  The purpose of this paper is to study the interaction between a two-level
system and two continuous-mode photons. Two scenarios are investigated: Case 1,
how a two-level system changes the pulse shapes of two input photons
propagating in a single input channel; and Case 2, how a two-level system
responds to two counter-propagating photons. By means of a transfer function
approach, the steady-state output field states for both cases are derived
analytically. In Example 1, a two-photon input state of Gaussian pulse shape is
used to excite a two-level atom. The simulation demonstrates that in the time
domain the atom tends to stretch out the two photons. Moreover, the striking
difference between the joint probability distribution of the output two-photon
state and that of the input two-photon state occurs exactly under the setting
when the two-level atom is most efficiently excited. In Example 2, a two-photon
input state of rising exponential pulse shape is used to excite a two-level
atom. Strong anti-correlation of the output two-photon state is observed, which
is absent in Example 1 for the Gaussian pulse shape. Such difference indicates
that different pulse shapes give rise to drastically different frequency
entanglement of the output two-photon state. Example 3 is used to illustrate
Case 2, where two counter-propagating single photons of rising exponential
pulse shapes are input to a two-level atom. The frequency-dependent
Hong-Ou-Mandel (HOM) interference phenomenon is observed. The simulation
results base on the analytic forms of output two-photon states are consistent
with those based on quantum master or filter equations [43,11]. Similar
physical phenomena have been observed in physical settings such as cavity
opto-mechanical systems and Keer nonlinear cavities.
