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

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106 Linear response - Probing the Bogoliubov band structure 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 8.4: Excitation strength to the lowest Bogoliubov band Z1(p) (8.8) at lattice depth s =10for gn =0.5ER (solid line), gn =0.02ER (dashed line) and gn =0(dash-dotted line). The excitation strength to the lowest Bogoliubov band - Analytic results The characteristic behaviour described above can be understood by considering the lowest Bogoliubov band in the tight binding regime where an analytic expression can be derived for Z1(p): Using the expression (6.26) with j =1,k =0for the condensate and the tight binding ansatz (7.25,7.26) for the Bogoliubov amplitudes, the expression for the strength Z1(p) (8.8) takes the form Z1(p) =Ntot|Up + Vp| 2 dz |f(z)| 2 e ipz/¯h 2 , (8.11) where we have neglected contributions due to the overlap of neighbouring Wannier functions. In chapter 7.3 we have found that Thus, expression (8.11) takes the form (Up + Vp) 2 = 2δ sin2 (pd/2) ¯hω(p) 2δ sin Z1(p) =Ntot 2 (pd/2¯h) ¯hω(p/¯h) dz |f(z)| 2 e ipz/¯h 2 (8.12) (8.13)
8.1 Dynamic structure factor 107 The integral mainly depends on the properties of the Wannier function near the center of the well and can hence be evaluated within the gaussian approximation discussed in section 6.2. Replacing the integral limits by −∞ and +∞, weobtain 2δ sin Z1(p) =Ntot 2 (pd/2) exp ¯hω(p) − 1 p2 σ2 2 ¯h 2 . (8.14) This expression for the excitation strength to the lowest band in a deep lattice exhibits all the characteristics mentioned above with respect to the numeric results: • The gaussian dependence on the momentum transfer p describes the overall decay of the excitation strength at large p. This behavior is also present in the absence of interactions (gn =0). Note that σ is smaller in a deeper lattice, yielding a broader envelope of the strength (8.14). This reflects that at larger s the excitations have more momentum components. • Provided that gn/ER = 0, the ratio between the Bloch and the Bogoliubov dispersion gives rise to the characteristic oscillations of Z1(p) in the vicinities of p = l2qB where the excitations contributing to the strength have phonon character: Near the points p = l2qB, wehave √ d Z1(p) ≈ Ntot δκ 2¯h |p − l2qB| exp(− 1 2 σ2 (2lqB) 2 ¯h 2 ) , (8.15) and hence the strength (8.14) vanishes approximately linearly. If gn/ER =0, Bloch and Bogoliubov dispersion coincide and we are left with the nonoscillating behavior Z1(p) = exp − 1 2σ2p/¯h 2 . Note that σ → 0 for s →∞and hence Z1(p) → 1. • Increasing gn/ER at fixed s or increasing s at fixed gn/ER = 0leads to an overall decrease of Z1(p): This behavior can be explained by considering the ratio between Bloch and Bogoliubov dispersion appearing in (8.14). Using Eq.(7.38) for the Bogoliubov dispersion, we can write 2δ sin 2 (pd/2) ¯hω(p) = δκ sin2 (pd/2¯h) 1+2δκ sin2 . (8.16) (pd/2¯h) Increasing gn/ER at fixed s leads to a decrease of κ which dominates the slight increase of δ. So, δκ → 0 and hence the ratio (8.16) and accordingly the strength (8.14) diminuish. Moreover, increasing s at fixed gn/ER = 0both δ and κ diminish, δκ → 0, and thus brings about an overall decrease of Z1(p). Hence, from (8.16) it follows that for increasing gn/ER or increasing s the excitation strength to the lowest band (8.14) is reduced according to the law Z1 ∼ √ δκ . (8.17) Looking at expression (8.11), we note that in the presence of interactions the strength is suppressed whenever Up ∼−Vp. This must always be the case in the vicinity of p = l 2π/d where the excitations are phonons. Yet, also the overall suppression of Z1 is a consequence of such a behavior of the Bogoliubov amplitudes since Up ∼−Vp can be achieved for all p if the lattice is made sufficiently deep.
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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
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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 and 108: 8.1 Dynamic structure factor 103 wi
Page 109: 8.1 Dynamic structure factor 105 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
Page 159 and 160: ∆ 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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