© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Ultrakalte Bosonen in optischen Gittern - der Einfluss spektraler Statistik auf Simulierbarkeit, Stabilität und Dynamik

Hannah Venzl

Published Jan 1, 2011• 0 citations

Abstract

In this thesis, we study interacting ultracold bosons in tilted optical lattices, a paradigm for complex many-body quantum systems. Our analysis is based on the tilted Bose-Hubbard Hamiltonian which incorporates inter-site tunneling, inter-particle interaction, and a static field. The competition between the three contributions induces the system’s complexity which, on the spectral level, is reflected in chaotic level statistics. We address the question how this spectral chaos influences the dynamical behavior of the system and, vice versa, which information we can infer from the system’s dynamics on its many-body spectrum. Employing methods from random matrix theory, we first perform a detailed analysis of the Hamiltonian’s regular and chaotic spectral regimes and determine their borders by means of a scaling analysis. In the main part of the thesis, we tackle the question raised above from three different perspectives: Firstly, we investigate how the Hamiltonian’s level statistics affects the simulability of the system and find a breakdown of the simulation efficiency at the crossover from the regular to the chaotic regime. Secondly, we analyze the stability of the Hamiltonian’s eigenstates within the chaotic spectral regime and identify robust states that are dynamically stable against perturbations. Thirdly, we study the fast and irreversible decay of the Bloch oscillations in the chaotic regime. We develop an approach that is based on a statistical analysis of the velocity operator and find a characteristic crossover in the Bloch oscillations’ decay, once again right at the transition from regular to chaotic level statistics.

Related papers

Many-body Landau-Zener tunneling in the Bose-Hubbard model

Andrea Tomadin, R. Mannella, Sandro Wimberger

We study a model for ultracold, spinless atoms in quasi-one-dimensional optical lattices and subjected to a tunable tilting force. Statistical tests are employed to quantitatively characterize the spectrum of the Floquet-Bloch operator of the system along the transition from the regular to the quantum chaotic regime. Moreover, we perturbatively include the coupling of the one-band model to the second energy band. This allows us to study the Landau-Zener interband tunneling within a truly many-body description of ultracold atoms. The distributions of the computed tunneling rates provide an independent and experimentally accessible signature of the regular-chaotic transition.

arXiv:0711.3230•Physical Review A•Jan 7, 2008•30 citations

Krylov complexity of a tilted extended Bose-Hubbard chain with Rydberg-dressed interactions

Yifan Chen, Tianyi Yan, Lu Qin +1 more

We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction $U$ and nearest-neighbor interaction $V$ are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near $U\simeq V$ in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.

arXiv:2610.00712•Sep 30, 2026•0 citations

Dynamical witnesses and universal behavior across chaos and non-ergodicity in the tilted Bose-Hubbard model

Carlos Diaz-Mejia, Sergio Lerma-Hernandez, Jorge G. Hirsch

Quantum chaos in isolated quantum systems is intimately linked to thermalization and the rapid relaxation of observables. Although the spectral properties of the chaotic phase in the tilted Bose-Hubbard model have been well characterized, the corresponding dynamical signatures across the transition to regularity remain less explored . In this work, we investigate this transition by analyzing the time evolution of the survival probability, the single-site entanglement entropy, and the half-chain imbalance. Our results reveal a clear hierarchy in the sensitivity of these observables: the relaxation value of the entanglement entropy varies smoothly as a function of the Hamiltonian parameters across the chaos-regular transition, while the imbalance exhibits a more pronounced distinction. Most notably, the survival probability emerges as the most robust indicator of the transition between chaos and regularity. When appropriately scaled, all three observables converge onto a common behavior as a function of the Hamiltonian parameters for different numbers of sites and bosons,enabling a universal characterization of the transition between chaotic and regular dynamics.

quant-pharXiv:2602.00369•Feb 3, 2026•0 citations

Statistics of Schmidt coefficients and the simulability of complex quantum systems

Hannah Venzl, Andrew J. Daley, Florian Mintert +1 more

We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the time-dependent density-matrix renormalization-group algorithm. We investigate this mechanism on the tilted Bose-Hubbard model; however the emergence of universal spectral properties allows translation of our conclusions to generic many-body quantum systems.

Physical Review E•May 26, 2009•16 citations

Characterization of the chaotic phase in the tilted Bose-Hubbard model

Pilar Martín Clavero, Alberto Rodríguez

The chaotic phase of the tilted Bose-Hubbard model is identified as a function of energy, tilt strength, and particle interaction, from the eigenstate structure and the statistical features of the energy spectrum. Our analysis reveals that the chaotic phase of the bare Bose-Hubbard Hamiltonian can be enhanced by the presence of a moderate tilt. We further unveil the development and scaling of the chaotic regime from the perspective of a homogeneous density configuration typically used in cold atom experiments, providing a valuable phase diagram for future theoretical and experimental studies of this system.

arXiv:2504.01462•Physical review. E•Jun 6, 2025•1 citation

Complex Dynamics of Ultracold Atoms

Patrick Plötz

Motivated by current experiments with ultracold atoms, the study of complex dynamics of Bose-Einstein condensates in optical lattices forms the central subject of this work. A lattice model of interacting bosons under the influence of an external force is motivated and derived from the experimental setup. Several dynamical regimes of this model are discussed in this thesis. In a first part we will develop a new measure for detecting avoided crossings in complex energy spectra and apply it to the quantum chaotic regime of the single-band system. The second and main part of this work is dedicated to the coupling between energy bands described in terms of a two-band model. The complex time evolution is already apparent in the horizontal and vertical population dynamics of the non-interacting problem. We find resonances in the interband transport, and, in a second step, study the effect of inter-particle interactions on these resonant oscillations. We are able to predict all time scales of the complex interband dynamics even in the presence of interactions. This is possible via the introduction of an effective model that is motivated and supported by a multitude of numerical results and proves exactly solvable.

•Jan 1, 2010•1 citation

In this thesis, we study interacting ultracold bosons in tilted optical lattices, a paradigm for complex many-body quantum systems. Our analysis is based on the tilted Bose-Hubbard Hamiltonian which incorporates inter-site tunneling, inter-particle interaction, and a static field. The competition between the three contributions induces the system’s complexity which, on the spectral level, is reflected in chaotic level statistics. We address the question how this spectral chaos influences the dynamical behavior of the system and, vice versa, which information we can infer from the system’s dynamics on its many-body spectrum. Employing methods from random matrix theory, we first perform a detailed analysis of the Hamiltonian’s regular and chaotic spectral regimes and determine their borders by means of a scaling analysis. In the main part of the thesis, we tackle the question raised above from three different perspectives: Firstly, we investigate how the Hamiltonian’s level statistics affects the simulability of the system and find a breakdown of the simulation efficiency at the crossover from the regular to the chaotic regime. Secondly, we analyze the stability of the Hamiltonian’s eigenstates within the chaotic spectral regime and identify robust states that are dynamically stable against perturbations. Thirdly, we study the fast and irreversible decay of the Bloch oscillations in the chaotic regime. We develop an approach that is based on a statistical analysis of the velocity operator and find a characteristic crossover in the Bloch oscillations’ decay, once again right at the transition from regular to chaotic level statistics.

We study a model for ultracold, spinless atoms in quasi-one-dimensional optical lattices and subjected to a tunable tilting force. Statistical tests are employed to quantitatively characterize the spectrum of the Floquet-Bloch operator of the system along the transition from the regular to the quantum chaotic regime. Moreover, we perturbatively include the coupling of the one-band model to the second energy band. This allows us to study the Landau-Zener interband tunneling within a truly many-body description of ultracold atoms. The distributions of the computed tunneling rates provide an independent and experimentally accessible signature of the regular-chaotic transition.

We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction

U

and nearest-neighbor interaction

V

are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near

U\simeq V

in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.

Quantum chaos in isolated quantum systems is intimately linked to thermalization and the rapid relaxation of observables. Although the spectral properties of the chaotic phase in the tilted Bose-Hubbard model have been well characterized, the corresponding dynamical signatures across the transition to regularity remain less explored . In this work, we investigate this transition by analyzing the time evolution of the survival probability, the single-site entanglement entropy, and the half-chain imbalance. Our results reveal a clear hierarchy in the sensitivity of these observables: the relaxation value of the entanglement entropy varies smoothly as a function of the Hamiltonian parameters across the chaos-regular transition, while the imbalance exhibits a more pronounced distinction. Most notably, the survival probability emerges as the most robust indicator of the transition between chaos and regularity. When appropriately scaled, all three observables converge onto a common behavior as a function of the Hamiltonian parameters for different numbers of sites and bosons,enabling a universal characterization of the transition between chaotic and regular dynamics.

We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the time-dependent density-matrix renormalization-group algorithm. We investigate this mechanism on the tilted Bose-Hubbard model; however the emergence of universal spectral properties allows translation of our conclusions to generic many-body quantum systems.

The chaotic phase of the tilted Bose-Hubbard model is identified as a function of energy, tilt strength, and particle interaction, from the eigenstate structure and the statistical features of the energy spectrum. Our analysis reveals that the chaotic phase of the bare Bose-Hubbard Hamiltonian can be enhanced by the presence of a moderate tilt. We further unveil the development and scaling of the chaotic regime from the perspective of a homogeneous density configuration typically used in cold atom experiments, providing a valuable phase diagram for future theoretical and experimental studies of this system.

Motivated by current experiments with ultracold atoms, the study of complex dynamics of Bose-Einstein condensates in optical lattices forms the central subject of this work. A lattice model of interacting bosons under the influence of an external force is motivated and derived from the experimental setup. Several dynamical regimes of this model are discussed in this thesis. In a first part we will develop a new measure for detecting avoided crossings in complex energy spectra and apply it to the quantum chaotic regime of the single-band system. The second and main part of this work is dedicated to the coupling between energy bands described in terms of a two-band model. The complex time evolution is already apparent in the horizontal and vertical population dynamics of the non-interacting problem. We find resonances in the interband transport, and, in a second step, study the effect of inter-particle interactions on these resonant oscillations. We are able to predict all time scales of the complex interband dynamics even in the presence of interactions. This is possible via the introduction of an effective model that is motivated and supported by a multitude of numerical results and proves exactly solvable.