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?
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
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.
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
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…]
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:
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.
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.
It’s the vibronic spectral density – redrawn (in a different unit convention, multiplied by ω2) 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
is known in analytic form:
〈ψ(t=0)|ψ(t)〉=2a2m/(2a2m+iℏt)exp(-a2F2t2/(4ℏ2)-iF2t3/(24ℏ m)). Continue reading Computational physics & GPU programming: Solving the time-dependent Schrödinger equation
From discussions I learn that while many physicists have heard of Graphics Processing Units as fast computers, resistance to use them is widespread. One of the reasons is that physics has been relying on computers for a long time and tons of old, well trusted codes are lying around which are not easily ported to the GPU. Interestingly, the adoption of GPUs happens much faster in biology, medical imaging, and engineering.
I view GPU computing as a great opportunity to investigate new physics and my feeling is that todays methods optimized for serial processors may need to be replaced by a different set of standard methods which scale better with massively parallel processors. In 2008 I dived into GPU programming for a couple of reasons:
As a “model-builder” the GPU allows me to reconsider previous limitations and simplifications of models and use the GPU power to solve the extended models.
The turn-around time is incredibly fast. Compared to queues in conventional clusters where I wait for days or weeks, I get back results with 10000 CPU hours compute time the very same day. This in turn further facilitates the model-building process.
Some people complain about the strict synchronization requirements when running GPU codes. In my view this is an advantage, since essentially no messaging overhead exists.
If you want to develop high-performance algorithm, it is not good enough to convert library calls to GPU library calls. You might get speed-ups of about 2-4. However, if you invest the time and develop your own know-how you can expect much higher speed-ups of around 100 times or more, as seen in the applications I discussed in this blog before.
This summer I will lecture about GPU programming at several places and thus I plan to write a series of GPU related posts. I do have a complementary background in mathematical physics and special functions, which I find very useful in relation with GPU programming since new physical models require a stringent mathematical foundation and numerical studies.
Efficient and fast transport of electric current is a basic requirement for the functioning of nanodevices and biological systems. A neat example is the energy-transport of a light-induced excitation in the Fenna-Matthews-Olson complex of green sulfur bacteria. This process has been elucidated by pump-probe spectroscopy. The resulting spectra contain an enormous amount of information about the couplings of the different pigments and the pathways taken by the excitation. The basic guide to a 2d echo-spectrum is as follows:
You can find peaks of high intensity along the diagonal line which are roughly representing a more common absorption spectrum. If you delay the pump and probe pulses by several picoseconds, you will find a new set of peaks at a horizontal axis which indicates that energy of the excitation gets redistributed and the system relaxes and transfers part of the energy to vibrational motion. This process is nicely visible in the spectra recorded by Brixner et al.
A lot of excitement and activity on photosynthetic complexes was triggered by experiments of Engel et al showing that besides the relaxation process also periodic oscillations are visible in the oscillations for more than a picosecond.
What is causing the oscillations in the peak amplitudes of 2d echo-spectra in the Fenna-Matthews Olson complex?
A purely classical transport picture should not show such oscillations and the excitation instead hops around the complex without interference. Could the observed oscillations point to a different transport mechanism, possibly related to the quantum-mechanical simultaneous superposition of several transport paths?
The initial answer from the theoretical side was no, since within simplified models the thermalization occurs fast and without oscillations. It turned out that the simple calculations are a bit too simplistic to describe the system accurately and exact solutions are required. But exact solutions (even for simple models) are difficult to obtain. Known exact methods such as DMRG work only reliable at very low temperatures (-273 C), which are not directly applicable to biological systems. Other schemes use the famous path integrals but are too slow to calculate the pump-probe signals.
Our contribution to the field is to provide an exact computation of the 2d echo-spectra at the relevant temperatures and to see the difference to the simpler models in order to quantify how much coherence is preserved. From the method-development the computational challenge is to speed-up the calculations several hundred times in order to get results within days of computational run-time. We did achieve this by developing a method which we call GPU-hierarchical equations of motion (GPU-HEOM). The hierarchical equations of motions are a nice scheme to propagate a density matrix under consideration of non-Markovian effects and strong couplings to the environment. The HEOM scheme was developed by Kubo, Tanimura, and Ishizaki (Prof. Tanimura has posted some material on HEOM here).
However, the original computational method suffers from the same problems as path-integral calculations and is rather slow (though the HEOM method can be made faster and applied to electronic systems by using smart filtering as done by Prof. YiJing Yan). The GPU part in GPU-HEOM stands for Graphics Processing Units. Using our GPU adoption of the hierarchical equations (see details in Kreisbeck et al[JCTC, 7, 2166 (2011)] ) allowed us to cut down computational times dramatically and made it possible to perform a systematic study of the oscillations and the influence of temperature and disorder in our recent article Hein et al [New J. of Phys., 14, 023018 (2012), open access] .
Solving the stationary Schrödinger (H-E)Ψ=0 equation can in principle be reduced to solving a matrix equation. This eigenvalue problem requires to calculate matrix elements of the Hamiltonian with respect to a set of basis functions and to diagonalize the resulting matrix. In practice this time consuming diagonalization step is replaced by a recursive method, which yields the eigenfunctions for a specific eigenvalue.
A very different approach is followed by wavepacket methods. It is possible to propagate a wavepacket without determining the eigenfunctions beforehand. For a given Hamiltonian, we solve the time-dependent Schrödinger equation (i ∂t-H) Ψ=0 for an almost arbitrary initial state Ψ(t=0) (initial value problem).
The reformulation of the determination of eigenstates as an initial value problem has a couple of computational advantages:
results can be obtained for the whole range of energies represented by the wavepacket, whereas a recursive scheme yields only one eigenenergy
the wavepacket motion yields direct insight into the pathways and allows us to develop an intuitive understanding of the transport choreography of a quantum system
solving the time-dependent Schrödinger equation can be efficiently implemented using Graphics Processing Units (GPU), resulting in a large (> 20 fold) speedup compared to CPU code
Recently, Christoph Kreisbeck has done a detailed calculations on the gate-voltage dependency of the conductance in Aharonov-Bohm nanodevices, taking full adventage of the simultaneous probing of a range of Fermi energies with one single wavepacket. A very clean experimental realization of the device was achieved by Sven Buchholz, Prof. Saskia Fischer, and Prof. Ulrich Kunze (RU Bochum), based on a semiconductor material grown by Dr. Dirk Reuter and Prof. Anreas Wieck (RU Bochum). The details, including a comparison of experimental and theoretical results shown in the left figure, are published in Physical Review B (arxiv version).
For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by
Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.
Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.
Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol.10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact.