arXiv subjects

Steven Tomsovic

Publications and source records attributed to Steven Tomsovic.

At least 37 records · Page 2Linked to original sources

arXiv2018

Ordered level spacing probability densities

Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3 \times 3$ matrix model. The closest neighbour density is generalized to the $k-$th closest neighbour spacing density, which allows for investigating long-range correlations. For larger $k$ the probability density of $k-$th closest neighbour spacings is well described by a Gaussian. Using these $k-$th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.

nlin.CD↗

arXiv2018

Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum

We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, $Λ$. The behaviors apply equally well to few- and many-body systems, e.g.\ interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading order behaviors depend on $\sqrtΛ$. The marginal case of the $1/2$ moment, which is related to the distance to closest maximally entangled state, is an exception having a $\sqrtΛ\ln Λ$ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charv{\' a}t-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e.\ the entanglement spectrum, show a transition from power laws and Lévy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

quant-ph↗

arXiv2018

Complex saddle trajectories for multidimensional quantum wave packet/coherent state propagation: application to a many-body system

A practical search technique for finding the complex saddle points used in wave packet or coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics, and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest time scale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.

cond-mat.stat-mech↗

arXiv2018

Post-Ehrenfest many-body quantum interferences in ultracold atoms far-out-of-equilibrium

Far out-of-equilibrium many-body quantum dynamics in isolated systems necessarily generate interferences beyond an Ehrenfest time scale, where quantum and classical expectation values diverge. Of great recent interest is the role these interferences play in the spreading of quantum information across the many degrees of freedom, i.e.~scrambling. Ultracold atomic gases provide a promising setting to explore these phenomena. Theoretically speaking, the heavily-relied-upon truncated Wigner approximation leaves out these interferences. We develop a semiclassical theory which bridges classical and quantum concepts in many-body bosonic systems and properly incorporates such missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it is shown that this theory captures post-Ehrenfest quantum interference phenomena very accurately, and contains relevant phase information to perform many-body spectroscopy with high precision.

quant-ph↗

arXiv2017

Partial local density of states from scanning gate microscopy

Scanning gate microscopy images from measurements made in the vicinity of quantum point contacts were originally interpreted in terms of current flow. Some recent work has analytically connected the local density of states to conductance changes in cases of perfect transmission, and at least qualitatively for a broader range of circumstances. In the present paper, we show analytically that in any time-reversal invariant system there are important deviations that are highly sensitive to imperfect transmission. Nevertheless, the unperturbed partial local density of states can be extracted from a weakly invasive scanning gate microscopy experiment, provided the quantum point contact is tuned anywhere on a conductance plateau. A perturbative treatment in the reflection coefficient shows just how sensitive this correspondence is to the departure from the quantized conductance value and reveals the necessity of local averaging over the tip position. It is also shown that the quality of the extracted partial local density of states decreases with increasing tip radius.

cond-mat.mes-hall↗

arXiv2017

Exact relations between homoclinic and periodic orbit actions in chaotic systems

Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulae expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This make possible the explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.

nlin.CD↗

arXiv2016

Generalized Gaussian wave packet dynamics: Integrable and Chaotic Systems

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent WBK method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle point trajectories at its foundation are found using a multi-dimensional, Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions which are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of $\hbar$ that comes with using the saddle point trajectories.

quant-ph↗

arXiv2016

Entanglement and localization transitions in eigenstates of interacting chaotic systems

The entanglement and localization in eigenstates of strongly chaotic subsystems are studied as a function of their interaction strength. Excellent measures for this purpose are the von-Neumann entropy, Havrda-Charv{\' a}t-Tsallis entropies, and the averaged inverse participation ratio. All the entropies are shown to follow a remarkably simple exponential form, which describes a universal and rapid transition to nearly maximal entanglement for increasing interaction strength. An unexpectedly exact relationship between the subsystem averaged inverse participation ratio and purity is derived that infers the transition in the localization as well.

quant-ph↗

arXiv2015

Action differences between fixed points and accurate heteroclinic orbits

A general relation is derived for the action difference between two fixed points and a phase space area bounded by the irreducible component of a heteroclinic tangle. The determination of this area can require accurate calculation of heteroclinic orbits, which are important in a wide range of dynamical system problems. For very strongly chaotic systems initial deviations from a true orbit are magnified by a large exponential rate making direct computational methods fail quickly. Here, a method is developed that avoids direct calculation of the orbit by making use of the well-known stability property of the invariant unstable and stable manifolds. Under an area-preserving map, this property assures that any initial deviation from the stable (unstable) manifold collapses onto themselves under inverse (forward) iterations of the map. Using a set of judiciously chosen auxiliary points on the manifolds, long orbit segments can be calculated using the stable and unstable manifold intersections of the heteroclinic (homoclinic) tangle. Detailed calculations using the example of the kicked rotor are provided along with verification of the relation between action differences. The loop structure of the heteroclinic tangle is necessarily quite different from that of the turnstile for a homoclinic tangle, its analogous partner.

nlin.CD↗

arXiv2015

A semiclassical reversibility paradox in simple chaotic systems

Using semiclassical methods, it is possible to construct very accurate approximations in the short wavelength limit of quantum dynamics that rely exclusively on classical dynamical input. For systems whose classical realization is strongly chaotic, there is an exceedingly short logarithmic Ehrenfest time scale, beyond which the quantum and classical dynamics of a system necessarily diverge, and yet the semiclassical construction remains valid far beyond that time. This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable. This begs the question, "how can an accurate approximation to a reversible/stable dynamics be constructed from an irreversible/unpredictable one?" The resolution of this incongruity depends on a couple key ingredients, a well-known, inherent, one-way structural stability of chaotic systems and an overlap integral not being amenable to the saddle point method.

quant-ph↗

arXiv2015

Universal scaling of spectral fluctuation transitions for interacting chaotic systems

The spectral properties of interacting strongly chaotic systems are investigated for growing interaction strength. A very sensitive transition from Poisson statistics to that of random matrix theory is found. We introduce a new random matrix ensemble modeling this dynamical symmetry breaking transition which turns out to be universal and depends on a single scaling parameter only. Coupled kicked rotors, a dynamical systems paradigm for such transitions, are compared with this ensemble and excellent agreement is found for the nearest-neighbor-spacing distribution. It turns out that this transition is described quite accurately using perturbation theory.

nlin.CD↗

arXiv2015

Wave Transport in One-Dimensional Disordered Systems with Finite-Size Scatterers

We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $ł_c$ which may contain an arbitrary number $δ/2π$ of wavelengths, where $δ= k l_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $δ$. For weak scatterers, we find: i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$, ii) a regime, to be called II, for $δ$ in the vicinity of $π$, where the system is almost transparent and less localized, and iii) right in the middle of regime II, for $δ$ very close to $π$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $δ$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.

cond-mat.dis-nn↗

arXiv2012

Universal Quantum Localizing Transition of a Partial Barrier in a Chaotic Sea

Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically the transport is suppressed if Planck's constant h is large compared to the classical flux, h >> Phi, such that wave packets and states are localized. In contrast, classical transport is mimicked for h << Phi. Designing a quantum map with an isolated partial barrier of controllable flux Phi is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter Phi/h. We find this curve to be symmetric to Phi/h=1, having a width of two orders of magnitude in Phi/h, and exhibiting no quantized steps. We establish the relevance of local coupling, improving on previous random matrix models relying on global coupling. It turns out that a phenomenological 2x2-model gives an accurate analytical description of the transition curve.

nlin.CD↗

arXiv2012

Constructing acoustic timefronts using random matrix theory

In a recent letter [Europhys. Lett. 97, 34002 (2012)], random matrix theory is introduced for long-range acoustic propagation in the ocean. The theory is expressed in terms of unitary propagation matrices that represent the scattering between acoustic modes due to sound speed fluctuations induced by the ocean's internal waves. The scattering exhibits a power-law decay as a function of the differences in mode numbers thereby generating a power-law, banded, random unitary matrix ensemble. This work gives a more complete account of that approach and extends the methods to the construction of an ensemble of acoustic timefronts. The result is a very efficient method for studying the statistical properties of timefronts at various propagation ranges that agrees well with propagation based on the parabolic equation. It helps identify which information about the ocean environment survives in the timefronts and how to connect features of the data to the surviving environmental information. It also makes direct connections to methods used in other disordered wave guide contexts where the use of random matrix theory has a multi-decade history.

math-ph↗

arXiv2012

Entanglement between two subsystems, the Wigner semicircle and extreme value statistics

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $ρ_{12}^{T_2}$. The density of states of $ρ_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $ρ_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.

quant-ph↗

arXiv2012

Wave transport in one-dimensional disordered systems with finite-width potential steps

An amazingly simple model of correlated disorder is a one-dimensional chain of n potential steps with a fixed width lc and random heights. A theoretical analysis of the average transmission coefficient and Landauer resistance as functions of n and klc predicts two distinct regimes of behavior, one marked by extreme sensitivity and the other associated with exponential behavior of the resistance. The sensitivity arises in n and klc for klc approximately pi, where the system is nearly transparent. Numerical simulations match the predictions well, and they suggest a strong motivation for experimental study.

cond-mat.dis-nn↗

arXiv2011

Random matrix theory for underwater sound propagation

Ocean acoustic propagation can be formulated as a wave guide with a weakly random medium generating multiple scattering. Twenty years ago, this was recognized as a quantum chaos problem, and yet random matrix theory, one pillar of quantum or wave chaos studies, has never been introduced into the subject. The modes of the wave guide provide a representation for the propagation, which in the parabolic approximation is unitary. Scattering induced by the ocean's internal waves leads to a power-law random banded unitary matrix ensemble for long-range deep ocean acoustic propagation. The ensemble has similarities, but differs, from those introduced for studying the Anderson metal-insulator transition. The resulting long-range propagation ensemble statistics agree well with those of full wave propagation using the parabolic equation.

physics.ao-ph↗

arXiv2011

On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems

The Kolmogorov-Sinai (K-S) entropy is a central measure of complexity and chaos. Its calculation for many-body systems is an interesting and important challenge. In this paper, the evaluation is formulated by considering $N$-dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem. This approach makes explicit a duality relation that is exactly analogous to one found in a generalized Anderson tight-binding model, and leads to a formally exact expression for the finite-time K-S entropy. Within this formalism there is a hierarchy of approximations, the final one being a diagonal approximation that only makes use of instantaneous Hessians of the potential to find the K-S entropy. By way of a non-trivial illustration, the K-S entropy of $N$ identically coupled kicked rotors (standard maps) is investigated. The validity of the various approximations with kicking strength, particle number, and time are elucidated. An analytic formula for the K-S entropy within the diagonal approximation is derived and its range of validity is also explored.

nlin.CD↗