Quantizing comets: semiclassical methods in action

One key aspect of theoretical physics is that the (rather few!) basic equations are not tied to a narrow field of applications, but that insights from celestial mechanics are equally relevant for the dynamics of quantum objects. I have written before how to interpret the Coulomb Green’s function of hydrogen or Rydberg molecules in terms of Lamberts theorem of cometary orbit determination.

Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.
Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.

Equally interesting is the dynamics of small celestials bodies in the vicinity of a parent body (the “restricted three body problem“). Near the Lagrange points the attraction of the sun and the effective potential in a coordinate system moving with the parent body around the sun cancel and the small objects can be trapped.
The comet Shoemaker-Levy 9 (SL9) is a prime example: SL9 was captured by Jupiter, broke apart at a close approach, and finally the string of fragments collided with Jupiter in 1994.

What would have happened with SL9 if Jupiter was contracted to a point mass?

Since SL9 was once captured, it should also have been released again. Indeed, in 2014 SL9 would have left Jupiter, as shown in the numerically integrated JPL orbit. To illustrate and simplify the transient dynamics, I have assumed in a recent publication a circular orbit of Jupiter and that SL9, Jupiter, and the sun are located in a plane. In reality the changing distance from the sun can open and close the entry points and in conjunction with precise location of the comet an escape or trapping becomes feasible:

Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped).
Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped). (C) Tobias Kramer.

The Shoemaker-Levy 9 case has been studied extensively in the astronomical literature (see for instance the orbital analysis by Benner et al). So what new insights are there for quantum objects? The goal is not to claim that comets have to be treated as quantum mechanical objects, but to realize that exactly the same dynamics seen in celestial mechanics guides electrons in magnetic fields through wave guides. I refer you for the details to my article Transient capture of electrons in magnetic fields, or: comets in the restricted three-body problem, but want to close by showing the electronic eigenfunctions, which show a real quantum feature absent in the classical case: electrons can tunnel through the forbidden area and thus will always escape from the parent body:

Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field.
Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field. (C) Tobias Kramer.

Random scattering: the small scale structure of the universe

Branched flow
Emergence of branched electron flow from weak scattering across a random potential. (C) Tobias Kramer

Our universe displays various mass concentrations of matter and is not a homogeneous density soup of particles as assumed in the simplest cosmological models where isotropy is assumed (cosmological principle). Interestingly recent supernova data (see Colin et al Evidence of anisotropy of cosmic acceleration A&A 631 L13, also on the arxiv) shows deviations of the angular distribution of matter as seen from Earth. One possibility (also not taken into account in the standard model) is the topology of the cosmos, discussed in this blog by Peter Kramer before. But back to the “small structures” (< 100 Mega parsec): How do mass concentrations arise and how get small fluctuations amplified?

One universal mechanism at work across many domains of physics is structure formation and concentration into branches by random weak scattering. The key-point is: despite randomness this type of scattering does not smear out the density as expected by diffusion processes. This important mechanism has been independently discovered in various domains of physics, but has been rarely discussed and further explored within a consistent framework. Interestingly the fundamental branching behavior can be seen in both quantum mechanics and classical physics. Together with Rick Heller and Ragnar Fleischmann we have written a short overview  article (available on the arxiv: “Branched flow”) which provides some background information and serves as a starting point for further exploration. We also provide a short python script to generate intricate patterns out of random deformations (script available on github).

pool
Sunlight creating a web of caustics at the bottom of a swimming poll. Picture (C) Tobias Kramer.

One example discussed in this blog is the formation of “dust concentrations” in the near nucleus coma of comets from a homogeneously emitting cometary surface. The structure formation of the universe (the cosmic web) is discussed by Y. Zeldovich and reviewed by P.J.E. Peebles in his monograph The Large-Scale Structure of the Universe (Princeton University Press, 1980) in the chapter “caustics and pancakes”. A more accessible incarnation of the caustics are the ripples of sunlight at a pool bottom.

How a wave packet travels through a quantum electronic interferometer

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?

cat
Transition between the resistivity of the nanoring with and without embedded quantum dot. The vertical axis denotes the Fermi energy (controlled by a gate), while the horizontal axis scans through the magnetic field to induce phase differences between the pathways.

Splitting the heat: the quantum limits of thermal energy flow

Device geometry. a) Scanning electron micrograph of the sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer, mode spacing is ħω = 5 meV.
Device geometry. a) Scanning electron micrograph of an the sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer (indicated by arrow). Mode spacing is ħω = 5 meV. © 2016 Kramer et al. Citation: AIP Advances 6, 065306 (2016); http://dx.doi.org/10.1063/1.4953812

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 ≤ π2 kB2 (T22-T12)/(3h) [h denotes Planck’s and kB 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.

When two electrons collide. Visualizing the Pauli blockade.

The upper panel shows two (non-interacting) electrons approaching with small relative momenta, the lower panel with larger relative momenta.
The upper panel shows two electrons with small relative momenta colliding, in the lower panel with larger relative momenta.

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:
φa(x,y)=e-[(x-o)2+(y-o)2]/(2a2)-ikx+iky and φb(x,y)=e-[(x+o)2+(y-o)2]/(2a2)+ikx+iky
This yields the two-electron wave function
ψa(x1,y1,x2,y2)=φa(x1,y1)*φb(x2,y2)-φa(x2,y2)*φb(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].

Computational physics & GPU programming: Solving the time-dependent Schrödinger equation

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.

autocorrelation function in a uniform force field
Autocorrelation function C(t) of a Gaussian wavepacket in a uniform force field. I compare the GPU and CPU results using the wavepacket code.

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(-(x2+y2)/(2 a2))
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

Time to find eigenvalues without diagonalization

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
Aharnov-Bohm Ring conductance oscillations
The Zebra stripe pattern along the horizontal axis shows Aharonov-Bohm oscillations in the conductance of a half-circular nanodevice due to the changing magnetic flux. The vertical axis denotes the Fermi energy, which can be tuned experimentally. For details see our paper in Physical Review B.

The determination of transmissions requires now to calculate the Fourier transform of correlation functions <Ψ(t=0)|Ψ(t)>. This method has been pioneered by Prof. Eric J. Heller, Harvard University, and I have written an introductory article for the Latin American School of Physics 2010 (arxiv version).

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).