Quantum dynamics of a hydrogen-like atom in a time-dependent box: non-adiabatic regime

TL;DR

This paper investigates the non-adiabatic quantum dynamics of a hydrogen-like atom confined in a time-dependent spherical box, analyzing how rapid boundary movements affect the atom's energy states and electron behavior.

Contribution

It introduces a model of a hydrogen atom in a dynamically changing spherical confinement, exploring non-adiabatic effects on atomic energy and electron states during rapid boundary movements.

Findings

Linearly expanding box causes atomic de-excitation.

Rapid contraction induces high pressure and ionization.

Harmonic breathing leads to diffusive excitation similar to microwave fields.

Abstract

We consider a hydrogen atom confined in time-dependent trap created by a spherical impenetrable box with time-dependent radius. For such model we study the behavior of atomic electron under the (non-adiabatic) dynamical confinement caused by the rapidly moving wall of the box. The expectation values of the total and kinetic energy, average force, pressure and coordinate are analyzed as a function of time for linearly expanding, contracting and harmonically breathing boxes. It is shown that linearly extending box leads to de-excitation of the atom, while the rapidly contracting box causes the creation of very high pressure on the atom and transition of the atomic electron into the unbound state. In harmonically breathing box diffusive excitation of atomic electron may occur in analogy with that for atom in a microwave field.

Tables

n	Confined atom ($r_0=10$)	Confined atom ($r_0=20$)	Confined atom ($r_0=100$)	Free atom
1	-0.4999992	-0.4999995	-0.5000000	-0.5000000
2	-0.1128062	-0.1249870	-0.1249999	-0.1250000
3	0.0914223	-0.0499180	-0,0555555	-0.0555555
4	0.4051543	0.0167131	-0.0312499	-0.0312500
5	0.8263889	0.1128777	-0.0199999	-0.0200000
6	1.3511207	0.2372833	-0.0138684	-0.0138888
7	1.9775747	0.3885954	-0.0095963	-0.0102040

States	$r_0=8$	$r_0=10$	$r_0=14$
$1s$	-0.499 975 100 445	-0.499 999 263 281	-0.499 999 999 498
	-0.499 975 100 446	-0.499 999 263 282	-0.499 999 999 498
$2s$	-0.084 738 721 356	-0.112 806 210 295	-0.124 015 029 431
	-0.084 738 721 357	-0.112 806 210 296	-0.124 015 029 432
$2p$	-0.104 450 066 406	-0.118 859 544 853	-0.124 540 597 990
	-0.104 450 066 406	-0.118 859 544 854	-0.124 540 597 990

Equations

$$\Biggl{[}-\frac{1}{2}\frac{\partial^{2}}{\partial r^{2}}-\frac{1}{r}\frac{\partial}{\partial r}+\frac{l(l+1)}{2r^{2}}-\frac{Z}{r}\Biggl{]}R_{nl}(r)=E_{nl}R_{nl}(r),$$

$$R_{nl}(r){|}_{r=r_0}=0.$$

$$i\frac{\partial R(r,t)}{\partial t}=\hat{H}R(r,t),$$

$$\hat{H}=-\frac{1}{2}\frac{{\partial }^2}{\partial r^2}-\frac{1}{r}\frac{\partial }{\partial r}+\frac{l(l+1)}{2r^2}-\frac{Z}{r}.$$

$$R(r,t){|}_{r=r_0(t)}=0.$$

$$y=\frac{r}{r_0(t)}.$$

$$\displaystyle i\frac{\partial R(y,t)}{\partial t}=\Biggl{[}-\frac{1}{2r_{0}^{2}}\frac{\partial^{2}}{\partial y^{2}}-\Biggl{(}\frac{1}{2r_{0}^{2}y}-i\frac{\dot{r}_{0}}{r_{0}}y\Biggl{)}\frac{\partial}{\partial y}$$

$$\displaystyle+\frac{l(l+1)}{2r_{0}^{2}y^{2}}-\frac{Z}{r_{0}y}\Biggl{]}R(y,t)\equiv\hat{\tilde{H}}R(y,t).$$

$$R(y,t)=\frac{1}{r_0(t{)}^{3/2}y}{e}^{\frac{i}{2}{r}_0(t){\dot{r}}_0(t)y^2}\Phi (y,t).$$

$$\tau ={\int }_0^t\frac{ds}{r_0(s{)}^2},$$

$$i\frac{\partial\Phi}{\partial\tau}=-\frac{1}{2}\frac{\partial^{2}\Phi}{\partial y^{2}}+\Biggl{(}\frac{1}{2}r_{0}^{3}\ddot{r}_{0}y^{2}+\frac{l(l+1)}{2y^{2}}-\frac{Zr_{0}}{y}\Biggl{)}\Phi.$$

$$\Phi (y,t){|}_{y=1}=0.$$

$$\hat{\tilde{H}}=e^{-iV}{e}^{-iU}(\hat{H}-i\frac{\partial }{\partial t})e^{iU}{e}^{iV},$$

$$U=i(r\frac{\partial }{\partial r}+\frac{3}{2})\ln r_0(t),$$

$$V=-\frac{1}{2}{r}_0\frac{d{r}_0}{d\tau }{y}^2.$$

$$\Phi (y,t)=\sum _{nl}{C}_{nl}(t){\phi }_{nl}(y),$$

$$i{\dot{C}}_{nl}(t)=r_0^{-2}\sum _{n^{\prime }{l}^{\prime }}{C}_{n^{\prime }{l}^{\prime }}(t)V_{nl{n}^{\prime }{l}^{\prime }}(t)+{\epsilon }_{nl}{r}_0^{-2}{C}_{nl},$$

$$V_{nl{n}^{\prime }{l}^{\prime }}(t)=<{\phi }_{n^{\prime }{l}^{\prime }}|-\frac{Z{r}_0}{y}+\frac{1}{2}{r}_0^3{\ddot{r}}_0{y}^2|{\phi }_{nl}>.$$

$$4\pi \sum _{nl}|C_{nl}(t){|}^2=1,$$

$${\int }_0^{r_0(t)}|\Psi (r,t){|}^2{d}^{3}r=1.$$

$$\displaystyle<E(t)>=4\pi\int_{0}^{r_{0}(t)}R^{*}(r,t)\Biggl{(}-\frac{1}{2}\frac{1}{r^{2}}\frac{\partial}{\partial r}\biggl{(}r^{2}\frac{\partial}{\partial r}\biggl{)}$$

$$\displaystyle+\frac{l(l+1)}{2r^{2}}-\frac{Z}{r}\Biggl{)}R(r,t)r^{2}dr,$$

$$\displaystyle<E_{kin}(t)>=4\pi\int_{0}^{r_{0}(t)}R^{*}(r,t)\Biggl{(}-\frac{1}{2}\frac{1}{r^{2}}\frac{\partial}{\partial r}\biggl{(}r^{2}\frac{\partial}{\partial r}\biggl{)}$$

$$\displaystyle+\frac{l(l+1)}{2r^{2}}\Biggl{)}R(r,t)r^{2}dr.$$

$$\hat{F}=-\frac{\partial \hat{H}}{\partial r_0(t)}.$$

$$<F(t)>=-\frac{\partial }{\partial r_0(t)}⟨R(r,t)|\partial \hat{H}|R(r,t)⟩.$$

$$<P(t)>=\frac{<F(t)>}{4\pi r_0^2(t)}.$$

Full text

I Introduction

Atoms and molecules confined in nanoscale domains have the physical properties which are completely different than those of free atoms. Such difference is caused by modification of the boundary conditions imposed for quantum mechanical wave equations. For free atoms the boundary conditions are imposed in whole space, while for confined atoms one should solve the wave equations with the boundary conditions imposed on a finite domain. Due to such modification, properties of the atoms, molecules and matter depend on the shape and size of a confining domain Sabin ; Sil ; Sen . Spatially confined atoms appear in many physical systems, such as optical traps Meystre ; Gardiner , low dimensional nanomaterials Tsurumi , fullerene Connerade1 ; Connerade2 , atom optic billiards Raizen -Douglas , metallic hydrogen MH ; MH1 and matter under high pressure Blank . A key issue in the study of such systems is the reaction of the electronic structure of an atom (molecule) on the confining force Sil ; Sen . A convenient model for the study of atoms under the spatial confinement is the atom in box system, where latter has spherical shape and impenetrable walls. Pioneering studies of atom-in-box system date back to the Refs. Michels ; Sommerfeld , where effect of the pressure on an atom was explored in quantum approach. Later, Wigner studied the problem within the Rayleigh-Schrödinger perturbation theory and showed that in the limit of infinite box size, the result does not converge into that for the free atom. Considerable number of papers on the atom-in-box problem (see, e.g., Weil -Burrows03 and review paper Jaskolski for more references) has been published, since from these pioneering works. In Weil the problem of hyperfine splitting in such system is treated, Ref.Rubinstein presents first numerical solution of the problem. More comprehensive treatment of atom-in-box system can be found in a series of papers by Burrows et.al Burrows0 ; Burrows ; Burrows03 ; Burrows04 , where the authors used different analytical and numerical methods for finding eigenvalues of the system. In Dragoslav ; Lumb ; Dragoslav1 the quantum dynamics of hydrogen atom confined in a spherical box and driven by external electric field is studied.

Experimentally, atom-in-box system can be realized, e.g., in co-called atom optic billiards which represent a rapidly scanning and tightly focused laser beam creating a time-averaged quasi-static potential for atoms Raizen -Montangero . By controlling the deflection angles of the laser beam, one can create various box (billiard) shapes. Another method is putting the hydrogen atom inside the fullerene Connerade1 ; Connerade2 . Different ways for confining of atom inside a cages in experiment are discussed in Jaskolski .

Usually, the studies of atom-in-box models are mainly focused on the static box boundaries. However, dynamical confinement appear in different low-dimensional and atom-in-trap systems and the response of the electronic structure of atom to the varying size and shape of the ) confinement boundaries may play important role in such systems. Some versions for experimental realization of dynamical confinement in different traps have already been considered in the Refs.TDT1 ; TDT2 ; TDT3 . In such cases the dynamics of atomic electron is completely different than that for fixed boundaries.

In this paper we study quantum dynamics of one electron atom confined in a spherical box with time dependent radius by focusing on the response of atomic electron to the effect of moving walls of the box. The time-dependence of the wall’s position is considered as non-adiabatic, i.e. we consider the cases of rapidly shrinking, expanding and harmonically breathing boxes.

The problem of moving boundaries in quantum mechanics is treated in terms of the Schrödinger equation with time-dependent boundary conditions. Earlier, the quantum dynamics of a particle confined in a time-dependent box was studied in different contexts (see Ref. doescher -Our02 ). Here we consider similar problem for an electron moving in a Coulomb field of the atomic nucleus, confined in a spherical box with rapidly varying radius. In such system central symmetry in the problem is not broken and moving wall of the box plays the role external driving potential for the hydrogen atom.

This paper is organized as follows. In the next section we give brief description of the atom-in-box problem for a static box and compare our numerical results with those by other authors. Section 3 presents detailed treatment of the quantum dynamics of the atom confined in a spherical box with moving walls. Analysis of the quantum pressure and force acting on the atom by moving wall of the box is also presented. Finally, section 4 provides some concluding remarks.

Table 1. The energy spectrum (first few $s-$states) of a hydrogen atom confined in a spherical box with radius $r_{0}$ (in atomic units).

II The Hydrogen atom in a static spherical box

Hydrogen atom confined in impenetrable spherical box was studied in different papers by using different approaches for computing of energy eigenvalues. In Capitelli ; Cabrera the eigenvalues atom-in-box system are computed by solving the radial Schrodinger equation numerically. Burrows and Cohen developed different algebraic and approximate analytic approaches for finding eigenvalues such system Burrows ; Burrows04 ; Burrows . In Lumb the authors used so-called Bernstein-polynomials for numerical solution of the radial Schrodinger equation for atom in spherical box. Review of the different approaches for finding of energy eigenvalues of confined atoms can be found in Jaskolski .

Consider the hydrogen-like (one-electron) atom confined in a spherical box with impenetrable walls with the radius $r_{0}$. Assuming that the nucleus of the atom is fixed at the center of box, for the dynamics of atomic electron in such system we have the stationary radial Schrödinger equation which is given as (atomic units, $m_{e}=\hbar=e=1$ are used throughout this paper):

[TABLE]

where $R_{nl}$ is the radial part of the wave function, $Z$ is the charge of the nucleus, $n$ and $l$ are the principal and orbital quantum numbers, respectively. The energy eigenvalues, $E_{nl}$ can be found from the boundary condition for $R_{nl}(r)$ Michels ; Dragoslav :

[TABLE]

Unlike the unconfined atom, the energy spectrum of atom-in-box system is completely discrete. It should be noted that Eq. (2) is not convenient for computing of the energy spectrum of this system, as it does not allow to compute high number of eigenvalues, especially, at small radii of the box. Here we compute the energy levels of the atom-in-box system by diagonalizing of the Hamiltonian (left part of Eq.(1))of the system over the spherical box eigenfunctions basis. This method allows to compute arbitrary high number of eigenvalues for any value of the box radius.

Table 1 compares first seven energy levels for the atom in a spherical box at different values of the box radius. Last column in this table presents energy levels of unconfined (free) atom. For the ground state level the eigenvalues of confined and free atoms are the same, while for excited states they become different. For confined atom the levels are higher than those for corresponding free atom energy levels. Moreover, for very small radius of the box the energy levels of atom-in-box system become positive. Shrinking of the box size increases confinement, force and pressure on the atomic electron, that causes the increase of the average kinetic energy of the atomic electron leading to positive eigenvalues in the spectrum of atom in box system.

Table 2 compares the energy levels obtained using our method with those from the Ref.Burrows0 . As it can be seen, the agreement is almost absolute (up to 11th order after the decimal place).

III Hydrogen atom in a time-dependent spherical box

Recent technological developments make possible trapping and manipulating of atoms and molecules in time-dependent potentials. Manipulation of the atomic Hamiltonians with both discrete and continuum spectra is of practical importance in such field as metrology and quantum information processing. Possibility for creating of time-dependent traps and confining there particles and atomic bound states have been discussed recently in different contexts TDT1 ; TDT2 ; TDT3 . Study of such systems requires using effective models providing simplified and highly accurate description of quantum dynamics and computing physically observable quantities. One of such models can be atom confined in a spherical box with time-dependent radius. Such time-dependence does not break spherical symmetry of the system and one can describe the whole system in terms of time-dependent radial wave equation. Confining and cooling of atoms in time-dependent optical traps was discussed in TDT1 . Creation of atomic Fock states in time-dependent trap at different regimes of wall’s motion was studied in TDT2 . Ref.TDT3 presents proposal for confining and driving an ultrafast dynamics in a time-dependent box. Quantum dynamics in externally manipulated time-dependent trap is treated in TDT4 .

Here we consider atom confined in a spherical box with time-varying radius given by $r_{0}=r_{0}(t)$. In this case the sphere retains its shape during the expansion(contraction), so that the the central symmetry is not broken. Therefore, if atomic nucleus is fixed at the center of sphere, the electron dynamics is described by the time-dependent radial Schrödinger equation which is given as

[TABLE]

where

[TABLE]

The boundary conditions for Eq.(3) are imposed as

[TABLE]

To solve Eq.(3) one should reduce the boundary conditions into time-independent form. This is can done by using the following transformation mak91 ; mak92 :

[TABLE]

In terms of new coordinate, $y$ Eq.(3) can be rewritten as

[TABLE]

In Eq.(5) the self-adjointness is broken, i.e., operator $\hat{\tilde{H}}$ is not Hermitian. In addition, due to the first-order derivative in this equation makes complicated its solution. Therefore, to restore Hermitticity and remove the first order derivative, one can use the transformation of the wave function which is given by

[TABLE]

Doing such transformation and introducing of the new time-variable defined as mak91 ; razavy ; Our01

[TABLE]

we reduce Eq.(5) into the Hermitian for which can be written as

[TABLE]

The boundary condition for $\Phi$ is imposed as

[TABLE]

We note that Eq.(7) can be obtained from Eq.(3) by using following unitary transformation for the Hamiltonian $H$ razavy :

[TABLE]

where

[TABLE]

and

[TABLE]

Eq.(7) is the Schrödinger equation for an electron moving in the field of Coulomb and time-dependent harmonic oscillator potentials. The whole system is confined in a spherical box with unit radius. Time and coordinate variables cannot be separated in Eq.(7) and one needs to solve it numerically. To do this we expand $R(y,t)$ in terms of the complete set of eigenfunctions of a spherical box with unit radius:

[TABLE]

where $\varphi_{nl}(y)=N_{nl}yj_{l}(\lambda_{nl}y)$ are the eigenfunctions of the stationary Schrödinger equation for a spherical box of unit radius, $j_{l}$ are the spherical Bessel functions. Inserting this expansion into Eq.(7) we get the system of first order differential equations with respect to $C_{nl}(t)$:

[TABLE]

where $\varepsilon_{nl}$ are the eigenvalues of the Schrodinger equation for spherical box and

[TABLE]

In solving Eqs.(9) numerically one should take into account the normalization condition for the expansion coefficients, $C_{nl}$:

[TABLE]

which follows from the normalization condition for the wave function:

[TABLE]

Having found the wave function, one can compute physical characteristics of the atomic electron, such as average total energy, force and pressure acting on atom by considering different regimes of the wall’s motion. In the following we will do that for linearly expanding, contracting and harmonically oscillating box wall.

Important physically observable characteristics of the atomic electron is its average total energy which is defined as

[TABLE]

and average kinetic energy given by

[TABLE]

In Fig. 1 the average total energy, $<E(t)>$ (a) and the average coordinate (expectation value of the electron’s position), $<r(t)>$ (b) of the atomic electron are plotted for the hydrogen atom confined in a linearly expanding spherical box ($r_{0}(t)=a+bt$). At $t=0$ the box is very shrink ($a=10$), i.e. before the expansion atom is highly compressed and kinetic energy of the electron is very large. The average total energy in Fig. 1(a) decreases in time, as box expands. This occurs due to the fact that expansion of the box causes decrease in the kinetic energy of electron. Decreasing is accompanied by oscillations. During long enough time oscillation amplitude asymptotically goes to zero and the value of $<E(t)>$ becomes close to that of the ground state energy of (free) hydrogen atom. Such oscillations are caused by the fact that at $t=0$, upon switching interaction between the highly compressed atom and the moving wall of the box, the transitions from the ground to excited states accompanied by mixing the states with different energies occur. As the box starts to expand rapidly ($b=1$), the atomic electron, having very high speed, starts to follow the moving wall. This can be seen from the Fig.1 (b) , where the average coordinate of electron is plotted for two initial conditions (the state of the atomic at $t=0$), $1s$ and $2s$ states of the hydrogen atom in a spherical box. For $2s$ state the growth is more rapid than that for $1s$. Thus in linearly expanding atom-in-box system where the expansion starts from very compressed state, de-excitation caused by decrease of the average kinetic energy of atomic electron occurs. When the box expands up to very large radius, such de-excitation leads to the transition of atom from highly excited to ground state. Therefore putting (confining)of atom in an linearly expanding box can be effective method for preparing ground state atoms.

Very important case which is attractive from the viewpoint of practical applications, is the contracting box. When box becomes shrink, the pressure on the atom, created by the wall of the shrinking box becomes higher. Such dynamic compression method creating time-dependent pressure is used, e.g. in metallic hydrogen formation experiments MH . Dynamic high pressure creates the extreme conditions for the atom when the force acting by the wall on the atomic electron becomes higher than that of atomic nucleus.

In Fig. 2 plots of the average total energy (a) and average coordinate (b) are plotted as functions of time for linearly contracting ($r_{0}(t)=a-bt$) box with the initial radius $a=100$ and contraction velocity $b=1$. In computing of the average energy, the initial state of atom is chosen as $2s$ state of spherically confined atom. The average total energy grows almost linearly during initial some stage, while upon shrinking up to some (critical)size, the growth becomes abrupt. Further shrinking leads to becoming the average total energy positive, which is caused by growing of the average kinetic energy of the atomic electron. Unlike the plot of Fig.1, there are no oscillations in $<E(t)>$ for shrinking box. This is caused by the fact that in this case, the ”sudden switching” of the interaction with the moving wall occurs at large distances from the atomic electron and Coulomb center, which does not cause mixing of the states. Plots of the average coordinate for two initial states, $1s$ and $2s$ show that for $1s$ state the average coordinate does not change, while for excited ($2s$) state $<r(t)>$ is not smooth and the curve passes through the minimum values during the shrinking.

The behavior of the average total energy presented in Figs. 1 and 2 can be understood, if one analysis the behavior of the average kinetic energy as sa function of time. In Fig.3, time-dependence of the average kinetic energy defined by Eq.(III) is plotted for linearly expanding (a) and contracting (b) boxes at the same values of the parameters as in Figs. 1 and 2. For linearly expanding box the average kinetic energy of the atomic electron decays in time and the decay is accompanied by oscillations. The curve of the average kinetic energy for linearly contracting box can be divided into two parts. In first part $<E_{kin}(t)>$ manifests linear, but slow growth during some initial time-interval, while in second part the growth becomes abrupt. Such rapid growth can be explained by increasing pressure on atomic electron by the wall of the box at small distances.

An important characteristics responsible for the behavior of atomic electron in time-dependent box, are the average force acting on the atomic electron by the moving wall and the pressure caused by this force. The force operator for time-dependent box is given by Our01 ; Our02

[TABLE]

The average force can be calculated as Our01

[TABLE]

where $R(r,t)$ is the wave function of the atomic electron in time-dependent box determined by Eq.(3).

Then the average pressure on the atom created by the moving wall can be written as

[TABLE]

Fig. 4 presents plots of the time-dependence of the average force (a) acting on the atomic electron by the box wall and the average pressure (b) for linearly expanding box. Figs. 4c and 4d present the similar plots for linearly contracting box. The values of the parameters are the same as those in Figs. 1 and 2. For linearly expanding box the modulus of the force acting on the electron by moving wall decays on time, asymptotically approaching zero. The value of the pressure for this system also decreases in time approaching zero in long time limit. Such decrease causes the decrease of the average kinetic energy of the atomic electron which subsequently causes the behavior of the average total energy observed in Fig. 1. For linearly contracting box the behaviors of the average force and pressure are completely different than those for expanding one. Sudden growth of the average force and pressure starts from some critical size of the box, $r_{0}(t)=25$. This is very high pressure (few GPa) which can be achieved dynamical compression method used, e.g. in metallic hydrogen formation MH ; MH1 . From the viewpoint of metallic hydrogen physics, such pressure can cause appearing of ”free electron gas” before the formation of metallic hydrogen. Increasing of the pressure for this case leads to the increase in the average kinetic energy and to the behavior of $<E(t)>$ observed in Fig.3.

Explicit time dependence of $r_{0}(t)$ determines the nature of confinement and is the key factor for the dynamics of atomic electron. Therefore by choosing or tuning explicit form of $r_{0}(t)$ one can achieve manifestation of different effects. The above linear compression and expansion of the trap leads to excitation and de-excitation of the atom. Another option for tuning of the confinement dynamics is choosing of $r_{0}(t)$ as time-periodic. In this case one deals with completely different (than the above) behavior of the atomic electron. One of the regimes belonging to this case, which is interesting from the viewpoint of experimental realization (e.g., in atom optic billiards), is harmonically breathing spherical box in where the time-dependence of the box radius is given by $r_{0}(t)=a+b\cos\omega t$, where $\omega$ and $b$ are the oscillating frequency and amplitude, respectively. In Fig. 5 the expectation values of the total energy, coordinate force and pressure are plotted for $\omega=1$, $b=10$ and $a=100$. The average total energy grows during some initial period after that suppression of the growth can be observed and $<E(t)>$ does not become positive. Similar ”saturation” can be observed in the time-dependence of the average force and pressure acting on atomic electron by moving wall. The motion of the electron, characterized by the expectation value of the coordinate, is localized around the value $1.7a.u.$ However, as shows Fig. 6, for higher oscillation amplitudes the interaction of atomic electron with the breathing wall leads to diffusive excitation, i.e., the average total energy grows and becomes positive after long enough time. Similar growth exhibit the average force and pressure as a functions of time. Linear growth (accompanied by oscillations)of the average coordinate can be observed in this regime. Thus there is a ”critical” value (approximately $12$a.u. in our case) of the breathing amplitude, at which the average total energy changes its sign during some time, It is clear that oscillation of the box’s wall leads to pumping of the energy into atom which causes its diffusive excitation. In other words, atom in a box with oscillating walls behaves itself as that in a monochromatic field, widely studied earlier in the context of chaotic (diffusive) ionization Casati ; Jensen ; Koch . Such model can be realized in atom optics billiard where oscillating billiard boundaries can be created by tightly focused laser beam with time-varying position Raizen ; Ariel2 ; Andersen ; Rohwedder .

Finally, to explain more clearly the behavior of the average total energy for harmonically breathing box, in Fig.7 we present the plots of the average kinetic energy, $<E_{kin}(t)>$ as a function of time. For the set of parameters $\omega=1$, $b=10$ and $a=100$, the average kinetic energy (a)grows linearly during initial period and the growth is suppressed after this period, while for high amplitude (b) regime ($b=15$) $<E_{kin}(t)>$ grows linearly in time Such linear growth causes, of course, means pumping of the energy into the atom that causes linear growth of the average total energy (see, Fig. 6) and diffusive excitation of atomic electron during long enough time interval. In the above figures characteristics of dynamic atom-in-box system as functions of time. However, it is useful to explore dependence these characteristics on the confinement radius, $r_{0}(t)$. For linearly expanding and contracting boxes the curves do not change their shapes, as such transformation imply linear deformation only. However, for harmonically breathing regime the shape of the curves are completely different, if one plots the dependence of the averages as a function of $r_{0}(t)$. Fig.(8) presents average total energy of hydrogen atom in harmonically breathing spherical box as a function of $r_{0}(t)$ for to sets of parameters. The curve consist of many cyclic lines whose origins and ends are connected at the same point.

IV Conclusions

In this work we studied behavior of a hydrogen-like atom under the dynamical confinement created by impenetrable spherical box with time-dependent radius. The main focus of the study is given to the response of the atomic electron’s dynamics to the moving wall of the box.

Time-dependence of the expectation values of the total energy, kinetic energy, force, pressure and average coordinate are analyzed for linearly expanding, contracting and harmonically oscillating (breathing) boxes. For rapidly expanding box de-excitation of the atomic electron leading to the decrease of the average total energy occurs. Rapidly contracting box causes creation of high pressure on atom and becoming the average total energy positive. In case of harmonically breathing sphere, atom behaves itself as that in microwave electric field, where the diffusive (multi-step) excitation of atom occurs.

The above models can be used in practically important problems, such as metallic hydrogen formation, atom cooling and modeling the behavior of matter under the extreme conditions caused by high pressures. In addition, atom in spherical box with rapidly varying radius can be effective model the study of different ultrafast phenomena in atom optics, and atomic attosecond physics with trapped atoms. We note that the above approach is applicable for the study of the adiabatic (slowly moving box walls) regime which is the subject for the future project. Very attractive from the practical viewpoint follow up of the above model is optical harmonic generation in such system. If the trap boundary is created by an optical field, interaction of such field with atomic electron may cause harmonic generation of different orders. In case of high harmonic generation, such a model with a dynamical confinement could be very effective tool for attosecond pulse generation.Moreover, earlier some results about the positive role of the confinement in high harmonic generation have been reported in the literature Dragoslav1 ; HHGA .

Acknowledgement

This work is partially supported by the grant of the Ministry of Innovation Development of Uzbekistan (Ref. No.F-2-003).