1. Introduction

Atomic and molecular systems confined in various plasma environments have occupied an important place in theoretical and experimental research fields ^{1,2,3,4,5,6,7,8}. It is well-known that in interpretation of various data associated with astrophysics, hot plasma etc., basic understanding of atomic excitation and ionization is required since these processes taking place in plasma environment provide important information about plasma. The processes which have recently attracted much attention are laser-assisted collisions in plasma ^9,10,11,12, and response of confined atoms to short and intense laser pulses ^13,14. The plasma environment considered here is Debye plasma where Debye screening length, λD=kBT/(4πe2n), being a function of the plasma temperature T and its density n, plays an important role. Various sets of plasma conditions involved in real plasma environments can be simulated for one value of λD as one can evaluate the plasma temperature for a particular number density and vice-versa by fixing λD ¹⁵.

The screening of interaction potential between the nucleus and the electrons moving in atomic orbitals plays an important role in a variety of processes. In case of atoms and molecules confined under various conditions, the potential is modelled by a so-called Debye-Hückel potential ¹⁶. The other form of screened potential called the exponential cosine screened coulomb potential, has long been used to describe an ionized impurity inside a semiconductor heterostructure ^13,15,17,18. These screened potentials are prototype for many physical processes such as atoms confined in Debye or Debye-cosine plasmas, where atomic properties change drastically compared to free atoms, depending on the screening parameter, in particular ^{13,15,19,20,21}. The spectrum of the atom becomes quite interesting as reflected in its response to external fields. If, in addition to the screening potentials, there happens to be a spherical confinement, then the atom shows a drastic change in the spectrum. For example, Lumb et al. ^14,21, have shown that there are very few bound states if confining radius r_c is very small (rc≪10a.u.). So, spherical confinement in addition to screened potential forms a new confining potential where atomic and molecular systems are yet to be explored in detail. Impurities present in quantum dots are an example of a confined system where the spherical boundary represents the cage radius. In such practical physical situations the existing potential is modified due to interaction with impurities. As reported earlier ¹⁸, there have been few studies on the scattering processes taking place in confined environment. Two-photon spectroscopy has been an important tool to study the excitation of atoms and molecules (²² and references therein). Two photon and three photon transitions in confined atoms under various kinds of confinements started receiving much attention since starting of 2000 ^{17,23,24,25,26,27,28,29} as tools to study such transitions became available.

In the present study, we focus on the two-photon transitions in atomic hydrogen embedded in spherically confined Debye plasma. The model employed in the present case includes the effect of spatial confinement as well as screened Coulomb potential. The hydrogen atom is assumed to be embedded in plasma environment consisting of a finite charge distribution. The boundary condition signifies the finite extent of the charge cloud.The Debye confinement considered here is more likely to occur in many practical situations. In general, plasma environment in space is represented by this potential as is known from available literature. Also, the artificially created short-lived laboratory plasmas always have a finite volume. Such environments simulate the plasma present in interstellar space. Using the present model of confined plasma, simple Debye plasma model can be studied by enlarging the radius of confining sphere to infinity. Understanding of hydrogen atom under such confinement is important as it helps in examining more complex systems. Here, we consider electron of confined hydrogen atom being excited to higher states through absorption of two photons. We study excitation from 1s to higher states. The effect of confinement on two-photon processes has been dealt with in detail. The spectrum of confined atom is evaluated using B-polynomial method ^30,31,32,33.

The paper is organized according to the following scheme. The relevant theoretical details are provided in Sec. 2. The results of the present work are discussed in Sec. 3. Finally the important findings are summarized in Sec. 4.

2. Theory

The model considered here comprises a hydrogen atom embedded in a Debye plasma environment. The atom is assumed to be at the center of an impenetrable spherical cavity of radius r₀. This geometrically symmetric arrangement is a special case of the more general possibility in which the atom may be present at any position within the cavity. For simplicity, we have chosen the special case. Our aim here is to study the effect of spherical confinement and surrounding plasma on the two-photon transition probability amplitudes D₂, transparency frequency ωt and resonance enhancement frequencies ωr. This in turn requires a knowledge of the energy spectrum and the dipole matrix elements of the system. The spectrum, oscillator strengths and other physical quantities of confined systems are known to be highly dependent on the chosen confinement parameters and hence need an accurate evaluation. The evaluation of energy spectra and oscillator strengths of confined hydrogen in Debye plasma environment has been carried out and the results for various confining radii and Debye lengths characterizing different plasma conditions have been reported in our earlier works ^15,21. The steps necessary for arriving at these results are summarized below for ready reference. We have used atomic units throughout our study.

The radial Schrödinger equation for the electron of the confined hydrogen atom is given by

where -e-r/λD/r is the Debye-Hückel potential, 1/λD being the Debye screening parameter ³⁴ and Vc(r) is the confinement potential defined as

The radial wave function R_{n, l} (r)= U_{n, l} (r)/r. U_n,l (r) is expanded in B-polynomial basis as

where c_i s are coefficients of expansion and B_{i, n} (r) are B-polynomials of degree n. The confinement potential being infinite at the boundary, i.e., r = r₀, forces the wave functions to vanish there. Under these restrictions the radial Schrödinger equation can be recast as a symmetric generalized eigenvalue equation in matrix form, given by

where matrix elements ai,j,fi,j,gi,j and d_{i, j} are defined as

The eigenvalues E provide the energy spectrum and eigenvectors C are used to calculate the corresponding radial wave functions using Eq. (3). We have used Fortran EISPACK library to solve Eq. (4).

The two-photon transition probability amplitude, D₂, of a hydrogen atom from the initial state 1s to a final state js can be evaluated by using ^17,24

where n represents the intermediate states including continuum, E_1s and E_js are the energy eigenvalues of the 1s and js states respectively and χ1n and χjn are the dipole matrix elements defined as follows

Where R_l (with l = n, k) represents the radial wave function. The corresponding formula for calculating the transition probability amplitude for 1s to jd state is given by

The values of ω0 for which D₂ approaches infinity, and also lie inside the interval ΔEif/2 and ΔEif are called the resonance enhancement frequencies, where ΔEif is the difference between final and initial (1s) state energies ¹⁷. Also, the frequencies for which the transition amplitude vanishes are called as the two-photon transparency frequencies ¹⁷.

3. Results and Discussions

The two-photon transition probability amplitudes of a hydrogen atom placed at the center of an impenetrable confining sphere of radius r₀ and embedded in a weak plasma characterized by Debye-Hückel potential are explored. Since Debye screening length λD is dependent on temperature and density of plasma, its different values represent different conditions of the system. We have explored the dependence of two-photon transitions on the extent of Debye screening as well as confinement radius. The probability amplitudes from 1s to js (j = 2,3,4) states have been calculated using Eq. (5) and to jd (j = 3,4) states using Eq. (7) for a range of incident photon frequencies. It may be mentioned that the Hamiltonian being discretized leads to discreet continuum states, hence the quantum nature of the system is retained even for small confinements. Therefore, irrespective of the states being free or bound, the nomenclature of states is assumed to be same. The range of frequencies selected for studying the variation of two-photon transition probability amplitudes, transparency frequencies and resonance enhancement frequencies for various values of confinement parameters r₀ and λD is ΔEif/2 to ΔEif as mentioned in Sec. 2.

The two-photon transition amplitudes as calculated by us show very close agreement with the previously available results in literature. For example, for a free hydrogen atom, the contributions to the 1s -2s transition amplitudes due to first few intermediate states taking ω0=0.375 Ryd., given by Bassani et al.³⁵ match well with our results calculated for r₀ = 50 a.u. and λD=∞. It may be noted that the spherical confinement radius r₀ = 50 a.u. is very large as compared to the size of hydrogen atom and hence corresponds to a nearly free atom. The total contribution calculated by them is -11.7805 and the value obtained in the present work is -11.7803. The energy levels, two-photon transition amplitudes, absorption coefficients, two-photon transparency frequencies and resonance enhancement frequencies as calculated by Paul and Ho ²⁴ for a Debye plasma screened hydrogen atom are in consonance with our results for various values of λD. For example, the value of two-photon absorption coefficient for 1s - 2s transition based on our calculations is 144.8617 for λD=10 a.u. and 138.8400 for λD=200 a.u. taking ω0=0.37 Ryd. This data matches exactly with the results of Paul and Ho ²⁴. This provides a check on our calculations. The aim of the present work is to analyze the effect of spherical confinement on such properties of the system.

Figures 1-3 show variation of the two-photon transition probability amplitudes for four different values of λD, viz., 10, 20, 30, and 200 a.u. and two values of r₀, viz., 10 a.u. and 50 a.u. It is found that the two-photon transition probability amplitudes D₂ depend on both Debye and spherical confinement. Figure 1 which shows D₂ elements for 1s -2s transitions clearly depicts the effect of change in Debye as well as spherical confinement. In Figs. 1(a) and (b), it is seen that if λD is varied over a wide range from 10 to 200 a.u., the resonance enhancement condition is achieved for small frequencies for smaller λD. Such variation is not much prominent for small change in λD as in Figs. 1(c) and (d). It is also evident from Fig. 1 that the nature of variation of D₂ changes with r₀. The resonance enhancement frequency shifts towards lower ω0 for weaker spherical confinement. The variation with r₀ and λD as observed in Fig. 1 is also present in the data plotted in Figs. 2 and 3. These features can be explained on the basis of the change in energy spectrum and radial matrix elements with Debye as well as spherical confinement. The detailed structure of energy spectrum for such a confined system has been described in our earlier works ^15,21. The decrease in number of bound states and increasing separation of the energy levels for smaller r₀ values are responsible for the observed behaviour. Figure 4 shows similar variation of D₂ for r₀ = 15, 20 and 40 a.u. and λD=∞ for1s - 2s, 1s - 3s, 1s - 3d and 1s - 4d transitions. It clearly depicts the effect of spherical confinement in absence of Debye screening.

Figure 1. Variation of two-photon 1s - 2s transition amplitude with frequency of incoming photons, ω0, for various values of screening length λ_D.  

Figure 2. Same as Fig. 1 for 1s - 3s transition. 

Figure 3. Same as Fig. 1 for 1s - 3d transition. 

The explicit values of transparency and resonance enhancement frequencies of two photon transition probability amplitudes are presented in Tables I-III. The frequencies corresponding to the transitions to 4s and 4d have also been included in the tables. The values for transitions to 2s, 3s and 3d are also implicit in the graphical representation of two photon transition probability amplitudes shown in Figs.1-4. The transparency frequency ^17,24 as observed for transitions to 3s, 4s and 4d states have been reported in Table I. The plasma screening has been found to considerably affect the transparency frequencies. With decrease in λD for a fixed r₀, the transparency frequency, ωt, also decreases as observed by Paul and Ho ²⁴. Similar effect has been observed for variation with confinement radius. That is, decrease in r₀ or increase in confinement leads to decrease in ωt as shown in Table I. For the case of tight confinement, r₀ = 10 a.u., only a single transparency condition is obtained for 1s to 4s transition for all λD as compared to two transparency frequencies for rest of the r₀ values. This is due to the fact that transparency frequency is limited only to the range ΔEif/2 and ΔEif.

Table I. Two-photon transparency frequencies for various Debye screening lengths, λD, and confinement radii, r₀. The data is in atomic units. The results for r₀ = 50 a.u. have been compared with those of free hydrogen. 

Table II. Two-photon resonance enhancement frequencies for various Debye screening lengths, λD, and confinement radii, r₀ = 10, 15 and 20 a.u. The data is in atomic units. 

Table III. Two-photon resonance enhancement frequencies for various Debye screening lengths, λD, and confinement radii, r₀ = 30, 50 a.u. The data is in atomic units. The results for r₀ = 50 a.u. have been compared with those of free hydrogen. 

Figure 4. Variation of two-photon transition amplitudes for 1s - 2s, 1s - 3s, 1s - 3d and 1s - 4d transitions with frequency of incoming photons, ω0, for screening length λD =∞. 

The resonance enhancement frequencies obtained for transitions to 2s, 3s, 4s, 3d and 4d levels are given in Tables II and III. These are found to be same for particular confinement radius and Debye length irrespective of the final state. The values for 1s - 3s transitions as calculated in the present work match very well with those quoted by Paul and Ho ²⁴. More frequencies correspond to resonance enhancement condition for transitions from 1s to 3s or 4s states on increasing the spherical confinement for some λD as can be seen from the data in Table III for r₀ = 30 and 50 a.u. An opposite trend is observed in the transitions from 1s to 3d or 4d states. The trend shows that confinement effect on probability of two-photon transition to a state depends on the shape of its orbital.

4. Conclusions

The effect of spherical confinement on two-photon transition probability amplitudes, transparency frequencies and resonance enhancement frequencies for a Debye plasma embedded hydrogen atom has been explored. The spectrum of the atom has been calculated using B-polynomials. It is understood that experimental data for the confining potential undertaken in the present study is not available. However, the obtained results for loose spherical confinement (r₀ = 50 a.u.) have been compared with the theoretical results reported earlier in literature for some values of Debye parameter and no spherical boundary in Tables I and III. A close agreement has been achieved. It is anticipated that with the advancement of technology the present model of confining potential may become a reality in future and the data presented in the paper would be useful for such experimental studies. The spherical confinement is found to play a role analogous to Debye confinement. In particular, the transparency and resonance enhancement frequencies decrease with increase in confinement.