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Abstract
We theoretically investigate the single photon scattering properties in a waveguide chirally coupling to a giant atom. The single photon transmission spectrum depends on the direction of the single photon incident when the energy loss of the giant atom can not be neglected. The difference between the transmission probabilities corresponding to opposite transport direction ΔT is calculated. It shows that both of the position and width of the ΔT are dependent on the size of the giant atom. Furthermore, the position of the maximum ΔT and the frequency width of ΔT can be modulated by a classical laser beam. Our results will be beneficial to control single photons in quantum devices design involving giant atoms.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Realization of strong interaction between quantum emitter and photons at single- photon level plays important roles in quantum information processing. With development of modern nanotechnology fabrication, the strong interaction between quantum emitters and optical modes in waveguide has been reported in several systems, including quantum dots coupling to photonic crystal waveguide [1–4], superconducting qubits coupled to transmission lines [5–7], and ultracold atoms coupled to optical fibers [8–10]. Based on the strong coupling, a new research field, named waveguide quantum electrodynamics (WQED) has been proposed [11]. The single photon scattering properties are investigated widely [12–40] and many quantum devices with high efficiency, such as quantum routers [36,37], frequency-comb [38], circulator [39], and quantum frequency converters [40,41], have been discussed. Furthermore, some quantum gates, like SWAP gate and CNOT gate have been investigated [42–45]. WQED is also a good platform to investigate many quantum phenomena such as supperradiance [46–48] and subradiance [49,50], quantum transition [51], and bound states in continuum [52] due to the low leakage of photons. Several excellent review papers [53–58] reported the advancements in WQED.
Recently, a new type of WQED structure has attracted much attention, where the quantum emitters are so called giant atoms (GAs). Different from the small quantum emitters, the GAs couple to waveguides with multiple coupling points due to its size [59–61], which provides a new channel for quantum interference. Many interesting quantum optical effects based on the multi-point couplings have been investigated, including decoherence-free subspace [60–62], non-exponential decay [63], creation of bound states [64–66], and Lamb shift [67]. The single photon scattering properties in the WQED have also been studied [68–74] and some quantum devices, such as single-photon frequency converters [69,70] and router [72] have been predicted.
It is well known that realization of incident direction dependent single photon scattering in WQED is crucial for constructing nonreciprocal optical element devices, such as diodes. Chiral coupling, which means that the coupling strengths between the quantum emitters and waveguide depend on the direction of incident photon, can be used to construct of nonreciprocal optical element devices [75]. Based on the chiral coupling, it has been discussed that single photon diode [76–78] and router [79–81] can be realized in the waveguide coupling to small atoms system. For the GA coupling to waveguide system, researches have discussed the chiral quantum optics [82]. Here, we discuss the single photon nonreciprocal scattering based on GA -waveguide chiral couplings. We show that one can realize a diode in this system. Comparing to small atoms, the performance of the diode based on GA can be manipulated by the size of the GA.
2. Configuration and theoretical model
The system we studied is shown in Fig. 1. A GA couples to the waveguide at point $x_{1}=0$ and $x_{2}=L$ [73]. The GA is a three-level system with ground state $|g\rangle$, excited state $|e\rangle$ and metastable state $|s\rangle$. The transition between states $|e\rangle$ and $|g\rangle$ is coupled by the modes of the waveguide. The coupling is chiral, where the coupling strength between the GA and the right (left) propagation photon is $g_{r} (g_{l})$. The transition between states $|e\rangle$ and $|s\rangle$ is decoupled from the waveguide but driven by a classical laser beam with Rabi frequency $\Omega$ and detuning $\Delta _{L}$.
Fig. 1. Sketch of the setup in this work, composed of a GA and a waveguide. The GA is a three-level system. One transition $|e\rangle \leftrightarrow |g\rangle$ chirally couples to the waveguide at $x=0$ and $x=L$. The coupling strength between the GA and the right (left) propagation photon is $g_{r} (g_{l})$. And the other transition $|e\rangle \leftrightarrow |s\rangle$ is driven by a classical laser beam with Rabi frequency $\Omega$ and detuning $\Delta _{L}$.
The Hamiltonian describing this system is given by [19,68]
Here, $H_{w}, H_{a}$ and $H_{int}$ represent the free propagation photon in the waveguide, the GA and the interaction between them, respectively. $v_{g}$ is the group velocity of the photon. $c^{\dagger }_{R}(x)(c^{\dagger }_{L}(x))$ denotes creating a right (left) propagation photon in the waveguide at position $x$. $\omega _{e}$ is the transition frequency between states $|e\rangle$ and $|g\rangle$. Here, we have set energy scale such that the energy of state $|g\rangle$ is zero. $\gamma _{e} (\gamma _{s})$ represents the energy loss of state $|e\rangle (|s\rangle )$. $\sigma _{m,n} (m,n=e,v,s)$ is the dipole transition operator. H.c. denotes Hermitian conjugate.
Since the excitation is conserved and only the single photon processing is considered, one can express the wave function of the system as
Here, $t_{R}$ and $r_{L}$ denote the single photon transmission and reflection amplitudes when the photon incident from the left of the waveguide. $h(x)$ is the Heaviside step function. $ae^{ikx}h(x)h(L-x)$ and $be^{-ikx}h(x)h(L-x)$ describe the wavefunction of the single photon between 0 and $L$. By solving the eigenvalue equation $H|\Psi \rangle =\omega |\Psi \rangle$, we obtain
By comparing Eqs. (8) to (11), one can find that $r_{L}=r_{R}$, and $t_{R}\neq t_{L}$. In the following, we will discuss how to manipulate nonreciprocal single photon transmission properties.
3. Single photon scattering by a giant atom with two-level system
Let us consider the case of $\Omega =0$ first. In this case, the three-level giant atom degenerates into a two-level system. And the two transmission amplitudes are given by
By analyzing the Eq. (14), we can obtain that when $\omega =\omega _{gf}$ and $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$, $\Delta T$ reaches the maximum value $\Delta T_{max}=(\Gamma ^{2}_{R}-\Gamma ^{2}_{L})/(\Gamma _{R}+\Gamma _{L})^{2}$.
Figure 2 shows the nonreciprocal single photon transmission probabilities. Fig. 2 (a) and (b) exhibit $T_{R}$ and $T_{L}$ versus detuning $\Delta _{e}$ with $\phi =0, 0.25\pi, 0.5\pi$, and $0.75\pi$. Both of $T_{R}$ and $T_{L}$ depend strongly on $\phi$. They reach the minimum value when $\Delta _{e}=(\Gamma _{R}+\Gamma _{L})\sin \phi$, but the minimum values of $T_{R}$ and $T_{L}$ are different due to the chiral coupling and energy loss of state $|e\rangle$. Figure 2 (c) shows $\Delta T$ as a function of $\Delta _{e}$. $\phi$ can change the line-shape of $\Delta _{e}$ strongly. The position of $\Delta T_{max}$ changes with $\phi$ taking different values. But the value of $\Delta T_{max}$ keeps fixed as 0.875 for the determined $\Gamma _{R}$ and $\Gamma _{L}$, as we discussed before. It should be pointed out that when $\phi =\pi$, both of $T_{R}$ and $T_{L}$ equal to 1. Thus, the system is transparent for the incident single photon. It can be explained as following: Different from the small atoms, the coupling strengths between the GA and waveguide are modified to be $\Gamma _{R/L}(1+\cos \phi )$. When $\phi =\pi$, $(1+\cos \phi )=0$, GA decouples to the waveguide and the $T_{R}=T_{L}=1$, thus, the single photon scattering properties are reciprocal for $\phi =\pi$.
Fig. 2. Nonreciprocal single photon transmission properties when $\Omega =0$. $T_{R}, T_{L}$ and $\Delta T$ versus $\Delta _{e}$ with different $\phi$ are exhibited in (a), (b), and (c), respectively. $T_{R}, T_{L}$ and $\Delta T$ versus $\Delta _{e}$ with different $\Gamma _{R}$ are shown in (d), (e) and (f), respectively. In (a),(b) and (c), $\Gamma _{R}=15\Gamma _{L}$. In (d), (e) and (f), $\phi =0.5\pi$. In all the calculations, $\Gamma _{L}=10^{-5}\omega _{e}$. $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$.
One can find that in Fig. 2(c) the width of $\Delta T$ decreases as $\phi$ increases from 0 to near $\pi$. To show this properties more clearly, we calculate the full width at half maximum (FWHM) of $\Delta T$, which is given by $\Delta \omega _{fw}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$. It is obviously that $\Delta \omega _{fw}$ decreases with $\phi$ increasing when $\phi$ is in the region of $[0, \pi )$. But the maximum value of $\Delta T$ ($\Delta T_{max}$) keeps fixed. When $\phi =2n\pi$, where $n$ is an integer, the FWHW reaches the maximum value $4(\Gamma _{R}+\Gamma _{L})$. However, when $\phi$ approaches $(2n+1)\pi$, FWHW is close to zero. Thus, one can choose appropriate $\phi$ to change the width of $\Delta T$. This property may be useful in design single photon diode with narrow band width.
Figure 2(d), (e), and (f) show $T_{R}$ and $T_{L}$ versus detuning $\Delta _{e}$ with different $\Gamma _{R}$. When $\Gamma _{R}=0$, the chiral coupling is perfect. the incident single photon transmits with unit probability since there is no interaction between the GA and the waveguide, thus $T_{R}=1$. $T_{L}$ depends on $\Delta _{e}$. When $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos 0.5\pi )=2\Gamma _{L}$, $T_{L}$ reaches the minimum value 0. So, $\Delta T_{max}$ reaches the maximum value of 1. Fig. 3(a) exhibits the influence of the ratio $\Gamma _{R}/\Gamma _{L}$ on $\Delta T_{max}$. It shows clearly that when $\Gamma _{R}=0$, $\Delta T_{max}$ reaches the maximum value of 1. However, when the coupling is non-chiral ($\Gamma _{R}=\Gamma _{L}$), the system can not show nonreciprocal scattering properties, and thus $\Delta T_{max}=0$. When $\Gamma _{R}/\Gamma _{L}$ further increases from 1 to a large value, $\Delta T_{max}$ increases. However, $\Delta T_{max}$ can not reach the maximum value of 1 due to energy loss and non-perfect chirally coupling. Figure 3(b) shows how the energy loss affects the single photon scattering properties. When $\gamma _{e}$ is zero, $T_{R}=T_{L}$, the nonreciprocal scattering properties disappear. However, with increasing of $\gamma _{e}$, $T_{R}$ and $T_{L}$ are quite different. When $\gamma _{e}$ increase to a value of $16\Gamma _{L}$, $\Delta T$ reaches the maximum value, which is consistent with theoretical analysis.
Fig. 3. $\Delta T_{max}$ versus $\Gamma _{R}/\Gamma _{L}$ with $\phi =0.5\pi$ and $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$ (a). $T_{R}$ (dotted-line), $T_{L}$ (dashed line) and $\Delta T$ (solid line) as a function of $\gamma _{e}$. In the calculations, $\phi =0.5\pi$, $\Gamma _{R}=15\Gamma _{L}$.
4. Single photon scattering controlled by a classical laser beam
Now, we discuss how to manipulate the nonreciprocal single photon scattering properties by the classical laser beam. From Eqs. (8) and (10), one can obtain that the $T_{R}$ and $T_{L}$ reach the minimum value at $\omega =\omega ^{'}_{gf\pm }$, where
andFigure 4 shows how to manipulate the nonreciprocal single scattering properties by the classic laser beam. $T_{R}, T_{L}$, and $\Delta T$ as a function of $\Delta _{e}$ when $\Delta _{L}=0$ are shown in Fig. 4 (a), (b), and (c), respectively. It exhibits clearly that there are two dips in the transmission spectra. The locations of the two dips shift with $\phi$, as implied by Eq. (15). It also shows that $\Delta T$ reaches the maximum value when $T_{R}$ and $T_{L}$ reach the minimum value, which are consistent with the theoretical analysis. The impedance matching condition of $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$ is used in calculating these figures, which leads to $\Delta T_{max}$ equaling to be 0.875 even though $\phi$ is different. It should be noted that there is a peak in the transmission spectrum at $\Delta _{e}=0$. This phenomenon does not exist when the classical laser beam is turned off. Fig. 4 (d), (e), and (f) show $T_{R}, T_{L}$, and $\Delta T$ as a function of $\Delta _{e}$ when $\Delta _{L}=20\Gamma _{L}$, respectively. The scattering spectra exhibit strong asymmetry. Furthermore, the locations of the peak in the transmission spectra shift from $\Delta _{e}=0$ to $\Delta _{e}=-\Delta _{L}$. But $\Delta T_{max}$ keeps fixed as 0.875 for the parameters chosen in the calculations.
Fig. 4. Nonreciprocal single photon transmission properties with $\Omega =30\Gamma _{L}$. $T_{R} (a), T_{L}$ (b) and $\Delta T$ (c) versus $\Delta _{e}$ with $\Delta _{L}=0$. $T_{R} (d), T_{L}$ (e) and $\Delta T$ (f) versus $\Delta _{e}$ with $\Delta _{L}=20\Gamma _{L}$. In all the calculations, $\Gamma _{L}=10^{-5}\omega _{e}$. $\Gamma _{R}=15\Gamma _{L}$, and $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$.
The physical mechanism for the appearance of two dips in the transmission spectrum can be intuitively explained by using dressed states. From the Eq. (3), one can obtain that the eigen-frequencies of the two dressed states are given by $\omega _{\pm }=\omega _{e}\pm (-\Delta _{L}+\sqrt {\Delta ^{2}_{L}+\Omega ^{2}})/2$, which are different from $\omega ^{'}_{gf\pm }$. The difference between $\omega _{\pm }$ and $\omega ^{'}_{gf\pm }$ results from the new quantum interference channel provided by the multi-point coupling between the GA and waveguide.
We further study the width of $\Delta T$. For the peak at $\omega ^{'}_{gf+}$, the FWHM of $\Delta T$ is given by $w_{+}=(4(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )-\sqrt {M-N}+\sqrt {P+Q})/2$, where $M\equiv (2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )+\Delta _{L}-(\Gamma _{R}+\Gamma _{L})\sin \phi )^{2}, N\equiv 8(\Gamma _{R}+\Gamma _{L})\Delta _{L}(1+\cos \phi )-\Omega ^{2}-4(\Gamma _{R}+\Gamma _{L})\Delta _{L}\sin \phi,P\equiv (2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )-\Delta _{L}+(\Gamma _{R}+\Gamma _{L})\sin \phi )^{2}, Q\equiv 8(\Gamma _{R}+\Gamma _{L})\Delta _{L}(1+\cos \phi )+\Omega ^{2}-4(\Gamma _{R}+\Gamma _{L})\Delta _{L}\sin \phi$. And the FWHM of $\Delta T$ for the peak at $\omega ^{'}_{gf-}$ is $w_{-}=(4(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )+\sqrt {M-N}-\sqrt {P+Q})/2$. It is obviously that both of $w_{+}$ and $w_{-}$ are strongly dependent on $\phi, \Delta _{L}$, and $\Omega$. Figure 5 (a) and (b) show $\omega _{+}$ and $\omega _{-}$ as a function of $\phi$ with $\Omega$ taking different values under the resonant excitation ($\Delta _{L}=0$). It shows that $\omega _{\pm }$ oscillates with $\phi$. When $\phi$ approaches to $(2n+1)\pi$, both of $\omega _{\pm }$ approach to zero. One can tune the width by modulating $\Omega$ for a fixed $\phi$, but the tuning region is not large. The influences of detuning $\Delta _{L}$ on the $w_{\pm }$ are exhibited in Fig. 5 (c) and (d). $w_{\pm }$ is sensitive to $\Delta _{L}$ when $\phi$ is far away from $(2n+1)\pi$.
Fig. 5. The FWHM of $\Delta T$ controlled by the classical laser beam. The $w_{+}$ and $w_{-}$ as a function of $\phi$ with $\Omega$ taking different values under the condition of resonant excitation $(\Delta _{L}=0$). (c) and (d) exhibit $w_{+}$ and $w_{-}$ as a function of $\phi$ with different $\Delta _{L}$ while $\Omega$ is fixed with $\Omega =20\Gamma _{L}$. In all the calculations, $\Gamma _{L}=10^{-5}\omega _{e}$. $\Gamma _{R}=15\Gamma _{L}$, $\gamma _{e}=2(\Gamma _{R}+\Gamma _{L})(1+\cos \phi )$.
5. Discussions and Conclusions
Realization of chiral coupling between giant atoms and waveguide at each coupling point is still a challenge. Luckily, there are many developments in this field. For example, Kannan et al. reported experimental realization of giant atoms coupling to waveguide at multiple points [59]. Furthermore, chiral coupling effect between giant atoms and waveguide has been proposed recently in Ref. [83] and Ref. [84] by using superconducting circuits, respectively. Our model may be realized by combining these two characteristics.
In summary, we have discussed single photon scattering properties in the giant atom chiral coupling to a waveguide system. The transmitted probabilities of single photon incident from opposite directions of the waveguide are calculated. The system can exhibit strong nonreciprocity mediated by the energy loss. $\Delta T$ depends strongly on the coupling strength and the size of the GA. For the perfect chiral coupling, $\Delta T$ can reach the ideal value of 1. For the non-perfect chiral coupling, the maximum value of $\Delta T$ is determined by the chiral coupling strengths. The position at which $\Delta T$ reaches the maximum value is determined by the size of GA. When the classical laser beam is turned on, the frequency of the incident photon to reach the maximum value of $\Delta T$ can be modulated by $\Omega$ and $\Delta _{L}$. Furthermore, the FWHM of the $\Delta T$ can be controlled by choosing the size of the GA, appropriate values of $\Omega$ and $\Delta _{L}$. Our results are helpful in designing quantum optical devices at the single-photon level.
Funding
National Natural Science Foundation of China (11975023); Education Department of Anhui Province (gxyqZD2020014); Key Research and Development Plan of Ministry of Science and Technology (2018YFB1601402); College Students' innovation and entrepreneurship training program of Anhui Province (S202110360273).
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. M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014). [CrossRef]
2. L. Scarpelli, B. Lang, F. Masia, D. Beggs, E. Muljarov, A. Young, R. Oulton, M. Kamp, S. Höfling, C. Schneider, and W. Langbein, “99% beta factor and directional coupling of quantum dots to fast light in photonic crystal waveguides determined by spectral imaging,” Phys. Rev. B 100(3), 035311 (2019). [CrossRef]
3. C. Carlson, D. Dalacu, C. Gustin, S. Haffouz, X. Wu, J. Lapointe, R. L. Williams, P. J. Poole, and S. Hughes, “Theory and experiments of coherent photon coupling in semiconductor nanowire waveguides with quantum dot molecules,” Phys. Rev. B 99(8), 085311 (2019). [CrossRef]
4. H. Le Jeannic, T. Ramos, S. F. Simonsen, T. Pregnolato, Z. Liu, R. Schott, A. D. Wieck, A. Ludwig, N. Rotenberg, J. J. García-Ripoll, and P. Lodahl, “Experimental reconstruction of the few-photon nonlinear scattering matrix from a single quantum dot in a nanophotonic waveguide,” Phys. Rev. Lett. 126(2), 023603 (2021). [CrossRef]
5. O. Astafiev, A. M. Zagoskin, A. Abdumalikov Jr, Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327(5967), 840–843 (2010). [CrossRef]
6. A. F. Van Loo, A. Fedorov, K. Lalumiere, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342(6165), 1494–1496 (2013). [CrossRef]
7. I.-C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. Wilson, “Generation of nonclassical microwave states using an artificial atom in 1d open space,” Phys. Rev. Lett. 108(26), 263601 (2012). [CrossRef]
8. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010). [CrossRef]
9. N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, “Large bragg reflection from one-dimensional chains of trapped atoms near a nanoscale waveguide,” Phys. Rev. Lett. 117(13), 133603 (2016). [CrossRef]
10. N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019). [CrossRef]
11. H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide qed: Many-body bound-state effects in coherent and fock-state scattering from a two-level system,” Phys. Rev. A 82(6), 063816 (2010). [CrossRef]
12. I. M. Mirza, J. G. Hoskins, and J. C. Schotland, “Chirality, band structure, and localization in waveguide quantum electrodynamics,” Phys. Rev. A 96(5), 053804 (2017). [CrossRef]
13. G.-Z. Song, E. Munro, W. Nie, L.-C. Kwek, F.-G. Deng, and G.-L. Long, “Photon transport mediated by an atomic chain trapped along a photonic crystal waveguide,” Phys. Rev. A 98(2), 023814 (2018). [CrossRef]
14. J.-T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30(15), 2001–2003 (2005). [CrossRef]
15. L. Zhou, Z. Gong, Y.-x. Liu, C. Sun, and F. Nori, “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]
16. T. Tsoi and C. Law, “Single-photon scattering on Λ-type three-level atoms in a one-dimensional waveguide,” Phys. Rev. A 80(3), 033823 (2009). [CrossRef]
17. N.-C. Kim, J.-B. Li, Z.-J. Yang, Z.-H. Hao, and Q.-Q. Wang, “Switching of a single propagating plasmon by two quantum dots system,” Appl. Phys. Lett. 97(6), 061110 (2010). [CrossRef]
18. W. Chen, G.-Y. Chen, and Y.-N. Chen, “Coherent transport of nanowire surface plasmons coupled to quantum dots,” Opt. Express 18(10), 10360–10368 (2010). [CrossRef]
19. D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” N. J. Phys. 12(4), 043052 (2010). [CrossRef]
20. C.-H. Yan, L.-F. Wei, W.-Z. Jia, and J.-T. Shen, “Controlling resonant photonic transport along optical waveguides by two-level atoms,” New J. Phys. 84(4), 045801 (2011). [CrossRef]
21. M.-T. Cheng and Y.-Y. Song, “Fano resonance analysis in a pair of semiconductor quantum dots coupling to a metal nanowire,” Opt. Lett. 37(5), 978–980 (2012). [CrossRef]
22. K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014). [CrossRef]
23. Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, “Single-photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92(2), 023806 (2015). [CrossRef]
24. 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]
25. S. Xu and S. Fan, “Generate tensor network state by sequential single-photon scattering in waveguide qed systems,” APL Photonics 3(11), 116102 (2018). [CrossRef]
26. G.-Z. Song, L.-C. Kwek, F.-G. Deng, and G.-L. Long, “Microwave transmission through an artificial atomic chain coupled to a superconducting photonic crystal,” Phys. Rev. A 99(4), 043830 (2019). [CrossRef]
27. D. Mukhopadhyay and G. S. Agarwal, “Multiple fano interferences due to waveguide-mediated phase coupling between atoms,” Phys. Rev. A 100(1), 013812 (2019). [CrossRef]
28. M.-C. Ko, N.-C. Kim, J.-S. Ryom, S.-R. Ri, and J.-B. Li, “Entanglement of two distant quantum dots with the flip-flop interaction coupled to plasmonic waveguide,” Int. J. Mod. Phys. B 33(21), 1950235 (2019). [CrossRef]
29. F. Dinc, I. Ercan, and A. M. Brańczyk, “Exact markovian and non-markovian time dynamics in waveguide qed: collective interactions, bound states in continuum, superradiance and subradiance,” Quantum 3, 213213 (2019). [CrossRef]
30. J. Dong, Q. Jiang, Q. Hu, B. Zou, and Y. Zhang, “Transport and entanglement for single photons in optical waveguide ladders,” Phys. Rev. A 100(1), 013840 (2019). [CrossRef]
31. D. Mukhopadhyay and G. S. Agarwal, “Transparency in a chain of disparate quantum emitters strongly coupled to a waveguide,” Phys. Rev. A 101(6), 063814 (2020). [CrossRef]
32. E. Stolyarov, “Single-photon switch controlled by a qubit embedded in an engineered electromagnetic environment,” Phys. Rev. A 102(6), 063709 (2020). [CrossRef]
33. J. Dong, B. Zou, and Y. Zhang, “Theoretical study of transparent peaks in a topological waveguide-cavity coupled system,” Appl. Phys. Lett. 119(25), 251101 (2021). [CrossRef]
34. S. A. Regidor and S. Hughes, “Cavitylike strong coupling in macroscopic waveguide qed using three coupled qubits in the deep non-markovian regime,” Phys. Rev. A 104(3), L031701 (2021). [CrossRef]
35. J.-S. Ryom, N.-C. Kim, M.-C. Ko, and S.-I. Choe, “Entanglement of two quantum dots with azimuthal angle difference in plasmonic waveguide system,” Plasmonics1–8 (2022).
36. L. Zhou, L.-P. Yang, Y. Li, and C. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111(10), 103604 (2013). [CrossRef]
37. X. Wang, W.-X. Yang, A.-X. Chen, L. Li, T. Shui, X. Li, and Z. Wu, “Phase-modulated single-photon nonreciprocal transport and directional router in a waveguide–cavity–emitter system beyond the chiral coupling,” Quantum Science and Technology 7(1), 015025 (2022). [CrossRef]
38. Z. Liao, H. Nha, and M. S. Zubairy, “Single-photon frequency-comb generation in a one-dimensional waveguide coupled to two atomic arrays,” Phys. Rev. A 93(3), 033851 (2016). [CrossRef]
39. X.-W. Xu, A.-X. Chen, Y. Li, and Y.-x. Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95(6), 063808 (2017). [CrossRef]
40. M. Bradford, K. C. Obi, and J.-T. Shen, “Efficient single-photon frequency conversion using a sagnac interferometer,” Phys. Rev. Lett. 108(10), 103902 (2012). [CrossRef]
41. H. Xiao, L. Wang, L. Yuan, and X. Chen, “Frequency manipulations in single-photon quantum transport under ultrastrong driving,” ACS Photonics 7(8), 2010–2017 (2020). [CrossRef]
42. H. Zheng, D. J. Gauthier, and H. U. Baranger, “Waveguide-qed-based photonic quantum computation,” Phys. Rev. Lett. 111(9), 090502 (2013). [CrossRef]
43. T. Li, A. Miranowicz, X. Hu, K. Xia, and F. Nori, “Quantum memory and gates using a Λ-type quantum emitter coupled to a chiral waveguide,” Phys. Rev. A 97(6), 062318 (2018). [CrossRef]
44. W. Konyk and J. Gea-Banacloche, “Passive, deterministic photonic conditional-phase gate via two-level systems,” Phys. Rev. A 99(1), 010301 (2019). [CrossRef]
45. G.-Z. Song, J.-L. Guo, Q. Liu, H.-R. Wei, and G.-L. Long, “Heralded quantum gates for hybrid systems via waveguide-mediated photon scattering,” Phys. Rev. A 104(1), 012608 (2021). [CrossRef]
46. A. Goban, C.-L. Hung, J. Hood, S.-P. Yu, J. Muniz, O. Painter, and H. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015). [CrossRef]
47. Y. Zhou, Z. Chen, and J.-T. Shen, “Single-photon superradiant emission rate scaling for atoms trapped in a photonic waveguide,” Phys. Rev. A 95(4), 043832 (2017). [CrossRef]
48. J.-H. Kim, S. Aghaeimeibodi, C. J. Richardson, R. P. Leavitt, and E. Waks, “Super-radiant emission from quantum dots in a nanophotonic waveguide,” Nano Lett. 18(8), 4734–4740 (2018). [CrossRef]
49. A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7(3), 031024 (2017). [CrossRef]
50. Z. Wang, H. Li, W. Feng, X. Song, C. Song, W. Liu, Q. Guo, X. Zhang, H. Dong, D. Zheng, H. Wang, and D.-W. Wang, “Controllable switching between superradiant and subradiant states in a 10-qubit superconducting circuit,” Phys. Rev. Lett. 124(1), 013601 (2020). [CrossRef]
51. L. Qiao, Y.-J. Song, and C.-P. Sun, “Quantum phase transition and interference trapping of populations in a coupled-resonator waveguide,” Phys. Rev. A 100(1), 013825 (2019). [CrossRef]
52. P. Facchi, M. Kim, S. Pascazio, F. V. Pepe, D. Pomarico, and T. Tufarelli, “Bound states and entanglement generation in waveguide quantum electrodynamics,” Phys. Rev. A 94(4), 043839 (2016). [CrossRef]
53. Z. Liao, X. Zeng, H. Nha, and M. S. Zubairy, “Photon transport in a one-dimensional nanophotonic waveguide qed system,” Phys. Scr. 91(6), 063004 (2016). [CrossRef]
54. X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017). [CrossRef]
55. D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89(2), 021001 (2017). [CrossRef]
56. D. Chang, J. Douglas, A. González-Tudela, C.-L. Hung, and H. Kimble, “Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons,” Rev. Mod. Phys. 90(3), 031002 (2018). [CrossRef]
57. P. Türschmann, H. Le Jeannic, S. F. Simonsen, H. R. Haakh, S. Götzinger, V. Sandoghdar, P. Lodahl, and N. Rotenberg, “Coherent nonlinear optics of quantum emitters in nanophotonic waveguides,” Nanophotonics 8(10), 1641–1657 (2019). [CrossRef]
58. A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, “Waveguide quantum electrodynamics: collective radiance and photon-photon correlations,” arXiv preprint arXiv:2103.06824 (2021).
59. B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. Frisk Kockum, J. Braumüller, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, A. Vepsäläinen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, “Waveguide quantum electrodynamics with superconducting artificial giant atoms,” Nature 583(7818), 775–779 (2020). [CrossRef]
60. 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]
61. 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]
62. A. Carollo, D. Cilluffo, and F. Ciccarello, “Mechanism of decoherence-free coupling between giant atoms,” Phys. Rev. Res. 2(4), 043184 (2020). [CrossRef]
63. G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, “Non-exponential decay of a giant artificial atom,” Nat. Phys. 15(11), 1123–1127 (2019). [CrossRef]
64. S. Guo, Y. Wang, T. Purdy, and J. Taylor, “Beyond spontaneous emission: Giant atom bounded in the continuum,” Phys. Rev. A 102(3), 033706 (2020). [CrossRef]
65. L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, “Oscillating bound states for a giant atom,” Phys. Rev. Res. 2(4), 043014 (2020). [CrossRef]
66. X. Wang, T. Liu, A. F. Kockum, H.-R. Li, and F. Nori, “Tunable chiral bound states with giant atoms,” Phys. Rev. Lett. 126(4), 043602 (2021). [CrossRef]
67. P. Wen, K.-T. Lin, A. Kockum, B. Suri, H. Ian, J. Chen, S. Mao, C. Chiu, P. Delsing, F. Nori, G. D. Lin, and I. C. Hoi, “Large collective lamb shift of two distant superconducting artificial atoms,” Phys. Rev. Lett. 123(23), 233602 (2019). [CrossRef]
68. 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]
69. L. Du and Y. Li, “Single-photon frequency conversion via a giant Λ-type atom,” Phys. Rev. A 104(2), 023712 (2021). [CrossRef]
70. L. Du, Y.-T. Chen, and Y. Li, “Nonreciprocal frequency conversion with chiral Λ-type atoms,” Phys. Rev. Res. 3(4), 043226 (2021). [CrossRef]
71. 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]
72. C. Wang, X.-S. Ma, and M.-T. Cheng, “Giant atom-mediated single photon routing between two waveguides,” Opt. Express 29(24), 40116–40124 (2021). [CrossRef]
73. S. Feng and W. Jia, “Manipulating single-photon transport in a waveguide-qed structure containing two giant atoms,” Phys. Rev. A 104(6), 063712 (2021). [CrossRef]
74. W. Zhao, Y. Zhang, and Z. Wang, “Phase-modulated autler-townes splitting in a giant-atom system within waveguide qed,” Front. Phys. 17(4), 42506–11 (2022). [CrossRef]
75. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017). [CrossRef]
76. C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015). [CrossRef]
77. W.-B. Yan, W.-Y. Ni, J. Zhang, F.-Y. Zhang, and H. Fan, “Tunable single-photon diode by chiral quantum physics,” Phys. Rev. A 98(4), 043852 (2018). [CrossRef]
78. L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99(4), 043833 (2019). [CrossRef]
79. M.-T. Cheng, X.-S. Ma, J.-Y. Zhang, and B. Wang, “Single photon transport in two waveguides chirally coupled by a quantum emitter,” Opt. Express 24(17), 19988–19993 (2016). [CrossRef]
80. C. Gonzalez-Ballestero, E. Moreno, F. J. Garcia-Vidal, and A. Gonzalez-Tudela, “Nonreciprocal few-photon routing schemes based on chiral waveguide-emitter couplings,” Phys. Rev. A 94(6), 063817 (2016). [CrossRef]
81. C.-H. Yan, Y. Li, H. Yuan, and L. Wei, “Targeted photonic routers with chiral photon-atom interactions,” Phys. Rev. A 97(2), 023821 (2018). [CrossRef]
82. A. Soro and A. F. Kockum, “Chiral quantum optics with giant atoms,” Phys. Rev. A 105(2), 023712 (2022). [CrossRef]
83. Y.-X. Zhang, C. R. i Carceller, M. Kjaergaard, and A. S. Sørensen, “Charge-noise insensitive chiral photonic interface for waveguide circuit qed,” Phys. Rev. Lett. 127(23), 233601 (2021). [CrossRef]
84. X. Wang and H.-R. Li, “Chiral quantum network with giant atoms,” Quantum Sci. Technol. 7(3), 035007 (2022). [CrossRef]
