The Phase Probability for Some Excited Binomial States

Transcription

1 Egypt. J. Sl., Vl. 5, N., 3 Th Pha Prbability fr S Excitd Biial Stat. Darwih Faculty f Educati, Suz Caal Uivrity at Al-Arih, Egypt. I thi papr, th pha prprti i Pgg-Bartt frali ar cidrd. Th pha ditributi i calculatd ad dicud fr th xcitd biial tat, th v-xcitd biial tat ad th dd-xcitd biial tat.

2 . Darwih 4. Itrducti: Claically thr ar tw thri fr light: wav Huyg ad crpucular Nwt. Sic th advt f quatu chaic th ccpt f th pht had b itrducd. Light cit f pht carryig rgy quata prprtial t th frqucy E hω, ad tu prprtial t th wav ubr p hc. Th quatu dcripti f th tat f light dpd th quatu dcripti f th pht. Th tat that dcrib pht i th Fc ubr tat, which i a igtat f th pht ubr pratr ˆ aˆ aˆ, i.. a ˆ aˆ whr â, a ˆ ˆ ad â, a ar th aihilati ad crati pratr f pht, rpctivly. Hwvr, athr tat wa itrducd by Glaubr 963 fr xapl [, ] aly th chrt tat which dcrib a fild with a fixd pha whil th ubr f pht i t fixd. Hwvr, thr i a avrag valu fr thi pht ubr. Th ubr f pht ha a Piia ditributi. Thi tat th chrt tat α, ha th pht ubr ditributi P, whr i th a pht ubr α, α i a cplx ubr, wh dulu i th aplitud f th fild d, ad it pha i th pha f th tat. Thi tat dcrib vry clly th lar fild. Thrtical dcripti fr thr tat fllwd. A tat wa itrducd which bridg th gab btw th Fc tat ad th chrt tat by taig athatical liit. Thi tat i th biial tat BS [3], whr th prbability ditributi fucti fr fidig $$ pht i giv by th biial ditributi, P fr fr > A ralizati f uch tat ca b thught f a a -lvl at i it xcitd tat that it a pht t g dw t it grud tat, if th prbability f dtctig a pht i th failur t dtct uch pht i. Hc th

3 Egypt. J. Sl., Vl. 5, N., 5 prbability t dtct cut ut f xprit i giv by quati. It i ay t that a th P, P,, whil wh th P, P,, whil wh, uch that a fixd ubr, w gt P ditributi. Th tat that dcrib thi ituati i th BS [3] by th Piia, giv, Athr tat itrducd t itrplat btw th thral ad th chrt tat, i th gativ biial tat NBS [4-6]. A furthr xapl, th gralizd gtric tat itrplat btw th ubr tat ad th -pur chatic tat [7-9]. Furthrr, th v dd BS itrplat, btw th v dd chrt tat ad th v dd ubr tat [,]. It wa rprtd that udr uitabl cditi, uprpiti f chrt tat culd b prducd if a chrt tat i allwd t prpagat thrugh a aplitud dipriv diu []. Thu quatu uprpiti f BS ca b prducd i th a way, ic th BS td t a chrt tat a a liitig ca. It wuld b itrtig t rfr t th xcitd biial tat EBS f th radiati fild, which ca b gratd by rpatd applicati f th pht crati pratr BS [3]. Thy rduc t Fc tat ad xcitd chrt tat ECS i crtai liit ad ca b viwd a itrdiat tat btw Fc tat ad th ECS [3]. Al, th dd-xcitd biial tat OEBS ad th v xcitd biial tat EEBS f th radiati fild which ar itrducd by rpatd applicati f th pht crati pratr EBS. Th tat itrplat btw th dd v ubr tat ad th dd v diplacd Fc tat [4]. Th ai f thi wr i t tudy th pha prprti fr bth EBS, EEBS ad OEBS i th Pgg-Bartt frali [5, 6]. I cti th pha ditributi fucti i itrducd. I cti 3 th pha ditributi rlatd t th EBS i calculatd. Al, fr th tat f EEBS ad OEBS th pha ditributi fucti ar calculatd i cti 4, 5. Fially cclui i giv i cti 6.

4 . Darwih 6. Th pha ditributi fucti: Th ti f th pha i quatu ptic ha fud rwd trg itrt bcau f th xitc f pha-dpdt quatu i. I thi cti, th pha prprti uig th Pgg-Bartt thd [5, 6] ar tudid. Thi i bad th pha tat Θ, which ar dfid a whr Θ xp i, 3 π ;,,..., 4 Th valu f i arbitrary; it idicat a pcific ba t f utually rthgal pha tat. I fact th pha tat Θ ar igtat f Hritia pha pratr Φˆ giv by Φˆ Θ Θ. 5 Th tat f th fr b b xp iψ 6 i calld a partial pha tat [5], whr b ar ral ad pitiv ad Ψ i a pha. Fr quati 3, 6, ca calculat th xpctati valu ad th variac fr th pha pratr Φˆ with rpct t th partial pha tat; w hav Φˆ Ψ 7 Φˆ π b b > Th pha prbability ditributi fr th partial pha tat i giv a P Θ b 9

5 Egypt. J. Sl., Vl. 5, N., 7 Sic th dity f pha tat i quati 9 rduc t π, thu i th ctiuu liit, P b [ ] b c π > I what fllw calculat thi fucti fr diffrt tat. 3. Th pha ditributi fucti f EBS: Th EBS i dfid a [3],, C with th ralizati ctat giv by ad C 3 Hr ad ar itgr. i i gral cplx with. By uig quati. ad th pha ditributi fucti fr thi tat i giv by P π c[ ] 4 * I Fig., P giv by quati. 4 i plttd agait th paratr, fr 5 ad. I thi figur w ca that thr i ly trtchig pa alg th axi at. At, th P, ad th pha i π

6 . Darwih 8 lt. But at icra, th pha tart t build up, ad th ifrati abut th pha ca b attaid. At bc, th P. π Fr Fig., w ca that a icra with ctat, th th axiu valu f P, at v t lwr valu f. Kpig ctat ad varyig, th axiu valu f P at icra lightly a icra. 4. Th pha ditributi fucti f EEBS: Th EEBS i itrducd thrugh th fllwig dfiiti uch that B,, 5 ar alway v. Thr ar th fllwig ca i v i..

7 Egypt. J. Sl., Vl. 5, N., 9,, B 6 whr B 7 With big th largt itgr l tha r qual t. i th ralizati ctat f EEBS fr v. B i th prbability aplitud f v xcitd biial ditributi. Th ralizati ctat f thi tat ca b writt a 4 8 By applyig quati ad 5, th pha ditributi fucti fr thi tat i giv by

8 . Darwih [ ] π c * P 9 ii dd i..,, B B i th ralizati ctat f EEBS fr dd. Al B i th prbability aplitud f v xcitd biial ditributi. Th ralizati ctat f thi ca i giv by 4 By applyig quati ad, th pha ditributi fucti fr thi tat rad, [ ] π c * P 3 I Fig. 3, P giv by quati 9, 3 i plttd agait th paratr, fr 8 ad. W ca that thr ar tw pa alg th axi at ad π. Calculati f th ffct f th valu f ad th axiu valu f P ar uarizd. I gral, th a trd i ticd a th arlir ca f th EBS.

9 Egypt. J. Sl., Vl. 5, N., 5. Th pha ditributi fucti f OEBS: Th OEBS ca b itrducd thrugh th fllwig dfiiti C,, 4 uch that ar alway dd [4]. Siilarly, w hav th fllwig tw cai v i.. whr C,, C 5 6 i th ralizati ctat f OEBS fr v, giv by

10 . Darwih 4 7 By uig quati ad 5, th pha ditributi fucti fr thi tat i giv by [ ] π c * P 8 ii dd i..,, C 9 whr C 3 i th ralizati ctat f OEBS fr dd, giv by 4 3 By uig quati ad 9, th pha ditributi fucti fr thi tat i giv by [ ] π c * P 3 I Fig. 4, P giv by quati 8, 3 i plttd agait th paratr, fr 5 ad. Agai, w that thr ar tw trtchig pa alg th axi at ad π. Th ffct f th valu f ad th axiu valu f P i th a with th ca f EBS ad EEBS.

11 Egypt. J. Sl., Vl. 5, N., 3 6. Cclui: I thi articl, th pha ditributi fucti i th f Pgg- Bartt frali ha b calculatd ad plttd fr th ca f th EBS, EEBS ad th OEBS. Thi ivtigati hw a l f pha ifrati P a r which rflct th fact f havig a Fc π tat. But fr < < pa accur arud, fr th ca f EBS, ad tw pa at ad ± π, fr th ca f EEBS ad OEBS which a that th pha i built up by addig r Fc tat. Thi a that th tat ar partially chrt pha tat.

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