2 where | x | $^{2}$+ | y | 2 = 1. We denote such a pulse as Z pulse hereafter FIG. 1: Diagram of a coupled two-level system subjected to ideal 2 π pulses. Z ( x | e 〉 + y | g 〉 ) = − x | e 〉 + y | g 〉 . (6) U (1) = ZU = ( d$_{1}$ d$_{2}$ − d$_{2}$ − d ∗ 1 ) . (7) In fact, Z = − σ$_{z}$ = | g 〉〈 g | − | e 〉〈 e | . The evolution operator for the 2 π pulse and the free evolution τ is For N such operations, the evolution operator at time t = Nτ becomes II. RABI OSCILLATIONS OF A TWO-LEVEL SYSTEM Let us consider a simple model of a two-level quan- tum system with constant coupling to its environment modes. Such a model serves as the base of further discus- sions for more complicated systems with mode-dependent coupling. The Hamiltonian of the system with detuning δ = ω − ω$_{0}$ (we set ℏ = 1 for convenience) and coupling strength ν is described by [2, 17] Let d$_{1}$ = d$_{1}$$_{r}$ + id$_{1}$$_{i}$ and d$_{2}$ = id$_{2}$$_{i}$ . After some straightfor- ward simplifications with the use of Pauli matrices, one easily obtains where sin λ = √ 1 − d 2 1 i and σ$_{α}$ = (1 / sin λ )( d$_{1}$$_{r}$σ$_{z}$ − d$_{2}$$_{i}$σ$_{y}$ ). Note that λ depends on τ instead of t = Nτ . The transition probability from the initial state | e 〉 to the ground state | g 〉 is U ( N ) = i $^{N}$(cos Nλ − iσ$_{α}$ sin Nλ ) (9) U ( N ) = ( d$_{1}$ d$_{2}$ − d$_{2}$ − d ∗ 1 ) N . (8) p ′ $_{eg}$( t = Nτ ) = |〈 g | U ( N $^{)}$| e 〉| 2 = p$_{eg}$ ( τ ) sin 2 Nλ sin 2 λ . (10) p ′ eg ≈ ν 2 δ ′ 2 tan 2 δ $^{′}$τ sin 2 Nδ $^{′}$τ (11) sin λ The above result is exact for any coupling strength and pulse delay τ . For weak coupling ν ≪ δ , Ω ≈ δ ′ and λ ≈ δ $^{′}$τ + π/ 2 + 2 kπ with k an integer, then for even N , which is exactly Eq. (9) in Agarwal et al. ’s paper [7]. IV. QUANTUM ZENO AND ANTI-ZENO EFFECTS providing the initial state is ψ (0) = | e 〉 . The system re- vives at times such that Ω t = kπ with k being an integer. The above results are applicable only to single mode bath/environment which couples to the central two-level system. In general the bath has multimodes and the cou- pling may depend on the mode, e.g., the dipolar coupling between an atom and an electric and magnetic field. For a many-mode bath, the transition probability of the two- level system subjected to control pulses is in general given by p ′ eg = ∑ n ν 2 Ω 2 n sin $^{2}$Ω$_{n}$ τ sin 2 Nλ$_{n}$ sin 2 λ$_{n}$ (12) with n the bath mode index [19]. Note that Ω$_{n}$ and λ$_{n}$ become mode dependent in the many-mode bath case By applying an ideal 2 π pulse (or parity kick, see Fig. 1), which is very strong in amplitude and short in time but gives a π phase shift solely to the excited state | e 〉 utilizing an auxiliary state | a 〉 [7], the state of the system changes according to III. CONTROLLED RABI OSCILLATIONS OF A TWO-LEVEL SYSTEM x | e 〉 + y | g 〉 2 π pulse − − − − → − x | e 〉 + y | g 〉 (5) p$_{eg}$ ( t ) = ν 2 Ω 2 sin $^{2}$(Ω t ) (4) H = δ $^{′}$σ$_{z}$ + νσ$_{x}$ (2) U = d$_{1}$ | e 〉〈 e | + d ∗ 1 | g 〉〈 g | + d$_{2}$ ( | e 〉〈 g | + | g 〉〈 e | ) (3) where ω$_{0}$ = E$_{e}$ − E$_{g}$ is the difference between eigenfre- quencies of the excited state | e 〉 and the ground state | g 〉 , with respectively eigenenergy E$_{e}$ and E$_{g}$ , and ω is the frequency of the external mode coupled to the two-level system [18]. Utilizing Pauli matrices, the Hamiltonian can be rewritten as H = δ 2 ( | e 〉〈 e | − | g 〉〈 g | ) + ν ( | e 〉〈 g | + | g 〉〈 e | ) , (1) with σ$_{z}$ = | e 〉〈 e |−| g 〉〈 g | , σ$_{x}$ = | e 〉〈 g | + | g 〉〈 e | , and δ ′ = δ/ 2. The evolution operator of the coupled two-level system is where d$_{1}$ = cos Ω t − i ( δ $^{′}$/ Ω) sin Ω t and d$_{2}$ = − i ( ν/ Ω) sin Ω t with Ω 2 = δ ′ 2 + ν $^{2}$. The transition prob- ability at time t from | e 〉 to | g 〉 is