Higher-order exceptional points

Higher-order exceptional points Ingrid Rotter Max Planck Institute for the Physics of Complex Systems Dresden (Germany)

Mathematics: Exceptional points Consider a family of operators of the form T(κ) = T(0) + κt κ scalar parameter T(0) unperturbed operator κt perturbation Number of eigenvalues of T(κ) is independent of κ with the exception of some special values of κ (exceptional points) where (at least) two eigenvalues coalesce Example: ( ) 1 κ T(κ) = κ 1 T(κ = ±i) eigenvalue 0 T. Kato, Perturbation theory for linear operators

What about Physics Do exceptional points exist? What about the eigenfunctions under the influence of an exceptional point? Can exceptional points be observed? Do exceptional points influence the dynamics of quantum systems?

Outline Hamiltonian of an open quantum system Eigenvalues and eigenfunctions of the non-hermitian Hamiltonian Second-order exceptional points Third-order exceptional points Shielding of a third-order exceptional point and clustering of second-order exceptional points

Hamiltonian of an open quantum system The natural environment of a localized quantum mechanical system is the extended continuum of scattering wavefunctions in which the system is embedded This environment can be changed by means of external forces, however it can never be deleted The properties of an open quantum system can be described by means of two projection operators each of which is related to one of the two parts of the function space The localized part of the quantum system is basic for spectroscopic studies

The localized part of the quantum system is a subsystem The Hamiltonian of the (localized) system is non-hermitian

2 2 non-hermitian matrix ( H (2) ε1 e = 1 + i 2 γ 1 ω ω ε 2 e 2 + i 2 γ 2 ) ω complex coupling matrix elements of the two states via the common environment: Re(ω)= principal value integral Im(ω) = residuum ε i complex eigenvalues of H (2) 0 ( H (2) ε1 e 0 = 1 + i 2 γ 1 0 0 ε 2 e 2 + i 2 γ 2 )

Eigenvalues Eigenvalues of H (2) are, generally, complex E 1,2 E 1,2 + i 2 Γ 1,2 = ε 1 + ε 2 ± Z 2 Z 1 (ε 1 ε 2 ) 2 + 4ω 2 2 E i energy; Γ i width of the state i

Level repulsion two states repel each other in accordance with Re(Z) Width bifurcation widths of two states bifurcate in accordance with Im(Z) Avoided level crossing two discrete (or narrow resonance) states avoid crossing because (ε 1 ε 2 ) 2 + 4ω 2 > 0 and therefore Z 0 (Landau, Zener 1932) Exceptional point two states cross when Z = 0

Eigenfunctions: Biorthogonality conditions for eigenfunctions and eigenvalues H Φ i = E i Φ i Ψ i H = E i Ψ i Hermitian operator: eigenvalues real Ψ i = Φ i non-hermitian operator: eigenvalues generally complex Ψ i = Φ i operator H (2) (or H (2) 0 ) : eigenvalues generally complex Ψ i = Φ i References (among others): M. Müller et al., Phys.Rev.E 52, 5961 (1995) Y.V. Fyodorov, D.V. Savin, Phys.Rev.Lett. 108, 184101 (2012) J.B. Gros et al., Phys.Rev.Lett. 113, 224101 (2014)

Eigenfunctions: Normalization Hermitian operator: Φ i Φ j real Φ i Φ j = 1 To smoothly describe transition from a closed system with discrete states to a weakly open one with narrow resonance states (described by H (2) ): Φ i Φ j = δ ij Relation to standard values Φ i Φ i = Re ( Φ i Φ i ) ; A i Φ i Φ i 1 Φ i Φ j i = i Im ( Φ i Φ j i ) = Φ j i Φ i ; B j i Φ i Φ j i 0 Φ i Φ j (Φ i Φ j ) complex phases of the two wavefunctions relative to one another are not rigid

Eigenfunctions: Phase rigidity Phase rigidity is quantitative measure for the biorthogonality of the eigenfunctions r k Φ k Φ k Φ k Φ k = A 1 k Hermitian systems with orthogonal eigenfunctions: r k = 1 Systems with well-separated resonance states: r k 1 (however r k 1) Hermitian quantum physics is a reasonable approximation for the description of the states of the open quantum system Approching an exceptional point: r k 0

Energies E i, widths Γ i /2 and phase rigidity r i of the two eigenfunctions of H (2) as a function of the distance d between the two unperturbed states with energies e i e 1 = 2/3; e 2 = 2/3 + d; γ 1 /2 = γ 2 /2 = 0.5; ω = 0.05i (left) e 1 = 2/3; e 2 = 2/3 + d; γ 1 /2 = 0.5; γ 2 /2 = 0.55; ω = 0.025(1 + i) (right) H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Energies E i, widths Γ i /2 and phase rigidity r i of the two eigenfunctions of H (2) as a function of a e 1 = e 2 = 1/2; γ 1 /2 = 0.5; γ 2 /2 = 0.5a; ω = 0.05 (left) e 1 = 0.55; e 2 = 0.5; γ 1 /2 = 0.5; γ 2 /2 = 0.5a; ω = 0.025(1 + i) (right) H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Numerical results show an unexpected behaviour: r k 1 at maximum width bifurcation (or level repulsion) Coupling strength ω between system and environment is constant in the calculations Evolution of the system between EP with r k 0 and maximum width bifurcation (or level repulsion) with r k 1 is driven exclusively by the nonlinear source term of the Schrödinger equation

Eigenfunctions: Mixing via the environment Schrödinger equation for the basic wave functions ( Φ 0 i : ) eigenfunctions of the non-hermitian H (2) ε1 0 0 = 0 ε 2 (H (2) 0 ε i ) Φ 0 i = 0 Schrödinger equation for the mixed wave functions ( Φ i ): eigenfunctions of the non-hermitian H (2) ε1 ω = ω ε 2 ( ) (H (2) 0 ω 0 ε i ) Φ i = Φ ω 0 i Standard representation of the Φ i in the {Φ 0 n } Φ i = b ij Φ 0 j ; b ij = Φ 0 j Φ i

Normalization of the b ij j (b ij) 2 = 1 j (b ij) 2 = Re[ j (b ij) 2 ] = j {[Re(b ij)] 2 [Im(b ij )] 2 } Probability of the mixing j b ij 2 = j {[Re(b ij)] 2 + [Im(b ij )] 2 } j b ij 2 1

Energies E i, widths Γ i /2 and mixing coefficients b ij of the two eigenfunctions of H (2) as a function of a e 1 = 1 a/2; e 2 = a; γ 1 /2 = γ 2 /2 = 0.5; ω = 0.5i (left); e 1 = 1 a/2; e 2 = a; γ 1 /2 = 0.53; γ 2 /2 = 0.55; ω = 0.05i (right) H. Eleuch, I. Rotter, Eur. Phys. J. D 68, 74 (2014)

Eigenfunctions: Nonlinear Schrödinger equation Schrödinger equation (H (2) ε i ) Φ i = 0 can be rewritten in Schrödinger equation with source term which contains coupling ω of the states i and j i via the common environment of scattering wavefunctions (H (2) 0 ε i ) Φ i = k=1,2 ( 0 ω ω 0 ) Φ j W Φ j Source term is nonlinear (H (2) 0 ε i ) Φ i = Φ k W Φ i Φ k Φ m Φ m m=1,2 since Φ k Φ m = 1 for k = m and Φ k Φ m = 0 for k m.

Most important part of the nonlinear contributions is contained in (H (2) 0 ε n ) Φ n = Φ n W Φ n Φ n 2 Φ n Far from an EP, source term is (almost) linear since Φ k Φ k 1 and Φ k Φ l k = Φ l k Φ k 0 Near to an EP, source term is nonlinear since Φ k Φ k = 1 and Φ k Φ l k = Φ l k Φ k = 0 Eigenfunctions Φ i and eigenvalues E i of H (2) contain global features caused by the coupling ω of the states i and k i via the environment Environment of an open quantum system is continuum of scattering wavefunctions which has an infinite number of degrees of freedom

The S-matrix χ E c V Ψ E S cc = δ cc c de E E χ E c V Ψ E = δ cc P c de 2iπ χ E E E c V Ψ E c = δ cc S (1) cc S(2) cc χ S (1) E cc = P c V Ψ E c de + 2iπ χ E E E c V PP ξ E c smoothly dependent on energy S (2) cc = i N γ c 2π χ E λ c V PQ Ω λ E z λ λ=1 resonance term

Resonance part of the S-matrix Calculation of the cross section by means of the S matrix σ(e) 1 S(E) 2 Unitary representation of the S matrix in the case of two resonance states coupled to one common continuum of scattering wavefunctions S = (E E 1 i 2 Γ 1) (E E 2 i 2 Γ 2) (E E 1 + i 2 Γ 1) (E E 2 + i 2 Γ 2) Reference: I. Rotter, Phys.Rev.E 68, 016211 (2003) Influence of EPs onto the cross section contained in the eigenvalues E i = E i i/2 Γ i reliable results also when r k < 1

Double pole of the S-matrix Double pole of the S matrix is an EP Line shape at the EP is described by Γ d S = 1 2i E E d + i Γ 2 d Γ 2 d (E E d + i 2 Γ d) 2 E 1 = E 2 E d Γ 1 = Γ 2 Γ d Deviation from the Breit-Wigner line shape due to interferences: linear term with the factor 2 in front quadratic term two peaks with asymmetric line shape

Cross section as a function of the coupling strength α between discrete states and continuum of scattering wavefunctions full lines: with interferences; dashed lines: without interferences α = 1 exceptional point M. Müller et al., Phys. Rev. E 52, 5961 (1995)

N N matrix ε 1 ω 12... ω 1N ω H = 21 ε 2... ω 2N...... ω N1 ω N2... ε N ε i e i + i/2 γ i energies and widths of the N states ω ik φ i H φ k Re(ω ik ) = principal value integral Im(ω ik ) = residuum i k: coupling matrix elements of the states i and k via the environment i = k: selfenergy of the state i (in our calculations mostly included in ε i )

Energies E i, widths Γ i /2, phase rigidity r i and mixing coefficients b ij for four eigenfunctions of H e 1 = 1 a/2; e 2 = a; e 3 = 1/3 + 3/2 a; e 4 = 2/3; γ 1 /2 = γ 2 /2 = 0.4950; γ 3 /2 = 0.4853; γ 4 /2 = 0.4950; ω = 0.01i (left) e 1 = 0.5; e 2 = a; e 3 = 2a 0.5; e 4 = 1 a; γ 1 /2 = 0.5; γ 2 /2 = 0.505; γ 3 /2 = 0.51; γ 4 /2 = 0.505; ω = 0.005(1 + i) (right) H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Open quantum systems with gain and loss H (2) = ( ε1 e 1 + i 2 γ 1 ω ω ε 2 e 2 i 2 γ 2 ω complex coupling matrix elements of the two states via the common environment ε i complex eigenvalues of H (2) 0 ( H (2) ε1 e 0 = 1 + i γ 2 1 0 0 ε 2 e 2 i γ 2 2 ) )

Energies E i, widths Γ i /2 and phase rigidity r i of the two eigenfunctions of H (2) as a function of a e 1 = 0.5; e 2 = 0.5; γ 1 /2 = 0.05a; γ 2 /2 = 0.05a; ω = 0.05 (left); e 1 = 0.55; e 2 = 0.5; γ 1 /2 = 0.05a; γ 2 /2 = 0.05a; ω = 0.025(1 + i) (right) H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Eigenvalues of H (2) E 1,2 E 1,2 + i 2 Γ 1,2 = ε 1 + ε 2 2 ± Z Z 1 2 (ε1 ε 2 ) 2 + 4ω 2 Condition for second-order EP Z = 1 2 (e 1 e 2) 2 1 4 (γ1 γ2)2 + i(e 1 e 2)(γ 1 γ 2) + 4ω 2 = 0

e 1,2 parameter dependent, γ 1 = γ 2, and ω = i ω 0 is imaginary (e 1 e 2 ) 2 4 ω 2 0 = 0 e 1 e 2 = ± 2 ω 0 two EPs (e 1 e 2 ) 2 > 4 ω 2 0 Z R (e 1 e 2 ) 2 < 4 ω 2 0 Z I width bifurcation between the two EPs γ 1,2 parameter dependent, e 1 = e 2 and ω is real (γ 1 γ 2 ) 2 16 ω 2 = 0 γ 1 γ 2 = ± 4 ω two EPs (γ 1 γ 2 ) 2 > 16 ω 2 Z I (γ 1 γ 2 ) 2 < 16 ω 2 Z R level repulsion between the two EPs

Energies E i and widths Γ i /2 as function of the distance d between two unperturbed states with energies e i (left) and, respectively, as function of a (right) e 1 = 2/3; e 2 = 2/3 + d; γ 1 /2 = γ 2 /2 = 0.5; ω = 0.05i (left) e 1 = e 2 = 1/2; γ 1 /2 = 0.5; γ 2 /2 = 0.5a; ω = 0.05 (right) H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Eigenfunctions of H (2) at an EP Φ cr 1 ± i Φcr 2 ; Φcr 2 i Φcr 1 References (among others): I. Rotter, Phys. Rev. E 64, 036213 (2001) U. Günther et al., J. Phys. A 40, 8815 (2007) B. Wahlstrand et al., Phys. Rev. E 89, 062910 (2014)

Phase rigidity in approaching an EP; Φ i Φ i r i 0 Mixing of the wavefunctions in approaching an EP b ij Under more realistic conditions, ω is complex simple analytical results cannot be obtained In any case 1 > r i 0 ; b ij > 1 under the influence of an EP

Phase rigidity in approaching maximum width bifurcation (or maximum level repulsion) r i 1 At this point, the wavefunctions are mixed strongly b ij 2 = 0.5 In approaching maximum width bifurcation (or maximum level repulsion) eigenfunctions Φ i are almost orthogonal; and strongly mixed in the set of {Φ 0 k }

Energies E i, widths Γ i /2, phase rigidity r i, and mixing coefficients b ij e 1 = 1 a; e 2 = a; γ 1 /2 = γ 2 /2 = 0.5; ω = 0.1i (left); e 1 = 1 a; e 2 = a; γ 1 /2 = 0.05; γ 2 /2 = 0.1; ω = 0.1(1/4 + 3/4 i) (right) H. Eleuch, I. Rotter, to be published

Energies E i, widths Γ i /2, phase rigidity r i, and mixing coefficients b ij e 1 = e 2 = 0.5; γ 1 /2 = 0.05a; γ 2 /2 = 0.05a; ω = 0.05 (left); e 1 = 0.5; e 2 = 0.475; γ 1 /2 = 0.05a; γ 2 /2 = 0.05a; ω = 0.05(3/4 + 1/4 i) (right) H. Eleuch, I. Rotter, to be published

Two different types of crossing points of three states two states show signatures of a second-order EP while the third state is an observer state the three states form together a common crossing point References (among others): G. Demange, E.M. Graefe, J. Phys. A 45, 025303 (2012) H. Eleuch, I. Rotter, Eur. Phys. J. D 69, 230 (2015) Formal-mathematical result versus observability an EP is a point in the continuum and is of measure zero

In difference to the eigenvalues, the eigenfunctions of H contain information on the influence of an EP onto its neighborhood Influence of a nearby state onto two states that cross at an exceptional point the states lose their individual character in a finite parameter range around the EP areas of influence of various second-order EPs overlap and amplify, collectively, their impact onto physical values More than two states of a physical system are unable to coalesce at a single point

Energies E i, widths Γ i /2, and mixing coefficients b ij as a function of a for N = 2 and N = 3 e 1 = 1 a/2; e 2 = a; e 3 = 1/3 + 3/2 a; γ 1 /2 = γ 2 /2 = 0.495; γ 3 /2 = 0.485; ω = 0.01 H. Eleuch, I. Rotter, Eur. Phys. J. D 69, 230 (2015)

Energies E i, widths Γ i /2, and mixing coefficients b ij as a function of a for N = 2 and N = 3 e 1 = 1 a/2; e 2 = a; e 3 = 1/3 + 3/2 a; γ 1 /2 = γ 2 /2 = 0.495; γ 3 /2 = 0.4853; ω = 0.01i H. Eleuch, I. Rotter, Eur. Phys. J. D 69, 230 (2015)

Instead of higher-order EPs a clustering of second-order EPs appears Clustering of second-order EPs causes a dynamical phase transition Here, eigenfunctions of H are strongly mixed and almost orthogonal; transition is non-adiabatic