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arXiv:1906.06404 · Physical Review A

Geometric decoherence in diffusive open quantum systems

Da-Wei Luo, Hai‐Qing Lin, J. Q. You, Lian-Ao Wu, Rupak Chatterjee, Ting Yu

Published Dec 10, 2019• 6 citations
arXivHTMLPDF

Abstract

Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing nonunitary, noncyclic, and nonadiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realization of quantum gates due to its robustness against dynamical errors; however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a nonperturbative control scheme.

DOI
10.1103/physreva.100.062112

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Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing nonunitary, noncyclic, and nonadiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realization of quantum gates due to its robustness against dynamical errors; however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a nonperturbative control scheme.
We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.
Nonadiabatic geometric quantum computation in decoherence-free subspaces has received increasing attention due to the merits of its high-speed implementation and robustness against both control errors and decoherence. However, all the previous schemes in this direction have been based on the conventional geometric phases, of which the dynamical phases need to be removed. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, of which the dynamical phases do not need to be removed. Specifically, by using three physical qubits undergoing collective dephasing to encode one logical qubit, we realize a universal set of geometric gates nonadiabatically and unconventionally. Our scheme not only maintains all the merits of nonadiabatic geometric quantum computation in decoherence-free subspaces, but also avoids the additional operations required in the conventional schemes to cancel the dynamical phases.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Trapped ions are among the most promising candidates for performing quantum information processing tasks. Recently, it was demonstrated how the properties of geometric phases can be used to implement an entangling two-qubit phase gate with significantly reduced operation time while having a built-in resistance against certain types of errors [M. Palmero et al., Phys. Rev. A 95, 022328 (2017)]. We investigate the influence of both quantum and thermal fluctuations on the geometric phase in the Markov regime. We show that additional environmentally induced phases as well as a loss of coherence result from the nonunitary evolution, even at zero temperature. We connect these effects to the associated dynamical and geometrical phases. This suggests a strategy to compensate for the detrimental environmental influences and restore some of the properties of the ideal implementation. Our main result is a strategy for zero temperature to construct forces for the geometric phase gate which compensate for the dissipative effects and leave the produced phase as well as the final motional state identical to the isolated case. We show that the same strategy also helps at finite temperatures. Furthermore, we examine the effects of dissipation on the fidelity and the robustness of a two-qubit phase gate against certain error types.
Quantum gates constructed by geometric phase are naturally robust to control errors due to the global nature of the geometric evolution path. Therefore, how to cope with the inevitable decoherence errors is worthy of serious attention for geometric quantum computation. Different from conventional nonadiabatic geometric quantum computation (NGQC), which needs the same evolution time for any geometric rotation angle, we have proposed and experimentally demonstrated nonadiabatic noncyclic geometric quantum computation (NNGQC) without the requirement of the cyclic evolution condition, which thus reduces decoherence errors due to shortened evolution time [J. W. Zhang et al., Phys. Rev. Lett. 127, 030502 (2021)]. In addition, the use of decoherence-free subspaces (DFS) is another effective strategy to protect quantum gates from dephasing error and has been studied in nonadiabatic holonomic quantum computation [G. F. Xu et al., Phys. Rev. Lett. 109, 170501 (2012)]. Motivated by previous studies, we propose a one-step scheme to implement single-qubit NNGQC operation in DFS constructed by two Rydberg atoms, and then generalize the scheme to the two-logical-qubit case in one step. The numerical results show that our scheme is robust against decoherence in contrast to the NGQC counterparts. The present scheme combines the features of NNGQC and DFS and can further reduce the impact of decoherence on gate infidelity, which may provide an option for the realization of Rydberg-atom-based quantum computation in the future.
Geometric phases have been shown to be feasible in implementing quantum gates to perform quantum information processing. For all the realistic applications, the environmental influence on the geometric phase and decoherence such as memory effects must be properly considered in order to achieve the required precision in geometric quantum computation. In this paper, we study the geometric phase for a generic open quantum system based on a microscopic model. A remarkable feature of the open system's geometric phase obtained from our theoretical formulation is that the geometric phase can be obtained regardless of the existence of the master equations, while the environmental noise features such as memory effects are fully accounted for. We demonstrate that the geometric phases for a general open quantum system can be fundamentally modified by its non-Markovian environments.