Transport and entanglement for single photons in optical waveguide ladders

Transfer and scattering matrix theories are derived for studying single-photon (SP) transport in optical waveguide ladders (OWLs). The OWLs consist of two one-dimensional waveguides connected by Jaynes-Cummings emitters (JCEs) and have two input and two output channels. The von Neumann entropy is introduced to describe the entanglement between the transmitted states from the two output channels. Two types of the OWLs are studied, i.e., the OWLs with two identical waveguides (i-OWLs) and those with two different waveguides (d-OWLs). When the OWLs contain only one JCE, the SP transport behavior in the i-OWLs is the same as that in the d-OWLs. When two JCEs are introduced, the quantum interference among the JCE-scattered waves can lead to the SP jumping with a 100% chance between the waveguides for the i-OWLs, while this is hard for the d-OWLs. As a result, the i-OWLs can serve as a SP router with respect to the d-OWLs. When the number of JCEs increases to a large value (e.g., 16), the transmission probabilities of the two output channels both tend to be 0.25 for the i-OWLs, but zero for d-OWLs. Correspondingly, the entanglements approximate a constant of 1 for the i-OWLs, but of zero for the d-OWLs. It shows that a large number of JCEs can suppress the influence of other system parameters including the SP frequency and JCE loss. Therefore, the i-OWLs with a large number of the JCEs show potential for a SP splitter and entanglement generator.

Figure 1

(a) Schematic drawing of an optical waveguide ladder: two 1D waveguides are connected by $N$ Jaynes-Cummings emitters (JCEs). The JCE is the optical cavity with an embedded two-level atom inside. The $n\mathrm{th}$ JCE located at $x_n(n=1,2,...,N)$ is coupled to waveguides 1 and 2 with strengths of $V_{1,n}$ and $V_{2,n}$, respectively. $t_{m,n}$ and $r_{m,n}(m=1,2)$ are the coefficients of the rightward and leftward moving photon in the $m\mathrm{th}$ waveguide between $x_n$ and $x_{n+1}$. (b) Example of dispersions for two different waveguides. They have the group velocities of $\pm 0.6c$ and $\pm 0.9c$ at the energy of ${\omega }_0$, corresponding to the wave vectors of $\pm k_0$ and $\pm 1.5k_0$, respectively, where $c$ and $k_0$ are the vacuum light speed and unit of the wave vector.

Figure 2

SP transmission and reflection spectra for the i-OWLs with one JCE ($N=1$). Parameters: ${\omega }_c={\omega }_a={\omega }_0$, $\mathrm{\Omega }=0.02{\omega }_0$, $J_1=J_2=0.005{\omega }_0$, $k_1^0=k_2^0=k_0$, $v_1^g=v_2^g=0.6c$, and ${\gamma }_a={\gamma }_c=0$.

Figure 3

(a)–(c) Contour maps of the transmission and reflection spectra for the i-OWLs with two JCEs ($N=2$), as functions of the photon energy, $\varepsilon$, and distance between the two JCEs, $\mathrm{\Delta }x$. Their values are normalized to their maxima denoted by “Max”. (d)–(f) Transmission and reflection spectra for $\mathrm{\Delta }x\in [0.3{\lambda }_0,0.7{\lambda }_0]$ with a step of $0.05{\lambda }_0$, located in the light green region in (a)–(c), respectively. For easy observation, lines are offset from the bottom with a step of 0.5. Parameters: ${\omega }_c={\omega }_a={\omega }_0$, $\mathrm{\Omega }=0.02{\omega }_0$, $J_1=J_2=0.005{\omega }_0$, $k_1^0=k_2^0=k_0$, $v_1^g=v_2^g=0.6c$, and ${\gamma }_a={\gamma }_c=0$.

Figure 4

Transmission and reflection spectra for the i-OWLs with two JCEs ($N=2$) under four large values of $\mathrm{\Delta }x$. Lines are offset from the bottom with a step of 0.5. Other parameters are the same as those for Fig. 3.

Figure 5

Transmission and reflection spectra for the i-OWLs with two JCEs ($N=2$) under four values of atom (left column) and cavity (right) losses. In calculation, $\mathrm{\Delta }x=0.5{\lambda }_0$ and other parameters are the same as those for Fig. 3.

Figure 6

Transmission, reflection, and absorption spectra for the i-OWLs with $N$ JCEs. All JCEs are uniformly spaced and the nearest-neighbor distance is $\mathrm{\Delta }x=0.5{\lambda }_0$. Lines are offset from the bottom with a step of 0.5. In calculation, ${\gamma }_a={\gamma }_c=0.001{\omega }_0$ and other parameters are the same as those for Fig. 3.

Figure 7

Contour maps of the transmission and reflection spectra for the d-OWLs with two JCEs ($N=2$), as functions of the photon energy, $\varepsilon$, and distance between the two JCEs, $\mathrm{\Delta }x$. Their values are normalized to their maxima denoted by “Max”. Parameters: ${\omega }_c={\omega }_a={\omega }_0$, $\mathrm{\Omega }=0.02{\omega }_0$, $J_1=J_2=0.005{\omega }_0$, $k_1^0=k_0$, $k_2^0=1.5k_0$, $v_1^g=0.6c$, $v_2^g=0.9c$, and ${\gamma }_a={\gamma }_c=0.001{\omega }_0$.

Figure 8

Transmission and reflection spectra for the d-OWLs with two JCEs ($N=2$) under four large values of $\mathrm{\Delta }x$. Lines are offset from the bottom with a step of 0.5. Other parameters are the same as those for Fig. 7.

Figure 9

Transmission, reflection, and absorption spectra for the d-OWLs with $N$ JCEs. All JCEs are uniformly spaced and the nearest-neighbor distance is $\mathrm{\Delta }x=0.5{\lambda }_0$. Lines are offset from the bottom with a step of 0.5. In calculation, ${\gamma }_a={\gamma }_c=0.001{\omega }_0$ and other parameters are the same as those for Fig. 7.

Figure 10

Entanglement spectra for the i-OWLs and d-OWLs with one JCE ($N=1$). The number after d-OWL denotes the incident channel of the SP. In calculation, ${\gamma }_a={\gamma }_c=0.001{\omega }_0$ and other parameters for the i-OWLs and d-OWLs are the same as those in Figs. 3 and 7, respectively.

Figure 11

Entanglement spectra for the (a) i-OWLs and (b), (c) d-OWLs with two JCEs ($N=2$) under several small and large $\mathrm{\Delta }x$. In (b) and (c) the SP is incident from the first and second channels, respectively. Lines are offset from the bottom with a step of 0.5. In calculation, ${\gamma }_a={\gamma }_c=0.001{\omega }_0$ and other parameters for the i-OWLs and d-OWLs are the same as those in Figs. 3 and 7, respectively.

Figure 12

Entanglement spectra for the (a) i-OWLs and (b)–(e) d-OWLs with $N$ JCEs ($N$ is given on the right side). In (b), (d) and (c), (e) the SP is incident from the first and second channels, respectively. Lines are offset from the bottom with a step of 0.5. In calculation, $\mathrm{\Delta }x=0.5{\lambda }_0$, ${\gamma }_a={\gamma }_c=0.001{\omega }_0$ and other parameters for the i-OWLs and d-OWLs in (b), (c) are the same as those in Figs. 3 and 7, respectively. For (d), (e), $v_1^g=0.6c$, $v_2^g=0.7c$, $k_1^0=k_0$, $k_2^0=7k_0/6$, and other parameters are the same as those in Fig. 7.