PHYSICAL REVIEW A 91, 022116 (2015) Decoherence speed limit in the spin-deformed boson model Sh. Dehdashti, 1, 2, 3 , * M. Bagheri Harouni, 4 , † B. Mirza, 3 , ‡ and H. Chen 1, 2 , § 1 State Key Laboratory of Modern Optical Instrumentations, Zhejiang University, Hangzhou 310027, China 2 The Electromagnetics Academy at Zhejiang University, Zhejiang University, Hangzhou 310027, China 3 Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran 4 Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib Street, Isfahan, 81746-73441, Iran (Received 6 November 2014; published 18 February 2015) In this paper, we study the role of the nonlinear environment on the bound passage time of dynamical quantum spin systems, which is of great interest in quantum control and has been applied to quantum metrology, quantum computation, and quantum chemical dynamics. We consider the decoherence speed limit for the spin-deformed bosonic model and the impacts of the nonlinear environment and its temperature on the decoherence speed limit. Moreover, we show that, at an early enough time, the parameters associated with the nonlinear environment exhibit important roles in controlling the decoherence process. In addition our results reveal that, in long times, these parameters do not affect the decoherence process. DOI: 10.1103/PhysRevA.91.022116 PACS number(s): 03.65.Yz, 03.67.−a I. INTRODUCTION Quantum mechanics as a fundamental law of nature im- poses limits to the evolution speed of quantum systems. Nowa- days, these limits have found remarkable roles in different scenarios including quantum communication, identification of precision bounds in quantum metrology, formulation of computational limits of physical systems, and development of quantum optimal control algorithms [1–5]. In fact, quantum mechanics acts as a legislative body that imposes speed limits, on the one hand, as a fundamental problem and, on the other hand, when considering the effects of environment on the evolution of quantum systems. In the first instance, as a fundamental problem, the quantum speed limit is imposed as a bound on the speed of evolution which is intimately related to the concept of passage time, τ min . This is the time required for a given pure state |χ 〉 to become orthogonal to itself under unitary dynamics [6]. Also, earlier studies have indicated that the passage time, τ min , can be the lower bound by the inverse of the variance in the energy of the system, i.e., τ min  π 2  H , (1) where H = ( 〈H 2 〉−〈H 〉 2 ) 1/2 , whenever the dynamics un- der study is governed by a Hermitian Hamiltonian, H [7–13]. If the passage time problem is considered as a quantum brachistochrone problem, it has been shown that, whenever the Hamiltonian is non-Hermitian PT -symmetric, the passage time can be made arbitrarily small without violating the time-energy uncertainty principle [14–17]. On the other hand, an analogous bound has been considered for some open quantum systems [18–29], since all systems are ultimately coupled to an environment [30–34]. In these cases, such a bound on the evolution of an open system would * shdehdashti@zju.edu.cn † m-bagheri@phys.ui.ac.ir ‡ b.mirza@cc.iut.ac.ir § hansomchen@zju.edu.cn help to address the robustness of the quantum system which is applied, for example, to simulators and computers against decoherence [35]. Therefore, this quantity (bound of speed limit) requires the effects associated with the environment to be quantified. In this case, the role of nonlinearity, such as con- finement and curvature as well as temperature, on the bound passage time is desirable. This motivates one to investigate the decoherence mediated by a structured environment through the passage time. In fact, the present contribution studies the impacts of temperature and nonlinearity of environment on the bound passage time for the spin system in contact with these environments. Along these lines we study, in this paper, the bound passage time τ ζ in a spin system, as a quantum system interacting with the deformed harmonic oscillators, as a nonlinear boson environment. In fact, the spin-boson model is one of the most important physical systems for both its theoretical aspects and its applications. With respect to theoretical aspects, the spin-boson model exhibits features characteristic of the decoherence process. Thus, it is an ideal candidate for the study of decoherence in two-level systems. The spin-boson model describes a single two-level system interacting with a large reservoir of boson field modes [36–39], i.e., a spin-1/2 particle coupled to an environment, which can be formulated by harmonic oscillators because of the central limit theorem [40,41]. This model has been widely studied in the context of decoherence and the dissipation process in quantum systems [31,42]. Also, the role of two-level (qubit) systems in quantum computing [31] and in experiments dealing with macroscopic quantum coherence has led to additional interest in the spin-boson model [43]. Two-level systems are also believed to be found in many amorphous materials [43,44] while the spin-boson model has been employed for some kinds of chemical reaction and the motion of defects in some crystalline solids and for analyzing the role of quantum decoherence in biological systems [45]. In addition, the effect of the nonlinear environment on the decoherence rate of a spin-boson model has been recently studied [46]. As already mentioned above, we study the effects of nonlinearity and temperature of the environment on the bound passage time of a spin system. We show that the bound 1050-2947/2015/91(2)/022116(7) 022116-1 ©2015 American Physical Society