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Chinese Control Conference (CCC)

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

Fang Gao, Yaoxiong Wang, Qing Zhong Gao, Dewen Cao, Feng Shuang

Published Jul 1, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.23919/ccc50068.2020.9189589

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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.
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.
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 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.
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.
In an open quantum system, the approach to some steady (equilibrium) state is controlled by the coupling of the system to its environment. In a system with a fixed number of particles, this coupling typically makes the steady-state occupation numbers for the energy eigenstates of the system follow the Maxwell-Boltzmann distribution, where the temperature is set by the properties of the environment. Along with the approach to the equilibrium, the coupling projects the system to a diagonal state in its energy eigenbasis. This dephasing makes the system forget all the information of the quantum coherences between the different energy states.
We report on the laboratory implementation of quantum-control-mechanism identification through Hamiltonian encoding and observable decoding (HE-OD). Over a sequence of experiments, HE-OD introduces a special encoded signature into the components of a previously determined control field expressed in a chosen representation. The outcome appears as a modulated signal in the controlled system observable. Decoding the modulated signal identifies the hierarchy of correlations between components of the control field in a particular representation. In cases where the initial quantum state and observable operator are fully known, then HE-OD can also identify the transition amplitudes of the various Dyson expansion orders contributing to the controlled dynamics. The basic principles of HE-OD are illustrated for second harmonic generation when the components of the field representation are simply taken as the pixels in the pulse shaper. The outcome of HE-OD agrees well with simulations, verifying the concept.