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

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134 Array of Josephson junctions The formalism developed in this chapter is useful in understanding the propagation of sound signals in a condensate subject to a lattice potential (see the subsequent chapter 11). In [102], we presented the dynamical equations for the occupations and phases of each site, as well as the result for the lowest Bogoliubov band. The discussion of the Josephson Hamiltonian is added here. 10.1 Current-phase dynamics in the tight binding regime In section 6.1 we have discussed Bloch state solutions of the stationary GP-equation in the periodic potential of the optical lattice. We have shown that the condensate Bloch functions Ψjk(z) can be written in terms of the condensate Wannier functions fj,l(z) in the following way Ψjk(z) = √ ndfj,l(z)e ikld . (10.1) l Note that Ψjk is normalized to the total number of atoms Ntot and apart from the normalization coincides with the Bloch function ϕjk (see Eq.(5.2)). In this section we will use an ansatz similar to (10.1) for the time-dependent condensate wavefunction Ψ(z,t), allowing for the time-dependence of site occupation and phase. In the tight binding regime the time-dependent phases and populations of each lattice site emerge as the dynamical variables. Let us consider a stationary Bloch state of the lowest band (j =1) in the tight binding regime. For convenience, we will choose the corresponding Wannier functions fl to take only real values. Since fl is well localized at site l wecansaythatthecondensateatsitelhas phase Sl = kld − µ(k) t. (10.2) ¯h The density of particles at site l is simply given by nl = n, (10.3) where, as previously, n is the average density of the system. Hence, in this state the population of each site is constant across the sample at any time t, while the phases Sl vary with the site index and have the simple time dependence µ(k)t/¯h. Furthermore, recall that two Wannier functions fl, fl ′ can be obtained from each other by a simple displacement and that the set {fl} is found for a given average density n Ansatz for the time-dependent wavefunction fl(z) =fl ′(z − (l − l′ )d) (10.4) fl(z) =fl(z; n) . (10.5) We are interested in time-dependent solutions Ψ(z,t) of (3.14). We focus on states whose form is obtained by releasing the restriction of the phases Sl and the average densities nl to
10.1 Current-phase dynamics in the tight binding regime 135 the values (10.2,10.3) and allow them to undergo a general time-dependent evolution Sl = kld − µ(k) ¯h t → Sl(t) , (10.6) nl = n → nl(t) . (10.7) Furthermore, we require the shape of the wavefunction at site l to be well approximated by the Wannier function fl obtained for a Bloch state (10.1) with average density n = nl(t). This presupposes an adiabatic adaptation of the shape of the wavefunction to the instantaneous value of nl(t). Given these assumptions the time-dependent state Ψ(z,t) is written in the form Ψ(z,t) = l fl(z; nl(t)) nl(t)de iSl(t) . (10.8) This ansatz must ensure that the quantities nl indeed have the meaning of the average density at site l. Thus, (10.8) must satisfy the equation ld+d/2 dz |Ψ(z,t)| 2 = nld. (10.9) ld−d/2 This is achieved by requring fl(z; nl) to be very well localized at the site l such that fl±1 have a negligible contribution to the population of site l ( ld+d/2 ld−d/2 dzf 2 l±1 =0). In addition, we require the orthogonalization condition dzfl(z; nl(t))fl ′(z; nl ′(t)) = δl,l ′ , (10.10) to be approximately satisfied. Exact orthogonality is guaranteed of course only for a stationary state where nl = nl ′. Dynamical equations phases and site occupations The wavefunction Ψ(z,t) evolves according to the time-dependent GP-equation (3.14). To obtain dynamical equations for the phases Sl and average densities nl, we insert (10.8) into the GPE. Upon multiplication of (3.14) by Ψ ∗ and integration over the total volume, using (10.10) we obtain where ˙nl = ˙Sl = − µl ¯h l ′ =l+1,l−1 µl = fl + δl,l′ √ nlnl ¯h ′ sin(Sl − Sl ′) , (10.11) l ′ =l+1,l−1 − ¯h2 ∂ 2 z 2m δl,l′ µ nl 2¯h ′ nl + V (z)+gnld|fl| 2 cos(Sl − Sl ′) , (10.12) fldz , (10.13) and δ l,l′ µ are directly related to the overlap while the time-dependent tunneling parameters δl,l′ between two neighbouring Wannier functions δ l,l′ = −2 dz fl − ¯h2 ∂2 z + V fl 2m ′ + gnldfl|fl| 2 fl ′ + gnl ′dfl|fl ′|2fl ′ , δ l,l′ µ = δ l,l′ − 4gnld fl|fl| 2 fl ′dz. (10.14)
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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
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6.2 Tight binding regime 79 2m¯v/q
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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 and 134: 9.6 Center-of-mass motion: Linear a
Page 135 and 136: 9.6 Center-of-mass motion: Linear a
Page 137: Chapter 10 Array of Josephson junct
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.
Page 185 and 186: [91] T. Stöferle, H. Moritz, C. Sc
Page 187 and 188: [125] E. Taylor and E. Zaremba, Bog
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[157] R.M. Bradley and S. Doniach,
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