Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A....

37
Keldysh Theory Cornelia Hofmann Department of Physics, Institute for Quantum Electronics, ETH Zurich, Switzerland

Transcript of Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A....

Page 1: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Keldysh Theory

Cornelia Hofmann

Department of Physics, Institute for Quantum Electronics,ETH Zurich, Switzerland

Page 2: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Goal

derive ionisation probability

P (t0) ∝ exp

(−2(2Ip(t0))3/2

3F (t0)

)exp

(v2⊥

2σ2⊥

)

Keldysh exponent

ADK transverse velocity probability

r

E

strong field→ tunnelling

ultrashort pulses→ FWHM ≈ fs regime

slow field compared toelectron dynamics

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Page 3: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Resources - Units

M. Y. Ivanov, M. Spanner and O. Smirnova, Anatomy of Strong Field Ionization,Journal of Modern Optics, 52:2-3, 165-184

A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011

Atomic units to make calculations/formulas easier:

~ = 1

|qe| = 1

me = 1

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Page 4: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Outline

1 Initial Situation

2 SFA, dipole and quasi-static approximation

3 Linear polarisation

4 Evaluate the probability amplitude

5 Keldysh exponent

6 End Result

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Experiments

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Page 6: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Laser-Atom system

-4 -2 0 2 4 6 8 10-2.0

-1.5

-1.0

-0.5

0.0

0.5

x�a.u.

E�a

.u.

V_effHrL

FHxL

VHrL

what we have to calculate:

i∂

∂t|Ψ〉 = H(t)|Ψ〉 with: H(t) = H0 + VL(t) (1)

field-free Hamiltonian of the atom

interaction with the laser field

Ψ = Ψ(t, ~r1, . . . ~rn)

full multi-electron wave function

explicitly time-dependant Hamiltonian (no separation ansatz)

general solution:|Ψ(t)〉 = e

−i∫ tt0

dt′H(t′)|Φi〉 (2)

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Page 7: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Laser-Atom system

-4 -2 0 2 4 6 8 10-2.0

-1.5

-1.0

-0.5

0.0

0.5

x�a.u.

E�a

.u.

V_effHrL

FHxL

VHrL

what we have to calculate:

i∂

∂t|Ψ〉 = H(t)|Ψ〉 with: H(t) = H0 + VL(t) (1)

field-free Hamiltonian of the atom

interaction with the laser field

Ψ = Ψ(t, ~r1, . . . ~rn)

full multi-electron wave function

explicitly time-dependant Hamiltonian (no separation ansatz)

general solution:|Ψ(t)〉 = e

−i∫ tt0

dt′H(t′)|Φi〉 (2)

5 / 20

Page 8: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations I

single active electron

Ψ(t, ~r1, . . . ~rn) ≈ Ψn−1(~r1, . . . ~rn−1)× Φ(~rn, t)

dipole approximation: λ >> r(atom)⇒ ~F (t)

SFA 1: neglect laser field while in bound state

adiabatic case (quasi-static approximation) ω << ωB

exact solution:

|Φ(t)〉 = −i∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ e−i

∫ tt0

dt′′H0(t′′)|Φi〉 (3)

-start t0

timet′ t

6-

-

VL

Φi H0

H Φ(t)

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Page 9: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations I

single active electron

Ψ(t, ~r1, . . . ~rn) ≈ Ψn−1(~r1, . . . ~rn−1)× Φ(~rn, t)

dipole approximation: λ >> r(atom)⇒ ~F (t)

SFA 1: neglect laser field while in bound state

adiabatic case (quasi-static approximation) ω << ωB

exact solution:

|Φ(t)〉 = −i∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ e−i

∫ tt0

dt′′H0(t′′)|Φi〉 (3)

-start t0

timet′ t

6-

-

VL

Φi H0

H Φ(t)

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Page 10: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations I

single active electron

Ψ(t, ~r1, . . . ~rn) ≈ Ψn−1(~r1, . . . ~rn−1)× Φ(~rn, t)

dipole approximation: λ >> r(atom)⇒ ~F (t)

SFA 1: neglect laser field while in bound state

adiabatic case (quasi-static approximation) ω << ωB

exact solution:

|Φ(t)〉 = −i∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ e−i

∫ tt0

dt′′H0(t′′)|Φi〉 (3)

-start t0

timet′ t

6-

-

VL

Φi H0

H Φ(t)

6 / 20

Page 11: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Solution Check

Subsitute (3) into Schrodinger’s Equation:

i∂|Φ(t)〉∂t

= i∂

∂t

(− i∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ e−i

∫ tt0

dt′′H0(t′′)|Φi〉

)

=∂

∂t

(∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ ie−i

∫ tt0

dt′′H0(t′′)|Φi〉

)

= −iH(t)

∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ VL(t)e−i

∫ tt0

dt′′H0(t′′)|Φi〉 + H0(t)e

−i∫ tt0

dt′′H0(t′′)|Φi〉

= H(t)|Φ(t)〉

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Page 12: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Solution CheckSubsitute (3) into Schrodinger’s Equation:

i∂|Φ(t)〉∂t

= i∂

∂t

(− i∫ t

t0

dt′[e−i

∫ tt′ dt′′H(t′′)

]VL(t′)

[e−i

∫ t′t0

dt′′H0(t′′)]|Φi〉

+ e−i

∫ tt0

dt′′H0(t′′)|Φi〉

)quasi-static:

=∂

∂t

(∫ t

t0

dt′[e−i(t−t

′)H]VL(t′)

[e−i(t

′−t0)H0

]|Φi〉

+ ie−i(t−t0)H0 |Φi〉

)

= −iH∫ t

t0

dt′[e−i(t−t

′)H]VL(t′)

[e−i(t

′−t0)H0

]|Φi〉

+ e−i(t−t)H VL(t)e−i(t−t0)H0 |Φi〉 + H0e−i(t−t0)H0 |Φi〉

= VL(t)|Φ(t)〉+ H0|Φ(t)〉 = H(t)|Φ(t)〉

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Page 13: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Basis in the Continuum

velocity basis |~v〉eigenstates: plane waves ei~κ~r for ~r large

no projection of ground state onto continuum states: 〈~v|Φi〉 = 0

projection of the evolved wave function onto a specific velocity ~v = (vx, vy, vz):

〈~v|Φ(t)〉 = Φ(~v, t)

= −i∫ t

t0

dt′⟨~v∣∣∣e−i ∫ t

t′ dt′′H(t′′)VL(t′)e−i

∫ t′t0

dt′′H0(t′′)∣∣∣Φi⟩ (4)

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

H |~v〉

9 / 20

Page 14: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Basis in the Continuum

velocity basis |~v〉eigenstates: plane waves ei~κ~r for ~r large

no projection of ground state onto continuum states: 〈~v|Φi〉 = 0

projection of the evolved wave function onto a specific velocity ~v = (vx, vy, vz):

〈~v|Φ(t)〉 = Φ(~v, t)

= −i∫ t

t0

dt′⟨~v∣∣∣e−i ∫ t

t′ dt′′H(t′′)VL(t′)e−i

∫ t′t0

dt′′H0(t′′)∣∣∣Φi⟩ (4)

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

H |~v〉

9 / 20

Page 15: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations IISFA 2: neglect ion field in continuum

→ Volkov propagator with HF = 12

(~p− q ~A

)2〈~v|e−i

∫ tt′ HF(t′′) dt′′ = e−i

∫ tt′

12 [~p+A(t′′)]2 dt′′〈~v′| = e−i

∫ tt′

12~v′′

2dt′′〈~v′|

ground state has energy −Ip

Φ(~v, t) = −i∫ t

t0

dt′⟨~v∣∣∣e−i ∫ t

t′ HF(t′′) dt′′ VL(t′)e−i

∫ t′t0

dt′′H0(t′′)∣∣∣Φi⟩

= −i∫ t

t0

dt′e−i∫ tt′

12~v′′

2dt′′ei(t

′−t0)Ip⟨~v′∣∣∣VL(t′)

∣∣∣Φi⟩ (5)

the canonical momentum ~p = ~v + q ~A is conserved

~v − ~A(t) = ~v∗ − ~A(t∗) ⇒ ~v∗ = ~v − ~A(t) + ~A(t∗)

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

×⟨~v − ~A(t) + ~A(t′)

∣∣∣VL(t′)∣∣∣Φi⟩ (6)

10 / 20

Page 16: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations IISFA 2: neglect ion field in continuum

→ Volkov propagator with HF = 12

(~p− q ~A

)2〈~v|e−i

∫ tt′ HF(t′′) dt′′ = e−i

∫ tt′

12 [~p+A(t′′)]2 dt′′〈~v′| = e−i

∫ tt′

12~v′′

2dt′′〈~v′|

ground state has energy −Ip

Φ(~v, t) = −i∫ t

t0

dt′⟨~v∣∣∣e−i ∫ t

t′ HF(t′′) dt′′ VL(t′)e−i

∫ t′t0

dt′′H0(t′′)∣∣∣Φi⟩

= −i∫ t

t0

dt′e−i∫ tt′

12~v′′

2dt′′ei(t

′−t0)Ip⟨~v′∣∣∣VL(t′)

∣∣∣Φi⟩ (5)

the canonical momentum ~p = ~v + q ~A is conserved

~v − ~A(t) = ~v∗ − ~A(t∗) ⇒ ~v∗ = ~v − ~A(t) + ~A(t∗)

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

×⟨~v − ~A(t) + ~A(t′)

∣∣∣VL(t′)∣∣∣Φi⟩ (6)

10 / 20

Page 17: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Approximations IISFA 2: neglect ion field in continuum

→ Volkov propagator with HF = 12

(~p− q ~A

)2〈~v|e−i

∫ tt′ HF(t′′) dt′′ = e−i

∫ tt′

12 [~p+A(t′′)]2 dt′′〈~v′| = e−i

∫ tt′

12~v′′

2dt′′〈~v′|

ground state has energy −Ip

Φ(~v, t) = −i∫ t

t0

dt′⟨~v∣∣∣e−i ∫ t

t′ HF(t′′) dt′′ VL(t′)e−i

∫ t′t0

dt′′H0(t′′)∣∣∣Φi⟩

= −i∫ t

t0

dt′e−i∫ tt′

12~v′′

2dt′′ei(t

′−t0)Ip⟨~v′∣∣∣VL(t′)

∣∣∣Φi⟩ (5)

the canonical momentum ~p = ~v + q ~A is conserved

~v − ~A(t) = ~v∗ − ~A(t∗) ⇒ ~v∗ = ~v − ~A(t) + ~A(t∗)

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

×⟨~v − ~A(t) + ~A(t′)

∣∣∣VL(t′)∣∣∣Φi⟩ (6)

10 / 20

Page 18: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Intuitive picture so far

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

× 〈~v − ~A(t) + ~A(t′)∣∣∣VL(t′)

∣∣∣Φi〉︸ ︷︷ ︸prefactor

(7)

-start t0

timet′ t

waiting in groundstate

kick from laser field, jump up to continuum state, exit tunnel with velocity ~v′

oscillating in the laser field

recorded on detector with velocity ~v

11 / 20

Page 19: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Intuitive picture so far

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

× 〈~v − ~A(t) + ~A(t′)∣∣∣VL(t′)

∣∣∣Φi〉︸ ︷︷ ︸prefactor

(7)

-start t0

timet′ t

-Φi H0

waiting in groundstate

kick from laser field, jump up to continuum state, exit tunnel with velocity ~v′

oscillating in the laser field

recorded on detector with velocity ~v

11 / 20

Page 20: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Intuitive picture so far

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

× 〈~v − ~A(t) + ~A(t′)∣∣∣VL(t′)

∣∣∣Φi〉︸ ︷︷ ︸prefactor

(7)

-start t0

timet′ t

6

-

VL

|~v′〉

Φi H0

waiting in groundstate

kick from laser field, jump up to continuum state, exit tunnel with velocity ~v′

oscillating in the laser field

recorded on detector with velocity ~v

11 / 20

Page 21: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Intuitive picture so far

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

× 〈~v − ~A(t) + ~A(t′)∣∣∣VL(t′)

∣∣∣Φi〉︸ ︷︷ ︸prefactor

(7)

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

HF

waiting in groundstate

kick from laser field, jump up to continuum state, exit tunnel with velocity ~v′

oscillating in the laser field

recorded on detector with velocity ~v

11 / 20

Page 22: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Intuitive picture so far

Φ(~v, t) = −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

× 〈~v − ~A(t) + ~A(t′)∣∣∣VL(t′)

∣∣∣Φi〉︸ ︷︷ ︸prefactor

(7)

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

HF |~v〉

waiting in groundstate

kick from laser field, jump up to continuum state, exit tunnel with velocity ~v′

oscillating in the laser field

recorded on detector with velocity ~v

11 / 20

Page 23: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Linear polarisation

�����

���:reuHe+

��:x@@Iy 6

z ����:−~F (t′)6

?@@R@@I

v⊥

re−laser field polarised along x→ vector potential:

~A(t) =F0

ωsin(ωt)x (8)

⇒(~v − ~A(t) + ~A(t′′)

)2=

(vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

)2

+ v2y + v2z

(9)

define action

S~v(t, t′) :=1

2

∫ t

t′

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

]2dt′′ +

v2y + v2z2

(t− t′)

−Ip(t′ − t0) (10)

Φ(~v, t) ∝ −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

⇒ Φ(~v, t) ∝ −i∫ t

t0

exp(−iS~v(t, t′)

)dt′ (11)

12 / 20

Page 24: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Linear polarisation

�����

���:reuHe+

��:x@@Iy 6

z ����:−~F (t′)6

?@@R@@I

v⊥

re−laser field polarised along x→ vector potential:

~A(t) =F0

ωsin(ωt)x (8)

⇒(~v − ~A(t) + ~A(t′′)

)2=

(vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

)2

+ v2y + v2z

(9)define action

S~v(t, t′) :=1

2

∫ t

t′

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

]2dt′′ +

v2y + v2z2

(t− t′)

−Ip(t′ − t0) (10)

Φ(~v, t) ∝ −i∫ t

t0

dt′e−i∫ tt′

12 (~v− ~A(t)+ ~A(t′′))2 dt′′ei(t

′−t0)Ip

⇒ Φ(~v, t) ∝ −i∫ t

t0

exp(−iS~v(t, t′)

)dt′ (11)

12 / 20

Page 25: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Recap

That was the hard part. ,Probability amplitude of finding an electron

with velocity ~v on the detector,from any ionisation time t′.

Φ(~v, t) ∝ −i∫ t

t0

exp(−iS~v(t, t′)

)dt′

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

HF |~v〉

13 / 20

Page 26: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Outlook

Now comes the fun part.Find the probability of ionisation depending on

ionisation time t′

assumption: instantaneous tunneling timeand tunnel exit velocity ~v′.

P (t′, ~v′) =?

-start t0

timet′ t

6-

-

VL

|~v′〉

Φi H0

HF |~v〉

set t0 = 0

push t towards t′ and all that towarts t0

we have to evaluate∫ t

t0

exp(−iS~v(t, t′)

)dt′

14 / 20

Page 27: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Saddle point approximation I

∫ x2

x1

f(x)eig(x) dx

f(x) slowly varying (in our case: f(x) ≡ 1)

main contribution where g′(x) = 0 (no fastoscillations which cancel contributions out)

S~v(t, t′) :=1

2

∫ t

t′

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

]2dt′′ +

v2y + v2z2

(t− t′)

−Ipt′

find a particular t′∗ such thatδS~v(t, t′)

δt′

t′∗

= 0

0 =−1

2

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′∗)

]2−v2y + v2z

2−Ip

15 / 20

Page 28: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Saddle point approximation I

∫ x2

x1

f(x)eig(x) dx

f(x) slowly varying (in our case: f(x) ≡ 1)

main contribution where g′(x) = 0 (no fastoscillations which cancel contributions out)

S~v(t, t′) :=1

2

∫ t

t′

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′′)

]2dt′′ +

v2y + v2z2

(t− t′)

−Ipt′

find a particular t′∗ such thatδS~v(t, t′)

δt′

t′∗

= 0

0 =−1

2

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′∗)

]2−v2y + v2z

2−Ip

15 / 20

Page 29: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Saddle point approximation II

0 =1

2

[vx −

F0

ωsin(ωt) +

F0

ωsin(ωt′∗)

]2+v2y + v2z

2+Ip

define a varied Keldysh parameter

(usually: γ =

√2Ip

F0ω

)

γ =

√2Ip + v2y + v2z

F0ω (12)

⇒ perpendicular velocity “adds” to the ionisation potential

final equation to solve in order to find the saddle point:

0 =

[vxω

F0− sin(ωt) + sin(ωt′∗)

]2+ γ2 (13)

16 / 20

Page 30: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Special Case: ~v = 0

0 =

[vxω

F0− sin(ωt) + sin(ωt′∗)

]2+ γ2

let’s find |Φ(~v = 0, ωt = nπ)|2

vx = 0 γ = γ sin(ωt) = 0

sin(ωt′∗) = ±iγ

ionisations only happen when the field is strong enough⇒ ωt′∗ << 1

ωt′∗ ≈ ±iγ

t′∗ ≈ iγ

ω= i

√2Ip

F0= iτKeldysh (14)

dominant contribution to ionisation happens (or starts) at imaginary time!(and there are more tunnelling times which come out imaginary from thecalculations . . . )

17 / 20

Page 31: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Special Case: ~v = 0

0 =

[vxω

F0− sin(ωt) + sin(ωt′∗)

]2+ γ2

let’s find |Φ(~v = 0, ωt = nπ)|2

vx = 0 γ = γ sin(ωt) = 0

sin(ωt′∗) = ±iγ

ionisations only happen when the field is strong enough⇒ ωt′∗ << 1

ωt′∗ ≈ ±iγ

t′∗ ≈ iγ

ω= i

√2Ip

F0= iτKeldysh (14)

dominant contribution to ionisation happens (or starts) at imaginary time!(and there are more tunnelling times which come out imaginary from thecalculations . . . )

17 / 20

Page 32: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Special Case: ~v = 0

0 =

[vxω

F0− sin(ωt) + sin(ωt′∗)

]2+ γ2

let’s find |Φ(~v = 0, ωt = nπ)|2

vx = 0 γ = γ sin(ωt) = 0

sin(ωt′∗) = ±iγ

ionisations only happen when the field is strong enough⇒ ωt′∗ << 1

ωt′∗ ≈ ±iγ

t′∗ ≈ iγ

ω= i

√2Ip

F0= iτKeldysh (14)

dominant contribution to ionisation happens (or starts) at imaginary time!(and there are more tunnelling times which come out imaginary from thecalculations . . . )

17 / 20

Page 33: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

The Keldysh exponent

P (~v = 0, t) = |Φ(~v = 0, t)|2 ∝∣∣∣∣−i ∫ t

0

e−iS~v(t,t′) dt′

∣∣∣∣2 SPA≈∣∣∣e−iS~v(t,t

′∗)∣∣∣2

substituting t′∗ = i

√2Ip

F0into S~v(t, t′), ωt = nπ, set n = 0:

S~v(0, t′∗) =1

2

∫ 0

t′∗

[F0

ωsin(ωt′′)

]2dt′′ − Ipt′∗

≈ 1

2

∫ 0

t′∗

(F0

ωωt′′)2

dt′′ − Ipt′∗ =−1

2F 20

(t′∗)3

3− Ipt′∗

≈ i

2

(2Ip)3/2

F0

(1

3− 1

)=−i2

2

3

(2Ip)3/2

F0(15)

Probability of ionisation at the peak of the field, with zero velocity, toexponential accuracy:

P (~v = 0, t = 0) ∝ exp

(−2(2Ip)3/2

3F0)

)(16)

18 / 20

Page 34: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

The Keldysh exponent

P (~v = 0, t) = |Φ(~v = 0, t)|2 ∝∣∣∣∣−i ∫ t

0

e−iS~v(t,t′) dt′

∣∣∣∣2 SPA≈∣∣∣e−iS~v(t,t

′∗)∣∣∣2

substituting t′∗ = i

√2Ip

F0into S~v(t, t′), ωt = nπ, set n = 0:

S~v(0, t′∗) =1

2

∫ 0

t′∗

[F0

ωsin(ωt′′)

]2dt′′ − Ipt′∗

≈ 1

2

∫ 0

t′∗

(F0

ωωt′′)2

dt′′ − Ipt′∗ =−1

2F 20

(t′∗)3

3− Ipt′∗

≈ i

2

(2Ip)3/2

F0

(1

3− 1

)=−i2

2

3

(2Ip)3/2

F0(15)

Probability of ionisation at the peak of the field, with zero velocity, toexponential accuracy:

P (~v = 0, t = 0) ∝ exp

(−2(2Ip)3/2

3F0)

)(16)

18 / 20

Page 35: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

The Keldysh exponent

P (~v = 0, t) = |Φ(~v = 0, t)|2 ∝∣∣∣∣−i ∫ t

0

e−iS~v(t,t′) dt′

∣∣∣∣2 SPA≈∣∣∣e−iS~v(t,t

′∗)∣∣∣2

substituting t′∗ = i

√2Ip

F0into S~v(t, t′), ωt = nπ, set n = 0:

S~v(0, t′∗) =1

2

∫ 0

t′∗

[F0

ωsin(ωt′′)

]2dt′′ − Ipt′∗

≈ 1

2

∫ 0

t′∗

(F0

ωωt′′)2

dt′′ − Ipt′∗ =−1

2F 20

(t′∗)3

3− Ipt′∗

≈ i

2

(2Ip)3/2

F0

(1

3− 1

)=−i2

2

3

(2Ip)3/2

F0(15)

Probability of ionisation at the peak of the field, with zero velocity, toexponential accuracy:

P (~v = 0, t = 0) ∝ exp

(−2(2Ip)3/2

3F0)

)(16)

18 / 20

Page 36: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

The generalised Keldysh exponent I

quasi-static idea: ionisations at other times t 6= 0 means we havelower field strength F (t) ≤ F0

account for Stark shift in the ionisation potential

Ip(t) = Ip(F (t)) = Ip,0 +1

2(αN − αI)F (t)2, (17)

allow transverse velocity at exit tunnel (adding to the ionisation potential), laserpropagation in z direction, elliptically polarised in x− y plane

P (v|| = 0, v⊥, vz, t) ∝ exp

{−2(2Ip(t) + v2⊥ + v2z

)3/23F (t)

}(18)

������

��:reuHe+ - x���y6

z ����:−~F (ti)6

?@@R

@@I

v⊥vz

re−

19 / 20

Page 37: Cornelia Hofmann Department of Physics, Institute …...Journal of Modern Optics, 52:2-3, 165-184 A. S. Landsman, Laser-Atom Interaction, Lecture notes FS 2011 Atomic units to make

Final result

first order taylor:(2Ip(t) + v2⊥ + v2z

)3/2 ≈ (2Ip(t))3/2 +3

2(2Ip(t))1/2(v2⊥ + v2z)

everything together:

P (v|| = 0, v⊥, vz, t) ∝ exp

(−2 (2Ip(t))3/2

3F (t)

)exp

(−v

2⊥ + v2z2σ2⊥

)(19)

withσ2⊥ =

F (t)

2(2Ip(t))1/2=

ω

2γ(20)

20 / 20