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

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16 Vortex nucleation term in (2.27) is the sum of the kinetic energy contribution m dr [vvortex · vQ] n(r) and the angular momentum term −m Ω · dr [r × vvortex] n(r) due to the vortex E3(d/Rx,δ, ¯ Ω) = m dr vvortex · [vQ − Ω × r] n(r) . (2.28) It turns out that this contribution can be calculated in a straightforward way. We find E3(d/Rx,δ, ¯ Ω) = m dr vvortex · [vQ − Ω × r] n(r) =−N¯hω⊥ ¯ Ω 1 − δ2 2 5/2 d 1 − , Rx (2.29) which is consistent with (2.8) in the case of an axi-symmetric condensate (δ =0). In deriving result (2.29) we have used the condition (2.22) for the quadrupole velocity field and we have integrated by parts using the expression (2.3) for vvortex. Note that the irrotational component ∇S of vvortex does not contribute to (2.29). The result (2.27) generalizes the one given in Ref. [49], which holds in the limit of small ¯ Ω where δ ∼ ε. 2.3 Critical angular velocity for vortex nucleation Within our model, the total energy of a quadrupolar deformed condensate in the presence of a vortex is given by (2.27). The values of the angular velocity Ω, the trap anisotropy ε, the average transverse oscillator frequency ω⊥, and the chemical potential µ are fixed by the experimental conditions. Hence the degrees of freedom of the system with which one can play in order to identify the optimal path for vortex nucleation are the vortex displacement d and the condensate deformation δ. Note that each of the three energy contributions in (2.27) has a different dependence on the chemical potential resulting in a non-trivial dependence of the critical angular velocity on the relevant parameters of the system (see Eq. (2.31) below). We assume that the system is initially in the state d/Rx =1, δ =0. This assumption adequately describes an experiment in which Ω and ε are switched on suddenly. In this case, the condensate is initially axisymmetric (δ =0) and vortex-free (d/Rx =1). Of course this configuration is not stationary and will evolve in time. In the following we will make use of energetic considerations in order to explore the possible paths followed by the system towards the nucleation of the vortex line. These paths should be associated with a monotonous decrease of the energy. In Figs. 2.2 and 2.3 we have plotted the energy surface Etot for two different values of the angular velocity ¯ Ω and fixed ε and µ/¯hω⊥. In both cases the angular velocity is chosen high enough to make the vortex state a global energy minimum. This minimum is surrounded by an energy ridge which, for δ =0, forms a barrier between the initial state (d/Rx =1) and the vortex state (d =0), as discussed in section 2.1. Moreover, the energy ridge exhibits a saddle point at non-zero deformation δ. The height of this saddle depends on ¯Ω, ε, andµ/¯hω⊥. In Fig. 2.2 the energy at the saddle point is higher than the energy of the initial state and the ridge can not be surpassed. However, at higher angular velocities ¯ Ω the situation changes. In Fig. 2.3 the saddle lies lower than the initial state. Hence in this case the system can bypass the barrier by crossing the saddle. The corresponding path is always associated with the occurrence of a strong intermediate deformation of the condensate. The critical angular velocity for the nucleation of vortices naturally emerges as the angular velocity ¯ Ωc at which the energy on the saddle point ((d/Rx)sp,δsp) isthesameastheenergy
2.3 Critical angular velocity for vortex nucleation 17 0.00 ⊕ +0.02 ⊘ d/Rx 0 -0.32 ⊗ 0.1 -0.01 ▽ ⊙ +0.01 0.2 0.3 δ Ω=0.64 ω⊥ 0.4 Figure 2.2: Below the critical angular velocity of vortex nucleation the plot shows the dependence of the total energy (2.27) minus the energy of the initial non-deformed vortexfree state E(d/Rx =1,δ =0)on the quadrupolar shape deformation δ and on the vortex displacement d/Rx from the center. Energy is given in units of N¯hω⊥. The dashed line corresponds to Etot − E(d/Rx = 1,δ = 0) = 0, while the solid curve refers to Etot − E(d/Rx =1,δ =0)=0.015N¯hω⊥. This plot has been obtained by setting ε =0.04, µ =10¯hω⊥ and Ω=0.64ω⊥. The initial state is indicated with ⊕, while ⊗ corresponds to the energetically preferable centered vortex state. The barrier ⊘ inhibits vortex nucleation in a non-deforming condensate (δ =0). The saddle point ⊙ lies lower than the barrier ⊘. However, at the chosen Ω the energy on the saddle is still higher than the one of the initial state ⊕. Note that the preferable vortex state is associated with a shape deformation δ>ε (see section 2.4). Note also the existence of a favorable deformed and vortex-free state labeled by ▽ [16]. of the initial state (d/Rx =1,δ =0): 0.5 E((d/Rx)sp,δsp, ¯ Ωc,ε,µ)=E(d/Rx =1,δ =0, ¯ Ωc,ε,µ) . (2.30) It is worth mentioning that crossing the saddle point is not the only possibility for the system to lower its energy. In fact Figs. 2.2 and 2.3 show the existence of stationary deformed vortex-free states which can be reached starting from the initial state. These are the states predicted in [16] and experimentally studied in [10, 13] through an adiabatic increase of either Ω or ε instead of doing a rapid switch-on. The energy ridge separates these configurations from the vortex state. Still, under certain conditions this stationary vortex-free state becomes dynamically unstable and a vortex can be nucleated starting out from it [10, 13, 17]. The study of this type of vortex nucleation is beyond the scope of the present discussion. The actual value of the critical angular velocity ¯ Ωc for vortex nucleation depends on the 0.6 0.7
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: 2.2 Role of Quadrupole deformations
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
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
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6.1 Bloch states and Bloch bands 67
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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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