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Strong correlation effects with atoms in optical lattices [Elektronische Ressource] / María Gracia Eckholt Perotti

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Technische Universit at Munc henMax-Planck-Institut fur QuantenoptikStrong correlation e ectswith atoms in optical latticesMar a Gracia Eckholt PerottiVollst andiger Abdruck der von der Fakult at fur Physikder Technischen Universit at Munc henzur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. R. GrossPrufer der Dissertation: 1. Hon.-Prof. I. Cirac, Ph.D.2. Univ.-Prof. Dr. H. FriedrichDie Dissertation wurde am 15.10.2009bei der Technischen Universit at Munc hen eingereichtund durch die Fakult at fur Physik am 11.12.2009 angenommen.I’ve become frogproofAbstractOur work concerns the e ects of strong correlations of ultracold atoms in optical lat-tices in three di erent scenarios. First, we analyze the in uence of dissipative processes onthe super uid{Mott insulator transition in the Bose{Hubbard model, observing a shift ofthe well-known phase transition. In a second setup, atoms are trapped in state-dependentlattices; we show that any asymmetry in the contact interaction produces a form of corre-lated hopping, which results in a pair super uid phase with interesting correlation prop-erties that di erentiate it from an ordinary atomic Bose{Einstein condensate. Finally,we investigate a two-species Bose{Hubbard model including a conversion term, whichcan be implemented experimentally through a Feshbach resonance.

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Published 01 January 2009
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Technische Universit at Munc hen
Max-Planck-Institut fur Quantenoptik
Strong correlation e ects
with atoms in optical lattices
Mar a Gracia Eckholt Perotti
Vollst andiger Abdruck der von der Fakult at fur Physik
der Technischen Universit at Munc hen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. R. Gross
Prufer der Dissertation: 1. Hon.-Prof. I. Cirac, Ph.D.
2. Univ.-Prof. Dr. H. Friedrich
Die Dissertation wurde am 15.10.2009
bei der Technischen Universit at Munc hen eingereicht
und durch die Fakult at fur Physik am 11.12.2009 angenommen.I’ve become frogproofAbstract
Our work concerns the e ects of strong correlations of ultracold atoms in optical lat-
tices in three di erent scenarios. First, we analyze the in uence of dissipative processes on
the super uid{Mott insulator transition in the Bose{Hubbard model, observing a shift of
the well-known phase transition. In a second setup, atoms are trapped in state-dependent
lattices; we show that any asymmetry in the contact interaction produces a form of corre-
lated hopping, which results in a pair super uid phase with interesting correlation prop-
erties that di erentiate it from an ordinary atomic Bose{Einstein condensate. Finally,
we investigate a two-species Bose{Hubbard model including a conversion term, which
can be implemented experimentally through a Feshbach resonance. We are particularly
interested in the exotic incompressible, yet super uid \super-Mott" phase.
iiiContents
Abstract iii
A short overview 1
Outline of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Background: Atoms in an optical lattice 9
1 Cold atoms in optical lattices 9
1.1 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Theory of AC-Stark shift trapping . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Single-particle states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Many-body Hamiltonian: the Bose{Hubbard model . . . . . . . . . . . . . 13
Phases of the Bose{Hubbard model . . . . . . . . . . . . . . . . . . . . . . 15
2 Description of theoretical methods 19
2.1 Quantum Rotor model: the phase approximation . . . . . . . . . . . . . . 20
2.2 Strong coupling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Mean- eld approach: the Gutzwiller ansatz . . . . . . . . . . . . . . . . . . 26
2.4 DMRG and the Matrix Product State algorithms . . . . . . . . . . . . . . 27
Matrix Product State ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 27
DMRG algorithm: optimization of MPS . . . . . . . . . . . . . . . . . . . 30
iTEBD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Atoms in a noisy environment 37
3 The physical system for dissipation 37
3.1 A mathematical toolbox for dissipative systems . . . . . . . . . . . . . . . 37
Master equation formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Heisenberg equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 A Hamiltonian with dissipation . . . . . . . . . . . . . . . . . . . . . . . . 40
Decoherence mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
v4 Description of theoretical results 43
4.1 Applying the Quantum Rotor model to a dissipative system . . . . . . . . 43
Hamiltonian for the full model . . . . . . . . . . . . . . . . . . . . . . . . . 44
Reduced density matrix of the system . . . . . . . . . . . . . . . . . . . . . 47
Characterizing the state of the system: correlation functions . . . . . . . . 48
A generalized ansatz for the master equation . . . . . . . . . . . . . . . . . 51
4.2 Strong coupling expansion around the interacting Mott . . . . . . . . . . . 54
4.3 A Gutzwiller ansatz for the system and reservoir . . . . . . . . . . . . . . . 55
Atoms with correlated hopping 59
5 The physical system for correlated hopping 59
5.1 Correlated hopping model . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 A family of correlated hopping Hamiltonians . . . . . . . . . . . . . . . . . 60
Relation to atomic parameters . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Description of theoretical results 63
6.1 A two-sites example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Toy model for pair-super uidity . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 No hopping limit: insulating phases . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Perturbation theory around the Mott phase . . . . . . . . . . . . . . . . . 68
6.5 Analysis of the pair-super uid with the phase model . . . . . . . . . . . . . 70
6.6 Phase diagram using a Gutzwiller ansatz . . . . . . . . . . . . . . . . . . . 73
6.7 Quasi-exact diagonalizations and long-range pair correlations . . . . . . . . 74
6.8 Some experimental considerations: detection of the phases . . . . . . . . . 76
Atoms in a Feshbach resonance 81
7 The physical system for atom-molecule resonances 81
7.1 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 A Hamiltonian for mixtures . . . . . . . . . . . . . . . . . . 82
8 Description of theoretical results 85
8.1 Adiabatic elimination of the molecular state . . . . . . . . . . . . . . . . . 85
8.2 Strong coupling expansion around the atomic solution . . . . . . . . . . . . 87
8.3 Phase diagram using a mean- eld approach . . . . . . . . . . . . . . . . . . 89
8.4 Quasi-exact diagonalizations and study of long-range correlations . . . . . 94
Appendix 103
A Writing a DMRG algorithm with MPS 103
A.1 Normalization of the state . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.2 E ective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105vii
B Derivation of the master equation 109
B.1 Number dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.1.1 The system as a damped harmonic oscillator . . . . . . . . . . . . . 109
B.1.2 The as a Mott insulator . . . . . . . . . . . . . . . . . . . . 110
B.2 Phase dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C Derivation of the model in superlattices 117
C.1 Dressed states trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C.2 State-changing collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.3 Final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
D Atom-molecule resonances: calculation of physical parameters 121
D.1 Table of physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
D.2 Derivation of the physical parameters . . . . . . . . . . . . . . . . . . . . . 122
Bibliography 133
Acknowledgments 135viii