Observing comet 21P/Giacobini-Zinner from the city center of Berlin proved to be challenging. I put up my 130 mm Newton telescope in the living room to catch a glimpse of Capella above the roof of the adjacent building around 3 am on Sept. 5. Not far from Capella, 21P/Giacobini-Zinner was barely visible, but a 10s exposure with the smartphone camera revealed the fuzzy coma of the comet. Since the comet is close to perihelion it moves swiftly across the stars within 1 hour, as seen in the 3 frames animation from my observations at 02:39, 03:04 and 03:14 local time.

It would be nice to knock out the street light or to block the city lights, but it turns out the new LED street lights are emitting with a broad spectrum across to the entire visible range.
I analyzed the spectra with a hand-build spectrometer from Astromedia to evaluate if a UHC filter might help, but unfortunately this is not the case.

The hopes are high for Christmas comet 46P/Wirtanen to be much brighter and better visible at the end of the year.
Wirtanen gets this time very close to Earth (0.071 AU, no danger of collision though) and will be extensively studied. For an overview of our professional work about comet 67P/Churyumov-Gerasimenko have a look at this ZIB feature.
Current comet light curves are compiled on the excellent COBS.SI webpage.

The lunar eclipse yesterday (July 27, 2018) provided an excellent opportunity to take pictures and to study the color of Earth’s shadow. I snapped some images with my smartphone (Huwai Mate 10 Pro, Leica camera) through a 130mm Newton telescope. The moon was pretty low and within the city barely visible. Just after leaving the central shadow cone, the bluish color of the umbra/penumbra region became very noticeable. Today, I analyzed the picture a bit and drew the relative red/green/blue components along the shadow. Indeed blue wins within a narrow stripe bordering the penumbra. According to the literature, this is a signature of the ozone layer (which absorbs orange and red light, while blue passes and arrives at the moon).

The Hayabusa 2 target asteroid (162173) Ryugu displays a strange diamond like shape, at least from the sunny side explored so far.
Time for me to reuse the code developed for comet 67P/Churyumov-Gerasimenko and to compute the gravitational potential (with added centrifugal term, assuming a 7 h rotation period). The rotation weakens the gravitational potential in the equator regions. Otherwise on the triangular faces the potential minima are close to the centers of the triangle.

The Rosetta spacecraft has come to rest on comet 67P/Churyumov-Gerasimenko. The comet is retreating from the sun, but the analysis of the scientific data is an ongoing endeavour with many discoveries yet to be made. As described before, the coma structure of 67P/C-G followed a surprisingly predictable pattern: dust is emitted from the entire sunlit surface and later in space forms intricate dust bundles and rays, directly reflecting the surface topography. The rotation of the nucleus leads to a bending of the dust trajectories which allows us to read of the velocity of the particles: around 3 m/s at distances 2-3 km from the surface. Our detailed prediction of the dust coma from May 2015 recently appeared in Advances in Physics: X, 3(1), 1404436, 2018 (open access), where we compare the model with Rosetta images such as these ones: (1, 2, 3).

The dust is propelled by the gas emitted from the surface by sublimation processes. It is a non-trivial task to back-out the surface ice distribution from the measured gas densities at Rosetta’s orbit via the COmetary Pressure Sensor (COPS), built by the ROSINA team (PI: Prof. Kathrin Altwegg, University of Bern, Switzerland). Using an analytical ansatz for the gas distribution, we managed to retrieve the “best-fit” distribution of the gas sources on the surface (T. Kramer, M. Läuter, M. Rubin, K. Altwegg: Seasonal changes of the volatile density in the coma and on the surface of comet 67P/Churyumov-GerasimenkoMonthly Notices of the Royal Astronomical Society, 469, S20, 2017).
Interestingly, the sources of higher gas emission are linked to reported short-lived outburst locations. A unified picture and modelling of gas and dust emission holds important clues about the composition of the cometary surface and we continue our investigation in that direction.

I was lucky to witness the total eclipse under perfect weather conditions in Salem, Oregon. Salem was very accessible with public transport (Amtrak trains arriving before the eclipse, special stop to accommodate people getting off to viewing locations). While this picture does little justice to the real impression of the “inverted Sun” (dark disk with bright corona around it), it shows the irregular “triangular” shape of the corona with three rays poking out. From Earth it is normally impossible to see the corona due to scattered light in the atmosphere, but solar satellites monitor the Sun constantly. I followed NASA’s example and overlaid a space satellite image with my image (after rotating it such that the ecliptic plane [conveniently indicated by Venus] aligns horizontally). If you want to try something similar yourself, I recommend the helioviewer website for easy selection and download of the solar satellite imagery. To match the scales on your pictures: Venus is 34.3 degrees away from the Sun, corresponding to about 130 solar radii. We are presently close to a minimum of solar activity and thus the corona less roundish.

The photosynthetic apparatus of the green sulfur bacteria (chlorobium tepdidum) spurred a long lasting discussion about quantum coherence in biology, mostly focused on its subunit, the Fenna Matthews Olson complex. An account of the discovery of the FMO Protein is given by Olson in Photosynth. Res 80, 2004.

Recent experiments by Dostál, Pšenčík, and Zigmantas (Nat. Chem 8, 2016) show measured time and frequency resolved 2d-spectra of the whole photosynthetic apparatus. These results allow one to trace the energy flow from the antenna down to the reaction center and relate it to theoretical models.

In our article [Kramer & Rodriguez Sci. Reports 7, 2017] , open access] we provide a model of the experimental results using the open quantum system dynamics code described previously. In addition we show how the different pathways in 2D spectroscopy (ground state bleaching, stimulated emission, and excited state absorption) affect the spectra and lead to shifts of “blobs” down from the diagonal places. This allows us to infer the effective coupling of the antenna part to the FMO complex and to assess the relative orientations of the different units. The comparison of theory and experimental result is an good test of our current understanding of the physical processes at work.

Together with Christoph Kreisbeck and Rafael A Molina I have contributed a blog entry to the News and Views section of the Journal of Physics describing our most recent work on Aharonov-Bohm interferometer with an imbedded quantum dot (article, arxiv). Can you spot Schrödinger’s cat in the result?

With ever shrinking sizes of electronic transistors, the quantum mechanical nature of electrons becomes more visible. For instance two electrons with the same spin orientation and velocities cannot be at the same location (Pauli blocking). At low temperatures, electronic waves travel many mircometers completely coherently, only reflected by the geometric of the confinement. A tight confinement leads to larger separation of quantized energy levels and restricts the lateral spread of the electrons to specific eigenmodes of a nanowire.

The distribution of the electronic current into various is then given by the geometrical scattering properties of the device interior, which are conveniently computed using wave packets. The ballistic electrons entering a nanodevice carry along charge and thermal energy. The maximum amount of thermal energy Q per time which can be transported through a single channel between two reservoirs of different temperatures is limited to Q ≤ π^{2} k_{B}^{2} (T_{2}^{2}-T_{1}^{2})/(3h) [h denotes Planck’s and k_{B} Boltzmann’s constant]. This has implications for computing devices, since this restricts the cooling rate (Pendry 1982).

In a collaboration with the novel materials group at Humboldt University (Prof. S.F. Fischer, Dr. C. Riha, Dr. O. Chiatti, S. Buchholz) and using wafers produced in the lab of A. Wieck, D. Reuter (Bochum, Paderborn) C. Kreisbeck and I have compared theoretical expectations with experimental data for the thermal energy and charge currents in multi-terminal nanorings (AIP Advances 2016, open access). Our findings highlight the influence of the device geometry on both, charge and thermal energy transfer and demonstrate the usefulness of the time-dependent wave-packet algorithm to find eigenstates over a whole range of temperature.

Have you ever thought of arXiv moderation in astro-ph being a problem? Did you experience a >5 months delay from submission of your pre-print to the arXiv to being publicly visible? Did this happen without any explanation or reaction from the arXiv moderators despite the same article being published after peer review in the Astrophysical Journal Letters?

Chances are high that your answer is no, to be precise the odds are 81404/81440=99.9558 percent that this did not happen to you. Lucky you! Now let me tell about the other 36/81440=0.0442043 percent. My computer based analysis of the last 80,000 deposited arxiv:astro-ph articles shows interesting results about the moderation patterns in astrophysics. To repeat the analysis

get the arXiv metadata, which is available (good!) from the arxiv itself. I used the excellent metha tools from Martin Czygan to download all metadata from the astro-ph and quant-ph sections since 5/2014.

parse the resulting 200 MB XML file, for instance with Mathematica. To get the delay from submission to arXiv publication, I took the time difference between the submission date stamp (oldest XMLElement[{http://purl.org/dc/elements/1.1/, date}) and the arXiv identifier, which encodes the year and month of public visibility.

Example: the article arxiv:1604.00876 went public in April 2016, 5 months after submission to the arXiv (November 5, 2015) and publication in the Astrophysical Journal Letters (there total processing time from submission to online publication, including peer review 1.5 months).

The analysis shows different patterns of moderation for the two sections I considered, quant-ph and astro-ph. It reveals problematic moderation effects in the arXiv astro-ph section:

Completely suitable articles are blocked, mostly peer reviewed and published for instance in the Astrophysical Journal, Astrophysical Journal Letters, Monthly Notices of the Royal Astronomical Society.

This might indicate a biased moderation toward specific persons and subjects. In contrast to scientific journals with their named editors, the arXiv moderation is opaque and anonymous. The metadata analysis shows that the moderation of the physics:astro-ph and physics:quant-ph use very different guidelines, with astro-ph having a strong bias to block valid contributions.

It makes the astro-ph arXiv less usable as a medium for rapid dissemination of cutting edge research via preprints.

This hurts careers, citation histories, and encourages plagiarism. New scientific findings are more easily plagiarized by other groups, since no arXiv time-stamped preprint establishes the precedence.

If we, the scientists, want a publicly funded arXiv we must ensure that it is operated according to scientific standards which serve the public. This excludes biased blocking of valid and publicly funded research.

Finally, the arXiv was not put in place to be a backup server for all journals, but rather to provide a space to share upcoming scientific publications without months of delay.

I will be happy to share comments I receive about similar cases. I am not talking about dubious articles or non-scientific theories, but about standard peer-reviewed contributions published in established physics journals, which should be on the astrophysical preprint arXiv.

Here follows the list of all articles which were delayed by more than 3 months from arxiv:physics/astro-ph (out of a total of 81,440 deposited articles) and if known where the peer reviewed article got published. I cannot exclude other factors besides moderation for the delay, but can definitely confirm incorrect moderation being the cause for the 2 cases I have experienced. Interestingly the same analysis on arxiv:physics/quant-ph did not reveal such a moderation bias of peer reviewed articles. This gives hope that the astrophysical section could recover and return to 100 percent flawless operation. Then the arXiv fulfils its own pledge on accountability and on good scientific practices (principles of the arXiv’s operation).

In general, any cometary activity is difficult to predict and many comets are known for sudden changes in brightness, break ups and simple disappearances. Fortunately, the Rosetta target comet 67P/Churyumov-Gerasiminko (67P/C-G) is much more amendable to theoretical predictions. The OSIRIS and NAVCAM images show light reflected from a highly structured dust coma within the space probe orbit (ca 20-150 km).

Is is possible to predict the dust coma and tail of comets?

Starting in 2014 we have been working on a dust forecast for 67P/C-G, see the previous blog entries. We had now the chance to check how well our predictions hold by comparing the model outcome to a image sequence from the OSIRIS camera during one rotation period of 67P/C-G on April 12, 2015, published by Vincent et al in A&A 587, A14 (2016) (arxiv version, there Fig. 13).

Our results appeared in Kramer & Noack, Astrophysical Journal Letters, 823, L11 (preprint, images). We obtain a surprisingly high correlation coefficient (average 80%, max 90%) between theory and observation, if we stick to the following minimal assumption model:

dust is emitted from the entire sunlit nucleus, not only from localized active areas. We refer to this as the “homogeneous activity model”

dust is entering space with a finite velocity (on average) along the surface normal. This implies that close to the surface a rapid acceleration takes place.

photographed “jets” are highly depending on the observing geometry:
if multiple concave areas align along the line of sight, a high imaged intensity results, but is not necessarily the result of a single main emission source. As an exemplary case, we analysed the brightest points in the Rosetta image taken on April 12, 2015, 12:12 and look at all contributing factors along the line of sight (yellow line) from the camera to the comet. The observed jet is actually resulting from multiple sources and in addition from contributions from all sunlit surface areas.

What are the implications of the theoretical model?

If dust is emitted from all sunlit areas of 67P/C-G, this implies a more homogeneous surface erosion of the illuminated nucleus and leaves less room for compositional heterogeneities. And finally: it makes the dust coma much more predictable, but still allows for additional (but unpredictable) spontaneous, 20-40 min outbreak events. Interestingly, a re-analysis of the comet Halley flyby by Crifo et al (Earth, Moon, and Planets 90 227-238 (2002)) also points to a more homogeneous emission pattern as compared to localized sources.

The conference focuses on theoretical and experimental new developments in the field of photoactive molecules. If you are working in this field, we encourage you to attend the meeting. Find out more and who already agreed to participate at photosynthesis2016.wordpress.com.

Please submit your abstract within the next two weeks by filling out the linked template from the conference site.

We (the scientific organizers, Tobias Kramer and Jörg Megow) are grateful for the full support by the Wilhelm and Else Heraeus Foundation, which covers the accommodation for all successful applicants.

Comet 67P/Churyumov–Gerasimenko is past its perihelion and is currently visible in telescopes in the morning hours. The picture is taken from Tenerife by the Bradford robotic telescope, where I submitted the request. The tail is extending hundred thousands kilometers into space and consists of dust particles emitted from the cometary nucleus, which measures just a few kilometers. In a recent work just published in the Astrophysical Journal Letters (arxiv version), we have explored how dust, which does not make it into space, is whirling around the cometary nucleus. The model assumes that dust particles are emitted from the porous mantle and hover over the cometary surface for some time (<6h) and then fall back on the surface, delayed by the gas drag of gas molecules moving away from the nucleus. As in the predictions for the cometary coma discussed previously, we are sticking to a minimal-assumption model with a homogeneous surface activity of gas and dust emission.

The movements of 40,000 dust particles are tracked and the average dust transport within a volumetric grid with 300 m sized boxes is computed. Besides the gas-dust interaction, we do also incorporate the rotation of the comet, which leads to a directional transport.
The Rosetta mission dropped Philae over the small lobe of 67P/C-G and Philae took a sequence of approach images which reveal structures resembling wind-tails behind boulders on the comet. This allowed Mottola et al (Science 349.6247 (2015): aab0232) to derive information about the direction of impinging particles which hit the surface unless sheltered by the boulder. Our model predicts a dust-transport inline with the observed directions in the descent region, it will be interesting to see how wind-tails at other locations match with the prediction. We put an interactive 3d dust-stream model online to visualize the dust-flux predicted from the homogeneous surface model.

Comet 67P/Churyumov–Gerasimenko has passed its nearest distance to the sun and its tail has been observed from earth. The comet emits dust and displays spectacular but short-lived outbreaks of localized jet activity. Very detailed OSIRIS pictures of the near-surface dust emission ready for stereo viewing have been posted by Brian May. The pictures also allow one to have a look at the prediction from the homogeneous dust emission model discussed previously. When you direct your attention in Brian May’s pictures to the background activity, you find very similar patterns as expected from the homogenous emission model. This activity is dimmer but steadily blowing off dust from the nucleus. Matthias Noack and I have generated and uploaded a visualization of the dust data obtained from the homogeneous activity model. In contrast to a localized activity models, collimated jets arise from a bundle of co-propagating dust trajectories emanating from concave surface areas. The underlying topographical shape model is a uniform triangle remesh of Mattias Malmer’s excellent work based on the release of Rosetta’s NAVCAM images via the Rosetta blog. The following video takes you on a flight around 67P/C-G, with 16 hours condensed into 90 sec.

The video is a side-by-side stereoscopic 3d rendering of 67P/Churyumov–Gerasimenko and the dust cloud, which can be viewed in 3d with a simple cardboard viewer. While the observer is encircling the nucleus, day and night passes and different parts of the comet are illuminated.

In the homogeneous activity model each sunlit triangle emits dust with an initial velocity component along the surface normal. Then dust is additionally dragged along within the outwards streaming gas, which is also incorporated in the model. In contrast to compact dust particles, the gas molecules are diffusing also in lateral directions and thus gas is not helping to collimate jets by itself. The Rosetta mission with its long term observation program offers fascinating ways to perform a reality check on various models of cometary activity, which differ considerably in the underlying physics and assumptions about the original distribution and lift-off conditions of the dust eventually forming the beautiful tails of comets.

Knowledge of GPGPU techniques is helpful for rapid model building and testing of scientific ideas. For example, the beautiful pictures taken by the ESA/Rosetta spacecraft of comet 67P/Churyumov–Gerasimenko reveal jets of dust particles emitted from the comet. Wouldn’t it be nice to have a fast method to simulate thousands of dust particles around the comet and to find out if already the peculiar shape of this space-potato influences the dust-trajectories by its gravitational potential? At the Zuse-Institut in Berlin we joined forces between the distributed algorithm and visual data analysis groups to test this idea. But first an accurate shape model of the comet 67P C-G is required. As published in his blog, Mattias Malmer has done amazing work to extract a shape-model from the published navigation camera images.

Starting from the shape model by Mattias Malmer, we obtain a re-meshed model with fewer triangles on the surface (we use about 20,000 triangles). The key-property of the new mesh is a homogeneous coverage of the cometary surface with almost equally sized triangle meshes. We don’t want better resolution and adaptive mesh sizes at areas with more complex features. Rather we are considering a homogeneous emission pattern without isolated activity regions. This is best modeled by mesh cells of equal area. Will this prescription yield nevertheless collimated dust jets? We’ll see…

To compute the gravitational potential of such a surface we follow this nice article by JT Conway. The calculation later on stays in the rotating frame anchored to the comet, thus in addition the centrifugal and Coriolis forces need to be included.

To accelerate the method, OpenCL comes to the rescue and lets one compute many trajectories in parallel. What is required are physical conditions for the starting positions of the dust as it flies off the surface. We put one dust-particle on the center of each triangle on the surface and set the initial velocity along the normal direction to typically 2 or 4 m/s. This ensures that most particles are able to escape and not fall back on the comet.

To visualize the resulting point clouds of dust particles we have programmed an OpenGL visualization tool. We compute the rotation and sunlight direction on the comet to cast shadows and add activity profiles to the comet surface to mask out dust originating from the dark side of the comet.

This is what we get for May 3, 2015. The ESA/NAVCAM image is taken verbatim from the Rosetta/blog.

Read more about the physics and results in our arxiv article T. Kramer et al.: Homogeneous Dust Emission and Jet Structure near Active Cometary Nuclei: The Case of 67P/Churyumov-Gerasimenko (submitted for publication) and grab the code to compute your own dust trajectories with OpenCL at github.org/noma/covis

The following post is contributed by Peter Kramer.

The new Planck data on the cosmic microwave background (CMB) has come in. For cosmic topology, the data sets contain interesting information related to the size and shape of the universe. The curvature of the three-dimensional space leads to a classification into hyperbolic, flat, or spherical cases. Sometimes in popular literature, the three cases are said to imply an inifinite (hyperbolic, flat) or finite (spherical) size of the universe. This statement is not correct. Topology supports a much wider zoo of possible universes. For instance, there are finite hyperbolic spaces, as depicted in the figure (taken from Group actions on compact hyperbolic manifolds and closed geodesics, arxiv version). The figure also shows the resulting geodesics, which is the path of light through such a hyperbolic finite sized universe. The start and end-points must be identified and lead to smooth connection.

Recent observational data seem to suggest a spherical space. Still, it does not resolve the issue of the size of the universe.
Instead of a fully filled three-sphere, already smaller parts of the sphere can be closed topologically and thus lead to a smaller sized universe. A systematic exploration of such smaller but still spherical universes is given in my recent article Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis.
In physics, it is important to give specific predictions for observations of the topology, for instance by predicting the ratio of the different angular modes of the cosmic microwave background. It is shown that this is indeed the case and for instance in a cubic (still spherical!) universe, the ratio of 4th and 6th multipole order squared are tied together in the proportion 7 : 4, see Table 5. On p. 35 of ( the Planck collaboration article) the authors call for models yielding such predictions as possible explanations for the observed anisotropy and the ratio of high and low multipole moments.

From time to time I get asked about the implications of the Pauli exclusion principle for quantum mechanical wave-packet simulations.
I start with the simplest antisymmetric case: a two particle state given by the Slater determinant of two Gaussian wave packets with perpendicular directions of the momentum:
φ_{a}(x,y)=e^{-[(x-o)2+(y-o)2]/(2a2)-ikx+iky} and φ_{b}(x,y)=e^{-[(x+o)2+(y-o)2]/(2a2)+ikx+iky}
This yields the two-electron wave function
ψ_{a}(x1,y1,x2,y2)=φ_{a}(x1,y1)*φ_{b}(x2,y2)-φ_{a}(x2,y2)*φ_{b}(x1,y1)
The probability to find one of the two electrons at a specific point in space is given by integrating the absolute value squared wave function over one coordinate set.
The resulting single particle density (snapshots at specific values of the displacement o) is shown in the animation for two different values of the momentum k (we assume that both electrons are in the same spin state).
For small values of k the two electrons get close in phase space (that is in momentum and position). The animation shows how the density deviates from a simple addition of the probabilities of two independent electrons.
If the two electrons differ already by a large relative momentum, the distance in phase space is large even if they get close in position space. Then, the resulting single particle density looks similar to the sum of two independent probabilities.
The probability to find the two electrons simultaneously at the same place is zero in both cases, but this is not directly visible by looking at the single particle density (which reflects the probability to find any of the electrons at a specific position).
For further reading, see this article [arxiv version].

I will resume posting about algorithm development for computational physics. To put these efforts in a more general context, I start with some observation about the current publication ranking model and explore alternatives and supplements in the next posts.

Working in academic institutions involves being part of hiring committees as well as being assessed by colleagues to measure the impact of my own and other’s scientific contributions.
In the internet age it has become common practice to look at various performance indices, such as the h-index, number of “first author” and “senior author” articles. Often it is the responsibility of the applicant to submit this data in electronic spreadsheet format suitable for an easy ranking of all candidates. The indices are only one consideration for the final decision, albeit in my experience an important one due to their perceived unbiased and statistical nature. Funding of whole university departments and the careers of young scientists are tied to the performance indices.

I did reflect about the usefulness of impact factors while I collected them for various reports, here are some personal observations:

A high h-index can be a sign of a narrow research field, since the h-index is best built up by sticking to the same specialized topic for a long time and this encourages serialised publications. I find it interesting that on the other hand important contributions have been made by people working outside the field to which they contributed. The discovery of three-dimensional quasicrystals discussed here provides a good example. The canonical condensed matter theory did not envision this paradigmatic change, rather the study of group theoretical methods in nuclear physics provided the seeds.

The full-text search provided by the search engines offers fascinating options to scan through previously forgotten chapters and books, but it also bypasses the systematic classification schemes previously developed and curated by colleagues in mathematics and theoretical physics. It is interesting to note that for instance the AMS short reviews are not done anonymously and most often are of excellent quality. The non-curated search on the other hand leads to a down-ranking of books and review articles, which contain a broader and deeper exposition of a scientific topic. Libraries with real books grouped by topics are deserted these days, and online services and expert reviews did in general not gain a larger audience or expert community to write reports. One exception might be the public discussion of possible scientific misconduct and retracted publications.

Another side effect: searching the internet for specific topics diminishes the opportunity to accidentally stumble upon an interesting article lacking these keywords, for instance by scanning through a paper volume of a journal while searching for a specific article. I recall that many faculty members went every monday to the library and looked at all the incoming journals to stay up-to-date about the general developments in physics and chemistry. Today we get email alerts about citation counts or specific subfields, but no alert contains a suggestion what other article might pick our intellectual curiosity – and looking at the rather stupid shopping recommendations generated by online-warehouses I don’t expect this to happen anytime soon.

On a positive note: since all text sources are treated equally, no “high-impact journals” are preferred. In my experience as a referee for journals of all sorts of impact numbers, the interesting contributions are not necessarily published or submitted to highly ranked journals.

To sum up, the assessment of manuscripts, contribution of colleagues, and of my own articles requires humans to read them and to process them carefully – all of this takes a lot of time and consideration. It can take decades before publications become alive and well cited. Citation counts of the last 10 years can be poor indicators for the long-term importance of a contribution. Counting statistics provides some gratification by showing immediate interest and are the (less personal) substitute for the old-fashioned postcards requesting reprints. People working in theoretical physics are often closely related by collaboration distance, which provides yet another (much more fun!) factor. You can check your Erdos number (mine is 4) or Einstein number (3, thanks to working with Marcos Moshinsky) at the AMS website.

How to improve the current situation and maintain a well curated and relevant library of scientific contributions – in particular involving numerical results and methods? One possibility is to make a larger portion of the materials surrounding a publication available. In computational physics it is of interest to test and recalculate published results shown in journals. The nanohub.org platform is in my view a best practice case for providing supplemental information on demand and to ensure a long-term availability and usefulness of scientific results by keeping the computational tools running and updated. It is for me a pleasure and excellent experience to work with the team around nanohub to maintain our open quantum dynamics tool. Another way is to provide and test background materials in research blogs. I will try out different approaches with the next posts.

Solving the interacting many-body Schrödinger equation is a hard problem. Even restricting the spatial domain to a two-dimensions plane does not lead to analytic solutions, the trouble-makers are the mutual particle-particle interactions. In the following we consider electrons in a quasi two-dimensional electron gas (2DEG), which are further confined either by a magnetic field or a harmonic oscillator external confinement potential. For two electrons, this problem is solvable for specific values of the Coulomb interaction due to a hidden symmetry in the Hamiltonian, see the review by A. Turbiner and our application to the two interacting electrons in a magnetic field.

For three and more electrons (to my knowledge) no analytical solutions are known. One standard computational approach is the configuration interaction (CI) method to diagonalize the Hamiltonian in a variational trial space of Slater-determinantal states. Each Slater determinant consists of products of single-particle orbitals. Due to computer resource constraints, only a certain number of Slater determinants can be included in the basis set. One possibility is to include only trial states up to certain excitation level of the non-interacting problem.

The usage of Slater-determinants as CI basis-set introduce severe distortions in the eigenenergy spectrum due to the intrusion of spurious states, as we will discuss next. Spurious states have been extensively analyzed in the few-body problems arising in nuclear physics but have rarely been mentioned in solid-state physics, where they do arise in quantum-dot systems. The basic defect of the Slater-determinantal CI method is that it brings along center-of-mass excitations. During the diagonalization, the center-of-mass excitations occur along with the Coulomb-interaction and lead to an inflated basis size and also with a loss of precision for the eigenenergies of the excited states. Increasing the basis set does not uniformly reduce the error across the spectrum, since the enlarged CI basis set brings along states of high center-of-mass excitations. The cut-off energy then restricts the remaining basis size for the relative part.

The cleaner and leaner way is to separate the center-of-mass excitations from the relative-coordinate excitations, since the Coulomb interaction only acts along the relative coordinates. In fact, the center-of-mass part can be split off and solved analytically in many cases. The construction of the relative-coordinate basis states requires group-theoretical methods and is carried out for four electrons here Interacting electrons in a magnetic field in a center-of-mass free basis (arxiv:1410.4768). For three electrons, the importance of a spurious state free basis set was emphasized by R Laughlin and is a design principles behind the Laughlin wave function.

In the previous post I described some of the computational challenges for modeling energy transfer in the light harvesting complex II (LHCII) found in spinach. Here, I discuss the results we have obtained for the dynamics and choreography of excitonic energy transfer through the chlorophyll network. Compared to the Fenna-Matthews-Olson complex, LHCII has twice as many chlorophylls per monomeric unit (labeled 601-614 with chlorophyll a and b types).
Previous studies of exciton dynamics had to stick to simple exponential decay models based on either Redfield or Förster theory for describing the transfer from the Chl b to the Chl a sites. The results are not satisfying and conclusive, since depending on the method chosen the transfer time differs widely (tens of picoseconds vs picoseconds!).

To resolve the discrepancies between the various approximate methods requires a more accurate approach. With the accelerated HEOM at hand, we revisited the problem and calculated the transfer rates. We find slower rates than given by the Redfield expressions. A combined Förster-Redfield description is possible in hindsight by using HEOM to identify a suitable cut-off parameter (Mcr=30/cm in this specific case).

Since the energy transfer is driven by the coupling of electronic degrees of freedom to vibrational ones, it of importance to assess how the vibrational mode distribution affects the transfer. In particular it has been proposed that specifically tuned vibrational modes might promote a fast relaxation. We find no strong impact of such modes on the transfer, rather we see (independent of the detailed vibrational structure) several bottleneck states, which act as a transient reservoir for the exciton flux. The details and distribution of the bottleneck states strongly depends on the parameters of the electronic couplings and differs for the two most commonly discussed LHCII models proposed by Novoderezhkin/Marin/van Grondelle and Müh/Madjet/Renger – both are considered in the article Scalable high-performance algorithm for the simulation of exciton-dynamics. Application to the light harvesting complex II in the presence of resonant vibrational modes (collaboration of Christoph Kreisbeck, Tobias Kramer, Alan Aspuru-Guzik).
Again, the correct assignment of the bottleneck states requires to use HEOM and to look beyond the approximate rate equations.

With increasing computational power of massively-parallel computers, a more accurate modeling of the energy-transfer dynamics in larger and more complex photosynthetic systems (=light-harvesting complexes) becomes feasible – provided we choose the right algorithms and tools.

The diverse character of hardware found in high-performance computers (hpc) seemingly requires to rewrite program code from scratch depending if we are targeting multi-core CPU systems, integrated many-core platforms (Xeon PHI/MIC), or graphics processing units (GPUs).

In my experience, it is not uncommon to develop a nice GPU application for instance with CUDA, which later on is scaled up to handle bigger problem sizes. With increasing problem size also the memory demands increase and even the 12 GB provided by the Kepler K40 are finally exhausted. Upon reaching this point, two options are possible: (a) to distribute the memory across different GPU devices or (b) to switch to architectures which provide more device-memory. Option (a) requires substantial changes to existing program code to manage the distributed memory access, while option (b) in combination with OpenCL requires (in the best case) only to adapt the kernel-launch configuration to the different platforms.

QMaster implements an extension of the hierarchical equation of motion (HEOM) method originally proposed by Tanimura and Kubo, which involves many (small) matrix-matrix multiplications. For GPU applications, the usage of local memory and the optimal thread-grids for fast matrix-matrix multiplications have been described before and are used in QMaster (and the publicly available GPU-HEOM tool on nanohub.org). While for GPUs the best performance is achieved using shared/local memory and assign one thread to each matrix element, the multi-core CPU OpenCL variant performs better with fewer threads, but getting more work per thread done. Therefore we use for the CPU machines a thread-grid which computes one complete matrix product per thread (this is somewhat similar to following the “naive” approach given in NVIDIA’s OpenCL programming guide, chapter 2.5). This strategy did not work very well for the Xeon PHI/MIC OpenCL case, which requires additional data structure changes, as we learnt from discussions with the distributed algorithms and hpc experts in the group of Prof. Reinefeld at the Zuse-Institute in Berlin.
The good performance and scaling across the 64 CPU AMD opteron workstation positively surprised us and lays the groundwork to investigate the validity of approximations to the energy-transfer equations in the spinach light-harvesting system, the topic for the next post.

This post provides a historical and conceptional perspective for the theoretical discovery of non-periodic 3d space-fillings by Peter Kramer, later experimentally found and now called quasicrystals. See also these previous blog entries for more quasicrystal references and more background material here.
The following post is written by Peter Kramer.

When sorting out old texts and figures from 1981 of mine published in Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells, Acta Cryst. (1982). A38, 257-264), I came across the figure of a regular pentagon of edge length L, which I denoted as p(L). In the left figure its red-colored edges are star-extending up to their intersections. Straight connection of these intersection points creates a larger blue pentagon. Its edges are scaled up by τ^{2}, with τ the golden section number, so the larger pentagon we call p(τ^{2} L). This blue pentagon is composed of the old red one plus ten isosceles triangles with golden proportion of their edge length. Five of them have edges t_{1}(L): (L, τ L, τ L), five have edges t_{2}(L): (τ L,τ L, τ^{2} L). We find from Fig 1 that these golden triangles may be composed face-to-face into their τ-extended copies as t_{1}(τ L) = t_{1}(L) + t_{2}(L) and t_{2}(τ L) = t_{1}(L) + 2 t_{2}(L).

Moreover we realize from the figure that also the pentagon p(τ^{2} L) can be composed from golden triangles as p(τ^{2} L) = t_{1}(τ L) + 3 t_{2}(τ L) = 4 t_{1}(L) + 7 t_{2}(L).

This suggests that the golden triangles t_{1},t_{2} can serve as elementary cells of a triangle tiling to cover any range of the plane and provide the building blocks of a quasicrystal. Indeed we did prove this long range property of the triangle tiling (see Planar patterns with fivefold symmetry as sections of periodic structures in 4-space).

An icosahedral tiling from star extension of the dodecahedron.

In the same paper, I generalized the star extension from the 2D pentagon to the 3D dodecahedron d(L) of edge length L in 3D (see next figure) by the following prescription:

star extend the edges of this dodecahedron to their intersections

connect these intersections to form an icosahedron

The next star extension produces a larger dodecahedron d(τ^{3}L), with edges scaled by τ^{3}. In the composition of the larger dodecahedron I found four elementary polyhedral shapes shown below. Even more amusing I also resurrected the paper models I constructed in 1981 to actually demonstrate the complete space filling!
These four polyhedra compose their copies by scaling with τ^{3}. As for the 2D case arbitrary regions of 3D can be covered by the four tiles.

The four elementary cells shown in the 1982 paper, Fig. 4. The four shapes are named dodecahedron (d) skene (s), aetos (a) and tristomos (t).

The paper models from 1981 are still around in 2014 and complete enough to fill the 3D space without gaps. You can spot all shapes (d,s,a,t) in various scalings and they all systematically and gapless fill the large dodecahedron shell on the back of the table.

Quasiperiodicity.

The only feature missing for quasicrystals is aperiodic long-range order which eventually leads to sharp diffraction patterns of 5 or 10 fold point-symmetries forbidden for the old-style crystals. In my construction shown here I strictly preserved central icosahedral symmetry. Non-periodicity then followed because full icosahedral symmetry and periodicity in 3D are incompatible.

The computation and prediction of two-dimensional (2d) echo spectra of photosynthetic complexes is a daunting task and requires enormous computational resources – if done without drastic simplifications. However, such computations are absolutely required to test and validate our understanding of energy transfer in photosyntheses. You can find some background material in the recently published lecture notes on Modelling excitonic-energy transfer in light-harvesting complexes (arxiv version) of the Latin American School of Physics Marcos Moshinsky.
The ability to compute 2d spectra of photosynthetic complexes without resorting to strong approximations is to my knowledge an exclusive privilege of the Hierarchical Equations of Motion (HEOM) method due to its superior performance on massively-parallel graphics processing units (GPUs). You can find some background material on the GPU performance in the two conference talks Christoph Kreisbeck and I presented at the GTC 2014 conference (recored talk, slides) and the first nanoHub users meeting.

for this tutorial we use the example input for “FMO coherence, 1 peak spectral density“.
You can select this preset from the Example selector.

we stick with the provided Exciton System parameters and only change the temperature to 77 K to compare the results with our published data.

in the Spectral Density tab, leave all parameters at the the suggested values

to compute 2d spectra, switch to the Calculation mode tab

for compute: choose “two-dimensional spectra“. This brings up input-masks for setting the directions of all dipole vectors, we stick with the provided values. However, we select Rotational averaging: “four shot rotational average” and activate all six Liouville pathways by setting ground st[ate] bleach reph[asing , stim[ulated] emission reph[asing], and excited st[ate] abs[orption] to yes, as well as their non-rephasing counterparts (attention! this might require to resize the input-mask by pulling at the lower right corner)

That’s all! Hit the Simulate button and your job will be executed on the carter GPU cluster at Purdue university. The simulation takes about 40 minutes of GPU time, which is orders of magnitude faster than any other published method with the same accuracy. You can close and reopen your session in between.

Voila: your first FMO spectra appears.

Now its time to change parameters. What happens at higher temperature?

If you like the results or use them in your work for comparison, we (and the folks at nanoHub who generously develop and provide the nanoHub platform and GPU computation time) appreciate a citation. To make this step easy, a DOI number and reference information is listed at the bottom of the About tab of the tool-page.

With GPU-HEOM we and now you (!) can not only calculate the 2d echo spectra of the Fenna-Matthews-Olson (FMO) complex, but also reveal the strong link between the continuum part of the vibrational spectral density and the prevalence of long-lasting electronic coherences as written in my previous posts

Two conferences are currently open for registration related to GPU and cloud computing. I will be attending and presenting at both, please email me if you want to get in touch at the meetings.

Over the last years, a debate is going on whether the observation of long lasting oscillatory signals in two-dimensional spectra are reflecting vibrational of electronic coherences and how the functioning of the molecule is affected. Christoph Kreisbeck and I have performed a detailed theoretical analysis of oscillations in the Fenna-Matthews-Olson (FMO) complex and in a model three-site system. As explained in a previous post, the prerequisites for long-lasting electronic coherences are two features of the continuous part of the vibronic mode density are: (i) a small slope towards zero frequency, and (ii) a coupling to the excitonic eigenenergy (ΔE) differences for relaxation. Both requirements are met by the mode density of the FMO complex and the computationally demanding calculation of two-dimensional spectra of the FMO complex indeed predicts long-lasting cross-peak oscillations with a period matching h/ΔE at room temperature (see our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes or arXiv version). The persistence of oscillations is stemming from a robust mechanism and does not require adding any additional vibrational modes at energies ΔE (the general background mode density is enough to support the relaxation toward a thermal state). But what happens if in addition to the background vibronic mode density additional vibronic modes are placed within the vicinity of the frequencies related electronic coherences? This fine-tuning model is sometimes discussed in the literature as an alternative mechanism for long-lasting oscillations of vibronic nature. Again, the answer requires to actually compute two-dimensional spectra and to carefully analyze the possible chain of laser-molecule interactions. Due to the special way two-dimensional spectra are measured, the observed signal is a superposition of at least three pathways, which have different sensitivity for distinguishing electronic and vibronic coherences. Being a theoretical physicists now pays off since we have calculated and analyzed the three pathways separately (see our recent publication Disentangling Electronic and Vibronic Coherences in Two-Dimensional Echo Spectra or arXiv version). One of the pathways leads to an enhancement of vibronic signals, while the combination of the remaining two diminishes electronic coherences otherwise clearly visible within each of them. Our conclusion is that estimates of decoherence times from two-dimensional spectroscopy might actually underestimate the persistence of electronic coherences, which are helping the transport through the FMO network. The fine tuning and addition of specific vibrational modes leaves it marks at certain spots of the two-dimensional spectra, but does not destroy the electronic coherence, which is still there as a Short Time Fourier Transform of the signal reveals.

I am preparing my presentation for the simGPU meeting next week in Freudenstadt, Germany, and performed some benchmarks.
In the previous post I described how to get an OpenCL program running on a smartphone with GPU. By now Christoph Kreisbeck and I are getting ready to release our first smartphone GPU app for exciton dynamics in photosynthetic complexes, more about that in a future entry.
Getting the same OpenCL kernel running on laptop GPUs, workstation GPUs and CPUs, and smartphones/tablets is a bit tricky, due to different initialisation procedures and the differences in the optimal block sizes for the thread grid. In addition on a smartphone the local memory is even smaller than on a desktop GPU and double-precision floating point support is missing. The situation reminds me a bit of the “earlier days” of GPU programming in 2008.
Besides being a proof of concept, I see writing portable code as a sort of insurance with respect to further changes of hardware (however always with the goal to stick with the massively parallel programming paradigm). I am also amazed how fast smartphones are gaining computational power through GPUs!

The comparison of OpenCL vs CUDA performance for the same algorithm can reveal some surprises on NVIDIA GPUs. While on our C2050 GPU OpenCL works a bit faster for the same problem compared to the CUDA version, on a K20c system for certain problem sizes the OpenCL program can take several times longer than the CUDA code (no changes in the basic algorithm or workgroup sizes).

The comparison with a CPU version running on 8 cores of the Intel Xeon machine is possible and shows clearly that the GPU code is always faster, but requires a certain minimal systems size to show its full performance.

I am looking forward to running the same code on the Intel Xeon Phi systems now available with OpenCL drivers, see also this blog.

[Update June 22, 2013: I updated the graphs to show the 8-core results using Intels latest OpenCL SDK. This brings the CPU runtimes down by a factor of 2! Meanwhile I am eagerly awaiting the possibility to run the same code on the Xeon Phis…]

[Update August 2013: Google has removed the OpenCL library with Android 4.3. You can find an interesting discussion here. Google seems to push for its own renderscript protocol. I will not work with renderscript since my priorities are platform independency and sticking with widely adopted standards to avoid fragmentation of my code basis.]
I recently got hold of a Nexus 4 smartphone, which features a GPU (Qualcomm Adreno 320) and conveniently ships with already installed OpenCL library. With minimal changes I got the previously discussed many-body program code related to the fractional quantum Hall effect up and running. No unrooting of the phone is required to run the code example. Please use the following recipe at your own risk, I don’t accept any liabilities. Here is what I did:

Download and unpack the Android SDK from google for cross-compilation (my host computer runs Mac OS X).

Download and unpack the Android NDK from google to build minimal C/C++ programs without Java (no real app).

Install the standalone toolchain from the Android NDK. I used the following command for my installation:

Put the OpenCL programs and source code in an extra directory, as described in my previous post

Change one line in the cl.hpp header: instead of including <GL/gl.h> change to <GLES/gl.h>. Note: I am using the “old” cl.hpp bindings 1.1, further changes might be required for the newer bindings, see for instance this helpful blog

Transfer the OpenCL library from the phone to a subdirectory lib/ inside your source code. To do so append the path to your SDK tools and use the adb command:

Cross compile your program. I used the following script, please feel free to provide shorter versions. Adjust the include directories and library directories for your installation.

Copy the executable to the data dir of your phone to be able to run it. This can be done without rooting the phone with the nice SSHDroid App, which by defaults transfers to /data . Don’t forget to copy the kernel .cl files:

Check the resulting data files. You can copy them for example to the Download path of the storage and use the gnuplot (droidplot App) to plot them.

A short note about runtimes. On the Nexus 4 device the program runs for about 12 seconds, on a MacBook Pro with NVIDIA GT650M it completes in 2 seconds (in the example above the equations of motion for 16*64=1024 interacting particles are integrated). For larger particle numbers the phone often locks up.

An alternative way to transfer files to the device is to connect via USB cable and to install the Android Terminal Emulator app. Next

cd /data/data/jackpal.androidterm
mkdir gpu
chmod 777 gpu

On the host computer use adb to transfer the compiled program and the .cl kernel and start a shell to run the kernel

Christoph Kreisbeck and I are happy to announce the public availability of the Exciton Dynamics Lab for Light- Harvesting Complexes (GPU-HEOM) hosted on nanohub.org. You need to register a user account (its free), and then you are ready to use GPU-HEOM for the Frenkel exciton model of light harvesting complexes. In release 1.0 we support

calculating population dynamics

tracking coherences between two eigenstates

obtaining absorption spectra

two-dimensional echo spectra (including excited state absorption)

… and all this for general vibronic spectral densities parametrized by shifted Lorentzians.

I will post some more entries here describing how to use the tool for understanding how the spectral density affects the lifetime of electronic coherences (see also this blog entry).
In the supporting document section you find details of the implemented method and the assumptions underlying the tool. We are appreciating your feedback for further improving the tool.
We are grateful for the support of Prof. Gerhard Klimeck, Purdue University, director of the Network for Computational Nanotechnology to bring GPU computing to nanohub (I believe our tool is the first GPU enabled one at nanohub).

If you want to refer to the tool you can cite it as:
Christoph Kreisbeck; Tobias Kramer (2013), “Exciton Dynamics Lab for Light-Harvesting Complexes (GPU-HEOM),” https://nanohub.org/resources/gpuheompop. (DOI:10.4231/D3RB6W248).
and you find further references in the supporting documentation.

I very much encourage my colleagues developing computer programs for theoretical physics and chemistry to make them available on platforms such as nanohub.org. In my view, it greatly facilitates the comparison of different approaches and is the spirit of advancing science by sharing knowledge and providing reproducible data sets.

Due to its known structure and relative simplicity, the Fenna-Matthews-Olson complex of green sulfur bacteria provides an interesting test-case for our understanding of excitonic energy transfer in a light-harvesting complex.

The experimental pump-probe spectra (discussed in my previous post catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex) show long-lasting oscillatory components and this finding has been a puzzle for theoretician and led to a refinement of the well-established models. These models show a reasonable agreement with the data and the rate equations explain the relaxation and transfer of excitonic energy to the reaction center.

However, the rate equations are based on estimates for the relaxation and dephasing rates. As Christoph Kreisbeck and I discuss in our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes(arxiv version), an exact calculation with GPU-HEOM following the best data for the Hamiltonian allows one to determine where the simple approach is insufficient and to identify a key-factor supporting electronic coherence:

It’s the vibronic spectral density – redrawn (in a different unit convention, multiplied by ω^{2}) from the article by M. Wendling from the group of Prof. Rienk van Grondelle. We did undertake a major effort to proceed in our calculations as close to the measured shape of the spectral density as the GPU-HEOM method allows one. By comparison of results for different forms of the spectral density, we identify how the different parts of the spectral density lead to distinct signatures in the oscillatory coherences. This is illustrated in the figure on the rhs. To get long lasting oscillations and finally to relax, three ingredients are important

a small slope towards zero frequency, which suppresses the pure dephasing.

a high plateau in the region where the exciton energy differences are well coupled. This leads to relaxation.

the peaked structures induce a “very-long-lasting” oscillatory component, which is shown in the first figure. In our analysis we find that this is a persistent, but rather small (<0.01) modulation.

2d spectra are smart objects

The calculation of 2d echo spectra requires considerable computational resources. Since theoretically calculated 2d spectra are needed to check how well theory and experiment coincide, I conclude with showing a typical spectrum we obtain (including static disorder, but no excited state absorption for this example). One interesting finding is that 2d spectra are able to differentiate between the different spectral densities. For example for a a single-peak Drude-Lorentz spectral density (sometimes chosen for computational convenience), the wrong peaks oscillate and the life-time of cross-peak oscillations is short (and becomes even shorter with longer vibronic memory). But this is for the experts only, see the supporting information of our article.

Are vibrations good or bad? Probably both… The pragmatic answer is that the FMO complex lives in an interesting parameter regime. The exact calculations within the Frenkel exciton model do confirm the well-known dissipative energy transfer picture. But on the other hand the specific spectral density of the FMO complex supports long-lived coherences (at least if the light source is a laser beam), which require considerable theoretical and experimental efforts to be described and measured. Whether the seen coherence has any biological relevance is an entirely different topic… maybe the green-sulfur bacteria are just enjoying a glimpse into Schrödinger’s world of probabilistic uncertainty.

In the second example of my series on GPU programming for scientists, I discuss a short OpenCL program, which you can compile and run on the CPU and the GPUs of various vendors. This gives me the opportunity to perform some cross-platform benchmarks for a classical plasma simulation. You can expect dramatic (several 100 fold) speed-ups on GPUs for this type of system. This is one of the reasons why molecular dynamics code can gain quite a lot by incorporating the massively parallel-programming paradigm in the algorithmic foundations.

Now to the coding of a two-dimensional plasma simulation, which is inspired by Laughlin’s mapping of a many-body wave function to an interacting classical ersatz dynamics (for some context see my short review Interacting electrons in a magnetic field: mapping quantum mechanics to a classical ersatz-system on the arxiv).

I start my series on the physics of GPU programming by a relatively simple example, which makes use of a mix of library calls and well-documented GPU kernels. The run-time of the split-step algorithm described here is about 280 seconds for the CPU version (Intel(R) Xeon(R) CPU E5420 @ 2.50GHz), vs. 10 seconds for the GPU version (NVIDIA(R) Tesla C1060 GPU), resulting in 28 fold speed-up! On a C2070 the run time is less than 5 seconds, yielding an 80 fold speedup.

The description of coherent electron transport in quasi two-dimensional electron gases requires to solve the Schrödinger equation in the presence of a potential landscape. As discussed in my post Time to find eigenvalues without diagonalization, our approach using wavepackets allows one to obtain the scattering matrix over a wide range of energies from a single wavepacket run without the need to diagonalize a matrix. In the following I discuss the basic example of propagating a wavepacket and obtaining the autocorrelation function, which in turn determines the spectrum. I programmed the GPU code in 2008 as a first test to evaluate the potential of GPGPU programming for my research. At that time double-precision floating support was lacking and the fast Fourier transform (FFT) implementations were little developed. Starting with CUDA 3.0, the program runs fine in double precision and my group used the algorithm for calculating electron flow through nanodevices. The CPU version was used for our articles in Physica Scripta Wave packet approach to transport in mesoscopic systems and the Physical Review B Phase shifts and phase π-jumps in four-terminal waveguide Aharonov-Bohm interferometers among others.
Here, I consider a very simple example, the propagation of a Gaussian wavepacket in a uniform potential V(x,y)=-Fx, for which the autocorrelation function of the initial state
⟨x,y|ψ(t=0)⟩=1/(a√π)exp(-(x^{2}+y^{2})/(2 a^{2}))
is known in analytic form:
⟨ψ(t=0)|ψ(t)⟩=2a^{2}m/(2a^{2}m+iℏt)exp(-a^{2}F^{2}t^{2}/(4ℏ^{2})-iF^{2}t^{3}/(24ℏ m)). Continue reading Computational physics & GPU programming: Solving the time-dependent Schrödinger equation