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If we consider a two-electron atom exposed to a linearly-polarized
laser field and treated within the electric dipole approximation
then the system retains an axial symmetry about the laser
polarization axis. Due to the presence of the single massive
force centre provided by the positively charged nucleus there
is great computational advantage in treating the five remaining
degrees of freedom in the electronic motion within spherical
geometry. In this the three angular degrees of freedom are
handled using basis-set methods and the two radial degrees
of freedom and the time using finite-difference techniques.
Such work was given a great stimulus by the advent of massively-parallel
computers (that actually worked!) in the mid-1990s. Serious
exploration of the time-dependent Schrodinger equation for
driven few-particle systems has been possible only by successfully
addressing the major challenges that have arisen in developing
theory, algorithms, numerical methods and of course efficient
computer code in harnessing the power of these machines. A
special feature has been the development of explicit time-propagator
methods [1]. These propagator
methods have now been taken around the world in their use
by former PhD students and visitors to Belfast. They have
also been kernel to methods developed here in Belfast to integrate
the time-dependent Schrodinger equation for the laser-driven
hydrogen molecular ion [2]
and the hydrogen molecule. The suite of associated computer
codes known as HELIUM has served as a benchmark code in the
latest UK Research Councils' High Performance Computing procurement
exercise leading to the purchase of a £113M Cray XT4
system and its installation at the Edinburgh Parallel Computing
Centre. In recent calculations we have run HELIUM over more
than 4,000 cores on this machine but it scales to run efficiently
over the full core count of more than 12,000.
Achievements
Over the years the ab initio approach outlined above has
led to many 'firsts' and scientific successes. We were the
only group to correctly predict [3]
in advance of experiment the emergence of doubly-ionizing
electron wave-packets unilaterally each laser half-cycle for
optical wavelengths. We were the first to accurately calculate
intense-field single- and double-ionization rates for two-electron
atoms at optical wavelengths [4]
and to the best of our knowledge this has not yet been achieved
by any other method. We were the first to predict [5]
and describe theoretically double-electron above-threshold
ionization (DATI), in which highly-correlated two-electron
wave-packets are ejected at integer multiples of the photon
energy. More recently HELIUM has made possible the first full
quantum mechanical calculation of wave-packet time-delays
in order to test the recollision model at Ti:sapphire wavelengths
[6]. It has also uncovered
an unexpected partition of kinetic energy in double ionization
at optical wavelengths [7].
This arose out of work with Professor Louis DiMauro's group
at Ohio State University in a joint experimental/theoretical
determination [7] of
accurate energy-resolved double ionization yields from helium
exposed to intense 390 nm laser light where a new ionization
law governing the cut-off energies in double ionization was
discovered. The accompanying figure corresponding to a peak
laser field intensity of 10.0 x 1014 W/cm2 displays the two-electron
joint probability density of doubly ionizing wavepackets (plotted
in momentum space) by the end of a 7-cycle pulse. Circular
arcs in this figure correspond to a fixed amount of escape
energy shared between the two ionizing electrons. A cutoff
in this is apparent and also, because of the notch, it is
clear there is a cutoff in the escape energy of an individual
electron together with asymmetric energy sharing at large
escape energies.
Recently, accurate single ionization rates have been calculated
[8] over very extensive
laser intensity ranges for both fundamental and frequency-doubled
Ti:sapphire wavelengths which should be useful to experimentalists
wishing to accurately intensity-calibrate their Ti:sapphire
laser sources. From these rates it proved possible to establish
the surprising applicability of a new intensity-dependent
perturbation theory. It was also found possible to discern
from the accurate rates the existence of a scaling law which
allows the calculation of rates at wavelengths intermediate
between 400 nm and 800 nm.
Recently also [9] the
theoretical formalism underlying HELIUM has been extended
in preparation for carrying out calculations for intense laser
light at x-ray wavelengths.
Past and present people
The work has so far produced five PhDs. Dr Daniel Dundas
who gained his PhD in 1998 is currently a RCUK Fellow in Atomistic
Simulation at QUB. Dr Edward Smyth having gained his PhD in
1999 moved to Numerical Algorithms Group (NAG) where he is
presently a Senior Technical Consultant with special expertise
in High Performance Computing. Dr Laura Moore graduating with
her PhD in 2001 joined local Northern Ireland industry but
missed the challenges and rejoined the group last year as
an EPSRC PDRA. Dr Karen Cairns (née Meharg) graduating
in 2003 is now a RCUK Fellow in Statistics applied to Epidemiology
at QUB. Finally Dr Barry Doherty graduating with his PhD in
2006 is now pursuing teacher training in Northern Ireland.
Present research
A particular research direction at present builds on several
significant theoretical findings [7]
of the collaboration our group completed with Professor DiMauro's
experimental group at Ohio State University. The Ohio State
experiment and the Belfast theoretical work studied rescattering-induced
double-ionization of helium in intense 390 nm light. The analysis
of the theoretical results obtained using HELIUM revealed
a new ionization law governing double-ionization and double-electron
above-threshold ionization that no-one had expected or correctly
predicted. Moreover, the analysis enabled us to develop a
new quantum theoretical model [7]
of rescattering-induced double-ionization that demonstrated
excellent agreement with the experimental DATI energy spectra.
A principal thrust of present research is to advance this
computational and theoretical work [7]
on rescattering-induced double-ionization into the UV and
XUV. A recent very substantial award of time on the new Cray
XT4 supercomputer at the Edinburgh Parallel Computing Centre
is enabling us to extend these high-integrity, full-dimensional
calculations into the near infra-red (800 nm). Dr Jonathan
Parker is engaged in this work.
Another research direction uses HELIUM to explore the interaction
of coherent x-rays with the K-shell electrons of matter. High-intensity
laser light of this kind will soon to be available from x-ray
Free Electron Lasers under construction at Stanford in the
USA and near Hamburg in Germany. Such an investigation is
important because filled K-shells are characteristic of most
forms of matter e.g. gasses, liquids, solids and most plasmas,
and x-ray light is absorbed most efficiently by K-shell, rather
than by outer-shell, electrons. The HELIUM code is being presently
augmented to include laser-electron interaction terms beyond
the electric dipole approximation as set out in [9]
with the intention of carrying out first calculations on the
helium-like ions of neon and argon. Dr Laura Moore is engaged in this work.
A yet further research direction is in carrying over the
time-propagator methods [1]
developed for use in HELIUM (and also subsequently used in
Belfast work on the laser-driven hydrogen molecular ion and
hydrogen molecule as well as being taken around the world
by visitors) so that these time-propagator methods can be
combined with the concepts of time-dependent R-matrix theory.
Such a combination should make possible efficient time-dependent
R-matrix calculations on massively-parallel supercomputers
where explicit time-propagators must be used in order to minimize
communications overhead. Dr Lampros Nikolopoulos is a EU Marie
Curie Fellow engaged in this work.
[1]
Smyth E, Parker J S and Taylor K T Comp Phys Comm 114, 1 (1998)
[2]
Dundas D, McCann JF, Parker JS and Taylor KT J Phys B: Atom
Mol Opt Phys 33 3261 (2000)
[3]
Taylor K T, Parker J S, Dundas D and Vivirito S Laser Physics
9, 98 (1999)
[4]
Parker J S, Moore L R, Dundas D and Taylor K T J Phys B: At
Mol Phys 33 L691 (2000)
[5]
Parker J S, Meharg K J, Moore L R, Dundas D and Taylor K T
J Phys B: At Mol Phys 34 L69 (2001)
[6]
Parker JS, Doherty BJS, Meharg KJ and Taylor KT J Phys B:
Atom Mol Opt Phys 36 L393 (2003)
[7]
Parker JS, Doherty BJS, Taylor KT, Schultz KD, Blaga CI and
DiMauro LF Phys Rev Lett 96 133001 (2006)
[8]
Parker JS, Meharg KJ, McKenna GA and Taylor KT J. Phys. B:
Atom Mol Opt Phys 40 1729 (2007)
[9]
Meharg K J, Parker JS and Taylor KT J Phys B: Atom Mol Opt
Phys 38 237 (2005)
[10]
Moore L R, Parker J S, Meharg K J, Armstrong G S J and Taylor K T
"Extensions to the helium code to handle intense X-ray light," J Mod Opt 55 (2008)
[11]
Nikolopoulos L A A, Parker J S and Taylor K T
"A combined R-matrix eigenstate basis set and finite-differences propagation method for the time-dependent Schroedinger equation: the
1-electron case," Phys Rev A 78 (2008)
[12]
G S J Armstrong, Parker J S, M Boca and K T Taylor,
"Intense-field ionization of helium: from the UV to the static-field limit"
Rutherford National Laboratory Annual Report (2008)
[13]
Parker J S, G S J Armstrong, M Boca and K T Taylor,
"From the UV to the static-field limit: rates and scaling
laws of intense-field ionization of helium", J Phys B 42 (2009)
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