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World Congress on Intelligent Control and Automation

Hamiltonian encoding-observable decoding methods for open quantum systems: Application to a single-qubit system

Fang Gao, Yaoxiong Wang, Xubing Tang, Lingmin Wei, Yi Zhao, Feng Shuang

Published Jun 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.1109/wcica.2014.7052894

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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.
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.
Many systems under control with an applied field also interact with the surrounding environment. Understanding the control mechanisms has remained a challenge, especially the role played by the interaction between the field and the environment. In order to address this need, here we expand the scope of the Hamiltonian-encoding and observable-decoding (HE-OD) technique. HE-OD was originally introduced as a theoretical and experimental tool for revealing the mechanism induced by control fields in closed quantum systems. The results of open-system HE-OD analysis presented here provide quantitative mechanistic insights into the roles played by a Markovian environment. Two model open quantum systems are considered for illustration. In these systems, transitions are induced by either an applied field linked to a dipole operator or Lindblad operators coupled to the system. For modest control yields, the HE-OD results clearly show distinct cooperation between the dynamics induced by the optimal field and the environment. Although the HE-OD methodology introduced here is considered in simulations, it has an analogous direct experimental formulation, which we suggest may be applied to open systems in the laboratory to reveal mechanistic insights.
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.
While closed-loop control of quantum dynamics in the laboratory is proving to be broadly successful, the control mechanisms induced by the fields are often left obscure. Hamiltonian encoding (HE) was originally introduced as a method for understanding mechanisms in quantum dynamics in the context of computational simulations, based on access to the system wavefunction. As a step towards laboratory implementation of HE, this paper addresses the issues raised by the use of observables rather than the wavefunction in HE. The goal of laboratory based HE is to obtain an understanding of control mechanism through a sequence of systematic control experiments, whose collective information can identify the underlying control mechanism defined as the set of significant amplitudes connecting the initial and final states. Mechanism is determined by means of observing the dynamics of special sequences of system Hamiltonians encoded through the control field. The proposed algorithm can handle complex systems, operates with no recourse to dynamical simulations, and functions with limited understanding of the system Hamiltonian. As with the closed-loop control experiments, the HE control mechanism identification algorithm performs a new experiment each time the dynamical outcome from an encoded Hamiltonian is called for. This paper presents the basic HE algorithm in the context of physical systems described by a finite dimensional Hilbert space. The method is simulated with simple models, and the extension to more complex systems is discussed.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
Optimal control of quantum dynamics in the laboratory is proving to be increasingly successful. The control fields can be complex, and the mechanisms by which they operate have often remained obscure. Hamiltonian encoding (HE) has been proposed as a method for understanding mechanisms in quantum dynamics. In this context mechanism is defined in terms of the dominant quantum pathways leading to the final state of the controlled system. HE operates by encoding a special modulation into the Hamiltonian and decoding its signature in the dynamics to determine the dominant pathway amplitudes. Earlier work encoded the modulation directly into the Hamiltonian operators. This present work introduces the alternative scheme of field based HE, where the modulation is encoded into the control field and not directly into the Hamiltonian operators. This distinct form of modulation yields a new perspective on mechanism and is computationally faster than the earlier approach. Field based encoding is also an important step towards a laboratory based algorithm for HE as it is the only form of encoding that may be experimentally executed. HE is also extended to cover systems with noise and uncertainty and finally, a hierarchical algorithm is introduced to reveal mechanism in a stepwise fashion of ever increasing detail as desired. This new hierarchical algorithm is an improvement over earlier approaches to HE where the entire mechanism was determined in one stroke. The improvement comes from the use of less complex modulation schemes, which leads to fewer evaluations of Schrodinger's equation. A number of simulations are presented on simple systems to illustrate the new field based encoding technique for mechanism assessment.