Saddle points for the multicolor fields

where ₂ and ₃ are small parameters for the 2ω and 3ω fields, respectively, φ is the phase delay between the fundamental field and the second harmonic, and φ_m is referred to as the multicolor phase or the multicolor delay.

The saddle point equations now become:

1

2(p||−A₀sin(ωt_i))²+ (4.35) +1

2(p⊥−₂A₀sin(2ωt+φ) −₃A₀sin(3ωt+φ+φm) )²+Ip = 0

Z t ti

(p||−A₀sin(ωτ))dτ = 0 (4.36)

Z t ti

(p⊥−₂A₀sin(2ωτ +φ) −₃A₀sin(3ωτ +φ+φ_m))dτ = 0 (4.37)

The procedure of solving the above saddle point equation 4.35, 4.36 and 4.37 is exactly the same as for the case of the two-color field.

Making the the substitutions k⁰_|| = ^p

0

||

A0, k_||⁰⁰ = ^p

00

||

A0, k⁰_⊥ = ^p_A^0⊥

0, k_⊥⁰⁰ = ^p_A^00⊥

0, ϕ⁰_i = ωt⁰_i, ϕ⁰⁰_i = ωt⁰⁰_i, ϕ_r =ωt_r, we obtain from Eqs.(4.36,4.37), after calculating the integral in Eq.(4.6),

k_||⁰ = k_||⁰⁰(ϕ_r−ϕ⁰_i)

ϕ⁰⁰_i +sinϕ⁰_isinhϕ⁰⁰_i

ϕ⁰⁰_i (4.38)

k_||⁰⁰= ϕ⁰⁰_i cosϕ⁰_icoshϕ⁰⁰_i −ϕ⁰_icosϕr−(ϕr−ϕ⁰_i) sinϕ⁰_isinhϕ⁰⁰_i

ϕ⁰⁰_i²+ (ϕ_r−ϕ⁰_i)² (4.39)

k⁰_⊥= k⁰⁰_⊥(ϕ_r−ϕ⁰_i) ϕ⁰⁰_i +₂

2

sin(2ϕ⁰_i+φ) sinh 2ϕ⁰⁰_i ϕ⁰⁰_i + +₃

3

sin(3ϕ⁰_i+φ+φ_m) sinh 3ϕ⁰⁰_i

ϕ⁰⁰_i (4.40)

4.3. SADDLE POINTS FOR THE MULTICOLOR FIELDS

k_⊥⁰⁰ =

= ₂ 2

ϕ⁰⁰_i cos(2ϕ⁰_i+φ) cosh 2ϕ⁰⁰_i −ϕ⁰_icos(2ϕ_r+φ)−(ϕ_r−ϕ⁰_i) sin(2ϕ⁰_i+φ) sinh 2ϕ⁰⁰_i ϕ⁰⁰_i²+ (ϕ_r−ϕ⁰_i)² + +₃

3

1

ϕ⁰⁰_i²+ (ϕ_r−ϕ⁰_i)² (ϕ⁰⁰_i cos(3ϕ⁰_i+φ+φ_m) cosh 3ϕ⁰⁰_i −ϕ⁰_icos(3ϕ_r+φ+φ_m)−

−(ϕ_r−ϕ⁰_i) sin(3ϕ⁰_i+φ+φ_m) sinh 3ϕ⁰⁰_i) (4.41)

As in the previous cases, we construct the surface:

F₁²(ϕ⁰_i, ϕ⁰⁰_i, ϕr, φ, φm) +F₂²(ϕ⁰_i, ϕ⁰⁰_i, ϕr, φ, φm) = 0 (4.42) whereF₁andF₂ are the real and imaginary parts of the left part of the equation 4.35. The values of real ϕ⁰_i and imaginary ϕ⁰⁰_i times are found as the solutions of the equation Eq.(4.42). Canonical momentum is calculated according to Eqs.(4.38, 4.39, 4.40) and Eq.(4.41). The obtained saddle-point solutions are shown in Figures 4.17 and 4.18.

0 2 4 6 0.5

0.6 0.7 0.8 0.9 1

0.02 0.04 0.06 0.08 0.1 0.12

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0.09 0.1 0.11 0.12 0.13

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0.2 0.4 0.6 0.8

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

−0.04

−0.02 0 0.02 0.04

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

−0.06

−0.04

−0.02 0 0.02 0.04

Recombination time, l.c Recombination time, l.c

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

Phase delay , rad. Phase delay , rad.

Phase delay , rad. Phase delay , rad.

Phase delay , rad. Phase delay , rad.

Real parallel canonical momentum, a.u.

Imaginary parallel canonical momentum, a.u.

Real lateral canonical momentum, a.u.

Imaginary lateral canonical momentum, a.u.

Figure 4.17: Complex saddle points solutions for the ionization time and canonical momentum for the combination of the fundamental sec-ond harmonic and the third harmonic fields. Calculation for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm², λ = 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between the second and the third harmonic φ_m = 0 rad.

4.3. SADDLE POINTS FOR THE MULTICOLOR FIELDS

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0.02 0.04 0.06 0.08 0.1 0.12

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0.2 0.4 0.6 0.8

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

Phase delay , rad. Phase delay , rad.

Phase delay , rad. Phase delay , rad.

Phase delay , rad. Phase delay , rad.

Real ionization time, l.c. Imaginary ionization time, l.c.

Real parallel canonical momentum, a.u.

Imaginary parallel canonical momentum, a.u.

Real lateral canonical momentum, a.u.

Imaginary lateral canonical momentum, a.u.

0.09 0.1 0.11 0.12

0 0.01 0.02 0.03 0.04 0.05

−0.02

−0.01 0 0.01 0.02

−5 0 5 x 10₋3

Figure 4.18: Complex saddle points solutions for the ionization time and canonical momentum for the combination of the fundamental, the second harmonic, and the third harmonic fields. Calcula-tion for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm²,λ= 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between second and third harmonicφ_m = 2.1 rad.

The difference between the figures 4.17 and 4.18 is the value of the phase between the second and the third perturbative harmonics: φ_m = 0 rad for the

plots in Fig.4.17 and φ_m = 2,1 rad for the plots in Fig. 4.18. We note the difference between these two cases, especially for the ionization time and the lateral canonical momentum.

Let us now look at the comparison between the saddle point solutions in the linear field and the multicolor field, for both situations. It is shown in Figures 4.19 and 4.20.

0. 20 0. 4 0. 6 0. 8 1

0. 02 0. 04 0. 06 0. 08 0. 1 0. 12

0. 2 0. 4 0. 6 0. 8 1

0. 12 0. 14 0. 16 0. 18 0. 2 0. 22

0. 20 0. 4 0. 6 0. 8 1

0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4

0. 20 0. 4 0. 6 0. 8 1

0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 One-colour eld

Multicolour eld

One-colour eld Multicolour eld

One-colour eld Multicolour eld

One-colour eld Multicolour eld

Real ionization time, l.c. Imaginary ionization time, l.c.

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

Real canonical momentum, a.u. Imaginary canonical momentum, a.u.

Cut off

Cut off

Cut off Cut off

Figure 4.19: Comparison of the complex saddle points solutions for the ion-ization time and the canonical momentum, obtained with (blue dotted line) and without (red line) perturbation field. Calcula-tion for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm²,λ= 1600 nm, parameters of the perturbation: λSH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between the second and the third harmonic φ_m = 0 rad.

4.3. SADDLE POINTS FOR THE MULTICOLOR FIELDS

0. 40 0. 6 0. 8 1

0. 02 0. 04 0. 06 0. 08 0. 1 0. 12

0. 4 0. 6 0. 8 1

0. 08 0. 09 0. 1 0. 11 0. 12 0. 13

0. 40 0. 6 0. 8 1

0. 5 1 1. 5 2 2. 5

0. 40 0. 6 0. 8 1

0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14

One-colour One-colour

Figure 4.20: Comparison of complex saddle points solutions for the ionization time and the canonical momentum, obtained with (blue dotted line) and without (red line) the perturbation field. Calculation for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm², λ = 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between second and third harmonicφ_m = 2,1 rad.

The difference between the saddle points solutions for the two values of the multicolor phaseφ_m is noticeable, when we examine the deviation of the saddle point solutions from the linear field case. These are presented in Figures 4.21 and 4.22.

For the multicolor phase φ_m = 2,1 rad we generally observe small deviations:

less than 1% for both real and imaginary parts of the ionization time and for the real canonical momentum and. The largest deviation in this case is for the

imaginary canonical momentum, of about 7% for the shortest trajectory, and again less than 1% for the plateau trajectories.

We observe higher deviations for the case with the multicolor phase φ_m = 0:

about 7% for the real part of ionization time, less than 9% for the imaginary ionization time, less than 3% for real canonical momentum and, and again the largest deviation of about 20% for the imaginary canonical momentum.

Let us look at the modulation of the amplitude of the harmonic spectrum, introduced by the exponential factor:

|D(t)|² ∼e^{2 ImS(ϕi,ϕr,k||,p⊥,φ,φm)} (4.43) The phase S(ϕ_i, ϕ_r, k||, p⊥, φ, φ_m) is written as:

S(ϕ_i, ϕ_r, k_||, k⊥, φ, φ_m) = 1 2

A²₀ ω

ϕr

Z

ϕi

dt k_||+ A_par(τ) A₀

!2

+

+¹₂A²₀ ω

ϕr

Z

ϕi

dt k⊥+A_per(τ) A₀

!2

+I_p

ω(ϕ_r−ϕ_i) (4.44) We integrate the expression 4.44 along the contour, shown in the figure 3.1.

The normalization of Eq. (4.43) for each recombination time to the maximum of the modulation is presented in Figure 4.23 for the values of the multicolor phase φ_m = 0 rad and φ_m = 2,1 rad.

We can now compare the positions of the optimal phaseφopt for the two differ-ent values of the multicolor phaseφ_m. We find the optimal delay numerically, as described in the section 4.1.1. Then, we will also calculate the modulation amplitude within the classical approach.

4.3. SADDLE POINTS FOR THE MULTICOLOR FIELDS

0 2 4 6

0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

0 2 4 6

0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

0 2 4 6

0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

0 2 4 6

0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Phase delay , rad. Phase delay , rad.

Phase delay , rad.

Phase delay , rad.

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

Real ionization time deviation, percent

Imaginary ionization time deviation, percent

Real canonical momentum deviation, percent

Imaginary canonical momentum deviation, percent 10

20 30 40 50 60 70 80

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8

100 200 300 400 500 600 700 800 0.5

0.6 0.7 0.8 0.9 1

0.5 0.6 0.7 0.8 0.9 1

0.5 0.6 0.7 0.8 0.9 1

0.5 0.6 0.7 0.8 0.9 1

Figure 4.21: Deviations of the saddle point solutions, obtained for the case with the perturbative multicolor field, from the solutions in the fundamental field only, expressed in percentages of the value of the unperturbed solutions. Calculation for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm², λ = 800 nm, parameters of the perpendicularly polarized multicolor field: λ = 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between second and third harmonic φ_m = 0 rad. Black line shows the recombination time of the cut off trajectory in the linear field case.

0 2 4 6 0 2 4 6

0 2 4 6 0 2 4 6

Phase delay , rad. Phase delay , rad.

Phase delay , rad.

Phase delay , rad.

Recombination time, l.c. Recombination time, l.c.

Recombination time, l.c. Recombination time, l.c.

deviation, percent Imaginary ionization time

deviation, percent

Real canonical momentum deviation, percent

Imaginary canonical momentum deviation, percent 0.5

0.6 0.7 0.8 0.9 1

2 4 6 8 10 12 14 16 18 20

0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.5 0.6 0.7 0.8 0.9 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.5 0.6 0.7 0.8 0.9 1

20 40 60 80 100 120 140 160 180 200

Figure 4.22: Deviation of the saddle point solutions, obtained for the case with the perturbative multicolor field, from the solutions in the fundamental field only, expressed in percentages of the value of the unperturbed solutions. Calculation for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm²,λ = 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between second and third harmonic φ_m = 2,1 rad. Black line shows the recombination time of the cut off trajectory in the linear field case.

4.3. SADDLE POINTS FOR THE MULTICOLOR FIELDS

0 2 4 6

0. 5 0. 6 0. 7 0. 8 0. 9 1

0. 65 0. 7 0. 75 0. 8 0. 85 0. 9 0. 95 1

Phase delay , rad.

Recombination time, l.c. Recombination time, l.c.

Phase delay , rad.

0 2 4 6

0. 5 0. 6 0. 7 0. 8 0. 9 1

0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Figure 4.23: The plot shows normalization of Eq. (4.43) to its maximum value for each recombination time. Calculation for the He atom.

Parameters of the driving pulse: I0 = 10¹⁴ W/cm², λ= 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀, delay between second and third harmonic φm = 0 rad on the left plot and φm = 2,1 rad on the right plot.

The classical gate is defined asG_cl(φ, φ_m) = exp^h−¹₂v_⊥²τⁱ, whereτ ≡t⁰⁰_i andv⊥

is the lateral velocity of the electron. It is derived from the classical condition on the zero lateral displacement of an the electron after ionization:

tr

Z

t⁰_i

v⊥dt=

tr

Z

t⁰_i

(k_⊥^cl+A⊥(t))dt = 0 (4.45)

where k_⊥^cl is a classical lateral canonical momentum. It has no imaginary part, because the classical return condition implies no imaginary parts for the ionization and the recombination times, as well as the absence of the tunnelling barrier (the trajectory starts and ends at the origin).

The results of the calculation of the optimal phases φopt for the two cases of the multicolor delays φ_m = 0 rad and φ_m = 2,1 rad are presented in Figure 4.24. Additionally, the plots in Figure 4.24 show such phases φ₀, for which the vector-potentials of the perpendicular fields at the instant of ionization are

zero:

A⊥(φ₀) = 0, A_2ω(φ₀) = 0, A_3ω(φ₀) = 0 (4.46)

We see significant difference between the values of the optimal delays φopt for the two different multicolor phases φ_m. Most importantly, the position of the optimal phaseφ_opt for the multicolor delay φ_m = 0 rad found by numerical cal-culation and as a maximum of classical gate coincide. However, it is clear, that in the case of φ_m = 2,1 rad the simplified classical approach is inapplicable.

The consequences of this fact are described further in the section 5.2.2.

0. 5 1 1. 5 2 2. 5 3 3. 5

0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

−2 −1 0 1 2 3 4 0. 4

0. 5 0. 6 0. 7 0. 8 0. 9 1

Phase , rad

Maximum of quantum gate Maximum of classical gate Delay , when

Delay , when Delay , when

Recombination time, l.c. Recombination time, l.c.

Phase , rad

Figure 4.24: Comparison between the optimal phases φ for the two values of the multicolor delay φ_m = 0 rad (left plot) and φ_m = 2,1 rad (right plot): black curve – numerical calculation of the optimal phase φ, red curve – maximum of the classical gate G_cl, green dotted curve – phasesφ, at which A⊥(φ) = 0, blue dotted curve – phasesφ, at whichA_2ω(φ) = 0, magenta dotted curve – phases φ, at which A_3ω(φ) = 0. Calculation for the He atom. Parameters of the driving pulse: I₀ = 10¹⁴ W/cm²,λ = 1600 nm, parameters of the perturbation: λ_SH = 800 nm, I_SH = 0.02I₀, λ_{T H} = 400 nm, I_{T H} = 0.01I₀.

5 Gating with two-color fields for

Im Dokument Multidimensional high harmonic spectroscopy (Seite 103-115)