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

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116 Macroscopic Dynamics analytical solution in the limit of a large initial displacement of the center-of-mass from the trap center. We comment on the instability of the center-of-mass motion which has been the subject of recent experimental and theoretical work. In [138] we have presented the hydrodynamic description of large current dynamics in the tight binding regime assuming that the effective mass is density-independent and that the change in the compressibility brought about by the lattice can be accounted for by the effective coupling constant ˜g. In [102] we reported the hydrodynamic equations for small currents which are valid at all lattice depths and allow for a density-dependent effective mass and a chemical potential which has a nonlinear dependence on density. Here, we generalize these hydrodynamic equations to account for large currents. They coincide with the ones we presented in [138] in the limit considered there and which, previously to [102], have been reported in [107]. Our result for the sound velocity in a condensate at rest is derived in [138] within the hydrodynamic framework developed there. The generalization to all lattice depth, general effective mass and compressibility, and to a condensate in slow motion is included in [102] (see also [107]). Our results for small amplitude collective oscillations were derived in [138] for the tight binding regime. In this thesis, their discussion is generalized to all lattice depths maintaining the assumptions made in [138] concerning the effective mass and the equation of state. The demonstration of the irrelevance of the equation of state for the dipole mode frequency and the derivation of the equations describing large amplitude center-of-mass oscillations at any lattice depth is added. 9.1 Macroscopic density and macroscopic superfluid velocity Our guiding idea is to retain from time-dependent GP-theory only the information necessary to describe the dynamics occurring on length scales much larger than the lattice spacing d. More precisely, we will assume that the state of the system in a window of size D>>d resembles a stationary state solution of the system without any external potential in addition to the optical lattice. Note that in order to describe non-stationary dynamics, D has to be much smaller than the size of the system. Accordingly, the dynamics we will be able to capture has non-stationary character only on length scales much larger than D. We set out by generalizing definition (5.21) of the average density at site l to the case of a time-dependent density nl(r⊥,t)= 1 ld+d/2 n(r⊥,z,t) dz , (9.1) d ld−d/2 where n(r⊥,z) = |Ψ(r⊥,z)| 2 is the density obtained by solving the time-dependent GPequation in the presence of optical lattice and an additional slowly varying external potential as for example a harmonic trap. With l replacing the continous variable z, expression (5.21) defines an average density profile of the condensate in the trap, as pointed out in section 5.4 for the static case. It is a smooth function of r⊥ and varies slowly as a function of the index l. We obtain a smooth macroscopic density profile nM(r⊥,z) by replacing the discrete index l by the continous variable z = ld l → z = ld ⇒ nl(r⊥,t) → nM(r⊥,z,t) . (9.2)
9.1 Macroscopic density and macroscopic superfluid velocity 117 In accordance with the basic idea introduced above, we assume nM to be approximately constant on the length scale D in the z-direction at any given time t. In analogy to the definition of the average density nl(r⊥), we introduce the average zcomponent of the superfluid velocity field at site l vzl(r⊥,t)= 1 ld+d/2 ¯h dz d ld−d/2 m ∂zS(r⊥,z,t) = ¯h 1 m d [S(r⊥,ld+ d/2,t) − S(r⊥,ld− d/2,t)] , (9.3) where S(r⊥,z,t) is the phase of the order parameter and (¯h/m)∂zS(r⊥,z,t) is the zcomponent of the superfluid velocity field obtained by solving the time-dependent GP-equation in the presence of optical lattice and an additional slowly varying external potential as for example a harmonic trap. The quantity (9.3) is required to be approximately constant over a distance D. Hence, we can rewrite (9.3) in the form vzl(r⊥,t)= ¯h 1 m D [S(r⊥,ld+ D/2,t) − S(r⊥,ld− D/2,t)] . (9.4) Now, we go a step further and require the approximate stationary state in the window of size D around the site l to be of the Bloch form ϕk = exp(iklz)˜ϕk (6.2) with quasi-momentum kl(r⊥,t). Note that kl(r⊥,t) must be a slowly varying function of the index l and the variable r⊥. The phase S in the window of size D around the site l is then approximately given by S(r⊥,z,t)=kl(t)z + ˜ Sk(r⊥,z,t) , (9.5) where the second contribution ˜ Sk is the phase of the Bloch wave ˜ϕk and thus ˜ Sk(r⊥,z,t)= ˜Sk(r⊥,z+ ld, t). Inserting (9.5) into (9.4) we find vzl(r⊥,t)= ¯hkl(t) m ¯h ˜Sjk(r⊥,ld+ D/2,t) − + m ˜ Sjk(r⊥,ld− D/2,t) . (9.6) D Since ˜ Sjk is periodic in z with periodicity d, the second contribution becomes small for D>>d. It can consequently be neglected for sufficiently large D and we are left with vzl(r⊥,t)= ¯hkl(r⊥,t) . (9.7) m We smooth (9.7) in the z-direction in the same way as the average density profile l → z = ld ⇒ vzl(r⊥,t) → vMz(r⊥,z,t) , (9.8) where the z-component of the macroscopic superfluid velocity field vM is given by vMz(r⊥,z,t)= ¯hk(r⊥,z,t) m . (9.9) The x, y-components vMx,vMy are given by the usual expressions (¯h/m)∂x,yS(r⊥,z,t). It is important to note that vMz =¯hk/m does not coincide with the group velocity ¯vk of a Bloch state condensate with quasi-momentum ¯hk. This is a very peculiar feature of a
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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
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 and 112: 8.1 Dynamic structure factor 107 Th
Page 113 and 114: 8.2 Static structure factor and sum
Page 119: Chapter 9 Macroscopic Dynamics In t
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
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
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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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