Physics posts
- September 8, 2010
- 12:59 PM
- 773 views
Yet more graphene transistors – it’s twins!
by Joerg Heber in All That Matters
Last week I blogged about a Nature paper on graphene transistors with a self-aligned nanowire gate. Well, as I gather from a blog post by Doug Natelson, largely the same UCLA researchers have now published a paper in Nano Letters that uses a rather similar idea, even though in the latest paper the nanowire gate is [...]... Read more »
Liao, L., Bai, J., Cheng, R., Lin, Y., Jiang, S., Qu, Y., Huang, Y., & Duan, X. (2010) Sub-100 nm Channel Length Graphene Transistors. Nano Letters, 2147483647. DOI: 10.1021/nl101724k
Liao, L., Lin, Y., Bao, M., Cheng, R., Bai, J., Liu, Y., Qu, Y., Wang, K., Huang, Y., & Duan, X. (2010) High-speed graphene transistors with a self-aligned nanowire gate. Nature. DOI: 10.1038/nature09405
- September 7, 2010
- 11:47 PM
- 620 views
The World of Tractography Where The White Matter Tracts Appear Colored
by Amiya Kumar Sarkar in Physiology physics woven fine
White matter tractography, a relatively new MRI based technique, can delineate fiber tracts and assist in surgical planning and research.... Read more »
P. Mukherjee,, J.I. Berman,, S.W. Chung,, C.P. Hess, & R.G. Henry. (2008) Diffusion Tensor MR Imaging and Fiber Tractography: Theoretic Underpinnings. AM J Neuroradiol . DOI: 10.3174/ajnr.A1051
- September 7, 2010
- 01:13 PM
- 1,020 views
Standard Cosmology Theory Is Confirmed By ACT For Smallest Scales In The Universe.
by Joseph Smidt in The Eternal Universe
It never ceases to amaze me how well standard cosmology theory fits the ever increasing amount of data with precision. The results just released from the Atacama Cosmology Telescope (ACT) confirm that, even on the smallest scales, the predictions of the Lambda CDM universe preceded by an epic of inflation are correct. This study extracts data for L modes of The CMB between 500 and 10,000. (For
... Read more »
Sudeep Das, Tobias A. Marriage, & et al. (2010) The Atacama Cosmology Telescope: A Measurement of the Cosmic Microwave Background Power Spectrum at 148 and 218 GHz from the 2008 Southern Survey. E-Print. arXiv: 1009.0847v1
- September 3, 2010
- 08:19 PM
- 1,376 views
In other news: shrinking computer chips, string theory
by Joerg Heber in All That Matters
This week two noteworthy papers have been published that I did not get around to highlight here. In terms of topic they could not be more different, one about a possible new data storage material, and the other one about string theory! The next big thing in computing could be silicon! It is not often [...]... Read more »
Yao, J., Sun, Z., Zhong, L., Natelson, D., & Tour, J. M. (2010) Resistive Switches and Memories from Silicon Oxide. Nano Letters. DOI: 10.1021/nl102255r
L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, & W. Rubens. (2010) Four-qubit entanglement from string theory. Phys.Rev.Lett.105:100507,2010. arXiv: 1005.4915v2
- September 2, 2010
- 04:33 PM
- 617 views
Free Kick Physics, Roberto Carlos Style
by Michael Gutbrod in A Scientific Nature
For all you soccer/football/fútbol/calcio fans out there, you may have been watching the 1997 Confederations Cup match between Brazil and France when Roberto Carlos lined up for a 35 meter (115 ft.), relatively long, free kick. Then you either screamed in unbridled joy or a crying disgust as Carlos appeared to botch the free kick [...]... Read more »
Guillaume Dupeux, Anne Le Goff, David Quéré and Christophe Clanet. (2010) The spinning ball spiral. New Journal of Physics. info:/
- September 2, 2010
- 06:04 AM
- 1,230 views
Solar system might be older than we thought…
by Kelly Oakes in Basic Space
Researchers from Arizona State University have found the oldest solar system object ever discovered. In fact, it’s so old that it formed up to two million years before the solar system did, according to current estimates. It might be time for a rethink of when and how our little place in the Universe came into [...]... Read more »
Audrey Bouvier, & Meenakshi Wadhwa. (2010) The age of the Solar System redefined by the oldest Pb–Pb age of a meteoritic inclusion. Nature Geoscience. info:/10.1038/ngeo941
- September 1, 2010
- 01:20 PM
- 1,414 views
This (Long) Week in the Universe: August 24th – September 1st
by S.C. Kavassalis in The Language of Bad Physics
What have people been talking about this week in high energy physics, astrophysics, gravitation, general relativity and quantum gravity?... Read more »
Lisa J. Kewley, David Rupke, H. Jabran Zahid, Margaret J. Geller, & Elizabeth J. Barton. (2010) Metallicity Gradients and Gas Flows in Galaxy Pairs. arXiv. DOI: 1008.2204
Mayer L, Kazantzidis S, Escala A, & Callegari S. (2010) Direct formation of supermassive black holes via multi-scale gas inflows in galaxy mergers. Nature, 466(7310), 1082-4. PMID: 20740009
Mikhail Gorchtein, Stefano Profumo, & Lorenzo Ubaldi. (2010) Probing Dark Matter with AGN Jets. arXiv. arXiv: 1008.2230v1
J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell, & M. B. Bainbridge. (2010) Evidence for spatial variation of the fine structure constant. arXiv. arXiv: 1008.3907v1
Harold V. Parks, & James E. Faller. (2010) A Simple Pendulum Determination of the Gravitational Constant. Phys. Rev. Let. arXiv: 1008.3203v2
L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, & W. Rubens. (2010) Four-qubit entanglement from string theory. Physical Review Letters. arXiv: 1005.4915v2
- September 1, 2010
- 01:03 PM
- 1,475 views
The thing with graphene transistors
by Joerg Heber in All That Matters
Graphene is one of the hottest research areas in nanotechnology, and it may seem slightly surprising it took me a month to write my first blog post on the topic. That moment has now come, with the advance publication of a Nature paper that presents highly attractive graphene transistor, even though in my humble opinion [...]... Read more »
Liao, L., Lin, Y.-C., Bao, M., Cheng, R., Bai, J., Liu, Y., Qu, Y., Wang, K. L., Huang, Y., & Duan, X. (2010) High-speed graphene transistors with a self-aligned nanowire gate. Nature. DOI: 10.1038/nature09405
- September 1, 2010
- 01:00 PM
- 1,210 views
The "Bad" Language of Physics
by S.C. Kavassalis in The Language of Bad Physics
One of the things I sometimes find myself writing about is the “bad” language used by physicists. Sometimes we say Riemannian when we really should say psuedo-Riemannian, sometimes we call something a metric when it really is a line element – the kind of nitpicky pet-peeves that practically everyone has about literature in their field. Today, I’m going to be talking about the bad language in physics in a totally different context however.... Read more »
Regge, T. (1961) General relativity without coordinates. Il Nuovo Cimento, 19(3), 558-571. DOI: 10.1007/BF02733251
Galassi, M. (1993) Lapse and shift in Regge calculus. Physical Review D, 47(8), 3254-3264. DOI: 10.1103/PhysRevD.47.3254
Kheyfets A, LaFave NJ, & Miller WA. (1990) Null-strut calculus. II. Dynamics. Physical review D: Particles and fields, 41(12), 3637-3651. PMID: 10012308
ALPER ÜNGÖR, & ALLA SHEFFER. (2002) PITCHING TENTS IN SPACE-TIME: MESH GENERATION FOR DISCONTINUOUS GALERKIN METHOD. International Journal of Foundations of Computer Science , 13(2). info:/10.1142/S0129054102001059
- August 31, 2010
- 09:00 PM
- 1,014 views
The Bad Language of Physics
by S.C. Kavassalis in The Language of Bad Physics
One of the things I sometimes find myself writing about is the “bad” language used by physicists. Sometimes we say Riemannian when we really should say psuedo-Riemannian, sometimes we call something a metric when it really is a line element – the kind of nitpicky pet-peeves that practically everyone has about literature in their field. Today, I’m going to be talking about the bad language in physics in a totally different context however.
Teepee Lattices, Future-Pointing Wigwams and Polish Numbers
My secret love for discrete spacetimes comes from a beautiful little sub-field of general relativity (that is experiencing a little bit of a revival right now thanks to loop quantum gravity) called Regge calculus. Regge calculus was a method suggested by John Wheeler and his student, Tullio Regge, in the early 1960s[1], to find approximate solutions to the Einstein Equations. Their basic idea was just to simplify spacetime and see what happened. Instead of having one complicated, curved, four-dimensional Lorentzian manifold to work in, we would piece our spacetime together out of four-dimensional simplices (the higher order word for triangle), that would have, nice, simple, flat interiors, so the entire picture would show curvature, but each individual section would be easy to describe and work in.
Consider this 2D example
This two-dimensional “simplexification” might be able to give you a better idea of the process. Here, we have triangulated a small “curved” surface. The interior of each triangle (a 2-simplex) is a flat, Minkowskian space, and the curvature is manifest at the vertices (0-simplices) where the triangles meet. When we scale this concept up to four dimensions, we end up with flat, Minkowskian, 4-simplices, and then our curvature is contained at the 2-simplices (curvature is now manifest on triangles, not points) where the 4-simplices meet¹.
So that is the basic idea behind Regge calculus – we break up spacetime into simple, triangular, chunks. The implementation of it is where the real difficulties arises.
Evolution
When we think of our favourite formulation of general relativity in the continuum limit, most of us probably think of the ADM formalism (No? I bet you’ve never even turned your alarm clock off in your sleep thinking that you still needed the lapse and shift to know what time it really was).
Creating a 4-geometry by "sandwiching" two 3-geometries (this is a "sandwich" of infinitesimal thickness): The 4-metric (ie. geometry) of the full 4D spacetime (which is what GR is all about) depends on the lapse and shift of the connectors ("sandwich filling") between the two 3-geometries as well as the 3-metric (the geometry of the "bread"). N is the lapse function (to get the proper time between the upper and lower surfaces you need Ndt) and Nⁱ is the shift function.
The ADM formalism says that we should foliate (slice) our continuous spacetime into three-dimensional spacelike surfaces that we can label by their time coordinates and then define our dynamics from there. It is an elegant, simple and well studied idea that is exceptionally powerful, so logically, it seems like a good set of concepts to work into Regge calculus.
Now, the original idea of Regge calculus was to model the four-dimensional Lorentzian manifold of spacetime by simplices (the Regge analogue of the tangent space), that have flat, Minkowskian interiors (although in his original paper [1] he actually used simplices with flat, Euclidean interiors, which effectively removed the ‘physics’ from the problem), but this didn’t leave much room for describing dynamics. To make Regge calculus more like our beloved ADM formalism, we approximate our differential three-manifolds² (our spacelike slices) by a collection of simplices, which allows us to preserve many important topological properties, while giving us room to describe dynamics (like the evolution of the universe, for example).
Simple Regge Evolution: (a) base triangulation (b) each vertex evolved (c) all connections to rightmost evolved vertex made.
So, the basic idea of evolution in Regge calculus is to take each vertex at a time t and “evolve” it up to some time t + dt (the lines connecting upper and lower vertices are in spirit with our connectors from the ADM foliation). Since we also want to maintain some sense of reasonableness in our model (ie. that spacetime is locally path-connected, and is thus also connected), we also connect each initial vertex to each evolved vertex, so, for a single 2-simplex, we get:
2-Simplex Evolved: A lattice in 2 + 1 dimensions.
It’s a pretty simple idea, but, if you made it through all of the above, you probably found yourself asking, “But how do we actually know how to evolve a vertex? How do we know how long/at what angle to make all of those connections?”. That’s a very good question, and one that doesn’t have a definite answer at this point (despite what certain individuals that have certain well known Regge evolution schemes named after them might think). There is a sizable body of work dedicated to figuring out how to make those connections as physical as possible.
One such example came from Mark Galassi, in 1992, in his PhD dissertation on this very topic in which we were introduced to the concept of a teepee lattice [2].
Teepee (public domain image from Wikipedia)
From Collins English Dictionary:
teepee, noun: a cone-shaped tent of animal skins used by certain North American Indians [A]
It’s pretty obvious where this is going…
From Galassi's "Lattice Geometrodynamics"
The resemblance is uncanny(ish).
What’s nice about Galassi’s “teepees” is that they make the connection to ADM’s lapse and shift more obvious (if you’re familiar with both Regge calculus and the ADM formalism, that is) [3]. Other than that, it’s a rather funny name. At least in Canada, gratuitous use of the word “teepee” makes a lot of people cringe. In many circles, “teepee” is considered to be politically incorrect, because it has been so overused as part of stereotyping the Native “Indians”. Political correctness be damned, says the physicist, it’s a fairly illustrative name for what’s going on in Regge evolution. Is it offensive to some? Maybe.
Interestingly, this Native American naming concept didn’t come from Galassi, but came from Kheyfets et al. a few years earlier during the “Null Strut calculus” craze, with the introduction of spacetime wigwams [4]. Null Strut calculus is a variant of Regge calculus that insists that the connection between a &... Read more »
Regge, T. (1961) General relativity without coordinates. Il Nuovo Cimento, 19(3), 558-571. DOI: 10.1007/BF02733251
Galassi, M. (1993) Lapse and shift in Regge calculus. Physical Review D, 47(8), 3254-3264. DOI: 10.1103/PhysRevD.47.3254
Kheyfets A, LaFave NJ, & Miller WA. (1990) Null-strut calculus. II. Dynamics. Physical review D: Particles and fields, 41(12), 3637-3651. PMID: 10012308
ALPER ÜNGÖR, & ALLA SHEFFER. (2002) PITCHING TENTS IN SPACE-TIME: MESH GENERATION FOR DISCONTINUOUS GALERKIN METHOD. International Journal of Foundations of Computer Science , 13(2). info:/10.1142/S0129054102001059
- August 31, 2010
- 01:51 PM
- 877 views
Seeing double: perhaps is simply optical diplopia
by Pablo Artal in Optics confidential
Changes in the optics of the eye can produce double or even multiple images... a real case is explained as an example and more... ... Read more »
Pérez, G., Abenza, S., De Casas, A., Marín, J., & Artal, P. (2010) Cause of Monocular Diplopia Diagnosed by Combining Double-pass Retinal Image Assessment and Hartmann-Shack Aberrometry. Journal of Refractive Surgery, 26(4), 301-304. DOI: 10.3928/1081597X-20100218-05
- August 31, 2010
- 07:21 AM
- 639 views
Solar cells brought into shape
by Joerg Heber in All That Matters
Solar energy is a huge market and any improvement to the efficiency of solar cells has a significant impact. In 2008, worldwide photovoltaic solar energy production was about 5 gigawatts, and this is expected to rise to 15 gigawatts in 2015. To put this figure in context, a nuclear reactor produces around 1 to 1.5 [...]... Read more »
Ferry, V., Verschuuren, M., Li, H., Verhagen, E., Walters, R., Schropp, R., Atwater, H., & Polman, A. (2010) Light trapping in ultrathin plasmonic solar cells. Optics Express, 18(S2). DOI: 10.1364/OE.18.00A237
Atwater, H., & Polman, A. (2010) Plasmonics for improved photovoltaic devices. Nature Materials, 9(3), 205-213. DOI: 10.1038/nmat2629
- August 30, 2010
- 03:44 PM
- 994 views
Hacking Commercial Quantum Cryptography Systems by Illumination
by Olexandr Isayev in olexandrisayev.com
Quantum hackers have performed the first 'invisible' attack on two commercial quantum cryptographic systems. By using lasers on the systems — which use quantum states of light to encrypt information for transmission — they have fully cracked their encryption keys, yet left no trace of the hack.... Read more »
Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., & Makarov, V. (2010) Hacking commercial quantum cryptography systems by tailored bright illumination. Nature Photonics. DOI: 10.1038/NPHOTON.2010.214
Feihu Xu, Bing Qi, & Hoi-Kwong Lo. (2010) Experimental demonstration of phase-remapping attack in a practical quantum key distribution system. Preprint. arXiv: 1005.2376v1
- August 30, 2010
- 12:49 PM
- 1,380 views
Indirect Excitation Control: Ultrafast Quantum Gates for Single Atomic Qubits
by Chad Orzel in Uncertain Principles
Last week, John Baez posted a report on a seminar by Dzimitry Matsukevich on ion trap quantum information issues. In the middle of this, he writes:
Once our molecular ions are cold, how can we get them into specific desired states? Use a mode locked pulsed laser to drive stimulated Raman transitions.
Huh? As far as I can tell, this means "blast our molecular ion with an extremely brief pulse of light: it can then absorb a photon and emit a photon of a different energy, while itself jumping to a state of higher or lower energy."
I saw this, and said "Hey, that's a good topic for a blog post." And on Friday, the new issue of Physical Review Letters included a new paper on just this topic (arxiv version for those without subscription access), making it a good topic for a ResearchBlogging post. So,
So, what's this all about? The paper reports on a new way of moving atoms from one state to another much faster than is possible with more typical methods. This is potentially useful for speeding up the operation of a quantum computer.
Transition speeds are critically important for quantum computing, because all quantum information processing systems are subject to some interactions with the environment that will eventually destroy the quantum character of the information through the process known as "decoherence." If you do a really good job, you can get decoherence times that are measured in seconds, which sets an upper limit on the number of operations you can do with a simple system before decoherence kills you (you can do quantum error correction to extend that, but then things start to get complicated). If you can do your state-change operations in 50 picoseconds rather than tens or hundred of microseconds, you can pack a lot more computing into that same amount of time.
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Campbell, W., Mizrahi, J., Quraishi, Q., Senko, C., Hayes, D., Hucul, D., Matsukevich, D., Maunz, P., & Monroe, C. (2010) Ultrafast Gates for Single Atomic Qubits. Physical Review Letters, 105(9). DOI: 10.1103/PhysRevLett.105.090502
- August 29, 2010
- 04:45 PM
- 694 views
A Mathematical Description of Cell Aggregate Mechanical Deformation
by Michael Long in Phased
Luigi Preziosi (Politecnico di Torino, Italy) and coworkers have developed a mathematical model for the mechanical stress experienced by cell aggregates, relevant to cellular function in normal health (blood flow) and disease (cancer). This news feature was written on August 29, 2010.... Read more »
Preziosi, L., Ambrosi, D., & Verdier, C. (2010) An elasto-visco-plastic model of cell aggregates. Journal of Theoretical Biology, 262(1), 35-47. DOI: 10.1016/j.jtbi.2009.08.023
- August 29, 2010
- 10:36 AM
- 1,903 views
the birth of a supermassive monster, revisited
by Greg Fish in weird things
We know that black holes can grow to become absolutely enormous in size, tipping the scales at billions and billions of times the mass of our sun. The numbers involved make the gravitational monsters in question very hard to visualize, and pose a big mystery. Did they form from the remnants of the first stars [...]... Read more »
Mayer, L., Kazantzidis, S., Escala, A., & Callegari, S. (2010) Direct formation of supermassive black holes via multi-scale gas inflows in galaxy mergers. Nature, 466(7310), 1082-1084. DOI: 10.1038/nature09294
- August 27, 2010
- 07:34 AM
- 1,220 views
Snapshots of magnetic fields
by Joerg Heber in All That Matters
In the absence of GPS, a compass is the best option to find your way around. However, although the earth’s magnetic field is a great way to find your own position, doing the reverse, measuring magnetic fields with a high accuracy — on an atomic scale — remains a challenge. Sure, there are electron microscopes, which are [...]... Read more »
Park, H., Baskin, J., & Zewail, A. (2010) 4D Lorentz Electron Microscopy Imaging: Magnetic Domain Wall Nucleation, Reversal, and Wave Velocity. Nano Letters, 2147483647. DOI: 10.1021/nl102861e
- August 26, 2010
- 12:42 PM
- 1,535 views
Measuring Gravity: Ain't Nothin' but a G Thing
by Chad Orzel in Uncertain Principles
There's a minor scandal in fundamental physics that doesn't get talked about much, and it has to do with the very first fundamental force discovered, gravity. The scandal is the value of Newton's gravitational constant G, which is the least well known of the fundamental constants, with a value of 6.674 28(67) x 10-11 m3 kg-1 s-2. That may seem pretty precise, but the uncertainty (the two digits in parentheses) is scandalously large when compared to something like Planck's constant at 6.626 068 96(33) x 10-34 J s. (You can look up the official values of your favorite fundamental constants at this handy page from NIST, by the way...)
To make matters worse, recent measurements of G don't necessarily agree with each other. In fact, as reported in Nature, the most recent measurement, available in this arxiv preprint, disagrees with the best previous measurement by a whopping ten standard deviations, which is the sort of result you should never, ever see.
This obviously demands some explanation, so:
What's the deal with this? I mean, how hard can it be to measure gravity? You drop something, it falls, there's gravity. It's easy to detect the effect of the Earth's gravitational pull, but that's just because the Earth has a gigantic mass, making the force pretty substantial. If you want to know the precise strength of gravity, though, which is what G characterizes, you need to look at the force between two smaller masses, and that's really difficult to measure.
Why? I mean, why can't you just use the Earth, and measure a big force? If you want to know the force of gravity to a few parts per million, you would need to know the mass of the Earth to better than a few parts per million, and we don't know that. A good measurement of G requires you to use test masses whose values you know extremely well, and that means working with smaller masses. Which means really tiny forces-- the force between two 1 kg masses separated by 10 cm is 6.6 x 10-9 N, or about the weight of a single cell.
OK, I admit, that's a bit tricky. So how do they do it? There are four papers cited in the Nature news article. I'll say a little bit about each of them, and how they figure into this story.
Read the rest of this post... | Read the comments on this post...... Read more »
Gundlach, J., & Merkowitz, S. (2000) Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback. Physical Review Letters, 85(14), 2869-2872. DOI: 10.1103/PhysRevLett.85.2869
Schlamminger, S., Holzschuh, E., Kündig, W., Nolting, F., Pixley, R., Schurr, J., & Straumann, U. (2006) Measurement of Newton’s gravitational constant. Physical Review D, 74(8). DOI: 10.1103/PhysRevD.74.082001
Luo, J., Liu, Q., Tu, L., Shao, C., Liu, L., Yang, S., Li, Q., & Zhang, Y. (2009) Determination of the Newtonian Gravitational Constant G with Time-of-Swing Method. Physical Review Letters, 102(24). DOI: 10.1103/PhysRevLett.102.240801
Harold V. Parks, & James E. Faller. (2010) A Simple Pendulum Determination of the Gravitational Constant. Physical Review Letters (accepted). arXiv: 1008.3203v2
- August 25, 2010
- 11:50 PM
- 1,034 views
Solid-state lighting: may not be magic bullet for energy savings
by Olexandr Isayev in olexandrisayev.com
The importance of artificial light to society has long been recognized with the utilization of fire thought of as the quintessential human invention. Now scientists have found that emerging, more energy efficient lighting technologies could be the key to a better quality of life. New research published on August 19 , in a special issue [...]... Read more »
Tsao, J., Saunders, H., Creighton, J., Coltrin, M., & Simmons, J. (2010) Solid-state lighting: an energy-economics perspective. Journal of Physics D: Applied Physics, 43(35), 354001. DOI: 10.1088/0022-3727/43/35/354001
- August 25, 2010
- 09:38 AM
- 1,435 views
Melting Simulated Insulators
by Chad Orzel in Uncertain Principles
The Joerg Heber post that provided one of the two papers for yesterday's Hanbury Brown Twiss-travaganza also included a write-up of a new paper in Nature on Mott insulators, which was also written up in Physics World.
Most of the experimental details are quite similar to a paper by Markus Greiner's group I wrote up in June: They make a Bose-Einstein Condensate, load it into an optical lattice, and use a fancy lens system to detect individual atoms at sites of the lattice. This lattice can be prepared in a "Mott insulator" state, where each site is occupied by a definite number of atoms. As the total number of atoms in the BEC increases, the number per site increases, and forms a set of "shells" with, say, exactly two atoms per site in the center, surrounded by a shell of one or two atoms per site, surrounded by a shell of exactly one atom per site, and so on.
The thing that sets this paper apart is a temperature-dependent effect, which appears as Figure 5, which I reproduce here:
So, what's this figure, besides really complicated and orange-y? It is pretty orange, isn't it? SteelyKid came downstairs while I was reading it, looked at the image on screen, and said "Fire, hot! Careful!"
This picture shows the "melting" of the Mott insulator as the temperature is increased. The three images at the top are pictures of the trapped atoms at different temperatures, increasing from left to right. You can see that the shells get less regular as the temperature increases-- there's still a clear shell structure in part c, but it's not as distinct as part a.
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Sherson, J., Weitenberg, C., Endres, M., Cheneau, M., Bloch, I., & Kuhr, S. (2010) Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature. DOI: 10.1038/nature09378
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