1 Introduction

The response of the unionized bound electrons to the incident electric field constitutes the core of the interaction mechanism between an ultrafast laser pulse and matter [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Since the laser pulse has a driving force that will make electronics even faster in the near future, studies on light-controlled petahertz electronics have increased recently [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Today, semiconductor technology has already been linked to optical ultrashort electric fields with attosecond temporal control of excited electrons by absorbing laser light [26,27,28,29,30]. Ultrafast light transients offer unique control over atomic-scale electronic dynamics where this on-demand control paves the way for the development of ultrafast quantum electronics [23, 25, 29, 30, 34, 35, 37]. Furthermore, studies on the control of optical fields have provided advancing contemporary electronics toward the petahertz regime [23,24,25,26,27,28, 30].

Understanding the ultrafast dynamic properties of electrons of solids under the excitation of ultrashort electric field of a laser pulse opens up new prospects of light–matter interaction studies and applications at the interface of photonics and multi-petahertz electronics [26]. In Ref. [24], the response of electrons in gallium arsenide is resolved for the first time at the attosecond timescale. In Ref. [23], it is managed to direct access for the first time to the nonlinear response of bound electrons where the measurement of the light–matter mechanism on timescales around attoseconds is enabled. It is measured that particles respond to the force of incident light with delay due to their inertia instead of an immediate response [23, 24]. In Ref. [23], for the measurement of this delayed response time of electrons, a waveform-controlled nearly isolated ultrashort half-cycle laser pulse is produced at visible wavelengths, thus suppressing the multiphoton excitation and ionization processes [23, 29, 33]. In the experiment, it was observed that the excited electrons of krypton atoms needed approximately 100 attoseconds until the electrons reacted to light pulses of approximately 380 as (FWHM) with a center wavelength of 530 nm [23]. It is important here to emphasize that the excited electrons are unionized and stay bound to their parent atoms in the experiment [23]. In other words, the unionized bound electrons of a specific atom, here it is the krypton atom, start responding to a specific input optical field of a laser pulse with the given wavelength and width, only after 100 attosecond time has passed [23].

The thought-provoking experimental laser-bound electron interaction without ionization results of Ref. [23] gave rise to the numerical research in this paper. The light–matter interaction depends strongly on the ratio of the ultrashort pulse width to the duration of the characteristic response time (polarization response) of the medium, in addition to the pulse intensity and energy [21, 38]. Due to its mass of inertia, the unionized bound electron will sense the applied field unconventionally at the leading edge district, which is indicated with a question mark in Fig. 1 below where this part of the pulse is earlier than the response time of the material.

Fig. 1
figure 1

Time scheme of the light–matter interaction

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The time duration region shown in Fig. 1, from the \(t=0\) up to the yellow point, is called the material-specific response time of the unionized bound electron that is measured in Ref. [23]. Each material has its own specific value. The material starts responding to the applied excitation field (laser pulse) only after this point (i.e., unionized bound electrons start polarization by oscillating). In other words, for a conventional interaction to occur, every material needs a specific amount of time to pass to respond or to sense that there is an excitation field in the environment being applied to it. Only after this much time has passed does the unionized bound electron start responding conventionally to the input optical field of the laser pulse. In Ref. [23], it was experimentally proven that the bound electrons need a specific amount of material-dependent response time for sensing that there is an excitation electric field in the environment being applied [21, 22]. Briefly, conventional light–matter interaction starts only after this response-time parameter, which is shown as time duration up to the yellow point in Fig. 1 above. On the other hand, it is not correct to assume that the material is completely transparent to the applied field in the response-time region due to the absence of the conventional light–matter interaction. The main question in this work we seek for its answer is then, what exactly happens in the means of light–unionized bound electron interaction dynamics before this yellow point. In other words, since there is no conventional interaction in the material-specific response-time region, there must be an unconventional light-matter interaction mechanism and how we must mathematically describe this unconventional light-matter interaction term in the Hamiltonian definition for the response-time region. Such an unconventional light-matter interaction mechanism depicted in Fig. 1 is poorly studied in the literature and is normally disregarded since the ratio of the duration of the response time to the duration of the applied light is relatively small enough to be ignored. However, this argument is not valid for an unionized excitation case with an ultrashort single-cycle (one optical cycle) light pulse since the ratio of the duration of the response time to the width of the applied pulse is not small enough to be ignored. Thus, for the ultrashort single-cycle light-bound electron interaction without ionization case, the traditional quantum mechanical analysis is questionable, and the current modeling needs to be rehandled for a new fundamental understanding of ultrafast nonequilibrium unconventional responses [40]. Moreover, not only is the polarization response of the bound electrons to the applied optical field not conventional, as explained above, but the response of the bound electrons will also differ for some specific time-slits in the scale of one period of the ultrashort single-cycle optical pulse [21, 22, 40]. A quite simple, straightforward and modest analytical approach is utilized in Refs. [21] and [22] to coarsely show this temporal effect. The refractive index of matter evolves differently with respect to various specific time-slits occurring in one period of the applied single-cycle pulse width due to the delayed conventional polarization response of the unionized bound electrons. The temporal management of the refractive index could lead to the development of new types of imaging technologies that are more precise than current techniques [40]. A similar temporal effect for a different optical property is shown experimentally where the reflectance of the material is changed for some specific time-slits in the width of the applied ultrashort pulse, allowing light to pass through at specific times [40].

On the basis of the aforementioned insights, in this work, we focused on the problem of how the ultrashort single-cycle light–unionized bound electron interaction without ionization phenomena can be mathematically modeled inside the time region of the applied optical field that is shorter than the response time of the material (see Fig. 2a). Understanding the mechanism of the unconventional interaction phenomenon inside the response-time region is extremely important for designing advanced petahertz electronics devices in the near future. Lightwave electronics will especially provide achieving the quantum information processing as well as increase the bandwidth by 6 orders of magnitude comparing the conventional gigahertz electronics [30, 41].

Fig. 2
figure 2

a Time scheme of the interaction of the ultrashort single-cycle moderate intensity laser electric field, b Applied ultrashort single-cycle moderate intensity laser pulse \(E(t)\) in this study

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There are two commonly used forms of light–matter interaction Hamiltonian of an atomic system with an external applied field [2, 5, 9,10,11,12,13,14, 16]. These are velocity gauge and length gauge forms, which are also known as minimal coupling and direct coupling, respectively [9,10,11,12,13,14]. In this work, the Hamiltonian for a single-active bound electron approximation is utilized using atomic units \((e=m=\hslash =1/4\pi {\varepsilon }_{0}=1)\) where the bound electron is coupled to an external semiclassical unquantized moderate intensity (not causing ionization) electric field of a single-cycle laser pulse \(\overrightarrow{E}(t)\) as follows:

$$i\frac{\partial \psi (\overrightarrow{r},t)}{\partial t}=\left[-\frac{1}{2}{\overrightarrow{\nabla }}^{2}+V\left(\overrightarrow{r}\right)-\overrightarrow{r}.\overrightarrow{E}\left(t\right)\right]\psi \left(\overrightarrow{r},t\right).$$
(1)

Here, the light–matter interaction in the electric-dipole approximation using the length gauge form is expressed with the \(\overrightarrow{r}.\overrightarrow{E}(t)\) term defining the electron interaction with the laser field \(\overrightarrow{E}(t)\) at time \(t\) [9, 11,12,13,14,15,16]. For simplicity, we assume a one-dimensional problem with a linear plane-polarized beam along the x-axis, disregarding transverse effects such as diffraction broadening and focusing [6]. Thus, the ultrashort single-cycle laser pulse \(\overrightarrow{E}(t)\) is defined as (see Fig. 2b):

$$\overrightarrow{E}\left(t\right)=\overrightarrow{\widehat{x}}{E}_{0}{Sin}^{2}\left(\frac{wt}{2{N}_{c}}\right)Sin\left(wt+\mathrm{\varnothing }\right),0\le t\le \tau ,$$
(2)

where \({E}_{0}=\sqrt{2I/c{\varepsilon }_{0}}=2.5x{10}^{11}\) V/m (= 0.49 a.u.) is the peak value of the amplitude of the electric field, \(\tau =({N}_{c}/f)=1.77\) fs (= 72.57 a.u.) is the pulse duration, \(\varnothing =0\) is the initial carrier envelope phase, \({N}_{c}=1\) is the number of pulse cycles, \(w=3.55x{10}^{15}\) rad/s is the carrier frequency, \(\lambda =530\) nm is the center wavelength and \(I=83x{10}^{14}\) W/cm2 is the moderate intensity (not causing ionization) of the laser pulse (see Fig. 2b).

In Eq. (2), the absence of the static component in the field of the laser pulse is taken into account where \({\int }_{-\infty }^{+\infty }\overrightarrow{E}\left(t\right)dt=0\) [6,7,8, 15]. Utilizing \(\overrightarrow{E}(t)\) and the assumptions explained above, we obtain:

$$i\frac{\partial \psi (x,t)}{\partial t}=\left[-\frac{1}{2}\frac{{\partial }^{2}}{\partial {x}^{2}}+V\left(x\right)-xE(t)\right]\psi (x,t)$$
(3)

where \(V\left(x\right)=-1/\sqrt{1+{(x-{x}_{center})}^{2}}\) is the real soft-Coulomb potential [1,2,3, 42], the Hamiltonian is self-adjoint, and \(V\left(x\right)\) is real [5]. Here, \({x}_{center}\) is the center point of the potential located on the spatial domain \(L\) of the numerical problem. In our case, we assumed \(L=1\) a.u. and \({x}_{center}=L/2\). In semiconductor physics, the soft-Coulomb potential is commonly used since it eliminates divergence issues for the case of a one-dimensional bare Coulomb potential [42]. Furthermore, it possesses additional advantages such that parity is a good quantum number, its bound states do not have permanent dipole moments and it preserves the qualitative aspects of the Rydberg and continuous spectrum of the true hydrogen atom [2, 3].

Due to the delay of the response of the unionized bound electron to the applied ultrashort single-cycle electric field in the response-time region of the material, the light–matter interaction dynamics for this region cannot be completely defined only by utilizing the conventional \(xE(t)\) term as in Eq. (3). Then, the question is what causes this unconventional light–matter interaction without ionization phenomenon in the delay region of the pulse corresponding to the response-time duration of the unionized bound electron (see Fig. 2a above) and how to mathematically define this phenomenon. From this point of view, it is obvious that the electric field of the laser pulse \(E(t)\) is not the only player in this region, and it must be modified by a time-varying phenomenon that occurs only in the response-time region of the material. In this study, we named this phenomenon the modifier function \(M\left(t\right)\), which is \({M}_{t=0}=0\) and starts occurring naturally at the very beginning of the applied field after \(t\cong 0\). Its effectiveness increases up to a specific time and then starts decreasing and completely (or almost) vanishes towards the close proximity of the response-time point for each material (yellow flash point in Figs. 1 and 2a). Thus, with the aforementioned unknown modifier function embedded, we propose the following upgraded Hamiltonian for the unconventional light–matter interaction:

$$i\frac{\partial \psi (x,t)}{\partial t}=\left[-\frac{1}{2}\frac{{\partial }^{2}}{\partial {x}^{2}}+V\left(x\right)-xM(t)E(t)\right]\psi (x,t)$$
(4)

where \(xM(t)E(t)\) is the modified expression for the unconventional ultrashort single-cycle laser-matter interaction term that is valid only inside the material-specific response-time region of the unionized bound electron since, as explained above, the existence of \(M(t)\) is assumed to be limited only for this region. Each material must have its own unique \(M(t)\) modifier function for a given specific input \(E\left(t\right).\) Furthermore, \(M(t)\) is assumed to be always real so that the assuring Hamiltonian is kept as self-adjoint. Thus, the time evolution of \(\psi (x,t)\) is obtained by the formal solution of Eq. (4) as \(\psi \left(x,t\right)={e}^{-i\left(t-{t}_{0}\right)H}\psi \left(x,{t}_{0}\right)\) [2,3,4,5, 15]. We can define this solution utilizing the time step \(\delta \) and the mesh width \(\varepsilon \) where the temporal variable goes into \(n\delta \) and the spatial variable becomes \(j\varepsilon \), respectively, as:

$${\psi }_{j}^{n+1}={e}^{-i\delta H}{\psi }_{j}^{n}$$
(5)

Which is correct in terms of order \((\delta )\) [2,3,4,5, 15]. Here, to preserve the norm of the wavefunction over time, instead of the term \({e}^{-i\updelta H}\), the Cayley form approximation is used [2,3,4,5, 15]:

$${\psi }_{j}^{n+1}=\left(1-\frac{1}{2}i\delta H\right)/\left(1+\frac{1}{2}i\delta H\right) {\psi }_{j}^{n}.$$
(6)

The scheme in Eq. (6) is unitary, unconditionally stable, and accurate up to order \({\left(H\delta \right)}^{2}\) [2,3,4,5, 15]. Utilizing the Hamiltonian definition in Eq. (4) and carrying the indicated algebra in Eq. (6), we have the following difference equation:

$$ \psi_{j}^{n + 1} + \frac{1}{2}i\delta \left\{ { - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varepsilon^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varepsilon^{2} }$}}} \right)\left[ {\psi_{j + 1}^{n + 1} - 2\psi_{j}^{n + 1} + \psi_{j - 1}^{n + 1} } \right] + V_{j} \psi_{j}^{n + 1} - x_{j} M^{n} E^{n} \psi_{j}^{n + 1} } \right\} = \psi_{j}^{n} - \frac{1}{2}i\delta \left\{ { - \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\varepsilon^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\varepsilon^{2} }$}}} \right)\left[ {\psi_{j + 1}^{n} - 2\psi_{j}^{n} + \psi_{j - 1}^{n} } \right] + V_{j} \psi_{j}^{n} - x_{j} M^{n} E^{n} \psi_{j}^{n} } \right\}, $$
(7)

and completing the necessary arrangements yields Eq. (8) as below:

$${\psi }_{j+1}^{n+1}+\left[i\Gamma -{\varepsilon }^{2}{V}_{j}-2+{\varepsilon }^{2}{x}_{j}{M}^{n}{E}^{n}\right]{\psi }_{j}^{n+1}+{\psi }_{j-1}^{n+1}= -{\psi }_{j+1}^{n}\left[i\Gamma +{\varepsilon }^{2}{V}_{j}+2-{\varepsilon }^{2}{x}_{j}{M}^{n}{E}^{n}\right]{\psi }_{j}^{n}-{\psi }_{j-1}^{n}$$
(8)

where \(\Gamma =2{\varepsilon }^{2}/\delta \) [5]. At this point, it is straightforward to define the discretized form of modifier function \(M(t)\) as:

$${M}^{n}=\{-{\psi }_{j+1}^{n}+\left[i\Gamma +{\varepsilon }^{2}{V}_{j}+2\right]{\psi }_{j}^{n}-{\psi }_{j-1}^{n}-{\psi }_{j+1}^{n+1}-\left[i\Gamma -{\varepsilon }^{2}{V}_{j}-2\right]{\psi }_{j}^{n+1}- {\psi }_{j-1}^{n+1}\} / \{{\varepsilon }^{2}{x}_{j}{E}^{n}\left({\psi }_{j}^{n+1}+{\psi }_{j}^{n}\right)\}$$
(9)

Once, we defined the discretized modifier function as in Eq. (9) In terms of the discretized wavefunction, input ultrashort single-cycle electric field and atomic potential, we can now move forward for the numerical solution of the modifier function. Here, we should remark that although \(M(t)\) is the co-player with \(E(t)\) for the unconventional light - matter interaction in the response-time region, as conceptually sketched in Fig. 2a, \(E(t)\) is also one of the components that defines \(M(t)\), as shown in Eq. (9). This situation is expected since \(M(t)\) must be characterized for each various input \(E(t)\). The actual algorithm for the numerical solution of Eq. (9) is adapted from [5], where the algorithm is introduced explicitly. Therefore, we will restrict the actual presentation to the most important parts. We upgraded the recursion relations given in [5] and modified them to be compatible with our specific case so that the equations include the discretized input electric field \({E}^{n}\) and the modifier function \({M}^{n}\). Thus, the upgraded and modified recursion relations of Ref. [5] are obtained for our case as follows:

$${\Omega }_{j}^{n}=-{\psi }_{j+1}^{n}+\left[i\Gamma +{\varepsilon }^{2}{V}_{j}+2-{\varepsilon }^{2}{x}_{j}{M}^{n}{E}^{n}\right]{\psi }_{j}^{n}-{\psi }_{j-1}^{n},$$
(10)
$${e}_{j}^{n}=2+{\varepsilon }^{2}{V}_{j}-{\varepsilon }^{2}{x}_{j}{M}^{n}{E}^{n}-i\Gamma -\frac{1}{{e}_{j-1}^{n}},$$
(11)
$${f}_{j}^{n}={\Omega }_{j}^{n}+{f}_{j-1}^{n}/{e}_{j-1}^{n},$$
(12)
$${\psi }_{j}^{n+1}=\left({\psi }_{j+1}^{n+1}-{f}_{j}^{n}\right)/{e}_{j}^{n}.$$
(13)

We also need to define the initial state of the particle impinging on the potential under the applied electric field. Here, we assumed a Gaussian wave packet as the initial state:

$${\psi }_{initial}={e}^{i{k}_{0}x}{e}^{-{\left(x-{x}_{0}\right)}^{2}/2{\sigma }_{0}^{2}}$$
(14)

where \({k}_{0}=\pi /(20\varepsilon ), {x}_{0}=L/4\) and \({\sigma }_{0}=L/20\). The values of \({k}_{0}\), \({x}_{0}\) and \({\sigma }_{0}\) are also adapted from [5]. In the calculations, we used reflecting boundary conditions by setting the wave function to zero outside the computational region [3]. The following is the simple pseudocode of the upgraded algorithm for our case describing how we numerically obtained the modifier function \(M(t)\):

  1. 1.

    State the initial wavefunction and the atomic potential

  2. 2.

    Start temporal loop

  3. 3.

    Start 1st spatial loop

  4. 4.

    Evaluate Eq. (10), Eq. (11), Eq. (12)

  5. 5.

    End 1st spatial loop

  6. 6.

    Set the current wavefunction as \({\psi }_{CURRENT}\)

  7. 7.

    Start 2nd spatial loop

  8. 8.

    Evaluate Eq. (13)

  9. 9.

    End 2nd spatial loop

  10. 10.

    Set the new wavefunction as \({\psi }_{NEW}\)

  11. 11.

    Evaluate Eq. (9) using \({\psi }_{CURRENT}\) and \({\psi }_{NEW}\)

  12. 12.

    End temporal loop

In Fig. 3, the result of the numerical solution of the modifier function \(M(t)\) is depicted. The numerical solution shows that, as explained previously, the hypothetical modifier function appears almost instantly at the very beginning (~ 1 au) of the leading edge of the applied field. Then, it fades immediately around ~ 3 au, which corresponds to 73 attoseconds, consistent with the 100 attoseconds found in [23]. Thus, this point must be the response-time point of the atomic system under consideration. Our proposition for the unconventional light–matter interaction phenomenon in the delay region of the pulse is exactly described with the result in Fig. 3, where our hypothesis states that the electric field must be modified by a time-varying function that only occurs in the response-time region of the material. The result shows clear proof that there is a hidden \(M(t)\) phenomenon that must be understood, and it is obvious that \(M(t)\) will differ for each atomic system (in other words, material) and for each applied field.

Fig. 3
figure 3

a Solution of modifier function \(M(t)\), b Zoomed-in view

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The modest analytical work in this paper is the first in the literature that proposes an approach for modeling the mechanism of unconventional interaction of light with bound electrons without ionization in the response-time region of the material where the interaction mechanism is not yet understood. We hypothesized this underestimated phenomenon by simply embedding an unknown time-varying function as a multiplier factor to the interaction term in the length gauge form in the Hamiltonian definition of the system. The numerical solution for the embedded unknown function clearly proved that there is a phenomenon that we named the modifier function existing only in the response-time region of the material.

We believe that although this work is just a very small step of a new beginning, it will encourage researchers in this field and open an unprecedented area of research, both analytical and experimental. Determining the modifier function phenomenon for materials will excite and increase the curiosity of both theoretician and experimentalist readerships. In this way, more precise interaction mechanism modeling will bring extreme advantages in petahertz electronics. Finally, we encourage the readership and declare it as our future work is to calculate the expectation value of the observables in the response-time region for the unconventional light–matter interaction of unionized bound electrons.