In quantum physics, exceptional points[1] are singularities in the parameter space where two or more eigenstates (eigenvalues and eigenvectors) coalesce. These points appear in dissipative systems, which make the Hamiltonian describing the system non-Hermitian.

Photonics

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The losses in photonic systems, are a feature used to study non-Hermitian physics.[2] Adding non-Hermiticity (such as dichroism) in photonic systems where exist Dirac points transforms these degeneracy points into a pair of exceptional points. It has been demonstrated experimentally in numerous photonic systems such as microcavities[3] and photonic crystals.[4] The first demonstration of exceptional points was done by Woldemar Voigt in 1902 in a crystal.[5]

Fidelity and fidelity susceptibility

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In condensed matter and many-body physics, fidelity is often used to detect quantum phase transitions in parameter space. The definition of fidelity is the inner product of the ground state wave functions of two adjacent points in parameter space,  , where   is a small quantity. After series expansion,  , the first-order correction term of fidelity is zero, and the coefficient of the second-order correction term is called the fidelity susceptibility. The fidelity susceptibility diverges toward positive infinity as the parameters approach the quantum phase transition point.

 

For the exceptional points of non-Hermitian quantum systems, after appropriately generalizing the definition of fidelity,

 

the real part of the fidelity susceptibility diverges toward negative infinity when the parameters approach the exceptional points.[6][7]

 

For non-Hermitian quantum systems with PT symmetry, fidelity can be used to analyze whether exceptional points are of higher-order. Many numerical methods such as the Lanczos algorithm, Density Matrix Renormalization Group (DMRG), and other tensor network algorithms are relatively easy to calculate only for the ground state, but have many difficulties in computing the excited states. Because fidelity only requires the ground state calculations, this approach allows most numerical methods to analyze non-Hermitian systems without excited states, and find the exceptional point, as well as to determine whether it is a higher-order exceptional point.

See also

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References

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  1. ^ Bergholtz, Emil J.; Budich, Jan Carl; Kunst, Flore K. (2021-02-24). "Exceptional topology of non-Hermitian systems". Reviews of Modern Physics. 93 (1): 015005. arXiv:1912.10048. Bibcode:2021RvMP...93a5005B. doi:10.1103/RevModPhys.93.015005. S2CID 209444748.
  2. ^ Miri, Mohammad-Ali; Alù, Andrea (2019-01-04). "Exceptional points in optics and photonics". Science. 363 (6422): eaar7709. doi:10.1126/science.aar7709. ISSN 0036-8075. PMID 30606818. S2CID 57600483.
  3. ^ Liao, Qing; Leblanc, Charly; Ren, Jiahuan; Li, Feng; Li, Yiming; Solnyshkov, Dmitry; Malpuech, Guillaume; Yao, Jiannian; Fu, Hongbing (2021-09-01). "Experimental Measurement of the Divergent Quantum Metric of an Exceptional Point". Physical Review Letters. 127 (10): 107402. arXiv:2011.12037. Bibcode:2021PhRvL.127j7402L. doi:10.1103/PhysRevLett.127.107402. ISSN 0031-9007. PMID 34533335. S2CID 227151509.
  4. ^ Kim, Kyoung-Ho; Hwang, Min-Soo; Kim, Ha-Reem; Choi, Jae-Hyuck; No, You-Shin; Park, Hong-Gyu (2016-12-21). "Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains". Nature Communications. 7 (1): 13893. Bibcode:2016NatCo...713893K. doi:10.1038/ncomms13893. ISSN 2041-1723. PMC 5187586. PMID 28000688.
  5. ^ Voigt, W. (1902-07-01). "VII. On the behaviour of pleochroitic crystals along directions in the neighbourhood of an optic axis". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4 (19): 90–97. doi:10.1080/14786440209462820. ISSN 1941-5982.
  6. ^ Tzeng, Yu-Chin; Ju, Chia-Yi; Chen, Guang-Yin; Huang, Wen-Min (2021-01-07). "Hunting for the non-Hermitian exceptional points with fidelity susceptibility". Physical Review Research. 3 (1): 013015. arXiv:2009.07070. doi:10.1103/PhysRevResearch.3.013015.
  7. ^ Tu, Yi-Ting; Jang, Iksu; Chang, Po-Yao; Tzeng, Yu-Chin (2023-03-23). "General properties of fidelity in non-Hermitian quantum systems with PT symmetry". Quantum. 7: 960. arXiv:2203.01834. doi:10.22331/q-2023-03-23-960.