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

Nov 11, 2022 115 likes •460 views

I ONIZATION OF RUBIDIUM WITH ULTRASHORT INTENSE LASER PULSES Mihly Andrs Pocsai 1 , 2 , Imre Ferenc Barna 1 1 Wigner Research Centre of the H.A.S. 2 University of Pcs, Faculty of Sciences, Department of Physics Budapest, 5 th of May, 2017 2

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SLIDE 1

IONIZATION OF RUBIDIUM WITH ULTRASHORT

INTENSE LASER PULSES

Mihály András Pocsai1,2, Imre Ferenc Barna1

1Wigner Research Centre of the H.A.S. 2University of Pécs, Faculty of Sciences, Department of Physics

Budapest, 5th of May, 2017 2nd Wigner/MPP AWAKE Workshop

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 1 / 28

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SLIDE 2

OUTLINE

1 OVERVIEW OF THE APPLIED THEORY 2 RESULTS AND FURTHER PLANS 3 REFERENCES

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 2 / 28

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SLIDE 3

Overview of the Applied Theory

OUTLINE

1 OVERVIEW OF THE APPLIED THEORY 2 RESULTS AND FURTHER PLANS 3 REFERENCES

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 3 / 28

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SLIDE 4

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 ˆ H = ˆ HRb + ˆ VI (2) Hamilton operator of the Rb atom [1]: ˆ HRb = −1 2∇2 − 1 r (1 − be−dr) (3) with b = 4.5 and d = 1.09993.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 4 / 28

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SLIDE 5

Overview of the Applied Theory

In Length gauge, the interaction operator has the form of ˆ HI = er · E(t, r) (4) The electric field E(t, r) contains terms of e±i(ωLt−kr) = e±iωLte∓ikr Consider the Taylor-expansion of the spatial term: e∓ikr = 1 ∓ ikr + . . . (5) note that kr ∼ kr and k ∼ λ−1. The atomic distances are much smaller than the laser wavelength: kr ≪ 1. e±i(ωLt−kr) ≈ e±iωLt (6) This is the dipole approximation.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 5 / 28

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SLIDE 6

Overview of the Applied Theory

The solution of the TDSE can be expanded in terms of the eigenstates

  • f the time-independent Schrödinger equation:

ˆ H

  • Φj(r)
  • = Ej
  • Φj(r)
  • (7)

We apply the following Ansatz for the TDSE: |Ψ(t, r) =

N

  • j=1

aj(t)

  • Φj(r)
  • e−iEjt

(8) Inserting (8) into (1), using (7), we get: i

N

  • j=1

˙ aj(t)

  • Φj(r)
  • e−iEjt =

N

  • j=1

ˆ VIaj(t)

  • Φj(r)
  • e−iEjt

(9) Multiple the above equation by Φk(r)| eiEkt. We get the following system of equations for the aj(t) coefficients: ˙ ak(t) = −i

N

  • j=1

VkjeiEkjtaj(t) (k = 1 . . . N) (10)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 6 / 28

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SLIDE 7

Overview of the Applied Theory

In (10), Ekj := Ek − Ej and Vkj := Φk(r)| ˆ VI

  • Φj(r)
  • is the couplings

matrix. The (highly) oscillatory term can be transformed out. Let ˜ ak(t) := ak(t)e−iEkt (11) Inserting (11) into (10), we get: i ˙ ˜ ak(t) =

N

  • j=1

Vkj˜ aj(t) + Ek˜ ak(t) (12) Initial conditions: ak (t → −∞) =

  • 1

k = 1 k = 1 (13) Final state probabilities: Pk(t → ∞) = |ak(t → ∞)|2 (14)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 7 / 28

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SLIDE 8

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: χn,l,m,κ( r) = C(n, κ)r n−1e−κrYl,m(θ, ϕ) (15) ϕk,l,m,˜

Z(

r) = N(k, ∆k)

k+∆k/2

  • k−∆k/2

Fl,˜

Z(k′, r)dk′Yl,m(θ, ϕ)

(16) Fl,˜

Z(k, r) =

  • 2k

π exp

  • π ˜

Z 2k

  • (2kr)l

(2l + 1)! exp (−ikr)

  • Γ(l + 1 − i ˜

Z/k)

  • ×

1F1(1 + l + i ˜

Z/k, 2l + 2, 2ikr) (17)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 8 / 28

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SLIDE 9

Overview of the Applied Theory

The variation principle yields the spectrum of the bound states. One gets a generalized eigenvalue problem: Hc = ESc (18) with Hij =

  • ψi
  • ˆ

H

  • ψj
  • (19)

and Sij =

  • ψi|ψj
  • (20)

Here ψj can refer either to a Slater-function or to a Coulomb

  • wavepacket. Finally:
  • Φj(r)
  • =

M

  • p=1

cj,p |ψp(r) (21)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 9 / 28

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SLIDE 10

Overview of the Applied Theory

The couplings matrix is proportional to the dipole matrix: Vkj = e

  • Φk(r) |rE(t)| Φj(r)
  • (22)

The electric field can be written as E(t) = εE0f(t) (23) The dipole matrix elements are therefore: Dkj = e

  • Φk(r) |rε

ε ε| Φj(r)

  • = e

M

  • p=1

ck,pψp(r) |rε ε ε|

M

  • q=1

cj,qψq(r)

  • =

M

  • p=1

M

  • q=1

c∗

k,pcj,q ψp(r) |rε

ε ε| ψq(r) (24)

  • stot. = 4π2αa2

0ω0

  • Ψ(f) |ε

ε ε · r| Ψ(i)

  • 2

(25)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 10 / 28

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SLIDE 11

Overview of the Applied Theory

Let d denote the dipole matrices corresponding the basis functions: dpq := ψp(r) |rε ε ε| ψq(r) (26) Using this notation, we get a compact formula for the dipole matrix elements: Dkj = Tr

  • c∗

k ◦ cj

  • d
  • (27)

For the electric field, I took the following form: E(t) = ezE0 cos2 t T

  • sin [ωL(t)t]

(28) with T being the pulse duration, ωL(t) = ω0 + σt (29) with ω0 being the central laser frequency and σ the chirp parameter.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 11 / 28

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SLIDE 12

Results and Further Plans

OUTLINE

1 OVERVIEW OF THE APPLIED THEORY 2 RESULTS AND FURTHER PLANS 3 REFERENCES

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 12 / 28

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SLIDE 13

Results and Further Plans 5 10 15 20 25 30 r [a.u.] 0.01 0.02 0.03 0.04 0.05 χn=4, l=2, m=0, κ=0.4656 (r) 100 200 300 400 500 r

  • 0.0003
  • 0.0002
  • 0.0001

0.0001 0.0002 0.0003 φk=0.2130, l=0, m=0, Z

∼

=1(r)

FIGURE : Slater-type orbital corresponding to the 4d state, and an s wavefunction of the electron after absorbing a photon.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 13 / 28

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SLIDE 14

Results and Further Plans

The Coulomb packets have been constructed according to Eγ(800nm) = 0.0570 a.u. (30) E7s = −0.0315 a.u. (31) ∆E = Eγ/4 (32) Emax = E7s + 3Eγ (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 1014W · cm−2 and 1012W · cm−2. The energy spectrum is defined by ∂P ∂E =

  • l
  • Φl

E(r)|Ψ(t = T, r)

  • 2

(34)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 14 / 28

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SLIDE 15

Results and Further Plans

FIGURE : Graphical overview of the spectrum of the Rubidium atom and ion.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 15 / 28

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SLIDE 16

Results and Further Plans

The spectrum of the bound states: κ [1] Eexp [a.u.] Eopt Ecalc [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

TABLE : The spectrum of the bound states: Ens. n = 5 . . . 13

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 16 / 28

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SLIDE 17

Results and Further Plans

The spectrum of the bound states: κ [1] Eexp [a.u.] Eopt Ecalc [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

TABLE : The spectrum of the bound states: Enp. n = 5 . . . 19

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 17 / 28

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SLIDE 18

Results and Further Plans

The spectrum of the bound states: κ [1] Eexp [a.u.] Eopt Ecalc [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

TABLE : The spectrum of the bound states: End. n = 4 . . . 10

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 18 / 28

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SLIDE 19

Results and Further Plans

The spectrum of the bound states: κ [1] Eexp [a.u.] Eopt Ecalc [a.u.] 0.247252

  • 0.0314329
  • 0.0312247
  • 0.031225

0.164201

  • 0.0201073
  • 0.019978
  • 0.0199783

0.115393

  • 0.0139554
  • 0.0138729
  • 0.0138734

0.108716

  • 0.0102476
  • 0.0101934
  • 0.0101937

0.187485

  • 0.00784234
  • 0.007803
  • 0.00780359

TABLE : The spectrum of the bound states: Enf. n = 4 . . . 8

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 19 / 28

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SLIDE 20

Results and Further Plans

5000 10000 15000t [a.u.] 0.2 0.4 0.6 0.8 1.0 |ak

2 [1]

|a5 s

2

|aC,p

2

FIGURE : Time evolution of the occupation probabilities. λ = 800 nm, T = 120 fs, I = 1012 W · cm−2, σ = −7.3133 · 10−7a.u.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 20 / 28

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SLIDE 21

Results and Further Plans

FIGURE : For details, see: M. Aladi et. al: Nucl. Intstr. Meth. Phys. Res. A 740, 203–207 (2014)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 21 / 28

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SLIDE 22

Results and Further Plans

FIGURE : For details, see: F . Morales et. al: PNAS 108, 16906–16911 (2011)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 22 / 28

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SLIDE 23

Results and Further Plans

FIGURE : Energy differential cross section.

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 23 / 28

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SLIDE 24

Results and Further Plans

FIGURE : Energy spectrum of the continuum electron. I = 1012W · cm−2 (left) and I = 1013W · cm−2 (right.)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 24 / 28

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SLIDE 25

Results and Further Plans

FIGURE : Energy spectrum of the continuum electron. I = 1014W · cm−2

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 25 / 28

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SLIDE 26

Results and Further Plans

Further plans: Discrectize the continuum more efficiently (in progress GENARGRID)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 26 / 28

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SLIDE 27

Results and Further Plans

Further plans: Discrectize the continuum more efficiently (in progress GENARGRID) Study photoionization in details (ATI, angular differential cross section, total cross section etc.)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 26 / 28

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SLIDE 28

Results and Further Plans

Further plans: Discrectize the continuum more efficiently (in progress GENARGRID) Study photoionization in details (ATI, angular differential cross section, total cross section etc.) Determine optimal laser parameters for ionisation

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 26 / 28

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SLIDE 29

Results and Further Plans

Further plans: Discrectize the continuum more efficiently (in progress GENARGRID) Study photoionization in details (ATI, angular differential cross section, total cross section etc.) Determine optimal laser parameters for ionisation Study propagation effects (with G. Demeter)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 26 / 28

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SLIDE 30

Results and Further Plans

Further plans: Discrectize the continuum more efficiently (in progress GENARGRID) Study photoionization in details (ATI, angular differential cross section, total cross section etc.) Determine optimal laser parameters for ionisation Study propagation effects (with G. Demeter) Provide quality input data for PIC simulations

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 26 / 28

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SLIDE 31

References

OUTLINE

1 OVERVIEW OF THE APPLIED THEORY 2 RESULTS AND FURTHER PLANS 3 REFERENCES

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 27 / 28

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SLIDE 32

[1] M. Z. Miloševi´ c, N. S. Simonovi´ c, Phys. Rev. A 91, 023424 (2015) [2] M. Aladi et. al: Nucl. Intstr. Meth. Phys. Res. A 740, 203–207 (2014) [3] I. F . Barna et. al: Eur. Phys. J. D 25, 239–246 (2003) [4] R. V. Ambartzumyan et. al: Zh. Eksp. Theor. Fiz. 70, 1660–1673 (1976)

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 28 / 28

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SLIDE 33

[5] F . Morales et. al: PNAS 108, 16906–16911 (2011)

Thank you for your attention!

M.A. Pocsai, I.F. Barna (Wigner/UP) Rubidium ionization 5th of May, 2017 28 / 28