Numerical Studies of Universality in FewBoson Systems

1
Numerical Studies of Universality in Few-Boson
Systems
Javier von Stecher
Department of Physics and JILA University of
Colorado
In collaboration with Chris H. Greene Jose P.
DIncao
Acknowledgements Doerte Blume Seth
Rittenhouse Nirav Mehta NSF
Probable configurations of a weakly-bound
four-boson state.
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How does Efimov physics and universality extends
beyond three bodies?
3
Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

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Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

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Solving the Schrödinger equation
model interactions
Hamiltonian
or combinations of Gaussians
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Correlated Gaussian basis set expansion

• remove CM and set L0

• basis functions

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

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J. von Stecher, C. H. Greene PRA (2009)
Hyperspherical representation
R overall size (collective motion)
Divide and conquer
typical potential curves
solved at fixed R.
Coupled 1D equations
Use correlated Gaussian to expand channel
functions
Find efficient way to evaluate matrix elements

• Accurate description of four-body systems!!

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Four-body Formulation
Hyperspherical Picture
Fragmentation thresholds
for scattering properties.
1111
R
U(R)
211
Interaction region
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31
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All particles close together Bound and
quasi-bound states
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J. von Stecher, C. H. Greene PRA (2009)
Hyperspherical Picture
D.S. Petrov, C. Salomon, G.V. Shlyapnikov PRL
(2004)
for studying universality
Four fermion system at unitarity

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

Connection to trapped system
trap frequency
Predictions
s02.0092 (CGH)
s02.0096 (CG)

• Appropriate framework to analyze universality

See also J. P. DIncao et al PRA (2009)
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Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Universality in four bosons
a?
Efimov scaling
Efimov scaling
,
3 body
4 body
,
Two four-body states
,
same scaling relations!!!
In agreement with Hammer Platter EJPA
(2007) (Hammer presentation)
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Hyperspherical Picture (a?)
3 bodies
4 bodies
R
R
Four-body potential curves follow the same
scaling of the Efimov states !!!
Four-body physics is universal!!
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Hyperspherical Picture (a?)
3 bodies
4 bodies
R
R
Four-body potential curves follow the same
scaling of the Efimov states !!!
Three-body parameter
Four-body physics is universal!!
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Hyperspherical Picture (a?)
3 bodies
4 bodies
R
R
Four-body potential curves follow the same
scaling of the Efimov states !!!
Three-body parameter
Four-body physics is universal!!
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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Four-body states
a?
Hammer Platter prediction
Agreement with Hammer Platter (2007)
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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Spectrum Extended Efimov plot

• Calculations at more than 500 scattering
  lengths!!!

dimer-two atoms threshold (red lines)
atom-trimer thresholds (green lines)
dimer-dimer threshold (red lines)
four-body states (black lines)
?1/a
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Universality away from unitarity
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Universality away from unitarity
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Universality away from unitarity
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Universality away from unitarity
alt0
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Universality away from unitarity
alt0
agt0

• Important for collisional properties
• Experimental observations

This afternoon
(Innsbruck, LENS)
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Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

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• How to extend the analysis of universality to
  larger systems?

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Extension to larger systems
Helium and spin-polarized tritium similar
behavior

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

Hanna Blume, PRA (2006)

• Linear correlations between cluster energies
  (like Tjon lines)

Blume et al., PRL(2002)
Other studies Lewerenz, JCP (1997). Blume
Greene, JCP(2000). Hanna Blume, PRA (2006).
Hammer Platter, EPJA (2007).
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Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

26
J. von Stecher arXiv0909.4056 (2009)
Model Hamiltonian

• Goal construct a minimal Hamiltonian that would
  lead to a universal ground state.

Hyperspherical potential at unitatity
3 bodies
dilute and weakly bound universal



controlled by short-range physics
short-range physics
Universal region
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J. von Stecher arXiv0909.4056 (2009)
Model Hamiltonian

• Goal construct a minimal Hamiltonian that would
  lead to a universal ground state.

Hyperspherical potential at unitatity
3 bodies
RC
dilute and weakly bound universal !!



Lowest state is universal !!!
short-range physics
Universal region
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J. von Stecher arXiv0909.4056 (2009)
Model Hamiltonian
Many-body Hamiltonian
controls three-body physics 3-body parameter
controls two-body physics scattering length
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J. von Stecher arXiv0909.4056 (2009)
Model Hamiltonian

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

Many-body Hamiltonian
3 bodies
Hard core three-body potential
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J. von Stecher arXiv0909.4056 (2009)
Numerical methods

• Correlated Gaussian basis set expansion up to
  N6.

• Gaussian potential for two and three-body
  interactions

• Diffusion Monte Carlo simulations up to N13.

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

• Trial wave function

HR correlations
3-body correlations
2-body correlations
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Outline

• Universality in four-boson systems
• Numerical formulation (CG and CGH)
• Four-bosons predictions
• Extension to larger systems
• Model Hamiltonian
• Cluster states

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J. von Stecher arXiv0909.4056 (2009)
Weakly-bound Clusters
Universal description
(just from dimensional analysis)
Universal function
Numerical predictions
Comparison with other predictions
Comparison between CG and DMC

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

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J. von Stecher arXiv0909.4056 (2009)
Weakly-bound Clusters
Properties

• Super Borromean states
• Linear Correlations (Tjon lines)
• N-body resonant enhancement of losses

This afternoon
Position of five-body resonance
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J. von Stecher arXiv0909.4056 (2009)
Dilute Clusters
Pair correlation function
Five-body cluster state
N6
N3
Sphere radius 4r0 Rc
Trayectories extracted from Quantum Monte Carlo
simulations
rm 7Rc 30 r0
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J. von Stecher arXiv0909.4056 (2009)
Hyperspherical Picture
sketching potentials
N bodies
Helium clusters
Monte Carlo Hyperspherical
111
R
31
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D. Blume C. H. Greene, JCP 2001

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

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Outlook

• 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

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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Four-body states
a?
Hammer Platter prediction
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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Four-body states
a?
Agreement with Hammer Platter (2007)

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

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alt0
agt0
Blue atom-trimer Red dimer-dimer Green
dimer-2-atoms
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J. von Stecher, J. P. DIncao and C. H. Greene,
Nat. Phys. (2009)
Deviations from universality

• Different potentials
• Different Efimov states

Non universal
Universal
less than 5 deviations
Testing universality

• Scattering length ratios for alt0 depends weakly
  on nonuniversal effects!!

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