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

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32 Introduction
Chapter 4 Single particle in a periodic potential This chapter reviews some standard results (see for example [19, 96]) concerning a single particle in one dimension subject to an external potential V (x) which is periodic in space with period d V (x) =V (x + d) . (4.1) The aim of this chapter is to provide some basic concepts such as quasi-momentum, band structure, Brillouin zone, Bloch functions and Wannier functions. Starting from the single particle case we can then conveniently extend and generalize these notions to the case of a Bose-Einstein condensate in the following chapters. 4.1 Solution of the Schrödinger equation Bloch states and Bloch bands The one-dimensional motion of a particle in the periodic potential (4.1) is described by the Schrödinger equation i¯h ∂ϕ(x) ∂t = − ¯h2 ∂ 2m 2 + V (x) ϕ(x) . (4.2) ∂x2 Due to the periodicity of the potential this equation is invariant under any tranformation x → x+ld where l is any integer. Thus, if ϕ(x) is the wavefunction of a stationary state, then ϕ(x + ld) is also a solution of the Schrödinger equation. This means that the two functions must be the same apart from a constant factor: ϕ(x + ld) = constant × ϕ(x). It is evident that the constant must have unit modulus; otherwise, the wave function would tend to infinity when the displacement through ld was repeated infinitely. The general form of a function having this property is ϕjk(x) =e ikx ˜ϕjk(x) , (4.3) where ¯hk is the quasi-momentum and ˜ϕjk(x) is a periodic function ˜ϕjk(x) = ˜ϕjk(x + d) . (4.4) 33
Page 1: Università degli Studi di Trento F
Page 4: 7 Bogoliubov excitations of Bloch s
Page 9 and 10: Chapter 1 Introduction Superfluids
Page 11 and 12: The second class of stationary solu
Page 13 and 14: Chapter 2 Vortex nucleation The pro
Page 15 and 16: 2.1 Single vortex line configuratio
Page 17 and 18: 2.2 Role of Quadrupole deformations
Page 19 and 20: 2.2 Role of Quadrupole deformations
Page 21 and 22: 2.3 Critical angular velocity for v
Page 23 and 24: 2.3 Critical angular velocity for v
Page 25: 2.4 Stability of a vortex-configura
Page 29 and 30: Chapter 3 Introduction Since the ac
Page 31 and 32: The order parameter’s temporal ev
Page 33 and 34: and spectroscopy were described in
Page 35: the external probe generates a dens
Page 39 and 40: 4.1 Solution of the Schrödinger eq
Page 47 and 48: 4.2 Tight binding regime 43 describ
Page 49 and 50: 4.2 Tight binding regime 45 paramet
Page 51 and 52: Chapter 5 Groundstate of a BEC in a
Page 53 and 54: 5.1 Density profile, energy and che
Page 55 and 56: 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
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Chapter 7 Bogoliubov excitations of
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7.2 Bogoliubov bands and Bogoliubov
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7.3 Tight binding regime of the low
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7.3 Tight binding regime of the low
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7.4 Velocity of sound 97 d|bjql| 2
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7.4 Velocity of sound 99 c(s)/c(s=0
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Chapter 8 Linear response - Probing
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8.1 Dynamic structure factor 103 wi
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8.1 Dynamic structure factor 105 Th
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8.1 Dynamic structure factor 107 Th
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8.2 Static structure factor and sum
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Chapter 9 Macroscopic Dynamics In t
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9.1 Macroscopic density and macrosc
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9.2 Hydrodynamic equations for smal
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9.4 Sound Waves 121 9.4 Sound Waves
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9.5 Small amplitude collective osci
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9.6 Center-of-mass motion: Linear a
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9.6 Center-of-mass motion: Linear a
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Chapter 10 Array of Josephson junct
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10.1 Current-phase dynamics in the
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10.2 Lowest Bogoliubov band 137 µ
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10.3 Josephson Hamiltonian 139 wher
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10.3 Josephson Hamiltonian 141 For
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Chapter 11 Sound propagation in pre
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11.2 Current-Phase dynamics 145 [14
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11.3 Nonlinear propagation of sound
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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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