Trimer and Tetramer Bound States in Heteronuclear...
Transcript of Trimer and Tetramer Bound States in Heteronuclear...
Trimer and Tetramer Bound States in Heteronuclear Systems
Christiane H. Schmickler
in collaboration withE. Hiyama
H.-W. Hammer
August 10, 2016
Outline
Efimov Effect
Efimov Effect for Mixtures
Gaussian Expansion Method
Results
Summary and Outlook
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Efimov Effect for Mixtures
• Similar effect appears for fermion-boson andboson-boson mixtures
• The factor between consecutive energies of statesdepends on the mass ratio
• If there is only one fermion, spin can be neglected
• We neglected interaction between atoms ofthe same type
• In heavy-heavy-light mixtures, the ratiobetween consecutive energies of states is smaller thanfor identical bosons→ easier to observe experimentally
m
M M
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Efimov Effect for Mixtures II
• Four-body states exist as well
M
MM
m
• As with identical bosons, there is typically a ground and an excitedstate at unitarity
Blume, Yan Phys. Rev. Lett. 113 (2014),213201
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Gaussian Expansion MethodImplementation by Hiyama, Kino, Kamimura PPNP 51 (2003),223
• Rayleigh-Ritz Variational Method
• Base functions are selected via geometric progression between aminimum and a maximum range
• Number of parameters used:
System Parameters
Dimer 3Trimer 18
Tetramer 45
• Parameter space increases rapidly
• Used a mixture of systematic scanning of parameter subspaces andrandom sampling within relatively broad boundaries to findoptimized base functions
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Interaction
• Use effective potentials for good performance, stable behaviourand easy parameter choice
• Combination of attractive 2-body and repulsive 3-body potential
ViN = V0V0V0 exp
(−r2iN2r20
), WijN = W0W0W0 exp
(−r2ij + r2jN + r2iN
16r20
)
• N − 1 atoms of mass M, Nth atom of mass m
• Natural energy scale of the problem: Es = 12r20
m+MMm
• Approximation valid, if binding Energies |E | � Es
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7Li-6Li Mixture
-1
-0.8
-0.6
-0.4
-0.2
0
0.075 0.0751 0.0752
√ |ED|−√ |E|
[10−
7√Es
]
r0/a
-9
-6
-3
0
0.0741 0.0751
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87Rb-7Li Mixture
-1
-0.8
-0.6
-0.4
-0.2
0
0.2215 0.2217 0.2219 0.2221
√ |ED|−√ |E|
[10−
7√Es
]
r0/a
-9
-6
-3
0
0.2201 0.2216
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133Cs-6Li Mixture
-1
-0.8
-0.6
-0.4
-0.2
0
0.2627 0.2629 0.2631
√ |ED|−√ |E|
[10−
7√Es
]
r0/a
-9
-6
-3
0
0.2612 0.2627
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133Cs-6Li Mixture
-1
-0.8
-0.6
-0.4
-0.2
0
0.2627 0.2629 0.2631
√ |ED|−√ |E|
[10−
7√Es
]
r0/a
-9
-6
-3
0
0.2612 0.2627
cd
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Relative Crossing Point Positions
0
0.5
1
1.5
2
0 5 10 15 20 25
c d×
104
M/m
7Li/6Li
87Rb/7Li
133Cs/6Li
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Cross-checking with Effective STM Treatment
• Before vanishing through the dimer threshold, the trimer becomesvery weakly bound
• Scattering length can be approximated by inverse binding energy
• Close to threshold, the dimer-atom scattering length should belarge (aDa/r0 ≈ 107)
• Solve STM-equation for effective 3-body system(dimer-atom-atom)
• Tune 3-body parameter Λto reproduce results
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Relative Crossing Point Positions II
0
0.5
1
1.5
2
0 5 10 15 20 25
c d×
104
M/m
7Li/6Li
87Rb/7Li
133Cs/6Li
Λ = 744Λ = 744Λ = 744
Λ = 426Λ = 426Λ = 426
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Summary and Outlook
• First investigation into behaviour of heteronuclear trimer andtetramer at dimer threshold
• Trimer and tetramer cross into the dimer at almost the same point
• Results for crossing difference seem inconsistent→ more data points are needed
• Next steps: investigate dependence on potential shape and 3-bodypotential strength
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