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arXiv:2501.03760 · New Journal of Physics

Dissipative evolution of a two-level system through a geometry-based classical mapping

D. Gil, Pedro Bargueño, Salvador Miret-Artés

Published Dec 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract In this manuscript, we introduce a geometry-based formalism to obtain a Meyer–Miller–Stock–Thoss mapping in order to study the dynamics of both isolated and interacting two-level systems (TLS). After showing the description of the isolated case using classical-like canonically conjugate variables, we implement an interaction model by bilinearly coupling the corresponding population differences à la Caldeira–Leggett, obtaining a Gross–Pitaevskii-like equation. This allows us to analytically find a transition between oscillatory and tunneling block dynamics by varying the coupling constant. Extending our model to the system plus environment case, where the environment is considered as a collection of TLSs, we reveal that the dynamics become chaotic, exhibiting tunneling-suppressed dynamics in the strong coupling limit and the usual damping effect similar to that of a harmonic oscillator bath in the weak coupling one. Finally, we observe that our interacting model turns an isolated symmetric TLS into an environment-assisted asymmetric one.

DOI
10.1088/1367-2630/ae2989

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Abstract In this manuscript, we introduce a geometry-based formalism to obtain a Meyer–Miller–Stock–Thoss mapping in order to study the dynamics of both isolated and interacting two-level systems (TLS). After showing the description of the isolated case using classical-like canonically conjugate variables, we implement an interaction model by bilinearly coupling the corresponding population differences à la Caldeira–Leggett, obtaining a Gross–Pitaevskii-like equation. This allows us to analytically find a transition between oscillatory and tunneling block dynamics by varying the coupling constant. Extending our model to the system plus environment case, where the environment is considered as a collection of TLSs, we reveal that the dynamics become chaotic, exhibiting tunneling-suppressed dynamics in the strong coupling limit and the usual damping effect similar to that of a harmonic oscillator bath in the weak coupling one. Finally, we observe that our interacting model turns an isolated symmetric TLS into an environment-assisted asymmetric one.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS–HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes–Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are\nfound: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.
The dynamical behaviors of a two-level system (TLS) coupled to a harmonic dissipative bath has been studied extensively using a variety of analytical and numerical methods. The focus of the vast majority of these studies has been on the properties of the TLS, averaged with respect to the bath degrees of freedom. In this work, we use real-time path integral methods to probe the behavior of select bath degrees of freedom during the dynamics of a symmetric two-level system (TLS) coupled to a dissipative bath by calculating system-bath densities (SBD) and coordinate expectation values. Overall, the SBD motion on each diabatic state is simpler than the motion of the total density. In the weak coupling regime, which characterizes the parameters of oscillators that comprise such a bath, the SBD on each TLS state remains primarily compact and Gaussian-like, such that its peak is well characterized by the mode expectation value. In the absence of a dissipative environment, nonadiabatic density depletion leads to spikes in coordinate expectation values. The evolution of the SBD peak trajectory for two discrete modes exhibits Lissajous patterns with frequency-dependent shapes that strongly resemble classical trajectory motion on a torus. These patterns become more complex when the coupling of the mode to the TLS is increased outside of this regime, leading to persistent small amplitude oscillations in the TLS populations characterized by a very slow decay and SBD trajectories that exhibit behaviors reminiscent of chaotic classical systems. Indirect coupling to a dissipative bath has a stabilizing effect on the dynamics, eliminating spikes, synchronizing the SBD motion on the two diabatic states and regularizing the SBD trajectory to simple rectangular Lissajous-like shapes with a slowly shrinking boundary, regardless of the mode frequencies.
In this manuscript, starting from a five-step algorithmic procedure for exactly classicalizing the dynamics of N -level quantum systems, we incorporate a classical bath of harmonic oscillators to model environmental interactions. Using the geometry of complex projective spaces CP N − 1 and a Langevin formalism, we obtain N − 1 Hamilton’s equations which encode both the quantum system and the bath degrees of freedom, representing a generalization of the Caldeira–Leggett model in a complex projective space. We demonstrate the efficacy of the method by applying it to two paradigmatic systems: a two-qubit system in CP 3 under entangling interactions, reproducing quantum observables such as state populations, quaternionic population differences and concurrence, and the seven-state Fenna–Matthews–Olson (FMO) complex in CP 6 , reproducing state populations in the picosecond timescale.
The dynamical behavior of a two-level system coupled linearly to a dissipative bath of two-level systems whose tunneling splittings and coupling parameters are distributed according to an Ohmic spectrum is studied using numerical path integral methods as well as the noninteracting blip approximation for an equivalent harmonic environment characterized by a scaled spectral density. Comparisons with the behaviors of the conventional spin-boson model of similar coupling strength reveal a weaker quenching of coherence in the present case. Finally, the boundary in parameter space between coherent and incoherent dynamics is mapped.
The dissipative quantum dynamics of a two-level system interacting with a structured reservoir consisting of a damped harmonic mode is investigated by means of the numerically exact path integral Monte Carlo method. This approach provides benchmark results in a broad range of parameter space, in particular, in those domains which are not accessible by approximate methods and alternative numerical schemes, that is, strong coupling between system and harmonic mode and from low to high temperatures. For low temperatures the numerical data are quantitatively in agreement with the noninteracting blip approximation only in the regimes of very weak and very strong coupling. It turns out that the entangled dynamics of the two-level system and the harmonic mode is relatively robust so that its signatures are observable up to relatively strong friction and high temperatures. Nonequilibrium initial preparations of the reservoir with respect to the initial state of the system give rise, for strong interactions, to a stepwise decay of the population, thus displaying coherent dynamics of the bath. The impact of an additional Ohmic bath coupled directly to the two-level system is studied as well, including the case where both reservoirs carry different temperatures.