Tunneling time in attosecond experiments and the time-energy uncertainty relation

In this work we present a theoretical model of the tunneling time and the tunneling process (in attosecond experiment for the He atom). Our model is supported with physical reasoning leading to a relation which performs an excellent estimation for the tunneling time in attosecond and strong-field experiments, where we address the important case of the He atom. Our tunneling time estimation is found by utilizing the time-energy uncertainty relation and represents a quantum clock. The tunneling time is also featured as the time of passage through the barrier similar to Einstein's photon-box Gedanken experiment. Our work tackles an important case study for the theory of time in quantum mechanics and is very promising for the search for a (general) time operator in quantum mechanics. The work can be seen as a fundamental step in dealing with the tunneling time in strong-field and ultrafast science and is appealing for more elaborate treatments using quantum wave-packet dynamics and especially for complex atoms and molecules.

Figure 1

Graphic display of the potential curves, the barrier width and the two inner and outer points $x_{e,\pm }=(I_p\pm {\delta }_z)/2F$, the “classical exit” point $x_{e,c}=I_p/F$, and $x_m\left(F\right)=\sqrt{Z_{\mathrm{eff}/F}}$, the position at the maximum of the barrier height [note $x_a=x_m(F=F_a)$]; see text.

Figure 2

T-time ${\tau }_{T,\mathrm{unsy}}$ [Eq. (10)] and ${\tau }_{T,\mathrm{sym}}$ [Eq. (12)] for two $Z_{\mathrm{eff}}$ models [41] and [42]. Time is in attosecond units vs laser-field strength in atomic units, corresponding to the tunneling ionization of the He atom in strong field (compare Fig. 3). Experimental values were kindly sent by Landsman [37].

Figure 3

T-time ${\tau }_{T,d}$ [Eq. (14)] and ${\tau }_{T,\mathrm{sym}}$ [Eq. (18)] for two different $Z_{\mathrm{eff}}$ models as in Fig. 2. Note that Eq. (18) is identical to Eq. (12) [see Fig. 2], with units as in Fig. 2. T-time corresponds to the tunneling ionization of the He atom in strong field, in excellent agreement with the experimental result [32, 33, 37]. Experimental values as in Fig. 2.

Figure 4

T-time ${\tau }_{T,d}$ Eq. (14) for two different $Z_{\mathrm{eff}}$ models as in Fig. 2 together with the FPI and Larmor clock results [37] and the experimental result [32, 33]. Experimental values, the FPI, and Larmor result were kindly sent by Landsman and Hofmann [37].

Figure 5

T-time ${\tau }_{T,d}$ [Eq. (14)] vs barrier width $d_B\left(F\right)$ [Eq. (13)] for two different $Z_{\mathrm{eff}}$ models as in Fig. 2. The lines at the bottom of the figure show the time spent by a particle (a photon) traversing the same barrier at the speed of light.