) is the unite step-function, which signifies that the second term of the right-hand side of the

Transcription

1 Casmr nteracton of excted meda n electromagnetc felds Yury Sherkunov Introducton The long-range electrc dpole nteracton between an excted atom and a ground-state atom s consdered n ref. [1,] wth the help of perturbaton theory. The result for the nteracton potental between two dssmlar atoms s as follows d d ur 4 ωω U(R) = u e du π ( ω + u )( ω ) ur ur ur ur ur Β + u (1.1) 4 d d ω θ 4 4 ( ω ). 9 ωr ωr ωr ( ω ω) Here ω and ω are transton frequences of the excted atom and the ground-state atom correspondngly, d s the matrx element of dpole moment, R s the dstance between the atoms, θ ( ω ) s the unte step-functon, whch sgnfes that the second term of the rght-hand sde of the equaton s not equal to zero only for the case of excted atom. If the atoms are n ther ground state the second term equals zero. If the dstance R between the atoms s less than the wavelength of atomc transtons R<<λ the van der Waals case the resonant term of Eq. (1.1) reads d ω 4 d U(R) =.(1.) 3 R ( ω ω) For the Casmr-polder case of large dstances (R>>λ), the Eq.(1.1) s as follows 4 d U(R) = 9 4 d ω ( ω ω) R.(1.3) The formulas (1.) and (1.3) correspond to resonance nteracton between the atoms. The nteracton can be ether attractve or repulsve dependng on frequency detunng of the atoms. If both atoms are n ther ground state, the equatons nteracton potentals dffers from (1.) and (1.3). For the van der Waals case we wll have the London formula [3] For the Casmr-Polder case we wll have [4] d d U(R) =.(1.4) 3 R ( ω + ω) 3 U( R) = 7 ( 0) ( 0) 4π R α α,(1.5) α 0 and α 0 are the polarzabltes of the atoms at zero frequences. We should stress here where ( ) ( ) that the dfference between the Eqs. (1.5) and (1.3) s sgnfcant. The potental of the nteracton between

2 7 two ground-state atoms drops as 1/ R wth the dstance, whle the potental of the nteracton between excted and ground-state atoms drops as 1/ R. In ths paper we are gong to nvestgate the nteracton between two dssmlar atoms one of whch s excted. We wll consder the nteracton between an excted atom and medum of dlute gas. It wll be shown that the perturbaton theory s not suffcent n ths case. Drect mplementaton of the perturbaton technque results n dvergence of ntrals. We wll show that the nteracton s suppressed due to absorpton of otons by the medum. To take nto account the absorpton we use a non-perturbatve method offered n Ref. [5]. Then we wll consder the nteracton between two meda of dlute gases under thermal equlbrum. The results of calculatons wll be compared wth the Lfshtz formula. We wll demonstrate the vloton of the Lfshtz formula for the temperature hgh enough to possess excted atoms. Interacton between an excted atom and a delectrc surface Perturbaton method Here we wll consder the Casmr-Polder nteracton (R>>λ) between an excted atom and a gas meda usng perturbaton technque. If the medum s dluted enough we can take nto account par nteractons only. In ths case the potental can be found by ntratng the equaton(1.3) (Fg.1) d d ω 8πn d d ω ρdρ U = ndv = dz = 9R V ( ω ω) 9( ω ω) (1.) z 0 ( ρ + z ) 0 Fg.1 The nteracton potental (1.) s dvergent. It means that one can not use perturbaton method to fnd the nteracton potental for an excted atom near a delectrc surface for the Casmr-Polder case. Interacton between an excted atom and a delectrc surface Non-perturbatve approach Here we wll use the method of quantum Green s functons (Ref.[5]) to take nto account the absorpton of otons n the medum. The Hamltonan of the system s as follows H ˆ = H ˆ + H ˆ + H ˆ + H ˆ + H ˆ, med nt Hˆ = ˆ ψ r R dˆ Eˆ r ˆ r R dr ϕˆ r R dˆ Eˆ r ϕˆ r R dr ν ν ν ν ( ) ( ) ψ ( ) ( ) ( ) ( ) nt ˆ χ r R ˆ ˆ r r R dr ν ν ( ) d E ( ) ˆ χ( ) m ˆ ψ = ψ b, ˆ ϕ= ϕ βˆ, ( r R ) ˆ ( r R ) m

3 ˆ 1 ( ) ˆ ˆ H = ω λ αkλα kλ + kλ Ĥ = ε bˆˆ b, Ĥ = ε βˆ βˆ Frs we wll consder a system of two atoms embedded n a delectrc medum. The nteracton potental can be found as the energy shft of one atom, say atom, due to the presence of the other atom 00 ( ) =Δ = ( ε ) U R E Re M 11 0 M Where ( ) The densty matrx s ε s the mass operator of atom [5]. ( x,x' ) = Tˆ ˆ ( x) ˆ ( x' ) Sˆ, wth scatterng matrx ˆ = ˆ ( 1) l ˆ ( ) ρ ψ ψ 1 c 1 c Sc Tcexp Hntl t dt l = 1, c. fter calculatons and standard substtuton 1 1 Nk λ +, whch takes nto account the thermal otons, we fnd U = U + U, nr r νν 1 νν ( ) ' ( ) ζ n U R nr = T ' αg ζn αe ζn ζn e , (1.7) n= 0 ζn R ζn R ζn R ζn R ζn R ζ = π nt n ( ) d ω ω ω ω d U(R) r = Coth 9 T γ ( ω ω) γ R + + exp θ ω 4 4 ωr ωr ωr The lfe-tme of the oton for the case of dlute gas medum s τ ( ) (1.8) 1 3γ = γ =. 8πω d n The non-resonance term (1.7) concdes wth the one obtaned n paper [] for the nteracton between two ground-state atoms. The resonance term generalzes the Power formula (1.3) for the case of two atoms n an absorbng medum. Now one can see that f the formula (1.8) s substtuted nto (1.) the result wll be dvergent no more. Interacton between two meda of excted atoms Here we wll consder the nteracton between two meda of dlute gases separated by a dstance L>>λ.

4 To fnd the force we wll ntrate the equatons (1.7) and (1.8) over the volume. To smplfy the calculatons we wll use the followng model. We wll omt the exponent n (1.8), but the wdth of the nteractng meda wll be restrcted by the free pass of a oton n these meda 1 3cγ L = cτ = cγ =. 8πω d n fter ntraton we should dfferentate the potental wth respect to L to fnd the force. ( ) ( ) F L = F L + Lf (( ω ω) ( γω) ) ( ) 8π d d ω ω ω ω n n ω ω c + + exp + exp T T ω ω 3 T ω 3 T ω L + L L + L ωe coth ωe coth ( L + L) Log LLog. T T L + L L The Lfshtz formula for the same case of hgh temperatures reads [7] πtd d ω ω F ( L) = tan tan n n Lf 3 9Lωω T T.(1.10) In Fg. we represent the dependence of the normalzed force on the dstance L. We see that the dependence obtaned wth the help of quantum electrodynamcs results n ether attracton or repulson dependng on the dstance between the meda. (1.9) Fg. The dependence of the normalzed force on dstance. The upper curve corresponds to the Lfshtz formula (1.10), the lower curve corresponds to Eq. (1.9). ( ω =1.1 ω,t = 04. ω ) The temperature dependences of the forces are gven n Fg.3 and Fg.4.

5 Fg.3 The typcal temperature dependence of the force obtaned wth the help of the Lfshtz formula (1.10) Fg.4 The typcal temperature dependence of the force obtaned wth the help (1.9) Conclusons We consdered Casmr-Polder nteracton between an excted and a ground-state atom. We showed that the perturbaton theory results n a dvergence n the case of nteracton between an excted atom and a gas medum. Usng non-perturbatve method, we found the nteracton potental between two atoms n a delectrc medum. The suppresson of nteracton due to absorpton of otons n the medum s demonstrated. It results n convergence of the ntrals for an excted atom nteractng wth a medum. Then we consdered a case of Casmr nteracton between two gases. The results of quantum electrodynamcs are compared wth the Lfshtz formula. References 1. E.. Power and T. Thrunamachandran, Phys. Rev. 47, 539 (1993).. E.. Power and T. Thrunamachandran, Phys. Rev. 51, 30 (1995).

6 3. F. London, Z. Phys. 3, 45 (1930). 4. H.. G. Casmr and D. Polder, Phys. Rev. 73, 30 (1948). 5. Yury Sherkunov Phys.Rev. 7, (005). 7. E. M. Lfshtz, Sov. Phys. JETP, 73 (195).. G.H.Goedecke and Roy C. Wood. Phys.Rev., 0, 577 (1999)