Multifractal energy spectra and their dynamical implications

Self acceleration from spectral geometry in dissipative quantum-walk dynamics

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry.

Multifractality of quantum wave functions in the presence of perturbations

We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model, and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the recent conjecture that multifractality breaks down following two scenarios. In the first one, multifractality is preserved unchanged below a certain characteristic length which decreases with perturbation strength. In the second one, multifractality is affected at all scales and disappears uniformly for a strong-enough perturbation. Our refined analysis shows that subtle variants of these scenarios can be present in certain cases. This study could guide experimental implementations in order to observe quantum multifractality in real systems.

Two scenarios for quantum multifractality breakdown

We expose two scenarios for the breakdown of quantum multifractality under the effect of perturbations. In the first scenario, multifractality survives below a certain scale of the quantum fluctuations. In the other one, the fluctuations of the wave functions are changed at every scale and each multifractal dimension smoothly goes to the ergodic value. We use as generic examples a one-dimensional dynamical system and the three-dimensional Anderson model at the metal-insulator transition. Based on our results, we conjecture that the sensitivity of quantum multifractality to perturbation is universal in the sense that it follows one of these two scenarios depending on the perturbation. We also discuss the experimental implications.

Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential

We report on the study of a polariton gas confined in a quasiperiodic one-dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.

Wavefunctions, quantum diffusion, and scaling exponents in golden-mean quasiperiodic tilings

We study the properties of wavefunctions and the wavepacket dynamics in quasiperiodic tight-binding models in one, two, and three dimensions. The atoms in the one-dimensional quasiperiodic chains are coupled by weak and strong bonds aligned according to the Fibonacci sequence. The associated d-dimensional quasiperiodic tilings are constructed from the direct product of d such chains, which yields either the hypercubic tiling or the labyrinth tiling. This approach allows us to consider fairly large systems numerically. We show that the wavefunctions of the system are multifractal and that their properties can be related to the structure of the system in the regime of strong quasiperiodic modulation by a renormalization group (RG) approach. We also study the dynamics of wavepackets to get information about the electronic transport properties. In particular, we investigate the scaling behaviour of the return probability of the wavepacket with time. Applying again the RG approach we show that in the regime of strong quasiperiodic modulation the return probability is governed by the underlying quasiperiodic structure. Further, we also discuss lower bounds for the scaling exponent of the width of the wavepacket and propose a modified lower bound for the absolute continuous regime.

Multifractality of quantum wave packets

We study a version of the mathematical Ruijsenaars-Schneider model and reinterpret it physically in order to describe the spreading with time of quantum wave packets in a system where multifractality can be tuned by varying a parameter. We compare different methods to measure the multifractality of wave packets and identify the best one. We find the multifractality to decrease with time until it reaches an asymptotic limit, which is different from the multifractality of eigenvectors but related to it, as is the rate of the decrease. Our results could guide the study of experimental situations where multifractality is present in quantum systems.

Cyclotron-Bloch dynamics of a quantum particle in a two-dimensional lattice. II. Arbitrary electric field directions

We study the quantum dynamics of a charged particle in a two-dimensional lattice, subject to constant and homogeneous electric and magnetic fields. We find that different regimes characterize these motions, depending on a combination of conditions, corresponding to weak and strong electric field intensities, rational or irrational directions of the electric field with respect to the lattice, and small or large values of the magnetic (Peierls) phase.

Generating a fractal butterfly Floquet spectrum in a class of driven SU(2) systems

A scheme for generating a fractal butterfly Floquet spectrum, first proposed by Wang and Gong [Phys. Rev. A 77, 031405(R) (2008)], is extended to driven SU(2) systems such as a driven two-mode Bose-Einstein condensate. A class of driven systems without a link with the Harper-model context is shown to have an intriguing butterfly Floquet spectrum. The found butterfly spectrum shows remarkable deviations from the known Hofstadter's butterfly. In addition, the level crossings between Floquet states of the same parity and between Floquet states of different parities are studied and highlighted. The results are relevant to studies of fractal statistics, quantum chaos, and coherent destruction of tunneling, as well as the validity of mean-field descriptions of Bose-Einstein condensates.

Butterfly Floquet spectrum in driven SU(2) systems

The Floquet spectrum of a class of driven SU(2) systems is shown to display a butterfly pattern with multifractal properties. The level crossing between Floquet states of the same parity or different parities is studied. The results are relevant to studies of fractal statistics, quantum chaos, coherent destruction of tunneling, and the validity of mean-field descriptions of Bose-Einstein condensates.

Quantum iterated function systems

An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators. This formalism is designed to describe certain problems of nonunitary quantum dynamics. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.

Regular and anomalous quantum diffusion in the Fibonacci kicked rotator

We study the dynamics of a quantum rotator, impulsively kicked according to the almost-periodic Fibonacci sequence. A special numerical technique allows us to carry on this investigation for as many as 10(12) kicks. It is shown that above a critical kick strength, the excitation of the system is well described by regular diffusion, while below this border it becomes anomalous and subdiffusive. A law for the dependence of the exponent of anomalous subdiffusion on the kick strength is established numerically. The analogy between these results and quantum diffusion in models of quasicrystals and in the kicked Harper system is discussed.

Generalized drude formula for the optical conductivity of quasicrystals

We derive a generalized Drude formula for the optical conductivity of quasicrystals, assuming a diffusion law in a perfect quasicrystal L(t) approximately t(beta), where L(t) measures the spreading of a wave packet in a time t. We show that the so-called Drude peak in the optical conductivity, characteristic of metals, is strongly modified. If beta<1/2 the Drude peak is even replaced by a dip and the dc conductivity increases when disorder increases. An interpretation of the experimental results on icosahedral and decagonal phases is proposed.

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.