We discuss a cavity with the nonlinear oscillator (corresponding to a Kerr medium) periodically kicked by a series of ultra-short laser pulses. This system is governed by the following Hamiltonian1 (in the interaction picture):
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% gacaWGubGaaiykaiaacYcaaaa!6585!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${H_{\operatorname{int} }} = \frac{{\bar hx}}{2}{({\hat a^\dag })^2}{\hat a^2} + \bar h( \in {\hat a^\dag } + \in *\hat a)\sum\limits_{n = 0}^\infty \delta (t - nT),$$
(1)
where χ denotes nonlinearity of the Kerr medium and c is a strength of the field-medium coupling. In addition, we assume that the field is initially in the vacuum state ∣0〉. Assuming that the field coupling is weak, i.e. ∈ ≪ χ, we apply the standard perturbation method of rediagonalization of the Hamiltonian’. In consequence, we obtain the analytical formulas for the probabilities corresponding to the vacuum ∣0〉, one-photon ∣1〉 and two-photon ∣2〉 Fock states. Thus, for the time t just after k-th laser pulse, we have
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% aaaaaa!8B5E!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{array}{*{20}{c}}
{|{a_o}(k){|^2} = {{(\cos (k \in ))}^2} + O({ \in ^2}),} \\
{|{a_1}(k){|^2} = {{(\cos (k \in ))}^2} + O({ \in ^2})} \\
{|{a_2}(k){|^2} = 2{ \in ^2}|B{|^2}{{(\sin (k \in ))}^2} + O({ \in ^4}),}
\end{array}$$
(2)
where
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