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Steven Tomsovic

Publications and source records attributed to Steven Tomsovic.

At least 19 recordsLinked to original sources
arXiv2026

Double-scaled bosonic and fermionic embedded ensembles, complex SYK, and the dual Hilbert space

We derive the density of states and $2$- and $4$-point functions of embedded ensembles for both fermions and bosons in the double-scaled limit. It is shown the models are equivalent to the double-scaled Sachdev-Ye-Kitaev model, expanding the double-scaled universality class to include both fermionic and bosonic systems. The models can be solved by introducing the Wick product of non-commuting Gaussian random variables. We show that deriving the Wick product is sufficient for computing the density of states, and properties of the Wick product can be used to compute $n$-point functions directly in the energy basis. In this context, the Wick product is equivalent to normal ordering of $q$-oscillators, which leads to the duality between moments of double-scaled models and expectation values in the chord Hilbert space. By considering operator probes as a second set of oscillators, we extend the duality to compute $n$-point functions. Embedded ensembles are equivalent to complex SYK at fixed charge, and we show working directly with embedded ensembles streamlines the derivations.

hep-th↗
arXiv2026

Hamiltonian Chaos

Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. The emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions.

quant-ph↗
arXiv2025

Quantum Chaos as an Essential Resource for Full Quantum State Controllability

Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and $\hbar^{-1}$. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.

quant-ph↗
arXiv2024

Controlling Many-Body Quantum Chaos: Bose-Hubbard systems

This work develops a quantum control application of many-body quantum chaos for ultracold bosonic gases trapped in optical lattices. It is long known how to harness exponential sensitivity to changes in initial conditions for control purposes in classically chaotic systems. In the technique known as targeting, instead of a hindrance to control, the instability becomes a resource. Recently, this classical targeting has been generalized to quantum systems either by periodically countering the inevitable quantum state spreading or by introducing a control Hamiltonian, where both enable localized states to be guided along special chaotic trajectories toward any of a broad variety of desired target states. Only strictly unitary dynamics are involved; i.e., it gives a coherent quantum targeting. In this paper, the introduction of a control Hamiltonian is applied to Bose-Hubbard systems in chaotic dynamical regimes. Properly selected unstable mean field solutions can be followed quite rapidly to states possessing precise phase relationships and occupancies. In essence, the method generates a quantum simulation technique that can access rather special states. The protocol reduces to a time-dependent control of the chemical potentials, opening up the possibility for application in optical lattice experiments. Explicit applications to custom state preparation and stabilization of quantum many-body scars are presented in one- and two-dimensional lattices (three-dimensional applications are similarly possible).

cond-mat.quant-gas↗
arXiv2023

Reduced Dimensional Monte Carlo Method: Preliminary Integrations

A technique for reducing the number of integrals in a Monte Carlo calculation is introduced. For integrations relying on classical or mean-field trajectories with local weighting functions, it is possible to integrate analytically at least half of the integration variables prior to setting up the particular Monte Carlo calculation of interest, in some cases more. Proper accounting of invariant phase space structures shows the system's dynamics is reducible into composite stable and unstable degrees of freedom. Stable degrees of freedom behave locally in the reduced dimensional phase space exactly as an analogous integrable system would. Classification of the unstable degrees of freedom is dependent upon the degree of chaos present in the dynamics. The techniques for deriving the requisite canonical coordinate transformations are developed and shown to block diagonalize the stability matrix into irreducible parts. In doing so, it is demonstrated how to reduce the amount of sampling directions necessary in a Monte Carlo simulation. The technique is illustrated by calculating return probabilities and expectation values for different dynamical regimes of a two-degree-of-freedom coupled quartic oscillator within a classical Wigner method framework.

cond-mat.stat-mech↗
arXiv2023

Controlling quantum chaos: time-dependent kicked rotor

One major objective of controlling classical chaotic dynamical systems is exploiting the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. In a recent letter [Phys.~Rev.~Lett. 130, 020201 (2023)], a generalization of this targeting method to quantum systems was demonstrated using successive unitary transformations that counter the natural spreading of a quantum state. In this paper further details are given and an important quite general extension is established. In particular, an alternate approach to constructing the coherent control dynamics is given, which introduces a new time-dependent, locally stable control Hamiltonian that continues to use the chaotic heteroclinic orbits previously introduced, but without the need of countering quantum state spreading. Implementing that extension for the quantum kicked rotor generates a much simpler approximate control technique than discussed in the letter, which is a little less accurate, but far more easily realizable in experiments. The simpler method's error can still be made to vanish as $\hbar \rightarrow 0$.

quant-ph↗
arXiv2022

Controlling Quantum Chaos: Optimal Coherent Targeting

One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a preeminent quantum chaotic paradigm, to illustrate the method, and investigate its accuracy. The method's error can be made to vanish as $\hbar \rightarrow 0$

nlin.CD↗
arXiv2022

Semiclassical roots of universality in many-body quantum chaos

Quantum chaos of many-body systems has been swiftly developing into a vibrant research area at the interface between various disciplines, ranging from statistical physics to condensed matter to quantum information and to cosmology. In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between classically chaotic dynamics and corresponding universal features at the quantum level. Recently, single-particle techniques dealing with ergodic wave interference in the usual semiclassical limit $\hbar \rightarrow 0$ have begun to be transformed into the field theoretical domain of N-particle systems in the analogous semiclassical limit $\hbar_{eff} = 1/N \rightarrow 0$, thereby accounting for genuine many-body quantum interference. This semiclassical many-body theory provides a unified framework for understanding random-matrix correlations of both single-particle and many-body quantum chaotic systems. Certain braided bundles of classical orbits, and of mean field modes, govern interference, respectively, and provide the key to the foundation of universality. Case studies presented include a many-body version of Gutzwiller's trace formula for the spectral density and out-of-time-order correlators along with brief remarks on where further progress may be forthcoming.

quant-ph↗
arXiv2022

Quantum coherence controls the nature of equilibration in coupled chaotic systems

A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interaction with a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax towards an equilibrium. It is found that quantum coherence of the initial product states in the uncoupled eigenbasis can be viewed as a resource for equilibration and approach to thermalization as manifested by the entanglement. Results are given for four distinct perturbation strength regimes, the ultra-weak, weak, intermediate, and strong regimes. For each, three types of initially unentangled states are considered, coherent random-phase superpositions, random superpositions, and eigenstate products. A universal time scale is identified involving the interaction strength parameter. Maximally coherent initial states thermalize for any perturbation strength in spite of the fact that in the ultra-weak perturbative regime the underlying eigenstates of the system have a tensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity as the interaction vanishes. In contrast to the widespread linear behavior, in this regime the entanglement initially grows quadratically in time.

quant-ph↗
arXiv2022

Resurgent revivals in bosonic quantum gases: a striking signature of many-body quantum interferences

Matter wave revivals depend on a delicate interplay of constructive many-body quantum interferences in the developing dynamics of an ultracold bosonic system in an optical lattice. It is shown that the interplay between weak intersite tunneling and strong onsite interactions can lead to the quantum dynamics of a density wave displaying several features not found in the mean-field limit: occupancy oscillations, resurgent revivals, and a (anti-) synchronization of revival peaks and occupancy oscillation peaks. This implies cooperative interference effects that alternate between constructive and destructive features leading to the peak revival behaviors. These many-body quantum interference phenomena create striking features in various observables, which are accessible in experimental measurements.

cond-mat.quant-gas↗
arXiv2021

A corrected Maslov index for complex saddle trajectories

Saddle point approximations, extremely important in a wide variety of physical contexts, require the analytical continuation of canonically conjugate quantities to complex variables in quantum mechanics. An important component of this approximation's implementation is arriving at the phase correction attributable to caustics, which involves determinantal prefactors. The common prescription of using the inverse of half a certain determinant's total accumulated phase sometimes leads to sign errors. The root of this problem is traced to the zeros of the determinants at complex times crossing the real time axis. Deformed complex time contours around the zeros can repair the sign errors that sometimes occur, but a much more practical way is given that links saddles back to associated real trajectories and avoids the necessity of locating the complex time zeros of the determinants.

quant-ph↗
arXiv2021

Semiclassical propagation of coherent states and wave packets: hidden saddles

Semiclassical methods are extremely important in the subjects of wave packet and coherent state dynamics. Unfortunately, these essentially saddle point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme to locate the $exposed$ complex saddle trajectories. The arguments for this approach were based mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theoretical foundation and then extended to incorporate saddles $hidden$ from Newton-Raphson searches initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes, but a hidden saddle can sometimes contribute just as much as or more than an exposed one. The distinctions between hidden and exposed saddles clarifies the interpretation of what constitutes tunneling for wave packets and coherent states in the time domain.

quant-ph↗
arXiv2020

Homoclinic orbit expansion of arbitrary trajectories in chaotic systems: classical action function and its memory

Special subsets of orbits in chaotic systems, e.g. periodic orbits, heteroclinic orbits, closed orbits, can be considered as skeletons or scaffolds upon which the full dynamics of the system is built. In particular, as demonstrated in previous publications [Phys. Rev. E 95, 062224 (2017), Phys. Rev. E 97, 022216 (2018)], the determination of homoclinic orbits is sufficient for the exact calculation of classical action functions of unstable periodic orbits, which have potential applications in semiclassical trace formulas. Here this previous work is generalized to the calculation of classical action functions of arbitrary trajectory segments in multidimensional chaotic Hamiltonian systems. The unstable trajectory segments' actions are expanded into linear combinations of homoclinic orbit actions that shadow them in a piece-wise fashion. The results lend themselves to an approximation with controllable exponentially small errors, and which demonstrates an exponentially rapid loss of memory of a segment's classical action to its past and future. Furthermore, it does not require an actual construction of the trajectory segment, only its Markov partition sequence. An alternative point of view is also proposed which partitions the trajectories into short segments of transient visits to the neighborhoods of successive periodic orbits, giving rise to a periodic orbit expansion scheme which is equivalent to the homoclinic orbit expansion. This clearly demonstrates that homoclinic and periodic orbits are equally valid skeletal structures for the tessellation of phase-space dynamics.

nlin.CD↗
arXiv2019

Entanglement production by interaction quenches of quantum chaotic subsystems

The entanglement production in bipartite quantum systems is studied for initially unentangled product eigenstates of the subsystems, which are assumed to be quantum chaotic. Based on a perturbative computation of the Schmidt eigenvalues of the reduced density matrix, explicit expressions for the time-dependence of entanglement entropies, including the von Neumann entropy, are given. An appropriate re-scaling of time and the entropies by their saturation values leads a universal curve, independent of the interaction. The extension to the non-perturbative regime is performed using a recursively embedded perturbation theory to produce the full transition and the saturation values. The analytical results are found to be in good agreement with numerical results for random matrix computations and a dynamical system given by a pair of coupled kicked rotors.

quant-ph↗
arXiv2019

Enhancement of many-body quantum interference in chaotic bosonic systems

Although highly successful, the truncated Wigner approximation (TWA) leaves out many-body quantum interference between mean-field Gross-Pitaevskii solutions as well as other quantum effects, and is therefore essentially classical. Turned around, this implies that if a system's quantum properties deviate from TWA, they must be exhibiting some quantum phenomenon, such as localization, diffraction, or tunneling. Here, we consider in detail a particular interference effect arising from discrete symmetries, which can lead to a significant enhancement of quantum observables with respect to the TWA prediction, and derive an augmented version of the TWA in order to incorporate them. Using the Bose-Hubbard model for illustration, we further show strong evidence for the presence of dynamical localization due to remaining differences between the TWA predictions and quantum results.

cond-mat.quant-gas↗
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

Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction

Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packet, coherent state, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentially with excursion complexity, and the corresponding cell areas decrease exponentially in size as well. With the specification of a desired precision, the exponentially proliferating set of homoclinic orbit actions is expressible by a slower-than-exponentially increasing set of cell areas, which may present a means for developing greatly simplified semiclassical formulas.

nlin.CD↗