Effect of a radiation field on electrons bound on liquid helium

L. C. M. Miranda, R. M. O. Galvão, C. A. S. Lima
1983 Physical Review B (Condensed Matter)  
Using the image-potential model, we discuss the influence of a radiation field on the distortion of the bound states of an electron trapped on a liquid-helium surface. It is shown that, for moderate field strengths, the ionization energy decreases with increasing field strength. After the first experimental work' demonstrating the existence of a potential barrier to extract electrons from liquid helium, the interaction between electrons and a liquidhelium surface has been the subject of intense
more » ... investigation in the last decade. ' ' The following description of the electron trapping is now generally accepted; an electron outside the liquid is attracted to it by the image force arising from the polarization of the surface, and is repelled by surface barrier as a result of the exclusion principle. The electron motion perpendicular to the surface of the liquid is thus quantized whereas along the liquid surface the transport is free. The original works2 4 on the bound states of an electron trapped according to the above picture used a onedimensional hydrogenlike model for the binding potential. Even though the agreement between this model and the spectroscopic data on the transition energies between the ground state and the first two excited states was reasonably good, subsequent improvements of the energy separation calculations had taken into account the repulsive barrier of the surface. More recently, Hipo1ito, Felicio, and Farias, based upon the results on the image potential of a metalvacuum interface, ' have produced excellent agreement between theory and experiment with use of a modified image potential model. 5 According to Refs. 4 and 5, and assuming that the liquid helium occupies the space z (0, the potential energy of the electron is given by where p = (t/i) r)/riz, A (t) =A singlet is the vector potential for the microwave beam in the dipole approximation, and V(z) is given by Eq. (1). To solve Eq. (2), we perform a unitary transformation, " ' namely, isf t) p//I~iv)( t)/t~ ( 3) plane electromagnetic wave of frequency co, linearly polarized along the normal to the liquid-helium surface (i.e., along the z direction). The radiation is also assumed to be nonresonant with the zero-field system; i.e. , we assume that the radiation frequency is smaller than the field-free binding energy. As the energies of our problem are of the order of Z2e2/2ao (or 6.6&&10 4 eV), we shall then assume a long wavelength radiation beam such as to a 10to 50-GHz rnicrowave. In the absence of the microwave field the electron trapping is assumed to be described by the image potential model of Eq. (1). Furthermore, we assume that the microwave field acts only on the electrons, having therefore no effect on the liquid-helium properties; i.e. , we assume that both e and Vo are not changed by the presence of the microwave, Accordingly, in the presence of the radiation, the electron motion is described by the solution to the onedimensional Schrodinger equation (Z & 0) V(z) = Vo, z(0 Zp 2 z &0, where e "', eA g(t) = -J Ck'A(r') = cos(ar) mc mc 0) where Vo is the repulsive barrier potential, Z is the strength of the image potential, namely, Z =(e -1)/4(a+1), where e is the dielectric constant, and P is the position of the center of mass of the induced charge. For liquid helium, e = 1.057 23, Vo= 1 eV, and the value of p found in Ref. 5 was 1.01 A. This value of p is in very good agreement with the estimates of Grimes, Brown, Burns, and Ziepfel4 (i.e. , p =1.04 A) using their experimental data. In this paper, we extend the works of Refs. 4 and 5 by reporting on the effects of a radiation field on the bound states of electrons bound to the surface of liquid helium. The same problem has recently been studied by Jensen' who show that this one-dimensional system is probably one of the best systems for the investigation of quantum stochasticity. The radiation beam is described by a classical 2 71(r) = -, J dr'A'(r') 2mc We find / A2 + V(z -S(r)) y . 2m (4b) Equation (5) shows that in the presence of a laser field, the electron motion may be alternatively described by the Schrodinger equation for an electron moving in a potential whose center of force appears to be oscillating with the frequency cu and amplitude eA/mcco, namely, the dressed potential V(z -5(t)). Since we are assuming that the radia-28 5313
doi:10.1103/physrevb.28.5313 fatcat:gxasnxlxmjfsnlhmazsaripe4y