Squeezing of a two-level atom off-resonantly coupled to a non-Markovian reservoir is investigated. We compared two different definitions of squeezing that are based on the Heisenberg uncertainty relation and the entropy uncertainty relation and found that information entropy squeezing is more precise than variance squeezing as a measure of squeezing. Then we utilized entropy squeezing to examine the influences of the non-Markovian effect and detuning on the atomic squeezing. It is interesting to note that atomic squeezing can be protected for a long time when both the non-Markovian effect and detuning are present simultaneously. The physical mechanism is that the decay rate is suppressed much more by detuning in the non-Markovian regime than in the Markovian case.

We explore the possibility of implementing a Heisenberg-limited Mach-Zehnder interferometry via an array of trapped ions, which obey a quantum Ising model within a transverse field. Based upon adiabatic processes of increasing the Ising interaction and then decreasing the transverse field, we demonstrate a perfect transition from paramagnetism to ferromagnetic states, which can be used as the beam splitter for the multi-ion Mach-Zehnder interferometry. The achieved NOON state of the ions enables the Heisenberg-limited interferometry. Using currently available techniques for ultracold ions, we discuss the experimental feasibility of our scheme with global operations.PACS numbers: 03.75. Dg, 03.67.Ac, 37.10.Ty Since the foundation of quantum theory, as a natural result of superposition principle, quantum interference has attracted continuous attentions in both theoretical and experimental studies. It has been widely used to implement high-precision measurement, quantum information processing and so on. A well-known scheme for performing quantum interferometry is the Mach-Zehnder (MZ) interferometry, which has a beam splitter for splitting the input states and another beam splitter for recombining the output states. Up to now, quantum MZ interferometry has been accomplished via photons Beyond the conventional quantum interferometry via unentangled states, it has been demonstrated that the measurement precision can be enhanced from the standard quantum limit (or the shot noise limit) to the Heisenberg limit by using multipartite entangled states [6]. An excellent candidate is the NOON state2, which is an equal-probability superposition of all N particles in one of two paths denoted by a and b. The entangled ions for highprecision metrology have been proposed theoretically [7] and demonstrated experimentally [8,9]. However, these schemes are subject to limited numbers of ions or the requirement for individual addressing. For an ensemble of thousands of neutral atoms, the possibility of performing a Heisenberg-limited MZ interferometry has been demonstrated via a quantized Bose-Josephson junction [10].In this article, based upon the adiabatic processes and global operations on an array of ultracold ions, we present a realizable scheme for performing a Heisenberg-limited MZ interferometry. The ion array is described by a quan- * Electronic address: huyanmin2003@163.com † Electronic address: mangfeng@wipm.ac.cn ‡ Electronic address: chleecn@gmail.com tum Ising model of ferromagnetic (FM) interaction J and transverse field B, which has been used to simulate quantum magnetism [11][12][13] and quantum phase transition (QPT) [14][15][16] by virtue of spin-dependent optical dipole forces [17]. In our scheme, if the system starts from a paramagnetic state dominated by B, we adiabatically increase J and then decrease B, and vice versa if the system starts from a FM state dominated by J. In contrast to tuning either J or B in previous schemes [11,16], the adiabatic processes in our scheme perfectly connects the two limits so...

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