Strong field ionization by ultrashort laser pulses: Application of the Keldysh theory

Physics Letters A 374 (2009) 386–390

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Physics Letters A www.elsevier.com/locate/pla

Strong field ionization by ultrashort laser pulses: Application of the Keldysh theory B.M. Karnakov a , V.D. Mur a , S.V. Popruzhenko a,∗ , V.S. Popov b a b

Moscow State Engineering Physics Institute, Kashirskoe Shosse 31, Moscow 115409, Russia Institute for Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, Moscow 117218, Russia

a r t i c l e

i n f o

Article history: Received 25 August 2009 Accepted 25 September 2009 Available online 27 October 2009 Communicated by V.M. Agranovich

a b s t r a c t Analytical expressions and numerical results describing ionization of atoms by intense linearly polarized ultrashort laser pulses are obtained in the frame of the Keldysh approach. Photoelectron spectra and total ionization probabilities are presented for several analytical models of a single-cycle laser pulse. In particular, strong left–right asymmetry of the spectra is shown for the case of odd pulses. © 2009 Published by Elsevier B.V.

1. Introduction During the last decade the new generation of infrared lasers came into operation which are capable to produce intense pulses of very short duration and highly stabilized shape. Typically, in modern experiments with Titan–Sapphire (Ti:Sa) lasers of the wavelength λ = 780–800 nm (the photon energy h¯ ω = 1.55–1.58 eV) short pulses with the duration 5–10 fs are being routinely used (see reviews [1,2] and references therein). Such pulses contain from two to four optical cycles only. Their shape is controlled via the so-called absolute phase which can be kept rather stable from shot to shot [2,3]. Via emission of high order harmonics due to the interaction with gaseous atomic and molecular media, ultrashort infrared laser pulses nowadays became a standard tool for generation of coherent attosecond pulses of ultraviolet and X-ray radiation [4]. In turn, the attosecond technique was successfully applied for the direct reconstruction of the field strength as a function of time for an optical-frequency few-cycle linearly polarized laser pulse [5]. In studies of the strong field laser-atom interactions, ultrashort optical and infrared pulses help to reveal new information concerning both the ionization dynamics and the properties of short pulses themselves. For example, the shape of photoelectron spectra appears to be sensitive to the absolute phase of the pulse [6]. In particular, this sensitivity can be used for reconstruction of the pulse shape. Pulses consisted of few optical cycles no longer have a well-defined frequency. Their carrier frequency simply indicates the center of a rather broad spectral distribution of the pulse. As a result, separate peaks corresponding to absorption of a certain

*

Corresponding author. E-mail address: [email protected] (S.V. Popruzhenko).

0375-9601/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.physleta.2009.10.058

number of photons with the fixed frequency can be broadened so much that the photoelectron spectrum becomes continuous. Quantum interference can modulate such spectra and this modulation appears to be quite sensitive to the pulse shape and the absolute phase too [7]. A number of theoretical work have already been devoted to the analysis of the ionization dynamics in few-cycle intense laser pulses (see references in [2,8]). Most of them are based on numerical calculations of the ionization probability performed via exact numerical solution of the time-dependent Schrödinger equation or within various approximations. Less attention was given to analytical approaches to the problem. Among the theories developed for approximate description of the strong field ionization, the wellknown Keldysh theory [9] or the strong field approximation [10] is widely accepted as the most fruitful analytical approach. The semiclassical theory of multiquantum ionization formulated for the first time in the work of Keldysh [9] for the simplest case of a monochromatic linearly polarized field was then widely developed and generalized [11–14]. The present status of the Keldysh theory is reviewed in [8,15]. Most of the earlier work was devoted to the case of a monochromatic infinitely long laser pulse. Generalization of the Keldysh result to the case of a linearly polarized ultrashort laser pulse consist of few or even a single optical cycle was made in [14] where the total rates of ionization from a short range potential were evaluated for several different pulse shapes at arbitrary values of the Keldysh parameter γ (see Eq. (3) below for the definition). In this work, we consider the same problem but calculate also the photoelectron spectra and analyze the dependence of the total ionization rate and the interference structure of photoelectron spectra on the absolute phase of the laser pulse. As was mentioned above, the dependence of photoionization spectra on the absolute phase can be used for reconstruction and control

B.M. Karnakov et al. / Physics Letters A 374 (2009) 386–390

of the pulse shape. Several recent experiments report considerable achievements on the subject (see, e.g., [7,16] and references therein). The Letter is organized as follows. In the next section the statement of the problem is formulated in more details and general expressions for the ionization amplitude and the momentum distribution of photoelectrons are derived. Section 3 presents the results for antisymmetric (odd) pulses where the electric field obeys the condition E(t ) = −E(−t ). In Section 4 a more general case of the pulse characterized by the value of the absolute phase θ0 is considered. The last section summarizes our conclusions.

M (q) = i

π

+∞ e i K 0 Φ(θ ) dθ.

Cκ K0

(1)

−∞

The amplitude (1) can be derived using the retarded timedependent Green function for the electron in a homogeneous electric field and the time-dependent Schrödinger equation in its integral form. Detailed derivations related to the case of a monochromatic linearly polarized field E(t ) = E 0 cos ωt are given in Ref. [12]. In Eq. (1) C κ is the asymptotic coefficient of the √ atomic wave function at large distances r  κ −1 [8], where κ = 2I is the characteristic momentum of the bound state with the ionization potential I . For ground states of neutral atoms and single-charged ions (both positive and negative) the coefficients C κ are close to unity (see the numbers in Ref. [8]). Finally, the phase Φ(θ) is

Φ(θ) = θ +

1

γ

θ

2

   2 a θ − a(+∞) + γ q dθ  .

(2)

0

The dimensionless photoelectron momentum is q = p/κ with p being the momentum measured by a detector. The multiquantum parameter K 0 and the Keldysh adiabaticity parameter γ in Eqs. (1), (2) are given by

K0 =

I

ω

γ=

,

κω E0

=

1 2K 0 F

F=

,

E0

κ3

.

(3)

Here ω and E 0 are the characteristic frequency of the laser pulse and its electric field strength amplitude, respectively; F is the reduced field. The electric field of the pulse and the respective vector potential are assumed in the form

A(t ) =

c

ω

E 0 a(θ),

E(t ) = −E 0 a (θ) ≡ E 0 ϕ (θ),

(4)

where the dimensionless time θ = ωt is introduced. In Eq. (2) we assume E(−∞) = a(−∞) = 0. Atomic units e = h¯ = m = 1 are used throughout the Letter. Eq. (1) is valid for arbitrary values of the multiquantum parameter K 0 including the case of single-photon ionization K 0  1. Indeed, when the field is weak enough so that γ  K 0 , the term quadratic with respect to the field amplitude E 0 can be safely omitted in the phase (2). Then, after two subsequent integrations by parts in Eq. (1) we arrive to the expression

√

M (q) = −

2 2

π

√ Cκ

F 1

(6)

which provide the inequality K 0 Φ(θ)  1 and specify the semiclassical domain where ionization is a multiquantum process, the integral over time θ in (1) can be evaluated by the saddle-point method at arbitrary values of the Keldysh parameter γ . Equation for a saddle-point θs has the form Φ  (θs ) = 0 which reads for the phase (2) as



2

a(θs ) − a(+∞) + γ q

= −γ 2 .

(7)

The momentum distribution of photoelectrons has the form



For an electron bound in the s-state in a short-range potential well the amplitude of ionization by the spatially homogeneous electric field E(t ) of a laser wave has the form

2

γ 2,

2

dw (q) =  M (q) d3 q

2. General equations

√

K 0  1,

387

κ

( p 2 + κ 2 )2

+∞

2 2 pE e i ( p +κ )t /2 dt

(5)

−∞

which is precisely the ionization amplitude calculated in the first order perturbation theory. Under the conditions

     −1/2 i K Φ(θ ) 2 3 s  0 Φ (θs ) = K0Cκ  e  d q, π 4

2

(8)

s

where the sum is taken over all solutions of Eq. (7) with positive imaginary parts, Im θs > 0. Summation over the stationary points in (8) takes the effect of interference into account. In a linearly polarized field the distribution possesses axial symmetry and depends on the longitudinal q and transversal q⊥ components of the photoelectron momentum. The ionization probability reaches its maximum at q = q0 = (q = q0 , q⊥ = 0), where the most probable longitudinal momentum q0 is a root of the equation

d  dq

=

 



Im Φ q , θs (q ) 

q =q0

2

γ

θ s (q0 )

    a θ − a(+∞) + γ q0 dθ 

Im 0

= 0.

(9)

It can be easily shown that for the monochromatic pulse, a(θ) = cos θ , the most probable momentum q0 = 0. In the tunneling limit, γ  1, an approximate solution of Eq. (9) is given by (the electron charge is equal to −1)

q0 =

 1 a(+∞) − a(θm ) ,

(10)

γ

where θm is the time instant where the absolute value of the electric field reaches its maximum, ϕ  (θm ) = 0. In the opposite limit of the multiphoton ionization, γ  1, Eq. (10) no longer holds. However, in this domain the value of the most probable momentum always vanishes as q0 ∼ c (γ )γ −1 with the coefficient c (γ ) only slowly dependent on γ . As a result, a simple formula (10) reproduces the solution of Eq. (9) with rather good accuracy for arbitrary values of the Keldysh parameter (see Fig. 1(b) in the next section). Near the maximum of the distribution, the phase Φ(q) can be expanded into the series with respect to the small deviation from the most probable momentum q0 . In this case the distribution takes a simple form





dw (q) = C κ2 P ( K 0 , γ ) exp −2K 0 f (γ ) + c ⊥ (γ )q2⊥

+ c (γ )(q − q0 )2



d3 q,

(11)

where the preexponential factor may contain interference. For the monochromatic laser field, a(θ) = cos θ , explicit expressions for the Keldysh function f (γ ), the preexponential factor P and the coefficients c ⊥ and c of the momentum distribution are given in Refs. [8,9,12]. Generalizations of these functions to the case of even pulses, ϕ (θ) = ϕ (−θ), −∞ < θ < ∞ were found in [14] for various pulse shapes. In particular, the pulse shape

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B.M. Karnakov et al. / Physics Letters A 374 (2009) 386–390

Fig. 1. Panel (a): Imaginary part f (γ ) (Keldysh function, solid lines) and real part χ (γ ) (dashed lines) of the phase Φ(q0 , θs (q0 )) calculated for the pulse shapes (13) (curves 1) and (14) (curves 2) versus the Keldysh parameter γ . The most probable momentum is found numerically from Eq. (9). The Keldysh function calculated with approximation (10) is shown by dotted lines. Panel (b): The most probable momentum q0 (γ ) calculated numerically from Eq. (9) (solid lines) and analytically along Eq. (10) (dotted lines).







ϕ (θ) = 1 − θ 2 exp −θ 2 /2



(12)

was considered in Refs. [14,17]. For even pulses the electric field strength is symmetric with respect to θ = 0 where it reaches the maximum. In this case a single stationary point makes a dominant contribution into the ionization amplitude and no considerable interference structure appears in the spectrum.1 The most probable momentum is q0 = 0, so that the spectrum possesses inverse symmetry, | M (q)|2 = | M (−q)|2 . 3. Odd pulses The case of odd laser pulses, ϕ (θ) = −ϕ (−θ), whose electric field is antisymmetric with respect to the time instant θ = 0 where the vector potential reaches its absolute maximum or minimum, is more diverse. For such pulses q0 = 0 and the momentum distribution does not obey inverse symmetry. Here we consider two analytical examples of antisymmetric pulse shapes:

a(θ) =

2 cosh θ

,

ϕ (θ) =

2 sinh θ

Photoelectron spectra in the polarization direction (i.e., for q⊥ = 0) calculated from (8) are shown in Fig. 2. The most prominent features of the spectra are their interference structure and strong asymmetry with respect to q = 0. Both effects are very pronounced in the tunneling domain, γ  1 and tend to vanish in the multiphoton limit, γ  1. It should be emphasized that the intervals between the interference maxima are not equal to the photon energy, but determined by the value of Re Φ(q ), so that no isolated peaks corresponding to absorption of a fixed number of photons with well-defined energy each are present in the spectrum, in contrary to the case of quasimonochromatic long pulses with a small spectral width. Another important observation which can be made from Fig. 2 is that the shape of the spectra including the position of the maximum, the width and the period of interference oscillations are seemingly sensitive to the pulse shape. This supports the idea to use photoelectron momentum distributions for control the shape of short laser pulses [6]. Interference structures of photoionization spectra in ultrashort laser pulses were observed in recent experiments [7,16].

(13)

cosh2 θ

4. Dependence on the absolute phase

and





a(θ) = exp 1 − θ 2 /2 ,





ϕ (θ) = θ exp 1 − θ 2 /2 .

(14)

In both cases two stationary points θs− = −(θs+ )∗ exist in the upper half-plane Im θs > 0 which contribute equally into the ionization probability. As a result, a pronounced interference pattern may appear in spectra. Fig. 1 shows the Keldysh function f (γ ) = Im Φ(q0 , θs+ ) = Im Φ(q0 , θs− ), the real part of the phase χ (γ ) = Re Φ(q0 , θs+ ) = − Re Φ(q0 , θs− ) and the most probable photoelectron momentum q0 calculated numerically from Eq. (9) and analytically from the √ approximate relation (10). In the latter case q0 = − 2/γ for (13) and q0 = −1/γ for (14). As is clearly seen from Fig. 1(b), approximation (10) works nicely for both pulse shapes. Moreover, the respective exact and approximate expressions for the Keldysh function are visually indistinguishable on the plot. Therefore, we conclude that approximation (10) provides good accuracy at arbitrary values of the Keldysh parameter. With this approximation closed analytical expressions for the Keldysh function can be obtained.

Consider now a pulse with the Gaussian envelope and the shape determined by the value of the absolute phase θ0 : 2 a(θ) = e −θ /2 cos(θ − θ0 ),

ϕ (θ) = e−θ

2

/2



For an arbitrary even pulse all the quantities entering Eq. (11) can be evaluated using the imaginary time method [12,18] which provides physically transparent description of the time-dependent tunneling.

(15)

Note that such field could, in principle, be technically obtained in experiments since it satisfies the condition

+∞ E(t ) dt = 0, −∞

which follows from Maxwell equations [19]. This is also correct for the pulses (13) and (14). The value of the absolute phase can be restricted by the interval −π /2 < θ0  π /2. For θ0 = 0 and θ0 = π /2 the pulse (15) is odd and even, respectively. With good accuracy, the most probable photoelectron momentum is given by Eq. (10) with θm to be found from

tan(θm − θ0 ) = 1



sin(θ − θ0 ) + θ cos(θ − θ0 ) .

2 2 − θm

2θm

.

(16)

Although Eq. (16) has an infinite number of solutions, only one or two corresponding to the major maxima of the electric field

B.M. Karnakov et al. / Physics Letters A 374 (2009) 386–390

389

Fig. 2. Photoelectron spectra along the polarization direction calculated from (8) for ionization of Xe (I = 12.1 eV) in the field of a Ti:Sa laser (h¯ ω ≈ 1.55 eV). Plots correspond to the pulse shape (13) (a)–(c) and (14) (d)–(f). The peak intensity is 4 × 1014 W/cm2 (a), (d), 1 × 1014 W/cm2 (b), (e) and 4 × 1012 W/cm2 (c), (f). In all cases K 0 ≈ 7.8. Values of the Keldysh parameter are shown on the panels.

should be taken into consideration while other solutions contribute exponentially little. The same statement holds for the extremal stationary points θs (q0 , θ0 ). Below we consider only one solution of (16) which corresponds to the absolute maximum of the electric field and the respective stationary point.2 Except for the even pulse (θ0 = π /2) the phase (2) is complex even in the extremal stationary points. Fig. 3 shows the real and the imaginary part (Keldysh function) of the phase versus the Keldysh parameter for several values of θ0 . It is rather unexpected that the Keldysh function appears to be basically independent on the absolute phase. This shows that the total rate of ionization

2 For θ0 = 0 (odd pulse) there are two equivalent maxima. In this case Fig. 3 shows the values belonged to the left stationary point, Re(θs ) < 0.

may be almost insensitive to a particular shape of the pulse (15). In turn, the real part of the phase shown in Fig. 3(b) depends upon the value θ0 considerably, particularly in the intermediate and tunneling regimes, γ  1. It shows again that the interference structure of photoionization spectra is quite sensitive to the absolute phase. 5. Conclusions In conclusion, applying the Keldysh method we have considered multiquantum ionization by ultrashort intense laser pulses. Our most important observation is that a single-cycle pulse will generate strongly asymmetric photoelectron spectra whose interference structure depends upon the pulse shape and, in particular, on its absolute phase. The obtained results may be of relevance

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B.M. Karnakov et al. / Physics Letters A 374 (2009) 386–390

Fig. 3. The Keldysh function f (γ ) (a) and the real part of the phase χ (γ ) (b) for the pulse (15) at different values of the absolute phase: θ0 = π /2 (even pulse, solid line), θ0 = π /3 (dashed line), θ0 = π /6 (dotted line) and θ0 = 0 (odd pulse, dash-dotted line).

for experimental studies of the strong filed ionization dynamics in ultrashort infrared laser pulses nowadays being performed worldwide. Ionization of an s-state by the pulse shape (14) was also considered in Ref. [17] within the Landau–Dykhne approach. Note, that Eqs. (19)–(22) of [17] are essentially erroneous: in particular, in the limit γ → 0 they do not reproduce well-known expressions for the static field. The numerical values of the ionization probability calculated in [17] for the pulse shape (14) are many orders in magnitude below the correct result, particularly for small γ . Besides, in Ref. [17] only one stationary point θs+ was taken into account so that the effect of quantum tunneling interference is missing as well. The results presented in the Letter are relevant, strictly speaking, for a system where the electron is bound by short range forces. Negative ions could serve as a realistic example of such systems. For atoms and ions with positive charge the ionization dynamics can be strongly modified by the long range Coulomb interaction between the outgoing photoelectron and the atomic core. This interaction influences both the total ionization probability and the momentum distributions including their interference structure (see review [8] and recent publications [20–22] for detail). Acknowledgements We acknowledge fruitful discussions with D. Bauer and S.P. Goreslavskii. This work was supported by the Russian Foundation for Basic Research (projects Nos. 07-02-01116 and 09-02-01201). References [1] T. Brabec, F. Krausz, Rev. Mod. Phys. 72 (2000) 545.

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