1. Introduction

Photons as ideal information carriers are widely applied to quantum information processing [1–4] due to their virtue of easy integration with traditional electrical communication network [5,6]. They can easily transit over a long distance at high velocity without disturbance. Therefore, the photon scattering by various nano-structures has been very extensively studied in both experiments [7,8] and theories [9–13]. Though no direct interaction between photons brings great difficulty to control photons, photon-atom interaction can help regulate photons, which makes it possible to realize quantum networks [14–16] and quantum communication [17–19]. Nevertheless, the intrinsic interaction between atom and photon is very weak in the three-dimensional free space, as a consequence of spatial mismatching between incident and scattered photon modes [20]. Purcell effect [21] is able to overcome this difficulty, i.e., the atom-photon interaction can be enhanced by confining the spatial environment of photons and atoms. One of the most widely used way is to encompass photons in a microcavity such as Fabry–Pérot cavity [22,23], photonic crystal nanocavity [24], and topological cavity [25]. The other way is to utilize a one-dimensional or quasi one-dimensional waveguide to confine photons [26–28]. Generally, quasi one-dimensional waveguides can be a line defect in photonic crystals [29,30], an optical nano-fiber [31], a surface plasmon nanowire [32–34], or a superconducting microwave line [35,36]. Based on the above two ways, a combination of cavity quantum electrodynamics and waveguide cavity quantum electrodynamics would be the more practical platform for quantum information processing. A typical model is a Jaynes-Cummings emitter (JCE) [37] coupled with a one-dimensional waveguide, which has been scrupulously considered due to its potential applications for quantum device and quantum communication [38–41]. The JCE is composed of an optical cavity with an embedded two-level atom. In particular, when a resonant single photon (SP) injected from the waveguide is scattered by the JCE, the emitted photon interferes with the incident wave and thus a 100% reflection is induced, which can function as a perfectly reflected mirror for the resonant photon [42–45]. A plane wave incidence [38,46], including only one wave component, is generally considered for the sake of simplicity. However, a pulse wave packet is the more practical case for actual signals in optical devices and quantum networks. A pulse wave packet contains many plane wave components and thus it is impossible to realize a 100% reflection even the width of pulse is considerably large [39,47]. Compared to the plane wave incidence, the wave form of the pulse wave incidence can be modified by controlling the coupling between the emitter and the waveguide [48].

With regard to the nature of the photon-photon interaction, the exploitation at the multi-photon level in a multi-waveguide system is a challenge [49–52]. In the process of quantum networks, the study of the two-photon (TP) transport is an important basis for the application of multi-photon states [27,38,47,53]. Compared with the SP packet, the TP transport is much more complicated due to the TP correlation and the nonlinearity of the JCE [54,55]. The study of the TP transport can provide important information about the JCE-induced photon-photon interaction, which is absent in the SP case. In addition, the TP scattering on a series of emitters has been typically investigated in single one-dimensional waveguide system [27,38,47]. The Hong-Ou-Mandel interferometer has been studied in the two semi-infinite one-dimensional waveguides terminally coupled together by a JCE [56], where two SPs are incident into the JCE from the two waveguides, respectively. The photon blockade effect has been suggested in the strong-coupling regime, where the TPs transport in a waveguide coupled to a JCE system [38]. Several theoretical methods have been proposed and developed, which can be roughly grouped into two types. One is the wave-function based approaches [57,58], and the other is the scattering matrix based ways, including the techniques of Bethe-ansatz method [43,59], Lehmann-Symanzik-Zimmermann reduction [38,60], input-output formalism [26,53,61], and Green’s function [62,63].

In this work, we focus on the waveguide cavity quantum electrodynamics system with two infinite one-dimensional waveguides coupled together by a JCE, which has four terminals, see Fig. 1. The four terminals are denoted as In, and Out 1 to Out 3, respectively. The exact wave function approach is utilized to exactly calculate the real-time dynamic evolution of the photon transport. Such a four-terminal optical architecture may show potential for tuning the transport of the TP states and the TP correlation in the waveguide cavity quantum electrodynamics system.

 

Fig. 1. Schematics of a four-terminal waveguide system, composed of two identical waveguides (Waveguides 1 and 2) and one JCE coupled with them with strengths $V_1$ and $V_2$. The JCE is made up of a cavity with eigenfrequency $\omega _c$ and an embedded two-level atom with transition frequency $\omega _a$. The top arrow line shows the $x$ axis, along the direction of the two waveguides. There are one incident terminal “In” and three output terminals “Out 1” to “Out 3”.

Download Full Size | PDF

The rest of this work is organized as follows. In Sec. 2, we firstly introduce the Hamiltonian of the four-terminal waveguide system in the momentum space, then derive the evolution of the incident TP state, and at last give the formulas for calculating the TP transport and correlation. Section 3 discusses the transmission characteristics of the scattered photons in the four terminals. The reflection and transmission spectra are obviously affected by the distance between the two photons and the pulse width of the incident TP wave. In Sec. 4, we present the TP correlation among the two waveguides. The non-normalized second-order correlation function in momentum space is used to demonstrate the TP scattering due to the JCE, while the normalized second-order correlation function in real space is used to describe the TP correlation between the terminals. Finally, we arrive at conclusions to summarize our work in Sec. 5.

2. Model and formulas

The four-terminal waveguide system in Fig. 1 is governed by the following Hamiltonian,

For the two-particle excitation, the state of the system corresponding to Eq. (1) takes the form,

Substituting Eqs. (1) to (5) into the Schrödinger equation of $i\hbar \partial _t|\psi \rangle = H|\psi \rangle$, one can find the coupled equation set of motion for the probability amplitudes of $\varphi _{j_1,j_2}(k_1,k_2,t)$, $\varphi _{c,j}(k,t)$, $\varphi _{a,j}(k,t)$, $C(t)$, and $A(t)$ as follows,

The numbers of the photons scattered into the terminal In ($R$) and the terminals Out 1 to Out 3 ($T_1$ to $T_3$) describe the reflection and transmission of the incident TP states, respectively. They are calculated by,

Fourier transformations of $\varphi _{j_1,j_2}(k_1,k_2)$, $\varphi _{c,j}(k)$, and $\varphi _{a,j}(k)$,

Without loss of generality, the following incident TP pulse is considered in this work, defined as the direct product of the two Gaussian waves,

The TP correlation in $k$-space is described by the equal-time second-order correlation function, defined as,

Here, $\delta$ measures the distance between the TPs in terminals $\alpha$ and $\beta$, $\varphi _{\alpha \beta }(x, x')$ is the TP wave function among the two terminals and can be directly drawn out of $\varphi _{j_1,j_2}(x_1,x_2)$, see Eq. (11). $n_{\alpha }(x)$ is the photon density on the terminal $\alpha$ and similarly can be drawn out of $n_j(x)$, see Eq. (12).

In calculation, we follow the linearization of the waveguide dispersion [46], that is, $\omega _k=\omega _0 +v_g[k- (\pm k_0)]$ for the rightward- and leftward-moving branches, where $v_g$ is the photon group velocity and $\pm k_0$ are the two centers of the linearization. The central wave vector of the incident TP pulse $k_p$ determines the energy of the incident photon by $\varepsilon _p=\omega _0 + v_g(k_p-k_0)$. $\omega _0$ and $k_0$ are taken as the units of the system energy and wave vector, respectively, with the corresponding units of time and length being $t_0 = 2\pi /\omega _0$ and $\lambda _0 = 2\pi /k_0$. The group velocity $v_g=\left.{d\omega \over dk}\right |_{k=k_0}$ is assumed as the unit of the velocity. The JCE is placed at the original point $x=0$ and its parameters are taken as follows: for the cavity $\tilde {\omega }_c = \omega _c - i\gamma _c$ with the energy $\omega _c=\omega _0$ and the loss $\gamma _c=0.001\omega _0$; for the two-level atom $\omega _g=0$ and $\tilde {\omega }_e = \omega _a - i\gamma _a$ with the transition energy $\omega _a=\omega _0$ and the loss $\gamma _a=0.001\omega _0$; the Rabi coupling between the cavity and the two-level atom equals $g=0.05\omega _0$. The coupling between the cavity and waveguides 1 and 2 are taken as $V_{1k} = V_{2k} \equiv V= 0.15\omega _0{\cdot }\lambda _0^{1/2}$. The parameter values are chosen mainly accordingly to Ref. [46] whose theoretical results are in good agreement with the experimental work [64]. The cavity-topological waveguide coupling system [25] also suggests the present adopted parameter values. In addition, the fourth-order Runge-Kutta algorithm is adopted to solve the time-dependent Eq. (6) in the momentum space. The considered $k$ space is confined in $[-k_p-10k_w, -k_p+10k_w]\cup [k_p-10k_w, k_p+10k_w]$ and the discrete step is $0.05k_w$. The availability of this kind of $k$-discretization can be confirmed from the time evolution of the photon densities in waveguides 1 and 2. The numerical tests indicate that the total particle number always remains 2 throughout the whole photon scattering process when the losses of the cavity and the two-level atom are neglected.

3. Scattering of the TP pulse

The photon probabilities scattered into the four terminals are plotted as functions of the incident photon energy $\varepsilon _p$ in Figs. 2(a-f). For small $k_w=0.01k_0$ the width of the incident TP pulse $x_w$ is much larger than the photon wavelength, so that their transmission and reflection spectra approach those of the SP plane wave. The two resonant scattering points are at the JCE eigenfrequencies, $\varepsilon _{\pm }=\omega _0\pm g = 0.95\omega _0\ \textrm {and}\ 1.05\omega _0$. For the incident SP plane wave, the scattered photon probabilities into the four terminals can be analytically found as [65]

The adoption of $V_{1}=V_{2}\equiv V$ is responsible for $R=T_2=T_3$, which still holds up for the TP scattering, referred to Figs. 2(a-f) where the solid light gray lines give the transmission and reflection for the SP case with the Gaussian incident wave function $\propto e^{-(k-k_p)^2/k_w^2}$. If neglecting the JCE losses, i.e., $\gamma _a$ and $\gamma _c$, the $\cal J$ becomes divergent at $\varepsilon _{\pm }$ and thereby the system has $T_1=R=T_2=T_3=0.25$ for the incident SP plane wave, referred to Figs. 2(a, b). They give an estimation for the incidence of the wide TP pulse, that is, all the probabilities scattered into the four terminals approach 0.25 when $\varepsilon _p=\varepsilon _{\pm }$ in Figs. 2(a, b). Since the JCE is resonantly excited when $\varepsilon _p=\varepsilon _{\pm }$, it provides the strongest scattering on the incident TP pulse. Accordingly, the photons scattered into the Out 1 are depressed, while those scattered into the terminals In, Out 2, and Out 3 are enhanced. That is, the spectra $T_1$ present transmission valleys (TVs) at $\varepsilon _{\pm }$, while the spectra $R$, $T_2$, and $T_3$ present transmission humps (THs). The two incident photons at $\varepsilon _p=\varepsilon _{\pm }$ are almost equally scattered into the four terminals. However, when $\varepsilon _p=\omega _0$ the two photons are almost fully scattered into the terminal Out 1, because the JCE is in non-resonant excitation.

 

Fig. 2. Left three columns: Transmission and reflection spectra with $k_w=0.01 k_0$, $0.05 k_0$, and $0.1 k_0$, respectively. The solid light gray lines give the transmission and reflection for the incident single photon (SP) pulse $\propto e^{-(k-k_p)^2/k_w^2}$ and others give those for the incident TP pulse defined in Eq. (13). Right two columns: Photon densities for the incident (light gray solid lines) and scattered (red solid and blue dashed lines) TP pulses with $k_w= 0.05 k_0$. Other parameters: $\omega _a = \omega _c = \omega _0$, $\gamma _a = \gamma _c = 10^{-3}\omega _0$, and $g = 0.05\omega _0$.

Download Full Size | PDF

As $k_w$ increases, $T_1$ becomes much larger than $R$, $T_2$ and $T_3$, comparing Figs. 2(e, f) with Figs. 2(c, d) or Figs. 2(a, b). This is also confirmed by the Figs. 2(g, j) where $k_w=0.05k_0$. Figures 2(g, j) both show that the photon density in terminal Out 1 is much larger than those in terminals In, Out 2, and Out 3. Since $R$, $T_2$, and $T_3$ are all from the scattering of the JCE, their photon densities are identical with each other.

For the very broad TP pulse the TP correlation shows weak influence on the photon transmission and reflection. This is reflected by Figs. 2(a, b) where all curves do not have an obvious change from the SP transmission and reflection, when $d$ increases from 0 to $2x_w$. For large $d$ the transmission and reflection spectra of the TP pulse tend to those of the SP one if they have the identical pulse width, comparing the green dotted curves and the gray ones in Figs. 2(a-f). This is due to that the TP correlation induced by the JCE almost disappears, for which there are two situations. In the first one, the average distance between the TPs is much larger than the pulse width $x_w$, that is, there exists a large time delay between the two photons reaching the JCE and therefore, the first photon pulse arriving at the JCE almost does not affect the second one. This is demonstrated in Figs. 2(i) and 2(l) where the photon densities of the scattered TP pulses have two peaks with almost the same heights. In the second situation, the TP pulse is much broader than the photon wavelength, i.e. $x_w\gg \lambda _0$. The probability of finding two photons at the same time-space point is very small and accordingly, the single excitation of the two-level atom shows a very weak influence on the TP transmission and reflection, see Fig. 2(a).

When $d$ is small and $x_w$ is about a few wavelengths, the single excitation of the two-level atom exerts a definite effect on the TP transmission and reflection, see Figs. 2(c, d) where $x_w={1\over 0.36 k_w}={1\over 0.36 \times 0.05k_0}\approx 8.8\lambda _0$. The transmissivity $T_1$ at $\varepsilon _\pm$ first decreases and then increases as $d$ increases from 0 to $2x_w$. If the two-level atom is in the ground state, the photon arriving at the JCE would be resonantly scattered into the four terminals. On the other hand, when the JCE has been resonantly excited by the first arriving photon, the second arriving photon would not be resonantly scattered. Note that the single-excitation levels of the JCE are just $\varepsilon _\pm =\omega _0\pm g$, while the double-excitation levels of the JCE are $\omega _0\pm \sqrt {2}g$. Therefore, the transitions between the single-excitation levels and the double-excitation ones are about $\omega _0\pm 0.4g$ and $\omega _0\pm 2.4g$, not equal to $\varepsilon _\pm$. This non-resonant scattering decreases as $d$ increases, leading to that $T_1$ first decreases for increasing $d$ from zero at $\varepsilon _\pm$. Moreover, the above resonant and non-resonant scattering processes by the JCE would interference with each other, resulting in the following increase of $T_1$ at $\varepsilon _\pm$ after its first decrease with increasing $d$, see the variation of $T_1$ at $\varepsilon _\pm$ in Fig. 2(c). At $\varepsilon _p=\varepsilon _0$ the two above scattering processes are both non-resonant and thus the incident photons prefer to be scattered into the terminals Out 1, but their mutual interference would influence $T_1$. The variations of $R$, $T_2$, and $T_3$ with increasing $d$ are just opposite with that of $T_1$, see Fig. 2(d). These characteristics of the four scattering probabilities are also held when the pulse width $x_w$ further decreases, see Figs. 2(e, f) where the transmission spectra for $d=0$ and $2x_w$ are not identical, referred to the enlarged insets. As $d$ increases to the values larger than $2x_w$, the TP transmission spectrum would further tend to that of the SP case. The TPs are identical and the incident TP wave has two pulses as $d\neq 0$, while the incident SP wave has only one. This difference is responsible for the deviation of the TP transmission from the SP’s in the large-$d$ case.

Figures 2(g-l) show the photon densities [i.e., $n_1(x)$ and $n_2(x)$] of the scattered TP pulses when $\varepsilon _p$ takses $\varepsilon _-$ and $\omega _0$, respectively. The photon densities for $\varepsilon _p=\varepsilon _+$ are similar with those for $\varepsilon _p=\varepsilon _-$. As $d=2x_w$, the transmitted TP pulses that correspond to $T_1$ have two peaks and show a weak difference with the incident TP pulses except the strength, see Figs. 2(i, l). As $d=x_w$, the two peaks in the transmitted TP pulses that correspond to $T_1$ have different strengths, see Figs. 2(h, k), indicating the influence of the TP correlation on the TP transmission. As $d=0$ the influence of the TP correlation on the TP transmission continues to increase but becomes hard to be directly observed, since these two peaks coincide together, see Figs. 2(g, j). When $\varepsilon _p=\varepsilon _-$, $T_1$ is just a little larger than $R$, $T_2$, or $T_3$. However, when $\varepsilon _p=\omega _0$, $T_1$ is much larger than $R$, $T_2$, or $T_3$. This point is consistent with the transmission and reflection spectra in Figs. 2(c, d). In all, the TP correlation can exerts a definite effect on the TP scattering when the TP distance is small and the width of the TP pulse is about a few wavelengths.

4. TP correlation

Although the transmission spectra depend on the integrated correlation, see Eqs. (7) and (14), the TP correlations contain more details in the photon scattering, described by the second order correlation function $G_{j_1j_2}(k_1,k_2)$ defined in Eq. (14). Since the single excitation of the two-level atom definitely presents strong influence on the TP transport when $k_w=0.05k_0$, see Figs. 2(c, d), this section focuses on the incident TP pulse with $k_w=0.05k_0$ and simultaneously, two energies of the incident photons are adopted, i.e., $\varepsilon _p=\varepsilon _-$ and $\omega _0$. The case for $\varepsilon _p=\varepsilon _+$ is similar with that for $\varepsilon _p=\varepsilon _-$.

Figure 3 shows the resonant excitation case with $\varepsilon _p=\varepsilon _-$ in the left half and the non-resonant one with $\varepsilon _p=\omega _0$ in the right half, where $d=0$, $x_w$, and $2x_w$ for the first, second, and third rows, respectively. The time evolutions of the photon distributions in the waveguides 1 and 2, i.e., $n_1(x)$ and $n_2(x)$, are plotted in the left two columns in both cases. When the TP pulse incident from the waveguide 1 arrives at the JCE, i.e., $x=0$, the photons would be partially scattered back into the terminal In and forward into the terminal Out 1, resulting in the leftward-moving component for the photon density $n_1(x)$, see the left column in both cases. In the waveguide 2, no photon exists before the incident TP pulse arrives at the JCE. After the JCE is exited, the photons can jump into the waveguide 2 and move toward the terminals Out 2 and Out 3, leading to the rightward- and leftward-moving components for the photon density $n_2(x)$, see the center column in both cases in Fig. 3. As the average distance of the two incident photons is two times larger than the pulse width, i.e., $d=2x_w$, the TP pulse presents two humps, see Figs. 3(g, h) and Figs. 3(p, q). The scattered waves are a little wider than the incident pulse, because the JCE can keep the excitation for a while. The forms of $n_1(x)$ and $n_2(x)$ do not change before and after the photon pulse reaches the JCE, indicating that the numerical scheme is convergent.

 

Fig. 3. Time evolution of photon densities in waveguides 1 and 2, $n_1(x)$ and $n_2(x)$, and corresponding TP correlation, $G_{11}(k_1,k_2)$, for the scattered TP pulses with $\varepsilon _p=\varepsilon _-$ (left half) and $\varepsilon _p=\omega _0$ (right half). The parameter $d$ takes $0$, $x_w$, and $2x_w$ for the first, second, and third rows, respectively. For easy observation, all photon densities and correlations are normalized to their maximums which are provided in each panel. Each correlation maps are divided into four subregions according to the moving directions of the photons, i.e., leftward (L) and rightward (R). All subregions have the identical side length $0.1k_0$ and their center points are denoted in their panels. $k_-$ is the wave vector of the photon with the energy $\varepsilon _-$. Other parameters: $\varepsilon _-=0.95\omega _0$, $\omega _a = \omega _c =\omega _0$, $\gamma _c = \gamma _a = 10^{-3} \omega _0$, $g = 0.05\omega _0$, and $k_w = 0.05k_0$.

Download Full Size | PDF

The right columns in both cases in Fig. 3 show the TP correlations $G_{11}(k_1, k_2)$ after the scattering of the JCE. $G_{j_1j_2}(k_1,k_2)$ describes the correlation between the two photons: one in the waveguide $j_1$ with the wave vector $k_1$ and the other in the waveguide $j_2$ with the wave vector $k_2$. According to their moving directions, i.e., the signs of $k_1$ and $k_2$, $G_{j_1j_2}(k_1,k_2)$ can be divided into the four types, i.e., ‘LR’, ‘RR’, ‘RL’, and ‘LL’. These notations are given in Figs. 3(c, l) as examples. The notation ‘L’ denotes the left-moving photon ($k<0$), while the notation ‘R’ denotes the right-moving one ($k>0$). For convenience, $G_{j_1j_2}(k_1,k_2)$ in these four regions are denoted as $G_{j_1j_2}^\textrm {LR}$, $G_{j_1j_2}^\textrm {RR}$, $G_{j_1j_2}^\textrm {RL}$, and $G_{j_1j_2}^\textrm {LL}$, respectively, whose plotted ranges for $(k_1,\ k_2)$ are provided in the figure caption. The TP correlations satisfy the following relations,

Equation (14) directly tells $G_{j_1j_2}(k_1,k_2)=G_{j_2j_1}(k_2,k_1)$, which leads to Eqs. (18a) and (18b). Since the reflection of the terminal In and the transmissions of the terminals Out 2 and Out 3 are all from the scattering of the JCE, the correlations of a channel with them are identical with each other, which is responsible for Eqs. (18c–18e). The Eq. (18) indicates that only the TP correlation $G_{11}(k_1, k_2)$ is independent, i.e., it can give the rest ones. As a result, only the distribution of $G_{11}(k_1, k_2)$ is shown in the right columns in both cases, see Fig. 3 where three different $d$ values are considered.

Since the spectra of $T_1$ present the TVs for $\varepsilon _p=\varepsilon _-$, see Fig. 2, the TP correlation $G_{11}(k_1,k_2)$ holds the weak correlation lines if the photons move rightward in the waveguide 1, denoted as the “TV” in Fig. 3(c). On the other hand, because $R$, $T_2$, and $T_3$ present the THs when $\varepsilon _p=\varepsilon _-$, $G_{11}(k_1,k_2)$ has the strong correlation lines, denoted as the “TH” in Fig. 3(c). These weak and strong correlation lines, henceforth, will be called as the TV and TH lines for convenience. They are, respectively, at $k=\pm k_-$ for $G_{11}(k_1,k_2)$, where $k_-$ is the wave vector of the photon with the energy $\varepsilon _-$. When the two photons that are both transmitted into the terminal Out 1 ($k_1>0,\ k_2>0$), the TP correlation $G_{11}^\textrm {RR}(k_1, k_2)$ holds two TV lines. On the contrary, the map of $G_{11}^\textrm {LL}(k_1, k_2)$ in Fig. 3(c) presents a bright spot when the two photons are both reflected back into the terminal In ($k_1<0,\ k_2<0$). For the LR and RL regions, one photon is reflected and the other is transmitted (i.e., $k_1>0,\ k_2<0$ or $k_1<0,\ k_2>0$). Therefore, $G_{11}^\textrm {LR}(k_1,k_2)$ and $G_{11}^\textrm {RL}(k_1,k_2)$ present one TV line and one TH line. Since $R=T_2=T_3\approx 0.15$ is a little smaller than the transmission $T_1\approx 0.5$, $G_{11}^\textrm {LL}(k_1,k_2)$ and $G_{11}^\textrm {RR}(k_1,k_2)$ have the smallest and largest strengths, respectively, and $G_{11}^\textrm {LR}(k_1,k_2)$ and $G_{11}^\textrm {RL}(k_1,k_2)$ measure the correlation strengths between them. As the average distance between the two photons, $d$, increases, the incident beam gradually becomes the two-hump shape, and so do the scattered beams, see Figs. 3(d, e) where $d=x_w$ and Figs. 3(g, h) where $d=2x_w$. For large $d$, the TP correlation presents an interference pattern, see Fig. 3(i). The interference patterns are splitted by the TV lines. This interference makes the strengths of the TP correlation increase, comparing Fig. 3(i) with Figs. 3(c, f).

For $\varepsilon _p=\omega _0$, $T_1\approx 0.75$ is much larger than $R=T_2=T_3\approx 0.05$, see Figs. 2(c, d). As a result, the right-moving photon number in waveguide 1 is much larger than those reflected into the terminal In and those jumping into waveguide 2, consistent with the left two columns in the right half in Fig. 3. This can also be seen from the TP correlation, i.e., the right column in the right half in Fig. 3, where the correlation $G_{11}^\textrm {RR}$ is much larger than $G_{11}^\textrm {LR}$, $G_{11}^\textrm {RL}$, and $G_{11}^\textrm {LL}$. The TH of $T_1$ at $\varepsilon _p=\omega _0$ is responsible for the bright spot of $G_{11}^\textrm {RR}$, referred to Figs. 2(c) and 3(l). This corresponds to the TH lines, opposite with the behavior of $G_{11}^\textrm {RR}$ in Fig. 3(c). On the other hand, the TV lines for $G_{11}^\textrm {LR}$, $G_{11}^\textrm {RL}$, and $G_{11}^\textrm {LL}$ are due to the TVs of $R$, $T_2$, and $T_3$ at $\varepsilon _p=\omega _0$, see Figs. 2(d) and 3(l). The TH and TV lines locate at $k=\pm k_0$, respectively. As $d$ increases, the interference pattern for the TP correlation $G_{11}(k_1,k_2)$ appears and the bright pattern area shrinks, which are similar with those in the case of $\varepsilon _p=\varepsilon _-$. Since $G_{12}(k_1,k_2)$, $G_{21}(k_1,k_2)$, and $G_{22}(k_1,k_2)$ can be derived by the relations in Eq. (18), they are not provided here. For the excitation case with $\varepsilon _p=\varepsilon _+$, the time evolution of the photon densities and the TP correlation $G_{11}(k_1,k_2)$ are similar with those of $\varepsilon _p=\varepsilon _-$. In both cases of $\varepsilon _p=\varepsilon _-$ and $\varepsilon _p=\omega _0$, the number of the photon scattered into waveguide 1 is always larger than that into waveguide 2, which can be seen from the transmission and reflection spectra with $k_w=0.05k_0$ showed in Figs. 2(c, d). Thus, $n_1(x)$ is larger than $n_2(x)$ in Fig. 3. Note that the case of $\varepsilon _p = \varepsilon _-$ belongs to a resonant scattering, while that of $\varepsilon _p = \varepsilon _0$ to a non-resonant one. As a result, the reflection in waveguide 1 for $\varepsilon _p=\omega _0$ is smaller than that for $\varepsilon _p=\varepsilon _-$, see Figs. 3(a, d, g) and Figs. 3(j, m, p). These resonant and non-resonant scatterings are also responsible for the pattern difference for $n_1(x)$ and $n_2(x)$ between the two cases.

The TP correlation $G_{j_1j_2}(k_1, k_2)$ can provide the scattering information of the TP pulse, but it is commonly not easy to measure. Thus, the normalized second order TP correlation in $x$ space, i.e., $g_{\alpha \beta }^{(2)}(\delta )$ in Eq. (15), is introduced to describe the correlation between the terminals. Similar with $G_{j_1j_2}(k_1, k_2)$, the normalized TP correlations between the terminal Out 1 and the other three ones are the same with each other. In addition, the normalized TP correlations between any two terminals (can be the same) of those In, Out 2, and Out 3 are identical too. As a result, we only consider the normalized TP correlation for the terminals Out 1 and Out 2, see Fig. 4 where the subscribes 1 and 2 in $g^{(2)}$ denote the terminals Out 1 and Out 2, respectively. For $d=0$ the normalized TP correlation of the incident wave is always equal to 0.5, being consistent with that in the bi-occupation Fock state (i.e., ${\langle 2|\hat a^\dagger \hat a^\dagger \hat a\hat a|2\rangle \over \langle 2|\hat a^\dagger \hat a|2\rangle ^2}={1\over 2}$), see Figs. 4(a, d). However, all $g_{\alpha \beta }^{(2)}$ for the scattered TP pulses present an oscillatory form. $g_{11}^{(2)}$ and $g_{22}^{(2)}$ are even with respect to $\delta$, while $g_{12}^{(2)}$ and $g_{21}^{(2)}$, satisfying $g_{12}^{(2)}(\delta )=g_{21}^{(2)}(-\delta )$, are not, because of the photon delay between the two terminals. $g_{11}^{(2)}$ presents a peak and a valley at $\delta =0$, respectively, for $\varepsilon _p=\varepsilon _-$ and $\varepsilon _p=\omega _0$, see Figs. 4(a, d). For $g_{22}^{(2)}$, since it always presents a peak at $\delta =0$, larger than 0.5, the two photons should prefer together scattering when they are scattered into the terminal Out 2.

 

Fig. 4. Normalized second-order correlations for the incident and scattered TP pulses. The upper and lower row present the correlation of $\varepsilon _p=\varepsilon _-$ and $\varepsilon _p=\omega _0$ denoted on the right side, respectively. The parameter $d$ takes 0, $x_w$ and $2x_w$ for the first, second and third columns.

Download Full Size | PDF

For increasing $d$, the normalized TP correlation for the incident TP pulse decreases to zero and tends to 1 for small and large $|\delta |$, respectively, see Figs. 4(b, c) or 4(e, f). When $d=2x_w$, all $g^{(2)}$ for the scattered TP pulse tend to that for the incident TP pulse and become more smoother than those in the case with $d=0$, see Figs. 4(c, f). This is physically reasonable, since the influence of the single excitation of the two-level atom on the TP scattering is weak when the distance between the two photons is large. $g_{11}^{(2)}$ and $g_{22}^{(2)}$ present (or approximately present) the minima at $\delta =0$, which are more different from the case with $d=0$. However, the photon delay between the terminals Out 1 and Out 2 can also be observed from $g_{12}^{(2)}$ and $g_{21}^{(2)}$, because they present right and left shifts with respect to the TP correlation of the incident TP pulse, see Figs. 4(c, f). When $d=x_w$, one key feature for $g^{(2)}$ is that $g_{22}^{(2)}$ has a large peak value at $\delta =0$ when $\varepsilon _p=\omega _0$, i.e., $g_{22}^{(2)}(0)\approx 3.5$, being much larger than the other $g^{(2)}$. This tells that the two photons more prefer together scattering when they are scattered into the terminal Out 2, with respect to the cases with $d=0$ or $d=2x_w$.

According to $G_{j_1j_2}(k_1,k_2)$ and $g_{\alpha \beta }^{(2)}(\delta )$, one can see that the TP correlation in the scattered TP state depends on the width of the Gaussian incident TP pulse, terminals, excitation energy, and distance between the two photons. These have potential for controlling the features of the TP pulses.

5. Discussion and conclusion

The transmission and correlation properties of the TP pulse and their dependence on the incident wave form were studied in a four-terminal waveguide system. The four-terminal waveguide system includes two identical one-dimensional waveguides (waveguides 1 and 2) and one JCE coupled with them and show potential for inducing photon-photon correlation and controlling TP transport in waveguide cavity quantum electrodynamics systems. The spectra of the photons scattered into the four terminals present the valleys or humps. Before the TP pulse arrives at the JCE, it freely transports in waveguide 1 and there is no photon in waveguide 2. As it arrives at the JCE, the TP pulse would be scattered, during which the photons have the opportunity not only to be reflected in the waveguide 1, but also to jump into the waveguide 2 and subsequently transmit to the two terminals of the waveguide 2. The TP shapes scattered into the four terminals change as the photon energy and the distance between the TPs. The scattered waves are wider than the incident pulse, because the excitation could be kept by the JCE for a while. When the average distance between the two photons is much larger than the TP pulse width or the TP pulse is much broader than the photon wavelength, the photon probabilities scattered into the four terminals are almost the same with those of the incident SP pulse. Otherwise, the single excitation of the two-level atom in the JCE exerts an obvious effect on the TP scattering. When the TP pulse is a few times the photon wavelength, the TP pulse contains plenty of the plane-wave components and thus the scattering spectra become smooth.

The equal-time second order correlation function $G_{j_1,j_2}(k_1,k_2)$ between waveguides $j_1$ and $j_2$ and normalized second-order correlation function $g_{\alpha \beta }^{(2)}(\delta )$ between two terminals $\alpha$ and $\beta$ are both utilized to describe the TP correlations. The characteristics of the photon scattering spectra are imprinted on the TP correlations. As the scattering spectra presents a valley, the TP correlations $G_{j_1,j_2}(k_1,k_2)$ have a weak strength. On the contrary, as the scattering spectra present a hump or a peak, the TP correlations $G_{j_1,j_2}(k_1,k_2)$ have a strong strength. When the average distance between the two photons increases, the effective photon-photon interaction due to the double-excitation levels of the JCE decreases, accompanying with an interference pattern in the distribution of the TP correlations. This interference from the distance between the two photons can enhance the TP correlations. When the distance between the TPs is small, the normalized TP correlations between any two terminals $g_{\alpha \beta }^{(2)}(\delta )$ for the scattered TP pulse are much different from those for the incident TP pulse. The TP correlations in the present four-terminal waveguide system show dependence on the photon energy, incident Gaussian pulse width, and the distance between the TPs and thus such a four-terminal waveguide system has potential to adjust the TP correlation.