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arXiv:1710.06690

Self-to-self transitions in open quantum systems: the origin and solutions

Yaoxiong Wang, Ling Yang, Ying Wang, Shouzhi Li, Dewen Cao, Qing Zhong Gao, Feng Shuang, Gao, Fang

Published Oct 18, 2017• 0 citations
arXivHTMLPDF

Abstract

The information of quantum pathways can be extracted in the framework of the Hamiltonian-encoding and Observable-decoding method. For closed quantum systems, only off-diagonal elements of the Hamiltonian in the Hilbert space is required to be encoded to obtain the desired transitions. For open quantum systems, environment-related terms will appear in the diagonal elements of the Hamiltonian in the Liouville space. Therefore, diagonal encodings have to be performed to differentiate different pathways, which will lead to self-to-self transitions and inconsistency of pathway amplitudes with Dyson expansion. In this work, a well-designed transformation is proposed to avoid the counter-intuitive transitions and the inconsistency, with or without control fields. A three-level open quantum system is employed for illustration, and numerical simulations show that the method are consistent with Dyson expansion.

DOI
10.48550/arxiv.1710.06690

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The information of quantum pathways can be extracted in the framework of the Hamiltonian-encoding and Observable-decoding method. For closed quantum systems, only off-diagonal elements of the Hamiltonian in the Hilbert space is required to be encoded to obtain the desired transitions. For open quantum systems, environment-related terms will appear in the diagonal elements of the Hamiltonian in the Liouville space. Therefore, diagonal encodings have to be performed to differentiate different pathways, which will lead to self-to-self transitions and inconsistency of pathway amplitudes with Dyson expansion. In this work, a well-designed transformation is proposed to avoid the counter-intuitive transitions and the inconsistency, with or without control fields. A three-level open quantum system is employed for illustration, and numerical simulations show that the method are consistent with Dyson expansion.
The information of quantum pathways can be extracted in the framework of the Hamiltonian- encoding and Observable-decoding method. For dosed quantum systems, only off-diagonal elements of the Hamiltonian in the Hilbert space is required to be encoded to obtain the desired transitions. For open quantum systems, environment-related terms will appear in the diagonal elements of the Hamiltonian in the Liouville space. Thus diagonal encodings have to be performed to differentiate different pathways, which will lead to self-to-self transitions and inconsistency of pathway amplitudes with Dyson expansion. In this work, a well-designed transformation is proposed to avoid the counterintuitive transitions and the inconsistency, with or without control fields. Numerical simulations show that the method are consistent with Dyson expansion.
Control mechanism is usually much more complex for open quantum systems than closed systems. In this article, the Hamiltonian encoding-observable decoding method for open quantum systems is formulated, starting from the Lindblad equation. The dipole-field-induced and dissipation-induced quantum pathways in Liouville space are analyzed with this method for a single-qubit open system. Different signatures can be found in the decoding results, and the amplitudes of dipole-field-induced quantum pathways change with the control field while those of dissipation-induced quantum pathways are constant. Analysis with this method should be helpful to understand the control mechanism for open quantum systems.
Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz \cite{abhra_1}, we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.
The exact treatment of Markovian open quantum systems, when based on numerical diagonalization of the Liouville super-operator or averaging over quantum trajectories, is severely limited by Hilbert space size. Perturbation theory, standard in the investigation of closed quantum systems, has remained much less developed for open quantum systems where a direct application to the Lindblad master equation is desirable. We present such a perturbative treatment which will be useful for an analytical understanding of open quantum systems and for numerical calculation of system observables which would otherwise be impractical.
We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.
In this work, the Dyson's equation formalism is outlined and applied toseveral open quantum systems. These systems are composed of a core,quantum-mechanical set of discrete states and several continua, representing macroscopic systems. The macroscopic systems introducedecoherence, as well as allowing the total particlenumber in the system to change.Dyson's equation, an expansion in terms of proper self-energy terms, isderived. The hybridization of two quantum levelsis reproduced in this formalism, and it is shown that decoherence followsnaturally when one of the levels is replaced by a continuum.The work considers three physical systems in detail. The first,quantum dots coupled in series with two leads, is presented in a realistic two-level model. Dyson's equation is used to account for the leads exactly to all ordersin perturbation theory, and the time dynamics of a single electron in the dotsis calculated. It is shown that decoherence from the leads damps the coherentRabi oscillations of the electron. Several regimes of physical interest areconsidered, and it is shown that the difference in couplings of the two leadsplays a central role in the decoherence processes.The second system relates to the decay-out ofsuperdeformed nuclei. In this case, decoherence is provided by coupling to theelectromagnetic field. Two, three, and infinite-level models are consideredwithin the discrete system. It is shown that the two-level model is usuallysufficient to describe decay-out for the classic regions of nuclearsuperdeformation. Furthermore, a statistical model for the normal-deformedstates allows extraction of parameters of interest to nuclear structure fromthe two-level model. An explanation for the universality of decayprofiles is also given in that model.The final system is a proposed small molecular transistor. TheQuantum Interference Effect Transistor is based on a single monocyclic aromatic annulene molecule, with twoleads arranged in the meta configuration. This device is shown to be completely opaque to charge carriers, due to destructive interference. Thiscoherence effect can be tunably broken by introducing new paths with a real orimaginary self-energy, and an excellentmolecular transistor is the result.