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

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74 Stationary states of a BEC in an optical lattice m ∗ (gn)/m ∗ (gn=0) 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0 5 10 15 20 25 30 Figure 6.9: Ratio between the effective mass m ∗ (gn) and the single particle effective mass for gn =0.5ER (solid line) and gn =0.1ER (dashed line) as a function of lattice depth s. ‘The condensate Wannier functions (6.25) form a complete orthonormal set and fulfill the relation fj,l(x) =fj(x − ld) . (6.27) As in the single particle case, the Wannier function fj,l of a condensate is localized at site l and spreads less and less over other sites the deeper the lattice. For convenience, we choose the Wannier functions to be real in the following. Momentum and quasi-momentum The momentum distribution of a Bloch state is obtained in complete analogy to the single particle case (see chapter 4.1). In fact, we find that apart from a slight screening effect interactions do not have a strong effect for typical values of gn. Swallow tails For gn ≥ s one encounters Bloch state solutions which lead to loops (“swallow tails”) at the edge of the lowest energy Bloch band [107, 106, 108, 109, 110, 111]. The appearance of these states goes along with a non-zero group velocity of the Bloch state at k =¯hqB, insharp contrast with the typical behavior of a single particle. For smaller values of gn > 0 such loops can also be found at the center of the first excited band [107]. For the values of gn we have s
6.2 Tight binding regime 75 considered, these swallow tails exist only for very small values of the lattice depth s and we will not discuss them in the following. Stability of condensate Bloch states It is important to note that Bloch states can be energetically or dynamically unstable. The stability can be analyzed by calculating the Bogoliubov excitation spectrum of a given ϕjk (see [1] chapter 5.6 and comments in section 7.1 below). Two types of instabilities can be encountered: An energetic instability is present if a small perturbation of the stationary solution ϕjk leads to a decrease of the energy. Hence in the presence of dissipative terms the system is driven to configurations with lower energy. In contrast, a dynamic instability is associated with the exponential growth in time of a small perturbation which does not require the inclusion of dissipation. As a general rule, for a given s and nonzero gn, Bloch states in the lowest band are stable for sufficiently small k. Then, there is a range of k
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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
Page 29 and 30: Chapter 3 Introduction Since the ac
Page 31 and 32: 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 77: 6.1 Bloch states and Bloch bands 73
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 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
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9.5 Small amplitude collective osci
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9.5 Small amplitude collective osci
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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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