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) + κ T ′ κ – scalar parameter T(0) – unperturbed operator κ T ′ – 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 ) = T( κ = ± i) → eigenvalue 0 − 1 κ 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
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
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
2 × 2 non-Hermitian matrix � ε 1 ≡ e 1 + i � 2 γ 1 ω H (2) = ε 2 ≡ e 2 + i ω 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 � ε 1 ≡ e 1 + i � 2 γ 1 0 H (2) = ε 2 ≡ e 2 + i 0 0 2 γ 2
Eigenvalues ◮ Eigenvalues of H (2) are, generally, complex E 1 , 2 ≡ E 1 , 2 + i 2Γ 1 , 2 = ε 1 + ε 2 ± Z 2 Z ≡ 1 � ( ε 1 − ε 2 ) 2 + 4 ω 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 + 4 ω 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. M¨ uller 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 � = A − 1 k � Φ k | Φ 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 . 05 i (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 . 5 a ; ω = 0 . 05 (left) e 1 = 0 . 55; e 2 = 0 . 5; γ 1 / 2 = − 0 . 5; γ 2 / 2 = − 0 . 5 a ; 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 Schr¨ odinger equation
Eigenfunctions: Mixing via the environment odinger equation for the basic wave functions Φ 0 ◮ Schr¨ i : � ε 1 � 0 eigenfunctions of the non-Hermitian H (2) = 0 0 ε 2 ( H (2) − ε i ) | Φ 0 i � = 0 0 ◮ Schr¨ odinger equation for the mixed wave functions Φ i : � ε 1 � ω eigenfunctions of the non-Hermitian H (2) = ω ε 2 � 0 � ω ( H (2) − ε i ) | Φ i � = − | Φ i � 0 ω 0 ◮ Standard representation of the Φ i in the { Φ 0 n } � b ij Φ 0 b ij = � Φ 0 ∗ Φ i = j ; j | Φ i �
◮ Normalization of the b ij j (b ij ) 2 = 1 � j (b ij ) 2 = Re [ � j { [ Re (b ij )] 2 − [ Im (b ij )] 2 } j (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 . 5 i (left); e 1 = 1 − a / 2; e 2 = √ a ; γ 1 / 2 = − 0 . 53; γ 2 / 2 = − 0 . 55; ω = 0 . 05 i (right) H. Eleuch, I. Rotter, Eur. Phys. J. D 68, 74 (2014)
Eigenfunctions: Nonlinear Schr¨ odinger equation ◮ Schr¨ odinger equation ( H (2) − ε i ) | Φ i � = 0 can be rewritten in Schr¨ odinger equation with source term which contains coupling ω of the states i and j � = i via the common environment of scattering wavefunctions � 0 ω � ( H (2) − ε i ) | Φ i � = − | Φ j � ≡ W | Φ j � 0 ω 0 ◮ Source term is nonlinear ( H (2) � � − ε i ) | Φ i � = � Φ k | W | Φ i � � Φ k | Φ m �| Φ m � 0 k=1 , 2 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 − ε n ) | Φ n � = � Φ n | W | Φ n � | Φ n | 2 | Φ n � ( H (2) 0 ◮ 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 c � dE ′ δ cc ′ − S cc ′ = E − E ′ � � χ E c ′ | V | Ψ E c � dE ′ − 2i π � χ E c ′ | V | Ψ E = δ cc ′ − P c � E − E ′ δ cc ′ − S (1) cc ′ − S (2) = cc ′ � � χ E c ′ | V | Ψ E c � dE ′ + 2i π � χ E S (1) c ′ | V PP | ξ E = P c � cc ′ E − E ′ smoothly dependent on energy N √ γ c S (2) � � χ E λ = i 2 π c ′ | V PQ | Ω λ � · cc ′ E − z λ λ =1 resonance term
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