Non-Markovian dynamics and quantum interference in open three-level quantum systems

Abstract

The exact analytical solutions for the dynamics of the dissipative three-level V-type and $\varLambda$-type atomic systems in the vacuum Lorentzian environments are presented. Quantum interference between the spontaneous emissions of different decaying channels for the V-type atomic system is observed. For the dissipative $\varLambda$-type atomic system, however, similar phenomenon of quantum interference does not exist. We demonstrate that quantum interference can be used to protect effectively the quantum entanglement and quantum coherence. The control of the transition from Markovian to non-Markovian processes is discussed.

Appendices

Appendix A: The inverse Laplace transformation of Eqs. (11), (12) for the degenerate cases

If the polynomials $p^{3}+h_{1}p^{2}+h_{2}p+h_{3}=0$ have a twofold root $b_{1}$ and a single root $b_{3}$, then the decomposition of Eqs. (11) and (12) becomes

where

with

$$\begin{aligned} {\hat{E}}_{1}= & {} \frac{b_{1}^{2}-(2b_{1}+M+\mathrm{i}\omega _{2})b_{3}-\mathrm{i}\omega _{2}M-B_{22}}{(b_{1}-b_{3})^{2}},\\ {\hat{F}}_{1}= & {} \frac{B_{12}}{(b_{1}-b_{3})^{2}},\\ {\hat{E}}_{2}= & {} \frac{(b_{1}+\mathrm{i}\omega _{2})(b_{1}+M)+B_{22}}{b_{1}-b_{3}},\\ {\hat{F}}_{2}= & {} -\frac{B_{12}}{b_{1}-b_{3}},\\ {\hat{E}}_{3}= & {} \frac{(b_{3}+\mathrm{i}\omega _{2})(b_{3}+M)+B_{22}}{(b_{1}-b_{3})^{2}},\\ {\hat{F}}_{3}= & {} -\frac{B_{12}}{(b_{1}-b_{3})^{2}}, \end{aligned}$$

and

$$\begin{aligned} {\hat{G}}_{1}= & {} \frac{b_{1}^{2}-(2b_{1}+M+\mathrm{i}\omega _{1})b_{3}-\mathrm{i}\omega _{1}M-B_{11}}{(b_{1}-b_{3})^{2}},\\ {\hat{H}}_{1}= & {} \frac{B_{21}}{(b_{1}-b_{3})^{2}},\\ {\hat{G}}_{2}= & {} \frac{(b_{1}+\mathrm{i}\omega _{1})(b_{1}+M)+B_{11}}{b_{1}-b_{3}},\\ {\hat{H}}_{2}= & {} -\frac{B_{21}}{b_{1}-b_{3}},\\ {\hat{G}}_{3}= & {} \frac{(b_{3}+\mathrm{i}\omega _{1})(b_{3}+M)+B_{11}}{(b_{1}-b_{3})^{2}},\\ {\hat{H}}_{3}= & {} -\frac{B_{21}}{(b_{1}-b_{3})^{2}}. \end{aligned}$$

The inverse Laplace transformation of Eqs. (A.1a) and (A.1b) gives the evolution

with

$$\begin{aligned} {\hat{E}}(t)= & {} ({\hat{E}}_{1}+{\hat{E}}_{2}t)\mathrm{e}^{b_{1}t}+{\hat{E}}_{3}\mathrm{e}^{b_{3}t},\\ {\hat{F}}(t)= & {} ({\hat{F}}_{1}+{\hat{F}}_{2}t)\mathrm{e}^{b_{1}t}+{\hat{F}}_{3}\mathrm{e}^{b_{3}t},\\ {\hat{G}}(t)= & {} ({\hat{G}}_{1}+{\hat{G}}_{2}t)\mathrm{e}^{b_{1}t}+{\hat{G}}_{3}\mathrm{e}^{b_{3}t},\\ {\hat{H}}(t)= & {} ({\hat{H}}_{1}+{\hat{H}}_{2}t)\mathrm{e}^{b_{1}t}+{\hat{H}}_{3}\mathrm{e}^{b_{3}t}. \end{aligned}$$

If the polynomials $p^{3}+h_{1}p^{2}+h_{2}p+h_{3}=0$ have only one threefold root b, then Eqs. (11) and (12) become

where

$$\begin{aligned} {\check{D}}_{3}= & {} [(b+\mathrm{i}\omega _{2})(b+M)+B_{22}]c_{1}(0)-B_{12}c_{2}(0),\\ {\check{D}}_{2}= & {} (2b+M+\mathrm{i}\omega _{2})c_{1}(0),\\ {\check{D}}_{1}= & {} 2c_{1}(0),\\ {\check{D}}'_{3}= & {} [(b+\mathrm{i}\omega _{1})(b+M)+B_{11}]c_{2}(0)-B_{21}c_{1}(0),\\ {\check{D}}'_{2}= & {} (2b+M+\mathrm{i}\omega _{1})c_{2}(0),\\ {\check{D}}'_{1}= & {} 2c_{2}(0). \end{aligned}$$

The inverse Laplace transformation of Eqs. (A.4a) and (A.4b) gives

with ${\check{E}}(t)=\{\frac{1}{2}[(b+\mathrm{i}\omega _{2})(b+M)+B_{22}]t^{2}+(2b+M+\mathrm{i}\omega _{2})t+2\}\mathrm{e}^{bt}$, ${\check{F}}(t)=-\frac{1}{2}B_{12}t^{2}\mathrm{e}^{bt}$, ${\check{G}}(t)=\{\frac{1}{2}[(b+\mathrm{i}\omega _{1})(b+M)+B_{11}]t^{2}+(2b+M+\mathrm{i}\omega _{1})t+2\}\mathrm{e}^{bt}$, ${\check{H}}(t)=-\frac{1}{2}B_{21}t^{2}\mathrm{e}^{bt}$.

Appendix B: Proof of no real root of Eqs. (34a), (34b)

If $\lambda \ne 0$, then $\chi \ne 0$. Multiplying Eq. (34a) by $\chi /2$ and then subtracting Eq. (34b), one has

$$\begin{aligned} (\delta _{1}+\delta _{2})\chi ^{2}-\left[ 2\lambda ^{2}-2\delta _{1}\delta _{2}+\frac{\lambda }{2}(\gamma _{1}+\gamma _{2})\right] \chi -\lambda (\gamma _{1}\delta _{2}+\gamma _{2}\delta _{1})=0 \end{aligned}$$

(B.1)

If $\delta _{1}+\delta _{2}\ne 0$, multiplying Eq. (34a) by $(\delta _{1}+\delta _{2})/2$ and then subtracting Eq. (B.1), one gets

$$\begin{aligned} \chi =\frac{\lambda \gamma _{1}(3\delta _{2}-\delta _{1})+\lambda \gamma _{2}(3\delta _{1}-\delta _{2})}{8\lambda ^{2}+2(\delta _{1}-\delta _{2})^{2}+2\lambda (\gamma _{1}+\gamma _{2})}. \end{aligned}$$

(B.2)

Plugging it into Eq. (34a), we can obtain the quadratic equation with respect to $\omega _{0}$,

$$\begin{aligned} a\omega _{0}^{2}+b\omega _{0}+c=0 \end{aligned}$$

(B.3)

where $a=4\lambda ^{2}(u+v)^{2}+4\lambda n(u+v)$, $b=-8\lambda ^{2}(u+v)(\omega _{1}u+\omega _{2}v)-2\lambda n[(3\omega _{1}+\omega _{2})u+(3\omega _{2}+\omega _{1})v]$ and $c=4\lambda ^{2}(\omega _{1}u+\omega _{2}v)^{2}+2\lambda n(\omega _{1}+\omega _{2})(\omega _{1}u+\omega _{2}v)+\frac{1}{2}\lambda n^{2}(u+v)$, with $u=\gamma _{1}-3\gamma _{2}$, $v=\gamma _{2}-3\gamma _{1}$ and $n=2(\omega _{1}-\omega _{2})^{2}+8\lambda ^{2}+2\lambda (\gamma _{1}+\gamma _{2})$. The discriminant of Eq. (B.3) is

$$\begin{aligned} \varDelta= & {} b^{2}-4ac\\= & {} -256\lambda ^{2}n^{2}[\gamma _{1}\gamma _{2}(\omega _{1}-\omega _{2})^{2}+\lambda ^{2}(\gamma _{1}+\gamma _{2})^{2}]<0, \end{aligned}$$

which implies that $\omega _{0}$ is a complex number.

If $\delta _{1}+\delta _{2}= 0$, then Eqs. (34a) and (B.1) reduce, respectively, to

$$\begin{aligned}&4\chi ^{2}-\lambda (\gamma _{1}+\gamma _{2})=0, \end{aligned}$$

(B.4)

$$\begin{aligned}&[4\lambda ^{2}+4\delta _{1}^{2}+\lambda (\gamma _{1}+\gamma _{2})]\chi -\lambda \delta _{1}(\gamma _{1}-\gamma _{2})=0. \end{aligned}$$

(B.5)

Combining them to eliminate $\chi$, we get the quadratic equation with respect to $\delta _{1}$

$$\begin{aligned} \pm \sqrt{(\gamma _{1}+\gamma _{2})\lambda }\delta _{1}^{2}-\lambda (\gamma _{1}-\gamma _{2})\delta _{1}\pm \sqrt{(\gamma _{1}+\gamma _{2})\lambda ^{5}} \pm \frac{1}{4}[(\gamma _{1}+\gamma _{2})\lambda ]^{3/2}=0. \end{aligned}$$

(B.6)

This equation also leads to $\delta _{1}$ only having complex roots.