Deák, L. & Fülöp, T. Reciprocity in quantum, electromagnetic and other wave scattering. Ann. Phys. 327, 1050 https://doi.org/10.1016/j.aop.2011.10.013 (2012).

Article 
ADS 
MathSciNet 

Google Scholar
 

Jalas, D. et al. What is — and what is not — an optical isolator. Nat. Photon. 7, 579 https://doi.org/10.1038/nphoton.2013.185 (2013).

Article 
ADS 

Google Scholar
 

Caloz, C. et al. Electromagnetic nonreciprocity. Phys. Rev. Appl. 10, 047001 https://doi.org/10.1103/PhysRevApplied.10.047001 (2018).

Article 
ADS 

Google Scholar
 

Yao, N. Y. et al. Topologically protected quantum state transfer in a chiral spin liquid. Nat. Commun. 4, 1585 https://doi.org/10.1038/ncomms2531 (2013).

Article 
ADS 

Google Scholar
 

Lemonde, M.-A., Peano, V., Rabl, P. & Angelakis, D. G. Quantum state transfer via acoustic edge states in a 2D optomechanical array. N. J. Phys. 21, 113030 https://doi.org/10.1088/1367-2630/ab51f5 (2019).

Article 
MathSciNet 

Google Scholar
 

Owens, J. et al. Chiral cavity quantum electrodynamics. Nat. Phys. 18, 1048–1052 https://doi.org/10.1038/s41567-022-01671-3 (2022).

Article 

Google Scholar
 

Lodahl, P. et al. Chiral quantum optics. Nature 541, 473 https://doi.org/10.1038/nature21037 (2017).

Article 
ADS 

Google Scholar
 

De Bernardis, D., Piccioli, F. S., Rabl, P. & Carusotto, I. Chiral quantum optics in the bulk of photonic quantum hall systems. PRX Quantum 4, 030306 https://doi.org/10.1103/PRXQuantum.4.030306 (2023).

Article 

Google Scholar
 

Kamal, A., Clarke, J. & Devoret, M. H. Noiseless non-reciprocity in a parametric active device. Nat. Phys. 7, 311 https://doi.org/10.1038/nphys1893 (2011).

Article 

Google Scholar
 

Lau, H.-K. & Clerk, A. A. Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing. Nat. Commun. 9, 4320 https://doi.org/10.1038/s41467-018-06477-7 (2018).

Article 
ADS 

Google Scholar
 

Chen, C., Jin, L. & Liu, R.-B. Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system. N. J. Phys. 21, 083002 https://doi.org/10.1088/1367-2630/ab32ab (2019).

Article 
MathSciNet 

Google Scholar
 

McDonald, A. & Clerk, A. A. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat. Commun. 11, 5382 https://doi.org/10.1038/s41467-020-19090-4 (2020).

Article 
ADS 

Google Scholar
 

Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907 https://doi.org/10.1038/nphys2063 (2011).

Article 

Google Scholar
 

Ringel, M., Pletyukhov, M. & Gritsev, V. Topologically protected strongly correlated states of photons. N. J. Phys. 16, 113030 https://doi.org/10.1088/1367-2630/16/11/113030 (2014).

Article 

Google Scholar
 

Peano, V., Houde, M., Marquardt, F. & Clerk, A. A. Topological quantum fluctuations and traveling wave amplifiers. Phys. Rev. X 6, 041026 https://doi.org/10.1103/PhysRevX.6.041026 (2016).

Article 

Google Scholar
 

Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 https://doi.org/10.1103/RevModPhys.91.015006 (2019).

Article 
ADS 
MathSciNet 

Google Scholar
 

Wang, Y.-P. et al. Nonreciprocity and unidirectional invisibility in cavity magnonics. Phys. Rev. Lett. 123, 127202 https://doi.org/10.1103/PhysRevLett.123.127202 (2019).

Article 
ADS 

Google Scholar
 

Zhu, N., Han, X., Zou, C.-L., Xu, M. & Tang, H. X. Magnon–photon strong coupling for tunable microwave circulators. Phys. Rev. A 101, 043842 https://doi.org/10.1103/PhysRevA.101.043842 (2020).

Article 
ADS 

Google Scholar
 

Sounas, D. L. & Alù, A. Non-reciprocal photonics based on time modulation. Nat. Photon. 11, 774 https://doi.org/10.1038/s41566-017-0051-x (2017).

Article 
ADS 

Google Scholar
 

Estep, N. A., Sounas, D. L., Soric, J. & Alú, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nat. Phys. 10, 923 https://doi.org/10.1038/nphys3134 (2014).

Article 

Google Scholar
 

Tzuang, L. D., Fang, K., Nussenzveig, P., Fan, S. & Lipson, M. Non-reciprocal phase shift induced by an effective magnetic flux for light. Nat. Photon. 8, 701 https://doi.org/10.1038/nphoton.2014.177 (2014).

Article 
ADS 

Google Scholar
 

Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782 https://doi.org/10.1038/nphoton.2012.236 (2012).

Article 
ADS 

Google Scholar
 

Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys. 13, 146–151 https://doi.org/10.1038/nphys3930 (2017).

Article 

Google Scholar
 

Rosen, I. et al. A synthetic magnetic vector potential in a 2D superconducting qubit array. Nat. Phys. 20, 1881–1887 https://doi.org/10.1038/s41567-024-02661-3 (2024).

Article 

Google Scholar
 

Regensburger, A. et al. Parity-time synthetic photonic lattices. Nature 488, 167 https://doi.org/10.1038/nature11298 (2012).

Article 
ADS 

Google Scholar
 

Miri, M.-A. & Alú, A. Exceptional points in optics and photonics. Science 363, eaar7709 https://doi.org/10.1126/science.aar7709 (2019).

Article 
MathSciNet 

Google Scholar
 

Hua, S. et al. Demonstration of a chip-based optical isolator with parametric amplification. Nat. Commun. 7, 13657 https://doi.org/10.1038/ncomms13657 (2016).

Article 
ADS 

Google Scholar
 

Huang, X., Lu, C., Liang, C., Tao, H. & Liu, Y.-C. Loss-induced nonreciprocity. Light Sci. Appl. 10, 30 https://doi.org/10.1038/s41377-021-00464-2 (2021).

Article 
ADS 

Google Scholar
 

Barzanjeh, S. et al. Mechanical on-chip microwave circulator. Nat. Commun. 8, 953 https://doi.org/10.1038/s41467-017-01304-x (2017).

Article 
ADS 

Google Scholar
 

Bernier, N. R. et al. Nonreciprocal reconfigurable microwave optomechanical circuit. Nat. Commun. 8, 604 https://doi.org/10.1038/s41467-017-00447-1 (2017).

Article 
ADS 

Google Scholar
 

Malz, D. et al. Quantum-limited directional amplifiers with optomechanics. Phys. Rev. Lett. 120, 023601 https://doi.org/10.1103/PhysRevLett.120.023601 (2018).

Article 
ADS 

Google Scholar
 

Mirhosseini, M., Sipahigil, A., Kalaee, M. & Painter, O. Superconducting qubit to optical photon transduction. Nature 588, 599 https://doi.org/10.1038/s41586-020-3038-6 (2020).

Article 
ADS 

Google Scholar
 

Kamal, A., Roy, A., Clarke, J. & Devoret, M. H. Asymmetric frequency conversion in nonlinear systems driven by a biharmonic pump. Phys. Rev. Lett. 113, 247003 https://doi.org/10.1103/PhysRevLett.113.247003 (2014).

Article 
ADS 

Google Scholar
 

Suárez-Forero, D., Jalali Mehrabad, M., Vega, C., González-Tudela, A. & Hafezi, M. Chiral quantum optics: recent developments and future directions. PRX Quantum 6, 020101 https://doi.org/10.1103/PRXQuantum.6.020101 (2025).

Article 
ADS 

Google Scholar
 

Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391 https://doi.org/10.1103/RevModPhys.86.1391 (2014).

Article 
ADS 

Google Scholar
 

Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 https://doi.org/10.1103/PhysRevA.69.062320 (2004).

Article 
ADS 

Google Scholar
 

Gu, X., Kockum, A. F., Miranowicz, A., xi Liu, Y. & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718–719, 1 https://doi.org/10.1016/j.physrep.2017.10.002 (2017).

Article 
ADS 
MathSciNet 

Google Scholar
 

Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 https://doi.org/10.1103/RevModPhys.93.025005 (2021).

Article 
ADS 
MathSciNet 

Google Scholar
 

Zhang, X., Galda, A., Han, X., Jin, D. & Vinokur, V. M. Broadband nonreciprocity enabled by strong coupling of magnons and microwave photons. Phys. Rev. Appl. 13, 044039 https://doi.org/10.1103/PhysRevApplied.13.044039 (2020).

Article 
ADS 

Google Scholar
 

Kim, M., Tabesh, A., Zegray, T., Barzanjeh, S. & Hu, C.-M. Nonreciprocity in cavity magnonics at millikelvin temperature. J. Appl. Phys. 135, 063904 https://doi.org/10.1063/5.0176462 (2024).

Article 
ADS 

Google Scholar
 

Koch, J., Houck, A. A., Hur, K. L. & Girvin, S. M. Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A 82, 043811 https://doi.org/10.1103/PhysRevA.82.043811 (2010).

Article 
ADS 

Google Scholar
 

Navarathna, R. et al. Passive superconducting circulator on a chip. Phys. Rev. Lett. 130, 037001 https://doi.org/10.1103/PhysRevLett.130.037001 (2023).

Article 
ADS 
MathSciNet 

Google Scholar
 

Landgraf, J., Peano, V. & Marquardt, F. Automated discovery of coupled-mode setups. Phys. Rev. X 15, 021038 https://doi.org/10.1103/PhysRevX.15.021038 (2025).

Article 

Google Scholar
 

Metelmann, A. & Clerk, A. A. Nonreciprocal photon transmission and amplification via reservoir engineering. Phys. Rev. X 5, 021025 https://doi.org/10.1103/PhysRevX.5.021025 (2015).

Article 

Google Scholar
 

Metelmann, A. & Clerk, A. A. Quantum-limited amplification via reservoir engineering. Phys. Rev. Lett. 112, 133904 https://doi.org/10.1103/PhysRevLett.112.133904 (2014).

Article 
ADS 

Google Scholar
 

Wang, Y.-X. & Clerk, A. A. Non-Hermitian dynamics without dissipation in quantum systems. Phys. Rev. A 99, 063834 https://doi.org/10.1103/PhysRevA.99.063834 (2019).

Article 
ADS 

Google Scholar
 

Kerckhoff, J., Lalumière, K., Chapman, B. J., Blais, A. & Lehnert, K. W. On-chip superconducting microwave circulator from synthetic rotation. Phys. Rev. Appl. 4, 034002 https://doi.org/10.1103/PhysRevApplied.4.034002 (2015).

Article 
ADS 

Google Scholar
 

Rosenthal, E. I., Chapman, B. J., Higginbotham, A. P., Kerckhoff, J. & Lehnert, K. W. Breaking Lorentz reciprocity with frequency conversion and delay. Phys. Rev. Lett. 119, 147703 https://doi.org/10.1103/PhysRevLett.119.147703 (2017).

Article 
ADS 

Google Scholar
 

Lecocq, F. et al. Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier. Phys. Rev. Appl. 7, 024028 https://doi.org/10.1103/PhysRevApplied.7.024028 (2017).

Article 
ADS 

Google Scholar
 

Abdo, B., Sliwa, K., Frunzio, L. & Devoret, M. Directional amplification with a Josephson circuit. Phys. Rev. X 3, 031001 https://doi.org/10.1103/PhysRevX.3.031001 (2013).

Article 

Google Scholar
 

Fang, K. et al. Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering. Nat. Phys. 13, 465 https://doi.org/10.1038/nphys4009 (2017).

Article 

Google Scholar
 

Ranzani, L. & Aumentado, J. Graph-based analysis of nonreciprocity in coupled-mode systems. N. J. Phys. 17, 023024 https://doi.org/10.1088/1367-2630/17/2/023024 (2015).

Article 

Google Scholar
 

Sliwa, K. M. et al. Reconfigurable Josephson circulator/directional amplifier. Phys. Rev. X 5, 041020 https://doi.org/10.1103/PhysRevX.5.041020 (2015).

Article 

Google Scholar
 

Naaman, O. & Aumentado, J. Synthesis of parametrically coupled networks. PRX Quantum 3, 020201 https://doi.org/10.1103/PRXQuantum.3.020201 (2022).

Article 
ADS 

Google Scholar
 

Kwende, R., White, T. & Naaman, O. Josephson parametric circulator with same-frequency signal ports, 200 mHz bandwidth, and high dynamic range. Appl. Phys. Lett. 122, 224001 https://doi.org/10.1063/5.0150427 (2023).

Article 
ADS 

Google Scholar
 

Carmichael, H. J. Quantum trajectory theory for cascaded open systems. Phys. Rev. Lett. 70, 2273 https://doi.org/10.1103/PhysRevLett.70.2273 (1993).

Article 
ADS 

Google Scholar
 

Stannigel, K., Rabl, P. & Zoller, P. Driven-dissipative preparation of entangled states in cascaded quantum-optical networks. N. J. Phys. 14, 06301 https://doi.org/10.1088/1367-2630/14/6/063014 (2012).

Article 

Google Scholar
 

Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 https://doi.org/10.1103/PhysRevX.5.031011 (2015).

Article 

Google Scholar
 

Hafezi, M. & Rabl, P. Optomechanically induced non-reciprocity in microring resonators. Opt. Express 20, 7672 https://doi.org/10.1364/OE.20.007672 (2012).

Article 
ADS 

Google Scholar
 

Ruesink, F., Miri, M.-A., Alù, A. & Verhagen, E. Nonreciprocity and magnetic-free isolation based on optomechanical interactions. Nat. Commun. 7, 13662 https://doi.org/10.1038/ncomms13662 (2016).

Article 
ADS 

Google Scholar
 

Kim, J., Kuzyk, M. C., Han, K., Wang, H. & Bahl, G. Non-reciprocal Brillouin scattering induced transparency. Nat. Phys. 11, 275 https://doi.org/10.1038/nphys3236 (2015).

Article 

Google Scholar
 

Shen, Z. et al. Experimental realization of optomechanically induced non-reciprocity. Nat. Photon. 10, 657 https://doi.org/10.1038/nphoton.2016.161 (2016).

Article 
ADS 

Google Scholar
 

Wanjura, C. C. et al. Quadrature nonreciprocity in bosonic networks without breaking time-reversal symmetry. Nat. Phys. 19, 1429 https://doi.org/10.1038/s41567-023-02128-x (2023).

Article 

Google Scholar
 

Slim, J. J. et al. Optomechanical realization of the bosonic Kitaev chain. Nature 627, 767 https://doi.org/10.1038/s41586-024-07174-w (2024).

Article 
ADS 

Google Scholar
 

Busnaina J. H. et al. Quantum simulation of the bosonic Kitaev chain. Nat. Commun. 15, 3065 https://doi.org/10.1038/s41467-024-47186-8 (2024).

Article 
ADS 

Google Scholar
 

Söllner, I. et al. Deterministic photon-emitter coupling in chiral photonic circuits. Nat. Nanotechnol. 10, 775–778 https://doi.org/10.1038/nnano.2015.159 (2015).

Article 
ADS 

Google Scholar
 

Petersen, J., Volz, J. & Rauschenbeutel, A. Chiral nanophotonic waveguide interface based on spin–orbit interaction of light. Science 346, 67 https://doi.org/10.1126/science.1257671 (2014).

Article 
ADS 

Google Scholar
 

Mitsch, R., Sayrin, C., Albrecht, B., Schneeweiss, P. & Rauschenbeutel, A. Quantum state-controlled directional spontaneous emission of photons into a nanophotonic waveguide. Nat. Commun. 5, 5713 https://doi.org/10.1038/ncomms6713 (2014).

Article 
ADS 

Google Scholar
 

Scheucher, M., Hilico, A., Will, E., Volz, J. & Rauschenbeutel, A. Quantum optical circulator controlled by a single chirally coupled atom. Science 354, 1577 https://doi.org/10.1126/science.aaj2118 (2016).

Article 
ADS 

Google Scholar
 

Potton, R. J. Reciprocity in optics. Rep. Prog. Phys. 67, 717 https://doi.org/10.1088/0034-4885/67/5/R03 (2004).

Article 
ADS 

Google Scholar
 

Fratini, F. et al. Fabry–Perot interferometer with quantum mirrors: nonlinear light transport and rectification. Phys. Rev. Lett. 113, 243601 https://doi.org/10.1103/PhysRevLett.113.243601 (2014).

Article 
ADS 

Google Scholar
 

Müller, C., Combes, J., Hamann, A. R., Fedorov, A. & Stace, T. M. Nonreciprocal atomic scattering: a saturable, quantum yagi-uda antenna. Phys. Rev. A 96, 053817 https://doi.org/10.1103/PhysRevA.96.053817 (2017).

Article 
ADS 

Google Scholar
 

Nefedkin, N., Cotrufo, M., Krasnok, A. & Alú, A. Dark-state induced quantum nonreciprocity. Adv. Quantum Technol. 5, 2100112 https://doi.org/10.1002/qute.202100112 (2022).

Article 

Google Scholar
 

Rosario Hamann, A. et al. Nonreciprocity realized with quantum nonlinearity. Phys. Rev. Lett. 121, 123601 https://doi.org/10.1103/PhysRevLett.121.123601 (2018).

Article 
ADS 

Google Scholar
 

Rieser, J. et al. Tunable light-induced dipole–dipole interaction between optically levitated nanoparticles. Science 377, 987 https://doi.org/10.1126/science.abp9941 (2022).

Article 
ADS 
MathSciNet 

Google Scholar
 

Herrmann, J. F. et al. Mirror symmetric on-chip frequency circulation of light. Nat. Photon. 16, 603 https://doi.org/10.1038/s41566-022-01026-7 (2022).

Article 
ADS 

Google Scholar
 

Zare Rameshti, B. et al. Cavity magnonics. Phys. Rep. 979, 1 https://doi.org/10.1016/j.physrep.2022.06.001 (2022).

Article 
ADS 
MathSciNet 

Google Scholar
 

Wang, Y.-Y. et al. Low-loss ferrite circulator as a tunable chiral quantum system. Phys. Rev. Appl. 16, 064066 https://doi.org/10.1103/PhysRevApplied.16.064066 (2021).

Article 
ADS 

Google Scholar
 

Nefedkin, N., Cotrufo, M. & Alú, A. Nonreciprocal total cross section of quantum metasurfaces. Nanophotonics 12, 589 https://doi.org/10.1515/nanoph-2022-0596 (2023).

Article 
ADS 

Google Scholar
 

Bergeal, N. et al. Phase-preserving amplification near the quantum limit with a Josephson ring modulator. Nature 465, 64 https://doi.org/10.1038/nature09035 (2010).

Article 
ADS 

Google Scholar
 

Macklin, C. et al. A near-quantum-limited Josephson traveling-wave parametric amplifier. Science 350, 307 https://doi.org/10.1126/science.aaa8525 (2015).

Article 
ADS 

Google Scholar
 

Naghiloo, M., Abbasi, M., Joglekar, Y. N. & Murch, K. W. Quantum state tomography across the exceptional point in a single dissipative qubit. Nat. Phys. 15, 1232 https://doi.org/10.1038/s41567-019-0652-z (2019).

Article 

Google Scholar
 

Abdo, B. et al. Josephson directional amplifier for quantum measurement of superconducting circuits. Phys. Rev. Lett. 112, 167701 https://doi.org/10.1103/PhysRevLett.112.167701 (2014).

Article 
ADS 

Google Scholar
 

Aumentado, J. Superconducting parametric amplifiers: the state of the art in Josephson parametric amplifiers. IEEE Microw. Mag. 21, 45 https://doi.org/10.1109/MMM.2020.2993476 (2020).

Article 

Google Scholar
 

Esposito, M., Ranadive, A., Planat, L. & Roch, N. Perspective on traveling wave microwave parametric amplifiers. Appl. Phys. Lett. 119, 120501 https://doi.org/10.1063/5.0064892 (2021).

Article 
ADS 

Google Scholar
 

Ranadive, A. et al. A travelling-wave parametric amplifier isolator. Nat. Electron. 8, 1089 https://doi.org/10.1038/s41928-025-01489-w (2025).

Article 

Google Scholar
 

Abdo, B., Jinka, O., Bronn, N. T., Olivadese, S. & Brink, M. High-fidelity qubit readout using interferometric directional Josephson devices. PRX Quantum 2, 040360 https://doi.org/10.1103/PRXQuantum.2.040360 (2021).

Article 
ADS 

Google Scholar
 

Lecocq, F. et al. Efficient qubit measurement with a nonreciprocal microwave amplifier. Phys. Rev. Lett. 126, 020502 (2021).

Article 
ADS 

Google Scholar
 

Mittal, S., Goldschmidt, E. A. & Hafezi, M. A topological source of quantum light. Nature 561, 502 https://doi.org/10.1038/s41586-018-0478-3 (2018).

Article 
ADS 

Google Scholar
 

Porras, D. & Fernández-Lorenzo, S. Topological amplification in photonic lattices. Phys. Rev. Lett. 122, 143901 https://doi.org/10.1103/PhysRevLett.122.143901 (2019).

Article 
ADS 

Google Scholar
 

Wanjura, C. C., Brunelli, M. & Nunnenkamp, A. Topological framework for directional amplification in driven-dissipative cavity arrays. Nat. Commun. 11, 3149 https://doi.org/10.1038/s41467-020-16863-9 (2020).

Article 
ADS 

Google Scholar
 

Wanjura, C. C., Brunelli, M. & Nunnenkamp, A. Correspondence between non-Hermitian topology and directional amplification in the presence of disorder. Phys. Rev. Lett. 127, 213601 https://doi.org/10.1103/PhysRevLett.127.213601 (2021).

Article 
ADS 
MathSciNet 

Google Scholar
 

Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221 https://doi.org/10.1103/PhysRevLett.78.3221 (1997).

Article 
ADS 

Google Scholar
 

Gheeraert, N., Kono, S. & Nakamura, Y. Programmable directional emitter and receiver of itinerant microwave photons in a waveguide. Phys. Rev. A 102, 053720 https://doi.org/10.1103/PhysRevA.102.053720 (2020).

Article 
ADS 

Google Scholar
 

Guimond, P. et al. A unidirectional on-chip photonic interface for superconducting circuits. npj Quantum Inf. 6, 32 https://doi.org/10.1038/s41534-020-0261-9 (2020).

Article 
ADS 

Google Scholar
 

Kannan, B. et al. On-demand directional microwave photon emission using waveguide quantum electrodynamics. Nat. Phys. 19, 394 https://doi.org/10.1038/s41567-022-01869-5 (2023).

Article 

Google Scholar
 

Joshi, C., Yang, F. & Mirhosseini, M. Resonance fluorescence of a chiral artificial atom. Phys. Rev. X 13, 021039 https://doi.org/10.1103/PhysRevX.13.021039 (2023).

Article 

Google Scholar
 

Wang, X. & Li, H.-R. Chiral quantum network with giant atoms. Quantum Sci. Technol. 7, 035007 https://doi.org/10.1088/2058-9565/ac6a04 (2022).

Article 
ADS 

Google Scholar
 

Soro, A. & Kockum, A. F. Chiral quantum optics with giant atoms. Phys. Rev. A 105, 023712 https://doi.org/10.1103/PhysRevA.105.023712 (2022).

Article 
ADS 
MathSciNet 

Google Scholar
 

Chen, Y.-T. et al. Nonreciprocal and chiral single-photon scattering for giant atoms. Commun. Phys. 5, 215 https://doi.org/10.1038/s42005-022-00991-3 (2022).

Article 

Google Scholar
 

Almanakly, A. et al. Deterministic remote entanglement using a chiral quantum interconnect. Nat. Phys. 21, 825 https://doi.org/10.1038/s41567-025-02811-1 (2025).

Article 

Google Scholar
 

Chen, W., Kaya Özdemir, Ş, Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192 https://doi.org/10.1038/nature23281 (2017).

Article 
ADS 

Google Scholar
 

Mao, W., Fu, Z., Li, Y., Li, F. & Yang, L. Exceptional-point-enhanced phase sensing. Sci. Adv. 10, eadl5037 https://doi.org/10.1126/sciadv.adl5037 (2024).

Article 
ADS 

Google Scholar
 

Langbein, W. No exceptional precision of exceptional-point sensors. Phys. Rev. A 98, 023805 https://doi.org/10.1103/PhysRevA.98.023805 (2018).

Article 
ADS 

Google Scholar
 

Duggan, R., Mann, S. A. & Alù, A. Limitations of sensing at an exceptional point. ACS Photon. 9, 1554 https://doi.org/10.1021/acsphotonics.1c01535 (2022).

Article 

Google Scholar
 

Ding, W., Wang, X. & Chen, S. Fundamental sensitivity limits for non-Hermitian quantum sensors. Phys. Rev. Lett. 131, 160801 https://doi.org/10.1103/PhysRevLett.131.160801 (2023).

Article 
ADS 
MathSciNet 

Google Scholar
 

Montenegro, V. et al. Quantum metrology and sensing with many-body systems. Phys. Rep. 1134, 1 https://doi.org/10.1016/j.physrep.2025.05.005 (2025).

Article 
ADS 
MathSciNet 

Google Scholar
 

Huang, J., Zhuang, M. & Lee, C. Entanglement-enhanced quantum metrology: from standard quantum limit to Heisenberg limit. Appl. Phys. Rev. 11, 031302 https://doi.org/10.1063/5.0204102 (2024).

Article 
ADS 

Google Scholar
 

Xiao, L. et al. Non-Hermitian sensing in the absence of exceptional points. Phys. Rev. Lett. 133, 180801 https://doi.org/10.1103/PhysRevLett.133.180801 (2024).

Article 
ADS 

Google Scholar
 

Gardiner, C. W. & Zoller, P. Quantum Noise (Springer, 2004).

Wang, Y.-Y. et al. Dispersive nonreciprocity between a qubit and a cavity. Sci. Adv. 10, eadj8796 https://doi.org/10.1126/sciadv.adj8796 (2024).

Article 

Google Scholar
 

Metelmann, A. & Clerk, A. A. Nonreciprocal quantum interactions and devices via autonomous feedforward. Phys. Rev. A 95, 013837 https://doi.org/10.1103/PhysRevA.95.013837 (2017).

Article 
ADS 

Google Scholar
 

Ahmadi, B., Mazurek, P., Horodecki, P. & Barzanjeh, S. Nonreciprocal quantum batteries. Phys. Rev. Lett. 132, 210402 https://doi.org/10.1103/PhysRevLett.132.210402 (2024).

Article 
ADS 
MathSciNet 

Google Scholar
 

Nagaosa, N. & Yanase, Y. Nonreciprocal transport and optical phenomena in quantum materials. Annu. Rev. Condensed Matter Phys. 15, 63 https://doi.org/10.1146/annurev-conmatphys-032822-033734 (2024).

Article 
ADS 

Google Scholar
 

Ando, F. et al. Observation of superconducting diode effect. Nature 584, 373 https://doi.org/10.1038/s41586-020-2590-4 (2020).

Article 
ADS 

Google Scholar
 

Zhang, H. et al. A Josephson diode. Phys. Rev. X 12, 041013 https://doi.org/10.1103/PhysRevX.12.041013 (2022).

Article 

Google Scholar
 

Viola, G. & DiVincenzo, D. P. Hall effect gyrators and circulators. Phys. Rev. X 4, 021019 https://doi.org/10.1103/PhysRevX.4.021019 (2014).

Article 

Google Scholar
 

Mahoney, A. C. et al. On-chip microwave quantum Hall circulator. Phys. Rev. X 7, 011007 https://doi.org/10.1103/PhysRevX.7.011007 (2017).

Article 

Google Scholar