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

More documents

Recommendations

Info

112 Linear response - Probing the Bogoliubov band structure S(p) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −6 −4 −2 0 2 4 6 p/qB Figure 8.7: Static structure factor (8.25) at lattice depth s =5for gn =0.5ER (solid line), gn =0.02ER (dashed line), gn =0(dash-dotted line) and at lattice depth s =0for gn =0.5 (dotted line; see also Eq.(8.22). S(p) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −6 −4 −2 0 2 4 6 p/qB Figure 8.8: Static structure factor (8.25) at lattice depth s =10for gn =0.5ER (solid line), gn =0.02ER (dashed line), gn =0(dash-dotted line) and at lattice depth s =0for gn =0.5 (dotted line; see also Eq.(8.22).
8.2 Static structure factor and sum rules 113 The behaviour of S(p) at small momenta can be described exactly using sum-rule arguments. Inserting (8.8) into (8.19) and (8.20) we can write p Zj(p)¯hωj(p) =Ntot j 2 , (8.26) 2m Zj(p) κ ¯hωj(p) = Ntot . (8.27) j 2 p→0 In lowest order in p the Bogoliubov bands ¯hωj(p) behave like Hence, the f-sum rule (8.26) can only be ensured at small p if ¯hω1(p)| p→0 = cp , (8.28) ¯hωj>1(p)| p→0 = const . (8.29) Z1(p) ∼ p, (8.30) Zj>1(p) ∼ p 2 . (8.31) Inserting this result and (8.28,8.29) into (8.18,8.27) we find that both the non-energy weighted sum rule (8.18) and the compressibility sum rule (8.20) are exhausted by the contribution from thefirstbandwhenp → 0, high energy bands giving rise to contributions of O(p 2 ). Thus, we can rewrite (8.18,8.20) in the form S(p)| p→0 = 1 Z1(p) ¯hω1(p) p→0 Ntot = Ntot Combining these two equations and using κ =1/m ∗ c 2 we obtain S(p) −→ p → 0 Z1(p) ∼ p, (8.32) κ . (8.33) 2 |p| 2m ∗ c (8.34) This shows that in the presence of 2-body interactions the low-p behaviour of the static structure factor is entirely determined by phonon correlations. This result holds for any value of s as long as gn = 0. In the absence of the optical lattice m ∗ = m and c coincides with the Bogoliubov sound velocity cuni = gn/m of the uniform system. Since we can write m ∗ c = √ m ∗ κ −1 and both m ∗ and κ −1 increase with s, we find that the presence of the lattice results in an enhanced suppression of the static structure factor at low values of p, as clearly shown in Figs.8.7 and 8.8.
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:
Page 43 and 44:
Page 45 and 46:
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 61 and 62:
Page 63 and 64:
Page 65 and 66: 5.4 Effects of harmonic trapping 61
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
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 115: 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 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

Create successful ePaper yourself

Delete template?

Save as template?