The action of intense nonresonance laser radiation on the phase equilibrium of systemswith coupled states *

V. K. Mukhomorov

Agrophysical Scientific Research Institute, St. Petersburg~Submitted December 5, 2003!Opticheski� Zhurnal71, 54–59~August 2004!

This paper analyzes the state of a bound two-electron formation~a bipolaron! in anelectromagnetic radiation field that is not considered weak. By using the Kramers–Hennebergerunitary transformation, the external, rapidly oscillating action is transferred to the argumentof the potential energy. It is shown that increasing the external field intensity produces a monotonicdecrease of the dissociation energy of a bound two-electron formation and reduces thenumber of its vibrational states. It is established that the external field produces attenuation ofthe electron–phonon interaction. The degree of dissociation of the bipolaron is computedas a function of the external field intensity, the temperature, and the concentration. The transitionbetween the bound two-particle state and the single-particle state is treated as a smearedphase transition. ©2004 Optical Society of America

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INTRODUCTION

This paper analyzes the action of an intense extenonresonance monochromatic light source on the dynamequilibrium constant that characterizes the transition betwtwo-electron coupled formations of bipolaron type and oelectron states~polarons! solvated in ammonia. An investigation of two-electron systems in polarizable polar medi1,2

showed that there is a fairly wide range of dielectric medincluding ammonia, in which stable bound two-electrquasi-molecular charged formations~bipolarons! can exist.The main effect of the interaction of electrons with the quatized polarization field of a condensed phase reduces tofact that each electron forms its own potential well for itseThe electrons in this case carry out a complex motion:one hand, they oscillate in fairly deep potential wells~thismotion corresponds to the appearance of the polaron opspectrum!. On the other hand, they participate in relatimotion, carried out by the centers of gravity of the polariztion potential wells relative to the equilibrium position ofbipolaron. In fact, the electronic properties of metaammonia systems depend on two parameters: temperand concentration. If one regards the states of bound etrons united into bipolarons~axially symmetric dimers! asnew quasi-particles with respect to one-electron polastates, the transitions in the system caused by bipolaronmation~decay! can be interpreted in terms of smeared phtransitions.3

In what follows, we shall assume that the frequencyv ofthe external radiation source is not in resonance withquencyV0 of the oscillations of the relative motion of thpolarons in a bipolaron around their equilibrium position bis less than the frequency needed to dissociate the bipolaV0,v,vD5D/\!ve , whereD is the dissociation energof the bipolaron, andve is the oscillation frequency of thevalence electrons intrinsic to the dielectric medium. Plapolarized laser radiation, given by vector potentialA(t)5A0x cosvt, is regarded as a classical plane electromagn

538 J. Opt. Technol. 71 (8), August 2004 1070-9762/2004/080

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wave with angular frequencyv propagating in thez direc-tion, wherex is the unit vector directed along thex axis, andA0 is the amplitude of the laser field, which is not considersmall. At the same time, in the low-frequency, or adiabalimit, in which v!V0 , the relative motion of the polarons irapid, and the external oscillating field can be regarded aperturbation, withvD!v i,v,I /\, wherev i is the oscil-lation frequency of the electron in the polarization potentwell, and I is the ionization potential of the polaron. It washown in Ref. 4 that the action of an intense external soubreaks down the polaron formation itself.

THE METHOD OF UNITARY TRANSFORMATIONS AND THEPRINCIPAL EQUATIONS

Wave functionC(r,t), which describes the relative motion of the polarons in potential fieldDE0(r), taking intoaccount the action of an external field characterized by vtor potentialA(t), satisfies the equation

i\]C~r,t !/]t2HC~r,t !, ~1!

where the effective one-particle HamiltonianH that de-scribes the motion of charged particles with reduced mmr** '0.01ac

4m* (ac is the electron–phonon coupling constant! has the following form:5

H5~1/2mr** !@Pr1~2e/c!A~ t !#21DE0~r!. ~2!

Here DE0(r) is the main centrosymmetric singlet termthe bipolaron, with the energy being measured from the toenergy of two polarons separated from each other by infinandr is the coordinate of the relative motion of the centeof gravity of the polarons. In the nonrelativistic dipole aproximation, the vector potential is independent of the spavariables. We allow such an approach if the wavelenof the external source,l5v/c, exceeds the effective localization size of the polaron,Rp'10a0* (a0* 5\2«* /e2m*is the effective Bohr radius, the isotropic effective maof the electron at the bottom of the conduction bandm* 51.73m, Ref. 1!, andc is the speed of light in ammonia

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Perturbation theory cannot be used to analyze Eq.~1! inthe case of an intense nonresonance field. We thereforean approach based on an application of the unitary KrameHenneberger transformation,6,7 which can be written in thedipole approximation as

U15exp@ i a~ t !•Pr /\#, U25exp@ ib~ t !/\#,

U1U51, U5U1U2 , ~3!

where we have used the following notation:

a~ t !5~2e/cmr** !E2`

t

A~ t !dt,

b~ t !52~2e2/c2mr** !E2`

t

A2~ t !dt.

The canonical transformations of Eq.~3! make is possible totranslate external fieldA(t) from a kinetic-energy operator inthe Hamiltonian to an argument of the potential functiDE0(r):

2~\2/2mr** !¹2w~r,t !1DE0@r1a~ t !#w~r,t !

5 i\]w~r,t !/]t,

DE0@r1a~ t !#5E0@r1a~ t !#22En`~a~ t !!

5E0@r1a~ t !#22En`~0!22En

`~a~ t !!

12En`~0!,

w~r,t !5UC~r,t !, ~4!

whereE0@r1a(t)# is the ground singlet term of the bipolaron, and 2En

`„a(t)… is the total energy of two polaron

separated from each other by infinity in an external intefield. Here it is taken into account that unitary operatorU1 isa translation operator, while quantitiesa(t) and b(t) arereal. The method of operator transformations maintainstranslational invariance of the Hamiltonian, including the eternal laser radiation. Equations~1! and ~4! are equivalent.We also took the following property of the translation opetor into account when deriving Eq.~4!:

exp~2 i j"p/\! f ~r !exp~ i j"p/\!5 f ~r2j!,

wherej is an arbitrary real vector.The equation that describes the polaron state in an

tense field has the form

2~\2/2m* !¹2x~r ,t !1V~r2a~ t !!x~r ,t !

5 i\]x~r ,t !/]t, ~5!

where

V~r2a~ t !!52~e2/2p«* !E ux~r1!u2dr1E q22

3exp~ iq•~r12r1a~ t !!dq. ~6!

Let us expand the exponential in Eq.~6! in a Fourier series,using

539 J. Opt. Technol. 71 (8), August 2004

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exp~ iq"a~ t !!5 (n52`

n5`

Jn~a0q!exp~ invt !, ~7!

where a(t)5a0p sinvt, a0

p52eA/m* cv, and Jn(x) is aBessel function.

For a rapidly oscillating external field, the main contrbution to the last sum comes from the term withn50. Thenthe polarization potential can be written as

V~r 2a0!5~e2/2p«* !E ux~r1!u2dr1E J0~a0•q!q22

3exp~ iq•~r12r !!dq. ~8!

Thus, to analyze the effect of an external intense field oncoupled states of two particles, it is necessary in generause Eqs.~4!, taking into account Eqs.~6!–~8!. However, inour case,a0 /a0

p52m* /mr** '1022ac4@1. Consequently,

Eqs.~4! can be replaced by

2~\2/2mr** !¹2w~r,t !1DE02@r1a~ t !#w~r,t !

5 i\]w~r,t !/]t, ~9!

whereDE00@r1a(t)#5E0@r1a(t)#22En

`(0).Potential energyDE0

0(r) for the ground singlet term othe bipolaron was calculated by a direct variational methin Ref. 1. As shown by analysis, the dependence ofground term of the bipolaron on the spacingr between thepolarons can be approximated by the following analytiformula:

DE00~r!52~e4m* /«* 2h2!~12«* /«`!a0* /r1Vef~r!.

~10!

The first term in this equation describes the long-ranscreened Coulomb repulsion of the polarons. The secterm, which determines the short-range attraction of silarly charged polarons, caused by the exchange of longitnal optical phonons, as well as by interelectron correlateffects, can be written as

Vef~r!5~e4m* /«* 2\2!$D1Cr21@A1B~r2r0!2#

3@12exp~2gr!#1a0* ~12«* /«`!/r%exp~2dr!,

~11!

where

A525.17531027, B56.88531023/a0*2,

C526.85731023/a0*2, D520.001, g50.245/a0* ,

d50.2/a0* , and r058.1731022a0* .

After applying the unitary transformations of Eqs.~3! toHamiltonianH, the energy of the ground term of the polarocan be written as

DE00~r1a~ t !!52~e4m* /«* 2\2!~12«* /«`!a0* /

@r1a~ t !#1Vef~r1a~ t !!,

539V. K. Mukhomorov

FIG. 1. Laser-dressed ground singlet term of a bipolaron.1—a050, 2—a052a0* , 3—a054a0* , 4—a056a0* , 5—a058a0* .

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Vef~r!5~e4m* /«* 2h2!$D1C~r1a~ t !!21@A1B

3~r2r01a~ t !!2#@12exp~2g~r1a~ t !!!#

1~12«* /«`!a0* /@r1a~ t !#%exp@2d~r1a~ t !!#,

~12!

wherea(t)5a0 sinvt. Parametera0522eA0 /mr** cv cor-responds to the laser intensityI 05cmr** 2a0*

2v4a02/

(8pe2). It was assumed in this case that the amplitude offorced charge oscillations in a bipolaron in a laser field wamplitudea0 is larger than the effective size of the biplaron. It thus proved to be possible to transfer the exteraction to the argument of the bipolaron’s external energyusing the unitary transformations in Eq.~3!. We shall expandthe potential energy of Eq.~12! in Fourier series, keepingonly those terms in the series expansion that make mcontributions. Then, in place of Eq.~12!, we get an effectiveinteraction potential that is time-independent but dependsthe laser radiation intensity only via parametera0 . It can berewritten as

DE00~r,a0!52~e4m* /«* 2\2!~12«* /«`!

3~2 f ~r!a0* /pr!$~12J0~ ida0!exp~2da0!!

2J0~ ida0!1C@r2J0~ ida0!22a0rJ1~ ida0!

1a02J2~ ida0!#exp~2dr!1AJ0~ ida0!

1B@~r2r0!2J0~ ida0!22a0~r2r0!

3J1~ ida0!1a02J2~ ida0!#exp~2dr!

2~AJ0~ i ~g1d!a01B@~r2r0!2J0~ i ~g

1d!a0!22a0~r2r0!J1~ i ~g1d!a0!

1a02J2~ i ~g1d!a0!#exp~2~g1d!r!!%,

540 J. Opt. Technol. 71 (8), August 2004

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HereJ0(x), J1(x), andJ2(x) are Bessel functions of imaginary argument of the first kind, of zeroth, first, and secoorders, respectively. In fact, the potential in Eq.~13! is thetime-averaged ‘‘laser-dressed’’ potential. The following weused in deriving Eq.~13!:

the definition

1

ur1a~ t !u5

1

2p E exp@ iq•~r1a~ t !!#

q2 dq, ~14!

the Jacobi–Anger expansion

exp~ iq"a~ t !!5 (n52`

`

Jn~a0q!exp~ invt !, ~15!

and also the integral representation of the Bessel functio

J0~a0q!5~1/2p!E0

2p

exp~ ia0q sinw!dw. ~16!

Thus, in terms of the approximations that are used,initial problem is reduced to the steady-state problem forstatic potential of Eq.~13!, which parametrically depends othe external field intensity, while the laser radiation intenscan be either small or large. This obviously has the effthat such bipolaron parameters as the vibrational frequethe dissociation energy, the equilibrium distance, andnumber of vibrational levels8 must also depend on the extenal field intensity.

DISCUSSION OF THE RESULTS

Figure 1 shows the variation of the laser-dressed grosinglet term of a bipolaron solvated in ammonia as a funct

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of the interpolaron spacingr for various values of the dimensionless parametera0 /a0* that characterizes the value of thexternal action. An external intense nonresonance fieldduces a monotonic shift of the ground term of the bipolaand lowers its binding energy—the bottom of the potenwell is increased in the energy scale, the repulsive barheight simultaneously decreases, and the equilibriumtancer0 between the centers of gravity of the polaronscreases. The laser-dressing effect reduces to the suppreof the interpolaron attractive forces. The dimer-dissociatenergy is thus a function of the external field intensity, athe equilibrium concentrations of polarons (np) and bipo-larons (nb) also obviously depend on the value of paramea0 . In accordance with the law of effective masses,9 theequilibrium constantK of the reactionnb�2np in a state ofthermodynamic equilibrium is determined by

n0K5nf

2

nf54S mf** kBT

2p\2 D 3S 2p\2

mb** kBTD 3/2exp~2D/kBT!

ZvibZrot,

~17!

whereZvib andZrot are the vibrational and rotational statiscal sums of the bipolaron,D is its dissociation energy, whichis determined by the energy gap between the minimum ofpotential and the maximum of the potential barrier forr.r0 . The first two factors in parentheses in Eq.~17! areassociated with successive statistical sums of polaronsbipolarons. We neglect the contributions to the statistisums associated with electron transitions.mp** andmb** arethe translational masses of the polaron and bipolaron, restively. The former ismp** 50.023ac

4m* , whereac513.4 is adimensionless electron–phonon coupling constant for etrons in ammonia. The effective translational mass ofbipolaron is a function of the interpolaron distance.1

We assume that the total number of electronsn0 in thesystem is conserved. Then we have for the total numbeelectrons the relationshipnp12nb5n0 . The system of elec-trons can be regarded as ideal ifn0L3,1, where L5e2/«skBT is the Landau length. We therefore restrict ouselves to a consideration of the concentration regionn0

<1020 cm23 and the temperature regionT'200– 240 K.For the chosen temperature and concentration interval,rotational energy levels of the bipolaron are located so clto each other that the rotational spectrum can be regardecontinuous. In this case, the rotational statistical sum isplaced by an integral and, in the approximationT.Tr , canbe written asZrot5T/Tr , whereTr5\2/2kBJ, the momentof inertia of the bipolaron isJ5mr** r0

2 , andr0 is the equi-librium distance in the bipolaron.

To compute the vibrational statistical sum, it is necessto find the intrinsic vibrational spectrum of the bipolaroFor a large number of vibrational states, the vibrational stistical sum is usually replaced by an integral. However,the case under consideration, the depth of the potentialwhena050 is only about 0.14 eV,1 while the ground term ofthe bipolaron is characterized by the presence of a potebarrier. This assumes that the number of vibrational statethe bipolaron will be limited. The vibrational statistical suwill therefore be computed by direct summation.

541 J. Opt. Technol. 71 (8), August 2004

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An exact analytical solution of Schro¨dinger’s equationwith the potential in Eq.~13! cannot be obtained, but satisfactory approximations of the solution of the radial equatican be found by the semiclassical method of a shifted 1Nexpansion.10 The analytical technique of a shifted 1/N expan-sion makes it possible to obtain the intrinsic spectrumsmooth potentials with fairly high accuracy, very close to tresults of an analytical solution. The method is based onuse of an expansion in a rapidly converging series indimensionless parameterk5N12l 2a, whereN is the num-ber of spatial measurements,l ( l 1N)\2 is the square of theeigenvalue of theN-dimensional orbital angular momentumanda is the shift parameter.

The wave function of the steady-state equation inapproximation of a shifted (1/N) expansion can be written inthe following form: wnlm(r)5(Rnl /r

(N21)/2)Ylm(q,w),whereYlm(q,w) are the spherical harmonics,n is the radialquantum number, and the principal quantum number eqn1 l 11. The radial partRnl(r) of the wave functionsatisfies10

2\2

2mr**d2Rnl

dr2 1H \2

8mr** r2 @12~12a!/k#

3@12~32a!/k#1DE0

0~r,a0!

Q J Rnl

5Wnl~a0!Rnl , ~18!

whereQ is a scale factor.10 Shift parametera is chosen so asto match the results of the technique of the shifted 1/N ex-pansion with the exact analytical results for the eigenvalof an N-dimensional harmonic oscillator.

In what follows, it is convenient in Eq.~18! to transformto the dimensionless variable

x5k1/2~r2rm!/rm , ~19!

whererm is the position of the local minimum of the effective potential of Eq.~18!.

Using variablex, we expand the potential in Eq.~18! inseries inx around x50 and in k. Then Eq.~18! can berewritten as

$2~\2/2mr** !d2/dx21~k\2/8mr** !3~113x2/k

24x3/k3/215x4/k22...!2~22a!\2~122x/k1/2

13x2/k2...!/4mr** 1~12a!~32a!h2~1

22x/k1/213x2/k2...!/8mr** 1rm2 k@DE0

0~rm ,a0!

1E~2!~rm ,a0!rm2 x2/2k1E~3!~rm ,a0!rm

3 x3/6k3/2

1...#Q%Rn,l~r!5Wn,l~a0!rm2 Rn,l~r!, ~20!

where

E~ j !~r,a0!5djDE00~r,a0!/r j ur5rm

.

541V. K. Mukhomorov

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Using the approach proposed in Ref. 10, we get from~20! the following rapidly convergent series for the eigenvues:

Wn,l~a0!5~k/rm!2@\2/8mr** 1rm2 DE0

0~rm ,a0!/

Q1g~1!/k21g~2!/k31O~1/k4!#

5Wn,l~0!~a0!1Wn,l

~1!~a0!1Wn,l~2!~a0!1..., ~21!

where the corrections to the main contribution to eneWn,l

(0)(a0), caused by quantum fluctuations and by anharmnicity effects, have an unwieldy form and are not giventhis paper. The analytical form of these corrections is shoin Ref. 10. Unlike the perturbation theory ordinarily usedquantum mechanics, this approach does not involve sming over an infinite set of unperturbed states of the systAs shown in computations using Eq.~21!, the number ofvibrational levels in the bipolaron potential is finite. Thufor example, whena050, there are only 18 vibrational levels that lie below the vertex of the repulsive potential barrThe number of vibrational states decreases as parametea0

increases, and only one vibrational level remains wha0 /a0* 58.8.

For parameter values ofa0 /a0* .7.5, the associated formation becomes quasi-stationary—the position of the vibtional levels is above the asymptotics of the ground tecorresponding tor→`, but below the top of the potentiabarrier. The action of external intense nonresonance radiathus reduces to an upward shift of the vibrational levelsenergy scale and a simultaneous reduction of their numb

Instead of the particle concentration, it is convenientuse a dimensionless quantity—the relative number of boformations,j5nb /np . From the condition of conservatioof the total number of electronsnp12nb5n0 , we find thatthe quantityj is associated with the chemical equilibriuconstant by the relationship

j50.5$11K/42@~11K/4!221#%1/2. ~22!

A sign of the existence of a smeared phase transitiothe presence of a finite derivativedj/da0 at the inflectionpoint of thej(a0) curve~Fig. 2!. The critical parameter corresponding to the inflection point isa0

(cr)57.2a0* . Functionj(a0) characterizes the kinetics of the phase transition. Twidth of the functiondj/da0 at its half-height can serve asquantitative characteristic of the smearing of the phase tsition. For a temperature of 200 K and a concentration1020 cm23 ~curve 1 of Fig. 2!, we get an estimate of th

542 J. Opt. Technol. 71 (8), August 2004

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smearing of the transmission ofDa051.8a0* , i.e., the num-ber of bound states varies in a wide interval of controlliparametera(t). In accordance with Ref. 3, such a transitiocan be regarded as a smeared phase transition of thelaron system into the polaron system. There actually arebound formations in the intensity regiona0.8.8a0* . At thesame time, at the same temperatures and concentrationthe absence of intense radiation, bound polaron formatidominate in the system of electrons.

Email: VMukhomorov@agrophys.ru

1V. K. Mukhomorov, ‘‘Singlet and triplet states of a continuum bipolarowith adiabatic and strong coupling,’’ Opt. Spektrosk.74, 1083 ~1993!@Opt. Spectrosc.74, 644 ~1993!#; V. K. Mukhomorov, ‘‘Ground andexcited states of a three-dimensional continual bipolaron,’’ Phys. StSolidi B 231, 462 ~2002!.

2V. K. Mukhomorov, ‘‘Ground state of an optical bipolaron with an intemediate strength of coupling,’’ J. Phys.: Cond. Matter13, 3630~2001!.

3B. N. Rolov,Smeared Phase Transitions~Znanie, Riga, 1972!.4V. K. Mukhomorov, ‘‘Bound–delocalized-state transition of an autolocization electron in an intense nonresonance laser field,’’ Opt. Spektr77, 869 ~1994! @Opt. Spectrosc.77, 779 ~1994!#.

5V. K. Mukhomorov, ‘‘Bipolaron states of electrons and magnetic propties of metal–ammonia systems,’’ Phys. Status Solidi B219, 71 ~2000!.

6W. C. Henneberger, ‘‘Perturbation method for atoms in intense libeams,’’ Phys. Rev. Lett.21, 838 ~1968!.

7T. C. Landgraf, J. R. Leite, N. C. Almeida, C. A. C. Lima, and L. C. MMiranda, ‘‘Spectrum of hydrogen atoms in intense laser field,’’ Phys. LA 92, 131 ~1982!.

8G. Iadonisi, V. Cataudella, G. De Filippis, and V. K. Mukhomoro‘‘Internal vibrational structure of the three-dimensional large bipolaroEurop. Phys. J.~B! 18, 67 ~2000!.

9R. Kubo,Statistical Mechanics~Mir, Moscow, 1967!.10T. Imbo and U. Sukhatme, ‘‘Improved wave functions for large-N expa

sion,’’ Phys. Rev. D31, 2655~1985!.

FIG. 2. Degree of dissociation of a bipolaron.1—n051020 cm23, T5200 K; 2—n051020 cm23, T5238 K; 3—n051019 cm23, T5200 K;4—n051020 cm23, T5238 K.

542V. K. Mukhomorov