Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

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104 Linear response - Probing the Bogoliubov band structure ¯hω/ER 12 10 8 6 4 2 .001 .01 .24 .15 0 −3 −2 −1 0 1 2 3 ¯hq/qB .20 .57 .10 .13 Figure 8.1: Bogoliubov bands for s =10and gn =0.5ER in the first three Brillouin zones. Excitation strengths Zj/Ntot (8.8) towards the states in the first three bands for p = −1.2qB, p =0.8qB and p =2.8qB. Zj(p)/Ntot 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −3 −2 −1 0 1 2 3 p/qB Figure 8.2: Excitation strength (8.9) to the first (j =1; solid line), second (j =2;dashed line) and third Bogoliubov band (j =3; dotted line) without lattice (s =0)forgn =0.5ER. .04
8.1 Dynamic structure factor 105 The excitation strength to the lowest Bogoliubov band - Numerical results The study of the excitation strength to the lowest Bogoliubov band Z1(p) requires the evaluation of (8.8) with the solutions u1q(z), v1q(z) and ϕ(z) for a given lattice depth s and interaction parameter gn/ER. Numeric results are depicted in Figs.8.3 and 8.4 for s =5, 10 and gn =0ER, 0.02ER and gn =0.5ER. In general, we find the following characteristics: • Z1(p) features an overall decay for increasing |p|. This is due to the fact that the momentum components of the created excitations are smaller at large momenta. • Provided that gn/ER = 0, Z1(p) exhibits characteristic oscillations: Z1(p) is suppressed in the vicinities of p = l2π/d where the excitations contributing to the strength have phonon character and is exactly zero at p = l2π/d (l integer) where the energy of the contributing excitation vanishes. • Increasing gn/ER at fixed s or increasing s at fixed gn/ER = 0leads to an overall decrease of Z1(p). Z1(p)/Ntot 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −6 −4 −2 0 2 4 6 p/qB Figure 8.3: Excitation strength to the lowest Bogoliubov band Z1(p) (8.8) at lattice depth s =5for gn =0.5ER (solid line), gn =0.02ER (dashed line) and gn =0(dash-dotted line).
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Università degli Studi di Trento F
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7 Bogoliubov excitations of Bloch s
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Chapter 1 Introduction Superfluids
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The second class of stationary solu
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Chapter 2 Vortex nucleation The pro
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2.1 Single vortex line configuratio
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2.2 Role of Quadrupole deformations
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2.2 Role of Quadrupole deformations
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2.3 Critical angular velocity for v
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2.3 Critical angular velocity for v
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2.4 Stability of a vortex-configura
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Chapter 3 Introduction Since the ac
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The order parameter’s temporal ev
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and spectroscopy were described in
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the external probe generates a dens
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Chapter 4 Single particle in a peri
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4.1 Solution of the Schrödinger eq
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4.2 Tight binding regime 43 describ
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4.2 Tight binding regime 45 paramet
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Chapter 5 Groundstate of a BEC in a
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5.1 Density profile, energy and che
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5.1 Density profile, energy and che
Page 57 and 58: 5.2 Compressibility and effective c
Page 59 and 60: 5.4 Effects of harmonic trapping 55
Page 67 and 68: Chapter 6 Stationary states of a BE
Page 69 and 70: 6.1 Bloch states and Bloch bands 65
Page 79 and 80: 6.2 Tight binding regime 75 conside
Page 81 and 82: 6.2 Tight binding regime 77 Compari
Page 83 and 84: 6.2 Tight binding regime 79 2m¯v/q
Page 85 and 86: 6.2 Tight binding regime 81 The lea
Page 87 and 88: Chapter 7 Bogoliubov excitations of
Page 89 and 90: 7.2 Bogoliubov bands and Bogoliubov
Page 97 and 98: 7.3 Tight binding regime of the low
Page 99 and 100: 7.3 Tight binding regime of the low
Page 101 and 102: 7.4 Velocity of sound 97 d|bjql| 2
Page 103 and 104: 7.4 Velocity of sound 99 c(s)/c(s=0
Page 105 and 106: Chapter 8 Linear response - Probing
Page 107: 8.1 Dynamic structure factor 103 wi
Page 111 and 112: 8.1 Dynamic structure factor 107 Th
Page 113 and 114: 8.2 Static structure factor and sum
Page 119 and 120: Chapter 9 Macroscopic Dynamics In t
Page 121 and 122: 9.1 Macroscopic density and macrosc
Page 123 and 124: 9.2 Hydrodynamic equations for smal
Page 125 and 126: 9.4 Sound Waves 121 9.4 Sound Waves
Page 127 and 128: 9.5 Small amplitude collective osci
Page 133 and 134: 9.6 Center-of-mass motion: Linear a
Page 135 and 136: 9.6 Center-of-mass motion: Linear a
Page 137 and 138: Chapter 10 Array of Josephson junct
Page 139 and 140: 10.1 Current-phase dynamics in the
Page 141 and 142: 10.2 Lowest Bogoliubov band 137 µ
Page 143 and 144: 10.3 Josephson Hamiltonian 139 wher
Page 145 and 146: 10.3 Josephson Hamiltonian 141 For
Page 147 and 148: Chapter 11 Sound propagation in pre
Page 149 and 150: 11.2 Current-Phase dynamics 145 [14
Page 151 and 152: 11.3 Nonlinear propagation of sound
Page 153 and 154: n(x, t)/n(x, t=0) 11.3 Nonlinear pr
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∆ n ( U / δ ) 1/2 1 0.8 0.6 0.4
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11.4 Experimental observability 157
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Chapter 12 Condensate fraction The
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12.2 Uniform case 161 The Bogoliubo
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12.2 Uniform case 163 The solution
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12.3 Shallow lattice 165 12.3 Shall
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12.4 Tight binding regime 167 3D th
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12.4 Tight binding regime 169 direc
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12.4 Tight binding regime 171 Quant
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Bibliography [1] L. Pitaevskii and
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[31] E. Lundh, C.J. Pethick, and H.
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[63] B.P. Anderson, P.C. Haljan, C.
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[91] T. Stöferle, H. Moritz, C. Sc
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[125] E. Taylor and E. Zaremba, Bog
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[157] R.M. Bradley and S. Doniach,
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Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

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