Higher-order exceptional points Ingrid Rotter Max Planck Institute - - PowerPoint PPT Presentation

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: T(κ) = 1 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

• f 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

• f 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

H(2) = ε1 ≡ e1 + i

2γ1

ω ω ε2 ≡ e2 + i

2γ2

• ω – complex coupling matrix elements
• f the two states via the common environment:

Re(ω)= principal value integral Im(ω) = residuum εi – complex eigenvalues of H(2)

H(2) = ε1 ≡ e1 + i

2γ1

ε2 ≡ e2 + i

2γ2

Eigenvalues

◮ Eigenvalues of H(2) are, generally, complex

E1,2 ≡ E1,2 + i 2Γ1,2 = ε1 + ε2 2 ± Z Z ≡ 1 2

• (ε1 − ε2)2 + 4ω2

Ei – energy; Γi – 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 (ε1 − ε2)2 + 4ω2 > 0 and therefore Z = 0 (Landau, Zener 1932)

◮ Exceptional point

two states cross when Z = 0

Eigenfunctions: Biorthogonality

◮ conditions for eigenfunctions and eigenvalues

H|Φi = Ei|Φi Ψi|H = EiΨi|

◮ Hermitian operator:

eigenvalues real → Ψi| = Φi|

◮ non-Hermitian operator:

eigenvalues generally complex → Ψi| = Φi|

◮ operator H(2) (or H(2) 0 ) :

eigenvalues generally complex →

Ψi| = 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: Φi|Φj real → Φi|Φj = 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 |Φj = δij ◮ Relation to standard values

Φi|Φi = Re (Φi|Φi) ; Ai ≡ Φi|Φi ≥ 1 Φi|Φj=i = i Im (Φi|Φj=i) = −Φj=i|Φi ; |Bj

i| ≡ |Φi|Φj=i| ≥ 0 ◮ Φ∗ i |Φj ≡ (Φi|Φj) 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 rk ≡ Φ∗

k|Φk

Φk|Φk = A−1

k ◮ Hermitian systems with orthogonal eigenfunctions:

rk = 1

◮ Systems with well-separated resonance states:

rk ≈ 1 (however rk = 1) → Hermitian quantum physics is a reasonable approximation for the description of the states of the open quantum system

◮ Approching an exceptional point:

rk → 0

Energies Ei , widths Γi /2 and phase rigidity ri of the two eigenfunctions of H(2) as a function of the distance d between the two unperturbed states with energies ei e1 = 2/3; e2 = 2/3 + d; γ1/2 = γ2/2 = −0.5; ω = 0.05i (left) e1 = 2/3; e2 = 2/3 + d; γ1/2 = −0.5; γ2/2 = −0.55; ω = 0.025(1 + i) (right)

• H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

Energies Ei , widths Γi /2 and phase rigidity ri of the two eigenfunctions of H(2) as a function of a e1 = e2 = 1/2; γ1/2 = −0.5; γ2/2 = −0.5a; ω = 0.05 (left) e1 = 0.55; e2 = 0.5; γ1/2 = −0.5; γ2/2 = −0.5a; ω = 0.025(1 + i) (right)

• H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

◮ Numerical results show an unexpected behaviour:

rk → 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 rk → 0 and maximum width bifurcation (or level repulsion) with rk → 1

is driven exclusively by the nonlinear source term of the Schr¨

• dinger equation

Eigenfunctions: Mixing via the environment

◮ Schr¨

• dinger equation for the basic wave functions Φ0

i :

eigenfunctions of the non-Hermitian H(2) = ε1 ε2

• (H(2)

− εi) |Φ0

i = 0 ◮ Schr¨

• dinger equation for the mixed wave functions Φi :

eigenfunctions of the non-Hermitian H(2) = ε1 ω ω ε2

• (H(2)

− εi) |Φi = − ω ω

• |Φi

◮ Standard representation of the Φi in the {Φ0 n}

Φi =

• bij Φ0

j ;

bij = Φ0∗

j |Φi

◮ Normalization of the bij

• j(bij)2 = 1
• j(bij)2 = Re[

j(bij)2] = j{[Re(bij)]2 − [Im(bij)]2} ◮ Probability of the mixing

• j |bij|2 =

j{[Re(bij)]2 + [Im(bij)]2}

• j |bij|2 ≥ 1

Energies Ei , widths Γi /2 and mixing coefficients |bij |

• f the two eigenfunctions of H(2) as a function of a

e1 = 1 − a/2; e2 = √a; γ1/2 = γ2/2 = −0.5; ω = 0.5i (left); e1 = 1 − a/2; e2 = √a; γ1/2 = −0.53; γ2/2 = −0.55; ω = 0.05i (right)

• H. Eleuch, I. Rotter, Eur. Phys. J. D 68, 74 (2014)

Eigenfunctions: Nonlinear Schr¨

• dinger equation

◮ Schr¨

• dinger equation

(H(2) − εi) |Φi = 0 can be rewritten in Schr¨

• dinger equation with source

term which contains coupling ω of the states i and j = i via the common environment of scattering wavefunctions (H(2) − εi) |Φi = − ω ω

• |Φj ≡ W|Φj

◮ Source term is nonlinear

(H(2) − εi) |Φi =

• k=1,2

Φk|W|Φi

• m=1,2

Φk|Φm|Φm since Φk|Φm = 1 for k = m and Φk|Φm = 0 for k = m.

◮ Most important part of the nonlinear contributions is

contained in (H(2) − εn) |Φn = Φn|W|Φn |Φn|2 |Φn

◮ Far from an EP, source term is (almost) linear since

Φk|Φk → 1 and Φk|Φl=k = −Φl=k|Φk → 0

◮ Near to an EP, source term is nonlinear since

Φk|Φk = 1 and Φk|Φl=k = −Φl=k|Φk = 0

◮ Eigenfunctions Φi and eigenvalues Ei 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

Scc′ = δcc′ − χE

c′|V|ΨE c

E − E′ dE′ = δcc′ − P χE

c′|V|ΨE c

E − E′ dE′ − 2iπχE

c′|V|ΨE c

= δcc′ − S(1)

cc′ − S(2) cc′

S(1)

cc′

= P χE

c′|V|ΨE c

E − E′ dE′ + 2iπχE

c′|VPP|ξE c

smoothly dependent on energy S(2)

cc′

= i √ 2π

N

• λ=1

χE

c′|VPQ|Ωλ ·

γc

λ

E − zλ 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

• f two resonance states coupled to one common

continuum of scattering wavefunctions S = (E − E1 − i

2Γ1) (E − E2 − i 2Γ2)

(E − E1 + i

2Γ1) (E − E2 + i 2Γ2)

Reference: I. Rotter, Phys.Rev.E 68, 016211 (2003)

◮ Influence of EPs onto the cross section contained in

the eigenvalues Ei = Ei − i/2 Γi → reliable results also when rk < 1

“Double pole” of the S-matrix

◮ Double pole of the S matrix is an EP ◮ Line shape at the EP is described by

S = 1 − 2i Γd E − Ed + i

2Γd

− Γ2

d

(E − Ed + i

2Γd)2

E1 = E2 ≡ Ed Γ1 = Γ2 ≡ Γd

◮ 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. M¨

uller et al., Phys. Rev. E 52, 5961 (1995)

N × N matrix

H =     ε1 ω12 . . . ω1N ω21 ε2 . . . ω2N . . . . . . ... . . . ωN1 ωN2 . . . εN    

εi ≡ ei + i/2 γi energies and widths of the N states ωik ≡ φi|H|φk Re(ωik) = principal value integral Im(ωik) = 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 εi)

Energies Ei , widths Γi /2, phase rigidity ri and mixing coefficients |bij | for four eigenfunctions of H e1 = 1 − a/2; e2 = a; e3 = −1/3 + 3/2 a; e4 = 2/3; γ1/2 = γ2/2 = −0.4950; γ3/2 = −0.4853; γ4/2 = −0.4950; ω = 0.01i (left) e1 = 0.5; e2 = a; e3 = 2a − 0.5; e4 = 1 − a; γ1/2 = −0.5; γ2/2 = −0.505; γ3/2 = −0.51; γ4/2 = −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 ≡ e1 + i

2γ1

ω ω ε2 ≡ e2 − i

2γ2

• ω – complex coupling matrix elements
• f the two states via the common environment

εi – complex eigenvalues of ˜ H(2)

˜ H(2) = ε1 ≡ e1 + i

2γ1

ε2 ≡ e2 − i

2γ2

Energies Ei , widths Γi /2 and phase rigidity ri of the two eigenfunctions of ˜ H(2) as a function of a e1 = 0.5; e2 = 0.5; γ1/2 = 0.05a; γ2/2 = −0.05a; ω = 0.05 (left); e1 = 0.55; e2 = 0.5; γ1/2 = 0.05a; γ2/2 = −0.05a; ω = 0.025(1 + i) (right)

• H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

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

◮ Eigenvalues of H(2)

E1,2 ≡ E1,2 + i 2Γ1,2 = ε1 + ε2 2 ± Z Z ≡ 1 2

• (ε1 − ε2)2 + 4ω2

◮ Condition for second-order EP

Z = 1 2

• (e1 − e2)2 − 1

4(γ1 − γ2)2 + i(e1 − e2)(γ1 − γ2) + 4ω2 = 0

◮ e1,2 parameter dependent, γ1 = γ2, and ω = i ω0 is imaginary

(e1 − e2)2 − 4 ω2 = → e1 − e2 = ± 2 ω0 → two EPs (e1 − e2)2 > 4 ω2 → Z ∈ ℜ (e1 − e2)2 < 4 ω2 → Z ∈ ℑ width bifurcation between the two EPs

◮ γ1,2 parameter dependent, e1 = e2 and ω is real

(γ1 − γ2)2 − 16 ω2 = → γ1 − γ2 = ± 4 ω → two EPs (γ1 − γ2)2 > 16 ω2 → Z ∈ ℑ (γ1 − γ2)2 < 16 ω2 → Z ∈ ℜ level repulsion between the two EPs

Energies Ei and widths Γi /2 as function of the distance d between two unperturbed states with energies ei (left) and, respectively, as function of a (right) e1 = 2/3; e2 = 2/3 + d; γ1/2 = γ2/2 = −0.5; ω = 0.05i (left) e1 = e2 = 1/2; γ1/2 = −0.5; γ2/2 = −0.5a; ω = 0.05 (right)

• H. Eleuch, I. Rotter, Phys. Rev. A 93, 042116 (2016)

◮ Eigenfunctions of H(2) at an EP

Φcr

1 →

± i Φcr

2 ;

Φcr

2 →

∓ i Φcr

1

References (among others):

• I. Rotter, Phys. Rev. E 64, 036213 (2001)
• U. G¨

unther et al., J. Phys. A 40, 8815 (2007)

• B. Wahlstrand et al., Phys. Rev. E 89, 062910 (2014)

◮ Phase rigidity in approaching an EP; Φi|Φi → ∞

ri → 0

◮ Mixing of the wavefunctions in approaching an EP

|bij| → ∞

◮ Under more realistic conditions, ω is complex

→ simple analytical results cannot be obtained In any case 1 > ri ≥ 0 ; |bij| > 1 under the influence of an EP

◮ Phase rigidity in approaching maximum width

bifurcation (or maximum level repulsion) ri → 1 At this point, the wavefunctions are mixed strongly |bij|2 = 0.5

◮ In approaching maximum width bifurcation (or

maximum level repulsion) eigenfunctions Φi are almost orthogonal; and strongly mixed in the set of {Φ0

k}

Energies Ei , widths Γi /2, phase rigidity ri , and mixing coefficients |bij | e1 = 1 − a; e2 = a; γ1/2 = γ2/2 = −0.5; ω = 0.1i (left); e1 = 1 − a; e2 = a; γ1/2 = −0.05; γ2/2 = −0.1; ω = 0.1(1/4 + 3/4 i) (right)

• H. Eleuch, I. Rotter, to be published

Energies Ei , widths Γi /2, phase rigidity ri , and mixing coefficients |bij | e1 = e2 = 0.5; γ1/2 = −0.05a; γ2/2 = 0.05a; ω = 0.05 (left); e1 = 0.5; e2 = 0.475; γ1/2 = −0.05a; γ2/2 = 0.05a; ω = 0.05(3/4 + 1/4 i) (right)

• H. Eleuch, I. Rotter, to be published

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

◮ 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

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

◮ In difference to the eigenvalues, the eigenfunctions

• f 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 Ei , widths Γi /2, and mixing coefficients |bij | as a function of a for N = 2 and N = 3 e1 = 1 − a/2; e2 = a; e3 = −1/3 + 3/2 a; γ1/2 = γ2/2 = −0.495; γ3/2 = −0.485; ω = 0.01

• H. Eleuch, I. Rotter, Eur. Phys. J. D 69, 230 (2015)

Energies Ei , widths Γi /2, and mixing coefficients |bij | as a function of a for N = 2 and N = 3 e1 = 1 − a/2; e2 = a; e3 = −1/3 + 3/2 a; γ1/2 = γ2/2 = −0.495; γ3/2 = −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