Transcript Slide 1

Semi-Classical Methods and N-Body
Recombination
Seth Rittenhouse
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
Efimov States in Molecules and Nuclei, Oct. 21st 2009
Hard Problems with Simple Solutions
Seth Rittenhouse
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
Efimov States in Molecules and Nuclei, Oct. 21st 2009
WKB is Smarter than You Think
Seth Rittenhouse
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
Efimov States in Molecules and Nuclei, Oct. 21st 2009
Chris H. Greene
Nirav Mehta
Jose P. D’Incao
Javier von Stecher
Review of Recombination Experiments
2006: First solid evidence of an Efimov State was seen in Innsbruck
Since then, several other groups have seen Efimov
states
Ottenstein et.al., PRL. 101, 203202 (2008)
Huckans et. al., PRL 102, 165302 (2009)
Since then, several other groups have seen Efimov
states
Zaccanti et. al., Nature Phys. 5, 586 (2009).
Ultra cold Li7 gas:
Rice group (soon to
be published)
More recently: Four body effects have been observed!
Ferlaino et. al., PRL 102, 140401 (2009)
Rice group
Hyperspherical Coordinates: the first step for easy
few body scattering.
General idea: treat the
hyperradius adiabatically
(think Born-Oppenheimer).
~R
Provides us with a
convenient view of the
energy landscape
Hyperspherical Coordinates: the first step for easy
few body scattering.
General idea: treat the
hyperradius adiabatically
(think Born-Oppenheimer).
~R
Provides us with a
convenient view of the
energy landscape
For example,
The energy landscape
3 Bodies 2-D
When the hyperradius is much different from all other
length scales, the adiabatic potentials become universal,
e.g.
which is the non-interacting behavior at fixed hyperradius.
The potentials for other length scale disparities look very
similar, but with l non-integer valued or complex.
Relevant examples of potential curves
Three bosons with negative scattering length:
Relevant examples of potential curves
Three bosons with negative scattering length:
Here be dragons!
Transition region
Repulsive universal
long-range tail
Attractive inner region
Relevant examples of potential curves
Four bosons with negative scattering length:
Relevant examples of potential curves
Four bosons with negative scattering length:
Repulsive four-body potentials
Broad avoided crossing
Efimov trimer threshold
Attractive inner wells
Not-so relevant examples of potential curves:
a cautionary tale
Sometimes things can get ugly, so be careful!
Let’s get quantitative
Once hyperradial potentials have been found, it might be nice to
have scattering crossections and rate constants.
Three-body:
Esry et. al., PRL 83, 1751 (1999); Fedichev et. al., PRL 77, 2921 (1996);
Nielsen and Macek, PRL, 83 1566 (1999); Bedaque et. al., PRL 85, 908 (2000);
Braaten and Hammer, PRL 87 160407 (2001) and Phys. Rep. 428,259 (2006);
Suno et. al., PRL 90, 053202 (2003).
Through some hyperspherical magic this can be
generalized to the N-body cross section and rate
Mehta, et. al., PRL 103, 153201 (2009)
This is messy, but there already is some good physics buried in
here.
At very low incident energies, only a single incident channel
survives. Using the unitary nature of the S-matrix, this
simplifies things quite a bit
At very low incident energies, only a single incident channel
survives. Using the unitary nature of the S-matrix, this
simplifies things quite a bit
This only depends on the incident channel!
If know about scattering in the initial channel, then we know
everything about the N-body losses!!!
Still a fairly nasty multi-channel problem, how can we solve this?
WKB to the rescue
Specify a little bit more, consider N-bosons with a negative two
body scattering with at least one weakly bound N-1 body state.
The lowest N-body channel will have a very generic form:
Approximate the incident channel S-matrix element using
WKB phase shift with an imaginary component.
= WKB phase inside the well
= WKB tunneling
= Imaginary phase (parameterizes losses)
Putting this all together gives the recombination rate constant
Putting this all together gives the recombination rate constant
Some things to note:
This only holds when the coupling to deep channels is with the scattering
length.
If coupling exists at large R, we must go back to the S-matrix, or find
another cleaver way to describe losses.
This assumes the S matrix element is completely controlled by the
behavior of the incoming channel. If outgoing channel is important, as in
recombination to weakly bound dimers, a more sophisticated
approximation of the S-matrix is needed.
Re-examine three bosons
Assume that all of the tunneling occurs in the universal large R region, and that
all phase accumulation occurs in the universal inner region.
Re-examine three bosons
Assume that all of the tunneling occurs in the universal large R region, and that
all phase accumulation occurs in the universal inner region.
This gives a recombination rate constant of
In agreement with known results
A little discussion of four-boson potentials
[Von Stecher et. al., Nature Phys. 5, pg 417]
Look at potentials in this region. Negative
scattering length with at least one bound
Efimov state.
Just after first Efimov
state becomes bound
Two four body bound states
are attached to each Efimov
threshold..
(Hammer and Platter, Euro.
Phys. J. A 32, 113;
von Stecher, D’Incao and
Greene Nature Phys. 5, 417).
Slightly larger
scattering length
Attractive region
becomes deep
enough to admit a
four-body state
Second Efimov state
becomes bound. Two
four-body states can
be supported for each
Efimov state.
Applying the WKB Recombination formula
Applying the WKB Recombination formula
4-body resonances
Second Efimov
state becomes
bound.
(Cusp?)
Can 4-body effects actually be seen?
Surprisingly, yes.
Measurable four-body recombination occurs to deeply bound dimer states:
(No weakly bound trimers)
More recently: Four body recombination to Efimov Trimers has
been measured.
N>4
Without potentials we can’t say too much, but recent work has
shown where we could expect resonances.
Can 5 or more body physics be seen,
Can 5 or more body physics be seen?
Without strong resonances, back of the envelope
approximation says, probably not.
N=4
N=5
N=6
Summary
• N-body recombination becomes intuitive
when put into the adiabatic hyperspherical
formalism
• Getting the potentials is hard, but even
without them, scaling behavior can be
extracted.
• Low energy recombination can be described
by the scattering behavior in a single
channel.
• WKB does surprisingly well in describing the
single channel S-matrix
• Four body recombination can actually be
measured in some regimes.
In 1970 a freshly-minted Russian PhD in theoretical nuclear
physics, Vitaly Efimov, considered the following natural question:
What is the nature of the bound state energy level spectrum
for a 3 particle system, when each of its 2-particle
subsystems have no bound states but are infinitesimally
close to binding?
Efimov’s prediction: There will be an
INFINITE number of 3-body bound states!!
En1  Ene2 / s0 , where s0  1.00624...is a universalconstant.
This exponential factor = 1/22.72=0.00194, i.e. if one bound state
is found at E0= -1 in some system of units, then the next level
will be found at E1= -0.00194, and E2= -3.8 x 10-6, etc…
.
The Efimov effect (restated) [Nucl. Phys. A. (1973)]
Qualitative and quantitative understanding of Efimov’s
result
At a qualitative level, it can be understood in hindsight,
because two particles that are already attracting each
other and are infinitesimally close to binding, just need a
whiff of additional attraction from a third particle in order
to push them over that threshold to become a bound
three-body system.
Quantitatively, Efimov (and later others) showed that a
simple wavefunction can be written down at each
hyperradius.
Lowest adiabatic hyperradial channel
a<0
Short range
stuff
Universal region
Transition region
for identical bosons
Universal region
Observing the Efimov effect: three-body recombination
a<0
K.E.
Observing the Efimov effect: three-body recombination
a<0
•Three-body recombination can be measured through trap
K.E.
losses.
•Shape resonance occurs when an Efimov state appears at 0
energy.
•Spacing of shape resonances is geometric in the scattering
length.
•Only one resonance, need two to show Efimov scaling
•Second resonance at
•Need low temperatures:
Other possible Efimov states
•He trimer
Other possible Efimov states
•Recently, three hyperfine states of 6Li
Ottenstein et.al., PRL. 101, 203202 (2008)
Huckans et. al., arXiv:0810.3288 (2008)
Real two-body interaction are multi-channel in nature.
Simplest thing: Zero-range model
How does this translate to three bodies?
Start by looking at a simplified model: no coupling.
Parameters for an excited threshold resonance
Make excited
bound state
resonant with
second threshold
Coupled
Uncoupled
Coupled
Full calculation looks a bit ugly.
First 300 potentials
[PRA, 78 020701 (2008)]
Simplified picture:
Cartoon of two important curves.
Efimov Diabat
Efimov states
Actually anleads
avoidedtocrossing
•Super-critical 1/R2 potential
geometrically spaced states.
•Coupling leads to quasi-stability: Three-body Fano-Feshbach
Three free particles
Resonances
•With no long-range coupling, widths scale geometrically
The Experiment
Photon
Three
Excite
If
photon
particles
the
and
energy
system
binding
come
iswith
degenerate
energies
together
RF photons.
are
at
with
low
released
Efimov
energy
as
state
with
kinetic
energy,
respect
energy
expect
to the first
strong
threshold.
coupling
to lower channels.
K.E.
K.E.
Cartoon three body loss spectrum.
many states
2nd state
1st state
Four Bosons and Efimov’s legacy
Figure from von Stecher et. al., eprint axiv/0810.3876
A little review of von Stecher’s work on four-boson potentials
eprint axiv/0810.3876
Look at potentials in this region. Negative
scattering length with at least one bound
Efimov state.
Just after first Efimov
state becomes bound
Slightly larger
scattering length
Attractive region
becomes deep
enough to admit a
four-body state
Second Efimov state
becomes bound. Two
four-body states can
be supported for each
Efimov state.
Simplest way to see four-body physics is through four-body
recombination.
N-body recombination rate coefficient, in terms of the T
matrix, is given by:
For four bosons in the low energy regime this reduces to
The behavior T matrix element is dominated by the lowest fourbody channel.
If a four-body state is present, a shape resonance occurs.
Using a simple WKB wavefunction gives the four-body
recombination rate coefficient up to an overall factor.
4-body resonances
Second Efimov
state becomes
bound
a7 scaling
(predicted by asymptotic
scaling potential)
Four-body behavior scales with the three-body Efimov parameter.
We can expect Log periodic behavior!
Position of four-body
resonances is
universal:
Observation of four-body resonances can give another handle on
identifying Efimov states
Summary
• 3-bodies and Efimov Physics: PRA 78, 020701
(2008)
– Zero-range multichannel interactions predict an Efimov potential
at an excited three-body threshold.
– Coupling to lower channels gives bound states coupled to the
three-body continuum: 3-body Fano-Feshbach resonances!
– Quasi-stable Efimov states may, possibly, be accessed via RF
spectroscopy allowing for the observation of multiple resonances.
• 4-bosons
– 4-body recombination shows universal resonance behavior.
– Postitions of 4-body resonances give a further handle on idetifying
an Efimov state.
Four-Fermions
Jacobi and “Democratic”
Hyperspherical Coordinates
“H” - type
1
3
2
1
4
Body-fixed “democratic” coodinates
(Aquilantii/Cavalli and Kuppermann):
Rotate Jacobi vectors
Into body-fixed frame:
2
Parameterize moments
Inertia with R, 1 and 2:
3
Parameterize body-fixed
Vectors with three-more
angles:
Variational basis for four particles:
(Assume L=0)
Note: There is no (shallow) three-body bound state for (up-up-down)
fermions .
Dimer+Dimer:
Dimer+Three-body
continuum:
Four-Body
continuum:
With
Afterpotentials,
just a few thousand
we can start
cpulooking
hours: at
Potentials!
scattering
Dimer-dimer scattering length
With effective range:
von Stecher, PRA (2008)
add (0)= 0.6 a
Petrov, PRL (2004)
Energy dependence means any finite collision
energy leads to deviation from the zero energy
results
What about dimer relaxation?
or
Unfortunately, there are an infinite number of final states!
Fermi’s golden rule leads to a simple expression for the rate:
is the WKB tunneling probability
is the WKB wave number
is the density of final states near R
is probability that three particles are close together at
hyerradius R.
By performing the integral over different hyperradial
regions, we can isolate different types of process.
Integration over only very small hyperradii isolates
relaxation channels where all four particles are involved.
Four-body
processes
Three-body
processes
influenced by
presence of
fourth particle
Three-body
only processes
Petrov (2004)
Small R
contribution
[arXiv:0806.3062]
Intermediate
scaling
behavior