Photoionisation processes are studied via ab initio calculations. - PowerPoint PPT Presentation

I ONIZATION OF RUBIDIUM WITH ULTRASHORT INTENSE LASER PULSES Mihály András Pocsai 1 , 2 , Imre Ferenc Barna 1 1 Wigner Research Centre of the H.A.S. 2 University of Pécs, Faculty of Sciences, Department of Physics Budapest, 5 th of May, 2017 2 nd Wigner/MPP AWAKE Workshop 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 1 / 28

O UTLINE 1 O VERVIEW OF THE A PPLIED T HEORY 2 R ESULTS AND F URTHER P LANS 3 R EFERENCES 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 2 / 28

Overview of the Applied Theory O UTLINE 1 O VERVIEW OF THE A PPLIED T HEORY 2 R ESULTS AND F URTHER P LANS 3 R EFERENCES 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 3 / 28

Overview of the Applied Theory Photoionisation processes are studied via ab initio calculations. One active electron approach has been applied. Consider the time-dependent Schrödinger-equation: i ∂ t | Ψ( t , r ) � = ˆ H | Ψ( t , r ) � (1) The Hamilton operator has the form of H = ˆ ˆ H Rb + ˆ V I (2) Hamilton operator of the Rb atom [1]: H Rb = − 1 2 ∇ 2 − 1 ˆ r ( 1 − be − dr ) (3) with b = 4 . 5 and d = 1 . 09993. 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 4 / 28

Overview of the Applied Theory In Length gauge, the interaction operator has the form of ˆ H I = e r · E ( t , r ) (4) The electric field E ( t , r ) contains terms of e ± i ( ω L t − kr ) = e ± i ω L t e ∓ i kr Consider the Taylor-expansion of the spatial term: e ∓ i kr = 1 ∓ i kr + . . . (5) note that kr ∼ kr and k ∼ λ − 1 . The atomic distances are much smaller than the laser wavelength: kr ≪ 1. e ± i ( ω L t − kr ) ≈ e ± i ω L t (6) This is the dipole approximation. 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5 / 28

Overview of the Applied Theory The solution of the TDSE can be expanded in terms of the eigenstates of the time-independent Schrödinger equation: ˆ � � � � H � Φ j ( r ) = E j � Φ j ( r ) (7) We apply the following Ansatz for the TDSE: N � e − iE j t � � | Ψ( t , r ) � = a j ( t ) � Φ j ( r ) (8) j = 1 Inserting (8) into (1), using (7), we get: N N e − iE j t = � � ˆ e − iE j t � � � � ˙ i a j ( t ) � Φ j ( r ) V I a j ( t ) � Φ j ( r ) (9) j = 1 j = 1 Multiple the above equation by � Φ k ( r ) | e iE k t . We get the following system of equations for the a j ( t ) coefficients: N � V kj e iE kj t a j ( t ) ˙ a k ( t ) = − i ( k = 1 . . . N ) (10) j = 1 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 6 / 28

Overview of the Applied Theory In (10), E kj := E k − E j and V kj := � Φ k ( r ) | ˆ � � V I � Φ j ( r ) is the couplings matrix. The (highly) oscillatory term can be transformed out. Let a k ( t ) := a k ( t ) e − iE k t ˜ (11) Inserting (11) into (10), we get: N i ˙ � ˜ V kj ˜ a j ( t ) + E k ˜ a k ( t ) = a k ( t ) (12) j = 1 Initial conditions: � 1 k = 1 a k ( t → −∞ ) = (13) 0 k � = 1 Final state probabilities: P k ( t → ∞ ) = | a k ( t → ∞ ) | 2 (14) 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 7 / 28

Overview of the Applied Theory The bound and the continuum states of the valence electron are described with Slater-type orbitals and Coulomb wavepackets [3], respectively: r ) = C ( n , κ ) r n − 1 e − κ r Y l , m ( θ, ϕ ) χ n , l , m ,κ ( � (15) k +∆ k / 2 � Z ( k ′ , r ) dk ′ Y l , m ( θ, ϕ ) Z ( � ϕ k , l , m , ˜ r ) = N ( k , ∆ k ) F l , ˜ (16) k − ∆ k / 2 � � � π ˜ ( 2 kr ) l 2 k Z � � � Γ( l + 1 − i ˜ F l , ˜ Z ( k , r ) = π exp ( 2 l + 1 )! exp ( − ikr ) Z / k ) � × � � 2 k 1 F 1 ( 1 + l + i ˜ Z / k , 2 l + 2 , 2 ikr ) (17) 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 8 / 28

Overview of the Applied Theory The variation principle yields the spectrum of the bound states. One gets a generalized eigenvalue problem: Hc = E Sc (18) with � � � � � ˆ H ij = ψ i H � ψ j (19) � � and � � S ij = ψ i | ψ j (20) Here ψ j can refer either to a Slater-function or to a Coulomb wavepacket. Finally: M � � � � Φ j ( r ) = c j , p | ψ p ( r ) � (21) p = 1 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 9 / 28

Overview of the Applied Theory The couplings matrix is proportional to the dipole matrix: � � V kj = e Φ k ( r ) | rE ( t ) | Φ j ( r ) (22) The electric field can be written as E ( t ) = ε E 0 f ( t ) (23) The dipole matrix elements are therefore: � M M � � � � � D kj = e Φ k ( r ) | r ε ε ε | Φ j ( r ) = e c k , p ψ p ( r ) | r ε ε ε | c j , q ψ q ( r ) p = 1 q = 1 (24) M M � � c ∗ ε = k , p c j , q � ψ p ( r ) | r ε ε | ψ q ( r ) � p = 1 q = 1 2 � ε · r | Ψ ( i ) �� � Ψ ( f ) | ε s tot . = 4 π 2 α a 2 0 ω 0 ε (25) � � � � 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 10 / 28

Overview of the Applied Theory Let d denote the dipole matrices corresponding the basis functions: d pq := � ψ p ( r ) | r ε ε ε | ψ q ( r ) � (26) Using this notation, we get a compact formula for the dipole matrix elements: �� c ∗ � � D kj = Tr k ◦ c j d (27) For the electric field, I took the following form: � t � E ( t ) = e z E 0 cos 2 sin [ ω L ( t ) t ] (28) T with T being the pulse duration, ω L ( t ) = ω 0 + σ t (29) with ω 0 being the central laser frequency and σ the chirp parameter. 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 11 / 28

Results and Further Plans O UTLINE 1 O VERVIEW OF THE A PPLIED T HEORY 2 R ESULTS AND F URTHER P LANS 3 R EFERENCES 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 12 / 28

Results and Further Plans χ n = 4, l = 2, m = 0, κ = 0.4656 ( r ) = 1 ( r ) φ k = 0.2130, l = 0, m = 0, Z ∼ 0.05 0.0003 0.04 0.0002 0.0001 0.03 r 0.02 100 200 300 400 500 - 0.0001 0.01 - 0.0002 30 r [ a.u. ] - 0.0003 5 10 15 20 25 F IGURE : Slater-type orbital corresponding to the 4 d state, and an s wavefunction of the electron after absorbing a photon. 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 13 / 28

Results and Further Plans The Coulomb packets have been constructed according to E γ ( 800 nm ) = 0 . 0570 a . u . (30) E 7 s = − 0 . 0315 a . u . (31) ∆ E = E γ / 4 (32) E max = E 7 s + 3 E γ (33) The total energy range given above can be split into ten ∆ E width parts. For every part there exists a package with s , p , d and f azimuthal quantum number, respectively (40 total). Note that the Keldysh-parameter runs from 0 . 5916 to 5 . 916 if the intensities lie between 10 14 W · cm − 2 and 10 12 W · cm − 2 . The energy spectrum is defined by ∂ P 2 � � �� � Φ l ∂ E = E ( r ) | Ψ( t = T , r ) (34) � � � � l 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 14 / 28

Results and Further Plans F IGURE : Graphical overview of the spectrum of the Rubidium atom and ion. 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 15 / 28

Results and Further Plans The spectrum of the bound states: κ [ 1 ] E exp [ a . u . ] E opt E calc [ a . u . ] 1.35789 -0.153507 -0.144606 -0.148578 0.747335 -0.0617762 -0.0590814 -0.0649904 0.162072 -0.0336229 -0.0317344 -0.0370498 0.285925 -0.0211596 -0.0196661 -0.0236445 0.563253 -0.0145428 -0.0132598 -0.0161758 0.0995939 -0.0106093 -0.0093473 -0.011617 0.177069 -0.00808107 -0.00685427 -0.00856655 0.192805 -0.00636018 -0.00463121 -0.00653549 T ABLE : The spectrum of the bound states: E ns . n = 5 . . . 13 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 16 / 28

Results and Further Plans The spectrum of the bound states: κ [ 1 ] E exp [ a . u . ] E opt E calc [ a . u . ] 0.934074 -0.0961927 -0.0961927 -0.102753 0.0606182 -0.0454528 -0.0454529 -0.0500378 0.593825 -0.0266809 -0.0266805 -0.0293907 0.360146 -0.0175686 -0.0175692 -0.0191912 0.257965 -0.0124475 -0.0124474 -0.0134602 0.10425 -0.00928107 -0.00928073 -0.00994177 0.117591 -0.00718653 -0.00718617 -0.00763469 0.0654625 -0.00572873 -0.00572882 -0.00604301 0.111322 -0.0046738 -0.004674 -0.00489973 0.0972476 -0.0038856 -0.00388597 -0.00405129 0.0594662 -0.00328125 -0.0032817 -0.00340454 0.091899 -0.00280771 -0.00280802 -0.00290064 0.0660554 -0.00242976 -0.00242971 -0.00250062 0.0886535 -0.00212316 -0.00212282 -0.00217684 0.0606625 -0.0018712 -0.00187064 -0.00190818 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 17 / 28 T ABLE : The spectrum of the bound states: E np . n = 5 . . . 19

Results and Further Plans The spectrum of the bound states: κ [ 1 ] E exp [ a . u . ] E opt E calc [ a . u . ] 0.465587 -0.0653178 -0.0543145 -0.0544901 0.260371 -0.0364064 -0.0306089 -0.0307153 0.142432 -0.0227985 -0.0196441 -0.019709 0.231017 -0.0155403 -0.0136696 -0.0137109 0.12431 -0.0112513 -0.0100561 -0.0100841 0.144521 -0.00851559 -0.00770051 -0.0077216 0.107533 -0.00666683 -0.00606339 -0.00608392 T ABLE : The spectrum of the bound states: E nd . n = 4 . . . 10 5 th of May, 2017 M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 18 / 28