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

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30 Introduction We develop simple theoretic frameworks for the description of statical and dynamical properties and discuss quantities which are crucial for their characterization. We also evaluate the validity of GP-theory in order to demonstrate the applicability of our results. In particular, after a short review of some standard results concerning a single particle in a one-dimensional periodic potential, we will discuss • the groundstate with and without harmonic trap (chapter 5) • stationary states of Bloch form (chapter 6) • the Bogoliubov excitations of the groundstate (chapter 7) • the linear response to a density perturbation (chapter 8) • the macroscopic dynamics with and without harmonic trap (chapter 9) • the description of the system as an array of Josephson junctions (chapter 10) • the propagation of sound signals (chapter 11) • the quantum depletion (chapter 12) Here is a more detailed overview of the thesis: Chapter 5 discusses the groundstate of a condensate confined in a one-dimensional optical lattice. We first consider a system without harmonic trapping which is uniform in the direction transverse to the lattice. We study the chemical potential, the energy, the density profile and the compressibility as a function of lattice depth s and the interaction parameter gn, where g is the two-body coupling constant (3.11) and n is the average density. In a second step, we allow for the additional presence of radial and axial harmonic trapping. We use the results obtained for the purely periodic potential as an input to calculate the groundstate properties in the combined trap. In chapter 6, we extend the discussion of stationary condensates in a one-dimensional lattice to non-groundstate solutions of the GP-equation. We focus on solutions which take the form of Bloch states and investigate the associated band spectra for the energy and the chemical potential in dependence on lattice depth and interaction strength carrying out a detailed comparison both with the non-interacting and the uniform cases. The energy Bloch band spectrum determines the current and therefore the group velocity and the effective mass. In the tight binding regime, the energy and chemical potential bands take a simple analytic form. From the expression for the Bloch energy bands we find equations for the current and the group velocity. Exploiting the tight binding formalism, we also derive simple expressions for the compressibility of the groundstate. Chapter 7 deals with small perturbations of a stationary Bloch state condensate in the periodic potential. We study in detail the Bogoliubv band spectrum of the groundstate both numerically for all lattice depths and analytically in the tight binding regime. Special attention is paid to the behavior of the sound velocity both for the groundstate and for a condensate moving with non-zero group velocity. The Bogoliubov band structure can be probed by studying the linear response of the condensate to a weak external perturbation. In chapter 8 we consider the particular case in which
the external probe generates a density perturbation in the system. We present results for the dynamic structure factor and the static structure factor of a condensate loaded into a one-dimensional lattice pointing out the striking effect of the periodic potential. In chapter 9, we show how to describe the long length scale GP-dynamics of a condensate in a one-dimensional optical lattice by means of a set of hydrodynamic equations for the density and the velocity field. Within this formalism, we can account for the presence of additional external fields, as for example a harmonic trap, provided they vary on length scales large compared to the lattice spacing d. As an application we derive an analytic expression for the sound velocity in a Bloch state condensate. In the combined presence of optical lattice and harmonic trap, the hydrodynamic equations can be solved for the frequencies of small amplitude collective oscillations. The results are compared with recent experimental data. We also discuss the large amplitude center-of-mass motion. In chapter 10 we describe the dynamics of the system in terms of the dynamics of the number of particles and the condensate phase at each lattice site. From this point of view, the system constitutes a realization of an array of Josephson junctions. The effect of a one-dimensional optical lattice on the propagation of sound signals is discussed in chapter 11. We devote special attention to the propagation in the nonlinear regime and distinguish different nonlinear effects in dependence on lattice depth. Finally, in chapter 12 we discuss the effect of the lattice on the condensate fraction within the framework of Bogoliubov theory. We provide estimates for the depletion, discuss the effective change of geometry induced by the lattice and set the limit of validity of our methods. This part of the thesis is essentially based on the following papers: • Macroscopic dynamics of a trapped Bose-Einstein condensate in the presence of 1D and 2D optical lattices M. Krämer, L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 88, 180404 (2002). • Dynamic structure factor of a Bose-Einstein condensate in a 1D optical lattice C.Menotti,M.Krämer, L. Pitaevskii, and S. Stringari: Phys. Rev. A 67, 053609 (2003). • Bose-Einstein condensates in 1D optical lattices: Compressibility, Bloch bands and elementary excitations M. Krämer, C. Menotti, L. Pitaevskii and S. Stringari, Eur. Phys. J. D 27, 247 (2003). • Sound propagation in presence of a one-dimensional optical lattice in preparation, with C. Menotti, A. Smerzi, L. Pitaevskii and S. Stringari 31
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 spectroscopy were described in
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
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
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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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