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arXiv:cond-mat/9910359 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Driven tunneling dynamics: Bloch-Redfield theory versus path-integral approach

Ludwig Hartmann, Igor Goychuk, Milena Grifoni, Peter Hänggi A

Published May 1, 2000• 93 citations
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Abstract

In the regime of weak bath coupling and low temperature we demonstrate numerically for the spin-boson dynamics the equivalence between two widely used but seemingly different roads of approximation, namely, the path-integral approach and the Bloch-Redfield theory. The excellent agreement between these two methods is corroborated by an efficient analytical high-frequency approach: it well approximates the decay of quantum coherence via a series of damped coherent oscillations. Moreover, a suitably tuned control field can selectively enhance or suppress quantum coherence.

DOI
10.1103/physreve.61.r4687

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In the regime of weak bath coupling and low temperature we demonstrate numerically for the spin-boson dynamics the equivalence between two widely used but seemingly different roads of approximation, namely, the path-integral approach and the Bloch-Redfield theory. The excellent agreement between these two methods is corroborated by an efficient analytical high-frequency approach: it well approximates the decay of quantum coherence via a series of damped coherent oscillations. Moreover, a suitably tuned control field can selectively enhance or suppress quantum coherence.
Quantum information processing has greatly increased interest in the phenomenon of environmentally-induced decoherence. The spin boson model is widely used to study the interaction between a spin-modelling a quantum particle moving in a double well potential-and its environment-modelled by a heat bath of harmonic oscillators. This paper extends a previous analysis of the static spin boson study to the driven spin boson case, with the derivation of an exact integro-differential equation for the time evolution of the propagator of the reduced spin density matrix. This is the first main result. By specializing to weak damping we then obtain the next result, a set of Bloch-Redfield equations for the equilibrium fixed spin initial condition. Finally we show that these equations can be used to solve the classic dissipative Landau-Zener problem and illustrate these solutions for the weak damping case. The effect of dissipation is seen to be minimised as the speed of passage is increased, implying that qubits need to be switched as fast as possible.
We present a novel approximation scheme to describe the influence of a harmonic bath on the dynamics of a two-level particle over almost the whole regime of temperatures and coupling to the environment, for a wide class of bath spectral densities. Starting from the exact path integral solution for the two-level system density matrix, effective intra-blip correlations are fully included, while inter-blip and blip-sojourn interactions are considered up to first order. In the proper regimes, an excellent agreement with conventional perturbative approaches and ab initio path-integral results is found.
The dissipative dynamics of a quantum bistable system coupled to a Ohmic heat bath is investigated beyond the spin-boson approximation. Within the path-integral approach to quantum dissipation, we propose an approximation scheme which exploits the separation of time scales between intra- and interwell (tunneling) dynamics. The resulting generalized master equation for the populations in a space localized basis enables us to investigate a wide range of temperatures and system-environment coupling strengths. A phase diagram in the coupling-temperature space is provided to give a comprehensive account of the different dynamical regimes.
The dynamics of a qubit under the decoherence of a two level fluctuator (TLF) in addition to its coupling to a bosonic bath is investigated theoretically. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. With the merit of our unitary transformation, non-adiabatic effect can be taken into account efficiently. And the other one is the numerically exact method, namely the quasi-adiabatic path-integral (QUAPI) propagator technique. We find that the analytical method works well for a wide parameter range and show good agreement with QUAPI. On the other hand, The enhancement and the reduction of quantum decoherence of the qubit is checked with varying bath temperature TTT and TLF-bath coupling.
We study the time evolution of the reduced density matrix for the Ohmic spin boson model out of an uncorrelated but otherwise arbitrary initial state. We consider arbitrary bias $\ensuremath{\epsilon}$ and tunneling $\mathrm{\ensuremath{\Delta}}$ at zero temperature for a weak coupling $\ensuremath{\alpha}$ to the bosonic bath. Using the real-time renormalization group method, we present a consistent weak-coupling expansion one order beyond the Bloch-Redfield approximation within a renormalized perturbation theory with analytical results covering the whole crossover regime from small times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\ll}1$ to large times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\gg}1$, where $\mathrm{\ensuremath{\Omega}}=\sqrt{{\ensuremath{\epsilon}}^{2}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}^{2}}$ denotes the Rabi frequency in terms of the renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$. In addition, for exponentially small or large times, we perform a nonperturbative resummation of all logarithmic terms. We show that standard Born approximation schemes calculating the effective Liouvillian of the kinetic equation up to first order in $\ensuremath{\alpha}$ are not sufficient to account for various important corrections one order beyond the Bloch-Redfield solution. (1) The resummation of all secular terms $\ensuremath{\sim}{(\mathrm{\ensuremath{\Gamma}}t)}^{n}$ is necessary to obtain the correct exponential decay of all terms of the time evolution with decay rate $\mathrm{\ensuremath{\Gamma}}$ or $\mathrm{\ensuremath{\Gamma}}/2$, together with the correct pre-exponential functions. (2) The resummation of all logarithmic terms at high and low energies leads to a renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$ and to pre-exponential functions of logarithmic and power-law form. (3) The fact that two eigenvalues of L(E)L(E)L(E) are close to each other by $O(\mathrm{\ensuremath{\Gamma}})$ requires degenerate perturbation theory for times $\mathrm{\ensuremath{\Gamma}}t\ensuremath{\sim}O(1)$, where certain terms of the Liouvillian in $O({\ensuremath{\alpha}}^{2})$ are needed to calculate the stationary state and the time evolution of the nonoscillating purely decaying modes up to $O(\ensuremath{\alpha})$. In contrast to the zero-bias case, we find two further interesting results for the time dynamics of the oscillating modes. (4) The terms of the pre-exponential functions with a strong time dependence show a leading long-time tail $\ensuremath{\sim}\ensuremath{\alpha}/(\mathrm{\ensuremath{\Omega}}t)$, besides other subleading terms $\ensuremath{\sim}\ensuremath{\alpha}/{(\mathrm{\ensuremath{\Omega}}t)}^{2}$ well-known from the zero-bias case. (5) The terms of the pre-exponential functions with a weak (logarithmic) time dependence vary according to a power law ${(\frac{1}{\mathrm{\ensuremath{\Omega}}t})}^{2\ensuremath{\alpha}\frac{{\ensuremath{\epsilon}}^{2}}{{\mathrm{\ensuremath{\Omega}}}^{2}}}$ for exponentially large times. The power-law exponent depends on the bias and has to be contrasted to the one at exponentially small times where it crosses over to the bias-independent result $2\ensuremath{\alpha}$. We discuss that the complexity to calculate one order beyond Bloch-Redfield approximation is rather generic and applies also to other models of dissipative quantum mechanics.