5 transition probability after many pulses and one half of the decay in the first pulse delay. The right column of Fig. 6 shows that ε decreases with increasing of 1 /τ (decreasing of τ ), confirming the analytical results of Eq. (27). In fact, decay freezing exists not only in the model system we consider in the paper, but also exists in many other pulse-controlled systems, such as gated semicon- ductor quantum dot [20–24], spin-boson model [25, 26], and nuclear spins [27]. The basic idea behind the de- cay freezing is that the control pulses creates an effective preferred direction along which the decay is frozen. In terms of Pauli matrix (c.f. Eq. 2), the preferred direction created by control pulses in the model we consider is z . By investigating three models of coupling spectrum intensity (Gaussian, Lorentzian and exponential), we demonstrate that a two-level system subjected to many ideal 2 π pulses exhibits both quantum Zeno and anti- Zeno effect, depending on the relative position of ω$_{0}$ to VI. CONCLUSION AND DISCUSSION FIG. 6: Decay freezing (left column) and dependence of the differential freezing value on pulse delay (right column) for the Gaussian (top row), Lorentzian (middle row), and exponential (bottom row) spectrum. Only results of odd number pulses are plotted for a better view. The dashed lines show decay without control pulses. Crosses of dashed lines and solid lines in the left column panels denote the specific pulse delay τ . 1 3 10 30 100 300 0 0.5 1 1.5 2 Number of pulses p $_{′}$eg  / p eg 1 3 10 30 100 300 0 0.5 1 1.5 2 2.5 3 Number of pulses p $_{′}$eg  / p eg 10 −2 10 0 10 2 10 −10 10 −8 10 −6 Time p $_{′}$eg 10 0 10 1 10 2 10 −4 10 −2 10 0 1 /  10 −2 10 0 10 2 10 −12 10 −10 10 −8 Time p $_{′}$eg 10 2 10 3 10 −6 10 −4 10 −2 1 /  10 −2 10 0 10 2 10 −11 10 −9 10 −7 Time p $_{′}$eg 10 1 10 2 10 −6 10 −4 10 −2 1 /  FIG. 5: (Color online) Same as Fig. 4 except that the spec- trum intensity is Lorentzian (top) and exponential (bottom). V. DECAY FREEZING By inspecting Eq. (13), the controlled transition prob- ability freezes if N λ ≫ 1 and the freezing value is Moreover, for small enough pulse delay τ , we have sin 2 λ ≈ 1 thus where we have replaced the rapidly oscillating integrand sin $^{2}$( N λ ) with its average 1 / 2. The left column of Fig. 6 from numerical calculation indeed shows that the tran- sitions freeze at long times for three different cases at small τ . The smaller the τ is, the smaller the freezing value is. Freezing of the transition probability is equiva- lent to freezing of the survival probability of the excited state, f = 1 − p ′ $_{eg}$. lim N $_{→∞}$p ′ eg = 1 2 ∫ dωρ ( ω ) ν 2 Ω 2 sin $^{2}$Ω τ 1 sin 2 λ , (26) lim N →∞ p ′ eg ≈ 1 2 ∫ dωρ ( ω ) ν 2 Ω 2 sin $^{2}$Ω τ = 1 2 p$_{eg}$ ( τ ) . (27) The above relation indicates that the freezing value of the transition probability at long times is one half of the free transition value at t = τ which is the pulse delay. Clearly, the road map to the quantum ZE is that the decay rate becomes zero after initial pulses (decay freezing) and the freezing value of the survival probability f approaches 1 as τ decreases. Define differential freezing value ε ( τ ) = p ′ $_{eg}$( N τ = 1000) − 1 $_{2}$p$_{eg}$ ( τ ), which describes the difference of the