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Science Technology and Engineering

Steady-state Evolution of Spin-boson Model of Single Model

Man Long

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The steady-state evolution of single-phonon spin-boson model is observed through the method of MATLAB numerical calculation,and analysed several evolutionary patterns while the coupling strength range from 0 to 2 as well as tunneling splitting equals 0.1.The result shows that the coupling strength has exerted a significant influence on states evolution,and there is an exclusive evolution discipline in a certain coupling strength interval,especially when the coupling strength is greater than 0.5 two-cycle phenomenon emerges.The average probability of each energy state in the time interval changes with the coupling strength is further calculated,and the result gives several evolution turning points.

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The steady-state evolution of single-phonon spin-boson model is observed through the method of MATLAB numerical calculation,and analysed several evolutionary patterns while the coupling strength range from 0 to 2 as well as tunneling splitting equals 0.1.The result shows that the coupling strength has exerted a significant influence on states evolution,and there is an exclusive evolution discipline in a certain coupling strength interval,especially when the coupling strength is greater than 0.5 two-cycle phenomenon emerges.The average probability of each energy state in the time interval changes with the coupling strength is further calculated,and the result gives several evolution turning points.
The existing studies of the temperature effect of the spin-boson model show that the tunneling splitting will increase with temperature, no matter how weak the couplings are between the bath and the spin. However, the small polaron theory said that in the weak coupling and low temperature regime, the tunneling is, in fact, dominated by the diagonal transitions whereas this diagonal contribution to the tunneling should be faded away with increasing temperature. Taking advantage of the analysis originated from the Feynman-Vernon's influence functional theory, the influence on the tunneling by the phonon bath can be expressed as a product of the effects of every single phonon mode, which can be studied by numerical exact diagonalization. We find that, in the weak coupling and low temperature regime, all the spin-single-mode systems exhibit the same decreasing dependence of tunneling on increasing temperature. In turn, with the conclusion of influence functional theory, the tunneling of the spin-boson model should decrease and it is independent of the bath structure. In the strong coupling regime, however, the temperature effect reverses from suppressing to enhancing the tunneling with the increase of temperature. Discrepancies between the old theories and the small polaron theory are also explained.
We present a systematic numerical iteration approach to study the evolution properties of the spin-boson systems, which works well in whole coupling regime. This approach involves the evaluation of a set of coefficients for the formal expansion of the time-dependent Schrödinger equation ∣t⟩=e−iH^t∣t=0⟩\vert t\rangle=e^{-i\hat{H}t}\vert t=0\rangle∣t⟩=e−iH^t∣t=0⟩ by expanding the initial state ∣t=0⟩\vert t=0\rangle∣t=0⟩ in Fock space. The main advantage of this method is that this set of coefficients is unique for the Hamiltonian studied, which allows one to calculate the time evolution based on the different initial states. To complement our numerical calculations, the method is applied to the Buck-Sukumar model. Furthermore, we pointed out that, when the ground state energy of the model is unbounded and no ground state exists in a certain parameter space, the unstable time evolution of the physical quantities is the natural results. Furthermore, we test the performance of the numerical method to the Hamiltonian that use anti-Hermitian terms for modeling open quantum systems.
The influence of dissipation on quantum tunneling in the spin-boson model with a sub-Ohmic bath is studied by a variational calculation. By examining the evolution of solutions of the variational equation with the coupling strength near the phase boundary, we are able to present a scenario of discontinuous transition in sub-Ohmic dissipation case in accord with Ginzburg-Landau theory. Based on the constructed picture, it is shown that the critical point found in the general way is not thermodynamically the critical point, but the point where the second energy minimum begins to develop. The true cross-over point is calculated and the obtained phase diagram is in agreement with the result of numerical renormalization group calculation.
Abstract We study the evolution properties of spin-boson systems by a systematic numerical iteration approach, which performs well in the whole coupling regime. This approach evaluates a set of coefficients in the formal expression of the time-dependent Schrödinger equation by expanding the initial state in Fock space. This set of coefficients is unique for the spin-boson Hamiltonian studied, allowing one to calculate the time evolution from different initial states. To complement our numerical calculations, we apply the method to the Buck–Sukumar model. We find that when the ground-state energy of the model is unbounded and no ground state exists in a certain parameter space, the time evolution of the physical quantities is naturally unstable.
When the coupling of a quantum system to its environment is nonnegligible, its steady state is known to deviate from the textbook Gibbs state. The Bloch-Redfield quantum master equation, one of the most widely adopted equations for solving the open quantum dynamics, cannot predict all the deviations of the steady state of a quantum system from the Gibbs state. In this paper, for a generic spin-boson model, we use a higher-order quantum master equation (in system-environment coupling strength) to analytically calculate all the deviations of the steady state of the quantum system up to second order in coupling strength. We also show that this steady state is exactly identical to the corresponding generalized Gibbs state, the so-called quantum mean force Gibbs state, at arbitrary temperature. All these calculations are highly general, making them immediately applicable to a wide class of systems well modeled by the spin-Boson model, spanning a diverse range of topics, from nanomaterials to various condensed-phase processes, and quantum computing (e.g., environment-induced corrections to the steady state of a superconducting qubit). As an example, we use our results to study the dynamics and the steady state of a solid-state double-quantum-dot system under physically relevant choices of parameters.