Dynamical control of two-level system’s decay and long time freezing Wenxian Zhang 1, 2 and Jun Zhuang 1 $^{1}$The State Key Laboratory for Advanced Materials and Devices, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China $^{2}$Ames Laboratory and Iowa State University, Ames, Iowa 50011, USA (Dated: November 9, 2018) We investigate with exact numerical calculation coherent control of a two-level quantum system’s decay by subjecting the two-level system to many periodic ideal 2 π phase modulation pulses. For three spectrum intensities (Gaussian, Lorentzian, and exponential), we find both suppression and acceleration of the decay of the two-level system, depending on difference between the spectrum peak position and the eigen frequency of the two-level system. Most interestingly, the decay of the two-level system freezes after many control pulses if the pulse delay is short. The decay freezing value is half of the decay in the first pulse delay. PACS numbers: 03.67.Pp, 03.65.Yz, 02.60.Cb It is of fundamental interests to study the quantum control of spontaneous emission which inspired the devel- opment of the quantum electrodynamics during the last century [1–3]. The unwanted spontaneous emission often sets the ultimate limit of precise quantum measurement and many proposals have been made to suppress it [4–7]. One way widely applied to control the spontaneous emis- sion of a small quantum system, such as a neutral atom, is to place it either into or nearby a microcavity such that only a single mode or several modes of the cavity are resonant to the eigenfrequency of the quantum sys- tem [8]. In this way the structure of the vacuum is modi- fied and the spontaneous emission could be controlled by the properties of the cavity. Another way to control the spontaneous emission is dynamical control of the quan- tum system such that the coupling between the quan- tum system and the vacuum is effectively modified. For example, dynamical decoupling of the quantum system from the vacuum via either phase modulation or ampli- tude modulation could in principle extend the coherent time of the system [9, 10]. Analogous to the quantum Zeno effect (ZE) which says frequent measurements of a quantum system would prevent the decay of an unsta- ble quantum system [9–11], the extension of the coherent time through coherent modulations of the quantum sys- tem are often called ZE as well. Under some unfavorable conditions, it is also possible that frequent modulations lead to acceleration of the spontaneous emission, the so- called quantum anti-Zeno effect (AZE) [12–15]. In practice people employ both ways separately or combination of them to realize the optimal control of a quantum system. G. S. Agarwal et al. ’s work [7, 13] is among many of such works. In their paper, a two- level model system with both structured vacuum and free space vacuum are investigated. They found significant suppression of spontaneous emission rate for structured vacua but either suppression or acceleration may appear in a free space vacuum, depending on the frequency of the control pulses which is 2 π phase modulation pulses. They The paper is organized as follows. Section II reviews briefly Rabi oscillation of a two-level system under a per- turbation and Sec. III develops an exact formula of Rabi oscillation under periodic pulses. In Sec. IV after estab- lishing a quasi-level picture of the pulsed two-level sys- tem with constant spectrum intensity, we investigate the quantum Zeno and anti-Zeno effect for three spectrum intensities, including the Gaussian, Lorentzian, and ex- ponential one. The boundary between quantum Zeno and anti-Zeno effect is given. We discuss the decay freez- ing as pulse delay τ getting small in Sec. V. Finally, conclusion and discussion are given in Sec. VI. Another puzzle is the ZE effect of a large number of pulses. According to the ZE, at a fixed evolution time t , the survival probability of the initially excited state approaches 1 with infinite number of pulses N → ∞ (the pulse delay τ = t/N approaches 0). It is unclear what the road map looks like as N increases. Two ways might be possible. One way is that the rate of decay depends solely on the pulse delay τ (independent on N ) and approaches zero if τ → 0 [16]. Another way is that the decay rate depends only on the number of pulses N (independent on τ ) and the decay rate becomes zero after initial several pulses. For the latter one, we expect to see decay freezing at large number of pulses. We will find which way is the correct one in this paper. A puzzle arises if one accepts the AZE condition be- cause neither ZE or AZE appear if one leaves the quan- tum system alone in which case obviously τ ≫ ω − 1 0 . In this paper we revisit this problem by adopting the ex- act solution of a two-level quantum system which does not require the weak coupling and short time approxi- mations compared to Ref. [7]. By investigating several typical structured vacuum, we show the conditions for quantum ZE and AZE and the boundary between them. demonstrated for a free space vacuum that ZE shows up for ω$_{0}$τ = 1 while AZE appears for ω$_{0}$τ = π where ω$_{0}$ is the eigenfrequency of the two-level system and τ is the delay time between control pulses. At the end they argued that AZE is possible for τ > ω − 1 0 [7]. I. INTRODUCTION arXiv:1001.1041v1 [quant-ph] 7 Jan 2010