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

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46 Single particle in a periodic potential δapprox/δnum 1.6 1.5 1.4 1.3 1.2 1.1 1 5 10 15 20 25 30 35 40 45 50 Figure 4.9: Ratio between the approximative δ (4.35) for j =1and the exact δ as obtained from Eq.(4.33) using the numerical data for m ∗ in the case of the lattice potential V = sERsin 2 (πx/d). s
Chapter 5 Groundstate of a BEC in an optical lattice We explore the groundstate of a condensate confined in a one-dimensional optical lattice within GP-theory. First, we consider a system which is uniform in the transverse direction (section 5.1,5.2 and 5.3). In this case, there are two tunable parameters: the lattice depth sER and the interaction parameter gn where n is the average density. In the lattice direction the density profile is strongly modulated by the periodic potential reflecting the concentration of the atoms at the bottom of the potential wells. Energy and chemical potential are significantly increased with respect to the uniform case, while the combined presence of lattice and repulsive interactions between atoms leads to a decrease in the compressibility. The inverse compressibility has a nonlinear dependence on average density since interactions counteract the compression of the atoms by the external potential. In a deep lattice however, the equation of state of the system takes the same linear dependence on density as in the uniform case and is charcterized by an effective coupling constant ˜g which grows as a function of lattice depth. The presence of repulsive interactions produces a screening effect since the particles resist more to being squeezed in the potential wells than in the absence of interaction. We find that increasing the interaction parameter gn corresponds to an effective decrease of the lattice depth s. In a second step, we allow for the additional presence of radial and axial harmonic trapping (section 5.4). We use the results obtained for the purely periodic potential as an input to calculate the groundstate properties in the combined trap. The scheme we develop adequately describes current experimental settings in which the optical potential is superimposed to a harmonically trapped TF-condensate leading to a distribution of atoms over many lattice sites. We derive simple analytic expressions for the chemical potential and the density profile averaged over each lattice period by using the effective coupling constant description. These expressions allow us to explicitly calculate the occupation of each site and the radius of the cloud. We discuss the dependence on lattice depth of the chemical potential, the condensate size and the density at the center of the harmonic trap averaged over one lattice period. Our results for the groundstate energy, chemical potential, compressibility and regarding 47
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 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: 4.2 Tight binding regime 45 paramet
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
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
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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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