Open Access
Add to BibSonomy
Abstract
We study the nonreciprocal excitation and entanglement dynamics of two giant atoms (GAs) coupling to a one-dimensional waveguide. With different positions of coupling points, three configurations of two separate GAs, two braided GAs, and two nested GAs are analyzed, respectively. The coupling strengths between different coupling points are considered as complex numbers with phases. For each coupling configuration, the nonreciprocal excitation dynamics and entanglement properties, which results from the phase differences of coupling strength and the phase induced by photon propagation between the two coupling points, are studied both in Markovian and non-Markovian regimes. The analytical solutions for nonreciprocal entanglement degree are given in the Markovian regime. It shows that the steady entanglement can be reached and strongly depends on the phases. Different from the case of the Markovian regime, the entanglement degree shows oscillating behavior in the non-Markovian regime. This work may find applications in the generation and controlling of entanglement in quantum networks based on waveguide quantum electrodynamics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Waveguide quantum electrodynamics has become a hot topic since the strong coupling between quantum emitters and waveguides has been realized in experiments. Several excellent reports reviewed the developments in this field [1–3], including photon scattering properties [4–9], collective radiations [10–14] and so on. The entanglement generation due to the spontaneous emission of quantum emitters mediated by waveguide has also been investigated [15–19]. The platforms for studying the photon scattering properties are including fiber-atom system [20], photonic crystal waveguide-quantum dots system [21], superconducting–system [22], metal nanowire-quantum emitter system [23,24] and so on. In most of previous waveguide QED studies, the size of quantum emitter is much smaller than the length of the radiated photon wave packet, so that the atoms can be regarded as point-like objects and the dipole approximation is valid.
Recently, the giant atom(GA), which means that the size of the atom is comparable with the length of the radiated photon wave, has been realized [25]. In such a system, the dipole approximation is broken. The multi-point coupling modes should be used to describe the interaction between the GAs and waveguides. The photon propagation between two coupling points supply a new path for the interaction between the GA and the waveguide. Thus, many quantum phenomena related to GAs have been reported, such as frequency-dependent Lamb shifts and relaxation rates [25–27], generation of bound states [28–33], decoherence-free interaction [34–36], and collective radiation [37]. Photons in one-dimensional waveguide scattered by GAs have also been investigated in many different configurations [38–40]. As an important source, the entanglement generation between GAs mediated by waveguide attracts much attention too [41,42]. For example, Yin et al. [43] investigated the entanglement dynamics of two GAs coupling to a waveguide. They show that the phase induced by photon propagation between two coupling points plays important roles in entanglement generation.
The study of nonreciprocal scattering is one of the hot topics in quantum optics. Physically, it usually requires that the time-reversal symmetry is broken for the nonreciprocal process. The broken time-reversal symmetry is usually achieved by using chiral coupling [44]. However, the time-reversal symmetry can also be broken by local coupling phases and the phase accumulation between the coupling points [45]. Optical nonreciprocity allows photons to pass through from one side but blocks it from the opposite direction, which is requisite for preventing the information back flow in quantum network. Furthermore, the nonreciprocal photon scattering phenomenon can be used to design many important quantum devices, such as isolators [46–49], circulators [50,51] and nonreciprocal frequency converter [52]. Recently, Du et al. reported that the nonreciprocal excitation transfer can be achieved by introducing a nontrivial coupling phase [38]. Motivated by these studies, we investigate the nonreciprocal excitation and nonreciprocal entanglement generation based on two GAs coupling to a common waveguide. In our model, the coupling strengths between the giant atoms and waveguide are complex numbers with phase [45], which is the key point for the manipulating the nonreciprocal excitation and nonreciprocal entanglement. Experimentally, this two GAs model can be achieved in the superconducting circuit and the phase can be introduced with Josephson-junction loops tunable by external fluxes [53]. In detail, the gauge-invariant phase difference across Josephson inductance in the loop strongly depends on the external flux bias. We show that by adjusting the phases difference between the coupling strengths and the phase induced by the photon propagation between different coupling points, one can realize the nonreciprocal excitations between the two giant atoms. Accordingly, the entanglement generations induced by spontaneous emission of the two giant atoms are nonreciprocal. The nonreciprocal entanglement between the two GAs can reach the steady value in the Markovin regime. Generally, this nonreciprocal excitation phenomenon is caused by the interference effect between the transmission of multiple quantum channels. Furthermore, this nonreciprocal excitation dynamics evolution behavior provides more options for the preparation of entangled states. Our work may find applications in quantum information processing and nonreciprocal component design.
The rest of the paper is organized as follows. In Sec.2, the model of the system and the Hamiltonian are introduced. In Sec.3, the dynamical equations of two GAs are deduced. The influence of coupling phase, phase accumulation and time delay on the nonreciprocal entanglement of two GAs are discussed. The conclusion is put in Sec.4.
2. Configuration and theoretical model
We consider two two-level GAs (GA$_{a}$ and GA$_{b}$) coupling to a waveguide. Each GA interacts with the waveguide through two separate connection points, which are labeled as $x_{jm} (j=a,b;m=1,2)$. Based on the different coupling arrangements of the two GAs with the waveguide, there are three different topologies for the system, which are called separate GAs, braided GAs, and nested GAs, respectively. The corresponding configurations are shown in Fig. 1(a)–1(c). The coupling coefficients for the GA$_{j} (j=a,b)$ and the waveguide are $g_{j}e^{i\theta _{1}} (j=a,b)$ and $g_{j}e^{i\theta _{2}}$, respectively. Here $g_{j}$ is a real number. $\theta _{1}$ and $\theta _{2}$ are two local coupling phases for different coupling points of the GA [53]. In the following discussions, for simplicity, we suppose $g_{a}=g_{b}=g$.
Fig. 1. The three different topologies of the two GAs coupling to a waveguide. The two GAs are labelled as $a$ and $b$. They couple to the waveguide at points $x_{jm}$, where $j=a,b$ denotes the GA and $m=1,2$ represents the connecting points. (a), (b) and (c) show the separate GAs, braided GAs and nested GAs, respectively. The coupling coefficient at each point is $g_{j}e^{i\theta _{m}}$.
The Hamiltonian describing the system is given as [54]
where $H_{A}$ is the Hamiltonian of the atom, which reads as ($\hbar =1$)Here $\omega _{0}$ is the transition frequency of the GAs. We suppose that the two GAs are identical with the same transition frequency. $\sigma ^{\dagger}_{j}$($\sigma _{j}$) is the raising (lowering) Pauli operators of the GAs. The Hamiltonian of the waveguide $H_{W}$ is given by
where $\omega _{k}$ is the frequency of propagating photons in the waveguide. $p^{\dagger}_{d}$ and $p_{d}$ $(d=R, L)$ represent the generation and annihilation operators of photons with wavevector $k$ propagating along right $(d=R)$ or left $(d=L)$ direction in the waveguide. Finally, the interaction Hamiltonian $H_{I}$ between the waveguide and the GAs readsSince we consider single-atom excitation in the system, the energy eigenstate at time $t$ can be expressed as
The first two terms at the right side of the above equations represents the self-excitation term, and the remaining terms denote the coherent terms. The value of $x_{jm} (j=a,b; m=1,2)$ is different for different models. In the following, we will discuss the two GAs entanglement dynamics evolution process through the above differential equations.
3. Nonreciprocal entanglement dynamics between the two GAs
In this section, we study the entanglement dynamics between the two GAs for three different topologies. The concurrence is adopted to quantitively measure quantum entanglement between the two GAs. In the bases of $\{|e_{a}\rangle |e_{b}\rangle,|e_{a}\rangle |g_{b}\rangle,|g_{a}\rangle |e_{b}\rangle,|g_{a}\rangle |g_{b}\rangle \}$, the concurrence is given by [55]
We consider two different initial states of the GAs. When $c_{a}(0)=1, c_{b}(0)=0$, the concurrence is expressed as $C(t)=C_{1}(t)$. While for the case of $c_{a}(0)=0, c_{b}(0)=1$, the concurrence is represented as $C(t)=C_{2}(t)$. To describe the nonreciprocal entanglement generation, we introduce
When $C_{1}(t)=C_{2}(t)$, $\Delta C(t)=0$, the nonreciprocal entanglement disappears. Once $\Delta C(t)$ and $C_{1}(t)$ ($C_{2}(t)$) are obtained, one can deduce $C_{2}(t)$ ($C_{2}(t)$) and then compare the magnitude relationship between $C_{1}(t)$ and $C_{2}(t)$. In the following, we will discuss the concurrence $C_{j}(t) (j=1,2)$ and $\Delta C(t)$ both in the Markovian and non-Markovian regimes.
3.1 Two separated GAs
We first consider the case of the two separate GAs, as shown in Fig. 1(a). The time-delayed differential equations of the probability amplitudes for the two separated GAs are given by
We consider the Markovian case firstly. The propagating times between different connecting points can be neglected compared to the lifetime of the GAs, which means $\tau \approx 0$. Supposing the GA$_{a}$ is in the excited state and GA$_{b}$ in the ground state at the initial time. Solving the Eqs. (10) and (11), one can get
Substituting the above results into Eq. (8), the concurrence can be written as
It shows that $C_{m}(t)(m=1,2)$ strongly depends on $\Delta \theta$ and $\varphi$ from Eqs. (14) and (17). Fig. 2 exhibits the influence of coupling phase difference $\Delta \theta$ and phase accumulation $\varphi$ on entanglement generation with $c_{a}(0)=1$ and $c_{b}(0)=0$. One can find that the concurrence $C_{1}(t)$ oscillates with $\Delta \theta$. The period is $2\pi$. When $\varphi =\Delta \theta =\pi /2$, the concurrence $C_{1}(0)=0$ at the initial moment. With the evolution of time going, the concurrence of the two GAs reaches the maximum value $C_{1}(t)\approx 0.74$. A similar case can be also found when $\varphi =\pi /4$, $\Delta \theta =3\pi /4$ and $\varphi =\pi /8$, $\Delta \theta =7\pi /8$ from Fig. 2(e) and (f). Tha maximum value of concurrence is also about 0.74. However, the time for generating the maximum concurrence value is different. When $\Delta \theta =n\pi (n=0,1,2\cdots )$ and $\varphi =\pi /2$, the two GAs probability amplitudes can be written as $|c_{a}(t)|=(1+e^{-4\Gamma t})/2$ and $|c_{b}(t)|=(1-e^{-4\Gamma t})/2$. When $\Gamma t\to \infty$, $|c_{a}(t\to \infty )|=|c_{b}(t\to \infty )|=C_{1}(t\to \infty )=0.5$, steady-stable concurrence is achieved, as shown by the red solid line in Fig. 2(d). When $\varphi =\pi /2$ and $\Delta \theta =3\pi /2$, $|c_{a}(t)|=e^{-2\Gamma t}$ and $|c_{b}(t)|=0$. GA$_{a}$ exhibits a standard exponential decay. It is interesting that the GA$_{b}$ is decoupled from waveguide and the concurrence between the two GAs $C_{1}(t)\equiv 0$. When $\varphi =\pi /4$ and $\Delta \theta =5\pi /4$, substituting these results into Eqs. (12) and (13), it can be obtained that $|c_{a}(t)|=e^{-\Gamma t}$ and $|c_{b}(t)|=0$. Similarly, when $\Delta \theta =9\pi /8$ and $\varphi =\pi /8$, $|c_{a}(t)|=e^{-[2+2\cos (9\pi /8)\cos (\pi /8)]\Gamma t}$ and $|c_{b}(t)|=0$. The concurrence of the two GAs $C(t)=0$ in all these cases, as shown by red solid line in Fig. 2(e) and 2(f). These phenomena can be understood as follows. When $|\varphi -\Delta \theta |=\pi$, Eq. (12) has the following form
with $c_{b}(t)\equiv 0$, the entanglement of two GAs will disappear, i.e. $C_{1}(t)=0$. On the other hand, from Eq. (15) one can find that when $\varphi +\Delta \theta =\pi$, $c_{a}(t)=0$. The concurrence between the two GAs $C_{2}(t)=0$. In particular, the concurrence dynamics $C_{2}(t)$ and $C_{1}(t)$ are symmetric with respect to $\Delta \theta =\pi$, which leads to the generation of nonreciprocal entanglement.Fig. 2. The concurrence of two separate GAs with different initial conditions of $\varphi$ for (a)-(c) when $GA_{a}$ is excited state and $GA_{b}$ is ground state at the initial time. The concurrence of two separated GAs as a function of $\Gamma t$ for (d) $\varphi =\pi /2$; (e) $\varphi =\pi /4$; (f) $\varphi =\pi /8$. The other parameters are shown in the figure.
Now, we investigate the nonreciprocal entanglement between the two GAs. Fig. 3(a)-(c) show the influence of coupling phase difference $\Delta \theta$ and $\Gamma t$ on nonreciprocal concurrence degree $\Delta C(t)$ with different initial conditions of $\varphi$. It exhibits that the nonreciprocal concurrence degree is symmetric with respect to $\Delta \theta =\pi$. The reason is that $C_{1}(t)$ is symmetric to $C_{2}(t)$ with respect to $\Delta \theta =\pi$, as we mentioned above. At the initial time, $\Delta C(t)=0$. Then the nonreciprocal concurrence becomes apparent due to the enhanced interaction between the GAs. As time goes on, it tends to be zero finally. Additionally, when $\Delta \theta =\pi \pm \varphi$, the maximum nonreciprocal concurrence value will be achieved in the evolution process, as shown in Fig. 3(a)-(c). It is worth noting that when $\Delta \theta =n\pi (n=0,1,2\cdots )$, the nonreciprocal concurrence can not be generated. To explain this property, we further analyzing Eqs. (14) and (17). The nonreciprocal concurrence can be generated due to the term $L_{s}$ by comparing Eqs. (14) and (17). When
Fig. 3. The nonreciprocal concurrence of the two separated GAs with different initial conditions of $\varphi$ for (a)-(c). (d)-(f) represent the probability amplitudes and concurrence of two separated GAs as a function of $\Gamma t$. The other parameters are shown in the figure.
Since the size of the GAs is large, it may lead to non-negligible time delay $\tau =d/v_{g}$. In the following, we show the influence of the time delay $\tau =d/v_{g}$ on the nonreciprocal concurrence at different values of the coupling phase difference $\Delta \theta$. By numerical calculations from Eqs. (10) and (11), we obtain the influence of time delay $\tau$ on the nonreciprocal concurrence, as shown in Fig. 4. In Fig. 4(a), when $\Delta \theta =0$ and $\varphi =\pi /4$, $C_{1}(t)$ evolves over time to the maximum value $C(t)=0.32$ and then decays to 0 gradually in the Markovian regime, as shown by blue solid line. In the non-Markovian regime, the concurrence $C(t)=0$ when $\Gamma t\le \Gamma \tau$. For $\Gamma t=\Gamma \tau$, the photon radiated from GA$_{a}$ reaches GA$_{b}$, so that it starts becoming involved in the dynamics of the system, $C(t)\neq 0$. Compared with the Markovian regime, the existence of time delay makes the concurrence $C(t)$ exhibit a non-exponential decay accompanied by oscillation and the maximum concurrence value decrease. This is caused by the two GAs interacting multiple times. The two GAs concurrence persists longer in the non-Markovian regime than that in the Markovian regime. These results are similar to the previous works which have been reported that the evolution process of giant atom dynamics produces oscillations and non-exponential decay in the non-Markovian regime [57]. The similar phenomenon can be also found from Fig. 4(b). Additionally, one can find that the original steady-stable concurrence between the two GAs is broken when $\Gamma \tau \neq 0$. Another important finding is that the existence of time delays can eliminate the decoupling between the atom and waveguide. For example, when $\varphi =\pi$ and $\Gamma \tau =0$, according to the previous analysis, the two GAs probability amplitudes can be written as $|c_{a}(t)|=1$ and $|c_{b}(t)|=0$. The GA$_{b}$ is decoupled from the waveguide and the concurrence $C(t)\equiv 0$, as shown by blue solid line in Fig. 4(c). Nevertheless, when $\Gamma \tau \neq 0$, the concurrence is no longer equal to 0.
Fig. 4. The concurrence $C(t)$ of two separated GAs changes with different initial conditions of $\Delta \theta$ and $\varphi$ for (a)-(c) in the non-Markovian regime. The nonreciprocal concurrence of two separated GAs as a function of $\Gamma t$ for (d) $\Delta \theta =\pi /4$; (e) $\Delta \theta =\pi /2$; (f) $\Delta \theta =\pi$ in the non-Markovian regime. In all panels, $\varphi =\pi /2$.
Furthermore, a completely non-Markovian behavior of nonreciprocal concurrence can be observed in Fig. 4(d) and 4(e). Similarly, the nonreciprocal concurrence appears when $\Gamma t>\Gamma \tau$. The maximum nonreciprocal concurrence value decreases with the increasing of time delay $\Gamma \tau$, but the vanishing time of the nonreciprocal concurrence is slowed down. And the revival peaks and the oscillating amplitudes of the $\Delta C(t)$ are different for different values of $\Delta \theta$ by comparing Fig. 4(d) and 4(e). The revival points of the nonreciprocal entanglement will appear at $\Gamma t=n\Gamma \tau$, where $n$ is a positive integer. In particular, it is worth noting that the time delay cannot be a condition for generating nonreciprocal concurrence. In detail, when $\Delta \theta =\pi$ and $\varphi =\pi /2$, the nonreciprocal concurrence $\Delta C(t)=0$ as the time delay changes, as depcited in Fig. 4(f).
3.2 Two braided GAs
For the case of two braided GAs [Fig. 1(b)], the equations of motion for the probability amplitudes are given by
The first two terms on the right-side of the above equations denote the dissipative term and the self-coherent interaction term, respectively. The remaining terms represent the coherent interactions between two GAs. In the Markovian regime, the two GAs probability amplitudes with the initial condition $c_{a}(0)=1$ and $c_{b}(0)=0$ can be written as
According to Eq. (8), the concurrence can be written as
When the $GA_{b}$ is in the excited state and $GA_{a}$ in the ground state at initial time, i.e. $c_{a}(0)=0$, $c_{b}(0)=1$. Eqs. (22) and (23) are reduced as
Similarly, the corresponding concurrence $C_{2}(t)$ can be written as
Below we focus on studying the two braided GAs entanglement dynamics for the case that the $GA_{a}$ is in the excited state and $GA_{b}$ is in the ground state. Fig. 5(a)-(c) exhibits the relation of concurrence with coupling phase difference $\Delta \theta$ and time $\Gamma t$, with different initial conditions $\varphi$. The concurrence $C_{1}(t)$ is a $2\pi$ periodic function of $\Delta \theta$. When $\varphi =\pi /2$ and $\Delta \theta =0$, the concurrence exhibits a distinct oscillatory behavior, which is different from the case of two separate GAs. It can be found that the concurrence strongly depends on phase accumulation $\varphi$ and coupling phase difference $\Delta \theta$. As $\varphi$ decreases, the time for the concurrence value to approach 0 increases in the whole range of $\Delta \theta$ except for $\Delta \theta =\pi$. To see these features more clearly, the profiles of Figs. 5(a)-(c) are shown by the curves in Figs. 5(d)-(f) for some special cases. For both cases that have been displayed in Figs. 5(d)-(e), the concurrence $C_{1}(t)=0$ at the initial time due to the $c_{b}(0)=0$. Indeed, when $\Delta \theta =0$ and $\varphi =\pi /2$, the two GAs probability amplitudes $|c_{a}(t)|=|\cos 2\Gamma t|$ and $|c_{b}(t)|=|\sin 2\Gamma t|$. Then $C_{1}(t)$ can be written as $C_{1}(t)=|\sin 4\Gamma t|$. The concurrence dynamics oscillates with a standard half-sine wave, as shown by red solid line in Fig. 5(d). This oscillation phenomenon is mainly caused by the coherent interaction of two GAs [16,43], which is obviously different from the phenomenon that the two separated GAs can reach a stable value in this case. The maximum value of concurrence can reach 1 at $\Gamma t=n\pi /8(n=1,3,5\cdots )$. For $\Delta \theta =\varphi =\pi /2$, $|c_{a}(t)|=|e^{-2\Gamma t}\cos 2\sqrt {2}\Gamma t|$ and $|c_{b}(t)|=|e^{-2\Gamma t}\sin 2\sqrt {2}\Gamma t|$. As $\Gamma t$ increases, the GA$_{b}$ is excited by GA$_{a}$ and the concurrence increases gradually. When $\Gamma t=\pi /4\sqrt {2}$, the concurrence decreases to 0 again due to the GA$_{a}$ in the ground state, i.e. $c_{a}(t)=0$ at this time. Then GA$_{a}$ is excited again by the photon radiated from GA$_{b}$ and finally both of the two GAs are in the ground state, which means $C_{1}(t)=0$. When $\Delta \theta =\pi$ and $\varphi =\pi /2$, the concurrence between the two GAs is small but not equal to 0, as shown by red dashed line in Fig. 5(d). In addition, when $\Delta \theta =\pi /2$ and $\varphi =\pi /4$, the concurrence $C_{1}(t)=0$ at the initial moment. As time goes, $C_{1}(t)$ reaches the maximum value $C_{1}(t)=0.65$ and then tends to be 0, as exhibited by red solid line in Fig. 5(e). When $\Delta \theta =\pi$ and $\varphi =\pi /4$, the two GAs concurrence can be written as $C_{1}(t)=0.5(e^{-(4-\sqrt {2})\Gamma t}-e^{-(4+\sqrt {2})\Gamma t})$. When $\Gamma t\approx 0.26$, the two GAs concurrence reaches the maximum value $C_{1}(t)=0.19$. These phenomena can be also observed in Fig. 5(f).
Fig. 5. The concurrence $C_{1}(t)$ of two braided GAs changes with different initial conditions of $\varphi$ for (a)-(c) in the Markovian regime. The profiles of (a)-(c) are shown by the curves in (d)-(e) at different coupling phase difference $\Delta \theta$.
It is interesting to note when the initial conditions change from $c_{a}(0)=1$ to be $c_{b}(0)=1$, the concurrence dynamics $C_{1}(t)$ and $C_{2}(t)$ are symmetric with respect to $\Delta \theta =0$. This symmetry phenomenon makes it possible to generate the nonreciprocal concurrence. In the following, we will use the similar procedures that have been mentioned above to investigate the nonreciprocal concurrence between the two braided GAs. As shown in Figs. 6(a)-(c), $\Delta C(t)$ has been introduced to express the nonreciprocal concurrence degree with different values of $\varphi$. It can be seen clearly that $\Delta C(t)$ is a periodic function symmetric with respect to $\Delta \theta =0$, with a period $\pi$. When $\Delta \theta =\pm \pi /4$ and $\varphi =\pi /2$, the nonreciprocal concurrence degree can be written as $\Delta C(t)=|(L_{1}-1/L_{1})e^{-(4-2\sqrt {2})\Gamma t}\sin 2\sqrt {6}\Gamma t|$ with $L_{1}=((1+2\sqrt {2}i)/3)^{1/2}$. At the initial moment, $\Delta C(t)=0$. Then $\Delta C(t)$ reaches the maximum value $\Delta C(t)\approx 0.82$ and finally it decays to zero as a sine wave. When $\Delta \theta =\pm \pi /2$ and $\varphi =\pi /4$, $\Delta C(t)=0.5|(L_{2}-1/L_{2})e^{-4\Gamma t}(e^{2\sqrt {2+4i}\Gamma t})-e^{-2\sqrt {2+4i}\Gamma t})|$ with $L_{2}=(1+2i)^{1/2}$. With the times going, $\Delta C(t)$ reaches the maximum value 0.6 and then tends to 0. As $\varphi$ decreases, the region that can generate nonreciprocal concurrence gradually approaches $\Delta \theta =\pm \pi$ by comparing Figs. 6(b) and 6(c). In particular, when $\Delta \theta =n\pi$, the phenomenon of nonreciprocal concurrence disappears. This phenomenon can be explained by analyzing Eqs. (24) and (27). According to Eqs. (24) and (27), one can find that
When $\Delta \theta =n\pi$ or $\varphi =n\pi (n=0,1,2\cdots )$, the above equation is true, which means $C_{1}(t)=C_{2}(t)=C(t)$ and corresponds to the vanishing of nonreciprocal concurrence. In the following, we further analyze the dynamical evolution of two GAs when the nonreciprocal concurrence disappears. It is remarkable that when $\varphi =n\pi$(n=0,1,2$\cdots$), $|c_{a(b)}(t)|=0.5(1\pm e^{-(4+4\cos \Delta \theta )\Gamma t})$. According to Eq. (24), the concurrence is given by
An interesting phenomenon is that the steady-stable concurrence can be generated in the whole range of $\Delta \theta$ except for $\Delta \theta =\pm \pi$ when $\Gamma t\to \infty$, i.e. $c_{a(b)}(t\to \infty )=C(t\to \infty )=0.5$. For example, when $\varphi =\pi$ and $\Delta \theta =0$, the two GAs probability amplitudes can be written as $|c_{a}(t)|=(1+e^{-8\Gamma t})/2$ and $|c_{b}(t)|=(1-e^{-8\Gamma t})/2$. Finally, a steady-stable concurrence is formed between the two GAs, as shown in Fig. 6(d). Fig. 6(e) shows that when $\Delta \theta =\pm \pi$, $c_{a}(t)\equiv 1$ and $c_{b}(t)\equiv 0$, which indicates that the nonreciprocal concurrence disappears due to the two GAs do not interact with each other in this situation. Additionally, when $\Delta \theta =0$ and $\varphi =\pi /2$, the two GAs probability amplitudes can be written as $|c_{a}(t)|=|\cos 2\Gamma t|$ and $|c_{b}(t)|=|\sin 2\Gamma t|$, respectively. The concurrence oscillates in the form of half-sine wave with a period of $\pi /4$, as depicted in Fig. 6(f). The above discussion shows that the coupling phase difference and phase accumulation can effectively affect the (nonreciprocal)concurrence dynamics behavior of two GAs by changing the radiation behavior of the two GAs.
Fig. 6. The nonreciprocal concurrence $\Delta C(t)$ of two braided GAs changes with different initial conditions of $\varphi$ for (a)-(c) in the Markovian regime. The probability amplitudes and concurrence of two braided GAs as a function of $\Gamma t$ for (d) $\Delta \theta =0$, $\varphi =\pi$; (e) $\Delta \theta =\varphi =\pi$; (f) $\Delta \theta =0$, $\varphi =\pi /2$.
Now, we study the influence of time delay on the entanglement dynamics. Here we only consider some special cases as examples, as depicted in Fig. 7. Fig. 7(a)-(c) exhibits the influence of time delay on concurrence with different values of $\Delta \theta$ and $\varphi$. When $\Delta \theta =0$ and $\varphi =\pi /2$, the concurrence oscillates in the form of half-sine wave as we mentioned above in the Markovian regime. For the case of $\Gamma \tau \neq 0$, it can be seen that the concurrence $C(t)=0$ when $\Gamma t<\Gamma \tau$. This is due to the effect of time delay caused by the distance between the coupling points. The existence of time delay will inhibit the oscillating amplitude, as shown by blue solid line in Fig. 7(a). The rate at which concurrence disappears is increased by further increasing the time delay. Another important feature is that the time delay makes that the steady-state value of concurrence decreases but not equal to zero when $\Delta \theta =0$ and $\varphi =\pi$, as depicted in Fig. 7(b). One can explain this phenomenon by using the final-value theorem. The final concurrence value can be reduced as
The above equation shows that the steady-stable value of concurrence strongly depends on the time delay. If the time delay $\Gamma \tau$ is small enough, the steady-stable entanglement can always be formed between the two GAs. For example, when $\Gamma \tau =0.2$, the concurrence value finally tends to 0.25. Specifically, when $\Delta \theta =\varphi =\pi$, the decoupling phenomenon between the GA and the waveguide disappears under the non-Markovian regime, which indicates that the concurrence $C(t)\neq 0$. As shown by red dashed line in Fig. 7(c), when $\Gamma \tau =0$, the concurrence $C(t)\equiv 0$. However, when $\Gamma \tau =0.2$, the concurrence tends to a stable value of 0.15 finally.
Fig. 7. The effect of time delay on two braided GAs concurrence $C(t)$ with different initial conditions for (a) $\Delta \theta =0$, $\varphi =\pi /2$; (b) $\Delta \theta =0$; $\varphi =\pi$; (c) $\Delta \theta =\varphi =\pi$ in the non-Markovian regime. The nonreciprocal concurrence of two braided GAs as a function of $\Gamma t$ for (d) $\Delta \theta =\pi /4$, $\varphi =\pi /2$; (e) $\Delta \theta =\varphi =\pi /2$; (f) $\Delta \theta =\pi$, $\varphi =\pi /2$ in the non-Markovian regime.
Below, we will discuss the effect of time delay $\Gamma \tau$ on the nonreciprocal concurrence degree $\Delta C(t)$. In Fig. 7(d), when $\Delta \theta =\pi /4$ and $\varphi =\pi /2$, the nonreciprocal concurrence degree $\Delta C(t)=0$ at the initial time. When $\Gamma t=\Gamma \tau$, the nonreciprocal concurrence starts to arise due to the interaction between the two GAs. Compared with Markovian regime, the existence of the propagating time makes the maximum value of nonreciprocal concurrence decrease. And the similar oscillatory phenomena can occur in the nonreciprocal concurrence degree. As the propagating time increases, the time for the nonreciprocal concurrence to stabilize increases. These phenomena can be also observed in Fig. 7(e).
3.3 Two nested GAs
We now turn to consider the case of two nested GAs. The model has been shown in Fig. 1(c). In this case, the corresponding time-delayed differential equations of the probability amplitudes for two nested GAs can be written as
In the Markovian regime, using the similar procedures that have been mentioned above, the analytical solutions of the two nested GAs amplitudes $c_{a(b)}(t)$ and the concurrence $C_{j}(t)$($j=1,2$) in different initial excited states are given by
We firstly study the influence of $\Delta \theta$ on two nested GAs entanglement with different values of phase accumulation in the regime where time delay is negligible. The concurrence dynamics $C_{1}(t)$ and $C_{2}(t)$ no longer have symmetric properties under different initial excitation conditions, which is different from the previous two structures. Figs. 8(a)-(c) represent the two GAs concurrence with $c_{a}(0)=1$ and $c_{b}(0)=0$. At the initial moment, the concurrence $C_{1}(t)=0$ for both cases due to the $GA_{b}$ is in the ground state. When $\Delta \theta =0$ and $\varphi =\pi /2$, the concurrence $C_{1}(t)$ can be obtained as $C_{1}(t)=0.16|e^{-4\Gamma t}[(2i+V_{1})e^{2V_{1}\Gamma t}+(2i-V_{1})e^{-2V_{1}\Gamma t}-4i]|$ with $V_{1}=-2i\sqrt {1+2i}$. Compared with the previous two model structures, the concurrence of two nested GAs $C_{1}(t)$ reaches the maximum value 0.48. Finally, it decays to zero in a non-exponential form due to the two GAs individual decay. For the case of $\varphi =\pi /4$ in Fig. 8(b), when $\Delta \theta =2n\pi (n=0,1,2\cdots )$, $C_{1}(t)=3.4|e^{-4\Gamma t}[(\sqrt {2}+V_{2})e^{2V_{2}\Gamma t}+(\sqrt {2}-V_{2})e^{-2V_{2}\Gamma t}-2\sqrt {2}]/V^2_{2}|$ with $V_{2}=2i(\sqrt {2}-(1+\sqrt {2})i)^{1/2}$. Similarly with Fig. 8(a), the concurrence can reach the maximum value $C_{1max}=0.72$ in this situation and then it decays exponentially to 0. By further decreasing the phase $\varphi$, the concurrence between two GAs is significantly enhanced in the whole range of $\Delta \theta$, as shown in Fig. 8(c). This is because of the reduction in the size of the GA, which increases the interaction between the two GAs. Furthermore, we simulate the two GAs concurrence when the initial conditions are changed as $c_{a}(0)=0$ and $c_{b}(0)=1$. The corresponding behaviors of $C_{2}(t)$ are shown in Figs. 8(d)-(f), which is much different from $C_{1}(t)$. It can be seen clearly that the concurrence is periodically symmetric with respect to $\Delta \theta =\pi$. The concurrence $C_{2}(t)$ can reach the maximum value in the evolution process when $\Delta \theta =\pi$. In detail, when $\Delta \theta =\pi$ and $\varphi =\pi /2$, according to Eq. (38), $C_{2}(t)$ is given as $C_{2}(t)=0.16e^{-4\Gamma t}|(-2i+V_{a1})e^{2V_{a1}\Gamma t}-(2i+V_{a1})e^{-2V_{a1}\Gamma t}+4i|$ with $V_{a1}=-\sqrt {8i-4}$. $C_{2}(t)$ reaches the maximum value 0.57 and finally decreases to 0. The similar phenomenon can be also found in Fig. 8(e)-(f). In contrast with $C_{1}(t)$, the concurrence generated in the case of the $GA_{b}$ in the initial excited state is more obvious.
Fig. 8. The concurrence diagram between two GAs. The upper column represents the concurrence with initial conditions $c_{a}(0)=1$ and $c_{b}(0)=0$; The lower column represents the concurrence with initial conditions $c_{a}(0)=0$ and $c_{b}(0)=1$. The other parameters are shown in the figure.
Based on the above analysis, one can find that the nonreciprocal concurrence can be also generated in this configuration because of different excitation initial conditions. In view of this, we further investigate the effect of $\Delta \theta$ on the nonreciprocal concurrence with different initial values of $\varphi$, as shown in Fig. 9(a)-(c). One can find that the nonreciprocal concurrence degree $\Delta C(t)$ is symmetric with respect to $\Delta \theta =\pi$. When $\Delta \theta =\pi$ and $\varphi =\pi /2$, the nonreciprocal concurrence degree is given as
$\Delta C(t)$ can reach the maximum value $\Delta C(t)=0.35$ in this situation. Then $\Delta C(t)$ decreases to 0 gradually due to the term $e^{-4\Gamma t}$ when $t\to \infty$. The similar behavior can be also found when $\Delta \theta =2n\pi (n=0,1,2\cdots )$, as depicted in Fig. 9(a). When $\Delta \theta =\pi$ and $\varphi =\pi /4$, one can get
When $\Delta \theta =n\pi /2(n=1,3,5\cdots )$ or $\varphi =m\pi (m=0,1,2\cdots )$, the above relation is satisfied and the nonreciprocal concurrence will disappear, as we discussed above. It is worth noting that when $\Delta \theta =n\pi /2$, the concurrence is given as
Fig. 9. The upper column represents the nonreciprocal concurrence of two nested GAs with different initial conditions of $\varphi$ for (a)-(c); The probability amplitudes and concurrence of two nested GAs as a function of $\Gamma t$ for (d)-(e). The other parameters are labled in the figure.
In the following, we will focus on studying the dynamics of the two GAs when the nonreciprocal concurrence vanishes. To simplify the research process, we define the concurrence $C_{1}(t)=C_{2}(t)=C(t)$ and consider some special cases. The corresponding results can be seen in a clearer way via the two-dimensional profiles in Figs. 9(d)-(f). When $\Delta \theta =0$ and $\varphi =\pi /2$, the probability amplitudes of the two GAs are given by $|c_{a}(t)|=1$ and $|c_{b}(t)|=0$, respectively. This means that GA$_{a}$ does not interact with the waveguide, which realizes the decoupling of the waveguide and GAs even if under different initial excitation conditions and corresponds to the (nonreciprocal) concurrence disappearance. For the case of $\Delta \theta =\pi /2$ and $\varphi =\pi$, similar to the previous models, the two GAs probability amplitudes can be written as $|c_{a}(t)|=(1+e^{-4\Gamma t})/2$ and $|c_{b}(t)|=(1-e^{-4\Gamma t})/2$, the concurrence $C(t)=(1-e^{-8\Gamma t})/2$. At the initial moment, the GA$_{a}$ is in the excited state and GA$_{b}$ is in the ground state, i.e. $c_{a}(0)=1$ and $c_{b}(0)=0$. When $\Gamma t\to \infty$, the two GAs probability amplitudes $|c_{a}(t)|=|c_{a}(t)|=0.5$. The steady-state concurrence is formed, as shown by black dashed line in Fig. 9(e). Moreover, when $\Delta \theta =\varphi =\pi /2$, $|c_{a}(t)|=|e^{-2\Gamma t}\cos 2\Gamma t|$ and $|c_{b}(t)|=|e^{-2\Gamma t}\sin 2\Gamma t|$. When $\Gamma t=\pi /4$, the GA$_{a}$ evolves from an excited state to a ground state. It is interesting that when $\Gamma t>\pi /4$, the GA$_{a}$ is excited again due to the interaction with GA$_{b}$ and the concurrence finally tends to 0,as shown in Fig. 9(f).
So far we have studied the two nested GAs entanglement dynamics in the Markovian regime. However, it is necessary to explore the two nested GAs entanglement dynamics in the non-Markovian regime. In detail, Figs. 10.(a)-(c) clearly show the evolution of the concurrence dynamics of two nested GAs when nonreciprocity disappears. For the both cases, the concurrence $C(t)=0$ in the range of $\Gamma t<\Gamma \tau$. And we can still observe the oscillatory decay phenomenon by comparing with Markovian regime, as depicted in Fig. 10(a). From Fig. 10(b) and 10(c), one can find that the steady-stable concurrence can also be achieved even under the influence of time delay $\Gamma \tau$. Therefore, we present the relationship between the time delay and the concurrence final value
It can be seen clearly that $C(t\to \infty )$ strongly depends on time delay. Eq. (41) indicates that the steady-stable concurrence can be formed with a cetain $\Gamma \tau$, as we mentioned above. Especially for the case of $\Delta \theta =0$ and $\varphi =\pi$, the concurrence $C(t)\equiv 0$ when $\Gamma \tau =0$. This result corresponds to the phenomenon exhibited in the Markovian regime. Similarly, we also simulate the influence of $\Gamma \tau$ on the nonreciprocal concurrence. For simplicity, let us consider some special cases, as displayed in Figs. 10(d)-(f). In Figs. 10(d) and 10(e), the blue solid line shows that when $\Gamma \tau =0$, the nonreciprocal concurrence decays exponentially to 0 after reaching its maximum value. In contrast, there appear more oscillations and peaks when $\Gamma \tau \neq 0$. With times going, $\Delta C(t)$ is characterized by a slow nonexponential oscillating decay process. By further analyzing, one can observe that the maximum nonreciprocal concurrence value decreases with the time delay increasing. These phenomena are caused by the quantum interference effect and the non-Markovian retarded effect. Finally, it should be pointed out that when $\Delta \theta =\pi /2$ and $\varphi =\pi /8$, the nonreciprocal concurrence cannot occur in either Markovian or non-Markovian regimes, i.e. $\Delta C(t)=0$, as depicted in Fig. 10(f).
Fig. 10. (a)-(c) represent the concurrence of two nested GAs in the non-Markovian regime; The nonreciprocal concurrence of two nested GAs as a function of $\Gamma t$ for (d)-(e) in the non-Markovian regime. The other parameters are labled in the figure.
4. Discussion and conclusions
To clearly see the entanglement dynamics between the two GAs under different coupling models, we organize the cases considered in the article in a tabular form, as shown in Table 1. For different coupling models, the maximum value of nonreciprocal entanglement $\Delta C(t)$ and the corresponding conditions for obtaining the maximum value are obviously different. The conditions for reaching the steady-state entanglement depend on the coupling configurations and phase values.
Table 1. The value of (nonreciprocal) concurrence $C(t)$ ($\Delta C(t)$) under different coupling models between two GAs and waveguide.
In summary, we have studied the nonreciprocal excitation and entanglement generation of two GAs with three different coupling configurations theoretically in both of Markovian regime and non-Markovian regime. The dynamical equations of two GAs are given by solving the Schrödinger equation. In the Markovian regime, we present analytical solutions for the two GAs probability amplitudes and concurrence. The results show that the generation of entanglement strongly depends on the coupling phase and the phase accumulation generated by the distance of the coupling points. Steady-stable entanglement can be achieved between the two GAs under certain conditions and the value of concurrence depends on the coupling phase. In particular, we show that nonreciprocal entanglement can be generated between two GAs under different initial excitation conditions. In the non-Markovian regime, the existence of time delay makes the nonreciprocal entanglement demonstrating non-exponential decay and oscillatory behavior, which delays the time for the nonreciprocal entanglement to disappear. These features are caused by the quantum interference effect and the non-Markovian retarded effect. Additionally, the decoupling phenomenon exhibited in the Markovian regime is destroyed. This work may contribute to construction of quantum network based on waveguide quantum electrodynamics.
Funding
National Natural Science Foundation of China (119075023); Major Project of Natural Science Foundation of Anhui Provincial Department of Education (2022AH040053); Key natural scientific research projects of universities in Anhui Province (2023AH051078); Zhejiang Province Key Laboratory of Quantum Technology and Device (20220101).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Z. Liao, X. Zeng, H. Nha, et al., “Photon transport in a one-dimensional nanophotonic waveguide qed system,” Phys. Scr. 91(6), 063004 (2016). [CrossRef]
2. D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89(2), 021001 (2017). [CrossRef]
3. A. S. Sheremet, M. I. Petrov, I. V. Iorsh, et al., “Waveguide quantum electrodynamics: collective radiance and photon-photon correlations,” Rev. Mod. Phys. 95(1), 015002 (2023). [CrossRef]
4. J.-T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30(15), 2001–2003 (2005). [CrossRef]
5. L. Zhou, Z. Gong, Y.-X. Liu, et al., “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]
6. D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010). [CrossRef]
7. Z. Liao, X. Zeng, S.-Y. Zhu, et al., “Single-photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92(2), 023806 (2015). [CrossRef]
8. H. Xiao, L. Wang, L. Yuan, et al., “Frequency manipulations in single-photon quantum transport under ultrastrong driving,” ACS Photonics 7(8), 2010–2017 (2020). [CrossRef]
9. I.-H. Choe, C.-M. Kim, N.-C. Kim, et al., “Single plasmon router with the two ingaas quantum dots placed at the junctions of h-type silver nanowire,” Plasmonics 18(1), 105–112 (2023). [CrossRef]
10. A. Lee, H. S. Han, F. K. Fatemi, et al., “Collective quantum beats from distant multilevel emitters,” Phys. Rev. A 107(1), 013701 (2023). [CrossRef]
11. K. Sinha, P. Meystre, E. A. Goldschmidt, et al., “Non-markovian collective emission from macroscopically separated emitters,” Phys. Rev. Lett. 124(4), 043603 (2020). [CrossRef]
12. Q. Bin and X.-Y. Lü, “Steady-state subradiance manipulated by the two-atom decay,” Phys. Rev. A 106(6), 063701 (2022). [CrossRef]
13. N. Li, H. Jiang, X. Xia, et al., “Collective radiation of multi-atom in cavity under the external driving field,” Phys. Lett. A 420, 127772 (2021). [CrossRef]
14. A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Quantum amplification by superradiant emission of radiation,” Phys. Rev. X 3(4), 041001 (2013). [CrossRef]
15. J.-S. Ryom, N.-C. Kim, M.-C. Ko, et al., “Entanglement of two quantum dots with azimuthal angle difference in plasmonic waveguide system,” Plasmonics 17(3), 949–956 (2022). [CrossRef]
16. A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, et al., “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011). [CrossRef]
17. A. González-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in solid-state quantum optics,” Phys. Rev. Lett. 110(8), 080502 (2013). [CrossRef]
18. B. Zhang, S. You, and M. Lu, “Enhancement of spontaneous entanglement generation via coherent quantum feedback,” Phys. Rev. A 101(3), 032335 (2020). [CrossRef]
19. I. M. Mirza and J. C. Schotland, “Multiqubit entanglement in bidirectional-chiral-waveguide qed,” Phys. Rev. A 94(1), 012302 (2016). [CrossRef]
20. M. Bajcsy, S. Hofferberth, V. Balic, et al., “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102(20), 203902 (2009). [CrossRef]
21. M. Arcari, I. Söllner, A. Javadi, et al., “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014). [CrossRef]
22. J.-Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474(7353), 589–597 (2011). [CrossRef]
23. A. Akimov, A. Mukherjee, C. Yu, et al., “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef]
24. A. Huck and U. L. Andersen, “Coupling single emitters to quantum plasmonic circuits,” Nanophotonics 5(3), 483–495 (2016). [CrossRef]
25. A. F. Kockum, P. Delsing, and G. Johansson, “Designing frequency-dependent relaxation rates and lamb shifts for a giant artificial atom,” Phys. Rev. A 90(1), 013837 (2014). [CrossRef]
26. Q. Cai and W. Jia, “Coherent single-photon scattering spectra for a giant-atom waveguide-qed system beyond the dipole approximation,” Phys. Rev. A 104(3), 033710 (2021). [CrossRef]
27. A. F. Kockum, “Quantum optics with giant atoms–the first five years,” in International Symposium on Mathematics, Quantum Theory, and Cryptography, vol. 33 (Springer Singapore, 2021), pp. 125–146.
28. X. Wang, T. Liu, A. F. Kockum, et al., “Tunable chiral bound states with giant atoms,” Phys. Rev. Lett. 126(4), 043602 (2021). [CrossRef]
29. C. Vega, M. Bello, D. Porras, et al., “Qubit-photon bound states in topological waveguides with long-range hoppings,” Phys. Rev. A 104(5), 053522 (2021). [CrossRef]
30. W. Zhao and Z. Wang, “Single-photon scattering and bound states in an atom-waveguide system with two or multiple coupling points,” Phys. Rev. A 101(5), 053855 (2020). [CrossRef]
31. H. Xiao, L. Wang, Z.-H. Li, et al., “Bound state in a giant atom-modulated resonators system,” npj Quantum Inform. 8(1), 80 (2022). [CrossRef]
32. X. Zhang, C. Liu, Z. Gong, et al., “Quantum interference and controllable magic cavity qed via giant atom in coupled resonator waveguide,” arXiv, arXiv:2303.16480 (2023). [CrossRef]
33. K. H. Lim, W.-K. Mok, and L.-C. Kwek, “Oscillating bound states in non-markovian photonic lattices,” Phys. Rev. A 107(2), 023716 (2023). [CrossRef]
34. L. Du, L. Guo, and Y. Li, “Complex decoherence-free interactions between giant atoms,” Phys. Rev. A 107(2), 023705 (2023). [CrossRef]
35. A. F. Kockum, G. Johansson, and F. Nori, “Decoherence-free interaction between giant atoms in waveguide quantum electrodynamics,” Phys. Rev. Lett. 120(14), 140404 (2018). [CrossRef]
36. A. Carollo, D. Cilluffo, and F. Ciccarello, “Mechanism of decoherence-free coupling between giant atoms,” Phys. Rev. Res. 2(4), 043184 (2020). [CrossRef]
37. Q.-Y. Qiu, Y. Wu, and X.-Y. Lü, “Collective radiance of giant atoms in non-markovian regime,” Sci. China Phys. Mech. Astron. 66(2), 224212 (2023). [CrossRef]
38. L. Du, M.-R. Cai, J.-H. Wu, et al., “Single-photon nonreciprocal excitation transfer with non-markovian retarded effects,” Phys. Rev. A 103(5), 053701 (2021). [CrossRef]
39. J. Zhou, X.-L. Yin, and J.-Q. Liao, “Chiral and nonreciprocal single-photon scattering in a chiral-giant-molecule waveguide-qed system,” Phys. Rev. A 107(6), 063703 (2023). [CrossRef]
40. X. Li, W. Zhao, and Z. Wang, “Controlling photons by phonons via giant atom in a waveguide qed setup,” Opt. Lett. 48(13), 3595–3598 (2023). [CrossRef]
41. H. Yu, Z. Wang, and J.-H. Wu, “Entanglement preparation and nonreciprocal excitation evolution in giant atoms by controllable dissipation and coupling,” Phys. Rev. A 104(1), 013720 (2021). [CrossRef]
42. A. C. Santos and R. Bachelard, “Generation of maximally entangled long-lived states with giant atoms in a waveguide,” Phys. Rev. Lett. 130(5), 053601 (2023). [CrossRef]
43. X.-L. Yin and J.-Q. Liao, “Generation of two-giant-atom entanglement in waveguide-qed systems,” Phys. Rev. A 108(2), 023728 (2023). [CrossRef]
44. P. Lodahl, S. Mahmoodian, S. Stobbe, et al., “Chiral quantum optics,” Nature 541(7638), 473–480 (2017). [CrossRef]
45. Y.-T. Chen, L. Du, L. Guo, et al., “Nonreciprocal and chiral single-photon scattering for giant atoms,” Commun. Phys. 5(1), 215 (2022). [CrossRef]
46. K. Xia, G. Lu, G. Lin, et al., “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014). [CrossRef]
47. L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic aharonov–bohm effect,” Opt. Lett. 40(22), 5140–5143 (2015). [CrossRef]
48. C. Liang, B. Liu, A.-N. Xu, et al., “Collision-induced broadband optical nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020). [CrossRef]
49. J.-C. Zheng and P.-B. Li, “Few-photon isolation in a one-dimensional waveguide using chiral quantum coupling,” Opt. Express 31(13), 21881–21898 (2023). [CrossRef]
50. Y. You, Y. Hu, G. Lin, et al., “Quantum nonreciprocity based on electromagnetically induced transparency in chiral quantum-optical systems,” Phys. Rev. A 103(6), 063706 (2021). [CrossRef]
51. F. Ruesink, J. P. Mathew, M.-A. Miri, et al., “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9(1), 1798 (2018). [CrossRef]
52. L. Du, Y.-T. Chen, and Y. Li, “Nonreciprocal frequency conversion with chiral Λ-type atoms,” Phys. Rev. Res. 3(4), 043226 (2021). [CrossRef]
53. X. Wang and H.-R. Li, “Chiral quantum network with giant atoms,” Quantum Sci. Technol. 7(3), 035007 (2022). [CrossRef]
54. X.-L. Yin, W.-B. Luo, and J.-Q. Liao, “Non-markovian disentanglement dynamics in double-giant-atom waveguide-qed systems,” Phys. Rev. A 106(6), 063703 (2022). [CrossRef]
55. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80(10), 2245 (1998). [CrossRef]
56. Y. Xue and Z. Wang, “Retardation effect and dark state in a waveguide qed setup with rectangle cross section,” Ann. Phys. 535, 2200458 (2023). [CrossRef]
57. L. Guo, A. Grimsmo, A. F. Kockum, et al., “Giant acoustic atom: A single quantum system with a deterministic time delay,” Phys. Rev. A 95(5), 053821 (2017). [CrossRef]
