OK, I also want to start out by thanking the organizers
for inviting me here.
And I want to thank Hugo for giving a very good introduction
to what I'm going to be talking about, which
is trying to access correlated electron motion
on attosecond timescales.
Just a short overview after the introduction,
I'm going to be talking about two main topics, mostly
attosecond streaking, where the goal is
to extract time delays in atomic photoionization,
where we are interested in figuring out
what the real origin of the measured time shifts are.
And then I'm going to be talking about the method
for doing an attosecond pump, attosecond probe experiment
using two-photon interferometry.
So just as a general introduction,
why are we interested in attosecond physics?
What happens on attosecond timescales?
The general or the most often given example
is the orbit time of a classical electron
in the ground state orbit of the Bohr model
is about 150 attoseconds.
So we already know that timescales of electron motion
in atoms or in molecules or in solids
are on the attosecond timescale.
The somewhat more complicated example
is actually from the group of Lorentz-Sederbaum
that I'm going to show you a short movie, the dynamics
of electron charge redistribution
inside this very complicated molecule
that I'm not going to try to pronounce.
And if I start this movie now, you
see that within, let's say, sub-phamptosecond timescales,
you really see a redistribution of the electron charge
inside the molecule.
And these are the kinds of dynamics
that you might want to access or even control directly.
And in all of these cases, many electrons involved
and the relevant information of dynamics
is encoded in the phases.
So what tools do we have to access these dynamics
or to control them?
Basically, there's two main things.
One is few cycle infrared laser pulses, which
are at least a few femtoseconds long.
And the other is attosecond extreme ultraviolet pulses.
And really, the point you have or the thing
you need to get down to these very sub-phamptosecond timescales
in terms of control is that you can control not only
the amplitude of these femtosecond pulses,
but that you can really control the electric field
so that you have a reproducible electric field.
In every shot you have, you have exactly the same electric field
in your femtosecond pulse.
So yes, that you have carrier envelope phase control,
basically.
And that allows you to also produce these attosecond pulses.
What we want to look at with these tools
then is actually multi-electron effects or correlation effects.
And what we mean by that is basically
that the wave function is not a single Slater determinant,
that it's not just a single product
of single particle orbitals.
And the atom that we use mostly for looking
at these kinds of effects is helium for a few reasons.
First of all, it's the simplest many electron
atom, which means that we can still treat it relatively
precisely with calculations.
And it already has rich correlation effects.
So we can do full numerical simulations
and look at all of these correlation effects
in quite a lot of detail.
The method we use for that, I don't want to go into too much
detail.
It's a much more straightforward method
than the very complicated R matrix.
Basically, we have the Hamiltonian
for the time-dependent Schrodinger
equation with the Hamiltonian for helium.
We use a non-relativistic Hamiltonian.
We assume that the nucleus is infinitely heavy.
We use the dipole approximation.
And then what we have is basically
six spatial dimensions, which reduce to five
if we have symmetry in linearly polarized fields.
And we have time, since we look at the time-dependent Schrodinger
equation, which means that overall, we
have a high dimensional object, which
makes it computationally quite demanding to solve this.
And what you want to do is choose this criticisation
and propagation schemes for these equations
that ensure that you can do this efficiently
and importantly, that you can parallelize well.
And the schemes that we have chosen,
I don't want to go into too much detail.
But basically, what we find is that we can parallelize well
enough that we get linear speedup.
We've tested that with up to 1,800 processors.
So that enables us really to do a lot of simulations
and to look at the results in detail
and still ensure that we have converged results.
So with that, I want to come to the first physical application,
which is etosecond streaking.
What's streaking or why streaking?
The classical streak camera, so to say,
is a method of mapping time information about the laser
pulse to position information on the screen,
where the idea is that you shoot some time-dependent laser
pulse onto a photo cathode.
The emitted photo electrons pass between these two plates.
You have a voltage ramp between the plates.
And depending on when the electrons pass,
they get deflected by a different amount
and then end up on the screen with a position shift that
depends on the time when they reach the plate.
So in this sense, you have a mapping of time to position.
And amazingly, in my opinion, they
can go down to around 100 femtoseconds time
resolution with this kind of classical setup.
But of course, that's not fast enough
to look at this etosecond dynamics that we want.
So what we are looking at is this method
of etosecond streaking, which is somehow
extending the same idea to this combination
of an etosecond extreme ultraviolet pulse that
will knock down out the electrons.
So that's the same photoelectric effect.
And then we accelerate the electron,
now not by the classical electric field
between these two plates, but by the electric field
of the synchronized infrared pulse.
There again, we need this perfect control
over really the electric field of the infrared pulse.
And when you now scan the delay between the etosecond pulse
and the infrared field, what you get
is that depending on when the electron is
emitted from the atom, it gets accelerated or decelerated
by the infrared field by different amounts.
And basically what you have, the electron
comes into free space out of the atom.
And then the momentum shift it acquires
is just the integral over the electric field
from the moment of ionization.
And that's just the vector potential of your light wave
at the moment of ionization.
So in that sense, you have a direct measurement actually
of the light wave or of the vector potential
of the light wave, which was the first application
of this kind of setup.
But what you also get is a mapping
of the time of ionization to momentum.
And you can use that to actually get information
about when your electron was kicked out
of the atom in some sense.
So these kinds of measurements were also
performed using these techniques.
And the first one was about 10 years ago
using proof of principle showing that you can actually
measure the lifetime of an O'Shea hole in Krypton
directly in the time domain, which is around 8 femtoseconds.
Then five years ago, it was shown
that in emission of electrons from a tungsten surface,
the core electrons from the 4F states
are actually delayed with respect to electrons
from the conduction band by around 110 heteroseconds.
And the application I want to focus on
is photo emission from atoms, from neon specifically,
while the same that Hugo was talking about,
where it was found that photo emission from the neon 2P
shell seems to occur somehow 21 heteroseconds later than photo
emission from the 2S shell.
And just quickly, how does this experiment work?
You have the neon atom with its 10 electrons.
The XUV pulse kicks out an electron either from the 2S
shell or from the 2P shell.
And what you get is, again, the streaking curves, where you here
have the delay between the heterosecond pulse
and the infrared pulse.
And the higher energies are the electrons
that were emitted from the 2P shell.
The lower energies, the more deeply bound ones,
are the electrons from the 2S shell.
You cannot see it on this graph, but if you look in detail,
these two curves are basically shifted horizontally
to each other.
And this shift is the time delay between the two emissions,
in some sense.
And surprisingly, what you find is
that actually the slower electron that's more deeply
bound seems to come first, which intuitively seems
a bit surprising, because it has to come out
from deeper out of the atom.
But the real question is, of course,
what are we actually measuring?
I mean, what I've been talking about are very simple pictures.
But to get some idea of what could we be measuring,
we have to think about scattering, which
photo ionization somehow is.
What actually is a time delay?
What could we access?
So in a very simple 1D picture, scattering
is described in quantum mechanics,
basically by a wave coming in and a wave going out.
And all the information about the scattering event
is encoded in the scattering phase.
And already, more than 50 years ago, it
was shown by Eisenbach, Wigner, and Smith,
that the time delay in scattering,
if you take a wave packet here around this energy,
you get a time delay, which is the energy
derivative of the phase.
So again, the dynamic information
is encoded in the phase of the matrix elements.
That's this Eisenbach-Wigner-Smith time delay
that you could hope to access in the streaking.
And that would be the time delay of photo ionization,
which just is scattering with the photon coming in
and an electron going out.
So if we now try to do this for hydrogen,
we say we take the simplest atom that we can.
We know we can solve that completely.
We see it doesn't work, at least not as you would expect it.
The Eisenbach-Wigner-Smith delay is this dashed line.
This is now as a scan of final electron energies,
or equivalently, photon energies.
And the streaking delays that we extract are this line.
So the sign is shifted.
Doesn't seem to work.
But that's not too surprising, in fact,
because the simple picture assumes
that the electron is in free space
when it gets accelerated by the electric field.
But of course, an ion has a long-range Coulomb potential.
So the electron is actually moving
in the combination of an electric field
from the laser plus a Coulomb potential.
So the momentum shift it acquires
is not just the integral of the electric field strength,
but it is modified by this Coulomb potential.
And it turns out that this so-called Coulomb laser
coupling shift can be reproduced already by classical dynamics.
You can also reproduce it by quantum dynamics
as the phase of basically the continuum metric element.
And what we can do is, since we know the Coulomb potential
is the same in all targets, we want
to know the interesting information, which
is the short-range interaction between the two electrons.
And then you always have the long-range Coulomb tail.
So we can just say we take this Coulomb laser coupling shift,
which should be the same in all targets, and subtract it out,
or add it to the model that we have for the streaking shift.
And what turns out is you can perfectly
reproduce the numerical results, which is this red line.
So if we test this now for, let's say,
a somewhat more interesting model, which is, in this case,
a model potential for neon.
So still on the single active electron description.
But using a model potential, we have here, again,
the calculator that isn't what Wigner-Smith time delays.
We have the numerical results.
And now if we add the Coulomb laser coupling term
to the Eisenberg-Wigner-Smith time delays,
we get perfect agreement between the two.
So it seems that we can actually, in fact,
access this Eisenberg-Wigner-Smith time
delays of the atoms as long as we take into account
we have this Coulomb laser coupling term.
But I want to point out, at least for this mean field
model of neon, which is not the only one that's out there,
but most of them are similar, the delay
we find at the photon energies, which
are used in the experiment is this difference, which
corresponds to around seven nanoseconds,
whereas in the experiment, there were around 21 nanoseconds
of time delay measured.
So the next question is, this is a single active electron
model.
Are there any correlation effects?
Are there any effects of the electron interaction
being somehow modified in the laser field that then give you
an additional time shift?
And for that, since we cannot do neon,
and Hugo was just talking about how difficult it
is to do these calculations for neon,
we use helium as a model system.
And there, we don't have different shells, obviously,
but we have so-called shake-up ionization.
So in helium, when we kick out an electron,
the normal process is that you kick out the electron,
and the ion remains in the ground state.
But because of electron correlation
or electron interaction, you can also excite the remaining ion
and leave it in an excited state, for example, 2s or 2p.
And what you get then in the energy spectrum
are the so-called shake-up peaks or correlation satellites,
which are offset in energy.
And if we now, again, do this streaking,
so these are now relatively heavy calculations, let's say,
but still not too bad, we get this streaking picture.
And we can, again, extract the time shifts
between these two different curves
and compare them to our model.
And if we do this, we find for the 1s electrons,
so for the ion left in the 1s state,
this works perfectly well.
So the line is, again, the model of Coulomb-Blasor
plus Eisenberg-Wigner-Smith time delays,
and the squares here, the numerical results.
But for the excited states, for 2p and 2s,
and also for the average of the n equal 2,
we find that this does not agree at all with the model
that we have.
So there's some additional shift in there,
which might be because of electron correlation.
And if we look into this in more detail and think about it,
we realize, of course, the helium plus ion, 2p, 2s,
are degenerate.
And this degeneracy is broken inside an electric field.
And what we're doing is exactly ionization
inside an electric field, because we
have the infrared pulse at the same time.
And the eigenstates of the ion in the infrared pulse,
if we say, since this is a relatively slow pulse,
this is a quasi-static field, the eigenstates
are 2 plus and 2 minus, instead of 2s and 2p.
What we see is, if we project on these as final ionic states,
we see a much larger effect.
And that's the first hint that we are on the right track.
And if we go into more detail, we
find out we can actually model this additional time shift
that we see very well by just taking
the approximations of saying that the infrared field, since
it's much slower than the dynamics that happens,
is basically a quasi-static field.
The star states have a permanent dipole moment,
which means that the ionic states have
an energy shift in the quasi-static infrared field that
depends on the moment of ionization.
And since the electrons are correlated and interacting,
the energy that the free electron gets
depends on the energy of the channel in which it's emitted.
So if I emit an electron into, let's say, the 2 plus ionic
state, the free electron will get either this additional
energy or lose the energy of the dipole in the electric field.
So you get an additional energy or momentum
shift on the free electron depending on the ionic state
because of this entanglement.
And this shift looks like a time delay
exactly because streaking is a mapping of energy shift
or momentum shift to time delay.
It turns out you can, if you include this,
you really get a perfect agreement
between the shift you extract from the numerical simulations
and the prediction from the model, where you just now
include this electron-electron Coulomb-Blasor coupling
somehow, where the electron-electron interaction is
modified by the presence of the infrared field.
And now, by taking into account the 2s and 2p channels,
such as coherent superpositions of the 2 plus and 2 minus
channels, we can reproduce also the numerical results for 2p
and 2s and also, of course, the average, the incoherent sum.
And interestingly, this effect survives even
for the incoherent sum of n equal 2.
So you have to take this term into account also
for this incoherent sum, where you would assume a priori,
maybe, that the directionality does not
matter that you would average out
of these directionality effects in the electric field.
But it turns out that you still get the sizable shift
because of this.
So with this, I'm already basically finished
with the streaking part.
What I showed you is that we managed
to basically quantify the multi-electron streaking time
shifts on the sub-attosecond level.
So really, the model agrees with the numerical results
with an error below one attosecond.
And what you just have to include
is this Coulomb laser coupling.
You have the Eisenberg-Migner-Smith time delay,
which is somehow the most interesting part in terms
of accessing new information that you cannot get
with any other kind of spectroscopy.
And you also have this laser modified electron-electron
coupling, at least if you have easily polarizably
generated final states.
One part that I don't want to talk about in too much detail
but that I want to mention is that for a two electron model
system with a model potential that somehow reproduces
the energy structure of neon, we don't
want to call it really a model for neon
because it's really quite far from that.
But the energies are roughly there.
We find that we don't see any additional correlation effects
apart from this laser modified electron-electron coupling,
which actually does not play a role here
because the final states are not polarizable.
So the next step here would be to try
to apply this model to neon.
And we've been working on this in collaboration
with Klaus Bartjardt and Oleg Saterini, who
have R-matrix structure calculations where
they can calculate this phase of the photoionization very well.
And so we can get the Eisenberg-Migner time
shift for this part.
The Coulomb-Blasor coupling is independent of the system
we look at.
And the final states are not easily polarizable,
so it should be possible.
And the streaking shift should be just the sum
of these first two terms.
And the results that we get are, again,
around 10 heteroseconds for the experimental photon energy.
And this is actually similar to the results
that Hugo and the other people in Belfast
obtained with the time-dependent R-matrix.
So I think by now we are sufficiently confident in our
results that we think that the experiment maybe
should be repeated to try to get better statistics
and maybe do a scan of electron over photon energies
and so on to really find out what's going on there.
And if I have some time, yes, I want
to go to my second part, which is about doing
heterosecond pump, heterosecond probe setups.
So why do we want to do this?
The standard nowadays in heterosecond physics
is, for example, in streaking to use a pump probe experiment
using a few cyclone-frared pulse with an heterosecond XUV.
The disadvantage of this is to really get
heterosecond time resolution.
You need to look at subcycle effects in the infrared pulse.
These are often strongly nonlinear effects,
which means that the dynamics get modified by the infrared
field and makes it very hard to interpret,
or can make it hard to interpret.
And for that reason, using two heterosecond pulses
has been called the holy grail of heterosecond physics
by some people.
The problem with that is that nowadays the pulse intensities
are too low to really get two photons absorbed
from heterosecond pulses, just because high harmonic
generation is not a very efficient process,
and free electron lasers don't produce heterosecond pulses
yet.
And there are not too many useful schemes out there
yet for doing heterosecond pump, heterosecond probe.
So not too many people have looked
at what could you actually do.
So the first part, the pulse intensity is being too low.
That's what the experimentalists are working at,
and there's quite a lot of progress there.
In fact, just last year there was already
a first paper doing heterosecond pump.
Not heterosecond, it was around one femtosecond,
but still pump probe in argon.
And these are other papers showing the increases
in high harmonic efficiency.
And of course, it's also the work
of getting free electron lasers down
to below femtosecond pulses.
So either of these could lead to enough pulse intensity
in heterosecond pulses to enable this kind of pump
probe with two heterosecond pulses.
But we want to look at this possible scheme
for looking at interesting correlated dynamics here.
So what we want to look at are the double excited
states of helium, which exist because the electron-electron
interaction is in the Hamiltonian.
And what that does is that you have in the spectrum of helium
these double excited states, which are almost all
autoionizing resonances, which are very strongly
correlated states.
So for these states, really, the single orbital picture
breaks down, and you have to use collective quantum
numbers to describe them.
So from this point of view, they are very interesting
in terms of correlation.
And the scheme that we want to use,
the simple idea for probing this is really straightforward.
You pump them.
You use a heterosecond pulse with the photon energy
or wavelengths that's tuned to a transition to more or less
the first double excited states in helium.
If this is broad enough, you create a wave packet
of these double excited states.
Then you wait for some time.
You let them evolve.
And then you want to probe the correlated dynamics
by using another pulse.
And in our case, we take an identical pulse
to kick out both electrons and look at double ionization.
The problem with this scheme is that you
have in both steps, both in the pump,
as well as in the probe, one photon transition that
has to talk to both electrons.
Because you have to excite both electrons
into the double excited states with one photon,
then you will have to ionize both electrons
with one photon again in the probing part.
But if you look at this scheme in more detail,
you actually realize it's not as simple as what I just wrote
down, because we're using two identical pulses, which
means that you actually have three paths for two photon
absorption that all lead to the same final states.
And these are either you can absorb two photons in the pump
pulse already and go directly to the double ionization.
Or you can actually stay in the ground state
and absorb two photons just in the probe pulse.
Or you can go through the path that you're actually
interested in, which is one photon from the pump
and one photon from the probe.
The dynamics you have then in this superposition
here contains actually quite a number of timescales.
You have here the superposition of double excited states,
which evolves on a scale of a few femtoseconds, which
is basically given by the bandwidth of your pulse,
so the bandwidth of the wave packet that you excite.
And you have a very fast beating,
which is basically given by the energy difference
between these double excited states and the ground state,
which both interfere then here to go to the final state.
And for reasons that I don't want
to go into too much detail on, the path alpha basically
just gives an incoherent background for most
of the observables that we look at.
So this is the electron energy distribution you get in the end.
And because of time constraints, I
don't want to talk about this in detail.
But basically the idea is this is the two electron energy
distribution, energy of the first electron, energy
of the second electron.
And what we propose to use is just this cut
of the electron energies and just
measure the single electron energy spectrum.
And this means it's a much easier experiment
than trying to measure both electrons, which you would
usually want to do for correlated electron dynamics.
So we only want to look at the single electron
and some energy selection.
And now the important idea is really
to use this two photon ionization, double ionization
out of the ground state, which is this pathway gamma,
as a reference wave, because this
is independent of the time delay.
This is always the same process as a reference
for the pump probe signal through the path beta,
where you have the interesting double excited state dynamics.
And what this gives you, the signal
that you get, the ionization probability,
as a function of time delay, you get this fast oscillation,
which is the interference between the ground state
and the double excited state wave packet, which is not really
the interesting part.
But then this fast oscillation is
modulated with an envelope, which is basically
given by this interference between the ground state
or the path gamma and the double excited state wave
packet, the path beta.
And this slow modulation of the envelope
on the femtosecond time scale is what actually
encodes the double excited state dynamics.
And the question is, of course, what kind of dynamics
can we see in this setup or in this scheme?
And what we show here is first, in the blue line,
the yield just from the double excited state, which
is what you won't actually see in the experiment,
because it's too small.
And then the red line is this interference term,
which modulates the envelope of the fast oscillation.
This, you see, this is around 100 times larger
than the signal from just the double excited states,
but otherwise reproduces it almost perfectly.
And the green line tells you what observable you are actually
sensitive to here, which is basically
the distance between the two electrons,
so the inverse distance modified by the dipole operator.
So you're basically sensitive to when the two electrons are
close together in this correlated electron,
two electron wave packet.
When they're close together, of course,
it's easier to ionize both of them at the same time.
So you get large double ionization
when the two electrons are close together.
And that's, for example, on this plot, you see this.
This is the electron distance for this point.
You see that they tend to be much closer.
And here, when they're further away from each other,
you get basically no double ionization.
So this is already the summary for this part of the talk.
So I've shown that this pump probe scheme gives success
to the correlated two electron dynamics
in the double excited states.
We only need to measure a single electron spectrum
with an energy filter.
And the main idea is to use the single-pulse two-photon
double ionization as a reference wave for the interferometry.
That's what gives you this increase of two orders
of magnitude in the signal, which might be enough
to do with an experiment, at least in a few years.
With that, I want to finish and thank all of my collaborators
and especially Stefan Nagel and Renate Patsevek,
who for the streaking part did all of the real work
in terms of numerical simulations and so on.
And thank you for your attention.
Thank you.
APPLAUSE