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

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162 Condensate fraction This expression can be evaluated analytically. One obtains the well known result ∆Ntot = Ntot 8 1 3 π1/2 (a3n) 1/2 . (12.20) In view of the discussion of the case s = 0below, it is important to point out that the integrand in (12.19) contributes most at p ∼ mc where the dispersion (12.17) passes from the phononic regime ¯hω ∼ p to the single particle regime ¯hω ∼ p2 . Yet the convergence is very slow and the integral is saturated by momenta much larger than mc [156], where elementary excitations have single particle character. Quantum depletion of the 2D uniform gas Let us consider a uniform system of axial size d and transverse size L and let us assume that themotionisfrozeninthez-direction. We set Ψ(r) = 1 √ d Ψ2(x, y). Under these conditions the stationary GPE takes the form − ¯h2 The 2D density is given by The 2D coupling constant emerges as 2m ∇2⊥ + g 2 |Ψ2| d Ψ2 = µΨ2 . (12.21) n2 = |Ψ2| 2 = Ntot/L 2 . (12.22) g2 = g ¯h2 a =4π . (12.23) d m d The groundstate solution is given by µ = g2n2 . (12.24) The calculation of the Bogoliubov spectrum is analogous to the 3D case. The result for the depletion changes only because of the change in the dimensionality of the integral over momenta. We rewrite the derivation here for clarity. The Bogoliubov equations for the system in the groundstate read p2 2m +2g2|Ψ2| 2 − µ up + g2|Ψ2| 2 vp , =¯hω(p)up (12.25) p2 2m +2g2|Ψ2| 2 − µ vp + g2|Ψ2| 2 up = −¯hω(p)vp , (12.26) where p 2 = p 2 x + p 2 y and we have assumed that the Bogoliubov amplitudes take the form 1 1 up(x, y) =Up √ d L ei(pxx+pyy)/¯h , (12.27) 1 1 vp(x, y) =Vp √ d L ei(pxx+pyy)/¯h . (12.28)
12.2 Uniform case 163 The solution for the excitation spectrum is found to be p ¯hω(p) = 2 p2 2m 2m +2g2n2 , and the square of the Vp-amplitude is given by (12.29) V 2 p = 1 ⎡ ⎣ 2 p2 4g2n2m +1/2 p2 p2 ⎤ − 1⎦ , (12.30) 4g2n2m ( 4g2n2m +1) where we have used the fact that the amplitudes should be normalized to dr |up| 2 −|vp| 2 = 1. We then find that in the thermodynamic limit the depletion of this 2D uniform gas is given by ∆Ntot Ntot = 1 L Ntot 2 (2π¯h) 2 ∞ 2πpdp 0 1 ⎡ ⎣ 2 = 1 4π m 2 g2 ¯h p 2 4g2n2m +1/2 ⎤ p2 4g2n2m ( p2 4g2n2m +1) − 1⎦ = a , (12.31) d where we have used (12.22) and (12.23). Note that the integral differs from (12.19) only due to the replacement of 4πp2dp by 2πpdp. If the axial profile of the order parameter is given by a gaussian of width σ which is normalized to one in the z-direction, the 2D coupling constant is given by g2 = g √ . (12.32) 2πσ In this case, the result for the depletion reads ∆Ntot Ntot Quantum depletion of the 1D uniform gas = a √ 2πσ , (12.33) In a one-dimensional uniform system the one-body density exhibits a power law decay at large distances. This rules out Bose-Einstein condensation in an infinite, but not in large but finite system. Considering the latter case and supposing that Bogoliubov theory is applicable we calculate the quantum depletion in analogy with the calculations above. This requires evaluating the expression with V 2 q = 1 2 ⎡ ⎢ ⎣ ∆Ntot Ntot = 1 Ntot ¯h 2q2 2m + g1dn1d 2 ¯h q2 2 ¯h q2 2m 2m +2g1dn1d q V 2 q , (12.34) ⎤ ⎥ − 1⎥ ⎦ , (12.35)
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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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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
Page 161 and 162: 11.4 Experimental observability 157
Page 163 and 164: Chapter 12 Condensate fraction The
Page 165: 12.2 Uniform case 161 The Bogoliubo
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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