Thank you, Henis. Good afternoon, everybody. And I would like to say thanks to Organizer
to organize a nice meeting for our invitation. I came from Dubna. You see that it's the right
place to study ultra-cold physics. This is a spring in Dubna. But our institute is more
known not researching of ultra-cold processes, but investigation of heavy ions and high-energy
physics. But I will tell about our result in ultra-cold few-body processes. This result
were obtained with a group of Peter. And some results were obtained in collaboration with
Innsbruck Group. This is an outline of my talk. The speaker of this morning, they simplify my
life, so I will not speak too much about why it's interesting, but just a few words about the
method which we used. And I will concentrate on some results we obtained with this method. But
nevertheless, I have to tell a few words about the setup. Setup was discussed this morning. We
will concentrate on such cicara-like traps and what happens in those trap. If atoms are in the trap,
they are under the influence of the confinement of the trap. And we can approximate the interaction
with the trap by harmonic oscillator and the parameter of the oscillator for a cesium atom
like this. This is a trap with about 100 nanometers. And we have to consider some few-body physics.
The simplest problem is two-body problem, but two-body problem, special one. And we have to
consider some peculiarities of this problem. First of all, we have to take into account the
quantization in the transverse direction, what happens due to interaction with the trap. Then
a more complicated problem, if it's two distinguishable particle, we have at least five
dimensional problem to radial variables and three angles. And there is a mixing term which makes
center of mass variable and relative variable. So it's a rather complicated problem. And we also
have to include into consideration in elastic processes like molecule formation. The problem
is interesting because this problem under control, we can control the interaction. Why this problem
is interesting from the physical point of view. We have listened this morning and I suppose in
the talk of Selim-Yohim, we also will listen new results. So if the two-particle interact in 3D
free space and there is a near threshold resonance by acting the external magnetic field, one can
tune the interaction. And the interaction can be very strong and this interaction one can describe
with pseudo-potential approximation by changing the magnetic field. Scattering lens can be
variated in a broad range. But if this two-body problem in the confining potential, this resonance
is shifted. But in literature, this resonance has known like confinement induced resonance. Because
in this place, there is no resonance in free space. But in the confining trap, there is a resonance.
It's a famous result of Maxima and Shani, which was obtained in quasi-potential approach. This
slide I will skip. And let's go to the theory. And the simple theory was suggested by Maxima and
Shani when he have considered two-body problem in pseudo-potential approximation. This two-dimensional
problem was projected to one dimension. And in this one dimension, we have renormalization of
interaction constant. And this constant has singularity at this point with respect to
inverse value of the scattering lens. This is a trap width. So near this point, we have to observe
some resonance behavior. In 3D space, we have no confinement. And if the scattering lens cross
zero, weakly bound state appear in the system. If you add the confinement like this one,
like cigar one, the energy is shifted to the value h bar omega. And which is important in
quasi-1D harmonic trap bound state exists for any scattering lens from negative scattering lens to
positive scattering lens. This is a principal difference from the scattering in 3D space. In 3D
space, a bound state exists only for the positive scattering lenses. But another peculiarity of
scattering in the trap, its appearance of excited level. And if excited level crossing the threshold,
then we have this point of confinement-induced resonance, exactly in the point which was
calculated by Mark Simmelstein in according to this formula. And in this case, interesting effect is
a p, which is well-known, so-called mapping of bosons to identical fermions. In this point,
transmission coefficient goes to zero in the point of confinement-induced resonance. And its result
was obtained in the pseudo-potential approximation. But pseudo-potential approximation has some
limitations. For example, you can consider only S-wave scattering or only P-wave scattering,
but cannot consider together S and P scattering as a limitation. So we have developed actually two
computational schemes. One computational scheme is multi-channel scattering problem, two-dimensional
one. And now we are working on three-dimensional one. And another one, it's splitting up method
for time-dependent 3D and 4D Schrodinger equation. Those results were obtained because we have extended
a famous discrete variable representation from 1D to 2D case. It's not a trivial problem also. So
here's some illustration of Hamiltonian for 4D quantum to body problem in quasi-1D harmonic trap.
In this case, Hamiltonian has three term. One term, it's center of mass motion, second term,
it's relative motion, and there is a mixing term. Here is a mixing term. So if omega 1 not equal
omega 2, if you consider scattering of two distinguishable particles, one cannot neglect
this term. One cannot neglect this term, and problem becomes a non-separable one, at least
four-dimensional. There is two radial variables, and there is two angles, phi and theta. So one has
to solve this four-dimensional problem. And this program was realized. And we have considered
2D scattering, 4D scattering, multi-channel scattering in 1D trap, and those problems are
in progress. So now some results. Here, first we have considered what happens with confinement-induced
resonance at arbitrary colliding energy. Olshani has considered the scattering at zero energy.
This is energy in relative units. This is a threshold. This is a first excited threshold,
second excited threshold, third excited threshold. And this is calculated transmission coefficient.
And the black curve corresponds to Olshani case. One can see that here transmission coefficient
is equal to zero. But if we change the scattering length, if we change the interaction between the
particles, so the position of the resonance is moving. Here, some resonance at non-zero colliding
energy. And here, in this part, we present a probability density which was calculated with
our code as a function of zeta, zeta in this direction, and transversal direction.
First result, it's just we are before the resonance in this region. So if one can look
in this transversal direction, one can see no nodes in this distribution, probability distribution.
This is a case of resonance. In this case, the scattering is blocked, but two nodes appear in
transversal direction. And then when we pass through the resonance, some scattering is possible
here. And in the language of Fiersbach resonance, it corresponds to the following. If we are below
the resonance, so there is no nodes in the transversal direction. If we are passing the resonance
here, two nodes appear. And then we have found a new effect in this system, so-called suppression
of quantum scattering near confinement-induced resonances. This effect is possible if we replace
the pseudo-potential, but some real interaction where there is S-wave and D-wave states. And if
such case appear in the 3D free space scattering, so we have the shape resonance,
P-wave shape resonance, and background due to the S-wave scattering. But if we add the waveguide,
so this P-wave resonance transforms to the, we call it dual confinement-induced resonance.
Here, repulsive coefficient has calculated, and you can see that here at this point,
at this point, the particles become transparent one to another. This is one effect what we have
found. Another effect, resonance mechanism of molecule formation in quasi-1D trap with center
of mass excitation. We have considered the collision of two particles, two distinguishable
particles. In this case, one cannot neglect by the coupling of center of mass motion and
relative motion due to this term. But initially, we just dropped this term by hand, and we decided
to consider scattering of two particles when there is no coupling with center of mass.
This is a bound state and two quantum numbers. This is a ground state with respect to transversal
excitation. First quantum number is excitation with respect to relative motion, and second
quantum number is excitation with respect to the center of mass motion. In this case,
we have calculated time evolution of the population of this state. One can see that
when particles are approaching one to another, one to another, this state is populated. This
population of the ground state is growing up, and then this state is decaying. It's a famous
effect like back decay in molecular physics. Then we turn on this term, coupling of center of mass
motion and relative motion. Due to this effect, we obtain the following. First, we see the population
of the ground state, this red curve, and then there is a transition from the ground state
due to this term, from the ground state to excited state. We see this blue curve, the population of
the excited state. This is a long-living state due to the back decay is forbidden, direct back decay
is forbidden, but decay is possible only like two-step decay. Because of this, this state is
low, long-living. Here's an evolution of the probability density distribution as a function
of time is given. Of course, it's a four-dimensional picture, but we did averaging over the angles and
remaining variables. This is a relative variables and the variable of center of mass.
This is like the probability density distribution. It looks like after one time units, and then after
three time units, you observe the population of the ground state. After eight time units,
one can observe that we have the system in excited state with respect to center of mass excitation.
This is the case if we drop the interaction between center of mass and relative motion.
Here, you see there is no the molecule. This is a mechanism which we have considered, but
the experimentalists so far did not observe this.
The next result is D-wave resonances in atomic traps. It is well known that
high partial wave interaction can lead to some novel many-body phenomena such as unconventional
superfluid or superconductivity, but experimental task is big how to observe very narrow
D-wave resonance at zero energy limit. Another possibility is to consider polar molecules.
There is this dipole-dipole interaction with anisotropic one, but the problem is that this
coefficient dipole-dipole interaction is terribly small. We decided to consider another
possibility for such anisotropy. We decided to consider this effect in the harmonic trap.
In the harmonic trap, due to the interaction with the trap, we have this mixing of S-wave and P-wave.
Such coupling may be rather strong and rather broad. One can demonstrate the transmission
coefficient near such D-wave resonance in the trap. One can see that strong anisotropy is
observing in a rather broad region near the D-wave resonance. We have calculated some
dependence of the shift of the S-wave scattering lens when we add the trap. We have observed that
this ratio of this shift with respect to CS6, the diffusion coefficient can be fit by a straight
line. It's some interesting effect. Finally, confinement splitting in 1D anisotropic harmonic
waveguide. This result was obtained with experimentalist from Innsbruck. The idea is
the following. By variating the magnetic field, we can variate the scattering lens. This is in
3D free space. Then in the trap, we have this picture which I have already discussed.
In the experiment, if the system is near the confinement induced resonance, such
three-body triple collision can be possible. The energy release can be taken by the short
particles and the molecule can form. This formation of the molecule near the resonance
will lead to increasing the atomic loss in the system. Such loss was analyzed
for different frequencies. But variating the frequency, we are variating the width of the trap.
And experimentally, they have observed this shift of the minimum of the atom in the trap.
And if one can look to the dependence of the scattering lens corresponding to
those minimum. Actually, it's not minimum, but the edge of these curves and fits this by straight
line. And this straight line, exactly the straight line, which was predicted by Elshani.
So one can treat this result like experimental evidence of confinement induced resonance.
Then we decided to look what happened if it's not anisotropic trap in the transversal direction. If
omega 1 not equal to omega 2, what happened in this case? It is clear that the
first excited state will be lift. And it's following from this simple picture that
this state will be split up on three components. But one component, this one component,
are not coupled with component third and component first. So the prediction was that
experimentally, they will observe two resonances in the case of anisotropic. And this is a result
for anisotropic case. And this is a result for anisotropic case. This result for anisotropic case.
So anisotropy of the trap leads to the splitting of the confinement induced resonance.
And this result has stimulated theorists to develop fuzzy development of the theory.
First two attempts were made with pseudo-potential approach, but pseudo-potential
approach, they did not observe the splitting. But it's clear because everybody knows that
observed in delta function potential can support only one bound state. So this result was not so
surprising. And because this result related with the splitting of excited state, we decided
to extend our consideration for the multichannel scattering. So we decided to consider the
scattering from the initial transversal state to all possible excited state, and also to consider
the scattering from excited state. And the effective transmission coefficient, it's the sum of
partial transmission coefficient multiplied to the initial population of the states.
And here, the result of that partial transmission coefficient is given for the ground state and for
the first excited state. First, for the case of anisotropic waveguide. What does this picture tell us?
This picture tells us that the position of the minimum of the transmission coefficient
from the ground state can be shifted with respect to the transmission coefficient from excited state.
And this is effective transmission coefficient calculated for a few different colliding energy.
But because the partial transmission coefficient from the ground state has a very deep minimum,
so the main contribution near minimum comes from excited state. So if the system initially even
have one or two percent population of first excited state, then experimentalists can feel it
near the confinement induced resonance. It was our idea. So in reality, we see that if it
comes to anisotropic waveguide, we see this appearance of two minimum in partial transmission
coefficient from excited state. And what happens with effective transmission coefficient? We see
that for one percent population of first excited state, we observe this splitting in transmission
coefficient. And this splitting becomes more visible if the population is increasing. But
necessary ingredient for this scheme for splitting of the minimum of the transmission coefficient
is a population at least a few percent of the transversely excited state. This is necessary
ingredient. When we finish this paper, we have found a paper of Kösner and Doan,
which when we have found such supporting words that if we consider
a pair collision near confinement induced resonance in the trap, it usually leads to
excitation of excited state. And also we did some calculation of transition probability near the
confinement induced resonance, which also support us that near confinement induced resonance, we
always have a system in excited state. And here I would like to show how the spectrum looks like
of two body system in the trap. This is real calculation. Before I have shown you some scheme,
but this is real calculated scheme. And we see that actually for the excited level, we don't see
the crossing of the threshold, but we see some rearrangement of the spectrum system goes this way
and then jump from this point to this point and new level appear in system. And this resonance
jumping from this point to this point, and so all the system have rearrangement. So this picture,
not correct. And in the anisotropic trap, we have such a splitting of this state.
And this part here, we have the quasi-crossing, which is in agreement to the distance between
the maxima of atomic loss in the Innsbruck experiment. But there is another scheme which
explains this splitting. This scheme based on the fact that if we take into consideration
unharmonicity, then we will have more complicated system. In addition to the harmonic spectrum,
some unharmonic part is appearing. This is a calculation for its effect of center of mass
coupling, but the same effect gives the unharmonicity term. So in addition to induce
resonances, harmonic induce resonance, which was considered by Olshani and which we consider in
our approximation, there is also unharmonicity part. And this unharmonicity part have been
considered in the paper of Alexandro Zients and in the group of Peter Dumont. And they have found
that this unharmonicity part in good agreement with the distance between the maximum of the
atomic loss in the Innsbruck experiment. But they neglect harmonic part. So our point of view is
that this problem is still uncompleted and demand other consideration. It's need to consider more
carefully the population of excited state, velocity distribution over the collision energy,
influence of inelastic channel of molecule formation for the shift and splitting of the
confinement induced resonance due to unharmonicity, and also tensorial structure of the
interatomic interaction. What I mean here, it's need also to include the rotational and spin
structure of the molecular state in the closed channel responsible for the Feshbach resonance,
and also includes the width of the Feshbach resonance. And also it's need to make this
direct step from the scattering parameters to come to atomic loss. This also was not done.
But part of this problem we already have realized this part, which I would like to discuss now.
So tensorial structure of the interatomic interaction, what I mean. It is known that in the
cesium atoms, there are many Feshbach resonances. And here in this graph,
three Feshbach resonances are given. S-wave resonance, D-wave resonance, and G-wave resonance.
Actually, Inzburg group, they were working near this D-wave resonance.
Olshani has considered simplified picture, S-wave resonance. Yes, I see. And Inzburg group,
they were working near this resonance. And they have this resonance and nearby resonance,
this one, and S-wave resonance have been analyzed in the two channel
scheme, which gives a good possibility to express the potential element of the
two channel potential through the position of the resonance, resonance widths, and magnetic field
corresponding to the resonance. We decided to extend this scheme for, this scheme was applied
in free space scattering. We decided to extend this scheme to scattering in the trap.
First, we repeat, we include this four channel potential. Why four channel potential? Because
first channel is entrance channel and three channel describes the Feshbach resonance of
different tensorial structure. And with this potential, we repeat the fitting which was done
in the paper of Lange. So we used those parameter in this paper and we fixed the potential of our
four channel potential, which describe inter-atomic interaction. And then we add the trap and did the
calculation of the transmission coefficient and the function of external magnetic field.
This is for calculation for the trap. And this is a calculation of Lange and at all for the case
of the absence of the trap. So for the scattering lens in the free space, you see for the S-wave
resonance, we see this shift of the resonance. This is Alshani resonance, broad S-wave resonance,
and two resonances, D-wave and G-wave resonances. Innsbruck group is working here. So we will look
in this region in more detail. This is some result for near this resonance.
It's transmission coefficient as a function of magnetic field for different omega, for different
omega. And we observe one maximum, which is independent on the parameter of the trap. And
we see this variation of the minimum of the transmission coefficient with variation of the
trap frequencies. The frequency is very close to the ones which were used in Innsbruck experiment.
When we fit those minimum, we see that this minimum described very well by Alshani formula.
So this is confinement-induced resonance. But what means this maximum? And we see that this
maximum corresponds to the crossing of the scattering lens, crossing of zero. And because
scattering lens equals zero in this point, so this maximum is independent on the magnetic field.
I again return to this famous picture, which people who are working in this field
are very familiar with this picture. So this point, those points C, correspond to
appearance of confinement-induced resonance. This maximum, this maximum point B, correspond
to appearance of new bound state. New bound state, this point B, corresponds this maximum.
Or the crossing of the scattering lens, crossing zero. And this point, this point, this point,
its Fechbach resonance, position of Fechbach resonance, in free space. So this is in free
space, the appearance of resonance. Everything is clear. And we have some unpublished data of
Elmar Haller from Innsbruck group. This part was published, but this part was not published.
Two stable resonance. Now we know what means this resonance. This resonance corresponds to
the crossing of the scattering lens, crossing the zero. But what with another peak? Here,
there are two possibilities. One possibility is the appearance of the bound state in excited state.
Because our preliminary calculation shows that this point will be more close to this point
than this one. Point B corresponds to this point, but B prime close to this. Another possibility is
some indication of appearance of the D-wave resonance. But it's not yet clear. We did not
see any structure here. And last slide. Here we have calculated also the molecular state
population, molecular state population as a function of magnetic field. Because we have this
four-channel potential, we can calculate the population during the collision of this
molecular state. Molecular state. This is a result for free space. Mainly, systems are
in the entrance channel. But then when we add the trap, we see that most dominant become
the population of D-wave states, this blue line. Other population of other state, they are
happened due to non-resonant interaction, and they are much less. And final slide. We have
present here the counterplot of this molecular state near this magnetic field. And one can see
that this is obviously not S-wave, but D-wave state. And here I will miss this. And finally,
last slide. The improvement of the model can be done in this direction, including
anharmonicity in elastic channel. And then one can consider anisotropic scattering in waveguide
to body problems in 2D traps and 3D traps and 3-body physics in the trap. And last slide.
Many experiments are in the world, and they need some theory, but time for simple model is over.
But we're ready for that. Thank you.