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 GordonVolkov solutions and beyond. Sndor Varr Wigner Research Centre for Physics Hungarian Academy of Sciences Institute for Solid State Physics and Optics,
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 f Outline of the talk.
exact solutions of the ‘Volkov problem’ in a medium. [5. Some perspectives of high-laser-field physics; ELI.]
Possible descriptions of photon-electron interactions
PHOTON ELECTRON
Trajectory, Ray (Geometric Optics) Field (Maxwell Theory)
Quantized Field (True Photon) Trajectory, current [Point, charged dust, M h i 1.
Electrodynamics Classical EM fi ld R di ti
current, Classical (Poisson) photon Mechanics, Hidrodynamics] fields, Radiation reaction Field, Transition Currents [Wave 4.
Theory
Optics Quantum Currents [Wave Mechanics] Theory. [Schrödinger, KG, Dirac, Maxwell]
transitions + General Photon Quantized Field 7
9 Full QED pair Quantized Field [Electron-Positron (Hole) Field,Solid State Physics] 7.
External EM Fields [e.g. e-e+ pair creation]
creation and back- 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
Roots of the non-perturbative analyses: go back to Gordon (1927) and Volkov (1935); Semiclassical States
] ) ( ) [(
2
rad rad
A i A i
( )
] ) ( [
rad
A i
state
) ( ) ( f A e A
x rad
) / ( c z t x k
Volkov 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,...
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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 ]
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Gordon W, Der Comptoneffekt nach der Schrödingerschen Theorie. Zeitschrift für Physik 40, 117-133 (1927). [ Application to strong-field: ~1960..]
Gordon’s solutions [ 1927 ] [ ]
] ) ( ) [(
2
rad rad
A i A i
) ( ) ( f A e A
x rad
) / ( c z t x k
) (
) (
e N x
S i p p
p
t in n n t iz
e z J e ) (
sin
Jacobi–Anger formula
] ) ( [ exp ) (
) (
d I x p i N
p p p p
) ( ) ( 2 )[ 2 / 1 ( ) (
2 2 ) (
A A p p k I p
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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 ]
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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 ] [ ]
) ( ] ) ( [ V A i
rad
) ( ) ( f A e A
x rad
) / ( c z t x k
2 )] ( )[ ( 1 ) (
) ( ) (
u p k A k x
ps ps
] ) ( [ exp 2
) (
d I x p i p k
p
p
) ( ) ( 2 )[ 2 / 1 ( ) (
2 2 ) (
A A p p k I p
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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
x ip p
e
) ( ) / ( c z t x k
p
2 2
p d k
2 2 2 2 2
p
d d k ) 2 ( 2
2 2 2 2
p p
A A p p d d p ik
In vacuum:
2
k
di S d d di diff i l First-order ordinary differential equation for p. Immediately integrable, yielding the Gordon-Volkov solutions.
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In a medium:
) 1 ( ) / (
2 2 2
m
n c k
Second-order ordinary differential equation for p
Orthogonality and completeness.
) ( ) ( 3
) (
3 ) ( ) ( 3
p p
p p
i r d
) (
3 ) ( ) ( 3
p p
s s ps
r d
c z t / c z t v /
) (
3
p p
s s s p ps
r d
) , , ( ) , , (
) ( ) ( 2
x x v v p d dp
ps ps v
c z t v /
) , , ( ) , , (
) ( ) ( 2 2 , 1
x x v v p d dp
ps ps v p s p
) ( ) ( 1 ) , , ( ) , , (
2 4 2 , 1
x x v v p p
ps s ps v
) ( ) (
2 4
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
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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
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E.g. application to high-intensity Compton scattering: A. I. Nikishov and V. I. Ritus, Zh. Eksperim. i
Kibble T W B, Physical Review 133, A705-A719 (1964).
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 )
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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 relativistic proportional to the proper time of the particle. Nonrelativistic and relativistic classical intensities.
) sin( ) , ( r k e r E t F t ) ( ) , ( ) sin( ) , ( r k e n r B t F t | | / , |, | k k n k e k c ) ), ( ( ) ( ) ), ( ( / ] / ) ( [ 1 / ) (
2 2
t t dt t d c e t t e c dt t d dt t d m dt d r B r r E r r t eF x m sin
I eF xosc
10
10 5 . 8 mc c
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
) ( ) , ( F t e n r B ) ( ) , ( F t e r E c t / r n . ) ( 1 const t c t d d r n
u F c m e d du ) / ( / ) (
2
F x d
2 2
1 dx d z d
Along the polarization x-direction one receives formally a Newton equation in dipole approximation !
) (
2
eF d m . ) / 1 ( const c vz
2
2 1 d dx d d c d z d
2y
d . ) / (
z
2
d y d
2 2 2
2 1 d dx m c m c m 2 d
Extreme radiation, from terahertz to xuv from ultrarelativistic motion
I mc eF
9
10 ) 4 / 1 /(
2
) (
t k k k J t z
k k
2 sin ] / ).] 1 /( [ [ ) 2 / ( ) (
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 state This is present in the A term in the Volkov state.
2 2 /
2 13
] [ 10 5 . 2 ) (
w r p
e eV I U
r ] [ 10 5 . 2 ) (
p
e eV I U r
Coupling parameters. Classical. The question of switching on/off Initial value problem! The question of switching-on/off. Initial value problem!
) cos( ) / ( ) , ( r k r n e r E t c t f F t / r e Ec
C c
eE
I mc eF x c
10
10 5 . 8
0.5 1.0
t = 1, j = -p2
1
/ c z t
2
/ c z t
0.0 F t 2 1 1 2
1 2 t 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 l manifest itself in the kinematics of the HHG spectrum. Quantized ponderomotive potential.
Relative depletion. From the
scatt laser
A A i ˆ ] ) ˆ ( [
From the quantized pondero- motive energy shift.
2 2 2
n n
C n
2 2 2 1
2
sin n
C
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Squeezing effect through the joint interaction in the “system of free electrons plus the quantized radiation modes”. I. Exact (stationary, squeezed) states. Measurement
■ 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 / 1 ˆ ˆ ( ) ( ˆ 1
2
A A e H r A p ) 2 / 1 ( ) ( 2 A A c m H
i i
r A pi
) ˆ ˆ ˆ ˆ ( r
) ˆ ˆ ( ) ( 2 ) ; (
; ;
A A A A A A r r r e n E
e D e S n D S
P
P
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 y electrons plus the quantized radiation modes”. II. Plasmon statistics.
squeezed plasmon excitation → SPO → spontaneous photon.
1 ; ; 2 1 )! ( 2 1
2 ) 1 ( 2
2 2
a a H n e W
n n n a sq n
2 )! ( 2 n ) /( ; ;
2 2 1 1
p p R p p N p N
sq
) 1 ( 2 ) 1 ( 1 ; 1
2 2 2 2 2 2 2
a a r 2 tanh
2 2 18 2
10
I
a ;
sq
R
Varró S, Theoretical aspects of Hanbury Brown and Twiss type correlations mediated by surface plasmon oscillations;
(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 ) (
r k r r z J e V
n n eff
) / ( 2
1
p c z
S f ( ) f S f ( ) f
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 .
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 .
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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 ns Photon Photons Photon detection
k n t t
k k g
) ( ) (
Electrons
Physics 10 053028 (35 pages) (2008) Varró S : Entangled states and entropy remnants of a photon electron system Physica Scripta
r t r r d t
g
) , ( ) (
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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
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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
Physica A 87, 601-613 (1977)
E.g. Mathieu–type solutions
) / ( c y n t x k
m
) 2 cos 2 (
2
w z h w
m
p eA h p k
2
,
Disposable parameter; band structure Fundamental parameter.
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[ 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 ].
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New class of exact solutions of the Dirac and Klein-Gordon equation in a strong laser field [ 2013 ].
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New class of exact solutions of the Dirac and Klein-Gordon equation in a strong laser field [ 2013 ].
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New exact solutions. page1.
) / ( c y n t x k
m
g
] ) [(
2 1 2 2
F A i
4 1 ) ( ) ( s s ps p
u
m
) 2 sin 2 4 cos 2 2 cos 2 (
) ( 1 2 1 2 ) ( 2
ps ps
z i z z dz d dz ) 2 cos exp(
2 / 1 2 ) (
z f
ps
2 / z
Hill equation. Narozhny and Nikishov (1974) for nm=0
) 2 cos ( 2 sin
2 2
f z qa if dz df z a dz f d
4
p
eF a
dz dz
) ( 4 2 /
) ( 2
k n p
k p k
2 0 1 m p
n k k
p x
k q p ) 1 ( 2
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[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].
New exact solutions. page2. g
n n k n n
D D D D n a n a n
2 1 ) ( 2 1 2 2
) 2 ( 4 ) 1 2 ( ) 1 ( ) 1 ( 4
n n k n n n
D D D D n a a n n a n a n
1 ) ( 1 2 2
4 ) 1 ( ) 1 2 ( ) 1 ( 4 ) 2 2 ( ) 2 2 (
n k r k n
ir n a D a g f
) (
) exp( ) 2 | ( ) | , (
n n
) (
4
p
eF a
n r 1
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[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].
New exact solutions. page3.
) / ( c y n t x k
m
p g
p x
nk p
m p x
k n p ) (
2 1
2 2 2
/ 1 ) (
p m
n
p ) ( 4 2 /
) ( 2
k n p
k p k c n k k
p m p
/ 1
2
p 0)
(
p m p
mc eF a
2
2 2 4
2 2
) ( ) (
y p y
ck k
p
10
10 5 8 I eF
10 5 . 8 I mc
(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.
) / ( c y n t x k
m
y
] ) [(
2 2
A i
m
) (
cos
g
e e
a x ip
) cos ( ) (sin g qa g a g
/
p
eF a
) | ( ] cos exp[ )] ˆ ˆ ( exp[ a IP a z p x p x p i
k
p x
k q p ) 1 ( 2
p x
k n p ) (
2 1
) ( 4 2 /
) ( 2
k n p
k p k ) | ( ] cos exp[ )] ( exp[ a IP a z p x p x p i
n z x p
) s , | ( ), c , | ( ) | ( a a a IP
k n k n k n
Even cosine and sine type
) s , | ( ), c , | ( ) | ( a a a IP
k n k n k n
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
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[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 l K-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
a g e e
k n a x p i e p
p p
mc eF a
2
2 2 4
‘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
[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
[1] S. V. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a
[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. , New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a
[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
Summary and conclusions
1 G d ’ d V lk ’ t l ti till i t t S l li
calculations have to be reconsidered (boundary-value problem, ultrashort pulses). Volkov states have a sort of ‘renaissance’, due to technological development.
field delivers an appropriate intuitive picture. ‘Quasi-classicality in many cases.’
demonstrated. p y Dirac 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
Acknowledgment. This work has been supported by the pp y Hungarian Scientific Research Foundation OTKA, Grant No. K 104260.
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
) / ( mc mc eE eE
cr C cr
e c m Ecr /
3 2
cm V sin Ecritical / 10 3 . 1 ~
16
% 100 / U U
2
2
mc
2
mc
eEx x V ) (
MeV mc 1 2
2
a T
2 / 3 2 2
8 8 r h P d
kT h Unruh
ck 2
/ 3
3 1 r e c dtd
kT h
Example for the ‘renaissance’ of the theoretical studies, initiated partly by the perspective of
ELI.
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From the „Topical issue on Fundamental physics and ultra-high laser fields.”
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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
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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 spanning an extremely broad range from the THz to the X-ray spectral regions. Femtosecond, near-infrared laser pulses with unprecedented parameter combinations will drive various secondary sources including terahertz (THz), mid-infrared (MIR), ultraviolet (UV), extreme ultraviolet (XUV), and X-ray pulses. These flashes
electromagnetic radiation will have durations from a few picoseconds (10-12 s)
g p ( ) femtoseconds (10-15 s) down to attoseconds (10-18 s), depending on the wavelength, thus constituting a unique research facility. The scientific infrastructure will be implemented in two stages. The lasers will be operating with a modest pulse 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 d experiments will be available early 2016. The duty-end laser amplifiers will be delivered, the secondary sources 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 of applications related to applied research and development, innovation, as well as multi- and interdisciplinary applications in biology/biophysics chemistry materials science energy research etc interdisciplinary 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-generation synchrotron 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
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developement at these large scale facilities, may be performed on a laboratory scale in the foreseeable future.
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paraméterei „ ]
Equations of motion in the semiclassical and in the quantum case
Semiclassical.
electron
H
t i V t c e m ) ( ) ( ˆ 2 1
2
r A p ] ) ( ) )[( 2 / 1 ( ) (
t i i t i i x
e e A e e A t
u A
Quantum.
i A A V e
1 ˆ ˆ ) ( ˆ ˆ 1
2
r A p
mode quantized electron
H H
t i A A V c m 2 ) ( 2 r A p ) ˆ ˆ ( ) / 2 ( ˆ
2 / 1 3 2
A A L c u A
1 ˆ ˆ ˆ ˆ
A A A A
) ( ) / 2 ( A A L c
x
u A
1 A A A A
Semiclassical description of multiphoton processes; Volkov: No true photon in it! Merely side-bands of e-waves. y
i V t e i ) ( ) ( 2 1
2
r A
t iz sin
Jacobi–Anger formula
t c m ) ( ) ( 2 r ) cos( ) ( ) / ( ) ( t t f cF t
x
u A
t in n n t iz
e z J e ) (
sin
)] ( [ ~
) (
n E E e e dt e dt
i f t in t E E i t in i f
i f
„The electron absorbs n photons”
t
n E E i t in t E E i
i f i f
) ( ) ( n E E
i f
„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 waves be 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.
) / ( c y n t x k
m
m
] [ A i ) ( ] [ ) (
) ( exp ) ( x ip
p
2 1 2 2
2 2 2 2 2 2 2
p p p
F A A p p d d p ik d d k
)] ˆ ˆ ( exp[ ) ( ) ( exp ) (
2 2 1 1 ) ( 2 ) (
x p x p x p i x k k p k p i
p p
2 1 2 / 1
) ( 2 2 2 2 2 2 2 2 ) ( 2
p p
F A A p p k p k d d
Varro_ECLIM_2010
Some details.
) / ( c y n t x k
m
m
1
m
n 1 1 / ) )( (
m m m x
n n k e k 1
m m
n
2
1
m
n
m
n u 1 1
1
m
n 1 1
m
n u 1 2
1
m
n u 1 2
3
Varro_ECLIM_2010
Di Kl i G d Dirac 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
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
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
1481-1484 (1973).
Mathieu type solutions due to Becker and Mitter [ 1979 ]. Electron in standing waves. FEL. Varro_ECLIM_2010
2413 (1979). See also the Kapitza–Dirac effect.