1. Introduction

On-chip optical nonreciprocal propagation in one-dimensional waveguide has attracted much attention in recent years due to its potential applications in quantum communications and quantum computations [1–10]. Due to the fast developments of waveguide quantum electrodynamics, the nonreciprocal single photon scattering in waveguide coupled directly to atoms or indirectly interacting with atoms mediated by resonators have been studied widely. Hafezi and Rabl firstly proposed an approach for on-chip optical non-reciprocity at the single-photon level by using strong optomechanical interaction in micro-ring resonators in 2012 [1]. After that, Xia et al. proposed a reversible nonmagnetic single-photon isolation using unbalanced quantum coupling in 2014 [4]. Their proposal was demonstrated by Scheucher et al. in 2016 [11]. Chiral atom-photon interactions in waveguide [12–15], which means the interaction strength between the atom and photon depends on the propagation direction of the photon, is considered as an important method to realize the single photon non-reciprocal propagation [16–28]. Controlling the phase differences between the coupling strengths is also a possible method to exhibit incident direction-dependent single photon scattering [29–31]. Concurrently, the optical non-reciprocity can be also achieved via diverse mechanisms such as introducing the non-Hermiticity [32–34] and dynamical modulations [35].

In traditional waveguide electrodynamics, the wavelength of light in waveguide is much larger than the size of atoms. So the atoms can be regarded as point-like objects and the dipole approximation can be used to deal with the interactions between them [36]. However, recent reports show that in some systems, such as superconducting qubits coupling to surface acoustic waves or microwave waveguide and ferro-magnetic spin ensemble interacting with meandering waveguide [37–40], the coupling points are multiple and thus the dipole approximation is broken. Quantum interference will take place due to the multiple coupling which can provide a new path for the interaction between the photon and the emitter. In fact, the quantum interference effect can also be generated in small atom systems due to coherent feedback loops generated by different waveguide coupling model structures [41,42]. In the giant atom system, the quantum interference effect can contribute to many interesting quantum optical phenomena, such as frequency-dependent Lamb shifts and relaxation rates [43], decoherence-free interaction [44–47], collective radiation [48], nonexponential decay [38,49] and bound states [50–53]. The entanglement generation [54–56] and disentanglement dynamics [57] in the giant atom-waveguide system have also been explored.

The single photon scattering properties in one-dimensional waveguide coupling to giant atoms are also investigated [58–63]. The quantum interference induced by multiple-point coupling plays important roles in controlling single photon scattering properties. Some important quantum devices at single-photon level, such as frequency converter, router have been proposed [64–66]. Recently, Chen et al. investigated nonreciprocal and chiral single-photon scattering for giant atoms [58]. In their model, the coupling phase between the giant atom and the waveguide are different. Together with dissipation, the nonreciprocal single photon scattering can be realized. Their model can work for the photon with fixed frequency. Here, we further study the single photon scattering properties in the waveguide coupling with a giant atom. The giant atom is a $\Lambda$ system with one transition driven by a classical field. We discuss how to control the single photon scattering properties in this system by the classical field. The time delay due to the size of the giant atom may play important roles in quantum devices based on giant atom, which may result in many interesting non-Markovian effects. We investigate detailedly the non-reciprocity of single photon propagation in the non-Markovian regime. Our results are helpful in designing nonreciprocal devices at single-photon level based on giant atom.

2. Configuration and theoretical model

The model we considered is shown in Fig. 1. A giant atom couples to a waveguide at $x_{1}=0$ and $x_{2}=d$. The giant atom is a three-level system with ground state $|g\rangle$, excited state $|e\rangle$, and metastable state $|s\rangle$. The transition between $|g\rangle$ and $|e\rangle$ is coupled by the modes of the waveguide with coupling strengths $g_{1}$ at $x=x_{1}$ and $g_{2}$ at $x=x_{2}$. The transition between $|e\rangle$ and $|s\rangle$ is decoupled from the waveguide but driven by a classical filed with Rabi frequency $\Omega$. In our previous studies [59], we considered that the coupling between the giant atom and waveguide is chiral, which means that the coupling strength between the giant atom and the photon depends on the propagation direction of the photon. In this paper, instead, we consider that the coupling is non-chiral but $g_{1}$ and $g_{2}$ are complex numbers, which can be given as $g_{1}=ge^{i\theta _{1}}$ and $g_{2}=ge^{i\theta _{2}}$ [58]. Here, $g$ is a real number.

 

Fig. 1. A three-level giant atom coupled to the waveguide, where the coupling points at $x=0$ and $x=d$. $ge^{i\theta _{1}}$ and $ge^{i\theta _{2}}$ are the coupling strength, $|g\rangle$ represents the ground state, $|e\rangle$ and $|s\rangle$ represents different excited states. $\gamma _{e}(\gamma _{s})$ represents the energy loss of state $|e\rangle (|s\rangle )$. The transition $|e\rangle \leftrightarrow |s\rangle$ is driven by a classical laser beam with Rabi frequency $\Omega$.

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The system can be modeled by the Hamiltonian

Since we only consider the single photon scattering and the excitation number is conserved, the wavefunction of the system is given as

For any given initial state $|\Psi \rangle$, the dynamics of the system can be obtained straight forwardly by numerically integrating the set of above equations. In this work, our target is to solve for the single photon transmission and reflection probability amplitudes. For this purpose, supposing at the initial moment, the single photon incident from the left of the waveguide, then $u_{R}(x)$ and $u_{L}(x)$ can be written as

Here, $t_{l}$ and $r_{l}$, respectively, denote the single photon transmission and reflection amplitudes when the photon incidents from the left of the waveguide. $h(x)$ is the Heaviside step function with $h(0)=1/2$. $ae^{ikx}h(x)h(d-x)$ and $be^{-ikx}h(x)h(d-x)$ describe the wavefunction of the single photon between 0 and $d$. Substituting Eqs. (10) and (11) into Eqs. (6–9), one can obtain that

For the single photon incident from the right of the waveguide, the propagation process is equivalent to the case that the left-incident one with exchanging the positions of two coupling points. Thus, one can get the transmission and reflection amplitudes for the right-incident photon as following:

It shows that $t_{r}\neq t_{l}$, which indicates that the nonreciprocal single photon scattering properties can be obtained in this system. And this property is mainly due to the broken time-reversal symmetry. In detail, when the photon input from the left-side, the coupling phase difference $\Delta \theta =\theta _{2}-\theta _{1}$ generated by different coupling points. Nevertheless, for the single photon incident from the right of the waveguide, one can find that the coupling phase factors are exchanged by comparing Eq. (12) and Eq. (14), i.e. $\Delta \theta \to -\Delta \theta$, which means that the broken time-reversal symmetry is realized.

In our previous studies, the delay time $\tau$ is always neglected, which means that only the Markovian regime is considered [59,60]. However, if $\tau$ is large, $\varphi$ is sensitive to the detuning $\Delta _{e}$, which results in non-Markovian features in the transmission and reflection spectra. In the following, we will discuss the frequency tunable single photon nonreciprocal transmission in the Markovian and non-Markovian regime, respectively.

3. Frequency tunable nonreciprocal transmissions in the Markovian regime

In this section, we consider the case with $\varphi \approx \varphi _{0}$. It is worth mentioning that when the classical field is turned off, i.e. $\Omega =0$, the transmission and reflection amplitudes are the same as those reported in Ref. [58]. $T_{j}=|t_{j}|^2$ and $R_{j}=|r_{j}|^2(j=L,R)$ are introduced to represent the transmission and reflection probabilities. In particular, we find that although photons are incident from different sides of the waveguide, reflection probabilities are identical, even if there is the slightest difference in the numerator by comparing Eq. (13) and Eq. (15). To show the nonreciprocity more clearly, $\Delta T=|T_{R}-T_{L}|$ is introduced as the nonreciprocal degree. When $\Delta T=0$, which means $T_{R}=T_{L}$, the transmission is reciprocal. When $\Delta T=1$, the system shows perfect non-reciprocity. After corresponding calculations, one can obtain

The above equation shows that $\Delta T$ is strongly dependent on Rabi frequency, phase factors, and frequency detuning. By analyzing the Eq. (16), one can get $\Delta T=1$ when

We first investigate the effect of phase factors on nonreciprocal scattering properties with a certain intensity of the classical field, as depicted in Fig. 2(a) and (b). It can be seen clearly that the nonreciprocal phenomenon is mainly manifested in the two regions that are symmetric about $\Delta _{e}=0$. In detail, Fig. 2(a) shows the effect of $\varphi _{0}$ on $\Delta T$ with $\Delta \theta =\pi /2$, $\gamma _{e}=4\Gamma$ and $\Omega =8\Gamma$. When the incident photon frequency is in resonance with the atomic energy transition frequency, i.e. $\Delta _{e}=0$, the transmission probability can reach the maximum value 1. The photon incident from both sides of the waveguide is completely transmitted, which means that the giant atom is decoupled from the waveguide, this is similar to the traditional electromagnetically induced transparency phenomenon [69–72]. The nonreciprocal degree $\Delta T$ decreases to 0, which is quite different from what happens with two-level giant atom [58]. When $\varphi _{0}=n\pi /2(n=1,3,5\cdots )$ and $\Delta _{e}=\pm 4\Gamma$, for the photon incident from the left of the waveguide, the system shows perfect absorption since $T_{L}=R_{L}=0$. The energy of the incident photon is absorbed by the giant atom and then emits it to the non-waveguide modes. The photon incident from the right-side of the waveguide is completely transmitted and $\Delta T$ reaches the maximum value $\Delta T=1$. When the frequency of the incident photon is far away from the resonance frequency, the nonreciprocal transmission disappears. Fig. 2(c) illustrates the above phenomenon more directly. $\Delta T$ is exhibited by the red dashed lines. In particular, when $\varphi _{0}=n\pi$, $\Delta T=0$, which means the transmission probability $T_{L}=T_{R}$, but not equal to 1. $T_{L}$ and $T_{R}$ can be written as $T_{L}=T_{R}=[(\Delta _{e}-16\Gamma ^2/\Delta _{e})^2+4\Gamma ^2]/[(\Delta _{e}-16\Gamma ^2/\Delta _{e})^2+16\Gamma ^2]$. The reflection probabilities $R_{L}=R_{R}=4\Gamma ^2/[(\Delta _{e}-16\Gamma ^2/\Delta _{e})^2+16\Gamma ^2]$. It can be found that $T_{L(R)}+R_{L(R)}\neq 1$ due to the dissipation effect. Furthermore, Fig. 2(b) depicts the effect of $\Delta \theta$ on nonreciprocal degree $\Delta T$. Similarly, when the detuning $\Delta _{e}=0$ and $\Delta \theta =n\pi (n=0,1,2\cdots )$, the nonreciprocal transmission phenomenon also disappears. When the Rabi frequency of the classical field and resonance frequency satisfy the relation $\Delta _{e}=\pm \Omega /2$ and $\Delta \theta =n\pi /2(n=1,3,5\cdots )$, as we mentioned above, $\Delta T$ increases to 1. The perfect nonreciprocal transmission is realized.

 

Fig. 2. The effect of phase factors on $\Delta T$ with a certainly Rabi frequency $\Omega$. (a) shows the influence of phase accumulation $\varphi _{0}$ on $\Delta T$ with $\Delta \theta =\pi /2$, $\gamma _{e}=4\Gamma$ and $\Omega =8\Gamma$; (b) shows the influence of coupling phase difference $\Delta \theta$ on $\Delta T$ with $\varphi =\pi /2$, $\gamma _{e}=4\Gamma$ and $\Omega =8\Gamma$; (c) The transmission probability and the nonreciprocal degree as a function of $\Delta _{e}$, the other parameters are shown in the figure; (d) shows the influence of dissipation $\gamma _{e}$ on $\Delta T$ with $\Delta \theta =\varphi _{0}=\pi /2$ and $\Omega =8\Gamma$.

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From Eq. (16), one can find that the dissipation of the giant atom plays important roles in realization the nonreciprocal single photon transmission. Fig. 2(d) shows the effect of dissipation and detuning on nonreciprocity with $\Delta \theta =\varphi _{0}=\pi /2$ and $\Omega =8\Gamma$. Eq. (16 can be written as $\Delta T=4\gamma _{e}\Gamma /[(\Delta _{e}-16\Gamma ^2/\Delta _{e})^2+(\gamma _{e}/2+2\Gamma )^2]$. When $\Delta _{e}=0$, the nonreciprocal degree $\Delta T=0$ over the whole range of $\gamma _{e}$ and the transmission probability $T_{L}=T_{R}=1$ even though the dissipation exists. It’s worth noting that when the system is in an ideal environment, i.e. dissipation $\gamma _{e}=0$, the nonreciprocal transmission will also disappear, $\Delta T=0$. This phenomenon can be explained as follows. Physically, when $\gamma _{e}\neq 0$ ($\gamma _{e}=0$), the giant atom can be treated as a non-Hermitian (Hermitian) scattering center. When $\gamma _{e}\neq 0$ and $\Delta \theta \neq n\pi$, the nonreciprocal scattering can be achieved, and the nonreciprocal scattering process of this system results from the combination of the non-Hermiticity and the broken time-reversal symmetry [58]. When $\Delta _{e}=\pm 4\Gamma$ and $\gamma _{e}=4\Gamma$, $\Delta T$ reaches the maximum value $\Delta T=1$. As the dissipation increases, the nonreciprocal degree $\Delta T$ decreases to 0 gradually.

In the following, we discuss how to manipulate the nonreciprocal single photon scattering properties by the classical field. To study the effect of Rabi frequency on nonreciprocal scattering more clearly, we assume that $\varphi _{0}=\Delta \theta =\pi /2$ and $\gamma _{e}=4\Gamma$. Then Eq. (16) can be reduced as

Figure 3(a) exhibits the effect of Rabi frequency and detuning on $\Delta T$. When $\Delta _{e}=0$, the nonreciprocal transmission does not exist. However, the perfect nonreciprocal transmission can be achieved in the vicinity of the single photon resonance frequency. Fig. 3(b) shows the partial value interception of Fig. 3(a). When the Rabi frequency $\Omega =0$, the three-level system degenerates to a two-level system and the nonreciprocal degree $\Delta T$ shows the standard Lorentzian linear distribution. With the increasing Rabi frequency, it can be seen intuitively that due to the introduction of classical field, the peak value of nonreciprocal scattering spectrum is shifted and the nonreciprocal degree $\Delta T$ has an obvious splitting phenomenon. The difference in the Rabi frequency $\Omega$ causes a change in peaks and evolutionary processes. This phenomenon can be explained by Eq. (19). From Eq. (19), one can obtain that when $\Delta _{e}=\pm \Omega /2$, $\Delta T$ reaches the maximum value $\Delta T=1$, corresponding to the multi-peaks in Fig. 3(b). In particular, for the resonant photon, one can also find that when $\Omega =0$, the perfect nonreciprocal transmission can be achieved. But when $\Omega \neq 0$, the nonreciprocal transmission degree $\Delta T\equiv 0$. This interesting phenomenon can be used to design a switch diode by controlling the classical field. According to the above analysis, it can be found that we can adjust the Rabi frequency to realize the perfect nonreciprocal transmission at arbitrary detuning frequency, which provides an alternative effective method besides changing the phase difference to realize perfect nonreciprocal scattering.

 

Fig. 3. (a) The effect of Rabi frequency $\Omega$ and detuning $\Delta _{e}$ on nonreciprocal degree $\Delta T$; (b) The nonreciprocal degree $\Delta T$ as a function of $\Delta _{e}$ with a certain Rabi frequency $\Omega$. The other parameters $\varphi _{0}=\Delta \theta =\pi /2$ and $\gamma _{e}=4\Gamma$.

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4. Frequency tunable nonreciprocal transmissions in the non-Markovian regime

Now we turn to consider the non-Markovian regime. Giant atom has been widely studied due to the self-interference effect caused by the structure of the multi-coupled point model. Experimentally, this regime corresponds to the case of superconducting transmon qubit coupled with surface acoustic waves on a piezoelectric substrate owing to the slow propagation speed of sound in solids [38–40,43]. In this work, when the input wave length is smaller than the atom size, the transmission delay time $\tau$ caused by the distance between the coupling points is non-negligible. The phase $\varphi =\varphi _{0}+\tau \Delta _{e}$ strongly depends on the detuning $\Delta _{e}$. Then Eq. (16) can be written as

The phenomenon of nonreciprocal transmission of single photon scattering caused by the non-Markovian effect is depicted in Fig. 4(a)-(b). To be specific, Fig. 4(a) shows the effect of detuning and time delay on the nonreciprocal degree with parameters $\varphi _{0}=\Delta \theta =\pi /2$, $\gamma _{e}=4\Gamma$ and Rabi frequency $\Omega =8\Gamma$. Similarly, when $\Delta _{e}=0$, the nonreciprocal degree $\Delta T=0$ and the incident photon is completely transmitted after interacting with the atom. Additionally, the nonreciprocal transmission can be achieved around $\Delta _{e}=\pm 4\Gamma$ and shows the characteristics of multiple segments when the time delay increases. This phenomenon is similar to the multiple excitations of the giant atom [73]. In order to see the impact of time delay directly, we investigate the $\Delta T$ with the changing of the detuning $\Delta _{e}$ when the time delay is fixed, as shown in Fig. 4(b). Indeed, as the time delay increases, the oscillation frequency becomes drastic and shows a certain periodicity. This phenomenon can be understood as follows. When $\varphi _{0}=\Delta \theta =\pi /2$, $\gamma _{e}=4\Gamma$ and $\Omega =8\Gamma$, Eq. (20) can be reduced as

According to Eq. (21), it is easy to find that the oscillating items of $\Delta T$ are mainly given by the numerator and the peaks are determined by the denominator. The oscillation frequency $\omega _{f}\propto \Gamma \tau$, which causes the oscillations to become more intense over time, as exhibited in Fig. 4(b). When $\tau \Delta _{e}=n\pi /2(n=1,3,5\cdots )$, $\Delta T=0$. The nonreciprocal phenomenon does not exist.

 

Fig. 4. (a) The nonreciprocal degree $\Delta T$ with the change of the detuning $\Delta _{e}$ and time delay $\Gamma \tau$ with $\Omega =8\Gamma$; (b) The nonreciprocal degree $\Delta T$ changes with $\Delta _{e}$ when the time delay $\Gamma \tau$ is fixed, with $\Omega =8\Gamma$; (c) The nonreciprocal degree $\Delta T$ as a function of $\Delta _{e}$ with different the Rabi frequencies $\Omega$. $\Gamma \tau$ is fixed as $\pi /2$. The other parameters $\Delta \theta =\varphi _{0}=\pi /2$ and $\gamma _{e}=4\Gamma$.

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Another important feature is that the system can still achieve perfect nonreciprocal transmission in the non-Markovian regime, as shown in Fig. 4(a). By analyzing Eq. (20), one can find that when

$\Delta T$ reaches the maximum value $\Delta T=1$. To simplify the analyzing process, we continue to assume that $\varphi _{0}=\Delta \theta =\pi /2$. The Eqs. (22) and (23) can be rewritten as

The above equations show that when $\Delta _{e}=\pm \Omega /2$, $\tau \Delta _{e}=n\pi$ and $\gamma _{e}=4\Gamma$, $\Delta T$ is equal to 1. The perfect nonreciprocal transmission is achieved. The locations that $\Delta T$ can reach the maximum value of 1 are shifted by the classical field. For example, Fig.4(c) depicts the effect of $\Delta _{e}$ on the nonreciprocal transmission degree $\Delta T$ when the time delay $\Gamma \tau =\pi /2$. Based on the above analysis, one can observe that the perfect nonreciprocal transmission can be realized when $\Delta _{e}= 2n\Gamma (n=1,2,3{\ldots })$ with different Rabi frequencies. Compared with the Markovian regime, the existence of time delay makes that realization of perfect nonreciprocal transmission becomes more difficult. But the resulting physical phenomena are more abundant.

5. Conclusions

In this paper, a frequency tunable single photon nonreciprocal scattering device in a waveguide coupling to a giant system has been presented. The scattering probability amplitudes of single photon incident from different directions of the waveguide are calculated in both the Markovian and non-Markovian regime. In the Markovian regime, the perfect nonreciprocal transmission $\Delta T=1$ can be realized at different single photon frequencies $\Delta _{e}$ by adjusting the Rabi frequency of the classical field. In the non-Markovian regime, the time delay resulting from the photon propagation between two coupling points leads to transmission spectrum showing periodic oscillations and reviving with multiple peaks. The oscillation period is determined by $\Gamma \tau$. The conditions for realization the perfect nonreciprocal transmission are also given based on the analytical results in the non-Markovian regime. The final results show that the nonreciprocal single photon transmission can be realized for the non-resonant photon by adjusting the classical field both in Markovian regime and non-Markovian regime. For the resonant photon, one can design a switchful diode by turning on/off the classical field.