Floquet Dressing and Bath Spectral Effects on the Geometric Phase of a Driven Dissipative Qubit

A periodically driven symmetric spin-boson model exhibits rich geometric phase dynamics arising from the interplay between driving and dissipation, which this study investigates using numerically exact methods to understand how Floquet restructuring steers bath spectral effects.

The gist

Periodic longitudinal driving reorganizes coherent qubit dynamics into a set of Floquet-dressed channels whose quasienergy differences and Fourier weights depend on the drive amplitude and frequency, allowing for the interpretation of how these dressed transitions sample the environmental spectral density at frequencies of the form εα − εβ + nomega, which provides a framework for interpreting drive-induced modifications to the dissipative geometric phase.

System Description and Driving

The study considers a symmetric two-level system (qubit) described by a bare Hamiltonian with intrinsic tunneling energy scale ∆, initially possessing inversion symmetry under the σz operator. This system is subjected to a classical monochromatic field coupled to the σz coordinate, defined by the time-dependent Hamiltonian HS(t) = −∆2/σx − A2/cos(omegat)σz. Although this instantaneous Hamiltonian breaks inversion symmetry when cos(omegat) ≠ 0, it retains a generalized dynamical symmetry under inversion combined with a half-period time translation. The drive produces micromotion that affects the trajectory of the Bloch vector, which is parametrized by its components r(t).

Theoretical Framework for Geometric Phase

The dissipative geometric phase (GP) is calculated using the kinematic mixed-state GP, defined as γD(t) = arg (Xk=± pλk(0)λk(t)⟨Ψk(0)Ψk(t)⟩ × e − R t0 ⟨Ψk(t′)Ψ˙ k(t′)dt′). For the qubit, this simplifies to γD(t) = −1/2 ∫ t0 dτ [1 − cos θ τ] ϕ˙ τ dτ (for initial state ρ(0) = +⟩⟨+). This expression explicitly shows that the dissipative GP phase depends on both the geometric leverage of the Bloch vector from the North pole, 1 − cosθ(t), and its azimuthal winding rate, ϕ˙(t).

Floquet Theory and Drive Dressing

The periodic driving is analyzed using Floquet theory, which describes the quasienergy structure generated by the drive. The longitudinal periodic drive can be viewed as a periodically rotating effective transverse tunneling field. This transformation leads to an effective Hamiltonian HeS(t) which contains a static component together with an infinite series of Floquet harmonics, expressed via the Jacobi–Anger expansion involving Bessel functions J0(κ) and J2m(κ). The dimensionless dressing parameter is defined as κ = A/omega, which controls the angular excursion of the effective transverse field. In the high-frequency limit (omega∆ ≫ 1), the leading Floquet effective Hamiltonian is given by H(0)F = −∆2/J0(κ)σx, where the effective tunneling amplitude is ∆eff = ∆J0(κ).

Drive-Bath Interaction and Spectral Steering

The drive modifies the system dynamics while leaving the physical system-bath coupling operator unchanged. However, by dressing the system, it changes the bath induced transition rates. The resulting Floquet sidebands sample different regions of the bath spectral density, as illustrated in Fig. 2. The competition between coherent Floquet dressing and multiphoton sidebands, alongside the frequency-dependent bath spectral weight J(ω), provides a natural framework for interpreting drive-induced modifications to the dissipative GP.

Control Parameters and Observables

The study introduces the difference measure δγD(t) = γA,omegaD (t) − γA=0D (t) − γA,omegaU (t) + γU0(t). This non-additive quantity quantifies the extent to which the field-dressed system’s trajectory is distorted by dissipation beyond the separate contributions of the undriven dissipative and driven unitary dynamics. The results demonstrate that periodic longitudinal driving can substantially modify the geometric phase acquired by the dissipative qubit through Floquet spectral steering of bath spectral density, showing that varying drive parameters (amplitude A and frequency omega) changes not only the coherent trajectory but also the frequencies through which the bath acts on that trajectory. Furthermore, increasing system-bath coupling strength α enhances spectral weight available at Floquet-dressed transition frequencies, providing a means to separate the role of drive-induced spectral structure from the overall magnitude of environmental perturbation. The cutoff frequency ωc/∆ acts as an additional control axis determining whether drive-induced spectral redistribution enhances or suppresses the dissipative deformation of the GP.

Conclusion and Outlook

The work establishes a direct connection between Floquet engineering and the GP of a dissipative quantum system, suggesting a general route toward GP control in which external fields are used to steer open-system dynamics spectrally.

Based on the provided scientific paper, here are specific improvements that can be made to AI systems, followed by a description of what those improved systems could achieve:

) Improved AI Systems

1. [] Develop and implement a novel Floquet Spectral Steering module for modeling open quantum systems (OQSs).

2. [] Integrate the process-tensor time-evolving matrix product operator (PT-TEMPO) method into a generalized dynamics simulation framework, allowing for numerically exact propagation of non-Markovian, driven dissipative trajectories.

3. [] Implement a Non-additive Geometric Phase (GP) Quantification algorithm that explicitly separates and quantifies the contributions of Floquet dressing, unitary evolution, and undriven dissipation to the total GP accumulation.

4. [] Develop a control architecture that maps external field parameters (amplitude/frequency, A/omega) to specific spectral regions of the environmental spectral density (J(ω)), enabling Spectral Steering of dissipative dynamics.

5. [] Create a high-dimensional parameter space explorer that systematically identifies regimes where the non-additive GP contribution is minimized or maximized, identifying conditions for GP protection.

) Capabilities of Improved AI Systems

The improved AI systems will be capable of performing the following specific tasks:

1. [] Simulate and predict the time evolution of quantum states in complex, periodically driven open quantum systems (like a spin-boson model) with high numerical accuracy, capturing non-Markovian effects over long timescales.

2. [] Analyze how external driving fields (frequency and amplitude) dynamically restructure the system's coherent dynamics into a set of Floquet channels that are specifically tailored to sample different regions of the environmental noise spectrum (spectral steering).

3. [] Determine precisely how the interplay between periodic driving and environmental dissipation leads to non-additive changes in geometric phases, allowing for a quantitative assessment of whether external driving helps or hinders the accumulation of phase information compared to separate unitary or dissipative evolution.

4. [] Predict optimal operating conditions (specific A/omega ratios and bath cutoff frequencies) that maximize the robustness (protection) of the system's geometric phase against environmental noise by steering it into spectral regions where dissipation is comparatively weak.

5. [] Provide a diagnostic tool to distinguish between the effects of coherent driving on system frequencies versus its effect on the coupling strength to specific bath modes, offering a deeper physical interpretation than current methods.