Well, thanks for all sticking around.
I kind of expected the room to be half empty by now,
but everyone seems to want to stay for the entire workshop.
So first, before I get started, I'd
like to mention my collaborators,
the people I've worked with a lot on this.
Argyvon Safavi Nayani, Naini, OK.
She's a graduate student at MIT that's
been working at ITAMP with Hossein quite a bit,
but I got to borrow her for this.
And then Javier von Stetcher, who was at GILLA,
and then Barbara Caprogosso and Sonne,
who was an ITAMP postdoc, but now is
at the University of Oklahoma.
Caprogoso, Capo, Caprogoso, Capo.
I can't spell words.
So yes, just throwing me off my game here, Hossein.
So I'm going to be talking about three-body interactions today
and RF fields and couplings.
I'm just really glad to be following Chris, actually,
because now that means I don't have
to give a description of the FMOV effect
and all the glorious, beautiful things that show up there.
Now, there is a bad downside to following Chris
when you're talking about universal three-body physics.
It's kind of like the act that follows the Beatles
or something like that.
So there's a lot of interesting work going on in FMOV physics
and in universal three-body physics,
and a lot of great things left to do.
But one of the questions that I wanted
to start answering and thinking about
is, we have all this glorious universal three-body states
available to us.
Is there some way that we can start using them
to produce new physics in a laboratory environment?
Is it beyond just probing FMOV physics
when we start using FMOV states to do something?
So that's kind of the picture I have here.
So I have three overall topics.
I'm going to talk about kind of RF association
of three-body states briefly, kind
of more of a review of the state of the art on that.
Then talk about how we can use these kind of RF coupling
schemes to create a tunable three-body interaction
on the lattice sites.
And finally, what the resulting phase diagram for the Bose-Hubbard
model comes out, including these.
So moving right on to RF association.
So this is really just a review of what's
happened in some ideas on how we can push this idea things
forward.
So FMOV physics, fairly well-established now
in some experiments.
You saw from Chris, there's a lot of experiments,
a lot of groups out there seeing all sorts of new things.
A lot of great work still going on
with this universal short-range cutoffs and things.
But one thing I was going to point out
is most of the things that have been done so far in trying
to probe FMOV physics, or universal three-body physics,
happens at zero energy.
It's all scattering properties.
So we've seen these three-body recombination
resonances here and here.
People have now seen the atom dimer scattering resonances
and these Stukelberg minima that have been seen.
But it's all at zero energy threshold.
So you can say maybe we've seen the zero energy spectrum.
But that's not much of a spectrum.
So the next question is, can we start
seeing more of the FMOV state?
Can we actually start probing and looking
at the binding energy of these things?
So there's some early things going on.
So recently, there have been some experiments
in RF spectroscopy recently.
I mean, I guess a couple of years ago.
That's old hat in this world, I guess, now.
So just to show a picture of what this RF Association
kind of process looks like, so for lithium-6,
it's actually not purely FMOV physics.
It's not three identical bosons interacting resonantly.
It's three distinguishable fermions
that can have a scattering length that
can be tuned to fairly large numbers.
But you can never have all three scattering lengths being
the exact same and resonant at the same time.
But this is kind of the overall energy diagram
that you would expect to see from this system.
So the experiment we imagine is we
start with a two-component Fermi gas, a mixture of dimers
and a single atom.
And we want to flip one or the hyperfine state
or the internal state of one of these
to the third distinguishable one.
And there's kind of three different things
that we can probe here.
We could go RF associate directly
to the FMOV state, the three-body bound state.
Or we could kind of do a spectator particle flip,
where the dimer doesn't get changed around
and the third particle just flips over,
so sort of a single-body process.
Or really, there's a third one where
we can break up the dimer and flip a spin at the same time.
Those are the three processes you can really see.
And then there's actually another threshold
that I'm showing in here that we're not really going to probe.
And you can go through and do the experiment.
This is from Nakajima at all, a Tokyo group is.
Now, sure enough, this is exactly what they see.
They see a nice little loss.
And once they go to these three distinguishable particles,
they can get very strong three-body losses.
So they can look at the loss rates
or the number of atoms remaining and see
these kind of resonances.
So they get one resonance that's a little bit lower here.
That's the FMOV resonance.
One very deep one that is just the single-particle flip,
and then one that's including the breakup.
And the thing that we're kind of concerned with
is now this difference in energy.
This is mapping out the FMOV energy.
So here's some of the experiments that have done this.
The first one I should have probably mentioned this
was done in Selim's group in Heidelberg.
That's this one here, where they started mapping out
this FMOV binding energies.
And they saw it fit the universal theory very nicely.
And then after that, the Tokyo group
did the same experiment and found
that it didn't fit the universal theory that nicely.
But still, I think it's pretty close.
And then more recently, in this kind of odd, like no spin flip
RF spectroscopy, kind of a wiggle spectroscopy method group
in Tel Aviv, Lev Kajkovich's group
has started mapping out the binding energies.
And I don't really understand this process very well.
It's kind of like using an RF wiggle
to change the scattering length in non-adiabatic couplings.
And I don't understand it well, but they
claim that they've got this working very well.
So this is kind of the state of the art now.
But one of the things that I can point out here
is still we're dealing with things that are still
very close to threshold.
All the lithium-6 experiments happened in a very near
an atom-dimer threshold.
And the Israel group has all done things
near this atom-dimer threshold.
And the reason that it has to go that way
is because these FMOP states, while they're
large universal three-body states,
they're really not that large.
They're pretty small states.
And if you want to do an RF association,
you have to have decent Franck-Condon overlap,
or you'll never get any formation of trimers.
So when the states are near here,
they have this very large kind of atom-dimer type
of wave function, where you have a dimer with a very weakly
bound atom.
So it's this huge kind of Feshbach-looking molecule
that has pretty good overlap with either the three-body
continuum, or in this case down here, the atom-dimer
continuum.
So we're still kind of limited to really near this atom-dimer
threshold.
So is there a way that we can extend
the regime in which our spectroscopy can
probe FMOP physics?
So this is a quick, simple idea.
Why not just use a tight trap?
Put three particles in a tight trap.
If I start, say, with three identical bosons,
and flip one spin, if we imagine that the scattering length
between the blue and the red here is very large,
you can get FMOP states attached to this excited threshold.
But the presence of this external trapping
limits the spatial extent of the original wave function,
and gives you fairly good overlap
with the final state, which can help us.
And in this picture, we have the bound state
and the initial kind of near-non-interacting three
particle state.
So we can kind of think about this as a two-state system.
Two-state system will have some Rabi frequency associated
with it.
So we can find out what that Rabi frequency is.
And we can actually get that analytically.
It's some poly gamma functions and the like all over.
But I'll show it just as the graph here.
And here I'm showing the three-body Rabi frequency.
I've assumed that the identical boson scattering length is 0
for this, and that the dissimilar scattering length
has gone off to infinity.
And as a function of the energy of the FMOP
trimer in the trap, minus the non-interacting energy,
we can map out this three-body Rabi frequency.
And I should also point out the Rabi frequency
here is kind of in the weak coupling limit.
It's exactly equal to the one-body spin-flip Rabi
frequency times the Franck-Condon overlap,
and then we get this nice little square root of 3,
kind of a boson enhancement, identical bosons.
And you can see that you can get all the way up
to like 70% of the single-particle Rabi
frequency, which is significant, which might give us a new probe
into mapping out FMOP state binding energies.
And so just to make that a little bit more concrete,
if we put in rubidium 85, say, or something like this,
there's a van der Waals length of about 82.
And I said the three-body cutoff that Chris had talked about,
too, something about that, doesn't exactly
give the right 9.5 three-body parameter,
but it gets pretty close.
So we use that as an estimate.
Now, with a 10 kilohertz isotropic trap,
you can find that the binding energy of this state
is about two trap units, 2 h-bar omega.
And the three-body Rabi frequency
is about 60% of the single-particle,
which is significant.
I mean, right away in an optical lattice,
10 kilohertz shouldn't be too hard to achieve.
Pretty reasonable number.
So we should be able to, right away, see bound FMOP states.
I mean, it depends on what this dissimilar-particle scattering
length looks like, but it shouldn't
be too big of a problem.
All right.
So that was all I had to say about RF spectroscopy.
So now we can have this way of coupling an FMOP state
to a ground state fairly strongly.
So can we start using it?
So that's what I'm going to talk about here.
And here I'm just kind of showing a schematic
that I'll show a little bit later,
kind of two, three, and four-body systems with an RF field
coupling it.
We're going to use the fact that these
are different at different energies in the near future.
So going back to this idea of using this two-state system,
we start out with a basically non-interacting identical
three-boson system.
FMOP state, up above with a single dissimilar particle,
can turn on some RF field that's detuned
from this transition a little bit.
And we can imagine that the ground state of this
absorbs one photon emits one photon, absorbs and emits,
and turns into kind of a mixed ground state.
We're just going to mix in a little bit of FMOP state
character to this.
And as a result of mixing that little bit of FMOP state
for a red detuned system, we decrease the energy
a tiny bit, which is nice.
Now when we have three particles on a single site here,
if we imagine this isotropic oscillator is actually
a lattice site, when three particles are on site,
we have an RF field tuned pretty close to this FMOP state.
And we can affect the energy just
by changing the detuning or the Rabi frequency.
So that's pretty neat.
Another thing that I'm going to point out here is, well,
if I can start affecting the energy of the three-body state,
what happens to the two-body states,
or the four-body states, or the five-body states,
and so on and so forth?
So here's a figure of the binding energy
of these different two, three, and four-body states
as a function of the scattering length, this boson
distinguishable particle scattering length,
in units of the oscillator length.
So A goes to infinity is over here, because it's 1 over A.
This is the negative scattering length side,
but we're still mostly concerned with the infinity.
So the way you can see is the three-body state
is about, for this particular case we're looking at,
about one oscillator unit below the two-body state.
And the four-body state actually drops off the map here
and goes down to about six oscillator units down.
So there's a fairly good gap here in the energies.
So the three-body state is fairly well isolated as well.
So going back to this picture.
So if we tune something fairly near this three-body transition,
well, we're pretty far tuned off from the two-body transition
and pretty far tuned away from the four-body transition.
So as a result, only three-body occupation
gets affected in any way.
So since only three-body occupation gets affected,
and see I can now point to Hans Christoff's talk earlier,
because he introduced the Bose-Hubbard model,
and I don't have to do that either.
Excellent.
Makes my life easier.
We can now imagine a new kind of Bose-Hubbard model.
Looks like this.
So we have this hopping terms, where it's just a hopping energy
and this ij goes over nearest neighbors, energy associated
with one particle moving to a neighboring lattice site.
And then we have this normal kind
of mean field interaction, u n times n minus 1 over 2
behavior, which really is kind of the energy of two particles
on the site.
And now with this external field on,
we have this additional three-body interaction.
But because the external field is
tuned to only that three-body transition energy,
it only affects the three-body states
when three particles are on site.
That's what this Kronecker delta is telling us.
And so here's kind of a schematic of it.
So one thing you want to know is, how big can we
make this u3?
Big enough to be useful in any way.
So you can do some quick kind of calculations.
And we find u3 in units of the two-body interaction.
Here, the two-body interaction is about 1 kilohertz,
fairly smallish, but still a doable number.
And I'm using a single particle transition frequency
of about 1.5 kilohertz, something that I think
is very doable for an RF transition.
I believe some of them are some.
Oh, there, hiding behind Peter.
You were doing about 10 kilohertz for your RF?
Yeah, 1.5 is doable.
OK, see?
So and then I'm just showing here,
this is kind of the lifetime or the width of the FMOV state
doing a quick semi-classical calculation, something
about 0.6 to 0.1-ish oscillator units in this picture.
So by changing the detuning to the red here,
you can see that we can get these three-body energies
of order, and attractive here, I'm
going to focus on the attractive side, of order,
the two-body energy.
And that's kind of the fundamental energy scale
that we care about in the Bose-Hubbard model.
So we can get some fairly strong interactions,
just to prove to you that this works.
So moving on, I'm moving way faster than I expected.
So you're going to be out of here all kinds of early.
Ah, maybe I should just make some jokes or something
for a little while.
Maybe I asked you to go back.
And if I would have four particles on the side,
would I not expect to have four counters to a key?
You would, and normally when you have a three-body interaction,
it's exactly kind of this.
You'd expect this u3n times n minus 1 times n minus 2 over 6.
But here, if we go back to this kind of picture,
to this picture, we find that because the four-body state is
detuned, not really anywhere near resonance with the three-body
RF field that we've put on here, it
doesn't really see that field at all.
The field is transparent to the four-body state.
And so it just sees the normal mean field energy
and doesn't get affected by the three-body
or by this additional tuning.
So really, we've selected out only the three-body states
or the three-body occupation that
gets pushed around at all by this external field.
Everything else follows the normal mean field.
So I mean, I guess some people have
argued that you'd expect to see something
like a three-body spectator system with a spectator.
But because this three-body state is so large in this oscillator
state, the FMOB state is so large,
you never have a spectator particle.
It's always interacting with the FMOB state,
and it gets pushed around.
That's what this is showing.
So here's the lowest four-body state.
And then you have another four-body state that's up here,
but it's still well separated from the three-body state.
And then actually, there's another one up here
and another one up here.
But they're always separated.
They're never a spectator system because this three-body state
is so large.
And same with the five-body, actually.
There's a five-body what Chris had shown from Javier's work
earlier.
There's a five-body state that's attached
to this three-body state as well.
But it's way down here, too, and a six-body, a seven-body,
and eight-body.
But they're very well separated from the system.
And also, again, because these FMOB states are so large,
you never have kind of a three-body with spectator
type of system.
It's always going to be pushed out of resonance with the RF
system.
Sorry, I have a question, too.
Do you have actually an outlet tone's doublet, right?
Yes.
But you're only showing one energy.
I'm only showing one energy.
That's the one that I'm concerned with.
And really, this graph was maybe a bit of a misleading statement
because we're not necessarily in the large detuning limit.
But in the strong detuning limit,
the energy that we're concerned with
is one where we only do fractional mixture
of this extra state and then strongly detune the limit.
And this is kind of just an initial calculation
here that I put together like three days ago for this talk.
But we've been playing around with parameters.
And we think that in the strong detuning limit,
we still should be able to get similar kind of numbers,
u3 over u, of order 1.
I don't want that.
It appears we could create a similar tone in different ways.
Like, for example, you could describe from the S
central keypad, let's say, of the optical matters,
and then use the two bodies it's using to single out the three
ways it's using things, right?
That would equally give you that term.
And then probably write down a number of ways to do that.
Is there something specific anti-molfish about it?
Or something specific that you need?
Or why should I use this F-molf way that you propose?
Well, because we know where F-molf states are.
And I can do the math analytically.
So that's why I care about it.
Or semi-analytically, that's one reason.
I don't know if there's any specific reason.
F-molf states have become well studied in both now
in rubidium and lithium and any number in cesium.
It's all over the place.
So we have a lot of systems that we
know F-molf physics exists in.
We can play around with it.
So it's a good reason maybe to use it in that way.
I don't know really.
I mean, I am a simple few-body physicist from Russian forest,
I guess, to paraphrase George Lyapnikov.
So this is kind of my early ventures into many-body physics.
I was actually at some in the next few slides,
you'll see that I did a Gutzweiler-Onsatz solution.
I was really proud of myself for that.
Woo!
So phase diagram, see?
Gutzweiler-Onsatz.
So we have our Bose-Hubbard model.
We make our Gutzweiler-Onsatz, where
we assume our superfluid order parameter has this kind of form.
We are making a kind of homogeneous assumption
where each site has the exact same sort of behavior
for a variational kind of Onsatz wave function.
Putting these two things together,
we can actually separate out the Hamiltonian,
or find the energy as a function of the wave function.
We get this nice Hamiltonian that
depends only on individual sites.
So everything is nice and separated.
This is standard Gutzweiler-Onsatz.
I just wanted to show it because I did it and I'm happy.
So the question is, what does this three-body interaction
at the level of mean field theory?
So here, I should point out, there's
this z number that shows everywhere.
Or that's the number of nearest neighbor sites for each system.
So in 1D, it's 2, and 2D square lattice, 4, so on and so forth.
And largely, we're going to care about 2D square lattices
in the future.
So when this U3 is 0, not too surprising.
We get exactly the normal mod insulator
to superfluid phase transition, where
we get the insulating lobes for one-particle occupation,
2, 3, 4, so on and so forth.
And we have a second-order phase transition
from the insulator to the superfluid, all fun and nice.
So now, I'm going to ask, what happens
when we make this three-body interaction attractive
and kind of small?
So here it is for U3 equals negative 0.5 times U.
And maybe it's not too surprising.
I thought it was kind of neat that when
we make this three-body interaction attractive,
we make it so that three-body occupation is enhanced.
So this three-body lobe gets much larger.
And simultaneously, the two-body and the four-body lobes
kind of get shrunk down.
So we're just making it so that two-body, four-body occupation
aren't very favorable, but three-body occupation
becomes quite favorable.
You can tune up the three-body interaction a little bit more,
make it more attractive.
And interestingly, at a critical value,
not surprisingly, U3 equals exactly negative U,
the two-body and four-body lobes exactly disappear,
completely get wiped out.
So that was kind of neat.
We were happy, and we're ready to start publishing papers.
But then the next question is, what
if we make it more attractive?
And so I thought what would happen is, OK,
we make the three-body interaction more attractive.
So we would be able to know if there's
a better spectrum for any type of two-body
to have a metastable state.
Yeah, there's almost certainly still a metastable state.
We're talking all ground state physics here.
I'm sure there's a metastable state that would hang around.
I don't know how long lived or any of these sort of things.
But so this was interesting.
So I assumed if we started making this more attractive,
what would happen is the three-body lobe would just
continue to grow and grow and grow,
and it would start eating up the one-body lobe,
and the one-body lobe would get, or one-particle lobe
would get smaller and smaller, and the five-particle lobe
would just get smaller and smaller.
That's kind of what I assumed would happen,
which is why I was ready to call it a day and get published.
But then when we decided to check anyway,
we found something rather interesting.
Instead of just shrinking down the one-body lobe,
the one-body lobe here actually doesn't change very much
at all whatsoever.
But what we do find is an overlap
between the three-body one-insulating lobe
and the one-body insulating lobe, which is kind of weird.
Like, you could imagine tracing this down,
and you'd find that if the lobe still existed,
it would go down here, and then the one-body would go up here,
and you have this kind of weird region of overlap
between the two.
Which on its own was very odd.
So we decided we needed to focus in
on this kind of overlap region.
There's another one up here that we won't talk about that much,
or I won't talk about it anymore, really, right there.
So we wanted to focus in on what was happening right there.
So if we look at the mean field energy
functional as a function of the order parameter.
So here we are in the middle of the mod-insulating,
n equals 1 lobe.
That's what this mu value does.
You can see that, of course, maybe not surprising.
This is an insulating state.
The order parameter, or the energy,
is minimized when the order parameter is 0.
Not too big of a surprise.
So then I start turning up the chemical potential getting
closer to this transition point, and we
see something kind of interesting.
We see that we're starting to develop a double minimum
structure in here.
So we turn up the chemical potential a little bit more,
go exactly to where this phase transition occurs,
and we find that it's not a smooth phase transition
from 0 psi to finite psi.
In fact, this double minimum shows
that when you go through this phase transition,
all of a sudden you have fully finite order psi, fully
finite order parameter.
This is a first order phase transition.
It's discontinuous, which was surprising to us,
because this is just insulator to superfluid behavior.
So when we look at the phase diagram close up,
this is what we find.
We find three first order phase transitions, one going directly
from insulator to insulator.
So we make a jump from n equals 1 to n equals 3,
and then two going from insulator to superfluid.
Further, we also have a triple point, which I find quite cute,
in fact, and using some approximations
within the mean field, a picture we
can find exactly where that triple point happens,
happens at 0.04 for this particular system,
for u3 equals negative 3 abs u.
And that's all good.
But then eventually, if you make j large enough,
you make it very favorable to start hopping around,
though the phase transition turns back to second order
and goes back to being normal.
So this is the phase diagram we've got in the mean field
theory.
If you check the inclusion of quantum fluctuations,
because it could be that this is just an artifact.
That's what we thought.
Maybe this is just an artifact.
Maybe there's nothing too interesting.
So we decided path integral Monte Carlo
was the right thing to do.
And that should solve the question.
Path integral Monte Carlo should find the exact ground state.
And I thought I'd give you a nice summary slide of everything
that I understand about path integral Monte Carlo.
So there we go.
We go to Barbara and Argyvon, and they know what they're doing.
And they run a bunch of different checks for us.
And this is what they find.
So here I'm showing, again, the mean field phase diagram,
and then the results from the Monte Carlo simulations.
And here black dots are first order phase transitions
that they found.
And then the open squares are second order phase transitions.
So they're really seeing that it follows pretty nicely
with the mean field results, which
I was quite gratified to see.
And we also found that this survives
at finite temperature.
When we go to a small, finite temperature in the Monte Carlo
results, we still see these kind of behaviors.
And to really verify that this was first order,
not some very, very sharp second order phase transition,
we thought first order phase transition
should exhibit some hysteresis behavior.
So in doing the Monte Carlo simulations,
this is actually how Argyvon ended up finding
the first order phase transitions,
just because everything gets shifted and abrupt
and is very hard at first order phase transition.
She ended up having to do hysteresis sweeps
to find where the phase transition was in general.
And so this is what she did.
Basically, in doing the hysteresis sweep,
we see that this is the one to superfluid phase transition,
and this is the three to superfluid phase transition.
So you can see we definitely see some sort of hysteresis
behavior by stability showing up.
So this really is saying that this looks like a first order
phase transition.
And then we also did hysteresis sweeps
for the one to three phase transition,
but it's a really boring hysteresis curve,
because it's a box.
So it's not really nearly as neat as those two.
So I'm going to summarize, and I actually
did a pretty good job.
All right, so just to summarize, I
think I showed that tight confinement and a nice optical
lattice really could open up a more efficient wave function
overlap and ability to probe ephemeral physics.
Using this kind of large Rabi frequency that could be there,
we can try and use an FMOV state
to provide a separately tunable three-body interaction.
And this three-body interaction has
some surprising and interesting consequences in the Bose-Hubbard
phase diagram.
And if you're interested, we have this archive paper
up and about.
And with that, thank you.