The octagon method for finding exceptional points, and application - - PowerPoint PPT Presentation

Institute of Theoretical Physics, University of Stuttgart, in collaboration with M. Feldmaier, F. Schweiner, J. Main, and H. Cartarius

The octagon method for finding exceptional points, and application to hydrogen-like systems in parallel electric and magnetic fields

G¨ unter Wunner

Institute of Theoretical Physics, University of Stuttgart, in collaboration with M. Feldmaier,

• F. Schweiner, J. Main, and H. Cartarius

Prague, 6 June 2016

The coworkers

• M. Feldmaier
• F. Schweiner
• J. Main
• H. Cartarius

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 2 / 20

A feature of open quantum systems:

Resonances exhibit avoided crossings on the real energy axis when the physical parameters of the systems are varied: example: hydrogen atom in parallel electric and magnetic fields

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 3 / 20

A feature of open quantum systems:

Resonances exhibit avoided crossings on the real energy axis when the physical parameters of the systems are varied: example: hydrogen atom in parallel electric and magnetic fields Indicative of an exceptional point in the complex energy plane.

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 3 / 20

A feature of open quantum systems:

Resonances exhibit avoided crossings on the real energy axis when the physical parameters of the systems are varied: example: hydrogen atom in parallel electric and magnetic fields Indicative of an exceptional point in the complex energy plane. How to find the exceptional point?

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 3 / 20

Motivation

Experiments on excitons in Cu2O in the group of D. Fr¨

• hlich and M. Beyer at

the University of Dortmund band structure of Cu2O Excitons: electron-hole pairs in semiconductors - hydrogen-like systems

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 4 / 20

Motivation

Experiments on excitons in Cu2O in the group of D. Fr¨

• hlich and M. Beyer:

0.0 0.5 1.0 0.4 0.6 2.17190 2.17192 2.17194 0.37 0.38 0.5 1.0 n = 3 n = 2

a

n = 13 n = 12 n = 25 n = 24 n = 23 n = 22 Photon energy (eV) n = 6 n = 5 O p t i c a l d e n s i t y ×6

d c b

5 mm 16 mm 30 mm 2.145 2.150 2.155 2.160 2.165 2.170 2.1716 2.1718 2.168 2.169 2.170 2.171 2.172 Tsumeb mine Namibia

• T. Kazimierczuk et al. ”

Giant Rydberg excitons in the copper oxide Cu2O.” Nature (2014) G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 5 / 20

Rydberg excitons in magnetic and electric fields

Reference field strengths: eB0 m = 2ERyd, eaBohrF0 = 2ERyd

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 6 / 20

Rydberg excitons in magnetic and electric fields

Reference field strengths: eB0 m = 2ERyd, eaBohrF0 = 2ERyd H Cu2O ERyd 13.6 eV 0.092 eV aBohr 0.0529 nm 1.04 nm B0 2.35 × 105 T 6.034 × 102 T F0 5.14 × 109 V/cm 1.76 × 106 V/cm

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 6 / 20

Rydberg systems in parallel electr. and magnet. fields

Hydrogen atom: Hhyd = p2 2m0 − e2 4πε0 1 r + e B 2m0 Lz + e2 B2 8m0

• x2 + y2

+ e F z mp/m0 ≈ 1800 → mp can be taken as approx. infinite

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 7 / 20

Rydberg systems in parallel electr. and magnet. fields

Hydrogen atom: Hhyd = p2 2m0 − e2 4πε0 1 r + e B 2m0 Lz + e2 B2 8m0

• x2 + y2

+ e F z mp/m0 ≈ 1800 → mp can be taken as approx. infinite Rydberg excitons: Hex = p2 2µ − e2 4πε0 εr 1 r + e B 2µ mh − me mh + me Lz + e2 B2 8µ

• x2 + y2

+ e F z me = 0.99m0, mh = 0.62m0 → both masses are important, µ = 0.38m0

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 7 / 20

Rydberg systems in parallel electr. and magnet. fields

Hex = p2 2µ − e2 4πε0 εr 1 r + e B 2µ mh − me mh + me Lz + e2 B2 8µ

• x2 + y2

+ e F z parallel fields:

→ angular momentum is conserved with respect to z-axis (Lz → m) → paramagnetic term: B-dependent shift of zero point energy H′ = H − HP

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 8 / 20

Rydberg systems in parallel electr. and magnet. fields

Hex = p2 2µ − e2 4πε0 εr 1 r + e B 2µ mh − me mh + me Lz + e2 B2 8µ

• x2 + y2

+ e F z parallel fields:

→ angular momentum is conserved with respect to z-axis (Lz → m) → paramagnetic term: B-dependent shift of zero point energy H′ = H − HP

in dimensionless units: lengths in aBohr, energies in 2ERyd, γ = B/B0 and f = F/F0

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 8 / 20

Rydberg systems in parallel electr. and magnet. fields

Hex = p2 2µ − e2 4πε0 εr 1 r + e B 2µ mh − me mh + me Lz + e2 B2 8µ

• x2 + y2

+ e F z parallel fields:

→ angular momentum is conserved with respect to z-axis (Lz → m) → paramagnetic term: B-dependent shift of zero point energy H′ = H − HP

in dimensionless units: lengths in aBohr, energies in 2ERyd, γ = B/B0 and f = F/F0

Hamiltonian of hydrogen-like systems in parallel fields

H′ = 1 2p 2 − 1 r + 1 8γ2 x2 + y2 + f z

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 8 / 20

In electric fields: bound states → resonances

Coulomb potential:

−9 −6 −3 3 x −6 −4 −2 2 V (x) −const

|x|

bound state

Coulomb-Stark potential:

−9 −6 −3 3 x −8 −4 4 V (x) −const

|x| + f x

resonance

” resonances” : non-stationary or quasi-bound states Ψ(r, 0) in position space time evolution: Ψ(r, t) = e−i E t

Ψ(r, 0)

complex energy E = Er − i Γ

2

resonances with complex energy eigenvalues can be calculated by solving the Schr¨

• dinger equation using the complex rotation method

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 9 / 20

Finding exceptional points: Octagon method

close to an EP: describe two related states by 2d matrix

Mkl = a(0)

kl + a(γ) kl (γ − γ0) + a(f) kl (f − f0),

γ0, f0 : initial guesses

for the eigenvalues E1 and E2 of M we set:

κ ≡ E1 + E2 = tr(M) = A + B (γ − γ0) + C (f − f0), η ≡ (E1 − E2)2 = tr2(M) − 4 det(M) = D + E (γ − γ0) + F (f − f0) + G (γ − γ0)2 + H (γ − γ0) (f − f0) + I (f − f0)2,

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 10 / 20

Finding exceptional points: Octagon method

close to an EP: describe two related states by 2d matrix

Mkl = a(0)

kl + a(γ) kl (γ − γ0) + a(f) kl (f − f0),

γ0, f0 : initial guesses

for the eigenvalues E1 and E2 of M we set:

κ ≡ E1 + E2 = tr(M) = A + B (γ − γ0) + C (f − f0), η ≡ (E1 − E2)2 = tr2(M) − 4 det(M) = D + E (γ − γ0) + F (f − f0) + G (γ − γ0)2 + H (γ − γ0) (f − f0) + I (f − f0)2, A = κ0 B = κ1 − κ5 2hγ C = κ3−κ7

2hf

D = η0 E = η1−η5

2hγ

F = η3 − η7 2hf G = η1+η5−2η0

2h2

γ

I = η3 + η7 − 2η0 2h2

f

H = η2−η4+η6−η8

2hγ hf

γ f 1 2 3 4 5 6 7 8 hγ hf

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 10 / 20

Octagon method: iterative algorithm

estimate the new position (γEP, fEP) of EP: 0 = η ≡ (E1 − E2)2 = D + E x + F y + G x2 + H x y + I y2 with x ≡ (γEP − γ0) and y ≡ (fEP − f0). Choose the correct one of four (complex) solutions.

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 11 / 20

Octagon method: iterative algorithm

estimate the new position (γEP, fEP) of EP: 0 = η ≡ (E1 − E2)2 = D + E x + F y + G x2 + H x y + I y2 with x ≡ (γEP − γ0) and y ≡ (fEP − f0). Choose the correct one of four (complex) solutions.

Iterative algorithm to find position (γEP, fEP) of EP

position estimate γ(n)

EP , f (n) EP in step n is taken as the centre point of a new

• ctagon in step n + 1

γ(n+1) = γ(n)

EP ;

γEP = lim

n→∞ γ(n)

f (n+1) = f (n)

EP ;

fEP = lim

n→∞ f (n)

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 11 / 20

Octagon method: example

γ/f = 80 levels labelled by principal quantum numbers n for γ, f → 0 initial parameters: γ0 = 1.481 × 10−3 f0 = 1.851 × 10−5 Re(E) = −6.90 × 10−3

2 4 6 8 10 12 14

iterations i

10−4 10−6 10−8 10−10 10−12 10−14

∆E

b)

|EEP,i − EEP,i−1| |E1,i − E2,i|

EEP,i = (E1,i + E2,i)/2

• est. position of EP with the OM:

γEP = 8.598 633 × 10−4 fEP = 2.005 076 × 10−5 Re(EEP) = −7.647 637 × 10−3 Im(EEP) = 8.461 814 × 10−7

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 12 / 20

Octagon method: verification of the EP found

−7.650 −7.645 −7.640 −7.635

Re(E) [10−3]

−1.6 −1.2 −0.8 −0.4 0.0

Im(E) [10−6] W (9)

η=0 = 1

W (7)

η=0 = 0

7 7 9 9

Two possibilities to verify EP:

1 solve Eq. (2) step-by-step while

encircling the EP → high numerical effort

2 use Eqs. (3) with the known

coefficients of the OM to check exchange behaviour of the resonances → numerically very cheap

H′ = 1 2p 2 − 1 r + 1 8γ2 x2 + y2 + f z (2) κ ≡ E1 + E2 = A + B (γ − γ0) + C (f − f0) η ≡ (E1 − E2)2 = D + E (γ − γ0) + F (f − f0) + G (γ − γ0)2 (3) +H (γ − γ0) (f − f0) + I (f − f0)2

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 13 / 20

Results: more EPs

γ/f = 120

−4 4 8 12

Re(E) [10−2]

50 75 100 125

γEP/fEP a)

−4 4 8 12

Re(E) [10−2]

2.5 5.0 7.5 10.0

γEP [10−2] b)

−4 4 8 12

Re(E) [10−2]

2.5 5.0 7.5 10.0

fEP [10−4] c)

0.0 2.5 5.0 7.5 10.0 12.5

fEP [10−4]

2.5 5.0 7.5 10.0

γEP [10−2] d)

EPs found (in material independent units)

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 14 / 20

Results: comparison hydrogen atom vs. Cu2O

Hydrogen atom: reduced mass: µ ≈ m0 dielectric constant: εr = 1 Cuprous oxide: reduced mass: µ = 0.38m0 dielectric constant εr = 7.50

Hydrogen atom Cu2O B [T] F [ V

cm ]

Er [eV] Ei [meV] B [T] F [ V

cm ]

Er [meV] Ei [µeV] 229.64 120250 −0.1904 −0.6209 0.590 41.16 −1.286 −0.419 561.26 140870 −0.1866 −0.2564 1.441 48.22 −1.261 −1.732 799.69 341940 −0.3886 −2.072 2.053 117.0 −2.625 −14.00 1261.3 668930 −0.3996 −0.5002 3.238 229.0 −2.699 −3.379 1506.7 686310 −0.5245 −4.402 3.868 234.9 −3.544 −29.74 2316.3 1096200 −0.6733 −0.5999 5.946 375.2 −4.549 −4.054 3595.7 2430880 −0.4788 −12.03 9.231 832.0 −3.234 −81.25

Energies for Cu2O need to be corrected by Egap = 2.17208 eV

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 15 / 20

Conclusion and outlook

Octagon method: an improved method for finding EPs:

converges to precise position of EP in parameter space and yields its precise complex energy

• nly initial parameters of an avoided crossing are needed

verification of the estimated EP without further time-consuming quantum-mechanical calculations

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 16 / 20

Conclusion and outlook

Octagon method: an improved method for finding EPs:

converges to precise position of EP in parameter space and yields its precise complex energy

• nly initial parameters of an avoided crossing are needed

verification of the estimated EP without further time-consuming quantum-mechanical calculations

Many EPs in Rydberg systems in parallel external fields:

appropriate units: results hold both for the hydrogen atom and for Rydberg excitons in Cu2O many EPs in Cu2O are in an experimentally accessible regime below 10 T experimental verification of our theoretical predictions: → measurement of photo absorption spectra → analysis by means of harmonic inversion

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 16 / 20

Conclusion and outlook

Octagon method: an improved method for finding EPs:

converges to precise position of EP in parameter space and yields its precise complex energy

• nly initial parameters of an avoided crossing are needed

verification of the estimated EP without further time-consuming quantum-mechanical calculations

Many EPs in Rydberg systems in parallel external fields:

appropriate units: results hold both for the hydrogen atom and for Rydberg excitons in Cu2O many EPs in Cu2O are in an experimentally accessible regime below 10 T experimental verification of our theoretical predictions: → measurement of photo absorption spectra → analysis by means of harmonic inversion

Further information and more details

• M. Feldmaier, J. Main, F. Schweiner, H. Cartarius and G. Wunner. ”

Rydberg systems in parallel electric and magnetic fields: an improved method for finding exceptional points” arXiv:1602.00909, in press (2016)

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 16 / 20

Complex rotation method

Re(E) Im(E) bound states continuous states

without complex rotation

Re(E) Im(E) bound states continuous states exposed resonances 2θ

complex rotation r → r eiθ bound states with real energy remain unaffected by complex rotation continuum states are rotated by −2θ into the complex plane for large enough θ resonances are exposed. Positions of the resonances are independent of rotation angle.

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 17 / 20

Computation of eigenstates

Introduce dilated semiparabolic coordinates: µ = µr b = 1 b √ r + z, ν = νr b = 1 b √ r − z, ϕ = arctan y x

• ,

b → |b| exp(iθ) induces the complex rotation of the real coordinates (µr, νr, ϕ). Schr¨

• dinger equation in dilated semiparabolic coordinates:
• 2H0 − 4b2 + 1

4b8 γ2 (µ4 ν2 + µ2 ν4) + b6 f (µ4 − ν4)

• Ψ = λ (µ2 + ν2) Ψ,

λ = 1 + 2b4 E′: the generalised eigenvalue H0 = Hµ + Hν sum of two 2d harmonic oscillators: Hρ =

• −1

2∆ρ + 1 2ρ2

• ,

∆ρ = 1 ρ ∂ ∂ρ ρ ∂ ∂ρ − m2 ρ2 , ρ ∈ {µ, ν}

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 18 / 20

Computation of eigenstates

Set up the Hamiltonian matrix in the product basis of the eigenstates of the 2d harmonic oscillator |Nµ, Nν, m = |Nµ, m ⊗ |Nν, m ΨNµ,Nν,m(µ, ν, ϕ) =

• Nµ! Nν!

(Nµ + |m|)! (Nν + |m|)!

• 2

π fNµ,m(µ) fNν,m(ν) eim ϕ, fNρ,m(ρ) = e− ρ2

2 ρ|m| L|m|

Nρ (ρ2)

for ρ ∈ {µ, ν}, Wave function of a state with (complex) energy Ei Ψi(µ, ν, ϕ) =

• Nµ,Nν,m

ci,Nµ,Nν,m ΨNµ,Nν,m(µ, ν, ϕ), with the expansion coefficients ci,Nµ,Nν,m of the associated eigenvector obtained from the diagonalization of the Hamiltonian matrix in the oscillator basis.

G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 19 / 20

Motivation

Rydberg excitons of Cu2O with a magnetic field added:

Aßmann et al. ” Quantum chaos of Rydberg excitons.” Preprint (2016) G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 20 / 20