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

More documents

Recommendations

Info

90 Bogoliubov excitations of Bloch state condensates Bogoliubov amplitudes The functions ũjq(z) and ˜vjq(z) in (7.11,7.12) are periodic with period d. Their Fourier expansion reads The normalization condition requires d l ũjq(z) = ˜vjq(z) = b ∗ jqlbj ′ ql − c ∗ jqlcj ′ ql l l 2πz il bjql e d , (7.17) 2πz il cjql e d . (7.18) = δjj ′ . (7.19) With q satisfying periodic boundary conditions this is sufficient to ensure the orthonormalization conditions (7.8,7.9) where the index σ is replaced by the band index j and the quasi-momentum q of the excitations. In Figs. 7.5 and 7.6 we plot the square modulus of the Fourier coefficients bjql and cjql at different values of q in the first and the second Bogoliubov band for gn =0.5ER at a lattice depth of s =1, s =5and s =10respectively. At all considered lattice depth, the relative importance of the Bogoliubov vjq-amplitude with respect to the ujq-amplitude diminishes in the transition from the lowest to the first excited band. In fact, the contribution of the vjq-amplitude is negligible in all considered cases in the second band. Hence, for gn =0.5ER, apart from an energy offset, essentially only the lowest Bogoliubov band differs from the Bloch bands of a single particle. This can be explained by the fact that for this choice of the parameter gn/ER, the healing length of the system is comparable to the lattice period d, a setting which is typical of current experiments. As a consequence, when the lattice is off the vjq-amplitude is relevant only at momenta ¯hq
7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes 91 as s →∞. The normalization (7.8) can then only be ensured if uq and vq grow to infinity in this limit. d|bjql| 2 ,d|cjql| 2 6 4 2 0 6 4 2 0 6 4 2 0 j =1, ¯hq =0.1qB −2 −1 0 1 2 j =1, ¯hq =0.5qB −2 −1 0 1 2 j =1, ¯hq =0.9qB −2 −1 0 1 2 6 4 2 0 6 4 2 0 6 4 2 0 j =2, ¯hq =0.1qB −2 −1 0 1 2 j =2, ¯hq =0.5qB −2 −1 0 1 2 j =2, ¯hq =0.9qB −2 −1 0 1 2 l l Figure 7.5: Square modulus of the Fourier coefficients bjql (white bars) and cjql (black bars) of ũjq(z) and ˜vjq(z) respectively as defined in (7.17,7.18) for the lowest Bogoliubov band (j =1, left column) and the first excited Bogoliubov band (j =2, right column) at q = 0.1qB, 0.5qB, 0.9qB as obtained from the numerical solution of Eqs.(7.14,7.15) with gn = 0.5ER and lattice depth s =5.
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 41 and 42:
4.1 Solution of the Schrödinger eq
Page 43 and 44: 4.1 Solution of the Schrödinger eq
Page 45 and 46: 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
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 91 and 92: 7.2 Bogoliubov bands and Bogoliubov
Page 93: 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 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 155 and 156:
Page 157 and 158:
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
Page 189:
[157] R.M. Bradley and S. Doniach,
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?