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Resonance-Dependent Dynamics in Driven Quantum Systems: A Normalized Simulation Framework for Frequency Sweeps, Phase Space Response, and Level Modeling

Dimosthenis K. Michopoulos, Spyridon K. Chronopoulos, Kostas P. Peppas

Published Jun 22, 2026• 0 citations
Publisher

Abstract

Abstract Externally driven quantum systems, operating under sensitive conditions of resonance, constitute an important class of physical platforms in quantum technologies, superconducting resonant architectures, and Josephson-junction-based environments. In this class of systems, an externally imposed electromagnetic driving frequency may influence oscillatory dynamics, coherence, spectral structure, and the frequency-dependent energy distribution. This work proposes an exploratory theoretical framework for interpreting resonance-dependent energy and spectral distributions in externally driven quantum systems, treating the driving frequency not only as an external perturbation but also as an active configuration parameter of the system's response. In the proposed formulation, resonance-sensitive behavior is described through a normalized resonance-response function. The formulation distinguishes between the non-driven reference spectrum and the externally driven effective spectrum, where resonance-dependent level shifts appear as deviations from the non-driven case. Two quantities, denoted by Λ and Δ, are introduced to describe the energy distribution associated with the externally driven response. Within the proposed interpretation, Λ represents the averaged potential and externally driven contribution, while Δ represents a cycle-averaged energy of the driven response. These quantities are not intended to replace the standard Hamiltonian description of quantum systems, but to provide an additional clarifying way to analyze how externally imposed conditions in resonance may reorganize effective energetic and spectral behavior. The proposed approach is relevant to periodically driven quantum systems, Floquet-type spectral descriptions (interpreting frequency-induced modifications), superconducting resonant environments, and Josephson-junction-based systems, while remaining complementary to standard time-dependent quantum and superconducting-circuit formulations. To support the theoretical interpretation, numerical simulations are presented for three representative microwave resonance areas at 6.1, 6.9, and 7.7 GHz. The simulations employ a normalized resonance-response function and include resonance-response profiles, spectral analysis, effective energy spectra, phase-space trajectories, averaged energy-distribution components, and Gaussian probability-density maps. Additional local- and global-sweep methodology relevant to phase-space simulations further illustrate how frequency deviation from resonance and externally imposed frequency variation may reshape the distribution of phase-space activity. The simulated trajectories represent the phase-space evolution of position and momentum coordinates in a driven-oscillator model, rather than literal quantum particle paths. Similarly, the Josephson effect motivated the use of microwave frequency sweeps, since driven Josephson systems exhibit phase-sensitive and resonance-dependent behavior. Therefore, the frequency-sweep simulations should be interpreted as a preparatory resonance-response analysis, rather than as a full Josephson model. The results suggest that resonance-dependent spectral and phase-space distribution may provide useful qualitative insight for frequency-controlled quantum systems, superconducting resonant architectures, and resonance-sensitive or control platforms. This study is intended as an exploratory theoretical and simulation-based contribution, presenting how externally controlled resonance conditions influence phase-space dynamics in driven quantum systems.

DOI
10.21203/rs.3.rs-10083963/v1

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Abstract Externally driven quantum systems, operating under sensitive conditions of resonance, constitute an important class of physical platforms in quantum technologies, superconducting resonant architectures, and Josephson-junction-based environments. In this class of systems, an externally imposed electromagnetic driving frequency may influence oscillatory dynamics, coherence, spectral structure, and the frequency-dependent energy distribution. This work proposes an exploratory theoretical framework for interpreting resonance-dependent energy and spectral distributions in externally driven quantum systems, treating the driving frequency not only as an external perturbation but also as an active configuration parameter of the system's response. In the proposed formulation, resonance-sensitive behavior is described through a normalized resonance-response function. The formulation distinguishes between the non-driven reference spectrum and the externally driven effective spectrum, where resonance-dependent level shifts appear as deviations from the non-driven case. Two quantities, denoted by Λ and Δ, are introduced to describe the energy distribution associated with the externally driven response. Within the proposed interpretation, Λ represents the averaged potential and externally driven contribution, while Δ represents a cycle-averaged energy of the driven response. These quantities are not intended to replace the standard Hamiltonian description of quantum systems, but to provide an additional clarifying way to analyze how externally imposed conditions in resonance may reorganize effective energetic and spectral behavior. The proposed approach is relevant to periodically driven quantum systems, Floquet-type spectral descriptions (interpreting frequency-induced modifications), superconducting resonant environments, and Josephson-junction-based systems, while remaining complementary to standard time-dependent quantum and superconducting-circuit formulations. To support the theoretical interpretation, numerical simulations are presented for three representative microwave resonance areas at 6.1, 6.9, and 7.7 GHz. The simulations employ a normalized resonance-response function and include resonance-response profiles, spectral analysis, effective energy spectra, phase-space trajectories, averaged energy-distribution components, and Gaussian probability-density maps. Additional local- and global-sweep methodology relevant to phase-space simulations further illustrate how frequency deviation from resonance and externally imposed frequency variation may reshape the distribution of phase-space activity. The simulated trajectories represent the phase-space evolution of position and momentum coordinates in a driven-oscillator model, rather than literal quantum particle paths. Similarly, the Josephson effect motivated the use of microwave frequency sweeps, since driven Josephson systems exhibit phase-sensitive and resonance-dependent behavior. Therefore, the frequency-sweep simulations should be interpreted as a preparatory resonance-response analysis, rather than as a full Josephson model. The results suggest that resonance-dependent spectral and phase-space distribution may provide useful qualitative insight for frequency-controlled quantum systems, superconducting resonant architectures, and resonance-sensitive or control platforms. This study is intended as an exploratory theoretical and simulation-based contribution, presenting how externally controlled resonance conditions influence phase-space dynamics in driven quantum systems.
We analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.
When an open quantum system is driven by an external time-dependent force, the coupling of the driving to the central system is usually included, whereas the impact of the driving field on the bath is neglected. We investigate the effect of a quantum bath of linearly driven harmonic oscillators on the relaxation dynamics of a quantum two-level system which is not directly driven. In particular, we calculate the frequency-dependent response of the system when the bath is subject to Dirac and Gaussian driving pulses. We show that a time-retarded effective force on the system is induced by the driven bath which depends on the full history of the perturbation and the spectral characteristics of the underlying bath. In particular, when a structured Ohmic bath with a pronounced Lorentzian peak is considered, the dynamical response of the system to a driven bath is qualitatively different than that of the undriven bath. Specifically, additional resonances appear which can be directly associated with a Jaynes-Cummings-like effective energy spectrum.
Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce numerical techniques that leverage classical microwave simulations, efficiently executable in finite-element solvers, to obtain the time-dependent Hamiltonian of microwave-driven superconducting circuits with arbitrary geometries under charge, flux, or mixed electromagnetic modulation. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.
Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micromotion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micromotion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time evolution of the system.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.