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arXiv:cond-mat/9605092 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Generalized Bloch equations for a strongly driven tunneling system

Peter Neu, Jochen Rau

Published Mar 1, 1997• 9 citations
arXivHTMLPDF

Abstract

Using the Robertson projection operator formalism, we derive generalized Bloch equations which describe the dynamics of a biased two-level tunneling system strongly driven by an external field and weakly coupled to a super-Ohmic heat bath. The generalized Bloch equations constitute a set of coupled nonlinear integro-differential equations. With their help we investigate the influence of phonons on the phenomenon of dynamical localization.

DOI
10.1103/physreve.55.2195

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Using the Robertson projection operator formalism, we derive generalized Bloch equations which describe the dynamics of a biased two-level tunneling system strongly driven by an external field and weakly coupled to a super-Ohmic heat bath. The generalized Bloch equations constitute a set of coupled nonlinear integro-differential equations. With their help we investigate the influence of phonons on the phenomenon of dynamical localization.
A detailed and unified account of the theory of the generalized Bloch equations is presented. The equations apply to a two-level system weakly coupled to a heat bath and subject to a monochromatic rotating field of arbitrary intensity. The relaxation tensor obtained is explicitly field-dependent. The derivation is valid for general coupling to a quantum heat bath. The generalized Bloch equations are shown to be thermodynamically consistent, as opposed to the standard Bloch equations. Different limits of the generalized Bloch equations are examined and related to previous studies. The potential use of the generalized Bloch equations as a probe of the bath spectral density is demonstrated for the case of a two-level system embedded in a Debye solid.
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
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.
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.
This paper investigates the dynamics of a two-level system coupled to a super-Ohmic boson bath, which is the so-called driven spin-boson model. Using the polaron transformation we derive the Bloch vector of a master equation where the bar tunneling parameter is used as the basis for a perturbation expansion. Our method is based on the canonical quantization provided by the Bloch-Redfield formalism. We apply quantum optimal control theory to the realization of a quantum ZZZ gate and also the coherent destruction of tunneling. To solve our optimal control problem, we use the techniques of automatic differentiation as an alternative to Pontryagin's minimum principle. The latter is less straightforward since the master equation is complex.
We consider one-dimensional chains of two-level quantum systems coupled via tunneling. The chain is driven by the superposition of dc and ac fields in the strong coupling regime. Based on the fundamental principles of electrodynamics and quantum theory, we have developed a generalized model of quantum dynamics for such interactions, free of rotating-wave approximation. The system of equations of motion was studied numerically. We analyzed the dynamics and spectra of the inversion density, dipole current density, and tunneling current density. In the case of resonant interaction with the ac component, the particle dynamics exhibits itself in the oscillatory regime, which may be interpreted as a combination of Rabi and Bloch oscillations with their strong mutual influence. Such scenario for an obliquely incident ac field dramatically differs from the individual picture of both types of oscillations due to the interactions. This effect is counterintuitive because of the existence of markedly different frequency ranges for such two types of oscillations. These dynamics manifest themselves in multiline spectra in different combinations of Rabi and Bloch frequencies. The effect is promising as a framework of a new type of spectroscopy in nanoelectronics and electrical control of nanodevices.