Retrtion otie Title of retrted rtile: Genertion of Bright Squeezed Light from Three-Level Atoms Pumped y Coherent Light: Open Quntum System Author(s): Gethew AGeru * Corresponding uthor Emil: gethewsmelsh@gmilom Journl: Journl of Quntum Informtion Siene (JQIS) Yer: 06 Volume: 6 umer: Pges (from - to): -4 DOI (to PDF): http://dxdoiorg/0436/jqis0660 Pper ID t SCIRP: 30085 Artile pge: http://wwwsirporg/journl/pperinformtionspx?pperid=67874 Retrtion e: 07-0- Retrtion inititive (multiple responses llowed; mr with X): Ⅹ All uthors Some of the uthors: Editor with hints from Journl owner (pulisher) Institution: Reder: Other: Dte inititive is lunhed: 07-0-06 Retrtion type (multiple responses llowed): Unrelile findings L error Inonsistent Anlytil error Bised interprettion Other: Irreproduile results Filure to dislose mjor ompeting interest liely to influene interprettions or reommenions Unethil reserh Frud Dt frition Fe pulition Other: Plgirism Self plgirism Overlp Redundnt pulition * Copyright infringement Other legl onern: Editoril resons Hndling error Unrelile review(s) Deision error Other: ⅩOther: Results of pulition (only one response llowed): re still vlid Ⅹwere found to e overll invlid Author's ondut (only one response llowed): Ⅹhonest error demi misondut none (not pplile in this se eg in se of editoril resons) * Also lled duplite or repetitive pulition Definition: "Pulishing or ttempting to pulish sustntilly the sme wor more thn one"
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Journl of Quntum Informtion Siene 06 6-4 Pulished Online June 06 in SiRes http://wwwsirporg/journl/jqis http://dxdoiorg/0436/jqis0660 Genertion of Bright Squeezed Light from Three-Level Atoms Pumped y Coherent Light: Open Quntum System Gethew A Geru Deprtment of Physis Addis A University Addis A Ethiopi Reeived Jnury 06; epted 7 June 06; pulished 30 June 06 Copyright 06 y uthor nd Sientifi Reserh Pulishing In This wor is liensed under the Cretive Commons Attriution Interntionl Liense (CC BY) http://retiveommonsorg/lienses/y/40/ Astrt The mnusript investigted the stedy-stte nlysis of the squeezing nd sttistil properties of the light generted y three-level toms ville in n open vity pumped oherent light nd the vity oupled to two-mode vuum reservoir The results indite tht s the frequeny inreses the lol qudrture squeezing of the two-mode vity light pprohes the glol qudrture squeezing The effet of the spontneous emission leds to n inrese in the qudrture squeezing ut to derese in the men photon numer of the system It is lso found tht unlie the men photon numer nd the vrine of the photon numer the qudrture squeezing does not depend on the numer of toms This implies tht the qudrture squeezing of the two-mode vity light is independent of the numer of photons Keywords Spontneous Emission Opertor Dynmis Photon Sttistis Power Spetrum Qudrture squeezing Qudrture Flututions Introdution Squeezed sttes of light hs plyed ruil role in the development of quntum physis Squeezing is one of the nonlssil fetures of light tht hve een extensively studied y severl uthors []-[8] In squeezed stte the quntum noise in one qudrture is elow the vuum-stte level or the oherent-stte level t the expense of enhned flututions in the onjugte qudrture with the produt of the unertinties in the two qudrtures stisfying the unertinty reltion [] [] [4] [9] Beuse of the quntum noise redution hievle elow the vuum level squeezed light hs potentil pplitions in the detetion of wee signls nd in How to ite this pper: Geru GA (06) Genertion of Bright Squeezed Light from Three-Level Atoms Pumped y Coherent Light: Open Quntum System Journl of Quntum Informtion Siene 6-4 http://dxdoiorg/0436/jqis0660
G A Geru low-noise ommunitions [] [3] Squeezed light n e generted y vrious quntum optil proesses suh s suhrmoni genertions []-[5] [0]-[] four-wve mixing [3] [4] resonne fluoresene [6] [7] seond hrmoni genertion [8] [5] nd three-level lser under ertin onditions [] [3] [4] [9] [6]-[7] Hene it proves useful to find some onvenient mens of generting right squeezed light A three-level lser is quntum optil devie in whih light is generted y three-level toms in vity usully oupled to vuum reservoir vi single-port mirror In one model of three-level lser three-level toms initilly prepred in oherent superposition of the top nd ottom levels re injeted into vity nd then removed from the vity fter they hve deyed due to spontneous emission [9] [6]-[] In nother model of three-level lser the top nd ottom levels of the three-level toms injeted into vity re oupled y oherent light []-[7] It is found tht three-level lser in either model genertes squeezed light under ertin onditions The superposition or the oupling of the top nd ottom levels is responsile for the squeezed of the generted light It ppers to e quite diffiult to prepre the toms in oherent superposition of the top nd ottom levels efore they re injeted into the vity In ddition it should ertinly e hrd to find out tht the toms hve deyed spontneously efore they re removed from the vity In order to void the forementioned prolems Fesseh [8] hve onsidered tht two-level toms ville in losed vity re pumped to the top level y mens of eletron omrdment He hs shown tht the light generted y this lser operting well ove threshold is oherent nd the light generted y the sme lser operting elow threshold is hoti light In ddition Fesseh [9] hs studied the squeezing nd the sttistil properties of the light produed y degenerte three-level lser with the toms in losed vity nd pumped y eletron omrdment He hs shown tht the mximum qudrture squeezing of the light generted y the lser operting fr elow threshold is 50% elow the vuum-stte level Alterntively the three-level toms ville in losed vity nd pumped y oherent light lso generted squeezed light under ertin onditions with the mximum glol qudrture squeezing is eing 43% elow the vuum-stte level [] It ppers to e prtilly more onvenient to pump the toms y oherent light thn eletron omrdment In this pper we investigte the stedy-stte nlysis of the squeezing nd sttistil properties of the light generted y oherently pumped degenerte three-level lser with open vity whih is oupled to two-mode vuum reservoir vi single-port mirror We rry out our lultion y putting the noise opertors ssoited with the vuum reservoir in norml order nd y ting into onsidertion the intertion of the three-level toms with the vuum reservoir outside the vity Model nd Dynmis of Atomi nd Cvity Mode Opertors Let us onsider system of degenerte three-level toms in sde onfigurtion re ville in n open vity nd interting with the two (degenerte) vity modes The top nd ottom levels of the three-level toms re oupled y oherent light When degenerte three-level tom in sde onfigurtion deys from the top level to the ottom levels vi the middle level two photons of the sme frequeny re emitted For the se of onvenient we denote the top middle nd ottom levels of these toms y nd respetively We wish to represent the light emitted from the top level y â nd the light emitted from the middle y â In ddition we ssume tht the two vity modes nd to e t resonne with the two trnsitions nd with diret trnsitions etween levels nd to e dipole foridden The intertion of one of the three-level toms with light modes nd n e desried t resonne y the Hmiltonin where nd ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H t ig () ˆ () ˆ (3) re lowering tomi opertors â nd â re the nnihiltion opertors for light modes nd nd g is the oupling onstnt etween the tom nd the light mode or light mode And the intertion of the
G A Geru three-level tom with the driving oherent light n e desried t resonne y the Hmiltonin in whih nd ˆ i H t ˆ ˆ ˆ (5) g (6) Here is the mplitude of the driving oherent light nd g is oupling onstnt etween the tom nd oherent light Thus upon omining Equtions () nd (4) the intertion of the three-level tom with the driving oherent light nd vity modes nd is desried t resonne y the Hmiltonin ˆ i H ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ s t ig On the other hnd the degenerte three-level toms ville in n open vity re oupled to two-mode vuum reservoir The mster eqution for the three-level tom interting with two-mode vuum reservoir hs the form [] [3] d ˆ ˆ t i H ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ s t t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ where is the spontneous emission dey onstnt ssoited with the two modes nd Hene with the id of Eqution (7) the mster eqution desriing the two-mode vity light of oherently pumped degenerte three-level tom would e d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ t g ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ We rell tht the lser vity is oupled to two-mode vuum reservoir vi single-port mirror In ddition we rry out our nlysis y putting the noise opertors ssoited with the vuum reservoir in norml order Thus the noise opertors will not hve ny effet on the dynmis of the vity mode opertors [] [8] In view of this we n drop the noise opertors nd write the quntum Lngevin eqution for the opertors â nd â s d ˆ t ˆ t i ˆ t Hˆ s t (0) d ˆ t ˆ t i ˆ t Hˆ s t () where is the vity dmping onstnt Then with the id of Eqution (7) we esily find d ˆ ˆ ˆ t t g t () d ˆ ˆ ˆ t t g t (3) The proedure of norml ordering the noise opertors renders the vuum reservoir to e noiseless physil entity We uphold the view point tht the notion of noiseless vuum reservoir would turn out to e omptile (4) (7) (8) (9) 3
G A Geru with oservtion [30] Furthermore ming use of the mster eqution nd the ft tht (where  is n opertor) it is not diffiult to verify tht d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g t d ˆ d ˆ A Tr Aˆ (4) (5) (6) (7) where d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g (8) d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ g (9) ˆ (0) ˆ () ˆ () We see tht Equtions (4)-(9) re nonliner nd oupled differentil equtions Therefore it is not possile to otin the ext time-dependent solutions We intend to overome this prolem y pplying the lrge-time pproximtion [8] Then using this pproximtion sheme we get from Equtions () nd (3) the pproximtely vlid reltions nd Upon sustituting (3) nd (4) into Equtions (4)-(9) we get g ˆ ˆ (3) g ˆ ˆ (4) d ˆ ˆ ˆ (5) d ˆ ˆ ˆ (6) d ˆ ˆ ˆ ˆ (7) where d ˆ ˆ ˆ ˆ (8) d ˆ ˆ ˆ (9) d ˆ ˆ ˆ ˆ ˆ (30) 4
G A Geru 4g (3) is the stimulted emission dey onstnt We next sum Equtions (5)-(30) over the three-level toms We then see tht d m ˆ m ˆ m ˆ (3) d mˆ ˆ ˆ m m (33) d mˆ ˆ ˆ ˆ m (34) in whih d ˆ ˆ ˆ ˆ m m (35) d ˆ ˆ ˆ d (36) t d ˆ ˆ ˆ ˆ ˆ m m (37) mˆ ˆ (38) mˆ ˆ (39) mˆ ˆ (40) ˆ ˆ (4) ˆ ˆ (4) ˆ ˆ (43) with the opertors ˆ ˆ nd ˆ representing the numer of toms in the top middle nd ottom levels In ddition employing the ompleteness reltion we esily rrive t ˆ ˆ ˆ ˆ (44) I ˆ ˆ ˆ (45) Furthermore pplying the definition given y () nd setting for ny ˆ (46) we hve Following the sme proedure one n esily find mˆ (47) mˆ (48) 5
G A Geru Moreover using the definition nd ting into ount Equtions (47)-(5) it n e redily estlished tht Upon dding Equtions () nd (3) we hve where mˆ (49) ˆ (50) ˆ (5) ˆ (5) mˆ mˆ mˆ (53) mˆ mˆ ˆ ˆ (54) mm ˆ ˆ ˆ ˆ (55) ˆ ˆ m m (56) d ˆ ˆ ˆ ˆ t t g t t (57) ˆ t ˆ t ˆ t (58) In the presene of three-level toms we n rewrite Eqution (57) s d ˆ t ˆ t m ˆ t (59) in whih is onstnt whose vlue remins to e fixed The stedy-stte solution of Eqution (57) is g ˆ t ˆ ˆ t t (60) Ting into ount of (60) nd its djoint the ommuttion reltion for the vity mode opertor is found to e nd on summing over ll toms we hve where ˆ ˆ ˆ ˆ (6) ˆ ˆ ˆ ˆ (6) ˆ ˆ ˆ ˆ (63) stnds for the ommuttor of ˆ ˆ when the superposed light mode is interting with ll the three-level toms On the other hnd using the stedy-stte solution of Eqution (59) one n verify tht Comprison of Equtions (6) nd (64) shows tht ˆ ˆ ˆ ˆ On ount of (65) one n put Eqution (59) in the form (64) g (65) 6
G A Geru 3 Photon Sttistis d g ˆ t ˆ t m ˆ t (66) Here we see to otin the glol (lol) men photon numer nd the glol (lol) vrine of the photon numer for the two-mode vity light em t stedy stte 3 The Glol Men Photon umer We wish to lulte the men photon numer of the two-mode vity light in the entire frequeny intervl The stedy-stte solution of Eqution (66) is given y g ˆ t mˆ t (67) On ount of (67) together with (54) the men photon numer of the two-mode vity light is expressile s in whih n ˆ t ˆ t light turns out to e n ˆ ˆ (68) With the id of (47) nd (48) the men photon numer of the two-mode vity 3 3 n We note tht the glol men photon numer tes for n 3 the form We oserve from the plots in Figure tht the presene of spontneous emission leds to derese in the glol men photon numer of the two-mode vity light em 3 Lol Men Photon umer We see to determine the men photon numer in given frequeny intervl employing the power spetrum for the two-mode vity light The power spetrum of two-mode vity light with entrl ommon frequeny 0 is defined s (69) (70) i 0 P Re d e ˆ tˆ t π (7) 0 ss On introduing (6) into Eqution (7) nd rrying out the integrtion we redily get π π P n 0 0 The men photon numer in the frequeny intervl etween nd is expressile s (7) n Pd (73) in whih 0 Thus upon sustituting (7) into Eqution (73) we find n n π n π d d (74) 7
G A Geru Figure Plots of the glol men photon numer [Eqution (69)] versus t stedy stte for 04 08 50 nd different vlues of nd on rrying out the integrtion over pplying the reltion we rrive t where z dx tn (75) x n nz (76) π π tn tn z 09889 for 0 And for 0 z Then omintion of these results with Eqution (76) yields n 05 0767n n 0936n nd n 09889n for 0 And we hve n 05 070n n 0905n nd n 0979n for 0 We therefore oserve tht lrge prt of the totl men photon numer is onfined in reltively smll frequeny intervl One n redily get from Figure tht z 05 0767 z 0936 nd we find z 05 070 z 0905 nd 0979 33 The Glol Vrine of the Photon umer The vrine of the photon numer for the two-mode vity light is expressile s nd using the ft tht n ˆ ˆ ˆ ˆ (77) (78) ât is Gussin vrile with zero men we rrive t n Employing one more (67) nd ting into ount (55) we redily get ˆ ˆ ˆ ˆ ˆ ˆ (79) 8
G A Geru Figure Plot of different vlues of z [Eqution (77)] versus for 04 08 3 nd ˆ ˆ ˆ ˆ On ount of (45) one n e put Eqution (80) in the form (80) ˆ ˆ ˆ (8) with the id of (68) nd (45) we rrive t ˆ ˆ n On the other hnd using (67) long with (56) we esily otin ˆ mˆ (83) so tht in view of (46) nd (69) there follows ˆ n ow on ount of Equtions (68) (83) nd (84) we redily find Eqution (79) to e This n e put in the form 4 n n n n (8) (84) (85) 9
G A Geru In view of (69) we rrive t 3 n nn 3 4 3 n 3 3 3 3 We immeditely see from the plots in Figure 3 tht the presene of spontneous emission leds to derese in the glol vrine of the photon numer of the two-mode vity light em In ddition the glol vrine of the photon numer of the two-mode vity light inreses with inresing Finlly we note tht the vrine of the photon numer tes for the form (86) (87) n (88) in whih n is given y (70) This represents the normlly ordered vrine of the photon numer for hoti stte 34 Lol Vrine of the Photon umer Here we wish to otin the vrine of the photon numer in given frequeny intervl employing the spetrum of the photon numer flututions for the superposition of light modes nd We denote the entrl ommon frequeny of these modes y 0 The spetrum of the photon numer flututions for the superposed light modes n e expressed s where i0 R Re d e nˆ t nˆ t π (89) 0 ss Figure 3 Plots of the glol vrine of the photon numer [Eqution (87)] versus t stedy stte for 04 08 50 nd different vlues of 30
G A Geru nˆ t ˆ t ˆ t (90) nˆ t ˆ t ˆ t (9) nd we hve used the nottion nˆ t nˆ t nˆ tnˆ t nˆ t nˆ t With the id of (90) nd (9) nd Eqution (4) the photon numer flutution n e expressed s i0 R Re d e ˆ ˆ ˆ ˆ π 0 t t t t ˆ t ˆ t ˆ tˆ t Upon introduing (6)-(65) into Eqution (9) nd on rrying out the integrtion over the spetrum of the photon numer flututions for the two-mode vity light is found to e where n R n π π ( ) 0 0 π 0 4 is given y (79) Upon integrting oth sides of (93) over one esily finds n (9) (93) R d (94) ss On the sis of Eqution (94) we oserve tht Rd represents the stedy-stte vrine of the photon numer for the two-mode vity light in the intervl etween nd d We thus relize tht the photonnumer vrine in the intervl etween nd n e written s n R d (95) in whih 0 Thus upon sustituting (93) into Eqution (95) we find n n π π d d ss π d 4 so on rrying out the integrtion over pplying the reltion desried y Eqution (75) we redily get where z is given y n n z (96) (97) π π 4 π z tn tn tn ( ) From the plots in Figure 4 tht we esily find z05 0765 z 09483 nd 0 And for 0 we find z05 0504 z 07893 nd 09496 of these results with Eqution (97) yields n n n n nd n 09934 n nd n n 05 0765 (98) z 09934 for z Then omintion 09483 we find n n n n for 0 And for 0 0504 07893 05 09496 We immeditely oserve tht lrge prt of the totl vrine of the photon numer is onfined in reltively smll frequeny intervl 3
G A Geru Figure 4 Plots of 4 Qudrture Squeezing z 0 nd different vlues of [Eqution (98)] versus for 04 08 In this setion we see to lulte the qudrture squeezing of the two-mode vity light in ny frequeny intervl 4 The Glol Qudrture Squeezing The squeezing properties of the two-mode vity light re desried y two qudrture opertors defined s It n e redily estlished tht It then follows tht ˆ ˆ ˆ t t t (99) ˆ ˆ ˆ t i t t (00) ˆ ˆ i ˆ ˆ (0) ˆ ˆ (0) ow upon repling the tomi opertors tht pper in Eqution (6) y their expettion vlues the ommuttion reltion for the two-mode light n write s in whih The vrine of the qudrture opertors is expressile s ˆ ˆ (03) ˆ ˆ (04) ˆ tˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t (05) 3
G A Geru In view of Eqution (4) one n put Eqution (05) in the form ˆ tˆ t ˆ t ˆ t (06) With the id of (68) (83) nd (04) together with (45) we otin ˆ ˆ m (07) ˆ ˆ m (08) Finlly on ount of (46) nd (48) the glol qudrture vrine of the two-mode vity light turns out t stedy stte to e nd 3 3 3 3 It is then not diffiult to oserve tht the two-mode vity light em is in squeezed stte nd the squeezing ours in the minus qudrture We next proeed to lulte the qudrture squeezing of the two-mode vity light reltive to the qudrture vrine of the two-mode vity vuum stte We define the qudrture squeezing of the two-mode vity light y Moreover upon setting 0 in Eqution (0) we see tht (09) (0) S () v () v whih represents the qudrture vrine of the two-mode vity vuum stte Hene on ount of Equtions (0) nd () we rrive t S 3 3 3 3 We note tht unlie the men photon numer the qudrture squeezing does not depend on the numer of toms This implies tht the qudrture squeezing of the two-mode vity light is independent of the numer of photons We see from the plots in Figure 5 tht the mximum glol qudrture squeezing of the two-mode vity light for 0 is 434% (nd ours t 077 ) nd for 0 is found 475% (nd ours t 033 ) And for 0 the mximum glol qudrture squeezing is oserved to e 50% elow the vuum-stte level nd this ours when the three-level lser is operting t 0303 Moreover upon setting 0 in Eqution (3) we note tht (3) S (4) 3 where Eqution (4) is indites tht the qudrture squeezing of the light produed y degenerte three-level lser with the three-level toms ville inside losed vity pumped to the top level y eletron omrdment whih hs een reported y Fesseh [] 33
G A Geru 4 Lol Qudrture Squeezing Figure 5 Plots of the glol qudrture squeezing [Eqution (3)] versus t stedy stte for 04 08 nd different vlues of Here we wish to otin the qudrture squeezing of the two-mode vity light in given frequeny intervl To this end we first otin the spetrum of the qudrture flututions of the superposition of light modes nd We define this spetrum for the two-mode vity light y in whih i0 S d e ˆ ˆ Re t t π 0 (5) ss ˆ ˆ ˆ t t t (6) ˆ t i ˆ t ˆ t (7) nd 0 is the entrl frequeny of the modes nd In view of Eqution (4) we otin nd ˆ t ˆ t ˆ t ˆ t (8) Then on ount of Equtions (99) (00) (6) nd (7) one n write Eqution (8) s ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ t t t t t t t t t t (9) Upon sustituting of (6)-(65) into Eqution (9) we rrive t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t ˆ t e e This n e put in the form ˆ e e ˆ t t ˆ t ˆ t e e (0) () () 34
G A Geru ow introduing () into Eqution (5) nd on rrying out the integrtion over we find the spetrum of the minus qudrture flututions for two-mode vity light to e S ss Upon integrting oth sides of (3) over we get π π 0 0 (3) S d (4) On the sis of Eqution (4) we oserve tht S d is the stedy-stte vrine of the minus qudrture in the intervl etween nd d We thus relize tht the vrine of the minus qudrture in the intervl etween nd is expressile s S d (5) in whih 0 On introduing (3) into Eqution (5) nd on rrying out the integrtion over employing the reltion desried y Eqution (75) we find where z z (6) π π tn tn We define the qudrture squeezing of the two-mode vity light in the frequeny intervl y S (7) (8) v Furthermore upon setting 0 in Eqution (6) we see tht the lol qudrture vrine of two-mode vity vuum stte in the sme frequeny is found to e in whih z (9) v π π zv tn tn (30) nd is given y () Finlly on ount of Equtions (0) () nd (9) long with (8) we v redily get v v 3 3 S zv z z zv This shows tht the lol qudrture squeezing of the two-mode vity light ems is not equl to tht of the glol qudrture squeezing Moreover we found from the plots in Figure 6 tht the mximum lol qudrture squeezing for 0 is 773% (nd ours t 006 ) nd for 0 is found 783% (nd ours t 006 ) And for 0 the mximum lol qudrture squeezing is oserved to e 788% ( nd ours t 006 ) Furthermore we note tht the lol qudrture squeezing pprohes the glol qudrture squeezing s inreses 5 Conlusions The stedy-stte nlysis of the squeezing nd sttistil properties of the light produed y oherently pumped degenerte three-level lser with open vity nd oupled to two-mode vuum reservoir is presented We (3) 35
G A Geru Figure 6 Plot of the lol qudrture squeezing [Equtions (3)] versus t stedy stte for 04 08 0 t 077 0 t 033 nd 0 t 0303 rry out our nlysis y putting the noise opertors ssoited with the vuum reservoir in norml order nd y ting into onsidertion the intertion of the three-level toms with the vuum reservoir outside the vity We oserve tht lrge prt of the totl men photon numer (vrine of the photon numer) is onfined in reltively smll frequeny intervl In ddition we find tht the mximum glol qudrture squeezing of the light produed y the system under onsidertion for 0 operting t 077 is 434% nd for 0 operting t 033 is 475% And for 0 the mximum glol qudrture squeezing is oserved to e 50% elow the vuum-stte level nd this ours when the three-level lser is operting t 0303 Furthermore results show tht the presene of spontneous emission leds to derese in the men photon numer nd to n inrese in the qudrture squeezing Moreover we find tht the mximum lol qudrture squeezing for 0 is 773% (nd ours t 006 ) nd for 0 is 783% (nd ours t 006 ) And for 0 the mximum lol qudrture squeezing is oserved to e 788% (nd ours t 006 ) In ddition we note from the plots in Figure 6 tht s inreses the lol qudrture squeezing pprohes the glol qudrture squeezing We oserve tht the light generted y this lser operting under the ondition is in hoti light And we hve lso estlished tht the lol qudrture squeezing of the two-mode light is not equl to the glol qudrture squeezing Furthermore we point out tht unlie the men photon numer nd the vrine of the photon numer the qudrture squeezing does not depend on the numer of toms This implies tht the qudrture squeezing of the two-mode vity light is independent of the numer of photons Anowledgements I would lie to thn Dr Fesseh Ksshun for introduing me to the fsinting filed of Quntum Optis I lso 36
G A Geru thn the ind referees for the positive nd invlule suggestions whih improve the mnusript gretly This wor ws supported y the Shool of Grdute Studies of Addis A University Addis A nd Mdwlu University Ble-Roe Ethiopi Referenes [] Ksshun F (04) Refined Quntum Anlysis of Light CreteSpe Independent Pulishing Pltform [] Wlls DF nd Milurn GJ (995) Quntum Optis Springer-Verlg Berlin [3] Sully MO nd Zuiry MS (997) Quntum Optis Cmridge University Press Cmridge http://dxdoiorg/007/cbo9780583993 [4] Meystre P nd Srgent III M (997) Elements of Quntum Optis nd Edition Springer-Verlg Berlin [5] Brnett SM nd Rdmore PM (997) Methods in Theoretil Quntum Optis Clrendon Press Oxford [6] Vogel W nd Welsh DG (006) Quntum Optis Wiley-VCH ew Yor http://dxdoiorg/000/35760854 [7] Collet MJ nd Grdiner CW (984) Squeezing of Intrvity nd Trveling-Wve Light Fields Produed in Prmetri Amplifition Physil Review A 30 386 http://dxdoiorg/003/physreva30386 [8] Leonhr U (997) Mesuring the Quntum Anlysis of Light Cmridge University Press Cmridge [9] Sully MO Wodieniz K Zuiry MS Bergou J Lu nd Meyer ter Vehn J (988) Two-Photon Correlted-Spontneous-Emission Lser: Quntum oise Quenhing nd Squeezing Physil Review Letters 60 83 http://dxdoiorg/003/physrevlett6083 [0] Dniel B nd Fesseh K (998) The Propgtor Formultion of the Degenerte Prmetri Osilltor Optis Communitions 5 384-394 http://dxdoiorg/006/s0030-408(98)00039-x [] Telu B (006) Prmetri Osilltion with the Cvity Mode Driven y Coherent Light nd Coupled to Squeezed Vuum Reservoir Optis Communitions 6 30-3 [] Drge TY nd Ksshun F (00) Coherently Driven Degenerte Three-Level Lser with Prmetri Amplifier PMC Physis B 3 [3] Anwr J nd Zuiry MS (99) Effet of Squeezing on the Degenerte Prmetri Osilltor Physil Review A 45 804 http://dxdoiorg/003/physreva45804 [4] Plimr LI nd Wlls DF (994) Dynmil Restritions to Squeezing in Degenerte Optil Prmetri Osilltor Physil Review A 50 67 http://dxdoiorg/003/physreva5067 [5] Drummond PD Meil KJ nd Wlls DF (980) on-equilirium Trnsitions in Su/Seond Hrmoni Genertion Opti At 7 3-335 http://dxdoiorg/0080/73806 [6] Sully MO nd Zuiry MS (988) oise Free Amplifition vi the Two-Photon Correlted Spontneous Emission Lser Optis Communitions 66 303-306 http://dxdoiorg/006/0030-408(88)9049- [7] Anwr J nd Zuiry MS (994) Quntum-Sttistil Properties of oise in Phse-Sensitive Liner Amplifier Physil Review A 49 48 http://dxdoiorg/003/physreva4948 [8] Lu nd Zhu SY (989) Quntum Theory of Two-Photon Correlted-Spontneous-Emission Lsers: Ext Atom- Field Intertion Hmiltonin Approh Physil Review A 40 5735 http://dxdoiorg/003/physreva405735 [9] Ansri A (993) Effet of Atomi Coherene on the Seond- nd Higher-Order Squeezing in Two-Photon Three- Level Csde Atomi System Physil Review A 48 4686 http://dxdoiorg/003/physreva484686 [0] Fesseh K (00) Three-Level Lser Dynmis with Squeezed Light Physil Review A 63 0338 http://dxdoiorg/003/physreva630338 [] Alehew E nd Fesseh K (006) A Degenerte Three-Level Lser with Prmetri Amplifier Optis Communitions 65 34-3 http://dxdoiorg/006/joptom0060307 [] Tesf S (008) Externlly Indued Continuous Vrile Entnglement in Correlted Emission Lser Journl of Physis B: Atomi Moleulr nd Optil Physis 4 Artile ID: 4550 http://dxdoiorg/0088/0953-4075/4/4/4550 [3] Ansri A Ge-Bnlohe J nd Zuiry MS (990) Phse-Sensitive Amplifition in Three-Level Atomi System Physil Review A 4 579 http://dxdoiorg/003/physreva4579 [4] Ansri A (99) Theory of Two-Mode Phse-Sensitive Amplifier Physil Review A 46 560 http://dxdoiorg/003/physreva46560 [5] Lu Zho F-X nd Bergou J (989) onliner Theory of Two-Photon Correlted-Spontneous-Emission Lser: A Coherently Pumped Two-Level-Two-Photon Lser Physil Review A 39 589 37
G A Geru http://dxdoiorg/003/physreva39589 [6] Svge CM nd Wlls DF (986) Squeezing vi Two-Photon Trnsitions Physil Review A 33 338 http://dxdoiorg/003/physreva3338 [7] Tesf S (006) Entnglement Amplifition in ondegenerte Three-Level Csde Lser Physil Review A 74 04386 http://dxdoiorg/003/physreva7404386 [8] Fesseh K (0) Stimulted Emission y Two-Level Atoms Pumped to the Upper Level Optis Communitions 84 357-363 http://dxdoiorg/006/joptom0006 [9] Fesseh K (0) Three-Level Lser Dynmis with the Atoms Pumped y Eletron Bom http://rxivorg/s/05438 [30] Fesseh K (0) Two-Level Lser Dynmis with oiseless Vuum Reservoir http://rxivorg/s/09473 38
G A Geru Appendix Solutions of the Expettion Vlues of the Cvity (Atomi) Mode Opertors In order to determine the men photon numer nd the vrine of the photon numer nd the qudrture squeezing of the two-mode vity light in ny frequeny intervl t stedy stte we first need to lulte the solution of the equtions of evolution of the expettion vlues of the tomi opertors nd vity mode opertors To this end the expettion vlues of the solution of Eqution (66) is expressile s t g t t t ˆ t ˆ 0 e e e mˆ t (3) o We next wish to otin the expettion vlue of the expression of pplying the lrge-time pproximtion sheme to Eqution (33) we get ˆ ˆ m upon sustituting the djoint of this into Eqution (3) we hve in whih m ˆm t tht pper in Eqution (3) Thus (33) d m ˆ ˆ t m t (34) We notie tht the solution of Eqution (34) for different from zero t stedy stte is (35) mˆ t 0 (36) In similr mnner pplying the lrge-time pproximtion sheme to Eqution (3) we otin ˆ ˆ m with the id of the djoint of Eqution (37) one n put Eqution (33) in the form m (37) d mˆ ˆ t mt (38) we lso note tht for different from zero the solution of Eqution (38) turns out t stedy stte to e Upon dding Equtions (34) nd (38) we find mˆ t 0 (39) d mˆ t mˆ t mˆ t (40) We note tht in view of Eqution (36) with the ssumption the toms initilly in the ottom level the solution of Eqution (40) turns out t stedy stte to e mt ˆ 0 (4) ow in view of (4) nd with the ssumption tht the vity light is initilly in vuum stte Eqution (3) goes over into t ˆ 0 (4) ât is Gussin vrile with zero men On ount of (4) together with Eqution (66) tht We finlly see to determine the solution of the expettion vlues of the tomi opertors t stedy stte Moreover the stedy-stte solution of Equtions (34)-(36) yields 39
G A Geru mˆ ˆ ˆ m m ˆ ˆ ˆ ˆ ˆ Solving these equtions simultneously one n esily otins 3 3 mˆ 3 3 ˆ ˆ 3 3 Finlly on ount of (47) nd (48) long with Eution (45) we find 3 3 3 ˆ (43) (44) (45) (46) (47) (48) (49) Two-Time Correltion Funtions Here we see to lulte the two-time orreltion funtions for the two-mode vity light To this end we relize tht the solution of Eqution (66) n write s g t ˆ t ˆ te e d e mˆ t (50) o On the other hnd one n put Eqution (40) in the form in whih Fˆm d mˆ t mˆ t mˆ ˆ t Fmt (5) t is noise opertor with zero men The solution of this eqution is expressile s ˆ mt mˆ te e d e mˆ ˆ t Fmt o (5) In ddition one n rewrite Eqution (34) s d mˆ ˆ ˆ t m t F t (53) where Fˆ t is noise opertor with vnishing men Employing the lrge-time pproximtion sheme to Eqution (53) we see tht mˆ ˆ t F t (54) on introduing this into Eqution (5) we hve ˆ mt mˆ te e d e F ˆ ˆ t Fmt o (55) 40
G A Geru ow omintion of Equtions (50) nd (55) yields g ˆ t = ˆ te e mˆ t de 0 On multiplying oth sides on the left y get d e d e F ˆ ˆ 0 0 t Fm t g ˆ tˆ t ˆ tˆ t e e ˆ t mˆ t de 0 â (56) t nd ting the expettion vlue of the resulting eqution we d e d e ˆ ˆ t F ˆ ˆ 0 0 t t Fm t Moreover pplying the lrge-time pproximtion sheme to Eqution (66) we otin With this sustituting into Eqution (57) there follows ˆ ˆ ˆ ˆ t t t t e e ˆ tˆ t de 0 (57) mˆ t ˆ t (58) g g d e d e ˆ ˆ t F ˆ ˆ 0 0 t t Fm t Sine the vity mode opertor nd the noise opertor of the tomi modes re not orrelted we see tht ˆ ˆ ˆ t F ˆ t t F t 0 (59) (60) ˆ ˆ ˆ t F ˆ m t t Fm t 0 (6) On ount of these results nd on rrying out the integrtion of Eqution (59) over we redily get It is not lso diffiult to verify tht ˆ t ˆ t ˆ t ˆ t e e ˆ t ˆ t ˆ t ˆ t e e ˆ t ˆ t ˆ t e e ˆ t ˆ t ˆ t e e (6) (63) (64) (65) 4