content_image/431127.jpg content_image/431147.jpg content_image/431136.jpg content_image/431148.jpg content_image/431126.jpg content_image/431143.jpg content_image/431135.jpg content_image/431144.jpg content_image/431146.jpg
<img src='content_image/431127.jpg'>

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 , 

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. 

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 

<img src='content_image/431147.jpg'>

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 ) ≈ 

<img src='content_image/431136.jpg'>

## A. Uniform spectrum intensity 

<img src='content_image/431148.jpg'>

<img src='content_image/431126.jpg'>

(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 

A  1 , otherwise 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. 

A Gaussian spectrum intensity has a form as 

<img src='content_image/431143.jpg'>

where Γ and ω$_{m}$ denotes the width and the position of the maximal intensity, respectively. 

In this case we expect only quantum Zeno effect to appear, A ′ ≤ A , because it is easy to check that ′ 

<img src='content_image/431135.jpg'>

## B. Gaussian spectrum intensity 

<img src='content_image/431144.jpg'>

<img src='content_image/431146.jpg'>

