A simple formula to predict the influence of the near-field in the optical control of confined electron systems

The time-dependent wave function ψ of a non-relativistic quantum particle with mass M and charge Q subjected to a laser field is described by the time-dependent Schrödinger equation

$$\begin{eqnarray}&&{\rm{i}}\hslash \displaystyle \frac{\partial \psi }{\partial t}=\left[\displaystyle \frac{1}{2M}{(-{\rm{i}}\hslash {\rm{\nabla }}-Q{\bf{A}})}^{2}+Q\varphi +V\right]\psi ,\end{eqnarray} \tag{ 1 }$$

where V, A, and φ represent, respectively, the electrostatic confining potential, and the vector and scalar potentials of the electromagnetic field. A and φ are determined in the Lorentz gauge by the following two coupled equations with vacuum permittivity and permeability, ε₀ and μ₀, respectively, as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{A}}}{\partial t}=-{\bf{E}}-{\rm{\nabla }}\varphi ,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \varphi }{\partial t}=-\displaystyle \frac{1}{{\varepsilon }_{0}{\mu }_{0}}{\rm{\nabla }}\cdot {\bf{A}}.\end{eqnarray} \tag{ 3 }$$

When the particle is excited by the laser pulse, it creates a polarization current density J, which is calculated from the time-dependent wave function ψ by

$$\begin{eqnarray}&&{\bf{J}}=\displaystyle \frac{Q\hslash }{2{\rm{i}}M}({\psi }^{* }{\rm{\nabla }}\psi -\psi {\rm{\nabla }}{\psi }^{* })-\displaystyle \frac{{Q}^{2}}{M}| \psi {| }^{2}{\bf{A}}.\end{eqnarray} \tag{ 4 }$$

This definition of J is derived by coupling the Schrödinger equation (1) to the equation of continuity for the probability density ∣ψ∣². Maxwell's equations with this polarization current density are described by the following equations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{E}}}{\partial t}=\displaystyle \frac{1}{{\varepsilon }_{0}}{\rm{\nabla }}\times {\bf{H}}-\displaystyle \frac{1}{{\varepsilon }_{0}}{\bf{J}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\bf{H}}}{\partial t}=-\displaystyle \frac{1}{{\mu }_{0}}{\rm{\nabla }}\times {\bf{E}}.\end{eqnarray} \tag{ 6 }$$

The Maxwell–Schrödinger hybrid simulation requires us to simultaneously solve equations (1)–(6). The physical situations described by these simultaneous equations are as follows [41–50]. The incident laser field described by E and H propagates in a vacuum towards the target system following Maxwell's equations ((5) and (6)). When the laser field reaches the target, the charged particle is excited through the electromagnetic potentials A and φ ((2) and (3)), whose propagation is described by the time-dependent Schrödinger equation (1). Then, the excited charged particle creates the polarization current density J defined by equation (4), which induces a near-field in the vicinity of the particle itself. This near-field interferes with the original incident laser field and thus redefines the electromagnetic fields.

In this subsection we explore the conditions to determine the relative strength of the near-field with respect to the incident laser field. We introduce the concept of the NFR, which can approximately evaluate using a simple calculation the ratio of the near-field amplitude over that of the incident laser field, and thus predict the significance of the near-field effect.

When an electron in its ground state is irradiated by a nearly resonant laser pulse, dynamic polarization in the charge-density distribution takes place owing to temporal oscillations of the probability density of the electron wave packet around its initial distribution of the ground state [41]. In the present problem of optically controlling quantum states the maximum polarization occurs when the initial state has been transferred completely to the excited state. This can happen when we employ a so-called light control pulse having a special spatio-temporal profile designed to achieve this complete transfer. The polarization in the charge-density distribution then creates a near-field whose maximum electric field can be estimated by the difference in the probability density distribution between the ground and the excited states. This idea allows us to evaluate the magnitude of the near-field and thus to define our proposed NFR.

Since charge neutrality is assumed in the present formulation, Gauss's law in Maxwell's equations is written with the polarization vector P, whose time-derivative gives the polarization current density J, as

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\varepsilon }_{0}{\bf{E}}=-{\rm{\nabla }}\cdot {\bf{P}}(\equiv {\rho }_{p}),\end{eqnarray} \tag{ 7 }$$

where the polarization charge density ρ_p has been introduced by the last equality. The spatio-temporal propagation of this polarization charge density can be calculated by the probability density of the particle wave function as

$$\begin{eqnarray}{\rho }_{p}({\bf{r}},\,t) & = & -\,{\rm{\nabla }}\cdot {\bf{P}}=-{\rm{\nabla }}\cdot \displaystyle \int {\bf{J}}{\rm{d}}t=-\displaystyle \int {\rm{\nabla }}\cdot {\bf{J}}{\rm{d}}t\\ & = & \displaystyle \int \displaystyle \frac{\partial Q| \psi {| }^{2}}{\partial t}{\rm{d}}t=Q| \psi ({\bf{r}},\,t){| }^{2}+C({\bf{r}}),\end{eqnarray} \tag{ 8 }$$

where C denotes the integration constant, which is determined by the initial condition for ρ_p and ψ. For example, the initial condition corresponding to unpolarized cases is described by ${\rho }_{p}(t=0)=-{\rm{\nabla }}\cdot {\bf{P}}(t=0)={\rm{\nabla }}\cdot {\varepsilon }_{0}{\bf{E}}(t=0)=0.$ Therefore, the polarization charge density ρ_p after an excitation by a laser field can be written as

$$\begin{eqnarray}&&{\rho }_{p}({\bf{r}},\,t)=Q| \psi ({\bf{r}},\,t){| }^{2}-Q| \psi ({\bf{r}},\,t=0){| }^{2}.\end{eqnarray} \tag{ 9 }$$

It is noted that the second term on the right-hand side of this equation always presents itself as a positive charge, exactly canceling the initial electric charge distribution $Q| \psi ({\bf{r}},\,t=0){| }^{2}$ because of the assumption of no initial polarization. In cases with nonzero initial polarization we also need to introduce additional conditions for the electric field E and the scalar potential φ, which are briefly described in the appendix.

When the target particle has been subjected to a light control pulse and has been transferred completely to the objective state ψ_O, a static electric field E_s appears thanks to the appearance of the polarization charge density along the polarization axis R of the incident pulse. Since this polarization occurs mainly along the R axis, the divergence of the electric field E in the left-hand side of equation (7) can be approximated by the R-derivative of E_s, namely,

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\varepsilon }_{0}{{\bf{E}}}_{s}\approx {\varepsilon }_{0}\displaystyle \frac{\partial {{\bf{E}}}_{s}}{\partial R}.\end{eqnarray} \tag{ 10 }$$

Therefore, from equations (7), (8) and (10) the following expression can be obtained for the static electric field E_s that can be used as a measure for the maximum strength of the near-field created by the particle excitation for arbitrary point R_a:

$$\begin{eqnarray}&&{{\bf{E}}}_{s}({R}_{a})\approx \displaystyle \frac{1}{{\varepsilon }_{0}}\displaystyle {\int }_{{R}_{0}}^{{R}_{a}}(Q| {\psi }_{O}(R){| }^{2}+C(R))\,{\rm{d}}R,\end{eqnarray} \tag{ 11 }$$

where R₀ indicating the start point for the integration must be chosen to coincide with the center of mass of the wave packet.

Since the intensity of E_s can be a measure of the near-field we define the NFR using the following equation

$$\begin{eqnarray}&&{\rm{N}}{\rm{F}}{\rm{R}}=\displaystyle \frac{{| {{\bf{E}}}_{s}| }_{{\rm{\max }}}}{{| {{\bf{E}}}_{i}| }_{{\rm{\max }}}},\end{eqnarray} \tag{ 12 }$$

where E_i is the electric field of the incident laser pulse. We note that this quantity NFR can be calculated without involving the time-dependent simulation but simply by the probability density distributions of the initial and objective states.

Figure 1 illustrates our studied system where a single electron is confined in a quasi-one-dimensional nanoscale potential well which extends along the z-axis like a nanotube. This confined electron system is irradiated by a light control pulse which is also polarized along this z-axis. In actual experiments a light pulse has a finite spot-size, like a Gaussian beam. On the other hand, since our target electron system is much smaller than the central wavelength of the light control pulses, we can safely assume that the electron feels a plane-wave form of the electric field. We thus excite the light control pulse as a plane wave in the present study. In the case when the system size is of the same order or even larger than the wavelength of the laser pulse, a light control pulse with also its spatial wave form optimized may be needed to provide higher efficiency than a simple plane-wave control pulse.

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The size and shape of this nanotube can be characterized by two parameters α and β; the α parameter specifies the length of the tube and has a unit of length in nanometers, while the dimensionless β parameter specifies the relative ratio between the length of the tube and the one-side length of its cross-section. For this system the time-dependent Schrödinger equation and the polarization current density ((1) and (4)) can be rewritten with mass m and charge q (= −e) of the electron as

$$\begin{eqnarray}&&{\rm{i}}\hslash \displaystyle \frac{\partial \psi }{\partial t}=\left[\displaystyle \frac{1}{2m}{\left(-{\rm{i}}\hslash \displaystyle \frac{\partial }{\partial z}-q{A}_{z}\right)}^{2}+q\varphi +V\right]\psi ,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\bf{J}}={J}_{z}{\hat{{\bf{a}}}}_{z}\\ &&\,=\left[\displaystyle \frac{q\hslash }{2{\rm{i}}m}\left({\psi }^{\ast }\displaystyle \frac{\partial \psi }{\partial z}-\psi \displaystyle \frac{\partial {\psi }^{* }}{\partial z}\right)-\displaystyle \frac{{q}^{2}}{m}| \psi {| }^{2}{A}_{z}\right]{\hat{{\bf{a}}}}_{z},\end{eqnarray} \tag{ 14 }$$

where ${\hat{{\bf{a}}}}_{z}$ denotes the unit vector along the z-axis. The wave function ψ in this Schrödinger equation is assumed to have finite probability density only within this tube. The incident light control pulse initially has only E_z and H_y components, before passing through the interaction region where the wave function spreads. All the other components of the electromagnetic field, however, need to be calculated, since immediately after the electron excitation the polarization current density creates the near-field, which  has all the field components.

By changing the α and β parameters we can adjust the shape, size, and thus the electron density of the confined electron system thanks to the normalization condition, denoted as

$$\begin{eqnarray}&&\displaystyle {\int }_{-\infty }^{\infty }\displaystyle {\int }_{-\infty }^{\infty }\displaystyle {\int }_{-\infty }^{\infty }| \psi (z){| }^{2}{\rm{d}}x{\rm{d}}y{\rm{d}}z={(\alpha \beta )}^{2}\displaystyle {\int }_{-\alpha }^{\alpha }| \psi (z){| }^{2}{\rm{d}}z=1.\end{eqnarray} \tag{ 15 }$$

The electrostatic confining potential V has been chosen to be a fourth power of z of the form

$$\begin{eqnarray}&&V={V}_{0}{\left(\displaystyle \frac{{10}^{-8}}{\alpha }\right)}^{2}{\left(\displaystyle \frac{z}{\alpha }\right)}^{4},\end{eqnarray} \tag{ 16 }$$

where V₀ is set as 50 eV. The potential energy curve of V is plotted by the solid green line as displayed in figure 2, with the solid blue and broken pink lines denoting the probability density distributions of the ground and first-excited states, ψ₀ and ψ₁, that have been chosen in the present simulation to be the initial and objective states, respectively. This kind of single-well potential can mimic the confining potential of nanoscale objects such as quantum dots. In all cases the potential energy curve and thus the probability density distributions of the eigenstates have been kept similar to each other for different values of α. This similarity allows us to study the near-field effects for different α on an equal footing. The energy difference between the ground and the first-excited states for α = 0.5 is 21 eV. This rather large energy gap corresponding to an EUV photon energy has been chosen rather arbitrarily, but so as to save computational time. Although a more realistic choice may be a few electron-volts, corresponding to a visible light, the observed trends of the near-field effects in the present study depend little on the choice of energy gap.

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The computational scheme we employed here to perform the hybrid simulation of the coupled Maxwell–Schrödinger equations is based on the so-called finite-difference time-domain (FDTD) method [41–52] in which the electromagnetic field and the wave function are represented on discretized space and time grids. The operations of the space- and time-derivatives on these discretized field and wave functions are performed relying on their finite-difference formulas. The spacing for the x, y, and z coordinates, namely, Δx, Δy, and Δz, are related to α in the present study, so as to maintain the similarity of the potential energy curve among cases with different values of α as

$$\begin{eqnarray}&&{\rm{\Delta }}x={\rm{\Delta }}y=\alpha \beta ,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}z=\displaystyle \frac{\alpha }{100}.\end{eqnarray} \tag{ 18 }$$

The smaller value of Δz in comparison to Δx and Δy defined above has been chosen to assure high accuracy in the calculation of the wave function, since in the present quasi-one-dimensional model the wave function extends only along the z-axis as displayed in figure 1. In the FDTD scheme these spatial grid sizes restrict us to choose the time spacing Δt in order to perform stable and reliable propagation. We have employed the following Δt value, which is smaller by a factor of 0.9 than the value allowed by the so-called Courant–Friedrichs–Lewy condition [41, 52]:

$$\begin{eqnarray}&&{\rm{\Delta }}t=\displaystyle \frac{0.9}{c\sqrt{\displaystyle \frac{1}{{\rm{\Delta }}{x}^{2}}+\displaystyle \frac{1}{{\rm{\Delta }}{y}^{2}}+\displaystyle \frac{1}{{\rm{\Delta }}{z}^{2}}}}.\end{eqnarray} \tag{ 19 }$$

One may refer to our previous study [41] where practical formulas to be used for coding the FDTD scheme in the hybrid simulation of the coupled Maxwell–Schrödinger equations were presented in detail in the appendix.

In this subsection we investigate the influence of the near-field on the control of quantum states and establish a relation between the results and the corresponding values of the NFR by changing the β parameter from 0.5 to 1.5 while α is fixed as 0.5 nm. With this β variation we can change the probability density of the electron wave packet and thus study the effect of electron density on near-field generation. The amplitude parameter E₀ in this subsection is fixed as 4.75 GV m⁻¹. As will be demonstrated in sections 3.2 and 3.3, the choice of E₀ is reflected in the pulse width. For large (small) values of E₀ the control pulse is short (long). Therefore, fast control can be achieved in general by a strong laser pulse. On the other hand, in real systems such strong laser pulses may damage the target system. In this study we have chosen this value of E₀ = 4.75 GV m⁻¹ rather arbitrarily such that the laser pulse is not too strong yet capable of finishing the control within a dozen cycles of oscillation of the electric field of the laser pulse.

The temporal profiles of the light control pulses designed for the two typical cases with β = 0.5 and 1.5 are displayed in figure 3. This figure shows that these two pulses overlap each other completely, indicating that the conventional light control pulse does not account for the difference in the electron density of the target system. This happens because the conventional pulse designing scheme does not take into account the local modification of the laser pulse by the near-field. We note, however, that a larger electron density in matter in general gives a larger near-field, which could disturb the incident laser pulse more significantly.

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The spatio-temporal propagation of the z-component of the electric field E_z (GV m^–1) in the narrow tube computed by the Maxwell–Schrödinger hybrid simulation is displayed in figure 4 for the two cases with β = 0.5 and 1.5 that correspond to the results of the light control pulses shown in figure 3. In the case of figure 4(a), corresponding to the case with β = 0.5, there appears to be a significant disturbance of the incoming plane-wave form of the electric field around the center of the tube within ∣z/α∣ < 0.25, where the electron wave packet and thus the polarization current density have appreciable amplitudes. This result indicates that the confined electron feels an electromagnetic field which is largely different from the original laser pulse, and that the light control pulse designed by the conventional scheme therefore shows a significantly low controllability. In the case of the result in figure 4(b), corresponding to the case with β = 1.5, on the other hand, there seems to be only a small modification of the incoming the electric field, which is distorted weakly at around ∣z/α∣ < 0.25. The origin of these two distinct results can be rationalized as the difference in the electron density, as has been observed in our previous study [41]: according to the normalization condition of equation (15) the amplitude of the wave packet is inversely proportional to the β parameter. Then, the amplitude of the wave packet is directly reflected in the amplitude of the polarization current density by equation (14). Therefore, the influence of the local modification to the incident laser pulse is strongly dependent on β.

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A more detailed investigation of the dependence of the near-fields on β was undertaken and the result is summarized in figure 5. In this figure we have plotted NFR and the time-averaged square-modulus of the overlap between the objective state ψ₁ and the electron wave packet after the pulse has gone (denoted by ${\bar{{\rm{\Omega }}}}_{1})$ for different values of β. The reason why we have taken a 'time-average' of this square-modulus of the overlap is due to the behavior of the electron wave packet in the Maxwell–Schrödinger hybrid scheme, whose components continue to oscillate if the control is incomplete. We note that this behavior is different from the oscillation of the time-dependent wave packet in the conventional framework of employing only the time-dependent Schrödinger equation. In the latter case the magnitude of each component remains unchanged after the pulse has gone and only their relative phases change in time, which results in the oscillation of the wave packet. In the present case, on the other hand, thanks to the coupling between Maxwell's and Schrödinger equations the oscillatory electron wave packet creates a near-field that modulates the relative strength in their components by repeated excitation and de-excitation as observed in our previous study [41]. A clear relation between ${\bar{{\rm{\Omega }}}}_{1}$ and the NFR can be established by the results in figure 5: as β increases, ${\bar{{\rm{\Omega }}}}_{1},$ which directly indicates the control accuracy with 1.0 for the highest and 0.0 for the lowest accuracies, is improved while the NFR decreases. More precisely, when the NFR is low, such as below 0.5, ${\bar{{\rm{\Omega }}}}_{1}$ is over 0.9, indicating a high control accuracy. On the other hand, when the NFR is high, such as over 1.5, ${\bar{{\rm{\Omega }}}}_{1}$ is in the range between 0.1 and 0.2, indicating that the control accuracy is quite low. This simple correspondence between the NFR and the control accuracy suggests that we can predict the near-field influence by simply calculating the NFR from equation (20). In the following subsections we consider other situations and show that this relation between the NFR and ${\bar{{\rm{\Omega }}}}_{1}$ applies commonly.

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In the previous subsection the dependence of the near-field on the β parameter, which controls the electron density of the target system, has been investigated. In this subsection the relation between the NFR and the control accuracy is demonstrated, focusing on its dependence on the characteristics of the laser pulse; more precisely, the dependence on the amplitude and thus the pulse width of the light control pulse. This can be achieved by introducing a dimensionless scaling parameter γ to the amplitude parameter of the laser pulse E₀ in equation (21) as

$$\begin{eqnarray}&&{E}_{0}=\gamma {E}_{0}^{n},\end{eqnarray} \tag{ 23 }$$

where ${E}_{0}^{n}$ represents the amplitude parameter for the 'normal' case with (α, β) = (0.5, 1.0) and has the value of 4.75 GV m⁻¹ as noted in the previous subsection. The geometry parameters α and β are kept constant throughout this subsection.

As was done in the previous subsection, we first demonstrate the results for two typical cases obtained by the Maxwell–Schrödinger hybrid simulation. Figure 6 shows the temporal profile of the light control pulses designed for γ = 0.2 and 2.0. In the case of γ = 0.2 the pulse width becomes longer than that displayed in figure 3 while in the case with γ = 2.0 it becomes shorter. This difference is due to the fact that any pulses designed to transfer the electronic state from its ground state into the first-excited state completely, have to contain the same amount of electromagnetic energy. Thus, the light control pulse for γ = 0.2 having a small amplitude has to be broad while that for γ = 2.0 having a large amplitude is short.

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As was done in figure 4, we have calculated the spatio-temporal propagation of E_z in the narrow tube for these two light control pulses and the results are displayed in figures 7(a) and (b) for γ = 0.2 and 2.0, respectively. The result for γ = 0.2 (figure 7(a)) shows that the incident plane-wave electric field is disturbed so strongly around the region ∣z/α∣ < 0.25 that even a clear node appears in this region as a result of interference between the original laser field and the near-field. In contrast, the result displayed in figure 7(b) for γ = 2.0 shows only a weak modification of the incident electric field, which becomes recognizable only after the pulse has started to decay after t > 1.0 fs. These observations can be rationalized as follows. As we noted in section 2.2, the near-field becomes largest when perfect control is achieved, namely, when the dynamic polarization in the charge-density distribution of the target electron system is largest. Therefore, in optical-control problems the upper bound of the near-field strength is governed purely by the electronic property of the matter and is independent of the characteristics of the laser pulses. The increase in the amplitude of the laser pulse, therefore, leads to a relative decrease in the amplitude of the near-field with respect to the incident laser field and thus a decrease in the near-field effect, as observed in figure 7(b).

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As has been done in figure 5, the correspondence between the NFR and the control accuracy ${\bar{{\rm{\Omega }}}}_{1},$ measured by the time-averaged square-modulus of the overlap between the electron wave packet and the objective state after the pulse has gone, is displayed in figure 8 for different values of γ ranging from 0.2 to 2.0. The results displayed in this figure support our reasoning: the control accuracy is improved for increasing γ and decreasing NFR and vice versa. The practically useful relationship between the NFR and the controllability of the light control pulse, which was explored in the previous subsection, can also be verified in the result of figure 8: when the value of the NFR is less than 0.5, the controllability of the light control pulse can be high with the ${\bar{{\rm{\Omega }}}}_{1}$ value >0.9. On the other hand, when the NFR value is over 1.5, the controllability becomes quite low with ${\bar{{\rm{\Omega }}}}_{1}$ as small as <0.1.

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In this subsection we investigate the influence of the near-field with respect to variation of the α parameter, which appears in equations (15)–(18), while keeping both β and γ as 1.0. By doing so we can examine its dependence on the length of the tube as displayed in figure 1. In the previous two subsections the dependence of the near-field effect on the electron density and on the characteristics of the light control pulse was studied by changing β and γ independently. However, when the α parameter is varied, as in this section, not only the electron density but also the energy gap between the ground and first-excited states, δ_E, changes since the confinement strength has been changed. A change in the energy gap between the target quantum states would naturally lead to a change in the frequency components involved in the light control pulse. Furthermore, we note that the integration range z_a to evaluate equation (20) would also be changed for different α since the extent of the electron wave packet become larger for larger α. This would result in another effect, since the electron wave packet oscillating in a wider spatial range would give a larger polarization and thus a larger near-field effect. Therefore, we can examine the interplay among the three different effects associated with the change in the electron density, the characteristics of the laser pulse, and the extent of the electron wave packet.

We note that the changes in the electron density, the control pulse, and the length of the electron wave packet along the polarization axis for different α contribute to the NFR in different ways. As α increases the confinement for the electron becomes weaker. At the same time, the cross-section of the tube becomes larger, as shown in figure 1. Therefore, the NFR would tend to decease for increasing α when considering only the change in the electron density. However, as α increases, the energy difference δ_E between the ground and the first-excited states becomes smaller owing to a weaker confinement, which results in the lowering of the resonance frequency. Then, the amplitude of the light control pulse decreases as its pulse width becomes longer. As has been demonstrated in section 3.2, such a decrease in the amplitude of the incident pulse leads to an increase in the NFR. Third, the increase in α leads to a longer electron wave packet along the z-axis and thus a larger polarization. This results in an increase in the value of the NFR.

The temporal profiles of the light control pulses designed for α = 0.5 and 1.5 are displayed in figure 9(a). The pink solid line corresponding to α = 0.5 in this figure shows a significantly larger amplitude and a higher central frequency than the blue line representing the case with α = 1.5. This difference is a consequence of the change in the strength of confinement, as has been explained above. Both pulses, however, can agree with each other when they are normalized such that the time t and the electric field ${E}_{z}^{(i)}$ are divided by the cycle T $(\equiv 2\pi \hslash /{\delta }_{E})$ and its maximum amplitude, respectively, as shown in figure 9(b). In order to maintain the shapes of the pulses as identical after this normalization we have employed the following form of the amplitude parameter that involves δ_E as

$$\begin{eqnarray}&&{E}_{0}=\displaystyle \frac{1}{9.5}{\left(\displaystyle \frac{{\delta }_{E}}{6.88}\right)}^{2}\times {E}_{0}^{n}.\end{eqnarray} \tag{ 24 }$$

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This agreement allows us to make a fair comparison among the cases with different α, focusing on the interplay between the effects of the electron density, the amplitude of the laser pulse or its pulse width, and the extent of the electron wave packet.

The spatio-temporal propagation of the resultant E_z obtained by the Maxwell–Schrödinger hybrid simulation for the cases with α = 0.5 and 1.5 is displayed in figures 10(a) and (b), respectively. In both figures the time and the field amplitude are normalized as in figure 9(b). The results displayed in figure 10(a) for the case with α = 0.5 show that the electric field of the incident laser pulse is modified weakly by the near-field after t/T ≥ 10, suggesting that the controllability of the pulse would be decreased only a little by this modification. On the other hand, the result in figure 10(b) corresponding to the case with α = 1.5, shows that the incident electric field is modified significantly around the electronic wave function, namely in the vicinity of the region ∣z/α∣ < 0.25, and that the amplitude of the electric field at around t/T ∼ 10 becomes very weak owing to destructive interference with the near-field. This result suggests that a larger α value would result in lower controllability, or in other words, a larger value of the NFR.

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We have performed similar simulations and calculated the NFR by changing the α value gradually in the range between 0.2 and 2.0. The controllability of the light control pulses has also been evaluated by calculating the time-averaged square-modulus of the projection of the final electron wave packet on the objective state ${\bar{{\rm{\Omega }}}}_{1}$, as was done in figures 5 and 8. The result is displayed together with the value of the NFR in figure 11. As α increases, the controllability of the laser pulse ${\bar{{\rm{\Omega }}}}_{1}$ decreases, while the NFR increases linearly. This result indicates that the dependence of the near-field effect on the characteristics of the light pulse, particularly its amplitude, or on the extent for the electron wave packet, is more dominant than that of the electron density in the present model system. As was done in the previous two subsections, we have examined the relationship between the values of the NFR and ${\bar{{\rm{\Omega }}}}_{1}.$ The result shows the same tendency as in the previous two cases: good controllability of the light control pulse $({\bar{{\rm{\Omega }}}}_{1}\gt 0.9)$ can be assured for NFR ≤ 0.5, while it becomes significantly low $({\bar{{\rm{\Omega }}}}_{1}\lt 0.1)$ for NFR ≥ 1.5. This result, together with those of the previous two subsections, suggests that the NFR can be an indicator to preliminarily estimate the influence of the near-field. This would enable us to judge whether to stay with the standard time-dependent Schrödinger equation (with low computational cost), or use the precise but complicated Maxwell–Schrödinger hybrid method, when using numerical simulations to design light control pulses for the optical control of quantum systems.

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Finally, we have summarized the ranges of parameters characterizing the light control pulses exploited in the present study. The field intensity of the light control pulses ranged from ∼10¹⁰ W cm⁻² (weak field) to ∼10¹⁵ W cm⁻² (strong field). The one-photon energy varied in the range between 1 to dozens of electron-volts depending on the strength of confinement for the target electron systems. These parameter ranges cover the photon energy and intensity ranges of interest in the current experiments using ultrashort laser pulses. Therefore, the present demonstration of the NFR can be of practical use in the estimation of the magnitude of the near-field effect in optical control of confined electron systems.