CNISM
Molecules in a laser field:
semiclassical & quantal treatment
E. Fiordilino*§, G. Castiglia*§, P. P. Corso*§,
R. Daniele, F. Morales, G. Orlando* and F. Persico*§
* CNISM and Dipartimento di Scienze Fisiche ed Astronomiche,
Via Archirafi 36, 90123 Palermo, Italy.
§ Consorzio COMETA - Research Line: Atoms in Strong Fields, Dipartimento di Scienze
Fisiche ed Astronomiche, Via Archirafi 36, 90123 Palermo, Italy.
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High Harmonic Generation
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Molecule
Diffused radiation
2
2q
2
P(t )  3 a(t )
3c
2
dS 4q
2
 3 a 
d 3c
Laser
Artistic spectrum
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Fullerene
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2

Lˆ2  VL f t  cos  sin  L t 
Hˆ 
2I
Lˆ2 , m    1 , m
  Lt
VL
L 
 L
, m  Y ,m  ,  

    1
2I

i t  Hˆ t
t

 
L
  1,    1   

t   c ,0

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 1


i
c


c


sin  c  1

 
L

2  12  3





c 1
  1c 1
ic    c   L 

 sin  

2  12  3 
 2  12  1
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Pulse duration 64 oc
0,0
I L  3.79 1011 W/cm 2
 L  0.2 eV
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0,0
I L  9.48 1012 W/cm 2
 L  0.2 eV
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0,0
I L  3.79 1013 W/cm 2
 L  0.2 eV
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5,0
I L  1.36 1013 W/cm 2
 L  1.2 eV
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5,0
I L  8 1014 W/cm 2
 L  1.2 eV
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5,0
I L  2.3 1015 W/cm 2
 L  1.2 eV
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i
 10
L
f
 22
L
5,0
I L  2.3 1015 W/cm 2
 L  1.2 eV
f
r t    r   exp  it d
i
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i
 10
L
f
 32
L
f
r t    r   exp  it d
i
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2

Hˆ 
Lˆ2  VL f t  cos  sin Lt 
2I
Cutoff law at low field
M  1, 0   L  L  1  0   VL
VL
1,0
M
1  0 
0,0
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Diatomic molecules in a
laser field
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Full hamiltonian




2
2


Hˆ  
 2R1   2R 2 
 r21   r22  U R1 , R 2 , r1 , r2   V R1 , R 2 , r1 , r2 , t 
2M
2m
2
e2
e2
e2
U R1 , R 2 , r1 , r2    


r1  r2 R1  R 2
i , j 1 ri  R j
V R1 , R 2 , r1 , r2 , t   e R1  R 2  r1  r2   Et 
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+
H2 Classical nuclei
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
Schroedinger equation for the electron
i   x, t   H 1 2 c t   x, t 
t
d 2 Ri
M 2  Fie t   Fij t   eE t 
Newton equation for the nuclei
dt
2  2
H 1 2 c t   
 eE t x 
2
2m x
Hamiltonian
e2
e2


2
2
a 2  R1 t   x 
a 2  R2 t   x 
Fie t    
e 2 x  Ri t   x, t 
2
A  x  R t  
2 3/ 2
dx
electron nuclei interaction
i
Fij t  
e 2 Ri t   R j t 
A  R t   R t  
nucleus-nucleus interaction
2 3/ 2
n
i
j
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I L  5 1014 W/cm 2
 L  2 eV
It is evident the presence of
regularly spaced sidebands
around the odd harmonic
peaks
The satellite peak spacing is
exactly given by the nuclei
vibration frequency
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1D model for various molecular isotopes
Different molecular isotopes
are characterized by
different satellite peak
spacing
Each satellite peak spacing is
directly related to the
corresponding nuclear vibration
frequency
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Fixed nuclei: spectrum for M=M0
No side-bands
around the odd
harmonic lines
of HHG are
present
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Schroedinger equation in 3D
The presence of the
satellite peaks in the
harmonic spectrum
does not depend upon
the dimensionality of
the model.
The harmonic lines
present side peaks
whose separation
scales as
N  1 M
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H2
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1D semiclassical model for various H2 molecular
isotopes
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1D full quantum calculations
+
for H2
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Conceptual considerations suggest the use of a full quantal calculation for the
molecule.

 R, x, t   H t  R, x, t 
t
2  2
e2
2  2
H t   


 eE t x 
2
2
2
M R 2
2
m

x
aN  R
i
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E t   E0 (t ) sin Lt
e2
R

a2    x 
2

2

e2
R

a2    x 
2

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2
I L  11014 W/cm 2
λ  780nm   L  1.6 ev
ΔωN
N  1 M
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  256 oc
____ M = M0
____ M = 4 M0
____ M = 16 M0
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1D full-quantum model for various H2+ isotopes
T = 4 o.c.
I L  2 1014 W/cm 2
  780nm
____ M = M0
____ M = 4 M0
____ M = 16 M0
For short pulses
heavy nuclei win
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1D full-quantum model for various H2+ molecular isotopes
T = 8 o.c.
____ M = M0
____ M = 4 M0
____ M = 16 M0
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1D full-quantum model for various H2+ molecular isotopes
T = 16 o.c.
____ M = M0
____ M = 4 M0
____ M = 16 M0
For long pulse
light nuclei win
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1D full-quantum model for various H2+ molecular isotopes
T = 32 o.c.
____ M = M0
____ M = 4 M0
____ M = 16 M0
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1D full-quantum model for various H2+ isotopes
T = 64 o.c.
____ M = M0
____ M = 4 M0
____ M = 16 M0
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1D full quantum calculations for molecular ions
oscillations of the nuclei
spread of the nuclear wavefunction
____ M = M0
____ M = 4 M0
____ M = 16 M0
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Ionization Signal
Nuclear wavefunction spread
Isotopic effect:
the intensity of the harmonics decreases by increasing the mass of the nuclei
Possible explanation:
Nuclear wavefunction for lighter nuclei easily spreads and crosses a region of R
with enhanced ionization
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ISOTOPIC EFFECT
SHORT LASER PULSE:
LONG LASER PULSE:
The intensity of harmonics
The intensity of harmonics
increases
decreases
by decreasing nuclear mass
by decreasing nuclear mass
The nuclear wavefunction does not
The nuclear wavefunction approaches
reach the critical ionization region
the critical ionization region
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CONCLUSIONS
• Fullerene should emit harmonics: I hope experiments will
come soon.
• The motion of the nuclei introduces in the HHG spectrum
satellite peaks of the odd-harmonics. The presence of the
side-peaks is a direct evidence of the nuclear vibration
motion.
• A full quantum model shows an isotopic effect in the
HHG spectra.
References:
N. L. Wagner, A. Wüest, I. P. Christov, T. Popmintchev, X. Zhou, M. M. Murnane,
H. C. Kapteyn, PNAS 103, 13279 (2006).
P. P. Corso, E. Fiordilino, F. Persico, JPB 40, 1383 (2007).
S. De Luca and E. Fiordilino, J. Phys. B: At. Mol. Opt. Phys. 29, 3277 (1996).
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