3 10 0 10 1 10 2 10 3 10 4 0 0.2 0.4 0.6 0.8 1 Number of pulses p $_{′}$eg  / p eg $_{0}$   =  $_{c}$   =  A B C $_{(}$ω) 0 0 0 0 c −2 −1 0 1 2 A 0 0 c −1 0 1 B 0 0 c 0 C 0 20 40 60 80 100 1 2 3 4 5 6 x 10 −6 Time p $_{′}$eg  = 10 4 3 2 1 FIG. 3: (Color online) Control pulse effect vs number of pulses N (main panel) and relative positions of resonant modes to ω 0 and ω c (upper three panels A, B, and C corresponds to three regions in the main panel, respectively). The parame- ters are N τ = 100 , ω 0 = 1 , ν 0 = 0 . 001, and ω c = 100. Two big steps shows two major peaks out of the lower and upper cutoff frequency, respectively. These small steps shows minor peaks going out of the cutoff frequencies. Circles are for the frequency-dependent coupling case. FIG. 2: Control pulse effect with fixed pulse delays. The parameters are ω 0 = 1 , ν 0 = 0 . 001 , ω c = 100, and τ = 10 , 4 , 3 , 2 , 1 from top to bottom for solid lines. The dashed line denotes the free decay. Time is in units of 1 /ω 0 here- after. and ν is constant here but will be considered as mode dependent in Sec. IV D. Assuming the bath spectrum is dense, we turn the summation over mode index n into the integration over mode frequency ω , (2 π/τ ) δ ( ω − ω$_{k}$ ) for large N . Note that the two nearest neighbor peaks around ω$_{0}$ with k = 0 and k = − 1 con- tribute equally about 40% among all the peaks [7, 15]. If we assume the bath has only positive frequency and concentrate on the two dominant peaks, we find  and p ′ $_{eg}$/p$_{eg}$ also shows step-like behavior as depicted in Fig. 3. Moreover, Fig. 3 exhibits only quantum Zeno effect, i.e., suppression of the decay by the control pulses. The decay are completely suppressed, A ′ = 0, once the control pulse frequency is larger than the cutoff frequency. where ρ ( ω ) is the density of states of the bath. For N = 1 we get the free decay results (no control pulse). Before considering the real physical systems which usually have mode dependent coupling, we study several toy models with constant coupling strength ν = ν$_{0}$ for all bath modes to gain some ideas about the effect of control pulses. A ′ A ≃   0 , ω$_{c}$τ < π 1 / 2 , ω$_{0}$τ < π < ω$_{c}$τ 1 , otherwise (16) p ′ eg = ∫ dωρ ( ω ) ν 2 Ω 2 sin $^{2}$Ω τ sin 2 N λ sin 2 λ (13) A. Uniform spectrum intensity B. Gaussian spectrum intensity A Gaussian spectrum intensity has a form as Taking ρ ( ω ) = ρ$_{0}$ ≡ 1 /ω$_{c}$ if ω ∈ [0 , ω$_{c}$ ] and ρ ( ω ) = 0 otherwise with ω$_{c}$ denoting the cutoff frequency of the bath, Fig. 2 shows typical free decay and controlled decay of the excited state. Except at very short times, Fig. 2 shows that the transition probability linearly depends on the total evolution time. Utilizing the linear dependence, one defines decay rate (Einstein constant) at long time t as for free decay [7] and where Γ and ω$_{m}$ denotes the width and the position of the maximal intensity, respectively. ρ ( ω ) = 1 √$_{2}$$_{π}$ Γ e − ( ω − ω m ) $^{2}$/ 2Γ $^{2}$, ω ≥ 0 (17) A ′ A = 4 π 2 ∑ k 1 (2 k + 1) 2 ρ ( ω$_{k}$ ) ρ ( ω$_{0}$ ) ≤ 1 (18) In this case we expect only quantum Zeno effect to appear, A ′ ≤ A , because it is easy to check that 1. ω 0 = ω m A ′ ≡ 1 τ ∂p ′ eg ∂N = 2 πρ$_{0}$ν 2 0 τ 2 ∑ k 1 ν 2 0 + ( ω$_{k}$ − ω$_{0}$ ) $^{2}$/ 4 (15) A ≡ ∂p$_{eg}$ ( t ) ∂t = 2 πρ$_{0}$ν 2 0 (14) for controlled decay, where ω$_{k}$ is the k th resonant mode frequency [15]. For weak coupling ν$_{0}$ ≪ π/τ , the k th resonant mode lies approximately at ω$_{k}$ ≈ ω$_{0}$ + (2 k + 1) π/τ where sin λ ≪ 1 and sin 2 N λ/ sin λ ≈ πδ ( λ ) ≈