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

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48 Groundstate of a BEC in an optical lattice the effects of harmonic trapping have been presented in [102]. In this thesis, also results for the density profile are included and the dependence on lattice depth of the density at the center of the harmonic trap averaged over one lattice period is described. We briefly comment on the generalization to 2D lattices. Based on an ansatz for the order parameter valid in deep lattices, the effective coupling constant ˜g and its effect on the harmonically trapped system have been previously discussed in [70]. 5.1 Density profile, energy and chemical potential When looking for stationary solutions, the GP-equation (3.14) takes the form − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz + g |Ψ(z)| d 2 Ψ(z) =µΨ(z) . (5.1) The groundstate is given by the solution of this equation with the lowest energy. Let us rewrite Eq.(5.1) in a more convenient form. First, we introduce the rescaled order parameter ϕ(z) = L2 Ψ(z) , N (5.2) with L the transverse size of the system and N the number of particles per lattice site such that d/2 dx |ϕ(z)| −d/2 2 The GPE (5.1) then reads =1. (5.3) − ¯h2 ∂ 2m 2 ∂z2 + sER sin 2 πz + dgn|ϕ(z)| d 2 ϕ(z) =µϕ(z) , (5.4) where we have introduced the average density n = N . (5.5) dL2 Casting Eq. (5.4) in dimensionless form, in dimensionless form it is possible to identify the governing parameters of the problem. We shall measure length in units of d/π, momentum in units of the Bragg-momentum qB =¯hπ/d and energy in units of the recoil energy ER = ¯h 2 π 2 /2md 2 . In this way, we obtain the dimensionless GPE − ∂2 ∂z 2 + s sin2 (z)+π gn |ϕ(z)| ER 2 ϕ(z) = µ ER ϕ(z) . (5.6) In contrast to the case of a single particle discussed in chapter 4, there are now two governing parameters: the lattice depth s and the interaction parameter gn/ER. The latter quantity can be changed by varying the average density or the scattering length by means of a Feshbach resonance.
5.1 Density profile, energy and chemical potential 49 Typical values of gn/ER in current experiments can be estimated by taking the density at the center of the harmonic trap in absence of the lattice and evaluating ER =¯h 2 π2 /2md2 for the lattice period d used in the respective experiment. In this way, one finds values ranging from gn =0.02ER to gn =1.1ER in the experiments [79, 73, 71, 87, 66]. For each choice of parameter s and gn/ER, we obtain a different groundstate wave function ϕ(z) and chemical potential d/2 µ = ϕ −d/2 ∗ (z) − ¯h2 ∂ 2m 2 2 πz + sERsin + gnd|ϕ(z)| ∂z2 d 2 ϕ(z)dz (5.7) and groundstate energy per particle d/2 ε = ϕ −d/2 ∗ (z) − ¯h2 ∂ 2m 2 2 πz + sERsin + ∂z2 d gnd 2 |ϕ(z)|2 ϕ(z)dz . (5.8) Calculating these quantities for different choices of s and gn/ER allows us to elucidate the role played by mean field interaction: Under what conditions does it play an important role ? How does it alter the effects produced by the lattice? Let us first discuss some solutions for the groundstate wavefunction ϕ(z) or, equivalently, the density nd |ϕ(z)| 2 . In Fig. 5.1, we report the results obtained at different values of lattice depth and interaction parameter. When s is increased for fixed gn/ER, the density becomes more and more modulated by the optical potential. Instead, when s is kept fixed while gn/ER is increased, the modulation of the density is reduced. Repulsive interactions screen off the lattice as mentioned above. In fact, at low lattice depth explicit formulas for the effective potential can be derived [103] and are found to be in agreement with experimental results [67]. Let us now proceed to the results obtained for the chemical potential and the energy per particle. In Fig. 5.2, we report the results obtained as a function of lattice depth s for different values of the interaction parameter gn/ER. Chemical potential and energy per particle coincide only in the absence of interactions. In general, they are linked by the relation µ = ∂ (nε) ∂n ∂ε = ε + gn . (5.9) ∂ (gn) The second term is necessarily positive for repulsive interactions and hence the chemical potential is always larger than the energy per particle (see Fig.5.2). When plotting the relative difference (µ−ε)/ε as a function of gn at fixed high s, weobserve that the curve is proportional to gn for sufficiently small gn (see Fig.5.3). The deviation from the linear behavior (see dotted line) for larger gn is due to the density dependence of ϕ. For gn small enough, the derivative ∂ε/∂ (gn) is essentially given by its gn =0-limit ∂ε d d/2 ≈ |ϕgn=0(z)| ∂ (gn) 2 −d/2 4 dz , (5.10) where ϕgn=0(z) is the groundstate solution in absence of interactions (gn =0). In the next section, we will show that the quantity (5.10) is closely related to the compressibility of the system.
Page 1: Università degli Studi di Trento F
Page 4: 7 Bogoliubov excitations of Bloch s
Page 9 and 10: Chapter 1 Introduction Superfluids
Page 11 and 12: The second class of stationary solu
Page 13 and 14: Chapter 2 Vortex nucleation The pro
Page 15 and 16: 2.1 Single vortex line configuratio
Page 17 and 18: 2.2 Role of Quadrupole deformations
Page 19 and 20: 2.2 Role of Quadrupole deformations
Page 21 and 22: 2.3 Critical angular velocity for v
Page 23 and 24: 2.3 Critical angular velocity for v
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: Chapter 5 Groundstate of a BEC in a
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 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
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7.4 Velocity of sound 99 c(s)/c(s=0
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Chapter 8 Linear response - Probing
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8.1 Dynamic structure factor 103 wi
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8.1 Dynamic structure factor 105 Th
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8.1 Dynamic structure factor 107 Th
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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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