Spontaneous and stimulated emission tuning characteristics. of a Josephson junction in a microcavity

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1 Sptaeus ad stimulated emissi tuig characteristics f a sephs jucti i a micrcavity Adrea T. seph, Rbi Whitig ad Rger Adrews Departmet f Physics, Uiversity f the West dies, St. Augustie, Triidad ad Tbag radrews@fas.uwi.tt We have ivestigated theretically the tuig characteristics f a sephs jucti withi a micrcavity fr e-pht sptaeus emissi ad fr e-pht ad tw-pht stimulated emissi. Fr sptaeus emissi, we have established the liear relatiship betwee the magetic iducti ad the vltage eeded t tue the system t emit at resat frequecies. Fr stimulated emissi, we have fud a scillatry depedece f the emissi rate the iitial Cper pair phase differece ad the phase f the applied field. Uder specific cditis, we have als calculated the values f the applied radiati amplitude fr the first few emissi maxima f the system ad fr the first five jucticavity resaces fr each prcess. Sice the emissi f phts ca be ctrlled, it may be pssible t use such a system t prduce phts demad. Such surces will have applicatis i the fields f quatum cryptgraphy, cmmuicatis ad cmputati. OCS cdes: ( ) sephs jucti; micrcavity; sptaeus emissi; stimulated emissi. trducti Sice the earlier half f the twetieth cetury, cfied atms have bee studied bth theretically ad experimetally. The iteracti f these atms r mlecules with cfied electrmagetic fields has always garered a great deal f atteti. 3,4 The field f Cavity

2 Quatum Electrdyamics (CQED) deals with the study f such iteractis i cavities. 5,6 Mrever, very fte, these ivestigatis ivlve the use f tw-level atms. 7,8 970 ad i 976 respectively, Tilley, 9 ad Rgvi ad Scully 0 had preseted theretical studies illustratig the aalgy betwee a sephs jucti ad a tw-level atm. The first experimetal cfirmati f their predictis came i 999. this study, measuremets were made f 50 GHz electrmagetic radiati emitted by three tw-dimesial sephs jucti arrays. Each array was cmprised f uderdamped Nb / Al / AlO x / Nb sephs tuel juctis ad each jucti was cupled t a high-q cell resace. The arrays emitted cheretly abve a threshld that icreased as the array size icreased ad the detected ac pwer exhibited a N depedece, where N represeted the umber f active juctis i the array. They als bserved sychrizati betwee icmig radiati ad radiati emitted by the juctis. Eve befre this publicati thugh, there had already bee research carried ut the iteracti f sephs juctis with cavities. Sice 999 ther articles detailig aalyses f sephs jucti-cavity systems have als bee published. 3,4 this paper, Rgvi ad Scully s mdel f the sephs jucti as a tw-level atm is used t ivestigate specifically the sptaeus ad stimulated emissi f radiati frm a sephs jucti withi a micrcavity. Fr sptaeus emissi, the Hamiltia emplyed uses the classical frm f the pair curret desity fr the jucti i the presece f a applied static magetic field ad a DC vltage. Fr stimulated emissi, with the jucti placed i a exteral electrmagetic field ad biased at a DC vltage, the frm f the pht-assisted pair curret desity fr the jucti is used. The sephs jucti is assumed t be much smaller tha the cavity legth. The e-pht prcess is ivestigated fr sptaeus emissi ad the e-pht ad tw-pht prcesses fr stimulated emissi. Relevat equatis are preseted t calculate the parameters ecessary t achieve resace fr sptaeus emissi ad

3 magitudes f parameters eeded t prduce maximum cut rates fr stimulated emissi are calculated.. teracti Hamiltia ad field quatisati fr a sephs jucti i a micrcavity The sephs jucti uder csiderati, is cmprised f a isulatig layer f thickess l sadwiched betwee tw supercductig layers ad has bee give a legth L. The resultat iteracti Hamiltia 5 describig the cuplig f the electr pairs t the electrmagetic field is give as where Hˆ 3 i d r jr Aˆ r, t c, () c is the speed f the electrmagetic radiati i the isulatig layer, j r curret desity, ad A ˆ r,t, the electrmagetic vectr ptetial., the pair The sephs jucti is placed i the micrcavity such that it is cetred at the rigi (Fig. ). a cavity, the psitive frequecy part f the quatised vectr ptetial is defied as 6 Aˆ 3 ck 6 aˆ exp ic kt 3 r, t d k k, ju raˆ U r j / kj kj kj kj. () k, j represets the uit vectr fr the trasverse plarizati directis where j, ad is the permittivity f the medium f the cavity ad isulatig layer. The wave vectrs ca be either k r k, where k is used fr mdes prpagatig t the right ad k, fr thse t the left. They are defied as 6 k si cs, si si, cs k, (3) 3

4 where 0 ad 0 are the spherical agles. U r ad r kj U are the kj spatial mde fuctis f the cavity. aˆ kj ad ˆ [ aˆ k j ad aˆ k j a k j ] are the creati ad destructi peratrs, respectively, fr the mde with spatial fucti U r r kj kj U. The gemetrical details f the Fabry-Pert micrcavity used i this ivestigati are give i Fig.. The mirrrs f the cavity, assumed t be ideal, with zer thickess ad absrpti, exted ifiitely i the x y plae. The z axis is perpedicular t the plae f the mirrrs ad the rigis f the x, y ad z axes are at the cetre f the cavity. The sephs jucti wave fucti,, is the direct prduct f the BCS wave fuctis fr the upper ad lwer supercductig layers. 0 Fr the lwer layer, q, q u v C C 0 q q q q q, (4) ad fr the upper supercductig layer, where s, s q m s, s u s vsc C 0 s s, (5) s q, represets the pair state f the electr pairs i the upper [lwer] supercductig layer, ad C s ad C C q ad C q are the creati peratrs fr the s electrs i the Blch states s ad s q ad q respectively. u is the s v s prbability amplitude that the pair state s, s is empty [filled]. is give as m, ad the part f that describes pair tuelig wuld iclude terms ctaiig C ad C C s s C q q. Fr electrs with eergy q, where q is the electr eergy relative t 4

5 the Fermi eergy, such that q, D D u q v q, where D is the Debye frequecy f the metal. Additially, the dyamics f the pair tuelig uder ivestigati are such that electrs are trasferred acrss the jucti with chage i their wave vectrs., ca therefre be writte as 0 C s,, mc s,, m Cs,, C s,, 0. (6) s 3. Sptaeus emissi The iitial state f the jucti is give i Eq. (6) ad phts are preset i the cavity. The iitial state f the system, sp, ca therefre be writte as sp,0, (7) where frm represets the iitial state f the sephs jucti. The fial state, F sp, has the Fsp,, (8) F where idicates the presece f a pht i the fial state ad represets the fial state F f the jucti. The prbability amplitude that the system, at time t is i the state F is t first rder i perturbati thery i t F Hˆ dt (9) t i where ad F are the iitial ad fial states respectively. 5

6 Substitutig Eq. () it Eq. (9), ad usig the expressis fr the iitial sp ad fial Fsp states fr sptaeus emissi, we btai i c t t d 3 r F r Aˆ r, t j 0 dt, (0) T geerate electrmagetic radiati frm the jucti thrugh the prcess f sptaeus emissi, a DC vltage, V, is established acrss the isulatig layer ad a small static magetic field 7 is applied i the y -directi H 0, H, 0. The presece f the vltage causes the supercurret t scillate at a frequecy f 8 ev ad electrmagetic waves are radiated frm the jucti. Fr a magetic field directed alg the y-directi, the sluti f the vectr ptetial i the xide ad supercductig layers are such that the dmiat cmpet lies i the directi f curret flw. This cclusi results frm the aalysis carried ut by Swihart, 9 ad Eck, Scalapi ad Taylr. 0 Due t the presece f the vltage ad the magetic field, the curret cmpet, F j r, f Eq. (0) ca be writte as 0 j is the pair curret desity amplitude ad the sephs frequecy k is defied by the equati 0 j si t k z. () ev. () k e c L lh, (3) ad represets the effect f the exterally applied magetic field the tuellig f the Cper pairs. L is the Ld peetrati depth f the supercductig material ad c is give by 6

7 c l L l / c. (4) The field cmpet f Eq. (0) is give as / 0 3 * *, ˆ ˆ exp 3 6 d k k k j U z a U z a ic kt kj kj kj kj c j, (5) where ly the egative frequecy part f the electrmagetic vectr ptetial makes a -zer ctributi. The frms f the mde fuctis, U k j z ad U kj z d z d are as give belw: 6 ad U U U z t exp ikz cs, i the regi j k j, (6) D j t j r j exp ik d z cs k j z, (7) D j z t r exp ik d z cs j j k j (8) D j U z t exp ikz cs j k j, (9) D j where r j ad t j [ r j ad t j ] are the cmplex reflecti ad trasmissi cefficiets fr mirrr [] respectively ad D j r j r j exp ikd cs. (0) The amplitude is calculated alg the z-axis, i.e. 0. Ather cditi impsed is that mirrr is perfectly reflectig ad mirrr highly reflectig, such that r, r, t 0, t 0. () j j j j 7

8 Apart frm cstat factrs, Eq. (5) is calculated t be ikz exp ikz d * r exp ikd 3 exp d k k exp ic kt. () Eq. () ad Eq. () are the substituted it Eq. (0) ad after carryig ut the spatial ad time itegratis, the steady-state amplitude wrks ut t be prprtial t r * 3 / L si k c i d exp k c c i d exp. (3) L c T calculate resat frequecies,, that ca be geerated by the sephs jucti, the relati c (4) L is used. By esurig that the width, d, f the micrcavity is a itegral multiple f the legth, L, f the sephs jucti, the resat frequecies f the jucti are resat frequecies f the micrcavity. Frm Eq. (3), there is resace i the emissi fr k. (5) c This gives B as B e L l, (6) where is the permeability f free space. Fr each value f calculated, the crrespdig vltage, V, is btaied frm V. (7) e 8

9 4. Stimulated emissi 4. The e-pht prcess T examie stimulated emissi frm the sephs jucti, a DC vltage, V, is maitaied acrss the jucti which is lcated i the micrcavity i which there already exists a applied electrmagetic field. There is static magetic field preset frm 0 j e si t V si H 0 ad r t which, expressed i terms f Bessel fuctis, is re-writte as ev j s si st s s j is w f the, (8). (9) represets the curret desity amplitude f the Cper pairs ad, the sephs frequecy. V is the magitude f the vltage iduced by the applied field,, the frequecy f the exteral radiati,, the phase f the applied radiati ad, the iitial Cper pair phase differece. ev. s ev represets the rdiary Bessel fucti f rder s ad argumet T calculate the amplitude fr stimulated emissi, the sephs jucti-micrcavity system is give a iitial state such that e pht is preset i the cavity ad tw phts are preset i the fial state. This is deted respectively as F st st F,,,. (30) st ad F st represet the iitial ad fial states fr stimulated emissi. 9

10 The amplitude fr stimulated emissi is i t 3 d r ψ j ψ Aˆ z,t dt c t F x, (3) After carryig ut the varius itegratis, Eq. (3) wrks ut t be prprtial t L d si exp i exp i c c d * r exp i c ev exp ev i exp i 0 (3) where represets resat frequecies f the jucti-cavity system. Here, agai, is give by Eq. (4) ad we are csiderig a situati where the resat frequecies f the bare jucti cicide with the resat frequecies f the micrcavity. is give as. (33) T btai Eq. (3), the applied micrwave frequecy is take t be equal t e f the resat frequecies f the system sice we are csiderig e pht prcesses. Fr e pht emissi, the curret i Eq. (9) is such that s 0 ad s. Uder the cditis f resace, the e pht stimulated emissi amplitude is prprtial t ev exp ev 0 i exp i (34) where The tw-pht prcess 0

11 Fr this prcess, the sephs jucti-micrcavity system is i a iitial state such that e pht is preset i the cavity ad three phts are preset i the fial state. The iitial ad fial states, therefre, are deted respectively as st F st F,,, 3. (35) T secd rder i perturbati thery, the prbability amplitude that the system at time t is i the state F st is F st i t t Hˆ i t dt st (36) grig cstats, the amplitude fr stimulated emissi f the tw-pht prcess is therefre calculated frm t t t t F t jt ψ 3 Aˆ z,t Aˆ z,t 3 3 dt dt d r d r ψ j The same cditis hld as fr the e-pht prcess. Hwever, x x. (37). (38) ( is the sephs frequecy, is the frequecy f the exteral radiati ad is the frequecy f the radiati geerated by the jucti.) Fr tw-pht emissi, the curret i Eq. (9) is such that s ad s 3. The amplitude fr stimulated tw-pht emissi therefre wrks ut t be prprtial t

12 . 4 4 exp 4 exp 4 exp exp exp 4 si 3 3 * ev ev i ev i ev i c d i r c d i c L (39) is give as. (40) At resace, the tw-pht stimulated emissi amplitude is prprtial t ev ev i ev i ev exp 4 exp 4, (4) where Results We aalyse a jucti-cavity system fr which values f, l, L, L ad d are take as 0, m, 0 m, 0.05 m ad 50 m, respectively. Frm Eqs. (6) ad (7), predicted umerical values f the magetic iducti, B, ad the vltage, V, may be calculated fr the first five resat frequecies f the jucti-cavity system. These values f B ad V ca be used t tue the jucti fr sptaeus emissi f radiati at these specific frequecies. the case f stimulated emissi, the amplitude i Eq. (34) is used t determie the cut rate fr the e-pht prcess ad the amplitude i Eq. (4) is used fr the tw-pht prcess. rder t fid the cditis f resace, we eed t determie the values f the argumet f the

13 Bessel fuctis X ev such that the cut rate is a maximum fr a give. Fr the e-pht prcess, Fig. 3 (a) is a plt f the mdulus square f the amplitude agaist X fr 0. The cut rate exhibits a decayig scillatry depedece X. The first three values that give a resace i the cut rate are X 3. 58, X ad X These values f X are used t fid V fr a particular resat frequecy f the system. Fig. 3 (b) gives the umerical values f V ad fr the three values f X. These predicted values f V ad ca be used t tue the jucti-cavity system. Fig. 4 (a) is a similar plt t 3 (a) fr the twpht prcess. This graph displays fur values f X fr which the cut rate is a maximum X.558, X 4.888, X ad X While there is als a decayig 3 scillatry depedece as fr the e-pht prcess, this begis ly after the secd psiti 4 f maximum cut rate. Fig. 4 (b) gives the umerical values f V that crrespd t the first five resat frequecies f the jucti-cavity system. The vltages V required t achieve the resat frequecies fr bth the e-pht ad tw-pht prcesses may be btaied usig Eq. (7). Figs. 5 (a) ad (b) shw the depedece f the stimulated emissi cut rate fr the e-pht prcess ad fr the tw-pht prcess. There is clearly a scillatry depedece f the cut rate bth parameters ad. Al-Saidi ad Strud s study 3 f a sigle small uderdamped sephs jucti i a large-q resat electrmagetic cavity, they emplyed the full Hamiltia fr the system whereas here, we are iterested ly i the iteracti part f the Hamiltia. Frm their aalysis they fud that fr special values f the gate vltage, there was strg iteracti betwee the jucti ad the resat pht mde. this paper it was fud that resace i the emissi depeded 3

14 the matchig f the vltages ad magetic iductis (Eqs. (5) t (7)) fr sptaeus emissi, ad fr stimulated emissi, the relati f the vltage iduced by the applied field t the ptetial differece acrss the jucti (which ca be determied frm the argumet f the Bessel fucti X ) ad the relatiship betwee the phase f the applied field ad the iitial Cper pair phase differece (Eq. (40)). 6. Cclusi We have preseted a mdel that describes the emissi characteristics f a sephs jucti i a micrcavity. Fr sptaeus emissi, we have shw that the magetic iducti, B, ad vltage, V, acrss the jucti must be prperly matched t achieve resace i the cavity. Fr emissi i the micrwave regi, typical values are f the rder f milliteslas fr B, ad millivlts fr V. Fr the stimulated emissi prcesses, we calculated the amplitude f the applied radiati, V, ecessary t prduce maximum cut rates fr the first five resat frequecies f the system. We have als shw that t achieve these maximum cut rates fr the e-pht prcess, the differece,, betwee the iitial Cper pair phase differece ad the phase f the applied field must be a itegral multiple f, while fr the tw-pht prcess, the differece,, betwee the iitial Cper pair phase differece ad twice the phase f the applied field must be a dd umber multiple f. The ability t ctrl the sephs jucti-cavity system t emit particular umbers f phts, t emit at particular frequecies, ad t prduce specific cut rates as required certaily makes it a versatile surce f phts. particular, sigle-pht surces will fid useful applicatis i quatum cryptgraphy ad ifrmati prcessig. 4

15 Refereces. H. B. G. Casimir, ad D. Plder, The ifluece f retardati the Ld-va der Waals frce, Phys. Rev. 73, (948).. K. H. Drexhage, teracti f light with mmlecular dye layers i Prgress i Optics X, E. Wlf, ed. (Nrth Hllad, Amsterdam, 974), p R. Ackerhalt, ad. H. Eberly, Quatum electrdyamics ad radiati reacti: relativistic atmic frequecy shifts ad lifetimes, Phys. Rev. D 0, (974). 4. D. Meschede, H. Walther, ad G. Müller, The e-atm maser, Phys. Rev. Lett. 54, (985). 5. R. Aru, ad G. S. Agarwal, Dark states ad iterfereces i cascade trasitis f ultracld atms i a cavity, Phys. Rev. A 66, 0438-(-9) (00). 6. A. Aiell, D. Fargi, ad E. Ciaci, Parametric flurescece ad secd-harmic geerati i a plaar Fabry-Pert micrcavity, Phys. Rev. A 58, (998). 7. Y. B. Xie, Tw-level atm ad multichrmatic waves i a lssless cavity,. Md. Optics 44, (997). 8. A. Carll, M. F. Sats, ad V. Vedral, Berry s phase i cavity QED: Prpsal fr bservig a effect f field quatisati, Phys. Rev. A 67, (-4) (003). 9. D. R. Tilley, Superradiace i arrays f supercductig weak liks, Phys. Lett. 33A, (970). 0. D. Rgvi, ad M. Scully, Supercductivity ad macrscpic quatum phemea, Phys. Rep. 5C, 75-9 (976).. P. Barbara, A. B. Cawthre, S. V. Shitv, ad C.. Lbb, Stimulated emissi ad amplificati i sephs jucti arrays, Phys. Rev. Lett. 8, (999). 5

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18 List f Figure Captis Fig.. Schematic f a tuel r S--S (S supercductr, isulatr) sephs jucti i a micrcavity. Fig.. Gemetry f the Fabry-Pert micrcavity. ad represet the left ad right mirrrs, respectively, f the micrcavity. Fig. 3. Graphs illustratig the umerical values fr the e-pht prcess f (a) X fr the first three psitis f maximum cut rate fr 0, ad (b) V fr the first five cavity resaces f the system fr X 3. 58, X ad X ad 0. Fig. 4. Graphs illustratig the umerical values fr the tw-pht prcess f (a) X fr the first fur psitis f maximum cut rate fr 0, ad (b) V fr the first five cavity resaces f the system fr X. 558, X , X ad X. 530 ad 0. 3 Fig. 5 Graph illustratig the variati f the cut rate with (a), where, fr the 4 e-pht prcess ad (b), where, fr the tw-pht prcess. 8

19 l L H m + - V z y x z d/ d/ Fig. 9

20 k - k z = 0 z x d/ d/ Fig. y z 0

21 X (X) 0(X) X 6.86 X (a) X V / mv X X X (b) Agular frequecy, ( 0 ) rads - Fig. 3

22 X ((-3 (X)) + (- (X)) -3 (X) - (X)) X.558 X X (a) X 40 0 X X V /mv X X (b) Agular frequecy, ( 0 ) rads - Fig. 4

23 0.7 -(X) exp ( i) 0(X) exp (i) / rad (a) 0.6 ((-3 (X)) exp(i) + (- (X)) exp(i) -3 (X) - (X)) (b) / rad Fig. 5 3