Few-Body Physics in Quantum Gases [Elektronische Ressource] / Kerstin Helfrich. Mathematisch-Naturwissenschaftliche Fakultät
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Few-Body Physics in Quantum Gases [Elektronische Ressource] / Kerstin Helfrich. Mathematisch-Naturwissenschaftliche Fakultät

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Few-Body Physics in Quantum GasesDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch{Naturwissenschaftlichen FakultatderRheinischen Friedrich{Wilhelms{Universitat Bonnvorgelegt vonKerstin HelfrichausAscha enburgBonn 2011Angefertigt mit Genehmigung derMathematisch-Naturwissenschaftlichen Fakult at derRheinischen Friedrich-Wilhelms-Universit at BonnDiese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unterhttp://hss.ulb.uni-bonn.de/diss online elektronisch publiziert.1. Gutachter: Prof. Dr. Hans-Werner Hammer2. Gutachter: Prof. Dr. Achim RoschTag der Promotion: 14.10.2011Erscheinungsjahr: 2011AbstractFew-body e ects play an important role for the understanding of ultracold quantum gases.We make use of an e ective eld theory approach to investigate various aspects of universalfew-body physics close to a Feshbach resonance. That is the regime where the scatteringlength is large compared to all other length scales of the system and thus determines theobservables. It is also the regime where the E mov e ect whose main characteristic is theoccurrence of a sequence of three-body bound states becomes important. At unitarity, i.e.,for diverging scattering length, the ratio of binding energies of neighboring states approachesa constant. The existence of those trimers can be captured by a single three-body parameterand in uences observables such as recombination rates.

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Published 01 January 2011
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Few-Body Physics in Quantum Gases
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch{Naturwissenschaftlichen Fakultat
der
Rheinischen Friedrich{Wilhelms{Universitat Bonn
vorgelegt von
Kerstin Helfrich
aus
Ascha enburg
Bonn 2011Angefertigt mit Genehmigung der
Mathematisch-Naturwissenschaftlichen Fakult at der
Rheinischen Friedrich-Wilhelms-Universit at Bonn
Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn unter
http://hss.ulb.uni-bonn.de/diss online elektronisch publiziert.
1. Gutachter: Prof. Dr. Hans-Werner Hammer
2. Gutachter: Prof. Dr. Achim Rosch
Tag der Promotion: 14.10.2011
Erscheinungsjahr: 2011Abstract
Few-body e ects play an important role for the understanding of ultracold quantum gases.
We make use of an e ective eld theory approach to investigate various aspects of universal
few-body physics close to a Feshbach resonance. That is the regime where the scattering
length is large compared to all other length scales of the system and thus determines the
observables. It is also the regime where the E mov e ect whose main characteristic is the
occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e.,
for diverging scattering length, the ratio of binding energies of neighboring states approaches
a constant. The existence of those trimers can be captured by a single three-body parameter
and in uences observables such as recombination rates.
Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that con-
tain all information of interest about the three-particle systems. With this method, we rst
investigate the scattering of atoms and dimers at nite temperature in the presence of the
E mov e ect. We calculate the dimer relaxation rate and obtain good agreement with avail-
able experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies
scattering length are studied in detail. If bosons are the majority species, the E mov ef-
fect occurs in an S-wave channel and we are able to compute three-body recombination and
dimer relaxation rates. The results are compared to the outcome of two existing experi-
ments. For mainly fermionic systems, the E mov e ect is only present in an overall P -wave
and three-body recombination at threshold vanishes. However, in mixtures of atoms and
dimers, scattering cross sections and dimer relaxation show the typical E mov behavior of
log-periodicity. We also consider two-dimensional Bose gases, where the E mov e ect does
not occur. For those systems, we derive an equation including the rst non-universal correc-
tions and deduce three-body observables such as binding energies and atom-dimer scattering
parameters.
Parts of this thesis have been published in the following articles and conference proceedings:
K. Helfrich and H.-W. Hammer, Resonant atom-dimer relaxation in ultracold atoms,
Europhys. Lett. 86, 53003 (2009).
K. Helfrich and H.-W. Hammer, The heteronuclear E mov e ect , Proceedings of the
19th International IUPAP Conference on Few-Body Problems in Physics, Bonn, Ger-
many, EPJ Web of Conferences 3, 02007 (2010).
K. Helfrich, H.-W. Hammer, and D.S. Petrov, Three-body problem in heteronuclear
mixtures with resonant interspecies interaction, Phys. Rev. A 81, 042715 (2010).
K. Helfrich and H.-W. Hammer, Resonant three-body physics in two spatial dimensions,
Phys. Rev. A 83, 052703 (2011).
K. Helfrich and H.-W. Hammer, Three bosons in two dimensions, in Strong interactions:
From methods to structures, Mini proceedings of the 474. WE-Heraeus-Seminar \Strong
interactions: From methods to structures", Bad Honnef, Germany, arXiv:1104.0847
[hep-ph] (2011).
K. Helfrich and H.-W. Hammer, On the E mov E ect in Higher Partial Waves , J.
Phys. B: At. Mol. Opt. Phys. 44, 215301 (2011).Contents
Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introduction 1
2 Motivation and Physical Background 5
2.1 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Cooling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Atom Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.5 Observables and Other Aspects . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Two-Body Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Three-Body Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 The E mov E ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Experimental Realization . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 E ective Field Theory 21
3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Two-Body Scattering in EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Three-Body Scattering in EFT . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
iii Contents
4 Atom-Dimer Scattering at Finite Temperatures 29
4.1 Atom-Dimer Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Atom-Dimer Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Heteronuclear Systems { Bosons in the S-Wave Channel 35
5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.1 Resonance Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.2 Three-Body Recombination for a> 0 . . . . . . . . . . . . . . . . . . . 38
5.2.3 Atom-Dimer Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.4 Three-Body Recombination for a< 0 . . . . . . . . . . . . . . . . . . . 43
5.3 Comparison to Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
40 875.3.1 The K- Rb Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
41 875.3.2 The K- Rb Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.3 Future Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Heteronuclear Systems { Fermions and Higher Angular Momenta 51
6.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 Scaling Factor and Resonance Positions . . . . . . . . . . . . . . . . . . . . . 53
6.3 Analytical Results for Atom-Dimer Scattering . . . . . . . . . . . . . . . . . . 55
6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.4.1 Atom-Dimer Observables Without E mov E ect . . . . . . . . . . . . 57
6.4.2 A Observables With E mov E ect . . . . . . . . . . . . . . 60
6.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Contents iii
7 Two-Dimensional Systems 65
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 Three-Body Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3.1 Three-Body Binding Energies . . . . . . . . . . . . . . . . . . . . . . . 70
7.3.2 Atom-Dimer Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.3.3 Three-Body Recombination . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8 Summary and Outlook 79
A Heteronuclear Integral Equation 83
A.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.2 Subtracted Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.3 Determining s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90L
B Integration Elements and Phase Space Factors 93
B.1 Integration Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.2 Phase Space Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
C Numerical Procedure 97
C.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C.2 Poles with Direct Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C.3 Poles with Contour Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography 101Notation and Conventions v
Notation and Conventions
Here, we introduce parameters and variables that are important throughout the whole thesis.
This should simplify the reading of the work and serve as reference.
Variable Description
a Scattering length
11a Bohr radius, a = 5:29177209 10 m0 0
a (a< 0) Scatt. length for which the E mov trimer hits the three-atom threshold
a (a> 0) for which the E mov hits the atom-dimer
a (a> 0) Scatt. length for which the three-body recombination rate is minimal0
a (a> 0) for which the three-body rate is maximal+
A Atom
A O -shell scattering amplitude
d Auxiliary dimer