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Abstract
In this work, the single photon scattering due to a giant atom coupled with a pair of waveguides is investigated theoretically. Using the real-space Hamiltonian, four scattering amplitudes are obtained, and the single photon routing properties are studied. Calculations reveal that the single photon routing properties are strongly dependent on the size of the giant atom. The possible physical mechanism is also discussed. To improve routing efficiency, the configuration where one waveguide is terminated is further studied. The calculated results indicate that an incident photon can be transferred to the other waveguide with unit efficiency by choosing the appropriate configuration for a fixed size of the giant atom. Our results may be used in quantum information processing and design quantum devices at single-photon level.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The interactions between atoms and one-dimensional confined electromagnetic fields, which are referred to as waveguide quantum electrodynamics (WQED), have garnered significant research attention owing to their potential applications in quantum computation and quantum information processing [1–14]. These interactions serve as a suitable platform for investigating many quantum optics phenomena, such as superradiance, subradiance [15–20], and bound states [21,22]. Several previous works have reported developments in WQED [23–28]. Given that waveguides can serve as quantum channels to transfer quantum information, the realization of a single photon router based on WQED, which can transmit quantum information to different nodes, remains an intriguing topic [29–48]. However, in most of these previous studies, the atom was considered to be considerably smaller than the wavelengths of the waveguide modes. Researchers have also employed dipole approximation to consider the atoms as single points.
In recent years, giant atoms, which disrupt dipole approximation, have been investigated in the field of quantum optics. Giant atoms are coupled with light at several points, leading to many interesting effects arising due to interference. Kockum et al. first studied quantum optics with a single giant atom [49]. Using the Markovian approximation, they derived the standard master equation for a giant atom and discussed the frequency-dependent relation. Subsequently, many studies have reported on the quantum optics effects induced by giant atoms, including the time delay [50,51], creation of bound states [52–54], and electromagnetically induced transparency [55,56]. The collective behaviors of giant atoms are also significantly different from those of small atoms [57–60], because decoherence-free coupling can occur between two giant atoms [57]. Studies have also reported experimental implementations of giant atoms using superconducting qubits coupled with either surface acoustic waves or transmissions and the coupling of small atoms with a waveguide [61–64].
Recently, quantum information processing based on giant atoms has been widely discussed [65–67]. Entanglement preparation [65], the establishment of a chiral quantum network [66], and a frequency converter at the single-photon level achieved using a giant atom [67] have been discussed. In this study, we investigate the single photon scattering caused by the coupling of a giant atom with two waveguides. Calculations reveal that the single photon routing properties are strongly dependent on the size of the giant atom. Furthermore, it is demonstrated that an incident single photon can be transferred to the other channel with 100$\%$ probability. The results of this work are expected to be applicable in quantum information processing using giant atoms .
2. Configuration and theoretical model
The configuration considered in this study is shown in Fig. 1. A giant atom is a two-level system with an excited state $|e\rangle$ and a ground state $|g\rangle$. The giant atom is connected to the two linear waveguides, waveguide M (WM) and waveguide N (WN), at $x=0$ and $x=L$[57,67], respectively. The coupling strength between the giant atom and WM(WN) is denoted as $V_{m}(V_{n})$. The Hamiltonian describing this system can be expressed as follows (hereafter, we set $\hbar =1$):
Here, $v_{g}$ is the group velocity of the photons traveling in the waveguides. We assumed that the two waveguides were identical with the same group velocity. $c^{\dagger}_{Rp}$ ($c^{\dagger}_{Lp}$) denotes the Bose creation operator for the right-going (left) photon at position $x$ in waveguide $p$. $\omega _{e}$ denotes the transition frequency between the states $|e\rangle$ and $|g\rangle$. $\sigma _{ij} (i,j=e,g)$ denotes the dipole transition operator of the giant atom. $\delta (x)$ is the Dirac-$\delta$ function.
Fig. 1. Configuration of giant atom-mediated single photon router (a). The giant atom is coupled with WM and WN at $x=0$ and $x=L$, respectively. Energy-level configuration of the two-level system (b).
Since only single photon scattering processing is considered and the excitation is converted, the wave function of the system can be written as
Assuming a single photon incident from the left of WM, the wavefunctions in Eq. (2) can be expressed as
where $h(x)$ is the step function, with $h(0)=1/2$. $e^{ikx}h(-x)$ in Eq. (3) denotes the single photon incident from the left of WM. $e^{ikx}ah(x)h(L-x) (e^{ikx}ch(x)h(L-x))$ in Eq. (3) (Eq. (5)) describes the wave function of the right-propagation single photon between $x=0$ and $x=L$ in WM(WN) and $a$($c$) denotes the probability amplitude. $e^{-ikx}bh(x)h(L-x) (e^{-ikx}fh(x)h(L-x))$ in Eq. (4) (Eq. (6)) represents the wave function of the left-propagation single photon between $x=0$ and $x=L$ in WM(WN). $b$($f$) denotes the probability amplitude. The last term $e^{ikx}t_{m}h(L-x)$ describes the single photon transmitted from WM with the transmission amplitude $t_{m}$. $r_{m}$ in Eq. (4) is the scattering amplitude that describes the single photon reflected back in WM; $t_{n}$ and $r_{n}$ denote the single photon transmission and reflection amplitude in WN, respectively.Substituting the expressions in Eqs. (3) to (6) into the eigenvalue equation $H|\omega \rangle =\omega |\omega \rangle$, we obtain
3. Single photon scattering due to giant atom
Before discussing the single photon routing properties mediated by the giant atom, a review of the case where a single photon is scattered by a small atom is presented. The four scattering amplitudes are given by $t_{ms}=(\Gamma _{n}-i\Delta )/(\Gamma _{m}+\Gamma _{n}-i\Delta )$, $r_{ms}=-\Gamma _{m}/(\Gamma _{m}+\Gamma _{n}-i\Delta )$, and $t_{ns}=r_{ns}=-2\sqrt {\Gamma _{m}\Gamma _{n}}/(\Gamma _{m}+\Gamma _{n}-i\Delta )$[37]. Here, the subscript $s$ for the four scattering amplitudes denotes the scattering amplitudes for the case of a small atom. When $\Gamma _{m}=\Gamma _{n}=\Gamma$ and the incident photon is resonant with the two-level system $\omega =\omega _{e}$, $t_{ms}=r_{ms}=t_{ns}=r_{ns}=1/2$. The single photon routing efficiency ($T_{ns}+R_{ns}$) reaches a maximum value of 0.5. Furthermore, there is no peak in the transmission spectrum ($T_{ms}$ as a function of $\omega$) [37].
Now, we discuss the properties of a single photon scattered by the giant atom. First, the simple case with $\Gamma _{m}=\Gamma _{n}=\Gamma$ is considered. The four scattering amplitudes in Eqs. (7) to (10) can be, respectively, degenerated into
Figure 2 shows the single photon scattering probability $T_{m}\equiv |t_{m}|^{2}$ and $T_{n}\equiv |t_{n}|^{2}$ with respect to detuning $\Delta$ and phase $\theta$. When $\theta =(2p+1)\pi$, where $p$ is an integer, $r_{m}=t_{n}=r_{n}=0$ and $t_{m}=1$, which implies that the incident photon can be transmitted directly. As stated before, this phenomenon does not occur in the case of a small atom coupled with two waveguides. The physical mechanism of this result can be explained based on the interference effect. For a giant atom, two mediated terms are introduced to describe the photon in the $x=0$ and $x=L$ regions; this is not possible for the case involving a small atom . Thus, the appearance of the new interference path induces the $\theta$-dependent scattering amplitudes. This phenomenon can be understood as follows. The scattering amplitude $t_{m}$ can be rewritten as
On comparing the expressions of $t_{ms}$ and $t_{m}$, it is found that, for a giant atom, an effective coupling strength $\Gamma ^{'}=4\Gamma \cos ^{2}\theta /2$ between the giant atom and the waveguide can be introduced. $\Gamma ^{'}$ is strongly dependent on $\theta$. When $\theta =(2p+1)\pi$, $\Gamma ^{'}$ reaches zero, which implies that the giant atom and the waveguide are decoupled. Thus, the incident single photon can be freely transmitted.
Fig. 2. $T_{m}$ (a) and $T_{n}$ (b) with respect to detuning $\Delta$ and phase $\theta$. (c) and (d) are cut from (a) and (b) at $\theta =\pi /2,\pi /4$ and $\pi /6$, respectively. In the calculations, $\Gamma =10^{-5}\omega _{e}$.
Figure 2 (d) also clearly indicates that the peak of $T_{n}$ shifts from the resonant frequency $\omega _{e}$, which is different from the case involving the small atom. By setting the denominator of the scattering amplitudes to be zero, it can be determined that the peak shifts to $\omega _{e}+4\Gamma \sin \theta$. Furthermore, the maximum value of $T_{n}$ is 0.25. As $R_{n}\equiv |r_{n}|^{2}$, which is equal to $T_{n}$, the maximum value of the possibility ($P_{n}=T_{n}+R_{n}$) of an incident single photon from WM being transferred to WN is 0.5.
To elucidate the effects of the coupling strength on the single photon scattering properties, we plot $T_{m}$, $T_{n}$, $R_{m}\equiv |r_{m}|^{2}$ and $R_{n}$ with respect to the detuning $\Delta$ and $\eta \equiv \Gamma _{n}/\Gamma _{m}$ with $\theta =\pi /6$, as shown in Fig. 3. When $\eta =1$, $T_{n}=R_{n}$ can reach a maximum value of 0.25. When $\eta$ is in the region of $[0.8, 1.2]$, $T_{n}$ and $R_{n}$ can still exceed 0.24. If $\eta$ is much less than 1, which means the coupling between the giant atom and the WN is very weak, $R_{m}$ can approach to 1 due to the weak influence of WN. The giant atom plays like a quantum mirror. However, when $\eta$ increases, the coupling between the giant atom and the WN becomes strong, and the interaction between the giant atom and the WM changes to be weak. Thus the single photon incident from the WM can pass the system easily, which leads to $T_{m}$ increasing.
Fig. 3. $T_{m}$ (a), $T_{n}$ (b), $R_{m}$ (c), and $R_{n}$ (d) with respect to the detuning $\Delta$ and $\eta \equiv \Gamma _{n}/\Gamma _{m}$. In the calculations, $\Gamma =10^{-5}\omega _{e}, \theta =\pi /6$.
4. Single photon scattering with one terminated waveguide
As described in the previous configuration, the single photon transfer efficiency from WM to WN was no less than 0.5. To improve efficiency, WM is considered to be terminated, as shown in Fig. 4. In this configuration, the wave functions are modified as follows:
The physical explanation of each part of the wave functions are similar with those in Eqs. (3) to (6). Based on the eigenvalue equation and the hard-wall boundary condition $\psi _{Rm}(L+d)+\psi _{Lm}(L+d)=0$, the following expressions are obtained:
Equation (20) indicates that $r_{m}$ is strongly dependent on $\theta$ and $\phi$. Nevertheless, appropriate values of $\theta, \phi$, and $\Delta$ can be determined to achieve $r_{m}=0$. We first analyze a simple case with $\theta =\pi /2$. Based on Eqs. (20)–(22), $r_{m}=0$, $t_{n}=-(1+i)/2$, and $r_{n}=(1-i)/2$ under the condition that $\phi =(q+1/2)\pi$ and $\Delta =6\Gamma$, where $q$ is an integer. The incident single photon is routed to WN with unit efficiency. The other solution is that, when $\phi =q\pi$ and $\Delta =2\Gamma$, $r_{m}=0$, $t_{n}=-(1-i)/2$, and $r_{n}=-(1+i)/2$. Figure 5 (a) and (b) present a plot of $R_{m}$ and $T_{n}$ with respect to $\phi$ and $\Delta$ with $\theta =\pi /2$. This is in agreement with the theoretical analysis. Figure 5(c) and (d) present the changes in $R_{m}$ and $T_{n}$ with respect to $\phi$ and $\Delta$ with $\theta =\pi /6$. This figure also indicates that, by choosing appropriate values for $\phi$ and $\Delta$, $R_{m}$ and $T_{n} (R_{n})$ can reach 0 and 0.5, respectively.
Fig. 4. Configuration with one terminated waveguide. The distance between the second coupling point and the end is $d$.
Fig. 5. $R_{m}$ (a) and $T_{n}$ (b) with respect to detuning $\Delta$ and phase $\phi$ with $\theta =\pi /2$. (c) and (d) show $R_{m}$ and $T_{n}$ with respect to detuning $\Delta$ and phase $\phi$ with $\theta =\pi /6$, respectively. In the calculations, $\Gamma =10^{-5}\omega _{e}$.
5. Conclusions
In this study , the properties of single photon scattering due to a giant atom coupled with a pair of waveguides are investigated. The scattering properties are found to be strongly dependent on the size of the giant atom. For the two infinite waveguides, when the size of the giant atom is appropriate, such that $kL=(2p+1)\pi$, the incident photon can be transmitted through the system with unit probability due to the interference effect. If $\theta \neq (2p+1)\pi$, the maximum routing efficiency from WM to WN is 0.5 when the incident photon frequency is $\omega _{e}+4\Gamma \sin \theta$ and the equal coupling strength . When WM is terminated, theoretical analyses show that the maximum routing efficiency from WM to WN can reach unity, provided appropriate values of $\theta, \phi$, and $\Delta$ are adopted for a fixed size of the giant atom. These results are expected to be applicable in quantum information processing and quantum device design at the single-photon level.
Funding
National Natural Science Foundation of China (11774262, 11975023); College Students' Innovation and Entrepreneurship Training Program of Anhui Province (S202010360264).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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