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Shuvalov, V. A.; Kochubei, G. S.; Priimak, A. I.; Gubin, V. V.; Tokmak, N. A.

doi:10.1023/a:1025018029138

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…In the particular case, where the two wave functions can be associated with two independent linear polarisations, the wave function at the EP would then be an elliptic or circular wave with a definite chirality. We note that a similar observation has been made in [9] for the treatment of damped acoustic waves in a solid medium. If the two wave functions can be associated with different parities, the superposition again has a definite chirality.…”

supporting

confidence: 82%

The chirality of exceptional points

Heiss

Harney

2001

The European Physical Journal D

207

189

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Exceptional points are singularities of the spectrum and wave functions which occur in connection with level repulsion. They are accessible in experiments using dissipative systems. It is shown that the wave function at an exceptional point is one specific superposition of two wave functions which are themselves specified by the exceptional point. The phase relation of this superposition brings about a chirality which should be detectable in an experiment.PACS numbers: 03.65. Bz, 02.30Dk, 05.45Gg Level repulsion is a well known pattern in virtually all aspects of quantum mechanics. It states that the levels of a selfadjoint Hamiltonian H generically do not cross as a function of a parameter λ on which H(λ) depends [1]. Its importance is particularly pronounced in the realm of quantum chaos [2,3]. The connection between exceptional points [4] and the occurrence of level repulsion has been discussed in [5].An exceptional point (EP) is a value λ c of the parameter λ, where two of the eigenvalues E k of H are equal to each other -say E ν (λ c ) = E ν+1 (λ c ) -but where the space of the corresponding eigenvectors is only onedimensional. We call this a coalescence of the eigenvalues and the eigenfunctions |ψ ν , |ψ ν+1 . It is well known that this cannot occur for a selfadjoint Hamiltonian, where E ν = E ν+1 entails a two-dimensional space of eigenvectors, in which case the phenomenon is called a degeneracy.Considerwhere H 0 , H 1 are real and symmetric N × N matrices, and let λ be a complex number. Then H is a complex symmetric matrix. At an EP there is always a singularity -namely a branch point -in the spectrum E k (λ) and the eigenfunctions |ψ k (λ) . The spectrum consists of the values that one analytic function assumes on N Riemannian sheets. The sheets are connected by N (N −1) square root branch points, the EP's. If an EP -connecting E ν and E ν+1 -occurs sufficiently close to the real λ-axis, the two levels undergo a level repulsion as λ sweeps over the real axis in the vicinity of the EP. Conversely, when two levels undergo repulsion, there is always a nearby EP, where the expansionsexist with a finite radius of convergence. Three major results have been shown in [6] and experimentally verified in [7] when an EP is encircled in the complex λ-plane:1. The two energy levels E ν and E ν+1 connected at the EP are interchanged by a complete turn in the λ-plane.2. The two wave functions |ψ ν and |ψ ν+1 are not just interchanged like their eigenenergies but one of them undergoes a change of sign. In other words, a complete loop in the λ-plane leads to {ψ ν , ψ ν+1 } → {−ψ ν+1 , ψ ν }. As an immediate consequence we conclude: (i) the EP is a fourth order branch point for the wave functions and (ii) different directions of going through the loop yield different phase behavior. In fact, encircling the EP a second time in the same direction we obtain {−ψ ν , −ψ ν+1 } while the next loop yields {ψ ν+1 , −ψ ν } and only the fourth loop restores the original pair {ψ ν , ψ ν+1 }. It follows that the opposite direction y...

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supporting

confidence: 82%

The chirality of exceptional points

Heiss

Harney

2001

The European Physical Journal D

207

189

View full text Add to dashboard Cite

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“…Here, I b is in A, and P -in Pa. The time constant of the probe is characterized by the approximation [1] (1) IMC housing, (2) permanent magnets, (3,4,15) nuts, (5) electrode (anode), (6) insulator, (7) electric leads, (8) magnet casing, (9) external magnetic shield, (10) inlet channel, (11) pole caps, (12) internal magnetic shield, (13) temperature sensor, (14) discharge chamber, (16) …”

Section: Pressure Probementioning

confidence: 99%

“…A similar dependence was observed for a cylindrical probe with a radius r p ≈ 4.5×10 -3 cm and length l p ≈ 0.45 cm. The ratio (θ = 0) /I i (θ = π/2) and the half-width of the peak of ion current are proportional to the degree of nonisothermality T i /T e of the flow of rarefied plasma [5,16,17].…”

Section: Experimental Stand and Measuring Proceduresmentioning

confidence: 99%