Application of variational method for three-color three

PHYSICS OF PLASMAS 16, 113301 共2009兲

Application of variational method for three-color three-photon transitions in hydrogen atom implanted in Debye plasmas S. Paula兲 and Y. K. Ho Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106, Republic of China

共Received 27 August 2009; accepted 15 October 2009; published online 11 November 2009兲 Multiphoton processes, where transparency appears, have long fascinated physicists. Plasma screening effects are investigated on three-color three-photon bound-bound transitions in hydrogen atom embedded in Debye plasmas; where photons are linearly and circularly polarized, two left circular and one right circular. All possible combinations of frequency and polarization are considered. Analytical wave functions are used for initial and final states along with the pseudostate wave functions for intermediate states. The analytical wave functions are obtained from the modified wave functions for screening Coulomb potential 共Debye model兲 using Ritz variational method. Here, we have found three-photon transparency for lower values of Debye length. This type of phenomenon occurs due to energy level shifting in the presence of Debye plasma environments. The description of resonance enhancements and three-photon transparency is reported in the present context along with the region of resonance enhancements and three-photon transparency. © 2009 American Institute of Physics. 关doi:10.1063/1.3258665兴 I. INTRODUCTION

Currently, there is a rapid growth in both experimental and theoretical studies of multiphoton processes and multiphoton spectroscopy of atoms, ions, and molecules in chemistry, physics, biology, material science, etc., for a collection of review papers in these areas we refer to the book edited by Lin et al.,1 where a comprehensive bibliography can be found. Nonlinear optical method such as two-photon and three-photon absorption are powerful tools to investigate electronic properties. Because of additional degree of freedom in experiments, where more than one photon is participating in an elementary absorption process, one expects additional information as compared to one-photon absorption. Due to different selection rules new electronic resonances can be excited. Plasma physics plays an enormous role both in the natural world and in the world of technology. We use plasmas for lighting 共candle flames, campfires, fluorescent lights, sodium vapor lights, etc.兲, industrial processing 共microchips, studies of chemical reactions, etc.兲, fusion energy research and neutron production, x-ray lasers, etc. Presently, considerable interest has been cultivated in the study of atomic processes in plasma environments2–18 because of the plasma screening effect on the plasma-embedded atomic systems. A number of studies have been conducted on the investigation of the influence of plasma on scattering processes. The screening effects have played a crucial and significant part in the investigation of plasma environments over the past several decades. Different theoretical methods have been employed along with the Debye screening to study plasma environments.12–29 Debye plasma is considered here, the concept of the Debye screening is valid only in a steady, thermodynamical equilibrium, and linear plasma. a兲

Electronic mail: [email protected].

1070-664X/2009/16共11兲/113301/10/$25.00

The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system which has yielded analytical solution in closed form. Many surveys have been conducted on the simplest atom, resulting in the vast accumulation of data and reports which have been systematically arranged and well documented in the literature. In the present context, linearly and circularly polarized three-color three-photon spectroscopy of the 1s-2p transition in hydrogen atom 共free and immersed in Debye plasmas兲 is studied theoretically for the first time, to our knowledge. The Hamiltonian of a hydrogen atom surrounded by Debye plasmas in atomic unit is 1 H = − ⵜ2 − e−r/␭D , r

共1兲

where −共1 / r兲e−r/␭D is the Debye screening potential for a hydrogen atom with ␭D being the Debye screening length. II. THEORY

Our goal is to calculate transition amplitudes of hydrogen atom in Debye plasma environment. For our present study, we have considered variational method for initial and final states and adopted pseudostate summation technique for intermediate states. Below we show the entire analytical calculation of the current context by three subsections. A. Linearly polarized three-color three-photon

The purpose of the work is a study of the excitation of ground state atomic hydrogen, by simultaneous absorption of three photons of frequencies ␻1, ␻2, and ␻3. Here, all the possible values of ␻1, ␻2, and ␻3 are considered. Six Feynman diagrams 共Fig. 1兲 of third order are associated with the

16, 113301-1

© 2009 American Institute of Physics

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113301-2

Phys. Plasmas 16, 113301 共2009兲

S. Paul and Y. K. Ho

second is s-p-d-p channel. The expressions of Cnn⬘ for different channels are varied as given below: for s-p-s-p channel, Cnn⬘ =

1

具R2p兩r兩Rns典具Rns兩r兩Rn⬘p典具Rn⬘p兩r兩R1s典, 3 冑3

共8兲

and for s-p-d-p channel, Cnn⬘ =

FIG. 1. 共Color online兲 Feynman diagrams associated with the process of excitation of the hydrogen atom initially in the ground state, by three-photon absorption, all of the different frequency.

process in which three photons are absorbed. In the dipole approximation, the amplitude of the process is constructed using a third-order tensor as follows:

⬘ 兲 + ⌸␻1␻3␻2共⍀1,⍀13 ⬘兲 M = 61 关⌸␻1␻2␻3共⍀1,⍀12 ⬘ 兲 + ⌸␻3␻2␻1共⍀3,⍀32 ⬘ 兲兴, + ⌸␻3␻1␻2共⍀3,⍀31

具R2p兩r兩Rnd典具Rnd兩r兩Rn⬘p典具Rn⬘p兩r兩R1s典.

共9兲

We consider variation method for initial and final states and pseudostate summation technique for intermediate states, which includes both discrete and continuum. R1s, R2p are radial part of the variational wave functions 关Eq. 共2兲 in Ref. 30兴 and Rnp, Rn⬘s, Rn⬘d are radial part of the pseudostate wave 共3兲L function 关Eq. 共3兲 in Ref. 27兴. We denote D1s-p-s-2p and 共3兲L D1s-p-d-2p as the transition amplitudes for 1s-2p transitions followed by s-p-s-p channel and s-p-d-p channel, respectively. Again we consider 共3兲L 共3兲L 共3兲L D1s-2p = D1s-p-s-2p + D1s-p-d−2p ,

共10兲

共2兲

where

B. Circularly polarized three-color three-photon

兺

具f兩⑀ជ j · rជ兩n典具n兩⑀ជ k · rជ兩n⬘典具n⬘兩⑀ជ l · rជ兩i典

n,n⬘

共En − ⍀⬘jk兲共En⬘ − ⍀ j兲

,

共3兲

共j , k , l兲 corresponds to the six permutations 共1, 2, 3兲, 共1, 3, 2兲, 共2, 3, 1兲, 共2, 1, 3兲, 共3, 1, 2兲, and 共3, 2, 1兲 for three different frequencies. Here, ⑀ជj is the unit polarization vector of the incident radiation field with frequency ␻ j. From energy conservation, we have E f − Ei = ប␻1 + ប␻2 + ប␻3, where Ei and E f are energies of initial state 兩i典 and final state 兩f典, respectively. In the above expression, 兩n典 and 兩n⬘典 are intermediate states and En, En⬘ are corresponding energies. The parameters ⍀ j and ⍀⬘jk corresponding to the six Feynman diagrams in Fig. 1 are ⍀ j = Ei + ប␻ j,

⍀⬘jk = Ei + ប␻ j + ប␻k .

共4兲

For simplicity, we consider all three photons are linearly polarized, i.e., ⑀ជj = ⑀ជk = ⑀ជl = ⑀ជ . Therefore, the expression 共3兲 reduces to an elementary form, given below, ⌸ ␻ j ␻k␻l =

15冑3

共3兲L where D1s-2-p are the transition amplitudes for 1s-2p transition. L indicates linearly polarized photons.

⬘ 兲 + ⌸␻2␻1␻3共⍀2,⍀21 ⬘兲 + ⌸␻2␻3␻1共⍀2,⍀23

⌸ ␻ j ␻k␻l =

4

Cnn

兺 共En − ⍀⬘ 兲共E⬘ n

n,n⬘

jk

⬘ − ⍀ j兲

,

共5兲

Here, we study 1s-2p transition in hydrogen atom, by simultaneous absorption of three photons ␥1, ␥2, and ␥3 of frequencies ␻1, ␻2, and ␻3. The orientations of three photons ␥1, ␥2, and ␥3 are left circular, right circular, and left circular, respectively. Here all possible values of ␻1, ␻2, and ␻3 are considered. We regard circular polarization, two photons are left circular and one is right circular. The process of two left and one right circular polarized photons is equivalent to the process of two right and one left circular polarized photons. There are two available angular momentum channels for three-photon 1s-2p transition, one is s-p-s-p channel and the second is s-p-d-p channel. All possible combinations of polarization and frequency are presented in Fig. 2 along with corresponding Feynman diagrams. There are three possible combinations of circular polarization, these are left-right-left 共LRL兲, right-left-left 共RLL兲, and left-left-right 共LLR兲. Selection rule indicates that LLR circular polarization arrangement is impossible for s-p-s-p channel. In the dipole approximation, the amplitude of the general process is constructed using a third-order tensor, followed by the Feynman diagrams 共Fig. 2兲, as follows:

where Cnn⬘ = 具f兩⑀ជ · rជ兩n典具n兩⑀ជ · rជ兩n⬘典具n⬘兩⑀ជ · rជ兩i典.

共6兲

1 LRL + ⌸␻RLL + ⌸␻RLL 关⌸ + ⌸␻LRL 2␻1␻3 2␻3␻1 3␻2␻1 n p ␻1␻2␻3 LLR + ⌸␻LLR ␻ ␻ + ⌸␻ ␻ ␻ 兴,

Here we consider linear polarization, so

⑀ជ · rជ = 共 34 ␲兲1/2rY 10共rˆ兲.

M=

1 3 2

共7兲

There are two available angular momentum channels for three-photon 1s-2p transition, one is s-p-s-p channel and the

3 1 2

共11兲

where superscripts denote sequence of circular polarization 共left and right兲 and n p is the total number of possibilities. Here

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113301-3

Phys. Plasmas 16, 113301 共2009兲

Application of the variational method…

⌸␻IJK␻ ␻ = FIJK 兺 i j k

n,n⬘

Cnn⬘

共En − ⍀⬘ij兲共En⬘ − ⍀i兲

共15兲

,

where FIJK is the angular integration factor and Cnn⬘ = 具R f 兩r兩Rn典具Rn兩r兩Rn⬘典具Rn⬘兩r兩Rg典.

共16兲

In the above expression, Rg and R f are radial parts of the initial and final state wave functions, respectively, and Rn, Rn⬘ are that of intermediate states. Analytical wave functions are used for initial and final states; for screening Coulomb potential 共Debye model兲, we use modified wave function29 based on Ritz variation method. Rn and Rn⬘ are calculated by using pseudostate summation technique30 with basis size of 30. The factor FIJK is the product of three angular integration, express as FIJK =

冉 冊 4␲ 3

3/2

具Y l1m1兩Y 1␮I兩Y l2m2典 ⫻ 具Y l2m2兩Y 1␮J兩Y l3m3典

⫻具Y l3m3兩Y 1␮K兩Y l4m4典,

共17兲

where ␮N = +1, ⌬l = ⫾ 1, ⌬m = +1 for left circular polarization and ␮N = −1, ⌬l = ⫾ 1, ⌬m = −1 for right circular polar共3兲C 共3兲C ization. We denote D1s-p-s-2p and D1s-p-d-2p as the transition amplitudes for 1s-2p transitions followed by s-p-s-p channel and s-p-d-p channel, respectively. The expressions of 共3兲C 共3兲C D1s-p-s-2p and D1s-p-d-2p are given below, LRL RLL RLL 共3兲C = − 41 关⌸␻LRL D1s-p-s-2p ␻ ␻ + ⌸␻ ␻ ␻ + ⌸␻ ␻ ␻ + ⌸␻ ␻ ␻ 兴, 1 2 3

FIG. 2. 共Color online兲 Combinations of circular polarizations 共left side of the figure兲 and corresponding Feynman diagrams 共right side of the figure兲, associated with the process of excitation of the hydrogen atom by absorption of three photons. Arrangement: diagram 1, left-right-left; diagram 2, rightleft-left; diagram 3, left-left-right. ␥1, ␥2, and ␥3 are three photons of frequencies ␻1, ␻2, and ␻3 and polarization left circular, right circular, and left circular, respectively.

2 1 3

3 2 1

2 3 1

共18兲 LRL RLL RLL 共3兲C = − 61 关⌸␻LRL D1s-p-d-2p ␻ ␻ + ⌸␻ ␻ ␻ + ⌸␻ ␻ ␻ + ⌸␻ ␻ ␻ 1 2 3

+

3 2 1

2 1 3

2 3 1

⌸␻LLR ⌸LLR 兴. 1␻3␻2 ␻3␻1␻2

共19兲

Again we consider 共3兲C 共3兲C 共3兲C = D1s-p-s-2p + D1s-p-d-2p , D1s−2p

⌸␻IJK␻ ␻ = i j k

兺

具f兩⑀ជ I · rជ兩n典具n兩⑀ជ J · rជ兩n⬘典具n⬘兩⑀ជ K · rជ兩g典

n,n⬘

共En − ⍀⬘ij兲共En⬘ − ⍀i兲

,

共12兲

where ⑀ជI is the unit polarization vector of the incident radiation field with frequency ␻i. I is either L or R corresponding to left and right circular polarization. From energy conservation, we have ⌬E = E f − Eg = ប␻1 + ប␻2 + ប␻3, where Eg and E f are energies of the initial state 兩g典 and the final state 兩f典, respectively. In the above expression, 兩n典 and 兩n⬘典 are intermediate states and En, En⬘ are the corresponding energies. The parameters ⍀i and ⍀⬘ij corresponding to the six Feynman diagrams in Fig. 2 are ⍀ i = E g + ប ␻ i,

⍀⬘ij = Eg + ប␻i + ប␻ j .

共13兲

Here we consider circular polarization, so

⑀ជ I · rជ = 共 34 ␲兲1/2rY 1␮I共rˆ兲,

共14兲

where ␮I = 1 for left circular polarization 共I = L兲 and ␮I = −1 for right circular polarization 共I = R兲. After evaluating the angular integrations, Eq. 共12兲 reduces to an elementary form, given below,

共20兲

共3兲C D1s-2p

where are the transition amplitudes for 1s-2p transition. C indicates circularly polarized photons. C. Method

We calculate the initial 共1s兲 and final 共2p兲 states by variation method, a short description of the approach is presented here 共for detail, see Ref. 29兲. The radial Schrödinger equation for hydrogen atom in dense plasma would be given by

再 冋 −

册

冎

ប2 d2 l共l + 1兲 Ze2 −r/␭ e D Pnl共r兲 = Enl Pnl , − − 2m dr2 r2 r

共21兲

where Pnl共r兲 is the radial wave function for the nlth shell. The numerical solutions and higher order perturbation calculation have been evaluated for Eq. 共21兲. Here we shall consider a simple analytical method to obtain the solution. Our approach is the same as the procedure of Jung12 but in a more general way. Jung has calculated for the 1s, 2s, and 2p states only. The solutions for Eq. 共21兲 are assumed to be in the hydrogenic form with a variation parameter. The trial wave function is considered, as follows:

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113301-4

Pnl共r兲 ⬅ rRnl共r兲 =

Phys. Plasmas 16, 113301 共2009兲

S. Paul and Y. K. Ho

冋

1 共n − l − 1兲! n ␣共n + l兲!

册冉 冊 1/2

2r n␣

l+1 2l+1 Ln−l−1

冉 冊

2r −r/n␣ e , n␣

共22兲

where ␣ is the variational parameter and

␣ → az

for ␭D → ⬁,

共23兲

az = a0 / z, a0 is Bohr radius, and ␭D → ⬁ indicates plasma free 2l+1 is the usual Laguerre polynomial. Substitute situation. Ln-l-1 the expression of trial wave function into the Schrödinger equation, we get 具Enl典 = −

冋

2 1 ប2 − 2m n2␣2 ␣

− Ze2

冕

⬁

0

冕

⬁

0

1 Pnl共r兲 Pnl共r兲dr r

册

1 Pnl共r兲 e−r/␭D Pnl共r兲dr. r

共24兲

After evaluating the integrations and simplification, we have ប2 1 Ze2 − · 2m n2␣2 n2␣

具Enl典 =

兺 冉 冊冉

n−l−1

⫻

冉

1 n␣ 1+ 2␭D

n+l

n−l−1

k

k

k=0

冊 冊冉 冊

共3兲L FIG. 3. 共Color兲 Transition matrix elements D1s-2p for 1s-2p transition in free hydrogen atom as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

2n

n␣ 2␭D

2k

共25兲

,

which is the expectation value of the nl state energy of hydrogen atom in plasmas. Equation 共25兲 shows that the energy level of hydrogen atom in plasmas depends on both n 共the principal quantum number兲 and l 共the orbital quantum number兲. The parameter ␣ is obtained from the minimization condition of 具Enl典, i.e., ⳵ / ⳵␣具Enl典 = 0 which gives

冉

␣ = az 1 +

n␣ 2␭D

冊

2n

冤冉

n−l−1

+

where Ak =

兺 k=0

共1 − 2k兲Ak

冉 冊冉

冉 冊

␣ ␭D n␣ 1+ 2␭D n2

冉 冊

n+l

n−l−1

k

k

n␣ 2␭D

冊

2k

n−l−1

冊

冥

兺 k=0

Ak

冉 冊 n␣ 2␭D

⌿n =

C共j,n兲␾ j 兺 j=0

具n兩H兩n⬘典 = En⌬nn⬘,

2k

共29兲

具n兩n⬘典 = ⌬nn⬘ .

共30兲

The finite-dimensional eigenvalue problem corresponding to the Hamiltonian of a hydrogen atom in plasmas field is the following: 共26兲

共H គ − En⌬គ 兲兩n典 = 0,

共31兲

where H គ and ⌬ គ are the Hamiltonian matrix and the overlap matrix, respectively 共for more detailed see the Ref. 27兲. III. RESULTS AND DISCUSSION

.

共27兲

The variational parameter ␣ is calculated by fixed point iteration method with the initial condition ␣0 = az 关by the help of condition 共23兲兴. We calculate the intermediate states by using the pseudostate summation method 共for detail, see Refs. 28 and 29兲. The basis functions of the pseudostate method are taken of the form 共Drachman et al.30兲

␾ j = e−arrl+jY lm共␪, ␾兲,

N−1

and we have

−1

,

adjustable parameter a = 1 for s-states, a = 0.5 for p-states, and a = 0.3 for d-states. The value of N is 30 in the present calculations. The wave functions are expanded in terms of linear combinations of the basis functions as

j = 0,1, . . . ,N − 1,

共28兲

where a is the basis parameter, l is the orbital angular momentum, and N is the basis size. Here it should be noted that the parameter a is adjustable. We consider the values of the

In the present section, we demonstrate some calculated data with a short description about resonance enhancement and three-photon transparency. A. Linearly polarized photons

Figure 3 shows three-photon 1s-2p transition amplitudes for free hydrogen atom as a function of ␻1 and ␻2, the frequencies of photons. In Figs. 4 and 5, we observe the partial transition matrix elements due to the s-p-s-p and s-p-d-p channels, respectively. These figures show that the partial transition amplitudes of s-p-s-p channel dominate that of for s-p-d-p channel. In Figs. 3–5, we observe three resonance enhancements when the pair of frequencies 共␻1 , ␻2兲,

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113301-5

Application of the variational method…

共3兲L FIG. 4. 共Color online兲 Partial transition matrix elements D1s-p-s-2p for 1s-2p transition in free hydrogen atom as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

共␻2 , ␻3兲, and 共␻3 , ␻1兲 each reaches minimum values. There is no three-photon transparency for free hydrogen. In the presence of Debye plasma field with Debye length of 7, we observe three-photon transparency 共shown in Fig. 6兲 and resonance enhancements, similar to the free hydrogen case, when frequencies of two photons are very small. As free hydrogen case, the magnitudes of transition amplitude for the s-p-s-p channel 共Fig. 7兲 are always greater than those of for s-p-d-p channel 共Fig. 8兲. In Figs. 6 and 7, we observe some stripe in the curve surfaces. Those are no physical signifi-

共3兲L D1s-p-d-2p

FIG. 5. 共Color online兲 Partial transition matrix elements for 1s-2p transition in free hydrogen atom as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

Phys. Plasmas 16, 113301 共2009兲

共3兲L FIG. 6. 共Color兲 Transition matrix elements D1s-2p for 1s-2p transition in hydrogen atom embedded in Debye plasma with Debye length of 7 as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

cance. To construct the prominent figures, the used software package automatically generates such type of ribbon. In the case of s-p-d-p channel transition amplitudes, there is no three-photon transparency. We calculate three-photon transition amplitudes for ␭D = ⬁, 100, 50, 10, 9, 8, and 7. Threephoton transparency remains absent from large values of Debye length up to ␭D = 10, for ␭ = 9, three photon transparency is started to appear. The shapes of the curves for threephoton 1s-2p transition amplitudes are more or less the same for ␭D = ⬁, 100, 50, and 10; only the bottom of the curves become flatter as ␭D decreasing. The diagrams of transition matrix elements for ␭D = 9, 8, and 7 are similar. In Tables I and II, we report some numerical data of partial transition 共3兲L 共3兲L matrix elements D1s-p-s-2p and D1s-p-d-2p for ␭D = ⬁ and ␭D

共3兲L FIG. 7. 共Color online兲 Partial transition matrix elements D1s-p-s-2p for 1s-2p transition in hydrogen atom embedded in Debye plasma with Debye length of 7 as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

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113301-6

Phys. Plasmas 16, 113301 共2009兲

S. Paul and Y. K. Ho

TABLE II. Linearly polarized three-photon transition amplitudes for hydrogen atom in the presence of Debye plasmas with Debye length of 7 as a function of incident photon frequencies in atomic unit.

共3兲L FIG. 8. 共Color online兲 Partial transition matrix elements D1s-p-d-2p for 1s-2p transition in hydrogen atom embedded in Debye plasma with Debye length of 7 as a function of photon frequencies ␻1 and ␻2. L indicates linearly polarized photons.

␻1

␻2

共3兲L D1s-p-s-2p

共3兲L D1s-p-d-2p

0.003 47

0.066 00

−6260.761 52

337.980 35

0.052 10 0.148 20

0.066 00 0.118 10

386.053 13 180.904 97

142.378 95 89.140 26

0.162 10

0.162 10

575.237 41

122.806 82

0.192 20

0.088 00

241.182 66

105.809 86

0.192 20 0.206 10

0.118 10 0.044 00

360.419 88 336.276 06

117.084 89 121.305 19

0.206 10

0.088 00

297.733 18

117.877 47

0.206 10 0.222 30

0.118 10 0.030 10

633.859 87 523.573 41

136.387 04 143.949 79

0.236 20

0.030 10

591.852 22

161.976 51

0.310 30

0.002 32

−7616.260 06

783.915 16

1 M = 兺 Cnn⬘ 6 n,n ⬘

冋兺 册 6

i=1

1 . Di

共32兲

The denominators Di 共i = 1 , 2 , . . . , 6兲 are given below, = 7, respectively. Here, we present transition matrix elements for ␭D = 7. The computed one-color three-photon transition amplitudes for ␻1 = ␻2 = ␻3 agree excellently with the results of Thayyullathil et al.31 and our earlier work.29 In our pervious calculation29 we used pseudostate summation method, the basis functions are different from the usual basis functions, to evaluate initial and final state wave functions. 1. Resonance enhancement

The process is said to be resonant whenever an intermediate state of the target spectrum is closely reached from the initial state by absorption of one or two photons. In the present section, we discuss about resonance enhancement for individual channel. From Eqs. 共2兲, 共4兲, and 共5兲 the transition matrix element M can be expressed as TABLE I. Linearly polarized three-photon transition amplitudes for free hydrogen atom as a function of incident photon frequencies in atomic unit.

␻1

␻2

共3兲L D1s-p-s-2p

共3兲L D1s-p-d-2p

0.045 0.120 0.150 0.165 0.180 0.180 0.180 0.195 0.210 0.240 0.270 0.300

0.3000 0.1500 0.1800 0.1500 0.0300 0.1200 0.1800 0.1500 0.1200 0.1200 0.0975 0.0600

737.330 62 122.556 03 202.814 57 161.793 66 284.645 05 146.923 97 541.814 47 290.180 01 220.975 99 629.175 09 1457.971 62 1120.215 70

194.871 28 65.392 57 78.431 27 72.728 05 85.116 59 71.293 30 95.038 02 86.710 52 84.371 90 111.625 89 154.650 08 201.912 44

D1 = 共⌬n⬘ − ប␻1兲共⌬n − ប␻1 − ប␻2兲, D2 = 共⌬n⬘ − ប␻1兲共⌬n − ⌬E + ប␻2兲, D3 = 共⌬n⬘ − ប␻2兲共⌬n − ⌬E + ប␻1兲, D4 = 共⌬n⬘ − ប␻2兲共⌬n − ប␻2 − ប␻1兲,

共33兲

D5 = 共⌬n⬘ − ⌬E + ប␻1 + ប␻2兲共⌬n − ⌬E + ប␻2兲, D6 = 共⌬n⬘ − ⌬E + ប␻1 + ប␻2兲共⌬n − ⌬E + ប␻1兲, where ⌬n = En − Eg and ⌬n⬘ = En⬘ − Eg. The six conditions of resonance, i.e., where the denominators become close to zero 共need not be zero兲 are presented below, ប ␻ 1 ⯝ ⌬ n⬘,

ប ␻ 2 ⯝ ⌬ n⬘,

ប␻1 ⯝ ⌬E − ⌬n,

ប␻1 + ប␻2 ⯝ ⌬E − ⌬n⬘ ,

ប␻2 ⯝ ⌬E − ⌬n,

ប␻1 + ប␻2 ⯝ ⌬n ,

共34兲

we observe resonance enhancement when ⌬n and ⌬n⬘ belong to the interval 共0 , ⌬E兲. We notice that for s-p-s-p channel ⌬1⬘ belongs to the interval 共0 , ⌬E兲 for free hydrogen atom 共it should be noted that ⌬1⬘ is very close to ⌬E, the difference is less than 10−30兲 and ⌬1⬘ and ⌬2 belong to 共0 , ⌬E兲 when ␭D = 7. The values of ⌬E for ␭D = ⬁ and ␭D = 7 are different. It follows from the set of inequalities 共34兲, there are three resonances for free hydrogen atom 关satisfy three conditions in Eq. 共34兲兴 and six resonances for ␭D = 7 关satisfy six conditions in Eq. 共34兲兴. In the case of s-p-d-p channel only ⌬1⬘ belongs to the interval 共0 , ⌬E兲 for both cases, free hydrogen and in the presence of Debye plasmas with Debye length of 7. Therefore we get only three resonances for s-p-d-p channel in both cases.

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113301-7

Phys. Plasmas 16, 113301 共2009兲

Application of the variational method…

共3兲C FIG. 9. 共Color兲 Transition matrix elements D1s-2p as a function of photon frequencies ␻1 and ␻2 for 1s-2p transition in 共a兲 free hydrogen atom and 共b兲 hydrogen atom embedded in Debye plasma with Debye length of 5. C indicates circularly polarized photons.

2. Three-photon transparency

Transparency is the material property of allowing light to pass through, i.e., photons are not absorbed. In case of three-photon transparency some values of M, the transition matrix element, in Eq. 共32兲 are negative. Thus the condition of transparency will be evaluated if we can determine the condition of M ⬍ 0. It is happened when any one of Di 共i = 1 , 2 , . . . , 6兲 in expressions 共33兲 is negative near a resonance point, the corresponding range of ប␻1 and ប␻2, the energies of two photons, are given below, ប ␻ ⬎ c⌬ n⬘,

ប␻⬘⬎c⌬E − ⌬n ,

ប ␻ ⬍ c⌬ n⬘,

ប␻⬘⬍c⌬E − ⌬n ,

ប ␻ ⬎ c⌬ n⬘,

ប ␻ + ប ␻ ⬘⬍ c⌬ n ,

ប ␻ ⬍ c⌬ n⬘,

ប ␻ + ប ␻ ⬘⬎ c⌬ n ,

ប␻ + ប␻⬘⬎c⌬E − ⌬n⬘,

ប␻⬘⬍c⌬E − ⌬n ,

ប␻ + ប␻⬘⬍c⌬E − ⌬n⬘,

ប␻⬘⬎c⌬E − ⌬n .

共35兲

In expressions 共35兲 ␻ and ␻⬘ are either ␻1 or ␻2 but for a particular range, as example 共35a兲, when we consider ␻ = ␻1, ␻⬘ must be ␻2. x⬎cy indicates that x is greater than and close to y. The energy conservation relation implies one more condition 0 ⬍ ប␻1 + ប␻2 ⬍ ⌬E. The three-photon transparency appears when ⌬n or ⌬n⬘ or both 共for one or more values of n and n⬘兲 belong into the interval 共0 , ⌬E − ␦兴, where ␦ should be greater than minimum possible values of photon energy. For ␭D = ⬁ 共in both the channel calculation兲 and ␭D = 7 共in s-p-d-p channel computation兲, we find ⌬E − ␦ ⬍ ⌬1⬘ ⬍ ⌬E 共in the present calculation ␦ is less than 10−30兲 and ⌬n, ⌬n⬘ ⬎ ⌬E for all n and n⬘ 艌 2 which does not satisfy the condi-

tion of transparency. Numerically we observe 0 ⬍ ⌬2 ⬍ ⌬1⬘ ⬍ ⌬E − ␦ for s-p-s-p channel, which satisfies the condition of transparency at ␭D = 7.

B. Circularly polarized photons

Figure 9共a兲 shows three-photon 1s-2p transition amplitudes for free hydrogen atom as a function of ␻1 and ␻2, the frequencies of two photons. There is no three-photon transparency for free hydrogen, although resonance enhancements are present. In the presence of Debye plasma field with Debye length of 5, we observe three-photon transparency 关shown in Fig. 9共b兲兴 and resonance enhancement for 1s-2p transition. No three-photon transparency occurs from ␭D = ⬁ 共free hydrogen atom兲 up to ␭D = 10. For ␭D = 9, transparency is started to appear. Three-photon transparency is observed for ␭D = 9 , 8 , . . . , 5 共integer value兲, when ␭D 艋 4 the bound state 2p is disappeared. Figure 9共a兲 shows three resonance peaks; the magnitudes of the peaks are different, one large, one medium, and the last one is very small 关inset of Fig. 9共a兲兴. In Fig. 9共b兲, seven resonances have been found with different magnitudes. In the resonance section, we shall discuss the phenomena briefly. In Figs. 10共a兲 and 10共b兲, we present the partial transition matrix element due to the s-p-s-p channel for ␭D = ⬁ and ␭D = 5, respectively. In the case of s-p-d-p channel transition amplitude, there are no three-photon transparency for both cases, ␭D = ⬁ 关Fig. 10共c兲兴 and ␭D = 5 关Fig. 10共d兲兴. We observe only three resonance peaks in two cases, Figs. 10共c兲 and 10共d兲. Those figures show that the partial transition amplitudes of s-p-s-p channel dominate that of s-p-d-p channel. Some numerical data of the transition matrix elements are presented in Table III.

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113301-8

Phys. Plasmas 16, 113301 共2009兲

S. Paul and Y. K. Ho

共3兲C FIG. 10. 共Color兲 Partial transition matrix elements as a function of photon frequencies ␻1 and ␻2 共a兲 D1s-p-s-2p for 1s-2p transition in free hydrogen atom, 共b兲 共3兲C 共3兲C for 1s-2p transition in free hydrogen atom, D1s-p-s-2p for 1s-2p transition in hydrogen atom embedded in Debye plasma with Debye length of 5, 共c兲 D1s-p-d-2p 共3兲C for 1s-2p transition in hydrogen atom embedded in Debye plasma with Debye length of 5. C indicates circularly polarized photons. 共d兲 D1s-p-d-2p

1. Resonance enhancement

TABLE III. Circularly polarized three-photon transition amplitudes as a function of incident photon frequencies in atomic unit. 共3兲C D1s-p-s-2p

共3兲C D1s-p-d-2p

␻1

␻2

␭D = ⬁

␭D = 5

␭D = ⬁

␭D = 5

0.1 0.1 0.2 0.005

0.1 0.2 0.1 0.3

118.181 214.619 153.126 4383.990

196.848 1 315.770 1 119.822 −12 773.338

50.533 35.77 54.475 63.290

73.272 69.219 109.904 621.964

The process is said to be resonant whenever an intermediate state of the target spectrum is closely reached from the initial state by absorption of one or two photons. In the present section, we discuss about resonance enhancement for individual channel. For s-p-s-p channel, from Eqs. 共15兲, 共16兲, and 共18兲 we get that the transition matrix elements 共3兲C D1s-p-s-2p are the sum of four different factions,

共3兲C = D1s-p-s-2p

1

兺 Cnn⬘ 12冑3 n,n⬘

冋

册

1 1 1 1 + + + . D1 D2 D3 D4

共36兲

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113301-9

Phys. Plasmas 16, 113301 共2009兲

Application of the variational method…

The denominators of those factions are given below, D1 = 共⌬n⬘ − ប␻1兲共⌬n − ប␻1 − ប␻2兲, D2 = 共⌬n⬘ − ប␻3兲共⌬n − ប␻3 − ប␻2兲, D3 = 共⌬n⬘ − ប␻2兲共⌬n − ប␻2 − ប␻1兲,

共37兲

2. Three-photon transparency

D4 = 共⌬n⬘ − ប␻2兲共⌬n − ប␻2 − ប␻3兲, where ⌬n = En − Eg and ⌬n⬘ = En⬘ − Eg. Using the energy conservation relation, the above set of divisors reduces to D1 = 共⌬n⬘ − ប␻1兲共⌬n − ប␻1 − ប␻2兲, D2 = 共⌬n⬘ − ⌬E + ប␻1 + ប␻2兲共⌬n − ⌬E + ប␻1兲, D3 = 共⌬n⬘ − ប␻2兲共⌬n − ប␻1 − ប␻2兲,

共38兲

D4 = 共⌬n⬘ − ប␻2兲共⌬n − ⌬E + ប␻1兲. Mathematically the condition of resonances, i.e., where the denominators become close to zero 共need not be zero兲 are given below, ប ␻ 1 ⬵ ⌬ n⬘,

ប␻1 ⬵ ⌬E − ⌬n,

ប ␻ 1 + ប ␻ 2 ⬵ ⌬ n,

ប ␻ 2 ⬵ ⌬ n⬘ ,

ប␻1 + ប␻2 ⬵ ⌬E − ⌬n⬘ ,

共39兲

where ⌬n and ⌬n⬘ belong to the interval 共0 , ⌬E兲. If ⌬n and ⌬n⬘ belong to the interval 共0 , ⌬E兲 for n = 1 , 2 , . . . , i and n⬘ = 1 , 2 , . . . , j; the total number of resonance surfaces is 2i + 3j, provided all the elements of the pairs 共⌬n⬘ , ⌬E − ⌬n兲 and 共⌬n , ⌬E − ⌬n⬘兲 are distinct. If ⌬m⬘ and ⌬E − ⌬m are the same for m⬘ 艋 j, m 艋 i, the two resonance surfaces corresponding ប␻1 ⬵ ⌬m⬘ and ប␻1 ⬵ ⌬E − ⌬m coincide. Similar things happen when ⌬k and ⌬E − ⌬k⬘ are the same for k 艋 i, k⬘ 艋 j. Numerically it is observed that ⌬1⬘ belongs to the interval 共0 , ⌬E兲 for plasma free situation. Therefore, there are three resonance surfaces 关Fig. 10共a兲兴 for plasma-free partial transition, followed by s-p-s-p channel. For ␭D = 5, we observe that ⌬1, ⌬2, and ⌬1⬘ belong to the interval 共0 , ⌬E兲 which implies seven resonance surfaces. Here it should be noted that the value of ⌬E is dissimilar for different ␭D. Considering only equal sign, the set of Eq. 共39兲 reduces to a branch of straight line equations. Using the similar procedure, we get the condition of resonances for s-p-d-p channel as follows: ប ␻ 1 ⬵ ⌬ n⬘,

ប␻1 ⬵ ⌬E − ⌬n,

⌬E − ⌬k⬘ are the same for k 艋 i, k⬘ 艋 j, the two resonance surfaces corresponding ប␻1 + ប␻2 ⬵ ⌬k and ប␻1 + ប␻2 ⬵ ⌬E − ⌬k⬘ turn into one surface. Numerically it is observed that ⌬1⬘ belongs to the interval 共0 , ⌬E兲 for plasma-free case and for ␭D = 5. Therefore, there are three resonance surfaces for both the cases.

ប ␻ 2 ⬵ ⌬ n⬘ ,

共40兲 ប␻2 ⬵ ⌬E − ⌬n, ប␻1 + ប␻2 ⬵ ⌬n, ប␻1 + ប␻2 ⬵ ⌬E − ⌬n⬘ , where ⌬n and ⌬n⬘ belong to the interval 共0 , ⌬E兲. If ⌬n and ⌬n⬘ belong to the interval 共0 , ⌬E兲 for n = 1 , 2 , . . . , i and n⬘ = 1 , 2 , . . . , j; the total number of resonance surfaces is 3共i + j兲, provided all the elements of the pairs 共⌬n⬘ , ⌬E − ⌬n兲 and 共⌬n , ⌬E − ⌬n⬘兲 are distinct. If ⌬m⬘ and ⌬E − ⌬m are the same for m⬘ 艋 j, m 艋 i, we observe two resonance surfaces instead of four resonance surfaces corresponding ប␻1 ⬵ ⌬m⬘, ប␻1 ⬵ ⌬E − ⌬m, ប␻2 ⬵ ⌬m⬘, and ប␻2 ⬵ ⌬E − ⌬m. If ⌬k and

Transparency is the material property of allowing light to pass through, i.e., photons are not absorbed. The detailed 共3兲C expression of partial transition matrix element D1s-p-s-2p for s-p-s-p channel, where three-photon transparency occurs for ␭D = 9 , 8 , . . . , 5, has been already presented in Eq. 共36兲. In the 共3兲C case of three-photon transparency some values of D1s-p-s-2p are negative. Thus the condition of transparency will be 共3兲C evaluated if we can determine the condition of D1s-p-s-2p ⬍ 0. 共3兲C From Eq. 共36兲 we can easily see that D1s-p-s-2p ⬍ 0, when denominator 共denominators兲 becomes 共become兲 negative near a resonance point. From the set of Eq. 共37兲, we obtain eight stipulations for which the denominators D1, D2, D3, and D4 are negative, respectively. The energy conservation relation implies one more condition 0 ⬍ ប␻1 + ប␻2 ⬍ ⌬E. The three-photon transparency appears when ⌬n or ⌬n⬘ or both 共for one or more values of n and n⬘兲 belong into the interval 共0 , ⌬E − ␦兴, where ␦ is greater than minimum possible values of photon energy. For ␭D = ⬁, we find ⌬E − ␦ ⬍ ⌬1⬘ ⬍ ⌬E and ⌬n, ⌬n⬘ ⬎ ⌬E for all n and n⬘ 艌 2 which does not satisfy the condition of transparency. Numerically we observe 0 ⬍ ⌬2 ⬍ ⌬1⬘ ⬍ ⌬E − ␦ which satisfies the condition of transparency at ␭D = 5. This type of phenomena happen because due to Debye plasma environments energy levels of hydrogen atom are shifted. IV. CONCLUSION

The transition matrix elements are presented in the context for three-color three-photon 1s-2p transition in hydrogen atom, free and in the presence of Debye plasma field. The three-photon transparency appears when hydrogen atom embedded in weakly coupled Debye plasmas. Here, we consider that three photons are linearly polarized and circularly polarized. Present results show that Debye plasma environments have a considerable effect on the three-photon bound-bound transitions. The dynamic motion of the plasma electrons has to be considered in order to investigation the plasma screening effect on hydrogen atom can be considered qualitatively by the introduction of the plasma dielectric function.32 The effects may be very important for high density plasma, but for low density plasma the effect can be neglected. The static plasma screening formula obtained by the Debye–Huckel model overestimates the plasma screening effects on the hydrogen atom in dense plasma. The static screening result presented here is subject to the condition that the plasma is a thermodynamically equilibrium plasma and neglect the contributions from ions in plasma since electrons provide more effective shielding than ions. In the static plasma screening, we observe that the two-photon transition amplitude is mainly determined by the Debye length, which in turn is determined

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113301-10

by the plasma temperature and density. With increase in plasma density at a given temperature, the Debye length decreases, and thus the effect from plasma temperature and density cannot be neglected. ACKNOWLEDGMENTS

The authors would like to thankfully acknowledge financial support by the National Science Council of Taiwan, ROC. 1

Phys. Plasmas 16, 113301 共2009兲

S. Paul and Y. K. Ho

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