trimer and tetramer bound states in heteronuclear...
TRANSCRIPT
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
2 / 16
Efimov Effect
√E 1
a
3 / 16
Efimov Effect
√E 1
a
Accumulation pointat unitarity
3 / 16
Efimov Effect
√E 1
a
Lowest stateconnected toeffective range
3 / 16
Efimov Effect
√E 1
a
×λ
×λ
3 / 16
Efimov Effect
√E 1
a
×λ ×λ
3 / 16
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
4 / 16
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
5 / 16
Efimov Effect in the Heteronuclear Four-Body System
√E 1
a
6 / 16
Efimov Effect in the Heteronuclear Four-Body System
√E 1
a
6 / 16
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
7 / 16
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
8 / 16
Efimov Plot for Heteronuclear Four-body System
√E 1
a
9 / 16
Efimov Plot for Heteronuclear Four-body System
9 / 16
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
10 / 16
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
11 / 16
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
12 / 16
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
12 / 16
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
13 / 16
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
14 / 16
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
15 / 16
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
16 / 16