In this paper we propose and demonstrate that the ultrafast nonlinear optical response of atoms may be accurately calculated in terms of metastable states obtained as solutions of the stationary Schrödinger equation including the quasi-static applied electric field. We first develop the approach in the context of an exactly soluble one-dimensional atomic model with delta-function potential, as this allows comparison between the exact ultrafast nonlinear optical response and our approximate approach, both in adiabatic approximation and beyond. These ideas are then applied to a three-dimensional hydrogen-like atom and yield similar excellent agreement between the metastable state approach and simulations of the Schrödinger equation for off-resonant excitation. Finally, our approach yields a model for the ultrafast nonlinear optical response with no free parameters. It can potentially replace the light–matter interaction treatment currently used in optical filamentation, and we present a numerical example of application to femtosecond pulse propagation.
© 2014 Optical Society of America
CorrectionsM. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, "Metastable electronic states and nonlinear response for high-intensity optical pulses: erratum," Optica 2, 509-509 (2015)
Propagation of ultrashort, off-resonant optical pulses in atomic gases produces a broad range of extreme nonlinear optical effects including high-harmonic generation , synthesis of attosecond pulse forms , and optical filamentation [3,4]. While it is generally accepted that the origin of the first two examples resides in the quantum mechanical nature of the light–matter interaction, the standard model of the nonlinearity underlying filamentation, comprising third-order nonlinearity for bound electrons plus a Drude model for freed electrons, has met with some success. However, numerical calculations of the nonlinear optical response of atoms using the time-dependent Schrödinger equation (TDSE) [5–10] make it abundantly clear that the quantum mechanical nature of the extreme nonlinearity involved is also required for a more complete and deeper understanding of filamentation. In particular, for off-resonant pulse excitation the quantum coherent nature of the light–matter interaction becomes key, and the distinction between bound and freed electrons employed in the standard model becomes problematic. In addition to the numerical simulations based on the TDSE, some model systems have been explored to gain insight. For example, some of the present authors have studied the nonlinear optical response via an exactly soluble one-dimensional (1D) atomic model with delta-function potential , and Richter et al. studied the role of the Kramers–Hennenberger atom, which displays bound states of a free electron, to elucidate the higher order Kerr effect. Models employing separable interactions  have also been studied in this context.
The above discussion highlights the urgent need for microscopically based and computationally economical descriptions of extreme nonlinear interactions that lend themselves to incorporation into full space–time field simulations. In particular, numerical solution of the TDSE coupled to the field propagation equations over the space and time scales relevant to filamentation in gases poses a formidable computational task that will not be realistic in the foreseeable future, although forefront simulations over restricted spatial domains have appeared [13,14]. The standard model is easily incorporated into existing propagation schemes but the third-order nonlinear and plasma contributions to the model are not self-consistently and microscopically linked as they are in the present unified approach.
In this paper we propose and demonstrate that the ultrafast, off-resonant nonlinear optical response of atoms may be accurately calculated in terms of metastable states as opposed to the more common bound and continuum states of the free atom. In particular, we consider solutions of the atomic time-independent Schrödinger equation including the quasi-static applied electric field, these states having complex energies, the imaginary parts being related to the metastable state lifetime. Meta-stability refers to the fact that while these states ionize and thus “decay” over relatively long times, when observed on shorter time-scales in the vicinity of the atom, they are difficult to distinguish from the normal (i.e., field-free) electronic states. These metastable states, also termed resonance, Siegert, or Gamow states, encapsulate the quantum coherent coupling between the bound and continuum states of the free atom, and provide a natural time-domain description of an atom exposed to strong fields . More specifically, although the single atom represents a closed system, we consider that the atomic wave function may be split into two components, , in which is expressed as a superposition of metastable states that is used to calculate the nonlinear optical response arising from the quantum coherent light–matter interaction. In contrast, accounts for electrons that are to all intents and purposes freed from their parent ion and contribute a classical current density. This portion of the wavefunction is populated by the losses from the metastable states, and is subsequently driven by the applied electric field according to Ehrenfest’s theorem. The metastable portion of the wavefunction, , gives rise to a complex, nonperturbative nonlinear response, which only reduces to the usual optical Kerr effect in the limit of a weak field. Using examples we demonstrate that our approach, which we term “the Metastable Electronic State Approach” or simply MESA, provides an extremely economical computational method of calculating the nonlinear optical response: already retaining only the ground metastable state in adiabatic approximation can provide a quantitative model for the nonlinear optical response and strong-field ionization, and further improvement is shown to result from retaining post-adiabatic corrections.
The remainder of this paper is organized as follows. We develop MESA in the context of an exactly soluble 1D atomic model with delta-function potential as a test bed as this allows for comparison between the exact ultrafast nonlinear optical response and our approximate approach, both in the adiabatic approximation and beyond. The method is then applied to a three-dimensional (3D) hydrogen-like atom, and this is key to showing that the approach can be employed for more realistic systems. We employ results for the nonlinear optical response from MESA to an example of optical filamentation as a demonstration of the computational economy and feasibility of the approach. Finally, a summary with conclusions and future research directions are given in Section 9.
2. EXACTLY SOLVABLE QUANTUM SYSTEM
An exactly solvable system proves to be of great utility for testing the method we propose. It also makes it easier to appreciate the structure of the theory, since all quantities we need can be evaluated analytically. We use a 1D, single-particle model with a Dirac-delta potential. Its time-domain Schrödinger equation can be written as11], and this we use to test our approximate solutions. The metastable states (Stark resonances) can also be obtained exactly:
The infinitely many solutions to this equation, denoted by , represent the complex-valued meta-stable energies. The most important resonance, , converges to the ground state when . Besides , there are two distinct families of resonances. Energies of the first are located just below the positive real axis, while family-two energies extend along the ray with the complex argument of . Meta-stable states constitute a bi-orthogonal system:1 as a function of . Its real part is superlinear for weak fields (Kerr effect), and it saturates and decays in very strong fields, whereas the imaginary part is intimately related to the ionization losses. As noted in our previous work, the saturation and decay of the real part is accompanied by rapid growth of the freed electron population, which produces a defocusing nonlinear response. This generic behavior turns out to be the same for more realistic 3D quantum systems.
3. GENERAL SCHEME
The central idea of the method we put forward, namely, MESA, is to abandon the traditional description of the quantum evolution in terms of Hamiltonian eigenstates. Instead, we argue that for a system exposed to a strong external field, such as due to a high-intensity optical pulse, it is more natural to utilize metastable states. They are superpositions of the (field-free) bound and free electronic states, reflecting the current strength of the field and the probability density “leakage” due to ionization .
We assume that it is possible to represent the wavefunction split into “metastable” and “free” components, , as described in the introduction. Our method is motivated by Berggren’s completeness relations  employed in nuclear scattering theory:
We are specifically interested in the case of off-resonant excitation, meaning that the instantaneous frequency of the applied field should remain well below the ionization frequency for all times, being the ionization energy. Our method should therefore be more accurate for longer wavelengths, and we find this to be the case. Moreover, for the case of the exactly solvable 1D model there is an analytic result that states that the lowest Siegert state exactly captures the quantum atomic dynamics in the adiabatic or quasi-static approximation for which .
Concentrating on the “metastable” component, we deal with an effectively open system. We write its evolution equation as an expansion into the system of metastable states slaved to the time-dependent field:20] for details of implementation). Such boundary conditions make this operator non-Hermitian, and the complex-valued metastable energies reflect the decaying nature of the resonant states. It can be shown that the adjoint has eigenfunctions which are just complex-conjugates of and that pairing Eq. (4) is nothing but a bi-orthogonality relation .
We use Eq. (4) to project out the evolution equations, normalizing all to unity for all times for simplicity of notation. The resulting evolution equations reflect the fact that are effectively time dependent and, therefore, contain matrix elements that show the change of metastable states with field strength :
We will first concentrate on the adiabatic approximation. For the initial condition in the ground state, the adiabatic wavefunction  becomes dominated by the ground-state resonance and the solution is5) is the main contribution to the induced polarization , . In pulse propagation simulations the linear medium properties are captured exactly by a spectral solver. To avoid double counting, only the nonlinear part of the atom polarization is used and coupled to the Maxwell equations. This we obtain as 11) is our first contribution to the nonlinear medium response. The second comes from , in the form of a classical current, because we assume that this is the “distant part” of the wave function that is no longer interacting with the atom. Its norm grows as a result of the metastable decay, so that the total probability is conserved as it should be in a closed system. Thus, the ionization rate is given by the imaginary part of the ground-state resonance complex energy, and the ionized fraction of atoms obey
4. POST-ADIABATIC CORRECTIONS
The purpose of this section is to describe three kinds of corrections that take MESA beyond the adiabatic approximation outlined above. In doing so, we will strive to formulate our theory in such a way that it will require the knowledge of only a single metastable state, namely, the one related to the ground state. This will greatly simplify practical applications, because it is relatively simple to characterize the ground-state resonance, while it may be more difficult to calculate properties of higher metastable states. The assumption underlying the following consideration is one of a weak field , which is changing slowly. This is the case for many regimes in extreme nonlinear optics, where typical intensities attained in hot spots are still 2 orders of magnitude weaker than atomic fields. Moreover, the propagation dynamics dynamically adjusts the evolving pulse such that the peak intensity is clamped. As a result the quantum state is strongly dominated by the contribution from the metastable ground state, and we construct the post-adiabatic corrections under this assumption.
The first correction of the adiabatic solution is obtained by solving Eq. (9) for the expansion coefficients driven by the dominant . Feeding that intermediate result back into the equation for the ground-state resonance, is obtained in the same form as , only with the resonant energy replaced by the corrected one:14). Next, because normalization to unity for all ensures that , the numerator in Eq. (14) can be rewritten, moving the action of onto and thus giving rise to . Our second assumption is that, at least within the space of the solutions that evolve from the ground state, the system of is complete, and this projection can be approximated by . Then we can simplify the above correction to 2 illustrates this effect, and shows ionization yields caused by a driving pulse in the adiabatic and post-adiabatic [i.e., with Correction (15)] approximations, compared to the exact solution. It is evident that the correction becomes negligible for longer wavelengths. At shorter wavelengths, it significantly decreases the gap between the adiabatic and exact solutions, thus justifying the approximations adopted in the derivation. Perhaps the most important observation to be made here is that even the uncorrected adiabatic solution provides rather accurate ionization rates. For practical purposes in the field of optical filamentation, especially in the near and mid-infrared, the adiabatic treatment should therefore be sufficient.
However, there is one post-adiabatic correction that should be included in such simulations; because it captures the losses, the optical field must suffer due to ionization. Interestingly, this correction is connected to the imaginary part of the metastable expectation value of the dipole moment. To derive the corresponding nonlinear polarization term, one has to include the first corrections of the wavefunctions when evaluating the expectation value of the dipole moment. Then, under the same approximation as in the derivation of the corrected resonant energy in Eq. (15), one obtains the following estimate:3] in order to salvage energy conservation. Our comparative simulations show that the amount of loss caused by is actually quite similar to that obtained in the phenomenological treatment. It is fair to say that Eq. (16) is an approximation, yet it is a first step beyond the current method. Naturally, it could be improved provided one can calculate higher resonant states.
Finally, the third correction originates in the split between and , and the fact that the response of the latter is approximated by a Drude-like classical current as if this portion of the wavefunction was completely free. A generalized version of Eq. (13) should contain an additional term,
5. COMPARISON WITH EXACT SOLUTIONS
The exactly solvable Dirac-delta system is an ideal test bed to assess accuracy of the proposed light–matter interaction description. We utilize the time-dependent exact solution for the nonlinear response given in Ref. . This is then compared with the nonlinear response calculated as described in the previous sections.
The sole adjustment we make before comparing these results concerns the low-frequency part of the response described above. It shows up as a constant current after the driving pulse vanishes, reflecting the net average velocity imparted on the ionized electrons. While it is possible to capture this effect in the resonance-response model, at present an estimate of requires one adjustable parameter. We therefore filter out the very low-frequency part of the exact response and add this to the MESA result to compare the nonlinear response.
Figure 3 illustrates that very good agreement can be achieved between the exact and approximated nonlinear response for a 2.5 μm wavelength driving pulse. One can see that the approximate solution follows the exact one while it filters out the very-high-frequency components. These are due to higher order resonances and are therefore absent in our post-adiabatic approximation.
The top panel shows that these response oscillations are too fast to affect the optical-frequency components of the driving pulse. For simulation of pulse propagation, it is therefore quite convenient that the model response does not follow them.
We thus can conclude that the proposed method captures precisely that frequency part of the total nonlinear response that is responsible for the back-reaction of the medium on the driving pulse. At the same time, even small-scale details in the nonlinear response shape are reproduced quite well.
6. 3D HYDROGEN-LIKE MODEL ATOM
Having seen that the nonlinear response of the exactly solvable model can be reproduced by the metastable-state response model quite accurately, the important question is if it still works for more realistic systems. The crucial difference is that one has to resort to approximate methods to find, and “measure,” properties of the ground-state resonance. In the following we describe the procedure that allows one to obtain parameterized nonlinear response from a series of TDSE simulations of a given system. The Hamiltonian represents a single-electron, hydrogen-like system,
To obtain the ground-state resonance as a function of the static-field strength , we start a TDSE simulation in the numerical ground state, and add transparent boundary conditions realized as a perfectly matched layer (PML) to the computational domain. Because the ground-state resonance is the longest living state, the initial wavefunction is driven toward it during the simulated real-time evolution, while the “unwanted” states decay as they leak through the PML layers. Once the solution stabilizes, we characterize this ground-state resonance by calculating the metastable expectation value for its dipole moment. We also extract its complex-valued resonance energy. This process is repeated for a range of field strengths, and the relevant data are tabulated. We envision that this kind of procedure will be applied when working with realistic (single-active-electron) models of atoms and perhaps even molecules.
Figure 4 shows the real and imaginary parts of metastable dipole moment . We have determined this quantity as a function of the field strength on different-size grids, and with three different implementations of the PML boundary. Two data sets, obtained for grid sizes of a.u. and a.u., are shown in the figure to match closely, thus indicating that convergence is already achieved on relatively small computational domains. It should be noted that, if one attempts to determine the standard quantum-mechanical expectation value of the dipole moment, no convergence can be reached because metastable wavefunctions diverge at infinity. It is thus crucial that our method works with the metastable generalization of the dipole moment [Eq. (5)].
Importantly, comparing Fig. 4 to Fig. 1, one can see that the behavior of the field-induced dipole is qualitatively the same in the 3D system as in the exactly solvable 1D model. This makes us believe that features of are quite generic.
Together with the field-dependent metastable ground-state energy , tabulation of the nonlinear of the dipole moment shown in Fig. 4 constitutes the core data that characterize the quantum system, and makes it possible to calculate its response to arbitrary pulsed excitation. Note that the TDSE simulations to obtain these data sets require significant, but only a one-time, numerical effort.
7. COMPARISON WITH TDSE SOLUTIONS
To demonstrate that the nonlinear response can be calculated the same way also for a more realistic system, we have generated time-dependent solutions of the Schrödinger equation for our hydrogen-like system, exposed to a near-infrared frequency pulse. For comparison purposes, we remove the linear polarization from our numerical solutions. We also remove a low-frequency background, which contributes only to the THz generation that would not affect dynamics of the driving infrared pulse.
Figure 5 shows an example in which both Kerr-like and plasma-like responses show up in the leading and trailing edge of the pulse, respectively. Regimes like this one, when counter-acting components of the nonlinear response manifest themselves on similar scales, are of utmost importance for extreme nonlinear optics, since the filamentation physics is naturally influenced by this kind of dynamic balance.
We have chosen the shape of the excitation pulse to have a flat middle portion in order to better visualize different processes that would control the dynamics of the propagating pulse. In the leading edge of the pulse, the polarization is in phase with the optical field. During this time, the self-focusing response dominates. In the trailing edge, in contrast, we see the response being out of phase, which is a sign that it acts mainly as a defocusing mechanism. In the temporal middle of the pulse one can see that the two compete. Importantly, the agreement between the numerically exact and the response calculated with the proposed model is rather good.
8. APPLICATION EXAMPLE: FEMTOSECOND FILAMENTATION
As an illustrative example of the utility of MESA, we consider a femtosecond filament created by a 30 fs optical pulse propagating in a model gaseous medium. The medium consists of atoms at atmospheric pressure, each responding to the optical field as described above. Besides the nonlinear atomic response, the medium has a linear susceptibility, which we choose to be the same as that of air. This is implemented as a part of the linear optical propagator . To execute the simulation, we utilize the generalized unidirectional pulse propagation equations framework (gUPPEcore)  with a medium module that implements the MESA response.
The parameters of our illustration are chosen to create a situation in which several processes affecting the dynamics act simultaneously. In particular, we choose the wavelength , for which the generation of new harmonics, along with their subsequent “temporal walk-off,” are more relevant than for 800 nm pulses.
Furthermore, we assume relatively tight focusing geometry, with . This is to verify that the defocusing properties of the model are sufficient to arrest the self-focusing collapse, which is made more severe by the external focusing. Figure 6, showing the on-axis energy fluence versus propagation distance for two different initial pulse intensities, demonstrates that it is indeed the case.
The defocusing effects are mainly due to free electrons. Figure 7 shows the linear density of generated freed electrons are versus propagation distance. The model accounts for the corresponding energy losses via the imaginary part of the induced dipole moment.
Figure 8 shows the spectrum for propagation distances just before and after the filament. Well-defined harmonic orders are obvious before the collapse: they give rise to an additional collapse regularization that is much stronger than for . Eventually, an extremely broad supercontinuum is generated in the filament (full line). In contrast to the more studied near-infrared regime, at this and still longer wavelengths the spectral dynamics coupled with the dispersion properties of the medium become the most important physical mechanism controlling filamentation. It thus becomes crucial that properties of third- and fifth-harmonic generation, together with free-electron generation and accompanying energy losses, are all modeled in a unified manner, so that, as the wavelength is varied, all processes are included in correct proportion.
Importantly for our demonstration, all relevant effects are captured in a self-consistent way, without the necessity and even possibility of parameter tuning. To our knowledge, this is the first demonstration of a filament simulation on an experimentally relevant scale in which the light–matter interaction description relies on first principles.
In summary, we have introduced the metastable state approach for calculating the ultrafast, off-resonant nonlinear optical response of gases, and demonstrated its accuracy and computational economy. As a test bed, we employed a 1D atomic model that allows for comparison of MESA against exact solutions, and excellent agreement between the exact and approximate results were obtained for the nonlinear optical response even using only a single metastable state in adiabatic approximation. We have also identified post-adiabatic corrections that are responsible for the slight increase of ionization rate and for losses that the optical field suffers as a consequence of ionization. The approach was also applied to a 3D hydrogen-like model with similarly good agreement between MESA and TDSE simulations. As a demonstration of the computational economy of MESA, an example of optical filamentation was presented.
In the present work, we required knowledge of the properties of only a single metastable state, namely, the one that arises from the system’s ground state. Higher order metastable states were included only indirectly with the help of additional approximations. It will be interesting to explore the feasibility of inclusion of several states in . This should make it possible to capture even more noninstantaneous effects, and help to understand how a driven quantum system starts to exhibit behavior dependent on its own history.
In future work we plan to apply MESA to simulation of optical filamentation in current experiments, with argon, for example, and also to explore the application to molecules to allow simulation of air. While the initial TDSE-based characterization of the given model atom or molecule becomes much more demanding, the application of the method to such realistic systems remains conceptually the same. Recent progress in direct experimental characterization of optical nonlinearities at ultrafast time scales brought quantitative results concerning the self-focusing nonlinearity  as well as free-electron generation  in various gases. Importantly, spatiotemporal profiles of the nonlinear responses can now be obtained essentially free of propagation effects, which complicate interpretation of dynamics in other, for example, filamentation regimes. Comparison with these and similar future experiments should provide a practical way to validate and benchmark the nonlinear response models based on the approach proposed here.
To finish, we note that MESA yields a model for the ultrafast nonlinear optical response with no free parameters for a given gas. Importantly, from a practical simulation point of view, the approach has a computational complexity comparable to that of the standard light–matter interaction model used in optical filamentation. In this sense, the method provides a microscopically founded replacement.
Air Force Office of Scientific Research (AFOSR) (FA9550-10-1-0561, FA9550-11-1-0144, FA9550-13-1-0228).
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