Jasper van Wezel - Research Highlights

Phase transitions and the breakdown of quantum dynamics

A phase transition is something you encounter every day when boiling water for tea. The dynamics of quantum particles involved in such a transition however, is not yet fully understood, and even violates the prescriptions of Schrödinger's equation.
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Many phase transitions in nature involve the breakdown of a symmetry. The fact that all directions in space look roughly the same, for example, is a symmetry that is broken when a piece of iron magnetises and singles out a specific direction for its magnetisation. The theory of spontaneous symmetry breaking provides a well-known explanation of how a magnet can survive in a state with less symmetry than the universe it is in. It does not, however, explain how a piece of iron can start from a non-magnetic state and then select, over the course of time, a single direction to magnetise in.

In a recently published article, we show that in fact, this selection of a single symmetry-breaking state is not possible for any collection of quantum particles obeying the rules of Schrödinger's equation. We explain why this makes the familiar everyday observation of phase transitions resulting in symmetry-breaking configurations eerily similar to the kind of quantum breakdown occurring during measurement, and suggest that both must involve instances of "spontaneous unitarity violation".

The theory of spontaneous unitarity violation is described in some of our previous works, including an older overview article, the identification of necessary ingredients in any such theory, and a recent extensions of the earlier theory.

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Highlight Picture   Spiral patterns in puddles of electrons

We show that electron puddles found in a particular electronic crystal spontaneously form swirling motifs of interlocking spirals.
The spiral patterns seen in human art since at least Celtic times, recently made a comeback in an unexpected place. The material TaS2 was already well known to contain sheets of atoms along which electrons spontaneously arrange themselves in rigid, ordered patterns. This is in fact not uncommon among layered materials, and usually results in striped, checkered, and triangular designs of electron density. As described in this article, however, we recently discovered that the electrons in TaS2 are organised in much more intricate arrangments, featuring a motif of interlocking spirals reminiscent of Celtic art.

The fact that these spirals all rotate in the same direction means that the electrons need to overcome the symmetries of the atomic lattice in which they reside. This is similar to the spontaneous formation of corkscrews in TiSe2 and elemental Se and Te, which we reported before. Unlike those earlier cases however, the electrons in TaS2 seem not form spirals because of their mutual repulsion, but rather because of their attraction to the surrounding atoms.

The intricate interplay of interactions among electrons and atoms, combined with the multitude of crystal lattices in which they can occur provide a vast playground for exotic patterns to emerge. Each of these have unique physical properties, that may be used in technological applications as well as guiding us to a better understanding of nature. The unexpected discovery of Celtic spirals in TaS2 exemplifies the richness of what may await us even in seemingly simple materials and haphazard puddles of electrons.

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Thermal Radiation from a Synthetic Black Hole Horizon

We show that the creation of a simulated black hole in one-dimensional models causes thermal radiation at the Hawking temperature of the simulated black hole.
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We recently suggested in this article that black hole physics can be simulated in simple one-dimensional materials. Further exploring the physics of these analogue black holes, we found in this recent publication that the simulated black hole emits thermal radiation in direct analogy to the Hawking radiation expected for real black holes.

The relative simplicity of the one-dimensional model allows us to trace the origin of the thermal radiation to the formation of the black hole. Starting from the analogue of a zero-temperature universe without masses, the sudden inclusion of a black hole horizon divides its space into mutually inaccessible regions. Quantum fields existing since before the black hole formation will not have time to adjust, and end up being entangled across the horizon. Looking at these states from just one side, and thus not seeing their entangled tails behind the horizon, the quantum fields appear to glow at the Hawking temperature.

These results bring the simulation of analogue Hawking radiation and black hole formation within experimental reach, and shed light on the precise conditions required for black hole radiation to be truly thermal.

The story of this article was featured in more than a hundred popular science outlets worldwide.

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Highlight Picture   Topology with rotational symmetry

Although our recent classification of topological crystals predicted the existence of new types of topological phases in materials with rotational symmetries, it did not provide a way of identifying these new phases, nor any of their physical properties. We now complete the picture by proposing a unified method for calculating topological invariants that can diagnose all known types of topology, including the newly predicted ones. It also indicates when these new phases will have excitations localised at either their edges or corners.
In topological insulators, the presence or absence of symmetries in the atomic lattice allows for many different types of topological phases. These are labelled by various types of topological numbers, ranging from the Chern number, to the two- dimensional Fu-Kane-Mele (FKM) invariants, the Lau-Brink-Ortix (LBO) or line invariant, as well as invariant features of the Wilson loop spectrum describing higher-order and fragile topological insulators. We recently proposed a unified approach describing all of these topological phases, both in topological crystals with time-reversal symmetry and in those without. This shows that the FKM, LBO, higher order, and similar invariants are all manifestations of the same topological structure of the Berry curvature. Moreover, being a complete classification of all possible such invariants, it predicted that additional, as yet unidentified, invariants of the same type exist in various crystals, for example those with rotational symmetries in two dimensions.

Although the classification predicted the existence of topological invariants for systems with a given symmetry, it did not give a way of identifying or evaluating them in any specific system. In our most recent article, we fill this hiatus by introducing a single, unified diagnostic that yields the values of all curvature-based invariants in two-dimensional topological materials with time-reversal and rotational symmetry, including the newly predicted ones. As a corollary, we show how the new invariant is related to the emergence of edge and corner states.

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Born’s rule cannot emerge from linear dynamics

We give a theoretical proof that one of the fundamental open problems in physics -how Born’s rule relating quantum dynamics to observed classical probabilities- cannot be solved using linear or unitary theories. As a corollary, we also show that an often-used constructing which seemingly predicts the natural emergence of Born’s rule, in fact contains a logical flaw.
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While quantum mechanics is the best-tested theory within physics to date, there is a well-known inconsistency in its axioms, known as the ‘measurement problem’. From the moment the framework of quantum mechanics was first formulated, this problem has fuelled a search for the connection between the experimentally verified probabilistic outcomes of measurement and the fundamentally deterministic time evolution prescribed by the Schrodinger equation. This quest for understanding how non-deterministic measurement arises from deterministic quantum dynamics continues unabated as an active field of research today. Approaches to addressing the measurement problem can be divided into two categories, either attempting to give alternative interpretations of the mathematical structures featured in the quantum theory, or attempting to supplement or alter the laws of quantum physics to include the emergence of measurement at macroscopic scales. The latter are known as objective collapse theories.

Several experimental techniques have recently been developed to explore the region between the microscopic realm, where quantum dynamics has been verified to proceed according to the Schrodinger equation to extremely high accuracy, and the macroscopic realm, where measurement devices yield probabilistic outcomes for quantum measurements. It is in this unexplored region intermediate between microscopic and macroscopic superpositions that objective collapse theories predict the quantum-classical crossover to take place, and yield observable differences in their physical predictions from interpretation-based approaches. The direct observation of the the mesoscopic realm thus necessitates a theoretical exploration of both the dynamics predicted by different classes of objective collapse theories, and the postulates underlying their predictions in this regime. Like the bounds imposed by experimental observation, consistency requirements on the theoretical postulates and dynamics may then be used to classify and constrain objective collapse theories.

In a recently published article (also available on arXiv) that was selected as Editor's suggestion by Physical Review A, we classify objective collapse theories according to the requirement that relative frequencies associated with measurement outcomes, known as Born’s rule, emerge without imposing them as part of any axiom. We show that imposing this physical constraint rules out theories for quantum measurement based on either linear or unitary generators of time evolution. We also formulate a minimal non-linear objective collapse model for the two-state system that does reproduce Born’s rule without assuming it at any point.

The story of this article was featured in several popular science outlets worldwide:
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Highlight Picture   Topological Bulk-Boundary correspondence in a non-Hermitian metamaterial

We demonstrate that a metamaterial of robots holding hands can realise a type of non-Hermitian topological state that has been hard to realise in natural matter. The metamaterial moreover establishes a bulk-boundary correspondence for the new phase, in which a localised mode moves from one side to the other as the robots undergo a topological phase transition.
In physics and mathematics, topology is the study of shapes and forms at large. Topology does not worry about fine details, but asks what can be learned about a system from its most global properties. For example, in topology, a donut and a wedding ring are essentially the same: both are solid shapes with a single hole. A pretzel, though, with two or three holes, would be considered a topologically different shape.

Topology holds the promise of revolutionizing technologies in many fields, from quantum electronics to acoustics, and mechanics. Topology also plays a role in many materials. A fundamental property of topological matter is the so-called bulk-boundary correspondence: a simple topological quantity observed in the material’s interior can predict the emergence of waves localised at the material’s edges.

A well-known law of physics states that energy is conserved: it can be transformed from one form into another (by rolling a ball down a hill for example, changing gravitational energy into energy of motion) but it does not get lost or just appear from nowhere. However, this law only holds true in idealized systems that are perfectly isolated from their surroundings. In real life physical systems, energy does get lost, for example simply because it leaves (dissipates from) the system. Conversely, in materials science one now constructs ‘active materials’, which actually gain energy from their surroundings.
Recently, there has been an explosion of activity with the aim of generalizing the concept of topology to such more real-life systems where energy can be lost or injected. However, despite intense efforts, any behavior of topological edge waves in systems that do not conserve energy
had not been observed experimentally. In a recent article we showed that we achieved two breakthroughs in this vibrant field.

First of all, the team discovered a novel form of bulk-boundary correspondence: a new relation between the interior of a material and what happens on its edge, particularly valid for these energy non-conserving systems. It was shown that a certain change in the topology inside the material, leads to a change of the locations of the wave-like effects on the edges.

Secondly, the team made this theoretical finding very concrete by constructing, out of cog wheels, rods, levers and tiny robots, a specific metamaterial with the theoretically predicted property. In fact, the most favorable media to see the effect of topology on wave propagation are such metamaterials, which are composite systems artificially made as arrangements of identical units.

In idealized scenarios, each identical unit in such a metamaterial talks to its neighbours in a symmetric way, resulting in the conservation of energy. However, in the material constructed by the researchers, the units talk to their left and right neighbours in a different way. This makes the system gain or lose energy from or to the environment. The physicists now managed to show that even in this case, one can let waves travel through the system and topology then explains how these waves in the interior affect waves on the boundary. In particular, the topology of the setup determines on which side of the material these edge waves occur.

The work can have significant impact on many branches of physics, ranging from quantum mechanics for systems that are not in equilibrium to the construction of new and interesting metamaterials for situations where it is useful to engineer wave properties by steering waves on demand. Potential applications are in sensing or energy harvesting, or for example in creating new materials that very effectively damp or mitigate shocks and vibrations.

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Synthetic Black Hole Horizon

We show that the physics of quantum mechanical wave packets approaching a black hole horizon can be simulated in simple one-dimensional materials.
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Interesting subjects in physics may emerge from an original combination of ideas belonging to seemingly different areas of research. During the last decade for instance, theoretical methods and concepts from the realm of quantum gravity have opened up new areas in condensed matter physics. Vice versa, condensed matter systems can be created that closely resemble general relativity objects such as black holes, providing a way to investigate aspects of black hole physics in an experimentally accessible setting.

In this context we consider in a recent publication the quantum evolution of wave functions near an event horizon in the simplified situation of two space-time dimensions. The advantage of considering (1+1)D space-time is that we can make precise connections to wave function dynamics of one-dimensional (1D) quantum lattice systems with fermions that can in principle be built and controlled in the laboratory.

We establish a direct relation between the geodesics of a space-time with a black hole, and the wave-packet dynamics in lattice models with hopping amplitudes that vary with the spatial coordinate. We show that in the low-energy limit, the quantum dynamics of the lattice models exactly follows that of a Dirac field in a particular space-time with a black hole. We can even explore the influence of the lattice, or equivalently the breaking of Lorentz invariance. This leads for example to the observation that wave packets with finite mean energy may bounce off the horizon, in stark contrast to the usual exponential slowdown.


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Highlight Picture   Charge Order: Details Matter

We show that all experimental observations on a particular charge ordered material can be quantitatively understood by taking into account the details of how electrons interact with the atomic lattice. We also show that this picture breaks down for monolayers of the same material, in which Coulomb interactions turn out to be more important than usual.
Electronic crystals are one of the most elementary types of emergent phenomenon known in physics. In spite of the relative simplicity of electronic crystals, experimental observations of actual electronic order in real materials often reveal unexpected features, that defy simple theoretical explanations.

In a recent theoretical study, we show for one particular material that quantitative agreement with all available experimental data can be achieved, but only when the detailed interactions between electrons and the atomic lattice are taken into account. This explains some outstanding and seemingly mysterious observations about this material, which have been debated for decades.

At the same time, we show in another recent publicationthat if the same material is made thinner and thinner to the point where it consists of only a single atomic layer, the physics of the electronic order changes drastically. The observed materials properties undergo a remarkable change when going from two to just one layer, which we show to be due to the effect of enhanced Coulomb interactions between electrons in the monolayer. These become as important as the interactions between electrons and atoms that determined the properties of the bulk material, giving rise to a unique situation with multiple types of electronic order existing side-by-side.

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Spontaneous Symmetry Breaking lecture notes

Together with Aron Beekman and Louk Rademaker, I published a set of lecture notes on Spontaneous Symmetry Breaking.
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Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In the lecture notes, starting from a careful definition of symmetry in physics, we introduce symmetry breaking and its consequences. Emphasis is placed on the physics of singular limits, showing the reality of symmetry breaking even in small-sized systems. Topics covered include Nambu-Goldstone modes, quantum corrections, phase transitions, topological defects and gauge fields. We provide many examples from both high energy and condensed matter physics, and aim the lecture notes especially towards graduate students.

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Highlight Picture   Excitonium discovered in real material

Excitons are pairs of electrons and holes inside a solid material that together behave like a single particle. It has long been suspected that when many such excitons exist in the same piece of matter, they can form a new state of matter, called excitonium. This new phase is essentially a single giant quantum state of excitons, called a Bose-Einstein condensate. If that state does exist, it is expected to hold important clues to the understanding of many other mysterious phases of matter, including even high-termperature superconductivity. However, observing an exciton condensate in any real material has remained a much sought-after goal of condensed matter physicists for decades. In a recent experiment, we now finally prove that this elusive state of matter really does exist.
In the early 20th century, physicists discovered that the world around us consists of two types of particles: bosons and fermions. The main difference between these particles is how they behave when one tries to bring them into the same physical state, with the same position, the same velocity, and so on. While for two fermions (such as electrons) it is fundamentally impossible to ever be in the exact same state, two or more bosons (such as photons, particles of light) can be in the same state at the same time without any problems. In fact, at low enough temperatures, collections of bosons will prefer such a situation: the particles have the tendency to all occupy the same state, in a process known as Bose-Einstein condensation.

For most types of bosons, Bose-Einstein condensation takes place at very low temperatures, near the absolute temperature minimum of 273 degrees below zero on the Celsius scale. An exception to this rule could be the behavior of excitons in a crystal. Excitons are combinations of negatively charged electrons and so-called holes - the absence of an electron somewhere in the crystal, leading to a local surplus of positive charge. Pairs of electrons and holes can be bound together and behave like a single bosonic particle, the exciton.

It was predicted in the 1960s that just like other bosons, excitons can form Bose-Einstein condensates. Moreover, this should happen at much higher temperatures than for most other particles – in theory it could happen even at room temperature. Since higher temperatures are much easier to reach in a laboratory setting, excitons could provide an accessible setting in which both the unusual quantum properties of the Bose-Einstein condensates themselves, as well as the unique material properties they bestow upon their host crystals, can be investigated. However, actually proving that Bose-Einstein condensation of excitons occurs in any real material has been a challenge for physicists for decades. An experiment done at the University of Illinois at Urbana-Champaign, carried out in collaboration with researchers at the Universities of Oxford and Amsterdam, published in Science, has now uncovered evidence that this elusive state of matter really does exist.

Despite the relatively high temperature at which the effect described in the Science article occurs (only 100 degrees centigrade or so below room temperature), and despite the presence of excitons having been suspected for many years, proving beyond doubt that excitons really do form a Bose-Einstein condensate turned out to be surprisingly difficult. The main reason is that there is a different physical phenomenon which is hard to distinguish from a Bose-Einstein condensate of excitons: the formation of a so-called Peierls state, where electrons inside a crystal structure spontaneously organize in a wave-like manner, with alternating peaks and troughs of electron density. Such a wave has many of the same physical characteristics expected for a Bose-Einstein condensate of excitons.

A new experiment carried out at the University of Illinois at Urbana-Champaign, in collaboration with researchers at the University of Oxford, and the University of Amsterdam, has now shown that the newly developed experimental technique of Momentum-resolved Electron Energy-loss Spectroscopy (M-EELS for short) does allow them to distinguish unique signatures of condensed excitons in a material called titanium diselenide. This technique for the first time allows scientists to measure low-energy bosonic particles made of electrons and holes, regardless of their momentum. With this unique capability, the researchers were able to prove that excitons in titanium diselenide spontaneously agglomerate into a Bose-Einstein condensate when the material is cooled down to below 100 degrees centigrade below room temperature.

These measurements for the first time give compelling evidence for the fact that excitons can form a Bose-Einstein condensate at relatively high, easily accessible temperatures. Moreover, they show that M-EELS is a powerful and versatile new technique with many potential future applications.

For more information, see the Science article, or the accompanying press releases from UIUC and the University of Amsterdam.

This discovery was picked up by several news sites and popular science blogs globally:

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Finding all possible topological insulators in crystals

Topology has over the past decade or so developed into a central organising principle in the characterisation of phases of matter. While all topological phases of fermions in free space have been fully worked out, taking into account what happens in real-life materials that have additional crystal symmetries remains an active field of research. We recently took a step forwards in this area, by developing a complete classification of all possible crystalline topological insulators, in any dimension, in the presence of only lattice symmetries.
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In our recent publication, we start from the concept of topological distinctness. A sphere and a donut, for example, are considered topologically distinct because the sphere has no holes and the donut has one. You cannot smoothly deform a sphere into a donut, if we agree that smooth deformations do not include puncturing holes on opposite sides of the sphere and reconnecting the edges. The Chern number in the integer quantum hall effect is similar in this regard. It counts the number of singular points that the electronic wave function must have in any given gauge. You cannot smoothly deform a state with no singularities into a state with one singular point, if we agree that the definition of smooth does not include creating singularities.

To define which states of electrons living in the periodic crystal structure of an actual insulating material should be considered topologically distinct, we employ a similar approach. We consider the band structure of electrons in crystals without any external symmetries such as time-reversal symmetry, particle-hole symmetry or chiral symmetry. A smooth deformations is then defined to be any continuous deformation of the band structure that does not break crystal symmetries and does not close the gap around the Fermi energy.

This definition of topology is actually slightly different from what is often used in the literature. We do not refer to the 'atomic limit', and do not consider anything topologically 'trivial'. We only claim certain insulators to be topologically distinct from one another. The approach towards classifying topological phases by starting from the definition of smooth deformations turns out to also be somewhat non-standard. A more common approach would be to generalise the notion of a Chern number into similar quantities. Doing that however, it is hard to definitively prove that such a generalised Chern number is truly a topological invariant, in the sense that states with different values of the number cannot be smoothly connected in any way. Moreover, approaches of this kind cannot claim to be complete, since there is no way to prove that further generalisations are not possible.

We therefore take the notion of smooth deformations as our starting point, which guarantees that all phases we consider will be truly distinct. In fact, the simple counting scheme we develop turns out to match precisely the perspective on topological classifications of the far more involved mathematical framework known as K-theory. This explicit connection to K-theory provides mathematical proof that our classification is also complete: using our definition of smooth deformations, there cannot be any more topological insulators in the types of crystals considered than the ones we find.

The classification we present introduces a large set of new topological invariants and new topological phases. It also gives a convenient way of studying what happens if you go from one topological phase to another, and to find out what sort of new physics may emerge on the boundary. Possibilities include the formation of Weyl semimetals, Dirac cones, and Fermi arcs, as well as other, more exotic states. We can now begin to try and find materials that host all of these new types of phases and phase transitions, which will make it possible to study all sorts of interesting new physics, as well as paving the way for applications of topology in technological devices to be developed.

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Highlight Picture   Spirals of electron density

Spirals are an intriguing shape to find in the natural world because of their inherent handedness – turning either to the left or right as you move along them. The recent discovery that electrons within a solid material can spontaneously form into a corkscrew shape was an unexpected example of a spiral emerging in physics. The surprisingly straightforward explanation for this phenomenon has recently attracted some attention in the popular science media.
A crystal is a material in which the atoms are organised into a regular, periodic structure. Electrons normally float freely among the atomic arrangement. But in some materials, the electrons spontaneously assemble into a rigid crystal structure of their own, unrelated to the surrounding atomic grid. These electronic crystals within crystals are an example of an emergent phenomenon in physics. Emergent phenomena occur regularly even in everyday life. For example, even though any two people will find it hard to applaud in the exact same rhythm, a large audience at a popular Broadway play will spontaneously clap in harmony almost as soon as the curtain drops. Likewise, a few electrons put together will always try to avoid marching to the same beat, but the enormous number of electrons within a single gram of solid material cannot avoid forming a collective arrangement.

We recently proposed a theoretical model describing the way in which collective arrangements of electronic charge can form in the shape of corkscrews. This is unexpected, since the materials in which the electrons form their helical structure, does not have a handedness, and the helices thus spontaneously pick one direction over the other. This unusual arrangement of electronic charge density into spiral shapes was suggested to be stabilised by a simultaneous spiral arrangement of the orbitals in these materials. The theory thus explains the experimental observations in materials like TiSe2, TaS2, and even the elements Se, Te, and Po.

This unexpectedly simple theory behind the spontaneous formation of spiral structures in these types of materials, accompanied by a new form of orbital order, was recently highlighted in a publication in popular science magazine Scientia, and accompanied by a podcast presenting the story in audible form.

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Patterned electrons in nanofilms

In the recent European Conference on NanoFilms, ECNF16, our group had three presentations on how electrons spontaneously arrange into microscopic patterns in sheets of material that are only one or a few atoms thick. These atomic layers turn out to be fruitful territory in which to look for exotic physics, such as spiral-shaped electron clouds and Bose-Einstein condensates of particle-hole pairs.
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The first presentation was given by Ana Silva, and showed how in certain materials, the Coulomb repulsion between individual electrons can cooperate with the attractive interaction between electrons and ions, to give rise to the formation of spiral-shaped regions with a high density of electrons. These electronic corkscrews break many symmetries of the crystal structure in which they appear, which opens up novel ways of studying and using these types of materials. Moreover, the way in which the electron-ion interaction depends on the specific shape of the orbital occupied by the electron, guarantees that the spiral charge ordered state will also be orbitally ordered. The prediction of this new type of spontaneously arranged orbitals allows for direct experimental tests of the theory.

In the second talk, I showed that it is possible to reconcile many apparently contradictory experimental observations on the particular nanofilm material NbSe2. We do this by considering how the interaction between an electron and the ions within the material is affected both by the shape of the orbital occupied by the electron, and by its momentum. In standard investigations of material properties these detailed dependences are often averaged over, which leads to apparent paradoxes when comparing different types of experimental observations. We show that a more careful analysis in fact captures all observations within a single theoretical framework.

Finally, Shan Zhu discussed in the third seminar how experiments on the nanofilm material TiSe2 give an indication for the existence of a special type of Bose-Einstein condensate. In this material, negatively charged electrons can pair up with positively charged anti-particles to form a type of neutral atom called a particle-hole pair. That many of these atoms can subsequently assemble into a Bose-Einstein condensate was predicted already five decades ago, but could never by confirmed experimentally. It is an interesting prediction, because this new type of Bose-Einstein condensate could potentially open the way for many types of electronic applications. We were now able to interpret recent experiments on TiSe2 in terms of the formation of a Bose-Einstein condensate of electron-holes pairs, with the caveat that the pairs are strongly deformed due to the TiSe2 crystal structure in which they occur. This interpretation has the advantage of naturally explaining the seemingly mysterious observation that superconductivity appears but also disappears again as pressure is varied within the charge ordered phase in this material.

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Highlight Picture   The shapes of electrons forming crystals within crystals

We show that all of the experimental results on a particular electronic crystal can be understood in terms of a single theory, in which the shapes of electrons and their effect on the surrounding atoms play a central part.
A crystal is a material in which the atoms are organised into a regular, periodic structure. Electrons normally float freely among the atomic arrangement. But in some materials, the electrons spontaneously assemble into a rigid crystal structure of their own, unrelated to the surrounding atomic grid. These electronic crystals within crystals are an example of an emergent phenomenon in physics. Emergent phenomena occur regularly even in everyday life. For example, even though any two people will find it hard to applaud in the exact same rhythm, a large audience at a popular Broadway play will spontaneously clap in harmony almost as soon as the curtain drops. Likewise, a few electrons put together will always try to avoid marching to the same beat, but the enormous number of electrons within a single gram of solid material cannot avoid forming a collective arrangement.

The electronic crystals are one of the most elementary types of emergent phenomenon known in physics. More involved examples include magnetism, superconductivity, and even the Higgs effect. In spite of the relative simplicity of the crystals within crystals, the detailed driving forces behind their occurrence in even the most well-studied materials have been clouded in mystery for decades. Several theories have been put forward, but none can explain the whole range of available experimental observations.

Now, a recent experimental study provides a clue about the origin of the electronic crystal in one representative material. Contrary to what is predicted by nearly all existing theories, the ordered arrangement turns out to arise in steps. First, small groups of electrons coalesce into tiny crystal islands. As these islands grow in size, they eventually touch, and finally lock together into a single electronic crystal. A theoretical study of the same material was published almost simultaneously with the experimental results. In it, we explain how the observed behaviour, and the disagreement with previous theories, arises from the detailed interplay between individual electrons and the atomic structure surrounding them.

A crucial role turns out to be played by the shapes electrons acquire within solid materials. Electrons in free space are generally described as point particles. But when they are embedded within a regular atomic array, the electrons are actually better thought of as extended clouds. Some electron clouds are round like beach balls, others elongated like cigars, and some resemble the petals of a four-leaved clover. The way in which differently shaped electrons affect both each other and their common background of atoms, has a profound influence on the shapes that the electronic crystal within a crystal can attain. Taking into account all of these effects, and thus going beyond the established models for the formation of electronic crystals, we can now reproduce, within a single theoretical model, all of the many experimentally observed properties of this material.

This breakthrough in the understanding of a single material opens up the way for studying other crystals within crystals. It is now clear that they too should generically be described by a theory accounting for the shapes of electrons and the resulting effect on the interactions between electrons and atoms. Such theories should eventually be able to predict the properties any material containing electronic crystals.

  • The experimental and theoretical articles were accompanied by a press release from the University of Amsterdam.

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    University of Amsterdam

    This October, I moved to the University of Amsterdam to take up a position as assistant professor. In Amsterdam I will continue my research on novel forms of charge and orbital order, singular limits in quantum mechanics and quantum dynamics, and other problems within condensed matter physics.
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    I will begin work at the University of Amsterdam in October 2014. As an assistant professor, I will contribute to both the teaching and research of the Institute for Theoretical Physics.

    My research at the University of Amsterdam will initially focus on the link between novel types of charge and orbital order, whose predicted properties have recently been confirmed experimentally, and phases with very similar characteristics that have recently been observed in high-temperature superconductors. The signatures include a breakdown of inversion symmetry and the formation of ordered patterns of electron density, but also the presence of a pseudogap phase, Fermi arcs, and an extended region of (quantum) fluctuations.

    I will also continue to explore the ways in which the laws of quantum mechanics may be expected to break down as the number of particles within a quantum object grows. The crossover between the quantum and classical worlds, like most other crossovers between qualitatively different physical regimes or laws, is governed by the presence of singular limits. Studying these, enables us to make predictions about how the classical world emerges from its underlying quantum theory, which may become testable by the very newest experimental techniques being developed now.

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    Highlight Picture   Chiral Charge Density Waves - part II

    Our theoretical description of how electronic charge and orbitals can organise themselves into spiral patterns has been successfully tested by three independent recent experiments. Using the theory, we can now understand the emergence of chirality in a wide class of materials.
    We recently proposed a theoretical model describing the way in which corkscrews of electronic charge can form in so-called transition-metal dichalcogenide materials. The unusual arrangement of electronic charge density (a scalar variable) into spiral shapes, was suggested to be stabilised by a simultaneous chiral arrangement of the orbitals in these materials. The theory thus explains the experimentally observed lack of inversion symmetry in the STM microscope images of the surfaces of materials like TiSe2 and TaS2.

    The theoretical model also made the prediction that there must exist two separate phase transitions in the chiral material TiSe2, leading from the semi-metallic phase, into the usual type of charge density wave state, and then finally into the novel chiral charge and orbital ordered phase. The presence of this sequence of phase transitions in TiSe2 has now been experimentally confirmed. The chiral phase transition was found to sit just 7 K below the main charge density wave transition, and can be identified in independent X-ray diffraction, electric resistivity, and specific heat measurements.

    Having identified the origin of the spiral charge patterns in a combined mechanism which incorporates the electronic orbitals as well as their charges, it is now possible to apply the theoretical description to related material classes as well. The elemental materials Te and Se for example are well known to have a chiral lattice structure, which can be understood as a consequence of the same type of simultaneous charge and orbital order as that observed in TiSe2. Similarly, Polonium was recently suggested to be susceptible to the formation of chiral charge and orbital order, although in this material the strong spin-orbit coupling prevents the onset of orbital order at low temperatures. This results in the unusual situation of a simple cubic lattice arrangement giving way to a less symmetric trigonal structure as temperature is increased.

    Most recently, investigations of high-Tc superconductors have revealed both the presence of incommensurate charge density waves, and the absence of inversion symmetry in their crystal lattice. It was suggested that these observations may be explained by the presence of chiral charge order, similar to what is observed in TiSe2. Whether the chiral state in these cuprate superconductors also arises from the same mechanism, involving the simultaneous ordering of both orbital arrangements and electronic charge density, is the topic of ongoing investigations.

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    University of Bristol

    This October, I will move to the University of Bristol to take up a position as lecturer. At Bristol I will continue my research on charge ordered phases, spontaneously broken unitarity, and other topics of condensed matter physics.
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    I will begin work at the University of Bristol in October 2012. As a lecturer, I will contribute to both the teaching and research of the School of Physics.

    My research at the University of Bristol will initially focus on the exploration of recently discovered new types of charge order, and the effects their presence has on its host materials. Charge order arises when the electrons in a material form a spatial pattern that does not match the underlying atomic lattice. Recently, it was found that in certain materials, the charge order can cooperate with other degrees of freedom (such as orbitals or plasmons) to qualitatively alter the properties of the materials in which they exist. Besides the fundamental interest in the entirely new state of matter constituted in this way, the effects on specific materials may also be of interest because of their possible use in applications which rely on the ability to tune electronic properties.

    Besides the work on electronically ordered materials, I will also continue to study the possible ways in which Einstein's theory of gravity may affect realizable quantum mechanical experiments. We recently suggested an experiment in which the interplay between these two seemingly distant realms of physics may become observable. Because Einstein's theory of gravity and the theory of quantum mechanics (the two most successful theories of physics ever known) contain mutually exclusive ingredients, they cannot be straightforwardly combined or reconciled which each other. Finding experimental evidence for which parts of these theories may survive, and which may not, in the regime where both play a role, is therefore an important step towards understanding the connection between the microscopic quantum world and the cosmic world of gravity.

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    Highlight Picture   Chiral Charge Density Waves

    It was recently discovered that electrons in certain materials can conspire to make spiral patterns in their density distribution. We explain this new phenomenon as a consequence of the interactions between the different orbitals occupied by the electrons in these materials.
    It is well known that chiral or helical patterns occur regularly in nature. The biological function of organic molecules for example often depends on their handedness, and certain types of magnets can have spiral shaped waves of local magnetization. Until recently however, no such chiral patterns were known in purely electronically ordered materials.

    The first chiral charge density wave was discovered in 2010 in the layered material TiSe2. In a viewpoint article, we compared its electronic chiral order to the well-known case of Tellurium, which has a helical lattice structure, but no charge order. In our most recent paper, the connection between these two types of chirality is further clarified by the discovery that both of them are due to the presence of so-called orbital order.

    The electron clouds surrounding the nucleus of an atom can take on many different shapes, or orbitals. In an orbitally ordered material, the electrons do not occupy all possible states with equal probability, but rather pick out one particular orbital configuration which is then collectively occupied. In chiral charge ordered materials, the orbital occupation follows a corkscrew pattern, rotating as it progresses from one layer in the material to the next. The result is the formation of a chiral charge density wave or a helical lattice structure.

    Because the chiral order involves orbitals as well as charge density modulations, we predicted that there will in fact be two charge ordered phases in TiSe2, one that is chiral, and one that is not. Recent X-ray, specific heat and transport experiments confirmed that a second phase indeed exists in TiSe2, in agreement with the theoretical predictions.

    Understanding the origin of the chiral charge order allows us to now start thinking about the implications of its presence for the properties of its host material. Although it is too early to tell what will come out, the unique properties associated with a helical charge distribution make this novel type of order a promising candidate for many possible applications.

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    Observing the Divergence of Quantum Critical Fluctuations

    Although it is well known that phase transition at zero temperature are accompanied by strong quantum fluctuations on all possible length scales, it is usually hard to directly observe these fluctuations in an experiment. We found that high resolution X-ray diffraction can be used to directly follow the divergence of the quantum critical fluctuations in the charge ordered material NbSe2.
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    Materials can be ordered in a variety of ways. The regular lattice structure of a crystal for example represents the translational and rotational order of its atoms. Likewise, a magnet is a material in which the magnetic moments of the individual atoms form an ordered array, and a superconductor has a macroscopically ordered phase. Charge order is the type of order that results when the electrons within a material arrange themselves in a regular pattern, that does not precisely match the underlying lattice.

    Like all ordered states, the charge order can be destroyed in two distinct ways. If the material is heated to sufficiently high temperatures, thermal fluctuations in the electronic density will wash out the regularity of the ordered state, and the charge order melts. Alternatively, the charge order can be destroyed even at zero temperature, for example by the application of pressure. In this case, the order is destroyed by the quantum fluctuations which arise from the competition of the high pressure ground state with the electronically ordered phase. At the quantum critical point, where the order disappears completely, quantum fluctuations of all strengths and sizes occur in the material.

    One particular material in which quantum fluctuations are expected to be able to destroy charge order, is NbSe2. This material is charge ordered at temperatures below 30 K, and the transition temperature into the charge ordered state can be made even lower by applying pressure to the material. At very high pressures, the charge order is absent even at the absolute zero of temperature. As in many other materials, the electronic quantum critical point at which the zero temperature charge order first disappears, is surrounded by a superconducting phase, making observations of the quantum critical fluctuations difficult. In a recent PNAS article, we show that it is possible to use X-ray scattering techniques under pressure to identify quantum critical charge fluctuations near the point where the charge order disappears. Because the scattered X-rays are insensitive to the superconducting order, it is possible to directly image the electronic order parameter even within the superconducting phase.

    This measurement provides a very rare, direct look at the spatial divergence of quantum critical fluctuations. Most experiments studying quantum critical points focus on the time or energy domain, and cannot directly assess the typical size of the fluctuations. At best, an indirect measurement in terms of critical exponents can be used. In the X-ray experiment described here however, we directly image the spatial structure of the charge ordered state close to the quantum critical point, as well as the diverging range over which its quantum critical fluctuations stretch.

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    Highlight Picture   A Nanoscale Experiment to Measure Gravity’s Influence on Quantum Mechanics

    We show that the conflict between the theories of quantum mechanics and gravity may well have consequences for nanoscale experiments in the very near future. An unexpected result of their interplay is the spontaneous reduction of quantum dynamics to classical physics.
    Many approaches towards uniting the theories of gravity and quantum mechanics have traditionally focused on the description of physics at the Planck scale. At this scale the typical dimensions of the curvature of spacetime and the wavelength of quantum particles become comparable, and a theory of quantum gravity is definitely required to describe the prevailing physics. However, there is also a second length scale at which the theories of gravity and quantum mechanics meet. Rather than looking at the size of a single, localized object, this scale is determined by the maximum distance over which a massive object can be superposed before the associated curvature of spacetime begins to affect its time evolution.

    We show in a recent paper, that this second realm of quantum-gravity interactions can be reached in nano-scale experiments which employ masses that are only slightly greater than the masses involved in existing experiments.The tendency of modern experiments to produce ever heavier superpositions (going from single electrons, via Bucky balls, to nanomechanical resonators) thus forces us to take into account the combination of gravity and quantum mechanics.

    In our article, we show that gravity will act as a “unitarity breaking field” in the presence of massive superpositions. This means that as a superposition becomes more and more massive, its dynamics will look less and less like that of a microscopic quantum particle. The most striking result of the altered dynamics is that for truly heavy objects (such as soccer balls, humans, tables and chairs), it will be impossible to form a quantum superposition state. The shear weight of the objects is enough to ensure that their dynamics will always be purely classical.

    Moreover, in another (open access) review article, we show that the altered dynamics of heavy objects also implies that they can be used as quantum measurement machines. That is, by coupling a microscopic quantum particle to a heavy enough measuring device, the quantum superposition will be reduced to just one of its components. Born’s rule (which gives the probability for which component prevails) is automatically recovered from the interplay between gravity and quantum mechanics.

    This way, the famous quantum paradoxes (such as Schrodinger’s cat, EPR, etc) can be resolved as consequences of the gravity-induced altered quantum dynamics of heavy objects. The line that separates “quantum weirdness” from everyday experience thus lies not in the consciousness of an observer, or even in any hidden variables describing elementary particles, but only in the shear size and weight of objects in our daily world.

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    Aneesur Rahman Fellowship

    Argonne National Laboratory (Illinois, USA) has appointed me as their Aneesur Rahman Fellow. At Argonne I will continue my research on quantum critical phenomena, the loss of unitarity, and other topics of condensed matter physics.
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    Starting October 2010, I will be the Aneesur Rahman Fellow at Argonne National Laboratory in Illinois, USA. During the fellowship I will do research with prof. Mike Norman at the Materials Science Devision.

    One of the main focusses of the research during the fellowship will be the search for observable experimental consequences of the interplay between Einstein’s theory of gravity and quantum mechanics. Although a lot of research on the quantum-gravity connection has been done in the realm of the so called Planck Scale, relatively little is known about the possible interactions between these theories in the more easily observable mesoscopic domain. Recently we proposed that the first consequences of the presence of gravity in mesoscopic quantum experiments may lead to the demise of the usual rules of quantum dynamics, and give rise to a process of quantum state reduction instead.

    Apart from these projects, I will also continue doing research on the physics of materials near an electronic quantum critical point. In these materials, the interplay between electronic interactions (such as exciton formation) and structural effects (like charge density wave formation) may lead to unexpected physics like the enhancement of order or even the emergence of novel phases of matter.

    Finally, I will continue various smaller research projects on (orbital) ordering phenomena in solid state materials; on the (de)coherence of different types of qubits and the associated quantum infomration theory; and on the consequences of driving different physical systems towards the dividing line between quantum and classical physics.

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    Highlight Picture   Excitons, Superconductivity and Charge Order

    Going against the popular belief in condensed matter theory that excitons and charge order are always bad for superconductivity, we show that in some cases the presence of excitons can actually enhance the superconducting transition temperature.
    It was pointed out decades ago by Ginzburg and Little that excitons (bound pairs of a particle and a hole) can act as mediators of superconductivity: just like the phonons in the standard theory of superconductivity they can bind together electrons to create a conductor with zero resistance. It was realised soon after this initial discovery that excitonic superconductors would not suffer from many of the limitations usually imposed by phonons. Dolgov and Maksimov even argued that room temperature superconductivity would be a real possibility.

    To date however, no excitonic superconductors have been found. The reason is that in real materials excitons are never alone. Phonons are unavoidable, and when excitons interact with phonons this usually leads to the formation of a so-called charge density wave. This charge ordered state binds up all the available electrons in specific locations, and none are left to form the superconducting state.

    In a recent article we show that this common theme of charge order precluding superconductivity is not the only possibility. Before the charge order sets in there is an opportunity for the excitons and phonons to work together in forming a superconducting state. What’s more, this superconducting state is more robust than it would have been without the influence of the excitons, and therefore has a higher transition temperature. There even turns out to be a material in which these effects may be seen experimentally. TiSe2 is normally charge ordered due to the interplay of excitons and phonons, but under pressure this charge order breaks down, and the new exciton-influenced type of superconductivity may have a chance to arise.

    Room temperature superconductivity is still a very long way off, but realising that excitons and phonons can cooperate as well as compete does open up new ways of thinking about the theme of high temperature superconductivity.

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    Broken Unitarity and Quantum Measurement

    Unitarity, the guiding principle of quantum physics, can spontaneously break down in macroscopic objects. In addition to explaining their classical dynamics, this loss of unitarity may also help to better understand the ability of large objects to function as quantum measurement machines.
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    Time evolution in quantum physics is an inherently unitary process. One consequence of its unitarity is that quantum information can never be lost, nor copied. Extending this result to macroscopic objects leads to an apparent paradox: even though all the things in our everyday world are ultimately built out of microscopic quantum mechanical particles, the macroscopic objects we see around us (tables, chairs, etc.) don’t seem to behave very quantum mechanically. Tables for example do not usually seem to occur in a state of superposition.

    Parts of the puzzle of how to connect the micro to the macro world have been understood during the past decades. Spontaneous symmetry breaking explains how a table can have a fixed position in the middle of the kitchen, even though it would really like to spread out in a quantum mist throughout the entire house. Decoherence on the other hand can be used to understand that the average effect of all the quantum particles that continuously bounce into the table, is just to make it seem more classical.

    The piece of the puzzle that remains unaddressed by both spontaneous symmetry breaking and decoherence is the question of what happens when a macroscopic object like a table is subjected to a single experiment which according to the unitary rules of quantum mechanics should force it into a state of superposition. This situation of individual-state quantum dynamics was addressed in a recent paper, in which I showed that the same properties which allow everyday objects to undergo spontaneous symmetry breaking, also allow them to spontaneously break the unitarity of quantum physics.
    This way of avoiding unitarity allows macroscopic objects like tables, chairs, etc. to interact with microscopic particles without having to absorb all their quantum information, and they can thus also avoid being forced into a state of superposition.

    Besides explaining the lack of superpositions in our everyday experience, the process of spontaneous unitarity breaking may also help to better understand what happens during quantum measurement. Following the unitarity of quantum mechanics to the letter, any measurement machine that is used to measure a quantum property of a superposed microscopic particle should end up in a state of superposition itself. However, if macroscopic objects can spontaneously break unitarity, they can avoid the superposition, and display only a single measurement result.

    In the paper, I show that the probability for finding a particular outcome in such a process coincides precisely with Born’s rule. That is, the probability of a particular result being displayed by the measurement machine is given by the (squared) amplitude of the corresponding component in the initial wave function.

    The quantum measurement problem is thus reduced to the problem of defining which objects are large enough to undergo spontaneous unitarity breaking, and which are small enough to evolve quantum mechanically. This new problem may be addressed within the near future by experiments on mesoscopic superposition states, bringing the quantum-classical crossover back into the realm of experimental physics, and freeing it from any metaphysical connection to conscience observers, parallel worlds, or gambling deities.

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    Highlight Picture   Thin Spectrum in Superconductors

    We have shown that finite-size superconductors have low-energy, in-gap states associated with their spontaneous breaking of a global phase symmetry. Because of this so-called thin spectrum of states, superconducting qubits can stay quantum coherent only for a limited time.
    Solid state qubits in general are based on materials that have some broken symmetry. In the case of superconducting qubits (such as Cooper pair boxes and flux qubits) the relevant broken symmetry is a phase symmetry.

    In a recent paper we have shown that the superconducting state corresponds to a state with a well defined global phase. The breaking of this global phase symmetry turns out to be precisely analogous to the breaking of a U(1) phase symmetry in a “canted” antiferromagnet (i.e. an antiferromagnet in a uniform magnetic field). This picture does not contradict Elitzur’s theorem which states that the gauge symmetry associated with local phases cannot be broken. Indeed, the superconducting state is still fully gauge invariant even after obtaining a well defined global phase.

    As a consequence of the spontaneous symmetry breaking in superconductors, there must exist a thin spectrum of states at very low energies within the superconducting gap. For finite-size superconducting qubits this thin spectrum of states gives rise to decoherence, and therefore to a finite lifetime of the qubits, in precisely the same way as in magnetic qubit systems (see our earlier PRL paper). In the case of superconducting Cooper pair box qubits the resulting limit to the lifetime of present-day setups is estimated to be of the order of 0.5 ms, which is still well beyond the limit set by the conventional sources of decoherence in these systems.

    The paper was also selected to be featured in three virtual journals:

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    Junior Research Fellowship

    Homerton College (Cambridge, UK) has appointed me in a Junior Research Fellowship position. At Homerton College I will continue my research on quantum coherence, symmetry breaking, and the connection between quantum mechanics and classical physics in general.
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    Starting October 2007, I will be a Junior Research Fellow at Homerton College in Cambridge (UK). During the fellowship I will do research with prof. Peter Littlewood at the theory of condensed matter group of the department of physics of Cambridge University.

    One of the main focusses of the research during the fellowship will be the search for a dynamical connection between the quantum theory and classical mechanics. In equilibrium situations, the connection between these two realms of physcics is well understood in terms of the process of spontaneous symmetry breaking. A dynamical description on the other hand, is still missing.

    Recently we have found a first consequence of the presence of spontaneous symmetry breaking on the dynamical process of decoherence: we have shown that there is a fundamental limit to the time that a (symmetry-broken) solid state qubit can stay quantum coherent.
    Other ideas about the quantum-classical connection include the suggestion made recently by Sir Roger Penrose that gravity may have a deteriorating influence on the unitarity of quantum mechanical time evolution. Some first ideas of how this suggestion could be put to the test experimentally can already be found in my PhD thesis.

    Apart from the projects aimed at findig a dynamical connection between quantum and classical physics, I will also continue doing research on (orbital) ordering phenomena in solid state materials; on the (de)coherence of different types of qubits and the associated quantum infomration theory; and on the realization and consequences of spontaneous symmetry breaking in different physical systems.

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    Highlight Picture   Quantum Mechanics & The Big World

    I have completed my PhD thesis, and obtained my PhD degree (with honors). The thesis describes our work on orbital ordering, decoherence due to spontaneous symmetry breaking and a proposed experimental test of gravity’s influence on quantum mechanics.
    On the 4th of April 2007 I have defended my PhD thesis ‘Quantum Mechanics & The Big World: order, broken symmetry and coherence in quantum many-body systems’. The thesis is available online. A hardcopy can be ordered through Leiden University Press, or at Amazon.com.

    Three main topics are covered in the thesis. The first part shows how a novel orbital-assisted Peierls transition can occur in the one-dimensional spin chains of NaTiSi2O6. In this material a combined spin dimerization and orbital ordering transition has been observed. We use a tight-binding model to describe this transition. Based on the model we predict that the crystal field seen by the titanium ions in this material cannot be larger than the size of the super-exchange interaction.

    In the second part our work on the relation between spontaneous symmetry breaking and decoherence in solid state qubits is described. In addition to the earlier results, we now also explicitly include a description of spontaneous symmetry breaking in superconducting qubits. The corresponding decoherence time of these qubits turns out to be only just out of reach for the experimental state of the art.

    Finally, in the third part we propose an experimental test for a recent idea by Sir Roger Penrose about the possible influence of gravity on quantum mechanics. Penrose suggested that because of the incompatibility of general covariance and unitarity, gravity might be involved in quantum state reduction. Based only on this assumption, a timescale for the gravity-induced collapse process can be found. We propose to test these ideas by pushing the well-developed technology of superconducting flux qubits just a bit further.

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    Decoherence due to Spontaneous Symmetry Breaking

    We have shown that there is a fundamental limit to the time that solid state qubits can stay quantum coherent. This limit is due to the decohering effect of the thin spectrum states associated with spontaneous symmetry breaking.
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    Solid state qubits are based on materials that have some broken symmetry; translation symmetry for example is broken in crystals, spin rotation symmetry in (molecular) magnets and (global) phase rotation symmetry in superconductors. Such a broken continuous symmetry implies the presence of a set of low-energy states called the thin spectrum states.

    We have shown in a recent PRL paper that the presence of these thin spectrum states in solid state qubits will cause them to slowly loose coherence. The fundamental timescale associated with this decoherence process does not depend on the parameters of the underlying system, and scales with the system size.

    The paper and the accompanying press releases (both in English and Dutch) attracted a great deal of media attention worldwide:


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