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## 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] 

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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 

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 

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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 

providing the initial state is ψ (0) = | e 〉 . The system re- vives at times such that Ω t = kπ with k being an integer. 

## III. CONTROLLED RABI OSCILLATIONS OF A TWO-LEVEL SYSTEM 

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 

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where | x | $^{2}$+ | y | 2 = 1. We denote such a pulse as Z pulse hereafter 

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 

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 

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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 

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 

with n the bath mode index [19]. Note that Ω$_{n}$ and λ$_{n}$ become mode dependent in the many-mode bath case 

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