- Email: [email protected]
Interaction of intense short laser pulses with positronium
Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Interaction of intense short laser pulses with positronium }kési b, L. Nagy a S. Borbély a,*, K. To a b
Faculty of Physics, Babesß-Bolyai University, Str. Koga˘lniceanu Nr. 1, 400084 Cluj-Napoca, Romania Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Debrecen, Hungary
a r t i c l e
i n f o
Available online 21 October 2008 PACS: 32.80.Rm 32.80.Fm 42.50.Hz Keywords: Over-the-barrier ionization Intense Ultrashort laser pulses Positronium
a b s t r a c t The ionization of the positronium is studied in the over-the-barrier regime. The Volkov and momentumspace strong-field approximation are applied to describe the ionization of the hydrogen-like atomic systems. Classical trajectory Monte Carlo calculations were also performed and a good agreement with quantum-mechanical calculations were found. The scaling of the ionization probability density as function of the reduced mass of the target is analyzed. We find that the maxima of the ionization probability density is determined by laser field parameters, and it’s width can be characterised by the charge of the core and reduced mass of the target. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The recent development of the positron physics [1,2] made available the experimental investigation of the interaction between the positronium and short laser pulses [1]. It is theoretically proved that the interaction of the positronium (Ps) with laser pulses is closely related to the hydrogen atom by scaling the laser frequency and intensity [3]. The ionization of positronium in intense laser fields was studied by several groups [3,4]. These studies are mainly focusing on the multiphoton [4] and above-threshold ionization (ATI) [4]. To the best of our knowledge, there is no theoretical study of the positronium ionization in the over-the-barrier (collisional) regime. The investigation of the ionization spectra in the collisional regime is a subject of interest, due to the fact that the underlying ionization mechanism is completely different from the multiphoton and ATI ionization mechanisms. The ionization in the over-the-barrier regime is considered to be a classical process, and it is believed that it can be described well by the classical trajectory Monte Carlo (CTMC) method [5,6]. There are also a few quantum-mechanical models, which are based on the approximate solution of the time dependent Schrödinger equation. The most simple one is the Volkov model [7], which is based on the sudden approximation and on the use of the Volkov wave functions to represent the final state. It is proved that the Volkov model provides good results only when the intensity of the laser field and the momentum transferred to the electron from the laser field is high. * Corresponding author. E-mail address: [email protected] (S. Borbély). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.041
The improved version of the Volkov model is the Coulomb–Volkov model (CV1) [8], which differs from the Volkov model by the use of Coulomb–Volkov wave functions for the final state. This model gives significantly more accurate results but this is still limited by the sudden approximation. It provides accurate results only if the pulse duration is less than two orbital periods of the active electron [8,9]. The momentum space strong-field approximation (MSSFA) eliminates this disadvantage of the CV1 model and it provides accurate results for high momentum transfer [7]. In the present work the Volkov and MSSFA models are extended to describe the ionization of the hydrogen-type atomic systems in the over-the-barrier regime. These extended models are applied to study the ionization cross section of the positronium. Based on the Volkov model an analytical scaling law between the ionization probability densities of the hydrogen atom and of the hydrogentype systems is established. This scaling law is also verified for the ionization probabilities obtained by MSSFA and CTMC method.
2. Theory 2.1. The laser field The linearly polarized laser pulse is represented by its electric component
( ~ E¼
^eE0 sinðxðt 2sÞ p2 Þ sin2 ðpstÞ if t 2 ½0; s 0
elsewhere
;
ð1Þ
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389
where ^e is the polarization vector along which the laser pulse is polarized, x is the frequency of the carrier wave and s is the pulse duration. Here the dipole approximation is implicitly applied when the spatial dependence of the external electric field ~ E is neglected in Eq. (1).
The studied hydrogen-type system consists of a core with mass mc , with charge Z eff and of an active electron with mass me . This system is investigated by separating the motion of the center of mass (with mass M ¼ me þ mc ) and the relative motion of the two particles reduced to the motion of a particle with the reduced mass l ¼ meMmc . The time dependent Schrödinger equation (TDSE) for the reduced mass particle is given as follows:
^2 Z eff p þ~ r~ EðtÞ WðtÞ; 2l r
o WðtÞ ¼ ot
ð2Þ
where WðtÞ is the time dependent wave function of the system. In order to determine the time evolution of the studied system one needs to know the WðtÞ wave function. In the present approach the wave function is assumed to have the following form:
WðtÞ ¼
Z
d~ kcð~ k; tÞWV ð~ k; tÞ;
ð3Þ
where WV ð~ k; tÞ are the Volkov wave functions. The Volkov wave functions are the solutions of the TDSE in dipole approximation for a free charged particle in radiation field and they can be expressed as i
WV ð~ k; tÞ ¼ e 2l
Rt 0
dt0 ð~ kþ~ Aðt 0 ÞÞ2 ið~ kþ~ AðtÞÞ~ r
e
:
ð4Þ
In the above expression
~ AðtÞ ¼
Z
t
0 ~ Eðt 0 Þdt
ð5Þ
0
is the vector potential of the electromagnetic field. The time dependent wave function is well defined by the expansion coefficients cð~ k; tÞ. Therefore the time evolution of the system can be calculated using these coefficients. By substituting the time dependent wave function from Eq. (3) into the TDSE given by Eq. (2), after elementary calculations the following equation for the expansion coefficients can be obtained
Z Rt 0 o i i dt ~ s ~ sþ2~ qþ2~ Aðt 0 ÞÞ cð~ q; tÞ ¼ d~ scð~ s þ~ q; tÞ e 2l 0 ð 3 ot ð2pÞ Z ~ drVð~ rÞei~s~r :
The initial condition for Eq. (6) can be obtained from the continuity of the wave function at t ¼ 0 as follows:
cð~ q; t ¼ 0Þ ¼
1 ð2pÞ3
hei~q~r jwi i:
ð7Þ
The transition probability from initial state wi to a free final state wf with a well defined momentum ~ p is given as (see for more details [7])
2.2. The hydrogen-type atomic system
i
387
2 Pi!f ¼ ð2pÞ3 cð~ p ~ AðsÞ; sÞ :
ð8Þ
2.3. The Volkov and MSSFA model The simplest way of solving Eq. (6) is by neglecting completely the Coulomb potential between the active electron and the nucleus (Z eff ¼ 0). This zero order solution of Eq. (6), also called the Volkov model, provides good results only for high laser field intensities, with high momentum transfer. Using initial condition given by Eq. (7) the Volkov model has the following analytical solution:
cð~ q; tÞ cð0Þ ð~ qÞ ¼
1 ð2pÞ3
hei~q~r jwi i:
ð9Þ
In most of the cases, the Volkov model does not provide results accurate enough, because the Coulomb interaction at moderate laser field intensities can not be totally neglected. At these intensities it is safe to assume [7] that the influence of the Coulomb interaction on the time evolution of the studied system during the laser field can be taken into account as a perturbation, and the expansion coefficients cð~ k; tÞ are close to those provided by the Volkov model (see Eq. (9)). Based on this arguments Eq. (6) can be simplified by replackÞ ing cð~ k; tÞ on the right-hand side by cð0Þ ð~
Z Rt 0 o ð1Þ i i dt ~ s ~ sþ2~ qþ2~ Aðt 0 ÞÞ c ð~ d~ scð0Þ ð~ q; tÞ ¼ s þ~ q Þ e 2l 0 ð 3 ot ð2pÞ Z ~ drVð~ rÞei~s~r :
ð10Þ
This method is closely related to the strong-field approximation (SFA) [10], the main difference is that our calculations are performed in the momentum-space. Therefore, to distinguish our scheme from the SFA, we call our present method momentumspace SFA (MSSFA) [7]. 2.4. The CTMC method
ð6Þ
In the present approach, Newton’s classical non-relativistic equations of motions are solved [5–7] numerically when an external laser field given by Eq. (1) is included. The initial position and
Fig. 1. Double differential ionization probability densities for E0 = 1 a.u., x ¼ 0:05 a:u:, s ¼ 5 a:u: kz – the component of the electron momentum parallel with the polarization vector ^e; kq – the component of the electron momentum perpendicular to ^e.
388
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389
momentum vectors of the active electron are distributed according to the microcannonical distribution
~ p2 f ð~ p;~ rÞ d Ei Vð~ rÞ ; 2l
ð11Þ
where Ei is the ionization potential. For these initial parameters, the equations of motion were integrated with respect to time by standard Runge–Kutta method until the real exit channels were obtained.
Fig. 2. Ionization probability densities as a function of the electron momentum for x ¼ 0:05 a:u: and E0 ¼ 1 a:u: at different pulse durations. Solid line – MSSFA. Dashed line – Volkov. Squares – CTMC.
389
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389 Table 1 The FWHM values of the ionization probability curves presented on Fig. 2.
3. Results Calculations were performed on the positronium target with Z eff ¼ 1 and l ¼ 0:5 using laser pulses with duration s of 3 a.u., 5 a.u. and 10 a.u. at laser field intensity E0 = 1 a.u. The carrier wave frequency is x ¼ 0:05 a:u:, which is close to the carrier wave frequency generated by Ti-sapphire lasers. These pulse parameters lead to the value of 0.05 for the Keldysh parameter, which is a characteristic value for the over-the-barrier ionization regime. The double differential ionization probability densities are calculated using the Volkov, MSSFA and CTMC models and they are presented on Fig. 1, where the ionization probability is plotted as a function of the momentum components ( pz – parallel with ^e; pq – perpendicular to ^e) of the ejected electron. At first sight all three models predict the same probability densities. In each case the electrons are ejected with maximum probability along the polarization vecAðsÞ, which is the value of the tor ^e, with momentum around ~ momentum gain from the external laser field. After detailed analysis, however, important differences can be identified. For the MSSFA and CTMC the maxima of the predicted probability densities are shifted toward smaller momenta. This shift is caused by the Coulomb attraction of the core during ionization. In a classical picture the electrons are decelerated by the Coulomb attraction. It is interesting to study how the ionization probability density scales in the case of hydrogen-type systems. For this purpose the Volkov model is an ideal tool, because it is completely analytical, allowing the derivation of exact scaling relations. In the framework of the Volkov model the ionization probability density for a hydrogentype system is given by
q 5 dP 1 ¼ ð2pÞ3 4 ; ~ p dk q2 þ ð~ k ~ AðsÞÞ2
ð12Þ
where ~ AðsÞ is the momentum transferred to the active electron from the laser field and q ¼ Zaeffl with al ¼ l1 the first Bohr radius. From Eq. (12) the scaling relations of the ionization probability density can be extracted. It can be observed that the electrons are ejected with maximum probability with momentum ~ k¼~ AðsÞ, which is determined by the laser pulse and it is not influenced by the parameters of the hydrogen-type systems. On the other hand the distribution of the electrons around this maximum is determined by the q parameter, which is defined by the parameters of the hydrogen-type system and it does not depend on the laser field parameters. For the characterization of the width of the distribution the full-width at half-maximum (FWHM) is used. In the Volkov model the FWHM can be written as
dFWHM q ¼ Z eff l:
ð13Þ
This means that the FWHM of the ionization probability densities in the case of hydrogen-type systems is directly proportional with the effective charge of the core (Z eff ) and with the reduced mass of the target (l). The same type of scaling of positron- and electron-impact ionization cross sections was observed experimentally [11], where the FWHM of the total ionization cross section as a function of the projectile energy was proportional with the ionization energy, i.e. with Z 2eff l. In our case, if the ionization probability is expressed as a function of electron energy, the FWHM is also proportional with Z 2eff l. The good agreement between the scaling relations can be a pure coincidence, or it can hide a basic physical cause. To decide which statement is valid, further investigations are needed, which can be the subject of a future investigation. For positronium with effective charge Z eff ¼ 1 and reduced mass l ¼ 0:5 the FWHM is the half of the FWHM for hydrogen atom. This finding can be verified on Fig. 2, where we have presented the MSSFA, Volkov and CTMC ionization probability densities as a func-
MSSFA
s¼3 s¼5 s ¼ 10
Volkov
CTMC
H
H/2
Ps
H
H/2
Ps
H
H/2
Ps
1.164 1.230 1.229
0.582 0.615 0.614
0.632 0.668 0.692
0.998 1.011 1.017
0.449 0.505 0.508
0.507 0.508 0.509
1.039 1.038 1.090
0.519 0.519 0.545
0.515 0.506 0.512
tion of the electron momentum for positronium and hydrogen target at different laser field parameters. According to Fig. 2, the maxima of the presented ionization probabilities for given pulse parameters is located around the same momentum value both for the positronium and hydrogen. The peaks of the ionization probability density calculated in a given theoretical framework for hydrogen are twice as wide as the peaks for positronium. To verify this qualitative observation we have calculated the FWHM for different pulse parameters for the hydrogen and positronium using the different model curves for the ionization probability densities. The obtained values are presented in Table 1, where we also presented the half of the FWHM for the hydrogen. An excellent agreement with the scaling law in the case of the Volkov model is observed. The good agreement with the scaling relation (see Eq. (13)) using CTMC and MSSFA approaches suggest that the scaling low is generally valid and it is not model dependent. 4. Conclusions The ionization of the hydrogen-type systems by intense ultrashort laser pulses was theoretically studied in the over-the-barrier regime. The Volkov and MSSFA models were successfully extended to describe the ionization of the positronium, and a good agreement between quantum-mechanical and classical calculations was found. The scaling of the ionization probability density as function of the parameters of hydrogen-type system was also investigated. We found that the maxima of the ionization probability density is determined by laser field parameters, and it’s width can be characterised by the charge of the core and reduced mass of the target. Acknowledgements This work was supported by the Romanian Academy of Sciences (Grant No. 31/2008), the Romanian National Plan for Research (PNII) under contract No. ID 539, the Hungarian National Office for Research and Technology, the grant ‘‘Bolyai” from the Hungarian Academy of Sciences, and the Hungarian research Found OTKA (K72172).
References [1] P. Baling, D. Fregenal, T. Ichioka, H. Knudsen, H.-P. E Kristiansen, J. Merrison, U.I. Uggerhoj, Nucl. Instr. and Meth. B 221 (2004) 200. [2] S.J. Brawley, J. Beale, S. Armitage, D.E. Leslie, A. Kover, G. Laricchia, Nucl. Instr. and Meth. B 266 (2008) 497. [3] L.B. Madsen, L. Lambropoulos, Phys. Rev. A 59 (1999) 4574. [4] V.D. Rodríguez, Nucl. Instr. and Meth. B 247 (2006) 105. [5] J.P. Hansen, J. Lu, L.B. Madsen, H.M. Nilsen, Phys. Rev. A 64 (2001) 033418. [6] C.R. Feeler, R.E. Olson, J. Phys. B: Atom. Mol. Opt. Phys. 33 (2000) 1997. }kési, L. Nagy, Phys. Rev. A 77 (2008) 033412. [7] S. Borbély, K. To [8] G. Duchateau, C. Illescas, B. Pons, E. Cormier, R. Gayet, J. Phys. B: Atom. Mol. Opt. Phys. 33 (2000) L571. [9] D.G. Arbó, J.E. Miraglia, M.S. Gravielle, K. Schiessl, E. Persson, J. Burgdrfer, Phys. Rev. A 77 (2008) 013401. [10] B.D. Milosevic, G.G. Paulus, D. Bauer, W. Becker, J. Phys. B: Atom. Mol. Opt. Phys. 39 (2006) R203. [11] M. Szluin`ska, P. Van Reeth, G. Laricchia, J. Phys. B: Atom. Mol. Opt. Phys. 35 (2002) 4059.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Interaction of intense short laser pulses with positronium }kési b, L. Nagy a S. Borbély a,*, K. To a b
Faculty of Physics, Babesß-Bolyai University, Str. Koga˘lniceanu Nr. 1, 400084 Cluj-Napoca, Romania Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Debrecen, Hungary
a r t i c l e
i n f o
Available online 21 October 2008 PACS: 32.80.Rm 32.80.Fm 42.50.Hz Keywords: Over-the-barrier ionization Intense Ultrashort laser pulses Positronium
a b s t r a c t The ionization of the positronium is studied in the over-the-barrier regime. The Volkov and momentumspace strong-field approximation are applied to describe the ionization of the hydrogen-like atomic systems. Classical trajectory Monte Carlo calculations were also performed and a good agreement with quantum-mechanical calculations were found. The scaling of the ionization probability density as function of the reduced mass of the target is analyzed. We find that the maxima of the ionization probability density is determined by laser field parameters, and it’s width can be characterised by the charge of the core and reduced mass of the target. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The recent development of the positron physics [1,2] made available the experimental investigation of the interaction between the positronium and short laser pulses [1]. It is theoretically proved that the interaction of the positronium (Ps) with laser pulses is closely related to the hydrogen atom by scaling the laser frequency and intensity [3]. The ionization of positronium in intense laser fields was studied by several groups [3,4]. These studies are mainly focusing on the multiphoton [4] and above-threshold ionization (ATI) [4]. To the best of our knowledge, there is no theoretical study of the positronium ionization in the over-the-barrier (collisional) regime. The investigation of the ionization spectra in the collisional regime is a subject of interest, due to the fact that the underlying ionization mechanism is completely different from the multiphoton and ATI ionization mechanisms. The ionization in the over-the-barrier regime is considered to be a classical process, and it is believed that it can be described well by the classical trajectory Monte Carlo (CTMC) method [5,6]. There are also a few quantum-mechanical models, which are based on the approximate solution of the time dependent Schrödinger equation. The most simple one is the Volkov model [7], which is based on the sudden approximation and on the use of the Volkov wave functions to represent the final state. It is proved that the Volkov model provides good results only when the intensity of the laser field and the momentum transferred to the electron from the laser field is high. * Corresponding author. E-mail address: [email protected] (S. Borbély). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.041
The improved version of the Volkov model is the Coulomb–Volkov model (CV1) [8], which differs from the Volkov model by the use of Coulomb–Volkov wave functions for the final state. This model gives significantly more accurate results but this is still limited by the sudden approximation. It provides accurate results only if the pulse duration is less than two orbital periods of the active electron [8,9]. The momentum space strong-field approximation (MSSFA) eliminates this disadvantage of the CV1 model and it provides accurate results for high momentum transfer [7]. In the present work the Volkov and MSSFA models are extended to describe the ionization of the hydrogen-type atomic systems in the over-the-barrier regime. These extended models are applied to study the ionization cross section of the positronium. Based on the Volkov model an analytical scaling law between the ionization probability densities of the hydrogen atom and of the hydrogentype systems is established. This scaling law is also verified for the ionization probabilities obtained by MSSFA and CTMC method.
2. Theory 2.1. The laser field The linearly polarized laser pulse is represented by its electric component
( ~ E¼
^eE0 sinðxðt 2sÞ p2 Þ sin2 ðpstÞ if t 2 ½0; s 0
elsewhere
;
ð1Þ
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389
where ^e is the polarization vector along which the laser pulse is polarized, x is the frequency of the carrier wave and s is the pulse duration. Here the dipole approximation is implicitly applied when the spatial dependence of the external electric field ~ E is neglected in Eq. (1).
The studied hydrogen-type system consists of a core with mass mc , with charge Z eff and of an active electron with mass me . This system is investigated by separating the motion of the center of mass (with mass M ¼ me þ mc ) and the relative motion of the two particles reduced to the motion of a particle with the reduced mass l ¼ meMmc . The time dependent Schrödinger equation (TDSE) for the reduced mass particle is given as follows:
^2 Z eff p þ~ r~ EðtÞ WðtÞ; 2l r
o WðtÞ ¼ ot
ð2Þ
where WðtÞ is the time dependent wave function of the system. In order to determine the time evolution of the studied system one needs to know the WðtÞ wave function. In the present approach the wave function is assumed to have the following form:
WðtÞ ¼
Z
d~ kcð~ k; tÞWV ð~ k; tÞ;
ð3Þ
where WV ð~ k; tÞ are the Volkov wave functions. The Volkov wave functions are the solutions of the TDSE in dipole approximation for a free charged particle in radiation field and they can be expressed as i
WV ð~ k; tÞ ¼ e 2l
Rt 0
dt0 ð~ kþ~ Aðt 0 ÞÞ2 ið~ kþ~ AðtÞÞ~ r
e
:
ð4Þ
In the above expression
~ AðtÞ ¼
Z
t
0 ~ Eðt 0 Þdt
ð5Þ
0
is the vector potential of the electromagnetic field. The time dependent wave function is well defined by the expansion coefficients cð~ k; tÞ. Therefore the time evolution of the system can be calculated using these coefficients. By substituting the time dependent wave function from Eq. (3) into the TDSE given by Eq. (2), after elementary calculations the following equation for the expansion coefficients can be obtained
Z Rt 0 o i i dt ~ s ~ sþ2~ qþ2~ Aðt 0 ÞÞ cð~ q; tÞ ¼ d~ scð~ s þ~ q; tÞ e 2l 0 ð 3 ot ð2pÞ Z ~ drVð~ rÞei~s~r :
The initial condition for Eq. (6) can be obtained from the continuity of the wave function at t ¼ 0 as follows:
cð~ q; t ¼ 0Þ ¼
1 ð2pÞ3
hei~q~r jwi i:
ð7Þ
The transition probability from initial state wi to a free final state wf with a well defined momentum ~ p is given as (see for more details [7])
2.2. The hydrogen-type atomic system
i
387
2 Pi!f ¼ ð2pÞ3 cð~ p ~ AðsÞ; sÞ :
ð8Þ
2.3. The Volkov and MSSFA model The simplest way of solving Eq. (6) is by neglecting completely the Coulomb potential between the active electron and the nucleus (Z eff ¼ 0). This zero order solution of Eq. (6), also called the Volkov model, provides good results only for high laser field intensities, with high momentum transfer. Using initial condition given by Eq. (7) the Volkov model has the following analytical solution:
cð~ q; tÞ cð0Þ ð~ qÞ ¼
1 ð2pÞ3
hei~q~r jwi i:
ð9Þ
In most of the cases, the Volkov model does not provide results accurate enough, because the Coulomb interaction at moderate laser field intensities can not be totally neglected. At these intensities it is safe to assume [7] that the influence of the Coulomb interaction on the time evolution of the studied system during the laser field can be taken into account as a perturbation, and the expansion coefficients cð~ k; tÞ are close to those provided by the Volkov model (see Eq. (9)). Based on this arguments Eq. (6) can be simplified by replackÞ ing cð~ k; tÞ on the right-hand side by cð0Þ ð~
Z Rt 0 o ð1Þ i i dt ~ s ~ sþ2~ qþ2~ Aðt 0 ÞÞ c ð~ d~ scð0Þ ð~ q; tÞ ¼ s þ~ q Þ e 2l 0 ð 3 ot ð2pÞ Z ~ drVð~ rÞei~s~r :
ð10Þ
This method is closely related to the strong-field approximation (SFA) [10], the main difference is that our calculations are performed in the momentum-space. Therefore, to distinguish our scheme from the SFA, we call our present method momentumspace SFA (MSSFA) [7]. 2.4. The CTMC method
ð6Þ
In the present approach, Newton’s classical non-relativistic equations of motions are solved [5–7] numerically when an external laser field given by Eq. (1) is included. The initial position and
Fig. 1. Double differential ionization probability densities for E0 = 1 a.u., x ¼ 0:05 a:u:, s ¼ 5 a:u: kz – the component of the electron momentum parallel with the polarization vector ^e; kq – the component of the electron momentum perpendicular to ^e.
388
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389
momentum vectors of the active electron are distributed according to the microcannonical distribution
~ p2 f ð~ p;~ rÞ d Ei Vð~ rÞ ; 2l
ð11Þ
where Ei is the ionization potential. For these initial parameters, the equations of motion were integrated with respect to time by standard Runge–Kutta method until the real exit channels were obtained.
Fig. 2. Ionization probability densities as a function of the electron momentum for x ¼ 0:05 a:u: and E0 ¼ 1 a:u: at different pulse durations. Solid line – MSSFA. Dashed line – Volkov. Squares – CTMC.
389
S. Borbély et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 386–389 Table 1 The FWHM values of the ionization probability curves presented on Fig. 2.
3. Results Calculations were performed on the positronium target with Z eff ¼ 1 and l ¼ 0:5 using laser pulses with duration s of 3 a.u., 5 a.u. and 10 a.u. at laser field intensity E0 = 1 a.u. The carrier wave frequency is x ¼ 0:05 a:u:, which is close to the carrier wave frequency generated by Ti-sapphire lasers. These pulse parameters lead to the value of 0.05 for the Keldysh parameter, which is a characteristic value for the over-the-barrier ionization regime. The double differential ionization probability densities are calculated using the Volkov, MSSFA and CTMC models and they are presented on Fig. 1, where the ionization probability is plotted as a function of the momentum components ( pz – parallel with ^e; pq – perpendicular to ^e) of the ejected electron. At first sight all three models predict the same probability densities. In each case the electrons are ejected with maximum probability along the polarization vecAðsÞ, which is the value of the tor ^e, with momentum around ~ momentum gain from the external laser field. After detailed analysis, however, important differences can be identified. For the MSSFA and CTMC the maxima of the predicted probability densities are shifted toward smaller momenta. This shift is caused by the Coulomb attraction of the core during ionization. In a classical picture the electrons are decelerated by the Coulomb attraction. It is interesting to study how the ionization probability density scales in the case of hydrogen-type systems. For this purpose the Volkov model is an ideal tool, because it is completely analytical, allowing the derivation of exact scaling relations. In the framework of the Volkov model the ionization probability density for a hydrogentype system is given by
q 5 dP 1 ¼ ð2pÞ3 4 ; ~ p dk q2 þ ð~ k ~ AðsÞÞ2
ð12Þ
where ~ AðsÞ is the momentum transferred to the active electron from the laser field and q ¼ Zaeffl with al ¼ l1 the first Bohr radius. From Eq. (12) the scaling relations of the ionization probability density can be extracted. It can be observed that the electrons are ejected with maximum probability with momentum ~ k¼~ AðsÞ, which is determined by the laser pulse and it is not influenced by the parameters of the hydrogen-type systems. On the other hand the distribution of the electrons around this maximum is determined by the q parameter, which is defined by the parameters of the hydrogen-type system and it does not depend on the laser field parameters. For the characterization of the width of the distribution the full-width at half-maximum (FWHM) is used. In the Volkov model the FWHM can be written as
dFWHM q ¼ Z eff l:
ð13Þ
This means that the FWHM of the ionization probability densities in the case of hydrogen-type systems is directly proportional with the effective charge of the core (Z eff ) and with the reduced mass of the target (l). The same type of scaling of positron- and electron-impact ionization cross sections was observed experimentally [11], where the FWHM of the total ionization cross section as a function of the projectile energy was proportional with the ionization energy, i.e. with Z 2eff l. In our case, if the ionization probability is expressed as a function of electron energy, the FWHM is also proportional with Z 2eff l. The good agreement between the scaling relations can be a pure coincidence, or it can hide a basic physical cause. To decide which statement is valid, further investigations are needed, which can be the subject of a future investigation. For positronium with effective charge Z eff ¼ 1 and reduced mass l ¼ 0:5 the FWHM is the half of the FWHM for hydrogen atom. This finding can be verified on Fig. 2, where we have presented the MSSFA, Volkov and CTMC ionization probability densities as a func-
MSSFA
s¼3 s¼5 s ¼ 10
Volkov
CTMC
H
H/2
Ps
H
H/2
Ps
H
H/2
Ps
1.164 1.230 1.229
0.582 0.615 0.614
0.632 0.668 0.692
0.998 1.011 1.017
0.449 0.505 0.508
0.507 0.508 0.509
1.039 1.038 1.090
0.519 0.519 0.545
0.515 0.506 0.512
tion of the electron momentum for positronium and hydrogen target at different laser field parameters. According to Fig. 2, the maxima of the presented ionization probabilities for given pulse parameters is located around the same momentum value both for the positronium and hydrogen. The peaks of the ionization probability density calculated in a given theoretical framework for hydrogen are twice as wide as the peaks for positronium. To verify this qualitative observation we have calculated the FWHM for different pulse parameters for the hydrogen and positronium using the different model curves for the ionization probability densities. The obtained values are presented in Table 1, where we also presented the half of the FWHM for the hydrogen. An excellent agreement with the scaling law in the case of the Volkov model is observed. The good agreement with the scaling relation (see Eq. (13)) using CTMC and MSSFA approaches suggest that the scaling low is generally valid and it is not model dependent. 4. Conclusions The ionization of the hydrogen-type systems by intense ultrashort laser pulses was theoretically studied in the over-the-barrier regime. The Volkov and MSSFA models were successfully extended to describe the ionization of the positronium, and a good agreement between quantum-mechanical and classical calculations was found. The scaling of the ionization probability density as function of the parameters of hydrogen-type system was also investigated. We found that the maxima of the ionization probability density is determined by laser field parameters, and it’s width can be characterised by the charge of the core and reduced mass of the target. Acknowledgements This work was supported by the Romanian Academy of Sciences (Grant No. 31/2008), the Romanian National Plan for Research (PNII) under contract No. ID 539, the Hungarian National Office for Research and Technology, the grant ‘‘Bolyai” from the Hungarian Academy of Sciences, and the Hungarian research Found OTKA (K72172).
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