Wignerian symplectic covariance approach to the interaction-time problem

Abstract

The concept of the symplectic covariance property of the Wigner distribution function and the symplectic invariance of the Wigner–Rényi entropies has been leveraged to estimate the interaction time of the moving quantum state in the presence of an absolutely integrable time-dependent potential. For this study, the considered scattering centre is represented initially by the Gaussian barrier. Two modifications of this potential energy are considered: a sudden change from barrier to barrier and from barrier to well. The scattering state is prepared in the form of a Schrödinger cat, and the Moyal equation governs its further time evolution. The whole analysis of the considered scattering problem is conducted in the above-barrier regime using the phase-space representation of quantum theory. The presented concept shifts the focus from the dynamics of the quantum state to a symplectically invariant state functions in the form of the Wigner–Rényi entropies of order one and one-half. These quantities serve as indicators of the beginning and end of the interaction of the non-Gaussian state with the sufficiently fast decaying potential representing the scattering centre. The presented approach is significant because it provides a new way to estimate the interaction time of moving quantum states with the dynamical scattering centre. Moreover, it allows for studying scattering processes in various physical systems, including atoms, molecules, and condensed matter systems.

Introduction

It is widely assumed that the scattering experiments provide valuable information about the quantum systems studied¹. The elementary model of the scattering process assumes the existence of a region of space (interaction zone) where a relevant scatterer effectively interacts with an incident beam or an impulse of light or particles. As a result of this interaction, the outgoing state of the beam or impulse changes depending on the mechanism of this interaction, and analysis of this state allows one to draw conclusions regarding the scatterer. One of the most intriguing issues arising from this description concerns the traversal time, i.e. the time it takes for a particle to traverse the considered region^2,3. Light and matter’s wave nature makes this issue complex, especially when the scattering description is beyond the plane-wave approximation^4,5. Such approximation is highly desired in real situations because the incident wave is rarely monoenergetic. This variety of energy spectrum of the state allows for simultaneous mixture of scattering and tunneling behavior leading to challenging analysis of these components of the dynamics on the experimental basis. Hence, the traversal time can be rather viewed as the time of interaction of the quantum state with the finite-range potential barrier. This problem has been arousing widespread interest for a long time, but also some controversy^6,7. Nevertheless, due to the development of measurement techniques^8,9, it seems advisable to generalize the question⁶ ‘How long does it take to tunnel through a barrier?’, and instead of asking about the tunnelling time, we should ask about the interaction time. This issue is now becoming particularly relevant because the results of Spierings and Steinberg’s experiments¹⁰ have paved the way for the comparison of measurements with the direct and indirect theoretical measures of barrier crossing times, see Ref.¹¹ and references therein. Those inspiring results create the possibility of searching methods and research tools enabling a quantitative determination of the interaction time, regardless of whether we are in the scattering or tunnelling regime. Analyzing this issue within Wigner’s phase-space formalism, we discovered a universally applicable method of assessing the interaction times based on well-established physical concepts like entropy or entropic measures. This method is deeply embedded in the symplectic covariance property of the Wigner distribution function (WDF)^12,13,14. This function is commonly used to describe the quantum states of the system in the phase space^{15,16,17,18,19,20,21,22}. Especially for pure states, the WDF is defined as follows^23,24

$$\begin{aligned} \displaystyle {\mathscr {W}}(\psi )(x,p) = \frac{1}{2\pi \hbar } \int _{{\mathbb {R}}}{dX \;} \psi \left( x+\frac{X}{2}\right) \psi ^{*}\left( x-\frac{X}{2}\right) \exp \left( {-\frac{i}{\hbar }pX}\right) , \end{aligned}$$

(1)

where $\psi (\cdot )\in L^2({\mathbb {R}},dx)$ is the wave function, and the rest of the symbols have the usual meaning. Surprisingly, if one is considering only pure states, then the interpretation of WDF not only as the phase-space distribution but rather the probability amplitude function is valid, as stated in²⁵, allowing for wider entropic analysis²⁶. It is worth noting here that the WDF, up to a multiplicative constant, is the only quasi-probability distribution function that is covariant under both translations and all symplectic transformations (including shear transformations, rotations and scalings)²⁷. The fundamental characteristic of these transformations lies in their ability to maintain the area occupied by the WDF regardless of the deformations they impose. Generally the area occupied by the WDF in the phase space is invariant with respect to the group of symplectic transformations²⁸

$$\begin{aligned} Sp(1,{\mathbb {R}}) = \left\{ \mathbf{{S}}\in {\mathscr {M}}_{2\times 2}({\mathbb {R}}):\mathbf{{S}}^T\mathbf{{J}} \mathbf{{S}}=\mathbf{{J}} \right\} = \left\{ \mathbf{{S}}\in {\mathscr {M}}_{2\times 2}({\mathbb {R}}):\det {\mathbf{{S}}}=1 \right\} , \end{aligned}$$

(2)

where ${\mathscr {M}}_{2\times 2}({\mathbb {R}})$ is the set of $2\times 2$ real matrices and $\mathbf{{J}}$ is the standard symplectic matrix ($\mathbf{{J}}^T=-\mathbf{{J}}$ and $\mathbf{{J}}^2=-\mathbf{{I}}$, where $\mathbf{{I}}$ is the identity matrix). On the other hand, a symplectic change of variables performed on arguments of the WDF is equivalent to calculating the WDF for the transformed wave function at previous points. This property is known as the symplectic covariance of the WDF²⁹, and formally it can be expressed by the following equality:

$$\begin{aligned} {\mathscr {W}}\left( \hat{{\mathscr {K}}}_{\mathbf{{S}}}\psi \right) (x,p) = {\mathscr {W}}(\psi )\left( \mathbf{{S}}^{-1}[x\phantom {m}p]^T\right) , \end{aligned}$$

(3)

where $\mathbf{{S}}\in Sp(1,{\mathbb {R}})$ is the aforementioned symplectic matrix and $\hat{{\mathscr {K}}}_{\mathbf{{S}}}$ is an associated operator belonging to the metaplectic group Mp(1) of unitary operators³⁰ that act in the Hilbert space of square-integrable functions $\psi (\cdot )\in L^2({\mathbb {R}},dx)$. Let us note that the relation between $\hat{{\mathscr {K}}}_{\mathbf{{S}}}$ and $\mathbf{{S}}$ is two-to-one, so the difference in the choice of the operator $\hat{{\mathscr {K}}}_{\mathbf{{S}}}$ is barely in its sign³¹. This issue can be omitted in the presented discussion due to the insensitivity of the WDF to phase disturbance of the wave function. The meaning of the metaplectic operators is revealed in the description of the WDF dynamics. For this purpose, we consider the time dependence of the WDF that results from the time evolution of the wave function $\psi (\cdot ;t)$ generated by the Cauchy problem in the form

$$\begin{aligned} \left\{ \begin{array}{l} i\hbar \partial _t\psi (\cdot ,t) = {\hat{H}}(t)\psi (\cdot ,t),\quad t>0 \\ \psi (\cdot ,0)=\psi _0(\cdot ), \end{array} \right. \end{aligned}$$

(4)

where the wave function $\psi _0(\cdot )$ represents the initial condition and ${\hat{H}}(t)$ is the time-dependent one-particle Hamiltonian defined as the sum of the kinetic and potential parts of the considered system. The formal solution of Eq. (4) can be written as follows,

$$\begin{aligned} \psi (\cdot ,t) = \hat{{\mathscr {U}}}(t,0)\psi _0(\cdot ), \end{aligned}$$

(5)

where $\hat{{\mathscr {U}}}(t,0)$ is the unitary time evolution operator³². If we assume the potential energy to be in the form of a polynomial of order two with time-dependent coefficients³³, then the operator $\hat{{\mathscr {U}}}(t,0)$ is a product of two unitary operators. The first one translates the wave function in position and momentum and also shifts the global phase of the wave function. The second one is a product of three metaplectic operators belonging to the group Mp(1) hence it is also a metaplectic operator as shown in Refs.^34,35. In particular, this statement holds in the case of motion in the free space. This observation is a starting point of our analysis; the time-evolution operator generated by the free-particle Hamiltonian is a metaplectic operator, and according to symplectic covariance property of WDF the following formula holds:

$$\begin{aligned} \int _{{\mathbb {R}}^2} {dx dp}\; f\left( {\mathscr {W}}\left( e^{-\frac{i}{\hbar }\frac{{\hat{p}}^2}{2m}t}\psi _0\right) (x,p)\right) = \int _{{\mathbb {R}}^2} {dx dp}\; f\left( {\mathscr {W}}(\psi _0)(x,p)\right) , \end{aligned}$$

(6)

where the function $f: [-1/(\pi \hbar ), 1/(\pi \hbar )]\rightarrow {\mathbb {R}}$ and both integrals are finite. This characteristic serves as the motivation for looking beyond the scenario of free propagation through the establishment of the time-dependent symplectically invariant measure $\zeta (t)$, according to the formula

$$\begin{aligned} \zeta (t):= \int _{{\mathbb {R}}^2} {dx dp}\; f\left( {\mathscr {W}}\left( \psi (t)\right) (x,p)\right) . \end{aligned}$$

(7)

Using these two equations we can conclude that the measure in question remains unchanged until a barrier potential disrupts the free motion. Owing to this property one can find a specific time interval for which the time evolution of the WDF differs from the free-space motion.

In Fig. 1, we present a qualitative explanation of the idea used.

Fig. 1

The deformation of patch defined over p$\hbar$ase space (x, p) under transformation generated by a free potential (a) and arbitrary potential (b) (marked with black line). For transformations generated by free particle potential (F), it is known that any patch defined on p$\hbar$ase space undergoes shearing transformation, that is linear, momentum-dependent stretching. By the contrast, for area of interaction (I), transformation is no longer of this type, so no symplectic transformation can be assigned to such an operator. Thus, symplectic covariance property is violated resulting in changes in the symplectically invariant measure. This allows finding certain times $t_i$, $t_f$, for which given invariant measure respectively begins and ends to vary with time. This area of interaction is marked with purple color.

First, the initial state evolves in free space (F), equivalently undergoing a symplectic shearing transformation. Following that, the result of this evolution is subjected to nonsymplectic deformations through the nonzero potential region (I). After that, such a deformed state evolves again as a state in free space (F), and its further evolution is symplectically invariant. In other words, we use the symplectic invariance of the quantum state resulting from the free motion in space to determine the interaction time through the non-zero potential energy region defining the interaction zone. One might pose the question of whether this concept, as per Eq. (7), can be extended to encompass cases involving harmonic or anharmonic potentials. While the answer is affirmative, it’s crucial to keep in mind that arbitrary smooth potentials primarily display anharmonic repelling or harmonic bounding behavior in localized regions, such as close neighborhood of local extrema of the potential. Considering this, and the fact that the wave function’s support is rarely compact, it leads us to the conclusion that these cases cannot serve as a reliable indicator of the interaction between a quantum object and a potential. In other words, there is always a portion of the wave function that interacts with the non-harmonic part of the potential, with the exception of cases involving pure $a x^2$ potentials, for $a\ne 0$.

In this paper, we use the symplectically invariant measures: Wigner–Rényi one-half entropy, Wigner–Shannon entropy to estimate the interaction time of the highly nonclassical state in the form of Schrödinger Cat (SC) state^{36,37,38,39,40} with the Gaussian potential barrier with time-modulated amplitude. It opens an engaging perspective for spectroscopic methods, especially in application to quantum-optical spectroscopy with the goal of enhancing the signal and narrow spectral lines, as presented in Ref.^41,42. To prove the usefulness, as well as to indicate the scope of applicability of the proposed method, we considered two limiting cases of the time-dependent scattering potential, i.e. we chose the potential amplitude in two ways. The first one corresponds to a single change in the value of the potential energy amplitude without changes in its sign, while the second one concerns its sign change without changes in the value. Based on both these cases, we directly determine the time-dependent Wigner–Shannon entropy^43,44,45 and the time-dependent Wigner–Rényi one-half entropy, recently examined as an equivalent measure of nonclassicality of the WDF²⁶ as two different examples of symplectically invariant measures. The careful analysis of these dynamical quantities allows us to estimate the interaction time in one of these cases. On the contrary, the second case is treated as a counterexample, illustrating the limitations of the offered method. We support these results by analyzing the dynamical transmission and capture coefficients which allow us to reinforce presented results.

Theoretical framework

Phase-space approach

In the following sections, we denote the time-dependent WDF as ${\mathscr {W}}(\psi (\cdot ,t))(x,p)=\varrho (x,p,t)$ if there is no need to explicitly specify what the wave function of the quantum state is. This shorter notation allows us to introduce the explicit time dependence of the WDF in a compact way. According to the general rules describing the quantum systems in the phase space, dynamical variables A(x, p) characterizing the considered system are expressed by the corresponding Weyl symbols, $a_W(x,p)$, which are functions defined on the phase space. The construction of the appropriate Weyl symbol is based on the Weyl transform²², according to which the Hermitian operator, ${\hat{A}}({\hat{x}}, {\hat{p}})$, acting on the Hilbert space, is assigned the real function $a_W(x,p)$ expressed by the formula⁴⁶

$$\begin{aligned} \displaystyle a_W(x,p) \displaystyle = \displaystyle \int _{{\mathbb {R}}}\; {dX}\; \left<x+\frac{X}{2}\right| {\hat{A}}({\hat{x}}, {\hat{p}}) \left| x-\frac{X}{2}\right> \displaystyle \displaystyle \exp \left( {-\frac{i}{\hbar }pX}\right) . \end{aligned}$$

(8)

In the case of the time-dependent Hamiltonian of the one-particle system, ${\hat{H}}={\hat{p}}^2/(2m)+W({\hat{x}},t)$, the corresponding Weyl symbol, $H_{\textrm{W}}(x,p,t)$, takes the form of the following real function

$$\begin{aligned} H_{\textrm{W}}(x,p,t) = \frac{p^2}{2m} + W(x,t), \end{aligned}$$

(9)

where W(x, t) is the potential term assumed to be separable, i.e. $W(x,t)=U(x)\Sigma (t)$, wherein U(x) is the short-range potential energy, i.e. it belongs to the class of absolutely integrable functions (cf. Supplementary Information for more details), and $\Sigma (t)$ is the time-dependent modulation factor of the potential amplitude. For the above-established Weyl symbol of the Hamiltonian, the time evolution of the WDF can be expressed by the Moyal equation,

$$\begin{aligned} \partial _t \varrho (x,p,t) = -\frac{p}{m}\partial _x\varrho (x,p,t)+\frac{1}{i\hbar } \left[ U\left( x+\frac{i\hbar }{2}\partial _p\right) - U\left( x-\frac{i\hbar }{2}\partial _p\right) \right] \Sigma (t)\; \varrho (x,p, t). \end{aligned}$$

(10)

The dynamics of the WDF generated by solving the initial value problem for the Moyal equation also requires establishing an initial condition. This initial condition is taken as the phase-space representation of the SC state for which the WDF is given by the formula

$$\begin{aligned} \begin{aligned} \varrho (x,p,0)&= A_1^2 \frac{\left( 1-\beta \right) }{\pi \hbar } \exp \left[ -\frac{\left( x-x_1\right) ^2}{2\delta _x^2} - \frac{2\delta _x^2\left( p-p_0\right) ^2}{\hbar ^2}\right] + A_1^2 \frac{\beta }{\pi \hbar } \exp \left[ -\frac{\left( x-x_2\right) ^2}{2\delta _x^2} - \frac{2\delta _x^2\left( p-p_0\right) ^2}{\hbar ^2}\right] \\&+ 2A_1^2 \frac{\sqrt{\beta \left( 1-\beta \right) }}{\pi \hbar } \cos \left[ \vartheta + \frac{p-p_0}{\hbar } \left( x_1 - x_2 \right) \right] \exp \left[ -\frac{\left( x-\frac{ x_1 + x_2 }{2}\right) ^2}{2\delta _x^2}-\frac{2\delta _x^2\left( p-p_0\right) ^2}{\hbar ^2} \right] , \end{aligned} \end{aligned}$$

(11)

where the normalization factor $A_1$ has the form

$$\begin{aligned} A_1 = \left[ 1+2\sqrt{\beta (1-\beta )} \exp \left( -\frac{(x_1-x_2)^2}{8\delta _x^2}\right) \right] ^{-\frac{1}{2}}. \end{aligned}$$

(12)

This state is further denoted as the SC-WDF. The first two terms of the SC-WDF correspond to the two Gaussian wave packets localized at the different phase-space points, namely $(x_{1}, p_0)$ and $(x_{2}, p_0)$. In turn, the last term represents the quantum interference pattern with a Gaussian envelope. This term is centered around the phase-space point $((x_1+x_2)/2,p_0)$ and rapidly oscillates along the x-axis. The set of parameters defining this SC-WDF is taken in the same form as follows: $\beta = 0.5$, $\vartheta =0$, $\delta _x^2 = 500$ a.u., $x_1 = -300$ a.u., $x_2 = -500$ a.u. and $p_0 = 0.15$ a.u. This choice of parameters means that the initial SC-WDF is located to the left of the Gaussian barrier at a distance greater than its initial spatial dimension, and it approaches the barrier with the expected value of the kinetic energy slightly greater than the top of the barrier. It means that our computational scattering experiments are conducted in the above-barrier regime⁴⁷.

Dynamical characteristics

Due to the time-dependent character of the performed analysis, we propose the following set of dynamical quantities that characterize the system described by Eq. (9). The remaining dynamical quantities considered by us are based on the fact that the expression

$$\begin{aligned} \zeta (t) = \int _{{\mathbb {R}}^{2}} {dx dp}\; f\left( \varrho (x,p,t)\right) \, \end{aligned}$$

(13)

is invariant under the time evolution governed by the Hamiltonian with the potential term in the form of a polynomial of the order less than or equal to 2, independent of the form of the function f (more details can be found in Supplementary Information). One of the quantities that can be understood in the sense of the above expression is the Wigner–Rényi entropic measure of order $\alpha$ which is defined for pure states as follows²⁶

$$\begin{aligned} S_\alpha (t) = \frac{1}{1-\alpha }\ln \left[ {(2\pi \hbar )^{2\alpha -1}\int _{{\mathbb {R}}^2}dx dp \left| \varrho (x,p,t)\right| ^{2\alpha } }\right] , \end{aligned}$$

(14)

where $\alpha$ is the Rényi index ($\alpha \ge 0$ and $\alpha \ne 1$). However, for further analysis, we focus our attention on the special case of this entropy, namely $\alpha =1$, which corresponds to the Wigner–Shannon entropy⁴⁵

$$\begin{aligned} S_1(t) = \lim _{\alpha \rightarrow 1}S_\alpha (t) =-\ln {(2\pi \hbar )}-2\pi \hbar \int _{{\mathbb {R}}^2} {dx dp}\; |\varrho (x,p,t)|^2\ln \varrho ^2(x,p,t) \,, \end{aligned}$$

(15)

where the coefficient $2\pi \hbar$ is a byproduct of normalization condition of the WDF in the $L^2({\mathbb {R}}^2,dxdp)$ norm.

Another dynamic quantity consistent with the expression given by Eq. (7) is the time-dependent Wigner–Rényi one-half entropy which is equal to

$$\begin{aligned} S_{1/2}(t) = 2\ln \left( \int _{{\mathbb {R}}^2}{dx dp}\;|\varrho (x,p,t)|\right) + \ln {(2\pi \hbar )}, \end{aligned}$$

(16)

The Wigner–Rényi one-half entropy can be related to the nonclassicality parameter

$$\begin{aligned} \delta (t) = \int _{{\mathbb {R}}^2}{dxdp}\;\left| \varrho (x,p,t)\right| \, - 1, \end{aligned}$$

(17)

which encapsulates information about the changes of the phase-space area occupied by the negative values of the WDF during the time evolution of this function^47,48,49,50. According to the discussions concerning the symplectic invariance measures and the symplectic covariance of the WDF presented in earlier parts of this paper, these two dynamic quantities, i.e. the Wigner–Shannon entropy and the Wigner–Rényi one-half entropy, can be applied to determine the interaction time of the state represented by the WDF with the potential given in the form of absolutely integrable functions because these both quantities should be stabilized if the WDF leaves the interaction zone and its motion is recognized as the free movement. In other words, the end of the interaction process should be manifested by the absence of changes in the values of one of these quantities over time. Based on this observation, the time of interaction between the potential and the WDF can be calculated according to:

Proposition 1

For a fixed Rényi index $\alpha > 0$ the interaction time $\tau$, between state represented by the Wigner distribution function and potential energy belonging to the class of absolutely integrable functions, based upon Wigner–Rényi entropic measure, can be defined as

$$\begin{aligned} \tau= & \textrm{min}\left\{ t\in {\mathbb {R}}_{+}: |S_\alpha (t)-S_\alpha (\infty )|> \varepsilon (M-m)\right\} \nonumber \\- & \textrm{max}\left\{ t\in {\mathbb {R}}_{+}: |S_\alpha (t)-S_\alpha (0)| > \varepsilon (M-m)\right\} \nonumber \\= & t_f - t_i, \end{aligned}$$

(18)

where $S_\alpha (t)$ is a Wigner–Rényi entropy with Rényi index $\alpha$, $S_\alpha (0)$ is the value of the entropy at the beginning of the evolution, $S_\alpha (\infty )$ is the value at the end of the evolution, $M = \textrm{max}_{t\in {\mathbb {R}}_+}S_\alpha (t)$ is the maximum of Wigner–Rényi entropic measure, $m = \textrm{min}_{t\in {\mathbb {R}}_+}S_\alpha (t)$ is its minimum, and $0<\varepsilon \ll 1$ is a dimensionless threshold parameter, that can be understood as the precision of the measuring device.

During the early evolution of the WDF, any $\zeta (t)$ is constant due to the lack of significant interaction between the potential and the initial condition; thus, one should expect nearly constant values of $\zeta (t)$ which implies a facilitated calculation of $t_i$. The problem arises when one tries to evaluate $t_f$ because, as mentioned earlier, if the SC-WDF happens to settle in the well, then formally the interaction continues and as a result a symplectically invariant measure is nonconstant in long-time behavior. The criterion we propose to find $t_f$ is the numerical analysis of the first derivative of the symplectically invariant measure with respect to time. The suitable choice of $\zeta (t)$ will be explained in detail within the next sections.

Results and discussion

Numerical algorithm

The following Fourier transforms,

$$\begin{aligned} {\mathscr {F}}_{1,x\rightarrow \lambda }\varrho (x,p,t) = \frac{1}{\sqrt{2\pi \hbar }}\int _{{\mathbb {R}}} {dx}\; e^{-\frac{i}{\hbar }\lambda x}\varrho (x,p,t)\, \end{aligned}$$

(19)

$$\begin{aligned} {\mathscr {F}}^{-1}_{1,\lambda \rightarrow x}\tilde{\varrho }(\lambda ,p,t) = \frac{1}{\sqrt{2\pi \hbar }}\int _{{\mathbb {R}}} {d\lambda }\; e^{\frac{i}{\hbar }\lambda x}\tilde{\varrho }(\lambda ,p,t)\, \end{aligned}$$

(20)

$$\begin{aligned} {\mathscr {F}}_{2,p\rightarrow y}\varrho (x,p,t) = \frac{1}{\sqrt{2\pi \hbar }}\int _{{\mathbb {R}}} {dp}\; e^{-\frac{i}{\hbar }py}\varrho (x,p,t)\, \end{aligned}$$

(21)

$$\begin{aligned} {\mathscr {F}}^{-1}_{2,y\rightarrow p}\tilde{\varrho }(x,y,t) = \frac{1}{\sqrt{2\pi \hbar }}\int _{{\mathbb {R}}} {dy}\; e^{\frac{i}{\hbar }py}\tilde{\varrho }(x,y,t), \end{aligned}$$

(22)

and the unitary equivalence of multiplication and derivative operators, i.e.,

$$\begin{aligned} {\mathscr {F}}^{-1}_{1,\lambda \rightarrow x} \lambda {\mathscr {F}}_{1,x\rightarrow \lambda } = \frac{i}{\hbar }\partial _x, \quad \quad {\mathscr {F}}^{-1}_{2,y\rightarrow p} y {\mathscr {F}}_{2,p\rightarrow y} = \frac{i}{\hbar }\partial _p, \end{aligned}$$

(23)

allow us to transform Eq. (10) to the following form

$$\begin{aligned} \partial _t \varrho (x,p,t) = \left( -\frac{i}{\hbar m}{\mathscr {F}}^{-1}_{1,\lambda \rightarrow x}p\lambda {\mathscr {F}}_{1,x\rightarrow \lambda }\right. + \left. \frac{i}{\hbar }\Sigma (t){\mathscr {F}}^{-1}_{2,y\rightarrow p}U_{\Delta }(x,y){\mathscr {F}}_{2,p\rightarrow y}\right) \varrho (x,p,t), \end{aligned}$$

(24)

where the variables $\lambda$ and y are the canonically conjugate variables to x and p, respectively, as it results from the definition of the Fourier transform pairs (19)–(22), and the auxiliary function $U_\Delta (x,y)$ is defined as

$$\begin{aligned} U_{\Delta } (x,y)=U\left( x+\frac{y}{2}\right) -U\left( x-\frac{y}{2}\right) . \end{aligned}$$

(25)

In the presented case, $\Sigma (t)$ is a piecewise constant function. For this class of functions, the time evolution can be obtained by solving the dynamical problem in consecutive intervals where $\Sigma (t)$ is constant. For each of those intervals, we impose a new initial condition, taken as the state resulting from the previous stage of calculations.

Let $[t_k,t_{k+1})$ be the interval where $\Sigma (t) = u_k$, and $\{t_{k_j}\}_{j\in J}$ be a set of equidistant, by time step $\Delta t$, points within that interval for some set of indexes J. Then, the evolution within a given interval can be calculated using the Strang approximation for the exponential operator^51,52,53,54, as

$$\begin{aligned} \begin{array}{l} \varrho (x,p,t_{k_{j+1}}) = \varrho (x,p,t_{k_j}+\Delta t) = \exp \left( -\frac{i}{\hbar m}{\mathscr {F}}^{-1}_{1,\lambda \rightarrow x}p\lambda {\mathscr {F}}_{1,x\rightarrow \lambda }+\frac{i}{\hbar }u_k{\mathscr {F}}^{-1}_{2,y\rightarrow p}U_{\Delta }(x,y){\mathscr {F}}_{2,p\rightarrow y}\right) \varrho (x,p,t_{k_j})\\ \displaystyle \approx {\mathscr {F}}^{-1}_{1,\lambda \rightarrow x} \exp \left[ {-\frac{i\Delta t}{2m}\lambda p}\right] {\mathscr {F}}_{1,x\rightarrow \lambda } {\mathscr {F}}^{-1}_{2,y\rightarrow p} \exp \left[ {i u_k\Delta t U_{\Delta }(x,y)}\right] {\mathscr {F}}_{2,p\rightarrow y} {\mathscr {F}}^{-1}_{1,\lambda \rightarrow x}\exp \left[ {-\frac{i\Delta t}{2m}\lambda p}\right] {\mathscr {F}}_{1,x\rightarrow \lambda }\varrho (x,p,t_{k_j}). \end{array} \end{aligned}$$

(26)

The result of Eq. (26) gives the final state in time $t_{k+1}$, which becomes the initial condition for the next interval $[t_{k+1},t_{k+2})$ with constant $\Sigma (t)= u_{k+1}$. This procedure is repeated to find the entire evolution of the WDF. The algorithm can be used for the single change in the amplitude of the potential at time $t_\textrm{s}$ if one assumes that $t_0=0$, $t_1 = t_\textrm{s}$, $t_2 = t_{\textrm{end}}$. Then, according to the formula above, the initial condition is iterated first within the interval $[0,t_\textrm{s})$ preceded by iteration within $[t_\textrm{s},t_\textrm{end})$. Due to the fact that the Fast Fourier Transform is applicable for periodic boundary conditions, we propose a uniform grid of size $N_x\times N_p$ with right half-open intervals, where the box size is chosen so that the WDF vanishes at or near the boundaries during the whole simulation.

Simulation parameters

To construct the explicit form of the Weyl symbol of the Hamiltonian given by Eq. (9), we propose the potential energy part of W(x, t) in the form of a Gaussian barrier,

$$\begin{aligned} U(x) = U_0\exp \left[ -\frac{(x-x_B)^2}{2w^2} \right] , \end{aligned}$$

(27)

with the following parameters: the center of the barrier localization is $x_B = 0$ a.u., the strength of this barrier is $U_0 = 0.008$ a.u., and its width is given by $w^2 = 50$ a.u. Let us note that the choice of the potential in the form of a Gaussian function guarantees that it is absolutely integrable. Moreover, this potential rapidly decays in the interval $L=\left[ x_B-3w,x_B+3w\right]$, where w can be interpreted as the standard deviation. Hence, we conclude that at infinity the scattered state can be regarded as a free state. In turn, the time-dependent modulation factor of the potential amplitude W(x, t), denoted by $\Sigma (t)$, is chosen in two ways. The first one illustrates a single change in the value of the potential energy amplitude without changes in its sign, while the second refers to its sign change without changes in the value. We discuss these two cases in consecutive subsections separately, because they clearly illustrate the presented idea, power of the method, and its limitations.

Simulations were performed on the grid with the size

• $x \in [-1500,1500)$ a.u. with $N_x = 1024$,

• $p \in [-1,1)$ a.u. with $N_p=1024$,

and $t \in [0,9000)$ a.u. with $N_t = 900$. That choice of parameters ensures that all points where the WDF is non-zero are always sufficiently far from the boundaries, so that no significant error propagates throughout the simulation.

One-time change in barrier height

As the first case, the single modulation of the amplitude is considered, resulting in lowering the barrier’s height in half. To this end, for a fixed potential energy, U(x), the potential term of the Moyal equation is given by the function $W(x,t)=U(x)\Sigma (t)$, where time-dependent modulation factor of the potential amplitude, $\Sigma (t)$, is assumed in the form

$$\begin{aligned} \Sigma (t) = 1-0.5 \, \theta (t-t_\textrm{s}), \end{aligned}$$

(28)

where $\theta (\cdot )$ is the Heaviside function and $t_\textrm{s}$ is the switching time corresponding to a sudden change in the amplitude of the potential by half of its initial value. This choice of the potential term allows us to find the time evolution of the WDF in the proposed model of the time-dependent potential by the numerical solution of the Moyal equation based on the previously discussed algorithm. The obtained results show that the initial state represented by the SC-WDF freely propagates towards the Gaussian barrier.

Fig. 2

The phase-space snapshot of the SC-WDF with a single potential change at $t_{\textrm{s}} = 2.1 \times 10^3 \text {a.u.}$ shown in snapshots taken at (a) $t=0.0 \times 10^3 \text {a.u.}$, (b) $1.9 \times 10^3 \text {a.u.}$, (c) $3.6 \times 10^3 \text {a.u.}$ and (d) $5.0 \times 10^3 \text {a.u.}$ The red lines stand for the isolines of Hamiltonian for barrier before and after the cut down.

Depending on the time instant $t_\textrm{s}$, a greater or smaller part of the WDF might interact with the barrier undergoing scattering, before the sudden drop in the amplitude of the barrier. Then, the motion of the WDF continues in the region of the lowered barrier. In the long time, it is expected that the WDF will evolve freely, without any significant disruption from the scattering center. Figure 2 displays snapshots of the time evolution of the WDF in the phase space of the considered system at selected time instants. Let us note that these figures also contain several isolines of the classical Hamiltonian identical to its Weyl symbol. Most important concept of the system considered is that the potential governing the evolution of the WDF changes at $t_s$ which disturbs the ongoing evolution of the WDF as seen in Fig. 2.

Fig. 3

The interaction time calculated as a function of switching time $t_\textrm{s}$. Results obtained are based on Wigner–Rényi one-half entropy $S_{1/2}(t)$, and Wigner–Shannon entropy $S_1(t)$.

The interaction time was calculated, according to Proposition 1 with $\varepsilon =0.01$, for different switching times $t_s$, as seen in Fig. 3. Apparently, one may notice some behavioral similarities between the two curves presented, especially for time $t_s < 1.4\times 10^3$ a.u. the interaction time $\tau (t_s)$ is constant in value but the amplitudes differ in value. This is due to the fact that the Wigner–Shannon entropy stabilizes quicker, according to Fig. 4 than the Wigner–Renyi one-half entropy resulting in lower interaction time values.

Fig. 4

The Wigner–Shannon entropy (a) and Wigner–Rényi one-half entropy (b) of the WDF state over time. Different colors stand for different potential switching times $t_{\textrm{s}}$ and the black dash-dotted line corresponds to the case of time-independent potential barrier.

Both figures depict a significant constant value of interaction time for $t_\textrm{s}\le 1.5\times 10^3$ a.u. For these times, the WDF interacts with the lower barrier of the 0.004 a.u. amplitude. As the time $t_\textrm{s}$ increases, a more significant part of the WDF interacts with a high-amplitude barrier, and as a result, the disturbance from the constant value of interaction time $\tau$ occurs. The overall character of the subsequent $t_\textrm{s}$-dependence of $\tau$ is out of the scope of this work. Lastly, the overall character suggests that the appearance of a lower barrier while the interaction is already taking place reduces the interaction time. Previously presented interaction time value can be supported by the time evolution of the symplectically invariant measures chosen for consideration, as shown in Fig. 4. The common characteristic of these two measures is that they are both a time-dependent functions that can be segmented into three different stages: before the interaction, during the interaction, and flattening after the interaction. The first part is a consequence of the rapid spatial decrease in potential energy, U(x), at long distances, where it can be assumed to be nearly constant ($U(x)\approx \textrm{const}$). Thus, the WDF undergoes a shearing transformation, that is, a momentum-dependent translation that elongates the WDF. This evolution continues as long as the WDF is away from the potential barrier. When it reaches the interaction zone, the Wigner–Shannon entropy and the Wigner–Rényi one-half entropy start to vary over time. This is the result of breaking of the assumptions for symplectic invariance: the potential is no longer in the form of a polynomial of at most quadratic order. Last but not least, we have the flattening of these curves. The time at which any of the given symplectically invariant measures begins to vary from a constant value is assumed to be the start of the interaction $t_\textrm{i}$ The time when the interaction stops, $t_\textrm{f}$, can be specified as the time when the symplectically invariant measure becomes flat (or nearly flat). This allows us to estimate the interaction time $\tau = t_f-t_i$ of the WDF with a potential W(x, t), which in the presented case has the form shown in Fig. 3. The argument of the lack of the WDF interaction with the potential can be reinforced by additional analysis based on the measures quantifying overall character of the dynamics: dynamical transmission and dynamical capture coefficients. The former provides the information about the amount of the SC-WDF that passes through the interaction zone according to the formula,

$$\begin{aligned} P(t,x_P) = \int _{{\mathbb {R}}}\int _{x_P}^\infty {dx dp}\; \varrho (x,p,t) . \end{aligned}$$

(29)

For a fixed $x_P$ we denote $P(t,x_P) = P(t)$. The position $x_P$ is chosen in the region where the influence of the potential is negligible or, preferably, zero. Complementary to P(t), we introduce the dynamical capture coefficient $C(t,\varepsilon )$, defined as

$$\begin{aligned} C(t,\varepsilon ) = \int _{{\mathbb {R}}}\int _{L(\varepsilon )} {dx dp}\; \varrho (x,p,t), \end{aligned}$$

(30)

which quantitatively describes the trapped part of the WDF. Analogically to P(t) if $\varepsilon$ is fixed, then $C(t,\varepsilon ) = C(t)$. The interval $L(\varepsilon )$ is the region where the absolute value of the potential U(x) is significantly nonzero, i.e. its values are greater than some fixed $\varepsilon >0$. Complementary to the previously defined dynamical measures is the dynamical reflection coefficient, defined as

$$\begin{aligned} R(t,x_R) = \int _{\mathbb {R}}\int _{-\infty }^{x_R} {dx dp}\; \varrho (x,p,t), \end{aligned}$$

(31)

where $x_R$ is chosen sufficiently far from the initial condition so that it does not contribute to the reflected part of the WDF. It is expected that the following condition is held: $\lim _{t\rightarrow +\infty } [R(t,x_R)+C(t,\varepsilon )+P(t,x_P)]= 1$. The P(t) and C(t), as already mentioned, quantify the dynamics of the WDF as the amount of the quasi-probability distribution function present within given subspace of the p$\hbar$ase-space of the system.

Fig. 5

(a) Dynamical transmission P, and (b) dynamical capture C coefficients as functions of time t in the case of barrier-to-barrier switching. Different colors stand for the different potential switching times $t_{\textrm{s}}$, and the black dash-dotted line corresponds to the case of time-independent potential barrier.

In Fig. 5a one can see the nondecreasing character of the dynamical transmission coefficient. As the switching time $t_\textrm{s}$ increases, due to scattering, the lower amount of the WDF is transmitted through the interaction zone. For time instants $t_\textrm{s}$ greater than the time for which purely scattering occurs, the dynamical transmission coefficient P(t) will follow the same course as in the case of scattering on the static Gaussian barrier with amplitude 0.008 a.u. However, for lower values of the switching time $\textrm{t}_s$, the WDF interacts with the low-energy barrier, thus nearly the entire WDF goes above the barrier, resulting in a long-time behavior of the dynamical transmission coefficient that is nearly equal to one ($P(t)\approx 1.0$). The complementary measure is the capture coefficient C(t), which indicates the amount of the WDF within the interaction zone with the potential. According to Fig. 5b, for long times (about $t>6\times 10^3$ a.u.) there is no interaction of WDF with the potential barrier. The black dash-dotted line presented in the Fig. 5 is associated with the dynamical transmission and capture coefficients in the case of the time-independent potential in the barrier state. This allows for juxtaposition of dynamical measures associated with stationary and time-varying barrier systems. Namely, one can visually track the behavior of quantum state parameters against the evolutionary properties of the same state governed by a simplified system. This curve of reference for the dynamical quantity of the simplified system is present in all figures to shed light on the behavior of considered potentials. Apart from strengthening of the arguments presented in previous figures, the dynamical capture coefficient opens a different perspective on the concept of the interaction time, which greatly differs in the physical interpretation. As C(t) measures the amount of WDF that interacts with a strictly significant part of the potential, the time when C(t) diverges from zero could be regarded as the start of the interaction. Similarly, the time when the dynamical capture coefficient approaches zero would be the end of the interaction between the WDF and the potential. While this insight might be correct for some classes of potentials, it tackles the problem of the interaction time from a different perspective. The interaction time defined by Proposition 1 is the distance in time space between two free propagating states. Physically means that there is no potential disturbing the evolution of the system. On the other hand, C(t) measures the absence of the quantum state within the potential considered, regardless of the form of the quantum state before, and after the interaction. Let us note that the parameters of the time-dependent Gaussian barrier determine the interaction with the quantum state for the scattering process in terms of energy scale and space-time scale.

The preceding discussion demonstrates that the transmission and capture coefficients can also be employed as complementary measures to ascertain the SC-WDF’s interaction time with the potential. This approach is commonly employed to characterize tunnelling and related phenomena through barriers^4,55. It can, in particular, be utilized to estimate the interaction time straightforwardly. Let us note that these times can be read directly from Fig. 5. For this purpose, the moment when the transmission coefficient reaches one and the capture coefficient simultaneously reaches zero must be indicated with the given accuracy. The read-off results from Fig. 5 are provided in tabular form in Table 1. As is noticed, the values of these interaction times are similar in magnitude to those obtained using entropic methods based on Proposition 1. Although both methods yield comparable order of magnitude results, it is noteworthy that the transmission and capture coefficients are not symplectic invariant quantities. This is evidenced by the fact that they do not satisfy the property given by Eq. (6).

In addition to the method mentioned above, we decided to compare our results with two other approaches that enable estimating the interaction time. The first one is the method proposed by Dragoman⁵⁶. It is a modified version of the Büttiker-Landauer semiclassical time, as presented in reference². Based on this method, the interaction time can be estimated according to the formula

$$\begin{aligned} \tau (t) = \frac{\displaystyle d \int _0^d {dx}\; \int _{{\mathbb {R}}} {dp}\; \varrho (x,p,t)}{\displaystyle \int _0^d {dx}\; \int _{{\mathbb {R}}} {dp}\; (p/m) \varrho (x,p,t)}. \end{aligned}$$

(32)

In our calculations we set $d = 6w$, where $w^2 = 50$ a.u. is the width of the Gaussian barrier. Direct application of Dragomon’s method requires some modifications because it was initially formulated for the time-independent problem with the scatterer defined on the compact support of size d. For our purpose, we performed additional averaging of Eq. (32) over the simulation time. Hence, the time $\tau (t)$ is replaced by the averaging quantity, $\langle \tau \rangle$, by the formula

$$\begin{aligned} \langle \tau \rangle = \frac{1}{t_{end}} \int _{0}^{t_{end}} dt \left[ \frac{\displaystyle 6w \int _{-3w}^{3w} {dx}\; \int _{{\mathbb {R}}} {dp}\; \varrho (x,p,t)}{\displaystyle \int _{-3w}^{3w} {dx}\; \int _{{\mathbb {R}}} {dp}\; (p/m) \varrho (x,p,t)} \right] , \end{aligned}$$

(33)

where $t_{end}$ is the time of the end of the simulation. The calculations were performed for the several switching times, $t_s$, and results are collected in Table 1. The estimated interaction times with this method are also comparable to those presented in this paper.

Pollak et al.^11,57,58 developed the second method of estimating the interaction time, with which we want to compare our results. This approach is referred to as the mean flight time method. According to this method, the interaction time can be determined as follows:

$$\begin{aligned} \langle t \rangle _T = \int _0^{\infty } dt\; t P(Y,t), \end{aligned}$$

(34)

where P(Y, t) is the flight-time distribution measured at the detector located at Y. The explicit expression for this quantity is given by the formula

$$\begin{aligned} P(Y,t) = \frac{\displaystyle |n(Y, t)|^2}{\displaystyle \int _0^{\infty } dt\; |n(Y,t)|^2}, \end{aligned}$$

(35)

in which n(Y, t) is the marginal distribution of the WDF with respect to the position Y. Its form is following

$$\begin{aligned} n(x=Y,t) = \int _{{\mathbb {R}}}{dp}\; \varrho (x=Y,p,t). \end{aligned}$$

(36)

As recommended by the authors of the research mentioned above, the detector distance from the scattering centre should be at least as far as the initial location of the incident wavepacket. In the simulations that were carried out, the initial SC-WDF is located at a distance of $d=500$ a.u. from the scattering centre. Thus, we have decided to evaluate mean flight time given by Eq. (34) for the detector, which is positioned at a distance $Y=600$ a.u. The estimated interaction times obtained with this assumption for the several switching times, $t_s$, are presented in Table 1. As can be seen, these times are also of the same order of magnitude as those estimated based on our proposed method.

Table 1 Comparison of interaction times based on different measures calculated in two ranges of the switching time $t_s$. Two ranges of the switching time $t_s$ were selected because for $t_s<1.2\times 10^3$a.u., all of the presented interaction times have constant values.

Finally, the presented comparative studies allow us to conclude that the entropic methods we propose estimate the interaction times range in a way consistent with that provided by other methods.

Fig. 6

Interaction time $\tau$ as a function of potential width $w^2$ for different switch times $t_s$, based upon (a) Wigner–Shannon entropy, (b) Wigner–Rényi one-half entropy.

Previously, we examined the influence of changing the barrier height on the interaction time as a function of the switching time. However, we also expect that the interaction time depends on the barrier width while maintaining the same characteristics of the sampling state of the considered system. The results of these calculations are presented in Fig. 6. For a fixed value of the switching time, we determined the interaction time as a function of the Gaussian barrier width, $w^2$, based upon the time-dependent Wigner–Shannon entropy and the Wigner–Rényi one-half entropy. In both cases, we observe that an increase in the barrier width causes an increase in the interaction time, but the interaction times differ in numerical values. The interaction time determined with the help of the Wigner–Rényi one-half entropy is greater than its counterpart obtained using the Wigner–Shannon entropy. These results are consistent with conclusions resulting from the analysis Fig. 3, i.e. the symplectically invariant measure based on the Wigner–Rényi one-half entropy remains more sensitive on nonsymplectic WDF deformation due to its interaction with a barrier. Ipso facto forms the best estimation of the interaction time using entropic measures.

Barrier-well switching

In the following case, we consider a second setup in which the amplitude of the Gaussian potential changes sign once, which means that the Gaussian barrier becomes the Gaussian well. For this purpose, we assume that the time-dependent part of the potential term, $\Sigma (t)$, has the following form,

$$\begin{aligned} \Sigma (t) = 1-2\theta (t-t_{\textrm{s}}), \end{aligned}$$

(37)

where $\theta (\cdot )$ is the Heaviside step and $t_s$ is the switching time. As in the previous case, we determine the time evolution of the WDF by the numerical solution of the Moyal equation based on the discussed algorithm. The example snapshots of the WDF during its time evolution in the phase space are shown in Fig. 7 for $t_{\textrm{s}}=2.1 \times 10^3$ a.u..

Fig. 7

The phase-space snapshot of the SC-WDF with a single potential change at $t_{\textrm{s}} = 2.1 \times 10^3 \text {a.u.}$ shown in snapshots taken at (a) $t=0.0 \times 10^3 \text {a.u.}$, (b) $1.9 \times 10^3 \text {a.u.}$, (c) $3.6 \times 10^3 \text {a.u.}$ and (d) $5.0 \times 10^3 \text {a.u.}$ The red and green lines stand for the isolines of Hamiltonian for barrier and well forms of potential energy, respectively.

Qualitatively, the initial dynamics of the WDF, before approaching the interaction zone, resembles a free evolution, as can be seen in Fig. 7a. Any changes inflicted in the shape of the WDF directly result from shearing transformation, leading to the symplectic deformation of this function. Then, the freely evolving WDF enters the interaction zone, where it begins interacting with the potential barrier, as seen in Fig. 7b. In this case, all three elements of the time evolution can be highlighted: free movement of the highly energetic parts of the WDF, scattering and tunnelling of the intermediate and low energetic parts of the WDF. These last two processes are responsible for destroying the symplectic deformation of the WDF resulting from the free evolution of this state in the phase space.

During interaction, the Gaussian barrier is suddenly switched to the Gaussian well, forming a state that on one hand evolves with slight disturbance, and on the other is submitted partially to the bounding within potential well, as visible in Fig. 7c. Finally, we observe the trichotomy of the WDF in the form of transmitted, reflected and trapped parts. The overall form of the WDF deformation strongly depends on its trapped part, see Fig. 7d, sparking our interest due to its role in the interaction with the potential. To gain better understanding in this matter, we support these findings by calculating the dynamical transmission and capture coefficients as a time function for different switching time values. The results of these calculations are shown in Fig. 8.

Fig. 8

(a) Dynamical transmission P, and (b) dynamical capture C coefficients as functions of time t, in the case of barrier-to-well switching. Different colors stand for the different potential switching times $t_{\textrm{s}}$ and the black dash-dotted line corresponds to the case of time-independent potential barrier.

The observed changes in the dynamical transmission coefficient [cf. Fig. 8a] allow us to conclude that the switching time influences the passage rate through the interaction zone. In particular, if the switch between the barrier and the well occurs when the state is inside the region of the interaction zone, then the dynamical transmission coefficient is lower. A natural implication would be that some amounts of the WDF are trapped in the created well. This statement can be supported by the dynamical capture coefficient, displayed in Fig. 8b. The non-zero values assigned to the dynamics of the WDF for $t>6\times 10^3$ a.u. suggest that there are indeed parts of the WDF bounded within potential well. The scenario is more complicated when the potential barrier changes to the potential well at the moment when the WDF is in the interaction zone. Namely, the nonsymplectic deformation of the WDF due to the interaction of the state with the Gaussian barrier results in the appearance of low-momentum tail of the WDF, which is sensitive to the presence of the potential well. Consequently, partial trapping of the WDF within potential well reduces the dynamical transmission coefficient through the interaction zone. As a result, the further time evolution of the WDF cannot be described as a free motion, and thus it is a limitation for determining the interaction time based on the method we proposed.

Fig. 9

(a) The Wigner–Shannon entropy and (b) the Wigner–Rényi one-half entropy of the WDF state against time. Different colors stand for different potential change time $t_{\textrm{s}}$ and the black dash-dotted line corresponds to the case of time-independent potential barrier.

Namely, WDF can be divided in two parts: one part trapped within potential well that is subjected to non-metaplectic transformations, and other freely evolving deformed state. The contribution of the former to the general picture of the evolution invalidates method proposed in Proposition 1, as a proper method for determining the interaction time in this scenario. To finalize this counter-exemplary analysis, we calculated the Wigner–Shannon entropy and the Wigner–Rényi one-half entropy as functions of time. These results are presented in Fig. 9a, b, respectively. Let us note that both these dynamical quantities do not stabilize over time during the performed simulations. This lack of long-time stability, in terms of constant, non-varying value of the symplectically invariant measure, is strictly connected to the reminder of the WDF trapped within potential well, deeming our method unsatisfactory in presented case. In this subsection, we presented an example of the time-dependent potential in which the method of determining the interaction time based on the symplectic covariance of the WDF fails. Consequently, this time is not defined correctly, although the Wigner–Shannon entropy and the Wigner–Rényi one-half entropy can be calculated. It is clear that despite the apparent stabilization of some of these curves, an attempt to determine the interaction time on this basis is challenging. Therefore, the example of the time-dependent potential presented here can be regarded rather as a counterexample, thus showing the limitations of the method mentioned above.

Concluding remarks

We have presented the idea of the symplectic covariance of the Wigner distribution function in an application to the problem of determining the interaction time of the quantum state, represented by the Wigner distribution function, with the potential in the form of an absolutely integrable function. The interaction time has been defined as the traversal time of this state through the interaction zone determined by the range of a sufficient decaying potential. To estimate this time, we used time-dependent entropic measures in the form of the Wigner–Rényi one-half entropy and the Wigner–Shannon entropy. We have examined this issue in the case of scattering of the Schrödinger cat state on the time-dependent Gaussian potential. For this purpose, we have determined the time evolution of the state initially taken as the Schrödinger cat state, by numerically solving the Moyal equation using the Strang splitting algorithm adapted to the time-dependent potentials. We have performed simulations considering two models of the switching of the potential amplitude. One corresponds to a one-time change in the barrier height value, and the second corresponds to a one-time switch between the barrier and the well (in this case, the sign of the Gaussian potential is changed). Although the first case fully complies with the proposed method for determining the interaction time, the second clearly shows its limitations, making it a great counterexample of the proposed method. We have investigated both cases in detail by performing augmented dynamical transmission and capture coefficients calculations. The results obtained have allowed us to state that the choice of the instant of switching between the barrier and the well significantly influences the Wigner distribution dynamics in the interaction zone. This is because the switching time is responsible for the amount of the Wigner distribution function reflected by the potential barrier or trapped in the interaction zone by the Gaussian well, depending on the case analyzed. Changing the height of the amplitude of the potential before the interaction allows for greater transmission of the Wigner distribution function, whereas delaying this change results in an increase in the amount scattered. Regardless of the complexity of the quantum dynamics, we have obtained the interaction time as a function of the switching time according to our proposition. We have also extended our analysis by changing the potential width, resulting in an insight of how the change of the barrier parameters affects the interaction time. On the other hand, changing the potential barrier to potential well at the specific moment of the interaction of the considered bimodal state temporarily traps it in the interaction zone. This trait of the time evolution, namely the appearance of the low-momentum tails of the Wigner function trapped within a potential well, affects the symplectically invariant measures used to determine the interaction time. As a result, both measures do not stabilize over time, suggesting an ongoing interaction between the Wigner distribution function and the potential. Summarizing this discussion, based on the symplectic covariance of the Wigner distribution function and the concept of symplectic invariant quantities, we have shown that for the rapidly decaying potential barriers, the time-dependent entropic measures: Wigner–Rényi one-half entropy, and Shannon entropy are suitable dynamical quantities for estimating the interaction time with varying sensitivity. However, when the state becomes partially localized in the potential wells, the presented method indicates that the interaction time becomes infinite, thus not providing a quantitative answer to the problem posed. The proposed entropic method of estimating the interaction time can be realized upon the tomographical methods of reconstructing the Wigner distribution function^59,60,61 with ultrafast sampling. We believe this approach can be improved by deep machine learning, reducing the efforts needed to reconstruct the Wigner distribution function accurately⁶².

One can use the entropic method in the case of states with finite lifetimes, such as resonances, but it requires great caution in interpreting the result and certainly longer simulation times. This thread will be developed in our forthcoming work.