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

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88 Bogoliubov excitations of Bloch state condensates ¯hω/ER 1.2 1 0.8 0.6 0.4 0.2 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ¯hq/qB Figure 7.2: Lowest Bloch band (dashed line) and lowest Bogoliubov band (dash-dotted line) for s =1and gn =0.5ER compared with the single particle Bloch band (solid line). In the case of the Bloch bands (solid and dashed line) the groundstate energy has been subtracted . The solid lines in Fig.7.3 show how the lowest Bogoliubov band changes when the lattice depth is increased at fixed interaction. At s =1(Fig.7.3a), apart from the formation of the energy gap close to q = qB, the curve still resembles the dispersion in the uniform case: both the phononic linear regime at small q and the quadratic regime at larger q are visible. When the potential is made deeper (s =5, 10; Fig.7.3b,c), the band becomes flatter. As a consequence, the quadratic regime disappears and the slope of the phononic regime decreases. This reflects the strong decrease of the velocity of sound as the lattice is made deeper (see discussion below in section 7.4). In Fig.7.4, we plot the energy gap between lowest and first excited band as a function of lattice depth for gn/ER =1. For comparison, we display the corresponding curve for the energy gap in the Bloch band spectrum of a single particle (see Fig.6.6). In addition, we plot the gap 2 √ sER between the vibrational levels obtained when approximating the bottom of a lattice well by a harmonic potential (see Eq.(6.48)). It turns out that the gap in the Bogoliubov band spectrum approaches the one in the single particle spectrum as s is tuned to large values. This is the case because in a very deep lattice the first excited band is almost not affected by interactions and is hence essentially given by the single particle Bloch band while the lowest Bogoliubov band becomes so flat that the excitation energy at the zone boundary is negligibly small compared to the energies of the second band. Note that the convergence to the gap 2 √ sER of the harmonic approximation is very slow, reflecting mainly the role of the anharmonicity of the potential wells.
7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes 89 ¯hω/ER 1.5 1 0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.5 1 0.5 a) b) c) 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ¯hq/qB Figure 7.3: Lowest Bogoliubov band at gn =0.5ER for different values of the potential depths: s =1(a), s =5(b) and s =10(c). The solid lines are obtained from the numerical solution of Eqs.(7.14,7.15) while the dashed lines refer to the tight-binding expression (7.38). ¯hω/ER 16 14 12 10 8 6 4 2 0 0 10 20 30 40 50 60 Figure 7.4: Energy gap at the zone boundary between lowest and first excited Bogoliubov band as a function of lattice depth for gn =0(solid line) and gn =1ER (dashed line). For comparison, we also plot the gap 2 √ sER between the vibrational levels obtained when approximating the bottom of a lattice well by a harmonic potential (dash-dotted line). s
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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
Page 41 and 42: 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
Page 87 and 88: Chapter 7 Bogoliubov excitations of
Page 89 and 90: 7.2 Bogoliubov bands and Bogoliubov
Page 91: 7.2 Bogoliubov bands and Bogoliubov
Page 95 and 96: 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
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
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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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