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 Cu 2 O in the group of D. Fr¨ ohlich and M. Beyer at the University of Dortmund band structure of Cu 2 O 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 Cu 2 O in the group of D. Fr¨ ohlich and M. Beyer: 1.0 b a n = 3 30 mm 0.5 n = 2 0.0 2.145 2.150 2.155 2.160 2.165 2.170 1.0 Tsumeb mine Namibia n = 6 n = 5 y 0.5 t i c s n e d l a 2.168 2.169 2.170 2.171 2.172 c i t 5 mm p 0.6 O n = 12 n = 13 16 mm 0.4 d 2.1716 2.1718 n = 25 n = 23 × 6 n = 22 n = 24 0.38 0.37 2.17190 2.17192 2.17194 Photon energy (eV) 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: � eB 0 m = 2 E Ryd , ea Bohr F 0 = 2 E Ryd 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: � eB 0 m = 2 E Ryd , ea Bohr F 0 = 2 E Ryd H Cu 2 O E Ryd 13 . 6 eV 0 . 092 eV a Bohr 0 . 0529 nm 1 . 04 nm 2 . 35 × 10 5 T 6 . 034 × 10 2 T B 0 5 . 14 × 10 9 V/cm 1 . 76 × 10 6 V/cm F 0 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: L z + e 2 B 2 H hyd = p 2 e 2 1 r + e B x 2 + y 2 � � − + e F z 2 m 0 4 πε 0 2 m 0 8 m 0 m p /m 0 ≈ 1800 → m p 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: L z + e 2 B 2 H hyd = p 2 e 2 r + e B 1 x 2 + y 2 � � − + e F z 2 m 0 4 πε 0 2 m 0 8 m 0 m p /m 0 ≈ 1800 → m p can be taken as approx. infinite Rydberg excitons: L z + e 2 B 2 H ex = p 2 e 2 1 r + e B m h − m e x 2 + y 2 � � 2 µ − + e F z 4 πε 0 ε r 2 µ m h + m e 8 µ m e = 0 . 99 m 0 , m h = 0 . 62 m 0 → both masses are important, µ = 0 . 38 m 0 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 L z + e 2 B 2 H ex = p 2 e 2 1 r + e B m h − m e x 2 + y 2 � � 2 µ − + e F z 4 πε 0 ε r 2 µ m h + m e 8 µ 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 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 L z + e 2 B 2 H ex = p 2 e 2 1 r + e B m h − m e x 2 + y 2 � � 2 µ − + e F z 4 πε 0 ε r 2 µ m h + m e 8 µ 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 2 E Ryd , γ = B/B 0 and f = F/F 0 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 L z + e 2 B 2 H ex = p 2 e 2 1 r + e B m h − m e x 2 + y 2 � � 2 µ − + e F z 4 πε 0 ε r 2 µ m h + m e 8 µ 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 2 E 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 x 2 + y 2 � 8 γ 2 � + 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: Coulomb-Stark potential: 2 4 − const | x | + f x 0 resonance 0 − 2 V ( x ) V ( x ) − 4 − const − 4 | x | bound − 6 − 8 state − 9 − 6 − 3 0 3 − 9 − 6 − 3 0 3 x 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 Schr¨ odinger 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 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 , 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 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 , 3 B = κ 1 − κ 5 A = κ 0 4 2 2 h γ h f C = κ 3 − κ 7 D = η 0 2 h f 5 0 1 F = η 3 − η 7 f E = η 1 − η 5 h γ 2 h γ 2 h f I = η 3 + η 7 − 2 η 0 G = η 1 + η 5 − 2 η 0 6 8 2 h 2 2 h 2 γ f 7 H = η 2 − η 4 + η 6 − η 8 γ 2 h γ h f 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 ( 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. 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 ( 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) = γ ( n ) n →∞ γ ( n ) EP ; γ EP = lim 0 0 f ( n +1) = f ( n ) n →∞ f ( n ) EP ; f EP = lim 0 0 G¨ unter Wunner (ITP1) Rydberg systems in parallel electric and magnetic fields Prague, 6 June 2016 11 / 20
Recommend
More recommend
