numerical studies of universality in few-boson systems javier von stecher department of physics and...

Numerical Studies of Universality in Few-Boson Systems

Numerical Studies of Universality in Few-Boson Systems

Javier von Stecher Department of Physics and JILA University of Colorado

Probable configurations of a weakly-bound four-boson state.

Acknowledgements:Doerte BlumeSeth RittenhouseNirav MehtaNSF

In collaboration withChris H. GreeneJose P. D’Incao

How does Efimov physics and How does Efimov physics and universality extends beyond universality extends beyond

three bodies?three bodies?

Outline:Outline:

Universality in four-boson systems

Numerical formulation (CG and CGH) Four-bosons predictions

Extension to larger systems

Model Hamiltonian Cluster states

Solving the Schrödinger equationSolving the Schrödinger equation

Hamiltonian:

Four-body Formulation

model interactions:

rr

V(r)

V(r) dd00

a<0a>0

VV00

aa

dimer energydimer energyscattering scattering lengthlength

-

or combinations of Gaussians

Analytical form of matrix elements (accurate and fast). Efficient Optimization: Stochastical variational method Good for bound states. Difficult to describe scattering properties

Correlated Gaussian basis set expansion:Correlated Gaussian basis set expansion:

r12

r13r24

r34

r14r23

1 2

34

• remove CM and set L=0

Symmetrization: spin statistic

• basis functions:


Hyperspherical representation:Hyperspherical representation:

Coupled 1D equations…solved at fixed R.

Use correlated Gaussian to expand channel functionsUse correlated Gaussian to expand channel functions

Divide and conquer:

Find efficient way to evaluate matrix elements

R

Accurate description of four-body systems!!Accurate description of four-body systems!!

R: overall size

(collective motion)

typical potential curves

Four-body Formulation J. von Stecher, C. H. Greene PRA (2009)

1+1+1+1

2+1+1

2+2

3+1

3+1

Hyperspherical PictureHyperspherical Picture

RR

v

v

Fragmentation thresholds

All particles close together:Bound and quasi-bound states

Interaction region

U(R

U(R

)

……for scattering properties.for scattering properties.


s0=2.0096 (CG) s0=2.0092 (CGH)

Universality arguments predict that the hyperspherical potential curves are of the type (Tan, Werner & Castin, …):

J. von Stecher, C. H. Greene PRA (2009)

J. von Stecher, D. Blume, C. H. Greene PRL 2007, PRA 2007, 2008

S.Y. Chang, and G. F. Bertsch PRA (R) (2007)

I. Stetcu, B. Barrett, U. van Kolck, J. Vary. PRA (2007)

Y. Alhassid, G. F. Bertsch, and L. Fang PRL (2008)

…..

Trap system predictions:

Four fermion system at unitarityFour fermion system at unitarity

Appropriate framework to analyze universality

D.S. Petrov, C. Salomon, G.V. Shlyapnikov PRL (2004)

Connection to trapped system:trap

frequency

Predictions:

See also: J. P. D’Incao et al PRA (2009)

Hyperspherical PictureHyperspherical Picture……for studying universalityfor studying universality


Outline:Outline:





Four-BosonsFour-Bosons

, …

, …

, …

3 body3 body

4 body4 body

Efimov scaling

Efimov scaling

Two four-body states

same scaling relations!!!

Universality in four bosonsUniversality in four bosons

a=

J. von Stecher, J. P. D’Incao and C. H. Greene, Nat. Phys. (2009)

In agreement with Hammer & Platter EJPA (2007)(Hammer presentation)

RR RR

3 bodies3 bodies 4 bodies4 bodies


Hyperspherical Picture (a=Hyperspherical Picture (a=))

v

v

v

Four-body potential curves follow the same scaling of the Efimov states !!!

Four-body physics is universal!!

RR RR

3 bodies3 bodies 4 bodies4 bodies


Hyperspherical Picture (a=Hyperspherical Picture (a=))

v

v

v

Four-body potential curves follow the same scaling of the Efimov states !!!

Four-body physics is universal!!

Three-body parameter:


Four-body statesFour-body states

a=


Hammer & Platter prediction:

Agreement with Hammer & Platter (2007)


v

Spectrum: Extended Efimov plotSpectrum: Extended Efimov plot

1/a


Calculations at more than 500 scattering lengths!!!

atom-trimer thresholds(green lines)

four-body states(black lines)

dimer-dimer threshold(red lines)

dimer-two atoms threshold(red lines)

v

~1/a


Universality away from unitarityUniversality away from unitarity

v

~1/a



v

1/a

Ener

gy

a*4b2 a*

3ba*4b1

v

~1/a



v

1/a

Ener

gy

a*4b2 a*

3ba*4b1

v

1/a

a*ad

a*dd,1

a*dd,2

acdd

Ener

gy



v

1/a

Ener

gy

a*4b2 a*

3ba*4b1

a<0



v

1/a

Ener

gy

a*4b2 a*

3ba*4b1

v

1/a

a*ad

a*dd,1

a*dd,2

acdd

Ener

gy

Universal four-body resonances in ultracold atomic and molecular gases J. P. D’Incao

This afternoon:Important for collisional propertiesImportant for collisional properties Experimental observationsExperimental observations

a>0a<0

(Innsbruck, LENS)

Outline:Outline:





N-BosonsN-Bosons

How to extend the analysis How to extend the analysis of universality to larger of universality to larger

systems?systems?

N-BosonsN-Bosons

Helium and spin-polarized tritium:similar behavior

Extension to larger systemsExtension to larger systems

Other studies:

Lewerenz, JCP (1997).Blume & Greene, JCP(2000).Hanna & Blume, PRA (2006).…

Blume et al., PRL(2002)

How many parameters are needed to describe N-boson systems?

Hammer & Platter, EPJA (2007).

• Linear correlations between cluster energies (like Tjon lines)

Hanna & Blume, PRA (2006):

Outline:Outline:





short-range physics

3 bodies3 bodies

Universal region

Model HamiltonianModel Hamiltonian

N-BosonsN-Bosons

Hyperspherical potential at unitatityHyperspherical potential at unitatity

dilute and weakly bound ~ universal

controlled by short-range physics

J. von Stecher arXiv:0909.4056 (2009)

RR

Goal: construct a minimal Hamiltonian that would lead to a

“universal ground state”.

short-range physics

3 bodies3 bodies

Universal region


N-BosonsN-Bosons

Hyperspherical potential at unitatityHyperspherical potential at unitatity

dilute and weakly bound ~ universal !!

RC

Lowest state is universal !!!Lowest state is universal !!!


Goal: construct a minimal Hamiltonian that would lead to a

“universal ground state”.


N-BosonsN-Bosons

controls three-body physics:3-body parameter

controls two-body physics:scattering length

Many-body Hamiltonian:Many-body Hamiltonian:



N-BosonsN-Bosons

Many-body Hamiltonian:Many-body Hamiltonian:

3 bodies3 bodies2 bodies2 bodies

rrijij

(r(r

ijij))

Bethe Peierls boundary condition

Two parameters (a and Rc) to independently control two and three-body physics.

U(R

U(R

)

RC


Hard core three-body potential

N-BosonsN-Bosons

Numerical methodsNumerical methods


Correlated Gaussian basis set expansion up to N=6.

Diffusion Monte Carlo simulations up to N=13.

• Gaussian potential for two and three-body interactions

• Hard Wall 3-body potential and an attractive square 2-body potential

• Trial wave function:

3-body correlations

2-body correlations

HR correlations

Outline:Outline:





N-BosonsN-Bosons

Weakly-bound ClustersWeakly-bound Clusters

Universal description: (just from dimensional analysis)

Universal functionUniversal function

Numerical predictions:


Comparison with other predictions

Comparison between CG and DMC

At least one N-body cluster state for each Efimov trimer!!

“Super” Borromean states

Linear Correlations (Tjon lines):

N-body resonant enhancement of losses:

N-BosonsN-Bosons

Weakly-bound ClustersWeakly-bound Clusters


Position of five-body resonance:Semi-Classical Methods and N-Body Recombination

S. Rittenhouse

This afternoon:

Properties:Properties:

Positions of resonances

N-BosonsN-Bosons

Dilute ClustersDilute Clusters

Pair correlation functionPair correlation function

Trayectories extracted from Quantum Monte Carlo simulations

Sphere radius: ~ 4r0 ~Rc

Five-body cluster state:Five-body cluster state:

rm~ 7Rc ~ 30 r0

~ 2Rc

N=3

N=6


RR

N bodiesN bodies

N-BosonsN-Bosons

Hyperspherical PictureHyperspherical Picture

Helium clustersHelium clustersMonte Carlo + Hyperspherical

D. Blume & C. H. Greene, JCP 2001

3+13+14+14+1

5+15+1

1+1+11+1+1

Potential curves scale with size and energy of cluster states All clusters follow Efimov scaling relations


……sketching potentialssketching potentials

• Extract width of four-body states

• Different species or/and spin statistic?– Universality?– Four-body states?

• Model Hamiltonian– with Quantum Monte Carlo

• Cluster states– Large N limit

Outlook:Outlook:

Energy per particle at unitarityEnergy per particle at unitarity



a=


Hammer & Platter prediction:



Weakly bound atom trimer state:Large and positive atom-trimer scattering length

Agreement with Hammer & Platter (2007)

a=


atom- trimer correlation

trimer correlation

Pair correlationsPair correlations

a>0

a<0

Blue: atom-trimerRed: dimer-dimerGreen dimer-2-atoms


Deviations from universalityDeviations from universality


Deviations:2nd order four-body corrections or numerical limitations?

Non universal Universal

Scattering length ratios for a<0 depends weakly on nonuniversal effects!!

less than 5%deviations

Different potentials Different Efimov states

Universal

Non Universal

Testing universality:

numerical studies of universality in few-boson systems javier von stecher department of physics and...

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