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

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108 Linear response - Probing the Bogoliubov band structure The excitation strength to the lowest Bogoliubov band - Conclusions In the presence of an optical lattice, it is possible to excite the lowest band ¯hω(q) with a momentum transfer p = q + l 2π/d lying outside the first Brillouin zone since the excitation has many momentum components. This is in contrast to the behaviour of a uniform system where the excitation ¯hω(q) can only be created provided p exactly equals q. Furthermore, we find dramatic effects due to the combined presence of lattice and interaction: First of all, the excitation strength to the lowest band is not only suppressed when the external perturbation couples to excitations of phononic type close to p =0,asinthe uniform case, but vanishes whenever p = l 2π¯h/d. This repeated suppression can be explained by the periodicity of the Bogoliubov excitations featuring a phononic regime in the vicinity of q = l 2π/d. Apart from the periodic suppression of Z1(p) close to p = l 2π/d, the strength diminishes rapidly for all values of p as s is increased and goes to zero in the limits s →∞. For a larger gn/ER this reduction is enhanced. This effect is a consequence of the excitations in the lowest band acquiring quasi-particle character associated with |uq| ∼|vq| at all values of q, even beyond the phononic regime. If ¯h/ξ ≪ qB, this implies that the presence of the lattice extends the role of correlations in the system to length scales below the healing length. The suppression of Z1(p) goes along with a suppression of the imaginary part of the response function and hence of the energy transfer between system and external probe. Hence, we can say that due to the lattice it becomes difficult to transfer energy to the system by means of a weak external probe in the range of energies below the second Bogoliubov band, in particular for momentum transfer p ≈ l 2π/d. In a recent experiment [91], the amplitude of a one-dimensional optical lattice was modulated to transfer momenta ±2qB to an ultracold Bose gas confined in the tubes of a twodimensional lattice. A strong response of the system was observed for energy transfers lying in the range of the lowest Bogoliubov band and within the gap between lowest and second band, in disagreement with our prediction. There are two reasons for this discrepancy: The one-dimensional lattice along the tubes was deep enough to enter a regime of strong coupling between the atoms, rendering a Bogoliubov theory approach for weakly interacing systems invalid. Furthermore, the excitation was strong enough to produce a nonlinear response of the system. The excitation strength to the higher Bogoliubov bands To study the excitation strength to the higher Bogoliubov bands we evaluate (8.8) with j>1 and the solutions uj>1,q(z), vj>1,q(z) of (7.14,7.15) for a given choice of the lattice depth s and the interaction parameter gn/ER. Numeric results for the excitation strength to the second band Z2(p) are depicted in Figs.8.5 for s =5and gn =0.5ER while Figs.8.6 display the strength to the third band Z3(p) for the same lattice depth and interaction parameter. These examples illustrate again that the j-th band can be excited by means of a momentum lying outside the j-th Brillouin zone, in contrast with the case s =0(see Fig. 8.2): Even though Zj still has its peak values in the j-th Brillouin zone, it takes non-zero values outside that zone with local maxima forming in the zones j +2l. At s =5the curves in the j-th
8.2 Static structure factor and sum rules 109 Brillouin zone still resembles the curve in the case s =0. Yet, they are smoothed out. This is more evident in the third band which is less affected by the presence of the optical lattice than the second band. The strengths Zj>1 decrease in the j-th Brillouin zone when s is increased, while increasing outside. This reflects the growth of momentum components of the excitations in the j-th band outside the respective Brillouin zone. Very differently from the case of the lowest band, there is no qualitative difference between the curves for gn =0and gn = 0. Also, an increase in gn/ER brings along only minor changes in the strengths: The values of the strength Zj outside the j-th Brillouin zone is reduced which corresponds to the screening of the lattice by interactions. These observations confirm the statement made in section 7 that interactions have a small effect on the higher Bogoliubov bands. Z2(p)/Ntot 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −6 −4 −2 0 2 4 6 p/qB Figure 8.5: Excitation strength to the second Bogoliubov band Z2(p) (8.8) for gn =0.5ER at lattice depth s =5(solid line) and s =0(dashed line). 8.2 Static structure factor and sum rules The integral of the dynamic structure factor provides the static structure factor S(p) = 1 Ntot S(p, ω)dω. (8.18) The static structure factor is a quantity of primary importance in many-body theory since it is closely related to the Fourier transform of the two-body correlation function (see [1] chapter 7.2). Eq.(8.18) is also referred to as non-energy weighted sum-rule of the dynamic structure
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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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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 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: 8.1 Dynamic structure factor 107 Th
Page 115 and 116: 8.2 Static structure factor and sum
Page 117 and 118: 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
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
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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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