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

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84 Bogoliubov excitations of Bloch state condensates The sound velocity in the groundstate condensate drops strongly as a function of lattice depth (see section 7.4). This behavior is directly linked to the increase of the effective mass which overcompensates the decrease of the compressibility. We also discuss the sound velocity in a condensate with non-zero group velocity. In contrast to a moving uniform system, the sound velocity in a condensate moving in a lattice is not simply given by the sum of the sound velocity in a condensate at rest and the group velocity of the condensate with respect to the lattice. In [102], we have reported the numerical results for the Bogoliubov band spectra and the sound velocity of the groundstate, as well as the analytical tight binding expressions for the lowest Bogoliubov band and the respective Bogoliubov amplitudes. There, we also discussed the hydrodynamic results for the sound velocity in a slowly moving condensate. This thesis adds the discussion of the numerical Bogoliubov amplitudes, their comparison with the tight binding expressions and the analysis of the ratio between the u and v-amplitude in the limit of a very deep lattice, as well as the discussion of the gap between first and second Bogoliubov band and the comparison of the heights of the lowest Bogoliubov and energy Bloch band. 7.1 Bogoliubov equations The dynamics of a coherent zero-temperature condensate in a 1D optical lattice is described by the time-dependent GPE (TDGPE) i¯h ∂Ψ(z,t) ∂t = − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz + g |Ψ(z,t)| d 2 Ψ(z,t) , (7.1) where we have excluded dynamics involving the transverse direction and L/2 −L/2 dr|Ψ|2 = Ntot. Stationary state solutions of Bloch form are given by N Ψjk(z,t) = = L 2 e−iµj(k)t/¯h ϕjk N L 2 e−iµj(k)t/¯h e ikz ˜ϕjk , (7.2) where L is the transverse size of the system, N is the number of particles per well and µj(k), ϕjk is a solution of the form (6.2) of the stationary GPE (5.4) (Recall that in sections 5 and 6 we have already made use of the rescaled order parameter (5.2)). To explore small time-dependent deviations from such stationary states, we write Ψ(z,t) =e −iµj(k)t/¯h ⎡ ⎤ ikz N e ⎣ ˜ϕjk(z)+δΨ(z,t) ⎦ , (7.3) L2 where δΨ is a small perturbation of the Bloch state Ψjk(z,t). We linearize the TDGPE (7.1) in δΨ(z,t) and obtain i¯h ∂δΨ(z,t) ∂t = − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz +2dgn| ˜ϕjk(z,t)| d 2 − µj(k) δΨ(z,t)+gnd˜ϕ 2 jkδΨ ∗ . (7.4)
7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes 85 Using the expansion δΨ(z,t) = σ uσ(z)e −iωσt + v ∗ σ(z)e iωσt , (7.5) where σ labels the elementary excitations of this stationary state, we find the set of equations − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz +2dgn| ˜ϕjk(z)| d 2 − µj(k) uσ(z)+gnd˜ϕ 2 jk(z)vσ(z) =¯hωσuσ(z) (7.6) − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz +2dgn| ˜ϕjk(z)| d 2 − µj(k) vσ(z)+gnd˜ϕ ∗2 jk(z)uσ(z) =−¯hωσvσ(z) (7.7) for the Bogoliubov amplitudes uσ(z), vσ(z) and the excitation energies ¯hωσ. We require dz |uσ| 2 −|vσ| 2 =1. (7.8) Moreover, the Bogoliubov amplitudes must satisfy the orthogonality relations dz (u ∗ σuσ ′ − v∗ σvσ ′)=0, (7.9) dz (u ∗ ′ σvσ − vσuσ ′)=0, (7.10) for σ = σ ′ . The normalization condition (7.8) implies that for an energetically stable state ϕjk the spectrum ¯hωσ is positive, while negative values ¯hωσ signify that the stationary state ϕjk is energetically unstable (see [1], chapter 5.6). The occurrence of complex frequencies ωσ with a negative imaginary part reveals the dynamical instability of the stationary state ϕjk. The stability analysis of condensate Bloch states commented on in the previous chapter is thus based on the solution of the Bogoliubov equations (7.6,7.7). The presence of a harmonic trap can changes significantly the low energy excitations of the system. In fact, in chapter 9 we will show that small amplitude collective oscillations occurring on a length scale of the system size are strongly affected both by the harmonic trap and the lattice. The excitation spectrum obtained from (7.6,7.7) does not account for the presence of a harmonic trap. Still, the solution of (7.6,7.7) is relevant also for trapped systems: It yields a local Bogoliubov band spectrum given by ¯hωσ(nM(r)) where nM(r) is the macroscopic density profile introduced above in chapter 5.4. This approach is valid for excitations whose characteristic length scale is small compared to the size of the system. 7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes Since ˜ϕjk(z) is periodic, we can apply the Bloch theorem as in section 4.1 and write the Bogoliubov amplitudes in the form uj ′ q(z) = 1 √ e Nw iqz ũj ′ q(z) , (7.11) vj ′ q(z) = 1 √ e Nw iqz ˜vj ′ q(z) , (7.12)
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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
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 and 84: 6.2 Tight binding regime 79 2m¯v/q
Page 85 and 86: 6.2 Tight binding regime 81 The lea
Page 87: Chapter 7 Bogoliubov excitations of
Page 91 and 92: 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
Page 135 and 136: 9.6 Center-of-mass motion: Linear a
Page 137 and 138: 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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