Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

More documents

Recommendations

Info

1. Interacting Bose Gases Tuning the interactions In contrast to electrons, the atoms are neutral and have only short-range interactions. Typically, the average particle distance is much larger than the scattering length, a length that describes the strength of s-wave scattering. For many atom species used in cold-atom experiments, the scattering length for atom-atom scattering can be tuned by means of a Feshbach resonance [43–46]. The basic idea is to tune the unbound scattering state into resonance with a bound state (molecule). This is done using the Zeeman shift induced by an external magnetic field. At the resonance, the scattering length has a pole. Together with the background scattering length, this allows the scattering length to be tuned to arbitrary positive or negative values. It is even possible to switch off the atom-atom interactions. 1.5. Cold atoms—History and key experiments Bose-Einstein condensation Bose and Einstein established the theory of Bose-Einstein statistics in 1924 and 1925 [47, 48]. The key idea is that quantum particles are indistinguishable, i.e., two states that differ only by the interchange of two particles are actually the same state. Considering a non-interacting Bose gas in three dimensions, Einstein realized that at a given temperature only a finite number of particles can populate the excited states. When more particles are added to the system, they condense into the lowest energy state, whose occupation number diverges (section 2.1). The critical particle number, however, turned out to be so large that for many decades it was impossible to reach sufficiently low temperatures and sufficiently high densities without the particles forming a liquid or a solid, due to their interactions. The first experiments that came close to Bose-Einstein condensation were experiments with superfluid helium [49]. However, neither 3 He nor 4 He can be regarded as a direct realization of Einstein’s condensate of the noninteracting gas. 3 He atoms are fermions and have to be described by BCS theory [50], and 4 He is dominated by interactions, which makes it more a liquid than an ideal gas. In the nineteen-eighties and nineties, a lot of effort was made to create a weakly interacting BEC of spin-polarized hydrogen and of gases of alkali atoms. In 1995, the alkali experiments were successful: a sodium BEC was realized at MIT [51], a rubidium BEC at JILA [38] and a lithium BEC at Rice University [52]. A few years later, also the hydrogen experiment was successful [53]. In the past years, alkali BECs have become the workhorses 6
1.5. Cold atoms—History and key experiments Table 1.1.: Typical temperatures and particle densities in BEC experiments T [µK] ρ[cm −3 ] Sodium, [51, MIT] 2 10 14 Rubidium, [38, JILA] 0.17 2.5 × 10 12 Lithium, [52, Rice] ∼ 0.2 not measured atmosphere 300 × 10 6 3 × 10 19 for all kinds of experiments. In 2001, E. A. Cornell (JILA), C. E. Wieman (JILA), and W. Ketterle (MIT) were awarded the Nobel prize in physics “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates”. The atom densities in BEC experiments are very limited because most elements form liquids or a solids at low temperatures, due to their interactions. At reduced densities, the temperatures required for Bose-Einstein condensation become even lower and demand sophisticated trapping and cooling techniques (evaporative cooling) [38, 51, 52]. Compared with atmospheric conditions, temperatures and densities in the alkali BECs are incredibly low (table 1.1). With these experiments, the phenomenon of Bose-Einstein condensation predicted 70 years earlier became directly accessible. The population of the ground state can be observed rather directly by taking time-of-flight absorption images [38, 51]. The trapping potential is switched off and the condensate expands, converting its momentum distribution to a real-space distribution, which can be observed by taking absorption images. In these images, a bi-modal distribution consisting of the condensate fraction around Figure 1.1: The time-of-flight images from [51] (taken from JILA web page) show the momentum-space portrait of the rubidium cloud: thermal cloud (left), bi-modal distribution (middle), condensate (right). 7
Page 1: Bogoliubov Excitations of Inhomogen
Page 5: Abstract In this thesis, different
Page 8 and 9: Contents 2.5. Exact diagonalization
Page 10 and 11: Contents Bibliography 141 List of F
Page 13 and 14: 1. Interacting Bose Gases—Complex
Page 15 and 16: 1.2. Disorder 1.2. Disorder Idealiz
Page 17: 1.4. Cold-atoms—Universal model s
Page 21 and 22: 1.5. Cold atoms—History and key e
Page 23 and 24: 1.6. Standing of this work are rest
Page 25: 1.6. Standing of this work The disc
Page 29 and 30: 2. The Inhomogeneous Bogoliubov Ham
Page 31 and 32: 2.1. Bose-Einstein condensation of
Page 33 and 34: 2.1. Bose-Einstein condensation of
Page 35 and 36: 2.2. Interacting BEC and Gross-Pita
Page 41 and 42: n(x) n∞ 1 0.8 0.6 0.4 0.2 0 -3 -2
Page 43 and 44: 2.3. Bogoliubov Excitations and pha
Page 45 and 46: 2.3. Bogoliubov Excitations At high
Page 47 and 48: „ Φq ′ ¯Φ q ′ « V „ Φq
Page 49 and 50: (2) k ′ q k = ξ2 � k ′′2 +
Page 51 and 52: k k ′ 2.4.1. Single scattering se
Page 53 and 54: 2.4. Single scattering event back a
Page 55 and 56: k + − w (1) k ′ k (a) 2.4. Sing
Page 57 and 58: 2.4. Single scattering event theory
Page 59 and 60: T 1 B 2 2.5. Exact diagonalization
Page 61 and 62: 2.5. Exact diagonalization of the B
Page 69 and 70:
3. Disorder In the previous chapter
Page 71 and 72:
3.1.1. Speckle amplitude 3.1. Optic
Page 73 and 74:
3.1. Optical speckle potential In c
Page 75 and 76:
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 81 and 82:
Page 83 and 84:
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 128 and 129:
6. Bloch Oscillations and Time-Depe
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
λTB t/TB 0.15 0.05 70 60 50 40 30
Page 148 and 149:
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
show all

Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

Create successful ePaper yourself

Delete template?

Save as template?