54 Laser Physics, Vol. 7, No. 1, 1997, pp. 54–57. Original Text Copyright © 1997 by Astro, Ltd. Copyright © 1997 by åÄàä ç‡Û͇ /Interperiodica Publishing (Russia). 1. INTRODUCTION During the past few years, much attention has been paid to generation of various nonclassical states of elec- tromagnetic field. Nonetheless, the experimental real- ization of those states is not a trivial problem. Among many papers concerning methods of generation of var- ious quantum states one can mention the Hong and Mandel paper [1], where they have shown that a one- photon Fock state is produced in the parametric down converter. Stoler and Yurke have studied theoretically the possibility of generation of antibunched light [2]. Another method of producing Fock states has been pro- posed by Brune et al. [3, 4], who have proposed the quantum nondemolition experiment in which detection of the atomic phase by the Ramsey method plays the role of a QND probe giving information on the cavity field energy. After a sequence of atomic measurements, the cavity field collapses into a Fock state with an unpre- dictable number of photons. A micromaser system in which two-level atoms are injected into a cavity gives also a possibility to generate highly excited Fock states. This model has been discussed by Filipowicz et al. [5]. A method of generation of various quantum states based on the interaction with a cavity electromagnetic field of N two-level atoms injected to a single-mode resonator has been presented by Vogel et al. [6]. Quite recently, Kozierowski and Chumakov [7] have shown that in the spontaneous emission of the partially inverted Dicke model Fock states can also be generated. The system discussed in this paper contains a cavity filled with a passive nonlinear “Kerr” medium, which is characterized by the (2q –1)th susceptibility χ (2q – 1) . The cavity is periodically kicked by a series of ultrashort laser pulses. Moreover, we assume that the field inside the cavity is initially in the vacuum state |0〉. We will show that for a sufficiently weak external excitation, res- onance effects start to play a significant role and lead, in effect, to Fock states. The effectiveness of the Fock state preparation is, of course, considerably diminished by the cavity losses. Nevertheless, it seems important to us that a cavity with a nonlinear Kerr medium and a field ini- tially in the vacuum state, kicked periodically by a train of classical pulses, can be, to high accuracy, a source of Fock states. In our previous paper [8] the problem of generation of the one-photon state has been discussed for such a model with the use of numerical approach and approximate representation for the unitary evolution operator. Here, we will derive approximate analytical formulas for the evolution of the probabilities corre- sponding to the particular Fock states in which we are interested. We will apply the Floquet states method [9– 11] and the perturbation theory to get analytical results. For comparison, we will perform calculations in which we solve the dynamics of our system numerically. 2. THE MODEL AND FLOQUET STATES The system we in which are interested is governed by the following Hamiltonian: (1) where (2) describes the cavity field interaction mediated by the nonlinear medium, whereas (3) is a time-dependent Hamiltonian corresponding to the driving of the cavity by the external classical field, which is a series of ultrashort pulses modeled by the Dirac delta functions. Moreover, we assume that this interaction is weak, i.e.,   χ (2q – 1) . Although the Hamiltonian (2) describes any order of the nonlinearity, in this paper we will restrict our considerations to the case q = 2 only. Since the Hamiltonian (1) is a periodic function of time t with the period T , it generates the uni- tary evolution operator that transforms the initial state of the system |Φ(0)〉 (the state for the time t = 0) H ˆ t () H ˆ NL H ˆ 1 t () , + = H ˆ NL  χ 2 q 1 – ( ) q ------------------- a ˆ † ( ) q a ˆ q = H ˆ 1 t ()  a ˆ † a ˆ + ( ) δ t nT – ( ) n 0 = ∞ ∑ = U ˆ HISTORY AND MODERN TRENDS IN QUANTUM ELECTRONICS Kicked Nonlinear Kerr Medium and Fock States Generation W. ´ Leonski*, S. Dyrting**, and R. ´ Tanas* * Institute of Physics, A. Mickiewicz University, Umultowska 85, Poznan, ´ 61-614 Poland e-mail: wleonski@phys.amu.edu.pl; tanas@phys.amu.edu.pl ** Department of Physics, University of Queensland, St. Lucia, 4072 Australia e-mail: dyrting@physics.uq.oz.au Received July 23, 1996 Abstract—We discuss a cavity filled with the “Kerr” medium with the (2q – 1)th nonlinearity χ (2q – 1) , period- ically kicked by a series of ultrashort laser pulses. Applying the Floquet state and perturbation methods, we find the analytic formulas for the probabilities of the n-photon states. We show that our system can produce pure Fock states. Moreover, we perform numerical calculations to validate our analytical results.