ARTICLE DOI: 10.1038/s42005-018-0085-5 OPEN Instantaneous ionization rate as a functional derivative I.A. Ivanov1,2, C. Hofmann 2, L. Ortmann2, A.S. Landsman2,3, Chang Hee Nam1,4 & Kyung Taec Kim1,4 1234567890():,; The notion of the instantaneous ionization rate (IIR) is often employed in the literature for understanding the process of strong field ionization of atoms and molecules. This notion is based on the idea of the ionization event occurring at a given moment of time, which is difficult to reconcile with the conventional quantum mechanics. We describe an approach defining instantaneous ionization rate as a functional derivative of the total ionization probability. The definition is based on physical quantities, such as the total ionization probability and the waveform of an ionizing pulse, which are directly measurable. The defi- nition is, therefore, unambiguous and does not suffer from gauge non-invariance. We com- pute IIR by numerically solving the time-dependent Schrödinger equation for the hydrogen atom in a strong laser field. In agreement with some previous results using attoclock methodology, the IIR we define does not show measurable delay in strong field tunnel ionization. 1 Center for Relativistic Laser Science, Institute for Basic Science, Gwangju 61005, Korea. 2 Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01187 Dresden, Germany. 3 Max Planck Center for Attosecond Science/Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea. 4 Department of Physics and Photon Science, GIST, Gwangju 61005, Korea. Correspondence and requests for materials should be addressed to I.A.I. (email: [email protected]) or to K.T.K. (email: [email protected]) COMMUNICATIONS PHYSICS | (2018) 1:81 | DOI: 10.1038/s42005-018-0085-5 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | DOI: 10.1038/s42005-018-0085-5 he notion of the instantaneous ionization rate (IIR) has Our definition does not show any appreciable tunneling delay in Tproved extremely fruitful for understanding the physics of strong field ionization for the case of hydrogen. tunneling ionization, where the ionization regime char- acterizedp byffiffiffiffiffiffiffi small values of the Keldysh parameter Results γ ¼ ω=E jjI ≲ 1,2 ω E I fi 0 2 1 . Here, , 0, and are the frequency, eld Total ionization probability can be considered as a functional P strength, and ionization potential of a target system expressed in [E] of the waveform E(t), which maps the electric field of the atomic units. pulse into a real number. For the regime of the tunneling ioni- Qualitatively, the fact that the IIR is a function which is sharply zation that we are interested in below, this functional is highly peaked near the local maxima of the electric field of the pulse can nonlinear and cannot be described in a closed form. A simplifi- be used to pinpoint the most probable electron trajectory. This cation is possible if we consider a waveform which can be provides a basis for the design and interpretation of the results of represented as E(t) = Ef(t) + δE(t), with fundamental field Ef(t) well-known experimental techniques, such as attosecond streak- fi δE t fi – and signal eld ( ). To be more speci c, let us assume that the ing, or angular attosecond streaking3 5 allowing one to follow fundamental field is linearly-polarized (along z-axis, which we electron dynamics at the attosecond level of precision. Recently, assume to be the axis of quantization) and is defined by the vector the temporarily localized ionization at the local maxima of the potential Af(t): fi laser eld has been used as a fast temporal gate to measure the E πt laser field6. A ðtÞ¼À^z 0 2 ωt; ð Þ f ω sin T sin 1 Quantitatively, the notion of IIR underlies many successful 1 simulations of tunneling ionization phenomena, relying on fi E ω Classical Trajectory Monte Carlo method7 or quantum with peak eld strength 0, carrier frequency , and total dura- T = NT T = π ω trajectories8,9 Monte-Carlo simulations. These methods become tion 1 , where 2 / is an optical cycle (o.c.) corre- ω N 2 N practically indispensable if the system in question is too com- sponding to the carrier frequency , with . fi T plicated to allow an ab initio treatment based on the numerical We assume the signal eld to vanish outside the interval (0, 1) fi δE t solution of the time-dependent Schrödinger equation (TDSE). of the fundamental pulse duration. If the signal eld ( )is fi Even if a numerical solution of the TDSE is possible, these suf ciently weak, we can write: T methods may provide a physical insight which is not obvious hihi Z1 from the TDSE wave-function. Accurate quantitative calculations δP δP ¼ PEf þ δE À PEf δEðtÞdt; ð2Þ using these approaches, which agree well with the ab initio TDSE, δEf ðtÞ have been reported in the literature7,10–15. In these approaches, 0 1,16–18 quantum-mechanical Keldysh-type theories are used to set δP 11–13 where δE ðtÞ is a functional derivative of the functional P[E] up initial conditions for the subsequent electron motion . IIR f in the well-known Ammosov–Delone–Krainov (ADK) form19,20, evaluated for E(t) = Ef(t). or more refined Yudin–Ivanov IIR21 provides a measure which On the other hand, using the customary definition of IIR, we allows us to assign probability to an ionization event occurring at can write the probability of ionization driven by the combined a given moment of time inside the laser pulse duration. field E(t) = Ef(t) + δE(t) as: The notion of the ionization event occurring at a given time T Z1 and, correspondingly, the notion of IIR, is not free from P ¼ W ðEðtÞÞdt; ð Þ ambiguity, however. An interpretation of the ionization event inst inst 3 which is often used is based on the imaginary time method18,22. 0 In this picture, an electron enters the tunneling barrier at some W E t fi complex moment of time with complex velocity. Upon des- where inst( ( )) is the IIR, which by de nition depends on the cending onto the real axis, the velocity and coordinates corre- instantaneous value E(t) of the electric field. We introduced the P sponding to the most probable electron trajectory become real, notation inst in Eq. (3) to emphasize that this expression pertains which can be interpreted as the electron’sexitfromunderthe to the notion of the IIR. Eq. (3) gives a very particular case of the barrier. This picture, however, cannot be taken unreservedly, functional P[E]—clearly not every functional can be represented since the path which descends onto the real axis is not unique in this way. and can be deformed, in principle, to cross the real time-axis at From Eq. (3), we obtain the change of the ionization prob- 22 ability P due to the presence of the weak signal field: almost any given point . An approach which allows us to inst fi T de ne IIR from the solution of the TDSE has been described Z1 dW E ðtÞ 23 fi inst f recently . In this approach, the IIR is de nedbyprojectingout δP δEðtÞdt; ð4Þ contributions of the bound states from the solution of the inst dEðtÞ TDSE. The authors found that the IIR thus defined lags behind 0 the local maxima of the electric field, which suggests a nonzero fi Taking into account that δE(t) in Eqs. (2) and (4) is arbitrary tunneling time. A shortcoming of this de nition of the IIR, T however, is its nongauge-invariant character. The projection of (provided it is small and vanishes outside the interval (0, 1)), one the solution of the TDSE onto the subspace of the bound states can see that, if the treatment based on the notion of the IIR is justified (i.e., using Eq. (3) with IIR Winst(E(t)) depending only on performed during the interval of the pulse duration generally fi depends on the gauge used to describe atom-field interaction. the instantaneous value of the electric eld gives an accurate value for the total ionization probability), one must have: Another approach which allows us to define IIR from the solution of the TDSE is based on the notion of the electron flux δP dW Ef ðtÞ and was given in ref. 24. inst : ð5Þ In the present work we describe a different approach to IIR, δEf ðtÞ dEðtÞ which is based on the notion of a functional derivative. This approach provides an unambiguous and gauge invariant defini- We note that the quantity on the right-hand-side (r.h.s.) of (5) tion of the IIR. We apply the IIR thus defined to the problem of is an ordinary derivative which depends on time only through the the tunneling time for the process of the tunneling ionization. instantaneous value of the electric field. The quantity on the left- 2 COMMUNICATIONS PHYSICS | (2018) 1:81 | DOI: 10.1038/s42005-018-0085-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | DOI: 10.1038/s42005-018-0085-5 ARTICLE hand-side (l.h.s.), on the other hand, is a functional derivative, a which depends on time (through the time moment t at which 0.06 E t 0 differentiation is performed), and on the waveform ( ), i.e., on E(t) (a.u.) the complete history, past, and future, of the pulse. Suppose, for –0.06 instance, that we represent the waveform (1) as a Fourier series: 0 0.2 0.4 0.6 0.8 1 X Time (t/T) EðtÞ¼ a eimΩt; m ð6Þ m Coulomb, 1 o.c.