Higher-order exceptional points - CASgemma.ujf.cas.cz/~krejcirik/AAMP13/slides/Rotter.pdf · 2016....

Higher-order exceptional points

Ingrid Rotter

Max Planck Institutefor 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 parameterT(0) – unperturbed operatorκT′ – perturbation

Number of eigenvalues of T(κ) is independent of κ with theexception of some special values of κ (exceptional points)where (at least) two eigenvalues coalesce

Example:

T(κ) =

(1 κκ −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 theinfluence of an exceptional point ?

Can exceptional points be observed ?

Do exceptional points influence the dynamicsof quantum systems ?

Outline

– Hamiltonian of an open quantum system

– Eigenvalues and eigenfunctions of thenon-Hermitian Hamiltonian

– Second-order exceptional points

– Third-order exceptional points

– Shielding of a third-order exceptional point andclustering of second-order exceptional points

Hamiltonian of an open quantum system

I The natural environment of a localized quantummechanical system is the extended continuum ofscattering wavefunctions in which the system isembedded

I This environment can be changed by means ofexternal forces, however it can never be deleted

I The properties of an open quantum system can bedescribed by means of two projection operatorseach of which is related to one of the two parts ofthe function space

I The localized part of the quantum system is basicfor spectroscopic studies

I The localized part of the quantum system is asubsystem

The Hamiltonian

of the (localized) system is

non-Hermitian

Outline






2× 2 non-Hermitian matrix

H(2) =

(ε1 ≡ e1 + i

2γ1 ω

ω ε2 ≡ e2 + i2γ2

)ω – complex coupling matrix elements

of the two states via the common environment:

Re(ω)= principal value integral

Im(ω) = residuum

εi – complex eigenvalues of H(2)0

H(2)0 =

(ε1 ≡ e1 + i

2γ1 0

0 ε2 ≡ e2 + i2γ2

)

Eigenvalues

I 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

I Level repulsion

two states repel each other in accordance with Re(Z)

I Width bifurcation

widths of two states bifurcate in accordance with Im(Z)

I Avoided level crossing

two discrete (or narrow resonance) states avoid crossingbecause (ε1 − ε2)2 + 4ω2 > 0 and therefore Z 6= 0(Landau, Zener 1932)

I Exceptional point

two states cross when Z = 0

Eigenfunctions: Biorthogonality

I conditions for eigenfunctions and eigenvalues

H|Φi〉 = Ei|Φi〉〈Ψi|H = Ei〈Ψi|

I Hermitian operator:eigenvalues real → 〈Ψi| = 〈Φi|

I non-Hermitian operator:eigenvalues generally complex → 〈Ψi| 6= 〈Φi|

I operator H(2) (or H(2)0 ) :

eigenvalues generally complex →

〈Ψi| = 〈Φ∗i |References (among others):M. Muller 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: Phase rigidity

I Phase rigidity is quantitative measure for thebiorthogonality of the eigenfunctions

rk ≡〈Φ∗k |Φk〉〈Φk|Φk〉

= A−1k

I Hermitian systems with orthogonal eigenfunctions:rk = 1

I Systems with well-separated resonance states:rk ≈ 1 (however rk 6= 1)→ Hermitian quantum physics is a reasonable

approximation for the description of thestates of the open quantum system

I 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 eie1 = 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)


as a function of ae1 = 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)


I Numerical results show an unexpected behaviour:

rk → 1

at maximum width bifurcation (or level repulsion)

I Coupling strength ω between system and environmentis constant in the calculations

I 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 ofthe Schrodinger equation

Eigenfunctions:Mixing via the environment

I Schrodinger equation for the basic wave functions Φ0i :

eigenfunctions of the non-Hermitian H(2)0 =

(ε1 00 ε2

)(H(2)

0 − εi) |Φ0i 〉 = 0

I Schrodinger equation for the mixed wave functions Φi :

eigenfunctions of the non-Hermitian H(2) =

(ε1 ωω ε2

)

(H(2)0 − εi) |Φi〉 = −

(0 ωω 0

)|Φi〉

I Standard representation of the Φi in the Φ0n

Φi =∑

bij Φ0j ; bij = 〈Φ0∗

j |Φi〉

I Normalization of the bij∑j(bij)2 = 1

∑j(bij)2 = Re[

∑j(bij)2] =

∑j[Re(bij)]2 − [Im(bij)]2

I 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 |of 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)

I Most important part of the nonlinear contributions iscontained in

(H(2)0 − εn) |Φn〉 = 〈Φn|W|Φn〉 |Φn|2 |Φn〉

I Far from an EP, source term is (almost) linear since〈Φk|Φk〉 → 1 and 〈Φk|Φl6=k〉 = −〈Φl6=k|Φk〉 → 0

I Near to an EP, source term is nonlinear since〈Φk|Φk〉 6= 1 and 〈Φk|Φl6=k〉 = −〈Φl6=k|Φk〉 6= 0

I Eigenfunctions Φi and eigenvalues Ei of H(2) containglobal features caused by the coupling ω of thestates i and k 6= 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|ΨEc 〉

E− E′dE′

= δcc′ − P∫ 〈χE

c′|V|ΨEc 〉

E− E′dE′ − 2iπ〈χE

c′|V|ΨEc 〉

= δcc′ − S(1)cc′ − S

(2)cc′

S(1)cc′ = P

∫ 〈χEc′|V|ΨE

c 〉E− E′

dE′ + 2iπ〈χEc′|VPP|ξE

c 〉

smoothly dependent on energy

S(2)cc′ = i

√2π

N∑λ=1

〈χEc′|VPQ|Ωλ〉 ·

γcλ

E− zλ

resonance term

Resonance part of the S-matrix

I Calculation of the cross section by means of the Smatrix

σ(E) ∝ |1− S(E)|2

I Unitary representation of the S matrix in the caseof two resonance states coupled to one commoncontinuum of scattering wavefunctions

S =(E− E1 − i

2Γ1) (E− E2 − i

2Γ2)

(E− E1 + i2Γ1) (E− E2 + i

2Γ2)

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

I Influence of EPs onto the cross section contained inthe eigenvalues Ei = Ei − i/2 Γi

→ reliable results also when rk < 1

“Double pole” of the S-matrix

I Double pole of the S matrix is an EP

I Line shape at the EP is described by

S = 1− 2iΓd

E− Ed + i2Γd

−Γ2

d

(E− Ed + i2Γd)2

E1 = E2 ≡ Ed

Γ1 = Γ2 ≡ Γd

I Deviation from the Breit-Wigner line shape due tointerferences:– 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 αbetween discrete states and continuum of scattering wavefunctions

full lines: with interferences; dashed lines: without interferencesα = 1←→ exceptional point

M. Muller 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 integralIm(ωik) = residuum

i 6= k: coupling matrix elements of the statesi 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 He1 = 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)


Open quantum systems

with gain and loss

H(2) =

(ε1 ≡ e1 + i

2γ1 ω

ω ε2 ≡ e2 − i2γ2

)ω – complex coupling matrix elements

of the two states via the common environment

εi – complex eigenvalues of H(2)0

H(2)0 =

(ε1 ≡ e1 + i

2γ1 0

0 ε2 ≡ e2 − i2γ2

)


as a function of ae1 = 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)


Outline






I Eigenvalues of H(2)

E1,2 ≡ E1,2 +i

2Γ1,2 =

ε1 + ε2

2± Z

Z ≡1

2

√(ε1 − ε2)2 + 4ω2

I Condition for second-order EP

Z =1

2

√(e1 − e2)2 − 1

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

I e1,2 parameter dependent, γ1 = γ2, and ω = iω0 is imaginary

(e1 − e2)2 − 4ω20 = 0 → e1 − e2 = ± 2ω0

→ two EPs

(e1 − e2)2 > 4ω20 → Z ∈ <

(e1 − e2)2 < 4ω20 → Z ∈ =

width bifurcation between the two EPs

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

(γ1 − γ2)2 − 16ω2 = 0 → γ1 − γ2 = ± 4ω

→ two EPs

(γ1 − γ2)2 > 16ω2 → Z ∈ =(γ1 − γ2)2 < 16ω2 → Z ∈ <

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)


I Eigenfunctions of H(2) at an EP

Φcr1 → ± i Φcr

2 ; Φcr2 → ∓ i Φcr

1

References (among others):I. Rotter, Phys. Rev. E 64, 036213 (2001)U. Gunther et al., J. Phys. A 40, 8815 (2007)B. Wahlstrand et al., Phys. Rev. E 89, 062910 (2014)

I Phase rigidity in approaching an EP; 〈Φi|Φi〉 → ∞

ri → 0

I Mixing of the wavefunctions in approaching an EP

|bij| → ∞

I Under more realistic conditions, ω is complex

→ simple analytical results cannot be obtained

In any case1 > ri ≥ 0 ; |bij| > 1

under the influence of an EP

I Phase rigidity in approaching maximum widthbifurcation (or maximum level repulsion)

ri → 1

At this point, the wavefunctions are mixed strongly

|bij|2 = 0.5

I In approaching maximum width bifurcation (ormaximum level repulsion)

eigenfunctions Φi are almost orthogonal;

and strongly mixed in the set of Φ0k

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






I Two different types of crossing points of threestates

– two states show signatures of a second-orderEP while the third state is an observer state

– the three states form together a commoncrossing 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)

I Formal-mathematical result versus observability

an EP is a point in the continuum

and is of measure zero

Outline






I In difference to the eigenvalues, the eigenfunctionsof H contain

information on the influence of an EP onto itsneighborhood

I Influence of a nearby state onto two states thatcross at an exceptional point

– the states lose their individual characterin a finite parameter range around the EP

– areas of influence of various second-orderEPs overlap and amplify, collectively, theirimpact onto physical values

More than two states of a physical systemare 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 = 3e1 = 1− a/2; e2 = a; e3 = −1/3 + 3/2 a; γ1/2 = γ2/2 = −0.495; γ3/2 = −0.485; ω = 0.01


Energies Ei , widths Γi/2, and mixing coefficients |bij | as a function of a for N = 2 and N = 3e1 = 1− a/2; e2 = a; e3 = −1/3 + 3/2 a; γ1/2 = γ2/2 = −0.495; γ3/2 = −0.4853; ω = 0.01i


Instead of higher-order EPsa clustering of second-order EPs appears

Clustering of second-order EPscauses a dynamical phase transition

Here, eigenfunctions of H are strongly mixedand almost orthogonal;

transition is non-adiabatic

Higher-order exceptional points - CASgemma.ujf.cas.cz/~krejcirik/AAMP13/slides/Rotter.pdf · 2016....

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