Anderson Transition in a One-Dimensional System with Three Incommensurate Frequencies

Casati, Giulio; Guarneri, Italo; Shepelyansky, Dima L.

doi:10.1103/physrevlett.62.345

Cited by 192 publications

(224 citation statements)

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“…The transition is characterized by a well defined critical point, a divergence of the localization length below the critical point (in the localized regime) and a vanishing of the diffusion constant above the critical point (in the diffusive regime). We have determined the scaling laws and the critical exponent ν of the Anderson transition, which is significantly larger than unity and very close to the value obtained in recent numerical experiments [12,23] on the Anderson model, enforcing the assumption [15] that the two systems are substantially equivalent. Whether this exponent is universal (i.e.…”

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confidence: 82%

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Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

et al. 2008

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We realize experimentally an atom-optics quantum chaotic system, the quasiperiodic kicked rotor, which is equivalent to a 3D disordered system, that allow us to demonstrate the Anderson metalinsulator transition. Sensitive measurements of the atomic wavefunction dynamics and the use of finite-size scaling techniques make it possible to extract both the critical parameters and the critical exponent of the transition, which is in good agreement with the value obtained in numerical simulations of the 3D Anderson model. The metal-insulator Anderson transition plays a central role in the study of quantum disordered systems. An insulator is associated with localized states of the system, while a metal generally displays diffusive transport associated with delocalized states. The Anderson model [1] describes such a metal-insulator transition, due to quantum interference effects driven by the amount of disorder in the system. Starting from the "tight-binding" description of an electron in a crystal lattice, Anderson postulated in 1958 that the dominant effect of impurities in the lattice is to randomize the diagonal, on-site, term of the Hamiltonian, and showed that this generally leads to a localization of the wavefunction, in sharp contrast with the Bloch-wave solution for a perfect crystal. This model has progressively been extended from its original solidstate physics scope [1,2,3,4] to a whole class of systems in which waves propagate in a disordered medium, as for example quantum-chaotic systems [5,6] and electromagnetic radiation [7,8,9]. However mathematically simple, the model predicts a wealth of interesting phenomena. In 1D, the wavefunction is always localized as recently observed in experiments using atomic matter waves in a disordered optical potential [10,11]; in 3D it predicts a phase transition between a localized (insulator) and a delocalized (metal) phase at a well defined mobility edge, the density of impurities or the energy being the control parameter.Despite the wide interest on the Anderson transition, few experimental results are available. In a crystal, it is very difficult to obtain the conditions for a clean observation of the Anderson localization. Firstly, one has no direct access to the electronic wavefunction and must rely on modifications of bulk properties like conductivity [2,12]. Secondly, it is difficult to reduce decoherence sources to a low enough level. We thus engineered a matter-wave system that is described by an Andersonlike model, which allows us to probe the physics of disordered systems in much better conditions than in condensed matter physics [13], namely: Almost no interaction between particles, weak absorption in the medium, no coupling with a thermal reservoir which could destroy localization and possibility of measuring the final quantum state of the system after a given interaction time. The system, the quasi-periodic kicked rotor [5,6,14,15], consists of cold cesium atoms exposed to a pulsed, offresonant, laser standing wave. The dynamics is thus effectively one-dim...

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confidence: 82%

“…[15] to their attention, and for fruitful discussions. They also acknowledge G. Beck for his help with the experiment.…”

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confidence: 99%

Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

et al. 2008

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“…A numerical analysis of the quantum diffusion [14] showed that, as in a 3d AM with a truly random potential, the system avoids dynamical localization for k > k c . Recently it has been possible to model this generalized kicked rotor by using cold atoms in an optical lattice [15].…”

Section: Antonio M García-garcíamentioning

confidence: 99%

Anderson transition in a three-dimensional kicked rotor

Wang

García-García²

2009

Phys. Rev. E

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We investigate Anderson localization in a three dimensional (3d) kicked rotor. By a finite size scaling analysis we have identified a mobility edge for a certain value of the kicking strength k = kc. For k > kc dynamical localization does not occur, all eigenstates are delocalized and the spectral correlations are well described by Wigner-Dyson statistics. This can be understood by mapping the kicked rotor problem onto a 3d Anderson model (AM) where a band of metallic states exists for sufficiently weak disorder. Around the critical region k ≈ kc we have carried out a detailed study of the level statistics and quantum diffusion. In agreement with the predictions of the one parameter scaling theory (OPT) and with previous numerical simulations of a 3d AM at the transition, the number variance is linear, level repulsion is still observed and quantum diffusion is anomalous with p 2 ∝ t 2/3 . We note that in the 3d kicked rotor the dynamics is not random but deterministic. In order to estimate the differences between these two situations we have studied a 3d kicked rotor in which the kinetic term of the associated evolution matrix is random. A detailed numerical comparison shows that the differences between the two cases are relatively small. However in the deterministic case only a small set of irrational periods was used. A qualitative analysis of a much larger set suggests that the deviations between the random and the deterministic kicked rotor can be important for certain choices of periods. Contrary to intuition correlations in the deterministic case can either suppress or enhance Anderson localization effects. The quantum kicked rotor [1, 2], namely, a particle in a circle periodically kicked (with period T ) by a smooth potential V (q) = k cos(q),has played a central role in the development of quantum chaos. This is hardly surprising since, despite its simplicity, it has a very rich and highly non-trivial dynamics. Moreover, it can be studied experimentally [3,4] by using cold atoms techniques. The classical dynamics undergoes a gradual crossover from quasi-integrable to fully chaotic as the kicking strength k is increased. Therefore it is possible to study in detail all types of intermediate behavior between chaos and integrability. Moreover it is specially suited for numerical simulations since the evolution matrix in a basis of plane waves has a simple form. This was important in the early days of quantum chaos when numerical simulations in conservative 2d chaotic systems were quite expensive.The quantum dynamics of the kicked rotor is also of interest. For short time scales and k sufficiently large quantum and classical motion are diffusive. However for longer time scales numerical simulations show [1] that quantum diffusion in momentum space eventually stops, namely, lim t→∞ p 2 (t) tends to a constant. By contrast classically p 2 (t) ∝ t for all t. This rather counterintuitive feature, usually referred to as dynamical localization [2], was fully understood [5] after mapping the kicked rotor problem ont...

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“…DL is expected to disappear if the time periodicity is broken [7], as experimentally observed in [3]. This can be done simply by adding a second series of kicks with a different period rT , giving…”

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Quantum diffusion in the quasiperiodic kicked rotor

et al. 2005

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PACS. 05.45.Mt -Quantum chaos; semiclassical methods. PACS. 32.80.Lg -Mechanical effects of light on atoms, molecules, and ions. PACS. 32.80.Pj -Optical cooling of atoms; trapping.Abstract. -We study the mechanisms responsible for quantum diffusion in the quasiperiodic kicked rotor. We report experimental measurements of the diffusion constant on the atomic version of the system and develop a theoretical approach (based on the Floquet theorem) explaining the observations, especially the "sub-Fourier" character of the resonances observed in the vicinity of exact periodicity, i.e. the ability of the system to distinguish two neighboring driving frequencies in a time shorter than the inverse of the difference of the two frequencies.Quantum chaos is the study of quantum systems whose classical limit is chaotic. A major challenge of quantum chaos is to understand the mechanisms that make quantum chaos different from classical chaos. An important difference between classical and quantum systems is the existence in the latter of interferences between various paths. At long times, a large number of complicated trajectories interfere. One could expect the contributions of the various paths to have uncorrelated phases, so that the interference terms vanish in the average after some time, implying that quantum and classical transport should be identical. This simple expectation is however too naive, because phases of the various paths are actually correlated; this is for example the case for the kicked rotor.The quantum kicked rotor has been extensively studied experimentally in recent years [1][2][3][4]. In its atomic version it consists of a cloud of laser-cooled atoms exposed to short pulses of a far detuned, standing laser wave, corresponding to the Hamiltonian (for the external motion of the atoms)where θ is the 2π−periodic position of the rotor, p the conjugate momentum, T the period and K is proportional to the strength of the kicks. In the classical limit, this system is chaotic for K 1 [5], and the motion is an ergodic diffusion in momentum space for K > 5. Dynamical localization (DL) is the suppression of such a diffusion in the quantum system by subtle quantum interference effects [6].c EDP Sciences

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Anderson Transition in a One-Dimensional System with Three Incommensurate Frequencies

Cited by 192 publications

References 10 publications

Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

Anderson transition in a three-dimensional kicked rotor

Quantum diffusion in the quasiperiodic kicked rotor

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