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

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130 Macroscopic Dynamics Large amplitude dipole oscillations Large amplitude dipole oscillations can be described using the large-current hydrodynamic equations (9.24,9.25). We consider the velocity field in z-direction to be given by (9.57) as in the case of small amplitudes, while vMx = vMy =0. The assumption of a density-independent effective mass is replaced by the more general requirement that the energy ε(k) is independent of density. This is equivalent to setting it equal to the single particle band ε(k, gn/ER) ≈ ε(k, gn/ER =0). (9.64) The hydrodynamic equations for large currents (9.24,9.25) for the center-of-mass as defined in (9.60) then take the form ∂ 1 Z − ∂t ¯h ∂kε(k) =0, (9.65) ¯h ∂ ∂t k + mω2 zZ =0. (9.66) containing the dispersion ε(k), or more precisely, the momentary group velocity ¯v(t) = ∂kε(k(t)) (see Eq.(6.18)) as a crucial ingredient. Small currents or, equivalently, small group velocities are associated with small deviations of k from k =0. In this case, the dispersion is given by ε =¯h 2 k2 /2m∗ and the equations (9.65,9.66) reduce to those for small amplitude dipole oscillations (9.61,9.62). In the tight binding regime, ∂kε(k) =δd sin(kd) (see Eq.(4.29)). Thus, the center-of-mass obeys the equations of motion [73] ∂ δd Z − sin (kd) =0, (9.67) ∂t ¯h ¯h ∂ ∂t k + mω2 zZ =0. (9.68) They describe a nonlinear pendulum. Its small amplitude oscillation is harmonic with frequency (9.63), as discussed previously (recall that δ is related to the effective mass through (4.33)). This regime is reached for δ Z ≪ . (9.69) mω 2 z Hence, the deeper the lattice the smaller the initial displacement has to be in order to stay in the regime of harmonic oscillations. For initial values of Z larger than 2δ/mω2 z the solutions of (9.67,9.68) correspond to a full rotation of k. If the initial value Z = Z0 is much larger than δ/mω2 z, one can treat the center-of-mass as constant and the solution takes the simple form k = −tZ0mω 2 z/¯h, (9.70) implying a constant increase of the quasi-momentum. This is analogous to the behavior of the quasi-momentum during Bloch-oscillations driven by a constant force. The time evolution for small displacement Z − Z0 from is given by δ Z = Z0 + dmω2 zZ0 cos t . ¯h (9.71) mω 2 zZ0 This solution corresponds to an oscillation of Z around Z0 with amplitude much smaller than Z0. Hence, the center-of-mass is always displaced from the center of the trap.
9.6 Center-of-mass motion: Linear and nonlinear dynamics 131 Breakdown of large amplitude dipole oscillations The above discussion of dipole oscillations associates the dynamics of the condensate quasimomentum k(t) with its dynamics in real space Z(t). Two limits have been considered explicitly: We have found that a harmonic center-of-mass motion goes along with a small amplitude oscillation of the condensate quasi-momentum around k =0. In contrast, very large initial displacements Z0 lead to a monotonic increase of k with time while in real space the condensate exhibits an off-centered oscillation. This latter case indicates that the stability analysis of condensate Bloch states commented on at the end of chapter 6.1 is relevant to understand the response of a condensate to the displacement of the harmonic trap in the lattice direction: Once k(t) takes values corresponding to unstable Bloch states we can’t be sure any more whether the condensate will actually exhibit the dynamics exemplified above. The breakdown of the superfluid current due to a dynamical instability has been predicted in [115]. The role played by dynamical instabilities has also been investigated by numerically solving the time-dependent Gross-Pitaevskii equation in the presence of both optical lattice and harmonic trapping potential. This has been done in [118] for a one-dimensional system and in [144, 145] including also the radial degrees of freedom, confirming that the occurence of dynamical instabilities leads to a breakdown of the center-of-mass motion. On the experimental side, it hs been found that beyond a critical displacement of the trap, which decreases as a function of lattice depth, the condensate does not exhibit oscillations and stops at a position displaced from the trap center [74, 76].
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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
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 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: 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
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 and 172: 12.4 Tight binding regime 167 3D th
Page 173 and 174: 12.4 Tight binding regime 169 direc
Page 175 and 176: 12.4 Tight binding regime 171 Quant
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.
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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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Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

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