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

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8 Introduction looking for solutions of the form δΨ(r,t)=e −iµ0t/¯h ilφ e unl(ρ, z)e iωnlt ∗ −iωnlt + vnl(ρ, z)e (1.16) corresponding to the collective modes of the initial axi-symmetric state. The modes are labeled by the number of nodes n of the amplitudes unl, vnl and the angular momentum ¯hl. For dr u ∗ nlun ′ l ′ − v∗ n ′ l ′vn ′ ,l ′ = δn,n ′δl,l ′ (1.17) they have energy ¯hωnl in the lab frame. In the rotating frame the excitation energy associated with the mode nl is instead given by The initial state is energetically unstable if for some nl ɛnl(Ω) = ¯hωnl − Ω¯hl . (1.18) ɛnl(Ω) < 0 . (1.19) This yields a critical angular velocity for the excitation of the mode nl ωcr(n, l) = ωnl (1.20) l corresponding to the Landau criterion for the creation of excitations nl in the rotating trap . It can most easily be satisfied for n =0. The respective excitations are called surface modes since they are associated with a perturbation which is concentrated at the surface of the condensate. They have been investigated experimentally in [5]. In the Thomas-Fermi (TF) limit [1] the critical angular velocities ωcr,nl for the surface modes take the simple form [43] ωcr,0l = ω⊥ √l . (1.21) In chapter 2 we will discuss the connection between the energetic instability towards the creation of surface modes and the nucleation of a single quantized vortex.
Chapter 2 Vortex nucleation The problem of the nucleation of quantized vortices in Bose superfluids has been the object of intensive experimental work with dilute gases in rotating traps [6, 7, 8, 12, 10, 11, 9, 13]. It has emerged clearly that the mechanism of nucleation depends crucially on the actual shape of the trap as well as on how the rotation is switched on. A first approach consists of a sudden switch-on of the deformation and rotation of the external potential which generates a non-equilibrium configuration [6, 7, 8, 12, 10, 11, 9]. For sufficiently large values of the angular velocity of the rotating potential one observes the nucleation of one or more vortices. In a second approach either the rotation or the deformation of the trap are switched on slowly [10, 13]. In the case of a sudden switch-on of a small deformation ε and rotation of the trap (1.3), the observed critical angular velocity turns out to be [6, 7, 8, 10, 9] 1 Ωc ≈ 0.7ω⊥ , (2.1) which is close to the value ω⊥/ √ 2 of the critical angular velocity (1.21) for the l =2surface mode in a TF-condensate. The purpose of this chapter is to gain further insight into the mechanism underlying the nucleation of a single vortex line in this setting. Using a simple semi-analytic model, we investigate the relevance of the quadrupole deformation of the condensate for the nucleation of the quantized vortex in the trap (1.3). In a non-deformed condensate the nucleation process is inhibited by the occurrence of a barrier located near the surface of the condensate. Using the TF-approximation to zero temperature GP-theory we show that this barrier is lowered by the explicit inclusion of ellipsoidal shape deformations and eventually disappears at sufficiently high angular velocities, making it possible for the vortex to nucleate and move to its stable position at the trap center. This chapter is based on the results presented in: Vortex nucleation and quadrupole deformation of a rotating Bose-Einstein condensate M. Krämer, L. Pitaevskii, S. Stringari, and F. Zambelli, Laser Physics 12, 113 (2002). 1 Note that in [6, 8] the rotating trap is actually switched on during the process of evaporative cooling above or around the condensation point. This does not change the results obtained for Ωc. 9
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: The second class of stationary solu
Page 15 and 16: 2.1 Single vortex line configuratio
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Page 21 and 22: 2.3 Critical angular velocity for v
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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
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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
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5.4 Effects of harmonic trapping 59
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5.4 Effects of harmonic trapping 61
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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
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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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