Effects of quantum interference in multiwaveguide systems bridged by Jaynes-Cummings emitters

Quantum interference, responsible for a number of resonant optical phenomena, always intrigues researchers because of its application in optical devices. This work studies it in multiwaveguide systems bridged by Jaynes-Cummings emitters (JCEs) based on the scattering matrix theory. Two types of quantum interference are distinguished here. The first is between the incident wave and those scattered from the JCEs, while the second is only among those scattered waves. The first type leads to the two transmission valleys at the two eigenfrequencies of the JCEs in single-waveguide–single-JCE coupled systems. However, the second type is responsible for the narrow transmission peaks in single-waveguide–multi-JCE coupled systems, locating in the above transmission valleys. This work first shows the properties of the second type of quantum interference in detail and then discusses its two applications. On the one hand, the second type of quantum interference can be used to tailor the transmission spectra to achieve the electromagnetically-induced-transparency-like line shapes with a large group delay. On the other hand, it can also lead to the single-photon jumping between two certain waveguides with a nearly $100\%$ chance in multiwaveguide systems by switching on or off some of the couplings of the JCEs to the waveguides. These applications with small peak widths commonly require a small loss and might have potential in quantum informatics.

Figure 1

(a) An example of a multi-WG model: four WGs with four input terminals (In-$m, m=1,2,3,4$) and four output terminals (Out-$m$) bridged by three JCEs (JCE-$n$ located at $x_n, n=1,2,3$). The JCEs are numbered from left to right. $t_{m,n}\left(r_{m,n}\right)$ represents the amplitude of the rightward-moving (leftward-moving) SP wave function in WG-$m$ between positions $x_n$ and $x_{n+1}$. (b) Slice of the $M$-WG system near JCE-$n$ that bridges WG-$m_n$ and WG-$m_n+1$. The amplitudes of the SP wave function on the left (right) side of JCE-$n$ are denoted by $t_{m,n-1}$ and $r_{m,n-1}$ ($t_{m,n}$ and $r_{m,n}$), with $m=1,2,\cdots ,M$. The coordinate axis $Ox$ is given above each panel.

Figure 2

(a) SP transmission spectra for a single WG coupled to a chain of $N$ identical JCEs. (b) Enlarged version of (a) near the two eigenfrequencies of the JCEs. The schematics of the single-WG system coupled to several identical JCEs is drawn at the top. The distances between any two adjacent JCEs are set to $d_1=d_2=\cdots =d_{N-1}=0.5{\lambda }_0$. For easy observation, the lines are offset from the bottom with a step of 1. In (a) and (b), $V=0.005{\omega }_0$ is adopted.

Figure 3

(a) Transmission spectra of a single WG coupled to two JCEs with different $d$. Lines are offset from the bottom with a step of 1. (b) Transmission spectra of a single WG coupled to three JCEs (red solid line). For comparison, the bottom and top lines in (a) are copied as the dashed black and dotted dark yellow lines, respectively. The steepness of the transmission valleys is tailored. (c) Transmission spectra of a single WG coupled to four JCEs. (d) Phase of the transmission coefficient corresponding to (c). The two insets plot the phase variations around the two narrow transmission peaks in (c). (e) Photon group delay corresponding to (c) or (d). In all panels, $V=0.005{\omega }_0$ is adopted, and the model structures are similar to that in Fig. 2.

Figure 4

(a) Transmission spectra of a single WG coupled to four JCEs with different $V$. The width of the transmission valley increases as $V$ increases. (b) Transmission spectra of a single WG coupled to five JCEs. Two narrow transmission peaks can be found in each transmission valley; see the insets. In (a), we set $d_1=0.3{\lambda }_0, d_2=0.5{\lambda }_0$, and $d_3=0.7{\lambda }_0$, the same as the values adopted in Fig. 3. In (b), $V=0.005{\omega }_0$ is adopted. Note that the model structures in (a) and (b) are drawn in Fig. 2.

Figure 5

Spectra (a) of $T_{11}=T_{22}$, of (b) $T_{21}=T_{12}$, and (c) of $R_{11}=R_{21}=R_{12}=R_{22}$. The schematics of the two WGs coupled to $N$ identical JCEs is drawn at the top. The distance between any two adjacent JCEs is set to $0.5{\lambda }_0$, i.e. $d_1=d_2=\cdots =d_{N-1}=0.5{\lambda }_0$. For convenience of observation, lines are offset from the bottom with a step of 1. Note that $V=0.005{\omega }_0$ is adopted in all cases.

Figure 6

Transmission spectra from In-1 to Out-$m$ ($m=1,2,3,4$) while switching on or off some of the couplings between the WGs and JCEs. The schematics of four WGs coupled to nine JCEs is drawn at the top, and nine JCEs are separated into three groups: G-1, G-2, and G-3. The distance between any two adjacent JCEs is set to $d=0.5{\lambda }_0$, and $V=0.005{\omega }_0$ is always adopted. Here, the index $m$ in $T_{m1}$ ($m=1,2,3,4$) also means that the former $m-1$ groups of the JCEs are switched on and the remaining groups are switched off.

Figure 7

(a) Transmission spectra of a single WG coupled to four JCEs with dissipative atoms. The three distances between two adjacent JCEs are set to $d_1=0.3{\lambda }_0, d_2=0.5{\lambda }_0$, and $d_3=0.7{\lambda }_0$, consistent with those used in Fig. 3. The value of the narrow peak decreases as the atom loss ${\gamma }_a$ increases. (b) Transmission spectra of the two-WG system coupled to three JCEs with the dissipative-atom case. The structure is drawn at the top of Fig. 6, and only the couplings of the G-1 JCEs are switched on. Note that $V=0.005{\omega }_0$ is adopted in all cases.