exact solutions of the relativistic wave equations in...

Exact solutions of the relativistic wave equations in strong laser fields:

the Gordon–Volkov solutions and beyond.

Sándor VarróWigner Research Centre for Physics

Hungarian Academy of SciencesInstitute for Solid State Physics and Optics, Budapest

Talk at Advances in Strong-Field Physics-ELTE. 03 February 2014.

O fOutline of the talk.

2. Classical (relativistic) consideations on trajectories.

1. General and historical notes. Gordon–Volkov states.

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

PHOTONELECTRON

Trajectory, Ray(Geometric Optics)

Field(Maxwell Theory)

Quantized Field (True Photon)

Trajectory, current [Point, charged dust, M h i

1. 2. Classical ElectrodynamicsClassical EM fi ld R di ti

3. Classical current, Classical (Poisson) photon

Mechanics, Hidrodynamics]

fields, Radiation reaction

Field, Transition Currents [Wave

4. 5. Semiclassical Theory

6. Quantum Optics QuantumCurrents [Wave

Mechanics]Theory. [Schrödinger, KG, Dirac, Maxwell]

Optics. Quantum transitions + General Photon

Quantized Field 7 8. QED in 9 Full QED pairQuantized Field [Electron-Positron (Hole) Field,Solid State Physics]

7. 8. QED in External EMFields [e.g. e-e+ pair creation]

9. Full QED, pair creation andback- reaction of 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

0])()[( 2 radrad AiAi

( )

0])([ radAi

stat

e

)()( 0 fAeA xrad )/( cztxk

Vol

kov

s

Ionization; ‘half-scattering’ 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,...

Varro_ECLIM_2010

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,...

Gordon’s solutions [ 1927 ]

Varro_ECLIM_2010

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

Gordon’s solutions [ 1927 ][ ]

0])()[( 2 radrad AiAi


)()( eNxSi

ppp

tin

n ntiz ezJe 00 )(sin

Jacobi–Anger formula

])([exp

)()( dIxpiN pp

pp

)()(2)[2/1()( 22)( AAppkI p

Varro_ECLIM_2010

Gordon W, Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik 40, 117-133 (1927). [ Application to strong-field and multiphoton processes: From ~1960..]

Volkov’s solutions [ 1935 ]

Varro_ECLIM_2010

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

Volkov states [ 1935 ][ ]

)(0])([ 0 VAi rad


2

)]()[(1)( )()( upkAkx psps

])([exp

2)( dIxpi

pk

p p

)()(2)[2/1()( 22)( AAppkI p

Varro_ECLIM_2010

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..]

Modulated de Broglie plane waves g p

xipp e )( )/( cztxk

p

22 pd

k

2222

2

pdd

k

0)2(2 2222

pp AApp

dd

pik

In vacuum:02 k

di S d d di diff i l

First-order ordinary differential equation for p.

Immediately integrable, yielding the Gordon-Volkov solutions.

Varro_ECLIM_2010

In a medium:0)1()/( 222 mnck

Second-order ordinary differential equation for p

Orthogonality and completeness.

)()(3

)(3)(

0)(3 pp

pp ird

)(3)(

0)(3 pp

sspsrdczt /

cztv /)(30 pp sssppsrd

),,(),,( )()(2

xx vvpddp pspsv

cztv /

),,(),,( )()(22,1

xx vvpddp pspsv

ps

p

)()(1

),,(),,(

240

2,1

xx

vv

pp pss

psv

)()( 240 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

Varro_ECLIM_2010

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.

First examples in the ‘laser era’: ‘Nikishov, Ritus’ [1963], ‘Brown and Kibble’ [1964]...

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

Varro_ECLIM_2010

E.g. application to high-intensity Compton scattering: A. I. Nikishov and V. I. Ritus, Zh. Eksperim. 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).

L. V. Keldysh [ 1965 ]. Multiphoton ionization and optical tunneling.

L V K ld h I i ti i th fi ld f t l t ti J E tl T t Ph (U S S R )

Varro_ECLIM_2010

L. V. Keldysh, Ionization in the field of a strong electromagnetic wave. J. Exptl. Teoret. Phys. (U.S.S.R.) 47, 1945-1957 (November, 1964). [ Soviet Physics JETP 20, 1307-1317 (May, 1965) ]

Classical considerations. The argument of the wave at the electron’s position is proportional to the proper time of the particle Nonrelativistic and relativisticproportional to the proper time of the particle. Nonrelativistic and relativistic

classical intensities.

)sin(),( 00 rkerE tFt )(),( 00

)sin(),( 00 rkenrB tFt

||/,|,| kknkek c

)),(()()),((

/]/)([1

/)(22

0 ttdt

tdcette

cdttd

dttdmdtd rBrrE

r

r

teFxm sin0

IeFxoscosc 1000 105.8

0

mcc 0

Classical considerations. The argument of the wave at the electron’s position is proportional to the proper time of the electron. This is the consequence of kk=0

)(),( Ft enrB )(),( Ft erE ct /rn

.)(1 consttc

tdd

rn

uFcmeddu )/(/ 0

)(2

Fxd 22 1

dxdzd

Along the polarization x-direction one receives formally a Newton equation in dipole approximation !

)(2

eFd

m

.)/1( constcvz

2 21

ddx

dd

cdzd

2 yd.)/( z02

dyd

2

02

02

0 21

ddxmcmcm 000 2

d

Extreme radiation, from terahertz to xuv from ultrarelativistic motion

ImceF

9

0

00 10

)4/1/( 200

0

)( 00 tkkkJtz

kk 2sin]/).]1/([[)2/()( 0

V. 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 . Lecture Notes (in Hungarian) Theor. Physics . SZTE (2012).

‘Ponderomotive potential’ for acceleration to the ultrarelativistic regime. This is present in the ‘A2 term’ in the Volkov stateThis is present in the A term in the Volkov state.

22 /213 ][105.2)( wrp eeVIU r ][105.2)(p eeVIU r

Coupling parameters. Classical.The question of switching on/off Initial value problem!The question of switching-on/off. Initial value problem!

)cos()/(),( 00 rkrnerE tctfFt 0/ reEc

CceE

ImceFx

coscosc 100

0 105.8

0.5

1.0t = 1, j = -p2

1/ czt

2/ czt

0.0Ft

2 1 0 1 2-1.0

-0.5

-2 -1 0 1 2t T

See: S. V. & F. Ehlotzky, Z. Phys. D 22, 691-628 (1992)

Squeezing in interaction of free electrons with quantised e.m. radiation fields

Multiphoton generalization of the Klein–Nishina formula for arbitrary intensity. This is an example which also shows that the nonclassical nature of the strong light field can even

if t it lf i th ki ti f th HHG t Q ti d d ti t ti lmanifest itself in the kinematics of the HHG spectrum. Quantized ponderomotive potential.

Relative depletion. From the

scattlaser AAi ˆ])ˆ([

From the quantized pondero-motive energy shift.

2200

200

nn C

n

2221 200 sinn

C

Varro_ECLIM_2010

Squeezing effect through the joint interaction in the “system of free electrons plus the quantized radiation modes”. I. Exact (stationary, squeezed) states. Measurement of non-classicality of decaying surface plasmon light.

■ Squeezing always shows up in photon – free electron interactions due to the “A2” term in ‘Q Volkov state’interactions due to the “A2” term in ‘Q-Volkov state’.

)2/1ˆˆ()(ˆ1 2

AAeH rAp )2/1()(

2

AAcm

H ii

rApi

)ˆˆˆˆ(r )ˆˆ()(

2);( ;; AAAAAAr

rrenE eDeSnDS

PP

S. Varró, N. Kroó, D. Oszetzky, N. Nagy and A. Czitrovszky, Hanbury Brown and Twiss type correlations with surface plasmon lihgt. Journal of Modern Optics 58, 2049-2057 (2011). [ Varró S, Theoretical aspects of Hanbury Brown and Twiss type correlations mediated by surface plasmon oscillations; Poster. Book of Abstracts PQE-2011, p. 253.. ]

Squeezing effect through the joint interaction in the “system of free

■ Distorted photon distribution of the “pump photons”; squeezed plasmon excitation → SPO → spontaneous photon

q g g j yelectrons plus the quantized radiation modes”. II. Plasmon statistics.

squeezed plasmon excitation → SPO → spontaneous photon.

10;0;2

1)!(2

1 2)1(2 22

aaHn

eW nn

nasq

n 2)!(2 n

)/(;; 2211 ppRppNpN sq

)1(2

)1(1;

1

2

22

2

2

22 aa

r2tanh2200

182 10 Ia 0;0 sqR

Varró S, Theoretical aspects of Hanbury Brown and Twiss type correlations mediated by surface plasmon oscillations; Poster. Book of Abstracts PQE-2011, p. 253., The 41st Winter Colloquium on the Physics of Quantum Electronics. (January 2-6, 2011 – Snowbird, Utah, USA)

Two-electron Volkov states. [Moller scattering in strong laser field.] Effective multiphoton potential 2)(potential. )]2/sin([|)|/()( 1

2)( rkrr zJeV nn

eff )/(21 pcz

S f ( ) f S f ( ) fFig. 7. Shows, on the basis of Eq. (1), the variation of the electron-electron effective potential along the propagation direction of the plasmon wave, in case of the four-photon absorption of the e-e pair ( ), for incoming laser intensity. We have also taken into account the assumed field-enhancement factor , and z=2 .

Fig. 8: Shows, on the basis of Eq. (1), the variation of the electron-electron effective potential along the propagation direction of the plasmon wave, in case of the four-photon absorption of the e-e pair ( ) for incoming laser intensity, by assuming the same field-enhancement factor as in Fig. 7a but here z=9 .

Varro_CEWQO_2009N. Kroó, P. Racz and S. V., Surface plasmon assisted electron pair formation in strong electromagnetic field.Submitted. arXiv-1311.6801 (2013) . Derivation of the effective potential in: BERGOU J., VARRÓ S. and FEDOROV M.V., J.Phys A 14, (1981) 2305.

Einstein- Podolsky-Rosen paradox with Entangled Photon – Electron Systems in High-Intensity Compton Scattering. I.

Electron detection

nsPh

oton

Photons

Photon detectionkntt

kkg 0)()(

Electrons

S. V. : Entangled photon-electron states and the number-phase minimum uncertainty states of the photon field. New Journal of Physics 10 053028 (35 pages) (2008) Varró S : Entangled states and entropy remnants of a photon electron system Physica Scripta

rtrrdtg

),()(

Varro_CEWQO_2009

Physics, 10, 053028 (35 pages) (2008). Varró S : Entangled states and entropy remnants of a photon-electron system. Physica Scripta T140 (2010) 014038 (8pp). [ Note: Recent results and ideas on short and long trajectory in HHG on atoms.. G. Kolliopoulos, et al, Revealing quantum path details in high-field physics., arXiv:1307.3859. Entanglement source. I. K. Kominis, G. Kolliopoulos, D. Charalambidis, P. Tzallas, Quantum Information Processing at the Attosecond Timescale. arXiv:1309.2902.]

‘Exotic example’ for ‘EPR’. Entangled Photon – Electron States. Photon statistics depends on the position of the detected electron after high-intensity Compton scattering IIon the position of the detected electron after high-intensity Compton scattering. II.

Varro_CEWQO_2009

Varró S : Entangled photon-electron states and the number-phase minimum uncertainty states of the photon field.New Journal of Physics, 10, 053028 (35 pages) (2008). Varró S : Entangled states and entropy remnants of a photon-electron system.Physica Scripta T140 (2010) 014038 (8pp)

Becker’s analysis on the ‘strong-field photon-electron interaction’ in a medium [ 1977 ].

Varro_ECLIM_2010

W. Becker, Relativistic charged particles in the field of an electromagnetic plane wave in a medium. Physica A 87, 601-613 (1977)

E.g. Mathieu–type solutions )/( cyntxk m

0)2cos2( 2 wzhw

m

peAhpk 02, 0

Disposable parameter; band structure

Fundamental parameter.

Varro_ECLIM_2010[ Figure taken from Arscott F M, Periodic differential equations (Pergamon Press, Oxford, 1964) p.123. ] . Nikishov & Ritus (1967), Nikishov (1970), Narozhny & Nikishov (1974), Becker (1977), Fedorov, McIver … FEL theories.

New class of exact solutions of the Dirac and Klein-Gordon equation in a strong laser field [ 2013 ].

Varro_ECLIM_2010

New exact solutions. page1. )/( cyntxk m g

0])[( 2122 FAi

41

)()(s spsp u

m

0)2sin24cos22cos2( )(12102

)(2

psps zizz

dz

d

dz

)2cosexp( 2/12

)( zfps 2/z Hill equation.

Narozhny and Nikishov (1974) for nm=0

0)2cos(2sin2

2

fzqaif

dzdfza

dzfd

0

04

peFa

dzdz)(42 /)(2 k

npkpk 20 1 mp nkk px kqp )1(2

0

Varro_ECLIM_2010

[1] S. V. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium. LaserPhysics Letters 10 (2013) 095301, E-print: arXiv:1305.4370 [quant-ph].

New exact solutions. page2. g

n

n

kn

n

DD

DD

nanan

2

1

)(2

12

2

00)2(4)12(000)1()1(4

n

n

kn

n

n

DD

DD

naann

anan

1

)(

12

2

4)1(000)12()1(400

0)22()22(0

n

kr

kn irnaDagf )( )exp()2|()|,(

nn)(

0

04

peFa

nr 1 0

Varro_ECLIM_2010

[1] S. V. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium. LaserPhysics Letters 10 (2013) 095301, E-print: arXiv:1305.4370 [quant-ph].

New exact solutions. page3. )/( cyntxk m p g

px nkp

m

px knp )( 21 2

02

02 /1)( pmn p

)(42 /)(2 knpkpk cnkk pmp /1 2

0

p 00 )( pm

p mceFa

2

00 224

22 )()( ypy ckk p 0

100 1058 IeF 000

0 105.8

Imc

(For optical frequencies) the new parameter “ a ” is 6 orders of magnitude larger

Varro_ECLIM_2010

(For optical frequencies) the new parameter a is 6 orders of magnitude larger than the usual intensity parameter ( ‘scaled vector potential’ )

K-G case. Ince polynomials. )/( cyntxk m y

0])[( 22 Ai

m

)(cos gee axip

0)cos()(sin gqagag 00 / peFa

)|(]cosexp[)]ˆˆ(exp[ aIPazpxpxpi k

px kqp )1(2 px knp )( 21 )(42 /)(2 k

npkpk

)|(]cosexp[)](exp[ aIPazpxpxpi nzxp

)s,|(),c,|()|( aaaIP kn

kn

kn

Even cosine and sine type

)s,|(),c,|()|( aaaIP kn

kn

kn

Odd cosine and sine type

[2] S V A l f t l ti f th Kl i G d ti f h d ti l i t ti ith l t ti l i

Varro_ECLIM_2010

[2] S. V., A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 11 (2014) 016001, E-print: arXiv:1306.0097 [quant-ph].

K G I l i lK-G case. Ince polynomials.

Klein-Gordon. k = 15. Klein-Gordon. k = 20.

Wave functions with negative eigenvalues (imaginary longitudinal momentum)

[2] S V A l f t l ti f th Kl i G d ti f h d ti l i t ti ith l t ti l[2] S. V., A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 11 (2014) 016001, E-print: arXiv:1306.0097 [quant-ph].

)|,(cos)4/()(2,1 agee k

naxpie

p

p

p mceFa

2

00

0 224

‘Void regions’ in the centre of the cycle. [ ‘Quantum bubble’ ]

‘Hyperfine splitting’ of the longitudinal momentum spectrum

Dirac Klein-Gordon

yp p g g p

[1] S. V. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 10 (2013) 095301, E-print: arXiv:1305.4370 [quant-ph].

[2] S V A l f t l ti f th Kl i G d ti f h d ti l i t ti ith l t ti[2] S. V., A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 11 (2014) 016001, E-print: arXiv:1306.0097 [quant-ph].

Double peak structure. Single peak structure. Oscillatory spectrum.

Dirac Klein-Gordon

p g p y p


[2] S V A l f t l ti f th Kl i G d ti f h d ti l i t ti ith l t ti[2] S. V., A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 11 (2014) 016001, E-print: arXiv:1306.0097 [quant-ph].

Double peak structure. Single peak structure. Oscillatory spectrum.

Dirac Klein-Gordon


[1] S V N t l ti f th Di ti f h d ti l i t ti ith l t ti l i[1] S. V. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium. Laser Physics Letters 10 (2013) 095301, E-print: arXiv:1305.4370 [quant-ph].

Summary and conclusions

1 G d ’ d V lk ’ t l ti till i t t S l li1. Gordon’s and Volkov’s exact solutions are still important. Several earlier calculations have to be reconsidered (boundary-value problem, ultrashort pulses). Volkov states have a sort of ‘renaissance’, due to technological development.

2. The classical description for both the electron and for the extreme EM radiation field delivers an appropriate intuitive picture. ‘Quasi-classicality in many cases.’

3. One has to go beyond the original (semiclassical) Volkov states. The appearance

4. Completely new kind of exact closed form solutions of the Klein–Gordon and

of non-trivial correlations, like plasmon anti-bunching or EPR has been demonstrated.

p yDirac equations have been presented, which are basically–periodic solutions in a medium (underdense plasma). These solutions also describe half-integer harmonics (4-periodic solutions) and ‘void regions’ in the electron density, a sort of ‘quantum bubble’ , which may be relevant in laser acceleration of particles.

Acknowledgment.This work has been supported by the pp yHungarian Scientific Research Foundation OTKA, Grant No. K 104260.

Appendicesppe d ces

O. Klein [1929], F. Sauter [1931], W. Heisenberg and X Euler [1936], J. Schwinger [1954]; Critical field, pair creation. A. I. Nikishov, V. I. Ritus, N. B. Narozhny [1970], E. Brezin, C.

Itzykson [1970] V S Popov [1972] L V KELDISH [1964] Itzykson [1970], V. S. Popov [1972]... L. V. KELDISH [1964], { S. W. Hawking [1974], P. C. W. Davies [1975], W. G. Unruh [1976] }

2)/( mcmceEeE crCcr ecmEcr /32

cmVsinEcritical /103.1~ 16%100/ UU2

2mc

0

2mc0

eExxV )(MeVmc 12 2

aT 2

0/3

22 88 rhPdkTh

Unruh

ck2 0/3 31r

ecdtd kTh

Example for the ‘renaissance’ of the theoretical studies, initiated partly by the perspective of ELI.

Varro_ECLIM_2010

From the „Topical issue on Fundamental physics and ultra-high laser fields.”

Varro_ECLIM_2010

ELI ALPS E t Li ht I f t t Att d Li ht P l S [ t b t t d i S d

Varro_ECLIM_2010

ELI-ALPS = Extreme Light Infrastructure – Attosecond Light Pulse Source [ to be constructed in Szeged, Hungary . The picture shows the bird’s view of the planned ELI facility. ]

ELI-ALPS = Extreme Light Infrastructure – Attosecond Light Pulse Source [ to be constructed in Szeged Hungary ][ to be constructed in Szeged, Hungary ] The ELI-ALPS facility in Szeged, Hungary will be a unique, versatile laser facility with its sources spanningan extremely broad range from the THz to the X-ray spectral regions. Femtosecond, near-infrared laserpulses with unprecedented parameter combinations will drive various secondary sources includingterahertz (THz), mid-infrared (MIR), ultraviolet (UV), extreme ultraviolet (XUV), and X-ray pulses. Theseflashes of electromagnetic radiation will have durations from a few picoseconds (10-12 s) overg p ( )femtoseconds (10-15 s) down to attoseconds (10-18 s), depending on the wavelength, thus constituting aunique research facility.

The scientific infrastructure will be implemented in two stages. The lasers will be operating with a modestpulse energy and somewhat longer pulses by the end of 2015. Secondary pulse generation as well as user

i t ill b il bl l 2016 Th d t d l lifi ill b d li d th dexperiments will be available early 2016. The duty-end laser amplifiers will be delivered, the secondarysources will be fine tuned, and the final design parameters will be realized by 2017.

After the construction phase of ELI-ALPS (2013-2017), the facility can be optimally used for a number ofapplications related to applied research and development, innovation, as well as multi- andinterdisciplinary applications in biology/biophysics chemistry materials science energy research etcinterdisciplinary applications in biology/biophysics, chemistry, materials science, energy research, etc.Because the facility will boast a unique parameter combination of compact high-brilliance photon sources,biological, medical, and industrial applications are envisaged. With the realization of highly brilliant laser-based X-ray sources, offering parameters partly comparable to those of large-scale third-generationsynchrotron radiation sources or even fourth-generation self-amplified spontaneous emission (SASE) free-electron lasers (FELs), many experiments and applications, which are currently running or under

Varro_ECLIM_2010

developement at these large scale facilities, may be performed on a laboratory scale in the foreseeablefuture.

Varro_ECLIM_2010Fig. 2.1. Layout of the scientific infrastructure of ELI-ALPS. [ Taken from: „Az ELI-ALPS tudományos felépítése és paraméterei „ ]

Equations of motion in the semiclassical and in the quantum case

Semiclassical. electronH

tiVt

ce

m

)()(ˆ

21 2

rAp

])())[(2/1()( 000000

tiitiix eeAeeAt uA

Quantum.

iAAVe

1ˆˆ)(ˆˆ1 2

rAp

modequantizedelectron HH

tiAAV

cm

2

)(2 0rAp

)ˆˆ()/2(ˆ 2/130

2 AALc uA 1ˆˆˆˆ AAAA)()/2( 0 AALcx uA 1 AAAA

Semiclassical description of multiphoton processes; Volkov: No true photon in it! Merely side-bands of e-waves.y

iVtei

)()(

21 2

rA tiz 0sin

Jacobi–Anger formula

tcm

)()(2 r

)cos()()/()( 0000 ttfcFt xuAtin

n n

tiz

ezJ

e0

0

)(

sin

)]([~ 0)(

00 nEEeedtedt iftintEEi

tinif

if

„The electron absorbs n photons”

tnEEitintEEiifif )()( 00

0nEE if

„The electron absorbs n photons

E.g. for photoeffect, the equation nh=A+Ekin expresses a quantum mechanical resonance. Planck’s constant enters as a ‘property’ of the electron, rather than being a property of the light. The resonance cannot be derived without the de Broglie Schrödinger wavesbe derived without the de Broglie – Schrödinger waves.

Usual argument for irrelevance of quantum description in strong-field physics: the l ti d i ti f (l ) fi ld i t l ll Th l fi ld i l i l ”relative deviations of (laser) fields is extremely small. „The laser field is classical.”

But ‘strong radiation fields’ can have infinitely many sort of photon distribution.

X-axis: A + A+, magnetic induction. Y-axis: (A – A+)/i, electric field strength.

Example for ‘EPR’ in strong-field physics. Entangled photon – elektron states. Entropy remnants after high-intensity Compton scattering III.

Some details. )/( cyntxk m m

0][ Ai )( ][ )(

)(exp)( xipp

02122 2222

2

22

p

pp FAAppd

dpik

dd

k

)]ˆˆ(exp[)()(exp)( 2211)(

2)( xpxpxpixk

kpkpi pp

0212/1 )(222222

22

)(2

pp FAAppkp

kdd

Varro_ECLIM_2010

Some details. )/( cyntxk m m

1000 mn

00100100

/))(( 0m

m

m

x nn

kek

0001 m

m

n

21 mn

mn

u0

11

1

mn0

11

mn

u

1021

mn

u

1023

Varro_ECLIM_2010

Di Kl i G dDirac Klein-Gordon

Wave functions

Redmond’s solutions [ 1965 ]. One classical plane wave and a constant magnetic field.

P J Redmond Solution of the Klein-Gordon and Dirac equations for a particle with a plane

Varro_ECLIM_2010

P. J. Redmond, Solution of the Klein-Gordon and Dirac equations for a particle with a plane electromagnetic wave and a parallel magnetic field. J. Math. Phys. 6, 1481-1484 (1965).

Fedorov and Kazakov [ 1973 ]. Quantized plane wave and a constant magnetic field.

Varro_ECLIM_2010

M. V. Fedorov and A. E. Kazakov, An electron in a quantized plane wave and in a constant magnetic field. Zeitschrift für Physik 261, 191-202 (1973).

Berson and Valdmanis’ solutions [ 1973 ]. Two classical or quantized plane waves.

I Bersons and J Valdmanis Electron in the field of two monochromatic waves J Math Phys 14

Varro_ECLIM_2010

I. Bersons and J. Valdmanis, Electron in the field of two monochromatic waves. J. Math. Phys. 14,1481-1484 (1973).

Mathieu type solutions due to Becker and Mitter [ 1979 ]. Electron in standing waves. FEL.

Varro_ECLIM_2010

W. Becker and H. Mitter, Electron in the field of two monochromatic waves. J. Phys. A: Math. Gen. 12, 2407-2413 (1979). See also the Kapitza–Dirac effect.

exact solutions of the relativistic wave equations in...

Documents