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KLaser-induced tunnel ionization:
tunneling time and relativistic effects
Nicolas Teeny∗, Enderalp Yakaboylu, Michael Kaliber,Heiko Bauke, Karen Z. Hatsagortsyan and Christoph H. Keitel
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg
Figure 1: Wave packet tunneling through a coulomb potential bent by a laser potential.
Relativistic tunneling
Non relativistic tunneling picture:
P̂2x
2+ V(x, y, z)− xE0 = −Ip −
(P̂2
y
2+
P̂2z
2
)
Constant total energy E = −Ip.
Relativistic tunneling picture:
P̂2x
2+ V(x, y, z)− xE0 = −Ip −
(P̂2
y
2+
(P̂z + xE0/c)2
2
)
Position dependent total energy:
E(x) = −Ip −(
P̂2y
2+
(P̂z + xE0/c)2
2
)
-5 0 5 10 15 20
-0.55
-0.50
-0.45
-0.40
-0.35
x Κ
Ε�Κ²
Figure 2: Schematic relativistic tunneling picture. The position dependent total energyE(x) is plotted (red-line), crossing the total potential V(x, y, z)− xE0 (blue-line) [1].
non-relativistic relativistic
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.0
0.2
0.4
0.6
0.8
1.0
pkc�Κ²
pro
bab
ilit
y
Figure 3: The momentum probability distribution of the tunneled electron at the bar-rier exit in the relativistic case is compared to the non-relativistic case [1].
Ionization time, exit momentum and asymptoticmomentum
Wigner
quasiclassical with tsub=0,px,0=0
quasiclassical with tsub≠0,px,0≠0
0 2 4 6 8 100
2
4
6
8
10
x/xe
t[x]Ip
Figure 4: Wigner trajectory compared to the classical trajectory in the deep tunnelingregime with the electric field strength E0 = Z3/30 [2, 3].
Wigner
quasiclassical
quasiclassical with tsub=0,px,0≠0
0 2 4 6 8 100.0
0.5
1.0
1.5
2.0
2.5
3.0
x/xet[x
]Ip
Figure 5: Wigner trajectory compared to the classical trajectory in the near-threshold-tunneling regime with the electric field strength E0 = Z3/17 [2, 3].
0.035 0.040 0.045 0.050 0.055E0/Z3 (a.u.)
−4
−2
0
2
4
6
8
10
time×Z
2(a.u.)
τA, γ = 0.25τA, γ = 0.35
τMT, γ = 0.25τMT, γ = 0.35
τ2, γ = 0.25τ2, γ = 0.35
τsub
Figure 6: Various times plotted for different electric field strengths E0 and differentKeldysh parameters γ. The ionization time τA determined by placing a virtual detectorat the tunneling exit. The Mandelstam and Tamm time τTMT measured at the instantof electric field maximum. The ionization time τ2 calculated from the asymptoticmomentum using the two-step model. The time spent under the barrier using Wignerformalism τsub is a good estimate for τA [4].
0.035 0.040 0.045 0.050 0.055E0/Z3 (a.u.)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p 0/Z
(a.u.)
method 1, γ = 0.25method 2, γ = 0.25
method 1, γ = 0.35method 2, γ = 0.35
Figure 7: The exit momentum at the instant of ionization at the tunnel exit for differ-ent electric field strengths E0 and different Keldysh parameters γ determined by twodifferent methods. Method 1 is based on the space resolved momentum distribution,while method 2 utilizes the velocity of the probability flow [4].
References
[1] M. Klaiber, E. Yakaboylu, H. Bauke, K. Z. Hatsagortsyan and C. H. KeitelPhys. Rev. Lett, vol. 110, p. 153004, 2013.
[2] E. Yakaboylu, M. Klaiber, H. Bauke, K. Z. Hatsagortsyan and C. H. KeitelPhys. Rev. A, vol. 90, p. 012116, 2014.
[3] E. Yakaboylu, M. Klaiber, H. Bauke, K. Z. Hatsagortsyanand C. H. Keitel Phys. Rev. A, vol. 88, p. 063421, 2013.
[4] N. Teeny, E. Yakaboylu, H. Bauke and C. H. KeitelarXiv:1502.05917
∗e-mail: [email protected]
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