Deterministic quantum controlled-PHASE gates based on non-Markovian environments

We suppose that both the two reservoirs stated before have Lorentzian spectrums

$$\begin{eqnarray}&&| g({\omega }_{1}){| }^{2}=\displaystyle \frac{\gamma }{\pi }\displaystyle \frac{{\lambda }_{1}^{2}}{{({\omega }_{1}-{\omega }_{{ab}})}^{2}+{{\lambda }_{1}}^{2}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&| \tilde{g}({\omega }_{1}){| }^{2}=\displaystyle \frac{\tilde{\gamma }}{\pi }\displaystyle \frac{{\tilde{\lambda }}_{1}^{2}}{{({\omega }_{1}-{\omega }_{{ac}})}^{2}+{{\tilde{\lambda }}_{1}}^{2}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&| \kappa ({\omega }_{2}){| }^{2}=\displaystyle \frac{\kappa }{\pi }\displaystyle \frac{{\lambda }_{2}^{2}}{{({\omega }_{2}-{\omega }_{0})}^{2}+{{\lambda }_{2}}^{2}},\end{eqnarray} \tag{ 27 }$$

where $\gamma (\tilde{\gamma })$ and ${\lambda }_{1}({\tilde{\lambda }}_{1})$ represent the strength and spectral width of the coupling between atomic transition ${\sigma }_{{ab}}({\sigma }_{{ac}})$ and reservoir 1, respectively. κ denotes coupling strength between intracavity and external fields, and ${\lambda }_{2}$ is the spectral width. By substituting $\kappa ({\omega }_{2})=\sqrt{\tfrac{\kappa }{\pi }}\tfrac{{\lambda }_{2}}{{\lambda }_{2}+{\rm{i}}({\omega }_{2}-{\omega }_{0})}$ into equation (17), the response function can be expressed as

$$\begin{eqnarray}v(t-t^{\prime} )=\left\{\begin{array}{ll}\sqrt{2\kappa }{\lambda }_{2}{{\rm{e}}}^{-{\lambda }_{2}(t-t^{\prime} )} & t^{\prime} \leqslant t\\ 0 & t^{\prime} \gt t.\end{array}\right.\end{eqnarray} \tag{ 28 }$$

We can also obtain the time correlation functions of the reservoirs by use of equations (25)–(27),

$$\begin{eqnarray}&&u(t-t^{\prime} )=\kappa {\lambda }_{2}{{\rm{e}}}^{-{\lambda }_{2}| t-t^{\prime} | },\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&g(t-t^{\prime} )=\gamma {\lambda }_{1}{{\rm{e}}}^{-{\lambda }_{1}| t-t^{\prime} | },\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\tilde{g}(t-t^{\prime} )=\tilde{\gamma }{\tilde{\lambda }}_{1}{{\rm{e}}}^{-{\tilde{\lambda }}_{1}| t-t^{\prime} | }.\end{eqnarray} \tag{ 31 }$$

It should be noted that we take ${\omega }_{2}^{{\prime} }={\omega }_{2}-{\omega }_{0}$(${\omega }_{1}^{{\prime} }={\omega }_{1}-{\omega }_{{ab}},{\omega }_{1}^{{\prime\prime} }={\omega }_{1}-{\omega }_{{ac}}$) and shift the integration limit ($0,+\infty$) in equation (22) (equations (23), (24)) to ($-{\omega }_{0},+\infty$)[($-{\omega }_{{ab}},+\infty$), ($-{\omega }_{{ac}},+\infty$)]. For high frequency optical systems, ${\omega }_{0}$(${\omega }_{{ab}},{\omega }_{{ac}}$) is so large that one can extend the lower limit to $-\infty$ as a good approximation [17, 21] and use the Fourier transformations to obtain the above equations. Specifically, the parameters ${\lambda }_{1},{\tilde{\lambda }}_{1}$ and ${\lambda }_{2}$ are inversely proportional to the reservoir correlation times. When ${\lambda }_{1},{\tilde{\lambda }}_{1}$ and ${\lambda }_{2}$ converge to infinite, the reservoirs become memoryless. Combining with ${\mathrm{lim}}_{x\to \infty }{{x}{\rm{e}}}^{-x| t-t^{\prime} | }=2\delta (t-t^{\prime} )$ and assuming $\gamma =\tilde{\gamma }=\tfrac{{\gamma }_{a}}{2},{\lambda }_{1}={\tilde{\lambda }}_{1}$ hereafter, one can describe the Markovian dynamics of the system as follows,

$$\begin{eqnarray} & & {\dot{c}}_{200}(t)=-{{\rm{i}}{g}}^{* }{c}_{300}(t)-\kappa {c}_{200}(t)+\sqrt{2\kappa }{a}_{\mathrm{in}}(t),\\ & & {\dot{c}}_{300}(t)=-{{\rm{i}}{g}{c}}_{200}(t)-{\gamma }_{a}{c}_{300}(t),\\ & & {\dot{c}}_{500}(t)=-\kappa {c}_{500}(t)+\sqrt{2\kappa }{\tilde{a}}_{\mathrm{in}}(t),\\ & & {a}_{\mathrm{in}}(t)+{a}_{\mathrm{out}}(t)=\sqrt{2\kappa }{c}_{200}(t),\\ & & {\tilde{a}}_{\mathrm{in}}(t)+{\tilde{a}}_{\mathrm{out}}(t)=\sqrt{2\kappa }{c}_{500}(t).\end{eqnarray} \tag{ 32 }$$

These equations are same as those in [11].

In the following, we consider the memory backflow from the reservoirs and discuss the realization of the quantum controlled-PHASE gates with these reservoirs. We suppose that the input single photon pulse has a Gaussian temporal shape

$$\begin{eqnarray}&&{a}_{\mathrm{in}}(t)[{\tilde{a}}_{\mathrm{in}}(t)]={{c}{\rm{e}}}^{-\tfrac{4{\rm{ln}}2{(t-{t}_{0})}^{2}}{{\rm{\Delta }}{t}^{2}}},\end{eqnarray} \tag{ 33 }$$

where ${\rm{\Delta }}t$ is its full-width-half-maximum temporal duration. t₀ represents the time the pulse peak enters the cavity. $c={(\sqrt{\pi }{\rm{\Delta }}t/2\sqrt{2{\rm{ln}}2})}^{-1/2}$ is a factor which ensures the Gaussian function normalized.

First, we focus on a simpler case, in which an atom is in state $| c\rangle$ initially. By substituting equation (33) and initial conditions stated before $[\int | {\tilde{a}}_{\mathrm{in}}(t){| }^{2}{\rm{d}}{t}=1,{a}_{\mathrm{in}}(t)={c}_{200}(0)={c}_{300}(0)={c}_{500}(0)=0]$ into equations (16) and (18)–(21), one can easily find that the values of ${c}_{200},{c}_{300}$ and ${a}_{\mathrm{out}}(t)$ are always zero, and therefore reservoir 1 is vacuum all the time. Hence, dynamics of the total system can be described only with equations (18), (21) and (28), (29). We just need to discuss the influences brought by the various couplings between intracavity and external fields, because atomic transition ${\sigma }_{{ac}}$ is decoupled from the cavity mode and reservoir 1 is vacuum at first. The detailed numerical results are shown in figures 2 and 3. For an ideal quantum controlled-PHASE gate, the output ${\tilde{a}}_{\mathrm{out}}(t)$ should be same as the input ${\tilde{a}}_{\mathrm{in}}(t)$ when an atom is in state $| c\rangle$. This perfect performance can be achieved in a Markovian environment (equation (32)), as plotted in figure 2. Here, the output field is a little delayed in relation to the input one, which is mainly due to the time spent to propagate in the cavity. We define parameter ${R}_{2}=2\kappa /{\lambda }_{2}$ to characterize the coupling between intracavity and external fields. For an input Gaussian pulse with ${\rm{\Delta }}t=1$ μs, when the coupling strength $\kappa =2\pi \times 1\,\mathrm{MHz}$, the output optical pulses under different R₂ are shown in figure 2(a), the corresponding probabilities to find a single photon in cavity ${P}_{500}(t)={| {c}_{500}(t)| }^{2}$ are illustrated in figure 2(d). When R₂ goes to zero, the reservoir 2 converges to a memoryless one. In this case, ${\tilde{a}}_{\mathrm{out}}(t)$ and ${P}_{500}(t)$ obtained in non-Markovian approximation will converge to the results in the Markovian limit. In our chosen parameters, the numerical results are almost same as the corresponding Markovian limit when R₂ equals 0.1. As R₂ gets larger, the Markovian approximation fails and one might expect the memory backflow induced by the non-Markovian reservoirs. The population ${P}_{500}(t)$ keeps oscillating and then decays to zero in this situation. The temporal shape of output pulse ${\tilde{a}}_{\mathrm{out}}(t)$ is no longer Gaussian and its phase-shift is finite, as shown in figures 2(a) and (d) and also in figures 2(c) and (f), in which the parameters $\kappa =2\pi \times 10\,\mathrm{MHz}$, ${\rm{\Delta }}t=0.1$ μs. We furthermore plot figures 2(b) and (e), the coupling strength κ is same as that in figures 2(c) and (f), but the time duration ${\rm{\Delta }}t$ of the optical pulse resembles the one in figures 2(a) and (d). One can find that the numerical results in non-Markovian approximation are almost same as the corresponding Markovian limit, even when R₂ is as large as 10. It is reasonable since the coupling strength κ is larger and therefore a wider spectral width ${\lambda }_{2}$ is necessary to keep the parameters R₂ same. Trade off between κ and ${\lambda }_{2}$ leads to the results in figures 2(b) and (e). For a shorter input optical pulse, as shown in figures 2(c) and (f), the characteristic time scale is shorter and become comparable with the reservoir correlation time as R₂ gets larger. Hence, the non-Markovian effects become more visible in the short pulse case. We concern about the output field ${\tilde{a}}_{\mathrm{out}}(t)$, as it can exhibit whether a quantum controlled-PHASE gate is successful. By setting the same parameters used in plotting figures 2(c) and (f), one can obtain the simulation results in figure 3. When R₂ is smaller than 1.6, the output light ${\tilde{a}}_{\mathrm{out}}(t)$ has exactly the same Gaussian temporal shape as the input one ${\tilde{a}}_{\mathrm{in}}(t)$, and its phase-shift is zero all the time. However, as R₂ keeps increasing and becomes larger than 1.6, the non-Markovian memory effect gets stronger, as previously said. It means that, when an atom is in state $| c\rangle$, we can always achieve an ideal output pulse by appropriately manipulating ${R}_{2}({\lambda }_{2})$ for a determined input pulse ${\tilde{a}}_{\mathrm{in}}(t)$ and coupling strength κ.

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Now we focus on the other case, in which an atom is in state $| b\rangle$ initially. By applying the same procedure as before, one can also find that ${\tilde{a}}_{\mathrm{out}}(t)$ and ${c}_{500}(t)$ are always zero. Dynamics of the total system can be described with equations (16), (19), (20) and (28)–(31). In this situation, both the interactions between the atom and reservoir 1, and intracavity-external fields coupling should be taken into account. Similarly, we define parameter ${R}_{1}=2{\gamma }_{a}/{\lambda }_{1}$ to characterize the interactions between an atom and reservoir 1. As the atom interacts with reservoir 1 and also with the cavity mode, the corresponding coupling strength are set as ${\gamma }_{a}=2\pi \times 6\,\mathrm{MHz}$ and $g=2\pi \times 13\,\mathrm{MHz}$, respectively, in figures 4 and 5. We first investigate how the non-Markovian reservoir 1 influences a quantum controlled-PHASE gate. When $\kappa =2\pi \times 1\,\mathrm{MHz}$, ${\rm{\Delta }}t=1$ μs, the output ${a}_{\mathrm{out}}(t)$ and populations ${P}_{200}(t)={| {c}_{200}(t)| }^{2},{P}_{300}(t)={| {c}_{300}(t)| }^{2}$ under different R₁ are shown in figures 4(a) and (b). The non-Markovian numerical results are almost same as the corresponding Markovian limit, even when R₁ is as large as 10. For a stronger coupling strength κ, a shorter input optical pulse can be considered, as illustrated in figures 4(c) and (d). The non-Markovian numerical results are in good agreement with those in Markovian approximation when ${R}_{1}=0.1$. As R₁ increases to 10, the memory effects of the non-Markovian environment appear apparently in this short input pulse case. Because the characteristic time scales become comparable with the correlation time of reservoir 1. The excitations which have been leaked into the reservoir 1 can flow back into the atom-cavity system. By the coupling between the single cavity mode and output fields, this memory effect can finally result in a much larger reflectivity, even when the atom-cavity system is in the weak coupling regime. It is quite good for high-speed quantum computation and communication. As we may acquire a short output pulse with 100% reflectivity and π phase-shift, through manipulating parameter ${R}_{1}({\lambda }_{1})$ appropriately, when the atom-cavity coupling is weak. This typical non-Markovian feature would never appear in a Markovian case. For the mutual information exchange between the atom and reservoir 1, more excitations are provided by the cavity field. Thus, as R₁ increases, the population ${P}_{200}(t)$ decreases while ${P}_{300}(t)$ increases in figure 4(d). Now we take into account the non-Markovian effects caused by the coupling between intracavity and external fields. The parameters in figure 5 are set as same as figure 4, except that we choose ${R}_{1}=10$ this time. A larger R₂ leads to a longer memory time of the external field, which enhances the back-flowing effects. Hence, the reflectivity increases and populations ${P}_{200}(t),{P}_{300}(t)$ decreases, as shown in figure 5.

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From the above, it is possible to achieve a deterministic fast controlled-PHASE gate by modulating the reservoirs with the Lorentz type spectral density. When one incidents a short input pulse, the corresponding coupling strength between the cavity and environment should be large. By appropriately increasing R₁(decreasing ${\lambda }_{1}$) and decreasing R₂(increasing ${\lambda }_{2}$), we can implement an ideal deterministic controlled-PHASE gate, even the atom-cavity coupling is weak. As demonstration, for a shorter input pulse with duration time ${\rm{\Delta }}t=0.05$ μs, the numerical results are illustrated in figure 6, under the parameters $\kappa =2\pi \times 20\,\mathrm{MHz}$, $g=2\pi \times 25\,\mathrm{MHz}$, ${R}_{1}=150$ and ${R}_{2}=1$. The simulation results of the corresponding Markovian limits are also shown here as comparison.

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In this part, we turn to the case in which the two reservoirs have following spectral densities

$$\begin{eqnarray}&&{| g({\omega }_{1})| }^{2}=2\pi {\eta }_{1}{\omega }_{1}{\left(\displaystyle \frac{{\omega }_{1}}{{\omega }_{c1}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{1}}{{\omega }_{c1}}},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{| \tilde{g}({\omega }_{1})| }^{2}=2\pi {\tilde{\eta }}_{1}{\omega }_{1}{\left(\displaystyle \frac{{\omega }_{1}}{{\tilde{\omega }}_{c1}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{1}}{{\tilde{\omega }}_{c1}}},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{| \kappa ({\omega }_{2})| }^{2}=2\pi {\eta }_{2}{\omega }_{2}{\left(\displaystyle \frac{{\omega }_{2}}{{\omega }_{c2}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{2}}{{\omega }_{c2}}},\end{eqnarray} \tag{ 36 }$$

where ${\eta }_{1}({\tilde{\eta }}_{1})$ represents the dimensionless coupling strength between ${\sigma }_{{ab}}({\sigma }_{{ac}})$ transition and reservoir 1, and ${\omega }_{c1}({\tilde{\omega }}_{c1})$ is the corresponding cutoff frequency of the spectrum. Similarly, the interaction between the single cavity mode and external field is described by the dimensionless coupling constant ${\eta }_{2}$ and cutoff frequency ${\omega }_{c2}$. The environment can be classified by the parameter s as sub-Ohmic $(0\lt s\lt 1)$, Ohmic $(s=1)$ and super-Ohmic $(s\gt 1)$. By applying the same procedure as before, we substitute $\kappa ({\omega }_{2})={\left[2\pi {\eta }_{2}{\omega }_{2}{\left(\tfrac{{\omega }_{2}}{{\omega }_{c2}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{2}}{{\omega }_{c2}}}\right]}^{1/2}$ into equation (17) and achieve the response function

$$\begin{eqnarray}&&v(t-t^{\prime} )=\displaystyle \frac{1}{\sqrt{2\pi }}{\int }_{0}^{+\infty }{\left[2\pi {\eta }_{2}{\omega }_{2}{\left(\displaystyle \frac{{\omega }_{2}}{{\omega }_{c2}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{2}}{{\omega }_{c2}}}\right]}^{1/2}{{\rm{e}}}^{-{\rm{i}}({\omega }_{2}-{\omega }_{0})(t-t^{\prime} )}{\rm{d}}{\omega }_{2}.\end{eqnarray} \tag{ 37 }$$

After substituting equations (34)–(36) into equations (22)–(24), respectively, we can get the following time correlation functions

$$\begin{eqnarray}&&u(t-t^{\prime} )={\int }_{0}^{+\infty }2\pi {\eta }_{2}{\omega }_{2}{\left(\displaystyle \frac{{\omega }_{2}}{{\omega }_{c2}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{2}}{{\omega }_{c2}}}{{\rm{e}}}^{{\rm{i}}({\omega }_{0}-{\omega }_{2})(t-t^{\prime} )}{\rm{d}}{\omega }_{2},\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&g(t-t^{\prime} )={\int }_{0}^{+\infty }2\pi {\eta }_{1}{\omega }_{1}{\left(\displaystyle \frac{{\omega }_{1}}{{\omega }_{c1}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{1}}{{\omega }_{c1}}}{{\rm{e}}}^{{\rm{i}}({\omega }_{{ab}}-{\omega }_{1})(t-t^{\prime} )}{\rm{d}}{\omega }_{1},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\tilde{g}(t-t^{\prime} )={\int }_{0}^{+\infty }2\pi {\tilde{\eta }}_{1}{\omega }_{1}{\left(\displaystyle \frac{{\omega }_{1}}{{\tilde{\omega }}_{c1}}\right)}^{s-1}{{\rm{e}}}^{-\tfrac{{\omega }_{1}}{{\tilde{\omega }}_{c1}}}{{\rm{e}}}^{{\rm{i}}({\omega }_{{ac}}-{\omega }_{1})(t-t^{\prime} )}{\rm{d}}{\omega }_{1}.\end{eqnarray} \tag{ 40 }$$

In the following, we simulate the quantum controlled-PHASE gate by setting the parameter s to 0.1, 1 and 2, respectively. Specifically, we take the single cavity mode frequency ${\omega }_{0}=20\,\mathrm{GHz}$ as the norm unit [28]. As stated before, the ${\sigma }_{{ab}}$ transition is resonantly coupled with the cavity mode, hence ${\omega }_{{ab}}={\omega }_{0}$. The frequency of ${\sigma }_{{ac}}$ transition is taken as ${\omega }_{{ac}}=0.5{\omega }_{0}$. In our numerical calculations, we may roughly set the cutoff frequency ${\omega }_{c1}={\omega }_{{ab}},{\tilde{\omega }}_{c1}={\omega }_{{ac}}$ and ${\omega }_{c2}={\omega }_{0}$, respectively. It should be noted that the Markov limits (equation (32)) are the same for these three spectral densities ($s=0.1,1$ or 2). Because the parameters ${\gamma }_{a}=\pi {| g({\omega }_{{ab}})| }^{2}+\pi {| \tilde{g}({\omega }_{{ac}})| }^{2}=2{\pi }^{2}{\eta }_{1}{\omega }_{{ab}}{e}^{-1}+2{\pi }^{2}{\tilde{\eta }}_{1}{\omega }_{{ac}}{e}^{-1}$ and $\kappa =\pi {| \kappa ({\omega }_{0})| }^{2}=2{\pi }^{2}{\eta }_{2}{\omega }_{0}{e}^{-1}$ are independent of s in this situation. Considering the decay rates of atomic levels, we set ${\eta }_{1}=0.00013$ and ${\tilde{\eta }}_{1}=0.00026$, which is corresponding to ${\gamma }_{a}=2\pi \times 6\,\mathrm{MHz}$ [4]. As the coupling between an atom and reservoir 1 is quite weak, we only simulate the quantum controlled-PHASE gate under different ${\eta }_{2},g$, ${a}_{\mathrm{in}}(t)$ or ${\tilde{a}}_{\mathrm{in}}(t)$(equation (33)). The detailed numerical results are shown in figure 7 through 9.

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In figure 7, we show the exact results of output ${a}_{\mathrm{out}}(t)$ and ${\tilde{a}}_{\mathrm{out}}(t)$ when the parameters ${\eta }_{2}=0.4,g=2\pi \times 2.5{\omega }_{0}$ and ${\omega }_{0}{\rm{\Delta }}t=2$. An ideal deterministic quantum controlled-PHASE gate can be achieved in the Markovian approximation, which is plotted in figures 7(a) and (e). However, in the non-Markovian case, the influences caused by the memory effects of structured environments become visible, when an atom is initially in state $| c\rangle$. The envelope shape of output pulse is therefore different from the input Gaussian pulse, as shown in figures 7(b)–(d). Moreover, the output ${\tilde{a}}_{\mathrm{out}}(t)$ have both real and imaginary parts, which implies the output light has a time-varied phase-shift changing from $-\pi$ to π. Although we can achieve the same results as its Markovian limit, when the coupling strength g is large and the atom is initially in state $| b\rangle$, as shown in figures 7(f)–(h). The deterministic quantum controlled-PHASE gates can not be implemented in this situation.

The detailed numerical results with a weaker coupling strength ${\eta }_{2}$ and a longer input pulse ${a}_{\mathrm{in}}(t)$ are illustrated in figures 8. The parameters are chosen as ${\eta }_{2}=0.04,g=2\pi \times 0.175{\omega }_{0}$ and ${\omega }_{0}{\rm{\Delta }}t=20$. When an atom is initially in state $| c\rangle$, the output pulse has exactly the same Gaussian shape as input, which is shown in figures 8(b)–(d). But it still has time-varied phase-shift. When the atom is initially in state $| b\rangle$, one can also get the same output as that in Markovian limit, see figures 8(f)–(h). We furthermore decrease the coupling strength ${\eta }_{2}$ to 0.004 and apply a longer input optical pulse ${\omega }_{0}{\rm{\Delta }}t=200$, the corresponding simulation results are plotted in figure 9. The characters of the output resemble those shown in figure 8.

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In our chosen parameters, in the weak coupling regime(${\eta }_{2}\leqslant 0.04$), the exact envelope shape of output pulse $| {a}_{\mathrm{out}}(t)|$[$| {\tilde{a}}_{\mathrm{out}}(t)| ]$ is almost same as the corresponding Markovian limit, i.e. as same as the shape of the input pulse $| {a}_{\mathrm{in}}(t)|$[$| {\tilde{a}}_{\mathrm{in}}(t)|$]. Though a π phase shift can be acquired as an atom initially in state $| b\rangle$, the phase shift of the output pulse is time-varied with the atom at state $| c\rangle$. Such phase shift in the output pulse hinders the implementation of an ideal deterministic controlled-PHASE gate in sub-Ohmic, Ohmic or super-Ohmic environments.

We have seen that the deterministic quantum controlled-PHASE gates can only be realized in the Lorentzian reservoirs. The key point in this realization is that the memory effects of the Lorentzian reservoir 1 can be utilized, especially when the atom-cavity coupling is weak, as shown in figures 4(c) and 6(a). To confirm the results above, here we present some insights into the physical mechanism, by making use of the pseudomode theory [42–44]. In the pseudomode method, the pseudomodes are defined by the positions and residues of the poles of the reservoir spectrum [42]. The dynamics of a system interacting with a structured reservoir can be equivalently described by the coherent coupling between the system and pseudomodes, and the latter one leaking into an independent Markovian reservoir [42].

When the reservoir 1 has a Lorentzian spectrum, equation (20) can be rewritten as

$$\begin{eqnarray}&&{\dot{c}}_{300}(t)=-{{\rm{i}}{g}{c}}_{200}(t)-{\rm{i}}{{\rm{\Omega }}}_{1}{b}_{1}(t)-{\rm{i}}{{\rm{\Omega }}}_{2}{b}_{2}(t),\end{eqnarray} \tag{ 41 }$$

where ${b}_{1}(t)$ and ${b}_{2}(t)$ represent the amplitudes of the two pseudomodes, respectively.

$$\begin{eqnarray}&&{b}_{1}(t)=-{\rm{i}}{\int }_{0}^{t}\sqrt{\gamma {\lambda }_{1}}{{\rm{e}}}^{-{\lambda }_{1}(t-t^{\prime} )}{c}_{300}(t^{\prime} ){\rm{d}}{t}^{\prime} ,\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{b}_{2}(t)=-{\rm{i}}{\int }_{0}^{t}\sqrt{\tilde{\gamma }{\tilde{\lambda }}_{1}}{{\rm{e}}}^{-{\tilde{\lambda }}_{1}(t-t^{\prime} )}{c}_{300}(t^{\prime} ){\rm{d}}{t}^{\prime} .\end{eqnarray} \tag{ 43 }$$

The oscillation frequencies of these two pseudomodes are ${\omega }_{{ab}}$ and ${\omega }_{{ac}}$, the decay rates are ${{\rm{\Gamma }}}_{1}=2{\lambda }_{1}$ and ${{\rm{\Gamma }}}_{2}=2{\tilde{\lambda }}_{1}$, respectively, which are defined by the poles of reservoir spectrums equations (25) and (26). ${{\rm{\Omega }}}_{1}=\sqrt{\gamma {\lambda }_{1}}$ and ${{\rm{\Omega }}}_{2}=\sqrt{\tilde{\gamma }{\tilde{\lambda }}_{1}}$ denote the coupling strengths between the atom and two pseudomodes, respectively. A smaller ${\lambda }_{1}$(${\tilde{\lambda }}_{1}$) implies a weaker coupling strength ${{\rm{\Omega }}}_{1}$(${{\rm{\Omega }}}_{2}$). As previously, we assume $\gamma =\tilde{\gamma }=\tfrac{{\gamma }_{a}}{2}$, ${\lambda }_{1}={\tilde{\lambda }}_{1}$(${{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{2}={\rm{\Gamma }}$). We can get the total population of the two pseudomodes

$$\begin{eqnarray}| b(t){| }^{2} & = & {| {b}_{1}(t)| }^{2}+{| {b}_{2}(t)| }^{2}\\ & = & {\left|{\displaystyle \int }_{0}^{t}\sqrt{{\gamma }_{a}{\lambda }_{1}}{{\rm{e}}}^{-{\lambda }_{1}(t-t^{\prime} )}{c}_{300}(t^{\prime} ){\rm{d}}{t}^{\prime} \right|}^{2}.\end{eqnarray} \tag{ 44 }$$

In our article, the atom is not only coupled with these two pseudomodes, but also interacts with the single cavity mode. After considering the effects of the pseudomodes leakage, one can directly analyze the energy flowing between the atom and reservoir 1, from the compensated rate of change of the total pseudomode population ${\rm{d}}| b(t){| }^{2}/{\rm{d}}{t}+{\rm{\Gamma }}| b(t){| }^{2}$ [44].

When R₁ equals 0.1, the compensated rate of change ${\rm{d}}| b(t){| }^{2}/{\rm{d}}{t}+{\rm{\Gamma }}| b(t){| }^{2}$ is almost as same as the decay rate of the excited state population of the atom in Markovian limit $2{\gamma }_{a}{| {c}_{300}(t)| }^{2}$, as illustrated in figure 10(a). A larger R₁(smaller ${\lambda }_{1}$) results in a weaker coupling strength ${{\rm{\Omega }}}_{1}$(or ${{\rm{\Omega }}}_{2}$) between the atom and pseudomodes, while the atom-cavity coupling is strong. In this condition, more excitations tend to transfer between the atom and single cavity mode, while less excitations are exchanged between the atom and pseudomodes, as shown in figure 10. When R₁ gets much larger, the atom is almost decoupled from the pseudomodes, and almost none excitations transfer to the pseudomodes, as depicted in figure 10(c), which brings about the excellent results in figure 6(a). This phenomenon can be viewed as the net effects due to the backflowing of the environments. Actually, the changes of parameter R₁ only affect the memory time(${\lambda }_{1}$) of reservoir 1, and the coupling strength γ between the atom and reservoir 1 is unchanged. It should be noted that the compensated rate of change has some negative values, which implies the population of the pseudomodes transfer back to the atom. It is a consequence of the memory effects of environments. As the atom-cavity coupling strength gets much weaker, the interaction between the atom and pseudomodes plays a more important role, the compensated rate of change oscillates in a more intuitive way in this time, as shown in figure 10(b).

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For a reservoir with a Lorentzian spectrum, one can utilize the above pseudomode theory to interpret the physics. However, this pseudomode method can not be used for sub-Ohmic, Ohmic or super-Ohmic reservoirs, as there exists no such poles in these reservoir spectrums. In our study of sub-Ohmic, Ohmic or super-Ohmic reservoirs, the phase shift of the output pulse is time-varied with the atom at state $| c\rangle$.