Quantum anti-Zeno effect without rotating wave approximation

In this article, we systematically study the spontaneous decay phenomenon of a two-level system under the influences of both its environment and repetitive measurements. In order to clarify some well-established conclusions about the quantum Zeno effect (QZE) and the quantum anti-Zeno effect (QAZE), we do not use the rotating wave approximation (RWA) in obtaining an effective Hamiltonian. We examine various spectral distributions by making use of our present approach in comparison with other approaches. It is found that with respect to a bare excited state even without the RWA, the QAZE can still happen for some cases, for example, the interacting spectra of hydrogen. However, for a physical excited state, which is a renormalized dressed state of the atomic state, the QAZE disappears and only the QZE remains. These discoveries inevitably show a transition from the QZE to the QAZE as the measurement interval changes.

Figure 1

Schematic of the difference between the overlap integrals with and without the RWA. The solid blue line is for the measurement function $F(\omega ,\Omega )$ centered at the original atomic level spacing $\Omega$ for the case with the RWA, the dotted red line is for the measurement function $F(\omega ,{\Omega }_1)$ centered at the modified frequency ${\Omega }_1$ for our current case without the RWA, and the green dashed line is for interacting spectrum $G(\omega )$ centered at ${\omega }_0$.

Figure 2

The decay rate vs measurement interval $\tau$ for the 2$p$-1$s$ transition of the hydrogen atom. Here the solid blue line is for $R$ and the dashed red line is for $\Delta R=|R-R_{\mathrm{rwa}}|$. $|{\Omega }_1-\Omega |/\Omega =1.71\times {10}^{-6}$, $\Omega =1.55\times {10}^{16}$ rad/s, and ${\omega }_c/\Omega =550$. Notice that $\Delta R$ is enlarged ${10}^6$ times. In all figures, the measurement interval $\tau$ is in units of atomic level spacing $1/\Omega$ and the decay rate $R$ is normalized with respect to the unperturbed one $R_0$.

Figure 3

The decay rate vs measurement interval $\tau$ for the 3$p$-1$s$ transition of the hydrogen atom. Here the solid blue line is for $R$ and the dashed red line is for $\Delta R=|R-R_{\mathrm{rwa}}|$. $|{\Omega }_1-\Omega |/\Omega =1.27\times {10}^{-6}$, $\Omega =1.83\times {10}^{16}$ rad/s, and ${\omega }_c^{\prime }/\Omega =412$. Notice that $\Delta R$ is enlarged by ${10}^6$ times.

Figure 4

The decay rate vs measurement interval $\tau$ for different spectra. Here the dotted green line (a) is for super-Ohmic spectrum $s=2$, the solid blue line (b) is for Ohmic spectrum $s=1$, the dashed red line (c) is for sub-Ohmic spectrum $s=0.5$, and the dot-dashed black line (d) is for sub-Ohmic spectrum $s=0.002$, which satisfies Eq. (21). For all spectra we set $A={10}^{-8}$, ${\omega }_c/\Omega =500$.

Figure 5

The decay rate difference $\Delta R=|R-R_{\mathrm{rwa}}|$ vs measurement interval $\tau$ for different spectra. Here the dot-dashed black line (a) is for sub-Ohmic spectrum $s=0.002$ with $|{\Omega }_1-\Omega |/\Omega =1.55\times {10}^{-3}$, the dashed red line (b) is for sub-Ohmic spectrum $s=0.5$ with $|{\Omega }_1-\Omega |/\Omega =9.41\times {10}^{-4}$, the solid blue line (c) is for Ohmic spectrum $s=1$ with $|{\Omega }_1-\Omega |/\Omega =6.57\times {10}^{-4}$, and the dotted green line (d) is for super-Ohmic spectrum $s=2$ with $|{\Omega }_1-\Omega |/\Omega =4.00\times {10}^{-4}$. Parameters are the same as those given in Fig. 4.

Figure 6

The frequency distance $\Delta \Omega ={\Omega }_1-s{\omega }_c$ vs the parameters $A$ and $s$ with ${\omega }_c/\Omega =500$.

Figure 7

The decay rate vs measurement interval $\tau$ for the 2$p$-1$s$ transition of the hydrogen atom. Here the solid blue line is for $R$ and the dashed red line is for $\Delta R=|R-R_{\mathrm{rwa}}|$. $|{\Omega }^{\prime }-\Omega |/\Omega =5.69\times {10}^{-8}$ and ${\omega }_c/\Omega =550$. Notice that $\Delta R$ is enlarged ${10}^8$ times.

Figure 8

The decay rate vs measurement interval $\tau$ for the 3$p$-1$s$ transition of the hydrogen atom. Here the solid blue line is for $R$, the dashed red line is for $\Delta R=|R-R_{\mathrm{rwa}}|$, and the dotted green line is for the result from Ref. [10]. $|{\Omega }^{\prime }-\Omega |/\Omega =5.32\times {10}^{-8}$ and ${\omega }_c/\Omega =412$. Notice that $\Delta R$ is enlarged ${10}^8$ times.