M AX -P LANCK -I NSTITUT FÜR K ERNPHYSIK Laser-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: ˆ P 2 x 2 + V ( x , y, z ) - xE 0 = - I p - ˆ P 2 y 2 + ˆ P 2 z 2 ! Constant total energy E = - I p . Relativistic tunneling picture: ˆ P 2 x 2 + V ( x , y, z ) - xE 0 = - I p - ˆ P 2 y 2 + ( ˆ P z + xE 0 / c ) 2 2 ! Position dependent total energy: E ( x )= - I p - ˆ P 2 y 2 + ( ˆ P z + xE 0 / c ) 2 2 ! - 5 0 5 10 15 20 - 0.55 - 0.50 - 0.45 - 0.40 - 0.35 x Κ ΕΚ² Figure 2: Schematic relativistic tunneling picture. The position dependent total energy E ( x ) is plotted (red-line), crossing the total potential V ( x , y, z ) - xE 0 (blue-line) [1]. non-relativistic relativistic - 0.6 - 0.4 - 0.2 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0 p k cΚ² probability 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 asymptotic momentum Wigner quasiclassical with t sub =0,p x,0 =0 quasiclassical with t sub ≠0,p x,0 ≠0 0 2 4 6 8 10 0 2 4 6 8 10 x/xe t[ x ] Ip Figure 4: Wigner trajectory compared to the classical trajectory in the deep tunneling regime with the electric field strength E 0 = Z 3 /30 [2, 3]. Wigner quasiclassical quasiclassical with t sub =0,p x,0 ≠0 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x/xe t[ x ] Ip Figure 5: Wigner trajectory compared to the classical trajectory in the near-threshold- tunneling regime with the electric field strength E 0 = Z 3 /17 [2, 3]. 0.035 0.040 0.045 0.050 0.055 E 0 /Z 3 (a.u.) -4 -2 0 2 4 6 8 10 time × Z 2 (a.u.) τ A , γ = 0.25 τ A , γ = 0.35 τ MT , γ = 0.25 τ MT , γ = 0.35 τ 2 , γ = 0.25 τ 2 , γ = 0.35 τ sub Figure 6: Various times plotted for different electric field strengths E 0 and different Keldysh parameters γ. The ionization time τ A determined by placing a virtual detector at the tunneling exit. The Mandelstam and Tamm time τ TMT measured at the instant of electric field maximum. The ionization time τ 2 calculated from the asymptotic momentum using the two-step model. The time spent under the barrier using Wigner formalism τ sub is a good estimate for τ A [4]. 0.035 0.040 0.045 0.050 0.055 E 0 /Z 3 (a.u.) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p 0 /Z (a.u.) method 1, γ = 0.25 method 2, γ = 0.25 method 1, γ = 0.35 method 2, γ = 0.35 Figure 7: The exit momentum at the instant of ionization at the tunnel exit for differ- ent electric field strengths E 0 and different Keldysh parameters γ determined by two different 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. Keitel Phys. Rev. Lett, vol. 110, p. 153004, 2013. [2] E. Yakaboylu, M. Klaiber, H. Bauke, K. Z. Hatsagortsyan and C. H. Keitel Phys. Rev. A, vol. 90, p. 012116, 2014. [3] E. Yakaboylu, M. Klaiber, H. Bauke, K. Z. Hatsagortsyan and C. H. Keitel Phys. Rev. A, vol. 88, p. 063421, 2013. [4] N. Teeny, E. Yakaboylu, H. Bauke and C. H. Keitel arXiv:1502.05917 * e-mail: [email protected]