Semi-Classical Methods and N-Body Recombination

Seth RittenhouseITAMP, 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

Jose P. D’IncaoNirav Mehta Javier von Stecher

Chris H. Greene

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

Ultra cold Li7 gas: Rice group (soon to be published)

Zaccanti et. al., Nature Phys. 5, 586 (2009).

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 thehyperradius adiabatically(think Born-Oppenheimer).

Provides us with a convenient view of the energy landscape

~ R

For example,The energy landscape 3 Bodies 2-D

Hyperspherical Coordinates: the first step for easy few body scattering.

General idea: treat thehyperradius adiabatically(think Born-Oppenheimer).

Provides us with a convenient view of the energy landscape

~ R

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 non-integer valued or complex.

Relevant examples of potential curves

Three bosons with negative scattering length:

Three bosons with negative scattering length:

Repulsive universal long-range tail

Attractive inner region

Transition regionHere be dragons!

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

Efimov trimer threshold

Attractive inner wells

Broad avoided crossing

Relevant examples of potential curves

Sometimes things can get ugly, so be careful!

Not-so relevant examples of potential curves:a cautionary tale

Let’s get quantitativeOnce 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

If know about scattering in the initial channel, then we know everything about the N-body losses!!!

This only depends on the incident channel!

Still a fairly nasty multi-channel problem, how can we solve this?

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:

WKB to the rescue

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 bosonsAssume 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 bosonsAssume 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!!

constant. universal a is ...00624.1 where, 0/2

10

seEE snn

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.

Transition region Universal regionUniversal regionShort range

stuff

Lowest adiabatic hyperradial channel a<0

for identical bosons

K.E.

a < 0

Observing the Efimov effect: three-body recombination

K.E.

a < 0

Observing the Efimov effect: three-body recombination

•Three-body recombination can be measured through trap 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.

Make excited bound state resonant with second threshold

Coupled

Coupled

Uncoupled

Parameters for an excited threshold resonance

Full calculation looks a bit ugly.First 300 potentials

[PRA, 78 020701 (2008)]

Simplified picture:

Cartoon of two important curves.

Efimov Diabat

Three free particles

Actually an avoided crossing

Efimov states

•Super-critical 1/R2 potential leads to geometrically spaced states.

•Coupling leads to quasi-stability: Three-body Fano-Feshbach Resonances

•With no long-range coupling, widths scale geometrically

K.E.

K.E.

Three particles come together at low energy with respect to the first threshold.Excite the system with RF photons.If photon energy is degenerate with Efimov state energy, expect strong coupling to lower channels.Photon and binding energies are released as kinetic energy

The Experiment

Cartoon three body loss spectrum.

1st state2nd statemany states

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 potentialseprint 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 four-body 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.

a7 scaling (predicted by asymptotic scaling potential)

4-body resonances

Second Efimov state becomes bound

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” - type1

2

3

4Body-fixed “democratic” coodinates (Aquilantii/Cavalli and Kuppermann):

Parameterize moments Inertia with R, 1 and 2:

21Rotate Jacobi vectorsInto body-fixed frame:

3Parameterize body-fixed Vectors with three-moreangles:

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:

After just a few thousand cpu hours:Potentials!

With potentials, we can start looking at scattering

Dimer-dimer scattering length

add (0)= 0.6 aPetrov, PRL (2004)

With effective range:von Stecher, PRA (2008)

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

Intermediate scaling behavior[arXiv:0806.3062]