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

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64 Stationary states of a BEC in an optical lattice Exploiting the tight binding formalism, we also derive simple expressions for the compressibility of the groundstate and the effective coupling constant introduced in chapter 5. Explicit formulars for the groundstate energy, chemical potential and compressibility in a very deep lattice are obtained based on a gaussian ansatz for the Wannier function of the lowest band. Using this ansatz, we also estimate the dependence on lattice depth of the peak density at the center of a harmonic trap added to the optical lattice. In [102], we have reported our numerical results for the Bloch band spectra, the group velocity, the effective mass and their analytical tight binding expressions, as well as the tight binding expressions for the groundstate compressibility. In this thesis, results for the Bloch state density profiles and the gap between first and second energy and chemical potential Bloch band are included. Also, the proof of the relation between the current of a Bloch state and the energy Bloch bands is added, as well as the discussion of the peak density and the onsite energy and chemical potential within the gaussian approximation to the Wannier function of the lowest Bloch band. 6.1 Bloch states and Bloch bands As discussed in chapter (4), the stationary state of a single particle in a periodic potential is described by a Bloch function (see Eq.(4.3)). Within GP-theory, the difference between a single particle and an interacting condensate is accounted for by the nonlinear term in the GPE (5.4). Now, suppose we are dealing with a solution whose density has period d. Under this condition, the GPE is invariant under any transformation z → z + ld and we can use the same arguments as in section (4.1) to show that the respective solution has the form of a Bloch function ϕjk(z) =e ikz ˜ϕjk(z) , (6.2) with the periodic Bloch wave ˜ϕjk(z) = ˜ϕjk(z + ld). Stationary states can be of this kind, but, due to the presence of the nonlinear term, they do not have to: Other classes of solutions are associated with density profiles of periodicity 2d, 4d, .... Such solutions have recently been found in [104, 105]. The groundstate discussed above in chapter 5 is of the form (6.2) with j =1,k=0. In the following we explore condensates in Bloch states (6.2). It is convenient to solve the GPE (5.4) for the periodic Bloch waves ˜ϕjk 1 2m (−i¯h∂z +¯hk) 2 + sER sin 2 πz + gnd| ˜ϕjk(z)| d 2 ˜ϕjk(z) =µj(k)˜ϕjk(z). (6.3) From the solution of Eq.(6.3) one gets the functions ˜ϕjk(z) and the corresponding chemical potentials µj(k). This section is devoted to such solutions. Density profile Let us first discuss some solutions for the density |ϕjk(z)| 2 . In Figs. 6.1 and 6.2, we report results obtained at s =5,gn =0.5ER and s =10,gn =0.5ER respectively for different bands and quasi-momenta (j =1, 2, 3, ¯hk =0, 0.5qB, 1qB). Density profiles of higher bands are more modulated and tend to allow for larger particle densities in high potential regions.
6.1 Bloch states and Bloch bands 65 Note also the qualitative resemblance of the profiles with the probability amplitude of a single particle in a harmonic oscillator potential. When the lattice is made deeper, density profiles of states within a certain band become more and more similar. This effect is more obvious the lower the band index. Overall, the behavior of the density profile for varying j and k is analogous to the single particle case. Fig.6.3 compares the density profiles at k =0.5qB for gn =0and gn =1ER. This shows that a change in gn/ER has the same screening effect as in the case of the groundstate (k =0). d|ϕjk(z)| 2 3 2 1 0 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 3 2 1 0 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 3 2 1 a) b) c) 0 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 z/d Figure 6.1: Density profiles d|ϕjk(z)| 2 obtained from (6.3) at s =5,gn=0.5ER for band index a) j =1,b)j =2and c) j =3at ¯hk =0(solid lines), ¯hk =0.5qB (dashed lines) and ¯hk = qB (dash-dotted lines)).
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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
Page 17 and 18: 2.2 Role of Quadrupole deformations
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Page 21 and 22: 2.3 Critical angular velocity for v
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Page 25: 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: Chapter 6 Stationary states of a BE
Page 71 and 72: 6.1 Bloch states and Bloch bands 67
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 and 112: 8.1 Dynamic structure factor 107 Th
Page 113 and 114: 8.2 Static structure factor and sum
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Chapter 9 Macroscopic Dynamics In t
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9.1 Macroscopic density and macrosc
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9.2 Hydrodynamic equations for smal
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9.4 Sound Waves 121 9.4 Sound Waves
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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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