Leggett-Garg Inequality for a Two-Level System under Decoherence Nasim Shahmansoori 1, ∗ and Afshin Shafiee 1, 2, † 1 Research Group on Foundations of Quantum Theory and Information, Department of Chemistry, Sharif University of Technology P.O.Box 11365-9516, Tehran, Iran 2 School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531 Tehran, Iran We consider a macroscopic quantum system subjected to an asymmetric double-well potential in a harmonic environment. By using a time-dependent approach, we calculate tunneling probabilities for the system which contain oscillation effects. To show how one can decide between quantum me- chanics and the implications of macrorealism assumptions, a given form of Leggett-Garg inequality is considered. The violation of this inequality occurs for a broader range of the system-environment interactions, compared to previous results obtained for two-level systems. Assuming that the cou- pling strength between the system and the environment can be controlled with time, one can see the violation even for strong decoherence effects. We also investigate the variation of the tilt/tunneling parameters on the violation of Leggett-Garg inequality. PACS numbers: 03.65.Xp, 03.65.Ta, 03.65.Yz INTRODUCTION Extrapolating the laws of quantum mechanics (QM ), up to the scale of everyday objects, means that objects composed of many atoms exist in quantum superpositions of macroscopically distinct states. In 1935, Schrodinger attempted to demonstrate the counter-intuitive implications of QM using a thought experiment in which a cat is put in a quantum superposition of alive and dead states [1]. The idea remained theoretical until 1980s, when much progress has been made in demonstrating the macroscopic quantum behavior of various systems such as superconductors [2–5], nanoscale magnets [6, 7], laser-cooled trapped ions [8], photons in a microwave cavity [9] and macromolecules [10]. A typical double-well potential system provides a unique opportunity to study the fundamental behavior of a macroscopic quantum system (MQS), specially macroscopic quantum tunneling. In the context of a double-well potential, Schrodinger’s cat describes a state in which macroscopic system (macrosystem) simultaneously occupies both wells. There are also studies focused on decoherence effects in double-well potentials. Huang et al. showed that decoherence due to the interactions of atoms with the electromagnetic vacuum can cause the defeating of Schrodinger cat-like states [11]. Thermal effects [12] and dissipation [13] constitute some sources of decoherence and can suppress tunneling between wells [14, 15]. In addition, double-well potentials have been extensively applied in many branches of physics. For example, it appears in the dynamics of Bose-Einstein condensates, the recent developments of ion trap technology, the ultracold trapped atoms theory and its applications [16–20]. Such a situation brings in mind the question of how the everyday macroscopic world works. The Leggett-Garg inequality (LGI) provides a criterion to investigate the existence of macroscopic coherence and thereby test the applicability of QM as we scale from the micro- to the macro-world [21, 22]. LGIs are based on two assumptions, macroscopic definiteness and noninvasive measurability. Violation of LGI implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. QM violates different forms of LGIs. A number of experimental tests and violations of these inequalities have been demonstrated in recent years [23, 24]. Leggett and Garg initially proposed an rf-SQUID flux qubit as a promising system to test their inequalities [22], which was later improved by Tesche [25]. The first measured violation of a type of LGI was reported by Palacious-Laloy et al. [26]. Palacios-Laloy et al. found that LGI is violated by their qubit with the conclusion that their system could not admit a realistic, non-invasively-measurable description. Recently, several experimental tests of LGIs were implemented, all of which confirm the predicted violations in accordance with the fundamental laws of QM [26–33]. Most of these experiments were weak measurements, where the effects of the measured back-action in a sequential set up are minimized [34]. In this article we examine LGI regarding an asymmetric double-well potential in a harmonic environment. To do this, we consider the effect of the environment as a perturbation on the system. For symmetric double-well potentials considered as a two-level quantum system, it has been shown that QM violates different forms of LGI [35]. Moreover, no violation occurs, when strong decoherence is at work. According to our calculations for an asymmetric double-well potential, it is possible to see the violation, even for significant effects of the decoherence. This can be achieved by controlling the strength of interaction between the system and it’s environment in different time domains. We also study the effects of the tilt/tunneling parameters on the violation of LGI. The structure of our paper is as follows. In section 2, we focus on an asymmetric double-well model. We introduce arXiv:1605.00394v3 [quant-ph] 1 Aug 2018