Phase-modulated Autler–Townes splitting in a giant-atom system within waveguide QED

Abstract

The nonlocal emitter-waveguide coupling, which gives birth to the so called giant atom, represents a new paradigm in the field of quantum optics and waveguide QED. We investigate the single-photon scattering in a one-dimensional waveguide on a two-level or three-level giant atom. Thanks to the natural interference induced by the back and forth photon transmitted/reflected between the atom-waveguide coupling points, the photon transmission can be dynamically controlled by the periodic phase modulation via adjusting the size of the giant atom. For the two-level giant-atom setup, we demonstrate the energy shift which is dependent on the atomic size. For the driven three-level giantatom setup, it is of great interest that, the Autler–Townes splitting is dramatically modulated by the giant atom, in which the width of the transmission valleys (reflection range) is tunable in terms of the atomic size. Our investigation will be beneficial to the photon or phonon control in quantum network based on mesoscopical or even macroscopical quantum nodes involving the giant atom.

Graphical abstract

Keywords

giant atom / single-photon scattering / quantum interference / Autler–Townes splitting

1 Introduction

In the field of quantum optics, the study of the interaction between light and matter is one of the long-lived subjects. Recently, the light–matter interaction in waveguide structures has attracted much attention, which leads to lots of theoretical and experimental works in waveguide QED community [1–8], such as dressed or bound states [9–15], phase transitions [16–18], single-photon devices [19–21], exotic topological and chiral phenomena [22–27], where the wavelength of light (or microwave field) is usually tens or hundreds of times larger than the size of the natural/artificial atoms constituting the matter [28–32]. Therefore, the light–matter interaction is usually modelled by the dipole approximation, where the atoms are regarded as point-like dipoles [33].

However, in recent years, the artificial superconducting transmon qubit coupled by the acoustic waves [34–36], of which the size is comparable to the wavelength of the phonons, is named as “giant atom” and has been successfully realized in experiments [37]. Alternatively, the giantatom model can also be realized in the superconducting transmission line setup, where the capacitive or inductive coupling allows more than one coupling points between the microwave field and the qubit [38, 39]. Moreover, using the cold atom system, a theoretical scheme for the realization of giant atom has been proposed in dynamical state-dependent optical lattices [40]. In the giant-atom community, a lot of new phenomena not existing in the conventional small atom system have been predicted, such as frequency dependent relaxation [41], non-exponential decay [42–44], tunable bound state [45, 46] as well as decoherence free subspace [47, 48] (For a recent review, see Ref. [49]). The underlying physics behind these phenomena is the interference and retarded effect during the photon/phonon propagating process between the different coupling points.

On the other hand, the dynamical control of the single-photon transmission is a hot topic in constructing quantum networks [50–52], motivated by the photon based quantum information processing. Photons provide a reliable transmission of quantum information and the waveguide is often seen as photonic channels in quantum network, with atoms (or artificial atoms) acting as quantum nodes. Along this line, people have proposed lots of schemes to realize single-photon device, in which the propagating of the photon in channels is controlled on demand by adjusting the nature of the quantum nodes [19–21, 53–57]. Furthermore, for a three-level quantum node, the electromagnetically induced transparency (EIT) [58] and Autler–Townes splitting (ATS) phenomena have been investigated both theoretically and experimentally [59–62]. Meanwhile, it is also possible to control photon by photon, to realize an all optical routing [63] or microwave-photon detector [64]. Combined with the interference effect, it motivates us to study how to modulate the single photon scattering by the giant atom.

In this paper, we tackle this issue in a one-dimensional waveguide with a two-level or three-level giant atom. For the two-level atom setup, we find that the change in the size of the atom can control its energy shift due to the interference effect between the backward and forward photon in the waveguide. In the driven three-level giant-atom system, we demonstrate the controllable phase-modulated ATS physics. Our results show that, the size of the giant atom, which serves as a controller, can be used to tune the width of the transmission valleys (reflection range) in a periodical manner. The underlying physical principal is further revealed in the viewpoint of quantum open system based on the dressed state representation.

The rest of the paper is structured as follows: In Section 2, we present the model and discuss the single photon scattering in a two-level giant-atom system. In Section 3, we demonstrate the Autler–Townes splitting behavior for a driven three-level giant-atom with Λ-type transition. In Section 4, we discuss the effects of the dissipation for both of two-level and three-level giant atom setups. At last, we end up with a brief conclusion in Section 5.

2 Two-level giant atom

As schematically shown in Fig. 1, the system we consider is composed of a linear waveguide and a giant atom, which is actually a two-level system. The giant atom is connected to the waveguide via two points with x = 0 and x = x₀, respectively. The Hamiltonian H₁ of the system can be divided into three parts, i.e., H₁ = H_s + H_ω + V₁. The first part H_s is the free Hamiltonian of the giant atom (Hereafter, we set \hslash =1),

\begin{array}{}\text{(1)}& H_s={\omega }_e|e\rangle \langle e|\mathrm{,}\end{array}

where ω_e is the transition frequency between the ground state |g〉 and the excited state |e〉. As a reference, we have set the frequency of the ground state |g〉 as ω_g = 0.

The second part H_ω of the Hamiltonian H₁ represents the free Hamiltonian of the waveguide, and is expressed as

\begin{array}{}\text{(2)}& \begin{array}{ll}{H}_{\omega }=\hfill & {\displaystyle \int }\text{d}x\{-\text{i}{v}_g{C}_R^{†}(x)\frac{\text{d}}{\text{d}x}{C}_R(x)\hfill \\ \hfill & +\text{i}{v}_g{C}_L^{†}(x)\frac{\text{d}}{\text{d}x}{C}_L(x)\}\mathrm{,}\hfill \end{array}\end{array}

where v_g is the group velocity of photons traveling in the waveguide. C_R^{†}(x)[C_L^{†}(x)] is the bosonic creation operator for the right-going (left-going) photon at position x.

For the third part V₁ of the Hamiltonian H₁, we describe the interaction between the waveguide and the giant atom. Within the rotating wave approximation, the Hamiltonian V₁ can be expressed as

\begin{array}{}\text{(3)}& \begin{array}{ll}{V}_1=\hfill & f{\displaystyle \int }\text{d}x\delta (x)[{\sigma }^{+}{C}_R\left(x\right)+{\sigma }^{+}{C}_L\left(x\right)+\text{H}\text{.c}\text{.}]\hfill \\ \hfill & +f{\displaystyle \int }\text{d}x\delta (x-x_0)[{\sigma }^{+}{C}_R\left(x\right)+{\sigma }^{+}{C}_L\left(x\right)+\text{H}\text{.c}\text{.}]\mathrm{,}\hfill \end{array}\end{array}

where f is the coupling strength between the waveguide and the two-level giant atom. σ⁺ = (σ⁻)^† = |e〉〈g| is the raising operator of the atom. The Dirac-δ function in the Hamiltonian V₁ indicates that the giant atom has a length of x₀ and connects to the waveguide via its head and tail, that is, x = 0 and x = x₀.

It is noted that, the total excitation of the atom and the photon in the waveguide is conserved. In the following section, we will restrict ourselves in the single excitation subspace, to investigate how to control the single photon scattering via adjusting the frequency of the photon and the size of the two-level giant atom.

In the single-excitation subspace, the eigenstate of the system can be written as

\begin{array}{}\text{(4)}& \begin{array}{ll}|E\rangle =\hfill & {\displaystyle \int }\text{d}x[{\varphi }_{R_1}(x)C_R^{†}(x)+{\varphi }_{L_1}(x)C_L^{†}(x)]|G\rangle \hfill \\ \hfill & +u_e{\sigma }^{+}|G\rangle \mathrm{,}\hfill \end{array}\end{array}

where |G〉 represents that the waveguide is in the vacuum states while the giant atom is in the ground state |g〉. {\varphi }_{R_1}(x) and {\varphi }_{L_1}\left(x\right) are single-photon wave functions of the right-going and left-going modes in the waveguide, respectively. u_e is the excitation amplitude of the giant atom. Solving the stationary Schödinger equation H₁ |E〉 = E |E〉, the amplitudes equation can be obtained as

\begin{array}{}\text{(5a)}& -\text{i}{v}_g\frac{\text{d}}{\text{d}x}{\varphi }_{R_1}\left(x\right)+f{u}_{e}M=E{\varphi }_{R_1}\left(x\right)\mathrm{,}\end{array}

\begin{array}{}\text{(5b)}& \text{i}{v}_g\frac{\text{d}}{\text{d}x}{\varphi }_{L_1}\left(x\right)+f{u}_{e}M=E{\varphi }_{L_1}\left(x\right)\mathrm{,}\end{array}

\begin{array}{}\text{(5c)}& {\omega }_e{u}_e+fN=E{u}_e\mathrm{,}\end{array}

where M = δ(x) + δ(x−x₀) and N={\varphi }_{R_1}(0)+{\varphi }_{R_1}(x_0)+ {\varphi }_{L_1}(0)+{\varphi }_{L_1}(x_0) .

Next, we consider the scattering behavior when a single photon with wave vector k is incident from the left side of the waveguide. In this case, the wave functions of {\varphi }_{R_1}(x) and {\varphi }_{L_1}(x) can be expressed as

\begin{array}{}\text{(6)}& \begin{array}{ll}{\varphi }_{R_1}\left(x\right)=\hfill & {\text{e}}^{\text{i}kx}\left\{\theta \right(-x)+A_1[\theta \left(x\right)-\theta \left(x-x_0\right)]\hfill \\ \hfill & +t_1\theta \left(x-x_0\right)\}\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(7)}& {\varphi }_{L_1}\left(x\right)={\text{e}}^{-\text{i}kx}\left\{r_1\theta \left(-x\right)+B_1\left[\theta \left(x\right)-\theta \left(x-x_0\right)\right]\right\}\mathrm{,}\end{array}

with

We explain the expression of the above wave functions physically as follows. When the right-going photon incident from the region x < 0 reaches the first connection point x = 0 between the giant atom and the waveguide, it can be transmitted or reflected, with the amplitudes of A₁ and r₁, respectively. The photon transmitted at the first connection point at x = 0 will travel freely in the waveguide until it reaches the second connection point at x = x₀, it will be then reflected or transmitted secondly, with the amplitudes B₁ and t₁, respectively.

Now, we substitute Eq. (6-8) into the amplitude Eq. (5), it yields the dispersion relation E = v_gk, and

\begin{array}{}\text{(9)}& t_1=\frac{\text{i}\text{Δ}{v}_g-2\text{i}{f}^2\sin \left(k{x}_0\right)}{\text{i}\text{Δ}{v}_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)}\mathrm{.}\end{array}

where ∆ = E−ω_e is the detuning between the atom and the propagating photon in the waveguide.

Furthermore, the transmission rate T₁ = |t₁|² can be obtained as

\begin{array}{}\text{(10)}& T_1=\frac{{\left[\begin{array}{c}\text{Δ}{v}_g-2f^2\sin \left(k{x}_0\right)\end{array}\right]}^2}{{\left[\begin{array}{c}\text{Δ}{v}_g-2f^2\sin \left(k{x}_0\right)\end{array}\right]}^2+4f^4{\left[1+\cos \left(k{x}_0\right)\right]}^2}\mathrm{,}\end{array}

In the small atom scenario (x₀ = 0) which is studied in Ref. [19], it is obvious that the incident photon will be completely reflected (T₁ = 0) when it is resonant with the atom, that is, ∆ = 0. However, for the giant atom in our setup, the incident photon will propagate back and forth in the spatial regime covered by the giant atom, leading to an interference effect. As a result, as shown in Fig. 2(a), the transmission rate can be controlled by adjusting the photon-atom detuning ∆ and the size of the giant atom x₀.

Furthermore, considering the atomic spontaneous emission, where we replace ω_e by ω_e−iγ_e, the transmission amplitudes t_1^{\prime } becomes

\begin{array}{}\text{(11)}& t_1^{\prime }=\frac{\text{i}\left(\text{Δ}+\text{i}{\gamma }_e\right)v_g-2\text{i}{f}^2\sin \left(k{x}_0\right)}{\text{i}\left(\text{Δ}+\text{i}{\gamma }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)}\mathrm{,}\end{array}

where γ_e is the spontaneous emission of the giant atom. As shown in Figs. 2(b) and (c), we plot the transmission rates T₁ and T_1^{\prime }=|t_1^{\prime }{|}^2 as functions of the detuning ∆, respectively, to compare the results with and without spontaneous emission. It shows that the spontaneous emission breaks the complete transmission and makes the transmission valleys become wider and shallower. However, the locations of the valleys are not changed. It implies that the environment nearly does not change the energy structure of the system.

More interestingly, in the giant-atom situation (x₀ ≠ 0), the detuning for complete reflection is determined by the transcendental equation

\begin{array}{}\text{(12)}& {\text{Δ}}_r=\frac{2f^2\sin (k_r{x}_0)}{v_g}\mathrm{,}\end{array}

where ∆_r is the detuning between the atom and the propagating photon when the incident photon is completely reflected, and k_r is the corresponding photonic wave vector, ∆_r is dependent on k_r via ∆_r = v_g k_r−ω_e. It is well known that the atomic frequency is determined by the location of the complete reflection of the incident photon [19]. Therefore, in Fig. 2(b), we can observe an atomic frequency shift which originates from the interference effect as discussed above, compared with the small atom system. Since |sin k_rx₀|≤1, Eq. (12) implies that only when the frequency of the incident photon satisfies |∆_r| ≤ 2f²/v_g, that is, ω_e + 2f²/v_g ≥ v_gk_r ≥ ω_e−2f²/v_g, it is possible to be completely reflected. Within this regime, we plot the dependence of ∆_r on the size of giant atom x₀ in Fig. 3(a) (solid line), which shows a sinusoidal shape. The exact sinusoidal shape by the numerical fitting (empty circles) is also shown in the figure and the fitting function is obtained by

\begin{array}{}\text{(13)}& {\text{Δ}}_r\approx S\sin ({\omega }_e{x}_0/v_g)\mathrm{,}\end{array}

where the fitting parameters S ≈ 2f²/v_g is plotted as a function of atom-waveguide coupling strength f in Fig. 3(b). We recall that, when two identical small atoms are coupled to the waveguide at the position x = 0 and x₀, the photon will also propagate back and forth between the two atoms. However, as shown in Appendix A, we cannot observe the shift, that is T = 0 occurs at ∆ = 0. The reason is that, small atoms do not interfere with themselves. However, in giant atoms, the interference within an atom promotes the apparent frequency shift in its energy level.

In experiments, the giant atom setup can be realized by the superconducting circuits, and the transition frequency is in the order of GHz [38, 39], while the intrinsic decoherence time of the superconducting qubit is achieved by 20 µs [65] or even longer [66]. The natural line width of the giant atom is then γ = 0.05 MHz. In Fig. 3(a), the energy shift is with the regime of |∆_r| = 5 ×10⁻³ω_e = 15 MHz, for some x₀, so that ∆_r ≫ γ, and ∆_r is about 300 times larger than γ. Therefore, the energy shift of the giant atom can be larger than its natural line width and observed experimentally by increasing the atom-waveguide coupling strength.

3 Three-level giant atom

In this section, let us consider the single-photon scattering on a driven three-level giant atom in Λ type. As schematically shown in Fig. 4, the atom is characterized by the ground state |g〉, excited state |e〉 and metastable state |f〉. Here, there are two types of physical processes in this system. One is similar to that in two-level giant atom system, that is, the interplay between the multiple backward and forward photons of the transmission and reflection in the waveguide; the other is peculiar for such a three-level atomic system, i.e., the strong driving field couples the states |e〉 and |f〉 and a weak field (the incident photon in the waveguide) probes the states |e〉 and |g〉 will induce the ATS phenomenon. The incorporation between these two processes may provide a possibility of the extra and interesting single-photon scattering spectrum for the giant-atom setup.

The Hamiltonian H₂ of the current system can be divided into three parts, i.e., H₂ = H_w + H_a + V₂. Here, H_w is the free Hamiltonian of the waveguide which is given in Eq. (2). H_a is the Hamiltonian of the giant atom

\begin{array}{}\text{(14)}& \begin{array}{ll}{H}_a=\hfill & {\omega }_e|e\rangle \langle e|+{\omega }_f|f\rangle \langle f|\hfill \\ \hfill & +\eta (|e\rangle \langle f|{\text{e}}^{-\text{i}{\omega }_{d}t}+|f\rangle \langle e|{\text{e}}^{\text{i}{\omega }_{d}t})\mathrm{,}\hfill \end{array}\end{array}

where ω_e and ω_f (ω_f < ω_e) are the frequencies of the state |e〉 and |f〉, respectively. As a reference, we have set ω_g = 0. η and ω_d are, respectively, the strength and frequency of the classical field, which drives |f〉 ↔ |e〉 transition. In the rotating frame, the time independent Hamiltonian becomes

\begin{array}{}\text{(15)}& {\tilde{H}}_a={\omega }_e|e\rangle \langle e|+\delta |f\rangle \langle f|+\eta (|e\rangle \langle f|+|f\rangle \langle e|)\mathrm{,}\end{array}

where δ = ω_f + ω_d.

The third part V₂ of the Hamiltonian H₂ describes the interaction between the waveguide and the giant atom. Within the rotating wave approximation, it can be expressed as

\begin{array}{}\text{(16)}& \begin{array}{ll}{V}_2=\hfill & f{\displaystyle \int }\text{d}x\delta (x)\left\{\left[C_R^{†}(x)+C_L^{†}(x)\right]|g\rangle \langle e|+\text{H}\text{.c}\text{.}\right\}\hfill \\ \hfill & +f{\displaystyle \int }\text{d}x\delta (x-x_0)\left\{\left[C_R^{†}(x)+C_L^{†}(x)\right]|g\rangle \langle e|+\text{H}\text{.c}\text{.}\right\}\mathrm{,}\hfill \end{array}\end{array}

where f is the coupling strength between the waveguide and giant atom.

In the single-excitation subspace, the eigenstate of the system can be expressed as

\begin{array}{}\text{(17)}& \begin{array}{ll}|E\rangle =\hfill & {\displaystyle \int }\text{d}x\left[{\varphi }_{R_2}(x)C_R^{†}+{\varphi }_{L_2}(x)C_L^{†}\right]|\varnothing \mathrm{,}g\rangle \hfill \\ \hfill & +{\lambda }_e|\varnothing \mathrm{,}e\rangle +{\lambda }_f|\varnothing \mathrm{,}f\rangle \mathrm{,}\hfill \end{array}\end{array}

where |\varnothing \mathrm{,}m\rangle (m = e, g, f) represents that the waveguide is in the vacuum state while the giant atom is in the state |m〉. {\varphi }_{R_2}\left(x\right) and {\varphi }_{L_2}\left(x\right) are single-photon wave functions of the right-going and left-going modes in the waveguide, respectively. λ_e and λ_f are the excitation amplitudes of the giant atom in the excited state |e〉 and metastable state |f〉, respectively. Similar to the discussion in two-level atom setup, for a right-going incident photon with wave vector k, the photon amplitude can be expressed as

\begin{array}{}\text{(18)}& \begin{array}{ll}{\varphi }_{R_2}\left(x\right)=\hfill & {\text{e}}^{\text{i}kx}\left\{\theta \right(-x)+A_2[\theta \left(x\right)-\theta \left(x-x_0\right)]\hfill \\ \hfill & +t_2\theta \left(x-x_0\right)\}\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(19)}& {\varphi }_{L_2}\left(x\right)={\text{e}}^{-\text{i}kx}\left\{r_2\theta \left(-x\right)+B_2\left[\theta \left(x\right)-\theta \left(x-x_0\right)\right]\right\}\mathrm{,}\end{array}

where t₂ and r₂ are the transmission and reflection amplitudes while A₂ (B₂) are the amplitudes for finding a right (left)-going photon inside the regime of the giant atom. Then, the Schödinger equation \tilde{H}|E\rangle =E|E\rangle (where \tilde{H}=H_w+{\tilde{H}}_a+V_2) yields

\begin{array}{}\text{(20)}& t_2=\frac{(E-\delta )\left[\text{i}\left(E-{\omega }_e\right)v_g-2\text{i}{f}^2\sin \left(k{x}_0\right)\right]-\text{i}{v}_g{\eta }^2}{(E-\delta )\left[\text{i}\left(E-{\omega }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)\right]-\text{i}{v}_g{\eta }^2}\mathrm{,}\end{array}

\begin{array}{}\text{(21)}& r_2=\frac{2f^2(E-\delta )\left[1+\cos \left(k{x}_0\right)\right]{\text{e}}^{\text{i}k{x}_0}}{(E-\delta )\left[\text{i}\left(E-{\omega }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)\right]-\text{i}{v}_g{\eta }^2}\mathrm{.}\end{array}

Then the transmission rate T₂ = |t₂|² and reflection rate R₂ = |r₂|² satisfy T₂ + R₂ = 1 due to the neglect of the natural relaxations in the atom.

In Fig. 5, we plot the transmission rate T₂ as functions of the detuning ∆ = E−ω_e between the incident photon and atomic |g〉 ↔ |e〉 transition and the size of the giant atom x₀. Here, we illustrate the result for resonantly driving in Figs. 5(a) and (d), and non-resonantly driving in Figs. 5(b), (c) and (e). All of the results are characterized by two narrow transmission valleys (T₂ ≃ 0) and a relatively wide transmission windows (T₂ ≃ 1) between them around the two-photon resonance.

First, we focus on the transmission window (T₂ ≃ 1, R₂ ≃ 0). It can be observed in Eq. (21) that R₂=0 when E−δ = E−ω_f−ω_d = 0, which implies the two-photon resonance condition ω_e−E = ω_e−ω_f−ω_d. This transmission window is caused by the usual ATS mechanism, which is true for both of the small and giant atom setup. Moreover, we can also observe that R₂ = 0 when 1 + cos kx₀ = 0, that is kx₀ = (2m + 1)π where m is an integer, and the dependence of the position is peculiar for the giant atom setup. Note that kx₀ is the accumulated phase as the travelling photon moves from one coupling point to the other. Therefore, such a ATS transmission is also related to the position-dependent phase introduced by the interference effect from the back and forth photons inside the regime covered by the giant atom.

Next, we discuss the two valleys, which represent the complete reflection (T₂ ≃ 0, R₂ ≃ 1). Based on Eq. (20), the complete reflection occurs when

\begin{array}{}\text{(22)}& \begin{array}{ll}{E}_{\pm }=\hfill & {\omega }_e+f^2\sin (k{x}_0)/v_g-\frac{{\text{Δ}}_2}{2}\hfill \\ \hfill & \pm \frac{\sqrt{{\left[{\text{Δ}}_2+2f^2\sin \left(k{x}_0\right)/v_g\right]}^2+4{\eta }^2}}{2}\mathrm{,}\hfill \end{array}\end{array}

where ∆₂ = ω_e−ω_f−ω_d is the detuning between the driving field and the atomic |f〉 ↔ |e〉 transition. This fact can be explained intuitively in the dressed state presentation. The eigenfrequencies of the driving Hamiltonian {\tilde{H}}_a in Eq. (15) are

\begin{array}{}\text{(23)}& {\omega }_{\pm }={\omega }_e-\frac{{\text{Δ}}_2}{2}\pm \frac{\sqrt{{\text{Δ}}_2^2+4{\eta }^2}}{2}\mathrm{.}\end{array}

and the corresponding states are

\begin{array}{}\text{(24)}& |{\psi }_{+}\rangle =\cos \frac{\theta }{2}|e\rangle +\sin \frac{\theta }{2}|f\rangle \mathrm{,}\end{array}

\begin{array}{}\text{(25)}& |{\psi }_{-}\rangle =-\sin \frac{\theta }{2}|e\rangle +\cos \frac{\theta }{2}|f\rangle \mathrm{.}\end{array}

For the resonantly driving (∆₂ = 0), we will have θ = π/2. For the case of non-resonantly driving, we will have θ = arctan(2η/∆₂) for ∆₂ > 0 and θ = π + arctan(2η/∆₂) for ∆₂ < 0.

Comparing Eq. (23) with Eq. (22), we find that the incident photon will be completely reflected when it is “nearly” resonant with ω_±. This fact is verified by the results shown in Fig. 5. Here, we use the phrase “nearly” to imply that E_± is not exactly equal to ω_±, but is slightly modulated by x₀. As a result, the photon transmission shows a periodic phase modulation in terms of x₀, which is clearly demonstrated in Figs. 5 (a), (b) and (c) for both of resonantly and non-resonantly driving situations.

It is also shown in Fig. 5 that the two valleys discussed above possess different widths. For example, see the horizontal lines labelled by “A”, “B”, “C” and “D” in Figs. 5 (a), (b) and (c). This can be explained by the different effective decay rates of the eigenstates with frequency ω_± in the viewpoint of quantum open system by regarding the waveguide as the effective environment. To this end, we rewrite the interaction Hamiltonian V₂ in the momentum space by performing the Fourier transformation (in terms of |ψ_±〉) as

\begin{array}{}\text{(26)}& \begin{array}{ll}{V}_2=\hfill & {\displaystyle \sum _k}\{[C_L^{†}(k)+C_R^{†}(k)]|g\rangle [G_{+\mathrm{,}k}\langle {\psi }_{+}|\hfill \\ \hfill & +G_{-\mathrm{,}k}\langle {\psi }_{-}|]+\text{H}\text{.c}\mathrm{.}\}\mathrm{,}\hfill \end{array}\end{array}

where

\begin{array}{}\text{(27)}& G_{+\mathrm{,}k}=f\cos \frac{\theta }{2}(1+{\text{e}}^{\text{i}k{x}_0})\mathrm{,}{G}_{-\mathrm{,}k}=-f\sin \frac{\theta }{2}(1+{\text{e}}^{\text{i}k{x}_0})\mathrm{,}\end{array}

characterize the coupling strength between the kth mode in the waveguide and the giant atom. We note that, |G_+,k| and |G_−,k| actually reflect the width of the two valleys when the value of k is taken to satisfy v_gk = ω_±.

In Table I, we list the values of |G_+,k | and |G_−,k | for the horizontal lines in Figs. 5(a), (b) and (c), along which the transmission rates are plotted as functions of the detuning ∆ in Figs. 5(d) and (e). It shows a good agreement for the valley width between the table and the curves. For the resonant driving, which valley is wider depends on the size of the giant atom x₀, as shown in Figs. 5 (a) and (d). However, for the non-resonant driving, as shown in Figs 5 (b), (c) and (e), the left valley is nearly always narrower than the right one when ∆₂ > 0 and wider when ∆₂ < 0. In this sense, the modification to the photon transmission by the atomic size is more sensitive for the resonant driving setup.

4 Effects of dissipation

When the natural relaxation of atoms is taken into consideration, the transmission amplitude is expressed as

\begin{array}{}\text{(28)}& t_2^{\prime }=\frac{{\text{Δ}}_f\left[\text{i}\left(E-{\omega }_e+\text{i}{\gamma }_e\right)v_g-2\text{i}{f}^2\sin \left(k{x}_0\right)\right]-\text{i}{v}_g{\eta }^2}{{\text{Δ}}_f\left[\text{i}\left(E-{\omega }_e+\text{i}{\gamma }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)\right]-\text{i}{v}_g{\eta }^2}\mathrm{,}\end{array}

where ∆_f = E−δ + iγ_f, and γ_e(f) is the spontaneous emission rate of the excited (metastable) state |e(f)〉. The corresponding transmission rate is plotted in Fig. 6. Compared with the ideal situation without relaxation, it is found that the valley becomes wider and shallower by comparing Figs. 5(d), (e) and Figs. 6(a), (b).

In the previous discussion, we have only considered the effect of spontaneous emission on the transmission rate. It should be noted that, in the presence of the natural relaxation, the conservation relation |t₁₍₂₎|² + |r₁₍₂₎|² = 1 does not hold on any more. Therefore, it is beneficial to discuss the reflection rate. Based on Eqs. (6) and (18), and the Schödinger equation, the reflection amplitudes of two- and three-level giant atom are expressed by

\begin{array}{}\text{(29)}& r_1=\frac{2f^2\left[1+\cos (k{x}_0)\right]{\text{e}}^{\text{i}k{x}_0}}{\text{i}\left(E-{\omega }_e+\text{i}{\gamma }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)}\mathrm{,}\end{array}

\begin{array}{}\text{(30)}& r_2=\frac{2f^2{\text{Δ}}_f\left[1+\cos \left(k{x}_0\right)\right]{\text{e}}^{\text{i}k{x}_0}}{{\text{Δ}}_f\left[\text{i}\left(E-{\omega }_e+\text{i}{\gamma }_e\right)v_g-2f^2\left(1+{\text{e}}^{\text{i}k{x}_0}\right)\right]-\text{i}{v}_g{\eta }^2}\mathrm{.}\end{array}

In Fig. 7, we plot the reflection rates R₁ = |r₁|² and R₂ = |r₂|² as functions of ∆, respectively. Comparing the results with and without spontaneous emission, it shows that the spontaneous emission breaks the complete reflection and makes the reflection peaks become wider. However, the locations of the peaks relative to the transmission valleys remain unchanged.

5 Conclusion and remarks

In this paper, we have studied the single-photon transmission in a one-dimensional linear waveguide system coupled with two-level or three-level giant atom. Generally, in the giant-atom regime, the backward and forward photons propagating in the waveguide can lead to an interference effect. Thus, for the case of two-level giant atom, we have shown that the complete reflection occurs, but not for the case that the incident photon is exactly resonant to the atom. In other words, the natural interference leads to an effective small but nonzero frequency shift, and the shift can be approximately regarded as a sinusoidal function of the atomic size. For the driven three-level giant atom, we obtain an ATS line shape, and the transparency window can be controlled by phase modulation in terms of the size of the giant atom. In addition, the location and width of the transmission valleys are tunable by adjusting the atomic size.

Throughout the paper, we have considered the tunability from the viewpoint of the changing coupling distance, i.e., the giant atomic size. Considering the superconducting transmission line as the linear waveguide, this distance can be tuned by the coupled inductance or capacitance [38, 39], to adjust the propagating phase kx₀. Alternatively, one can also tune the frequency of the atom, so that the wave vector k of the resonant mode in the waveguide will be shifted. As a result, the propagating phase will be changed, and produce the similar single-photon scattering spectrum. However, the period on x₀, which is relative to k [as shown in Eqs. (10), (20), and (21)] will be correspondingly modulated.

Beyond the specific model here, our work demonstrates how to use the interplay between (among) the different physical processes to modify the photon transmission in the waveguide. In the giant atom scenario, one of the processes is provided by interference effect during the photon propagation, while the other arises from the inner energy-level coupling, which is induced by, for example, the classical driving. Motivating by the interplay mechanism, we hope that the giant atom can be useful in designing photon or phonon based quantum device, which goes beyond the conventional small atom setup.

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