Schwinger effect of QED in strong fields

Transcription

1 Schwinger effect of QED in strong fields 谢柏松 北京师范大学核科学与技术学院 中国科学技术大学交叉学科理论研究中心

2 Contents 1 Introduction 2 Semiclassical treatment 3 Quantum kinetic methods 4 Summary and outlook

3 Contents 1 Introduction 2 Semiclassical treatment 3 Quantum kinetic methods 4 Summary and outlook

4 1. QED and new situation QED has been developed during the first half of the 20 century. In 1965, physics Nobel Prize awarded to J. Schwinger, R. Feynman and S. Tomonaga. Experimental test of QED theory has high precision, by high energy and low intensity with accelerator and Perturbation theory is a well-suited tool.

5 In strong-field regime: by high intensity laser system (due to chirped pulse amplification technology, CPA). X-ray free electron laser (XFEL). Extreme Light Infrastructure (ELI) plan in Europe in near future. Electron Characteristic Energy

6 Nonlinear quantum electrodynamics Even before the event of QED, Weisskopf, Heisenberg, Euler etc. derived nonlinear effects based on vacuum fluctuations. Schwinger presented derivation of vacuum polarization. Schwinger, On gauge invariance and vacuum polarization, PR82, 664(1951).

7 Nonlinear quantum electrodynamics New fectures Nonequilibrium and nonperturbative problem of QED vacuum in the strong fields. Depending on the laser s frequency and intensity. Various effects ranging from photon-photon scattering over vacuum birefringence to vacuum pair creation might arise. Delbrück scattering in strong static inhomogeneous external fields, photon splitting in strong external static fields, and elastic scattering among real photons.

8 Some vacuum effects Casimir effect: Predicted 1948, measured I atm at 10 nm, important in nanomechanics. T. Emig, PRL98, (2007). Photon interactions

9 2. Methods of QED research Semiclassical methods such as: (1)WKB approximation E. Brezin and C. Itzykson, PRD 2, 1191 (1970). V. S. Popov, JETP34, 709 (1972). A. DiPiazza, PRD70, (2004). (2)Instanton techniques S. P. Kim and D. N. Page, PRD65, (2002). H. Gies and K. Klingmuller, PRD72, (2005). G. V. Dunne and C. Schubert, PRD72, (2005). These methods relate the imaginary part of the effective action for a given background field to the vacuum persistence amplitude, which in turn can be related to the pair creation rate.

10 Quantum kinetic methods such as (1)Quantum Vlasov equation Y. Kluger et al., PRL67,2427(1991); PRD45,4659 (1992); PRD58,125015(1998). (2)Wigner function formalism I. Bialynicki-Birula et al., PRD44, 1825(1991). F. Hebenstreit, R. Alkofer and H. Gies, PRD82, (2010). Account for both the pair creation and the subsequent transport process. In addition to the pair production rate, quantum kinetic methods yield valuable phase-space information such as momentum distribution.

11 Fundamental physics with lasers Relativity + uncertainty principle quantum vacuum (virtual pair plasma). Even before Schwinger field is reached, vacuum becomes weakly nonlinear Stimulated processes. Schwinger field 1998, SLAC experiment

12 3. Schwinger mechanism Proper-time technique Gauge invariance Nonperturbative Constant fields Plane wave Schwinger, PR82, 664(1951)

13 Cases studied 1951, Schwinger: constant field+a single plane wave field. 1970, Brezin et al. : simple time-dependent electric field. 2000, more complicated time-dependent electric fields or space-dependent electric fields. As well as collinear electric and magnetic fields, e.g. S. P. Kim and D. N. Page, PRD73, (2006). Schwinger mechanism for fields with spatial and temporal variation. Temporal compression: increased production rate. Spatial compression: lower production rate. Laser fields production rate unknown.

14 Contents 1 Introduction 2 Semiclassical treatment 3 Quantum kinetic methods 4 Summary and outlook

15 1. Proper-time methods The proper-time representation of the effective action: iw (1) δw δa L (1) (1) μ = i =< i 2 d 4 0 tr xl j γ 0 (1) μ = ( x)0 ds s e 2 1 ds s e ism tr x, γ [ e is( γπ) μ >= ie tr[ γ G( x, x So the unrenormalized Lagrangian : K( x, x'; s A) ism 2 =< < x x e e is( γπ) is( γπ) 2 2 ] A)] For the propagator G, as well as for the effective Lagrangian, we need x > x' >

16 As the coordinate representation of the proper- time evolution operator ihs U ( s) = e Hamiltonian H = ( γπ) K is equal to a transformation amplitude < x U ( s) x' > = < x, s x', 0 > 2 = Π 2 e 2 σ μν F μν Schrodinger equation i t U ( s) = HU ( s) To obtain the transformation amplitude, we have to solve the dynamical Problem formulated by oneparticle equation of (proper-time) motion of Heisenberg type: dxμ dπ μ = i[ xμ, H ] = 2Π μ; = i[ Π μ, H ] ds ds

17 In 1951 Schwinger paper: (1) first solve the (operator) equations of motion; (2) secondly, insert these into the Halmiltonian; (3) thirdly, solve the Schrodinger equation Analytical results can be obtained for some simple field configuration, e.g., for constant and plane wave field. The other WKB method: W. Dittrich and H. Gies, Probing the Quantum Vacuum, Springer2000, pp.3-12.

18 Quantum Electrodynamics (FourthEdition) Pp The Effective Lagrangian of the Electromagnetic Field

19

20 Taylor expansion:

21 The integral for the energy density can be given a value by choosing a contour in the complex τ plane:

22 Recall that complex energies characterize the decay of a quantum mechanical state. The probability of a timedependent state this means that the vacuum state, which originally is free of particles, decays spontaneously in a strong electric field by creation of electron positron pairs. The particle creation rate per unit volume and time is

23 2. Worldline instantons Schwinger s formula QED pair creation by an external field can be concisely described in terms of the imaginary part of the effective Lagrangian. In scalar QED at one-loop and for a constant field this imaginary part is given by Schwinger s formula: G. V. Dunne and C. Schubert, Phys. Rev. D72, (2005).

24 Worldline effective action The Euclidean one-loop effective action for a scalar particle in an Abelian gauge background A μ is given by the worldline path integral expression

25 The new, nonlocal, worldline action : weak-field condition

26 S is stationary if the path x α (u) satisfies: Worldline instanton A highly nontrivial set of coupled nonlinear (ordinary) differential equations. The stationary instanton paths satisfy:

27 Pair production rate In a background electric field the fluctuations about the worldline instanton paths lead to an imaginary part in the effective action Γ[A], and the leading behavior is worldline action S evaluated on the worldline instanton This imaginary part of the effective action gives the pair production rate.

28 3. Our work on the pair production rate in a general polarized electric field Theoretical Frame

29 Assume: From the motion equations we have: The worldline action in the stationary conditions:

30 Assume: elliptic eccentricity Gauge fields in Euclidean space reads The stationary orbit is determined by integrating Eq.(3): Keldysh adiabatic parameter

31 Assume: If we change the integral variable from u toτ = ωx 4 the constrain conditions can be rewritten: The turning point τ 0 can be given by the stationary condition:

32 Linear polarized Stationary action In some publications, a convenient function is given:

33 By using the asymptotic approximate expressions of g in two limiting cases: nonperturbative (Schwinger mechanism) processes perturbative (multiphoton mechanism) processes

34 Circular polarized

35 similarly as in linearly polarized case:

36 Compare with the result get by WKB method : Same!

37 Elliptic polarized Two limiting case In terms of theorem of integral-mean-value for Eq.(13) and combined an algebraic equation (14),they are approximated as:

38 g function in two limiting case:

39 Further discussion in circular polarized field Rewritten Eq.(21) Combined with Eq.(22) one gets:

40 Contents 1 Introduction 2 Semiclassical treatment 3 Quantum kinetic methods 4 Summary and outlook

41 1. Quantum kinetic methods Recent papers (1) quantum Vlasov equation R. Alkofer et al., PRL87,193902(2001); C.D.Roberts, PRL89, (2002) (2) Wigner function formalism I. Bialynicki-Birula et al., PRD44, 1825(1991). F. Hebenstreit, R. Alkofer and H. Gies, PRD82, (2010).

42 A key quantity in the description of nonequilibrium particle production processes is the single-particle momentum distribution, f(p,t). For Dirac particles coupled to an Abelian gauge field it can be obtained by solving a quantum Vlasov equation. Once f(p,t) is known, the calculation of the produced particle number density and total particle number is straightforward. In general the Vlasov equation involves source and collision terms, and its coupling to Maxwell s equation provides for the field-current feedback typical of plasmas. The absolute and relative importance of these terms depends on the magnitude of the background field and the mass of the produced particles.

43 For the relatively weak XFEL-like fields it is expected that the produced particles number density to be small and hence collision of particles can be omitted. For only time-dependent electric field: df ( p, t) dt 2 ee( t) ε 2 2ω ( p, t) = t 0 ee( t')[1 2 f ( p, t')] dt' cos[2 2 ω ( p, t') t t' dτω( p, τ )] d p p = ω ( p) df ( p, t) e E( t) ε E int 2e f ( p, t) (2π ) ω( p) ee( t) p dt 8ω ( p) p with the three-vector momentum p = ( p, p ) the transverse mass-squared ε = m p and the total energy-squared ω ( p, t) = ε + p ; p = p3 ea( t)

44 2.Wigner function formalism Some papers F. Hebenstreit et al., PRD82, (2010); PRL107, (2011). Account for both the pair creation and the subsequent transport process. In addition to the pair production rate, quantum kinetic methods yield valuable phase-space information such as momentum distribution.

45 Contents 1 Introduction 2 Semiclassical treatment 3 Quantum kinetic methods 4 Summary and outlook

46 summary The worldline instantons method has been successfully extended to get the electron-positron pair production rate (PPR) in an elliptic polarized time alternating field. It is found that the PPR is reduced for elliptic polarization field compared to linear polarization field, in particular in multiphoton processes. However in nonperturbative processes of high field, there is almost same PPR as a constant field which indicates the insensitivity of polarization of field. Our results obtained has not only a consistency with previous works for involving PPR by other methods but also an implication that the polarized field effects on PPR may become more important in perturbative multiphoton regime in near future ELI possible experimentations.

47 Recent developments New developments in extreme plasma systems and lasers. [M. Dunne, Nature Phys. 2, 2 (2006); M. Buchanan, Nature Phys.2, 721 (2006)] Relativistic optics & nonlinear QED. [G. Mourou et al., RMP78, 309 (2006); M. Marklund & P. K. Shukla, ibid., 591 (2006);Y.I. Salamin, S.X. Hu, K.Z. Hatsagortsyan, & C.H. Keitel, Phys. Rep. 427,41 (2006)] New developments in numerical and experimental tools, [e.g. R.A. Fonseca, Proc. HPCPAST 2002; R.Trines et al., PRL 94, (2005)] Near future importance: laboratory astrophysics, e.g. hydro, nuclear astro. [B.A. Remington et el., RMP 78, 755 (2006)] Questions in fundamental physics, e.g. * Photon -photon scattering. [J.S D 10, 141 (2000);E.Lundstr6m et al., PRL 96,83602 (2006)] Hawking-Unruh effect. [S.W. Hawking, Nature 248,30 (l974);w.unruh, PRD 14,870( 1976); P. Chen & T.Tajima, PRL 83,256 (1999); R. Schutzhold et al., PRL (2006)] * Holographic Schwinger effect [G. W. Semenoff K.Zarembo, PRL107, (2011)]

48 Thanks for your time and attention