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where we have used the large N and weak coupling as- sumptions. By considering the two dominant peaks, we further obtain that 

for small τ such that Γ τ ≲ π . Equation (19) shows that the decay rate decreases rapidly with the control pulse frequency in a Gaussian form. For exceedingly small τ which satisfies Γ τ ≪ π , A ′ is essentially zero, which means the transition is inhibited and the survival prob- ability of the initial state saturates. The red dashed line in Fig. 4 demonstrates the quantum Zeno effect and the prohibition of the transition. 

For | ω$_{0}$ − ω$_{m}$ | < Γ, the results are similar to ω$_{0}$ = ω$_{m}$ and exhibit only quantum Zeno effect. To observe sub- stantially enhancement of the transition, i.e., the quan- tum anti-Zeno effect [12], | ω$_{0}$ − ω$_{m}$ | ≳ Γ is required. More specifically, by taking the biggest peak around ω$_{0}$ , one obtains the necessary condition for quantum anti-Zeno effect as 

The blue solid line in Fig. 4 shows the quantum Zeno and anti-Zeno effects for | ω$_{0}$ − ω$_{m}$ | = 2. Both quantum Zeno (large N region) and anti-Zeno (small N region) effects are observed. At fixed time t = N τ , large τ (small N) gives anti-Zeno effect while small τ (large N) gives Zeno effect. The boundary between quantum Zeno and anti-Zeno effect is about 2 | ω$_{m}$ − ω$_{0}$ | τ ≈ π if | ω$_{m}$ − ω$_{0}$ | ≫ Γ. 

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Strongest anti-Zeno effect is obtained at | ω$_{0}$ − ω$_{m}$ | τ = π where one of the two main resonant modes lies near ω$_{m}$ . 

## C. Lorentzian and exponential spectrum intensity 

The spectrum intensity for Lorentzian and exponential are, respectively, 

Similar to Gaussian spectral intensity, only the quantum Zeno effect is observed if ω$_{0}$ = ω$_{m}$ for the Lorentzian and exponential shape (Fig. 5). The transition probability is essentially inhibited once τ ≪ τ$_{c}$ where τ$_{c}$ ∼ 1 /ω$_{c}$ . Once | ω$_{0}$ − ω$_{m}$ | ≳ Γ, we find both quantum Zeno and anti-Zeno effects and the boundary between Zeno and anti-Zeno regime is determined approximately by 2 | ω$_{m}$ − ω$_{0}$ | τ ≈ π . The peak position of the quantum anti-Zeno effect lies about at | ω$_{0}$ − ω$_{m}$ | τ ≈ π . As shown in Fig. 5, the reduced transition probability in the exponential case has a narrower peak width than that in the Lorentzian case. Two small secondary peaks are noticeable in the anti- Zeno region of the exponential spectrum case. These peaks are due to the second and third quasi-level resonant to ω$_{m}$ . They satisfy the condition, respectively, | ω$_{0}$ − ω$_{m}$ | τ ≈ 3 π, 5 π . 

The widely adopted dipolar coupling in spontaneous emission of a two level atom or molecule has a frequency dependence as ν ( ω ) = $_{ν$_{0}$}$√$_{ω}$ where ν$_{0}$ is taken as a con- stant, which depends on the dipole matrix elements [2]. We will consider the same coupling spectrum intensity as before, i.e., 

for Gaussian, Lorentzian, and exponential density of state, respectively. As shown in Figs. 4 and 5, the fre- quency dependent coupling has little effect on the per- formance of the control pulses. 

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## D. Frequency dependent coupling 

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