UNIVERSITA` DEGLI STUDI DI FIRENZE DIPARTIMENTO DI FISICA Facolt`a di Scienze Matematiche, Fisiche e Naturali PhD Thesis in PHYSICS XXI cycle - FIS/03 Anderson localization of a weakly interacting Bose-Einstein condensate presented by Chiara D’Errico Supervisor ........................... prof. Giovanni Modugno Coordinator ........................ prof. Alessandro Cuccoli Referees.............................. prof. Thierry Giamarchi prof. Anna Vinattieri Contents Introduction 1 1 Anderson Localization 3 1.1 Localization in a one-dimensional system . . . . . . . . . . . . 5 2 Ultra-cold gases in disordered potentials 13 2.1 Control of interaction via Feshbach resonances . . . . . . . . . 14 2.2 Optical dipole potentials . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Dipole forces . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Optical lattices . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Periodic potentials . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Dynamics of a Bloch wavepacket . . . . . . . . . . . . 22 2.3.3 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . 23 2.4 Disordered optical potentials . . . . . . . . . . . . . . . . . . . 29 2.4.1 Laser speckles . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Quasi periodic one-dimensional optical lattices . . . . . 31 3 Anderson localization in incommensurate lattices 37 3.1 Theory of localization of non-interacting particles . . . . . . . 38 4 Experimental realization ofa weaklyinteracting Bose-Einstein condensate 49 4.1 Realization of BEC of 39K with tunable interaction . . . . . . 50 4.1.1 Weakly interacting 39K condensate . . . . . . . . . . . 55 4.2 Interferometric determination of the zero crossing position . . 57 5 Experimental observation of Anderson localization with a i CONTENTS non-interacting BEC 63 5.1 Observation of Anderson localization . . . . . . . . . . . . . . 63 5.1.1 Realization of the quasi-periodic lattice . . . . . . . . . 64 5.1.2 Diffusion dynamics . . . . . . . . . . . . . . . . . . . . 64 5.1.3 Exponential distribution of the localized BEC . . . . . 71 5.1.4 Analysis of the momentum distribution . . . . . . . . . 74 5.1.5 Interference of multiple localized states . . . . . . . . . 80 5.2 Interacting one dimensional disordered system . . . . . . . . . 83 5.2.1 Experimental observation of effects of weak interaction 84 Conclusions 91 A Scattering theory 93 B Aubry-Andr´e Hamiltonian in momentum space 97 Bibliography 101 ii Introduction Localization of particles and waves in disordered media is one of the most intriguing phenomena in modern physics. This phenomenon has been origi- nally studied by P. W. Anderson, fifty years ago, in the paper ”Absence of diffusion in some random lattices” [1], in the contest of transport of electrons in crystals. For this study, in 1977 Anderson was awarded the Nobel Prize in physics. Anderson studied the transport of non-interacting electrons in a crystal lat- tice, described by a single particle with random on-site energy. In his model he showed that when the amplitude of the disorder becomes higher than a criticalvalue, the diffusionin thelatticeof aninitially localized wavepacket is suppressed. He predicted a transition between extended and localized states, that, due to the presence of electron-electron and electron-phonon interac- tions, has not been directly observed for electrons in a crystal. The interplay between disorder and interaction, in fact, is still an interesting open question in the modern condensed matter physics. First effect of weak nonlinearities have been recently shown for light waves in photonic lattices [2, 3]. The Anderson transition is a much more general phenomenon and has been studied in many other systems where interactions or non-linearities are al- most absent. This term, in fact, can be generalized to electromagnetic waves, acoustic waves, quantum waves, etc. However Anderson transition was never observed for matter waves. Ultracold atoms offer a new possibility for the study of disorder-induce localization. The physics of disorder on this kind of systems has been accessible thanks to the introduction of laser speckles [4] and quasi-periodic optical lattices [5]. Preliminary investigations have be done in regimes where the observation of the localization was precluded either by the size of the disorder or by delocalizing effects of nonlinearity 1 CONTENTS [4, 6, 7, 8, 9]. Only recently the Anderson localization has been observed for matter-waves [10, 11], and this thesis describes one of such studies. In particular, in this thesis we report on the study of the disorder induced localization of a Bose-Einstein condensate in a lattice system, following the original idea of Anderson [1]. The atom-atom interaction in the condensate can be tuned to zero independently of the other parameters [12]. We intro- duce disorder on the structure of the lattice by using a weaker incommensu- rate secondary lattice, which produces a quasi-periodic potential. This kind of system corresponds to an experimental realization of the so called Harper [13] or Aubry-Andr´e model [14], which displays a transition from extended to localized states analogous to the Anderson transition. The main advan- tage of using this kind of disorder is the fact that it offers the possibility to observe the transition already in one dimension [15], whereas in the case of pure random disorder, a system with more than two dimensions would be needed [16]. We clearly observed Anderson localization by investigating transport proper- ties, spatial and momentum distributions. We studied, in fact, the diffusion of the BEC in the bichromatic lattice and we observed that disorder is able to stop the transport into the lattice, when its strength is high enough to localize the system. We studied also the spatial distribution and we found that while the condensate after the diffusion in the single lattice has a gaus- sian profile, when the disorder is strong enough to localize the system the distributions present an exponential behaviour, emblematic characteristic of Anderson localization. The other possibility we exploited to observe the An- derson localization is the investigation of the momentum distribution, whose width is inversely proportional to the width of the spatial wavefunction and gives important information on the eigenstates of the system. 2 Chapter 1 Anderson Localization In the last decades a great interest in the study of disordered structures has grown. This is mainly due to the fact that disorder is everywhere, since in nature perfect ordered systems do not exist. Any system, in fact, is charac- terized by a disordered structure if it is observed in a sufficiently small scale (crystalswithimpurities, amorphoussubstances, fractalessurfaces, etc). One of the main properties of disordered potentials is the fact that they are char- acterized by localized eigenstates, with a localization length ℓ << L smaller than the size of the system. One of the most interesting phenomena in solid-state physics, related to the study of disordered potentials, is Anderson localization that describes the absence of diffusion induced by disorder for electrons in crystals. Anderson presented in 1958 a model [1] in which he supposed to have a periodic array of sites j, that he called ”lattice”, regularly or randomly distributed in three- dimensional space. He assumed to have generic ”entities” occupying these sites, thatcouldbeelectronsoranyotherkindofparticles. Themodelsimply assumes to have an energy E , which can randomly vary from site to site, for j the particle that occupies site j and to have an interaction matrix element V (r ), which transfers the electrons from one site to the next. Anderson jk jk studied the behavior of the wave function of a single particle on site n at an initial time, as a function of the time. He found that there is no transport at all, in the sense that even increasing the time the amplitude of the wave function around the site n falls off rapidly with the distance. An Anderson localized state is characterized by an exponential decay of the amplitude of 3 1. ANDERSON LOCALIZATION the wave function, as the distance from the localization point increases, over a spatial extension larger than the mean distance between two fluctuations of the potential. The presence of interactions between particles can strongly influence the pos- sibility to observe the disorder induced localization. So the ideal system for this kind of physics is a non interacting sample. For this reason, the in- triguing phenomenon of Anderson localization has never been observed in atomic crystals, where thermally excited phonons and electron-electron in- teractions represent deviations from the Anderson model [1]. After realizing that Anderson localization is a wave phenomenon relying on interference, the Anderson’s idea was extended to optics [17, 18]. During the ’80, the localization was initially observed for photons (naturally non-interacting) in non-absorbing scattering media. The first prediction [19] and observation of coherent backscattering [20, 21] (weak localization), have been followed by the observation of strong localization of light in highly scattering dielectric media [22, 23, 24, 25, 26, 27]. However in all these studies the potential was fully random without the periodic structure of the lattice that characterizes the original Anderson’s model. The first observation of Anderson localization in a perturbed periodic po- tential has been the transverse localization of light caused by random fluc- tuations on a two dimensional photonic lattice [2] (Fig.1.1). Measuring the transverse diffusion(intheplaneperpendicular tothepropagationdirection), they demonstrated how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. More recently in 1D disordered photonic lattices the tran- sition from free ballistic wave packet expansion to exponential localization has been observed [3]. The first observation of Anderson localization in matter waves arrived only recently in two complementary experiments [10, 11], one of which is the sub- ject of this thesis. Ultra-cold atoms are a perfect system for the study of disorder-induced localization, mainly for the possibility to control the in- teraction strength. In the first work [10] an exponential localization has been observed for aBose-Einstein condensate released into a one-dimensional waveguide (where interactions are negligible as an effect of the low atomic 4 1.1. Localization in a one-dimensional system Figure 1.1: Transverse localization scheme [2]. (a) A probe beam entering a disordered lattice, which is periodic in two transverse dimensions but invariant in the propagation direction. (b,c) Experimentally observed diffraction pattern after propagation in the completely periodic lattice (b) and in one particular realization of disordered lattice (c). density) in the presence of a controlled random disorder created by laser speckle (Subsection 2.4.1). Conversely, in our experiments [11] we observe Anderson localization in a one-dimensional quasi-periodic lattice of a BEC where interactions are nulled via a Feshbach resonance (Section 2.1). We demonstrated that for larger enough disorder this kind of system is charac- terized by the presence of exponentially localized states, analogous to the Anderson ones. We clearly observed the localization by investigating trans- port properties, spatial and momentum distributions. 1.1 Localization in a one-dimensional system The quantum transport properties of a system are intimately related to the underlying symmetries of the Hamiltonian. In a perfectly periodic system all the eigenstates are extended Bloch waves [28]. For a random potential in a one-dimensional system, where there is no trace of translational symmetry, we instead expect to have an opposite behavior and the eigenfunctions must be spatially localized. This phenomenon can be produced from two different causes, depending on the degree of disorder of the system. In the description of a one-dimensional crystal, in fact, Lifshitz introduced for the first time the distinction between strong and weak disorder [29]. In his original defini- tion it was considered weakly disordered a crystal with low concentration of impurities, where the mean distance between two consecutive impurities was of the order of many lattice constants. The other extreme of strong disorder was associated to an high concentration of impurities. A weakly disordered 5 1. ANDERSON LOCALIZATION system presents Lifshitz localization, where a single fluctuation of the poten- tial is enough to induce localization. On the contrary, Anderson localization occurs in strongly disordered systems and it is produced really by the high concentration of the impurities distributed in the system. Even if the phenomenon of localization is generally present with a one- dimensional random disorder, it has to be discussed and analyzed for each model. In some cases, in fact, localization is present only with particular parameters [30]. We can deduce the behavior of the Anderson localized wavefunctions in a simple model, as done by Mott [31]. With this simple problem of quantum mechanics, we can deduce the emblematic characteristic of Anderson local- ization: the exponential decreasing of the wave function fromthe localization point. Let us start considering a one-dimensional periodic potential of length L, characterized by a series of barriers equally spaced, with the same width b and the same high V : 0 0 if x D i V(x) = ∈ (1.1)  V if x E  0 ∈ i where we defined the domains:  D [ia,(i+1)a b] i ≡ − (1.2) E [ia b,ia] i ≡ − in which V(x) is respectively 0 and V . The Schr¨odinger equation can be 0 given separately for regions D and E : i i ∂2ψ 2m + Eψ = 0 x D ∂x2 h¯2 ∈ i ∂2ψ 2m + (E V )ψ = 0 x E (1.3) ∂x2 h¯2 − 0 ∈ i If we solve the system for E < V we have: 0 ψ(x) = Aieiαx +A′ie−iαx+ϕi if x ∈ Di (1.4)  Bieβx +Bi′e−βx+φi if x ∈ Ei with α2 = 2mE/h¯2 and β2 = 2m(V E)/h¯2. By considering the periodicity 0 − of the potential, the Bloch’s theorem asserts that the wavefunction solution 6