Inspiration from Nature: Ultralight Fractal Designs for High Mechanical Efficiency

[From Left to Right] Daniel Rayneau-Kirkhope, Robert Farr, Yong Mao



Authors: Daniel Rayneau-Kirkhope¹, Robert Farr^2,3, Yong Mao⁴

Affiliation:
¹Open Innovation House, School of Science, Aalto University, Finland,
²Unilever R&D, Colworth House, Sharnbrook, Bedford, UK
³London Institute for Mathematical Sciences, Mayfair, London, UK
⁴School of Physics and Astronomy, University of Nottingham, UK

Hierarchical design is ubiquitous in nature [1]. Material properties can be tailored by having structural features on many length scales. The gecko, a lizard ranging from 2 to 60 cm in length, has a remarkable ability to walk on vertical walls and even upside-down on ceilings. This ability is brought about through the repeated splitting of the keratinous fibres on the bottom of the gecko’s foot, which increases the contact area so effectively that even the very weak van der Waals interactions can support the entire weight of the gecko [2].

A more specific form of hierarchical design is self-similar design, where one structural feature is found to be repeated on a number of different length scales. A natural example is the trabecular or spongy bone found around the joints in animals [3]. Here, a series of small beams are arranged in such a way that the stiffness and strength requirements are met while using minimal material. Regardless of the level of magnification, the same patterns are found in the structure. Interestingly, the exact configuration of the constituent beams in the trabecular bone is constantly changing: it is the result of a continuous opitimisation process that goes on throughout the lifetime of the bone and responds to change in stress levels [4]. It is found that when the animal’s bones support only small loads, many very slender pillars are present, and when the loading increases, fewer but stouter pillars are employed [5].

In our recently published work [6], we demonstrate that through the use of hierarchical, self-similar design principles, advantageous structural properties can be obtained. We show that the scaling of the amount of material required for stability against the loading can be altered in a systematic manner. A particular structure is fabricated through rapid prototyping, and we obtain the optimal generation number (for our specific structure) for any given value of loading.

Scaling

The volume of material required for stability can be related to the loading through a simple power law relationship. That is, the volume required is given by a dimensionless loading parameter raised to some power with a pre-factor (that is dependent only on material properties and specifics of the geometry). When the loading is small, it is the scaling (power) of the loading that dominates the relationship. Under tension, this power is one and a structure requires an amount of material that is proportional to the loading it must withstand; for a solid beam under compression, due to elastic buckling, the power is one-half. Given, for all realistic applications, the non-dimensional loading parameter is much less than 1, this means more material is required to support compressive than tensional loads. This one-half power law has direct consequences when one considers optimal structure: if a beam is bearing a compressive load, it is more efficient to use one beam rather than two, whereas in the case of tension, due to the linear relationship, splitting a tension member into more than one piece has no effect on the volume required for stability.

Fractal design

Our work centers on a very simple, iterative procedure that can be used to create designs of great complexity. The “generation” of a structure describes the number of iterations used to create the geometry. The simplest compression bearing structure is a solid slender beam. When loaded with a gradually increasing force, the beam will eventually buckle into a sinusoidal shape known as an “Euler buckling mode”. We can suppress this by using a hollow tube, but we introduce a second mode of a local failure of the tube wall – Koiter buckling. After optimizing for tube diameter and thickness, it is found that the scaling power increases to two-thirds, and the volume of material required for stability is reduced.

Figure 1: Showing the iterative process from low generation numbers to higher for structures bearing compression along their longest axis. At each step, all beams that are compressively loaded are replaced by a (scaled) generation-1 frame.

The next step is to replace the hollow beam with a space frame of hollow beams. The space frame used here is made up of n octahedra and two end tetrahedra. Optimising the number of octahedra, the radius and the wall thickness of the component beams (which are all assumed be identical) we find a new power law, and again, an improvement over the hollow beam design.

Continuing this procedure of replacing all beams under compressive load with (scaled) space frames constructed from hollow beams (figure 1), we find that the scaling law is always improved by the increased level of hierarchy. In general, the scaling is described by a (G+2)/(G+3) power-law relating non-dimensional volume to non-dimensional loading. Thus, as the generation number tends to infinity, the scaling relating material required for stability to loading approaches that of the tension member.

3D Printing

Working with Joel Segal, of the University of Nottingham, we fabricated an example of a generation-2 structure with solid beams, shown in figure 2. This was done through rapid prototyping technologies: micrometer-layer-by-micrometer-layer the structure was printed in a photosensitive polymer with each beam a fraction of a millimeter in radius. This structure shows the plausibility of the design and the extent to which modern manufacturing techniques allow for an increased creativity in design geometry. Through a process of 3-d printing and electro less deposition, it is believed that a metallic, hollow tubed structure could be created.

Figure 2: Showing a structure fabricated through rapid prototyping techniques. The inset shows the layering effect of the 3D printing technique. The structure shown in constructed in RC25 (Nanocure) material from envisionTEC on an envisionTEC perfactory machine.

Optimal generations

Although the scaling is always improved by increasing the generation number of the structure, the prefactor isn’t. The optimal structure is then obtained by balancing the scaling relationship with the prefactor in the expression. Generally, as the loading decreases (or the size of the structure increases), the scaling becomes more important and the optimal generation number increases. For large loads (or small structures) it can even be the case a simple, solid, beam is optimal.

Our work also formalises this relationship, for a long time engineers have created chair legs from hollow tubes or cranes out of space frames, Gustave Eiffel used three levels of structural hierarchy in designing the Eiffel tower. We show formally, that the optimal generation number has a set dependence on the loading conditions and allow future structures to be designed with this in mind. A further consequence of the alteration of the scaling law is that the higher the generations, the less difference it makes as to whether you have one structure holding a given load or two structures holding half the load each.

Reference:
[1] Robert Lakes, "Materials with structural hierarchy", Nature, 361, 511 (1993). Abstract.
[2] Haimin Yao, Huajian Gao, "Mechanics of robust and releasable adhesion in biology: Bottom–up designed hierarchical structures of gecko", Journal of the Mechanics and Physics of Solids, 54,1120 (2006). Abstract.
[3] Rachid Jennanea, Rachid Harbaa, Gérald Lemineura, Stéphanie Bretteila, Anne Estradeb, Claude Laurent Benhamouc, "Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection", Medical Image Analysis, 11, 91 (2007). Abstract.
[4] Rik Huiskes, Ronald Ruimerman, G. Harry van Lenthe, Jan D. Janssen, "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone", Nature, 405, 704 (2000). Abstract.
[5] Michael Doube, Michał M. Kłosowski, Alexis M. Wiktorowicz-Conroy, John R. Hutchinson, Sandra J. Shefelbine, "Trabecular bone scales allometrically in mammals and birds", Proceedings of the Royal Society B, 278, 3067 (2011). Abstract.
[6] Daniel Rayneau-Kirkhope, Yong Mao, Robert Farr, "Ultralight Fractal Structures from Hollow Tubes", Physical Review Letters, 109, 204301 (2012). Abstract.

Quantum Statistics From Outside Space And Time

[From left to right] 
Front row: Yeong-Cherng Liang (Geneva), Jean-Daniel Bancal (then Geneva, now Singapore); 
Middle row: Antonio Acin (Barcelona), Nicolas Gisin (Geneva); 
Back row: Valerio Scarani (Singapore), Stefano Pironio (Brussels).







Authors:
Jean-Daniel Bancal^1,5, Stefano Pironio², Antonio Acin^3,4, Yeong-Cherng Liang¹, Valerio Scarani⁵, Nicolas Gisin¹

Affiliations:
¹Group of Applied Physics, University of Geneva, Switzerland
²Laboratoire d’Information Quantique, Université Libre de Bruxelles, Belgium
³ICFO-Institut de Ciències Fotòniques, Castelldefels (Barcelona), Spain
⁴ICREA-Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
⁵Centre for Quantum Technologies and Department of Physics, National University of Singapore, Singapore

Quantum theory predicts what nature does with unparalleled scope and accuracy. In the new perspective of quantum information, one can add that quantum theory predicts what nature can do for you. But the theory is silent about how nature does it. This silence is at the heart of the uneasiness that some scientists and most laypersons have felt, and continue feeling, when confronted with quantum physics.

Einstein, for one, believed that this silence was only a temporary feature of the then-newly born formalism. He was confident that, with time, someone would find an explanation for quantum phenomena in terms of mechanisms in space-time. However, the passing of time brought physics further from Einstein’s wishes: in 1964, John Bell proved that quantum theory is incompatible with pre-established agreement. Indeed, some predictions of the theory violate a criterion called Bell inequality. Whenever this happens, we are certain that the results of measurements were certainly not pre-recorded in the physical systems. This violation has been clearly observed in several experiments, since Aspect’s pioneering one in 1982.

Past 2Physics article by Antonio Acín:
August 19, 2012: "Testing the Dimension of Classical and Quantum Systems" by Martin Hendrych, Rodrigo Gallego, Michal Mičuda, Nicolas Brunner, Antonio Acín, Juan P. Torres

The violation of Bell inequalities puts severe constraints on Einstein’s hope: any mechanism in space-time that reproduces quantum predictions must involve superluminal influences – exactly what Einstein wanted to avoid at all cost! The faster-than-light character comes from the fact that the predictions of quantum theory, as well as the observed experimental results, are unchanged if the measurement events are space-like separated.

For many physicists, once the obscene word “superluminal” is uttered, all speech must cease. Others, less definite at the moment of telling nature how to behave, were ready to give some credit to a mechanistic explanation, but under one condition: it must not be possible for us to use these influences to send actual messages faster than light. In other words, to be acceptable by the physics community, the hypothetical influences must at least be “hidden”. Abner Shimony coined the expression “peaceful coexistence” between relativity and the violation of Bell inequalities. There did not seem to be much more to be said on the topic, apart from noticing that the Bohmian presentation of quantum theory uses precisely hidden influences in the updating of its “quantum potential”.

In our recent work, we show that the constraint can be strengthened to the extreme: any finite-speed influence would lead to the possibility of sending messages faster than light, in blatant violation of relativity. The price to pay for peaceful coexistence is to accept influences that propagate at infinite speed, making the quantum universe fully connected, while conspiring to remain hidden from us.

In order to understand the generality of our result, let us first stress one point: if the goal would be to reproduce all the predictions of quantum theory as we know it, it is already obvious that the speed must be infinite. Indeed, finite-speed influences would certainly lead to deviations from quantum predictions, depending on the space-time configuration: for instance, if the measurement events cannot be connected by the influence, one should never observe a violation of Bell inequalities. In this work, we decided not to erect quantum theory to the rank of untouchable truth: we accepted finite-speed influences as working assumption, together with the required deviations from the predictions of our present-day formalism. We only assumed that, if the influences have the time to propagate from one measurement event to the next, quantum statistics are produced (because this is what is observed). Even in such a flexible scenario, we were able to prove that ultimately finite-speed influences cannot be hidden. Further, the conclusion can be reached using only observable statistics: it is not only theory-independent, but also “device-independent”.

Illustration of how to obtain a constraint on finite-speed models with three particles

The trick consists in devising a suitable multi-partite configuration, because no conclusion can be reached in the more familiar bipartite scenario. The idea of the argument can be sketched with three parties (see figure). The diagrams are drawn in the preferred frame in which the hypothetical influences propagate. Configuration (a) is such that the influences propagate from A to B, then from B to C: in this case, we request that quantum statistics are recovered. In configuration (b), A and B do not receive each other’s influence, while C receives the influences from both A and B. Now:

• On the one hand, according to the role of influences in this model, a possible violation of Bell inequalities by A and B in configuration (a) should cease if the observers change to configuration (b) e.g. by advancing B’s measurement. 
• On the other hand, for the influences to be hidden, the statistics AC and BC must be the same in both configurations.

The contradiction can be reached by finding marginal statistics AC and BC, whose possible three-partite extension are necessarily such that AB violates some Bell inequality. As it turns out, we have not found such an example with three parties, but a similar one involving four particles.

In conclusion: assuming that some quantum statistics can be observed, a classical mechanism that explains them must not only use superluminal influences: either it uses infinite-speed influences, or it allows us to send messages faster than light. Both alternatives are mind-boggling? Well, there is a reason why physicists prefer to skip the issue of “how nature does it”.

The history of an idea, with three references:

•  The idea is already ten years old, but at that time we did not have the mathematical tools to find a complete proof: Valerio Scarani, Nicolas Gisin, "Superluminal influences, hidden variables, and signaling", Physics Letters A, 295, 167 (2002). Abstract.
• We resumed working on this project in February 2011. Two weeks later appeared in the arXiv a nice breakthrough: Sandro Coretti, Esther Hänggi, Stefan Wolf, "Nonlocality is transitive", Physical Review Letters, 107, 100402 (2011).  Abstract. Wolf and coworkers had found a proof for “no-signaling statistics”. Those statistics cannot be obtained with quantum physics, so the final challenge remained open; nevertheless, this work showed that there is real hope of finding a proof. 
• Our proof for quantum statistics is: J-D. Bancal, S. Pironio, A. Acín, Y-C. Liang, V. Scarani, N. Gisin, "Quantum non-locality based on finite-speed causal influences leads to superluminal signaling", Nature Physics, 8, 867 (2012). Abstract.

Disordered photonics: A New Strategy for Light Trapping in Thin Films

Left to right: Kevin Vynck, Matteo Burresi, Francesco Riboli, and Diederik S. Wiersma

Authors: 
Kevin Vynck, Matteo Burresi, Francesco Riboli, and Diederik S. Wiersma

Affiliation: 
European Laboratory for Non-linear Spectroscopy (LENS) &
National Institute of Optics (CNR-INO), Florence, Italy

Thin-film solar cells nowadays represent a promising alternative to more conventional, thick, silicon panels. Using less material for solar cells allows for a significant saving of natural resources and a lowering of the production costs. A counter-effect of using thinner films is that the amount of light that is absorbed and eventually converted into electricity is significantly reduced. For this reason, improving the absorption of light by thin dielectric films constitutes a challenge of paramount importance in the development of high-efficiency, cost-effective, photovoltaic technologies [1].

Significant efforts have been made in recent years to design structures on the scale of the wavelength that are able to efficiently “trap” light in thin films. Among the various techniques proposed [2], great attention has been given to so-called photonic crystals, made, for instance, by creating a periodic array of holes in the film. At well-defined frequencies and angles of incidence, light impinging on the film can couple to the optical modes created by the nanostructuring and be trapped in the absorbing medium for a long time, thereby significantly increasing the light absorption [3]. Alternatively, randomly textured surfaces have been designed to efficiently spread light in the film on broad spectral and angular ranges, leading as well to an overall increase of the light absorption [4].

In a recent Letter published in Nature Materials [5], our team has presented a new strategy for light trapping thin films that takes advantage of both the efficient light trapping of photonic structures and the broadband/wide-angle properties of random media. The solution that we proposed relies on the use of two-dimensional disordered photonic structures, such as that shown in Figure 1 (higher panel), which exhibit complex electromagnetic modes to which coupling from free space is possible.

Figure 1: (Upper panel) Schematic view of a thin film containing a random pattern of holes. (Lower panel) Top and side views of the electromagnetic energy density in a randomly-nanopatterned film at two different frequencies (where t is the film thickness and λ, the wavelength of light). Light is efficiently trapped in the film, due to the light coupling to disordered optical modes. (Figures adapted from Ref. [5])

To understand the physical process involved, it is instructive to consider how light behaves in such a film. The dielectric film naturally acts as a waveguide for light, confining it in the plane of the film and preventing any out-of-plane loss. Placing holes at random positions in the film makes such that light is multiply-scattered in the plane, in a similar way as a two-dimensional random walk. Multiple scattering and wave interference lead to the formation of optical modes, the characteristics of which (e.g., their spatial extent) are intimately related to the structural properties of the disordered system. A key feature of these modes is that they are leaky, due to the finite thickness of the film, meaning that they are accessible from the third dimension, and thus, can be used for light trapping purposes.

For illustration, the electromagnetic energy density produced by a plane wave at normal incidence on a thin film containing a random pattern of holes (air filling fraction of 30%) is shown in Figure 1 (lower panel) at two different frequencies. The very high energy density in the film is a clear indication of an efficient light trapping effect. The speckle patterns observed arise from the interference between the multiply-scattered waves in the plane of the film, as described above.

The main results of our work are given in Figure 2, showing the absorption spectra of a bare (unpatterned) film with a moderate absorption efficiency (< 5%) and of the same film containing the random pattern of holes considered above. A strong enhancement of the absorption efficiency is observed over a broad range of frequencies, as well as for wide incidence angles and both polarizations of light (see the inset). These are very important properties for solar panels since they should ideally be efficient in all circumstances.

Figure 2: Absorption spectra of the bare (unpatterned) film (black curve) and the films containing random and amorphous patterns of holes (blue and gray curves, respectively). The inset shows the angular dependence of the absorption of the randomly-nanopatterned film at t/λ=0.15 for both polarizations of light. The random pattern of holes leads to a large absorption of the incident light over broad spectral and angular ranges. Disorder correlations in the amorphous pattern allow for a fine-tuning of the absorption spectrum. (Figure adapted from Ref. [5])

Since, as stated above, the coupling process is mediated by the optical modes, which intrinsically depend on the type of disorder considered, we further investigated the possibility to tune the light absorption by engineering the disorder. More particularly, we considered the case of an “amorphous” structure, characterized by a short-range correlation in the position of the holes, as periodic patterns, yet lacking any long-range order. The results on the absorption efficiency, shown in Figure 2, are remarkable: while the absorption is diminished at lower frequencies, becoming quite close to that of the bare slab, it is significantly increased at higher frequencies. The absorption enhancement occurs when the wavelength in the material approximately equals the typical distance between holes, proving that disorder correlations provide us with an important degree of control over the light absorption spectrum.

A final test to conclude our work has been to simulate the absorption of a film of amorphous silicon in the red part of the solar spectrum, where efficient light trapping is generally needed. We observed that the absorption efficiency of the films containing the disordered hole patterns (random and amorphous) was at least as high as that of the film containing the periodic hole pattern. This is an important result as it shows that periodic nanostructuring does not necessarily guarantee the best possible outcome. The lack of periodicity in photonic structures and the robustness of the properties of the films to structural imperfections could lead to the development of low-cost solar panels with a higher efficiency.

References:
[1] Albert Polman and Harry A. Atwater, "Photonic design principles for ultrahigh-efficiency photovoltaics", Nature Materials, 11, 174-177 (2012). Abstract.
[2] Shrestha Basu Mallick, Nicholas P. Sergeant, Mukul Agrawal, Jung-Yong Lee and Peter Peumans, "Coherent light trapping in thin-film photovoltaics", MRS Bulletin 36, 453-460 (2011). Abstract.
[3] Xianqin Meng, Guillaume Gomard, Ounsi El Daif, Emmanuel Drouard, Regis Orobtchouk, Anne Kaminski, Alain Fave, Mustapha Lemiti, Alexei Abramov, Pere Roca i Cabarrocas, Christian Seassal, "Absorbing photonic crystals for silicon thin-film solar cells: design, fabrication and experimental investigation", Solar Energy Materials and Solar Cells, 95, S32-S38 (2011). Abstract.
[4] C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, F. Lederer, "Comparison and optimization of randomly textured surfaces in thin-film solar cells", Opics Express, 18, A335-A341 (2010). Abstract.
[5] Kevin Vynck, Matteo Burresi, Francesco Riboli, Diederik S. Wiersma, "Photon management in two-dimensional disordered media", Nature Materials, 11, 1017-1022 (2012). Abstract.

Topological States and Adiabatic Pumping in Quasicrystals

Left to right: Kobi E. Kraus, Oded Zilberberg, Yoav Lahini, Zohar Ringel and Mor Verbin

Authors: 
Yaacov E. Kraus¹, Yoav Lahini^2,3, Zohar Ringel¹, Mor Verbin², Oded Zilberberg¹

Affiliation:
¹Dept. of Condensed Matter Physics, Weizmann Institute of Science, Israel
²Dept. of Physics of Complex Systems, Weizmann Institute of Science, Israel
³Department of Physics, Massachusetts Institute of Technology, USA

The materials that make up our world have a variety of electrical properties. Some materials, such as metals, conduct electricity extremely well, while others are insulators, and are very efficient as shields from electric currents.

Recently, a new discovery revolutionized the prevailing paradigm of electrical properties of materials, when a new type of material was discovered [1, 2]. These materials are termed “topological insulators”, and have very unique electrical properties. For example, electricity would flow smoothly on the surface of a topological insulator, while the interior will be completely insulating. Interestingly, if one would cut this material in half, the new surface that is created, which was previously buried within the insulating interior, will suddenly become conducting. If the material is cut repeatedly, the same will happen each time.

Past 2Physics article by Yoav Lahini:
November 07, 2010: "Hanbury Brown and Twiss Interferometry with Interacting Photons"
by Yoav Lahini, Yaron Bromberg, Eran Small and Yaron Silberberg

In addition to this peculiar property, the electrical behavior on the surface itself reveals unique phenomena that are even expected to simulate bizarre new particle excitations [3]. As a result, these fascinating materials generated much activity in the condensed matter physics community, in an attempt to find new topological materials and to study their intriguing properties.

In a recent paper [4], we found that other unconventional materials, known as quasicrystals, are in fact also members of the topological materials family. Moreover, the topological behavior that they exhibit is similar to that of usual topological materials in some aspects, but differs from them in others.

Quasicrystals are materials in which the atoms are arranged in a distinct way. In most solid materials, the atoms are arranged in space either periodically or in a completely random fashion. Quasicrystals are an intermediate type of solid - they are neither periodic nor random. Rather, there is some non-repeating (i.e. non periodic) but well defined rule to the arrangement of their atoms [5, 6]. Despite the fact that quasicrystals have been experimentally observed already in 1982 [5], for a long time there was a debate between crystallographers whether they exist at all, as it was assumed that all crystalline materials are necessarily periodic. The conclusion that quasicrystals are a new type of solid revolutionized material science, updated the physical definition of what is a crystal, and culminated in the awarding of the Nobel Prize in Chemistry to its discoverer, Dan Shechtman from the Technion - Israel Institute of Technology [7].

Yet, many of the physical properties of quasicrystals, such as their electrical conductance, are not fully understood. The work recently published by our group in Physical Review Letters [4], discusses the electrical properties of surfaces of quasicrystals, and finds a new and surprising connection between quasicrystals and topological states of matter. Specifically, we show that a one-dimensional quasicrystal behaves, to some extent, like two-dimensional topological matter known as quantum Hall systems. We prove this claim theoretically and measure it experimentally.

The experiments were done on a novel type of quasicrystals, known as photonic quasicrystals [8, 9]. These systems are made of quasi-periodic arrangements of transparent materials, rather than atoms. In these systems, one studies the optical properties, rather than the electrical, but the underlying physics is very much the same. A major advantage of using photonic quasicrystals is the ability to fabricate one-dimensional materials, and to directly image the propagation of light within them.

In our experiments, we have realized a one-dimensional photonic quasicrystal, and measured the boundary (the surface of a one-dimensional system) properties of these quasicrystals. We found that the photonic states that reside at the boundary are localized -- meaning that light that is injected to that boundary will stay there. This is analogous to the electric currents on the surface of topological matter, which do not penetrate the interior of the material, but remain confined to the surface. This finding was surprising, as common wisdom was that -- generally, such a behavior is not supposed to occur in one-dimensional systems.

Our theory explains how that becomes possible in quasicrystals. In brief, the arrangement of atoms in a quasicrystal can be mathematically described as some type of projection of a periodic system on a system of lower dimension – for example, projection of a two-dimensional square lattice onto a one-dimensional line [10]. Note that this description defines the position of the atoms of the quasicrystal, but do not imply the properties of any electrons (or photons) moving through it. In our case, the one-dimensional quasicrystalline models we worked with can be described as another type of one-dimensional projection of a quantum Hall system, known as “dimensional reduction” [3]. Most importantly, the novel projection used to define our one-dimensional quasicrystals preserves the topological properties! Thus, we find that beyond their mere structure, quasicrystals can, in some sense, also “inherit” nontrivial topological properties from their higher-dimensional periodic “ancestors”.

Taking things a step forward, we have shown that the boundary states observed in the experiments indeed possess nontrivial topological properties, by demonstrating a topological “pumping” of light from one side of the quasicrystal to the other [4].

Figure 1: Experimental observation of adiabatic pumping via topologically protected boundary states in a photonic quasicrystal. (a) An illustration of the adiabatically modulated photonic quasicrystal, constructed by slowly varying the spacing between the waveguides along the propagation axis z. Consequently, the injected light is pumped across the sample. (b) Experimental results: Light was injected into the rightmost waveguide. The measured intensity distributions as a function of the position are presented at different stages of the adiabatic evolution, i.e., different propagation distances. It is evident that along the adiabatic evolution the light crossed the lattice from right to left.

This fascinating discovery appears to be just the beginning. Our results suggest that additional quasicrystals should exhibit topological states [11, 12], and that these states will always be linked to systems of a higher dimension. This approach might mean that three-dimensional quasicrystalline materials -- either photonic or electronic -- would exhibit strange surface properties, which can be explained as originating from a six-dimensional topological system. These subjects are currently under active investigation.

References:
[1] “Colloquium: Topological Insulators”, M.Z. Hasan and C.L. Kane, Reviews of Modern Physics, 82, 3045 (2010). Abstract.
[2] “Topological insulators and superconductors”, Xiao-Liang Qi and Shou-Cheng Zhang , Reviews of Modern Physics, 83, 1057 (2011). Abstract.
[3] “Topological field theory of time-reversal invariant insulators”, Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang, Physical Review B 78, 195424 (2008). Abstract.
[4] "Topological States and Adiabatic Pumping in Quasicrystals”, Yaacov E. Kraus, Yoav Lahini, Zohar Ringel, Mor Verbin, and Oded Zilberberg, Physical Review Letters, 109, 106402 (2012). Abstract.
[5] “Metallic Phase with Long-Range Orientational Order and No Translations Symmetry”, D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn, Physical Review Letters, 53, 1951 (1984). Abstract.
[6] "Quasicrystals: A New Class of Ordered Structures", Dov Levine and Paul Joseph Steinhardt, Physical Review Letters, 53, 2477 (1984). Abstract.
[7] See http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html .
[8] “Wave and defect dynamics in nonlinear photonic quasicrystals”, Barak Freedman, Guy Bartal, Mordechai Segev, Ron Lifshitz, Demetrios N. Christodoulides and Jason W. Fleischer, Nature, 440, 1166 (2006). Abstract.
[9] “Observation of a Localization Transition in Quasiperiodic Photonic Lattices”, Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson and Y. Silberberg, Physical Review Letters, 103, 013901 (2009). Abstract.
[10] “Algebraic theory of Penrose's non-periodic tilings of the plane”, N.G. de Bruijn, Kon. Nederl. Akad. Wetensch. Proc. Ser. A (1981).
[11] “Topological Equivalence Between The Fibonacci Quasicrystal and The Harper Model”, Yaacov E. Kraus and Oded Zilberberg, Physical Review Letters, 109, 116404 (2012). Abstract.
[12] “Observation of Topological Phase Transitions in One-Dimensional Photonic Quasicrystals”, M. Verbin, Y. E. Kraus, O. Zilberberg, Y. Lahini and Y. Silberberg, in preparation. 

Making Use of the Audible Hiss Around the Second Law of Thermodynamics

Felix Ritort

Author: Felix Ritort 

Affiliation:
Small Biosystems Lab, Departament de Fisica Fonamental, Facultat de Fisica, Universitat de Barcelona, Spain.
Ciber BBN de Bioingenieria, Biomateriales y nanomedicina, Instituto de Sanidad Carlos III, Madrid, Spain.

The second law of thermodynamics pervades the natural world. Using a non-specialized jargon we may say that systems (and the universe as a whole) tend to become unstructured and disordered by the sole action of the naturally occurring physical forces.

From Clausius to Boltzmann the second law of thermodynamics has emerged around the concept of entropy. Entropy is a quantity that characterizes the thermodynamic state of a system and quantifies the degree of usefulness of its energy content. The combination of energy and entropy gives the free energy, a thermodynamic state function that determines, in concert with the second law, the natural fate of thermodynamic transformations.

A useful formulation of the second law states that for any irreversible transformation of a system kept in contact with a thermal bath, the amount of work performed on the system by an external agent is greater or equal to the free energy difference between the initial and final states. In other words, any thermodynamic transformation requires the expenditure of a minimum amount of work typical of reversible or quasi-static (i.e. slow enough) transformations. Faster transformations always require larger amounts of work incurring in finite amounts of energy that are dissipated to the environment. Such transformations are generally called irreversible: the larger the amount of work, the more the process is irreversible. The second law stands amongst the most fundamental laws of nature, yet some hiss around it has started to emerge over the past years. That hiss -- already known to Boltzmann, Smoluchowski, Schroedinger and others -- is now experimentally accessible.

Indeed, the second law inequality has to be understood in terms of averages. For very small systems (often called mesoscopic) or enough short times some realizations of the irreversible transformation transiently violate the second law, whereas others (the vast majority) preserve it. Such variability stems for the thermal or Brownian forces in the environment that produce noise and fluctuations in the dynamical evolution of the system. Every time the same experiment is repeated a different outcome for the actual performed work is observed. Only by averaging over many (actually infinite) repetitions of the same transformation, the second law inequality is preserved.

Over the past years a series of remarkable results that go under the generic name of fluctuation theorems have shown that such hiss around the second law can be used to recover free energy differences during irreversible transformations (for a review see [1]). This is unexpected: the second law -- being mathematically expressed as an inequality -- does not provide a clue about how to extract free energy differences in irreversible transformations. In 1997 Chris Jarzynski derived a beautiful equality that shows how free energy differences can be recovered from repeated work measurements along irreversible processes [2]. A few years later Gavin Crooks extended Jarzynski’s results and proved a fluctuation relation by considering a thermodynamic process and its time reversal [3].

Fluctuation theorems provide an elegant way to extract free energy differences by repeating the same experiment many times back and forth. Single molecule manipulation offers excellent opportunities for exploring such theoretical predictions. One can use AFM (Atomic Force Microscope), magnetic tweezers or optical tweezers to repeatedly pull back and forth a folded biomolecule (e.g. a nucleic acid hairpin or a protein) to unfold the native structure and refold the unfolded structure many times. In 2005 the first experimental test of the fluctuation theorem by Crooks and the measurement of free energy of folded RNA native structures from irreversible mechanical experiments were carried out [4].

Since then and until recently all applications of fluctuation theorems to recover free energy differences only considered native states. However, much less known but equally relevant in biophysics are kinetic states, i.e. states that are metastable in the thermodynamics sense (i.e. they have free energies that are higher than the lowest free energy of the native state). The paper recently published in Nature Physics by Anna Alemany et al. [5] addresses the recovery of free energy of kinetic states in DNA molecular structures from irreversible pulling experiments with optical tweezers.

Figure 1. Illustration of the transformation between two sets containing partially equilibrated states. Configurations inside each colored phase-space region are sampled according to the Boltzmann factor. In contrast, the statistical weight of the regions is not determined by its equilibrium free energy. λ(t) denotes the pulling protocol in the experiments, whereas A,B,C stand for any native or kinetic (intermediate or misfolded) state.

We apply an extended version of the fluctuation relation originally introduced by our group in 2010 to recover free energy branches [6]. In a nutshell such extended version requires that the initial state during the thermodynamic transformation be partially (rather than globally) equilibrated (Figure 1). Physically this means that the system is locally equilibrated in one or more metastable states for a finite time. The application of the extended relation then follows the same route as the standard Crooks relation. The new feature is that the different unfolding and folding trajectories must be classified into different sets -- based upon which state they start and end with; and the fraction of trajectories belonging to each set together with the partial work distributions for each set must be measured accordingly. We applied the extended relation to DNA structures exhibiting kinetic intermediates and misfolded structures (Figure 2). In all cases we showed that the extended fluctuation relation provides free energy estimates of the different kinetic states even in cases where equilibrium methods are not applicable (i.e. when full equilibrium cannot be reached in the experimentally accessible timescales).

Figure 2. Pulling experiment. (Left panel) Schematics of the pulling experiment with optical tweezers. The bead captured in the optical trap (red colored region) is used to measure the force acting on the molecular construct (hairpin plus handles). The position of the trap relative to the bead in the pipette defines the control parameter λ in the experiment. (Right panels) Force-distance curves.

The extended fluctuation relation paves the way to future free energy measurements of short-lived or inaccessible states that are not affordable with other bulk techniques. Applications are abound in RNA, proteins, DNA-protein, DNA-peptide, protein-protein systems. It is remarkable that that hiss around the second law finally turned out to be so useful for the measurements of pure thermodynamic quantities.

References
[1] Felix Ritort, "Nonequilibrium fluctuations in small systems: from physics to biology", Advances in Chemical Physics, 137, 31-123 (2008). Abstract.
[2] C. Jarzynski, "Nonequilibrium equality for free energy differences", Physical Review Letters, 78, 2690–2693 (1997). Abstract.
[3] G. E. Crooks, "Path-ensemble averages in systems driven far from equilibrium", Physical Review E 61, 2361–2366 (2000). Abstract.
[4] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, Jr and C. Bustamante, "Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies", Nature, 437, 231–234 (2005). Abstract.
[5] Anna Alemany, Alessandro Mossa, Ivan Junier, Felix Ritort, "Experimental free-energy measurements of kinetic molecular states using fluctuation theorems", Nature Physics, 8, 688–694 (2012). Abstract.
[6] Ivan Junier, Alessandro Mossa, Maria Manosas, Felix Ritort, "Recovery of free energy branches in single molecule experiments", Physical Review Letters, 102, 070602 (2009). Abstract.

Imperfections, Disorder and Quantum Coherence

Steve Rolston [Image courtesy: University of Maryland, USA]

A new experiment conducted at the Joint Quantum Institute (JQI, operated jointly by the National Institute of Standards and Technology in Gaithersburg, MD and the University of Maryland in College Park, USA) examines the relationship between quantum coherence, an important aspect of certain materials kept at low temperature, and the imperfections in those materials. These findings should be useful in forging a better understanding of disorder, and in turn in developing better quantum-based devices, such as superconducting magnets. The new results are published in the New Journal of Physics [1].

Most things in nature are imperfect at some level. Fortunately, imperfections---a departure, say, from an orderly array of atoms in a crystalline solid---are often advantageous. For example, copper wire, which carries so much of the world’s electricity, conducts much better if at least some impurity atoms are present.

In other words, a pinch of disorder is good. But there can be too much of this good thing. The issue of disorder is so important in condensed matter physics, and so difficult to understand directly, that some scientists have been trying for some years to simulate with thin vapors of cold atoms the behavior of electrons flowing through solids trillions of times more dense. With their ability to control the local forces over these atoms, physicists hope to shed light on more complicated case of solids.

That’s where the JQI experiment comes in. Specifically, Steve Rolston and his colleagues have set up an optical lattice of rubidium atoms held at temperature close to absolute zero. In such a lattice atoms in space are held in orderly proximity not by natural inter-atomic forces but by the forces exerted by an array of laser beams. These atoms, moreover, constitute a Bose Einstein condensate (BEC), a special condition in which they all belong to a single quantum state.

This is appropriate since the atoms are meant to be a proxy for the electrons flowing through a solid superconductor. In some so called high temperature superconductors (HTSC), the electrons move in planes of copper and oxygen atoms. These HTSC materials work, however, only if a fillip of impurity atoms, such as barium or yttrium, is present. Theorists have not adequately explained why this bit of disorder in the underlying material should be necessary for attaining superconductivity.

The JQI experiment has tried to supply palpable data that can illuminate the issue of disorder. In solids, atoms are a fraction of a nanometer (billionth of a meter) apart. At JQI the atoms are about a micron (a millionth of a meter) apart. Actually, the JQI atom swarm consists of a 2-dimensional disk. “Disorder” in this disk consists not of impurity atoms but of “speckle.” When a laser beam strikes a rough surface, such as a cinderblock wall, it is scattered in a haphazard pattern. This visible speckle effect is what is used to slightly disorganize the otherwise perfect arrangement of Rb atoms in the JQI sample.

In superconductors, the slight disorder in the form of impurities ensures a very orderly “coherence” of the supercurrent. That is, the electrons moving through the solid flow as a single coordinated train of waves and retain their cohesiveness even in the midst of impurity atoms.

In the rubidium vapor, analogously, the slight disorder supplied by the speckle laser ensures that the Rb atoms retain their coordinated participation in the unified (BEC) quantum wave structure. But only up to a point. If too much disorder is added---if the speckle is too large---then the quantum coherence can go away. Probing this transition numerically was the object of the JQI experiment. The setup is illustrated in figure 1.

Figure 1: Two thin planes of cold atoms are held in an optical lattice by an array of laser beams. Still another laser beam, passed through a diffusing material, adds an element of disorder to the atoms in the form of a speckle pattern. [Image courtesy: Matthew Beeler]

And how do you know when you’ve gone too far with the disorder? How do you know that quantum coherence has been lost? By making coherence visible.

The JQI scientists cleverly pry their disk-shaped gas of atoms into two parallel sheets, looking like two thin crepes, one on top of each other. Thereafter, if all the laser beams are turned off, the two planes will collide like miniature galaxies. If the atoms were in a coherent condition, their collision will result in a crisp interference pattern showing up on a video screen as a series of high-contrast dark and light stripes.

If, however, the imposed disorder had been too high, resulting in a loss of coherence among the atoms, then the interference pattern will be washed out. Figure 2 shows this effect at work. Frames b and c respectively show what happens when the degree of disorder is just right and when it is too much.

Figure 2: Interference patterns resulting when the two planes of atoms are allowed to collide. In (b) the amount of disorder is just right and the pattern is crisp. In (c) too much disorder has begun to wash out the pattern. In (a) the pattern is complicated by the presence of vortices in the among the atoms, vortices which are hard to see in this image taken from the side. [Image courtesy: Matthew Beeler]

“Disorder figures in about half of all condensed matter physics,” says Steve Rolston. “What we’re doing is mimicking the movement of electrons in 3-dimensional solids using cold atoms in a 2-dimensional gas. Since there don’t seem to be any theoretical predictions to help us understand what we’re seeing we’ve moved into new experimental territory.”

Where does the JQI work go next? Well, in figure 2a you can see that the interference pattern is still visible but somewhat garbled. That arises from the fact that for this amount of disorder several vortices---miniature whirlpools of atoms---have sprouted within the gas. Exactly such vortices among electrons emerge in superconductivity, limiting their ability to maintain a coherent state.

Another of the JQI scientists, Matthew Beeler, underscores the importance of understanding the transition from the coherent state to incoherent state owing to the fluctuations introduced by disorder: “This paper is the first direct observation of disorder causing these phase fluctuations. To the extent that our system of cold atoms is like a HTSC superconductor, this is a direct connection between disorder and a mechanism which drives the system from superconductor to insulator.”

Reference:
[1] M C Beeler, M E W Reed, T Hong, and S L Rolston, "Disorder-driven loss of phase coherence in a quasi-2D cold atom system", New Journal of Physics, 14, 073024 doi:10.1088/1367-2630/14/7/073024 (2012). Abstract. Full Article.

First Material with Longitudinal Negative Compressibility

Adilson E. Motter (Left) and Zachary G. Nicolaou (Right)
















Authors: Zachary G. Nicolaou^1,2 and Adilson E. Motter¹
Affiliation:
¹Department of Physics and Astronomy, Northwestern University, USA
²Department of Physics, California Institute of Technology, USA

Conventional materials deform along the direction of the applied force in such a way that they expand when the force is tensional and contract when it is compressive. But our new paper [1] published this month in Nature Materials demonstrates that not all materials have to be that way. We explored network concepts to design metamaterials exhibiting negative compressibility transitions, during which a material undergoes contraction when tensioned (or expansion when pressured). This effect is achieved through destabilizations of metastable equilibria of the constituents of the material. These destabilizations give rise to a stress-induced phase transition associated with a twisted hysteresis curve for the stress-strain relationship. The proposed materials are the first to exhibit longitudinal negative compressibility at zero frequency.

Negative compressibility surface. When pressured, the surface expands instead of contracting [Image copyright: Adilson E Motter]

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The motivation for this work comes from our previous research on networks. It has been known that some networks respond in a surprising way to various types of perturbations. For example, in previous research our group has shown that the removal of a gene from the metabolic network of a living cell can often be compensated by the removal (not addition) of other genes [2]. Our hypothesis was that, with the right design, similarly counter-intuitive responses could occur in materials as well, which are essentially networks of interacting particles. The idea of using network concepts to design a material that could contract longitudinally when tensioned was particularly attractive because no existing material (natural or engineered) had been found to exhibit that property.

Negative compressibility cube. When tensioned, the cube contracts instead of expanding [Image copyright: Adilson E Motter]. 

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There are numerous potential applications for materials with negative compressibility transitions. They include the development of new actuators, microelectromechanical systems, and protective devices---from ordinary ones, such as seat belts, to devices that reduce the consequences of equipment failure. These materials may also lead to force amplification devices, which could be used to sense minute forces and transform them into large ones. Indeed, the strain-driven counterpart of negative compressibility transitions is a force amplification phenomenon, where an increase in deformation induces a discontinuous increase in response force. Other potential applications would be to improve the durability of existing materials, such as in crack closure of fractured materials. In fact, we expect other researchers to come up with yet different applications that we have not even thought about.

Negative compressibility material. The material at the center of the image expands vertically as it is squeezed [Image copyright: Adilson E Motter].

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The most surprising aspect of this research is the very finding that you can create a material that contracts when it would be expected to expand and expands when it would be expected to contract. Think of a piece of rod that you tension by pulling its ends with your fingers. It would normally get longer, but for these materials it can get shorter. This has been generally assumed not to be possible for the excellent reason that no known material behaves that way. Moreover, it is easy to show that this is indeed impossible if we assume that the material will respond continuously to the applied force. Our work shows, however, that this unfamiliar form of compressibility can occur by means of an abrupt change---a phase transition. A posteriori, perhaps another surprising aspect of our research was the simplicity of the system once we understood how it works, and this can have practical implications for the fabrication of the material.

This work illustrates rather dramatically how deceiving it is to assume that a material’s property will be limited by those of existing ones, the reason being that existing materials explore only a tiny fraction of the space of all possibilities. Previous research has pushed the boundaries of electromagnetic properties and led, for example, to materials with negative refractive index [3]. Our research shows that even mechanical properties that have no immediate analogs in electromagnetic metamaterials can be tailored and even inverted. At the end, the material’s properties are only limited by how different interacting parts can be assembled together. For a related discussion in the context of networks, see Ref. [4].

References:
[1] Z. G. Nicolaou and A. E. Motter, Mechanical metamaterials with negative compressibility transitions, Nature Materials 11, 608-613 (2012). Abstract.
[2] A. E. Motter, "Improved network performance via antagonism: From synthetic rescues to multi-drug combinations", BioEssays 32, 236-245 (2010). Full Article.
[3] R. A. Shelby, D. R. Smith and S. Schultz, "Experimental verification of a negative index of refraction", Science 292(5514), 77-79 (2001). Abstract.
[4] A. E. Motter and R. Albert, "Networks in motion", Physics Today 65(4), 43-48 (2012). Abstract. 

Transformational Thermodynamics: Cloaks to Keep You Cool

Sébastien Guenneau (left) and Claude Amra (right)












Authors: Sebastien Guenneau and Claude Amra

Affiliation: Institut Fresnel, Centre National Recherche Scientifique, Aix-Marseille University and Ecole Centrale Marseille, France

 In 2006, two papers -- published in the same issue of 'Science' -- revolutionized the world of classical optics with the concept of transformation optics (TO). One was by Ulf Leonhardt of University of St Andrews, Scotland, UK [1] and the other was by John Pendry of Imperial College, London, UK and David Schurig and David Smith of Duke University, USA [2]. The concept of transformation optics allows coordinate changes in an isotropic homogeneous dielectric medium that can lead to a anisotropic heterogeneous metamaterial described by tensors of permittivity and permeability, a fact foreseen twenty years ago in two visionary papers on computational electromagnetic [3,4].

The conceptual breakthrough in 2006 was to note that one can blow-up a point into a finite region using a change of coordinates known from mathematicians working on inverse problems [5], in order to conceal this region from electromagnetic waves, and to demonstrate the feasibility of an invisibility cloak for microwaves [6], thereby bringing perhaps the most stunning electromagnetic paradigm of all times into reality!

However, similar changes of coordinates can also be applied to other wave equations, such as linear water waves propagating at the surface of a fluid [7], pressure waves propagating in a fluid [8], coupled pressure and shear waves propagating in a solid material [9,10], or flexural waves in thin elastic plates [11,12]. Such metamaterials designed using transformation acoustics (TA) could be used to protect regions from tsunamis (in general from ocean waves) or earthquakes (especially from surface elastic waves known as Rayleigh waves) on a larger scale! They could also be used to improve sound in opera theaters or to hide submarines from sonars (silence cloak).

But that’s not the end of the invisible story, as one can also leave the world of TO and TA and enter the brave new world of transformation thermodynamics (TT, or T²), whereby one now wishes to control diffusion processes, such as heat. Some precursory theoretical and numerical studies on the conduction equation in anisotropic media [5,13,14] have shown that one can control the diffusive heat flow in new ways in the static limit, a fact experimentally demonstrated this year [15].

Figure 1: [Click on the image to view high resolution version] Numerical simulation showing the distribution of temperature in a region heated from the left (temperature=100 degrees Celsius, red color). The temperature gradually decreases away from the source, until it reaches a temperature of 0 degree Celsius on the left hand side (blue color). Importantly, one sees that the temperature vanishes inside the inner disc of the thermal cloak (annulus containing an anisotropic heterogeneous conductivity). This thermal protection is achieved by curving the isothermal curves (black lines).

Our group at the Fresnel Institute in Marseille and the Ecole Centrale in Paris has shown, under the umbrella of the French National Center for Scientific Research (CNRS), that TT is a valid concept: one can control the flow of heat when time flows [16], and it is enough to use concentric layers with isotropic homogeneous conductivity to design invisibility cloaks (to protect a region from heat, see figure 1) and concentrators (to enhance heat exchange in a region). This opens unprecedented routes towards heat insulators -- for instance, for Green houses and also for heat harvesting in photovoltaics.

References:
[1] U. Leonhardt, “Optical Conformal Mapping”, Science 312, 1777 (2006). Abstract.
[2] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science, 312, 1780 (2006). Abstract. 
[3] A. Nicolet, J.F. Remacle, B. Meys, A. Genon and W. Legros, “Transformation methods in computational electromagnetic“, Journal of Applied Physics, 75, 6036-6038 (1994). Abstract.
[4] A.J. Ward and J.B. Pendry, “Refraction and geometry in Maxwell’s equations“, Journal of Modern Optics, 43, 773-793 (1996). Abstract.
[5] A Greenleaf, M Lassas and G Uhlmann, “On nonuniqueness for Calderon’s inverse problem“, Mathematical Research Letters, 10, 685–693 (2003). Full Article.
[6] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies”, Science, 314, 977 (2006). Abstract.
[7] M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband Cylindrical Acoustic Cloak for Linear Surface Waves in Fluid”, Physical Review Letters, 101, 134501 (2008). Abstract.
[8] S. Zhang, C. Xia, and N. Fang, “Broadband Acoustic Cloak for Ultrasound Waves”, Physical Review Letters, 106, 024301 (2011). Abstract.
[9] G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant from”, New Journal of Physics, 8, 248 (2006). Abstract.
[10] M. Brun, S. Guenneau and A.B. Movchan, "Achieving control of in-plane elastic waves". Applied Physics Letters, 94, 061903 (2009). Abstract.
[11] M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Ultrabroadband Elastic Cloaking in thin Plates”, Physical Review Letters, 103, 024301 (2009). Abstract.
[12] N. Stenger, M. Wilhelm, and M. Wegener, “Experiments on Elastic Cloaking in Thin Plates”, Physical Review Letters, 108, 014301 (2012). Full Article. 2Physics Article.
[13] C. Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity", Applied Physics Letters, 92, 251907 (2008). Abstract.
[14] T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction", Applied Physics Letters, 93, 114103 (2008). Abstract.
[15] Supradeep Narayan and Yuki Sato, “Heat flux manipulation with engineered thermal materials", Physical Review Letters 108, 214303 (2012). Abstract.
[16] S. Guenneau, C. Amra, and D. Veynante, ‘’Transformation thermodynamics: cloaking and concentrating heat flux’’, Optics Express, 20, 8207 (2012). Abstract.

Free Randomness Can Be Amplified

Author: Roger Colbeck

Affiliation: Institute for Theoretical Physics, ETH Zurich, Switzerland

Are there fundamentally random processes in Nature? 150 years ago, scientists with a classical world-view would have likely answered in the negative: the laws of classical mechanics state that if one knew all the physical properties of every particle at some particular instant in time, then, in principle, the future evolution could be calculated. However, from the beginning of the 20th Century, this world-view started being challenged as quantum theory was born. This profoundly different theory asserts that the outcomes of measurements are fundamentally random. So, when a single photon is emitted from a source and sent through a half-silvered mirror, for example, all quantum theory tells us is that with probability 1/2 the photon is reflected, and with probability 1/2 it passes through, a distribution that can be confirmed statistically.

But is that the whole story? How can we be sure that the destiny of the photon (whether it will pass the mirror, or be reflected) wasn't already determined in such a way that observations on many photons nevertheless give the same statistics as if random?

In 1964, the work of John Bell [1] shed some light on the question of whether there could be higher explanations for the quantum statistics. By studying an extended experiment, involving two entangled photons sent towards two half-silvered mirrors, he showed that if the source determined the behaviour of the photons then the resulting correlations could not be those predicted by quantum mechanics. Experiments later confirmed the quantum predictions, e.g. [2].

It is tempting to use the above to argue for the existence of fundamentally random processes, but there is a catch. Bell's argument relies on different configurations of the half-silvered mirrors, and he assumes that these are chosen at random. Thus, for the purpose of arguing that there are truly random processes (note that this wasn't Bell's aim), the argument is circular. If we can randomly choose the configurations, then the outcomes are random, as was previously stressed by Conway and Kochen [3].

In our paper [4], we show that if we have access only to some weak randomness to choose the configurations, then the outcomes of certain quantum experiments are nevertheless completely random. To capture the idea of weak randomness, imagine that you write down a string of 0s and 1s choosing them as randomly as you can. However, before you write each bit, a sophisticated machine is asked to guess your next choice. If your choices are weakly random, then the machine can guess the next one with some probability greater than 1/2. What we show in our paper, is that provided the machine's guessing probability is not too high, it is possible to make random bits for which the machine knows nothing: this is randomness amplification.

[Image credit: T. Neupert]: Illustration of randomness amplification: A moving die about to strike an assembled tower. The tower falls yielding random numbers on several dice, thus amplifying the randomness of the original. In classical physics, the apparently random way the dice fall is in principle predictable, while, in quantum theory, there are ways to make this amplification fundamental.

It is interesting to note that this is not possible within classical mechanics. There, given a source of weak randomness, there is no protocol that can improve the quality of the randomness [5]. Thus, the task we present gives a new example of the improved power of using quantum systems over classical ones for information processing.

We conjecture that our result is extendible such that, provided the machine cannot guess the choices perfectly, it is possible to generate perfectly random bits. This would provided the strongest possible evidence in the existence of random processes in Nature: it would show that either the world is completely deterministic, or there are perfectly random processes.

This work also has applications in virtually any scenario that relies on randomness. For example, a casino that doesn't completely trust its random number generators could in principle use a protocol of the type we suggest to improve the quality of the randomness.

References:
[1] Bell, J. "On the Einstein Podolsky Rosen Paradox". Physics, 1, 195--200 (1964). Full Article.
[2] Aspect, A., Grangier, P. & Roger, G. "Experimental Realization of Einstein-Rosen-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities". Physical Review Letters 49, 91--94 (1982). Abstract.
[3] Conway, J. & Kochen, S. "The free will theorem", Foundations of Physics, 36, 1441--1473 (2006).Abstract.
[4] Colbeck, R. & Renner, R. "Free randomness can be amplified", Nature Physics,  doi:10.1038/nphys2300, (Published online May 6, 2012). Abstract.
[5] Santha, M. & Vazirani, U. V.  "Generating Quasi-Random Sequences From Slightly-Random Sources",  in Proceedings of the 25th IEEE Symposium on Foundations of Computer Science (FOCS-84) 434--440 (1984). Abstract.