Thermodynamic Properties of the Harmonic Oscillator and a Four Level System

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1 Appld Physcs Rsarch Vol. 3, No. ; May Thrmodynamc Proprs of h Harmonc Oscllaor and a Four Lvl Sysm Oladunjoy A. Awoga Thorcal Physcs Group, Dparmn of Physcs, Unvrsy of Uyo, Uyo, Ngra E-mal: ola.awoga@yahoo.com Akpan N. Iko Thorcal Physcs Group, Dparmn of Physcs, Unvrsy of Uyo, Uyo, Ngra E-mal: ndmko5@yahoo.com Ansua A. Ess Thorcal Physcs Group, Dparmn of Physcs, Unvrsy of Uyo, Uyo, Ngra E-mal: ass@yahoo.com Lous E. Akpabo Thorcal Physcs Group, Dparmn of Physcs, Unvrsy of Uyo, Uyo, Ngra E-mal: labo@yahoo.com Rcvd: January, Accpd: January, do:.5539/apr.v3np47 Absrac Th hrmodynamcs proprs of a quanum harmonc oscllaor and four lvl oscllaor sysms ar valuad. Ths rsuls lad o h xac valu for h nropy of h sysm whch corrsponds o h scond law of hrmodynamcs. W also show h numrcal rsuls for h harmonc and four lvl oscllaors and s n good agrmn wh h on oband bfor n h lraur. Kywords: Dnsy Marx, Four Lvl Sysm, Harmonc Oscllaor, Ha Capacy, Enropy. Inroducon Th formaon of hrmodynamcs rss on h nropy, mpraur and h hr laws of hrmodynamcs, namly h frs law, scond law and hrd law, rlang hs sa varabls (Shna 6, Landau and Lfshz 98, Kl 988). Howvr, h sascal mchancal foundaon of hrmodynamcs rls srongly on h quanum mchancal proprs of mar spcally hr characrscs a h low mpraur rgm. Howvr, h sascal mchancs gvs rs o som subly whn gong from a closd dscrpon of all dgrs of frdom, ncludng hos of larg nvronmns o a rducd dscrpon of an opn sysm whr all bah dgrs of frdom ar racd ou. In rcn ms dffrn dfnons of spcfc ha hav bn dscussd and proposd (Hangg and Ingold 6). And n addon h scond law of hrmodynamcs n h quanum rgon by calculang h nropy S for a quanum oscllaor n an arbrary ha boh a fn mpraur hav bn xamnd (Hangg and Ingold 8, Ingold al 9, Hangg and Ingold 5). Th harmonc oscllaor has playd a sgnfcan rol n physcs and chmsry. Th spcfc ha of crysals has bn calculad for boh Ensn and Dby approxmaon (Fynman 97, Ingold al 9). Th Dby hory prdcs ha h ladng rm n h ha capacy of all dlcrc solds a suffcnly low mpraur s o h ordr of T 3. Thr ar many physcal proprs ha can b calculad n h harmonc modls. Som of hs ncluds hrmal conducvy, hrmal xpanson oband from h anharmonc rms n h crysal and many ohrs (Km al 3, Hjar and d Zara ). Publshd by Canadan Cnr of Scnc and Educaon 47

2 Appld Physcs Rsarch Vol. 3, No. ; May Th smpls way of analyzng h harmonc oscllaor s o valua h paron funcon of h sysm. Many xs (Davydov 99, Mrzbachr 976, Mssah 97) hav valuad h paron funcon of harmonc sysms bcaus onc h paron funcon of a sysm s known hr hrmodynamc proprs can hn b oband. In hs papr, our objcv s o valua h hrmodynamcs of h harmonc oscllaor and a four lvl sysm by h mhod of dnsy marx and compar our rsuls wh hos oband va h paron funcon. Th organzaon of h papr s as follows; n scon w rvw h dnsy marx and scon 3 w rprsn h dynamcs of dsspav quanum sysm, n scons 4 w valua h hrmodynamc proprs of h harmonc oscllaor. In scon 5 w nroduc h dynamcs of a four lvl sysm; whl n scon 6 w calcula h hrmodynamc proprs of a four-lvl sysm whl scon 7 gvs a brf dscusson.. Dnsy Marx In quanum mchancs, h sa of an solad sysm s rprsnd by a sa vcor whch can b xpandd as a () whr a and Th nsmbl avrag of h xpcaon valu of Â s Â P A P Aˆ () whr p rprsn h probably. Th dnsy opraor s dfnd as ˆ P (3) and on oban h nsmbl avrag n rms of h dnsy opraor as (Hjar H, ) A p A ˆ A Tr ˆA ˆ (4) In ordr o normalz h dnsy opraor w wr (4) n h form Tr ˆ Aˆ A Tr ˆ (5) Th canoncal nsmbl spulas a dnsy opraor of h form ˆ Ĥ (6) whr and (5) bcoms K B T Tr ˆ Hˆ A Tr ˆ 48 ISSN E-ISSN

3 Appld Physcs Rsarch Vol. 3, No. ; May A n En En A (7) Whr Z En n and Pn z In rms of canoncal dnsy opraor, w wr E n Pˆ z H Hˆ, Z Tr Th nsmbl avrag for any obsrvabl s hn gvn by (Landau and Lfshz 98 and Davydov 99) f Tr ˆ f R Trf = (8) R Tr Havng sablshd a rlaon for h canoncal dnsy opraor, w hn oban h nrnal nrgy, nropy and Hlmholz fr nrgy n rms of hs opraor as E In Tr Ĥ (9) S K In[ˆ] B 3. Dynamcs of Dsspav Quanum Sysms k TTr F Tr ˆ In[ ˆ ] () ˆ Hˆ TTr ˆˆ s () Th wav funcon s h sa of a sysm n a pur quanum mchancs whch s an lmn of a Hlbr spac H. Howvr, for a dsspav quanum sysm, a quanum sascal formulaon s mployd snc dsspav ffcs can and do convr pur sas no sascal nsmbls. As was nod bfor, h sa of h sysm s usually rprsnd by a dnsy opraor ˆ whos dagonal lmns drmn h populaon of h nrgy gnsa whl h off dagonal lmns drmn h cohrnc bwn nrgy gnsas whch dsngush cohrn suprposon sas. In a non-dsspav sysm h m voluon of h dnsy marx ˆ ( ) wh ˆ ( ) s govrnd by (Mrzbachr 97) ˆ ( ) uˆ( ) ( ) u ( ) () whr u ˆ( ) s h m-valuaon opraon sasfyng our Schrödngr quaon d d uˆ ( ) Hˆ ( ) uˆ( ) (3) û () ˆ whr ˆ s h dny opraor. Th dnsy opraor also sasfs h quanum Louvll quaon d d ˆ ( ) H ˆ, ˆ ( ) (4) whr H s h oal Hamlonan of h sysm whch dpnds on a s of conrol fld f n Publshd by Canadan Cnr of Scnc and Educaon 49

4 Appld Physcs Rsarch Vol. 3, No. ; May H ˆ Hˆ W ( ) ˆ fn H n n wh Ĥ bng h nrnal Hamlonan and Ĥ s h nracon Hamlonan for h fld f n n for Now subsung (5) no (4) w g d d I n N. (5) N Hˆ, ( ) f ( ) Hˆ ˆ ( ) ˆ ( ) (6 ) ˆ ( ) n n n whr s h dsspaon supr-opraor. 4. Th Harmonc Oscllaor Consdrng an oscllaor wh dgr of frdom of uny, mass m and frquncy, w wr h Hamlonan of h sysm as p H m x m (7) Usng (7) w wr h paron funcon as Z Tr Hˆ (8) yldng h wll-known xprsson Z Cosc (9) wh (9) w fnd h nrnal nrgy as E kt () Fgur () shows ha hr s a lnar rlaonshp bwn h avrag nrgy and mpraur. Ths s n concordanc wh quanum mchancs wh n kt () gvng E whr h ladng rm s h ground sa nrgy. Th Hlmhoz fr nrgy s oband usng () as w n () F In (3) Th fr nrgy ncrass ngavly wh mpraur whch agan agrs wh h prdcon of hrmodynamcs as shown n fg. () Smlarly, h nropy s calculad from (9) S kb In (4) T For low mpraurs n h rgon, h nropy approachs zro lk (Landau and Lfshz 98) 5 ISSN E-ISSN

5 Appld Physcs Rsarch Vol. 3, No. ; May S T (5) Th nropy approachs a consan valu as mpraur ncrass. Ths obys h scond law of hrmodynamcs whch sa ha for a closd sysm S as shown n fgur 3. Th sam as (E. Kozlak and F. L. Lambr, 8). W also calcula h spcfc ha capacy from () as Cv T E A low mpraurs, h C v bhavs as k (6) B kbt B K BT C v k and hs characrsc s no analyc n mpraur and corrsponds o Ensn s modl for low-mpraur bhavour of h C V of a sold. For hgh mpraur ˆ w fnd (P. Hangg, 8) Cv k T (7) 4 B (8) kbt and hs shows ha h lm of fr parcl s no oband from h harmonc oscllaor by sng Th C v approachs a consan valu.. C v k B as mpraur ncrass. Ths s h Rul of Dulong and P n classcal lm. Equaon (6) can also b wrn as an nfn srs as n B kt B n Cv k n Th bhavour of ach sa s shown n fgur 5 for quaon (9). 5. Dynamcs of a Four-Lvl Sysm Th Hamlonan for a drvn for lvl sysm wh nrgy lvl s (Mrzbachr 976) Hˆ ( ) Hˆ ( ) ˆ ( ) ˆ f H f H (3 ) whr Ĥ s h nrnal Hamlonan of h sysm and Ĥ rprsn nracon Hamlonan wh ndpndn ral valu conrol flds f () and f () (9) H (3) H (3 ) wh and bng dpol momns for h ranson and h ranson frquncy Publshd by Canadan Cnr of Scnc and Educaon 5

6 Appld Physcs Rsarch Vol. 3, No. ; May ISSN E-ISSN Usng (), (4), (6) and (3) w wr our Louvll quaon n h form whr ) (34 ) ( ) ( ), ( T nm H (35) r r r r wh r dfnng h ra of populaon rlaxaon from o whl r s h ra of populaon from o and dfns h dphasng ra. 6. Th Thrmodynamcs of a Four-Lvl Sa Sysm W now consdr a sysm conssng of N ndpndn componns wh four nrnal sas dscrbd by a smpl parcl Hamlonan of h form (36) whr h sysm s n hrmal qulbrum wh a rsrvor a mpraur T. W consruc h dnsy marx of h sysm as H

7 Appld Physcs Rsarch Vol. 3, No. ; May (37) In ordr o normalz (37) w ak s rac as Tr ( ) (38) whch ylds ( ) ( ) ( ) ( ) C. (38) Smplfyng (38) gvs h normalzaon consan C as and h dnsy marx (37) bcoms C 4cosh 4cosh (39) Th nrnal nrgy of h sysm s E Tr H (4) yldng E anh (4) Th nrgy ncrass wh mpraur and approachs a consan valu as dpcd n fgur 6. Th Hlmholz fr nrgy s oband as F In 4Cosh (4) Fgur 7 shows a plo of Hlmholz nrgy wh mpraur and shows ha F ncrass ngavly and ndfnly wh mpraur. Th nropy s calculad usng () as S k B In4Cosh an (43) T Th nropy has a mnmum valu of.695k B. Ths shows ha h four lvl sa sysms s always n a sa of dsordr. Wh ncrasng mpraur, h nropy approachs a consan valu of.38k B. Ths also obys h scond law of hrmodynamcs S as shown n fgur 8. Publshd by Canadan Cnr of Scnc and Educaon 53

8 Appld Physcs Rsarch Vol. 3, No. ; May Fnally, w oband h ha capacy as Th bhavour of quaon (44) s shown n fgur 9 for C V and ncrass bwn T and dcrass bwn T. Equaon (44) can also wrn as an nfn srs as, and h bhavor s dpcd n fgur. 7. Concluson n 4 n Cv k n n kt B n, (45) W hav valuad h hrmodynamcs proprs of a quanum harmonc oscllaor and a four lvl sysm va dnsy marx mhod. Th rsulng spcfc ha capacy approachs h classcal (P-Dulong law) rsul for hgh mpraurs and gos o zro for vanshng mpraur. Th nrops of boh sysms also oby h scond law of hrmodynamcs as wll as h hrd law of hrmodynamcs. Th Hlmholz fr nrgs of h sysms ncras ngavly wh mpraur whch s h prdcon of hrmodynamcs. Fnally, w oban h numrcal rsuls for h harmonc and four lvl sysm and hy conform o h known rsuls usng h paron funcon mhods. Rfrncs Davydov A. S. (99). Quanum Mchancs, Prgamon Prss, Nw York, pp E. Kozlak and F. L. Lambr. (8). Rsdual Enropy, h Thrd law and Lan Ha. Enropy,. pp do:.339/37, hp://dx.do.org/.339/37 Fynman R.P. (97). Sascal Mchancs, Addson-Wsly. Hangg P and Ingold L. (5). Fundamnal Aspcs of Quanum Brownan Moon. Chaos. pp. -. do:.63/.85363, hp://dx.do.org/.63/ Hangg P. and Ingold G. (6). Quanum Brownan Moon and h Thrd Law of Thrmodynamcs. Aca Phys.Pol., B, 37, No. 5, Hjar H and d Zara J. O. (). Jarzynsk Equaly Illusrad by Smpl Exampls. Euro J. Phys, 3, pp Ingold G, Hangg P and Talknr P. (9). Spcfc Ha Anomals of Opn Quanum Sysms. arxv: quan-ph/ do:.3/physrve.79.65, hp://dx.do.org/.3/physrve Ingold G, Lambr A and Rynaud S. (9). Quanum Dsspav Brownan Moon and h Casmr Effc. arxv: quan-ph/ do:.3/physrve.8.43, hp://dx.do.org/.3/physrve.8.43 Km S. P, Sanana A. E and Khana F. C. (3). Dcohrnc of Quanum Dampd Oscllaors. J. Koran Phys. Soc., 43. No. 4, pp Kl C. (988). Inroducon o Sold Sa Physcs. Wly & Sons, Inc. Nw York, pp. -5. Landau L. D. and Lfshz E. M. (98). Sascal Physcs, Prgamon Prss, Nw York, pp Mrzbachr E. (976). Quanum Mchancs; Wly & Sons Inc: Nw York, pp Mssah A. (97). Quanum Mchancs Vol. ; Wlly & Sons Inc: Nw York, pp P. Hangg and G. L. Ingold. (8). Aca Phys., Pol. B37 6. Shna J. P. (6). Enropy, Ordr Paramrs and Complxy, Cladvon Prss, Oxford, pp ISSN E-ISSN

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Supplementary Figure 1. Experiment and simulation with finite qudit. anharmonicity. (a), Experimental data taken after a 60 ns three-tone pulse.

Supplementary Figure 1. Experiment and simulation with finite qudit. anharmonicity. (a), Experimental data taken after a 60 ns three-tone pulse. Supplmnar Fgur. Eprmn and smulaon wh fn qud anharmonc. a, Eprmnal daa akn afr a 6 ns hr-on puls. b, Smulaon usng h amlonan. Supplmnar Fgur. Phagoran dnamcs n h m doman. a, Eprmnal daa. Th hr-on puls s