Controllable single-photon routing between two waveguides by two giant two-level atoms

We investigate the single-photon quantum routing composed of two infinite waveguides coupled to two giant two-level atoms. The exact expressions of the single-photon transmission and reflection amplitudes are derived with the real-space approach. It is found that the single photon scattering behavior is strongly dependent on the phase difference between the two adjacent atom-waveguide coupling points, the frequency detuning, the coupling strength between the two giant atoms, and the interaction strengths between the giant atoms and the waveguides. Our studies show that an ideal single photon router with unit efficiency can be realised by designing the size of the giant atom, and the frequency detuning or adjusting the interaction strengths between the atoms and the waveguides. The results suggest the potential to effectively control the single-photon quantum routing based on the giant-atom setup.

1 Introduction

The design and realization of scalable quantum information processing systems rely on quantum networks [1]. As an essential ingredient in a quantum network, quantum routers can be used to transfer the quantum information from one channel to others, distributing it in the network. Due to their fastness, very low loss and long-playing coherence [2–4], photons are considered as ideal candidates of carriers of quantum information and thus quantum routing of photons has drawn more attentions in recent years. Numerous theoretical and experimental researches on quantum routing or photon transport have been made based on several different structures, such as coupled-resonator waveguides [5–16], whispering-gallery-mode resonators [17–20], waveguide-emitter systems [21–29], superconducting circuit [30–36].

Recently, giant atoms, as an emerging playground in quantum optics, have attracted a plethora of research interest because they exhibit several striking new phenomena, such as frequency-dependent relaxation rate and Lamb shift [37,38], oscillating bound states [39–43], decoherence-free interaction [44–46], chiral quantum optics [47,48], and phase-controlled frequency conversions [49,50]. For a traditional small atom, the size of the atom is very smaller than the wavelength of the field, and thus it can be well described with the dipole approximation. However, a giant atom couples to a waveguide at multiple points, and the distance between these points is no longer negligible compared with the waveguides of propagating fields in the waveguide. Such a giant-atom structure can be realized in several systems, such as superconducting quits coupled to either surface acoustic waves [51–54] or microwave transmission lines [4,44,55–57], cold atoms interacting with optical lattices [58]. In the giant-atom setup, the multiple coupling points lead to additional interference effects that are not present in quantum optics with conventional small atoms [37,39,45,59,60].

It is very interesting to study the issue that how to mediate single-photon scattering by the giant atoms. Some pioneering works have been reported in this area [42,50,61–65]. Reference [42] investigated the single-photon scattering properties in a one-dimensional coupled-resonator waveguide, where a giant two-level atom is nonlocally coupled to the waveguide via two or multiple resonators. The results showed that a giant atom can be treated as a controller to manipulate the propagation of the photon by introducing the interference effect. In a giant-molecule waveguide-QED system consisting of two coupled giant atoms and an infinite waveguide, the single-photon scattering in both the Markovian and non-Markovian regimes was studied. In this system, the asymmetric Fano line-shapes and the Rabi splitting-like phenomenon were observed [61]. Zhao et.al., reported that single photon scattering in a one-dimensional waveguide can be dynamically controlled by the periodic phase modulation via changing the size of the giant atom and a tunable Autler-Townes splitting can be achieved with the giant atom [62]. Furthermore, a chiral giant-atom model in both the Markovian and the non-Markovian regimes was focused, and it was shown that intriguing interference effects induced by a giant-atom structure result in exotic quantum phenomena such as ultranarrow scattering window [50].

Very recently, the strong dependence of single photon routing properties on the size of a giant atom coupling to two waveguides was revealed. To improve the routing efficiency, a perfect mirror was placed into the non-target waveguide to form a boundary and then the routing efficiency can reach unity [63]. An alternative approach was proposed in Ref. [21] to achieve a high transfer-rate routing, which is to place an atomic mirror aside the input channel but not to terminate it. Inspired by these works, we show an extension of the single photon router based on a single giant atom to the router with two coupled giant atoms, one of which serves as an atomic mirror. With the real-space approach, the exact expressions of the single-photon transmission and reflection amplitudes are derived. We explore the dependence of the single photon properties on the phase shift between two atom-waveguide coupling points, the frequency detuning between the incident photon and the atoms, the coupling strength between the two giant atoms, and the interaction strengths between the atoms and the waveguides. It is found that the single photon incident from one waveguide can be redirected in the other waveguide with a 100% probability by well designing the size of the giant atom and the frequency detuning or adjusting the ratio between the two atom-waveguide coupling strengths.

The paper is organized as follows: In Section 2, we present theoretical model and give the exact expressions of the single-photon transmission and reflection amplitudes with the real-space approach. In Section 3, we discuss the effects of the phase shift between two adjacent coupling points, the frequency detuning, the atomic interaction, and the coupling strengths between the giant atoms and the waveguides on the single photon routing properties. Finally, we conclude with a brief summary of the results in Section 4.

2 Theoretical model

As shown in Figure 1, we consider a single-photon router composed of two infinite linear waveguides and two coupled giant two-level atoms. The giant atom a simultaneously couples to the two waveguides, waveguide m and waveguide n, at two points x = 0 and x = x₀, respectively, while the giant atom b only interacts the waveguide m at x = 2x₀ and x = 3x₀. For simplicity, we assume here that the distances between any two neighboring connection points are identical. With rotating-wave approximation [66,67], the real-space Hamiltonian of the system could be written as (ℏ = 1)

$H=H_w+H_{ab}+H_I,(1)$

where H_w represents the free Hamiltonian of the two waveguides, H_ab describes the two coupled giant atoms, H_I is the interaction Hamiltonian between the atoms and the waveguides. The expressions of these Hamiltonian are written as follows:

$H_w=\sum _{s=m,n}\int dx\left[c_{R_s}^{+}\left(x\right)\left(-i{\upsilon }_g\frac{\partial }{\partial x}\right)c_{R_s}\left(x\right)+c_{L_s}^{+}\left(x\right)\left(i{\upsilon }_g\frac{\partial }{\partial x}\right)c_{L_s}\left(x\right)\right],(2)$

$H_{ab}={\omega }_a{\sigma }_a^{+}{\sigma }_a^{-}+{\omega }_b{\sigma }_b^{+}{\sigma }_b^{-}+\mathit{\text{g}}\left({\sigma }_a^{+}{\sigma }_b^{-}+{\sigma }_b^{+}{\sigma }_a^{-}\right),(3)$

$\begin{array}{ll}\hfill H_I& =\sum _{s=m,n}{V}_{as}\int dx\left[\delta \left(x\right)+\delta \left(x-x_0\right)\right]\left[c_{R_s}^{+}\left(x\right){\sigma }_a^{-}+c_{L_s}^{+}\left(x\right){\sigma }_a^{-}+H.c.\right]\hfill \\ \hfill & +V_{bm}\int dx\left[\delta \left(x-2x_0\right)+\delta \left(x-3x_0\right)\right]\left[c_{R_m}^{+}\left(x\right){\sigma }_b^{-}+c_{L_m}^{+}\left(x\right){\sigma }_b^{-}+H.c.\right],\hfill \end{array}(4)$

where $c_{R_s}^{+}(x)$ $(c_{R_s}(x))$and $c_{L_s}^{+}(x)$ $(c_{L_s}(x))$ (s = m, n) are the bosonic creation (annihilation) operators of the right- and left-propagating photons at position x in the waveguide s, respectively. υ_g is the group velocity of the photons. ω_a (ω_b) is the atomic transition frequency for the atom a (b). ${\sigma }_a^{+}$ $({\sigma }_b^{+})$ and ${\sigma }_a^{-}$ $({\sigma }_b^{-})$ are the raising and lowering operators for the atom a (b). g characterizes the intensity of the interaction between the two giant atoms. V_as (V_bs) is the coupling strength between the atom a (b) and the waveguide s (s = m, n). The Dirac delta functions δ(x) and δ(x − x₀) indicate that the giant atom a interacts with the two waveguides via two points x = 0 and x = x₀, respectively. The functions δ(x − 2x₀) and δ(x − 3x₀) represent that the giant atom b couples to the waveguide m at x = 2x₀ and x = 3x₀. H. c. stands for the Hermitian conjugate.

FIGURE 1

FIGURE 1. Schematic configuration of routing single photons in two channels made of two infinite linear waveguides. The giant two-level atom a couples twice with the waveguide m (waveguide n) at x = 0 and x = x₀ with strength V_am (V_an), and the giant two-level atom b only interacts the waveguide m at x = 2x₀ and x = 3x₀ with strength V_bm. The purple dashed line indicates an interatomic interaction with strength g. Initially, a single photon is launched from the left side of waveguide m and then is reflected, transmitted, or transferred to waveguide n.

We assume that initially a single photon with energy υ_gk is injected from the far left of the waveguide m. In the single-excitation subspace, the eigenstate of the system can be written as

$|\omega \rangle =\sum _{s=m,n}\int dx\left[{\varphi }_{R_s}\left(x\right)c_{R_s}^{+}\left(x\right)+{\varphi }_{L_s}\left(x\right)c_{L_s}^{+}\left(x\right)\right]|\varnothing \rangle +u_a{\sigma }_a^{+}|\varnothing \rangle +u_b{\sigma }_b^{+}|\varnothing \rangle ,(5)$

where |∅⟩ is the vacuum state, which indicates zero photon in any waveguide and the two giant atoms in their ground states. u_a (u_b) is excitation amplitude of the atom a (b) in the excite state |e⟩_a (|e⟩_b). ${\varphi }_{R_s}(x)$ and ${\varphi }_{L_s}(x)$ (s = m, n) are, respectively, the probability amplitudes of the right- and left-propagating photons in the waveguide s. They have the following forms:

${\varphi }_{R_m}\left(x\right)=e^{\mathit{ikx}}\left[\theta \left(-x\right)+t_{m1}\theta \left(x\right)\theta \left(x_0-x\right)+t_{m2}\theta \left(x-x_0\right)\theta \left(2x_0-x\right)+t_{m3}\theta \left(x-2x_0\right)\theta \left(3x_0-x\right)+t_m\theta \left(x-3x_0\right)\right],(6)$

${\varphi }_{L_m}\left(x\right)=e^{-ikx}\left[r_m\theta \left(-x\right)+r_{m1}\theta \left(x\right)\theta \left(x_0-x\right)+r_{m2}\theta \left(x-x_0\right)\theta \left(2x_0-x\right)+r_{m3}\theta \left(x-2x_0\right)\theta \left(3x_0-x\right)\right],(7)$

${\varphi }_{R_n}\left(x\right)=e^{\mathit{ikx}}\left[t_{n1}\theta \left(x\right)\theta \left(x_0-x\right)+t_n\theta \left(x-x_0\right)\right],(8)$

${\varphi }_{L_n}\left(x\right)=e^{-ikx}\left[r_n\theta \left(-x\right)+r_{n1}\theta \left(x\right)\theta \left(x_0-x\right)\right],(9)$

where θ(x) is the Heaviside step function. t_m (t_n) and r_m (r_n) are, respectively, the transmission and reflection coefficients in the waveguide m (n). t_mj (t_nj) and r_mj (r_nj) represent the probability amplitudes for right-going and left-going photons between x = (j − 1)x₀ and x = jx₀ (j = 1, 2, 3) in the waveguide m (n), respectively.

Solving the eigenvalue equation H|ω⟩ = ω|ω⟩ with the expressions in Eqs 2–9, one can obtain a set of equations

$V_{am}{u}_a-i{\upsilon }_g\left(t_{m1}-1\right)=0,$

$V_{am}{u}_a-i{\upsilon }_g{e}^{i\theta }\left(t_{m2}-t_{m1}\right)=0,$

$V_{am}{u}_a+i{\upsilon }_g\left(r_{m1}-r_m\right)=0,$

$V_{am}{u}_a+i{\upsilon }_g{e}^{-i\theta }\left(r_{m2}-r_{m1}\right)=0,$

$V_{an}{u}_a-i{\upsilon }_g{t}_{n1}=0,$

$V_{an}{u}_a-i{\upsilon }_g{e}^{i\theta }\left(t_n-t_{n1}\right)=0,$

$V_{an}{u}_a+i{\upsilon }_g\left(r_{n1}-r_n\right)=0,$

$V_{an}{u}_a-i{\upsilon }_g{e}^{-i\theta }{r}_{n1}=0,$

$V_{bm}{u}_b-i{\upsilon }_g{e}^{2i\theta }\left(t_{m3}-t_{m2}\right)=0,$

$V_{bm}{u}_b-i{\upsilon }_g{e}^{3i\theta }\left(t_m-t_{m3}\right)=0,$

$V_{bm}{u}_b+i{\upsilon }_g{e}^{-2i\theta }\left(r_{m3}-r_{m2}\right)=0,$

$V_{bm}{u}_b-i{\upsilon }_g{e}^{-3i\theta }{r}_{m3}=0,$

$\mathit{\text{g}}{u}_a-{\mathrm{\Delta }}_b{u}_b+V_{bm}\left(t_{m2}{e}^{2i\theta }+r_{m2}{e}^{-2i\theta }+t_m{e}^{3i\theta }\right)=0,$

$\mathit{\text{g}}{u}_b-{\mathrm{\Delta }}_a{u}_a+V_{am}\left(1+r_m+t_{m1}{e}^{i\theta }+r_{m1}{e}^{-i\theta }\right)+V_{an}\left(r_n+t_{n1}{e}^{i\theta }+r_{n1}{e}^{-ik\theta }\right)=0,(10)$

where Δ_a = ω − ω_a and Δ_b = ω − ω_b are the frequency detunings between the incident photon and the atom a and b, respectively. θ = kx₀, describes an accumulated phase of single photon traveling between any two adjacent coupling points. Then one can get the transmission and reflection amplitudes

$t_m=\frac{\left({\mathrm{\Delta }}_b-2{\mathrm{\Gamma }}_b\sin \theta \right)Q_1-2\mathit{\text{g}}\sqrt{{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b}{Q}_2-{\mathit{\text{g}}}^2}{i\left(1+e^{i\theta }\right)Q_3+{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b-{\mathit{\text{g}}}^2}(11)$

$r_m=\frac{-{\left(1+e^{i\theta }\right)}^2\left({\mathrm{\Gamma }}_a{\mathrm{\Delta }}_b+{\mathrm{\Gamma }}_b{\mathrm{\Delta }}_a{e}^{4i\theta }+2\mathit{\text{g}}\sqrt{{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b}{e}^{2i\theta }+2i{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b{Q}_4\right)}{\left(1+e^{i\theta }\right)Q_3-i\left({\mathrm{\Delta }}_a{\mathrm{\Delta }}_b-{\mathit{\text{g}}}^2\right)},(12)$

$t_n=r_n{e}^{-i\theta }=\frac{e^{-i\theta }{\left(1+e^{i\theta }\right)}^2\left({\mathrm{\Gamma }}_a{\mathrm{\Delta }}_b+\mathit{\text{g}}\sqrt{{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b}{e}^{2i\theta }-i{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b{Q}_5\right)}{\left(1+e^{i\theta }\right)Q_3-i\left({\mathrm{\Delta }}_a{\mathrm{\Delta }}_b-{\mathit{\text{g}}}^2\right)},(13)$

where ${\mathrm{\Gamma }}_a=V_{am}^2/{\upsilon }_g=V_{an}^2/{\upsilon }_g$ and ${\mathrm{\Gamma }}_b=V_{bm}^2/{\upsilon }_g$, respectively, are the decay rates from the atom a and b into the waveguides. Then Γ_a (Γ_b) characterizes the coupling strength between the atom a (b) and the waveguides. Here, we assume V_an = V_am for simplicity. We introduce the notations Q_j (j = 1, 2, 3, 4, 5) as follows: Q₁ = Δ_a + 2iΓ_a (1 + cos θ + 2i sin θ), Q₂ = sin θ + 2 sin 2θ + sin 3θ, $Q_3=2\mathit{\text{g}}\sqrt{{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b}{e}^{i\theta }(1+e^{i\theta })+2({\mathrm{\Gamma }}_b{\mathrm{\Delta }}_a+2{\mathrm{\Gamma }}_a{\mathrm{\Delta }}_b)+i{\mathrm{\Gamma }}_a{\mathrm{\Gamma }}_b(1+e^{i\theta })[8-e^{2i\theta }{(1+e^{i\theta })}^2]$, Q₄ = 1 + e^3iθ(2 cos 2θ − 1), $Q_5=e^{3i\theta }{(1+e^{i\theta })}^2-2(1+e^{i\theta })$.

Based on Eqs 11–13, the single-photon transmission and reflection rates could be defined by

$T_s=|t_s{|}^2,R_s=|r_s{|}^2,\left(s=m,n\right).(14)$

One can obtian the relations R_n = T_n and R_m + T_m + R_n + T_n = 1 for probability conservation.

3 Single photon scattering mediated by two giant atoms

Now, we explore the single photon routing properties between the two waveguides mediated by the two giant atoms. Here, we interest in the quantum transfer efficiency R_n + T_n from the waveguide m to n, and only focus on T_n due to the relation R_n = T_n. The simple case with Δ_a = Δ_b = Δ, Γ_a = Γ_b = Γ and g = 0 is considered firstly. In this case, the transmission and reflection amplitudes in Eqs 11–13 can be simplified as

$t_m=\frac{\left(\mathrm{\Delta }-2\mathrm{\Gamma }\sin \theta \right)Q_1^{\prime }}{i\left(1+e^{i\theta }\right)Q_3^{\prime }+{\mathrm{\Delta }}^2}(15)$

$r_m=\frac{-\mathrm{\Gamma }{\left(1+e^{i\theta }\right)}^2\left[\mathrm{\Delta }\left(1+e^{4i\theta }\right)+2i\mathrm{\Gamma }{Q}_4\right]}{\left(1+e^{i\theta }\right)Q_3^{\prime }-i{\mathrm{\Delta }}^2},(16)$

$t_n=r_n{e}^{-i\theta }=\frac{\mathrm{\Gamma }{e}^{-i\theta }{\left(1+e^{i\theta }\right)}^2\left(\mathrm{\Delta }-i\mathrm{\Gamma }{Q}_5\right)}{\left(1+e^{i\theta }\right)Q_3^{\prime }-i{\mathrm{\Delta }}^2},(17)$

where $Q_1^{\prime }=\mathrm{\Delta }+2i\mathrm{\Gamma }(1+\cos \theta +2i\sin \theta )$, $Q_3^{\prime }=6\mathrm{\Gamma }\mathrm{\Delta }+i{\mathrm{\Gamma }}^2(1+e^{i\theta })[8-e^{2i\theta }{(1+e^{i\theta })}^2]$.

Figure 2 displays the transmission rates T_m and T_n versus the detuning Δ between the incident photon and the atoms, and the phase shift θ between two adjacent atom-waveguide coupling points. Here, the coupling strength between the two atoms is not taken into account, i.e., g = 0. Obviously, these spectra vary periodically as θ with a period of 2π. Note that the value of T_m is always zero when Δ = 2Γ sin θ, except for some special phases θ = (2j − 1)π (j = 1, 2, 3 … ) by analyzing Eq. 15. A dark purple sinusoidal shape appears at Δ/Γ = 2 sin θ, indicating T_m = 0, as shown in Figure 2A. Similar transmitted behavior have also been demonstrated in other giant-atom setups [50,62,68]. In the case of θ = (2j − 1)π (j = 1, 2, 3 … ), we have R_m = T_n = R_n = 0 (R_m and R_n not shown in Figure 2) with the factor 1 + e^iθ = 0 in the expressions of Eqs 16, 17 but T_m = 1. The perfect transmission of the single photon arises from the quantum interferences among the multiple connection points [45]. Figure 2B displays that T_n can reach a maximum value of 0.48 (R_n = T_n = 0.48), which can be verified by numerical calculation based on Eq. 17. The results indicate that the maximum total quantum routing probability (R_n + T_n) from waveguide m to waveguide n is about 0.96.

FIGURE 2

FIGURE 2. Transmission rates T_m (A) and T_n (B) as a function of the detuning Δ and the phase shift θ, in which the frequency detunings are equal (Δ_a = Δ_b = Δ), and the two atom-waveguide coupling strengths are the same (Γ_a = Γ_b = Γ). A dark purple sinusoidal shape implies T_m = 0, and T_n can reach its maximum value of 0.48. The transferred rate T_n + R_n (R_n = T_n) of the photon from the input waveguide m into the targeted waveguide n is 0.96. The other parameter is g = 0.

As shown in Figures 3, 4, we plot the effect of the coupling strength between the two giant atoms on single-photon routing properties. We also assume Δ_a = Δ_b = Δ and Γ_a = Γ_b = Γ. Figure 3 exhibits the transmission rates T_m and T_n as a function of the detuning Δ and the coupling strength g. One can see that there are two valleys or peaks, which are more clearly demonstrated by the profiles of Figure 3, as depicted in Figure 4. When g = 0, there is a single valley or peak in these scattering spectra, but for g ≠ 0 two valleys or peaks appear in these spectra. As g increases, the separation of the two valleys or peaks gradually increases and the intensities also change. The results indicate that the coupling strength between the two atoms can control the single-photon scattering in the two waveguide channels. However, an increase of the coupling strength leads to decreasing of T_n. Consequently, when the coupling strength g = 0 the maximum total quantum routing probability with 0.88 can reach.

FIGURE 3

FIGURE 3. Transmission rates T_m (A) and T_n (B) as a function of the detuning Δ and the coupling strength g between the two atoms under the same atom-waveguide interaction strengths (Γ_a = Γ_b = Γ). The spectra show V-type shapes, indicating the existence of two valleys or peaks. The phase shift between two adjacent coupling points is fixed to θ = π/4.

FIGURE 4

FIGURE 4. The transmitted curves T_m (A) and T_n (B) show the profiles of Figure 3. The effect of the coupling strength g between the atom a and b on the single-photon scattering is revealed. The number of valleys or peaks varies from one to two as g changes from zero to non-zero.

Next, the influence of the atom-waveguide coupling strengths on single-photon routing properties is investigated. Here, we also set Γ_a = Γ, serving as the unit of other parameters, but change the value of Γ_b by adjusting the ratio Γ_b/Γ_a = η. Moreover, the interatomic coupling is turned off, i.e., g = 0. To explore a possible unit transfer efficiency from the waveguide m to n, we let Δ = 2Γ_b sin θ = 2Γη sin θ [θ ≠ (2j − 1)π, j = 1, 2, 3 … ] to ensure T_m = 0, meanwhile, seek the condition of R_m = 0. After analyzing Eqs 11, 12, both T_m = 0 and R_m = 0 are guaranteed when the following conditions are satisfied simultaneously:

$\mathrm{\Delta }=2\mathrm{\Gamma }\eta \sin \theta ,(18)$

$\mathrm{\Delta }=\frac{-2\mathrm{\Gamma }\left[1+\cos 3\theta \left(2\cos 2\theta -1\right)\right]}{\sin 4\theta },(19)$

$\eta =\frac{1+\cos 3\theta \left(2\cos 2\theta -1\right)}{\mathrm{\Gamma }\left[\cos \theta \left(1-2\cos 2\theta \right)-\cos 4\theta \right]},(20)$

To clearly demonstrate possible extremum values of T_n and R_m, we plot T_n and R_m as a function of θ and η by substituting g = 0 and Δ = 2Γ_b sin θ into Eqs 11–13. One can see that there are six extremum points with T_n = 0.5 and R_m = 0 marked with white dots in Figures 5A,B, respectively, and they are symmetric about θ = π. The exact locations of the six extreme points are obtained based on the numerical analysis of Eqs 18–20, as shown in Table 1. We plot T_m, R_m and T_n versus Δ where the values of the other parameters are provided in the Table. It is worth noting that each pair of spectra (i.e., Figures 6A–F) with the same values of η are symmetric with respect to the vertical line Δ = 0. In fact, we have T_m [η, − Δ, 2π − (2j − 1)π/8] = T_m [η, Δ (2j − 1)π/8], R_m [η, − Δ, 2π − (2j − 1)π/8] = R_m [η, Δ (2j − 1)π/8] and T_n [η, − Δ, 2π − (2j − 1)π/8] = T_n [η, Δ (2j − 1)π/8] (j = 2, 3, 4, 5, 6, 7). In addition, two peaks appear in the reflected spectra R_m but not in other spectra. The locations of the two peaks are derived as ${\mathrm{\Delta }}_{\pm }=(\eta +2)\sin \theta \pm \sqrt{{(\eta +2)}^2{\sin }^2\theta +\eta Q}$ with Q = 8 + 16 cos θ + 7 cos 2θ − 4 cos 3θ − 6 cos 4θ − 4 cos 5θ − cos 6θ, where the values of the parameters θ and η are given by Table 1. These results mean that the single photon transfer efficiency from the waveguide m to n can reach 100% (i.e., R_n + T_n = 1) by designing a proper size of the giant atom, and adjusting the atom-waveguide coupling strength and the frequency detuning between the atoms and the single photon. Different from the single-photon router containing a small-atom mirror in Ref. [21], we introduce two coupled giant atoms to mediate the single-photon routing between two infinite waveguides, where a giant atom is located between the two waveguides to transfer the single photon from the input waveguide m to the output waveguide n, and the other one only interacts with the non-target waveguide m to serve as a giant-atom mirror.

FIGURE 5

FIGURE 5. The extreme values of T_n (A) and R_m (B) for various phase shift θ and the ratio η between the two atom-waveguide coupling strengths. White dots represent extremum points with T_n = 0.5 or R_m = 0, showing symmetric about θ = π. The values of the parameters at the six extreme points are listed in Table 1. The interatomic interaction is fixed to g = 0.

TABLE 1

TABLE 1. The values of the parameters θ, η and Δ at extremum points with T_n = R_n = 0.5 and T_m = R_m = 0 in Figures 5, 6.

FIGURE 6

FIGURE 6. Transmission and reflection rates T_m (dark solid line), T_n (blue dashed line) and R_m (red dotted line) as a function of Δ around the six extreme points shown in Figure 5. (A) η = 3.50, θ = 3π/8, (B) η = 1.33, θ = 5π/8, (C) η = 2.20, θ = 7π/8, (D) η = 2.20, θ = 9π/8, (E) η = 1.33, θ = 11π/8, (F) η = 3.50, θ = 13π/8. The transmitted spectra of T_m and T_n show a single valley or peak, while the reflected spectrum of R_m exhibits two peaks. The values of η and θ are provided by Table 1 and the interatomic interaction is not considered (g = 0).

Finally, we explore the effects of the detunings Δ_a and Δ_b on the single photon quantum routing. After analyzing Eqs 11–13 and letting g = 0, Γ_a = Γ_b = Γ, a single photon router with unit efficiency may be achieved when the following equations are fulfilled

${\mathrm{\Delta }}_b-2\mathrm{\Gamma }\sin \theta =0,(21)$

${\mathrm{\Delta }}_a\sin 4\theta +2\mathrm{\Gamma }\left[1-\cos 3\theta \left(1-2\cos 2\theta \right)\right]=0,(22)$

${\mathrm{\Delta }}_b+2\mathrm{\Gamma }\left(1-2\cos 2\theta \right)\left(\sin 3\theta +\cos 3\theta \cot 4\theta \right)-2\mathrm{\Gamma }\cot 4\theta =0,(23)$

The calculation and analysis for the above equations show that there are eight solutions of θ meeting the conditions, specially, θ = (2j − 1)π/8, (j = 1, 2, 3, 4, 5, 6, 7, 8), and the other parameters are listed in Table 2. One can find that there is the relation T_n [ − Δ_a, − Δ_b, 2π − (2j − 1)π/8] = T_n [Δ_a, Δ_b, (2j − 1)π/8]. As a consequence, any two transmitted spectra with θ = (2j − 1)π/8 and 2π − (2j − 1)π/8 show central symmetry with respect to the origin (Δ_a = 0, Δ_b = 0). Figure 7 shows the transmission rate T_n versus Δ_a and Δ_b with some given values of θ provided by Table 2. This indicates that a single photon router with unit efficiency can be also realised by tuning the phase shift of two adjacent connection points and the frequency detuning between the single photon and the atoms.

TABLE 2

TABLE 2. The values of the parameters θ, Δ_a and Δ_b with T_n = R_n = 0.5 and T_m = R_m = 0 in Figure 7.

FIGURE 7

FIGURE 7. Transmission rate T_n as a function of Δ_a and Δ_b, in which the four values of the phase shift θ are given by Table 2. Specially, (A) θ = π/8, (B) θ = 3π/8, (C) θ = 5π/8, (D) θ = 7π/8. The transmitted spectra with the other four values of θ are not presented here. The total transfer efficiency from the waveguide m to n can achieve 100% by adjusting the detunings. The other parameters are Γ_a = Γ_b = Γ, g = 0.

4 Conclusion

We have studied the influence of two coupled giant two-level atoms on the modulation of single photon quantum routing in two channels composed of two infinite waveguides, and derived the exact expressions of the single-photon transmission and reflection amplitudes with the real-space approach. Our studies show the single photon scattering can be mediated by the atomic phase difference, the frequency detuning, the interatomic interaction, and the coupling strengths between the giant atoms and the waveguides. In the two-giant-atom scheme, it is found that the giant atom located between the two waveguides plays a role in transferring the single photon from the input waveguide to the output waveguide, and the other one interacting with the non-target waveguide is equivalent to a giant-atom mirror, inducing additional interferences. Importantly, a single photon router with unit efficiency can be realised by designing a proper size of the giant atom and adjusting the ratio of the coupling strength and the frequency detuning. Our work may provide a feasible approach to design an efficient quantum router and manipulate photon transport at the single-photon level using the giant-atom setup.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

YZ and ZZ conceived the physical model and the idea of the study, and ZZ and ZP performed the numerical calculation and numerical analysis. ZZ and WY contributed to writing the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

This work was partly supported by Scientific Research Fund of Hunan Provincial Education Department under Grant Nos 19B206, 21B0485, the National Natural Science Foundation of China under Grant Nos 11704115, 11504104, 51867002, 61835013, and the Hunan Provincial Natural Science Foundation of China Grant No. 2021JJ40188.

Acknowledgments

The authors thank C. X. Jia and C. L. Jiang for helpful discussions.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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