Dynamics of Particle in a Box in Time Varying Potential Due to Chirped Laser Pulse

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1 Joural of Moder Physics, 21, 1, doi:1.4236/jmp Published Olie December 21 (hp:// Dyamics of Paricle i a Box i Time Varyig Poeial Due o Chirped Laser Pulse Absrac Brijeder Dahiya, Viod Prasad Deparme of Physics, Swami Shraddhaad College, Uiversiy of Delhi, Delhi, Idia brijeder.dahiya@gmail.com, vprasad@ss.du.ac.i Received Augus 13, 21; revised Ocober 15, 21; acceped November 19, 21 We describe a compuaioal mehod for simulaig he ime depede quaum mechaical sysem ieracig wih exeral field. I his mehod he Schrödiger equaio is solved by expadig he wave fucio i he basis se of uperurbed Hamiloia. The expasio yields a se of coupled firs order differeial equaio. For expasio coefficies, he coupled chael mehod is applied o a paricle i a box ieracig wih exeral field i he form of chirped laser pulse. The pulse shape is ae as Gaussia. We sudy he effec of differe pulse parameers i.e. chirp rae, iesiy, ceer frequecy, box legh ad laser duraio o he dyamics of he paricle. May ieresig resuls are obaied ad explaied. Keywords: Trasiio Probabiliy, Chirp Rae, Box Legh 1. Iroducio Quaum mechaics [1,2] is he fudameal base for several braches i Physics ad paricle i a box is oe of he fudameal problems of quaum mechaics. I his paper, we have sudied dyamics of a paricle i a box i ime varyig exeral field/poeial. This ime varyig field is achieved usig a chirped laser pulse. Wag ad Champage [3] have sudied he ieracio of Gaussia laser pulse wih he paricle i a box. I heir sudy hey foud ha by usig laser cere frequecy o be resoa wih he rasiio bewee firs wo saes i.e. = 1 ad = 2, rasiio probabiliy of = 2 sae is appreciable ad all oher higher sae have o zero bu egligible probabiliy. Here, we are preseig umerical simulaio of he sysem. We umerically solve he coupled differeial equaios obaied for a paricle i a box, i he presece of a chirped laser pulse. I is show ha high-efficiecy populaio rasfer is possible for several values of chirp rae ad box legh. We also explore he depedece of populaio rasfer o chirp rae. The resuls idicae ha we ca have large rasiio probabiliies for higher saes also wih he help of chirped laser pulse. Wih he developme i laser echology [4-6] i rece years, various echiques have bee developed o modulae ad eve shape he laser pulse. The mos exploied feaure of modulaed pulse is he chirp which describes he variaio of carrier frequecy wih ime. If he frequecy icreases wih he ime, he pulse is posiively chirped ad if he frequecy decreases wih ime, i is egaively chirped. Ruhma ad Kosloff [7] used egaively chirped pulse o achieve large ampliude of vibraioal moio i higher vibraioal saes of he groud elecroic surface of CsI hrough a effecive ira pulse pump dump mechaism. Cao e al. [8] have sudied ha a posiively chirped pulse is very efficie i populaio iversio. So boh posiive ad egaive chirps are impora ad ca be used as per requireme of he problem or he sysem. The selecio also depeds o he sysem wih which he laser pulse is ieracig ad oher parameers of pulse e.g. iesiy, ceer frequecy, pulse widh ec. No wor for such ype of problem has bee published earlier. Recely cohere corol for box poeial wih laser fields has bee sudied by Imre F. Bara ad PéerDombi [9]. I his paper we have used umerical mehod ad compuer simulaio o express all resuls. I Secio 2, we have described a geeral coupled chael mehod o solve he Schrödiger equaio uder ime depede perurbaio codiio. The Schrödiger equaio is reduced o a se of firs order coupled differeial equaios which are solved usig efficie ad commoly used fourh order Ruge-Kua mehod. Secio 3 briefly describe paricle i a box ad is ieracio wih chirped laser pulse. We apply he coupled chael mehod o fid he soluio. I Secio 4, we briefly describe he compuer simulaio wor. Fially i Secio 5, we describe he umerical resuls for he Copyrigh 21 SciRes.

2 B. DAHIYA ET AL. 373 rasiio probabiliies for differe saes as a fucio of ime, chirp rae, box legh ad laser ceer frequecy. Paricularly, effec of liear chirp is examied i deail. I Secio 6, we coclude he paper. 2. The Coupled Chael Mehod The coupled chael mehod [1,11] is used o solve he ime depede Schrödiger equaio. Cosider a quaum mechaical sysem havig uperurbed Hamiloia Ĥ for a eige sae wih eige eergy values E which saisfy he equaio Hˆ E (1) Here forms a complee orhogoal se of eige vecors i.e. 1, m Cosiderig he wave fucio as exp a ie (2) (3) Wihou perurbaio wave fucio will have usual ime depedece bu coefficie a is idepede of ime. Le us cosider ha perurbaio is ured o isaaeously a =. The he full Hamiloia will be described as Hˆ Hˆ V (4) Hˆ ˆ H V (5) V() is he perurbaio par (which is a fucio of ime) havig codiio V () (6) V () Now he ime depede Schrödiger equaio is (as explaied i Secio 5, box uis are used hroughou he paper) i Hˆ (7) The soluio of (7) ca be wrie as a expie (8) Here a (probabiliy ampliudes) acquire ime depedece. I he absece of exeral perurbaio, if sysem is i oe of he eige sae of Ĥ, he i will remai i ha sae forever. However by he presece of a small perurbaio, he sysem maes rasiio bewee is uperurbed eige saes. So wih he perurbaio, he probabiliy of fidig he sysem i h sae a ay ime will vary wih ime ad P () is give by P a 2 (9) Usig (4) i (7) i Hˆ V (1) Usig (1) ad (8) i (1), we ge he followig equaio a i expie Ea expie exp exp a ie E a ie V (11) Or Operaig i wih a i expie a expie V (12) from lef, we ge a i exp ie a expie V (13) a i expie a expie V (14) Le he rasiio marix eleme be V V (15) The rasiio marix eleme V may be eiher zero or o-zero depedig upo he selecio rules. Cosiderig rasiio frequecy as E E, Equaio Copyrigh 21 SciRes.

3 374 B. DAHIYA ET AL. (14) will become a i a expiv (16) Thus marix differeial equaio for co-efficiesa is a 1 V11 V12 exp i12 a1 a2 V21 expi21 V22 a2 i a V a (17) Ad by solvig his se of coupled differeial equaios we ge value of a () for differe saes ad hece probabiliy of fidig he sysem i ay paricular sae a ay ime. We ca wrie a R ii (18) where R () ad I () are real ad imagiary par of a () respecively. Usig (18) i (16) we ca have R I i i R ii cos isi V (19) Separaig real ad imagiary pars of a () we ge followig equaios R Rsi V I cosv (2) I R cos VI si V (21) Equaio (2) ad (21) ca be solved by usig umerical mehods e.g. Ruge-Kua Mehod. 3. Paricle i a Box i Chirped Laser Field Cosider a paricle i.e. elecro i a deep poeial well from which i cao escape ad loses o eergy whe i collide wih walls. So he poeial is defied as V( x) x L (22) V( x) x, x L Now he Schrödiger equaio becomes 2 2 x E x (23) 2 2m x By solvig his equaio, he uperurbed eige fucio ad he eige eergy value are 2 si x x / L (24) L ad E mL 2 (25) where L is he legh of he box ad = 1, 2, 3,, ad as i will give probabiliy of fidig he paricle everywhere equal o zero, which is o possible. Cosider ha he sysem is ieraced wih chirped laser pulse [12]. Here he laser field is defied as E E si 2 / cos (26) Here E, τ, ω ad β are he ampliude, laser duraio, laser ceer frequecy ad he chirp rae respecively. The laser ieracio wih he paricle (i.e. elecro) is defied as V x, ee. xl 2 (27) Here e is elecro charge ad zero poi poeial is chose a L/2. Now he rasiio marix V ca be wrie as * V ee x xl 2 x dx * ee 2 x x L x dx (28) ee D where D if = or + is eve, D L 2 if + is odd Usig his rasiio marix V i Equaio (2) ad (21), we ca fid exciaio/rasiio probabiliy of ay sae a ay ime uder chirped laser field. By usig liear chirp, we will ge some exciig ew resuls which is uexpeced by simple laser pulse. These are described i he resuls secio. 4. Compuaioal Aalysis The deermiaio of coefficies a () of differe saes allows us o calculae he rasiio probabiliy for various laser pulse parameers. For he prese calculaios, we have sudied he sysem wih = 1 levels. Furher, we have checed he covergece by variaio i umber of levels, ad have obaied excelle covergece wih = 1 levels. For maig he calculaios raceable we have separaed he real ad imagiary pars of he coefficies a () [see (18-21)]. Thus we obai 2 real coupled differeial equaios o be solved. Ay ime propagaor scheme ca be used o solve hese equaios. For solvig hese equaios we have used he efficie fourh order Ruge- Kua mehod by assumig ha he sysem is iiially i he groud sae. I ex secio, we discuss he resuls hus obaied. Copyrigh 21 SciRes.

4 B. DAHIYA ET AL Resuls Figure 1 shows ha rasiio probabiliy of = 2 sae is sufficiely large bu here is some probabiliy I is worh meioig, here, ha we have used he box uis hroughou he paper. I his sysem of ui, he ui of mass is elecro mass (m e ), ui of legh is 1 1 m, ui of ime is s, ui of elecric field is V/m ad ui of eergy is groud sae eergy of elecro i box of legh 1 1 m i.e ev. Cosider ha iiially he paricle is i groud sae i.e. P 1 () = 1.. A plo of rasiio probabiliy of paricle i a box of legh 2 box uis as a fucio of ime i opical cycle i.e. /τ is show i fig.1 for zero chirp, posiive chirp ad egaive chirp raes. Here laser ceer frequecy is se a ω = 3/4 i.e. resoa frequecy bewee = 1 ad = 2 sae, he laser field sregh i.e. E is 1/8 ad laser duraio i.e. τ is 4.. flow o higher saes also ad eve here is probabiliy flow o = 1 sae. A he ed of he pulse he rasiio probabiliies becomes cosa. I is clear from figure ha he variaios i rasiio probabiliies are compressed ad sreched wih he help of chirp. The rasiio probabiliies show oscillaory behavior durig he pulse. The higher saes shows more oscillaory behavior as compared o = 1, 2 or 3 sae. I Figure 2 we have preseed he variaio of rasiio probabiliy for differe saes as a fucio of laser chirp. The resuls here are ae a he ed of he pulse. I his case ceral frequecy ω has bee ae as 1/8 for a, b ad 3/8 for c, d. Also laser duraio τ is 4. for a, d ad 8. for b, c subfigures. This is useful i com- Figure 1. Trasiio probabiliy of paricle i a box i laser field as a fucio of ime i opical cycle. Box widh is m. Laser ceer frequecy is 3/4, laser duraio τ = 4 ad field ampliude E = 1/8. I Figure 1 (a, b, c) β =., Figure 1 (d, e, f) β = ad Figure 1 (g, h, i) β = Here, he ey used explais ha 1-5 are for a, d, g ad 6-1 are for b, e, h subfigures. Subfigures c, f, i refers o he correspodig fields. Copyrigh 21 SciRes.

5 376 B. DAHIYA ET AL. Figure 2. Trasiio probabiliy for a paricle i a box i chirped laser field as a fucio of chirp rae. Field sregh E = 1/2 ad box legh = 2.. The cere frequecy ω = 1/8 for a, b ad ω = 3/8 for c, d subfigures. While he pulse widh τ = 4. for a, d ad τ = 8. for b, c subfigures. The resuls are ae a he ed of he pulse. parig he resuls for variaio of rasiio probabiliy of differe saes as a fucio of chirp rae (β) for wo differe frequecies ad laser duraios simulaeously. As i ca be see from figure ha rasiio probabiliies of = 3 sae is higher for egaive chirp ha posiive chirp for τ = 4. bu i chages behviour for τ = 8., alhough rasiio probabiliy for = 1 ad = 2 saes icreases as frequecy ω chages from 1/8 o 3/8 ad rasiio probabiliy remais greaer for posiive chirp ha egaive chirp. This chage i probabiliy shows ha probabiliy also depeds o exposer ime. Furher, Figure 2 also shows ha almos complee populaio iversio ca be achieved by chirped laser pulses. There is subsaial populaio rasfer o higher saes also. Figure 3 represes he rasiio probabiliy of 5 saes as a fucio of box legh. The resuls are ae a he ed of he pulse. I is clear from he figure ha egaively chirped pulse coecs higher saes more effecively as compared o o-chirped pulse for he variaio i box legh. Also (d) par shows ha rasiio probabiliy of differe saes chages drasically for he exposed ime. I Figure 4, we compare he rasiio probabiliy for 5 saes as a fucio of laser frequecy. The resuls are ae a he ed of he pulse. I par (b) he resoace bewee = 1 ad = 2 saes occurs a heir aural frequecy bu i par (a) he chirp shifs he resoace ad also i coecs he higher order erms more efficiely. 6. Coclusios The effec of chirped laser pulse o paricle i a box has bee sudied ad we have come o he coclusio ha frequecy modulaed laser pulse is a effecive approach o ge sysem i excied sae. The rasiio probabiliy of differe saes as a fucio of differe pulse parameers ad he box legh have bee sudied. The sys- Copyrigh 21 SciRes.

6 B. DAHIYA ET AL. 377 Figure 3. Trasiio probabiliy of 5 saes for paricle i a box i chirped laser field as a fucio of box legh. Chirp rae β = (excep for (a) where β is zero), laser duraio τ = 4 (excep for (d) where τ is 8) ad field ampliude E = 1/2 (excep for (b) where E is 1/4).The resuls are ae a he ed of he pulse. Figure 4. Trasiio probabiliy of 5 saes for paricle i a box i chirped laser field as a fucio of ceral frequecy. Chirp rae i par (a) is β = ad zero i par (b). Laser duraio is τ = 4, box legh is 2. ad field ampliude E = 1/2.The resuls are ae a he ed of he pulse. Copyrigh 21 SciRes.

7 378 B. DAHIYA ET AL. em is o merely wo level shifig bu wih he help of chirped laser pulse we ca shif he sysem o higher saes, eve = 1 sae has some rasiio probabiliy (whaever small, i may be). The mai resul of he sudy is he variaio i he dyamics due o ime varyig exeral poeial ad large rasiio probabiliies for higher saes i he presece of chirped laser pulse which ca be achieved wih o-chirped pulse. The sysem remais i higher sae for appreciable ime ha may help i may physical ad chemical aspecs. 7. Acowledgemes The auhors are graeful o he uow referee for valuable suggesios for he improveme of he paper. 8. Refereces [1] D. Bouwmeeser, A. K. Eer ad A. Zeiliger, The Physics of Quaum Iformaio, Spriger Berli, NY, 2. [2] D. J. Griffihs, Iroducio o Quaum Mechaics, 2d Ediio, Pearso Preice Hall, Upper Saddle River, New Jersey, 25. [3] J. Wag ad J. D. Champage, Simulaio of Quaum Sysems wih Coupled Chael Mehod, America Joural of Physics, Vol. 76, No. 4-5, 28, pp [4] W. Koecher, Solid Sae Laser Egieerig, 6h Ediio, Spriger, New Yor, 26. [5] H. Kawashima, M. M. Wefers ad K. A. Nelso, Femosecod Pulse Shapig, Muliple-Pulse Specroscopy ad Opical Corol, Aual Review of Physical Chemisry, Vol. 46, Ocober 1995, pp [6] P. Krehli, Characerizaio of Semicoducor Laser Frequecy Chirp Based o Sigal Disorio i Dispersive Opical Fiber, Opo-Elecroics Review, Vol. 14, No. 2, Jue 26, pp [7] S. Ruhma ad R. Kosloff, Applicaio of Chirped Ulrashor Pulses for Geeraig Large-Ampliude Groud- Sae Vibraioal Coherece: A Compuer Simulaio, Joural of he Opical Sociey of America B, Vol. 7, No. 8, Augus 199, pp [8] J. S. Cao, C. J. Bardee ad K. R. Wilso, Molecular _ pulses: Populaio Iversio wih Posiively Chirped Shor Pulses, Joural of Chemical Physics, Vol. 113, No. 5, Augus 2, pp [9] I. F. Bara ad P. Dombi, Cohere Corol for he Spherical Symmeric Box Poeial i Shor ad Iesive XUV Laser Fields, Ceral Europea Joural of Physics, Vol. 6, No. 2, Jue 28, pp [1] J. P. Hase ad K. Taulbjerg, Coupled-Chael Calculaios of Parial Capure Cross Secios i Muliply Charged Io Collisios wih Hydroge, Physical Review A, Vol. 4, No. 7, Ocober 1989, pp [11] A. P. Hicma, Coupled-Chael Calculaios of Excied-Aom Collisios, Ieraioal Reviews i Physical Chemisry, Vol. 16, No. 2, 1997, pp [12] M. H. Milema Iroducio o he Theory of Laser Aom Ieracio, Pleum Press, New Yor, Copyrigh 21 SciRes.