1 Introduction

The QSL time [1,2,3], defined as the minimal evolution time between two states, is of crucial importance in many fields of quantum physics, such as quantum computation [4], quantum control [5] and quantum metrology [6]. The QSL bound has been derived for closed systems [7, 8], driven quantum systems [9] and more recently for open quantum systems [10, 11].

In particular, it has been found that the non-Markovian effect [12, 13] characterized by the non-Markovianity can speed up the evolution of open quantum systems [14,15,16,17,18,19]. The non-Markovian dynamics [20, 21] can be triggered by many factors, such as strong system-environment coupling, structure of environment and low temperatures. As reported in Ref. [14] increasing coupling strength will increase the non-Markovianity of the environment and thus accelerate the quantum evolution. On the experimental side, the non-Markovianity-assisted speedup in a weakly driven optical cavity QED system has been reported in Ref. [22].

In this study, we will investigate the quantum evolution speed of an atom interacting with a PBG environment [23, 24]. Due to the change of mode density of the PBG reservoir, an atom embedded in this reservoir will exhibit non-Markovian dynamics. This non-Markovian atom-field interaction has broad applications in quantum information processing, including creating steady entanglement [25], fractionalized single-atom inversion [26] and population trapping. Motivated by these potential applications, explorations are made regarding the QSL time in PBG environment. In Ref. [16], it has been found that the atomic frequency changes can lead to the transition from no-speedup to speedup of evolution, and the speedup phenomenon mainly occurs inside the band-gap region. Two questions naturally arise: (1) Are there other environmental changes which will speed up the evolution in PBG environment? (2) What sort of changes enable quantum systems to have faster evolution speed outside the band-gap region, where the non-Markovianity is very small?

In this paper, we focus on these questions. We consider a two-level atom embedded in a two-band PBG reservoir, where a defect mode is in the gap. It is shown that the embedded position of the atom, the width of the band gap and the defect mode can be helpful to accelerate the quantum evolution. By changing the position of the atom and the coupling strength with the defect mode, the speedup phenomenon can exist outside the band-gap region. The reason for the speedup is also studied. Our results would be useful for experimental exploration of speedup evolution in quantum systems composed of quantum dots in PBG materials.

This paper is organized as follows. The physical model is given in Sect. 2. In Sect. 3, the effects of the embedded position, the width of the band gap and the defect mode on the QSL time are studied. In Sect. 4 , we summarize our results and discuss the possible experimental realization.

2 Physical model

We consider a two-level atom embedded in a photonic crystal with the location \({\mathbf {r}}\). The main characteristic of the photonic crystal is the existence of the band gap, where the photonic mode density is zero. Due to the presence of the band gap, the dispersion relationship and the mode density of the electromagnetic radiation field are both changed in comparison with that for free space vacuum field. The photon dispersion relationship can be given by

$$\begin{aligned} \omega =\left\{ \begin{array}{c} \omega _{c_{1}}+A_{1}(k-k_{0})^{2}\;\ \omega _{{}}\ge \omega _{c_{1}} \\ \omega _{c_{2}}-A_{2}(k-k_{0})^{2}\;\;\omega _{{}}\le \omega _{c_{2}}, \end{array}\right. \end{aligned}$$
(1)

where \(\omega _{c_{1}(c_{2})}\) is the upper (lower)-band edge frequency, \( k_{0}\) is wave number corresponding to the band edge and \(A_{j}=\omega _{c_{j}}/k_{0}^{2}\) \((j=1,2).\) The density of states can be expressed as

$$\begin{aligned} \rho _{1}\left( \omega \right)= & {} \frac{V_{0}}{(2\pi )^{3}}\frac{k_{0}^{2}}{2 \sqrt{A_{1}}}\frac{\varTheta \left( \omega -\omega _{c_{1}}\right) }{\sqrt{ \omega -\omega _{c_{1}}}}\;\;\omega \ge \omega _{c_{1}}, \end{aligned}$$
(2)
$$\begin{aligned} \rho _{2}\left( \omega \right)= & {} \frac{V_{0}}{(2\pi )^{3}}\frac{k_{0}^{2}}{2 \sqrt{A_{2}}}\frac{\varTheta \left( \omega _{c_{2}}-\omega \right) }{\sqrt{ \omega _{c_{2}}-\omega }}\;\;\omega \le \omega _{c_{2}}, \end{aligned}$$
(3)

where \(\varTheta \left( x\right) \) is the Heaviside step function. Clearly, the whole radiation field can be divided into three parts: the photonic band gap with gap with \(\omega _{c_{1}}-\omega _{c_{2}}\), the upper-band reservoir with mode density \(\rho _{1}\left( \omega \right) \), and the lower-band reservoir with mode density \(\rho _{2}\left( \omega \right) \).

Here, we consider the atom is coupled to the radiation field and a defect mode with frequency \(\omega _{d}\) inside the gap. The Hamiltonian in the rotating-wave approximation reads (\(\hbar =1\))

$$\begin{aligned} H= & {} \omega _{0}\left| 1\right\rangle \left\langle 1\right| +\omega _{d}a_{d}^{\dag }a_{d}+\sum _{l}\omega _{l}b_{l}^{\dagger }b_{l}+\sum _{\mu }\omega _{\mu }c_{\mu }^{\dagger }c_{\mu }+g_{d}(a_{d}^{+}\left| 0\right\rangle \left\langle 1\right| +H.c.) \nonumber \\&+\sum _{l}(g_{l}\left( \mathbf {r}\right) b_{l}^{+}\left| 0\right\rangle \left\langle 1\right| +H.c.)+\sum _{\mu }(g_{\mu }\left( \mathbf {r} \right) c_{\mu }^{+}\left| 0\right\rangle \left\langle 1\right| +H.c.), \end{aligned}$$
(4)

where \(\omega _{0}\) is the atomic transition frequency, \(\left| 0\right\rangle \) and \(\left| 1\right\rangle \) are the ground and the excited states of the atom. The field operators \(a_{d}\) and \(a_{d}^{\dagger } \) correspond to the defect mode, which are coupled to the atom with coupling constant \(g_{d}\). We assume that \(g_{d}\) is real and independence of the atomic position. \(c_{\mu }\) (\(c_{\mu }^{\dagger }\)) and \(b_{l}\)(\( b_{l}^{\dagger }\)) are, respectively, the annihilation (creation) operators for the upper- and lower-band reservoirs with coupling constants depending on the location of the embedded atom. The spatial dependence coupling constants can be given by [27]

$$\begin{aligned} g_{\mu (l)}\left( {\mathbf {r}}\right) =\omega _{0}d_{0}\left( \frac{1}{2\varepsilon _{0}\omega _{\mu (l)}V_{0}}\right) ^{1/2}{\mathbf {u}}_{d}\cdot {\mathbf {E}}_{\mu (l)}^{*}\left( {\mathbf {r}}\right) . \end{aligned}$$
(5)

Here, \({\varepsilon }_{0}\) is the free space permittivity, \(V_{0}\) is the quantization volume, \({\mathbf {u}}_{d}\) and \(d_{0}\) refer to the unit vector and the magnitude of atomic dipole moment. In photonic crystals, the eigenmodes \({\mathbf {E}}_{\mu (l)}\left( {\mathbf {r}}\right) \) can be characterized by Bloch modes. That is to say, the electric field varies from point to point within a unit cell of photonic crystals. Here, we assume that the electric fields \({\mathbf {E}}_{\mu (l)}^{*}\left( {\mathbf {r}}\right) \) are [28]

$$\begin{aligned} {\mathbf {E}}_{u}^{*}\left( {\mathbf {r}}\right) =E_{\mathbf {k}}\cos \theta \left( {\mathbf {r}}\right) {\mathbf {e}} \end{aligned}$$
(6)

and

$$\begin{aligned} {\mathbf {E}}_{l}^{*}\left( {\mathbf {r}}\right) =E_{\mathbf {k}}\sin \theta \left( {\mathbf {r}}\right) {\mathbf {e,}} \end{aligned}$$
(7)

where \({\mathbf {e}}\) and \(E_{\mathbf {k}}\) are, respectively, the unit vector and the amplitude of the electric field. \(\theta \left( {\mathbf {r}}\right) \) is the angle parameter seen by the atom located at \({\mathbf {r}}\). Thus, the fields of the double-band reservoir are two coherent modes with phase difference \(\pi /2\). The \(g_{\mu (l)}\left( {\mathbf {r}}\right) \) can be rewritten as \(g_{u}\left( {\mathbf {r}}\right) \cong g_{\mathbf {k}}\cos \theta \left( {\mathbf {r}}\right) \) and \(g_{l}\left( {\mathbf {r}}\right) \cong g_{ \mathbf {k}}\sin \theta \left( {\mathbf {r}}\right) \) with \(g_{\mathbf {k }}=\omega _{0}d_{0}(\frac{1}{2\varepsilon _{0}\omega V_{0}})^{1/2}E_{\mathbf { k}}({\mathbf {u}}_{d}\cdot {\mathbf {e}})\).

In this work, we use the discretization method [29] to solve the problem of the non-Markovian dynamics in photonic crystals. The core of this method is to divide the density of modes into discrete and perturbance parts. The discrete part which is near the band edge is replaced by a finite (but large) number of discrete harmonic oscillators, and the perturbance part which is far from the band edge can be treated perturbatively. The non-Markovian dynamics in this system is essentially arising due to the existence of the finite number of discrete harmonic oscillators, i.e., the discrete part. The coupling constant to the discrete modes of the double-band reservoir can be given by

$$\begin{aligned}&g_{m}\left( {\mathbf {r}}\right) \approx \sqrt{\frac{2\beta }{M\pi }\sqrt{ \omega _{v_{1}}-\omega _{c_{1}}}}\cos \theta \left( {\mathbf {r}}\right) , \end{aligned}$$
(8)
$$\begin{aligned}&g_{j}\left( {\mathbf {r}}\right) \approx \sqrt{\frac{2\beta }{M\pi }\sqrt{ \omega _{c_{2}}-\omega _{v_{2}}}}\sin \theta \left( {\mathbf {r}}\right) , \end{aligned}$$
(9)

where \(\omega _{v_{1(2)}}\) is the upper (lower) limit of the discretized part of the density of states, \(\beta \) is the effective coupling between the atom and the PBG reservoir, M is the number of discrete modes. The detail derivation process of \(g_{j(m)}\left( {\mathbf {r}}\right) \) is shown in our work [25].

We assume that the atom is initially excited. In the interaction picture, the state vector for the full system reads

$$\begin{aligned} \left| \varphi \left( t\right) \right\rangle= & {} a\left( t\right) e^{-i\omega _{0}t}\left| 1,0_{d},\tilde{0}_{m},\tilde{0} _{j}\right\rangle +b(t)e^{-i\omega _{d}t}\left| 0,1_{d},\tilde{0}_{m}, \tilde{0}_{j}\right\rangle \nonumber \\&+\sum _{m}c_{m}\left( {\mathbf {r}},t\right) e^{-i\omega _{m}t}\left| 0,0_{d},\tilde{1}_{m},\tilde{0}_{j}\right\rangle \nonumber \\&+\sum _{j}c_{j}\left( {\mathbf {r}},t\right) e^{-i\omega _{j}t}\left| 0,0_{d},\tilde{0}_{m},\tilde{1}_{j}\right\rangle , \end{aligned}$$
(10)

where the radiation states \(\left| 0,1_{d},\tilde{0}_{m},\tilde{0} _{j}\right\rangle \) and \(\left| 0,0_{d},\tilde{1}_{m},\tilde{0} _{j}\right\rangle \) \((\left| 0,0_{d},\tilde{0}_{m},\tilde{1} _{j}\right\rangle )\) account for the mode of defect and the upper (lower)-band reservoir having one excitation.

The equations for the amplitudes are governed by the Schrőinger equation, and after eliminating the off-resonant modes with frequency \(\omega >\omega _{v_{1}}\) and \(\omega <\omega _{v_{2}}\) , we obtain [29]

$$\begin{aligned} \dot{a}\left( t\right)= & {} i\sum _{\omega _{\mu }\!>\!\omega _{v_{1}}}\frac{ \left| g_{\mu }\left( {\mathbf {r}}\right) \right| ^{2}}{\omega _{\mu }\!-\!\omega _{0}}a\left( t\right) \!+\!i\sum _{\omega _{l}\!<\!\omega _{v_{2}}}\frac{ \left| g_{l}\left( {\mathbf {r}}\right) \right| ^{2}}{\omega _{l}\!-\!\omega _{0}}a\left( t\right) \!-\!ig_{d}b\left( t\right) e^{-i(\omega _{d}-\omega _{0})t} \nonumber \\&-i\sum _{m=1}^{M}g_{m}\left( {\mathbf {r}}\right) c_{m}\left( {\mathbf {r}} ,t\right) e^{-i(\omega _{m}\!-\!\omega _{0})t}\!-\!i\sum _{j=1}^{M}g_{j}\left( {\mathbf {r}}\right) c_{j}\left( {\mathbf {r}},t\right) e^{-i(\omega _{j}\!-\!\omega _{0})t},\qquad \end{aligned}$$
(11)
$$\begin{aligned} \dot{b}\left( t\right)= & {} -ig_{d}a\left( t\right) e^{i(\omega _{d}-\omega _{0})t}, \end{aligned}$$
(12)
$$\begin{aligned} \dot{c}_{m}\left( {\mathbf {r}},t\right)= & {} -ig_{m}\left( {\mathbf {r}}\right) a\left( t\right) e^{i(\omega _{m}-\omega _{0})t}, \end{aligned}$$
(13)
$$\begin{aligned} \dot{c}_{j}\left( {\mathbf {r}},t\right)= & {} -ig_{j}\left( {\mathbf {r}}\right) a\left( t\right) e^{i(\omega _{j}-\omega _{0})t}. \end{aligned}$$
(14)

By numerically solving the above set of equations, we can obtain the evolution of amplitudes.

3 Quantum speed limit time

In this section, we study the PBG reservoir effects on the QSL time. The QSL time in the evolution of open systems is given by [14]

$$\begin{aligned} \tau _\mathrm{QSL}=\max \left\{ \frac{1}{E_{op}},\frac{1}{E_{tr}},\frac{1}{E_{hs}}\right\} \sin ^{2}\left[ B\left( \rho _{0},\rho _{\tau }\right) \right] , \end{aligned}$$
(15)

where \(B\left( \rho _{0},\rho _{\tau }\right) =\arccos \sqrt{\left\langle \varphi \left( 0\right) \right| \rho _{\tau }\left| \varphi \left( 0\right) \right\rangle }\) is the Bures angle between the target state \(\rho _{\tau }\) and the initial pure state \(\rho _{0}=\left| \varphi \left( 0\right) \right\rangle \left\langle \varphi \left( 0\right) \right| \). \( E_{op,tr,hs}=\frac{1}{\tau }\int _{0}^{\tau }\mathrm{d}t\left\| \dot{\rho }(t) \right\| _{op,tr,hs}\), with \(\left\| \dot{\rho }(t)\right\| _{op,tr,hs}\) denoting the operator norm, trace norm and Hilbert-Schmidt norm of \(\dot{\rho }(t) \), respectively.

To study the environmental effects on QSL time, one method is to evaluate the characteristic of the intrinsic speed of the quantum evolution. By this method, one can explore the QSL time \(\tau _\mathrm{QSL}\) by first fixing the actual driving time \(\tau \) (or the actual evolution time). When the calculated QSL time \(\tau _\mathrm{QSL}\) is equal to the driving time \(\tau \), i.e., \(\tau _\mathrm{QSL}=\tau \), the speed of quantum evolution is the fastest, or say the speedup evolution cannot appear. For intrinsic speedup evolution, it requires \(\tau _\mathrm{QSL}<\tau \), and the shorter the \(\tau _\mathrm{QSL}\), the faster the intrinsic speed of evolution (or equivalently, the greater the capacity for potential speedup) will be.

We assume that the atom is initially in the state \(\left| \varphi \left( 0\right) \right\rangle =\xi \left| 1\right\rangle +\eta \left| 0\right\rangle \) with \(\left| \xi \right| ^{2}+\left| \eta \right| ^{2}=1\). The reduced density operator of the atom at time t can be given by

$$\begin{aligned} \rho _{AB}\left( t\right) =\left( \begin{array}{cc} \left| \xi \right| ^{2}\left| a\left( t\right) \right| ^{2} &{} \xi \eta ^{*}a\left( t\right) \\ \xi ^{*}\eta a^{*}\left( t\right) &{} 1-\left| \xi \right| ^{2}\left| a\left( t\right) \right| ^{2} \end{array} \right) . \end{aligned}$$
(16)

Using the definition of QSL time in Eq. (15), we can obtain

$$\begin{aligned} \tau _\mathrm{QSL}=\frac{\left| \xi \right| \left( 1-a\left( t\right) \right) \left[ 1-\left( 1-2\left| \xi \right| ^{2}\right) a\left( t\right) \right] }{\frac{1}{\tau }\int _{0}^{\tau }\mathrm{d}t\left| \dot{a}\left( t\right) \sqrt{1-\left| \xi \right| ^{2}\left( 1-4\left| a\left( t\right) \right| ^{2}\right) }\right| }. \end{aligned}$$
(17)

Clearly, the QSL time is related to the amplitude \(a\left( t\right) \) under a given driving time. Meanwhile, the amplitude \(a\left( t\right) \) is affected by the atomic embedded position \( \theta \left( {\mathbf {r}}\right) \), the width of the band gap \(\varDelta _{c}=\) \( \omega _{c_{1}}-\omega _{c_{2}}\) and the parameters of the defect mode as well. In what follows, we shall investigate the effects of \(\theta \left( {\mathbf {r}}\right) \), \(\varDelta _{c}\) and the defect mode on the QSL time, respectively.

Fig. 1
figure 1

The QSL time \(\tau _\mathrm{QSL}\) and the non-Markovianity N (in the unit of \(1/\beta \)) as a function of \(\omega _{0}/\beta \) for various values of \(\theta \left( {\mathbf {r}} \right) \) with \(\omega _{c_{1}}=101\beta \) and \(\omega _{c_{2}}=99\beta \). a, c for driving time \(\tau =2 \), b, d for driving time \(\tau =10 \). The shaded area illustrates the photonic-band-gap region. In the numerical simulation, the number of the discrete quantum harmonic oscillators is chosen as \(M=150\)

Full size image

In order to study the embedded position effects, the situation of \(\xi =1\) is chosen, and we first neglect the defect mode, i.e., \(g_{d}=0\). In Fig. 1, we plot the QSL time \(\tau _\mathrm{QSL}\) (in the unit of \(1/\beta \)) as a function of \(\omega _{0}/\beta \) for various values of \(\theta \left( {\mathbf {r}} \right) \) together different driving time \(\tau \). We find that the embedded position effects on the QSL time are rather strong and differ significantly for different driving time. For the case of \(\tau =2\), the quantum evolution is accelerated significantly when the \(\theta \left( {\mathbf {r}}\right) \) changes from \(\theta \left( {\mathbf {r}} \right) =\pi /4\) to \(\theta \left( {\mathbf {r}}\right) =\pi /3\) and \(\theta \left( {\mathbf {r}}\right) =\pi /6\), and the speedup only occurs within the band-gap region. For the case of \(\tau =10\), as increasing the \(\theta \left( {\mathbf {r}}\right) \) value (from \(\pi /6\) to \(\pi /3\)), we observe that the \(\tau _\mathrm{QSL}\) curve moves to the low atomic frequency zone, but does not have much change in the values. The increase in the \(\theta \left( {\mathbf {r}}\right) \) value causes the stronger coupling strength between atom and the lower-band reservoir, as seen from Eqs. (3) and (4). Thus, the stronger coupling with the lower-band reservoir can push the \(\tau _\mathrm{QSL}\) curve to the lower-band region. Furthermore, for long driving time \(\tau =10\), the speedup region lies even outside the band gap by changing the atomic embedded position, which is quite different from that for \(\tau =2\). That is, by changing the atomic embedded position, the speedup can present outside the band-gap region for the case of long driving time.

In order to explain the above phenomena, we also depict the non-Markovianity N for \(\tau =2\) and \(\tau =10\) cases. The non-Markovianity is defined as the total information flowing back from the environment. The total amount of backflow information can be given by

$$\begin{aligned} \underset{\rho _{1,2}\left( 0\right) }{\max }\int _{\sigma >0}dt\sigma \left( t,\rho _{1,2}\left( 0\right) \right) , \end{aligned}$$
(18)

where \(\sigma \left( t,\rho _{1,2}\left( 0\right) \right) =\frac{d}{dt}D\left( \rho _{1}\left( t\right) ,\rho _{2}\left( t\right) \right) \) denotes the changing rate of the trace distance. The trace distance is defined as \(D\left( \rho _{1}\left( t\right) ,\rho _{2}\left( t\right) \right) =\frac{1}{2}tr\left| \rho _{1}(t)-\rho _{2}(t)\right| \) with \(\left| o\right| =\sqrt{o^{+}o}\). Clearly, the dynamical process is non-Markovian if there exists a pair of initial states such that \(\sigma \left( t,\rho _{1,2}\left( 0\right) \right) >0\), and the maximum is taken over all pairs of initial states. It has been found that [30] the pair of optimal states is proved to be the states \( \rho _{1}\left( 0\right) =\left| 0\right\rangle \left\langle 0\right| \) and \(\rho _{2}\left( 0\right) =\left| 1\right\rangle \left\langle 1\right| \). This allows us to derive the rate of change of the trace distance in the form \(\sigma \left( t,\rho _{1,2}\left( 0\right) \right) =\partial _{t}\left| a\left( t\right) \right| ^{2}\).

When the initial state is \(\left| \varphi \left( \xi =1\right) \right\rangle =\left| 1\right\rangle \), the QSL time yields

$$\begin{aligned} \tau _\mathrm{QSL}=\frac{\tau }{2N/\left( 1-\left| a\left( t\right) \right| ^{2}\right) +1}. \end{aligned}$$
(19)

The details on the above equation was given in Ref. [16]. Clearly, the QSL time is not only related to the population of atomic excited state, i.e., \(p=\left| a\left( t\right) \right| ^{2}\), but also related to the non-Markovianity N within the driving time. It is the interplay between the two quantities that ultimately determines the intrinsic speedup of quantum evolution. For open quantum systems, if the environmental changes can increase the non-Markovianity or the atomic excited population, the intrinsic speedup of evolution (or say, the potential speedup) can be achieved. As an illustration, the non-Markovianity N as a function of \(\omega _{0}/\beta \) for various values of \(\theta \left( {\mathbf {r}} \right) \) is drawn in Fig. 1c, d. Obviously, the \(\tau _\mathrm{QSL}\) curve change is inversed with the non-Markovianity curve, which means that increasing non-Markovianity will decrease the QSL time and therefore lead to a faster speed of the intrinsic evolution. Therefore, the speedup region is the same as the non-Markovian region (see Fig. 1). It is to say, the embedded position of the atom plays an important role in changing the non-Markovianity, which directly affects the QSL time of the system.

Fig. 2
figure 2

a The QSL time \(\tau _\mathrm{QSL}\), b the non-Markovianity N and c the atomic excited population p as a function of \(\omega _{0}/\beta \) for various values of \(\varDelta _{c}\). The other parameters are \(\theta \left( {\mathbf {r}} \right) =\pi /4\) and \(\tau =10 \). In the numerical simulation, the number of the discrete quantum harmonic oscillators is chosen as \(M=150\)

Full size image

From Fig. 2, one can see the effects of the width of the band gap \(\varDelta _{c}\). Clearly, increasing the \(\varDelta _{c}\) will accelerate the evolution of the system (see Fig. 2a). By contrasting Fig. 2b, c, we can easily find that the reason for the acceleration is not due to the non-Markovianity but mostly to the population of the atomic excited state. Therefore, atomic excited population is also a factor for evolution speedup as that given in Eq. (19). Surprisingly, the non-Markovianity decreases with increasing \(\varDelta _{c}\), while for population p, it is obviously different. Namely, it is not always true that the non-Markovianity and the population trapping follow the same trend.

Fig. 3
figure 3

a The QSL time \(\tau _\mathrm{QSL}\), b the non-Markovianity N and c the atomic excited population p as a function of \(\delta /\beta \) for various values of \(g_{d}\). The other parameters are \(\theta \left( {\mathbf {r}} \right) =0\) and \(\tau =10 \)

Full size image

Figure 3 shows the effects of the defect mode on the QSL time, the non-Markovianity and the population trapping, respectively. For illustration, we consider the embedded position \(\theta \left( {\mathbf {r}} \right) =0\) as example, i.e., we neglect the lower-band reservoir. When the atomic transition frequency is deeply inside the band-gap edge (\(\delta /\beta <0\)), the atomic excited population p decreases with the increase in coupling strength \(g_{d}\) (see Fig. 3c). However, when the atomic transition frequency is going outside the band edge (\(\delta /\beta >0\)), the population will begin to increase with increasing \(g_{d}\). Thus, the coupling with the defect mode can reduce the atomic population trapping in \(\delta /\beta <0\) case, but increase it in \(\delta /\beta >0\) case. A different phenomenon is shown in Fig. 3b, when the \(g_{d}\) increases, the non-Markovianity N also increases in the two cases of \(\delta /\beta <0\) and \(\delta /\beta >0\) . Therefore, with the \(g_{d}\) increasing, the notable acceleration only occurs outside the band-gap edge, where both the non-Markovianity N and population p increase, as seen from Fig. 3a.

In concluding this section, we would like to emphasize that, the embedded position of the atom, the width of the band gap and the defect mode all play an important role in shortening the QSL time or, say, enhancing the intrinsic speed of evolution. That is to say, we provide three environmental changes which will speed up the evolution in PBG environments. In order to observe the potential speedup phenomenon in PBG environment, we can simply change the above three factors, such as increase the width of the band gap and the atom-defect coupling strength.

4 Discussion and conclusion

In summary, we have studied the QSL of a two-level atom coupled to the PBG reservoir and a defect mode in the band gap. By making use of the discretization method, we have found that the embedded position of the atom, the width of the band gap and the defect mode can be helpful to decrease the QSL time, i.e., accelerate the evolution of the system. Our results have also demonstrated the linkage between the QSL time and the population trapping as well as the non-Markovianity for a given driving time. With the variation of atomic embedded position [\(\theta \left( {\mathbf {r}}\right) \)], we could find the obvious changes of the non-Markovianity, which directly affects the speed of the quantum evolution. Moreover, increasing the band-gap width can increase the population p, which is correlated with the speedup of the evolution. We have also considered the effects of the defect mode on the QSL time. We show that the obvious acceleration only occurs outside the band-gap edge. Thus, the speedup evolution can be moved to the region outside the band gap, where the memory effect of the reservoir is weak.

The phenomenon we illustrated in this work can be observed in the system of InAs quantum dots embedded in a GaAs photonic crystal membrane [31]. In experiment, the coupling between only one quantum dot and PBG materials can be realized [32, 33], and the photonic crystal membranes made of GaAs have proven very well suited for quantum-optics experiments [34]. To control the atomic embedded position in order to observe the speedup phenomenon, we can use the method of the electrohydrodynamic jet printing in experiment [35]. In particular, through precise band-gap alignment and guided-mode design, the strong coupling of quantum dots and defect modes can be achieved. In experiment, we can extract the quantum speed from the rate of change of the antibunching when we change the environment [22]. By this way, we can verify our prediction. Thus, this composite system is a good candidate to test the results of our work.