Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt

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1 Appled Mathematcs Volume 2, Artcle ID 4539, pages do:.55/2/4539 Research Artcle A Treatment of the Absorpton Spectrum for a Multphoton V -Type Three-Level Atom Interactng wth a Squeezed Coherent Feld n the Presence of Nonlneartes F. K. Faramawy Mathematcs Department, Faculty of Scence, Al-Azhar Unversty, Nasr Cty, Caro 884, Egypt Correspondence should be addressed to F. K. Faramawy, fkf54@yahoo.com Receved 5 May 2; Revsed 9 October 2; Accepted 27 October 2 Academc Edtor: Ch. Tstouras Copyrght q 2 F. K. Faramawy. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We study the nteracton of a three-level atom wth a sngle mode feld through multphoton transton n a cavty, takng explctly nto account the exstence of forms of nonlneartes of both the feld and the ntensty-dependent atom-feld couplng. The analytcal forms of the absorpton spectrum s calculated usng the dressed states of the system. The effects of photon multplctes, mean photon number, detunng, Kerr-lke medum, and the ntensty-dependent couplng functonal on the absorpton spectrum are analyzed.. Introducton The spectrum of spontaneous emsson of a V -confguraton three-level atom, whose two upper levels are coupled by a classcal feld and ther energy spacng s much larger than the spontaneous emsson wdths has been nvestgated. It has been shown that the spontaneously generated nterference can nduce the spectrum to exhbt sx peaks and depend on the phase of the classcal feld. The effects of a broadband squeezed vacuum on three-level atoms at dfferent confguratons Λ, V,andΞ confguratons have also been nvestgated 2 5. Further work has also been done to study the resonance fluorescence spectra of three-level atoms nteractng wth two coherent lasers and two ndependent squeezed vacuum 3 5. The fluorescence spectrum for a strongly drven three-level system n whch one of the two one-photon transton s coupled to a fnte-bandwdth squeezed vacuum feld has been examned 4. Quantum nterference effects n resonance fluorescence and absorpton spectra of a V -type three-level atom damped by a broadband squeezed vacuum studed n 6.

2 2 Appled Mathematcs In recent years, there has been tremendous progress n the ablty to generate states of the electromagnetc feld wth manfestly quantum or nonclasscal characterstcs expermentally 7 9. Because the squeezed coherent states are expermentally avalable, we use t n ths paper. Squeezed states of lght are nonclasscal states for whch the fluctuatons n one of two quadrature phase ampltudes of the electromagnetc feld drop below the level of fluctuatons assocated wth the vacuum state of the feld. Squeezed states therefore provde a feld whch s n some sense queter than the vacuum state and hence can be employed to mprove measurement precson beyond the standard quantum lmts 7. The goal of ths paper s to shed some lght on the absorpton spectrum for a general three-level system. The model we wll consder s consstng of a sngle V -type three-level atom nteractng wth a multphoton one mode feld n a perfect cavty, ncludng acceptable knds of nonlneartes of both the feld and the ntensty-dependent atom-feld couplng. To reach our goal t s more convenent to use exact expresson for the untary operator U t n the frame of the dressed state formalsm. Ths wll be consdered n Secton 2. In Secton 3 we employ the analytcal results obtaned, by usng the fnte double-fourer transform of the two-tme feld correlaton functon, to fnd an analytcal expresson for the absorpton spectrum. By a numercal computaton, we examne the nfluence of photon multplctes, mean photon number, detunng parameters, the functonal dependence of the couplng as well as the nonlnearty parameter on the absorpton spectrum n Secton 4. Fnally, the conclusons are summarzed n Secton Formulaton of the Problem The Hamltonan of the system n the rotatng-wave approxmaton s of the form ħ H H H n, 3 H ω σ, Ωâ â. 2. The operators â and â are the boson operators for the feld satsfyng â, â. Where ω,ω 2,andω 3 are the atomc energy levels ω >ω 2 >ω 3 and Ω s the feld frequency, wth the detunng parameters Δ and Δ 2 gven by Δ kω ω ω 3, Δ 2 kω ω 2 ω The nteracton part of the Hamltonan n the presence of an arbtrary nonlnear medum, va multphoton process k can be wrtten as ( ) ( ( ) H n R â â λ (σ 3 f â â )â k â k f â â )σ 3 ( ( ) λ 2 σ 23 f 2 (â â )â k â k f 2 â â )σ

3 Appled Mathematcs 3 ω ω kω kω ω 3 3 Fgure : Energy level dagram for a V -type three-level atom wth k-photon detunng Δ, Δ 2. R â â and f â â are Hermtan operators functons of photon number operators, such that λ f â â and λ 2 f 2 â â represent arbtrary ntensty-dependent atom-feld couplng, whle R â â denotes the one-mode feld nonlnearty whch can model Kerr-lke medum nonlnearty as wll be dscussed later. The operators σ satsfy the followng commutaton relatons σ,σ κl σ l δ κ σ κ δ l, â, σ Fgure. The ntal state Ψ AF of the combned atom-feld system may be wrtten as Ψ AF Ψ A Ψ F, 2.4 where Ψ A the ntal state of the atom and Ψ F Θ Θ s the ntal state of thefeld.thentalstate Θ p n n where the probablty ampltude p n s defned n the usual manner as p n n Θ. The tme evoluton between the atom and the feld s defned by the untary evoluton operator generated by H. ThusU t s gven by U t exp Ht. 2.5 Ths untary operator U t s wrtten as k ( ) U t exp E s Φ 3 t s Φ s s 3 n k ( exp E n ) Ψ n t Ψ n, 2.6 where, 2, 3, and the egenvalues θ E s 3 ω 3 Ωs R s, s,,...,k, ( ) X 2 3X 2 cos ( ) θ, E n X cos 9X X 2 2X 3 27X 3 2 ( X 2 3X ) 3/2 2 ( )2π 3, 2.7

4 4 Appled Mathematcs wth X r r 2 r 3, ] X 2 [V 2 V 2 2 r r 2 r r 3 r 2 r 3, X 3 r 2 V 2 r V 2 2 r r 2 r 3, r ω Ωn R n, r 2 ω 2 Ωn R n, 2.8 r 3 ω 3 Ω n k R n k, V λ f n n k! n k!, V 2 λ 2 f 2 n, n! n! and Φ s, Ψ n are the dressed states of the system assocated wth the egenvalues E s,, 2, 3 E n 3 and Φ s s, 3, Ψ n α n s,,...,k, n, β n n, 2 γ n n k, 3, 2.9 where α n β n γ n M ( ( ) V r 2 E n ( ) V 2 r E n r E n )( r 2 E n ) ( ) M r E n 2 ( ) r 2 E n 2 ( ) V 2 r 2 E n 2 ( ) V r E n 2. Havng obtaned the explct form of the untary operator U t, the egenvalues and the egenfunctons for the system under consderaton, we are therefore n a poston to dscuss some propertes related to the atom or the feld, especally the absorpton spectrum.

5 Appled Mathematcs 5 3. Absorpton Spectrum In ths secton we calculate the absorpton spectrum as 5, T T A ν Γ dt dt 2 exp Γ ν T t Γ ν T t 2 σ 3 t σ 23 t, σ 3 t 2 σ 32 t 2, 3. where T s the nteracton tme and Γ s the bandwdth of the flter. The Fourer transform, of the two-tme commutator averaged dpole-dpole correlaton, s drectly related to the absorpton spectrum, where σ 3 t σ 23 t, σ 3 t 2 σ 32 t 2 σ 3 t σ 23 t σ 3 t 2 σ 32 t 2 σ 3 t 2 σ 32 t 2 σ 3 t σ 23 t. 3.2 The frst term σ 3 t σ 23 t σ 3 t 2 σ 32 t 2 s assocated wth emsson processes whle the second term σ 3 t 2 σ 32 t 2 σ 3 t σ 23 t, whch has an opposte sgn, corresponds to stmulated absorpton. In order to calculate the absorpton spectrum we need to calculate the two-tme commutator of 3.. The probe absorpton coeffcent s gven by the dfference between a stmulated absorpton and the emsson component. However, n what follows we analyze the case when the atom s ntally prepared n t s most excted state, and consderng the feld s beng ntally a squeezed coherent state. After carryng out the varous operatons we obtan the absorpton spectrum n the form: k A ν Γ s 3 p s 2 s α 2 ( Υ [ α s 2 α s β s E s 3,E s α s β s ) β s 3 3 Γ p n 2 n α 2 n k γ 2 ( Υ n k [ α n 2 α n β n 3 3 Γ p n 2 n α n [ α n 2 α n α n β n 2 n γ 2 Υ β n β n ( E n 2] E n k 2] α n β n,e n,e n ) ) β n 2], 3.3 where Υ ( x, y ) [ ( ) ] exp 2ΓT 2 exp ΓT cos ν x y T Γ 2 ( ν x y )

6 6 Appled Mathematcs Thus the tme-averaged absorpton spectrum conssts of resonant structures whch arse from transtons among dfferent dressed states. The fnal structure of the tme-averaged absorpton spectrum wll depend on the form of the nput photon dstrbuton p n. Due to the quantum nterference between component states the oscllatons n the cavty feld become composed of dfferent component states. 4. Numercal Results and Dscusson On the bass of the analytcal soluton presented n a prevous secton, we wll study numercally the absorpton spectrum n a a squeezed coherent ntal feld. The photon number dstrbuton for a squeezed coherent state 7, can be wrtten as ( ) P n 2 tanh r n s 2 n n! cosh r H ε 2 [ n exp ε 2 tanh r Re ε 2], 2 cosh r snh r 4. where ε α cosh r α snh r, α α exp ς,andh n s the Hermte polynomal. We suppose here the mnor axs of the ellpse, representng the drecton of squeezng, parallel to the coordnate of the feld oscllator. The ntal phase ς of α s the angle between the drecton of coherent exctaton and the drecton of squeezng. The mean photon number of ths feld s equal to n α 2 snh 2 r. Puttng r we get the photon dstrbuton for an ntal coherent state wth n α 2, whereas for α the photon dstrbuton for an ntal squeezed vacuum state wth n snh 2 r s recovered. The latter dstrbuton s oscllatory wth zeros for odd n. In general, the spectrum attans negatve and postve values: a negatve value represents amplfcaton and a postve value represents absorpton. Snce the expresson n 3.4 depends on the dfference n populaton between the upper and the lower energes of the dressed states, one can easly predct that at hgh ntenstes the absorpton spectrum exhbts an equal number of absorbng and amplfyng components. As we mentoned before the total components appearng spectrum are proportonal to the dfference between the absorpton and emsson processes occurrng durng the transt tme T. Transtons to atomc levels and the dressed states exhbt a structure n absorpton spectrum whose components beng ether absorbng or amplfyng. 4.. Effect of Multplcty and Mean Photon Number For k, we observe that the central peak surrounded by two hole-burnng and symmetrc sde peaks gvng absorpton and hole-burnng gvng amplfcaton degenerate around the central lne for small mean photon number as shown Fgure 2 A - a. But, the stuaton s changed for k >, where the central structure dsappear, and we observe a number of symmetrc sde peaks and hole burnng, whch demonstrate absorpton and amplfcaton, respectvely, degenerate around the central lne, see Fgures 2 A - b,c. As the mean photon number and varance ncreases not only the depth of all sde amplfcaton and the heght of all sde absorpton peaks decrease but also all amplfcaton and absorpton peaks move away from the central lne see Fgure 2 B. Furthermore, not only the range of the spectrum and the number of the spectral component ncreases as the mean photon number ncreases but also as k ncreases compare frames n Fgure 2. Also, the depth of all sde amplfcaton and

7 Appled Mathematcs (A) Squeezed coherent feld (B) Squeezed coherent feld Fgure 2: The evoluton of the functon A ν n a perfect cavty as a functon of ν kω / λ λ 2 wth λ,2, Δ,2, χ, Γ., ς, f,2 n, T and a k, b k 2, c k 3wth r, α, 2 r.7,α 5. the heght of all sde absorpton peaks decrease as soon as the mean photon number ncrease for all values of k Effect of Detunng In Fgure 3 we dsplay the absorpton spectrum A ν for dfferent values of the detunng parameters Δ, Δ 2. We observe that the detunng plays a crucal role n the behavor of the absorpton spectrum. For all values of k Fgure 3 A exhbts asymmetrc amplfcaton and absorpton elements for small values of the detunng parameters Δ and Δ 2. Whle nterestng modfcatons are observed, for large detunng parameters, there s strong absorpton wth mnmal amplfcaton, and the peaks are shfted to rght sde, and we have only strong absorpton elements, see Fgure 3 B - a. Ths phenomena dsappears clearly as k ncrease, where at k greater than one we stll note strong absorpton and amplfcaton n rght sde whle mnmal absorpton and amplfcaton n the left sde, Fgures 3 B - b,c. Fnally, the heghts of the spectral component depend on the values of the detunng parameters and the photon multplctes k Effect of Kerr Medum Now we wll turn our attenton to the effect on the spectrum A ν of the nonlnearty of the feld wth a Kerr-type medum due to the term R n beng taken n the form χn n, where χ s related to the thrd-order nonlnear susceptblty. In fact the optcal Kerr effect s one

8 8 Appled Mathematcs (A) Squeezed coherent feld (B) Squeezed coherent feld Fgure 3: The same as Fgure 2 B but wth Δ 5, Δ 2 4, 2 Δ Δ 2 2. of the most extensvely studed phenomenon n the feld of nonlnear optcs because of ts applcatons. The addton of the Kerr-lke medum parameter to the problem adds asymmetry to the spectrum as can be seen from comparson of the cases consdered n Fgure 4 wth Fgure 2 B. Also, t s to be remarked that the ampltude of the sde-bands decrease. As χ ncreases the amplfcaton and absorpton element n the left sde ncrease and emerge whle on the rght sde hand elements decrease, and the central lne s pushed away to the left sde. For large values of χ there are no amplfcaton and the spectrum tends to a sngle absorpton element n the left hand sde see Fgure 4 B. Ths phenomenon dsappears clearly as k ncreases. Whle for large k.e., k 3 we stll observe a number of strong amplfcaton and absorpton peaks n left sde, whle mnmal amplfcaton and absorpton peaks redstrbuted n rght sde, see Fgure 4 B - c. Fnally, t s nterestng to note that Kerr medum has an effect opposte to the effect of the detunng compare Fgure 4 to Fgure Effect of Intensty-Dependent Couplng Functonal Comparng Fgure 2 B, where we set the ntensty couplng functonal f n f 2 n wth cases consdered n Fgure 5 A where we take f n f 2 n n, we note that the shape of the spectrum s changed on both sdes of the central frequency. Also, there s no absorbton or amplfcaton at the central lne. Furthermore, the range of the spectrum s larger and numbers of absorpton and amplfcaton component are ncreased, and the dstance between them becomes larger than the case of f n f 2 n compare Fgure 5 A to Fgure 2 B. Ths due to, n ths case the Rab frequency s larger than that n the case f n f 2 n. But the stuaton s completely changed, when we take f n f 2 n / n. Snce the Rab frequency n ths case s smaller than that n f n f 2 n, hence the range of

9 Appled Mathematcs (A) Squeezed coherent feld (B) Squeezed coherent feld Fgure 4: The same as Fgure 2 B but wth χ. and 2 χ (A) Squeezed coherent feld (B) Squeezed coherent feld Fgure 5: The same as Fgure 2 B but f n f 2 n n, 2 f n f 2 n / n.

10 Appled Mathematcs the spectrum s smaller than that when we take f n f 2 n. Also, we observe that, the shape of the spectrum s completely changed where for k we note a deep hole burnng gvng a strong amplfcaton at the center of the spectrum surrounded by one absorpton and one amplfcaton peak, see Fgure 5 B - a. Ths central hole burnng dsappearng for k>, where the spectrum becomes rch and exhbts swtch from amplfcaton to absorpton as k ncrease see Fgures 5 B - b,c. Fnally, absorpton spectrum can be controlled by choosng a rght form of ntensty-dependent couplng functonal. 5. Concluson We have nvestgated the absorpton spectrum for a multphoton nteracton wth a V -type three-level atom, takng nto account arbtrary forms of nonlneartes of both the feld and the ntensty-dependent atom-feld couplng. The spectrum s calculated when the feld s ntally n a squeezed coherent state. We have explored the nfluence of varous parameters of the system on the absorpton spectrum. It s observed that as the photon multplctes number ncrease, the number of allowed transton between the dressed states ncreases and hence number of absorpton and amplfcaton peaks appearng n the spectrum ncreases as k ncreases; the poston of absorpton and amplfcaton peaks s assocated wth not only the photon number n and the photon multplcty number k but also the ntenstydependent atom-feld couplng λ f n ; the heghts of the spectrum components becomes shorter and the dstances between them s larger as the mean photon number ncreases; v the symmetry shown n the standard three-level atom model for the spectra s no longer present once Kerr effect or detunng s consdered; v the effect of detunng on the absorpton spectrum s twofold. The frst effect s the shft of the spectrum to the rght hand sde. The second effect s the dependence of the ampltudes and heghts of the peaks on Δ ; v the Kerr medum has an effect opposte to the effect of the detunng, where the earler has shorter elements. Also, the heghts and wdths of the peaks not only depend on the photon multplcty but also depend on the value of χ. Consequently, changes n the detunng and the Kerr medum parameters can show n the spectra, and hence the heghts of the peaks; ther shfts and wdths are altered when compared wth the case of resonance; v the strong feld effects can be produced by choosng the rght parameters for these nonlneartes. References C. Hooer, G. X. L, K. Allaart, and D. Lenstra, Spontaneous emsson n a V-type three-level atom drven by a classcal feld, Physcs Letters, Secton A, vol. 263, no. 4 6, pp , Z. Fcek and P. D. Drummond, Three-level atom n a broadband squeezed vacuum feld I. General theory, Physcal Revew A, vol. 43, no., pp , S. Smart and S. Swan, Three-level atom n a squeezed vacuum II. Resonance fluorescence, Journal of Modern Optcs, vol. 4, no. 6, pp , 994.

11 Appled Mathematcs 4 B. J. Dalton, M. R. Ferguson, and Z. Fcek, Resonance fluorescence spectra of three-level atoms n a squeezed vacuum, Physcal Revew A, vol. 54, no. 3, pp , B. J. Dalton, M. Bostcky, and Z. Fcek, Probe absorpton spectra for drven atomc systems n a narrow bandwdth squeezed vacuum, Physcal Revew A, vol. 53, no. 6, pp , F. Carreño,M.A.Antón, and O. G. Calderón, Quantum nterference effects n resonance fluorescence and absorpton spectra of a V-type three-level atom damped by a broadband squeezed vacuum, Optcs Communcatons, vol. 22, no. 4-6, pp , D. F. Walls and G. J. Mlburn, Quantum Optcs, Sprnger, Berln, Germany, 2nd edton, C. M. Caves and B. L. Schumaker, New formalsm for two-photon quantum optcs I. Quadrature phases and squeezed states, Physcal Revew A, vol. 3, no. 5, pp , V. V. Dodonov, Nonclasscal states n quantum optcs: a squeezed revew of the frst 75 years, Journal of Optcs B. Quantum and Semclasscal Optcs, vol. 4, no., pp. R R33, 22. J. Peřna, Quantum Statstcs of Lnear and Nonlnear Optcal Phenomena, D. Redel, Dordrecht, The Netherlands, 984.

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