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

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 Günter 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 Günter 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 Günter 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. Günter 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? Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 3 / 20

] k 2 ) + ([A 2 (I 21 13 ) 2m I2 + B 2 0 2 Motivation ] {k1, k 2 } + B 3 2 (I 1S h2 + I 2 S h1 ) 2m 0 ( I 1 S h1 1 )] ) k 2 3 IS 1 h + c.p. 2m 0 ) + c.p. H so (1) Experiments on excitons in Cu 2 O in the group of D. Fröhlich and M. Beyer at the University of Dortmund tron mass m 0 and spin-orbit coupling ). (2) ], the energy shift -band is set to zero e spin matrices of he Pauli matrices. are defined as in,, band structure of Cu 2 O FIG. 1. (Color online) Band structure of Cu 2O [16]. Due to the spin-orbit coupling (2) the valence band splits into a lower lying fourfold-degenerated band ( ) Γ + 8 of and a higher lying twofold-degenerated band ( ) Γ + 7. The lowest lying conduction band has Γ + 6 symmetry. Depending on the bands involved, one distinguishes between the yellow, green, blue and violet Excitons: electron-hole pairs in semiconductors - hydrogen-like systems Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 4 / 20

Motivation ARCH LETTER Experiments on excitons in Cu 2 O in the group of D. Fröhlich and M. Beyer: Optical density 1.0 a 0.5 0.0 1.0 0.5 0.6 0.4 0.38 n = 3 b n = 2 2.145 2.150 2.155 2.160 2.165 2.170 n = 5 n = 6 Tsumeb mine Namibia c 2.168 2.169 2.170 2.171 2.172 n = 12 n = 13 d 2.1716 2.1718 n = 22 n = 23 n = 24 n = 25 6 5 mm 30 mm 16 mm 0.37 2.17190 2.17192 2.17194 Photon energy (ev) 1 High-resolution absorption spectra of yellow P excitons T. Kazimierczuk in Cu 2 O. et al. crystal Giantfrom Rydberg which excitons samples in theofcopper different oxidesize Cuand 2 O. crystal Nature orientation (2014) w tra are measured with a single-frequency laser on a natural sample of c, A large crystal and a thin crystal mounted strain-free in a brass ho ss 34 mm at 1.2 Günter K. Peaks Wunner correspond (ITP1) to resonances Rydberg withsystems different in parallel electric d, Wavefunction and magnetic of fields the P exciton withprague, n 5 25. 6To June visualize 2016 the 5 / giant 20 e

Rydberg excitons in magnetic and electric fields Reference field strengths: e B 0 m = 2E Ryd, ea Bohr F 0 = 2E Ryd Günter 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: e B 0 m = 2E Ryd, ea Bohr F 0 = 2E Ryd H Cu 2 O E Ryd 13.6 ev 0.092 ev a Bohr 0.0529 nm 1.04 nm B 0 2.35 10 5 T 6.034 10 2 T F 0 5.14 10 9 V/cm 1.76 10 6 V/cm Günter 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: H hyd = p2 e2 1 2m 0 4πε 0 r + e B L z + e2 B 2 ( x 2 + y 2) + e F z 2m 0 8m 0 m p /m 0 1800 m p can be taken as approx. infinite Günter 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: H hyd = p2 e2 1 2m 0 4πε 0 r + e B L z + e2 B 2 ( x 2 + y 2) + e F z 2m 0 8m 0 m p /m 0 1800 m p can be taken as approx. infinite Rydberg excitons: H ex = p2 2µ e2 1 4πε 0 ε r r + e B 2µ m h m e L z + e2 B 2 ( x 2 + y 2) + e F z m h + m e 8µ m e = 0.99m 0, m h = 0.62m 0 both masses are important, µ = 0.38m 0 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 7 / 20

Rydberg systems in parallel electr. and magnet. fields H ex = p2 2µ e2 1 4πε 0 ε r r + e B 2µ m h m e L z + e2 B 2 ( x 2 + y 2) + e F z m h + m e 8µ parallel fields: angular momentum is conserved with respect to z-axis (L z m) paramagnetic term: B-dependent shift of zero point energy H = H H P in dimensionless units: lengths in a Bohr, energies in 2E Ryd, γ = B/B 0 and f = F/F 0 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 8 / 20

Rydberg systems in parallel electr. and magnet. fields H ex = p2 2µ e2 1 4πε 0 ε r r + e B 2µ m h m e L z + e2 B 2 ( x 2 + y 2) + e F z m h + m e 8µ parallel fields: angular momentum is conserved with respect to z-axis (L z m) paramagnetic term: B-dependent shift of zero point energy H = H H P in dimensionless units: lengths in a Bohr, energies in 2E Ryd, γ = B/B 0 and f = F/F 0 Hamiltonian of hydrogen-like systems in parallel fields H = 1 2 p 2 1 r + 1 8 γ2 ( x 2 + y 2) + f z Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 8 / 20

In electric fields: bound states resonances Coulomb potential: V (x) 2 0 2 4 6 const x bound state 9 6 3 0 3 x Coulomb-Stark potential: V (x) 4 0 4 8 const x + f x resonance 9 6 3 0 3 x 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 = E r i Γ 2 resonances with complex energy eigenvalues can be calculated by solving the Schrödinger equation using the complex rotation method Günter 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 M kl = a (0) kl + a (γ) kl (γ γ 0) + a (f) kl (f f 0), γ 0, f 0 : initial guesses for the eigenvalues E 1 and E 2 of M we set: κ E 1 + E 2 = tr(m) = A + B (γ γ 0) + C (f f 0), η (E 1 E 2) 2 = tr 2 (M) 4 det(m) = D + E (γ γ 0) + F (f f 0) + G (γ γ 0) 2 + H (γ γ 0) (f f 0) + I (f f 0) 2, Günter 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 M kl = a (0) kl + a (γ) kl (γ γ 0) + a (f) kl (f f 0), γ 0, f 0 : initial guesses for the eigenvalues E 1 and E 2 of M we set: A = κ 0 κ E 1 + E 2 = tr(m) = A + B (γ γ 0) + C (f f 0), η (E 1 E 2) 2 = tr 2 (M) 4 det(m) = D + E (γ γ 0) + F (f f 0) B = C = κ 3 κ 7 2h f D = η 0 E = η 1 η 5 2h γ G = η 1+η 5 2η 0 2h 2 γ H = η 2 η 4 +η 6 η 8 2h γ h f F = I = + G (γ γ 0) 2 + H (γ γ 0) (f f 0) + I (f f 0) 2, κ1 κ5 2h γ η3 η7 2h f η3 + η7 2η0 2h 2 f f 5 4 6 3 0 1 7 γ h f h γ 2 8 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 10 / 20

Octagon method: iterative algorithm estimate the new position (γ EP, f EP ) of EP: 0 = η (E 1 E 2 ) 2 = D + E x + F y + G x 2 + H x y + I y 2 with x (γ EP γ 0 ) and y (f EP f 0 ). Choose the correct one of four (complex) solutions. Iterative algorithm to find position (γ EP, f EP ) of EP position estimate γ (n) EP, f (n) EP in step n is taken as the centre point of a new octagon in step n + 1 γ (n+1) 0 = γ (n) EP ; f (n+1) 0 = f (n) EP ; γ EP = lim n γ(n) 0 f EP = lim f (n) n 0 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 11 / 20

Octagon method: example b) E 10 4 10 6 10 8 10 10 10 12 10 14 E EP,i E EP,i 1 E 1,i E 2,i 2 4 6 8 10 12 14 iterations i γ/f = 80 levels labelled by principal quantum numbers n for γ, f 0 initial parameters: γ 0 = 1.481 10 3 f 0 = 1.851 10 5 Re(E) = 6.90 10 3 E EP,i = (E 1,i + E 2,i )/2 est. position of EP with the OM: γ EP = 8.598 633 10 4 f EP = 2.005 076 10 5 Re(E EP ) = 7.647 637 10 3 Im(E EP ) = 8.461 814 10 7 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 12 / 20

Octagon method: verification of the EP found Im(E) [10 6 ] 0.0 0.4 0.8 1.2 1.6 (7) Wη=0 = 0 η=0 = 1 W (9) 7.650 7.645 7.640 7.635 Re(E) [10 3 ] 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 2 p 2 1 r + 1 8 γ2 ( x 2 + y 2) + f z (2) κ E 1 + E 2 = A + B (γ γ 0) + C (f f 0) η (E 1 E 2) 2 = D + E (γ γ 0) + F (f f 0) + G (γ γ 0) 2 (3) +H (γ γ 0) (f f 0) + I (f f 0) 2 Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 13 / 20

Results: more EPs a) 125 γep/fep 100 75 50 c) 10.0 fep [10 4 ] 7.5 5.0 2.5 4 0 4 8 12 Re(E) [10 2 ] 4 0 4 8 12 Re(E) [10 2 ] b) 10.0 γep [10 2 ] 7.5 5.0 2.5 d) 10.0 γep [10 2 ] 7.5 5.0 2.5 4 0 4 8 12 Re(E) [10 2 ] 0.0 2.5 5.0 7.5 10.0 12.5 f EP [10 4 ] γ/f = 120 EPs found (in material independent units) Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 14 / 20

Results: comparison hydrogen atom vs. Cu 2 O Hydrogen atom: reduced mass: µ m 0 dielectric constant: ε r = 1 Cuprous oxide: reduced mass: µ = 0.38m 0 dielectric constant ε r = 7.50 B [T] F [ V cm Hydrogen atom V ] Er [ev] Ei [mev] B [T] F [ cm Cu 2O ] 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 Cu 2 O need to be corrected by E gap = 2.17208 ev Günter 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 only 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 Cu 2O many EPs in Cu 2O 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 Günter 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 only 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 Cu 2O many EPs in Cu 2O 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) Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 16 / 20

Complex rotation method Im(E) Im(E) exposed resonances Re(E) Re(E) bound states continuous states bound states 2θ continuous states without complex rotation complex rotation r r e iθ 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. Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 17 / 20

Computation of eigenstates Introduce dilated semiparabolic coordinates: µ = µ r b = 1 r + z, b ν = ν r b = 1 b r z, ϕ = arctan ( y x b b exp(iθ) induces the complex rotation of the real coordinates (µ r, ν r, ϕ). Schrödinger equation in dilated semiparabolic coordinates: ( 2H 0 4b 2 + 1 ) 4 b8 γ 2 (µ 4 ν 2 + µ 2 ν 4 ) + b 6 f (µ 4 ν 4 ) Ψ = λ (µ 2 + ν 2 ) Ψ, λ = 1 + 2b 4 E : the generalised eigenvalue H 0 = H µ + H ν sum of two 2d harmonic oscillators: H ρ = ( 12 ρ + 12 ) ρ2, ρ = 1 ρ ρ ρ ρ m2, ρ {µ, ν} ρ2 ), Günter 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 Ψ Nµ,N ν,m(µ, ν, ϕ) = N µ, N ν, m = N µ, m N ν, m N µ! N ν! (N µ + m )! (N ν + m )! 2 π f N µ,m(µ) f Nν,m(ν) e im ϕ, f Nρ,m(ρ) = e ρ2 2 ρ m L m N ρ (ρ 2 ) for ρ {µ, ν}, Wave function of a state with (complex) energy E i Ψ i (µ, ν, ϕ) = c i,nµ,n ν,m Ψ Nµ,N ν,m(µ, ν, ϕ), N µ,n ν,m with the expansion coefficients c i,nµ,n ν,m of the associated eigenvector obtained from the diagonalization of the Hamiltonian matrix in the oscillator basis. Günter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 19 / 20

Motivation Rydberg excitons of Cu2 O with a magnetic field added: Energy (ev) 2.166 2.170 2.174 2.178 2.182 0 45000 40000 1 35000 30000 25000 2 Magnetic Field (T) 20000 15000 3 10000 4 5 6 7 et al. Quantum chaos of up Rydberg excitons. Preprint (2016) Figure 1: Transmission spectrum of the Cu2 O yellowaßmann series measured in magnetic fields to 7 T starting from the state with principal quantum number n=4. The red dashed line marks the band gap at 2.17208 ev. Gu nter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 20 / 20