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Physical Review A

Perturbative dynamics in the pseudocoherent phase of the spin-boson model

Jiarui Zeng, Yao Yao

Published Jul 12, 2024• 2 citations
Publisher

Abstract

We propose a time-dependent perturbation theory to study a newly discovered pseudocoherent phase of the spin-boson model. Effectiveness of the method is confirmed by comparing its results with that calculated from the numerically exact quantum dynamics method, which exhibits good agreements. We then describe how an incoherent nature of the pseudocoherent phase emerges from explicit expressions of the proposed method, and properties of invariant time of extreme values and scaling of amplitudes resulting from a second-order approximation. Correction effects of higher-order terms are also discussed. In addition, we utilize trace distance to investigate non-Markovianity and influences of model parameters. We find propagations in the pseudocoherent phase are always non-Markovian because of bath-driving oscillations. The non-Markovianity shows different dependence on cutoff frequency of bosonic baths from coherent and incoherent phases, which can be applied for qualitative identifications of different dynamic phases in realistic detections.

DOI
10.1103/physreva.110.012212

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We propose a time-dependent perturbation theory to study a newly discovered pseudocoherent phase of the spin-boson model. Effectiveness of the method is confirmed by comparing its results with that calculated from the numerically exact quantum dynamics method, which exhibits good agreements. We then describe how an incoherent nature of the pseudocoherent phase emerges from explicit expressions of the proposed method, and properties of invariant time of extreme values and scaling of amplitudes resulting from a second-order approximation. Correction effects of higher-order terms are also discussed. In addition, we utilize trace distance to investigate non-Markovianity and influences of model parameters. We find propagations in the pseudocoherent phase are always non-Markovian because of bath-driving oscillations. The non-Markovianity shows different dependence on cutoff frequency of bosonic baths from coherent and incoherent phases, which can be applied for qualitative identifications of different dynamic phases in realistic detections.
Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the Ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by α\sqrtαα​, where ααα is the coupling strength to the environment. These calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.
The dynamics of a driven spin-boson model is studied by means of the perturbation approach based on a unitary transformation. Analytical expressions for the population difference and the coherence of the two state system are obtained. The results show that for weak driving, the population difference displays damping coherent oscillation and/or damping quantum beat, depending on the initial preparation. The coherence exhibits damped oscillation with Rabi frequency. When driving is strong enough, the population difference exhibits undamped large-amplitude coherent oscillation. In addition, our theory leads to correct results in two limiting cases: without dissipation and without driving field.
Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the Ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ is the coupling strength to the environment. We also explore the calculation of the next-order Born approximation: we show that, at the expense of very large computational complexity, interesting physical quantities can be rigorously computed at fourth order using computer algebra, presented completely in an accompanying MATHEMATICA file. We compute the $O(\ensuremath{\alpha})$ corrections to the long time behavior of the system density matrix; the result is identical to the reduced density matrix of the equilibrium state to the same order in $\ensuremath{\alpha}$. All these calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.
The non-Markovian dynamics of the driven spin-boson model at zero and finite temperature is investigated theoretically. Based on a unitary transformation and convolution theorem, the analytical expression of population difference in Laplace space is obtained. This method provides a tool to help us to understand the interference between driving and dissipation within the non-Markovian frame. We study the transient and the steady-state non-Markovian dynamics of population difference P ( t ) in the case of different temperature, driving and spin–bath coupling (SBC) strength. It has been shown that it is necessary to use non-Markovian methods when SBC is strong. At zero temperature, the curve of steady-state amplitude P lt versus Rabi driving amplitude Ω is N-like shaped. While at finite temperature, the curve becomes Λ-like shaped. These curves come from the interference between driving and dissipation. Also, from the interference pattern above, we know that large coherent oscillations can be obtained by optimizing the parameters: Rabi driving amplitude, temperature and SBC strength.
We investigate memory effects in the spin-boson model using a recently proposed measure for non-Markovian behavior based on the information exchange between an open system and its environment. Employing the numerical exact multilayer multiconfiguration time-dependent Hartree approach, we simulate the dynamics of the spin-boson model at zero temperature for a broad range of parameters. For a fast bath, i.e., in the scaling limit, we find non-Markovian dynamics for a coherently decaying spin at weak system-bath coupling, whereas memory effects are absent for stronger coupling in the regimes of incoherent decay and localization. If the timescales of system and bath are comparable, a complex, nonmonotonic dependence of non-Markovianity on the system-bath coupling strength is observed.
We develop a semianalytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($\ensuremath{\alpha}\ensuremath{\le}1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the noninteracting blip approximation that is a well-known good solution in this limit.