PHYSICAL REVIEW A 86, 053407 (2012) Dichotomy between tunneling and multiphoton ionization in atomic photoionization: Keldysh parameter γ versus scaled frequency Turker Topcu and Francis Robicheaux Department of Physics, Auburn University, Auburn, Alabama 36849-5311, USA (Received 27 September 2012; published 9 November 2012) It is commonly accepted in the strong-laser physics community that the dynamical regime of atomic ionization is characterized by the Keldysh parameter γ . Two distinct cases, where γ<1andγ>1, are associated with ionization mechanisms that are predominantly in the tunneling and in the multiphoton regimes, respectively. We perform fully three-dimensional quantum simulations for the ionization of the hydrogen atom by solving the time-dependent Schrodinger¨ equation for a wide range of laser parameters encoded by the Keldysh parameter. We find that the meaning of the Keldysh parameter γ changes when the laser frequency ω is changed, and demonstrate that γ is useful in determining the dynamical ionization regime only when coupled with the scaled laser frequency when a large range of laser frequencies and peak intensities are considered. The scaled frequency relates the laser frequency ω to the classical Kepler frequency ωK of the bound electron. Together with the Keldysh parameter, the pair (γ,) refers to a more realistic picture of the dynamical ionization regime. We refer to final momentum distributions of the ionized electrons at several interesting points on the (γ,) landscape in order to infer whether the tunneling or the multiphoton mechanism is dominant in these regions. DOI: 10.1103/PhysRevA.86.053407 PACS number(s): 32.80.Rm, 05.45.Pq I. INTRODUCTION This is the step that incorporates the conservation of energy into the argument which ultimately decides that the ionization It has been ubiquitous in strong-laser physics that the happens through photon absorption. dynamical regime of atomic ionization, whether it be tunneling Several shortcomings of the Keldysh-like theories are evi- or ionization through absorption of photons, is associated with dent from the approximations made to the S-matrix elements to the Keldysh parameter γ [1,2]. In strong lasers, the Coulomb allow for closed analytical expressions. For example, theories field experienced by the atomic electron is depressed by the such as the strong-field approximation (SFA) involve no strong laser field. Depending on the peak field strength of dynamics within the potential barrier, are not gauge invariant, the laser field, this depression can be substantial so as to and the result usually depends on the choice of the origin result in a potential barrier oscillating at the laser frequency. [3–6]. Less obvious shortcomings of the applicability of the Quantum mechanically, there is nonvanishing probability that Keldysh parameter as an index for assessing ionization regimes the electron can tunnel through this barrier and escape into have also been demonstrated and it has been shown to be the continuum. The likelihood of ionization in this manner is unsuitable to describe laser-induced ionization when a wide quantified by the conditions in which the Keldysh parameter range of frequencies is considered. Even when γ<1, the is less than unity, γ<1. This is commonly referred to as the laser frequency can be so high as to allow for ultraintense quasistatic limit, in which the depressed Coulomb barrier is fields, while keeping the Keldysh parameter small. In such essentially static as seen by the electron. When γ>1, the atom instances, the γ → 0 limit converges to the fully relativistic 2 2 ionizes by absorption of a number of photons and the electron conditions, where the ponderomotive energy Up = F /4ω escapes through either direct or indirect paths of ionization. becomes comparable to the rest energy of the electron [7](we However, γ>1 does not necessarily mean that there is no use atomic units throughout this paper). This invalidates the tunneling contribution to the ionization; it implies that the γ → 0 limit as the static field limit in which ionization simply tunnel ionization is least likely, and the electron is more likely occurs through field ionization (tunneling or over the barrier). to escape by absorbing photons. In another study [8], it was shown that high-energy plateau Keldysh theory is strictly a theory of tunneling [1,2]. structures regularly attributed to tunneling in above-threshold In Keldysh-like strong-field theories, the classical action is ionization (ATI) and high-order harmonic generation (HHG) always complex regardless of γ , therefore its description is spectra can be seen in the multiphoton regime, provided that always contained within the tunneling picture. In other words, the photon energy is much smaller than the ponderomotive ionization through a classically allowed path does not occur potential. This suggests that rescattering can occur even though in Keldysh-like theories. Therefore, the statement that γ>1 the electron initially ends up in the continuum with substantial corresponds to multiphoton ionization is not a statement made kinetic energy, as opposed to having no initial kinetic energy directly by the Keldysh theory, but rather is a deduction which if it was to escape through tunneling. incorporates conservation of energy with the prediction that The reasoning behind the Keldysh parameter is the fol- γ<1 refers to ionization dynamics governed predominantly lowing: In the standard Keldysh theory, the tunneling length is by tunneling. It predicts the ionization rates when tunneling L ∼ Ip/F , where Ip and F are the ionization potential and the is most likely, and when there is ionization with γ>1, we peak electric-field strength, respectively. The velocity in the deduce that it must have followed an ionization path that is not classically forbidden region, where the combined Coulomb tunneling, i.e., one characterized by absorption of photons. and electric-field potential is larger than the total energy of 1050-2947/2012/86(5)/053407(10)053407-1 ©2012 American Physical Society TURKER TOPCU AND FRANCIS ROBICHEAUX PHYSICAL REVIEW A 86, 053407 (2012) the electron, can be obtained using the WKB approximation We perform two sets of calculations for the ionization of a to be roughly v ∼ 2Ip/2. Then the time it takes for the hydrogen atom out of 1s,2s,8s, and 16s states in laser fields: electron to tunnel through the depressed Coulomb potential one for the ionization rate and one for the ionization probability is essentially τ ∼ L/v = 2Ip/F . The ratio of this tunneling for a large set of (γ,) pairs. We map out a landscape in time to the laser period is a measure of how fast the barrier (γ,) space, which shows regions bearing characteristics that oscillates compared to the time it takes for the electron to tunnel can be attributed to either tunneling or multiphoton features. ionize, i.e., ωτ, where ω is the laser frequency. This ratio is Then, calculating final momentum distributions of the ionized γ = ω 2Ip/F and is referred to as the Keldysh parameter. electrons at a select few points on the (γ,) map for large It tells how static the oscillating potential is as seen by the and small γ , we investigate whether tunneling or multiphoton bound electron. The barrier is effectively static with regard ionization is dominant in these regions. Unless explicitly stated to the time it takes the electron to tunnel ionize when γ<1 otherwise, we use atomic units throughout this paper. (or T>τ), and oscillating if γ>1(orT<τ). The problem with exploiting the Keldysh parameter for a wide range of II. NUMERICAL SIMULATIONS frequencies is immediately evident from this definition. Note that γ is linearly proportional to ω, whereas it is inversely In our simulations, we drive the atom using a continuous- proportional to F . Therefore, for a fixed laser wavelength, one wave (cw) laser field in the rate calculations and a laser can vary the laser intensity in ways such that any value for the pulse in the probability calculations. All of our simulations Keldysh parameter can be attained. Conversely, fixing the field are based on ab initio solutions of the three-dimensional intensity, one can vary the wavelength to get any γ desired. time-dependent Schrodinger¨ equation in the length gauge. This point of view does not necessarily take into account We represent the total wave function on an (l,r) grid with conservation of energy and the possibly relevant time scales a square-root mesh in the r direction. We use the lowest-order other than the laser period, such as the classical orbital period split operator technique for the time propagation of the of the electron. As a result, incorrect deductions can be made Schrodinger¨ equation, where each split piece is propagated for ionization dynamics for certain sets of laser parameters. using an O(δt3) implicit scheme. This is an exactly unitary For example, assuming a fixed γ 1, we can keep choosing propagator and enables us to use larger time steps during the smaller and smaller intensities for a laser pulse, which may time propagation compared to those needed for an explicit push the photon energy well into the x-ray region. However, scheme. We use a mask function, which runs from 2/3ofthe it would be unreasonable to expect that ionization by such a distance from the origin to the box edge, to remove the ionized laser pulse would happen through tunneling. part of the wave function in order to evade spurious reflections Experiments such as the ones in Refs. [9,10] have suggested from the box edge. A detailed account of the O(δt3) implicit that the ratio of the laser frequency ω to the classical orbital method and the split operator technique employed in this work frequency ωK of the bound electron, i.e., ω/ωK , could be just can be seen in Ref.