electron wavefunction in strong and ultra-strong laser field
DESCRIPTION
Electron wavefunction in strong and ultra-strong laser field. One- and two-dimensional ab initio simulations and models. Jacek Matulewski Division of Collision Theory and Nonlinear Systems Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University in Toruń, Poland - PowerPoint PPT PresentationTRANSCRIPT
Electron wavefunction in strong and ultra-strong laser field
One- and two-dimensional ab initio simulations and models
Jacek Matulewski
Division of Collision Theory and Nonlinear SystemsFaculty of Physics, Astronomy and Informatics
Nicolaus Copernicus University in Toruń, Poland
Marburg, 20 I 2004
2/38
Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)
Authors: Andrzej Raczyński, Jarosław Zaremba and Jacek Matulewski
3/38
Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)
4/38
Language
l
ln
n dEEltEntt0
;),()()(
1)0(
2)()( ttP nn
2),E(lim)E( tS l
tl
Space built with :
Time depended quantities we’re interested in
Initial state = ground bound state of quantum system
5/38
Simulation in 1D - typical result for strong laser fieldEvolution of wavefunction (fast oscillations with frequency of laser field was removed)2
),( tx
Potential well:2a = 0.244 a.u. = 1.3·10-11 mV0 = 2.049 a.u. = 55.7 eV
Electric field: = 1 a.u. = 6.6·1015 s-1
= 1 a.u. = 5.1·1011 V/m(I = 2 = 1016 W/cm2)
Phase and pulse shape must be chosen carefully because of possibility of electron escape due to fast drift(Newton’s equations)
6/38
Simulation in 1D - typical result for strong laser field
Population of initial state:
Photoelectron spectra:
1e-08
1e-071e-061e-05
0.00010.001
0.010.1110
0 1 2 3 4 5 6 7 8 9 10 11 12
E
S(E)
ATI phenomena
fast oscillations
7/38
Simulation in 1D - slow drift and stabilisation
Evolution of wavefunction in = 5 a.u.(precisely: of its part located near potential well)
Wavefunction properties• slowly changes position• remains its shape
8/38
Simulation in 1D - slow drift and stabilisation
Some photos of wavefunction in = 5 a.u.(precisely: of its part located near potential well)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
-20 -15 -10 -5 0 5 10
0T8T16T20T24T40T
x
|(x,t)|2
Wavefunction properties• slowly changes position• remains its shape
The idea is to find the potential which keeps shape of the wavefunction unchanged and model its movement due to perturbation of free oscillating electron by well potential
9/38
Simulation in 1D - Generalised KH modelPhys Rev A 61, 043402 (2000)
),()cos(),()(),(2
1),(
2
2
txtxtxxUtxx
txt
i
),())()((),())((),(21
),( 12
2
tytyttytyUtyy
tyt
i
Transformation to Kramers frame , whereand is putted together fast oscillations and slow drift
),(),( tytx )(txy )()()( 10 ttt
),()]()()[(
])()[)((),())((),(21
),(
101
0100002
2
tyttyt
xtxtyUtyxtyUtyy
tyt
i
Spreading of oscillating potential around (x0 is average value of 0(t))
))()(( 10 ttyU 00 )( xty
One dimensional Schrödinger equation for one electron in laser field (dipole approximation):
10/38
0
0.01
0.02
0.03
0.04
-20 -15 -10 -5 0 5
x
0.5952 |x+20T)|2
UKH(x+20T))
|(x,20T)|2
),())((),(21
),( 002
2
tyxtyUtyy
tyt
i
Simulation in 1D - Generalised KH modelPhys Rev A 61, 043402 (2000)
Approximation:Function describing slow drift is well fitted by solution of Ehrenfest equation1
)),(),(( ),()(
),()()(22
0*
1 tataVtxdx
xdUtxdxxU
dxd
t
The reason of slow drift is interaction of wavefunction with the well edges
),()(),(21
),(2
2
tyyUtyy
tyt
i KH
Replacing time depended potential with HK wellwhich is zeroth term of its Fourier expansion:
Tt
t
KH dxyUT
yU ))((1
)( 00
In our parameters HK well has only one bound state
11/38
One period of oscillations - marks of the interaction with the potential well
Simulation in 1D - Generalised KH model (results)Phys Rev A 61, 043402 (2000)
12/38
Simulation in 1D - Generalised KH model (results)Phys Rev A 61, 043402 (2000)
-15
-10
-5
0
5
10
15
50 100 150 200 250 300 350 400
t
(t)
xmax(t)
Position of the electron
13/38
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 50 100 150 200 250 300 350 400
t
|(t)|2
Simulation in 1D - Generalised KH model (results)Phys Rev A 61, 043402 (2000)
Occupation of the potential well ground state (initial state)
14/38
Simulation in 1D - Generalised KH model (results)Phys Rev A 61, 043402 (2000)
What else influence the final population of the initial state?
)2
exp( )0( 1 t
The way of switching on the pulse
Time of live in HK-well(Volkova et all, Zh. Eksp. Teor. Fiz. 106, 1360 (1994))
Transformed back to laboratory frame wavefunction of HK eigenstate
Slow drift and fast oscillations
)))cos()(cos(()2
)()(exp( ),(
21
0
ttxdtt
xttxt
Stabilisation is not permanent because of HK state is ionised (higher terms of Fourier expansion)
15/38
0
0.01
0.02
0.03
0.04
-20 -15 -10 -5 0 5
x
0.5952 |x+20T)|2
UKH(x+20T))
|(x,20T)|2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1 1.5 2 2.5 3 3.5 4 4.5 5
|(16T)|2
|(16.5T)|2
|(16T)|2
-15
-10
-5
0
5
10
15
50 100 150 200 250 300 350 400
t
(t)
xmax(t)
1. Stabilisation in Kramers- Henneberger well
2. Slow drift of almost free electrons (interaction with well)
Simulation in 1D - SummaryPhys Rev A, 61, 043402 (2000)
16/38
Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)
17/38
Typical simulation set in 3D:wave propagation direction: yelectric field polarisation direction: xmagnetic field polarisation direction: z
),(ˆ),( tyExtrE ) cos(ˆ),( 0 ty
cExtrE
t
rEdtrA ),( ),( c
ykyk ˆˆ
)sin(ˆ),( 0 tyc
ExtyA
),(ˆ
),(ˆ
),(),(
tyBz
tyAz
tyAtrB
y
t
yEdx ),( ˆ
)cos(ˆ),( 0 tycc
EztyB
18/38
Atom + laser field in 3D without dipole approximation:classical description of free electron (Lorentz force)quantum description of atom in classical external field
dipole approximation
Atom + pole lasera w 1D:stabilisation - HK wellslow drift
Atom + laser field in 2D:stabilisation - HK well in 2Dmagnetic drift
Omitting free evolution in z directionSimplifying the y dependence of hamiltonian
Possible description approaches
19/38
Classical description in 3D (de facto 2D)
),(ˆ),( tyBztyB
cykyk
ˆˆElectromagnetic field:
),(ˆ),( tyExtyE
0 , ,)( BvBvEqBvEqF xy
Lorentz force acting on free electron:
Equations of motion:
))(cos()()(
))(cos())(
1()(
0
0
ttkytxmcqE
ty
ttkycty
mqE
tx
and trivial equation for z(t)
20/38
Classical description in 3D (de facto 2D)
Solution:10 20 30 40 50 60
-30
-25
-20
-15
-10
-5
x(t) and
1) cos(20
t
qE
10 20 30 40 50 60
5
10
15
20
25
y(t)
Equations of motion:
))(cos()()(
))(cos())(
1()(
0
0
ttkytxmcqE
ty
ttkycty
mqE
tx
21/38
Quantum description in 3D
)(2
)),(ˆ(ˆ2
rVm
tyAqpH
in Coulomb gauge
)(),(2
),()(2
ˆ 22
2222
rVtyAm
qtyA
mqi
mH xzyx
Schrödinger equation(omitting evolution in z direction):
),,(),(2
)(),,(),(),,()(2
),,(ˆ
22
222
tyxtyAm
q
rVtyxtyAmqi
tyxm
tyxH xyx
Dipole approximation+ unitary transformation
One-dimensional calculations dE(oscillations, stabilisation, slow drift)
Two-dimensional calculation pA(no dipole approximation!!!)
22/38
Solutions of two-dimensional Schrödinger equationsin pA gauge, no dipole approximation, ultra-strong laser field
tx
-30
-20--10
0
10
0 10 20 30 40 50 60 70
0
4
8
12
16
0 10 20 30 40 50 60 70
ty
2)10,,( Ttyx
Stabilisation (HK well)Magnetic driftin propagation direction
atom: radial well a = 1 a.u., V0 = 2 a.u.field: = 1 a.u., E0 = 15 a.u.
Reiss, Phys. Rev 63 013409 (2000)
23/38
Wavefunction - phenomenology of stabilisationPhys Rev A 68, 013408 (2003)
atom: radial well a = 1 a.u., V0 = 2 a.u.field: = 1 a.u.
2)10,,( Ttyx
two scattering centresin x = 0 and x = 30 a.u.
E0 = 15 a.u.
E0 = 20 a.u.4T, 7T, 10T,
24/38
Characteristics of wavefunction1. Regardless the phase of laser field magnetic drift is always directed to +y2. Drift is linear in time + 2 oscillations3. Constant drift velocity depends on E0
2
4. Wave of electron finding probability has double frequency in y direction
Wavefunction - phenomenology of stabilisationPhys Rev A 68, 013408 (2003)
2)6,,( Ttyx
E0 = 20 a.u.
25/38
(1) Schrödinger equation with no dipole approximation (pA gauge)
Quantum model of stabilisation and magnetic drift in 2DSimplifying the hamiltonian
spreading of vector potential in seriesPatel et al., Phys.Rev.A 64 013411
(2) Lowest order approximation of nondipol Schrödinger equation (additional term proportional to y in hamiltonian describing magnetic field)
(3) Equation in Kramers frame with additional magnetic field
transformation to frame of oscillating electrontransformation removing A2 term
averaging of oscillating potential
(4) Electron motion equation in HK well and coupling with continuum by magnetic field
26/38
Quantum model of stabilisation and magnetic drift in 2DLooking for hamiltonian with simplier dependence on y
(1)
) () (...) (
) (
...)(
21)(
)()() ( 2
2
2
2
tEcy
tAdt
tdAcy
tA
dttfd
cy
dttdf
cy
tfcx
tfkytA
xxx
x
x
)(2
)) ( ˆ ˆ(),(ˆ
2
rVm
kytAxqptxH x
) ( ˆ)) ( ˆˆ ()(2
)) ( ˆ ˆ()(ˆ
2
tExtAxpcy
rVm
tAxqptH xx
x
)()( )(2
))(ˆˆ()(ˆ
2
tEtAcy
rVm
tAxqptH xx
x
In high frequency regime the distribution of wavepacket in momentum spaceis concentrated around the zero
(2)
Term describingmagnetic field
27/38
) () ()(2
))(ˆ ˆ()(ˆ
2
tEtAcx
rVm
tAxqptH xx
x (2)
Quantum model of stabilisation and magnetic drift in 2DTransformation to the Kramers frame
)()(),()(2
)()(2
)(ˆ 22
222
tAtAcy
yxVtAm
qtA
mqi
mtH xxxxxyx
Transformation removing the A2 term
)')'(2
exp( 2t
dttAi
)')'(exp(ˆ t
dttAU
Transformation to framemoving with an electron
In Kramers frame potential oscillates
) () (),,()(2
)(ˆ 222
tAtAcy
tyxVm
tH xxyx (3)
28/38
Quantum model of stabilisation and magnetic drift in 2DFourier expansion of the time depended potential of the oscillating well
In high frequency regime one can replace fast oscillating potential withtime averaged potential of Kramers-Henneberger well:
)(ˆ) () ();,()(2
)(ˆ0
222
tHHtAtAcy
EyxVm
tH IKHxxKHyx (4)
„typical” time independed part of hamiltonianwith bounding potential
Term similar to dE (but here all in pA)
T
xKH dyxVT
yxV0
)),((1
),()),(( ytxV
)(),( 2220 yxrVyxV
where
1)cos()( 20
t
Etand
29/38
Quantum model of stabilisation and magnetic drift in 2DBounded states of Kramers-Henneberger well VHK(x, y) in 2D
Potential and ground state (E1 = – 0.0192 a.u.). Only excited state (E2 = – 0.0158 a.u.)
E0 = 15 a.u.
);,()();,(ˆ000 EyxEEEyxH KH
KH
Eigenvalue problem:
30/38
Quantum model of stabilisation and magnetic drift in 2DHK well bound states coupling with continuum
)2sin(2
ˆ)() (ˆ)(ˆ20 tc
EyHtAtA
cy
HtH KHxxKH
(4)
Approximated hamiltonian for linear polarised laser field propagating in y direction:
Characteristics of wavefunction1. Regardless the phase of laser field magnetic drift is always directed to +y2. Drift is linear in time + 2 oscillations3. Constant drift velocity depends on E0
2
4. Wave of electron finding probability has double frequency in y direction
Classical model in 2D/3D
]0 ),,(),(),,(),(),([),( tyBtyvtyBtyvtyEtyF zxzyx
]0 ),2sin(2
),cos([)(20
0 tc
EtEtF
Dipole approxination (regardingy dependence in Ex(y,t) and Bz(y,t))
) 2sin(2
)(20 tc
EytVy
31/38
Stabilisation and magnetic drift in ultra-strong laser fieldSummary of 2D simulations
Probability of finding electron inside the o box with size 100 a.u. around the well
Field intensity
with dipole app.
without dipole app.
Ryabikin, Sergeev, Optics Express 417, 7 12 (2000)
32/38
Saving the stabilisation using constant magnetic fieldfor quantum model and details see Phys Rev A 68, 045401 (2003)
Wavefunction without and with constant magnetic field
33/38
Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)
34/38
Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)
35/38
( ) ( ) c o s ( )t f t t 0
I
02
8
0 0 0 0 1
1 0
. j . a .
W / c m 2
( ) ( ) c o s ( )t f t t 0 ,
I
02
1 6
1
1 0
j . a .
W / c m 2
( ) ( ) c o s ( )
( ) c o s ( )
t f t t
f t t t
0 0
1 1 1
I 02 71 0 j . a .
( ) c o s (
s i n ( ) )
t t
t
0 0
1
I 02 31 0 j . a .
Other projects (quantum engineering)
36/38
Control of ionisation of hydrogen atom by two-colour pulsePhysics Letters A 25 205-211 (1999)
detuning
37/38
Control of Rydberg atom by chirped pulsePhys Rev A 57, 4561 (1998)
Chirped pulse:
n
n tnJtt ]2)cos[()()2sincos( 1010
-1
-0.8
-0.6
-0.4
-0.2
0
-4 -2 0 2 4
n=1
n=2
n=3
n=4
By changing the depth of modulation one can control the coupling amplitudes depending on )(nJ
0 = 13 = 0.518 a.u.
1 = 34 = 0.0587 a.u.
= 0.05 a.u.
Rochester atom: 22/)( xaqxV
38/38
Control of Rydberg atom by chirped pulsePhys Rev A 57, 4561 (1998)
Chirped pulse:
n
n tnJtt ]2)cos[()()2sincos( 1010
1e-081e-071e-061e-050.00010.0010.010.1110100
0.13 0.93 1.73 2.53 3.33 4.13
1e-071e-06
1e-05
0.0001
0.0010.01
0.1
1
10100
0.13 0.93 1.73 2.53 3.33 4.13
photoelectron energy
(no modulation)0
1.0
Photoelectron spectra
39/38
Control of Rydberg atom by chirped pulsePhys Rev A 57, 4561 (1998)
n
n tnJtt ]2)cos[()()2sincos( 1010Chirped pulse:
0
0.2
0.4
0.6
0.8
1
0 250 500 750 1000 1250 1500
n = 1n = 2
= 3.8
0
0.1
0.2
0.3
0.4
0 250 500 750 1000 1250 1500
n = 4 = 3.8 = 1.9
Bound states population
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Outline
1. Stabilisation and related phenomena in 1D strong field ionisation
2. Beyond the dipole approximation - magnetic drift in 2D ultra-strong field
3. Brief report on other projects (control of wavefunction by pulses)