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Cited by 225Quantum Mechanics and Non-Hermitian PhysicsSpectral Theory in Mathematical PhysicsQuantum Mechanics and Applications

Optical Realization of Relativistic Non-Hermitian Quantum Mechanics

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Longhi, Stefano·2010·Physical Review Letters
DOI 10.1103/physrevlett.105.013903
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Abstract

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.105.013903

Cites

42 of 42 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 21 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    For the same DFB structure with a homogeneous lossy region (γ(x)=λ), the transfer matrix is given by Eq.(7) with σ=-E-iλ.

    “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 .”
    Quote in the paperSubstantial claim
    • 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.
  • Findingcitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    The imaginary parts of the eigenvalues E for λ>λc 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.

    “Note that the imaginary parts of the eigenvalues E for λ &gt; 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].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [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.

    • cites [19] J. Carroll, J. Whiteaway, and D. Plumb, Distributed feedback semiconductor lasers (The Institution of Electrical Engineers, London, 1998). source
    • 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.
  • Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    The envelopes ψ1 and ψ2 of counterpropagating waves in the DFB structure satisfy Eq.(1) with a real mass m and a complex-valued vector potential V defined by Eq.(2).

    “the envelopes ψ 1 and ψ 2 satisfy Eq.(1) with a real mass m and a complex-valued vector potential V defined by”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 2

    For a PT-non-invariant Hamiltonian, a spectral singularity could arise from the vanishing of M11 but not of M22, i.e. it might not correspond to a divergence of the spectral transmission or reflection coefficients.

    “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.”
    Quote in the paperSubstantial claim
    • 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.
  • Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4

    The transfer matrix of the PT-invariant structure can be calculated as M = M2 M1, where M1 and M2 are the transfer matrices of the uniform gain and lossy sections.

    “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)].”
    Quote in the paperSubstantial claim
    • 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.
  • May combine two claims. 16
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    At λ>λc 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.

    “At λ &gt; λ c the resonances cross the real energy axis and bound states with Im( E ) &gt; 0 emerge; correspondingly, the spectrum of H ceases to be real-valued.”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    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=λc two resonances cross the real energy axis, i.e. they become two spectral singularities.

    “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)].”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingown data
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    In the Hermitian limit λ=0, the spectrum of H is purely continuous, and a number of resonances (zeros of M22 in Im(E)<0) and anti-resonances (zeros of M11 in Im(E)>0) exist.

    “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 ) &lt; 0 half plane) as well as of anti-resonances (i.e. zeros of M 11 in the Im( E ) &gt; 0 half plane) do exist [Fig.1(b)].”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    In the lossy 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=λc, two antiresonances (rather than resonances) cross the real energy axis, i.e. they become two spectral singularities.

    “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.”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 2

    The zeros of M22(E) on the Im(E)>0 half plane, together with the zeros of M11(E) on the Im(E)<0 half plane, define the point spectrum of H; at such energies, φ(2)E(x) [for Im(E)>0] and φ(1)E(x) [for Im(E)<0] are bound states of H.

    “(ii) Point spectrum . The zeros of M 22 ( E ) on the Im( E ) &gt; 0 half plane, together with the zeros of M 11 ( E ) on the Im( E ) &lt; 0 half plane, define the point spectrum of H ; at such energies, the function φ (2) E ( x ) [for Im( E ) &gt; 0] and φ (1) E ( x ) [for Im( E ) &lt; 0] are bound states of H .”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4

    Owing to the PT-invariance of H, M11(E)=M*22(E*), and thus the zeros of M11 are simply the complex conjugates of those of M22.

    “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 .”
    Quote in the paperNot a substantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    The transition from λ<λc to λ>λc is accompanied by a narrowing of the resonance widths in the transmission spectrum as λ→λc- and to the threshold for self-oscillation at λ=λc.

    “Such a transition, from λ &lt; λ c to λ &gt; 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 .”
    Quote in the paperSubstantial claim
    • 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.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4

    Figure 2(b) shows the loci of zeros of M22 in the complex energy plane for increasing values of λL and for m0L=1.

    “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.”
    Quote in the paperNot a substantial claim
    • 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.

  • May combine two claims.

    Findingcitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 2

    For a PT-invariant Hamiltonian, M11(E)=M*22(E*) and hence at a spectral singularity both M11 and M22 vanish, the resolvent is unbounded for E=E0±i0+, and the transmission and reflection spectral coefficients diverge.

    “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].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [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.

    • 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.

  • May combine two claims.

    Findingown data
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 3

    The critical value λc can be expressed as λc = f(m0L)/L, where f(m0L) can be calculated numerically; for m0L=1, λcL≈1.755.

    “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.”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 2

    The zeros of M22(E) [M11(E)] on the Im(E)<0 [Im(E)>0] half plane correspond to the resonances [antiresonances] of the scattering problem.

    “(iii) Resonances . The zeros of M 22 ( E ) [ M 11 ( E )] on the Im( E ) &lt; 0 [Im( E ) &gt; 0] half plane, i.e. the poles of the analytic continuation of G + [ G -] on the Im( E ) &lt; 0 [Im( E ) &gt; 0] half plane, correspond to the resonances [antiresonances] of the scattering problem.”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 2

    The continuous spectrum of H is the entire real energy axis, where G(E) has a branch cut.

    “(i) Continuous spectrum . The continuous spectrum of H is the entire real energy axis ( -∞ &lt; E &lt; ∞ ), where G ( E ) has a branch cut.”
    Quote in the paperSubstantial claim
    • 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.

  • May combine two claims.

    Findingown data
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 4

    As the point of PT symmetry breaking is approached, narrowing of the transmission resonances is observed, and the onset of lasing at λ=λc+ corresponds to the breaking of the PT symmetry.

    “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.”
    Quote in the paperSubstantial claim
    • 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.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    If every eigenfunction of a PT-invariant Hamiltonian is also an eigenfunction of the PT operator, the PT symmetry of H is unbroken and its spectrum is real-valued.

    “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].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [4]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    The Dirac equation in one spatial dimension for a two-component spinor wave function with time-independent vector and scalar couplings reads Eq.(1) in natural units.

    “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]”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [11]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [12]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    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, but these previous studies did not consider non-Hermitian interactions.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [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).

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian correspond to simple observable physical quantities and are related to well-known physical phenomena like resonance narrowing and laser oscillation.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    Optical structures in media with a complex refractive index have been proposed to test and visualize non-Hermitian features rooted in the non-relativistic Schrödinger equation with a complex potential.

    “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].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [8]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [9]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [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.

    • cites [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).

      Not checked in the free run.

  • May combine two claims.

    Premisenone given
    § Optical realization of relativistic non-Hermitian quantum mechanics · p. 1

    If either V or m assume complex values, the Dirac Hamiltonian H is non-Hermitian and its spectrum is, in general, complex-valued.

    “If either V or m assume complex values, the Dirac Hamiltonian H is non-Hermitian and its spectrum is, in general, complexvalued.”
    Quote in the paperSubstantial claim
19 of 19

  • 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 | &gt; L/ 2.”
1 of 1

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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
DOI
10.1103/physrevlett.105.013903
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
Licence
link.aps.org/licenses/aps-default-license (vor)
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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