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

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80 Stationary states of a BEC in an optical lattice that the effective coupling constant (5.15) in the tight binding regime reads L/2 ˜g = gd −L/2 f 4 gn=0(z) . (6.45) A quantitative estimate of the validity of this expression will be given in the following section using a gaussian ansatz for the Wannier function. Gaussian approximation to the Wannier function of the lowest band In the limit of zero tunneling (s →∞), Wannier functions become eigenstates. In particular, a single particle in the groundstate can be described by the corresponding harmonic oscillator wavefunction. If s ≫ 1 and as long as the single-well condensate is far from being correctly described by the TF-approximation inside each well, it is reasonable to approximate the Wannier function by a gaussian ϕ(z) ≡ f(z) = 1 π 1/4√ σ exp(−z2 /2σ 2 ) . (6.46) The gaussian (6.46) is a useful variational ansatz for the calculation of on-site quantities as the compressibility, but fails to describe properties which crucially depend on the overlap between neighbouring Wannier functions, as the effective mass. In fact, when tunneling is possible, but its effects only small, we expect the Wannier function of the lowest band to be still similar to the gaussian (6.46) inside a well, but to have oscillating tails in the barrier region in order to ensure orthogonality. At a given lattice depth s and interaction gn/ER, the value of the gaussian width σ is fixed by requiring the ansatz (6.46) to minimize the energy of the system. Since contributions due to tunneling go beyond the accuracy of this description, it is consistent to consider only the on-site energy ε0 (see Eq.(6.30)). Moreover, we expand the lattice potential (3.6) around its minima V (z)/ER ≈ s 2 πz − d s 3 4 πz , (6.47) d where the first term corresponds to a harmonic potential of oscillator frequency ˜ω =2 √ sER/¯h, (6.48) while the second term allows for anharmonicity effects of O(z4 ).Thesolutionσhas to satisfy the equation − d3 π3 1 + sπ σ − sπ3 σ3 d d3 σ3 − 1 gn π d 2 ER 2 2 π2 1 =0. (6.49) σ2 Since in we are interested in the large-s limit, we neglect the interaction term. Using s ≫ 1, σπ/d
6.2 Tight binding regime 81 The leading contribution yields the single particle harmonic oscillator groundstate where σ = s−1/4d/π = ¯h/m˜ω for the harmonic well with frequency ˜ω =2 √ sER/¯h, while the second term is a small correction arising from the anharmonicity which slightly increases the width σ of the groundstate. Inserting the gaussian ansatz into the expressions (6.30) and (6.33) for the on-site contribution to the energy and the chemical potential, we obtain ε0 ER µ0 ER = d2 2π 2 1 1π2 + s σ2 2d2 σ2 − 1π4 σ4 4d4 + 1 gn d 1 √ , (6.51) 8π ER π σ = ε0 + ER 1 gn d 1 √ . (6.52) 8π ER π σ Within the gaussian approximation, the inverse compressibility (6.44) can be rewritten as κ −1 = gnd √ 2πσ and the tight binding effective coupling constant (6.45) takes the form ˜g = gd √ 2πσ (6.53) (6.54) with σ given by (6.50). For s =10,gn=0.5ER the approximation κ −1 =˜gn with ˜g given by Eq.(6.54) differs from the exact value of κ −1 by less than 1%. In section 5.4, we have found that the average density at the central site of a condensate loaded in the combined potential of optical lattice and harmonic trap decreases like nl=0 ∼ (˜g/g) −3/5 ∼ s −3/20 . As already mentioned there, the decrease of the average density at the trap center (5.31) is to be contrasted with the increase of the non-averaged peak density n(r⊥ =0,z =0)∼ (˜g/g) 2/5 ∼ s 1/10 . We can now prove the latter statement using the gaussian ansatz (6.46) with σ given by (6.49): The peak density then reads n(r⊥ =0,z =0)=f(z =0) 2 nl=0(r⊥ =0;s) = 1 π1/2 (µ − µgn=0) , (6.55) σ ˜g where we have used the approximative solution (µ − µgn=0)/˜g for the average density profile (see Eq.(5.31)) with µ − µgn=0 and ˜g given by Eqs.(5.29) and (6.54) respectively. At s =0, the density at the center is given by n(r⊥ =0,z =0)=µ/g where µ is the usual TF-value of the chemical potential (see Eq.(5.29) with ˜g = g and µgn=0 =0). It follows that the ratio between the peak densities at large s and at s =0equals √ 2(˜g/g) 2/5 .Fors =20and s =50, this amounts to a 32% and 46% increase respectively.
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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
Page 33 and 34: and spectroscopy were described in
Page 35 and 36: the external probe generates a dens
Page 37 and 38: Chapter 4 Single particle in a peri
Page 39 and 40: 4.1 Solution of the Schrödinger eq
Page 47 and 48: 4.2 Tight binding regime 43 describ
Page 49 and 50: 4.2 Tight binding regime 45 paramet
Page 51 and 52: Chapter 5 Groundstate of a BEC in a
Page 53 and 54: 5.1 Density profile, energy and che
Page 55 and 56: 5.1 Density profile, energy and che
Page 57 and 58: 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: 6.2 Tight binding regime 79 2m¯v/q
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 and 110: 8.1 Dynamic structure factor 105 Th
Page 111 and 112: 8.1 Dynamic structure factor 107 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
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9.6 Center-of-mass motion: Linear a
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Chapter 10 Array of Josephson junct
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10.1 Current-phase dynamics in the
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10.2 Lowest Bogoliubov band 137 µ
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10.3 Josephson Hamiltonian 139 wher
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10.3 Josephson Hamiltonian 141 For
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Chapter 11 Sound propagation in pre
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11.2 Current-Phase dynamics 145 [14
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11.3 Nonlinear propagation of sound
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n(x, t)/n(x, t=0) 11.3 Nonlinear pr
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∆ 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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