1. Introduction
Single-photon scattering has been a hot topic in quantum optics and quantum communications since photons can be regarded as ideal candidates for carriers of information. In the last decades, waveguide QED, which refers to the interactions between atoms and one-dimensional waveguides, has become one of the most popular platforms to realize tunable single-photon scattering [1-21]. Today, quantum routers [22, 23], circulators [24], and quantum frequency combs [25] have been realized in hybrid waveguide systems. Further, due to the strong light-matter interaction in waveguide QED, many other interesting quantum phenomena such as super-radiance [26-28] and sub-radiance [29, 30], bound states [31-36], and long-distance entanglement [37-42] have been studied in this systems.
However, those fruitful achievements are obtained based on the conventional waveguide QED, in which the atoms are commonly treated as pointlike dipoles as their size is much smaller than the wavelength of light. However, this dipole approximation will be invalid in superconducting systems [43], in which the superconducting artificial transmon qubit [44] can be coupled to the surface acoustic waves in a waveguide at multiple points [45-49]. In these systems, the transmon qubit serves as a so-called giant atom, the size of which can be larger than the wavelength of phonons for a given frequency. Hence, compared to the small-atom setup, propagating photons in the giant-atom setup can experience multiple transmissions and reflections, leading to the interference and retarded effects. Those effects have enabled the giant-atom community to present some unique quantum phenomena not existing in the conventional small atomic system, such as the Lamb shift of a giant atom [48], decoherence-free interaction between multiple giant atoms [47], and tunable bound states [49].
On the other hand, a chiral waveguide, which refers to a scenario where the interactions between the atoms and the waveguide modes are direction-dependent, has been achieved experimentally [50-59]. These chiral light-emitter couplings have provided magnificent advantages in modulating single-photon transport. For example, without chiral interaction, the conventional waveguide QED is usually symmetric, thus, photon transmission is always reciprocal. However, once the chiral coupling is introduced, the time-reversal symmetry of the system is naturally broken. In this situation, if the dissipation of the atomic system is considered simultaneously, the single-photon nonreciprocal scattering effect can be obtained in waveguide-emitter systems [60, 61]. Further, in the absence of the chiral coupling, the photon routing probability can only reach 50% at most. In contrast, in chirally coupled waveguide QED, this value can be significantly improved to as high as 100%, which implies that the photon has been deterministically routed to the targeted direction [62-64]. It is worth indicating that optical non-reciprocity can be also achieved via diverse mechanisms such as introducing the non-Hermiticity in the non-Hermitian systems [65-67], optical nonlinearities [68, 69], and dynamical modulations [70, 71].
Recently, a hybrid waveguide system, which overlaps the chiral coupling and the giant atom, has attracted lots of attention. For example, [72] studied the chiral quantum optics in a waveguide chirally coupled to multiple giant atoms [73] reported that the frequency conversion can be realized in a one-dimentional waveguide chirally coupled to a three-level giant atom. Inspired by those works, in this paper, we propose a chiral giant atom-waveguide model in which a driven three-level giant atom is chirally coupled to two one-dimensional waveguides simultaneously. The properties of the single-photon transport are investigated in both the Markovian and the non-Markovian regimes, which are defined depending on whether the propagation time of photons between different atom-waveguide coupling points is negligible or not. In the Markovian regime, we show that when the atomic dissipation is considered, the chiral coupling can be used to achieve the non-reciprocity in a single waveguide and the single-photon router with 100% probability in two different waveguides. Moreover, the presence of the driving field and the size of the giant atom can provide a more tunable parameter to manipulate the single-photon behavior. We also investigate how the nonreciprocal scattering effect and the routing capability in both the giant- and small-atom setups are influenced by atomic dissipation and imperfections. Our numerical results show that in the case of imperfect chirality and a larger atomic dissipation, the nonreciprocal scattering effect and the routing capability in the giant-atom setup are more robust than that in the small-atom setup. In the non-Markovian regime, we show that the transmission behaviors exhibit non-Markovian features. The non-Markovicity induced non-reciprocity and photon routing are studied in more detail.
The paper is organized as follows. In section 2 , we present the theoretical model, including the system Hamiltonian, and derive the single-photon transmission and reflection amplitudes in two waveguides. In section 3 , we show the nonreciprocal scattering effect and the routing capability in the Markovian regime. In section 4 , we study the non-Markovian induced non-reciprocity and quantum photon routing. Finally, we conclude our work in section 5 .
2. Model
The system considered in this paper is shown in figure 1. A giant atom is chirally coupled to the waveguide a (b) at ${x}_{a}^{1}=0$ and ${x}_{a}^{2}={d}_{a}$ (${x}_{b}^{1}=0$ and ${x}_{b}^{2}={d}_{b}$), respectively. The giant atom is a three-level atom characterized by the ground state ∣g⟩, the metastable state ∣f⟩, and the excited state ∣e⟩. As shown in figure 1, we use ${g}_{\alpha \beta }^{j}$ to denote the chiral coupling strengths between the atomic transition ∣e⟩ ↔ ∣g⟩ and the waveguide α at the point ${x}_{\alpha }^{j}$. Here, the subscripts α = a, b, β = R, L, and j = 1,2 are presented throughout the paper.
Figure 1. Schematic configuration of a Λ-type three-level giant atom coupled to two waveguides simultaneously. ${g}_{\alpha \beta }^{j}$ (α = a, b, β = R, L and j = 1, 2) represents the chiral coupling strength between the atomic transition ∣e⟩ ↔ ∣g⟩ and the waveguide α at the point ${x}_{\alpha }^{j}$. |
The real-space Hamiltonian for the hybrid system can be written as H = Hw + Ha + HI. Here, Hw describes the free energy of the propagating photons in the waveguide, which can be expressed as
$\begin{eqnarray}\begin{array}{rcl}{H}_{w} & = & (-{\rm{i}}{\upsilon }_{{\rm{g}}})\left\{{\int }_{-\infty }^{\infty }{\rm{d}}x\left[{a}_{R}^{\dagger }(x)\displaystyle \frac{\partial }{\partial x}{a}_{R}(x)-{a}_{L}^{\dagger }(x)\displaystyle \frac{\partial }{\partial x}{a}_{L}(x)\right]\right.\\ & & \left.+{\int }_{-\infty }^{\infty }{\rm{d}}x\left[{b}_{R}^{\dagger }(x)\displaystyle \frac{\partial }{\partial x}{b}_{R}(x)-{b}_{L}^{\dagger }(x)\displaystyle \frac{\partial }{\partial x}{b}_{L}(x)\right]\right\},\end{array}\end{eqnarray}$
where νg is the group velocity of the photons. ${a}_{R}^{\dagger }/{b}_{R}^{\dagger }$ (aR/bR) and ${a}_{L}^{\dagger }/{b}_{L}^{\dagger }$ (aL/bL) are the bosonic creation (annihilation) operators of the right- and left-going photons in the waveguides a and b.The second term Ha denotes the Hamiltonian of the atom,
$\begin{eqnarray}\begin{array}{rcl}{H}_{a} & = & ({\omega }_{e}-{\rm{i}}{\gamma }_{e})| e\rangle \langle e| +({\omega }_{f}-{\rm{i}}{\gamma }_{f})| f\rangle \langle f| \\ & & +{\rm{\Omega }}({\sigma }_{\mathrm{ef}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{d}t}+{\sigma }_{\mathrm{fe}}{{\rm{e}}}^{{\rm{i}}{\omega }_{d}t}),\end{array}\end{eqnarray}$
where ${\sigma }_{\mathrm{ef}}={\left({\sigma }_{\mathrm{fe}}\right)}^{\dagger }$ is the transition operator between states ∣e⟩ and ∣f⟩. ωe and ωf are the frequencies of the states ∣e⟩ and ∣f⟩, respectively. γe (γf) represents the energy loss of state ∣e⟩ (∣f⟩). The classical control field with strength Ω and frequency ωd is used to induce the transition between ∣e⟩ and ∣f⟩. In the rotating frame, Ha can be written as the time-independent form $\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{a} & = & ({\omega }_{e}-{\rm{i}}{\gamma }_{e})| e\rangle \langle e| \\ & & +({\omega }_{f}-{\rm{i}}{\gamma }_{f})| f\rangle \langle f| +{\rm{\Omega }}({\sigma }_{\mathrm{fe}}+{\sigma }_{\mathrm{ef}}),\end{array}\end{eqnarray}$
with δ = ωf + ωd.The chiral interaction between the waveguide a/b and the giant atom is described by the third term HI in a Hamiltonian. The detailed expressions are described as follows:
$\begin{eqnarray}\begin{array}{rcl}{H}_{I} & = & \int {\rm{d}}x\{\delta (x)[{g}_{\mathrm{aR}}^{1}{a}_{L}^{\dagger }(x)+{g}_{\mathrm{aL}}^{1}{a}_{L}^{\dagger }(x)]| g\rangle \langle e| \\ & & +\delta (x-{d}_{a})[{g}_{\mathrm{aR}}^{2}{a}_{L}^{\dagger }(x)+{g}_{\mathrm{aL}}^{2}{a}_{L}^{\dagger }(x)]| g\rangle \langle e| \\ & & +\delta (x)[{g}_{\mathrm{bR}}^{1}{b}_{L}^{\dagger }(x)+{g}_{\mathrm{bL}}^{1}{b}_{L}^{\dagger }(x)]| g\rangle \langle e| \\ & & +\delta (x-{d}_{b})[{g}_{\mathrm{bR}}^{2}{b}_{L}^{\dagger }(x)+{g}_{\mathrm{bL}}^{2}{b}_{{\rm{L}}}^{\dagger }(x)]| g\rangle \langle e| +{\rm{H}}.{\rm{c}}.\}.\end{array}\end{eqnarray}$
In the single-excitation subspace, the eigenstate of the system can be written as
$\begin{eqnarray}\begin{array}{rcl}| {E}_{k}\rangle & = & \int {\rm{d}}x[{\phi }_{\mathrm{aR}}(x){a}_{R}^{\dagger }(x)+{\phi }_{\mathrm{aL}}(x){a}_{L}^{\dagger }(x)\\ & & +{\phi }_{\mathrm{bR}}(x){b}_{R}^{\dagger }(x)+{\phi }_{\mathrm{bL}}(x){b}_{L}^{\dagger }(x)]| \unicode{x000F8},g\rangle \\ & & +{\mu }_{e}| \unicode{x000F8},e\rangle +{\mu }_{f}| \unicode{x000F8},f\rangle ,\end{array}\end{eqnarray}$
where ∣ø, n⟩ (n = e, g, f) represents that the waveguide is in the vacuum state while the atom is in the states ∣n⟩. $\phi$αR(x) and $\phi$αL(x) are the single-photon wave function of the right-going and left-going modes in the waveguide α. μe and μf are the three-level atom in the excited state ∣e⟩ and metastable state ∣f⟩, respectively. Solving the stationary Schrödinger equation H∣Ek⟩ = E∣Ek⟩, the series of coupled equations can be obtained as follows: $\begin{eqnarray}\begin{array}{rcl}E{\phi }_{\mathrm{aR}}(x) & = & -{\rm{i}}{\upsilon }_{{\rm{g}}}\displaystyle \frac{\partial }{\partial x}{\phi }_{\mathrm{aR}}(x)+{g}_{\mathrm{aR}}^{1}{\mu }_{e}\delta (x)+{g}_{\mathrm{aR}}^{2}{\mu }_{e}\delta (x-{d}_{a}),\\ E{\phi }_{\mathrm{aL}}(x) & = & -{\rm{i}}{\upsilon }_{{\rm{g}}}\displaystyle \frac{\partial }{\partial x}{\phi }_{\mathrm{aL}}(x)+{g}_{\mathrm{aL}}^{1}{\mu }_{e}\delta (x)+{g}_{\mathrm{aL}}^{2}{\mu }_{e}\delta (x-{d}_{a}),\\ E{\phi }_{\mathrm{bR}}(x) & = & -{\rm{i}}{\upsilon }_{{\rm{g}}}\displaystyle \frac{\partial }{\partial x}{\phi }_{\mathrm{bR}}(x)+{g}_{\mathrm{bR}}^{1}{\mu }_{e}\delta (x)+{g}_{\mathrm{bR}}^{2}{\mu }_{e}\delta (x-{d}_{b}),\\ E{\phi }_{\mathrm{bL}}(x) & = & -{\rm{i}}{\upsilon }_{{\rm{g}}}\displaystyle \frac{\partial }{\partial x}{\phi }_{\mathrm{bL}}(x)+{g}_{\mathrm{bL}}^{1}{\mu }_{e}\delta (x)+{g}_{\mathrm{bL}}^{2}{\mu }_{e}\delta (x-{d}_{b}),\\ E{\mu }_{e} & = & ({\omega }_{e}-{\rm{i}}{\gamma }_{e}){\mu }_{e}+\displaystyle \sum _{\alpha =a,b}[{g}_{\alpha {\rm{L}}}^{1}{\phi }_{\alpha {\rm{L}}}(0)+{g}_{\alpha {\rm{R}}}^{1}{\phi }_{\alpha {\rm{R}}}(0)\\ & & +{g}_{\alpha {\rm{L}}}^{2}{\phi }_{\alpha {\rm{L}}}({d}_{\alpha })+{g}_{\alpha {\rm{R}}}^{2}{\phi }_{\alpha {\rm{R}}}({d}_{\alpha })],\\ E{\mu }_{f} & = & (\delta -{\rm{i}}{\gamma }_{f}){\mu }_{f}+{\rm{\Omega }}{\mu }_{e}.\end{array}\end{eqnarray}$
We now consider the transport behavior when a single photon with energy E = νgk is incident from the left side of the waveguide a. The wave functions $\phi$αR(x) and $\phi$αL(x) can be written as6 ) with (7 ), one can obtain the expressions of ta, tb, ra, and rb:
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\mathrm{aR}}(x) & = & {{\rm{e}}}^{{\rm{i}}{kx}}\{\theta (-x)+A[\theta (x)-\theta (x-{d}_{a})]\\ & & +{t}_{a}\theta (x-{d}_{a})\},\\ {\phi }_{\mathrm{aL}}(x) & = & {{\rm{e}}}^{-{\rm{i}}{kx}}\{{r}_{a}\theta (-x)+B[\theta (x)-\theta (x-{d}_{a})]\},\\ {\phi }_{\mathrm{bR}}(x) & = & {{\rm{e}}}^{{\rm{i}}{kx}}\{M[\theta (-x)-\theta (x-{d}_{b})+{t}_{b}\theta (x-{d}_{b})\},\\ {\phi }_{\mathrm{bL}}(x) & = & {{\rm{e}}}^{-{\rm{i}}{kx}}\{{r}_{a}\theta (-x)+N[\theta (x)-\theta (x-{d}_{b})]\},\end{array}\end{eqnarray}$
where θ(x) is the Heaviside step function with θ(0) = 1/2; A and B (M and N) are the probability amplitudes of finding right-moving and left-moving photons in the region of ${x}_{a}^{1}\lt x\,{\lt }_{a}^{2}$ (${x}_{b}^{1}\lt x\,{\lt }_{b}^{2}$), respectively. ta/tb (ra/rb) denotes the single-photon transmission (reflection) coefficient in waveguide a/b. Combining equations ( $\begin{eqnarray}{t}_{a}=\displaystyle \frac{2{\rm{i}}{{\rm{\Delta }}}_{e}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}{{\rm{e}}}^{-{\rm{i}}{\phi }_{1}}-\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})+2({{\rm{\Gamma }}}_{\mathrm{aR}}^{1}+{{\rm{\Gamma }}}_{\mathrm{aR}}^{2})-[{{\rm{\Gamma }}}_{t}+(\sqrt{{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{2}}]}{2{\rm{i}}{{\rm{\Delta }}}_{e}-[{{\rm{\Gamma }}}_{t}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{2}}]},\end{eqnarray}$
$\begin{eqnarray}{t}_{b}=\displaystyle \frac{2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}{{\rm{e}}}^{-{\rm{i}}{\phi }_{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}{{\rm{e}}}^{{\rm{i}}({\phi }_{1}-{\phi }_{2})})}{2{\rm{i}}{{\rm{\Delta }}}_{e}-[{{\rm{\Gamma }}}_{t}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{2}}]},\end{eqnarray}$
$\begin{eqnarray}{r}_{a}=\displaystyle \frac{2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}}+(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+\sqrt{{{\rm{\Gamma }}}_{2{\rm{R}}}^{2}{{\rm{\Gamma }}}_{2{\rm{L}}}^{2}}{{\rm{e}}}^{2{\rm{i}}{\phi }_{1}})}{2{\rm{i}}{{\rm{\Delta }}}_{e}-[{{\rm{\Gamma }}}_{t}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{2}}]},\end{eqnarray}$
$\begin{eqnarray}{r}_{b}=\displaystyle \frac{2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}{{\rm{e}}}^{{\rm{i}}{\phi }_{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}{{\rm{e}}}^{{\rm{i}}({\phi }_{1}+{\phi }_{2})})}{2{\rm{i}}{{\rm{\Delta }}}_{e}-[{{\rm{\Gamma }}}_{t}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{aR}}^{1}{{\rm{\Gamma }}}_{\mathrm{aR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{aL}}^{1}{{\rm{\Gamma }}}_{\mathrm{aL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{1}}+2(\sqrt{{{\rm{\Gamma }}}_{\mathrm{bR}}^{1}{{\rm{\Gamma }}}_{\mathrm{bR}}^{2}}+\sqrt{{{\rm{\Gamma }}}_{\mathrm{bL}}^{1}{{\rm{\Gamma }}}_{\mathrm{bL}}^{2}}){{\rm{e}}}^{{\rm{i}}{\phi }_{2}}]},\end{eqnarray}$
in which ${{\rm{\Delta }}}_{e}={\rm{\Delta }}+{\rm{i}}{\gamma }_{e}-\tfrac{{{\rm{\Omega }}}^{2}}{{{\rm{\Delta }}}_{f}+{\rm{i}}{\gamma }_{f}}$, Δ = E − ωe, Δf = E − δ = E − ωf − ωd, ${{\rm{\Gamma }}}_{\alpha \beta }^{j}=\tfrac{{\left({g}_{\alpha \beta }^{j}\right)}^{2}}{{\upsilon }_{{\rm{g}}}}$ (α = a, b and β = R, L), and ${{\rm{\Gamma }}}_{t}={{\rm{\Sigma }}}_{j=\mathrm{1,2}}({{\rm{\Gamma }}}_{\mathrm{aL}}^{j}+{{\rm{\Gamma }}}_{\mathrm{aR}}^{j})+{{\rm{\Sigma }}}_{j=\mathrm{1,2}}({{\rm{\Gamma }}}_{\mathrm{bL}}^{j}+{{\rm{\Gamma }}}_{\mathrm{bR}}^{j});$ $\phi$1 = kda = (ωe + Δ)τa ($\phi$2 = kdb = (ωe + Δ)τb) is the phase of photons accumulated between the atom and the waveguide a (b) at the two coupling points, with τa = da/νg (τb = db/νg) the corresponding propagation time. By rewriting $\phi$1 and $\phi$2 as $\phi$1 = $\phi$10 + τaΔ and $\phi$2 = $\phi$20 + τbΔ with $\phi$10 = ωeτa and $\phi$20 = ωeτb, we can study single-photon scattering in both the Morkovian and the non-Markovian regimes depending on whether the time delay τα can be negligible or not. In the markovian regime, the time delay is much smaller than the lifetime of the atom, such that $\phi$1 ≈ $\phi$10 and $\phi$2 ≈ $\phi$20 are approximately constant. However, in the non-Markovian regime, the time delay is comparable to the relaxation time of the atom, thus it is nonnegligible. Consequently, $\phi$1 and $\phi$2 strongly depend on the detuning Δ. The properties of the single-photon transport in both the Morkovian and the non-Markovian regimes are discussed more detail in below.3. Single-photon transport in the Markovian regime
3.1. Nonreciprocal scattering effect
We first ignore the waveguide b and consider the single-photon scattering effect in waveguide a. Note that in the markovian regime, $\phi$1 ≈ $\phi$10. For simplicity, in the following, we assume ${{\rm{\Gamma }}}_{\alpha L}^{j}={{\rm{\Gamma }}}_{L}$ and ${{\rm{\Gamma }}}_{\alpha {\rm{R}}}^{j}={{\rm{\Gamma }}}_{R}$, then the coefficient of the transmission shown in equation (8 ) can be simplified as6 ), one can obtain
$\begin{eqnarray}\begin{array}{rcl}{t}_{a} & = & \displaystyle \frac{{\rm{i}}{{\rm{\Delta }}}_{e}+{{\rm{\Gamma }}}_{R}(1+{{\rm{e}}}^{-{\rm{i}}{\phi }_{1}})-{{\rm{\Gamma }}}_{L}(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})}{{\rm{i}}{{\rm{\Delta }}}_{e}-({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})}\\ & = & \displaystyle \frac{{\rm{i}}\left[{\rm{\Delta }}-\tfrac{{{\rm{\Omega }}}^{2}}{{{\rm{\Delta }}}_{f}+{\rm{i}}{\gamma }_{f}}-\sin {\phi }_{1}({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})\right]+({{\rm{\Gamma }}}_{R}-{{\rm{\Gamma }}}_{L})(1+\cos {\phi }_{1})-{\gamma }_{e}}{{\rm{i}}({\rm{\Delta }}-\tfrac{{{\rm{\Omega }}}^{2}}{{{\rm{\Delta }}}_{f}+{\rm{i}}{\gamma }_{f}})-({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})-{\gamma }_{e}}.\end{array}\end{eqnarray}$
The corresponding transmission rate is TaR = ∣ta∣2. If the incident photon is from right to left, the probability amplitudes of the waveguide modes can be written as $\begin{eqnarray}{\phi }_{\mathrm{aR}}(x)={{\rm{e}}}^{{\rm{i}}{k}_{a}x}\{{\tilde{r}}_{a}\theta (x-{d}_{a})+\tilde{A}[\theta (x)-\theta (x-{d}_{a})]\},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{\mathrm{aL}}(x) & = & {{\rm{e}}}^{-{\rm{i}}{k}_{a}x}\{\theta (x-{d}_{a})\\ & & +\tilde{B}[\theta (x)-\theta (x-{d}_{a})]+{\tilde{t}}_{a}\theta (-x)\},\end{array}\end{eqnarray}$
where ${\tilde{t}}_{a}$ and ${\tilde{r}}_{a}$ represent the corresponding transmission and reflection coefficients, respectively. Combing equation ( $\begin{eqnarray}\begin{array}{rcl}{\tilde{t}}_{a} & = & \displaystyle \frac{{\rm{i}}{{\rm{\Delta }}}_{e}+{{\rm{\Gamma }}}_{L}(1+{{\rm{e}}}^{-{\rm{i}}{\phi }_{1}})-{{\rm{\Gamma }}}_{R}(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})}{{\rm{i}}{{\rm{\Delta }}}_{e}-({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})}\\ & = & \displaystyle \frac{{\rm{i}}\left[{\rm{\Delta }}-\tfrac{{{\rm{\Omega }}}^{2}}{{{\rm{\Delta }}}_{f}+{\rm{i}}{\gamma }_{f}}-\sin {\phi }_{1}({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})\right]+({{\rm{\Gamma }}}_{L}-{{\rm{\Gamma }}}_{R})(1+\cos {\phi }_{1})-{\gamma }_{e}}{{\rm{i}}({\rm{\Delta }}-\tfrac{{{\rm{\Omega }}}^{2}}{{{\rm{\Delta }}}_{f}+{\rm{i}}{\gamma }_{f}})-({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})(1+{{\rm{e}}}^{{\rm{i}}{\phi }_{1}})-{\gamma }_{e}}.\end{array}\end{eqnarray}$
We use ${T}_{\mathrm{aL}}=| {\tilde{t}}_{a}{| }^{2}$ to describe the transmission rate in this case. In addition, the isolation depth I = ∣TaL − TaR∣ is introduced to quantitatively describe the non-reciprocality, which is given by $\begin{eqnarray}I=\displaystyle \frac{4{\gamma }_{e}| {{\rm{\Gamma }}}_{R}-{{\rm{\Gamma }}}_{L}| (1+\cos {\phi }_{1}){{\rm{\Delta }}}_{f}^{2}}{{\left[({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})(1+\cos {\phi }_{1})+{\gamma }_{e}\right]}^{2}{{\rm{\Delta }}}_{f}^{2}+{\left[{\rm{\Delta }}{{\rm{\Delta }}}_{f}-({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L}){{\rm{\Delta }}}_{f}\sin {\phi }_{1}-{{\rm{\Omega }}}^{2}\right]}^{2}}.\end{eqnarray}$
Here, we have assumed that the lifetime of the state ∣f⟩ is very long such that γf ≈ 0.In the two-photon resonant condition Δ = Δf, we plot the transmission rates TaR, TaL, and the isolation depth I versus the detuning Δ in figure 2. By fixing $\phi$1 = 0, we first exhibit the effect of the Rabi frequency Ω on the properties of the single-photon transmission. As shown in figure 2(a), one can find that in the absence of a driving field (i.e., Ω = 0), the three-level atom is simplified as a two-level atom. In this situation, if the left-incident photon is resonant with the transition frequency between ∣e⟩ ↔ ∣g⟩, it will be completely reflected. When the driving field is applied with Ω = 0.5ΓR, the control field can induce the transition between ∣g⟩ ↔ ∣e⟩, leading to two dressed states with energies E± = ωe ± Ω. The perfect reflection (T = 0) occurs whenever the incident photon is resonant with the two dressed states. Thus, for $\phi$10 = 0, two absorption valleys located at Δ = ± Ω can be observed from figure 2(a). This is similar to the traditional electromagnetically induced transparency phenomenon [74]. Further, once Ω > ΓR, the system enters into the Autler-Townes Splitting [75] regime. Hence, the line width and the shape of the transmission peak shown in figure 2(a) become widen and smooth. Comparing figures 2(a) to (b), one can find that the transmission rates TaR and TaL have similar responses to Ω. While the minimum values of TaR and TaL are different for a certain Ω. Consequently, as shown in figure 2(c), the isolation depth I reaches the maximum value when TaL and TaR reach the minimum value.
Figure 2. Nonreciprocal single-photon transmission properties. TaR, TaL, and I versus Δ/ΓR with different Ω are shown in (a), (b), and (c), respectively. TaR, TaL, and I versus Δ/ΓR with different $\phi$10 are shown in (d), (e), and (f), respectively. In (a), (b), and (c), $\phi$10 = 0. In (d), (e), and (f), Ω = ΓR. In all the calculations ΓL = 0.1ΓR, Δ = Δf, γe = ΓR and γf = 0.01ΓR. |
Then, we fix Ω = ΓR to explore the influence of the phase $\phi$1 ≈ $\phi$10 on the transmission behaviors of the left- and right-incident photon in figures 2(d-f). It is clearly shown that the symmetry of the transmission spectra is broken. Moreover, by analysing equations (12 ), (15 ), and (16 ) we can obtain that when ${\rm{\Delta }}={{\rm{\Delta }}}_{f}=\tfrac{({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})\sin {\phi }_{1}\pm \sqrt{{\left[({{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L})\sin {\phi }_{1}\right]}^{2}\,+\,4{{\rm{\Omega }}}^{2}}}{2}$, TaR (TaL) and I reach the minimum and the maximum values, respectively. Therefore, the position of the two absorption valleys and the isolation depth peak shown in figure 2(d) (figures 2(e)) and 2(f), respectively, are modulated by the phase $\phi$1. In particular, when $\phi$10 = π, we have TaR = TaL = 1 and I = 0. This can be explained by using equations (12 ) and (15 ), in which one can find that due to the giant-atom structure, the coupling strengths have been modified to be ${{\rm{\Gamma }}}_{R/L}={{\rm{\Gamma }}}_{R/L}(1+\cos {\phi }_{10})$. When $\phi$10 = (2k + 1)$\phi$, we have ΓR/L = 0, which means that the system is transparent for the incident photon. Hence, the single-photon transmission is reciprocal for $\phi$10 = π.
It is worth indicating that, except for the chiral photon-atom interactions, the atomic dissipation is also a necessary condition to realize the nonreciprocal scattering effect. Based on the ideal chiral case (ΓL = 0), we explore how the isolation depth I in the giant- and small-atom setups is influenced by the atomic dissipation γe in figure 3(a). It is shown that when γe = 0, I in both setups is equal to 0, which means that the transmission becomes reciprocal. While, with increasing of γe, I in both setups can be significantly improved. More interestingly, the isolation depth I in the small-atom setup will first reach the maximum value of 1 and then decrease approximately linearly on γe. Especially, when γe = ΓR, I in the small-atom setup decreases to 0.89; in contrast, I reaches the maximum value of 1 in the giant-atom setup. Moreover, once γe > 0.7ΓR, the value of I in the giant-atom setup is bigger than that in the small-atom setup. Those results imply that for a larger atomic dissipation, the non-reciprocity in the giant-atom setup is more robust than that in the small-atom setup. Additionally, by fixing γe = ΓR, we show the influence of the ratio ΓL/ΓR on TaR, TaL and I in figure 3(b). It shows clearly that in the perfect chiral case (ΓL = 0), the right-incident photon will be completely transmitted to the left, i.e., TaL = 1; while the transmission to the opposite direction is totally blocked (TaR = 0). Thus I can reach as high as 1. In contrast, when the coupling is non-chiral (ΓL = ΓR), we have TaR = TaL and I = 0, which suggests that the transmission is reciprocal.
Figure 3. The isolation depth I varying with (a) γe/ΓR and (b) ΓL/ΓR. Other relevant parameters are Ω = 2ΓR and $\phi$10 = 0.5π. |
3.2. Single-photon routing
Now, by taking the chiral coupling between the transition ∣e⟩ ↔ ∣g⟩ and waveguide b into consideration, we show how to route the photon into a desired direction, for example, route the single-photon from port aL deterministically to port bR. Compared to the single waveguide case, the atom now has an additional channel decaying into the waveguide b. This indicates that the left incident photon can be reemitted into either of the two waveguides. In all traditional routing schemes, the probability of successful routing is typically limited. In figure 4, we show that the chiral photon-atom couplings can effectively improve the routing capability, since the balance between the reemitting photons to the right and left directions of the waveguide b is broken. As shown in figure 4, under the ideal condition with ${{\rm{\Gamma }}}_{{\rm{\alpha }}{\rm{R}}}^{j}={{\rm{\Gamma }}}_{R}$, ${{\rm{\Gamma }}}_{{\rm{\alpha }}{\rm{L}}}^{j}={{\rm{\Gamma }}}_{{L}}=0$ and γe = γf = 0, when Ω = 0, the resonant single-photon incident from port aL can be routed into port bR with probability 100%. Further, as shown in figures 4(b)-(c), in the presence of the driving field with frequency ωd = ωe − ωf, the photon with frequency ω = ωe ± Ω can still be deterministically transmitted into port bR. This shows that targeted photon routing can be achieved deterministically for an incident photon with a nonresonant frequency. This router can thus overcome the limitations of the previous routers, wherein only the photon with the same frequency as the atom can be deterministically routed to the desired direction. Note that, by setting ${{\rm{\Gamma }}}_{\mathrm{aR}}^{j}={{\rm{\Gamma }}}_{\mathrm{bL}}^{j}={{\rm{\Gamma }}}_{R}$ and ${{\rm{\Gamma }}}_{\mathrm{aL}}^{j}={{\rm{\Gamma }}}_{\mathrm{bR}}^{j}=0$, we have ra = tb = 0. In this case, if $\phi$10 = $\phi$20 = 0 and Δ = ± Ω, one can obtain rb = 1. This means that the incident photon is transferred to waveguide b and output from the left of waveguide b with a probability of 100%.
Figure 4. Transmission rates Ta and Tb as a function of Δ/ΓR with different Ω. Other parameters are: ${{\rm{\Gamma }}}_{\alpha R}^{j}={{\rm{\Gamma }}}_{R}$, ${{\rm{\Gamma }}}_{\alpha L}^{j}={{\rm{\Gamma }}}_{L}=0$, Δ = Δf, $\phi$10 = $\phi$20 = 0 and γe = γf = 0. |
The results shown in figure 4 are unremarkable since a similar conclusion can be also obtained in the small-atom setup. So what are the special results for the routing scheme in the giant atom setup? In order to answer this question, we first investigate the effect of the phases $\phi$10 and $\phi$20 on the routing probability. In figure 5, by fixing $\phi$10 = − $\phi$20 = $\phi$ and Ω = 2ΓR, we plot the routing probability Tb versus $\phi$ and the detuning Δ. We find that when $\phi$ is in −0.5π < $\phi$ < 0.5π, the single-photon incident from port aL can be targeted and transferred to the port bR. However, once ∣$\phi$∣ > 0.5π, the routing probability declines. Particularly, as we mentioned above, when $\phi$ = ± π, due to the destructive interference, the incident single-photon can be directly transmitted into port aR with Ta = TaR ≡ 1. Consequently, the incident photon cannot be routed to waveguide b, which indicates that the routing scheme fails. Figure 5 clearly indicates that the routing probability can be controlled by adjusting the phase.
Figure 5. Quantum routing probability Tb varying with Δ/ΓR and $\phi$. Other parameters are the same as that in figure 4. |
The numerical results shown in figures 4 and 5 are obtained based on the ideal case. In the following, we examine the routing capability in a realistic case, in which the chiral coupling is not perfect and the atomic dissipation is also considered. Here, we define the chirality in the waveguide α as ${S}_{\alpha }=\tfrac{{{\rm{\Gamma }}}_{\alpha {\rm{R}}}-{{\rm{\Gamma }}}_{\alpha {\rm{L}}}}{{{\rm{\Gamma }}}_{\alpha {\rm{R}}}+{{\rm{\Gamma }}}_{\alpha {\rm{L}}}}$. Since we have assumed ΓαR = ΓR and ΓαL = ΓL, the values of the chirality Sa and Sb are considered to equal S, i.e., ${S}_{\alpha }=S=\tfrac{{{\rm{\Gamma }}}_{R}-{{\rm{\Gamma }}}_{L}}{{{\rm{\Gamma }}}_{R}+{{\rm{\Gamma }}}_{L}}$. Obviously, S = 1 and S = 0 correspond to the ideal chiral and the non-chiral cases, respectively. The dependence of the routing probability Tb on the chirality S is studied in figure 6(a). It is shown that with the same atomic dissipation, the routing capability in the giant atom setup is better than that in the small atom setup. The other relevant parameter is the Purcell factor $P=\tfrac{{{\rm{\Gamma }}}_{a}+{{\rm{\Gamma }}}_{b}}{{\gamma }_{e}+{\gamma }_{f}}$ with ${{\rm{\Gamma }}}_{\alpha =a,b}={\sum }_{j}({{\rm{\Gamma }}}_{{\rm{\alpha }}{\rm{R}}}^{j}+{{\rm{\Gamma }}}_{{\rm{\alpha }}{\rm{L}}}^{j})$, which accounts for the modification of the total decay rate of an emitter when placed in the vicinity of a nanostructure. In figure 6(b), we explore the dependence of Tb on P. It is shown that the routing probability Tb firstly increases with P and then it tends to a fixed value. In particular, for a certain S, the routing probability in the giant-atom setup is higher than that in the small-atom setup. Specifically, for achievable values P = 40 and S = 0.95, Tb in the giant-atom setup can reach as high as 0.9, while it only can reach 0.85 in the small-atom setup. The results revealed by figure 6 indicate that when the imperfections are considered, the routing capability in the giant-atom setup is better than that in the small-atom setup.
Figure 6. Quantum routing probability Tb as a function of (a) S and (b) P. In (a) and (b), we fix Ω = 2ΓR, $\phi$10 = − $\phi$20 = 0.5π, Δ = Ω and γf = 0.01ΓR. Other parameters are the same as that in figure 4. |
4. Single-photon transport in the non-Markovian regime
Next, we investigate the nonreciprocal single-photon scattering and the single-photon router in the non-Markovian regime, in which the time delay between different atom-waveguide coupling points is comparable to or larger than the lifetime of the atom. Therefore, the accumulated phases $\phi$j = $\phi$j0 + ταΔ strongly depend on the detuning Δ. Such detuning-dependent phases will definitely induce the non-Markovian features in the transmission spectra. The non-Markovicity induced non-reciprocity and single-photon router are investigated in figures 7 and 8, respectively. Here, we choose $\phi$j = $\phi$j0 = π and ταΓR = 3. As we mentioned above, in the Markovian regime when $\phi$j = $\phi$j0 = π, the non-reciprocity will disappear and the routing scheme will be destroyed. However, in the non-Markovian regime, this conclusion can be broken. As shown in figure 7, we find that for a certain Ω, the nonreciprocal scattering effect can still survive in the case of $\phi$10 = π. Further, compared with the Markovian regime, this phase accumulated effect for detuned photons give rise to more complicated transmission spectra with multiple peaks and dips. Figure 8 demonstrates that the routing scheme is also efficient in the case of $\phi$10 = $\phi$20 = π. It is shown that the single-photon with different frequencies is allowed to transport to waveguide b except for the specific values of Δ. Moreover, the frequency of the routing photons also depends on the Rabi frequency Ω. As shown in figures 8(a) and (b), for different Ω, the routing capabilities for the photons with specific frequencies are different.
Figure 7. Transmission rates TaR, TaL, and the ratio I as a function of Δ/ΓR with τa = τb = 3/ΓR in (a) Ω = 0 and in (b) Ω = 3ΓR. In all calculations $\phi$10 = π. Other parameters are the same as that in figure 4. |
Figure 8. Routing probability in the non-markovian regime. Transmission spectra Ta and Tb as a function of Δ with (a) Ω = 0 and (b) Ω = 3ΓR. Here τa = τb = 3/ΓR and $\phi$10 = $\phi$20 = π. Other parameters are the same as that in figure 4. |
5. Conclusion
We proposed a chiral giant atom-waveguide model, in which two one-dimensional waveguides are mediated by a driven three-level giant atom. The single-photon scattering behaviors in both the Markovian and the non-Markovian regimes have been explored. Under the Markovian limit, we find that due to the chiral photon-atom coupling, the non-reciprocity in a single waveguide and the quantum routing with high probability in two waveguides can be achieved. The presence of the driving field and the giant atom structure provide more feasibilities to modulate the scattering behaviors. We also studied the effect of the imperfection and the atomic dissipation on the transmission spectra and the routing capability in both the giant-atom model and the corresponding small-atom model. The numerical results showed that in the imperfect chiral coupling and the larger atomic dissipation cases, the values of the difference between the transmission rates corresponding to opposite transport direction I and the routing probability Tb in the giant-atom model are larger than that in the small-atom model. This implies that the non-reciprocity and routing capabilities in the giant-atom setup are better than that in the small-atom setup. Further, we simulated the transmission spectra and the routing probability in the non-Markovian regime, in which the propagation time of photons between different atom-waveguide coupling points cannot be neglected. We find that the phase accumulated is detuning dependent, which leads to more complicated transmission behaviors. In particular, the non-Markovicity induced nonreciprocal scattering and photon routing are discussed in more detail.