I. INTRODUCTION Recetly, the hoto creatio i a emty cavity with movig boudaries, so called the dyamical Casimir eect has attracted much attetio esecial

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1 SNUTP Productio of hotos by the arametric resoace i the dyamical Casimir eect Jeog-Youg Ji, Hyu-Hee Jug y, Jog-Woog Park z,kwag-su Soh x Deartmet of Physics Educatio, Seoul Natioal Uiversity, Seoul , Korea Abstract We calculate the umber of hotos roduced by the arametric resoace i a cavity with vibratig walls. We cosider the case that the frequecy of vibratig wall is!1( = 1; 2; 3; :::) which is a geeralizatio of other works cosiderig oly 2!1, where!1 is the fudametal-mode frequecy of the electromagetic eld i the cavity. For the calculatio of time-evolutio of quatum elds, we itroduce a ew method which is borrowed from the time-deedet erturbatio theory of the usual quatum mechaics. This erturbatio method makes it ossible to calculate the hoto umber for ay ad to observe clearly the eect of the arametric resoace Ca, w, Dv Tyeset usig REVTE Electroic address: jyji@hyb.su.ac.kr y Electroic address: hhjug@gmc.su.ac.kr z Electroic address: jwark@gmc.su.ac.kr x Electroic address: kssoh@hya.su.ac.kr 1

2 I. INTRODUCTION Recetly, the hoto creatio i a emty cavity with movig boudaries, so called the dyamical Casimir eect has attracted much attetio esecially i a vibratig cavity [1{3]. It was also roosed that the high-q electromagetic cavities may rovide a ossibility to detect the hotos roduced i the dyamical Casimir eect [5,6]. Therei, they cosidered the vibratig wall with the frequecy = 2! 1 ad foud the resoace excitatio of the electromagetic modes. For the solutios of Mathieu dieretial equatio x + 2 (1 + cos t)x =0; (1.1) it is well kow that arametric resoaces occur whe the frequecy with which the arameter oscillates is close to ay value 2= with itegral [7]. Sice there are may mode frequecies for the dieretial equatio describig the electromagetic elds i a cavity, itis atural to cosider other frequecies with =! 1 : Now the method itroduced i Ref. [6] caot be used i these geeral cases because the geeratig fuctio ca be made oly for the secial case = 2: Therefore we itroduce a ew erturbatio method which leads to easily solvable equatios. I ricile, we may solve the equatios to ay order of exasio arameter ; but we calculate u to the rst order of ; which is suciet to see the eect of the arametric resoace. The orgaizatio of this aer is as follows. I Sec. II we review the scheme of the eld quatizatio i the case of movig boudaries. I Sec. III we itroduce the ew erturbatio method to d time evolutio of quatum electromagetic eld. Here we write the domiat art of the solutio of wave equatio which results from the arametric resoace. I Sec. IV we calculate the umber of hotos created by the vibratio of the boudary. The last sectio is devoted to the summary ad discussio. Here, we discuss the hysical roerties of the arametric resoace i the couled dieretial equatios, ad we estimate the hoto umber created i realistic situatio. Fially, the higher order calculatios are cosidered briey. II. QUANTUM ELECTROMAGNETIC FIELDS IN A CAVITY WITH MOVING WALLS Let us cosider a emty cavity formed by two erfect coductig walls, oe beig at rest at x = 0 ad the other movig accordig to a give law of motio L(t) whe 0 <t<t: The eld oerator i the Heiseberg reresetatio A(x; t) associated with a vector otetial obeys the wave equatio (c =1) ad ca be writte 2 2 A =0 A(x; t) = [b (x; t)+b y (x; t)]: (2.2) 2

3 Here b y ad b are the creatio ad the aihilatio oerators ad (x; t) is the corresodig mode fuctio which satises the boudary coditio (0;t)=0= (L(t);t): For a arbitrary momet of time, followig the aroach of Ref. [8{10], we exad the mode fuctio as (x; t) = k Q k (t)' k (x; t) (2.3) with the istataeous basis ' k (x; L(t)) = s 2 kx si L(t) L(t) : (2.4) Here Q k (t) obeys a iite set of couled dieretial equatios [4]: Q k +! k 2 (t)q k =2 j g kj _Q j + _ j g kj Q j + _ 2 j;l g jk g jl Q l (2.5) where = _ L=L ad g kj = ad the time-deedet mode frequecy is ( ( 1) k j 2kj j 2 k 2 (j 6= k) 0 (j = k) : (2.6)! k (t) = k L(t) : (2.7) For t 0; the right had side of Eq. (2.5) vaishes ad the solutio i this regio is chose to be Q k (t) = e i! kt k : (2.8) The the quatum eld ca be writte as A(x; t 0) = e i!t [b ' (x; L 0 )+H:c:] (2.9) 2! L 0 is the iitial distace betwee the walls ad! = L 0 : From the form of the Hamiltoia H = P! (b y b ); we ca iterret by b as the umber oerator associated with the hoto with the frequecy! : After the oscillatio of the wall (t T ); the solutio of Eq. (2.5) with the iitial coditio (2.8), ca be writte as Q k (t T )= k e i! kt + k e i! kt : (2.10) 3

4 From (2.2) ad (2.3), we have A(x; t T )= e i!t [a ' (x; L 0 )+H:c:]; (2.11) 2! where the ew creatio ad aihilatio oerators a y ad a are writte i terms of b y ad b, usig the followig Bogoliubov trasformatios a k = a y k = [b k + b y k] [b y k + b k ]: (2.12) Further, it follows from H = P! (a y a ) that ay a is the ew umber oerator at t T: If we start with a vacuum state j0 b i such that b j0 b i = 0, the exectatio value of the ew umber oerator is N k = h0 b j a y k a k j0 b i = 1 =1 j k j 2 ; (2.13) which ca be iterreted as the umber of created hotos. (Oe should ote that the quatum state does ot evolves i time i the Heiseberg icture.) III. TIME EVOLUTION OF QUANTUM ELECTROMAGNETIC FIELDS IN A CAVITY WITH AN OSCILLATING WALL I this sectio we d the time evolutio of quatum eld oerator (2.2) by solvig Eq. (2.5) with the motio of the wall give by L(t) =L 0 [1 + si(t)]: (3.1) Here =! 1 = =L 0 ad is a small arameter characterized by the dislacemet ofthe wall. This is a geeralizatio of the revious work i Ref. [6] where the secial case =2 was treated. For 1; havig i mid that (t) ad takig the rst order of i the mode frequecy (2.7)! k (t) = k L 0 [1 + si(t)] 1 ; (3.2) we ca relace Eq. (2.5) by a air of couled rst-order dieretial equatios _Q k = P k P_ k =! 2 L k (1 2 si t)q k +2 _ L + L L j j g kj P j g kj Q j : (3.3) 4

5 Let us ow itroduce or iversely, ;k = r!k 2 Q k i P k! k (3.4) The (3.3) reads s 1 Q k = [ ;k + ;k+ ] 2! k P k = i r!k 2 [ ;k + ;k+ ]: (3.5) _ ;k = i! k ;k i! k si t[ ;k + ;k+ ] cos t j i 2 2 si t j g kj s!j! k [ ;j + ;j+ ] g kj 1!j! k [ ;j + ;j+ ]: Itroducig the iite dimesioal colum vector ~ (t) = 0 ;1 ;1+ ;2 the above equatio ca be writte as a matrix form. 1 C (3.6) A ; (3.7) d ~ (t) =V (0) ~ (t)+v (1) ~ (t) (3.8) dt ad V (0) ad V (1) are matrices. The comoets of the matrices are ad V (0) k;j 0 = i! k kj 0 (3.9) V (1) k;j 0 = s=! 1 v s k;j 0esi! 1t ; (3.10) where v s k;j 0 = g kj s j k s 4j! s k 2 kj (3.11) with s; ; 0 =+; :Here we used =! 1 ad! k = k! 1 : To d the solutio of Eq. (3.8), we itroduce a erturbatio exasio: 5

6 ~ = ~ (0) + ~ (1) + 2 ~ (2) + : (3.12) The iteratio method used to solve this roblem is similar to what we did i time-deedet erturbatio theory. By isertig (3.12) ito Eq. (3.8), idetifyig owers of yields a series of equatios: d dt d dt ~ (0) = V (0) ~ (0) (3.13) ~ (1) = V (1) ~ (0) + V (0) ~ (1) : (3.14) From the iitial coditio (2.8), we have the solutio to zeroth order equatio (3.13) Further, the Eq. (3.14) is easily solved (1) ;k(t) =e i! kt (0) ;k = k e i! kt : (3.15) Z t 0 dt 0 e i! kt 0 Usig (3.15) ad (3.10), it ca be writte exlicitly as that is, (1) ;k (t) =! 1e ik! 1t Z t 0 ;j j; 0 V k;j 0 (0) dt 0 e ik! 1t 0 +v + k;j 0ei! 1t 0 ) j 0e ij! 1t 0 =! 1 e ik! 1t Z t 0 j; 0 (v k;j 0e i!1t0 dt 0 (v k; e i(k++)! 1t 0 0: (3.16) +v + k; e+i( k+ )! 1t 0 ); (3.17) (1) (t) = ;k+ v k ; E k (t) ++k v+ k ; E k +k (t); (1) ;k (t) =v k ; E k + k(t)+v + k ; Ek k(t); (3.18) where E k m(t) = (! 1 te ik! 1t ; for m =0; e ik! 1t ); for m 6= 0: i m (e i(m+k)! 1t (3.19) Therefore we have, usig (3.18) ad (3.5), Q k (t) = 1 e i! kt k + [ v k ; E k ++k(t) v + k ; E k +k(t) + v k ; E k + k(t)+v + k ; Ek k(t)] + O( 2 ): (3.20) 6

7 Oe should ote that Q (1) k icludes terms roortioal to! 1 t which are the eects of arametric resoace. I the usual situatio, sice! 1 t 1, oly the resoace terms are domiat ad the solutio (3.20) becomes by retaiig oly them: Q k (t) 1 e i! kt k +! 1t [ v + k ; ei! kt k; + v k ; e i! kt k;+ + v + k ; e i! kt k; ]: (3.21) IV. NUMBER OF PHOTONS CREATED BY THE PARAMETRIC RESONANCE After some time iterval T the wall stos at x = L 0. The the wave fuctio becomes (x; t > T )= k " k e i! kt + k e i! kt ' k (x): (4.1) The mode fuctio (2.3) ad its time-derivative should be cotiuous at t = T : k Q k (T )' k = k k e i! kt + k e i! kt! ' k k k _ Q k (T )' k + Q k (T )_' k = i! k k e i! kt + i! k k e i! kt! ' k ; (4.2) where we used (2.10) for the mode fuctio at t>t:by multilyig ' l to both sides of the above equatios ad itegratig, we ca get k ad k, which are k = k = i! k Q k _Q k + _ L L i! k Q k + _Q k _ L L Retaiig oly the domiat terms (! 1 T 1) Usig (2.6) ad (3.11), ally we have l l g kl Q l! e i! k T i 2! k g kl Q l! e i! k T i 2! k : (4.3) k =! 1 Tv + k ; k; : (4.4) j k j 2 = 1 4 k(! 1T ) 2 k; : (4.5) Therefore the total umber of hotos created i the k th mode from the emty cavity is 7

8 N k = 1 =1 j k j 2 = ( 1 4 ( k)k(! 1T ) 2 k< 0; otherwise: (4.6) This result is a geeralizatio of Ref. [6] i the short time limit (! 1 T 1) ad it agrees with that result for =2ad k =1:It should also be oted that the maximal umber of hotos are created at the mode frequecy k = 2 or! k = 2 : (4.7) for =eve ad at its earest eighbor frequecies k =(1)=2 for =odd: V. DISCUSSION We chaged the secod order couled dieretial equatio (2.5) to the rst order dieretial equatio (3.6) by itroducig the ew variables (3.4). This makes it easy to deal with the dieretial equatio ad to d the erturbatio series by virtue of diagoalizatio of V (0) : The 1 -order solutio is foud exlicitly ad it icludes the terms roortioal to time ad this term is relatively large comared to other terms which iclude oly oscillatig arts. Cosiderig oly those domiat terms, we calculated the umber of hotos created after stoig of the wall vibratio. The results show that the eect of arametric resoace is the largest at the half of the frequecy of the vibratig wall (! k ==2): This ca be uderstood by cosiderig the Mathieu equatio (1.1) where the arametric resoace takes lace most strogly for = 2: While, i the case > 2; we see the other resoace eects i additio to! k ==2;which is due to the eect of couligs with other mode frequecies i the cavity. This ca be iterreted as the arametric resoace for the couled dieretial equatio. Followig the discussios of Ref. [6], we estimate the rate of hoto geeratio i the several modes. Our results (4.6) which are valid oly i the limit! 1 T 1 may be used to estimate the relative hoto umbers deedig o the mode frequecy. For a log time, we assume that the hoto umbers are roortioal to time [3,5] or to exoetial fuctio of time [2]. The we have a hoto distributio roortioal to Eq. (4.6). For examle, takig =4wehave N 2 =4N D ad N 1 = N 3 =3N D where N k is the umber of created hotos with! k : Here we used the same exerimetal arameters as cosidered i Ref. [6]: max v s max =c ; v s max 50 m=s;! 1 =2 10 GHz for L 0 2 cm; ad N D 300 hotos durig T =1s:For these exerimetal arameters, we exect that N 2 = 1200 hotos ad N 1 = N 3 = 900 hotos: The higher order calculatios ca be erformed by takig the dieretial equatio to higher order i ad iteratig the itegral. Here we ote that whe we make the -order dieretial equatio like (3.6) from (2.5) we cosider the higher order comig from (3.2). However if we limit the roblem to the domiat terms which result from the arametric resoace, the roblem is simle. Let us exlai these by cosiderig the 2 -order. The dieretial equatio ca be writte, i the 2 -order, as d dt ~ (2) = V (2) ~ (0) + V (1) ~ (1) + V (0) ~ (2) : (5.1) 8

9 By itegratig this dieretial equatio it is clear that the V (2) term gives at most the term roortioal to 2! 1 T: This is very small comared to (! 1 T ) 2 which comes from itegratig the resoace term of (1) : Therefore it is eough to write d dt to cosider -th order calculatio. By iteratios, we have ~ () = V (1) ~ ( 1) + V (0) ~ () ; (5.2) () ;k(t) 1! (! 1t) e ik! 1t 1 ;:::; 1; s ;:::;s 1 0; k+ v s k;; 1 ( 1 ); 1 v s 1 1 ( 1 ); 1 ; 2 ( 2 ); 2 v s 2 2 ( 2 ); 2 ; 1 ( 1 ); 1 v s 1 1 ( 1 ); 1 ;; ; (5.3) where k = s 1 +:::+s k : For large t, we should be careful about the covergecy of the series. We exect that the covergecy roerty may be rovided by the factor 1=!. However the exlicit calculatios are too dicult to d the simle geeral form of the solutio ad this will be our cocer of future work [11]. ACKNOWLEDGMENTS This work was suorted by the Ceter for Theoretical Physics (S.N.U.), Korea Research Ceter for Theoretical Physics ad Chemistry, ad the Basic Sciece Research Istitute Program, Miistry of Educatio Project No. BSRI Oe of us (JYJ) is suorted by Miistry of Educatio for the ost-doctorial fellowshi. 9

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