Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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6. Bloch Oscillations and Time-Dependent Interactions Figure 6.2: Sketch of the dispersion relation in a lattice. The dashed line shows the free dispersion ɛ 0 k = �2 k 2 /(2m), folded back into the first Brillouin zone. The lattice with period d couples the degenerate states k = +π/d and k = −π/d at the edges of the Brillouin zone, which leads to band gaps (avoided crossings). In a deep lattice potential, the band gap becomes large and excitations from the lowest band to higher bands can be neglected at low temperatures. − k π ǫ d n = 3 n = 2 n = 1 on the particles. Even if the lattice is weak, in the sense that each site is only a weak scatterer, the coherent superposition of waves reflected at the periodic lattice can lead to total reflection if the scattered waves interfere constructively. 6.1.1. Bloch oscillation of a single particle The lattice is symmetric under discrete translations by integer multiples of the lattice vectors. As a consequence, the Bloch theorem [131] states that every energy eigenstate ψ(r) can be written as the product of a plane wave eik·r and a lattice-periodic function u (n) k (r). The energy states are organized in energy bands labeled with the index n (figure 6.2). If the lattice potential is deep, the gap separating the lowest band from higher bands becomes large, such that tunneling to higher bands (Landau- Zener tunneling [132]) can be neglected and a description restricted to the lowest band is sufficient. The dispersion relation ɛ(k) is proportional to − cos(kd) (subsection 6.2.1). The basic concept of Bloch oscillations can be understood within a semiclassical reasoning. Let us consider a wave packet with a narrow momentum distribution in a lattice. If the wave packet is accelerated by a uniform force −F (like gravity or an electric field for charged particles), then its momentum �k = −F t will increase linearly. In a lattice, however, momentum and velocity do not coincide. Indeed, the velocity is given by the group velocity, i.e. the first derivative of the dispersion relation vg = ∂kɛ(k) ∝ − sin(kd) = sin(F t). Instead of accelerating uniformly, the 116 π d
6.1. Introduction Figure 6.3: Sketch of the intensity of two counterpropagating laser beams, which creates the lattice potential for the ultracold atoms. Taken from [37]. wave packet oscillates in space. This phenomenon is known as Bloch oscillation and was predicted quite some time ago by Bloch and Zener [131, 133]. Notably, the edge of the Brillouin zone π/d coincides with the momentum, where the de Broglie wave length equals the lattice period. Simply speaking, Bragg reflection of the particle leads to the oscillating motion. In subsection 6.2.2, we will consider Bloch oscillations in a more rigorous way than in this introductory paragraph. 6.1.2. Experimental realization Bloch oscillations rely on the coherent reflection of waves and are very sensitive to any kind of dephasing, as interaction effects or impurities in the lattice. In solid-state systems, the lattice spacing d is given by atomic distances, which is so short that the electrons suffer from scattering events long before they reach the edge of the Brillouin zone π/d. For the experimental observation of Bloch oscillations, it was necessary to increase artificially the lattice spacing, which was achieved in semiconductor superlattices [134, 135]. Later, Bloch oscillations were observed in cold atomic gases in optical lattice potentials [7, 8]. In cold-atom experiments, the atoms are trapped in the optical dipole potential (subsection 3.1.3, or [39]) of a laser standing wave, figure 6.3. The strength of the lattice can be adjusted via the laser intensity and detuning, and the lattice spacing can be selected via the laser wave length and the relative angle of the beams [136]. There is also an optical analog of Bloch oscillations. In the experiments [137, 138], an array of weakly coupled wave guides was prepared with an index of refraction that increased linearly across the array. Then, a light beam propagating along the wave-guide array oscillates along the transverse direction. The longitudinal direction of the array takes the role of the time axis c t. 117
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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
Page 77 and 78: 3.2. A suitable basis for the disor
Page 79 and 80: 3.3. Effective medium and diagramma
Page 85 and 86: 3.4. Deriving physical quantities f
Page 87 and 88: 3.4. Deriving physical quantities f
Page 89 and 90: Fixed particle number 3.4. Deriving
Page 91 and 92: ρµξ d 0.2 0.15 0.1 0.05 1 2 3 4
Page 93 and 94: 4. Disorder—Results and Limiting
Page 95 and 96: 4.1. Hydrodynamic limit I: ξ = 0 t
Page 97 and 98: 4.1. Hydrodynamic limit I: ξ = 0 T
Page 99 and 100: 4.1. Hydrodynamic limit I: ξ = 0 I
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 130 and 131: 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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