Semi-Classical Methods and N-Body Recombination
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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. 21 st 2009. Hard Problems with Simple Solutions

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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


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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


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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


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Jose P. D’Incao

Nirav Mehta

Chris H. Greene

Javier von Stecher


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Review of Recombination Experiments

2006: First solid evidence of an Efimov State was seen in Innsbruck


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Since then, several other groups have seen Efimov states

Ottenstein et.al., PRL. 101, 203202 (2008)

Huckans et. al., PRL 102, 165302 (2009)


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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)


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More recently: Four body effects have been observed!

Ferlaino et. al., PRL 102, 140401 (2009)

Rice group


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

General idea: treat the

hyperradius adiabatically

(think Born-Oppenheimer).

Provides us with a

convenient view of the

energy landscape

~ R


Hyperspherical coordinates the first step for easy few body scattering10 l.jpg
Hyperspherical Coordinates: the first step for easy few body scattering.

  • General idea: treat the

  • hyperradius adiabatically

  • (think Born-Oppenheimer).

  • Provides us with a

  • convenient view of the

  • energy landscape

For example,

The energy landscape

3 Bodies 2-D

~ R


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When the hyperradius is much different from all other scattering.

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.


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Relevant examples of potential curves scattering.

Three bosons with negative scattering length:


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Relevant examples of potential curves scattering.

Three bosons with negative scattering length:

Transition region

Here be dragons!

Repulsive universal long-range tail

Attractive inner region


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Relevant examples of potential curves scattering.

Four bosons with negative scattering length:


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Relevant examples of potential curves scattering.

Four bosons with negative scattering length:

Repulsive four-body potentials

Broad avoided crossing

Efimov trimer threshold

Attractive inner wells


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Not-so relevant examples of potential curves: scattering.

a cautionary tale

Sometimes things can get ugly, so be careful!


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Let’s get quantitative scattering.

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).


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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.


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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


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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?


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WKB to the rescue channel survives. Using the unitary nature of the S-matrix, this simplifies things quite a bit

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:


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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)




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Some things to note: constant

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.


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Re-examine three bosons constant

Assume that all of the tunneling occurs in the universal large R region, and that all phase accumulation occurs in the universal inner region.


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Re-examine three bosons constant

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


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A little discussion of four-boson potentials constant

[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.


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Just after first Efimov state becomes bound constant

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).





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Applying the WKB Recombination formula be supported for each Efimov state.


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Applying the WKB Recombination formula be supported for each Efimov state.

4-body resonances

Second Efimov state becomes bound.

(Cusp?)


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Can 4-body effects actually be seen? be supported for each Efimov state.

Surprisingly, yes.

Measurable four-body recombination occurs to deeply bound dimer states:

(No weakly bound trimers)



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N>4 been measured.

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,


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Can 5 or more body physics be seen? been measured.

Without strong resonances, back of the envelope approximation says, probably not.

N = 4

N = 5

N = 6


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Summary been measured.

  • 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.


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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!!

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…

.


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The Efimov effect (restated) physics, Vitaly Efimov, considered the following natural question:[Nucl. Phys. A. (1973)]


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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.


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Lowest adiabatic hyperradial channel result

a<0

Short range

stuff

Universal region

Transition region

Universal region

for identical bosons



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Observing the Efimov effect: three-body recombination result

a < 0

K.E.

  • 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.


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Other possible Efimov states result

  • Recently, three hyperfine states of 6Li

Ottenstein et.al., PRL. 101, 203202 (2008)

Huckans et. al., arXiv:0810.3288 (2008)


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Real two-body interaction are multi-channel in nature. result

Simplest thing: Zero-range model


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How does this translate to three bodies? result

Start by looking at a simplified model: no coupling.


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Parameters for an excited threshold resonance result

Make excited bound state resonant with second threshold

Coupled

Uncoupled

Coupled


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Full calculation looks a bit ugly. result

First 300 potentials

[PRA, 78 020701 (2008)]


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Simplified picture: result

Cartoon of two important curves.

Efimov Diabat

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

Actually an avoided crossing

Three free particles


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The Experiment result

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

K.E.

K.E.


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Cartoon three body loss spectrum. result

many states

2nd state

1st state


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Four Bosons and Efimov’s legacy result

Figure from von Stecher et. al., eprint axiv/0810.3876


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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.






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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




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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)


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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


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Summary parameter. We can expect Log periodic behavior!

  • 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.


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Four-Fermions parameter. We can expect Log periodic behavior!


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“H” - type parameter. We can expect Log periodic behavior!

1

2

3

1

3

Rotate Jacobi vectors

Into body-fixed frame:

Parameterize moments

Inertia with R, 1 and 2:

Parameterize body-fixed

Vectors with three-more

angles:

2

4

Jacobi and “Democratic” Hyperspherical Coordinates

Body-fixed “democratic” coodinates

(Aquilantii/Cavalli and Kuppermann):


Variational basis for four particles assume l 0 l.jpg

Note: There is no (shallow) three-body bound state for (up-up-down) fermions .

Dimer+Dimer:

Dimer+Three-body continuum:

Four-Body continuum:

Variational basis for four particles: (Assume L=0)


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After just a few thousand cpu hours: (up-up-down) fermions .

Potentials!

With potentials, we can start looking at scattering


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Dimer-dimer scattering length (up-up-down) fermions .

With effective range:

von Stecher, PRA (2008)

add (0)= 0.6 a

Petrov, PRL (2004)


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Energy dependence means any finite collision energy leads to deviation from the zero energy results


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What about dimer relaxation? deviation from the zero energy results

or



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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.


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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.


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Three-body processes influenced by presence of fourth particle

Four-body processes

Three-body only processes


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Petrov (2004) particle

Small R contribution

Intermediate scaling

behavior

[arXiv:0806.3062]