The DOI matches its Crossref record; 42 of the 42 references it cites were found in a registry or an index; none of the 21 sources its claims cite is checked for support yet.
Optical Realization of Relativistic Non-Hermitian Quantum Mechanics
Open accessAbstract
Light propagation in distributed-feedback optical structures with gain or loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and are related to well-known physical phenomena such as resonance narrowing and laser oscillation.
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Written from the paper’s full text.
TL;DR
Light propagation in distributed-feedback optical structures with gain/loss regions is proposed as a classical optical simulator of the one-dimensional non-Hermitian Dirac equation. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are mapped to observable DFB phenomena such as resonance narrowing and laser oscillation.
Key findings
- For a PT-non-invariant uniform-gain DFB structure, the critical normalized gain is λ_c L ≈ 1.755 for m₀L = 1; at λ_c two resonances cross the real energy axis and become spectral singularities, and above λ_c bound states emerge and the spectrum ceases to be real-valued.
- For the same structure with uniform loss, at λ_c two antiresonances cross the real axis; the spectral transmission does not diverge, and the bound states have Im(E) < 0, so perturbations are damped rather than amplified.
- For a PT-invariant DFB structure with symmetric gain and loss regions, PT symmetry breaking occurs at λ_c ≈ 4.46/L for m₀L = 1, with two pairs of complex-conjugate eigenvalues appearing from resonances and antiresonances crossing the real axis.
- The paper reports that spectral singularities and PT symmetry breaking of the Dirac Hamiltonian correspond to measurable DFB quantities, including resonance narrowing and the onset of lasing.
Methods
- Theoretical mapping of the 1D Dirac equation with real mass m(x) and complex vector potential V(x) to coupled-mode equations for counterpropagating optical envelopes in a DFB grating.
- Scattering theory and transfer-matrix analysis: the transfer matrix M(E), Green function/resolvent, and zeros of M₁₁ and M₂₂ are used to classify continuous spectrum, point spectrum, resonances, and spectral singularities.
- Two examples are analyzed numerically for m₀L = 1: a uniform index grating with homogeneous gain (PT-non-invariant) and a uniform grating with symmetric homogeneous gain/loss regions (PT-invariant).
Limitations
- The paper states no limitations.
- Evident from the text: this is a theoretical proposal for a classical optical simulator, not an experimental realization; no fabricated device or measured data are reported.
- Evident from the text: the analysis is restricted to the 1D Dirac equation and two specific DFB configurations with uniform grating and piecewise-constant gain/loss, with numerical examples given for m₀L = 1.
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Claims the paper makes
0 of 21 cited sources checked · 0 supported · 0 flagged
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let us consider now the same DFB structure of Fig.1(a) but with γ ( x ) = λ , corresponding to a uniform grating with a homogeneous lossy region. The transfer matrix of the structure is given again by Eq.(7), but with σ = -E -iλ .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
None of the sources cited for this claim have been checked yet.
In the paper
“Note that the imaginary parts of the eigenvalues E for λ > λ c are precisely the growth rates of the two detuned unstable modes at the onset of lasing found in the theory of DFB lasers with a uniform grating [18, 19].”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“the envelopes ψ 1 and ψ 2 satisfy Eq.(1) with a real mass m and a complex-valued vector potential V defined by”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, for a PT -non-invariant Hamiltonian, a spectral singularity could arise from the vanishing of M 11 but not of M 22 , i.e. it might not correspond to a divergence of the spectral transmission or reflection coefficients.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The transfer matrix of the structure can be calculated as M = M 2 M 1 , where M 1 and M 2 are the transfer matrices of the uniform gain and lossy sections [see Eq.(7)].”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This work suggests that well-known phenomena occurring in DFB structures, like spectral narrowing of resonances and self-oscillation, are the measurable quantities associated to the onset of spectral singularities and PT symmetry breaking of Dirac Hamiltonians with certain complex couplings.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“As a first example, let us consider a PT -non-invariant potential corresponding to θ ( x ) = 0 and m ( x ) = m 0 , γ ( x ) = -λ in the interval | x | < L/ 2 [Fig.1(a)], where λ is the dimensionless gain coefficient. This case corresponds to the simplest version of a DFB laser with a uniform index grating and a uniform gain region [19].”
What the sources say
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The main motivation in the study of such quantum-optical analogs is that concepts like exceptional points, spectral singularities and PT symmetry breaking become measurable quantities in an optical experiment [8, 9].”
What the sources say
- Not matchedcites [8]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
In the paper
“An optical realization of the Dirac equation (1) is provided by propagation of light waves in an effective one-dimensional DFB structure [18].”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Not matched
None of the sources cited for this claim have been checked yet.
In the paper
“Light propagation in distributed feedback optical structures with gain/loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions.”
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1 · plain
“Here it is shown that light propagation in distributed-feedback (DFB) optical structures with gain and/or loss regions, which is at the heart of such important devices as DFB semiconductor lasers [18, 19], can provide a fertile ground to test the spectral properties of non-Hermitian Dirac Hamiltonians.”
What the sources say
- Not matchedcites [18] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 , 2327 (1972).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1 · plain
- It may state two separate claims as one. 16
- Probably not a substantial claim. 4
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“At λ > λ c the resonances cross the real energy axis and bound states with Im( E ) > 0 emerge; correspondingly, the spectrum of H ceases to be real-valued.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As λ is increased, the spectrum remains real-valued, the resonances (antiresonances) rigidly shift parallel to the imaginary axis, until at a critical value λ = λ c two resonances cross the real energy axis, i.e. they become two spectral singularities [Fig.1(b)].”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the Hermitian limit λ = 0, the spectrum of H is purely continuous, and a number of resonances (i.e. zeros of M 22 in the Im( E ) < 0 half plane) as well as of anti-resonances (i.e. zeros of M 11 in the Im( E ) > 0 half plane) do exist [Fig.1(b)].”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this case, as the loss coefficient λ is increased from zero, the spectrum remains real-valued shifting parallel to the imaginary axis. At the critical value λ = λ c , two antiresonances (rather than resonances) now cross the real energy axis, i.e. they become two spectral singularities.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(ii) Point spectrum . The zeros of M 22 ( E ) on the Im( E ) > 0 half plane, together with the zeros of M 11 ( E ) on the Im( E ) < 0 half plane, define the point spectrum of H ; at such energies, the function φ (2) E ( x ) [for Im( E ) > 0] and φ (1) E ( x ) [for Im( E ) < 0] are bound states of H .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that, owing to the PT -invariance of H , M 11 ( E ) = M ∗ 22 ( E ∗ ), and thus the zeros of M 11 (not shown in the figure) are simply the complex conjugates of those of M 22 .”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Such a transition, from λ < λ c to λ > λ c , is accompanied by a narrowing of the resonance widths in the transmission spectrum as λ → λ -c [see Fig.1(c)] and to the threshold for self-oscillation at λ = λ c .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Figure 2(b) shows the loci of zeros of M 22 in the complex energy plane for increasing values of λL and for m 0 L = 1.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
In the paper
“Note that, for a PT -invariant Hamiltonian, M 11 ( E ) = M ∗ 22 ( E ∗ ) and hence at a spectral singularity both M 11 and M 22 vanish, the resolvent is unbounded for E = E 0 ± i 0 + , and the transmission and reflection spectral coefficients diverge according to Eq.(4) [22].”
What the sources say
- Not matchedcites [22] Unitarity of the scattering matrix, implying | t | 2 + | r ( l,r ) | 2 = 1 and thus bounded values of transmission and reflection coefficients, is satisfied for Hermitian Hamiltonians, however in non-Hermitian systems unitarity may be broken [see, for instance: F. Cannata, J.-P. Dedonder, and A. Ventura, Ann. Phys. 322 , 397 (2007)].
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
- Not matched
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The critical value λ c can be expressed in the form λ c = f ( m 0 L ) /L , where the function f ( m 0 L ) can be calculated numerically. For instance, for the case of Fig.1 ( m 0 L = 1) one has λ c L ≃ 1 . 755.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(iii) Resonances . The zeros of M 22 ( E ) [ M 11 ( E )] on the Im( E ) < 0 [Im( E ) > 0] half plane, i.e. the poles of the analytic continuation of G + [ G -] on the Im( E ) < 0 [Im( E ) > 0] half plane, correspond to the resonances [antiresonances] of the scattering problem.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(i) Continuous spectrum . The continuous spectrum of H is the entire real energy axis ( -∞ < E < ∞ ), where G ( E ) has a branch cut.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As the point of PT symmetry breaking is approached, narrowing of the transmission resonances is observed [see Fig.2(c)], and the onset of lasing at λ = λ + c corresponds to the breaking of the PT symmetry.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
In the paper
“If every eigenfunction of a PT -invariant Hamiltonian is also an eigenfunction of the PT operator, the PT symmetry of H is said to be unbroken and its spectrum is real-valued [4].”
What the sources say
- Not matchedcites [4]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Let us consider the Dirac equation in one spatial dimension for a two-component spinor wave function ψ ( x, t ) = ( ψ 1 , ψ 2 ) T with time-independent vector ( V ) and scalar ( S ) couplings, which in natural units ( ℏ = c = 1) reads [11, 12]”
What the sources say
- Not matchedcites [11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [12]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
- Not matched
In the paper
“Similarities between light propagation in DFB structures and relativistic wave equations were noticed in early studies on gap solitons in connection with the massive Thirring model of field theory [20], however these previous studies did not consider non-Hermitian interactions.”
What the sources say
- Not checked yetcites [20] A.B. Aceves and S. Wabnitz, Phys. Lett. A 141 , 37 (1989); D.N. Christodoulides and R.I. Joseph, Phys. Rev. Lett. 62 , 1746 (1989).
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and related to well-known physical phenomena like resonance narrowing and laser oscillation.”
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4 · hedged
“This work suggests that well-known phenomena occurring in DFB structures, like spectral narrowing of resonances and self-oscillation, are the measurable quantities associated to the onset of spectral singularities and PT symmetry breaking of Dirac Hamiltonians with certain complex couplings.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4 · hedged
None of the sources cited for this claim have been checked yet.
In the paper
“Recently, optical structures in media with a complex refractive index have been proposed to test and visualize non-Hermitian features rooted in the non-relativistic Schr¨ odinger equation with a complex potential [8, 9, 14, 15].”
What the sources say
- Not matchedcites [8]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [14] A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38 , L171 (2005).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked yetcites [15] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32 , 2632 (2007); K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, Phys. Rev. Lett. 100 , 103904 (2008); S. Longhi, Phys. Rev. Lett. 103 , 123601 (2009).
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If either V or m assume complex values, the Dirac Hamiltonian H is non-Hermitian and its spectrum is, in general, complexvalued.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The authors state this as an assumption.
Both m(x) and V(x) have limited support, i.e. m = V = 0 for |x| > L/2.
“In the following, we will assume that both m ( x ) and V ( x ) have a limited support, i.e. m = V = 0 for | x | > L/ 2.”
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Wu, J., & Xie, X.-T. (2012) · Physical Review A · doi:10.1103/physreva.86.032112 · Cited by 14
Coherent perfect absorption in one-dimensional photonic crystal with a PT-symmetric defect
Zhang, Z., Li, H., Chen, H., Hu, C.-L., & Zhou, P. (2014) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/105/47008 · Cited by 14
Phase-shifted PT -symmetric periodic structures
Raja, S. V., Govindarajan, A., Mahalingam, A., et al. (2020) · Physical Review A · doi:10.1103/physreva.102.013515 · Cited by 13
Ren, J., Franke, S., & Hughes, S. (2022) · ACS Photonics · doi:10.1021/acsphotonics.1c01848 · Cited by 14
Less Interpretation and More Decoherence in Quantum Gravity and Inflationary Cosmology
Crull, E. M. (2014) · Foundations of Physics · doi:10.1007/s10701-014-9847-4 · Cited by 13
Chaotic dynamics of complex trajectory and its quantum signature*
Zhao, W.-L., Gong, P., Wang, J., & Wang, Q. (2020) · Chinese Physics B · doi:10.1088/1674-1056/abc0dc · Cited by 15
Scattering singularities of optical waveguides under complex modulation
Liu, Q., Qin, C., Wang, B., et al. (2020) · Physical Review A · doi:10.1103/physreva.101.033818 · Cited by 13
Xu, Y.-L., Feng, L. H., Lu, M., & Chen, Y. (2012) · Physics Letters A · doi:10.1016/j.physleta.2011.12.043 · Cited by 12
Effervescence in a binary mixture with nonlinear non-reciprocal interactions
Saha, S., & Golestanian, R. (2025) · Nature Communications · doi:10.1038/s41467-025-61728-8 · Cited by 11
“So far, photonicsystems have been primarily used for experimental realisations of non-Hermitian quantum mechanics 24,25 .”
Alexandre, J., Ellis, J. R., & Millington, P. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.106.065003 · Cited by 11
Effects of a non-Hermitian potential on the Landau quantization
Ramos, B. F., Pedrosa, I. A., & Bakke, K. (2019) · International Journal of Modern Physics A · doi:10.1142/s0217751x19500726 · Cited by 11
Le, C., Yang, Z., Cui, F., Schnyder, A. P., & Chiu, C. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.045126 · Cited by 12
Baghshahi, H. R., Tavassoly, M. K., & Behjat, A. (2016) · The European Physical Journal Plus · doi:10.1140/epjp/i2016-16080-0 · Cited by 11
Optical differential operation near exceptional points
Yang, J., Xia, D., & Zhi, Q. (2023) · Optics & Laser Technology · doi:10.1016/j.optlastec.2023.109982 · Cited by 10
Non-Hermitian topological phase transitions in superlattices and the optical Dirac equation
Longhi, S. (2021) · Optics Letters · doi:10.1364/ol.440052 · Cited by 10
Self-citation$\mathcal {PT}$ symmetric Hamiltonian model and Dirac equation in 1+1 dimensions
Yeşiltaş, Ö. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/46/1/015302 · Cited by 9
Longhi, S. (2013) · Physical Review A · doi:10.1103/physreva.88.062112 · Cited by 9
Self-citationNon-Hermitian model for resonant cavities coupled by a chiral mirror
Santos, R. B. B. (2012) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/100/24005 · Cited by 9
Spontaneous PT symmetry breaking in Dirac-Kronig-Penney crystals
Longhi, S., Cannata, F., & Ventura, A. (2011) · Physical Review B · doi:10.1103/physrevb.84.235131 · Cited by 9
Self-citationYang, Y., Jia, H., & Yang, J. J. (2020) · The Journal of the Acoustical Society of America · doi:10.1121/10.0001513 · Cited by 9
“Because of the similarity between Schrödinger equation and classic wave equation, non-Hermitian systems with PT-symmetric potential have been widely discussed in fields of classic waves [2-30].”
BackgroundQuantum criticality at the boundary of the non-Hermitian regime of a Floquet system
Zhao, W.-L., & Liu, J. (2024) · Physical Review A · doi:10.1103/physreva.109.052215 · Cited by 9
Parity-Time Symmetry in Hartree–Fock Theory
Burton, H. G. A., Thom, A. J. W., & Loos, P. (2019) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b00289 · Cited by 8
N -channel comb filtering and lasing in PT -symmetric superstructures
Raja, S. V., Govindarajan, A., Mahalingam, A., & Lakshmanan, M. (2021) · Physical Review A · doi:10.1103/physreva.103.013503 · Cited by 8
Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation
Meza, L. E. A., DUTRA, A. D. S., Hott, M., & Roy, P. (2015) · Physical Review E · doi:10.1103/physreve.91.013205 · Cited by 8
PT-symmetric non-Hermitian AB-stacked bilayer honeycomb photonic lattice
Zhang, D., Li, X., Zhang, L.-L., Fan, D.-Z., Jin, Z., & Gong, W. (2020) · Journal of the Optical Society of America B · doi:10.1364/josab.398686 · Cited by 7
Complete absorption of topologically protected waves
Baardink, G., Cassella, G., Neville, L., Milewski, P. A., & Souslov, A. (2021) · Physical review. E · doi:10.1103/physreve.104.014603 · Cited by 10
Phase transitions and virtual exceptional points in quantum emitters coupled to dissipative baths
Longhi, S. (2025) · Journal of Applied Physics · doi:10.1063/5.0299681 · Cited by 8
“…associated with resonance coalescence – conceptually related to virtual EPs of resonance origin 62 – and are therefore distinct from discrete EP degeneracies of NH Hamiltonians, Liouvillian EPs 67,68 , or spectral singularities in the continuous spectrum of infinite-dimensional NH systems 85–88 .”
Self-citationLi, W., & Leng, Y. (2024) · Laser Physics · doi:10.1088/1555-6611/ad515a · Cited by 7
Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems
Ghosh, P. K. (2011) · Physics Letters A · doi:10.1016/j.physleta.2011.07.025 · Cited by 7
Anomalous Anderson localization behavior in gain‐loss balanced non‐Hermitian systems
Jiang, T., Fang, A., Zhang, Z.-Q., & Chan, C. T. (2020) · Nanophotonics · doi:10.1515/nanoph-2020-0306 · Cited by 7
PT -symmetric quantum field theory
Bender, C. M. (2020) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1586/1/012004 · Cited by 7
Qiong, W., Cheng, L.-Z., & He, Z. (2024) · Laser Physics · doi:10.1088/1555-6611/ad7c3a · Cited by 6
Dirac equation with complex potentials
Ho, C.-L., & Roy, P. (2014) · Modern Physics Letters A · doi:10.1142/s0217732314502101 · Cited by 6
Nonlinear eigenvalue problems andPT-symmetric quantum mechanics
Bender, C. M. (2017) · Journal of Physics Conference Series · doi:10.1088/1742-6596/873/1/012002 · Cited by 6
“These experiments have been performed in such diverse areas as optics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], nuclear magnetic resonance [17], superconductivity [18, 19], microwave cavities [20, 21], laser physics [22, 23], atomic diffusion [24], electronic and mechanical systems [26, 27, 28, 29], optical grap…”
MethodComplex Langevin dynamics and supersymmetric quantum mechanics
Joseph, A., & Kumar, A. (2021) · Journal of High Energy Physics · doi:10.1007/jhep10(2021)186 · Cited by 5
“Such theories have found applications in many areas such as optonics [59, 60] and phase transitions [61, 62].”
BackgroundScattering singularities in Fano-resonant multilayer grating structure with phase-change materials
Wang, T., Liu, Q., Li, T., et al. (2024) · Optics & Laser Technology · doi:10.1016/j.optlastec.2024.111602 · Cited by 5
Dragoman, D. (2012) · Journal of the Optical Society of America B · doi:10.1364/josab.29.001528 · Cited by 5
Stable lossless polaritons on non-Hermitian optical interfaces
Xu, Y., Jiang, J., & Chen, H. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.95.041409 · Cited by 5
Kaur, H., Pal, R., Raju, T. S., & Kumar, C. N. (2016) · Annals of Physics · doi:10.1016/j.aop.2016.09.004 · Cited by 5
Zhang, Y. (2022) · Scientific Reports · doi:10.1038/s41598-022-15860-w · Cited by 5
“A critical condition for the existence of the infinitely high potential barriers is that the denominator of the effective potential has zero point [see Eq.(7)], i.”
BackgroundTopological invariant for multiband non-Hermitian systems with chiral symmetry
Liu, C.-C., Li, L.-H., & An, J. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.107.245107 · Cited by 5
Parity-Time Symmetry and Exceptional points: A TutorialPreprint
Krasnok, A., Nefedkin, N. E., & Alù, A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.08135 · Cited by 5
Lian, H., Yang, X., Fan, Y., et al. (2023) · Optics Communications · doi:10.1016/j.optcom.2023.129992 · Cited by 4
Non-Hermitian non--symmetric Dirac Hamiltonians with real energy eigenvalues
Alhaidari, A. D. (2013) · Physics Letters A · doi:10.1016/j.physleta.2013.06.022 · Cited by 4
Pseudo Hermitian Interactions in the Dirac Equation
Panella, O., & Roy, P. (2014) · Symmetry · doi:10.3390/sym6010103 · Cited by 4
Non-Hermitian wave packet approximation for coupled two-level systems in weak and intense fields
Puthumpally-Joseph, R., Sukharev, M., & Charron, É. (2016) · The Journal of Chemical Physics · doi:10.1063/1.4947140 · Cited by 4
Chamorro‐Posada, P. (2014) · Journal of the Optical Society of America B · doi:10.1364/josab.31.002728 · Cited by 3
Engineering wavefront caustics trajectories in PT-symmetric lattices
Bender, N., Ramezani, H., & Kottos, T. (2015) · Optics Letters · doi:10.1364/ol.40.002138 · Cited by 3
Smolyaninov, I. I. (2014) · Physica B Condensed Matter · doi:10.1016/j.physb.2014.03.063 · Cited by 3
Controlling Excitation Inversion of a Cooper Pair Box Interacting with a Nanomechanical Resonator
Valverde, C., de Oliveira, H. C. B., Avelar, A. T., et al. (2012) · Chinese Physics Letters · doi:10.1088/0256-307x/29/8/080303 · Cited by 3
Non-Hermitian Dirac-Fermions and configuration-entropy: A two regime system
Flores, J. C. (2020) · Physics Letters A · doi:10.1016/j.physleta.2020.126987 · Cited by 3
Zhang, Y., Yang, X., & Huang, M. (2024) · Scientific Reports · doi:10.1038/s41598-024-78871-9 · Cited by 3
The stabilizer group of honeycomb lattices and its application to deformed monolayers
Hernández-Espinosa, Y., Rosado, A., & Sadurní, E. (2016) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/49/48/485201 · Cited by 3
Goldshlag, W., Su, R., Park, S., Reboli, T., & Eden, J. G. (2021) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/1361-6455/ac19f6 · Cited by 3
Quantum Decoherence and Interlevel Relations
Crull, E. M. (2022) · Figshare · doi:10.7274/q524jm23h39 · Cited by 3
Chiral-reversing vortex radiation from a single emitter by eigenstates phase lockingPreprint
Wang, X., Chen, H., Wang, S., Ge, L., Zhang, S., & Ma, R. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.01055 · Cited by 3
“Here, for the first time, we utilize such a similarity for the direct generation of the vortex radiation, which is free from any spatial phase extraction/modulation technology reported before [15, 17, 66-67, 73-78].”
BackgroundMethodNon-Hermitian electron–positron annihilation under thermal effects
Cabral, D. S., Santos, A. F., Bufalo, R., & Xavier, N. B. (2025) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-025-06229-9 · Cited by 2
Bakke, K. (2025) · Foundations of Physics · doi:10.1007/s10701-025-00855-0 · Cited by 2
Decoherence in a PT -Symmetric Qubit
Bhat, J. M., Lone, M. Q., Datta, S., et al. (2023) · Ukrainian Journal of Physics · doi:10.15407/ujpe68.2.101 · Cited by 2
Quantum entanglement and coherence in a system of four-level atom interacting with a nonlinear field
Raffah, B. M., Berrada, K., Abdel‐Khalek, S., & Alruqi, A. B. (2023) · Modern Physics Letters A · doi:10.1142/s0217732323500943 · Cited by 2
Improving the fidelity of teleportation through noisy channels using parity-time symmetric operation
Leng, Y., & Chen, F. (2025) · Laser Physics · doi:10.1088/1555-6611/adf0f3 · Cited by 2
Flores, J. C. (2022) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/ac39eb · Cited by 2
Non-Hermitian quantum networks and configuration entropy: Negative temperatures
Flores, J. C. (2020) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/128/46001 · Cited by 2
Non-Hermitian Physics and Engineering in Silicon Photonics
Wang, C., Fu, Z., & Yang, L. (2021) · Topics in applied physics · doi:10.1007/978-3-030-68222-4_7 · Cited by 2
On the presence of electric dipole and toroidal moments in non-Hermitian QED
Bufalo, R., & Xavier, N. B. (2024) · Physics Letters B · doi:10.1016/j.physletb.2024.139167 · Cited by 1
Chen, F., & Leng, Y. (2025) · Laser Physics · doi:10.1088/1555-6611/adebf0 · Cited by 1
Wen, Y.-T., & Leng, Y. (2025) · Laser Physics · doi:10.1088/1555-6611/ae0285 · Cited by 1
Does gravity render probability quasilocal?
Trivedi, O. (2026) · Physics of the Dark Universe · doi:10.1016/j.dark.2026.102303 · Cited by 1
“…operator η such that ˆ H † = η ˆ Hη − 1 (7) then ˆ H is pseudo-Hermitian and the modified inner product This structure underlies PT -symmetric quantum mechanics and related frameworks in which real spectra and probabilistic consistency are retained in a generalized Hilbert space geometry [13,14].”
Does cosmology require Hermiticity in quantum mechanics?
Trivedi, O., & Gurrola, A. (2026) · Physics Letters B · doi:10.1016/j.physletb.2026.140691 · Cited by 1
“This has led to a broad program in which non-Hermitian models are treated both as effective theories of open dynamics and as candidates for generalized fundamental dynamics equipped with modified inner products and biorthogonal structures [10–14].”
Coupled Parity-Time symmetric cavities: Results from Transmission Line Modelling simulations
Phang, S., Vuković, A., Creagh, S. C., et al. (2015) · International Conference on Transparent Optical Networks · doi:10.1109/icton.2015.7193643 · Cited by 1
“If the system has gain/loss below the threshold point the eigenvalues are real, whilst above the threshold point the eigenvalues form a complex conjugate pair [1,3–6].”
BackgroundCPT symmetry in honeycomb lattices and quantum brachistochrone problem
Yeşiltaş, Ö. (2013) · Open Physics · doi:10.2478/s11534-013-0319-9 · Cited by 1
Solitons supported by defects of PT-invariant potentials with real part of dual-frequency lattices
Meng, Y., & Liu, Y. (2012) · Optics Communications · doi:10.1016/j.optcom.2012.07.001 · Cited by 1
RelativisticZitterbewegungin non-Hermitian photonic waveguide systems
Wang, G., Xu, H., Huang, L., & Lai, Y.-C. (2017) · New Journal of Physics · doi:10.1088/1367-2630/aa5127 · Cited by 1
Vortices in 𝒫𝒯-symmetric non-Hermitian superfluid
Begun, A. M., Chernodub, M. N., & Molochkov, A. V. (2022) · EPJ Web of Conferences · doi:10.1051/epjconf/202225810005 · Cited by 1
Effect of relativistic motion on superconducting quantum bits under decoherence
Al-Ghamdi, M. S., Berrada, K., Abdel‐Khalek, S., et al. (2022) · Results in Physics · doi:10.1016/j.rinp.2022.105402 · Cited by 1
Solving non-Hermitian Dirac equation in the presence of PDM and local Fermi velocity
Ghosh, R. (2022) · International Journal of Modern Physics A · doi:10.1142/s0217751x22502220 · Cited by 1
Effect of parity-time symmetric operation on quantum entanglement
Wang, Q., & He, Z. (2024) · Laser Physics · doi:10.1088/1555-6611/ad9151 · Cited by 1
A dimer $\mathcal{PT}$ -symmetric model simulated in GaAs/AlGaAs quantum wells
Meng, L.-C., Zhang, W., Liu, J.-B., et al. (2016) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/114/34001 · Cited by 1
Non-Hermitian off-diagonal magnetic response of Dirac fermions
Kiss, R. Z., Sticlet, D., Moca, C. P., & Dóra, B. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.165411 · Cited by 1
“Photonic waveguides[8, 31, 32, 36] are also used to emulate the non-hermitian Dirac equation, with the complex refractive index due to losses representing the nonhermitian term.”
MethodExploiting Wave Transport in PT Symmetric Media
Lin, Z. (2012) · doi:10.14418/wes01.1.856 · Cited by 1
Beyond the Petermann limit: can exceptional points increase sensor precision?Preprint
Smith, D. D., Chang, H., Mikhailov, E. E., et al. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.12213 · Cited by 1
“Prominent examples of these non-conservative systems in the field of optics include coupled resonators (CRs) and ring laser gyroscopes (RLGs) [1-5], materials with periodic potentials [6-9], and systems with parity-time (PT) symmetry [10-15].”
BackgroundKimuya, A. M. (2025) · Preprints.org · doi:10.20944/preprints202311.1310.v7
Defect solitons in defective PT potentials with real part of dual-frequency latticesPreprint
Meng, Y., Liu, Y., & Li, P. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1103.2881
“ssibility of optical realization of relativistic non-Hermitian quantum mechanics, provided by the distributedfeedback optical structures with gain and/or loss regions - has been proposed by S. Longhi [17]. One may note that all the PT potentials considered above are periodic. A question natural arises: how does light…”
BackgroundResonance Trapping and Dynamical Phase Transitions
Rotter, I. (2010) · International Journal of Theoretical Physics · doi:10.1007/s10773-010-0532-x
Metadata
- Authors
- Longhi, S.
- Affiliations
- Politecnico di Milano (IT)
- Corresponding author
- Longhi, Stefano
- ORCID
- Longhi, Stefano
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2010-06-30
- Volume
- 105
- Issue
- 1
- Pages
- 013903-013903
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1974418311
- Cited by
- 225
- References
- 41
- Citations by year
- 2026: 8 · 2025: 13 · 2024: 10 · 2023: 9 · 2022: 17 · 2021: 17 · 2020: 18 · 2019: 10 · 2018: 6 · 2017: 10 · 2016: 20 · 2015: 15 · 2014: 18 · 2013: 13 · 2012: 20
- FWCI
- 9.28
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Virtual Community of Pathological Anatomy (University of Castilla La Mancha) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- non-Hermitian quantum mechanics, Dirac equation, PT symmetry breaking, spectral singularities, optical gain and loss, laser oscillation
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