Thank you very much for saying for the nice introduction and of course thank you very
much also for the invitation to come here and for organizing this wonderful workshop
and actually I have to admit also the breaks are really quite valuable so we've been discussing
so much and in such a good way that we could even have more breaks.
But let's start with the talk.
So what I'm going to be telling you about is deterministically prepared few fermion
systems and of course in the course of this workshop I mean there have been many talks
which actually focus exactly on this atomic systems, molecular systems, of course always
deterministically prepared then the ground state will always be exactly the same.
Every single implementation that you realize will be exactly the same.
So now what is it all special about this thing here in my talk?
I hope I will convince you that there are a few things that are kind of peculiar about
it.
First of all let me give you a brief motivation why we actually are interested in these kind
of systems so this is kind of ultra cold systems that we are talking about here.
What we really realize in our lab is an isolated quantum system in which we really can control
all the degrees of freedom.
And that is the interaction strength, the confinement and the motion of the particles.
And you can really picture our systems to be looking very much like this cartoon here
like more or less harmonic confinement and then the lowest vibrational levels of a harmonic
oscillator filled one by one with a spin up and spin down particle each for example.
And of course what we would like to do here is study pairing in this system kind of various
regimes, weak coupling, strong coupling, so weak coupling you could think of this system
kind of being if you had like for example three up particles and one down particle this
would weakly pair to just the particle which is sitting in the same motion state.
But if you have strong pairing you know this one minority particle will somehow be bound
to all the other ones paired with all the other ones forming something that looks a
little bit like polar on and this kind of systems can of course be studied with quite
big flexibility and also precision in this isolated quantum system.
And of course you can think even further and can ask the question once you have a real
isolated quantum system you would want to see dynamics, you want to see the time evolution
of such a system and typically we are dealing with isolated systems but usually we can still
describe them very well as being in a thermal bath with a particular chemical potential and
so on because the systems are so large that we cannot make the difference between an isolated
and a system and a system that's in a thermal bath.
But this should be very well possible in this finite deterministic systems that we are studying.
Okay so for the motivation so what I'm going to talk to you about is first of all I'll
explain you very shortly how we actually prepare these systems and what are the tools that
we can use to manipulate them and if you look at these kind of pictures here the cartoons
that I'm drawing for you here for the second part where we study repulsive interactions
between the particles it looks a little bit like the cartoons that people from the ultra
short community draw when they draw their coolant potentials with electrons bound in
them so we can also tilt our potentials in our case not with short laser pulses but just
applying magnetic field gradients and we can look at tunneling of individual atoms out
of our potential well.
And so in that sense the physics came many places be quite similar than the one in the
ultra short community and the third part will then be devoted to similar experimental techniques
but we will study attractive systems and here we will study correlated tunneling of particles
that's also a topic that's quite common in the ultra short community but we are mostly interested
here in kind of analogies with nuclear physics where you have for example tunneling of pairs
in the sense that you have alpha decays or alpha particles tunneling out of your nuclear
potential and what I will also show you that we can observe a strong or even effect so
it depends very strongly of whether we have an odd or even number of particles in our
system how strongly the particles are bound together.
Okay, so first of all how do we prepare these systems?
We start with a conventional ultra cold Fermi gas like many of us have this in laboratories
a typical size will be 0.1 millimeter maybe even a little bit larger with a temperature
on the order of 250 nano Kelvin Fermi temperature half a micro Kelvin so mildly the generate
gas yeah so something that's relatively easy to prepare it's actually a lithium six atoms
but it doesn't really matter but what we do then is we superimpose to this kind of conventional
ultra cold Fermi gas a very tiny trap which in its size is in the order of a micrometer
or a few micrometers and it has a depth that's on the order of it's also a few micro caravans
so much deeper than the Fermi temperature of our reservoir ensemble yeah which suddenly
gives us quite a strong enhancement of the degeneracy because now the Fermi temperature
will be the depth of this combined potential on the order of three micro Kelvin whereas the
temperature is still if they're in perfect thermal equilibrium a fraction of a micro Kelvin yeah and
so we end up with something below a tenth of the of the Fermi temperature as as the temperature
of this combined system and if you now calculate what the occupation probability will be for the
lowest energy states in this tiny trap here you get some some value that's very very close to one
point zero point nine nine nine four nines yeah so that's uh that's very close to one so close
that we cannot even measure this and we of course not limited by this policy we cannot ever
we have no idea how to re-measure such an occupation probability in the laboratory
okay so then we remove our reservoir and this is a sample uh we are studying and so in this tiny
trap we have maybe something like a few hundred atoms and now we start tilting this trap so
basically do exactly what you do with ultra short laser pulses you tilt the trap and remove atoms so
with your ultra short laser pulse you would remove electrons so we're removing atoms
and by tilting this trap in a very controlled way so that's kind of a dc field that we apply
we can just choose how many vibrational levels we will keep in our potential well
yeah for example in this configuration i'm drawing you here there will be five vibrational levels
you would expect to uh to have if every vibration level is occupied by one spin up and one spin down
atom in this case for example you would uh have 10 atoms remaining okay so that's kind of the idea
behind the technique so what this looks like in a classical system is shown here so we call this
kind of the classical atom culling machine so what's done here is that's actually uh you know
when when when you when you obtain your phd in germany you get this nice hat with all this
little things built on top of it that kind of illustrate what you've been doing during your
phd and that's uh uh free time savannah's head what you see up here is um a kind of reservoir
with lots of atoms in it and so you can fill from this reservoir like a tiny shot glass with atoms
and you can tilt this it actually really works yeah so there's some servos that control the
motion of all these uh different parts and there's also a laser that you need to focus and so on
and what you can see here is in the end there's four atoms remaining now in this case
if we tilt the potential back yeah and so that's basically the classical way of uh doing things
but of course we are interested in the quantum world so there's a little bit more difficult
than this but it's not so much more difficult so what you can see uh in uh in uh in the real
experiment what's plotted here is um the the histogram of many many implementations of the
same experiment and the histogram of the fluorescence that you measure in a magneto optical
trap after we prepared such a system and what you can see there's a distinct peak of fluorescence
at so that's the fluorescence signal normalized to atom numbers so if there's if it means one
there's the fluorescence of one atom and you can see that in basically unity probability we really
prepare if you want two atoms in our trap kind of kind of which kind of uh corresponds to this
simple cartoon picture and we can do the same thing also for eight atoms for example you see
eight atoms so there the fidelity is a little bit worse in preparation but it's still reasonably
good so this really looks like this and um so we can reappear the number of atoms with very good
precision and what you should also i mean what you might probably now uh question is whether this
simple cartoon picture that i'm giving you here just drawing simple harmonic oscillator levels
well this is it all valid for the systems we are looking at yeah and it's very well
valid for one single simple reason the aspect ratio of our trap is about one to ten so it's a
cigar shaped trap with a one to ten aspect ratio so uh this will be the axial vibrational levels
here and the first radial vibrational level will be about at the height of the tenth axial level
yeah so the first radial excitations will appear much higher in energy so you really can think of
this system of being a one-dimensional system with more or less evenly spaced harmonic oscillator
vibrational levels so that's really the the system we are studying yeah and so now i told you we can
control the number of particles we can control the motion of the particle because we can really
prepare them in the ground state and now we can also control the interactions so for all these
experiments we have a knob that we can control the scattering length between the particles
so we can tune the scattering between non-identical particles using a so-called feshbar
resonance so it has been explained in many talks before so we're doing the scattering length here
with the magnetic field and there's some value where the scattering has become exactly zero
that's where we prepare the systems so we know exactly we are actually preparing a non-identical
system we know exactly what kind of system we prepare and what you should also note that for
all our purposes identical fermions so fermions with the same spin they will not be interacting
because we're in the s wave limit of scattering and the s wave scattering length between identical
fermions is zero yeah so we can control the interactions between non-identical particles
whereas the interaction between identical particles will for all our purposes be zero
okay so this is kind of tuning the scattering length using a magnetic field but now i told you
we're kind of in this 1d regime and we've had we've had a very nice talk by vladimir in the beginning
of this workshop explaining us how our interactions are modified in 1d so basically we have to take
into account that we have some radial confinement and then by this is formula for by maxime or
shani we can actually calculate from the 3d 3d scattering length we can really get obtain a 1d
scattering length which basically shifts the the resonance interaction strength to a little bit
lower value in magnetic field yeah but we can still basically access the whole interesting
range of interaction strength in this one-dimensional system
okay so this kind of the preparation and uh of our system and now we really want to do some physics
with this the first thing is uh we study the repulsive few particle system so this kind of
system we tune in some repulsive interaction so basically tune this one-dimensional coupling
strength to positive values here and so the two words that i'm giving you is feminization
i'm going to explain you uh to you what this means and magnetism and now you can ask yourself
so if you're a condensed matter theorist most of you will tell immediately okay the ground state
of a one-dimensional system without uh pinning the spins should be not magnetized yeah and
there are several theory papers on this uh proving this and one that i like particularly
well is this paper here by leap and mattis um where they actually calculate the energy
of a system of n electrons in one dimension with an arbitrary symmetric potential
and they basically determine what the energy difference will be if you flip one of the spins
and what they find is that if you have an unmagnetized system if you have
if the total spin of the system is smaller the energy of the system should always be smaller
than the case of the spin polarized system yeah so if the total spin is smaller the energy should
be smaller that's basically a conclusion from this paper and so you can nicely read that paper
and you can immediately understand that the proof is correct but so we'll see in the course of this
talk that there's some way around this yeah that we can still observe some magnetic ferromagnetic
effects even though this paper is right okay so first of all what's our major tool we basically
observe tunneling dynamic out of a tilted strap yeah and first of all i'll explain to you all of
this with just two particles interacting because this makes our life much easier to understand
the system yeah so we put in two particles one spin up one spin down we have them interact
repulsively and then we tilt the potential so much that particles can tunnel out of the trap
and from the observed tunneling time scales we then can infer the total energy of the system
so this looks kind of like this this is a measurement where we plot here on the vertical
axis the mean observed atom number so remember every time we make a measurement it will be an
integer number of atoms because we simply count atoms so there's always the mean of many many
measurements so basically you can say this is the expectation value of atom number at a particular
point in time so here's the time that we tilt the potential and then there's two different for
different interaction strengths these are these different curves we also have different dynamics
different time scales and the first thing that i want you to see from this measurement in the
first place is so we start with two atoms smaller so it's a little bit less that's the preparation
fidelity we always end up with one atom so basically what's happening in this experiment
so we apply some repulsive interaction between the atoms we end up with one atom remaining in
the trap and so why is that so what happens when we apply some repulsive interactions if there's
if the interaction is basically zero the atoms both sit down from the ground vibrational level
of the strap and tunneling is quite weak because the energy is very low but now we apply some
repulsive interactions that the energy is increased which helps tunneling which favors
the tunneling but as soon as one particle has left the other one will be in the ground state
and will be protected from further tunneling of the trap yeah so that's how we can in in very
simple ways analyze this kind of behavior we simply fit some a simple exponential decay
to this one particle leaving the trap and this time this decay constant that we fit we plot as
a function of magnetic field which of course tunes the repulsion between the particles
has kind of this measurement so we apply some repulsive interaction increase the energy from
the ground state such a tunneling occurs yeah and what you can see here at low magnetic field
the repulsion is very very weak you observe very long time scales on the order of a second
and then you see as the magnetic field is increased the tunneling time becomes
shorter and shorter because the energy of the system is increased now what's also
applied here is another curve and this curve is some very nice calibration of our system
because what we do for this curve is we fill our trap not with two distinguishable particles
like they spin up and spin down but we put in our trap two spin polarized particles they're
not interacting and because they're fermions they will occupy the two lowest vibrational levels in
the trap yeah and so why we tune the magnetic field this system is not tuned yeah and so this
gives us a calibration measurement because now what we see from this measurement from these data
here this will be the tunneling time of this single atom in the first vibrational level of
our potential yeah and so this is on the order of hundred milliseconds so that's where how we
tuned the tilt that this is kind of in the in the medium range for us to be observed yeah
and now what you can see and what's quite interesting this particular system with two
distinguishable particles it will be first have a lower energy that's why the tunneling time is
longer and then when the coupling strength diverges to infinity then this system has the
same tunneling dynamics as the spin polarized system yeah and so this also means it will have
the same energy and in one dimension it also has the same wave function square that's why this
system is called fermionized right where the coupling strength diverges and where these
tunneling times with these two systems are the same and what you can also see in this graph here
is that you can even go further and increase the energy of this system beyond the energy of two
spin polarized particles yeah so you can make the repulsion between the two particles
stronger than the fermi repulsion between two identical particles yeah and that's the so-called
super tongs regime that was first prepared in this paper here with identical bosons
and so we are starting kind of the two particle analog here of the tongs regime which is the
strongly interacting regime around this resonant interaction strength in g1d
okay so what you can see and that's some somehow what you should memorize if i go
right on this diverging coupling strength i get the same energy for the two non-identical particles
as for two identical particles and if i go beyond the energy grows even bigger that's something we
should remember for the upcoming um measurements but first of all let me explain you why this
should be the case yeah that's actually some sketch that has been shown for example also
like this of nega before and also by vladimir so um what we basically do is what's plotted here
is one minus one over the one dimensional coupling strength it's kind of weird because units but why
we plot it this way is because we increase the repulsion as we go from left to right on this graph
so here is basically zero repulsion and then the repulsion increases more more more and right on
this axis where one over g1d becomes zero that will be the resonant interaction strength yeah
and we increase more more the repulsion as we go this way so we start here with zero interaction
so that's kind of our harmonic oscillator potential with harmonic oscillator levels
and now as we increase the repulsion we change the magnetic field this way and we increase
basically all the energy levels per little bit and right here on resonance we've increased the
energy by one h bar omega in our trap and if we go even further all the way across this resonance
then we basically something doesn't shift right here yeah we basically shifted all our energy
levels by two h bar omega and there will be new energy level with the same interaction strength
with at again zero interaction strength will be the same one as this one but as we tune our
interaction strength continuously we actually go to an excited state in this in this system in
this way we don't end up in the ground state of the system but we end up in an excited state
that's the important thing to know yeah so we increase the energy of these two atoms
by two h bar omega and we are no longer in the ground state of the system
so this is for two particles for this you can have an either an exact calculation 3d solution
for example from this paper here or you can simply also use for all purposes that we are
interested in here simply tomaz-busch derivation of the scattering of two particles in in one
dimension they were perfectly fine for this case and now we should go with this kind of insight
go and add more particles than two yeah so this maybe i said already yeah so this would be
normal this is called the repulsive side where g1d is positive and it would be called the
attractive side uh but really we make the system more and more attractive because we go to an
excited branch in in in this energy level diagram so this would be kind of the attractive case
down here but because we're in an excited state it's still the energy is still growing as we go
to the attractive side of this resonance that's something you should also keep in mind so now
let's go back to this graph here just to give you an idea what really happens so these are
these curves that i showed you before where we said okay there's only one particle of the two
particles tunneling so we have one particle remaining and so if we have weak interaction
strength so we have slower tunneling dynamics we increase the repulsion we get this curve and you
see these other data points here these are the two identical particles and you see the dynamics
exactly the same this is because if you draw the wave functions this the relative wave function
between the two particles so with this one will be more or less looking still almost looking like
like non-interacting system but the particles slightly repel each other and right for this
curve here the wave functions look like this so this will be the wave function of two identical
fermions so this every one of us can immediately calculate and this will be the wave function
corresponding to two distinction particles but we have the same energy as the two identical
fermions and that's why the wave function looks almost the same but the sign is flipped at the
at the origin okay and then if you go even further we recover this kind of wave function
which almost looks like the one of of the second excited state in the harmonic potential
okay so now let's go for the for more particles this is now the simplest case that we can do is
just add one extra particle yeah it looks like it's a very simple problem so we add one more
particle here and we can now ask the question what's the probability for the for this spin
down particle here to be the first particle to leave the draft if we tilt it yeah whether this
happens or this happens and of course very naturally we'd always expect that somehow this
guy up here yeah the majority particle should leave our trap first yeah that's what we expect
and let's see what happens uh that's what what we plot here is the probability to have this
minority particle leave first so that we obtain this kind of a system and this is plotting it
again here as a function of interaction strength and so this is kind of zero repulsion on this
side and as you go further to the right on this graph it becomes more and more repulsive
yeah and the middle here this would be the case of infinite interaction strength so what you can
see here as we increase the repulsion that we get some finite probability to for this minority
particle to tunnel first it's kind of interesting and it increases more more more and if we go for
very large repulsion we obtain a probability of larger than 50 percent for this minority particle
to tunnel which is kind of weird yeah so um and also on right on the resonance if you kind of
extrapolate interpolate between these two points you obtain something like one third yeah and
that's actually something one should expect because right on on this resonance i told you
the wave function should somehow look more or less between two non-identical particles should
look the same as between as the one between identical particles and so the energy of these
all these three particles should be about the same and that's why the probability for each
particle to leave the trap should be the same yeah and so if you go even further then somehow
the minority particle has a preference to leave first yeah this kind of weird you can try to do
the same experiment with even more particles for example with uh uh four particles or five
particles where we each have one minority particle so the extreme case would be this one
and these are the corresponding data here what you can see is it there's actually some dashed lines
here where it doesn't work so well but if you go all across the resonance you can see that the
probability for this minority particle to leave will still remain around 50 and that's even more
striking because imagine i mean if you look at this picture and you want to try to remove the
minority particle first it doesn't look very intuitive that should happen but now what you
can see really is the repulsion is so strong that you can that this still occurs yeah
and the last measurement on this series that i would like to show you now this is all kind
of strongly imbalanced system where there's only one minority particle
yeah you could think of this also like as a polar on problem if you were not to increase
the majority number of majority particle more and more but the another system that we are
interested in that we that we study in this way was this a balanced system so two spin up particles
two spin down particles so that's this kind of uh the smallest uh spin balanced fermionic
system that you can make somehow and we now we tilt the drop so much that there's only two
particles remaining and what we look at now is the probability of having a spin polarized system
remaining basically remove two of the four particles and we we just count whether we have
two spin ups or two spin downs or a mixture of both yeah and of course if you look at this kind
of picture and i were just you know to draw to tilt this potential you'd always argue there should
always be one spin up and one spin down particle leaving yeah and we should have a spin balanced
system if we have two particles remaining yeah so let's try it here now the data is on now this blue
point and what you can see as we cross this resonance go to very strong repulsion there's
about 60 percent probability of obtaining a spin polarized system
and this is for us a very clear manifestation of ferromagnetic correlations in the system
so somehow the different spins separate from each other in this case now in energy but this
must immediately also have a consequence on the spatial distributions of the particles yeah so
what we really observe here is ferromagnetic correlations in this finite system in a one
dimensional finite system yeah and so somehow we have now kind of we went around this proof that
ferromagnetism should not be present in one dimensional system for just for the reason being
that all the theories so far only took into account that you can go to
a g1d equals infinity which will be this vertical line here but if you can go beyond yeah create an
excited state of the system then you can still obtain ferromagnetic correlations i mean that's
the finding of this experiment and you can now of course ask the question what does this really mean
yeah do we really have kind of a tiny ferromagnet here yeah and then just i mean the few points that
one should kind of consider to in the way one thinks about this the first thing is the system
that you prepare is obvious obviously not the ground state yeah and so of course many of these
assumptions i mean you typically want you want to make the assumption that the system you look at
is the ground state of the system the second thing is this magnetic state occurs only kind of in the
attractive regime of this confinement induced resonance not in the on the repulsive side but
that's also kind of a consequence of the system not being in the ground state and because the
system is finite yeah there's no phase transition that occurs between this case where the system
is kind of anti-ferromagnetic where on every vibrational level there's exactly one spin up
and one spin down and the ferromagnetic case there's no phase transition and just a crossover
because the finite system but of course if you think of magnetism you always think of
phase transitions and they're not present here in this system yeah but what's also kind of important
to remember is the system is we can prepare the system in such a way that is completely stable
even though it's not the ground state it will still be completely stable against any relaxation
in that sense you can still somehow call it a ferromagnet yeah so we we try to observe
the stability of the system we couldn't go for some reason beyond about 0.6 seconds but that's
really long compared to the 1.5 kilohertz strap frequency in the in the axial direction that
governs the dynamics of the system okay so that's the part of repulsive interaction
let me shortly go for the case of attractive interactions in this system and there we look
for correlated tunneling or even effect i told you already before so this system is actually
much more difficult to study so now what we do is let's take first of all two particles
we make the atoms attract each other so we reduce the energy of the system and so we get similar
dynamics as i've shown you before we start with two particles and you look at the mean atom number
and the first thing i told you to look at when i showed you the dynamics for the repulsive case
was that we will always have one atom remaining now if i look at this curve here you see very
clearly it goes to zero so that number now goes to zero so we no longer have single particle
tunneling in our system yeah which complicates situation quite a bit because now we have to
always consider the tunneling of two parties at the same time somehow yeah so and if we now change
the the attraction so first this was kind of this blue curve here was relatively weak
attraction now we make the attraction stronger yeah then what happens the system lives for longer
times as is as we increase the attraction in the system yeah so that's what you can see from these
curves here but it's a little bit harder to analyze now because there's always two particles
turning at the same time so we don't have very clear way how to think of this yet but we try to
analyze how exactly can we see now whether the two particles leave the trap together or the way
they just leave one after the other and we try to get some measurement on this and that's
basically what we can do is we can only count the number of particles remaining in the trap
yeah so basically we can after some time wait time we can basically just count there will be
like for example this case would be both particles are still present in the trap we can count you
know whether there's one particle left in the trap or we can have the situation there's no
particle left in the trap but this could have happened in two ways you know we can have first
one particle leave and then the other one or they could have left together and now how to find out
what's the real process that's happening do these two particles leave together
or do they live just one after the other so we do for this purpose we plot the probability
of observing one single atom so this case here as a function of the expectation value so the mean
observed number of particles in the system so we start basically here that that would be in the
beginning initially and then as time progresses the mean number of particles is being reduced
and first so this is kind of first uncorrelated tunneling so basically zero attraction between
the particles and then of course the probability for observing one particle so this case increases
and then decreases again as we go for a zero number of particles yeah which makes sense
and so now if we increase the attraction strength between the particles what happens is the
probability of observing one particle goes down so it is two different interaction strength
and you can see for relatively strong interaction strength we basically never
observe a single particle in the trap which tells us that the particles have to tunnel in pairs
okay so that's still relatively preliminary data and we still have to figure out exactly
what we can learn from this but you can already see that we can observe somehow particles tunneling
in pairs and the last thing i would like to show you uh today is this odd even effect i promised
you and there we basically would like to study how the energy of the system evolves as we add
more than two particles and what's plotted in this diagram it's a little bit involved
so this is actually a plot for different atom numbers and we plot we measure now
the time scale for a particle to leave the trap for two different cases either for a non-interacting
system or for an attractive system and so the first point here that's basically just one single
particle and so that's now the ratio of the of the time scale for an interacting system versus the
time scale for a non-interacting system and so for one single particle if it's interacting it
doesn't matter because it can only that the energy will not be changed but as you have two
particles and we make these two particles attract each other then the time scale for tunneling
is increased by 60b so the stability of the system is increased by a large fraction already
as we add a third particle the non-interacting system is of course again less stable but the
stability decreases compared to the two particle case because this one single particle in the first
vibrational level is not paired to any other one and you can go on so this would be like the
even number of particles again and you can go up we can we did this up to six particles and you can
see then the stability of the system has almost increased by a factor of 10 so the time scale for
tunneling is almost 10 times longer for the interacting system as compared to the non-interacting
system and i'm more or less done yes yeah so uh here's a summary first of all we can prepare
these few fermion systems with quite good fidelity and we have very nice control over
both repulsive and attractive interactions we can look at fermionization of uh distinguishable
particles we can can observe magnetic correlations in the repulsive few body systems and we can
observe correlated tunneling and an odd even effect for attractive interactions and with this
i would like to thank first of all my team for doing all this great work yesterday we've been
working really hard to make all of this possible and i would like to thank you for your attention