Parametric Excitation of Edge Waves in the Quantum Hall Effect.

We consider the process of parametric excitation of gapless edge modes in a Hall bar by an alternating current. We find that such a process can be realized provided both an inter-edge interaction and a constant current are present in the bar. The expected experimental signatures of this phenomenon are discussed. PACS numbers: 73.40.Hm, 72.15.Nj Typeset using REVTEX 1 Parametric excitation of waves is a phenomenon shared by many nonlinear systems. Generation of spin waves (magnons) in magnetically ordered dielectrics by microwave fields and the excitation of standing surface waves in liquids by vertical vibrations are just two examples out of a wealth of such instances (see, e.g. [1]). A prerequisite for the existence of parametric processes in a system is a dependence of the dispersion law of its excitations on some externally controllable parameter. If such a condition holds a suitable temporal and spatial variation of this parameter may result in the generation of excitations. The instrumental role of edges in forming the gapless modes of both the integer and fractional quantum Hall effects (IQHE,FQHE) was noted theoretically a long time ago [2,3]. Subsequently these, so called edge magnetoplasmons, were observed in a series of experiments [4–9]. In classical [10] and quantum [11,12] treatments of the problem it was shown that inter-edge interaction in a Hall bar causes mixing of the single edge density waves. The resulting excitations remain gapless and have components on both edges propagating in the same direction. In a recent paper we considered these gapless modes and noted that their dispersion relation depends on the total current that flows in the bar [13]. This observation led us to suggest the possibility of parametrically exciting these edge waves by the application of an alternating current through the bar. It is the purpose of this letter to show how this may be realized and to discuss the experimental signatures that such a phenomenon is expected to exhibit. Although our previous work [13] was carried out in the framework of the Chern-Simons theory we find it more convenient to explore the present subject using the Luttinger model description of the QHE. We will consider a Hall bar of width W and length L(≫ W ) under the conditions of the IQHE or the FQHE. Furthermore, we restrict our discussion to the case where only the first Landau level is filled or to FQHE plateaus with filling factors ν = 1/n where n is an odd integer. Under these conditions each edge is described by one branch of single particle states which is obtained by linearizing the energy of the Landau level for non interacting electrons around the Fermi energy at the edge [14]. Each of the linear edge branches (Left and Right) ends at a lower cutoff in energy corresponding to the points where 2 a degenerate level of bulk states commences. The Hamiltonian of the Luttinger model which describes the edge excitations in this case is given by [14,15]