Exact solutions of the relativistic wave equations in strong laser - PowerPoint PPT Presentation

Exact solutions of the relativistic wave equations in strong laser fields: the Gordon–Volkov solutions and beyond. Sándor Varró Wigner Research Centre for Physics Hungarian Academy of Sciences Institute for Solid State Physics and Optics, Budapest Talk at Advances in Strong-Field Physics-ELTE. 03 February 2014.

O Outline of the talk. f 1. General and historical notes. Gordon–Volkov states. 2. Classical (relativistic) consideations on trajectories. 3. Interaction with a quantized EM radiation. Plasmons are squeezed. Photon–electron entanglement; an example for ‘EPR’ 4. Interaction with a classical EM plane wave in a medium. New exact solutions of the ‘Volkov problem’ in a medium. [5. Some perspectives of high-laser-field physics; ELI.]

Possible descriptions of photon-electron interactions Trajectory , Ray Field PHOTON Quantized Field ELECTRON (Geometric Optics) (Maxwell Theory) (True Photon) Trajectory , 2. Classical 1. 3. Classical Electrodynamics current [Point, current, Classical charged dust, Classical EM (Poisson) photon M Mechanics, h i fi ld fields, Radiation R di ti Hidrodynamics] reaction Field , Transition 5. Semiclassical 6. Quantum 4. Theory. Theory Optics. Quantum Optics Quantum Currents [Wave Currents [Wave Mechanics] [Schrödinger, KG, transitions + Dirac, Maxwell] General Photon Quantized Field Quantized Field 8. QED in 8. QED in 7. 7 9. Full QED, pair 9 Full QED pair External EM [Electron-Positron creation and Fields [e.g. e-e+ (Hole) Field,Solid back- reaction of State Physics] pair creation] charges Figure based on Table 1. of Varró S; Intensity effects and absolute phase effects in nonlinear laser-matter interactions; In Laser Pulse Phenomena and Applications (Ed. Duarte F J); Chapter 12, pp 243-266 (Rijeka, InTech, 2010) ISBN: 978-953-307-242-5.

Perturbation theory, Feynman graphs; Higher-order corrections. [Do we want to sum up contributions of hundreds of graphs? Of course not! ] sum up contributions of hundreds of graphs? Of course, not! ] Figure Appendix 1a . The fifty-six topologically distinct eight-order diagrams which provide the third correction to two-photon absorption. From Fahrad H. M. Faisal, Theory of multiphoton processes (Plenum Press, New York and London, 1987) p. 386.

Roots of the non-perturbative analyses: go back to Gordon (1927) and Volkov (1935); Semiclassical States ( )          2   [( ) ( ) ] 0 i A i A   rad rad          [ ( ) ] 0 i A  rad state Volkov s          ( ) ( ) A e A f ( / ) k x t z c 0  rad x Scattering Ionization; ‘half-scattering’ Volkov state Gordon W, Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik 40, 117-133 (1927). [ Application to strong-field: ~1964..] Schrödinger, dipole case:e.g. Keldish,... Wolkow D M, Über eine Klasse von Lösungen der Diracschen Gleichung. Zeitschrift für Physik 94, 250-260 (1935). [Application to strong-field: ~1964..] e.g. Nikishov and Ritus,... Varro_ECLIM_2010

Gordon’s solutions [ 1927 ] Gordon W, Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik 40, 117-133 (1927). [ Application to strong-field: ~1960..] Varro_ECLIM_2010

Gordon’s solutions [ 1927 ] [ ]          2   [( ) ( ) ] 0 i A i A   rad rad          ( ) ( ) ( / ) A e A f k x t z c  0 rad x Jacobi–Anger formula i  S      sin   iz t in t ( ) p 0 0  ( )  e J z e  ( ( ) ) n x N e n p p p p        ( )  d   exp [ ( ) ] N i p x I p p            ( ) 2 2 ( ) ( 1 / 2 )[ 2 ( ) ( ) I p k p p A A Gordon W, Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik 40, 117-133 (1927). [ Application to strong-field and multiphoton processes: From ~1960..] Varro_ECLIM_2010

Volkov’s solutions [ 1935 ] Wolkow D M, Über eine Klasse von Lösungen der Diracschen Gleichung. Zeitschrift für Physik 94, 250-260 (1935). [Application to strong-field: ~1960..] Varro_ECLIM_2010

Volkov states [ 1935 ] [ ]            [ ( ) ] 0 ( ) i A V  0 rad          ( ) ( ) ( / ) A e A f k x t z c  0 rad x         ( )[ ( )] k A      ( ) ( ) ( )  1  x u ps ps       2 2 k k p p            d  ( )  exp [ ( ) ] i p x I p p            ( ) 2 2 ( ) ( 1 / 2 )[ 2 ( ) ( ) I p k p p A A Wolkow D M, Über eine Klasse von Lösungen der Diracschen Gleichung. Zeitschrift für Physik 94, 250-260 (1935). [Application to strong-field and multiphoton processes: from ~1960..] Varro_ECLIM_2010

Modulated de Broglie plane waves g p            (  ip x ( / ) k x t z c ) e  p p  p 2 d  2 k k  2 d   d d p p             2 2 2 2 2 2 2 2 2 ( 2 ) 0 ik p p p A A p  d 2  First-order ordinary differential equation for  p. In vacuum: 0 k Immediately integrable, yielding the Gordon-Volkov solutions. In a medium: di Second-order ordinary differential S d d di diff i l 2   2  2  ( / ) ( 1 ) 0 k c n equation for  p m Varro_ECLIM_2010

Orthogonality and completeness.                 ( ( ) )    3 3 ( ( ) ) ( ) d r i p p  0 3 p p    / t z c      3   ( )     ( )       ( ( ) ) d d r r p p p p         / / v v t t z z c c 0 0 3 3 ps ps s s s s p s        2  ( )   ( )   ( , , ) ( , , ) dp d p v x v x   v ps p ps p    1 , 2 s               2 ( ) ( ) ( ( , , , , ) ) ( ( , , , , ) ) x x dp p d p p v v     v v ps ps ps ps    1 , 2 s          1 ( ( ) ) ( ( ) ) v v x x     0 0 4 4 2 2 Eberly J H 1969 Interaction of very intense light with free electrons Progress in Optics VII . (Ed. E. Wolf) pp 359-415 (North-Holland, Amsterdam) .Neville R A and Rohrlich F 1971 Quantum field theory on null planes. Il Nuovo Cim. A 1 625-644. Ritus V I and Nikishov A I 1979 Quantum electrodynamics of phenomena in intense fields Works of the Lebedev Physical Institute 111 5-278 (in Russian) . Bergou J and Varró S 1980 Wavefunctions of a free electron in an external field and their application in intense field interactions: II. Relativistic treatment. J. Phys. A: Math. Gen . 13 2823-2837 . Boca M and Florescu V 2010 The completeness of Volkov spinors. Rom. J. Phys . 55 511-525 . Boca M 2011 On the properties of the Volkov solutions of the Klein-Gordon equation J. Phys. A: Math. Theor. 44 445303. Varro_ECLIM_2010

First examples in the ‘laser era’: ‘Nikishov, Ritus’ [1963], ‘Brown and Kibble’ [1964]... E.g. application to high-intensity Compton scattering: A. I. Nikishov and V. I. Ritus, Zh. Eksperim. i E li ti t hi h i t it C t tt i A I Niki h d V I Rit Zh Ek i i Teor. Fiz. 46, 776 (1963) [English transl. : Soviet Phys.—JETP 19, 529 (1964)]. Brown L S and Kibble T W B, Physical Review 133, A705-A719 (1964). Varro_ECLIM_2010

L. V. Keldysh [ 1965 ]. Multiphoton ionization and optical tunneling. L. V. Keldysh, Ionization in the field of a strong electromagnetic wave. J. Exptl. Teoret. Phys. (U.S.S.R.) L V K ld h I i ti i th fi ld f t l t ti (U S S R ) J E tl T t Ph 47, 1945-1957 (November, 1964). [ Soviet Physics JETP 20, 1307-1317 (May, 1965) ] Varro_ECLIM_2010

Classical considerations. The argument of the wave at the electron’s position is proportional to the proper time of the particle. Nonrelativistic and relativistic proportional to the proper time of the particle Nonrelativistic and relativistic classical intensities.        ( ( , , ) ) sin( ( ) ) E r t e F t k r 0 0 0 0        ( , ) sin( ) B r t n e F t k r 0 0   c   | |, , / | | k e k n k k     ( ) / ( ) d m d r t dt e d r t       0 ( ( ), ) ( ( ), ) e E r t t B r t t   dt  c dt 2 2 1 [ ( ) / ] /   d r t dt c    x osc eF              10   sin 0 m x eF t osc 8 . 5 10 I 0 0 0 0    c mc