Ionization of Atoms in a Strong Laser Radiation Field and the Imaginary Time Method


  • V. M. Rylyuk Odessa State Academy of Civil Engineering and Architecture
  • V. A. Nastasyuk K.D. Ushynskyi South-Ukrainian National Pedagogical University



relativistic tunnel and multiphonon ionization, imaginary time method, Keldysh parameter


The phenomenon of nonlinear relativistic ionization induced by a strong electromagnetic wave has been considered. The relativistic version of the imaginary time method is used to calculate the probability for an electron with an energy of the order of its rest energy to tunnel through a potential barrier under the action of a strong electromagnetic wave. Besides the exponential factor, the Coulomb and pre-exponential ones are also obtained with regard for the electron spin and the ionization probability. Simple analytical formulas for the momentum distributions of relativistic photo-electrons are derived. The relativistic effects are shown to result in a nonzero drift velocity of an electron, when it quits the barrier. In the nonrelativistic limit, the well-known Keldysh exponent and the Landau–Lifshitz formula for the ionization probability of a hydrogen atom in the ground state are obtained.


T.-M. Yan and D. Bauer, Phys. Rev. A 86, 053403 (2012).

K. Krajewska, I.I. Fabrikant, and A.F. Starace, Phys. Rev. A 86, 053410 (2012).

V.M. Rylyuk, Phys. Rev. A 86, 013402 (2012).

H.R. Reiss, Phys. Rev. Lett. 102, 143003 (2009).

S.V. Popruzhenko, V.D. Mur, V.S. Popov, and D. Bauer, Zh. Eksp. Teor. Fiz. 135, 1092 (2009).

M.V. Frolov, N.L. Manakov, and A.F. Starace, Phys. Rev. A 78, 063418 (2008).

L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1945 (1964).

S.-W. Bahk, P. Rousseua, T.A. Planchon et al., Opt. Lett. 29, 2837 (2004), Appl. Phys. B 80, 823 (2005); 81, 727 (2005).

S.V. Bulanov, F. Califano, G.I. Dudnikova et al., Rev. Plasma Phys. 22, 227 (2001).

T. Tajima and G. Mourou, Phys. Rev. Spec. Top. – Ac. 5, 031301 (2002).

A.I.Nikishov and V.I.Ritus, Zh. Eksp. Teor. Fiz. 50, 255 (1966).

A.M. Perelomov, V.S. Popov, and M.V. Terentyev, Zh. Eksp. Teor. Fiz. 50, 1393 (1966); 51, 309 (1966).

A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor. Fiz. 50, 255 (1966).

D.P. Crawford and H.R. Reiss, Phys. Rev. A 50, 1844 (1994).

N.B. Delone and V.P. Krainov, Usp. Fiz. Nauk 168, 531 (1998).

V.P. Krainov, Opt. Express 2, 268 (1998).

D.P. Crawford and H.R. Reiss, Opt. Express 2, 289 (1998).

V.P. Krainov, J. Phys. B 32, 1607 (1999).

V.S. Popov, V.D. Mur, and B.M. Karnakov, Pis'ma Zh. Eksp. Teor. Fiz. 66, 213 (1997).

V.D. Mur, B.M. Karnakov, and V.S. Popov, Zh. Eksp. Teor. Fiz. 114, 798 (1998).

V.S. Popov, V.P. Kuznetsov, and A.M. Perelomov, Zh. Eksp. Teor. Fiz. 53, 331 (1967).

L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1983).

J. Schwinger, Phys. Rev. 82, 664 (1951).

G. Breit, Nature 122, 649 (1928).

V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Relativistic Quantum Theory (Pergamon Press, Oxford, 1982).

L.D. Landau and E.M. Lifshitz, Quantum Mechanics. nonRelativistic Theory (Pergamon Press, New York, 1977).



How to Cite

Rylyuk, V. M., & Nastasyuk, V. A. (2018). Ionization of Atoms in a Strong Laser Radiation Field and the Imaginary Time Method. Ukrainian Journal of Physics, 59(2), 116.



Optics, lasers, and quantum electronics