Generalized Delta Functions and Their Use in Quasi-Probability Distributions

Transcription

1 Geealzed Delta Fuctos ad The Use Quas-Pobablty Dstbutos RA Bewste ad JD Faso Uvesty of Maylad at Baltmoe Couty, Baltmoe, MD 5 USA Quas-pobablty dstbutos ae a essetal tool aalyzg the popetes of quatum systems, especally quatum optcs The Glaube-Sudasha P-fucto P( α ) s especally useful fo calculatg the desty matx of a system, but t s ofte assumed that P( α ) may ot exst fo hghly quatum-mechacal systems due to ts sgula atue Hee we defe a geealzed delta fucto wth a complex agumet ad deve ts popetes, whch ae vey dffeet fom those of a covetoal Dac delta fucto The geealzed delta fucto s the used to calculate P( α ) fo a Schödge cat state a supsgly smple fom The geealzed delta fucto effectvely covets the dagoal elemets that appea P( α ) to off-dagoal elemets the desty opeato Smla techques ca be used fo moe geeal quatum states I INTRODUCTION Quas-pobablty dstbutos ae wdely used to descbe the popetes of quatum-mechacal systems [-8] Ulke a tue pobablty dstbuto, quas-pobablty dstbutos ca have egatve values that demostate the oclasscal popetes of the coespodg system As a esult, plots of quas-pobablty dstbutos ca povde sght to the popetes of a system a mxed state I addto, quas-pobablty dstbutos ca be used to calculate the popetes of quatum systems that would be dffcult o mpossble to calculate othe ways Ceta quas-pobablty dstbutos ae ofte sgula, whch s aothe dcato of the oclasscal atue of a system I ths pape, we toduce a geealzed deltafucto δ ( z) wth a complex agumet z ad use t to chaacteze the sgula atue of these quas-pobablty dstbutos We deve the popetes of geealzed delta fuctos ad show that they ae vey dffeet fom a covetoal Dac delta fucto Delta-fuctos wth a complex agumet have bee befly dscussed pevously [9- ] but wth seveal dffeet deftos ad o goous poof of the popetes, as s dscussed moe detal the ext secto The Glaube-Sudasha P-fucto P( α ) s a quaspobablty dstbuto that s especally useful fo calculatg the desty matx of a system usg the elato () ˆ P( α) α α dα dα Hee α s a coheet state wth complex ampltude α wth eal ad magay pats α ad α The P-fucto mplctly defed by Eq () coespods to a dagoal epesetato of the desty opeato a bass of coheet states [-3], wheeas the desty opeatos fo hghly oclasscal states have offdagoal elemets that ae fequetly as lage as the dagoal elemets As a esult, the P-fucto ca be hghly sgula ad t s ofte assumed that P( α ) may ot exst fo hghly oclasscal states such as a Schödge cat state [,] Hee we use the popetes of the geealzed delta fucto to calculate P( α ) fo a cat state a emakably smple fom The popetes of the geealzed delta fucto povde a staghtfowad explaato of how the dagoal P-fucto ca epeset off-dagoal desty opeatos The ma goals of ths pape ae to toduce the popetes of geealzed delta fuctos ad the use them to calculate hghly sgula quas-pobablty dstbutos Geealzed delta fuctos wth complex agumets ae defed ad the popetes deved the ext secto Secto III befly evews quas-pobablty dstbutos ad some of the applcatos, cludg the elatoshp betwee the most commoly used quas-pobablty dstbutos The Glaube- Sudasha P-fucto fo a Schödge cat state s deved Secto IV, whch llustates the usefuless of the geealzed delta fucto as well as povdg a smple fom fo P( α ) As a futhe example, the effects of a lea amplfe o a Schödge cat state ae cosdeed Secto V, whee t s show that the geealzed delta fucto ases atually the lmt of low ga ad decoheece A summay ad coclusos ae povded Secto VI II GENERALIZED DELTA FUNCTIONS I ths secto, we dscuss two equvalet epesetatos fo a geealzed delta fucto ad deve ts popetes Ulke a oday Dac delta fucto, geealzed delta fuctos ca oly be appled to tegals volvg a aalytc fucto f( z ) wth complex agumet z I addto, the popetes of geealzed delta fuctos wll be see to be moe smla to a tegal aoud a cotou the complex plae tha the popetes of a covetoal Dac delta fucto A Defto ad basc popetes We defe a geealzed delta fucto δ ( z) wth a complex agumet z by z / σ ( z) lm e () δ σ σ Hee t s udestood that δ ( z) wll be cluded a tegal alog the eal axs as s the case fo a oday Dac delta fucto δ ( x), as llustated Fg a Eq () s equvalet to oe of the usual epesetatos fo a Dac delta fucto except that hee z ca be a complex umbe To be moe

2 pecse, Eq () coespods to a dstbuto athe tha a tue fucto Delta fuctos wth complex agumets have also bee defed [] as δ( z + z ) δ( z ) δ( z ), (3) whee z ad z ae the eal ad magay pats of z As dscussed moe detal Appedx B, ths defto of δ ( z) coespods to the poduct of two covetoal Dac delta fuctos, whch s completely dffeet fom the popetes of the geealzed delta fucto defed Eq () Oe of the goals of ths pape s to elmate ths ambguty alog wth devg the popetes of δ ( z) To avod cofuso, we wll use the otato δ ( z + z) δ( z) δ( z) [] to dstgush ths fom δ ( z) It wll be show below that δ ( z) has the fudametal popety that Moe specfcally, t wll be show that δ ( z) has the popety that ad f ( α, α) dα ( ( x + y)) dα f ( x + y, α) (6) f ( α, α) dα ( ( x + y)) dα f ( α, x + y) (7) Hee α ad α ae the eal ad magay pats of the agumet of a complex aalytc fucto f ( α ) Itegals of ths kd wll appea Secto IV whee we calculate the Glaube-Sudasha P-fucto fo a cat state, fo example f ( x) d ( x z) dx f ( z), (4) whee f( z ) s a aalytc fucto of the complex vaable z Although Eq (4) s supefcally smla to the popetes of a covetoal Dac delta fucto, δ ( z) dffes seveal mpotat espects, cludg the fact that z ca be complex The geealzed delta fucto δ ( z) s ozeo ad sgula ove a exteded age of the eal axs, ulke a Dac delta fucto, as s dscussed moe detal Appedx A The tegal Eq (4) eplaces the value of the eal agumet x f( x ) wth a complex umbe z, whch s moe smla to the popetes of a cotou tegal aoud a pole the complex plae tha a oday delta fucto, as llustated Fg b Oe mpotat dffeece s that the tegal of the fucto f( z ) eed ot vash alog a path at fty as s equed fo a cotou tegal aoud a pole, ad the tegals of teest below would dvege alog such a path Ulke a covetoal Dac delta-fucto, Eq (4) oly holds f f( z ) s a aalytc fucto, whch was ot metoed eale dscussos of delta fuctos wth complex agumets [9,] To see ths, cosde the case whch a oaalytc fucto f( z ) s defed by f( z) fo Im( z) < f( z) f( z) fo Im( z), whee f ( z ) ad f ( z ) ae two dffeet aalytc fuctos The t s appaet that the tegal Eq (4) wll gve a value of f ( ) ad ot f ( ) whe z, sce the tegal alog the eal axs s cofed to the ego whee f( z) f( z), eve though the oaalytc fucto f( z ) s defed as f ( ) at that pot (5) Fg Compaso of a geealzed delta fucto wth a covetoal Dac delta fucto (a) Itegato of a covetoal Dac delta fucto alog the eal axs, whee δ ( x a) s ozeo oly at the pot x a (b) Itegato of a geealzed delta fucto δ ( z) alog the eal axs gves the value of the tegad at the pot x z, whch does ot le alog the eal axs fo complex z Ths s smla to the popetes of a cotou tegal aoud a pole, except that the path eed ot be exteded to fty to fom a closed loop I addto, δ ( x z) s ozeo ad sgula ove a exteded ego alog the eal axs B Devato fo aalytc fuctos f(z) I ode to deve Eq (4), we make use of the fact that f( z ) has bee assumed to be aalytc, whch allows t to be expaded a Taylo sees as ( f ) () f( z) z (8)! ( Hee f ) ( z ) s the th devatve of f Isetg Eq (8) to Eq (4) gves ( ) f () F( z) x! d ( x z) dx, (9)! whee we have let F( z ) deote the esult of the tegal If we defe the tegal I Eq (9) as

3 3 I x d ( x z) dx, () the t ca be see that Eq (4) wll hold povded that I z Fom the defto of the geealzed delta fucto Eq (), I ca be ewtte as ( x z) / σ I lm x e dx () σ σ If we deote the eal ad magay pats of z by the Eq () becomes zz σ ( z )/ z ad z, e ( x z ) / / l m σ xz σ I x e e dx () σ σ The tegal Eq () has the fom of a vese Foue tasfom wth espect to the paamete z Evaluatg the tegal Eq () gves I / / z Γ + σ L / σ lm eve σ (3) + ( )/ / z lm Γ σ L( )/ odd σ σ Hee Γ ( x) s the gamma fucto ad L a ae the modfed Laguee polyomals defed by a m + a x L ( x) ( ) m m! (4) m We ca ow cosde the lmt of Eq (3) as σ Fom Eq (4), the Laguee polyomals Eq (3) have tems m that volve σ m whe s eve ad σ whe s odd, whee m s a tege betwee ad The coespodg factos Eq (3) ae σ fo eve ad σ fo odd As a esult, the oly tems Eq (3) that do ot go to as σ ae the oes wth m Usg the fact that Γ ( ) ( )! fo educes Eq (3) to I m z (5) Isetg Eqs (5) ad () to Eq (9) gves the Taylo sees expaso of Eq (8), whch shows that F( z) f( z) as desed It ca be see fom Eqs () ad (5) wth that d ( x z) dx (6) Eq (6) demostates that the geealzed delta fucto s a popely omalzed dstbuto C Itegal epesetato Eq () epesets the geealzed delta fucto as a sequece of Gaussa fuctos wth a complex agumet We ow show that the geealzed delta fucto ca also be epeseted by the Foue tasfom of a eal expoetal as zp xp d ( x z) e e dp (7) p A somewhat dffeet poof of ths popety was suggested Ref [] based o the assumpto that the vese Foue tasfom of ceta sgula fuctos must exst It wll be coveet to defe a fucto F ( x ) as ad a tegal I gve by zp xp Fz ( x) e e dp (8) p I f ( x) F ( x) dx (9) Hee f( z ) s oce aga assumed to be a aalytc fucto wth the Taylo sees expaso of Eq (8) As befoe, we eed oly cosde the tegal I defed hee as z I x F ( x) dx () Isetg Eq (8) to Eq () gves zp xp I x e e dpdx p z () Itechagg the ode of tegato allows ths to be ewtte as by zp xp I e x e dxdp p () The Foue tasfom of x s well kow ad s gve ( ) x e xp dx p d ( p), (3) ( whee δ ) ( p) s the th devatve of the usual Dac delta fucto Isetg Eq (3) to Eq () gves zp ( I e d ) ( p) dp (4) z

4 4 Ths tegal ca be evaluated usg tegato by pats to gve I ( z) (5) Fom the Taylo sees expaso of f( z ) t follows that f ( x) Fz ( x) dx f ( z) (6) Eq (6) has the same sftg popety as Eq (4), fom whch t ca be cocluded that magal pobablty dstbuto Pm ( x ) fo the vaable x s gve by Pm ( x) Wxp) (, dp (3) Eq (3) would hold classcal pobablty theoy f Wxp (, ) wee the jot pobablty dstbuto fo x ad p The oclasscal atue of ths state eques, howeve, that Wxp (, ) take o egatve values A smla elato holds fo the magal pobablty Pm ( p ) fo the vaable p Fz ( x ) δ ( x z ) (7) Combg Eqs (7) ad (8) thus establshed the tegal epesetato of the geealzed delta fucto Eq (7) By makg a chage of vaables fom p to p' p, the tegal Eq (7) ca be cast to the equvalet fom zp xp d ( x z) e e dp (8) p These esults establsh the fudametal sftg popety of the geealzed delta fucto Eq (4) As metoed above, Ref [] suggested a etely dffeet defto fo delta fuctos wth a complex agumet As a esult, we have toduced the tem geealzed delta fucto to elmate ay possble cofuso III QUASI-PROBABILITY DISTRIBUTIONS We befly evew some of the popetes of quaspobablty dstbutos ths secto As metoed above, plots of quas-pobablty dstbutos ca be vey useful vsualzg the atue of oclasscal states [5-6, 8] Quaspobablty dstbutos ae also a essetal tool may calculatos quatum optcs [4,7-8] Oe of the most commoly used quas-pobablty dstbutos s the Wge dstbuto Wxp (, ) defed as W ( x, p) x + q x q e dq p ˆ pq ρ (9) Hee x ad p ae two cojugate quatum vaables, such as the two quadatues of a sgle-mode electomagetc feld The Wge dstbuto has the popety that t ca have egatve values, ulke a tue pobablty dstbuto Negatve values of the Wge dstbuto ofte ase fom the supeposto of two quatum states [8], ad the egatve egos of the Wge dstbuto dcate oclasscal behavo A example of a Wge dstbuto wth egatve values s show Fg, whch coespods to a plot of Wxp (, ) fo a umbe state cotag two photos Although Wxp (, ) has egatve values ad thus caot coespod to a tue pobablty dstbuto, t has the popety that the Fg The Wge dstbuto Wxp (, ) fo the umbe state ψ, plotted as a fucto of the quadatues x ad y A poto of the plot has bee emoved ode to show the ego whee the Wge dstbuto becomes egatve, whch llustates the oclasscal atue of ths state Two othe commoly used quas-pobablty dstbutos ae the Glaube-Sudasha P-fucto P( α ) ad the Husm-Kao Q-fucto Q( α ) [-] Hee α s a complex umbe coespodg to the ampltude ad phase of a coheet state The Q-fucto s defed by [-] Q( α) α ˆ ρ α (3) Q( α ) has the advatage that t ca be eadly calculated fo a gve desty opeato dectly fom Eq (3) The Q-fucto ca, tu, be used to calculate the P-fucto, whch s mplctly defed [-3] by Eq () The P-fucto has the advatage that the desty opeato ca be eadly calculated fom t usg Eq () The P-fucto teds to be hghly sgula fo quatum mechacal systems As we shall see secto IV, P( α ) fo a Schödge cat state ca be expessed a smple fom usg geealzed delta fuctos By coveto, α s elated to the eal quadatue paametes x ad y by α (x y) + (3) Moe geeally, each of these quas-pobablty dstbutos ca be calculated fom oe aothe [-] as

5 5 llustated Fg 3 Eqs () ad (3) ca be combed ad veted to gve [] P( α) e Q( ξ) e dξ dξ ( ) ξ /4 ( αξ + αξ ), (33) whee Q ( ξ ) s the Foue tasfom of the Q-fucto ad ξ ξ + ξ s the cojugate vaable of α Eq (33) acts as a tasfomato fom the Q-fucto to the P-fuctos, whch coespods to oe of the aows Fg 3 ot the oly oes that exst but they epeset the smplest set of tasfomatos betwee the vaous quas-pobablty dstbutos IV SCHRÖDINGER CAT STATES Havg deved the popetes of geealzed delta fuctos ad toduced the theoy of quas-pobablty dstbutos, we wll ow use these esults to vestgate the sgula atue of the Glaube-Sudasha P-fucto As a example, we wll deve P( α) fo a Schödge cat state Ths wll be doe by fst calculatg Q( α ) ad the tasfomg that to P( α ), whch coespods to two of the aows Fg 3 As a cosstecy check o the esults, we wll also calculate the desty matx coespodg to P( α ) usg Eq () ad compae t wth the ogal desty matx, whch completes a closed loop of tasfomatos Fg 3 A Exstece of the P-fucto Fg 3 Possble tasfomatos fom oe quas-pobablty dstbuto to aothe Hee ψ epesets the state of the system o the coespodg desty matx fo a mxed state, Q epesets the Q-fucto, P epesets the P-fucto ad W epesets the Wge dstbuto The aows epeset possble tasfomatos fom oe quas-pobablty dstbuto to aothe Othe equvalet tasfomatos may also exst but ae ot cluded hee The Wge dstbuto ca be foud fom the P- fucto by covolvg t wth a Gaussa [] α β W( α) P( β) e dβdβ (34) The Q-fucto ca the be calculated fom the Wge dstbuto by covolvg t wth yet aothe Gaussa [] α β Q( α) W ( β) e dβdβ (35) As metoed above, the P-fucto teds to be sgula fo systems that ae quatum atue [,8] Covolvg the P- fucto wth a Gaussa smooths the fucto so as to elmate ay sgulates the Wge fucto The fucto s futhe smoothed whe the Wge fucto s covolved wth yet aothe Gaussa Eq (35) to obta the Q-fucto Covolvg wth a Gaussa also teds to smea out some of the quatum mechacal featues that ca be see a plot of the Wge fucto, makg them less vsble a plot of the Q-fucto I those cases whee the P-fucto ca be plotted, t teds to dsplay moe fomato tha ca be see eve the Wge fucto Eqs (), (3) ad (33)-(35) all coespod to the tasfomatos show dagammatcally Fg 3 As metoed the capto of Fg 3, these tasfomatos ae Because of ts hghly sgula atue, t s ofte assumed that P( α ) does ot exst fo hghly oclasscal states such as a Schödge cat state [,] To the best of ou kowledge, thee has oly bee oe pevous calculato of P( α ) fo a Schödge cat state, wth the esult that [3] P( α) A δ ( α α) + ζ δ ( α α) + e ( α ( α + α )/ (/)( α α) / ( α ( α+ α)/) ζ e (/)( α α) / ( α ( α+ α)/) e δ ( α ( α + α ) / ) + ζ e (/)( α α) / ( α ( α+ α)/) (/)( α α) / ( α ( α+ α)/) e δ + ) ( α ( α α ) / ) (36) Eq (36) volves expoetals of dffeetal opeatos ad coespods to a fte expaso Asde fom ts complexty, the covegece of ths expaso s ot appaet ad t s of lmted usefuless Fo example, t would be dffcult to use Eq (36) to calculate the ogal state of a Schödge cat as a cosstecy check The defto of P( α ) Eq () coespods to a dagoal epesetato of the desty opeato a bass of coheet states, wheeas the desty matx of a Schödge cat state has lage off-dagoal tems A epesetato of ths kd s possble because of the ove-completeess of the coheet states Oe advatage of ou appoach s that the use of geealzed delta fuctos explctly shows how the dagoal P( α ) ca epeset off-dagoal desty opeatos I patcula, the ecostucto of the desty opeato wll volve tegals of the fom f( α, α) dα ( z ) dα ( z ) dαdα, (37) whch ca be evaluated usg Eqs (6) ad (7)

6 6 B Q-fucto ad ts Foue tasfom A Schödge cat state coespodg to a supeposto of coheet states ca be defed as ψ A( α + ζ α ) (38) Hee ζ s a complex umbe ad A s a omalzato costat The coespodg desty opeato fo the Schödge cat state of Eq (36), a pue state, s gve by ( A + ˆ ρ α α + ζ α α ζα α ζ α α ) (39) Each of the fou tems the desty opeato of Eq (39) ca be wtte geeal as ρ κ γ β (4) Hee κ s a appopate complex costat ad β ad γ ae equal to α o α Sce the tasfomatos of Eqs (), (3) ad (33)-(35) ae lea, we ca calculate each tem of the vaous quas-pobablty dstbutos fo the ρ sepaately ad combe the esults at the ed Isetg Eq (4) to the defto of the Q-fucto Eq (3) gves Usg the fact that κ Q ( α) α γ β α (4) ( α + β αβ ) α β e, (4) we ca wte the geeal tem the Q-fucto as Ths ca be ewtte as ( β γ ) α + β + κ + Q ( ) e α αγ α e (43) ( β γ ) ( ( α + βα+ αγ )) κ + Q ( ) e α e (44) Eq (4) ca be used oce aga to gve A α α α α Q( α) e + ζ e ( α + αα ( αα+ α α)) + ζ α α e ( α + α α ( α α+ α α)) + ζ α α e (46) Eq (46) coespods to the Q Fg 3 fo the Schödge cat state of Eq (38) A calculato of the coespodg P-fucto wll eque the Foue tasfom of the Q-fucto, whch s gve by [] ( αξ αξ ) ( ξ) + ( α) α α (47) Q Q e d d Isetg the value of Q( α ) fom Eq (45) gves Q κ β γ ( ξ ) e α + βα+ αγ ( αξ + αξ ) e e dα dα The tegal Eq (48) ca be evaluated to gve (48) α + βα+ αγ ( αξ + αξ ) e e dαdα (49) ξ /4 ( β + γ) ξ / ( β γ) ξ / e e e e Isetg Eq (49) to (48) gves the expesso fo ( ξ ) as Q ξ /4 ( β + γ ) ξ / ( ) / (5) Q ( ξ) κe β γ e e β γ ξ C Calculato of the P-fucto The P-fucto fo a Schödge cat state ca ow be calculated by setg the Foue tasfom of the Q-fucto fom Eq (5) to Eq (33), whch gves ( β γ) ξ/ αξ Pβ γ ( α) κ β γ e + e dξ ( β γ) ξ / αξ e e dξ (5) It ca be see that Eq (5) volves the vese Foue tasfoms of complex expoetals Usg the tegal epesetato of a geealzed delta fucto Eq (8) educes Eq (5) to κ ( α + ( βα+ αγ)) Q ( α) β γ e (45) Ths gves the full Q-fucto as P ( α) β + γ β γ δ α κ β γ δ α (5)

7 7 Note that γ β the fst two tems Eq (39), whch case P ββ ( α) κδα ( Re{ β}) δα ( Im{ β}), (53) whee δ s aga the usual Dac delta fucto Usg the otato [, ] followg Eq (3) gves δα ( Re{ β}) δα ( Im{ β}) δ ( α β), (54) Combg all fou tems allows the full P-fucto fo the Schödge cat of Eq (38) to be wtte as P( α) A δ α α + ζ δ α α ( ) ( ) α + α α α + ζ α α δ α δ α α + α α α + ζ α α δ α δ α, (55) whch s fa smple ad moe useful tha Eq (36) Eq (55) s oe of the ma esults of ths pape D Cosstecy of the esults As a cosstecy check o these esults, we wll ow use the P-fucto of Eq (55) to calculate the coespodg desty opeato ad vefy that t s equal to the ogal desty opeato I patcula, we would lke to udestad moe detal how the dagoal α α tems P( α ) Eq () ca epeset off-dagoal desty opeatos If the geealzed delta fuctos Eq (55) had the same popetes as covetoal Dac delta fuctos, the the effect sde the tegal would be to make the eplacemet α α' whee α ' s aothe complex umbe I that case, the desty opeato of Eq () would have to ema dagoal because α α', α α' (56) But the effects of the geealzed delta fucto ae ot at all equvalet to Eq (56), as ca be see usg the defto of a coheet state the fom α / ( α + α) α e (57)! Hee s a umbe state cotag photos The adjot of ths equato gves α / ( α α) α e (58)! Fo ow, we wll cosde oly the thd tem Eq (55), whch has the effect of makg the eplacemet α + α ( α + α )/ + [( α α )/] α α α ( α + α )/ [( α α )/] α (59) Asde fom a omalzg costat, we see fom Eqs (57)-(59) that α α, α α, (6) cotast wth Eq (56) The pot s that the geealzed delta fucto has dffeet effects o α ad α, as ca be see fom Eq (6) The et effect of ths s to covet dagoal tems the desty opeato of Eq () to off-dagoal tems I moe detal, we ca ewte the dagoal tems Eq () usg α α ( α ) ( ) ( ) + α α + α α α e j k jk!! j k j k (6) Isetg Eqs (5) ad (6) to the defto of the P-fucto gves each of the tems ρ ˆ the fom ˆ ρ k β γ e j k ( ( β γ ) /4 ( β γ ) /4 ) + + β + γ + ( β γ) (6) β + γ + ( )( β γ) j k jk!! Eq (6) ca be smplfed to gve j k k j k ˆ γ β ρ k β γ e j k (63) jk!! Usg Eq (4) Eq (63) gves j k j k ˆ ( β + γ )/ γ β ρ ke j k (64) jk!! Fom the defto Eqs (57) ad (58) j k j k ( β + γ )/ γ β γ β e j k (65) jk!! Compag Eqs (64) ad (65) shows that j ˆ ρ κ γ β (66)

8 8 Usg the appopate vaables ad costats equato (66) gves a desty opeato that s exactly the same as the ogal desty opeato Eq (39) Ths esult shows that the P-fucto does gve back the coect off-dagoal desty matx whe followg the cycle llustated by the aows Fg 3, as would be expected Nevetheless, ths s a stkg esult as ca be see by cosdeg a cat state defed by e φ ψ A( α + α ), (67) whee α s ow a eal umbe What Eq (6) shows ths example s that, whe tasfomg fom the P-fucto back to the ogal state, the geealzed delta fuctos eplace α wth α whle α s eplaced wth α tself, eve though α s a eal umbe Although we have oly cosdeed the use of the geealzed delta fucto to calculate the P-fucto of a Schödge cat state hee, smla techques ca be used to calculate the P-fucto fo moe geeal states The elevace of geealzed delta fuctos to the theoy of quatum ose amplfes wll be dscussed the ext secto V LINEAR AMPLIFICATION I ths secto we cosde a stuato whch a Schödge cat state s passed though a lea phasesestve amplfe wth ga g Ths poduces a mxed state wth a o-sgula P( α ) that ca be expessed as a sum of Gaussas wth a fte wdth σ As the ga appoaches uty, σ ad these Gaussas become geealzed delta fuctos as descbed the pevous secto These esults show that geealzed delta fuctos ca appea atually physcal systems the lmt of small decoheece It has pevously bee show by Caves et al [4] that the effect of a lea phase-sestve amplfe o ay optcal sgal s to smply scale the paamete α the Q-fucto by Q out α ( α) Q g g (68) Hee Q ( α ) ad Q out ( α ) ae the Q-fuctos befoe ad afte the amplfcato pocess Ths s a supsgly smple esult that allows the effects of quatum ose to be aalyzed a staghtfowad way Applyg ths tasfomato to the geeal Q-fucto tem of a Schödge cat Eq (45) gves Q βg ( α g β g g ( βα α g )) + + κ β g g ( α) e (69) g Isetg Eq (69) to the Foue tasfom Eq (47) gves Q βg κ β g ( ξ ) e g βg α ( βα αg ) g g ( αξ+ αξ) + + e e dα dα Evaluatg the tegal Eq (7) gves (7) α + ( βα+ αg ) g g ( αξ + αξ ) e e dαdα (7) ξ g g ( βξ+ ξg) βg 4 ge e e Combg Eqs (7) ad (7) gves the Foue Tasfom of the Q-fucto as ξ g g ( β ξ+ ξg) 4 Q ( ξ) κ β g e e (7) βg Eq (7) ca ow be seted to Eq (33) to obta the P-fucto the fom P βg κ β g ( α) ( ) ξ g ( g ) ( βξ+ ξg ) 4 ( αξ+ αξ ) e e e dξdξ The tegal Eq (73) ca be evaluated to gve ( g ) 4 ( βξ+ ξg) 4 e g ξ g ( αξ + αξ ) e e e d d ( α g βg g ( βα αg )) + + g ξ ξ (73) (74) Isetg Eq (74) to Eq (73) gves the geeal tem the P- fucto as P βg κ β g ( α) e ( g ) ( α g βg g ( βα αg )) + + g (75) Combg all fou tems P ( α ) gves the full P-fucto fo the amplfed Schödge cat state as P out ( α) A e + ζ e ( g ) + ζ α α e α gα g ) α gα g ) /( /( g g /( g ( α + α α ( α α+ α α )) ) ( α + g α α g( α α+ α α)) /( g ) + ζ α α e (76) We ow defe the paamete σ by

9 9 g σ (77) Isetg Eq (77) to Eq (75) allows the fou tems P( α ) to be ewtte the fom ( α g( β + g)/) /σ Pβg ( α) κ β g e σ ( α g( β g)/) /σ e σ (78) Fom the defto of δ ( z) Eq (), t ca be see that the two expoetal tems Eq (78) become geealzed delta fuctos the lmt as σ, whch coespods to takg the lmt g I that lmt, Eq (76) educes to the P- fucto fo a Schödge cat wthout amplfcato as gve Eq (55) Plots of Q( α ) ad P( α ) fo the amplfed Schödge cat state ae show Fg 4 fo seveal values of the ga The P-fucto caot be plotted fo the case of g, sce t educes to a combato of geealzed delta fuctos that lmt The fuctoal fom of a geealzed delta fucto the complex plae s gaphcally llustated, howeve, Appedx A These esults show that geealzed delta fuctos ca ase atually the descpto of quatum-mechacal systems the lmt of small decoheece, whch suggests that they may occu a umbe of othe applcatos as well Fg 4 Plots of the Q ad P fuctos gve by Eqs (69) ad (76) espectvely fo dffeet values of the ga g These coespod to a Schödge cat state fom Eq (38) wth the paametes show the uppeleft coe Note that whe g appoaches uty the P-fucto becomes the geealzed delta fucto ad caot be plotted It ca also be see that the P-fucto ad Q-fucto become appoxmately the same whe the ga s elatvely lage VI SUMMARY AND CONCLUSIONS We have defed a geealzed delta fucto δ ( z) as a sequece of Gaussa dstbutos that s smla to oe of the usual epesetatos of a Dac delta fucto but wth a complex agumet z A equvalet tegal epesetato was gve Eq (7) It was show that f ( x) d ( x z) dx alog the eal axs gves the value f( z ) povded that the fucto f s aalytc As llustated Fg, ths popety of geealzed delta fuctos s moe smla to a cotou tegal aoud a pole tha t s to a covetoal Dac delta fucto I addto, δ ( z) s ozeo ad sgula ove a exteded ego of the eal axs, ulke a covetoal Dac delta fucto Delta fuctos wth complex agumets have bee befly dscussed pevously [9-] but wth seveal dffeet deftos ad o goous poof of the popetes Pat of ou motvato fo cosdeg geealzed delta fuctos s the uque ablty to descbe the sgula atue of the Glaube-Sudasha P-fucto P( α ) The P-fucto s vey useful fo calculatg the desty matx of a system usg Eq (), but t s ofte assumed that P( α ) does ot exst fo vaous quatum states due to ts hghly sgula atue Hee we calculated P( α ) fo a Schödge cat state a smple fom that volves geealzed delta fuctos As a cosstecy check o ths esult, we used P( α ) to calculate the coespodg desty opeato, whch ageed wth that of the ogal Schödge cat state A addtoal motvato fo ths appoach s that t clealy shows how the dagoal P- fucto ca epeset desty opeatos wth off-dagoal tems We also showed that the Gaussa epesetato of the geealzed delta fucto ases atually whe a Schödge cat state s passed though a lea phase-sestve amplfe The P-fucto that case coespods to a set of Gaussas wth a fte wdth σ fo ga g >, whle σ the lmt of g Thus geealzed delta fuctos may be expected to play a ole a vaety of physcal systems the appopate lmt Fo example, t has pevously bee oted that delta fuctos wth complex agumets ca be used to smplfy the aalyss of asymptotc tegals ecouteed classcal electomagetsm [9] We ae cuetly usg geealzed delta fuctos to calculate the P-fucto of othe quatum supeposto states ad we have foud them to be valuable aalyzg the effects of lea amplfes o etagled states, fo example As a esult, we expect that geealzed delta fuctos wll be of use a vaety of futue applcatos Ths wok was suppoted pat by the Natoal Scece Foudato ude gat No 478

10 REFERENCES [] WP Schlech, Quatum Optcs Phase Space (WILEY-VCH, Bel, ) [] KE Cahll ad RJ Glaube, Phys Rev 77, 88-9 (969) [3] ECG Sudasha Phys Rev Lett, (963) [4] CM Caves, J Combes, Z Jag, ad S Padey, Phys Rev A 86, 638 () [5] K Vogel ad H Rske, Phys Rev A 4, (989) [6] V Bužek ad MS Km, Phys Rev A 48, (993) [7] J Eselt ad H Rske, Phys Rev A 43, (99) [8] C Fee, Rep Pog Phys 74, 6 () [9] IV Ldell, Am J Phys 6, (993) [] AD Poulaks, Tasfoms ad Applcatos Hadbook (CRC Pess, Boca Rato, ) d ed [] JW Has ad H Stocke, Hadbook of Mathematcs ad Computatoal Scece (Spge-Velag, New Yok, 998) [] B Sades, Phys Rev A 45, (99) [3] L Madel, Physca Scpta, 34-4 (986) APPENDIX A: FUNCTIONAL FORM The geealzed delta fucto ca be hghly sgula ove a exteded ego of the eal axs Hee we cosde the fuctoal fom of the geealzed delta fucto ad povde a gaphc llustato of the atue of ts sgulaty We beg by cosdeg the epesetato of the geealzed delta fucto gve by Eq () the text [9] z / σ ( z) lm e (A) δ σ σ As was doe the ma text, we let z z + z whch gves ( z z )/ σ zz/ σ ( z) lm e e (A) δ σ σ If we goe the complex expoetal Eq (A) fo the momet, t s appaet that the lmt goes to zeo wheeve z > z whle t dveges whe z z Retug to the complex expoetal, t ca be see that the magay pat of δ ( z) s zeo whe ethe z o z s equal to zeo, whle t s udefed othewse We kow that the value of the complex expoetal Eq (A) must le somewhee o the complex ut ccle The ete expesso must go to zeo whe the fst facto Eq (A) goes to zeo Howeve, the locato of the secod facto o the complex ut ccle becomes udefed whe the fst expoetal tem goes to Ths stuato s kow as complex fty, whch we deote by Combg these agumets, the fuctoal fom of the geealzed delta fucto ca be summazed as f Re{ z} d ( z) f Re{ z} ad Re{ z} > Im{ z} f Re{ z} ad Re{ z} Im{ z} (A3) By compaso, the usual Dac delta fucto ca be descbed by f x δ ( x) f x (A4) It ca be see that Eq (A3) educes to Eq (A4) whe the magay pat of z s equal to zeo Eq (A3) ad Fg A epeset the fom of the geealzed delta fucto tself Whe t s cluded a tegal wth a complex agumet, as Eq () the text, the cete pot of Fg A s shfted to a pot the complex plae As a esult, the tegad s sgula ove a exteded ego of the eal axs As dscussed the text, t s udestood that δ ( z) s oly defed whe t s cluded such a tegal Fg A Schematc epesetato of the fom of the geealzed delta fucto δ ( z) as summazed Eq (A3) The og of ths plot would be shfted to the complex plae whe the geealzed delta fucto s cluded a tegal wth a complex agumet These esults ae llustated schematcally Fg A

11 APPENDIX B: ALTERNATIVE DEFINITIONS As metoed the ma text, a delta fucto wth a complex agumet has bee defed two dffeet ways [9- ] Ou defto of the geealzed delta fucto Eq () was motvated pat by the eed to dstgush betwee these two deftos, as s dscussed moe detal ths Appedx A delta fucto wth a complex agumet has pevously bee defed [] by δ[( x + y) ( a + b)] δ[( x a) + ( y b)] δ( x a) δ( y b) (B) I ou otato [] ths would be wtte stead as δ Fom Eq (B) oe ca see that ths s actually a poduct of two oday Dac delta fuctos Ths delta fucto has a epesetato gve by z / / ( ) lm σ z σ δ z + z e e, (B) σ σ whch s qute dffeet fom the defto of δ ( z) Eq () the text The geealzed delta fucto δ ( z) s used sde oedmesoal tegals, wheeas δ ( z) s teded to be cluded a double tegal ove both the eal ad magay pat of the agumet It has the popety that f( z) d ( z zd ) zdz f( z) (B3) The Foue tasfom epesetato of δ ( z) s gve by d ( z + z ( z + z)) ( z z) ξ ( z z) ξ e e dξ, dξ ( ) (B4) whch ca be cotasted wth Eq (7) We have defed the geealzed delta fucto δ ( z) ode to dstgush t fom the poduct of two oday Dac delta fuctos Both δ ( z) ad δ ( z) ae useful, howeve, ad both appea ou aalyss of Schödge cat states the text