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

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152 Sound propagation in presence of a one-dimensional optical lattice n φ / π 1.2 1 0.8 −200 −100 0 100 200 1 0.5 0 0.5 −1 −200 −100 0 100 200 x / d Figure 11.7: Sound signals with shock waves in the back for a deep lattice with δ/U ≪ 1/3. Upper panel: Relative density nℓ(t)/nℓ(t =0)obtained from the DNLS-simulation. Lower panel: Relative phase (11.11) obtained from the DNLS-simulation n φ / π 1.2 1 0.8 −200 −100 0 100 200 1 0.5 0 0.5 −1 −200 −100 0 100 200 x / d Figure 11.8: At an early stage of the simulation depicted also in Fig. 11.7 the relative phase touches π/2.
11.3 Nonlinear propagation of sound signals 153 3. Saturation Regime in deep lattices Provided that the lattice is deep enough to satisfy δ ≪ U/3, shock waves form only in the back of the signal, as discussed above. If under this condition we further increase the strength of the external perturbation, the relative phase φℓ+1/2 at some site reaches π (see Fig.11.9, upper two panels) and there the current starts flowing in the opposite direction (see Eq.(11.10)). As a consequence, a wake of noise is left behind the sound packets and saturation in the amplitude of the propagating signal is found (see Fig.11.9, lower two panels). The interesting feature is that the noise has average zero velocity, since the oscillations of population between different wells are completely dephased. Hence it never overtakes the sound signal, which is always able to “escape” from the noise. The system is in an interesting state, where the outer part is phase coherent, while the part between the two sound signals has lost coherence. We stress that in the uniform case, in presence of strong nonlinearities, we can observe a strong deformation of the signal, but in the central region the system is always able to recover the ground state after the sound wave has passed by (see Fig.11.4 a)). To demonstrate the saturation of the signal, we let the system evolve for a long time after the external perturbation and look at the amplitudes ∆n and ∆φ which are defined as indicated in Fig. 11.2. We find an interesting scaling behavior that helps to distinguish the three regimes mentioned above in the case of deep lattices and renders evident the saturation of the signal amplitude. Both in the GPE simulations (for relatively deep lattices) and in the DNLS simulations, we find a very interesting scaling law of the results, shown in Fig.11.10: The effect of the barrier is universal when: • the barrier parameters are combined in the form TP b/w which reflects the capability of the system to react to an external perturbation; • the relative density variation is rescaled as ∆n U/δ, while the amplitude of the relative phase signal ∆φ need not be rescaled. The phenomenology of the system as summarized in Fig.11.10 shows that for small barrier parameter Tpb/w, the perturbation produced in the system is small, and depends linearly on Tpb/w. Increasing the barrier parameter Tpb/w, the signal amplitude ∆n saturates quite suddenly. The three regions indicated in Fig.11.10 correspond to the three regimes (1-3) listed above. Based on the DNLS equation (11.10) one can give a quantitative estimate of the saturation value for ∆n. Assuming small density and relative phase variations, the first of Eqs.(11.10) becomes ˙nℓ ≈ nδ ¯h sin(φℓ+1/2) − sin(φℓ−1/2) ≈ nδ ¯h d∂φ ∂ℓ . (11.14) Now, we make the ansatz n =1+∆n[f+(ℓd − ct)+f−(ℓd + ct)] and φ = −∆φ[f+(ℓd − ct)+ f−(ℓd+ct)] for the density and relative phase variation, where f+ and f− describe respectively the packets moving to the left and to the right. Using that c =(d/¯h) √ δU,we get ∆n = δ ∆φ. U (11.15)
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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
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Chapter 7 Bogoliubov excitations of
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7.2 Bogoliubov bands and Bogoliubov
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7.3 Tight binding regime of the low
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7.3 Tight binding regime of the low
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7.4 Velocity of sound 97 d|bjql| 2
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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
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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
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Page 125 and 126: 9.4 Sound Waves 121 9.4 Sound Waves
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Page 137 and 138: Chapter 10 Array of Josephson junct
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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
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Page 159 and 160: ∆ n ( U / δ ) 1/2 1 0.8 0.6 0.4
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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
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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
Page 189: [157] R.M. Bradley and S. Doniach,
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Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

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