Imaginary time method
and
nonlinear ionization by
powerful free electron lasers
S.V. Popruzhenko
Moscow Engineering Physics Institute, Moscow
EMMI workshop “Particle dynamics under extreme matter conditions”
Speyer, September 26-29, 2010
Outline
I. Imaginary time method for time dependent QM problems
• ITM for time-independent tunneling
• generalization to the time-dependent problem
• Coulomb correction to the rate of ionization
II. Nonlinear ionization at short wavelengths
• modern free electron lasers
• experiments on nonlinear ionization at short wavelengths
• nonlinear ionization rate for arbitrary frequencies
• can we explain the data?
• Time-independent tunneling
U r   U 0 r   E 0 r
M  r     exp iW (  ) ,
W   


2   U  r dr
a
r
b
U 0 r 
U r 
  I p
W in    i 
2 U  r    dr
a
w  exp 2 iW in 
For a short-range well
2 

w SR  I p F exp  

 3F 
F 
a
b
E0
E at
,
E at  ( 2 I p )
3/2
• Time-independent tunneling
U r   U 0 r   E 0 r
M  r     exp iW (  ) ,
W   


2   U  r dr
a
r
b
U 0 r 
U r 
  I p
W in    i 
2 U  r    dr
a
w  exp  2 iW in 
For a short-range well
2 

w SR  I p F exp  

 3F 
F 
a
b
E0
E at
,
E at  ( 2 I p )
3/2
Can we generalize this description on the time-dependent case?
• Time-dependent tunneling
M  exp  iW (  ) ,
E 0  E t 
W   


2   U  r dr
a
W   
r2
t2
 p  , r dr    L r , r , t    dt
r1
L  r , r , t  
t1
r  
2
r
2
U
+ initial conditions
r
r t 0   b
 U r , t 

r t s 
v t 0   0
t s  t 0  i
r t 0   b
v t 0   0
the initial time is complex
v t   i 2 I p  E 0 t  t s ,
 
2I p / E0
r t   i 2 I p t  t s   E 0 t  t s  / 2
2
• Time-dependent tunneling: imaginary time method
Popov, Kuznetsov, Perelomov, 1967
M p   exp  iW ( p ) 
W p  

  L r , r , t    dt
ts
L  r , r , t  
2
r
2
 U r , t 
The trajectory has to be found
from Newton equation
x(t)
t s = t 0 (p )+ i  (p )
r   U
2
v (t)
t
t0
with the initial conditions
x0
r t s   0 ,
1
2
v t     p
v
2
t s   
 0,
• Imaginary time method
• nonlinear (multiphoton) ionization and excitation by intense
electromagnetic fields
• strong field QED processes in the semiclassical domain
• laser-assisted decay of quasienergy states
•…
• Imaginary time method
• nonlinear (multiphoton) ionization and excitation by intense
electromagnetic fields
• strong field QED processes in the semiclassical domain
• laser-assisted decay of quasienergy states
•…
ITM gives the solution in terms of classical complex trajectories.
This provides a particularly convenient way to account for
perturbations which cannot be considered within the
conventional PT
• Coulomb correction to the SFA

Strong Field Approximation
M
SFA
p    i 
V 
p
VˆF t   0


i
 p r , t   exp  i p  r 
2

M SFA p  

t
2




p

A

d 

V  E t   r
L.V. Keldysh, 1964
F.H.M. Faisal, 1973
H.R. Reiss, 1980


 C p , t   exp  iW p , t  
s
s
x(t)

t s = t 0 (p )+ i  (p )

W 
1 2



v
t

I
p
  2
 dt

ts
v t   p  A t 
2
v (t)
t
t0
v t s 
2
p  0
2
 I p
r t s   0
x0
• Coulomb correction to the SFA
p  0

p  
Zr
r
3

CCSFA:
(0)
(1)
x (t)+ x (t)
M CCSFA p  


~
~
~
C p 0  , ts   exp i W p , ts 


~
W 
t s = t 0 (p 0 )+ i  (p 0 )

Z
1 2



v
t


I
dt
p
~  2

r

ts
v t   p  A t 
t 0 (p 0 )
p  
v t s 
2
Zr
r
3

 W1  Z
2
dt
 r t  ,
 I p
r t s   0
 W 2  W r   r , v   v   W r , v 
x0
ts
A.M. Perelomov, V.S. Popov, 1967
S.V. Popruzhenko, V.D. Mur, V.S. Popov, D. Bauer, 2008
(0)
x (t)
• Strong field ionization rates
Short-range well, static field
w SR  I p Fe

2
3F
,
F  E 0 / 2 I p 
3/2

Coulomb well, static field
 W1  Z
dt
 r t 
ts
w  Q1 w SR
 2 
Q1   
F 
2 n
,
n*  Z /
Hydrogen, ground state, 1014W/cm2:
2I p
Perelomov, Popov,1967
F  0 . 05 ,
n *  1  Q 1  400
II. Nonlinear ionization at short wavelengths
• modern free electron lasers
• experiments on nonlinear ionization at short wavelengths
• nonlinear ionization rate for arbitrary frequencies
• can we explain the data?
• Modern free electron lasers
FLASH – Free electron LASer in Hamburg
2002: photon energy 12.7eV (100nm), intensity up to 1013W/cm2, pulse
duration 100fs;
2007: photon energy 92.8eV (13nm), intensity up to 1016W/cm2, pulse
duration 10fs;
Currently: photon energy 200eV (6nm) is approached (6нм); its fifths
harmonic is already in the KeV domain
SPring-8 based SASE Source in Japan
Currently: 50-62nm (photon energy around 20eV),
intensity up to (1-3)*1014W/cm2
LINAC Coherent Light Source in Stanford
• New regime of laser-matter interaction
Optical and infrared lasers
 
 1
2 mI p 
E0
Intense XUV lasers
  1,
 10
2
• Nonlinear ionization at short wavelengths
• Nonlinear ionization at short wavelengths
A.A. Sorokin et al., 2007
Ionization of Xe by 10fs 13nm
(93eV) pulses with intensity up
to 1016W/cm2
ions up to Xe21+ were recorded
7 photons for ionization of Xe20+
and >57 photons in total
K. Motomura et al., 2009
Ionization of Ar by 100fs 62nm
(20eV) pulses with intensity up
to 2*1014W/cm2
ions up to Xe6+ were recorded
• Nonlinear ionization at short wavelengths
1. Do we understand the mechanism?
2. If yes, can we provide a quantitative description?
• Nonlinear sequential ionization rates
2 mI 
 
E0
Rate
Static   0
w  F   Q1 I p Fe

 2 
Q1   
F 
2
3F
Low-frequency
  1
Arbitrary (high)
frequency
  1
w  F ,    Q1 I p
3F


3
e
2
2  
 1
3 F  10




2 n
n* 
 1
F 
E0
E at
,
Z


E at  ( 2 I p )
Z
2I
3/2
• Nonlinear ionization rates: intense XUV fields
 
2 mI p 
 50
E0
w  F ,    Q1 I p

Xe
8
 Xe
3F


3
e
9
W(Xe10+)=10-24!
2
2  
 1
3 F 
10




• Nonlinear ionization rates

w  F ,    exp  2 Im W  F , 

W 

 L  I dt
 v r

ts
t s p 
r t   r0 t   r1 t   ...
W  W 0   W1   W 2  ...


ts
ts
 W 1    U C  r0 t dt  Z 
dt
r0 t 
 2 
Q 1  exp  2 Im  W 1    
F 
A.M. Perelomov, V.S. Popov, 1967
 W 2 p   W 0 r0  r1   W 0 r0 
Q 2  exp  2 Im  W 2 
S.V. Popruzhenko, V.D. Mur, V.S. Popov, D. Bauer, 2008
w  Q1  Q 2  w sr
2 n
• Ionization rate for arbitrary frequencies
Rate
Static
Lowfrequency
w  F   Q1 IFe

w  F ,    Q1 I
 2 
Q1   
F 
2
3F
3F


3
e
2
2  
 1
3 F  10




2 n
 1
2 

Q 2  exp  2 Im W 2    1 

e


2 n
Arbitrary
(high)
frequency
 2 
w  Q C  w sr   
F 
n* 
Z
2I
2 n*
2 

 1 

e 

F 
E0
E at
 1
2 n*
 w sr
• Comparisons with numerical results
K e ld y s h p a ra m e te r 
40
80
20
10
5
Ionization rate of Xe17+ in the field of
an XUV laser with the photon
energy 93eV (13nm);
4p0 state with I=434эВ;
5-photon ionization.
The rate calculated from the TDSE
numerical solution is shown by
triangles.
new
0
sr
L o g 1 0 (w )
-4
tu n n e l
-8
QC
-1 2
-1 6
15
10
10
16
10
In te n s ity , W /c m
Q1  10
18
Q 2  10
,
Q C  9  10
8
9
17
2
10
18
U r   
K 0  4 . 67
Z
r

54
 Z e
r
n *  3 . 19
 r
,
Z  18 ,
  7 . 93
• Comparisons with the data
The data of K. Motomura et al.: Ar at 62nm and Kr at 51nm
• Comparisons with the data
Ar at 62nm, 2•1014W/cm2, ions up to Ar6+ have been observed
Ion state
Ion. potential,
eV
Min. number of
photons
Probability,
Theory 1
Probability,
Theory 2
Ar+
27.6
2
10
102
Ar2+
40.7
3
0.15
20
Ar3+
59.7
3
5•10-2
10
Ar4+
75.1
4
1•10-3
0.3
Ar5+
91.0
5
4•10-6
10-3
Ar6+
124.3
7
2•10-12
10-9
 2 
 
F 
2 n*
 2 
w 2  Q C  w sr   
F 
2 n*
w1  Q C  w sr
2 

 1 

e


2 

 1 

e 

2 n*
 w sr
 2 n *  1
 w sr

Theory 1

Theory 2
Excitation
is accounted
into Rydberg states
for in the rate
• Comparisons with the data
Kr at 50nm, 2•1014W/cm2,
ions up to Kr7+ have been observed
Ion state
Ion. potential,
eV
Min. number of
photons
Probability,
Theory 1
Probability,
Theory 2
Kr2+
36.9
2
20
4•103
Kr3+
52.5
3
0.1
30
Kr4+
64.7
3
0.3
1•102
Kr5+
78.5
4
1•10-3
0.5
Kr6+
111.0
5
4•10-7
3•10-4
Kr7+
126
6
1•10-9
7•10-7
 2 
 
F 
2 n*
 2 
w 2  Q C  w sr   
F 
2 n*
w1  Q C  w sr
2 

 1 

e


2 

 1 

e 

2 n*
 w sr
 2 n *  1
 w sr

Theory 1

Theory 2
Excitation
is accounted
into Rydberg states
for in the rate
Concluding remarks
1. Imaginary Time Method can be used for solving strong field problems
difficult to approach by other methods: strong field sequential ionization
rate of atoms and ions is a proper example when ITM works efficiently.
2. The nonlinear rate of ionization we derived provides much better
qualitative agreement with the data obtained with intense XUV lasers
than previously known rates.
3. However, it remains unclear if our rate is sufficient for quantitative
description or the phenomenon.
4. Incorporation of Rydberg states may essentially improve the theory.
5. If with the Rydberg states accounted for we are still unable to reproduce
the data this gives a strong support to the idea that electron-electron
correlations are also important at short wavelengths.
Where to learn more?
Theory:
SVP and D. Bauer, Journal of Modern Optics 55, 2573 (2008)
Applications:
SVP, G.G. Paulus and D. Bauer, PRA 77, 053409 (2008)
SVP, V.D. Mur, V.S. Popov and D. Bauer, PRL 101, 193003 (2008);
JETP 108, 947 (2009)
Tian-Min Yan, SVP, M.J.J. Vrakking and D. Bauer, http://www.arxiv.org
Collaboration
D. Bauer, Tian-Min Yan
University of Rostock, Germany
V.D. Mur
Moscow Engineering Physics Institute, Russia
A.Palffy, H.M.C. Cortes
Max Planck Institute for Nuclear Physics, Heidelberg, Germany
V.S. Popov
Institute for Theoretical and Experimental Physics, Moscow, Russia
M.J.J. Vrakking
Max Born Institute, Berlin, Germany