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Non-adiabatic Quantum Vlasov Equation for Schwinger Pair Production Sang Pyo Kim Department of Physics, Kunsan National University, Kunsan 573-701, Korea Instituto de F´ısica y Matema´ticas, Universidad Michoacana de San Nicolás de Hidalgo, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, Mexico Christian Schubert Instituto de F´ısica y Matema´ticas, Universidad Michoacana de San Nicolás de Hidalgo, Apartado Postal 2-82, C.P. 58040, Morelia, Michoacán, Mexico Using Lewis-Riesenfeld theory, we derive an exact non-adiabatic master equation describing the time evolution of the QED Schwinger pair-production rate for a general time-varying electric field. Thisequationcanbewrittenequivalentlyasafirst-ordermatrixequation,asaVlasovtypeintegral 2 equation, or as a third-orderdifferential equation. In the last version it relates to the Korteweg-de 1 Vries equation, which allows us to construct an exact solution using the well-known one-soliton 0 solutiontothatequation. Thecaseoftime-likedeltafunctionpulsefieldsisalsoshortlyconsidered. 2 PACSnumbers: 11.15.Tk,12.20.Ds,13.40.-f n a J I. INTRODUCTION is of relevance for a recent proposal to apply Ramsey 7 interferometry to the Schwinger effect [17]. ] Vacuumpairproductionbyastrongelectricfield,pre- h dicted by Schwinger in 1951 [1], may now finally be seen -t due to the construction of ultra-strong laser systems [2]. II. DERIVATION OF THE MASTER EQUATION p e However,the correspondingfields areverydifferentfrom h thefewspecialconfigurationsforwhichanexactcalcula- We will give the derivation of the master equation for [ tion of the pair creation rate is possible. Thus, recently the scalar QED case; the derivation for the spinor QED there has been increased interest in the development of case is similar, and will be included in a forthcoming, 2 v approximation schemes, such as semiclassical methods more detailed publication [18]. A scalar particle with 0 [3–5] and Monte Carlo simulations [6]. charge q and mass m in a homogeneous time-dependent 0 A case that is relatively amenable to an exact treat- electric field with the gauge potential Ak(t) has the 9 Fourier decomposed Hamiltonian of time-dependent os- ment is the one ofa purely time-dependent electric field. 0 cillators [in units of ~=c=1] . Here the spatial momentum is a good quantum num- 0 ber, which allows one to reduce the time evolution of 11 the system to a collection of mode equations labeled by Hˆ(t)= (2dπ3k)3 πk†πk+ωk2(t)φ†kφk , (1) 1 the fixed momentum k. The pair production calculation Z h i : can then be further reduced to a one-dimensional scat- v where teringproblem,suitable for standardnumericalorWKB i X methods [7–9]. Alternatively, the mode equation can be ω2(t) = (k qA (t))2+k2 +m2. (2) transformedtothequantumVlasovequation,anintegral k k− k ⊥ r a equationfor k(t),the totalexpectednumberofcreated We will quantize the theory in the Schrödinger picture, N pairs in the mode k [10–14] (see Ref. [15] for a compari- wherethetime-dependentquantumstateobeysthefunc- son of the two approaches). tional Schrödinger equation In this paper, we reconsider the time evolution of the QED Hamiltonian in a time-varying field using Lewis- i∂Ψ(t) =Hˆ(t)Ψ(t). (3) Riesenfeld invariant theory [16] and a suitable operator ∂t basis forming a spectrum generating algebra SU(1,1). We derive an exact non-adiabatic master equation for Inthispicturethefieldoperatorsφˆ(x)andπˆ(x)=φˆ˙† are thetimeevolutionoftheSchwingerpair-productionrate. time-independent with the momentum space commuta- Thisequationcanbewrittenequivalentlyasafirst-order tion relations matrix equation, as a quantum Vlasov equation, or as a third-order differential equation. For a specific solution [φˆk,πˆk′]=[φˆ†k,πˆk†′]=i(2π)3δkk′, (4) ansatzthisthird-orderequationrelatestothe Korteweg- deVries(KdV)equation,whichallowsustoconstructan but the corresponding creation and annihilation opera- exact solution using the well-known one-soliton solution tors with the equal-time commutators tothatequation. Wealsoconsiderthecaseofalternating time-likedeltafunctionpulsefields,atypeoffieldswhich [aˆk(t),aˆ†k′(t)]=[ˆb−k(t),ˆb†−k′(t)]=(2π)3δkk′ (5) 2 are generally time-dependent [19–22], the mean number of pairs present at time t assuming that this number was nk at the initial time t0, can now φˆk = aˆk(t)ϕk(t)+ˆb†−k(t)ϕ∗k(t), be read off from πˆk = aˆ†k(t)ϕ˙∗k(t)+ˆb−k(t)ϕ˙k(t). (6) nk,taˆ†k(t0)aˆk(t0)nk,t = νk(t0,t)2(2nk+1)+nk. h | | i | | Hereϕk is anauxiliaryfieldsatisfyingtheclassicalmode (15) equation Thus ϕ¨k(t)+ωk2(t)ϕk(t)=0, (7) as well as the Wronskian constraint k(t):= νk(t0,t)2(2nk+1) (16) N | | Wr[ϕk,ϕ∗k] ϕk(t)ϕ˙∗k(t) ϕ∗k(t)ϕ˙k(t)=i. (8) is the number of pairs spontaneously produced from the ≡ − initial vacuum by the electric field. The Eq. (7) and the Wronskian determine ϕk(t) up to a To obtain a time evolution equation for this quantity, phasefactor,whichwefixbyrequiringthatϕk(t)bereal weobservethatthe time-dependentHamiltonian(1)has at the initial time t . Thus if t is finite, then for t t the spectrum generating algebra SU(1,1). Choosing the 0 0 0 ≤ one has Hermitian basis ϕk(t)= e−iωk(0)(t−t0) (9) Mˆ(k0)(t0) = (2π1)3 aˆ†k(t0)aˆk(t0)+ˆb−k(t0)ˆb†−k(t0) , 2ωk(0) 1 (cid:2) (cid:3) (for t0 = this should hpold in the asymptotic sense). Mˆ(k+)(t0) = (2π)3 aˆk(t0)ˆb−k(t0)+aˆ†k(t0)ˆb†−k(t0) , −∞ tWiaenncootnejtuhgaattetsheaorepeLreawtoisr-sRaîke(ste)n,fˆbe−ldk(itn)vaanrdianthtse,irthheartmisi-, Mˆk(−)(t0) = (2πi)3(cid:2)aˆk(t0)ˆb−k(t0)−aˆ†k(t0)ˆb†−k(t0)(cid:3), they fulfill the Liouville-von Neumann equation (cid:2) (cid:3)(17) i∂Iˆk(t) +[Iˆk(t),Hˆk(t)]=0, (10) this algebra becomes ∂t as can be easily checked. ˆ(k0)(t0), ˆk(±)(t0) = 2i ˆk(∓)(t0), M M ± M latTedhebygrbooutnhdaˆskt(at)tean|0dkˆ,bt−ik(fot)ratnhdetkh-ethn-mthodexeciisteadnsntiahtie- (cid:2)Mˆ(k+)(t0),Mˆk(−)(t0)(cid:3) = −2iMˆ(k0)(t0). (18) is The corre(cid:2)lators are the expec(cid:3)tation values of Eq. (17) [aˆ†(t)ˆb† (t)]nk withrespectto|nk,ti,thatis,ofthenumberofproduced nk,t := k −k 0k,t . (11) pairs and of pair creation and annihilation: | i nk! | i 1+2 k(t) = (2νk(t0,t)2+1)(2nk+1), Thus the total time-dependent vacuum state is given by N | | (k+)(t) = (µk(t0,t)νk(t0,t)+µ∗k(t0,t)νk∗(t0,t)) M 0,t = 0k,t . (12) (2nk+1), | i | i × Yk k(−)(t) = i(µk(t0,t)νk(t0,t) µ∗k(t0,t)νk∗(t0,t)) M − In the free theory, the time-dependent vacuum state re- (2nk+1). (19) × ducestotheMinkowskivacuum,asexpected. Thescalar productforthe quantizedfields andtheir hermitiancon- Note that all three correlators are real and proportional jugates allows us to find the Bogoliubov transformation to the quantum number 2nk+1, and thus proportional between the past time t and the present time t, which to the ones defined by the vacuum state. 0 is given by UsingEq. (14)andthemodeequation(7),wefindthe first order master equation aˆk(t0) = µk(t0,t)aˆk(t)+νk(t0,t)ˆb†−k(t), ˆb†−k(t0) = µ∗k(t0,t)ˆb†−k(t)+νk∗(t0,t)aˆk(t), (13) ddt1M+k2(−N)k=Ωk(0−) Ωk(0−) Ω(k0+)1M+k(2−N)k, where M(k+)  0 −Ω(k+) 0  M(k+) µk(t0,t) = iWr[ϕ∗k(t0),ϕk(t)],     (20) νk(t0,t) = iWr[ϕ∗k(t0),ϕ∗k(t)]. (14) where TheBogoliubovcoefficientssatisfytherelationforbosons (±) ωk2(t) ωk2(t0) Ω (t):= ± , (21) |µk(t0,t)|2−|νk(t0,t)|2 =1. Ourmainobjectofinterest, k ωk(t0) 3 with the initial conditions k = nk, k(±) = 0 at t = with the initial condition f(t0) = f˙(t0) = 0. Then, the N M t (where t may be ). An immediate consequence correlatorsare given by 0 0 −∞ of the master equation (20) is the conservation of the t quantity 1+2 = 1+ω dt′f(t′)Ω(−)(t′), 0 N t Z0 (1+2 k)2 ( (k+))2 ( k(−))2 =(1+2nk)2. (22) (−) = ω0f(t), N − M − M M t This relates to the conservation of charge, as well as to (+) = ω0 dt′f(t′)Ω(+)(t′). (28) M − t the invariance of the Casimir operator for the SU(1,1) Z0 algebra [18]. The spinor QED case can be treated anal- Alternatively the integral equation (27) can, taking one ogously [18]. As far as the master formula (20) is con- derivative, be converted into a third order linear differ- cerned, the generalization to the fermionic case requires ential equation, only changing 1+2 k(t) to 1 2 k(t), and replacing N − N ... (ω2)˙ ωk2(t) by F +4ω2F˙ +2(ω2)˙F = (29) ω2 0 ωk2(t)=(kk−qAk(t))2+iqE(t)+k2⊥+m2. (23) where t F(t):= dt′f(t′) (30) t III. ALTERNATIVE FORMULATIONS OF THE Z0 MASTER EQUATION andtheinitialconditionsareF(t )=F˙(t )=F¨(t )=0. 0 0 0 Observe that F¨ is absent in Eq. (29), which by Abel’s The first order matrix equation (20) can be equiva- theorem implies that the Wronskian of the solutions of lently rewrittenbothasa singleintegralequationandas the corresponding homogeneous equation is constant. a third order linear differential equation. Since we work The differential equation (29) bears an interesting re- withafixedmodek,inthissectionwewillgenerallysup- lationshiptotheKdVequation. Theformoftheintegral press the index k and abbreviate ω := ω(t ). We will equation (27) suggests the ansatz 0 0 now also set nk =0. (ω2)˙(t) ω2(t) ω2 First, we combine the equations for (±) to a second f(t)= , F(t)= − 0. (31) order inhomogeneous equation for (−M), 8ω04 8ω04 M Definingr(t):=ω2(t)/ω2andthenu(x,t):= r(x 10t), d2M(−) Ω˙(+)dM(−) +(Ω(+))2 (−) one can show that u sat0isfies the KdV equati−on, − dt2 − Ω(+) dt M d Ω(−) uxxx−6uux+ut =0. (32) =Ω(+) (1+2 ) . (24) dt Ω(+) N ThuswecanusecertainsolutionsoftheKdVequationto h i calculatepaircreationratesforthecorrespondingelectric ThehomogeneouspartofEq. (24)hastheexactsolutions fields. (−)(t)=C±e±iRtt0dt′Ω(+)(t′) (25) M IV. EXACTLY SOLVABLE CASES with integration constants C±. Using those in the usual Wewillnowstudytwoexactlysolvablecases. First,we waytoconstructthesolutionoftheinhomogeneousequa- consider the following soliton-type solution of the KdV tionwiththeappropriateinitialconditions,weobtainthe equation (see, e.g., Refs. [23–25]) quantum Vlasov equation as the integral equation 2 t u(x,t)= 1 , (33) d(1+2 (t)) = Ω(−)(t) dt′ Ω(−)(t′)(1+2 (t′)) − − cosh2(x 10t) dt N t N − Zt0 h which corresponds to cos( dt′′Ω(+)(t′′)) . (26) 2 1 × t′ r(t)=1+ , F(t)= . (34) Z i cosh2(ω t) 4ω2cosh2(ω t) 0 0 0 Second, inspection of the master equation (20) shows, This is a solution to Eq. (29) with the appropriate thatits generalsolutioncanbe parameterizedby afunc- boundary conditions at t = . The gauge potential 0 tion f(t) fulfilling the integral equation −∞ is Ω(−)(t) t 2ω2 f˙(t)= ω0 −2Zt0 dt′f(t′) ω2(t)+ω2(t′) (27) qA(t)=kk−skk2+ cosh2(0ω0t). (35) (cid:0) (cid:1) 4 From Eq. (28) we get the exact pair creation rate, each describing the exact time evolution of the cumula- tive pair creation variable k(t) for an electric field that 1 N (t)= . (36) depends only ontime, but is arbitraryotherwise. To the N 8cosh4(ω0t) bestofourknowledge,theseequationsarenew. Wehave concentrated here on scalar QED, leaving the details of Note that (t) returns to zero for t , which is N → ∞ thespinorQEDcasetoamoreextensivepublication[18]. due to the solitonic character that makes the scattering In future work, we also plan to study the precise con- reflectionless. In fact, the mode solution to Eq. (7) is ditions under which a non-adiabatic treatment is really given by necessary. To define the adiabatic approximation, we e−iω0t write the mode solution in terms of the adiabatic basis ϕ(t)= A(t), (37) [11] √2ω 0 where the amplitude is e−iθ(t) eiθ(t) ϕk(t)=αk(t) +βk(t) , (42) A(t)=(e2ω0t+1)2 F (2,2 i;1 i; e2ω0t) (38) 2ωk(t) 2ωk(t) 2 1 − − − with 2F1 the hypergeometric function, and it does not where θ(t) = tt dt′pωk(t′) and the Bpogoliubov relation 0 hgoalvieubaovnecgoaetffiivceiefnrteq(u1e4n)ciys part in the future. The Bo- |αβkk(|2t)−2|iβnkt|2he=Rde1fihnoitlidosn. (W16e) tohfen(rte)p.lace |νk(t0,t)|2 by | | N From Eq. (14) one can easily show that for this ap- ν(t)= e2iω0tA˙∗(t), (39) proximation to hold it is sufficient to assume that 2ω 0 which approximately leads to (t) = 2e4ω0t for ω0t ωk(t) ωk(0) ω˙k(t) −1 and N(t)=2e−4ω0t for ω0tN≫1, the leading appro≪x- sωk(0) −sωk(t) , ωk2(t) ≪|βk(t)| (43) imation to the exact formula (36). Thus there is no pair (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) creationinthiscase,contrarytothesomewhatsimilarly- throug(cid:12)houtthe time evolut(cid:12)ion.(cid:12)This c(cid:12)riteriumis similar, looking Sauter field case [26]. This example also shows althoughnot strictly equivalent, to the one givenin [11], clearly that, as emphasized in Ref. [14], no direct physi- cal meaning should be ascribedto k(t) at intermediate ω˙ ω¨ N times. 1, 1. (44) ω2 ≪ ω3 ≪ Second, we consider an electric field consisting of two opposite delta function pulses, In any case, all the inequalities in (43),(44) are certainly fulfilled for even the strongest laser sources which are E(t)=E0δ(t) E0δ(t t1), (40) presently existing or in development. Those have a field − − strength still much lower than the critical strengthEc = which has the gauge potential of a potential well [17]: m2/e andthe characteristictime scale much longerthan Ak = 0 for t < 0 and t > t1, corresponding to ωk(0), the Compton time [2]. and Ak =−E0 for 0<t<t1, corresponding to ωk. The Concerning the relation of the master equation to the master equation (20) together with continuity at t = 0 KdVequation,althoughthereisawell-knownconnection leads to the pair production for the period 0<t<t 1 between the latter equation and one-dimensional quantum mechanical scattering (see, e.g., Refs. [23–25, 27]), (+) Ω 2 1+2 k(t) = (2nk+1) k it appears not to have been previously applied to the N 2ωk Schwingerpaircreationproblem. Itwillbeinterestingto (cid:16) (cid:17) Ω(−) 2 see whether also the multi-soliton solutions of the KdV k 1 cos(2ωkt) , (41) equation may be used in this context. × − Ω(+) h (cid:16) k (cid:17) i Finally, let us mention that it is straightforward to extendourmasterequationtothe caseofaninitialstate andfortheperiodt>t itnowremainsconstant,retain- 1 which is a thermal state at temperature T. As will be ing its value for t . Note that for a single delta function 1 shown in Ref. [18], such a change leads again only to an pulse k(t) keeps oscillating, so that the limit t cannotNbe defined. 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