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

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100 Bogoliubov excitations of Bloch state condensates where c is the sound velocity (7.48) in the condensate at rest and m∗ µ is the effective mass obtained from the chemical potential band spectrum (see Eq.(6.23)). Hence, in the lab frame the sound velocity is given by ck = c ± ¯h|k| m∗ . (7.53) µ The plus- and minus-sign hold when the sound wave propagates in the same and in the opposite direction as the condensate respectively. Since the quantity ¯hk/m∗ µ is generally different from the group velocity ¯v = ¯hk/m∗ , the sound velocity is not generally obtained by adding ¯v and c in the case of copropagating sound waves, or by subtracting ¯v from c in the case of counterpropagation. This is the case only for s =0where m∗ = m∗ µ = m. It is interesting to note that the result (7.53) gives a physical meaning to the quantity m∗ µ. In the tight binding regime, we can write ck = c ± ¯v +(δµ − δ)d2k/¯h , where δµ − δ = n∂δ/∂n (see Eqs.(6.38,6.36)), in order to show that the density-derivative of δ plays an important role. We remind that, in contrast, contributions arising from this quantity are negligible as far as excitations of groundstate condensate are concerned (see discussion in section 7.3). Expression (7.53) has been derived assuming a small value of the condensate quasimomentum k. The hydrodynamic formalism can also be employed to calculate the sound velocity for any value of k. The result reads [107, 125] (see also section 9 below) n ck = m∗ ∂µ(k) (k) ∂n ± ∂µ(k) ∂k , (7.54) where m ∗ (k) is the generalized effective mass (6.22) with j =1. Expanding this expression up to O(k) at k =0, we recover result (7.53).
Chapter 8 Linear response - Probing the Bogoliubov band structure The Bogoliubov band structure can be probed by exposing the condensate to a weak external perturbation: The linear response of the system involves transitions between the initial groundstate and the excited states of the Bogoliubov spectrum. The spectrum of excitations is determined by scanning both transferred energy and momentum and by recording the encountered resonances. In this chapter we will consider the particular case in which the external probe generates a density perturbation in the system. This can be achieved experimentally by doing Bragg spectroscopy where the system is illuminated with two laser beams. The absorption of a photon from one beam initiates the stimulated emission of a photon into the second beam. The difference between the wavevectors of the two beams and their detuning fixes the momentum and the energy transferred to sample. So far, this technique has been used successfully to investigate condensates without lattice [126, 127, 128, 129, 130, 131]. A method equivalent to Bragg spectroscopy is offered by the possibility to manipulate the lattice itself in a timedependent way [132, 79, 91]: A modulation of the lattice depth transfers momentum 0, ±2qB and energy ¯hω, where ω is the frequency of the modulation. The presence of the lattice brings about significant changes with respect to the linear response of the uniform system: Not only does the spectrum of excitations change (see previous chapter 7) and therefore the resonance conditon for a probe to transfer momentum and energy, but also the excitation strengths for a particular momentum transfer feature a strong dependence on lattice depth and density. In section 8.1 we present results for the dynamic structure factor of a condensate loaded into a one-dimensional lattice: We show that when keeping the momentum transfer fixed while scanning the energy transfer a resonance is encountered for each Bogoliubov band. This implies that the j-th band can be excited even if the transferred momentum does not lie in the j-th Brillouin zone. Due to phononic correlations the excitation strength towards the lowest Bogoliubov band develops a typical oscillating behaviour as a function of the momentum transfer, and vanishes at even multiples of the Bragg momentum. Even though the excitation energies ¯hωj(p) are periodic as a function of p, this is not true for the excitation strength to the j-th band. 101
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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
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 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: 7.4 Velocity of sound 99 c(s)/c(s=0
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
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
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11.3 Nonlinear propagation of sound
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11.3 Nonlinear propagation of sound
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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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