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

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168 Condensate fraction depletion (12.62) can be rewritten in the form ∆Ntot/Ntot = a/ √ 2πσ in coincidence with the expression for the depletion in a disc (12.33). This means that we are not dealing with an array of separated two-dimensional condensates, but with one condensate only being distributed over several sites. Equivalently, we can refer to the system as a coherent array of two-dimensional condensates, each of them containing a non-condensed fraction ∆N N giving rise to a non-condensed fraction of the whole system ∆Ntot Ntot = 1 ∆N Nw N l = ã , (12.63) d = ã , (12.64) d where the fact that we can simply added up the number of non-condensed atoms at each site is a non-trivial step presupposing the coherence of the system as a whole. It is interesting to note that the depletion (12.62) is obtained by letting the tunneling parametergotozero(δ → 0). Still, the resulting depletion takes values much smaller than one provided that the confinement within each well does not become infinitely large giving rise to a diverging ˜g and hence ã 1 . The reason for this is that before letting δ → 0 we have taken the limit Ntot, L→∞,n= const.. This forces the system as a whole to be coherent since the ratio between the Josephson tunneling energy EJ and the Josephson charging energy EC (see Eqs.(10.34,10.49)) diverges l l EJ/EC = Nκδ/2 →∞ (12.65) for fixed κ, δ as N →∞, while the average density n is kept constant. This indicates that coherence is maintained across the whole sample (see discussion in section 10.1). In conclusion, we can say that the result (12.62) does not tell us what happens to the condensate fraction when the system size and the total number of particles are kept fixed while the lattice is made very deep. Surely, we expect the true result to differ from (12.62) in this case: In connection with the occurence of number squeezing and the approach to the superfluid-insulator transition the condensed fraction should become small as a signature of the decoherence of the system breaking up into disconnected parts each one of them occupying one lattice site. 1D thermodynamic limit We have seen that in the thermodynamic limit, the quantum depletion is upper-bounded by the usually very small quantity ã/d. If the continuum approximation (12.58) in the radial 1 If δ → 0 is achieved by letting s →∞,also˜g →∞and thus ã →∞(see Eq.(6.54) with σ given by (6.50)). Yet, there are other ways of taking the limit δ → 0 which do not affect ˜g: For example, the tunneling rate can be made zero by increasing d or by raising the barriers of the periodic potential while keeping the potential unchanged close to its minima.
12.4 Tight binding regime 169 direction is not applicable and it is crucial to take into account the discreteness of the sum over the quantum numbers px and py in Eq.(12.57), the results are different. This is the case if the number of particles in each well is sufficiently small, or if the longitudinal size of the system, fixed by the number of wells Nw, is sufficiently large. The limiting case is obtained when the contribution arising from the term with px = py =0,q= 0is the dominant one in Eq.(12.57) and we are thus left with ∆Ntot Ntot = 1 ⎡ 1 ⎢ 2δ sin ⎢ Ntot 2 ⎣ q=0 2 ( qd 2 )+κ−1 2δ sin2 ( qd 2 ) 2δ sin2 ( qd 2 )+2κ−1 ⎤ ⎥ − 1⎥ ⎦ . (12.66) Supposing that the system is very long, we make use of the continuum approximation in the z-direction. This yields ∆Ntot Ntot = 1 L Ntot 2π 2 π/d dq qmin 1 ⎡ ⎢ 2 ⎣ 2δ sin2 ( qd 2 )+κ−1 ⎤ ⎥ − 1⎥ ⎦ , (12.67) 2δ sin2 ( qd 2 ) 2δ sin2 ( qd 2 )+2κ−1 with qmin =2π/L. Since we are interested in particular in what happens in a very deep lattice, we expand the integrand to lowest order in the ratio δ/κ −1 . Replacing q by G(b) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 s = q d , (12.68) π 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 12.1: The function (12.61) involved in the result for the quantum depletion (12.60) obtained by considering the thermodynamic limit of the tight binding expression (12.57). b
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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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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
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Page 163 and 164: Chapter 12 Condensate fraction The
Page 165 and 166: 12.2 Uniform case 161 The Bogoliubo
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: 12.4 Tight binding regime 167 3D th
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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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