Wigner equation of motion for time-dependent potentials LORENZO GALLEANI* and LEON COHEN City University of New York, 695 Park Ave, New York, NY 10021 USA (Received 1 April 2001; revision received 5 June 2001) Abstract. The equation of motion is obtained for the Wigner distribution for time-dependent potentials. It is shown that one cannot obtain an equation of motion for the standard Wigner distribution but one can do so for the four- dimensional distribution of the variables position, momentum, time, and frequency. Three forms are given and it is shown that for time-independent potentials the new form reduces to the equations originally obtained by Wigner and Moyal. 1. Introduction The Wigner distribution has found many applications in a variety of ®elds and in particular quantum optics [1, 2]. It is a prototype of other distributions and the class of generalized Wigner distribution functions [3]. Also, the Wigner distri- bution of time and frequency and its generalization have been routinely used in engineering in the ®elds of optical, signal, image, and speech processing, among other areas [4±8]. This ®eld of application, which is over 60 years old, is called `time-frequency analysis’. In Wigner’s original paper he obtained the equation of motion for the Wigner distribution and subsequently Moyal obtained a dierent form which is very close to the Liouville form in classical mechanics [9, 10]. The equation of evolution for the Wigner distribution has been used in many ways to study both quantum and classical systems [11±13]. In the derivations of Wigner and Moyal only the case of time-independent potentials was considered. It is the aim of this paper to obtain the equation of motion of the Wigner distribution for time-dependent potentials. As we will see, one cannot obtain an equation for the ordinary Wigner distribution but one must use a four dimensional Wigner distribution. The four- dimensional Wigner distribution was introduced in [14] and will be discussed in section 3. Journal of Modern Optics ISSN 0950±0340 print/ISSN 1362±3044 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500340110088515 journal of modern optics, 2002, vol. 49, no. 3/4, 561±569 * Permanent address: Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy.