So I also want to thank the organizers for inviting me and giving me the
opportunity to present our results from Innsbruck. So thanks a lot. Let me start
right away. Time is short so my idea is that I will give you a little slower
introduction and then perhaps towards the end I have to speed up a little bit. I
hope you excuse. I know that you've gotten an introduction to the world of
ultra-cold atoms in fact yesterday already so I can perhaps be a little
bit fast on some of the things. Just want to mention I'm from Innsbruck.
Innsbruck is a natural place to study systems with strong correlations. As you
can see the systems here is very strongly correlated as Innsbruck is very
strongly confined. It is in fact essentially one dimensional and it's
ultra-cold. I mean this picture could have been taken 10 days ago and this
can happen at any time of the year when the weather is sufficiently cold. So now
let me start with acknowledging who did the work that I will present today. In
fact this work done was a former student of mine who's now a postdoc in my group
in collaboration with Andrew Daley and Hans-Peter Bufler from Stuttgart. Andrew
Daley is now in Pittsburgh and whatever I'll be talking about today I am
interested in the properties of bosons and optical lattices. So think of this
sketch here and you've seen very beautiful experiments on similar
systems actually with high resolution imaging from Stefan Kueh yesterday. So
what I will be focusing on today is I will be looking at systems where our
coherent interaction part, let's call it u, can be of any value in particular can
be negative and can depend on the various parameters of the system in
particular on the number of particles on each lattice side. So I will show you
what I mean by that in a moment. So that's sort of the idea of what I will
be talking about. Let me just briefly say where this comes from, what can
we do, why can we tune in an interaction parameter and I will give you some more
details in a moment. Basically just to make that very short if three atoms that
scatter a couple to a molecular state the scattering properties of the two
atoms can be changed. We call this a Feshbach resonance. So this is just a
schematic diagram. What it means is that the scattering length sort of the radius
for the cross section can diverge at a specific value of the magnetic field. We
tune the states via the Zeeman effect so the control parameter in the end is a
magnetic field and basically it all boils down to such a behavior on top
of some background scattering length. There is as a resonance, dispersive type
resonance with some width that is given by the strength of how strongly the
free atom state couples to the molecular state at a specific value of the
magnetic field that depends on the details of a molecular structure. So we
just take this as a given fact. We have this tunability as we call it.
Scattering length is a function of the magnetic field and in the simplest case let me
just walk back. This interaction parameter depends linearly on the
scattering length. What would you expect? Now in an experiment I just want to show
you that just briefly without walking you through any of the details. The
scattering length here in units of thousands of Bohr radii depends on the
magnetic field. We have a whole series of these Feshbach resonances. I will just
zoom in around this very convenient range from 0 to about 100 Gauss or so.
And let me zoom in. This is the scattering length now from 0 to 150 Gauss.
And what you can now see is you can see some narrow resonances. And I'm showing
you this to show you basically that there's a finite number of resonances so
they're very well separated. Some higher order resonances but basically that's it
and I'm not walking you through the quantum numbers at all. What I just want
to tell you is that there is a zero crossing for the scattering length which
is well separated from all the resonances so one can really sit
essentially on that zero crossing. And I should say usually we have in principle
we have magnetic field control at the level of a milli Gauss here. So this is
now a hundred Gauss so we can sit here really very precisely on one of these
let's say zero crossings. Typically we cool our atomic sample called the
Bohr-Einstein condensate at some specific value of the scattering length at
some convenient repulsive interaction. We'll not walk you through the physics
of ultracode systems in all detail. Just want to say that this means
that all the atoms sit in the ground state of our system. But the
message is we can tune interactions. We can actually tune them up along one
of these Feshbach resonances. And certainly one has to compare length scales.
I will not again walk you through all the details of that but one has to compare.
This is a thousand Bohr radii and thousand Bohr radii easily compares
with the confinement length in a typical confined setting. And in particular what
I will also need as an analysis tool one can use these
resonances to couple to the molecular states and one can make molecules there.
So I will need that later on. So just one more ingredients that you need.
Now again so far I've talked about the elastic part of the of the scattering.
There's also an elastic part which is summarized by so-called three-body
recombination. And basically sorry I'm too quick. Basically what it means is on
that inelastic processes set in and here I'm focusing on the three on
three-body processes. So three particles combining form a molecule and a free
atom giving up some binding energy. And what I just want to show you here
that there are resonances which we call epimorph resonances because we couple
to so-called three-body epimorph states. The physics of that is from today's talk
completely irrelevant. I just want to say tell you that we have an inelastic
process a decoherence process at a specific value of the scattering length.
In this particular case it's around 900 minus 900 Bohr radii. So just keep that
in mind because we will need it later. And it's actually something that point
is that we not only can tune the Korean interactions but also the inelastic
processes. You see this is actually a drastic change in this inelastic
process. This is the fourth root of the three-body loss rate coefficient and you
see it really goes through the roof here if you plug in the numbers. Now let me
come back to the lattice physics and as I said I'm sure you have gotten a
good introduction yesterday. Basically if you look at the dynamics of particles
bosons in our case on the lattice the situation the following you have three
terms you have tunneling interaction energy the U that I just mentioned
before and typically some additional gentle slowly varying trap and one can
calculate or for cold atoms one can calculate these rates coherent rates I
should say yeah these are Korean processes coherent tunneling from one
side to the other just by using one year functions for example so also there
the interaction we can just calculate by using one year functions I
should already mention here that this assumes that these one year functions
are single part of one year function so non-interacting one year functions and
the external energy shift is just set by some potential. Now what approximations
does what do for this kind of system yeah one first of all assumes that there
are no long-range interactions yeah they are only near they are only on
side and when particles sit on top with each other one assumes that there's no
long-range tunneling so this is just to the next neighbor one assumes in fact
for writing down this kind of Hamiltonian one assumes that there are no
higher bands in the in the considered yeah just the lowest block band and and
also one just assumes that the interaction is just contact and just
writes down a delta function type of interaction and here you see the
dependence the linear dependence of u on the scattering length a so this is very
simple in principle and certainly one has to make a lot of approximations let
me just remind you of what then the system does when one does that when one
takes all this for granted for sufficiently large tunneling the system
is delocalized it is super fluid in the lattice and it is sort of it's all
coherent and if one lets interactions dominate the system is is a system of
localized particles it in particular shows so no phase covariance or would
not expect any any interference patterns and the system then has a gap that is on
the order of u now and this has been well much discussed over the last 10
years in the context of cold atoms one then expects this kind of phase
diagram super fluid phase here as a function so what what's plotted here
should perhaps back up a little bit so this is chemical potential and this is
a tunneling strength here and if one reduces the tunneling strength or
increases interactions when interest into this localized phase with which
depending on the chemical potential of the density of the system can have one
particle per site or two particles per site and so on and in the experiment with a
harmonic confinement one would see in one trap once it would see that all at
the same time one would get a wedding cake structure and I'm sure I wasn't
there yesterday but you showed all this in situ and very nice pictures that one
can actually measure that directly and the list of groups that have been
working on this is getting longer and longer now the experimental setup is
such that we do this in a 3d setting we shine in laser beams to generate the
potentials by optical dipole forces we can control all these parameters in
principle in independently just to remind you the tunneling rate we control
by the depth of the potential the scattering length controls the
interaction and some external dipole trap to some extent also controls that
external potential now how does one probe that and there are various ways
and one of the standard techniques is actually to go into this this localized
phase by by changing the depth of the lattice and then seeing how the
coherence of the system disappears so that's basically shown here in time of
light images and I'm sure you've seen that before so I don't have to walk you
through that let me just to warm up show you what we do we look at these one
of these time of flight images so this is a system is still superfluid will the
pattern looks like that because we look at it up from some funny angle that's
irrelevant for what what I will be talking about today just look at the
width of this coherent peak here at the center and that's what we plot as a
function of the lattice depth the width of that peak as we increase interaction
as we increase the lattice depth for a different strength of the interaction
so we just dial in our scattering length value and then we just go through this
phase transition and basically what we see is that this width has basically
stays constant for some time and then increases drastically and we associate
the phase transition with this kink in this in this curve here and as you would
expect for strong interactions this the localization happens quicker so this
kink happens earlier as one varies lattice depth one can plot that as a
function the stress transition kink point as a function of scattering length
and it actually turns out that basically it depends a little bit how you define
where this kink is I mean turns out this is a somewhat difficult to really
determine where you really would determine this kink to be but it
basically agrees with various theories that are able to calculate the phase
transition point in fact Monte Carlo simulations or your mean field
calculations so this is an unreasonable agreement so this is just a warm-up to
show you that we basically think we understand this phase transition now
let's do further analysis of the phase transition one can look at the mod
insulated excitation spectrum what does that mean one excites the system by
modulation techniques I'll show you in a moment and basically one then moves
particles on top of each other's and has to pay a price and energy for that and
what would one expect naively one would expect well for example if you move a
particle from here to here you have to pay the price of this energy u or if
you move a particle to where you have already two particles sitting you have
to pay a price of 2u in this case you have to pay the price of u to you but
you get the price of 1u so in difference is u so you would expect peaks
at u in the excitation spectrum and at 2u when you excite in fact between the
shell so if you do excitations here between the one at the motel and the one
at the mod shell and the two at the mod shell the technique is modulation amplitude
modulation this has been well established in fact Henning here worked on
this was one of the pioneers with the Aslinger group many years ago doing
these techniques over many years in various contexts so basically let me
just remind you of how this works we take our sample which is a both
answer condensate at some value of the interaction parameter we then ramp up
the lattice we then usually at that point then we change interactions to
some specific value then we modulate and then we go back and we recapture the
system and we then look at the the condensate fraction basically we heat
the system during this process because we cause defects and our one way of
measuring that is just looking at how much of the condensate is left over
yeah so one looks at either the thermal or the non condensed fraction and the
condensed fraction and in the experiments where I say especially this
is just a sort of the non condensed part that's spatially separate in in these
measurements time of actually it's a time of light measurements so now how
does it look like a typical experiment so we measure this fraction as a
function of the modulation frequency and what you can see you can see a clear
resonances in fact I showed you that and it's too quick and in fact this is the
u resonance for our specific values is that at 1 kilohertz and this is a 2 u
resonance which is then at about 2 kilohertz now this is at a conservative
weak interactions now let's increase interactions and here for about 320
bore radii now and now interestingly what happens is that this u peak starts
to become asymmetric and in fact develop some side lobe and if one increases that
further let's move in into that peak then this double structure completely
splits yeah so so these are three curves for different densities that I will walk
you through in a moment but you see a very strong splitting now of this peak
which for strong interactions by now has shifted to 2 kilohertz now this so it had
to has shifted certainly because we have increased the the scattering length here
but it's now completely it's two different peaks so now what is happening
here well I said earlier some approximations have been made to for the
simple theory in particular assuming that we only take the lowest block band
that we just take a simple approximation for the interaction potential and that's
not valid for strong interactions and that has been discussed in the literature
over the last 10 years or even more so particular by Thomas Bush and
collaborators many years ago this is what happens for the energy of the bound
states and just taking just two particles in a well this dash line without walking
through the details is the Bose-Habat what you would need for the Bose-Habat
model but you see this modification is due to the the trap structure so this is
you would expect yeah for two particles and one has to take the right delta
function potential actually one has to regularize it and then the next step is
actually one has to watch watch the situation what happens when three
particles are present yeah they change actually these particles interact and
there the density is changed and so effectively this means one has effective
three-body interactions that that shift the energy of the system and this has
been discussed some years ago in this publication and last but not least just
to make a little connection because that will be the future if I have one minute
later on I will show you that one also has to really take into account the
three-body bound state that might appear at some point in particular for for
attractive interactions yeah but this in fact for the lattice work is open it's
work in progress so but let's see what happens yeah in in the in this Bose-Habat
model and so what one generally would expect is that this U now depends on the
number of particles yeah so it's it's not independent of how many particles one
has in each side and naively you would expect that because if particles
actually do interact you cannot simply take non-interacting functions one-year
functions that describe the dynamics at each side yeah they the repulsive
interaction or attractive interaction would shift would shift which will lead
to a shift in energy so so now one can go walk go back and then say well okay
we have a process here that depends on two particles but now if one does a
process like that or let's say let's as an example take this one yeah one takes
away to you that depend on two particles at each side yeah and but but
has to pay the price of three you with three particles each of these particles
a dash with the with each other yeah so that would be now the right energy that
one has to calculate yeah so and in fact one can probe that because I said well
these processes have happened at higher density so in the two-atom short shell
these happen at lower density in fact that's what we do we test for low
densities yeah low densities in our case means we have essentially just a one
atom module we almost have no occupation in the two atom module
unfortunately we cannot look at it in situ we don't know exactly but we can
certainly test that and show you actually how in a moment and for high
densities in particular we see that both of these different peaks appear so
these are the two peaks that that appear in this particular case now one technique
there are two ways of probing all this one technique is actually looking at double
occupancy by turning the doubly occupied sites yeah for example a site like this
into a molecule yeah so this would be a molecular peak here yeah and one can
look at triply occupied sites by looking at loss so in fact we see then on this
particular resonance where we have strictly occupied sites we in fact see a
three-body loss which we don't see on the other resonance because there are
no three particles on each side so okay without going through all the details of
the theory that's the expectation here and our data sorry too quick the
dashed lines are the Bose Hubbard the simple Bose Hubbard correction it turns
out this is the spring that I just showed you here so this is actually
quite sizable splitting is very large it's on the order up to a kilohertz or
so and it turns out this just as a one comment on the theory the theory is a
perturbative theory and it turns out if it's actually not known how to sum up
that series and what if one can just some one does some assumptions one can
try summing up the theory and then the red line appears the dashed line would
be that what can theory says so in fact it turns out that the this this
three-body process is in detail not understood yet so then the educated guess
of what's happening here and you'll see and show you in a moment we see
deviation from that one those two attractive interactions and I should
mention there's been nice work from the amount of loss group some years back
well actually this splitting was seen down here at about a hundred bore ADI in
the sort of beat measurements with with this mod insulator state yeah so what
did this effect now is much more drastic because we can tune the interactions to
much larger values should mention the typical experiments are done with rubidium
and the scheduling length is about a hundred bore ADI now let's for the last
few minutes let's let me switch now to the attractive case yeah so this is
actually was in that sense a warm-up so and then with the attractive case
three-body loss plays a role as I showed you earlier this huge epimorph resonance
so what happens if one goes to that resonance one would expect naively at
least in free space won't expect the three-body pocket part of three
three-body loss characterized by this rate and in general if one goes into
attractive interactions one would not expect that the super fluid is stable
yeah it should collapse yeah and then in the end it will decay due to three-body
loss and the ground state even the mod insulator case would be basically all
particles sitting on top of each other which is ridiculous certainly because
they undergo this molecular process so now certainly what we can do is we can
first go into the track the repulsive case and then into attractive case and
then ask ourselves how stable is this mod insulator state yeah is it is it for
a track of interaction is it has it some stability or does it decay immediately
and it's for sure not the ground set anymore yeah because one hand now has
attractive interactions and that would be the ground state of the system now how
do we do this in the experiment we load the lattice for some to certain depth
with repulsive interactions indicated in red then we switch to attractive
interactions we go through the zero crossing and go to attractive
interactions the system certainly is frozen over that time scale yeah I mean
the fire at this depth of the lattice the the turning rate is not zero yeah so
the turning rate has still the time constant on the order of 100 milliseconds
you'll see that in a moment and then you can even leave the system or you can
again use modulation techniques to see what happens and then we can measure as
before the excitation that we introduce in the system and observe overall heating
of the system when we go back to the repulsive case let's take a way of doing
it so let me show you data basically what we do is for the following we just
vary the interactions we start as I said with the repulsive case and then switch
to the attractive case and you see we are quite attractive here a few up to a
few thousand bore radii we see that some that something happens first of all
there are some narrow resonance that showed you earlier which are irrelevant
for today's work but they are a little bit nasty if we hit one of these it's a
problem okay we can discard those there's a zero crossing here that because
we course it with the finite speed it causes some problem yeah the system
becomes briefly superfluid yeah and it seems to so we see we see that we always
seem to lose the particles in particular we seem to lose BC fraction as a second
the BC fraction but now we can just stay sit here at the some attractive value and
see how long the system lives and you would now say for the given value of the
depth the system should decay on the time scale of let's say a tenth of a
second and it turns out it's not the case it decays on the tenth time scale
of seconds yeah actually the time scale here is about 15 seconds yeah for even
very attractive interactions so for sure there's a blockade effect and it's
probably two things it's the fact that that we have very strong repulsive
directions attractive interactions actually also lead to a blocking of
particles moving to a site where they can be lost and and one has actually
this dissipative process that does something I come to that in the moment
just to make that data a little bit more complete we also plot atom number as a
function of a scattering length yeah so we do lose some atoms here so there are
some there's some loss happening and okay perhaps we are not fully in the
mod insulator state completely perfectly and so on and so on there's some some
defects nevertheless we the system is not quite is even more stable certainly
with respect to atom number the time scale is somewhat longer you see up to
10 seconds here so we for sure lose coherence faster than we lose particles
okay anyways we do the same measurements as before we do the same
resonances I will not the same technique as before we can look again at the pair
production or the Dublin if you wish efficient creation efficiency with
making turning pairs into molecules so we see three-body loss resonances and
two-body enhancement here so we can analyze these resonances again for
different densities and it's again similar processes as before as you would
expect we can take a whole series of these kind of spectra for different
attractive interactions now so this this this gray part here is this two-body
resonance the dash the red part is the three-body resonance and for the last
few minutes I will focus on that and there's in fact a resonance coming down
from a higher band yeah which on which I will not talk about today it's it has
to do is higher higher higher specially I mean excited states yes band structure
so now this is the upshot I showed you this data already before the splitting
here for the repulsive case now let's go to the attractive case here the theory
fits much less we don't know why admittedly yeah could have it could be
several reasons we do not think that we have a problem with the calibration of
the scattering length that's not the problem we think we still the theory is
still the even the two-body physique it misses yeah now but for the for now I
just will stick with the three-body problem here so the three-body resonance
it turns out this resonance here what's happening with that again the theory
even that that that that educated guess doesn't fit anymore and it this
resonance here shows some funny behavior show you that this is the resonance here
yeah to the real data it shifts as expected it has to shift yeah it turns
out it disappears it broadens yeah and in fact what we think is happening
there's three-body loss as I mentioned before yes this is the three-body
resonance and the stronger the stronger the loss yeah we so we really are moving
up towards this resonance here now this epimop resonance we are way away away
from the top yet yeah we are still at 400 bore area which is about here but
this three-body loss is setting in so and what what what we think is happening
is that in fact this big loss here is blocking any the Korean processes yeah
so in fact it blocks out blocks itself out of existence so what's the analogy
it's actually known as the Zeno effect if you have a three-body level three
level system so I'm excited stay as some some states couple to excited state with
some Korean coupling and you have a loss process from that you can ask yourself
the question what is the effective loss from here to there yeah and I'm sure
many of you have seen that and the calculation is trivial to calculate that
that effective loss rate yeah given some detuning here what's the analogy in our
system in our system the analogy is atoms at these wells coupled via the
coherent tunneling the loss is three-body loss and the process is then
its particles get lost here so what's the effective rate and one can do these
associations particular J is equal to Omega here and the detuning corresponds
to the coherent evolution and for sufficiently large three-body losses
should look like that yeah so you have this very characteristic suppression
well known for anybody who has looked at the Zeno effect that it squares with J
is J squared over the over the old loss rate yeah so you see that large
three-body loss should stabilize the system so one can look at these widths
of these resonances yeah that I should just showed you before sorry let me just
go back of these rest of this resonance here yeah we can plot that we don't have
a theory of how it should look like for the attractive case so if we go more and
more attractive this width increases we it becomes hard to measure a width
because it gets so broad yeah that disappears yeah and we can also look
for the repulsive case the increase here is not so drastic because we don't have
this enhancement in near this fm of resonance it would be nice to understand
the details of that so I'm just showing you the this very general data now the
last two minutes let me ask you why what could happen now we have a process like
this where we have let's say doublons yeah not molecules just pairs yeah and
they move in this letters yeah the question is could that be stable for
attractive interactions yeah in fact it should not again when this this particle
moves here this double on particle moves it for particles is already three and
more and you have again the three-body loss process so that also should be
blocked yeah so you could certainly have a process like that where there's an
empty site so the question is could that system be stable and in fact Andrew
Daly a few years from Innsbruck suggested that one could have a very
interesting system that would have not a mod insulator of dimers but for the
attractive case one have would have a superfluid of these doublons yeah so
there would be a very peculiar state because it's not the ground state it's
in a highly excited state but it could be super fluid and could be stable due
to this quantum zero effect and that's actually the ongoing work how do we
prepare that how do we prepare such a system we start with the one at the mod
insulator state so I can only show these pictures Stefan sorry don't look at
this yeah we can I can only do a comic yeah because we can see in situ these
pictures but for what I will show you it's at the moment irrelevant what we
actually do now we create doublons in a very very nice trick where we tilt the
letters briefly and tilt it back yeah this is known as quantum magnetism just
recently this was used a pioneered in the grinder group that one just has one
particle move to the side this imagine you tilt along one of these directions
forget about the movement dynamics transversally just along one
directions and each of these 1D systems independent and one creates then this
kind of situation where one has a spin up spin down spin up spin down to this
two-room magnetic state in fact we can do that in the experiment and that would
be our starting state yeah let me just show you that we are working on this
right now and this double on creation works very well yeah this is the energy
shift the tilt that we've introduced for different interaction strength and
we see that the singly occupied sites we initially have about in this case about
70,000 atoms and the single one atom mod shell actually gets drastically reduced
yeah and and the two W occupied sites get get strongly enhanced for various
values of the interactions you if you have a weaker interactions you happen
you expect that tilt to do induce that transition first quicker yeah so this is
just the raw data is very very preliminary if one puts in numbers it
turns out we can turn 75 percent of our atoms into double ons and the question
is what's the limit yeah can it be I mean basically we should be able apart
from the tiny super fluid fraction that's around the one at a module one we
should be able to turn all of that into into double ons yeah so and that would
allow us to to then look at the dynamics of such a system and see whether this is
this is blocked or not yeah so my time's up I'll stop here I will not go into but
I know some of you that I would like to talk to about now what happens with
these epimorph resonances in the in the context of the lattice in fact we have a
lot of data that we don't understand but I want to start here stop here and thank
you for your attention