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Abstract
We study the coherent transport of one or two photons in a one-dimensional waveguide chirally coupled to a nonlinear resonator. Analytic solutions of the one-photon and two-photon scattering is derived. Although the resonator acts as a non-reciprocal phase shifter, light transmission is reciprocal at one-photon level. However, the forward and reverse transmitted probabilities for two photons incident from either the left side or the right side of the nonlinear resonator are nonreciprocal due to the energy redistribution of the two-photon bound state. Hence, the nonlinear resonator acts as an optical diode at two-photon level.
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1. Introduction
A diode that allows unidirectional propagation of signal is an indispensable circuit element for information processing. In the design of the circuit for optical information processing, optical isolators serving as optical diodes [1] are often used to realize nonreciprocal propagation of light. Conventional optical isolators are designed based on magneto-optic effect and work in a strong magnetic field [2]. The design and fabrication of a magnetic-free optical isolator on chip have attracted a great interest in the past decades. A number of alternative schemes have been proposed theoretically and realized experimentally based on diverse mechanisms, such as nonlinear optics [3–11], chiral quantum optics [12–16], optomechanics [17–28], atomic gases in motion [29–34], and non-Hermitian optics [35–38]. It has to be stressed that the previous studies of optical nonreciprocity have mainly focused on transmission rates of the incident classical light. Very recently, nonreciprocal quantum effects have been explored theoretically [39], including nonreciprocal photon blockade [39–44] and nonreciprocal quantum entanglement [45,46], which open up a way to create and manipulate one-way nonclassical light.
Here, we propose to explore the controllability of the nonreciprocal transport light in quantum regime. Quantum switches that control the transport of a confined single photon have been proposed in a one-dimensional (1D) waveguide [47–49]. Strong photon-photon quantum correlations can also be created by few-photon transport in a 1D waveguide coupled to nonlinear scatterers, such as a two-level atom [50–52], a multi-level atom [53–58], multiple two-level atoms [59–65], a resonator coupling to a two-level atom [66–68], a multi-level atom [69], or a mechanical resonator [70–72]. The transport properties of a few photons inside a 1D waveguide coupled to a nonlinear resonator have been explored theoretically, and the bunching or anti-bunching behavior as a signature of strong quantum correlations at the few-photon level has been reported [73,74]. However, most of the schemes are proposed based on the configuration that a 1D waveguide coupling to a nonlinear scatterers symmetrically, i.e., the coupling strengths are the same for photons propagating in different directions, which cannot provide quantum non-reciprocity at the few-photon level in these systems.
In this paper, we propose a scheme to realize an optical diode at the few-photon level based on the chiral coupling between a 1D waveguide and a Kerr-type nonlinear resonator, i.e., the coupling strengths between the resonator and the waveguide are different for photons propagating in different directions. The Laplace transform is applied to solve the photon scattering problems, and an analytic solution is obtained. It is found that, for a single photon coming from either side of the resonator, the transmission amplitudes are the same in magnitude, but their phases are different, which indicates that the resonator can act as a non-reciprocal phase shifter. However, the forward and reverse transmitted probabilities for two photons incident from either the left side or the right side of the nonlinear resonator are nonreciprocal due to the energy redistribution of the two photons bound state. The chiral coupling configuration therefore open a new avenue toward optical diode in few-photon level for quantum information processing on-chip.
This paper is organized as follows. In Sec. 2, we introduce the model and establish the notation. Then we derive the analytic solutions of the one-photon and two-photon scattering in Sec. 3. and 4, and the forward and reverse transmitted probabilities for photon incident from either the left side or the right side of the nonlinear resonator are discussed at few-photon levels. We make a conclusion in Sec. 5.
2. Chiral waveguide coupled to a nonlinear resonator
The system we consider is a 1D waveguide of infinite length coupled to a nonlinear resonator of the single-mode field at the origin, which is schematically shown in Fig. 1. The 1D continuum of the waveguide are formed by the right-going and left-going modes, the right-going mode are represented by the canonical creation and annihilation operators $\hat {r}_{\omega }^{\dagger }$ and $\hat {r}_{\omega }$ , and the left-going mode are represented by $\hat {l}_{\omega }^{\dagger }$ and $\hat {l}_{\omega }$. Both modes obey the equal-time bosonic commutation relation $\left [ \alpha _{\omega },\alpha _{\omega ^{\prime }}^{\dagger }\right ] =\delta \left ( \omega -\omega ^{\prime }\right )$ with $\alpha _{\omega }=\hat {r}_{\omega },\hat {l}_{\omega }$. Photons traveling in the 1D waveguide can tunnel into and out of a resonator with eigenfrequency $\omega _{c}$. The mode of the resonator is denoted by the annihilation operator $\hat {a}$. The resonator is filled with a Kerr medium with nonlinear interaction strength $U$. We assume that the boson velocity $v$ is frequency independent, and introduce the symbol $\omega$ to denote the difference between the frequency of photon modes and $\omega _{c}$. Then the Hamiltonian of the resonator-waveguide system reads
The model we studied is motivated by the studies on light-matter chiral couplings. Its orignal model is a two-level system (TLS) chiral coupled to the waveguide modes. We replace the TLS by an additional bosonic lattice site $U\hat {a}^{\dagger }\hat {a}^{\dagger }\hat {a}\hat {a}$, i.e., the ground and excited states of the TLS correspond to none and a single boson on this additional site, respectively. The Hamiltonian of our model and that of the orignal model are equivalent in the limit $U\rightarrow \infty$. Similar trick has been employed [75].
Fig. 2. The transmittance (a) and reflectance (b) as a function of $\omega$. Blue solid, black dotted, and red dashed curves are shown for $(\gamma _1,\gamma _2)=(1,0), (0.8,0.2),(0.5,0.5)$, respectively. All parameters are in units of $\Gamma$.
The light-matter chiral couplings can be realized for a nonlinear resonator coupling to a chiral waveguide based on broken time-reversal symmetry. According to the Fermi’s golden rule, the coupling rates $\gamma _{1}$ and $\gamma _{2}$ between the nonlinear resonator and waveguide are inversely proportional to the group velocities of the left- and right going modes respectively. Usually the couplings are achiral for the time-reversal symmetry of the waveguide, which ensures the same group velocities for the left- and right going modes. However, if the time-reversal symmetry of the waveguide is broken by magnetic or synthetic magnetic field, the group velocities of the left- and right going modes become different from each other, then the chiral couplings $\gamma _{1}\mathrm {\neq }\gamma _{2}$ is achieved. We note that chiral waveguide has been demonstrated experimentally in gyromagnetic photonic crystals for enough strong time-reversal breaking [76–78], and the chiral chiral dissipation $\gamma _{1}\mathrm {\neq }\gamma _{2}$ has been proposed or demonstrated in atom chains [79–81], superconducting circuit QEDs [82–85] and optomechanical crystals [86–88], based on spin-orbit coupling [79] or synthetic magnetic field [82–86]. So the few-photon optical diode based on chiral couplings proposed here can be implemented in a wide variety of experimental platforms, e.g., magneto-optical photonic crystals [76–78], atom chains [79–81], superconducting circuit QEDs [82–86] and a optomechanical crystal [86–88].
Now, we introduce the following operators of the waveguide
The $b$ modes couple to the resonator and the $c$ modes evolve freely. The total number of excitations is conserved in both the $b$ and $c$ spaces separately.
3. Scattering of single photon
We study the single-photon scattering with photons incoming from either left or right of the resonator [48,49,89]. As the photons in the $c$ space propagate freely, the scattering process mainly occurs in the $b$ space.
States $\hat {a}^{\dagger }\left \vert 0\right \rangle$ and $\{\hat {b}_{\omega }^{\dagger } \left \vert 0\right \rangle \}$ are the basis in the $b$ space with one excitation, where the ground state $\left \vert 0\right \rangle$ is a vacuum state of both the waveguide and the resonator. $\hat {a}^{\dagger }\left \vert 0\right \rangle$ is a state for the single excitation in the resonator, $\hat {b}_{\omega }^{\dagger }\left \vert 0\right \rangle$ is a state for single excitation in the $\omega$th $b$ mode. Then, the time-dependent wave function of the system reads
By performing Laplace transform, Eq. (6) is transformed to a system of algebraic equations
The scattering process indicates that initially there is no excitation in the resonator, i.e. $A\left ( 0\right ) =0, A_{\omega }\left ( 0\right )\neq 0$. For the convenience of later discussion, we assume the probability amplitude at the initial time
is symmetric with respect to the center $\delta$ with width $\epsilon$. After a long time with $t>>\Gamma ^{-1},\epsilon ^{-1}$, the resonator is in the vacuum state and the scattered phonon travels freely in the waveguide, i.e.,Let us return to the scattering process in terms of the right and left-going modes. We first consider the single-photon wavepacket is in the right-going modes, traveling towards the resonator at origin from the left. Its state is described by
Since the chiral coupling between the waveguide and the resonator introduces an imbalance of the probabilities between the right- and left-moving emitted photon, we plot the scattering coefficients as a function of the incident frequency $\omega$ for a fixed $\Gamma$. A Lorentzian line shape can be observed around the resonance, however, a single photon at resonance frequency is completely reflected only for symmetric coupling. As $\gamma$ increases, the reflection becomes lower and lower. If the chiral coupling is fixed, the propagating single photon get nearly total transmitted as its center is far way from the resonator mode.
4. Two-phonon scattering
We now consider the scattering process in the double-excitation subspace. For photons incident from either side of the resonator in the $c$ space, they get transmitted directly. For two photons incident from either side of the resonator in the $b$ space, the two-photon wave functions have a complicated form due to the nonlinear interaction of the resonator. The general two-excitation state at arbitrary time in the $b$ space reads
During the scattering process the incident photons have a chance to jump into the resonator, however, We are mainly interested in the longtime behavior with $t\gg \gamma _i^{-1}, \epsilon _i^{-1}$. The asymptotic solution of $C_{\omega \omega ^{\prime }}$ is obtained by first Laplace transform of Eqs. (15) and then via the inverse Laplace transform up to the forth order of $\sqrt {\Gamma }$ after some algebra calculations. It has an overall phase factor $\exp \left [ -i\left ( \omega +\omega ^{\prime }\right ) t\right ]$ multiplying an time-independent part
For two incoming right-going photons from the left of the resonator with the initial state
For two incoming left-going photons from the right of the resonator with the initial state
We now assume that two photons are initially independent of each other, i.e., $\delta _1=\delta _2=\delta$ and $\epsilon _1=\epsilon _2=\epsilon$. In this case, the initial state of the two photons becomes
Fig. 3. The probability of the initial state (red solid line and green dashed line in (a)) and the norm square of $D_{\omega \omega ^{\prime }}$ as function the one photon frequency $\omega$ for given $U=10, \epsilon =0.1$ in (a),(c) and $U=2, \epsilon =0.1$ in (b). In (a), the red solid line and the blue dotted line is for $(E,\delta )=(0,0)$, the green dashed line is for $(E,\delta )=(10,5)$. $(E,\delta )=(2,1)$ $(0,1)$ and $(2,0)$ for red solid line,green dashed line and blue dotted line in (b), and $(E,\delta )=(10,5)$, $(8,5)$ and $(10,4)$ for red solid line,green dashed line and blue dotted line in (c) respectively. All parameters are in units of $\Gamma$.
Fig. 4. The norm square of the amplitudes of (a) the NP part, (b) $D_{\omega \omega '}$, and (c) $\alpha _{\omega \omega '}^{rr}$, (d) $\beta _{\omega \omega '}^{ll}$ of the two transmitted photons as functions of $\omega,\omega '$ with $U=10, \delta =0, \epsilon =0.1, (\gamma _1,\gamma _2)=(0.6,0.4)$. All parameters are in units of $\Gamma$.
In Fig. 3(b-c), we have set $E=U=2\delta$ for the red solid line and $E\neq U=2\delta$ for green dashed line, $E=U\neq 2\delta$ for blue dotted line, where all $\delta \neq 0$. It can be found that the maximum value of the IP part is achieved at $E=2\delta =U$, i.e., the center frequency of each photon is far away from the resonator mode, the sum of the centers of the two photons is equal to the energy of the resonator containing two photons and the total energy of the incident photon satisfies the two-photon resonance. When the nonlinear strength $U$ is in the order of magnitude of $\Gamma$, e.g. panel (b), the IP part has its maximum value around the $(\omega,\omega ')=(1,1)$ where the incident state also achieves its maximum value, but the width of the IP part is larger than that of the incident state. When the nonlinear strength $U$ is much larger than $\Gamma$, e.g. panel (c), the bound state achieves its maximum value around the $(\omega,\omega ')=(0,10), (10,0)$ however, the incident wave achieves its maximum value at $(\omega,\omega ')=(5,5)$. Hence, the IP part gives rise to a redistribution of the energy of the photons. It is known in the previous section that the propagating single photon with the center around $\delta >\Gamma$ and width $\epsilon \ll \Gamma$ gets nearly total transmitted, in other word, the NP part is the domination of the amplitude of the both photons transmitted, however, the IP part plays the important role in the amplitude of the both photons reflected [73,89,90].
To show the effect of the asymmetric coupling on the scattering wave of the two photons coming from the left or right of the nonlinear resonator, we have plotted the norm square of the amplitudes of $\alpha _{\omega \omega '}^{rr}$ and $\beta _{\omega \omega '}^{ll}$ as functions of $\gamma _2$ by fixed $U=10, \epsilon =0.1$ at the single photon resonance $\omega =\omega '=\delta =0$ and the two-photon resonance $(\omega,\omega ')=(0,10)$ or $(0,10), \delta =5$ as well as the two reflected photons as functions of $\gamma _2$ in Fig. 5. The asymmetric light transmission can be obtained as long as $\gamma _1\neq \gamma _2$. At the single photon resonance in Fig. 5(a), the probabilities for two transmitted photons first decreases as $\gamma _2$ increases from 0 to $0.5$, then they increases as $\gamma _2$ increases, they only differ each other in magnitude, we note that this phenomenon also appears when the position of the maximum value of the IP part is coincidence with the center of the incident wave. At the two-photon resonance in Fig. 5(b), the probabilities for two transmitted photons $\alpha _{\omega \omega '}^{rr}$ incident from the left of the nonlinear resonator decreases as $\gamma _2$ increases, but the probabilities for two transmitted photons $\beta _{\omega \omega '}^{ll}$ incident from the right side increases as $\gamma _2$ increases. The monotonicity in Fig. 5(b) is produced merely by the IP part because the incident photons are localized around $(\omega,\omega ')=(5,5)$ under the condition $\epsilon \ll \Gamma < \delta$. Since the asymmetric coupling permits unidirectional propagation of light and forbids the transmission in the reverse direction, the nonlinear resonator accomplishes the diode activity for two identical photons at the two-photon resonance with each energy far away from the incident center.
Fig. 5. The norm square of the amplitudes of $\alpha _{\omega \omega '}^{rr}$ and $\beta _{\omega \omega '}^{ll}$ as functions of $\gamma _2$ for (a) $\omega =\omega '=\delta =0$, (b) $(\omega,\omega ')=(0,10)$ or $(10,0), \delta =5$. (c) The probability for the two reflected photons as functions of $\gamma _2$. The other parameters are set as follow: $U=10, \epsilon =0.1$. All parameters are in units of $\Gamma$.
5. Conclusion
In conclusion, we have studied the coherent transport of one or two photons in a 1D waveguide asymmetrically coupled to a nonlinear resonator. We first construct the exact single-photon scattering states after a long time, and find that the asymmetric coupling leads to a phase difference on the amplitude of transmittance, a non-reciprocal phase shifter can be formed by the nonlinear resonator. Later, an analytic solution of two-photon scattering is derived, it is found that 1) the two-photon scattering state incident from either the left side or the right side of the nonlinear resonator contains the contribution of the NP part and the IP part. The different phases of transmission amplitude caused by the chiral coupling produce the different interference pattern of two-photon transmission at single-photon resonance. 2) The forward and reverse transmitted probabilities change with the asymmetric coupling. The forward and reverse transmitted probabilities have the same monotonicity as the asymmetric coupling changes and only differ each other in the magnitude as the position of the maximum value of the IP part is coincidence with the center of the incident wave. However, the forward and reverse transmitted probabilities have different monotonicity under the condition $\epsilon \ll \Gamma < \delta$ when the position of the maximum value of the IP part is far away from the center of the incident wave, so a nonreciprocal two-photon transmission through a nonlinear resonator is found due to the energy redistribution of the two photons by the bound state, which indicates that the nonlinear resonator acts as an optical diode. Two-photon optical diode provides us a controlled generation of chiral heralded single photons [94], which are crucial resources for various quantum information processing [95,96] and quantum metrology application [97,98]. Moreover, the strongly photon-photon interaction in the chiral waveguide could be used to perform quantum simulations of chiral strong correlations and quantum phase transitions in many-body physics [99,100].
Funding
National Natural Science Foundation of China (11935006, 11975095, 12075082, 12064010); Natural Science Foundation of Hunan Province (2021JJ20036); Science and Technology Program of Hunan Province (2020RC4047).
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. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is — and what is not — an optical isolator,” Nat. Photonics 7(8), 579–582 (2013). [CrossRef]
2. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011). [CrossRef]
3. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An All-Silicon Passive Optical Diode,” Science 335(6067), 447–450 (2012). [CrossRef]
4. Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical Nonreciprocity in Asymmetric Optomechanical Couplers,” Sci. Rep. 5(1), 8657 (2015). [CrossRef]
5. A. S. Zheng, G. Y. Zhang, H. Y. Chen, T. T. Mei, and J. B. Liu, “Nonreciprocal light propagation in coupled microcavities system beyond weak-excitation approximation,” Sci. Rep. 7(1), 14001 (2017). [CrossRef]
6. 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]
7. Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017). [CrossRef]
8. L. N. Song, Z. H. Wang, and Y. Li, “Enhancing optical nonreciprocity by an atomic ensemble in two coupled cavities,” Opt. Commun. 415, 39–42 (2018). [CrossRef]
9. X. Zhang, Q. T. Cao, Z. Wang, Y. X. Liu, C. W. Qiu, L. Yang, Q. Gong, and Y. F. Xiao, “Symmetry-breaking-induced nonlinear optics at a microcavity surface,” Nat. Photonics 13(1), 21–24 (2019). [CrossRef]
10. S. R. K. Rodriguez, V. Goblot, N. C. Zambon, A. Amo, and J. Bloch, “Nonreciprocity and zero reflection in nonlinear cavities with tailored loss,” Phys. Rev. A 99(1), 013851 (2019). [CrossRef]
11. P. Yang, X. Xia, H. He, S. Li, X. Han, P. Zhang, G. Li, P. Zhang, J. Xu, Y. Yang, and T. Zhang, “Realization of Nonlinear Optical Nonreciprocity on a Few-Photon Level Based on Atoms Strongly Coupled to an Asymmetric Cavity,” Phys. Rev. Lett. 123(23), 233604 (2019). [CrossRef]
12. 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]
13. D. Roy, “Few-photon optical diode,” Phys. Rev. B 81(15), 155117 (2010). [CrossRef]
14. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016). [CrossRef]
15. 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]
16. Z. Wang, L. Du, Y. Li, and Y. X. Liu, “Phase-controlled single-photon nonreciprocal transmission in a one-dimensional waveguide,” Phys. Rev. A 100(5), 053809 (2019). [CrossRef]
17. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical Nonreciprocity in Optomechanical Structures,” Phys. Rev. Lett. 102(21), 213903 (2009). [CrossRef]
18. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672 (2012). [CrossRef]
19. X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91(5), 053854 (2015). [CrossRef]
20. A. Metelmann and A. A. Clerk, “Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering,” Phys. Rev. X 5(2), 021025 (2015). [CrossRef]
21. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2(7), 635 (2015). [CrossRef]
22. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017). [CrossRef]
23. Z. Shen, Y. L. Zhang, Y. Chen, C. L. Zou, Y. F. Xiao, X. B. Zou, F. W. Sun, G. C. Guo, and C. H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016). [CrossRef]
24. F. Ruesink, M. A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016). [CrossRef]
25. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017). [CrossRef]
26. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Phonon Laser,” Phys. Rev. Appl. 10(6), 064037 (2018). [CrossRef]
27. D. G. Lai, J. F. Huang, X. L. Yin, B. P. Hou, W. Li, D. Vitali, F. Nori, and J. Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020). [CrossRef]
28. Y. Chen, Y. L. Zhang, Z. Shen, C. L. Zou, G. C. Guo, and C. H. Dong, “Synthetic Gauge Fields in a Single Optomechanical Resonator,” Phys. Rev. Lett. 126(12), 123603 (2021). [CrossRef]
29. D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110(9), 093901 (2013). [CrossRef]
30. S. A. R. Horsley, J. H. Wu, M. Artoni, and G. C. La Rocca, “Optical Nonreciprocity of Cold Atom Bragg Mirrors in Motion,” Phys. Rev. Lett. 110(22), 223602 (2013). [CrossRef]
31. K. Xia, F. Nori, and M. Xiao, “Cavity-Free Optical Isolators and Circulators Using a Chiral Cross-Kerr Nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018). [CrossRef]
32. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12(12), 744–748 (2018). [CrossRef]
33. E. Z. Li, D. S. Ding, Y. C. Yu, M. X. Dong, L. Zeng, W. H. Zhang, Y. H. Ye, H. Z. Wu, Z. H. Zhu, W. Gao, G. C. Guo, and B. S. Shi, “Experimental demonstration of cavity-free optical isolators and optical circulators,” Phys. Rev. Res. 2(3), 033517 (2020). [CrossRef]
34. C. Liang, B. Liu, A. N. Xu, X. Wen, C. Lu, K. Xia, M. K. Tey, Y. C. Liu, and L. You, “Collision-Induced Broadband Optical Nonreciprocity,” Phys. Rev. Lett. 125(12), 123901 (2020). [CrossRef]
35. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
36. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of Asymmetric Transport in Structures with Active Nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013). [CrossRef]
37. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
38. L. Chang, X. S. Jiang, S. Y. Hua, C. Yang, J. M. Wen, L. Jiang, G. Y. Li, G. Z. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014). [CrossRef]
39. R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121(15), 153601 (2018). [CrossRef]
40. B. Li, R. Huang, X. W. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630 (2019). [CrossRef]
41. K. Wang, Q. Wu, Y. F. Yu, and Z. M. Zhang, “Nonreciprocal photon blockade in a two-mode cavity with a second-order nonlinearity,” Phys. Rev. A 100(5), 053832 (2019). [CrossRef]
42. H. Z. Shen, Q. Wang, J. Wang, and X. X. Yi, “Nonreciprocal unconventional photon blockade in a driven dissipative cavity with parametric amplification,” Phys. Rev. A 101(1), 013826 (2020). [CrossRef]
43. X. W. Xu, Y. J. Zhao, H. Wang, H. Jing, and A. X. Chen, “Quantum nonreciprocality in quadratic optomechanics,” Photonics Res. 8(2), 143 (2020). [CrossRef]
44. X. W. Xu, Y. Li, B. Li, H. Jing, and A. X. Chen, “Nonreciprocity via Nonlinearity and Synthetic Magnetism,” Phys. Rev. Appl. 13(4), 044070 (2020). [CrossRef]
45. Y. F. Jiao, S. D. Zhang, Y. L. Zhang, A. Miranowicz, L. M. Kuang, and H. Jing, “Nonreciprocal Optomechanical Entanglement against Backscattering Losses,” Phys. Rev. Lett. 125(14), 143605 (2020). [CrossRef]
46. F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017). [CrossRef]
47. L. Zhou, S. Yang, Y. X. Liu, C. P. Sun, and F. Nori, “Quantum Zeno switch for single-photon coherent transport,” Phys. Rev. A 80(6), 062109 (2009). [CrossRef]
48. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3(11), 807–812 (2007). [CrossRef]
49. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable Scattering of a Single Photon inside a One-Dimensional Resonator Waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]
50. J. T. Shen and S. H. Fan, “Strongly Correlated Two-Photon Transport in a One-Dimensional Waveguide Coupled to a Two-Level System,” Phys. Rev. Lett. 98(15), 153003 (2007). [CrossRef]
51. J. T. Shen and S. H. Fan, “Strongly correlated multiparticle transport in one dimension through a quantum impurity,” Phys. Rev. A 76(6), 062709 (2007). [CrossRef]
52. S. Xu and S. H. Fan, “Fano interference in two-photon transport,” Phys. Rev. A 94(4), 043826 (2016). [CrossRef]
53. D. Roy, “Two-Photon Scattering by a Driven Three-Level Emitter in a One-Dimensional Waveguide and Electromagnetically Induced Transparency,” Phys. Rev. Lett. 106(5), 053601 (2011). [CrossRef]
54. H. X. Zheng, D. J. Gauthier, and H. U. Baranger, “Cavity-Free Photon Blockade Induced by Many-Body Bound States,” Phys. Rev. Lett. 107(22), 223601 (2011). [CrossRef]
55. H. X. Zheng, D. J. Gauthier, and H. U. Baranger, “Strongly correlated photons generated by coupling a three- or four-level system to a waveguide,” Phys. Rev. A 85(4), 043832 (2012). [CrossRef]
56. T. Y. Li, J. F. Huang, and C. K. Law, “Scattering of two distinguishable photons by a $\Xi$-type three-level atom in a one-dimensional waveguide,” Phys. Rev. A 91(4), 043834 (2015). [CrossRef]
57. Y. Pan and G. Zhang, “Scattering of few photons by a ladder-type quantum system,” J. Phys. A: Math. Theor. 50(34), 345301 (2017). [CrossRef]
58. 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]
59. J. Dai, A. Roulet, H. N. Le, and V. Scarani, “Rectification of light in the quantum regime,” Phys. Rev. A: At., Mol., Opt. Phys. 92(6), 063848 (2015). [CrossRef]
60. A. RosarioHamann, C. Muller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity Realized with Quantum Nonlinearity,” Phys. Rev. Lett. 121(12), 123601 (2018). [CrossRef]
61. C. Muller, J. Combes, A. R. Hamann, A. Fedorov, and T. M. Stace, “Nonreciprocal atomic scattering: A saturable, quantum Yagi-Uda antenna,” Phys. Rev. A: At., Mol., Opt. Phys. 96(5), 053817 (2017). [CrossRef]
62. E. Rephaeli, S. E. Kocabas, and S. H. Fan, “Few-photon transport in a waveguide coupled to a pair of colocated two-level atoms,” Phys. Rev. A 84(6), 063832 (2011). [CrossRef]
63. D. Roy, “Two-photon scattering of a tightly focused weak light beam from a small atomic ensemble: An optical probe to detect atomic level structures,” Phys. Rev. A 87(6), 063819 (2013). [CrossRef]
64. D. Roy, “Cascaded two-photon nonlinearity in a one-dimensional waveguide with multiple two-level emitters,” Sci. Rep. 3(1), 2337 (2013). [CrossRef]
65. J. F. Huang, J. Q. Liao, and C. P. Sun, “Photon blockade induced by atoms with Rydberg coupling,” Phys. Rev. A 87(2), 023822 (2013). [CrossRef]
66. T. Shi, S. H. Fan, and C. P. Sun, “Two-photon transport in a waveguide coupled to a cavity in a two-level system,” Phys. Rev. A 84(6), 063803 (2011). [CrossRef]
67. T. Shi and S. H. Fan, “Two-photon transport through a waveguide coupling to a whispering-gallery resonator containing an atom and photon-blockade effect,” Phys. Rev. A 87(6), 063818 (2013). [CrossRef]
68. K. B. Joanesarson, J. Iles-Smith, M. Heuck, and J. Mork, “Few-photon transport in Fano-resonance waveguide geometries,” Phys. Rev. A 101(6), 063809 (2020). [CrossRef]
69. W. B. Yan, Q. B. Fan, and L. Zhou, “Control of correlated two-photon transport in a one-dimensional waveguide,” Phys. Rev. A 85(1), 015803 (2012). [CrossRef]
70. W. Z. Jia and Z. D. Wang, “Single-photon transport in a one-dimensional waveguide coupling to a hybrid atom-optomechanical system,” Phys. Rev. A 88(6), 063821 (2013). [CrossRef]
71. L. Qiao, “Single-photon transport through a waveguide coupling to a quadratic optomechanical system,” Phys. Rev. A 96(1), 013860 (2017). [CrossRef]
72. J. Liu, W. Zhang, X. Li, W. Yan, and L. Zhou, “The Correlated Two-Photon Transport in a One-Dimensional Waveguide Coupling to a Hybrid Atom-Optomechanical System,” Int. J. Theor. Phys. 55(10), 4620–4630 (2016). [CrossRef]
73. J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82(5), 053836 (2010). [CrossRef]
74. X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90(3), 033832 (2014). [CrossRef]
75. P. Longo, P. Schmitteckert, and K. Busch, “Few-Photon Transport in Low-Dimensional Systems: Interaction-Induced Radiation Trapping,” Phys. Rev. Lett. 104(2), 023602 (2010). [CrossRef]
76. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009). [CrossRef]
77. Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental Realization of Self-Guiding Unidirectional Electromagnetic Edge States,” Phys. Rev. Lett. 106(9), 093903 (2011). [CrossRef]
78. S. A. Skirlo, L. Lu, Y. Igarashi, Q. Yan, J. Joannopoulos, and M. Soljacic, “Experimental Observation of Large Chern Numbers in Photonic Crystals,” Phys. Rev. Lett. 115(25), 253901 (2015). [CrossRef]
79. T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum Spin Dimers from Chiral Dissipation in Cold-Atom Chains,” Phys. Rev. Lett. 113(23), 237203 (2014). [CrossRef]
80. 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]
81. R. Jones, G. Buonaiuto, B. Lang, I. Lesanovsky, and B. Olmos, “Collectively Enhanced Chiral Photon Emission from an Atomic Array near a Nanofiber,” Phys. Rev. Lett. 124(9), 093601 (2020). [CrossRef]
82. P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, P. J. J. O’Malley, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, E. Kapit, H. Neven, and J. Martinis, “Chiral ground-state currents of interacting photons in a synthetic magnetic field,” Nat. Phys. 13(2), 146–151 (2017). [CrossRef]
83. E. Sanchez-Burillo, C. Wan, D. Zueco, and A. Gonzalez-Tudela, “Chiral quantum optics in photonic sawtooth lattices,” Phys. Rev. Res. 2(2), 023003 (2020). [CrossRef]
84. 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]
85. L. Du, Y. Zhang, J.-H. Wu, A. F. Kockum, and Y. Li, “Giant Atoms in a Synthetic Frequency Dimension,” Phys. Rev. Lett. 128(22), 223602 (2022). [CrossRef]
86. A. Seif, W. DeGottardi, K. Esfarjani, and M. Hafezi, “Thermal management and non-reciprocal control of phonon flow via optomechanics,” Nat. Commun. 9(1), 1207 (2018). [CrossRef]
87. X.-L. Dong, P.-B. Li, T. Liu, and F. Nori, “Unconventional Quantum Sound-Matter Interactions in Spin-Optomechanical-Crystal Hybrid Systems,” Phys. Rev. Lett. 126(20), 203601 (2021). [CrossRef]
88. X.-L. Dong, C.-P. Shen, S.-Y. Gao, H.-R. Li, H. Gao, F.-L. Li, and P.-B. Li, “Chiral spin-phonon bound states and spin-spin interactions with phononic lattices,” Phys. Rev. Res. 4(2), 023077 (2022). [CrossRef]
89. J. T. Shen and S. Fan, “Coherent Single Photon Transport in a One-Dimensional Waveguide Coupled with Superconducting Quantum Bits,” Phys. Rev. Lett. 95(21), 213001 (2005). [CrossRef]
90. J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30(15), 2001 (2005). [CrossRef]
91. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62(5), 7389–7404 (2000). [CrossRef]
92. J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79(2), 023837 (2009). [CrossRef]
93. J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. II. Coupling to a whispering-gallery resonator containing a two-level atom,” Phys. Rev. A 79(2), 023838 (2009). [CrossRef]
94. S. A. Castelletto and R. E. Scholten, “Heralded single photon sources: a route towards quantum communication technology and photon standards,” Eur. Phys. J. Appl. Phys. 41(3), 181–194 (2008). [CrossRef]
95. N. A. Peters, K. J. Arnold, A. P. VanDevender, E. R. Jeffrey, R. Rangarajan, O. Hosten, J. T. Barreiro, J. B. Altepeter, and P. G. Kwiat, “Toward a quasi-deterministic single-photon source,” Proc. SPIE 6305, 630507 (2006). [CrossRef]
96. S. Buckley, K. Rivoire, and J. Vuckovic, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75(12), 126503 (2012). [CrossRef]
97. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum Metrology,” Phys. Rev. Lett. 96(1), 010401 (2006). [CrossRef]
98. E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, “Photonic quantum metrology,” AVS Quantum Sci. 2(2), 024703 (2020). [CrossRef]
99. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]
100. J. Tangpanitanon and D. G. Angelakis, “Many-Body Physics and Quantum Simulations with Strongly Interacting Photons,” arXiv:1907.05030 [quant-ph].
