Tunable non-Markovian dynamics with a three-level atom mediated by the classical laser in a semi-infinite photonic waveguide

In this work, we investigate the non-Markovian (NM) dynamical evolution of a three-level atom coupled to a semi-infinite one-dimensional photonic waveguide and driven by a classical driving field. The single-end of waveguide behaves as a perfect mirror while light can pass through the opposite end with no backreflection. We derive the exact analytical expressions for the wave function of the driven three-level atom by solving a set of delay differential equations and give the conditions of atom-photon bound state formation that can inhibit the dissipation, which appears in the atom-mirror interspace in order to trap a considerable amount of initial atomic excitation. We show that when the evolution time of the system is shorter than the time delay taken by a photon to perform a round trip between the atom and mirror, the atomic dynamics exhibits Markovian properties, which depend on the driving strength and rescaled parameters. If the evolution time of the system exceeds the time delay, the photon can be reabsorbed by the atom with the photon emitted by the atom having completed the round trip, which exhibits NM memory effects, where the population eventually convergences to the finite-steady values. We also study the influence of the loss and dephasing on the formation of a bound state. Finally, the above results are extended to a more general quantum system involving single-end photonic waveguide coupled to an arbitrary number of noninteracting three-level atoms driven by the driving fields. The presented formalism might open a way to better understand exactly the NM dynamics of driven multilevel atoms coupled to a semi-infinite waveguide.

Figure 1

Schematic of the setup. A $\mathrm{\Lambda }$-type three-level atom driven by a classical driving field with Rabi frequency $\mathrm{\Omega }{e}^{-i{\omega }_{l}t}$ is coupled to a semi-infinite waveguide at $x=x_0$ with the coupling coefficient $g_k$.

Figure 2

Probability amplitudes given by Eqs. (13) and (14) as functions of time $t$ (in units of ${\mathrm{\Gamma }}^{-1}$) when the semi-infinite waveguide system evolves in the NM regime for the resonance case. In Fig. 2 and 2, we take $\mathrm{\Omega }=\alpha =1.8\pi \mathrm{\Gamma }$. In Figs. 2 and 2, we take $\mathrm{\Omega }=-\alpha =-1.8\pi \mathrm{\Gamma }$. The red-star, blue-dotted, and purple-dot-dashed lines correspond to $\mathrm{\Phi }=0.48\pi$, $1.60\pi$, and $2.00\pi$, respectively. The green-solid line corresponds to the Markovian case that the mirror is absent (i.e., the infinite waveguide with $t_d=\infty$). The coordinates of points $A$, $B$, $C$, and $D$ are (15.85,0.137), (10.88,0.412), (15.74,0.137), and (10.77,0.412), respectively. The other parameters chosen are ${\omega }_{\mathrm{g}}=0.4\pi \mathrm{\Gamma }$, ${\omega }_l=2.4\pi \mathrm{\Gamma }$, $C_x\left(0\right)=\sqrt{0.8}$, and $C_e\left(0\right)=\sqrt{0.2}$.

Figure 3

The figure shows the influence of the time delay $t_d$ on the atomic probability amplitudes $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ given by Eqs. (13) and (14) with ${\omega }_g=\frac{{\omega }_l}{2}=0.2\mathrm{\Gamma }$ (corresponding to the phase $\mathrm{\Phi }=0$) for the resonance case. For the first row, we set $\mathrm{\Omega }=\alpha =0.7\mathrm{\Gamma }$. For the second row, we set $\mathrm{\Omega }=-\alpha =-0.7\mathrm{\Gamma }$. The red-dashed, blue-dotted, purple-solid, and green-dot-dashed lines correspond to $\mathrm{\Gamma }{t}_d=1.4$, 1.6, 1.8, and 2.0, respectively. The other parameters chosen are $C_x\left(0\right)=\sqrt{0.8}$ and $C_e\left(0\right)=\sqrt{0.2}$. The insets in Fig. 3 magnify the probability amplitudes in the regime $t<t_d$.

Figure 4

The figure shows the influence of the driving strength $\mathrm{\Omega }$ on the atomic probability amplitudes $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ given by Eqs. (17) and (18) with ${\omega }_g=\frac{{\omega }_l}{2}=0.2\mathrm{\Gamma }$ (i.e., the phase $\mathrm{\Phi }=0$) and $\mathrm{\Gamma }{t}_d=1.5$ under the off-resonance case. For the first column, the red-stars, blue-solid line and green-dot-dashed line correspond to $\mathrm{\Omega }=\sqrt{0.2}\mathrm{\Gamma }$, $\sqrt{0.7}\mathrm{\Gamma }$, and $\sqrt{2.0}\mathrm{\Gamma }$, respectively. For the second column, the red-stars, blue-solid and green-dot-dashed lines correspond to $\mathrm{\Omega }=-\sqrt{0.2}\mathrm{\Gamma }$, $-\sqrt{0.7}\mathrm{\Gamma }$, $-\sqrt{2.0}\mathrm{\Gamma }$, respectively. The other parameters are set at ${\alpha }_1=\mathrm{\Gamma }$, ${\alpha }_2=0.7\mathrm{\Gamma }$, $C_x\left(0\right)=\sqrt{0.8}$, and $C_e\left(0\right)=\sqrt{0.2}$.

Figure 5

The figure shows the influence of the driving frequency ${\omega }_l$ on the atomic probability amplitudes $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ given by Eqs. (17) and (18) as a function of the time with ${\omega }_l=2{\omega }_g$ (i.e., the phase $\mathrm{\Phi }=0$) under the off-resonance case. We set the time delay $\mathrm{\Gamma }{t}_d=1.5$. The red-dashed, blue-dotted, purple-solid and green-dot-dashed lines correspond to ${\omega }_l=0.1\mathrm{\Gamma }$, $0.2\mathrm{\Gamma }$, $0.3\mathrm{\Gamma }$, and $0.4\mathrm{\Gamma }$, respectively. For the first row, we set $\mathrm{\Omega }=\sqrt{{\alpha }_1{\alpha }_2}$. For the second row, we set $\mathrm{\Omega }=-\sqrt{{\alpha }_1{\alpha }_2}$. The other parameters chosen are ${\omega }_x=1.2\mathrm{\Gamma }$, ${\omega }_e=0.5\mathrm{\Gamma }$, and $C_x\left(0\right)=C_e\left(0\right)=1/\sqrt{2}$.

Figure 6

The figure shows the influence of different initial states on the steady probability amplitudes $|C_x(t\to \infty )|$ and $|C_e(t\to \infty )|$ given by Eqs. (42) and (43) with $\mathrm{\Omega }=\alpha =\mathrm{\Gamma }$ and $\mathrm{\Phi }=2j\pi$ in the case of resonance. We set the time delay $\mathrm{\Gamma }{t}_d=2$.

Figure 7

The figure shows the influence of different initial states on the steady probability amplitudes $|C_x(t\to \infty )|$ and $|C_e(t\to \infty )|$ given by Eqs. (42) and (43) with $\mathrm{\Omega }=-\alpha =\mathrm{\Gamma }$ and $\mathrm{\Phi }=2j\pi$ in the case of resonance. We set the time delay $\mathrm{\Gamma }{t}_d=2$.

Figure 8

The figure shows the influence of the time delay $t_d$ on the steady probability amplitudes $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ with ${\omega }_g=\frac{{\omega }_l}{2}=0.2\mathrm{\Gamma }$ (corresponding to the phase $\mathrm{\Phi }=0$) in the case of resonance. In (a), the coordinates of points $A_3$, $B_3$, $C_3$, $D_3$, and $E_3$ are (2000,0.698), (2000,0.692), (2000,0.676), (2000,0.631), and (2000,0.579), respectively. In (b), the coordinates of points $A_4$, $B_4$, $C_4$, $D_4$, and $E_4$ are (2000,0.698), (2000,0.692), (2000,0.676), (2000,0.631), and (2000,0.579), respectively. The other parameters chosen are $\mathrm{\Omega }=\alpha =\mathrm{\Gamma }$, $C_x\left(0\right)=\sqrt{1/2}$, and $C_e\left(0\right)=-\sqrt{1/2}$, which corresponds to Fig. 6.

Figure 9

Evolution in time of the atomic probability amplitudes $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ given by Eqs. (13) and (14) with the different driving strength $\mathrm{\Omega }$ in the case of resonance. The parameters chosen are ${\omega }_g=\frac{{\omega }_l}{2}=0.2\mathrm{\Gamma }$ (i.e., the phase $\mathrm{\Phi }=0$), $\mathrm{\Gamma }{t}_d=1.5$, $\mathrm{\Omega }=\alpha$, $C_x\left(0\right)=\sqrt{0.8}$, and $C_e\left(0\right)=\sqrt{0.2}$. The insets in (a) and (b) magnify the probability amplitudes in the regime $t<t_d$.

Figure 10

Steady probability amplitudes $|C_x(t\to \infty )|$ and $|C_e(t\to \infty )|$ given by Eqs. (42) and (43) against time delay with $\mathrm{\Omega }=\sqrt{{\alpha }_1{\alpha }_2}$ [(a), (b)] and $\mathrm{\Omega }=-\sqrt{{\alpha }_1{\alpha }_2}$ [(c), (d)] for the off-resonance case. The other parameters chosen are $\mathrm{\Phi }=0$, ${\omega }_e=0.5\mathrm{\Gamma }$, ${\omega }_x=1.2\mathrm{\Gamma }$, $C_x\left(0\right)=\sqrt{0.8}$, and $C_e\left(0\right)=\sqrt{0.2}$.

Figure 11

The figure shows the influence of driving frequency ${\omega }_l$ on the steady probability amplitudes $|C_x(t\to \infty )|$ and $|C_e(t\to \infty )|$ given by Eqs. (42) and (43) with $\mathrm{\Omega }=\sqrt{{\alpha }_1{\alpha }_2}$ [(a), (b)] and $\mathrm{\Omega }=-\sqrt{{\alpha }_1{\alpha }_2}$ [(c), (d)] for the off-resonance case. We set the time delay $\mathrm{\Gamma }{t}_d=1.5$. The other parameters in Figs. 11 are the same as Figs. 10, respectively.

Figure 12

The figure shows the influence of the dephasing processes and external decay on $|C_x\left(t\right)|$ and $|C_e\left(t\right)|$ given by Eqs. (47) and (48) vs time with ${\mathrm{\Gamma }}_{\mathrm{tot}}\equiv {\mathrm{\Gamma }}_{\mathrm{ext}}+\mathrm{\Gamma }$ and $r=R+i\sqrt{R(1-R)}$ with $\mathrm{\Omega }=\sqrt{{\alpha }_1{\alpha }_2}$ and ${\omega }_l=2{\omega }_g$ (i.e., the phase $\mathrm{\Phi }=0$). For the first row, we set $\mathrm{\Gamma }{t}_d=1.5$, ${\omega }_x=1.2\mathrm{\Gamma }$, ${\omega }_e=0.5\mathrm{\Gamma }$. For the second row, we set ${\mathrm{\Gamma }}_{\mathrm{tot}}{t}_d=1.5$, ${\omega }_x=1.2{\mathrm{\Gamma }}_{\mathrm{tot}}$, and ${\omega }_e=0.5{\mathrm{\Gamma }}_{\mathrm{tot}}$. For (a) and (b), we fix $R=1$, ${\mathrm{\Gamma }}_{\mathrm{ext}}=0$, and vary the dephasing rate $\delta \omega$. For (c) and (d), we set $R=0.98$, $\delta \omega =0.25{\mathrm{\Gamma }}_{\mathrm{tot}}$, and vary the ratio between ${\mathrm{\Gamma }}_{\mathrm{ext}}$ and ${\mathrm{\Gamma }}_{\mathrm{tot}}$ with keeping ${\mathrm{\Gamma }}_{\mathrm{tot}}$ fixed. The other parameters chosen are $C_x\left(0\right)=\sqrt{0.8}$ and $C_e\left(0\right)=\sqrt{0.2}$.

Figure 13

One-dimensional semi-infinite waveguide coupled to an array of noninteracting three-level atoms placed at $q{x}_0$ ($q=1,2,\cdots ,N$), which is driven by the driving fields ${\mathrm{\Omega }}_q{e}^{-i{\omega }_{l}t}$ with the different Rabi frequency ${\mathrm{\Omega }}_q$ but a same driven frequency ${\omega }_l$ being assumed for all the atoms.