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

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160 Condensate fraction 12.1 Quantum depletion within Bogoliubov theory Let us consider the problem in a 3D box of size L in x, y-direction and an optical lattice with Nw sites oriented along z Note that the quantum depletion of the condensate has been calculated in [123, 124] for different geometries. The quantum numbers of the elementary excitations are the band index j and the quasi-momentum q along the z direction and the momenta px and py in the transverse directions. Within the framework of Bogoliubov theory, the quantum depletion of the condensate is given by ∆Ntot = Ntot 1 dz dx dy |vj,q,px,py(r)| Ntot j q,px,py,|p|=0 2 , (12.1) where Ntot denotes the total number of atoms, ∆Ntot is the number of non-condensed parti- cles, |p| = p 2 x + p 2 y +¯h 2 q 2 and vj,q,px,py(r) are the Bogliubov v-amplitudes of the elementary excitations. The sum runs over all bands j, over the quasi-momenta q in the first Brillouin zone and the momenta of elementary excitations in the transverse directions px,py allowed by the periodic boundary conditions q = 2π ν, Nwd ν=0, ±1, ±2, ..., ±Nw , 2 (12.2) px,py =¯h 2π ν, L ν=0, ±1, ±2, ... . (12.3) The Bogoliubov amplitudes vj,q,px,py(r) in Eq. (12.1) solve the Bogoliubov equations − ¯h2 2m ∇2 + sER sin 2 − ¯h2 2m ∇2 + sER sin 2 πz d πz d +2dgn| ˜ϕ(z)| 2 − µ ujqpxpy(r)+gnd˜ϕ 2 vjqpxpy(r) =¯hωj(q)ujqpxpy (12.4) +2dgn| ˜ϕ(z)| 2 − µ vjqpxpy(r)+gnd ˜ϕ ∗2 ujqpxpy(r) =−¯hωj(q)vjqpxpy .(12.5) They can be obtained from the three-dimensional time-dependent GPE i¯h ∂Ψ(r,t) = − ∂t ¯h2 2m ∇2 + sER sin 2 πz + g |Ψ(r,t)| d 2 Ψ(r,t) , (12.6) where the order parameter Ψ fulfills the normalization condition dr |Ψ(r,t)| 2 = Ntot , (12.7) by considering small time-dependent perturbations δΨ(r,t) of the groundstate Ψ0(r) where Ψ(r,t)=e −iµt/¯h [Ψ0(r)+δΨ(r,t)] , (12.8) δΨ(r,t)=uσ(r)e −iωσt + v ∗ σ(r)e iωσt as exemplified in section 7 for the case px = py =0. (12.9)
12.2 Uniform case 161 The Bogoliubov amplitudes must comply with the normalization condition dz dx dy |uq,px,py(r)| 2 −|vq,px,py(r)| 2 =1. (12.10) Eq. (12.1) adquately describes the quantum depletion provided ∆Ntot Ntot Otherwise it is necessary to go beyond Bogoliubov theory. 12.2 Uniform case ≪ 1 . (12.11) At lattice depth s =0, Eq. (12.1) yields the quantum depletion of the weakly interacting uniform Bose gas. In this case the sum j,q with q belonging to the first Brillouin zone can be replaced by q with q = 2π ν, L ν=0, ±1, ±2,... . (12.12) The correctly normalized Bogoliubov amplitudes are given by e uq,px,py(r) =Uq,px,py i(pxx+pyy)/¯h L3/2 e iqz , (12.13) e vq,px,py(r) =Vq,px,py i(pxx+pyy)/¯h L3/2 e iqz , with (12.14) Uq,px,py = Vq,px,py = p2 2m +¯hωuni p2¯hωuni 2 2m p2 2m − ¯hωuni p2¯hωuni 2 2m , (12.15) , (12.16) where p2 = p2 x + p2 y +¯h 2 q2 and ω is the dispersion relation of the elementary excitations p ¯hωuni = 2 p2 2m 2m +2gn . (12.17) We insert (12.13,12.14) into Eq.(12.1). To obtain the result in the thermodynamic limit Ntot, L→∞,n= const. we make use of the continuum approximation → px,py,q L3 (2π) 3 1 ¯h 2 dpx dpy dq . (12.18) This leads to ∆Ntot Ntot = 1 L Ntot 3 (2π) 3 1 ¯h 2 dpx dpy dq|Vq,px,py| 2 = 1 L Ntot 3 (2π) 3 1 ¯h 2 ∞ 4πp 0 2 dp 1 ⎡ p ⎣ 2 2 4g2n2m +1/2 p2 4g2n2m ( p2 4g2n2m +1) ⎤ − 1⎦ . (12.19)
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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
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5.4 Effects of harmonic trapping 55
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Chapter 6 Stationary states of a BE
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6.1 Bloch states and Bloch bands 65
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6.2 Tight binding regime 75 conside
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6.2 Tight binding regime 77 Compari
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6.2 Tight binding regime 79 2m¯v/q
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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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Page 119 and 120: Chapter 9 Macroscopic Dynamics In t
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Page 133 and 134: 9.6 Center-of-mass motion: Linear a
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Page 137 and 138: Chapter 10 Array of Josephson junct
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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
Page 161 and 162: 11.4 Experimental observability 157
Page 163: Chapter 12 Condensate fraction The
Page 167 and 168: 12.2 Uniform case 163 The solution
Page 169 and 170: 12.3 Shallow lattice 165 12.3 Shall
Page 171 and 172: 12.4 Tight binding regime 167 3D th
Page 173 and 174: 12.4 Tight binding regime 169 direc
Page 175 and 176: 12.4 Tight binding regime 171 Quant
Page 179 and 180: Bibliography [1] L. Pitaevskii and
Page 181 and 182: [31] E. Lundh, C.J. Pethick, and H.
Page 183 and 184: [63] B.P. Anderson, P.C. Haljan, C.
Page 185 and 186: [91] T. Stöferle, H. Moritz, C. Sc
Page 187 and 188: [125] E. Taylor and E. Zaremba, Bog
Page 189: [157] R.M. Bradley and S. Doniach,
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