Instantaneous multiphoton ionization rate and initial distribution of electron momentum

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Instantaneous multiphoton ionization rate and initial distribution of

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Bondar, Denys I.

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Instantaneous multiphoton ionization rate and initial distribution of

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Bondar, Denys I.

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Instantaneous multiphoton ionization rate and initial distribution of electron momentum

Denys I. Bondar

*

Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 and National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A 0R6

共Received 2 May 2008; published 22 July 2008兲

The Yudin-Ivanov formula关Phys. Rev. A 64, 013409 共2001兲兴 is generalized such that the most general analytical expression for single-electron spectra, which includes the dependence on the instantaneous laser phase, is obtained within the strong field approximation. No assumptions on the momentum of the electron are made. Previously known formulas for single-electron spectra can be obtained as approximations to the general formula.

DOI:10.1103/PhysRevA.78.015405 PACS number共s兲: 32.80.Rm, 32.80.Fb

Although an initial theoretical understanding of strong field ionization was put forth by Keldysh 关1兴 as early as 1964, many questions remain unresolved. As far as single-electron ionization in the presence of a linearly polarized laser field is concerned, there are two important topics. The first, namely the ionization rate as a function of instanta-neous laser phase, was studied in depth by Yudin and Ivanov 关2兴 共see also 关3,4兴兲. However, their result assumes zero initial momentum of the liberated electron. Effects due to nonzero initial momentum have yet to be included. The second topic pertains to the single-electron spectra, that is, the ionization rate as a function of the final momentum of an electron. Despite much work on this topic 共see, for example, Refs. 关5–7兴 and references therein兲 a universally accepted formula is lacking and the discussion is still ongoing. Among the most accurate results, Goreslavskii et al. 关6兴 have obtained an expression for the complete single-electron ionization spectrum, but without consideration of laser phase. The present paper derives a more general formula 关see Eq. 共8兲兴 that includes the dependences on both the instantaneous laser phase and the final electron momentum.

It is very convenient to formulate the Keldysh theory in terms of the Dykhne adiabatic approximation, as was presented in 关8兴. According to the Dykhne method 关9–11兴 共see also 关8,12兴兲, if the Hamiltonian of a system Hˆ共t兲 is a slowly varying function of time t, and Hˆ 共t兲␺n共t兲 = En共t兲␺n共t兲

共n = i , f兲, then the probability ⌫ of the transition ␺i→␺f is

given by共atomic units ប = me=兩e兩 = 1 are used throughout兲

⌫ = exp关− 2 Im共S兲兴, S=

冕

t₁ t₀

关Ef共t兲 − Ei共t兲兴dt, 共1兲

where t1 is any point on the real axis of t, and t0 is the

transition point, i.e., a complex root of the equation

Ei共t0兲 = Ef共t0兲, 共2兲

which lies in the upper half-plane. If there are several roots, we must choose one that is the closest to the real axis of t.

Moreover, there are no assumptions regarding the form of the Hamiltonian.

Now we shall apply the Dykhne approach to the problem of ionization of a single electron under the influence of a linearly polarized laser field with the frequency ␻ and the strength F. The initial and final energies for such a process are given by Ei共t兲 = − Ip, 共3a兲 E_f共t兲 =1 2关k + A共t兲兴 2 , 共3b兲

where Ip is the ionization potential, k is the canonical

mo-mentum共measured on the detector兲, and A共t兲 = −F

␻sin共␻t兲.

According to Eq.共1兲, the probability of one-electron ion-ization ⌫ can be written as

⌫ = exp关− 2 Im S共k储,k⬜,Ip兲兴, S共k储,k_⬜,Ip兲 =

冕

t₁ t₀

冉

1 2关k储+ A共t兲兴 2 +1 2k⬜ 2 + Ip

冊

dt, 共4兲

where S is the action. Equation共2兲 can be rewritten in terms of S as

⳵

⳵_t0S共k储,k⬜,Ip兲 = 0. 共5兲

Note that the analogy between the saddle point S-matrix cal-culations关13兴, where transitions are calculated using station-ary points of the action, and the Dykhne approach can be seen from Eqs.共4兲 and 共5兲. The transition point is given by

␻t0= Arcsin

冋

␥

冉

k储

冑

2Ip + i

冑

1 +k⬜ 2 2Ip

冊

册

, 共6兲

where␥ is the Keldysh parameter

␥=␻ F

冑

2Ip=

冑

I_p 2Up and Up=共 F 2␻兲 2

is the ponderomotive potential. In order to extract the imaginary and real parts of this solution, the fol-lowing equation关14兴 can be used:

*dbondar@sciborg.uwaterloo.ca

PHYSICAL REVIEW A 78, 015405共2008兲

Arcsin共x + iy兲 = 2K␲+ arcsin␤+ i ln共␣+

冑

␣2_{− 1兲, 共7兲}

where K is an integer and

再

␣ ␤

冎

= 1 2

冑

共x + 1兲 2 + y2⫾1 2

冑

共x − 1兲 2 + y2. Using Eq.共7兲 in Eq. 共4兲, we obtain

⌫共␥,k储,k_⬜兲 ⬀ exp

冉

− 2Ip ␻ f共␥,k储,k⬜兲

冊

, 共8兲 where f共␥,k储,k⬜兲 =

冉

1 + 1 2␥2+ k2 2Ip

冊

arccosh␣−

冑

␣2_{− 1}

冉

␤ ␥

冑

2 I_pk储+ ␣关1 − 2␤2_兴 2␥2

冊

,

再

␣ ␤

冎

= ␥ 2

冉

冑

k2 2Ip +2 ␥ k储

冑

2Ip + 1 ␥2+ 1 ⫾

冑

k2 2Ip − 2 ␥ k储

冑

2Ip + 1 ␥2+ 1

冊

, k2_{= k} 储 2 + k_⬜2.

Note that␣⬎ 1. It must be stressed here that no assumptions

on the momentum of the electron have been made.However, Eq.共8兲 has an exponential accuracy because the influence of the Coulomb field of a nucleus cannot be accounted for by the strong field approximation. The correct exponential prefactor has been obtained within the Perelomov-Popov-Terent’ev共PPT兲 approach 关15–18兴.

Similarly to the Yudin-Ivanov formula, Eq.共8兲 is valid if the strength of the laser field F depends on time, F → E0g共t兲, where the envelope g共t兲 of the pulse is assumed to

be nearly constant during one-half of a laser cycle.

Equation共8兲 is the central result of this paper. In the fol-lowing, this equation is applied to some special cases in or-der to establish connections with previous results.

In the case of zero final momentum共k = 0兲, we have that

␣=

冑

1 +␥2 _{and ␤= 0. In this limit we recover the original}

Keldysh formula关1兴

f共␥,0,0兲 =

冉

1 + 1

2␥2

冊

arcsinh␥−

冑

1 +␥2

2␥ .

In the tunneling limit 共␥Ⰶ 1兲 the following formulas can be obtained. Expanding the function f共␥, k储, k⬜兲 in a Taylor

series up to third order with respect to ␥ and setting k⬜= 0,

we obtain ⌫共␥,k储,0兲 ⬇ ⌫共␥,0,0兲exp

冋

− k储 2 3␻␥ 3

册

. 共9兲

Equation共9兲 has been derived by a classical approach in Ref. 关19兴 共see also Ref. 关20兴兲. Discussions regarding the physical

origin of Eq. 共9兲 are presented in Ref. 关13兴. Performing the same expansion and setting k储= 0, we obtain

⌫共␥,0,k_⬜兲 ⬀ exp

冋

−2共k⬜

2

+ 2Ip兲3/2

3F

册

. 共10兲

This equation has been derived in Ref.关20兴. For small values of k_⬜, Eq.共10兲 can be approximated by

⌫共␥,0,k⬜兲 ⬇ ⌫共␥,0,0兲exp

冋

−

冑

2Ipk⬜ 2

F

册

. 共11兲

Let us fix k_⬜= 0 and continue working in the tunneling regime. For the case of high kinetic energy, such as k储

2_/ 2 ⬎ 2Upand

冑

k储 2_/共4U p兲 − 1 Ⰷ␥, we obtain ␣⬇

冑

k储 2 4Up , ␤⬇ 1, and the ionization rate ⌫ is given by

⌫ ⬀ exp

再

−2Up ␻

冋

冉

k储 2 2Up + 1

冊

arccosh

冑

k储 2 4Up − 3

冑

k储 2 4Up

冉

k储 2 4Up − 1

冊

册

冎

. 共12兲

Equation共12兲 has been obtained in Ref. 关5兴.

Calculating the asymptotic expansion of the function

f共␥, k储, 0兲 for k储

2_/

2 Ⰷ 2Up, we obtain

A B

FIG. 1. The single-electron spectrum of a hydrogen atom at 800 nm;共a兲 at 1 ⫻ 1013_{W / cm}2_{共␥ = 3.376兲; 共b兲 at 6 ⫻ 10}14_{W / cm}2

共␥ = 0.4357兲.

BRIEF REPORTS PHYSICAL REVIEW A 78, 015405共2008兲

⌫ ⬀

冉

F 2_exp共3兲 4␻2_k 储 2

冊

k_储2/2␻ 共k储 2_/ 2 Ⰷ 2Up, k⬜= 0兲. 共13兲

Equation 共13兲 has been obtained for tunneling ionization in Ref. 关5兴. Here, we have proved that Eq. (13) is valid for arbitrary␥. A similar formula can be derived for k_⬜⫽_0,

⌫ ⬀

冉

F 2_exp共1兲 4␻2_k ⬜ 2

冊

k_⬜2/2␻ 共k_⬜2_/ 2 Ⰷ 2Up, k储= 0兲, 共14兲

which is also valid for arbitrary values of the Keldysh pa-rameter␥.

Consider the asymptotic expansion of Eq. 共8兲 for large values of Ip共IpⰇ k2/2兲. In this case␣and␤can be

approxi-mated by ␣⬇

冑

1 +␥2 , ␤⬇

_冑

␥ 1 +␥2 k储

冑

2Ip .

Using these equations, we obtain

f共␥,k储,k⬜兲 ⬇ f共␥,0,0兲 + k2 2Ip arcsinh␥−

_冑

␥ 1 +␥2 k储 2 2Ip . 共15兲 Equation 共15兲 has been reached within the PPT approach 关15–18兴 共see also 关7兴兲.

As mentioned above, Goreslavskii et al.关6兴 have derived an expression for the spectral-angular distribution of single-electron ionization without any assumptions on the momen-tum of the electron. However, they have summed over saddle points, i.e., the contribution from previous laser cycles has been taken into account. On the contrary, we have not per-formed any summation because we are interested in the most recent contribution to ionization. Therefore, our result does account for the phase dependence of the ionization rate, un-like that of Ref.关6兴.

To make the phase dependence explicit in Eq. 共8兲, we apply the substitution

k储→ k储− A共t兲. 共16兲

The analytical expression for the ionization rate as a function of a laser phase when k = 0 has been achieved by Yuding and Ivanov关2兴. Thus, Eq. 共8兲 is seen to be a generalization of the Yudin-Ivanov formula.

Note that generally speaking, there is no unique and con-sistent way of defining the instantaneous ionization rates within quantum mechanics, and such a definition is a topic of an ongoing discussion共see, e.g., Refs. 关21,22兴 and references therein兲. However, the instantaneous ionization rates are in-deed rigorously defined within the quasiclassical approxima-tion共the Yudin-Ivanov formula兲, and we have employed this approach in the current paper. Alternatively, one can approxi-mate the instantaneous ionization rates by the static ioniza-tion rates at each point in time using the instantaneous value of the laser field.

Lastly, we illustrate Eq. 共8兲 for the case of a hydrogen atom. The single-electron ionization spectra in the multipho-ton regime 共␥Ⰷ 1兲 and in the tunneling regime 共␥Ⰶ 1兲 are plotted in Fig. 1共a兲 and Fig. 1共b兲, respectively. One con-cludes that the smaller ␥, the more elongated the single-electron spectrum. We can notice that the maxima of both the spectra are at the origin. Nevertheless, a dip at the origin has been observed experimentally 关23,24兴 in the parallel-momentum distribution for the noble gases within the tun-neling regime, and afterwards it has been investigated theo-retically in Ref.关25兴 and references therein. However, such a phenomenon is beyond Eq. 共8兲. The phase dependence of ionization for different initial momenta, recovered by means of Eq. 共16兲, is illustrated in Fig.2 for selected positive mo-menta. The curves for negative momenta are mirror reflec-tions 共through the axis ␾= 0兲 of the corresponding positive curves. Figure3 shows that the cutoff of the single-electron spectrum in the tunneling regime 共the dashed line兲 corre-sponds exactly to the kinetic energy 2Up, which is the

maxi-mum kinetic energy of a classical electron oscillating under the influence of a linearly polarized laser field.

In summary, we have derived an expression for strong field ionization including both the dependence on momenta and instantaneous laser phase. Previous results关1,2,5,20兴 re-garding strong field ionization can be recovered as special cases of the general formula. The present result concerns only the exponential dependence of the ionization process.

B A

FIG. 2. The plot of ⌫共␥ , k +F

␻sin ␾ , 0兲 for a hydrogen atom at

800 nm;共a兲 at 1 ⫻ 1013_{W / cm}2_{共␥ = 3.376兲; 共b兲 at 6 ⫻ 10}14_{W / cm}2

共␥ = 0.4357兲.

FIG. 3. The plot of ⌫共␥ , k储, 0兲 / ⌫共␥ , 0 , 0兲 for a hydrogen atom at

800 nm; the solid line: 1 ⫻ 1013W / cm2共␥ = 3.376兲; the dashed line: 6 ⫻ 1014_{W / cm}2_{共␥ = 0.4357兲.}

BRIEF REPORTS PHYSICAL REVIEW A 78, 015405共2008兲

However, supplementing our approach with the pre-exponential factor taken from PPT关7,15–18兴, the present re-sult is the most general formula obtained within the quasi-classical approximation.

The author is grateful to M. Spanner, G. L. Yudin, and M. Yu. Ivanov for many valuable discussions and remarks, and for outstanding comments on the manu-script.

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