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

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164 Condensate fraction where g1d = g , L2 (12.36) n1d = nL 2 , (12.37) are the effective 1D coupling constant and 1D density in a system with radial extension L and 3D density n whose motion is frozen in the radial direction. Supposing that the system is very long, we make use of the continuum approximation in the z-direction. This yields ∆Ntot Ntot = 1 Ntot L 2π 2 ∞ dq qmin=2π/L 1 2 We rewrite this expression in the form ∆Ntot Ntot = 1 Ntot ⎡ ⎢ ⎣ L 1 2π ξ 2 ∞ dq 2πξ/L 1 2 ¯h 2q2 2m + g1dn1d 2 ¯h q2 2 ¯h q2 2m 2m +2g1dn1d ξ 2 q 2 +1 ξ 2 q 2 (ξ 2 q 2 +2) − 1 ⎤ ⎥ − 1⎥ ⎦ . (12.38) , (12.39) with the healing length ξ = √ 2mg1dn1d. The integral can be solved analytically yielding ∆Ntot Ntot = 1 ⎡ L 1 ⎣− Ntot 2π ξ 4π ξ 1 +2+2π + √ arctanh L2 L 2 2 ξ2 Using ξ ≪ L this expression can be recast in the form ∆Ntot Ntot = 1 L 1 1 2 ξ2 √ arctanh 1 − 4π Ntot 2π ξ 2 4L2 = 1 √ L 1 1 2L √ ln Ntot 2π ξ 2 πξ Using the formula c = g1dn1d/m for the sound velocity and defining we rewrite this result in the form ∆Ntot Ntot 1 1+4π 2 ξ 2 2L 2 ⎤ ⎦ . (12.40) (12.41) ν = mc . (12.42) 2π¯hn1d = ν ln √ 2L πξ (12.43) This result is valid provided that the depletion is small which can be ensured by making ν sufficiently small. In this respect, it is interesting to note that the quantity ν becomes smaller when the 1D density n1d = Ntot/L is made larger. The depletion diverges for L →∞.This follows from the power law decay behavior of the 1-body density: The condensate depletion measures to what extent the 1-body density drops on a distance of the system length.
12.3 Shallow lattice 165 12.3 Shallow lattice In chapter 5 and 9 we have seen that a condensate loaded in a lattice can be effectively described as a uniform system of atoms whose motion in the z-direction occurs with effective mass m∗ and whose interaction, at sufficiently low average density, is described by an effective coupling constant ˜g. We can then pose the question under what condition this holds also for the calculation of the quantum depletion. According to the above discussion of the uniform case, the dispersion of the relevant elementary excitations must be well approximated by ¯hω(px,py,q)= p2 x 2m + p2y 2m + ¯h2 q2 2m∗ p2 x 2m + p2y ¯hq2 + 2m 2m∗ +2˜gn . (12.44) Note that this dispersion implies different sound velocities in the axial z and transverse x, ydirections respectively: cz = ˜gn , (12.45) m∗ cx,y = ˜gn . (12.46) m As pointed out in section 12.2, the range of relevant momenta px, py and quasi-momenta q reaches up to values much larger than mcx,y and much larger than m∗cz respectively. Hence, a necessary condition for the applicability of the approximation (12.44) is that m ∗ cz
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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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8.2 Static structure factor and sum
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8.2 Static structure factor and sum
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Page 119 and 120: Chapter 9 Macroscopic Dynamics In t
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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
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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
Page 161 and 162: 11.4 Experimental observability 157
Page 163 and 164: Chapter 12 Condensate fraction The
Page 165 and 166: 12.2 Uniform case 161 The Bogoliubo
Page 167: 12.2 Uniform case 163 The solution
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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