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

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52 Groundstate of a BEC in an optical lattice 5.2 Compressibility and effective coupling constant From the solutions µ(s, gn) for the chemical potential, the compressibility κ of the system can be immediately calculated using the relation κ −1 = n ∂µ (5.11) ∂n It is useful to evaluate this quantity in order to understand better the combined effects of lattice and interactions. In particular, analyzing the behavior of the compressibility helps to study in more detail the dependence of the chemical potential on density in a lattice of fixed depth. Knowing the form of this density-dependence is crucial to determine for example the frequencies of collective oscillations, the density profile and the chemical potential obtained when adding a harmonic trap to the optical lattice (see sections 5.4 and 9). In the uniform system, the chemical potential is simply linear in the density µ = gn leading to an increase of the inverse compressibility that is proportional to the density κ−1 = gn. In the presence of the lattice, we expect deviations from this behaviour due to the localization of the particles near the bottom of the well centers. In Fig.5.4, the inverse compressibility κ −1 is plotted as a function of gn for different potential depths s. As a general rule, the condensate becomes more rigid as gn/ER or s is increased. Yet, in contrast to the uniform case, the monotonic growth of κ −1 is linear only at small densities. At larger values of gn/ER, the slope of the curve tends to decrease and develop a non-linear functional dependence. At small values of gn where the inverse compressibility is approximately linear in gn, letus denote the proportionality constant by ˜g such that κ −1 =˜g(s)n. (5.12) corresponding to the chemical potential µ = µgn=0 +˜g(s)n, (5.13) where µgn=0 depends on the lattice depth, but not on density. Since the condensate is more rigid in the lattice relative to the uniform case, we have ˜g >g. Actually, ˜g is a monotonically increasing function of s (see increase of the slopes at gn =0for increasing s in Fig. 5.4). The quantity ˜g can be considered as an effective coupling constant: In a situation in which Eq.(5.12) is valid, the compressibility of the condensate in the lattice with coupling constant g is the same as the compressibility of a uniform condensate with coupling constant ˜g. So,as far as the compressibility is concerned we can deal with the problem as if there was no lattice by simply replacing g → ˜g. Below, we will see that this is a useful approach when describing macroscopic properties, both static (see section 5.4) and dynamic (see section 9), which do not require a detailed knowledge of the behavior on length scales of the order of the lattice spacing d. In fact, the properties of the compressibility discussed in this section will prove useful in devising a hydrodynamic formalism for condensates in a lattice. The concept of an effective coupling constant ˜g is applicable when the influence of 2-body interaction on the wavefunction ϕ(z) is negligible in the calculation of the chemical potential (5.7). Provided this condition is fulfilled, we obtain using Eq.(5.7) κ −1 = n ∂µ d/2 = gnd |ϕgn=0(z)| ∂n 4 dz , (5.14) −d/2
5.2 Compressibility and effective coupling constant 53 where ϕgn=0(z) is the groundstate solution in absence of interactions (gn =0). Comparison with Eq.(5.12) yields d/2 ˜g = gd |ϕgn=0(z)| −d/2 4 dz . (5.15) This explicit formula makes again visible that the decrease of the compressibility described by ˜g is due to the concentration of the particles close to the well centers. In Fig.5.5 we compare the exact results for κ−1 with the approximate formula (5.12) as a function of s for different values of gn/ER. This plot allows to determine how deep the lattice has to be made for a given value of gn/ER in order to be able to use the effective coupling constant description. Given, for example, gn =0.5ER we find that at s =10and s =20the expression (5.12) is valid within ≈ 10% and ≈ 7% respectively. A larger value of gn requires larger values of s to achieve the same accuracy. κ −1 /ER 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 gn/ER Figure 5.4: The inverse compressibility (5.11) as a function of gn/ER for s =0(solid line), s =5(dashed line) and s =10(dash-dotted line).
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Università degli Studi di Trento F
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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 and 52: Chapter 5 Groundstate of a BEC in a
Page 53 and 54: 5.1 Density profile, energy and che
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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
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Page 83 and 84: 6.2 Tight binding regime 79 2m¯v/q
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Page 87 and 88: Chapter 7 Bogoliubov excitations of
Page 89 and 90: 7.2 Bogoliubov bands and Bogoliubov
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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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