Strong-field approximation for intense-laser–atom processes: The choice of gauge

Abstract

The strong-field approximation can be and has been applied in both length gauge and velocity gauge with quantitatively conflicting answers. For ionization of negative ions with a ground state of odd parity, the predictions of the two gauges differ qualitatively: in the envelope of the angularresolved energy spectrum, dips in one gauge correspond to humps in the other. We show that the length-gauge SFA matches the exact numerical solution of the time-dependent Schrödinger equation.PACS numbers: 42.50. Hz, 32.80.Rm, 32.80.Gc Quantum mechanics is gauge invariant: it is easily proven that a given physical quantity can be evaluated in any gauge with the same result [1]. In nonrelativistic quantum mechanics, when the dipole approximation is adopted, the interaction of an atom with a timedependent field such as a laser field is usually described in either one of two gauges: the length gauge (L gauge) or the velocity gauge (V gauge). In numerical solutions of the time-dependent Schrödinger equation (TDSE), gauge invariance has been confirmed many times. In analytical work, however, some approximations almost always have to be adopted. There is no formal reason of why after such approximations the resulting theory should still be gauge invariant. Indeed, the lack of gauge invariance after what seems like very well justified approximations has driven many a researcher to despair [2].In this paper, we will address one of the most glaring manifestations of this "gauge problem": the lack of gauge invariance of the strong-field approximation (SFA) in intense-laser-atom physics [3]. The SFA underlies almost any analytical approach to total ionization rates, above-threshold ionization, high-order harmonic generation, and nonsequential double ionization, both of atoms and of molecules. Briefly, it assumes that the initial bound state of the atom or molecule is unaffected by the laser field while the final state, which is in the continuum, does not feel the presence of the binding potential. The lack of gauge invariance of the SFA has been noted many times; see, e.g., Ref. [4]. Comparisons that have been carried out indeed have exhibited significant disagreements between the results obtained from L gauge and V gauge [5]. Different authors have preferred different gauges. The question of which gauge is superior for which problem has often been raised, but never led to any consensus about its answer. Below, we will give an answer for the case of a short-range binding potential, where the SFA is expected to be most accurate [6], by comparing the SFA in L gauge and V gauge with the numerical solution of the TDSE.For a fixed nucleus and in the single-active-electron approximation, where the effects of all electrons but one are absorbed into an effective binding potential, the complete Hamiltonian in the presence of an external electromagnetic field can be decomposed aswhere the subscript x specifies the gauge (x = L, V) andThis operator contains the binding potential V (r) and is independent of the gauge. With the dipole approximat...

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