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

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60 Groundstate of a BEC in an optical lattice is related to the fact that in a 1D system one has µ ∝ Nl. In contrast, for a 3D system with radial trapping one obtains µ ∝ √ Nl from Eq.(5.23). As mentioned above, the average density profile nl(r⊥) varies slowly as a function of the site index l. We obtain a smooth macroscopic density profile nM(r⊥,z) by replacing the discrete index l by the continous variable z = ld This is convenient in devising a hydrodynamic formalism (see section 9): The particular profile (5.24) becomes nM(r⊥,z)= 1 µ − µgn=0 − ˜g m 2 ω2 zz 2 − m 2 ω2 ⊥r 2 ⊥ , (5.32) with µ given by Eq.(5.29). The extension of the condensate along z is given by 2(µ − µgn=0) Z = mω2 . (5.33) z Obviously, the profile (5.32) mimics a TF-profile of a condensate without lattice. This confirms the statement made in section 5.2 that if µopt =˜gn + µgn=0 the system can be described as if there was no lattice as long as g is replaced by ˜g. Before concluding this section, we would like to emphasize that it is important to do the average (5.21) before applying the LDA. The strong modulation of the wavefunction generated by the lattice contributes a large kinetic energy which would be, mistakenly, discarded within a LDA. The situation is different in a system made up of only very few wells (produced for example by raising a few barriers in a TF-condensate). In this case, the condensate in each well might be well described by the TF-approximation (see for example [57]). In the LDA proposed in this section, the large kinetic energy caused by the lattice, is contained in the chemical potential µopt(nl(r⊥)). The effects described in this section, in particular the dependence of µ (5.29) and R0 (5.28) on s, have been investigated by [71]. In this experiment, the chemical potential and the radius of the groundstate in the combined potential of harmonic trap and one-dimensional optical lattice is determined from the measurement of the radial size of the cloud after a time of free flight. The results for the chemical potential are depicted in Fig.5.10 together with the theoretical prediction (5.29). The generalization of the results presented in this section to two-dimensional cubic lattices is straightforward. For a two-dimensional lattice in the x, y-directions, the smoothed macroscopic density profile is given by nM(r⊥,z)= 1 µ − µgn=0 − ˜g m 2 ω2 zz 2 − m 2 ω2 ⊥r 2 ⊥ , (5.34) with the effective coupling constant and ˜g = gd 2 d/2 |ϕgn=0(x, y)| −d/2 4 dxdy , (5.35) µ = ¯h¯ω 2 a 15Ntot aho 2/5 ˜g + µgn=0 , (5.36) g
5.4 Effects of harmonic trapping 61 µ k=0 / µ 0 1.6 1.4 1.2 1.0 0 5 10 U 0 / E rec Figure 5.10: The density-dependent contribution to the chemical potential µ − µgn=0 of a condensate in the combined potential of harmonic trap and one-dimensional optical lattice divided by its value in the absence of the lattice as a function of lattice depth U0 ≡ s. Experimental data from [71] together with the theoretical prediction (5.29) (solid line). Figure taken from [71]. in complete analogy to the case of a one-dimensional lattice. Yet, note that the 2D effective coupling constant (5.35) increases more strongly with s since the condensate is compressed in two directions. These predictions have proved useful in the preparation of a one-dimensional Bose gas [88]: They allow to estimate the 1D density in the tubes produced by a two-dimensional lattice which is superimposed to a harmonically trapped condensate. Moreover, the calculation of the chemical potential allows to determine the lattice depth needed to satisfy the condition 15 20 µ ≪ ¯h˜ωr , (5.37) for the one-dimensionality of the gas in the tubes, where ˜ωr is the characteristic radial trapping frequency in each tube.
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Università degli Studi di Trento F
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7 Bogoliubov excitations of Bloch s
Page 9 and 10:
Chapter 1 Introduction Superfluids
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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 and 36: the external probe generates a dens
Page 37 and 38: Chapter 4 Single particle in a peri
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 61 and 62: 5.4 Effects of harmonic trapping 57
Page 63: 5.4 Effects of harmonic trapping 59
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 and 108: 8.1 Dynamic structure factor 103 wi
Page 109 and 110: 8.1 Dynamic structure factor 105 Th
Page 111 and 112: 8.1 Dynamic structure factor 107 Th
Page 113 and 114: 8.2 Static structure factor and sum
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8.2 Static structure factor and sum
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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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