential. Superconductivity lives in the condensate of paired electrons, called Cooper pairs. It took years for the various groups working in the charge regime to get to the stage where single electrons were suppressed and only Cooper pairs counted. In our field there was an excellent open spirit of collaboration to the mutual benefit. Ideas were shared and results communicated.

In the years around 1990 several groups (including Delft) fabricated and studied small Josephson junctions. These junctions are characterized with the energy it costs to add a Cooper pair with charge 2e to an island between junctions (the charging energy *E _{C}*=4

*e*

^{2}/2

*C*) and the strength of the tunnel coupling (the Josephson energy

*E*). To reach the quantum regime one needs a charging energy that equals or exceeds

_{J}*E*. In 1989 we published a paper [4] on two-dimensional arrays of junctions with varying

_{J}*E*/

_{C}*E*ratio. At low temperatures the high

_{J}*E*/

_{C}*E*samples became insulating, with low

_{J}*E*/

_{C}*E*superconducting. This was in fact a quantum phase transition. The arrays were easier than single or double junctions, because the environment had little effect. We also studied double junctions and linear arrays[5]. The double junction in the single electron regime was called the single electron transistor (SET); in the superconducting state this became the SSET. The Tinkham group at Harvard [6] demonstrated beautiful 2e-periodicity as a function of the gate charge in an SSET up to 300 mK. The Quantronics group (avant la lettre) in Saclay confirmed this [7]. Several years later that group performed a beautiful experiment on a Cooper pair box, a gated island with capacitive coupling to an electrometer. They demonstrated the existence of a superposition between two charge states on the island [8].

_{J}The experiment that convinced me deep inside that a Josephson circuit could be “quantum” was done in 1994 in our group. It was performed by two PhD students: Wiveka Elion and Marco Matters [9]. They made a sample as indicated in the figure. Our goal was to investigate Cooper pair transport through a low capacitance island under Coulomb blockade conditions, while we could vary the environment in situ. We expected a minor modification of the current-voltage characteristic near zero voltage. The measurement yielded a much stronger effect, a very clear modulation of the supercurrent through the device. This supercurrent is leakage through the Coulomb blockade but the charging energy is not influenced by the flux in the SQUID. Once you accept quantum behavior, the effect is easy to explain. In quantum mechanics the Heisenberg uncertainty between position and velocity is well known. If position is measured with a very high accuracy,it inevitably leads to a large uncertainty in the velocity. A similar Heisenberg relation exists for charge (number of Cooper pairs) and phase in a superconductor. When the Josephson energy of the SQUID is high, fluctuations of the island phase are suppressed. Heisenberg tells us that fluctuations of the charge must increase. We later made a special sample where we could also influence the charge by means of a gate. The results were perfectly in line with quantum calculations.

# Flux qubit phase 1, design and spectroscopic measurement

From the experiments in Delft and elsewhere it became clear that the charge noise in small junction circuits is very large. In contrast, flux states in loops are much cleaner, due to the absence in nature of magnetic monopoles. We had done a long series of experiments on quantum properties of vortices in Josephson junction arrays. A vortex or fluxoid in these circuits is a local singularity in the phase surrounded by a circulating current. Such a vortex has macroscopic dimensions, but from previous experiments we had learned that it could behave as a quantum particle to a certain extent. Once we started thinking seriously about Josephson qubits it was logical for us to choose this direction. The equivalent of the Cooper pair box in this regime is a single closed loop with a weak junction that allows the tunneling in and out of a vortex. If the loop is biased with half a magnetic flux quantum, states with zero and one vortex have the same energy. The accompanying persistent currents have opposite directions. This could perhaps become the flux qubit. It sounds reasonable now, but tunneling of a macroscopic object such as a vortex was not known or even accepted.

Until that time all measurements in our group were done in transport, applying a current and measuring the voltage or vice versa (hence the name Quantum Transport). For a qubit one needed a static measurement on a single object. Measurement and operation by circuit quantum electrodynamics started only many years later. For measurements on the flux qubit we used a sensitive SQUID that could measure the persistent currents in the qubit by inductive coupling.

I had many administrative duties in those years. To give me some room for research I was offered the possibility to spend time at MIT. I went there three months per year in 1998, 1999 and 2000. I joined the group of Terry Orlando. I knew him from Stanford and he had been in Delft for a sabbatical. I also talked much with Seth Lloyd, to learn more about quantum computing. In 1999 at MIT we worked out a serious design for a flux qubit. For the calculation of the tunneling rates we had help from Leonid Levitov. We published the design in Science in 2000 [10]. Caspar van der Wal, now professor in Groningen, was the PhD student in Delft who was involved from a distance (we had email by that time). The design called for a closed loop with three or four junctions. One of them is smaller to allow the phase to slip (the vortex to cross), the others provide enough inductance to stabilize the two states. We could do the quantum calculation of the tunneling rates and the numbers looked practical for the junctions that we could fabricate. Terry Orlando wrote a follow-up paper with more extensive explanations [11].

In Delft, Caspar went to work to fabricate the flux qubit as designed. We knew the numbers required and we had the technology. The sample was a three-junction loop with a SQUID tightly around it. Obviously, it was essential to reduce as much as possible all decoherence that the SQUID circuitry brought in. It took Caspar about a year to create a sample where the tunnel strength was strong enough to induce a visible gap in the energy spectrum. The tunnel splitting at a flux bias of exactly half a flux quantum could not be seen directly because in both the symmetric and the antisymmetric superposition states the SQUID signal was zero. By extrapolation from spectroscopic data away from the symmetry point we could establish the presence of an avoided crossing of about 0.7 GHz. The two underlying classical states had a persistent current of plus or minus 0.5 µA. We published the results in Science, with the title “Quantum superposition of macroscopic persistent-current states” [12]. A few months before, the Lukens group at Stony Brook published a paper in Nature with a very similar title [13]. Their sample had a large loop with one junction that could be tuned by means of a magnetic flux. They observed a superposition of the 3^{rd}state in the potential well below half a flux quantum with the 6

^{th}state in the other well. The two papers drew much attention; they were discussed intensely at the APS March meeting of spring 2000 even before our paper came out. This was not so much because of quantum computing (the Stony Brook sample was certainly not suitable for that), but because of the macroscopic nature of the quantum object. In a Cooper pair box the actual tunneling is done by one Cooper pair. In the flux loops, billions of Cooper pairs reverse direction when the state is flipped. The current is macroscopic. This felt like a Schrödinger cat, and indeed the accompanying introductions in Nature (Stony Brook) and Science (Delft) had the titles: “Schrödinger’s cat is now fat”, and “Schrödinger’s cat is out of the hat”, respectively. We had many discussions about this, but our point of view was that our system had only one degree of freedom. No cat.

# Developments outside Delft

During my time at MIT in 1999 we were surprised by a thunderbolt from Japan. The NEC group of Tsai and Nakamura submitted a paper to Nature titled “Coherent control of macroscopic quantum states in a single-Cooper-pair box” [14]. Normally speaking we knew everything that was going on in the labs of the groups in our field well before it was published. The NEC group was certainly known but more distant, not only literally. They had managed to perform a fantastic experiment where a two-state Cooper pair box was driven from the ground state with a gate signal of varying amplitude, ending up in the excited state for a certain fraction. They had beaten the short coherence time (2 ns) by using a superfast signal generator allowing shaped DC pulses of 100 ps. It was not quite a qubit yet, but it was certainly very close. A bit later they observed clear Rabi oscillations [15].

It was clear that the straightforward charge qubit would not be the best solution for a practical qubit, due to the strong charge noise. The Quantronics group came up with a modified charge-phase qubit that they called the quantronium [16]. It has a charge island that is part of a closed loop with two small junctions on either side and one large junction. The large junction is used for readout by a transport current. Coherence is optimal when both the charge on the island and the flux in the loop are tuned to a “sweet spot”. The Martinis group, still at NIST in Colorado, developed the phase qubit [17], which utilized two levels in the potential well of a current-biased large Josephson junction. Both these new qubits became operational in the year before we demonstrated Rabi oscillations in our flux qubit.

I should perhaps comment on the names. In all superconducting quantum systems both phase and charge are important. The NIST phase qubit was based on an energy oscillation between Coulomb charging energy and Josephson phase energy. In our qubit, the magnetic flux generated by the persistent currents is a small fraction of a flux quantum and the name flux qubit is misleading. I would have liked to stick to our original name persistent-current qubit. However, history and public opinion have decided otherwise

# Flux qubit phase 2, coherent quantum dynamics

After the APS March meeting in 2000 where the news of the quantum superposition states of macroscopic SQUID rings in Stony Brook and Delft came available, Science published an editorial note about it [18]. It gave a description of the results and expressed the hope that this could be a very good basis for a qubit. However, at the end Eli Yablonovich was quoted saying that “SQUID researchers face the same daunting challenge as others trying to develop practical qubits: preserving the mixed quantum states in a hostile world.

They‘ve got to get down in the trenches with the rest of us to solve that problem”. This is what we experienced in the following two years. Caspar left to the Lukin group at Harvard and from there to Groningen. New people came in to optimize our sample and our experimental setup. Irinel Chiorescu had obtained his PhD with Barbara in Grenoble before he joined us in 2000. He had to learn our fabrication and measurement techniques. He exhibited tremendous pushing power. Kees Harmans switched from semiconductors to superconductors. Yasunobu Nakamura came from NEC for a year’s sabbatical with us in 2001-2002. He brought all his experience and insight with him. Yasu hoped with us that the flux qubit would have better coherence than the charge qubit. He worked out the new design with a SQUID in direct galvanic contact to achieve a strong measuring signal. Much care was taken to reduce the back-action from the SQUID circuit.

Many improvements were made to the set-up. In May 2002 I was driving back from a short vacation in northern France when I received a phone call from Irinel, telling me that they had observed clear Rabi oscillations. It was an important moment. After I came back we improved the data and wrote the paper that was submitted to Science [1]. The flux qubit was born. When our paper was published, it was accompanied by a Perspectives paper titled “Flux qubit completes the hat trick” written by John Clarke. The superconducting qubit community had three complementary qubits at its disposal: charge, phase and flux qubits.

# The flux qubit in later perspective

A few concluding remarks can be made. The flux qubit worked quite well. The coherence times were similar to those of the quantronium and phase qubits, around 1 microsecond. The two levels that are used for the qubit are well separated from the third level, which makes very fast excitation possible. Coupling to a second qubit can be made strong. We developed the first superconducting controlled-not gate [20]. We could also realize ultra-strong coupling to a harmonic oscillator and observe the Bloch-Siegert shift [21]. We realized a practical way to control the qubit level splitting on a ns time scale [22]. After my retirement our work on flux qubits stopped. In other places flux qubits have been made with coherence times around 100 microseconds [23].

In 2007 the transmon came along which is a degenerate version of the Cooper pair box. The Josephson energy is made much larger than the charging energy to suppress the influence of charge noise. At this time the transmon is the prevailing superconducting qubit for quantum computer applications. Transmons are used also in Delft in the new team headed by Leo DiCarlo.

**About the Author:**

Hans Mooij started in the predecessor of the Quantum Transport group in 1971, he was appointed as professor in 1980. He worked on superconducting electronic systems, gradually focussing on quantum aspects. He retired gradually, starting 2006.

*References*

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