Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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6. Bloch Oscillations and Time-Dependent Interactions We investigate the robustness of the stable points by varying both δ and ω slightly (by 10 −4 ) and determining the maximum Lyapunov exponent λ = maxk λk from a numerical integration of (6.25). The inverse Lyapunov exponent is a measure for the robustness and is mapped to the radius in the graphical representation in the parameter space figure 6.12. In the dissymmetry of the points + cos(F t) and − cos(F t), we recover the result from above. The stable points are arranged on a regular pattern, with the most stable points arranged on lines. With increasing denominator ν2 the robustness drops very rapidly. 6.7. Conclusions (Part II) We have treated the problem of Bloch oscillations with a time-dependent interaction in the framework of the one-dimensional tight-binding model, i.e. for a deep lattice potential with a strong transverse confinement. For smooth wave packets, the bounded-time argument allows identifying a class of interactions g(t) that lead to periodic dynamics. Beyond the mere existence of these periodic solutions, we have set up two complementary methods for the quantitative description of stability and decay of the Bloch oscillating wave packet. The collective-coordinates approach is valid as long as the wave packet is essentially conserved. This approach is capable of describing on the one hand the centroid and the breathing dynamics in the periodic cases, and on the other hand the beginning of the decay in the unstable cases, for example at constant interaction. The other approach, the linear stability analysis of the infinite wave packet, is suitable for the quantitative description of the decay of wide wave packets. More precisely, the relevant excitations have to be well decoupled in k-space from the original wave packet around k = 0. Perturbations like the off-phase perturbation in subsection 6.6.4 are well described. Other perturbations, like a constant interaction, are missed, because the width of the wave packet is not properly included in the ansatz of an infinitely wide wave packet. Together, the two approaches provide a rather complete picture of the wave-packet dynamics. The most remarkable physical results of this part are firstly the existence of long-living Bloch oscillations despite non-zero interaction if the interaction g(t) respects the time-reversal symmetry (6.16). Secondly, a modulation of the interaction that enhances the breathing of the wave packet can make the Bloch oscillation more robust with respect to perturbations. 136
A. List of Symbols and Abbreviations Bogoliubov part ak Coefficient for Bogoliubov transformation (2.32), ak = � ɛ 0 k /ɛk A(kξ, θ) Elastic scattering envelope function Ad(k) Angular integral in the self-energy in section 4.1 BEC Bose-Einstein condensate β Inverse temperature (kBT ) −1 ˆβν Bogoliubov quasiparticle operator (in eigenbasis) Cd Disorder correlation function c Speed of sound nnc Condensate depletion d Dimension ɛk Bogoliubov energy (2.35) ɛ 0 k Kinetic energy of a free particle F Anomalous Green function ˆF Grand canonical Hamiltonian g Interaction parameter G Green function ˆγk Bogoliubov mode γk Scattering rate γ(k) Complex degree of coherence (only section 3.1) 137
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Bogoliubov Excitations of Inhomogen
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Abstract In this thesis, different
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Contents 2.5. Exact diagonalization
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Contents Bibliography 141 List of F
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1. Interacting Bose Gases—Complex
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1.2. Disorder 1.2. Disorder Idealiz
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1.4. Cold-atoms—Universal model s
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1.5. Cold atoms—History and key e
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1.5. Cold atoms—History and key e
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1.6. Standing of this work are rest
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1.6. Standing of this work The disc
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2. The Inhomogeneous Bogoliubov Ham
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2.1. Bose-Einstein condensation of
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2.1. Bose-Einstein condensation of
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2.2. Interacting BEC and Gross-Pita
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n(x) n∞ 1 0.8 0.6 0.4 0.2 0 -3 -2
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2.3. Bogoliubov Excitations and pha
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2.3. Bogoliubov Excitations At high
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„ Φq ′ ¯Φ q ′ « V „ Φq
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(2) k ′ q k = ξ2 � k ′′2 +
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k k ′ 2.4.1. Single scattering se
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2.4. Single scattering event back a
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k + − w (1) k ′ k (a) 2.4. Sing
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2.4. Single scattering event theory
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T 1 B 2 2.5. Exact diagonalization
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2.5. Exact diagonalization of the B
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3. Disorder In the previous chapter
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3.1.1. Speckle amplitude 3.1. Optic
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3.1. Optical speckle potential In c
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3.2. A suitable basis for the disor
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3.2. A suitable basis for the disor
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3.3. Effective medium and diagramma
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3.4. Deriving physical quantities f
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3.4. Deriving physical quantities f
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Fixed particle number 3.4. Deriving
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ρµξ d 0.2 0.15 0.1 0.05 1 2 3 4
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4. Disorder—Results and Limiting
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4.1. Hydrodynamic limit I: ξ = 0 t
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Page 101 and 102: ǫk 0.25 0.5 0.75 1 1.25 1.5 1.75 k
Page 103 and 104: gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5
Page 105 and 106: 4.2. Hydrodynamic limit II: towards
Page 107 and 108: ∆ǫk ǫk �0.1 �0.2 �0.3 �
Page 109 and 110: 1 0.5 0 -0.5 0 20 40 60 80 100 x/σ
Page 111 and 112: ΛN �0.1 �0.2 �0.3 �0.4 4.3
Page 113 and 114: nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
Page 115 and 116: 4.4. Particle regime which coincide
Page 117 and 118: 4.4. Particle regime regime, becaus
Page 119 and 120: 0.02 0.015 0.01 0.005 MSg 0.01 0.1
Page 121 and 122: 5. Conclusions and Outlook (Part I)
Page 123 and 124: 5.3. Theoretical outlook potential.
Page 125: Part II. Bloch Oscillations 113
Page 128 and 129: 6. Bloch Oscillations and Time-Depe
Page 146 and 147: λTB t/TB 0.15 0.05 70 60 50 40 30
Page 150 and 151: A. List of Symbols and Abbreviation
Page 152 and 153: 140 A. List of Symbols and Abbrevia
Page 154 and 155: Bibliography [11] P. A. Lee and T.
Page 156 and 157: Bibliography [35] M. Lewenstein, A.
Page 158 and 159: Bibliography [58] J. Stenger, S. In
Page 160 and 161: Bibliography [81] V. I. Yukalov and
Page 162 and 163: Bibliography [108] A. Mostafazadeh,
Page 164 and 165: Bibliography [131] F. Bloch, Über
Page 167 and 168: Acknowledgments I thank everybody w
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