.comment-link {margin-left:.6em;}


2Physics Quote:
"Over the past three decades, the promise of exponential speedup using quantum computing has spurred a world-wide interest in quantum information. To date, there are three most prominent quantum algorithms that can achieve this exponential speedup over classical computers. Historically, the first one is quantum simulation of complex systems proposed by Feynman in 1980s. The second one is Shor’s algorithm (1994) for factoring large numbers – a killer program to break the widely used RSA cryptographic codes... Very recently, the third one came as a surprise."
-- Chao-Yang Lu and Jian-Wei Pan
(Read their article: "Quantum Computer Runs The Most Practically Useful Quantum Algorithm")

Sunday, August 23, 2015

One Clock in Two Places Simultaneously

Ron Folman (photo credit: Naomi Weisman)

Author: Ron Folman

Affiliation: Ben-Gurion University of the NegevBeer Sheva, Israel.

Link to atom chip group >>


This story is about Einstein’s theory of general relativity (GR) [1] and quantum mechanics (QM) [2]. These two theories constitute the physics revolutions of the 20th century, and there are numerous attempts to unify them or, at the very least, understand how they work together. This story is also about time. Some would claim that we are still far from really understanding time [3].

GR came to life in 1915, and was successful in explaining minute anomalies in the orbits of Mercury and other planets. In 1919, Sir Arthur Eddington was able, during a total solar eclipse, to verify a prediction of the theory when he headed an expedition which confirmed the deflection of light by the Sun. Some predictions, such as gravitational waves, remain unobserved. Another of the verified predictions of GR is that time is affected by gravity; it ticks at different rates depending on how close you are to a massive object. This has been confirmed for several decades now by sending rockets with clocks to large heights and recently, as atomic clocks became much more accurate, even by simply moving a clock a few centimeters higher in a lab. This effect was nicely displayed in the recent movie 'Interstellar'.

GR was always considered strange and hard to fathom. It is rumored that in one of his lectures on GR, Sir Eddington was asked by Ludwik Silberstain: "Professor Eddington, you must be one of three persons in the world who understands general relativity." Eddington paused, unable to answer. Silberstein continued "Don't be modest, Eddington!" Finally, Eddington replied "On the contrary, I'm trying to think who the third person is." It is quite amazing to see how less than a century later GR is an important part of our day-to-day life, as it is an essential part in the GPS navigation system.

QM is thought by many to be even weirder. It took away determinism, it allowed non-locality, and it even allowed objects to be in two places (or two states, e.g., of energy or of spin) at the same time. This seemed so ridiculous that one of the fathers of QM, Erwin Schrödinger came up with a thought experiment in which a cat is alive and dead at the same time, theoretically made possible due to QM. Other contributors to the birth of QM disliked it just as much. Einstein said that “God does not play with dice” and Louis de-Broglie believed determinism will come back in a more comprehensive theory. Niels Bohr said: “For those who are not shocked when they first come across quantum theory cannot possibly have understood it”. Yet, the theory is successfully withstanding endless testing for almost a century now.

It is quite strange that these two successful theories have not been put together into one simple description of the universe. It remains to be seen what exactly the interplay between the two is: for example, some claim that the crucial process of decoherence, which is responsible for defining the border between the quantum realm and the classical world (as our day-to-day world of large objects is known), is powered by GR [4]. Some even claim that the process of “wave-function collapse” is driven by GR [5]. This process is responsible for the fact that out of many possible outcomes which exist according to QM, we see only one when we make a measurement.

The experiment :

In the experiment we conducted, published recently in Science [6], we wanted to study the interplay between the two theories. Specifically we wanted to see what would happen if we take one clock and put it in two places in which time ticks at different rates, simultaneously. We then put the clock together again and observed it.

In order to do this we used a device called an atom chip (see our website for many relevant papers). It is a powerful tool in manipulating the internal and external degrees of freedom of ultra-cold atoms. Every atom in a cloud of ultra-cold atoms (a Bose-Einstein condensate at -273 deg. Celsius) was put in two places simultaneously (known in scientific language as a spatial quantum superposition [7]), by utilizing a device called an interferometer. In parallel, every atom was turned into an atomic clock by manipulating its internal (spin) degrees of freedom.

As we did not have sensitive enough clocks to observe the minute effect of GR, we induced an artificial time difference between the two locations occupied by the single clock (these two copies of the single clock are referred to in scientific language as wave-packets). Our proof-of-principle experiment indeed shows (as predicted by [8]) that time differences have a major effect on the quantum outcome expected from an interferometer (called an “interference pattern”). Specifically, we showed that time may act as a “which path” witness as if we measured in which of the two locations the clock is. Once such a measurement is made, QM tells us that the special state of quantum superposition is destroyed and the clock can no longer be in two places at the same time (this is called “complementarity”). However, we have also showed that if the hands of the clock in one wave-packet are rotated enough so that they again overlap the hands of the clock in the other wave-packet, we again cannot differentiate between the two locations and the superposition, and consequently the interference pattern is restored.

Technically, we used very strong magnetic gradients to construct a Stern-Gerlach type of interferometer, and also used similar gradients to induce the relative time lag (using the Zeeman effect). The atoms are laser-cooled and then additionally cooled by forced evaporation. The atoms are isolated from the environment by the fact that the whole experiment is conducted in a vacuum chamber. The final outcome of the experiment is registered with a CCD camera observing the atoms.

We now have a new tool at our disposal to investigate time and the interplay between GR and QM. What we will find is an open question. Usually, new tools bring new insight. The next step should be to use more accurate clocks and to separate the clock into positions farther apart so that we are able to see the effect of GR.

The Team :

The team included Ph.D. student Yair Margalit, post-doctoral fellows Dr. Zhifan Zhou and Dr. Shimon Machluf, and researchers Dr. Yonathan Japha and Dr. Daniel Rohrlich. In the picture are the first two authors of the paper: PhD student Yair Margalit (left) and post-doctoral fellow Zhifan Zhou (right).
Additional Thoughts :

It is perhaps appropriate to end a popular article with some broad thoughts and even speculations. It stands to reason that human kind knows much less than what it doesn’t know. Namely, we are just beginning to explore and understand the structure and dynamics of our world. History and Philosophy of science teach us that our understanding is constantly changing and no theory is the “end of the road”. Will we see extensions to GR or QM in our lifetime? Will they somehow be unified?

The atoms in our lab are our teachers; they are teaching us new things every day. As we build a better dictionary from atom language to human language we will be able to understand more. They are teaching us about our physical universe and about how to understand the strange mysteries of QM. They are teaching us technology (clocks, magnetic sensors, gravitational sensors, navigation, quantum communications and quantum computing). Eventually they may even teach us things that have to do directly with who we are. For example, is there such a thing as free will? On this latter topic I invite the reader to see a very short and simplified film I made [9].

It makes me envious but also very happy to know that some of the young readers of this article will know much more than I ever will. I encourage the young reader to take up the adventure wholeheartedly.

[1] “General relativity turns 100”, Special issue, Science, 347 (March 2015). Table of contents.
[2] “Foundations of quantum mechanics”, Insight issue, Nature Physics, 10 (April 2014).  Table of contents.
[3] Lee Smolin, "Time Reborn" (Mariner Books, 2014).
[4] Igor Pikovski, Magdalena Zych, Fabio Costa, Časlav Brukner, “Universal decoherence due to gravitational time dilation”, Nature Physics, 11, 668-672 (2015). Abstract.
[5] Roger Penrose, "The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics", Chapter 6 (New York: Oxford U. Press, 1989) ; Lajos Diósi, “Gravity-related wave function collapse: mass density resolution”, Journal of Physics: Conference Series, 442, 012001 (2013). Full Article ; Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P. Singh, Hendrik Ulbricht, “Models of wave-function collapse, underlying theories, and experimental tests”, Review of Modern Physics, 85, 471 (2013). Abstract.
[6] Yair Margalit, Zhifan Zhou, Shimon Machluf, Daniel Rohrlich, Yonathan Japha, Ron Folman, “A self-interfering clock as a 'which path' witness”, published online in 'Science Express' (August 6, 2015). Abstract.
[7] This new terminology is meaningful as all attempts to interpret the reality of this situation with day-to-day language have led to some sort of contradiction. The strange reality of the quantum world can perhaps only be accurately described by mathematics. Using words, which originate in our day-to-day (“classical”) experience, always fails those who attempt to harness them for the description of the quantum world. Perhaps one may alternatively state that the clock is in a state where it somehow “feels” how time “ticks” in several places at once.
[8] Magdalena Zych, Fabio Costa, Igor Pikovski, Časlav Brukner, “Quantum interferometric visibility as a witness of general relativistic proper time”, Nature Communications, 2, 505 (2011). Abstract. 2Physics Article.
[9] “ ‘Are we alive?’ – a thought by Ron Folman”: YouTube Link.  

Labels: , ,

Sunday, August 16, 2015

"Secrets of the Universe"

2Physics.com is partnering with K2 Communications, a global leader in the production and distribution of IMAX films, which is currently working on the production of a IMAX 3D documentary film titled "Secrets of the Universe". K2 Communications has been commissioned by the National Science Foundation (NSF) by way of a substantial grant to promote the pursuit of an education and/or career in STEM (Science, Technology, Engineering & Mathematics) by inspiring an appreciation for science through the amazing discoveries in the world of particle physics.

The plan is to do this through a spectacular IMAX 3D experience titled "Secrets of the Universe" that will explore some of the most elusive and fascinating phenomena in the cosmos under investigation at the Large Hadron Collider (LHC) in Geneva, Switzerland, and explain today’s advance scientific discoveries in layman's terms. With greater understanding and enthusiasm from the general public, we can expect greater support and progress for STEM.

The film will address, more specifically, the exciting possibilities for discovery during the second run currently under way at the LHC: research on dark matter, antimatter, and parallel universes that has the potential to turn science fiction into science fact. Perhaps most importantly, it will feature the science and engineering feat that is the Large Hadron Collider - arguably humankind’s greatest machine - as a monument to how technology and research build off each other to create perpetual progress.

Multiple award-winning Director Stephen Low – a veteran of more than 15 giant screen documentaries – will lead the film team that includes the highly acclaimed IMAX cinematographer Sean Phillips.

Here is the Link to the "Secrets of the Universe" campaign >>

Labels: , ,

Sunday, August 09, 2015

Computational High-Resolution Optical Imaging of the Living Human Retina

Some authors of the paper in Nature Photonics (reference [3]). From left to right: Yuan-Zhi Liu, Fredrick A. South, and Stephen A. Boppart.

Authors: Yuan-Zhi Liu, Fredrick A. South, Stephen A. Boppart

Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA.

Link to Biophotonics Imaging Laboratory >>

Eye imaging is very important not only because the vision is the primary human sense but also because the eye is the direct viewing window of the central nervous system. Acquiring the detailed information of the individual cells, blood vessels, and nerves at the retina of the living human eye could enable earlier diagnosis and better treatment for degenerative eye and neurological diseases. Optical coherence tomography, OCT, is a medical imaging technique analogous to ultrasound imaging, but has a higher resolution because it uses light rather than sound. Based on low coherence interferometry, OCT decouples the transverse and axial resolution, and can reject the out-of-focus background to achieve the depth-resolved image. Since the first application in eye imaging, OCT has become the standard of eye care for diagnosing and tracking disease, such as glaucoma and age-related macular degeneration [1].

The presence of ocular aberrations in the eye, however, limits cellular resolution tomography of the human retina. Combined with OCT, hardware-based adaptive optics (HAO) has pushed the limits of eye imaging, enabling diffraction-limited imaging of previously unresolvable structures, such as the three-dimensional visualization of photoreceptor distributions and individual nerve fiber bundles in the living human retina [2]. However, the sophisticated hardware and controlling software of HAO adds considerable complexity and cost to the standard OCT system, limiting the number of researchers and medical professionals who could benefit from the technology.

In our recent publication [3], we developed a phase stabilization technique and fully-automated computational adaptive optics (CAO) approach in conjunction with a standard en face optical coherence tomography (OCT) setup to computationally correct strong optical aberrations even in a highly-dynamic sample such as the human eye. This is the first time clear images of individual human photoreceptors, both near and far from the fovea, have been obtained in the living retina without the need of HAO.
Figure 1: (click on the image to view with higher resolution). Cone photoreceptors in the foveal region of the retina. (a) SLO image with the location of the photoreceptor mosaic outlined. (b) En face OCT mosaic before aberration correction. (c) The same OCT mosaic after automated correction. The insets are zoomed by 2X.

In HAO, a wavefront modulation device, e.g. deformable mirror, is used to physically modulate the pupil function to correct wavefront aberrations. CAO is a post-processing technique based on the complex signal from the interferometric measurement of OCT. The digital aberration correction filter is applied to the computational pupil plane, which is calculated from the Fourier transform of the focal plane image. The frequent and unavoidable motion of eye can severely interrupt the post-processing of the interference signal. In combination with the high-speed image acquisition from an en face OCT setup, the phase stabilization algorithm corrects even the sub-wavelength motion by comparing the phase differences of the adjacent frames. Afterward, a computational technique simulating the operation of the hardware Shack-Hartmann wavefront sensor is used to correct large optical aberrations which revealed individual photoreceptor cells. A technique named guide-star-based CAO is then applied to utilize the individual photoreceptors to sense and correct smaller, but still significant, aberrations.

CAO opens a new way for high-resolution imaging in ophthalmology. In addition to lower cost and simpler system configuration than HAO, the post-processing nature of CAO also allows more flexible aberration correction. Consider spatially-variant aberrations as an example. HAO needs to sense the aberrations in each small volume and then execute each correction while imaging the subject. By using CAO, researcher/doctors can shorten the image acquisition time and apply multiple aberration corrections afterward. Released from the physical limitations of the deformable mirror, e.g. the number and the stroke range of actuator, CAO has the potential to correct higher order and larger amplitude aberrations. Because CAO manipulates the complex interference signal, in principle better image quality can be achieved by applying not only the phase but also the amplitude corrections.

[1] D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography", Science, 254 (5035), 1178–1181 (1991). Abstract.
[2] Yan Zhang, Barry Cense, Jungtae Rha, Ravi S. Jonnal, Weihua Gao, Robert J. Zawadzki, John S. Werner, Steve Jones, Scot Olivier, Donald T. Miller, "High-speed volumetric imaging of cone photoreceptors with adaptive optics spectral-domain optical coherence tomography", Optics Express, 14, 4380-4394 (2006). Abstract.
[3] Nathan D. Shemonski, Fredrick A. South, Yuan-Zhi Liu, Steven G. Adie, P. Scott Carney, Stephen A. Boppart, "Computational high-resolution optical imaging of the living human retina", Nature Photonics, 9, 440-443 (2015). Abstract.

Labels: ,

Sunday, August 02, 2015

Bending Electric Discharges with Lasers

Matteo Clerici

Authors: Matteo Clerici1,2, Yi Hu1,3, Philippe Lassonde1, Carles Milián4, Arnaud Couairon4, 
Demetrios N. Christodoulides5, Zhigang Chen3,6, Luca Razzari1, François Vidal1, François Légaré1, Daniele Faccio2, Roberto Morandotti1,7

1Institut National de la Recherche Scientifique–Énergie Matériaux Télécommunications, Varennes, Québec J3X 1S2, Canada.
2School of Engineering and Physical Sciences, Scottish Universities Physics Alliance, Heriot-Watt University, Edinburgh, UK.
3The MOE Key Laboratory of Weak Light Nonlinear Photonics, School of Physics and TEDA Applied Physics School, Nankai University, Tianjin, China.
4Centre de Physique Théorique CNRS, École Polytechnique, Palaiseau, France.
5College of Optics, Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Florida, USA.
6Department of Physics and Astronomy, San Francisco State University, California, USA.
7Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, China.


Electric discharges are part of our everyday life. On a large scale, lightning is one of the most spectacular atmospheric phenomena, while on a much smaller scale -- cars, for instance -- rely on electric discharges for igniting the fuel. Small-scale controlled discharges find relevant industrial applications, e.g. in cutting and milling, and are also investigated as a mean to control gas flows and pollution.

Since the seventies, it has been realized that intense laser pulses may trigger and guide electric discharges [1], and ground breaking experimental results started to appear few years later (see e.g. [2–4] and references therein). From these successes, scientists have theoretically and experimentally investigated further on the laser guiding effect, also stimulated by the possibility of applying this concept to lightning control and protection [5].

While at standard air temperature and pressure conditions an electric field of nearly 34 kV/cm is required to start an electric arc between two electrodes, this threshold decreases with decreasing gas density. An intense laser focused in air ionizes the gas and locally increases the temperature. To this initial stage, a localized decrease in the density follows, which allows the discharge to start at lower than a standard electric field. This is the dominant effect responsible for the laser triggering and guiding mechanism [6,7].

Curved and healing beams:

In last few years researchers have demonstrated that the intense peak of laser pulses may travel in free space along complex trajectories, instead of the well-known straight line [8]. To enable this new degree of control the laser wavefront must be manipulated with proper optical elements. For instance, to generate curved laser beams a cubic phase modulation is required, and can be obtained either by designing a specific phase mask or by exploiting the aberrations of standard lenses. The beam generated this way is named Airy beam and the generalization of its behavior recently resulted in a vast available literature. Now, curved beams that follow an almost circular path or more complex trajectories can be obtained [9,10].

The research into laser shaping also brought to attention the ability of particular beams to self-heal, i.e. to recreate their intense part even after being partially obstructed. This happens, for instance, for Airy beams, and also for Bessel beams, another class of laser beams obtained by focusing a standard laser with a conical lens, instead of an ordinary one.

Bending the currents:

In the present work, recently published on Science Advances [11], we have combined the concept of laser guided electric discharge with the possibilities allowed by the recent results on beam shaping, and we have shown how electric sparks can be guided along a curved path avoiding an obstacle (see Fig. 1). Furthermore we have demonstrated that the self-healing property of laser beams allows the discharge to go back on track also when the obstacle cannot be avoided.

In our work we have employed a high-voltage generator able to deliver a potential difference of up to 35 kV applied to two wire electrodes suspended in air. At a distance of 5 cm no discharge occurs since the applied field is, at most, 5 times lower than the air breakdown field.
Figure 1. Sketch of the experiment. A phase mask and a lens reshape the laser beam into an Airy beam, which bend along the propagation and generate plasma on a parabolic shape. The region of air heated by the plasma has a reduced density, hence allowing the discharge to occur and guiding it along its curved path, overcoming an obstacle that is in the line of sight of the two electrodes. The inset on the top-right shows the actual measurement, acquired with a photo camera (top view). The obstacle is an opaque glass.

We then focus a shaped, intense laser pulse (intensity > 1012 W/cm2) between the electrodes, ionizing the air and therefore heating the gas. We compared different shapes: a Gaussian beam (the most common beam shape), a Bessel beam, an Airy beam, and an S-shaped beam, obtained inserting in the laser path i) a standard lens, ii) an axicon (conical lens), iii) a tilted cylindrical lens system and iv) a phase mask, respectively. This way we have seen that electric discharge occurs only when the laser pulse ignites the plasma and the discharge path indeed follows the trajectory of the shaped laser pulse. For the Gaussian beam we observe a quite erratic behavior, consequence of the poor spatial localization of the heated gas; the discharge appear more deterministic when employing a Bessel beam, due to the high spatial localization of the Bessel peak. Finally, the discharge is clearly curved when an Airy beam is employed, as shown in the inset of Fig. 1. This allows conducting the charge from one electrode to the other while avoiding an obstacle that is placed in the direct line of sight.

Current self-healing:

Bessel beams and Airy beams share the ability to regenerate after hitting on an obstacle. This is mainly due to the fact that the energy responsible for the build-up of the intense laser peak flows from a reservoir that is off-axis with respect to the intense portion of the beam.

We have shown that the beam self-healing allows guiding the electric discharge along a predefined trajectory even in case of an obstacle blocking part of the beam. This is shown in Fig. 2 for the case of a Bessel beam. The laser propagates from left to right and is focused by the conical lens visible in the left of the picture. A tiny plasma plume develops over few centimeters and regenerates after hitting the obstacle (see inset in Fig.2). When the potential difference is applied to the electrodes, a discharge occurs following the path defined by the laser plasma. It performs a jump in a non-controlled way, right after the obstacle, and then goes back on track as soon as the laser plasma reappears (see Fig. 2).
Figure 2. Self-healing and electric discharges. We show that Bessel beam induced plasma reappears after the beam is obstructed by an obstacle, due to the Bessel beam self-healing property (see inset, the laser propagates from left to right). This effect finds a straightforward application in guiding electric currents: even if the dielectric obstacle cannot be avoided, the current can be delivered on target following the predefined path.


We have shown that shaping of laser beams allows extensive control of the path over which electric currents can be guided in air. Our results have shown that a reduction of up to a factor 10 in the breakdown voltage can be obtained with the proper laser beam, and this opens the possibility to realize complex discharge path and extreme curvatures that will be investigated in the future and may find applications e.g., in the machining industry.

R.M. and F.L. gratefully acknowledge Ministère de l’Enseignement Supérieur, de la Recherche, de la Science et de la Technologie (MERST), the Natural Sciences and Engineering Research Council of Canada (NSERC) and Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT) for funding this work. M.C. acknowledges the support from the People Program (Marie Curie Actions) of the European Union’s Seventh Framework Program (FP7/2007-2013) under REA grant agreement no. 299522. The work of D.N.C. was supported by the Air Force Office of Scientific Research (AFOSR, MURI grant no. FA9550-10-1-0561), and of Z.C. by the AFOSR and National Science Foundation (NSF). D.F. acknowledges financial support from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC grant agreement no. 306559, and the Engineering and Physical Sciences Research Council (EPSRC, UK, grant EP/M006514/1). A.C. and C.M. acknowledge financial support from the French Direction Générale de l’Armement (DGA). Experiments were carried out at the Advanced Laser Light Source (ALLS) facility located at the Institut National de la Recherche Scientifique.

[1] David W. Koopman, T.D. Wilkerson, "Channeling of an ionizing electrical streamer by a laser beam", Journal of Applied Physics, 42, 1883–1886 (1971). Abstract.
[2] X.M. Zhao, J.C. Diels, "Femtosecond pulses to divert lightning", Laser Focus World, 29, 113 (1993). 
[3] M. Rodriguez, R. Sauerbrey, H. Wille, L. Wöste, T. Fujii, Y.-B. André, A. Mysyrowicz, L. Klingbeil, K. Rethmeier, W. Kalkner, J. Kasparian, E. Salmon, J. Yu, J.-P. Wolf, "Triggering and guiding megavolt discharges by use of laser-induced ionized filaments", Optics Letters, 27, 772–774 (2002). Abstract.
[4] D. Comtois, C. Y. Chien, A. Desparois, F. Génin, G. Jarry, T. W. Johnston, J.-C. Kieffer, B. La Fontaine, F. Martin, R. Mawassi, H. Pépin, F.A.M. Rizk, F. Vidal, P. Couture, H. P. Mercure, C. Potvin, A. Bondiou-Clergerie, I. Gallimberti, "Triggering and guiding leader discharges using a plasma channel created by an ultrashort laser pulse", Applied Physics Letters, 76, 819–821 (2000). Abstract.
[5] Jérôme Kasparian, Roland Ackermann, Yves-Bernard André, Grégoire Méchain, Guillaume Méjean, Bernard Prade, Philipp Rohwetter, Estelle Salmon, Kamil Stelmaszczyk, Jin Yu, André Mysyrowicz, Roland Sauerbrey, Ludger Wöste, Jean-Pierre Wolf, "Electric events synchronized with laser filaments in thunderclouds", Optics Express, 16, 5757–5763 (2008). Abstract.
[6] F. Vidal, D. Comtois, Ching-Yuan Chien, Alain Desparois, B. La Fontaine, T.W. Johnston, J. Kieffer, Hubert P. Mercure, H. Pepin,  F.A. Rizk, "Modeling the triggering of streamers in air by ultrashort laser pulses", IEEE Transaction Plasma Science 28, 418–433 (2000). Abstract.
[7] S. Tzortzakis, B. Prade, M. Franco, A. Mysyrowicz, S. Hüller, P. Mora, "Femtosecond laser-guided electric discharge in air", Physical Review E, 64, 057401 (2001). Abstract.
[8] G.A. Siviloglou, J. Broky, A. Dogariu, D.N. Christodoulides, "Observation of accelerating Airy beams", Physical Review Letters, 99, 213901 (2007). Abstract.
[9] F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P.A. Lacourt, M. Jacquot, J.M. Dudley, "Sending femtosecond pulses in circles: highly nonparaxial accelerating beams", Optics Letters, 37, 1736–1738 (2012). Abstract.
[10] Ido Kaminer, Rivka Bekenstein, Jonathan Nemirovsky, Mordechai Segev, "Nondiffracting accelerating wave packets of Maxwell’s equations", Physical Review Letters, 108, 163901 (2012). Abstract.
[11] Matteo Clerici, Yi Hu, Philippe Lassonde, Carles Milián, Arnaud Couairon, Demetrios N. Christodoulides, Zhigang Chen, Luca Razzari, François Vidal, François Légaré, Daniele Faccio, Roberto Morandotti, "Laser-assisted guiding of electric discharges around objects", Science Advances, 1, e1400111 (2015). Article.


Sunday, July 26, 2015

Space-borne Gravitational Wave Detector LISA/eLISA

Yan Wang

[Yan Wang is the recipient of the 2014 Stefano Braccini Thesis Prize administered by the Gravitational Wave International Committee (GWIC) for his PhD thesis “On inter-satellite laser ranging, clock synchronization and gravitational wave data analysis” (PDF). His thesis work was carried out at Leibniz University of Hannover, Germany.

The Stefano Braccini Thesis Prize was established to honor the memory of a talented gravitational wave physicist whose promising career was cut short. Stefano worked with the French-Italian Virgo project, and contributed to the superattentuator design, to the integration and commissioning of Virgo and to its data analysis efforts. -- 2Physics.com]

Author: Yan Wang

Affiliation: School of physics, University of Western Australia, Perth, Australia.

Observations of electromagnetic radiation have revolutionized our understanding of the Universe and fundamental physics during the last century. With the advent of the expected first detection of gravitational waves (GWs) in near future, a completely new window onto the Universe will soon be opened by GW astronomy. GWs are spacetime ripples, predicted by Einstein’s theory of General Relativity. Their existence has been indirectly proven by measurements of the orbital decay due to gravitational radiation of the binary pulsar PSR 1913+16 [1], for which Hulse and Taylor won the 1993 Nobel Prize, but, due to their weak coupling with matter (i.e. GWs pass through stars, galaxies, Earth, Sun, and everything), direct detection of GWs has been beyond our technological capabilities until now. This weak coupling does mean, however, that GWs carry uncorrupted physical, astrophysical and cosmological information, enabling us to probe deep into the very early Universe, to test general relativity (GR) with unprecedented precision, and to measure the masses and spins of black holes (BHs) with exquisite accuracy.

Currently, large laser interferometers are the most sensitive GW detectors. There are several existing ground-based interferometric GW detectors: LIGO (Hanford and Livingston) [2], VIRGO [3], GEO600 [4], either operating or being upgraded; and KAGRA [5] under construction. During the past few decades, scientists took all efforts to isolate or mitigate various kinds of disturbances on the Earth, in order to increase the sensitivity of the detectors. These detectors are expected to detect GWs in near future.
Figure 1: Classic LISA configuration.

There are also space-borne interferometric GW detector (planned) missions. Among them, the most mature one is the Laser Interferometer Space Antenna (LISA) [6-7] ‘family’ (e.g. classic LISA, eLISA, and variations). LISA/eLISA consists of three spacecrafts (Fig. 1), each individually following a slightly elliptical orbit around the Sun, trailing the Earth by about 20 degree. These orbits are chosen such that the three spacecrafts retain an equilateral triangular configuration with an arm length of a few million kilometers as much as possible. This is accomplished by tilting the plane of the triangle by about 60 degree out of the ecliptic. Graphically, the triangular configuration does a cartwheel motion around the Sun.

The benefits of sending an interferometric GW detector to space are mainly: (i) Less noise disturbances, (ii) More GW sources. Roughly speaking, laser interferometric GW detectors are most sensitive to celestial systems of a size comparable to the interferometers’ arm length. LISA/eLISA is sitting in a frequency band, where there are most abundant GW sources (Fig. 2). There are several known white dwarf binaries in our galaxy directly visible to LISA/eLISA. In addition, LISA/eLISA can resolve thousands of other white dwarf binaries in our galaxy. LISA/eLISA can also observe massive black hole mergers throughout the entire universe [8]. Extreme mass ratio inspirals (i.e. stellar mass compact object orbiting a massive black hole) and primordial GWs from the birth of the universe add great scientific values to the mission as well [8].
Figure 2: (Click on the image to view with higher resolution) The GW spectrum from extremely low frequency to high frequency [Image courtesy: Chris Henze]

For many years, scientists have been spending great effort -- both theoretically and experimentally -- in preparation of LISA/eLISA. Some of the key techniques required by LISA/eLISA cannot be tested on the ground. LISA pathfinder satellite is going to be launched in this November (2015) to test the drag-free altitude control system in space, laser interferometry with picometer resolution at mHz band, the reliability of the instruments in the space environment, etc.

Unlike the ground-based interferometric GW detectors, the arm lengths of LISA/eLISA are varying significantly with time due to celestial mechanics in the solar system. As a result, the arm lengths differ by about one percent (i.e. tens of kilometers), and the dominating laser-frequency noise will not cancel out. The remaining laser-frequency noise would be stronger than other noises by about 8 orders of magnitude. Fortunately, the coupling between distance variations and the laser-frequency noise is very well known and understood. Therefore, we can use time-delay interferometry (TDI) techniques [9], which combine the measurement data series with appropriate time delays, in order to cancel the laser-frequency noise to the desired level.

However, the performance of TDI depends largely on the knowledge of arm lengths and relative longitudinal velocities between the spacecrafts, which are required to determine the correct delays to be adopted in the TDI combinations. In addition, the raw data are referred to the individual spacecraft clocks, which are not physically synchronized but independently drifting and jittering. This timing mismatch would degrade the performance of TDI variables. Therefore, they need to be referred to a virtual common constellation clock which needs to be synthesized from the inter-spacecraft measurements. Simultaneously, one also needs to extract the inter-spacecraft separations and synchronize the time-stamps properly to ensure the TDI performance. This has been a long existing gap.

Recently [10-11], we have tried to bridge this gap by designing sophisticated first stage data analysis algorithms for LISA/eLISA. The following are the main steps involved in the algorithms: (i) different types of inter-spacecraft measurements (e.g. the pseudo ranging measurements, the beat-notes of the carrier frequencies of the lasers emitted from one spacecraft and a remote spacecraft, the beat-notes of the laser sidebands) are precisely formulated as functions of the system state variables; (ii) several precise and effective dynamic models are designed for the system state variables (these models basically describe how the state variables evolve with time); (iii) the measurement data are pre-processed so that they can be used by a optimal filtering algorithm; (iv) the information of the measurements and the information from the dynamic models of the system state variables are optimally combined via a Kalman-like optimal filter, in order to reduce the noise in the measurements and the clock recording time stamps.

Simulation shows that our algorithms can successfully calibrate and synchronize the phasemeter raw data, estimate the inter-spacecraft distances and the clock errors, hence making the raw measurements usable for TDI techniques and astrophysical data analysis algorithms. This result can significantly increase the robustness of the LISA/eLISA project. The flexible design structure of our algorithms also provides a general framework of first stage LISA/eLISA data preparation, which can be easily extended to deal with various emergent scenarios in the future.

[1] Joel M. Weisberg, Joseph H. Taylor, "The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis", ASP Conference Series on 'Binary Radio Pulsars', vol. 328, p25 (2005). Article.
[2] http://www.advancedligo.mit.edu/
[3] http://www.cascina.virgo.infn.it/advirgo/
[4] http://www.geo600.org
[5] http://gwcenter.icrr.u-tokyo.ac.jp/en/
[6] https://www.elisascience.org/
[7] LISA International Science Team 2011 (European Space Agency), "LISA Unveiling a hidden universe", LISA Assessment Study Report (Yellow Book), ESA/SRE(2011) 3. Link.
[8] The eLISA Constortium, "The Gravitational Universe", Whitepaper submitted to ESA for the L2/L3 Cosmic Vision call. arXiv:1305.5720 [astro-ph.CO] (2013).
[9] Massimo Tinto, Sanjeev V. Dhurandhar, "Time-Delay Interferometry", Living Review Relativity 17 (2014), 6. Article.
[10] Y. Wang, Thesis: ‘On inter-satellite laser ranging, clock synchronization and gravitational wave data analysis’ (2014). Link.
[11] Yan Wang, Gerhard Heinzel, Karsten Danzmann, "First stage of LISA data processing: Clock synchronization and arm-length determination via a hybrid-extended Kalman filter", Physical Review D, 90, 064016 (2014). Abstract.

Labels: ,

Sunday, July 19, 2015

A Quantum Gas Microscope for Fermionic Atoms

The Fermi gas microscope group: (from left) graduate students Katherine Lawrence and Melih Okan, postdoc Thomas Lompe, graduate student Matt Nichols, Professor Martin Zwierlein, and graduate student Lawrence Cheuk. Photo credit: Jose-Luis Olivares/MIT.

Authors: Lawrence Cheuk and Martin Zwierlein

Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.

Link to Ultracold Quantum Gases Group >>

What do electrons, protons, neutrons and even quarks have in common? They all are fermions, particles with half-integer spin. Unlike their bosonic counterparts, integer spin particles, fermions cannot occupy one and the same quantum state. This simple fact leads to the structure of our elements, where electrons have to avoid each other and occupy different orbits around the atomic nucleus, or at least differ in their spin orientation.

When many fermions interact strongly with each other, they can form complex matter with exotic properties, from atomic nuclei to solid state materials, to distant neutron stars. Their collective behavior leads to diverse phenomena such as the structure of the elements, high-temperature superconductivity and colossal magneto-resistance.

Yet our understanding of strongly-interacting Fermi systems is limited. In recent years, ultracold atomic Fermi gases have emerged as a pristine platform to study many-fermion systems. In particular, fermionic atoms trapped in an optical lattice formed by standing waves of light can simulate the physics of electrons in a crystalline solid, shedding light on novel physical phenomena in materials with strong electron correlations.

Yet our understanding of strongly-interacting Fermi systems is limited. In recent years, ultracold atomic Fermi gases have emerged as a pristine platform to study many-fermion systems. In particular, fermionic atoms trapped in an optical lattice formed by standing waves of light can simulate the physics of electrons in a crystalline solid, shedding light on novel physical phenomena in materials with strong electron correlations.

In the present work, recently published in Physical Review Letters [3], we have realized quantum gas microscope that images ultracold fermionic 40K atoms with single-lattice-site resolution. Similar results have also been achieved at about the same time by researchers at University of Strathclyde and Harvard University [4,5].
Figure Caption: Fermionic 40K atoms in a 2D optical lattice with 541nm spacing imaged using Raman sideband cooling. Image taken from [3].

In our experiment, we prepare a two-dimensional layer of 40K atoms via laser cooling and forced evaporation. The atoms are then trapped in an optical lattice formed by retro-reflected laser beams, which form a standing wave with 541nm spacing. In order to resolve atoms with single-lattice-site resolution, we utilize a novel setup that incorporates a solid immersion lens into the vacuum window. This allows an enhancement in the numerical aperture, leading to higher resolution and enhanced light collection. In addition, optical aberrations that arise from a planar window are minimized in this setup.

In order to detect the atoms, we perform fluorescence imaging while simultaneously cooling the atoms. To make the atoms fluoresce, they are illuminated with near-resonant light. However, as the atoms emit photons, they experience heating from the recoil of photons. As the atoms are heated up, they hop between lattice sites and can even hop out of the lattice. In order to faithfully measure the occupation of the lattice sites, one must therefore eliminate the heating that arises when atoms fluoresce. We accomplish this via a technique known as Raman sideband cooling.

Raman sideband cooling, a technique first demonstrated in the 1990s, selectively transfers atoms from high-energy states to lower energy states via a two-photon Raman process. Atoms that are already in the lowest energy state, however, remain “dark” to the Raman light. By collecting photons that are scattered during this cooling process, we extract the position of the atoms while cooling the atoms. Hopping and atom loss are thus avoided. Furthermore, we have found that even after imaging the atoms with Raman sideband cooling, the atoms are predominantly in the lowest energy state. This invites the possibility of assembling low-entropy many-fermion states atom by atom.

The advent of fermion microscope will allow new studies of many-fermion systems in optical lattices, such as measurement of high order correlations and detection of magnetic ordering. Such studies could shed light on the behavior of other fermions, in particular, electrons. This may one day advance our understanding of the diverse phenomena that arise in complex solid-state systems.

[1] Waseem S. Bakr, Jonathon I. Gillen, Amy Peng, Simon Fölling, Markus Greiner, "A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice", Nature, 462, 74 (2009). Abstract.
[2] Jacob F. Sherson, Christof Weitenberg, Manuel Endres, Marc Cheneau, Immanuel Bloch, Stefan Kuhr, "Single-atom-resolved fluorescence imaging of an atomic Mott insulator", Nature, 467, 68 (2010). Abstract.
[3] Lawrence W. Cheuk, Matthew A. Nichols, Melih Okan, Thomas Gersdorf, Vinay V. Ramasesh, Waseem S. Bakr, Thomas Lompe, Martin W. Zwierlein, "Quantum-Gas Microscope for Fermionic Atoms", Physical Review Letters, 114, 193001 (2015). Abstract.
[4] Maxwell F. Parsons, Florian Huber, Anton Mazurenko, Christie S. Chiu, Widagdo Setiawan, Katherine Wooley-Brown, Sebastian Blatt, Markus Greiner, "Site-Resolved Imaging of Fermionic 6Li in an Optical Lattice", Physical Review Letters, 114, 213002 (2015). Abstract.
[5] Elmar Haller, James Hudson, Andrew Kelly, Dylan A. Cotta, Bruno Peaudecerf, Graham D. Bruce, Stefan Kuhr, "Single-atom imaging of fermions in a quantum-gas microscope", arXiv:1503.02005v2 [cond-mat.quant-gas] (2015).

Labels: , , ,

Sunday, July 12, 2015

Metamaterial Shrinks Integrated-Photonics Devices

Rajesh Menon

Author: Rajesh Menon

Affiliation: Department of Electrical & Computer Engineering, University of Utah, USA.

Integrated electronics is the driving force behind the information revolution of the last 6 decades. A similar revolution is happening in photonics, where devices that manipulate the flow of light (or photons) are being miniaturized and integrated. The main challenge for integrated photonics is that the wavelength of light is far larger than the equivalent wavelength of electrons. This is the main reason that devices fundamental to integrated electronics are significantly smaller than those used in integrated photonics. Furthermore, no one had come up with a way to design devices close to this limit for integrated photonics.

We recently solved this problem by first coming up with a new design algorithm and then experimentally verifying that our devices work as intended [1-3]. One crucial advantage of our method is that our fabrication process is completely compatible with the very mature processes already developed for silicon electronics. This means that we can exploit the vast existing manufacturing infrastructure to enable integrated photonics.

In our recent publication, we demonstrated the smallest polarization beam-splitter to date [1]. This device (shown in the figure below) has 1 input and 2 outputs. The 2 outputs correspond to the 2 linear polarization states of light. The device is designed to take either polarization of light (or both) as the input and separate the 2 polarizations into the 2 outputs. We input light into our device one polarization at a time and measured the transmission efficiency into the correct output. This allowed us to verify that the device performs as designed. This is analogous to separating two channels of communication (for example, a video stream from PBS and another from Netflix). Previously such separation would have required time and power-consuming electronics or if photonics devices were used, they would have been much larger (so much harder to integrate onto a chip).
Figure 1: (a) Scanning-electron micrograph of fabricated polarization beamsplitter. Simulated intensity distribution at (b) TE polarization and (c) TM polarization showing the separation of the beams.

In the big picture, our research has the potential to maintain Moore's law for photonics. By enabling integrated photonics devices to be much smaller (in fact, close to their theoretical limits), we allow the integration of more devices in the same area (which increases functionality) and also enable the devices to communicate faster (since they are closer together; light has to travel shorter distances). Finally, by packing more devices into the same chip, one also exploits economies of scale to reduce the cost per chip (similar to what has happened in electronics). The practical impact for customers is that one can expect to drastically reduce power consumption and enable faster communications and computing. Data centers today consume over 2% of the total global electricity. Reducing power consumption in data centers and other electronics can go a long way to reduce our CO2 emissions and stem global climate change.

Our vision is to create a library of ultra-compact devices (including beamsplitters, but also other devices) that can then be all connected together in a variety of different ways to enable both optical computing and communications. The first devices were fabricated at a University. Next, we need to fabricate these in a standard process at a company, and then provide this library of devices to designers and hopefully, unleash their creativity. We believe that these devices will usher in unpredictable, but unbelievably exciting applications.

References :
[1] Bing Shen, Peng Wang, Randy Polson, Rajesh Menon, “An integrated-nanophotonic polarization beamsplitter with 2.4 × 2.4 μm2 footprint”, Nature Photonics, 9, 378-382 (2015). Abstract.
[2] Bing Shen, Peng Wang, Randy Polson, Rajesh Menon, “Integrated metamaterials for efficient, compact free-space-to-waveguide coupling”, Optics Express, 22, 27175-27182 (2014). Abstract.
[3] Bing Shen, Randy Polson, Rajesh Menon, “Integrated digital metamaterials enables ultra-compact optical diodes”, Optics Express, 23, 10847-10855 (2015). Abstract.

Labels: , ,

Sunday, July 05, 2015

Site-Dependent Evolution of Electrical Conductance from Tunneling to Atomic Point Contact

Howon Kim (left) and Yukio Hasegawa

Authors: Howon Kim and Yukio Hasegawa

Affiliation: The Institute for Solid State Physics, University of Tokyo, Japan.

In our recent work [1], the evolution of electrical conductance was investigated from tunneling to atomic point contact whose atomic geometry was precisely defined using scanning tunneling microscopy (STM). We found that the conductance evolution depends on the contact site; for instance, on-top, bridge, or hollow ('hexagonal close packed, hcp' and 'face centered cubic, fcc') site in the close-packed lattice of the substrate, indicating the importance of the atomic configuration in the conductance of the atomic junctions.

Electronic conduction through atomic-sized metal contacts is of fundamental interest as a transport mechanism though the ultimately squeezed conductor [2]. Several seminal phenomena, such as quantization and step-wise variation [3, 4] in the conductance, have been reported using a method called break junction in which the conductance is measured just before the breaking moment of a nanometer-width thin wire. Since atomic geometry of the point contact cannot be controlled and the measured conductance fluctuates at every breaking, therefore, the obtained conductances are usually analyzed in a statistical manner. Scanning tunneling microscopy (STM) has also been utilized for the study of atomic point contacts, in which the contact is formed by pushing the probe tip toward the sample surface. Because of plastic deformation of the tip by the contact formation, however, quantitatively reliable and reproducible measurements have been difficult.

Figure 1. schematic showing the atomic geometry of the atomic point contacts formed at an on-top site (left) and a 3-fold hollow site (right) of a close-packed surface.

Here, in our study, making most of the capability of the atomically resolved imaging of STM, we measured the conductance of the atomic point contact in an atomically controlled manner. We first positioned the probe tip on a specific site, for instance, on-top, bridge, or hollow (fcc and hcp) site, in the crystallographic lattice of the substrate surface (Fig. 1), and then measured the conductance while moving the tip toward the substrate from tunneling to contact regimes. It is found that the conductance evolution depends significantly on the contact site. When the contact is formed, the hollow site has the largest conductance, and among the two hollow sites the hcp site is more conductive than fcc. When the tip is pulled from the contact by just 20-30 pm, a crossover occurs and the conductance at on-top site becomes the largest.

Figure 2: electrical conductance measured from tunneling (Δz = -20 pm) to contact (Δz = -60 pm) regimes. The measured conductance G is normalized by the quantum conductance G0 given by 2e2/h (~77.5 μS). For each conductance trace, 10 traces taken at the corresponding sites marked in the atomically-resolved STM image (inset) are averaged.

The traces of the electrical conductance measured from tunneling to contact at on-top, bridge, fcc, and hcp sites of the Pb(111) surface are shown in Fig. 2. For each plot, 10 traces obtained from the corresponding marked sites in the inset STM image are averaged. At the tip displacement Δz of -50 ~ -60 pm from the tunneling (Δz = 0), the atomic contact is formed as the conductance shows saturation around the quantum conductance G0 given by 2e2/h (~77.5 μS). The contact conductance shows strong site dependence; the conductance at the hcp site is largest and more than 50 % larger than the one measured at the on-top site. Around Δz = -30 ~ -40 pm, that is, when the tip is located above the substrate by 20 ~ 30 pm from the contact, the plot indicates the largest conductance at on-top site.
Figure 3. Spatial mappings of the conductance at various tip displacements (upper left) topographic STM image (3.0 X 3.0 nm2) taken simultaneously with 64 X 64 conductance traces. (lower left) conductance mapping at Δz = -32pm, where the largest conductance at on-top site is enhanced (Lower right) conductance mapping at Δz = -60 pm, that is, the contact regime, where hollow site, particularly hcp site, has large conductance. (upper right) schematics explaining the site dependence of the conductance. The atoms on which the chemical interaction is exerted are marked red.

In order to spatially demonstrate the site dependence, we performed real-space mappings of the conductance in the on-top enhancement region and in the contact regime. The upper-left panel of Fig. 3 is an STM image showing the atomic contrast taken simultaneously with the conductance traces. At a tip displacement Δz of -32pm, the conductance mapping (lower-left of Fig. 3) exhibits bright contrast at the on-top site, similarly to that in the topographic image. As the conductance mapping at Δz = 0 does not have any contrast, the bright contrast indicates the conductance enhancement at the on-top site. On the other hand, the conductance mapping in the contact regime (lower right of Fig. 3, Δz = -60pm) has its contrast reversed from that of the topographic one, indicating a larger conductance at the hollow site than at the on-top site. These results clearly demonstrate that the point contact conductance is quite sensitive to the atomic configuration.

When the distance between the tip and substrate is reduced, the attractive chemical interaction is exerted between the surface and tip apex atoms. This interaction presumably opens up the conduction channel and contributes to the development of the conductance. Schematics in the upper right panel of Fig. 3 show how the chemical interaction works in the case of contacts formed at on-top and hollow sites. When the tip approaches from the tunneling regime, the attractive interaction is exerted first at the on-top site (the force-exerted atoms are marked red in the schematics) because the substrate atoms are closer at on-top site than at hollow sites, thus making the on-top conductance enhanced. In the contact regime, however, the attractive force becomes stronger at hollow sites because of the greater number of involved atoms than the on-top site. This is probably the reason why conductance becomes larger there at the contact. Obviously, theoretical studies [5], simultaneous measurements of force and conductance by atomic force microscopy [6], and/or conduction channel analysis of the atomic point contact [7] are needed to elucidate the observed conductance behaviors.

References :
[1] Howon Kim and Yukio Hasegawa, "Site-dependent evolution of electrical conductance from tunneling to atomic point contact", Physical Review Letters, 114, 206801 (2015). Abstract.
[2] Nicolás Agraı̈t, Alfredo Levy Yeyati, Jan M. van Ruitenbeek, "Quantum properties of atomic-sized conductors", Physics Reports, 377, 81 (2003). Abstract.
[3] J. M. Krans, J. M. van Ruitenbeek, V. V. Fisun, I. K. Yanson, L. J. de Jongh; "The signature of conductance quantization in metallic point contacts", Nature, 375, 767 (1995). Abstract.
[4] L. Olesen, E. Lægsgaard, I. Stensgaard, F. Besenbacher, J. Schiøtz, P. Stoltze, K. W. Jacobsen, J. K. Nørskov; "Quantized conductance in an atom-sized point contact", Physical Review Letters, 72, 2251 (1994). Abstract.
[5] Jose Manuel Blanco, Cesar González, Pavel Jelínek, José Ortega, Fernando Flores, Rubén Pérez, "First-principles simulations of STM images: From tunneling to the contact regime", Physical Review B, 70, 085405 (2004). Abstract.
[6] Yoshiaki Sugimoto, Keiichi Ueda, Masayuki Abe, Seizo Morita "Three-dimensional scanning force/tunneling spectroscopy at room temperature", Journal of Physics : Condensed Matter, 24, 084008 (2012). Abstract.
[7] Howon Kim and Yukio Hasegawa, "Site-dependent conduction channel transmission in atomic-scale superconducting junctions", arXiv:1506.05528 [cond-mat.mes-hall].

Labels: , ,