Dynamical symmetry breaking of lambda and vee-type three-level systems on
quantization of the field modes
Mihir Ranjan<EMAIL_ADDRESS>and Surajit
<EMAIL_ADDRESS>
Department of Physics
Guru Charan College
Silchar 788004, India
Asoke Kumar<EMAIL_ADDRESS>
Department of Physics
Assam University
Silchar 788011, India
Gautam<EMAIL_ADDRESS>
S N Bose National Centre for Basic Sciences
JD Block, Sector III
Salt Lake City, Kolkata 700098, India

###### Abstract

We develop a scheme to construct the Hamiltonians of the lambda, vee and
cascade type of three-level configurations using the generators of $SU(3)$
group. It turns out that this approach provides a well defined selection rule
to give different Hamitonians for each configurations. The lambda and vee type
configurations are exactly solved with different initial conditions while
taking the two-mode classical and quantized fields . For the classical field,
it is shown that the Rabi oscillation of the lambda model is similar to that
of the vee model and the dynamics of the vee model can be recovered from
lambda model and vice versa simply by inversion. We then proceed to solve the
quantized version of both models introducing a novel Euler matrix formalism.
It is shown that this dynamical symmetry exhibited in the Rabi oscillation of
two configurations for the semiclassical models is completely destroyed on
quantization of the field modes. The symmetry can be restored within the
quantized models when the field modes are both in the coherent states with
large average photon number which is depicted through the collapse and revival
of the Rabi oscillations.

I.Introduction

Quantum Optics gives birth to many novel proposals which are within reach of
present-day ingenious experiments performed with intense narrow-band tunable
laser and high-Q superconducting cavity [1]. Major thrust in the atomic,
molecular and optical experiments primarily involves the coherent manipulation
of the quantum states which may be useful to verify several interesting
results of quantum information theory and also the experimental realization of
the quantum computer [2,3]. The actual number of the quantum mechanical states
of atoms involved in the interaction with light is of much importance in these
days since many coherent effects are due to the level structure of the atom.
It is well-known that the two-level system and its quantized version, namely,
the Jaynes-Cummings model (JCM), have been proved to be an useful theoretical
laboratory to understand many subtle issues of the cavity electrodynamics
[4,5]. The two-level system is modeled using the Pauli’s spin matrices - the
spin-half representation of $SU(2)$ group, where apart from the level number,
the spectrum is designated by the photon number as the quantum number. A
natural but non-trivial extension of the JCM is the three-level system and it
exhibits plethora of optical phenomena such as, two-photon coherence [6],
resonance Raman scattering [7], double resonance process [8],population
trapping [9], three-level super radiance [10], three-level echoes [11], STIRAP
[12], quantum jump [13], quantum zeno effect [14], Electromagnetically Induced
Transparency [15,16] etc. There are three distinct schemes of the three-level
configurations which are classified as the lambda, vee and cascade systems
respectively. The Hamiltonians of these configurations are generally modeled
by two two-level systems coupled by the two modes of cavity fields of
different frequencies [17,18]. Although these Hamiltonians succeed in
revealing several phenomena [19,20], however, their $ad$ $hoc$ construction
subsides the underlying symmetry and its role in the population dynamics of
these systems. The connection between the $SU(N)$ symmetry and the $N$-level
system in general, was investigated extensively in recent past [21-27]. These
studies not only mimic the possible connection between quantum optics with the
octet symmetry, well-known paradigm of particle physics, but for $N=3$, it
also reveals several interesting results such as the realization of the eight
dimensional Bloch equation, existence of non-linear constants [18,22],
population transfer via continuum [28], dynamical aspects of three-level
system in absence of dissipation [29] etc. However, inspite of these progress,
a general formalism as well as the $ab$ $initio$ solutions of all three
configurations are yet to be developed for the reasons mentioned below.

The generic model Hamiltonian of a three-level configuration with three well-
defined energy levels can be represented by the hermitian matrix

$H=\left[{\begin{array}[]{*{20}c}\Delta_{3}&h_{32}&h_{31}\\\
h_{32}&\Delta_{2}&h_{21}\\\ h_{31}&h_{21}&\Delta_{1}\\\
\end{array}}\right],\hfill(1)$

where $h_{ij}(i,j=1,2,3)$ be the matrix element of specific transition and
$\Delta_{i}$ be the detuning which vanishes at resonance. We note that from
Eq.(1), the lambda system, which corresponds to the transition
$1\leftrightarrow 3\leftrightarrow 2$ shown in Fig.1a, can be described by the
Hamiltonian with elements $h_{21}=0$, $h_{32}\neq 0$ and $h_{31}\neq 0$.
Similarly the vee model, characterized by the transition $3\leftrightarrow
1\leftrightarrow 2$ shown in Fig.1b, corresponds to the elements $h_{21}\neq
0$, $h_{32}=0$ and $h_{31}\neq 0$ and for the cascade model we have transition
$1\leftrightarrow 2\leftrightarrow 3$, we have $h_{21}\neq 0$, $h_{32}\neq 0$
and $h_{31}=0$ respectively. Thus we have distinct Hamiltonian for three
different configurations which can be read off from Eq.(1) shown in Fig.1.
This definition, however, differs from the proposal advocated by Hioe and
Eberly, who argued the order of the energy levels to be $E_{1}<E_{3}<E_{2}$
for the lambda system, $E_{2}<E_{3}<E_{1}$ for the vee system and
$E_{1}<E_{2}<E_{3}$ for the cascade system respectively [18,21,22]. In their
scheme, the level-2 is always be the intermediary level which becomes the
upper, lower and middle level to generate the lambda, vee and cascade
configurations respectively. It is worth noting that, if we follow their
scheme, these energy conditions map all three three-level configurations to a
unique cascade Hamiltonian described by the matrix with elements $h_{12}\neq
0$, $h_{23}\neq 0$ and $h_{13}=0$ in Eq.(1). Thus because of the similar
structure of the model Hamiltonian, if we start formulating the solutions of
the lambda, vee and cascade configurations, then it would led to same spectral
feature. Furthermore, due to the same reason, the eight dimensional Bloch
equation always remains same for all three models [18,22]. Both of these
consequences go against the usual notion because wide range of coherent
phenomena mentioned above arises essentially due to different class of the
three-level configurations. Thus it is worth pursing to formulate a
comprehensive approach, where we have distinct Hamiltonian for three
configurations without altering the second level for each model.

The problem of preparing multilevel atoms using one or more laser pulses is of
considerable importance from experimental point of view. Thus the completeness
of the study of the three-level systems requires the exact solution of these
models to find the probability amplitudes of all levels, the effect of the
field quantization on the population oscillation and, most importantly, the
observation of the collapse and revival effect. In recent past, the three-
level systems and its several ramifications were extensively covered in a
general framework of the $SU(N)$ group having $N$-levels [21-27,30,31]. Also,
the semiclassical model [24,32,33] and its fully quantized version [23,34,35]
are studied, but to our knowledge, the pursuit of the exact solutions of
different three-level systems in the spirit of the theory of Electron Spin
Resonance (ESR) model and JCM, are still to be facilitated analytically.

In a recent paper, we have studied the exact solutions of the equidistant
cascade system interacting with the single mode classical and quantized field
with different initial conditions [36]. It is shown that for the semiclassical
model the Rabi oscillation exhibits a symmetric pattern of evolution, which is
destroyed on quantization of the cavity field. We also show that this symmetry
is restored by taking the cavity mode to be the coherent state indicating the
proximity of the coherent state to the classical field. We have further
studied the equidistant cascade four-level system and obtain similar
conclusions [37]. To extend above studies for the lambda and vee models we
note that the vee configuration can be obtained from the lambda configuration
simply by inversion. However, it is worth noting that, the lambda
configuration is associated with processes such as STIRAP [12], EIT [15,16]
etc, while the vee configuration corresponds to the phenomena such as quantum
jump [13], quantum zeno effect [14], quantum beat [3] etc indicating that both
the processes are fundamentally different. It is therefore natural to examine
the inversion symmetry between the models by comparing their Rabi oscillations
and study the effect of the field quantization on that symmetry. The
comparison shows that the inversion symmetry exhibited by the semiclassical
models is completely spoiled on quantization of the cavity modes indicating
the non-trivial role of the vacuum fluctuation in the symmetry breaking.

The remaining part of the paper is organized as follows. In Section-II, we
discuss the basic tenets of the $SU(3)$ group necessary to develop the
Hamiltonian of all possible three-level configurations. Section-III deals with
the solution of the lambda model taking the two field modes as the classical
fields and then in Section IV we proceed to solve the corresponding quantized
version of the model using a novel Euler matrix formalism. Section-V and VI we
present similar calculation for the vee model taking the mode fields to be
first classical and then quantized respectively. In Section-VII we compare the
population dynamics in both models and discuss its implications. Finally in
Section-VIII we conclude our results.

II.The Models

The most general Hamiltonian of a typical three-level configuration is given
by Eq.(1) which contains several non-zero matrix elements showing all possible
allowed transitions. To show how the $SU(3)$ symmetry group provides a
definite scheme of selection rule which forbids any one of the three
transitions to give the Hamiltonian of a specific model, let us briefly recall
the tenets of $SU(3)$ group described by the Gell-Mann matrices, namely,

$\lambda_{1}=\left[{\begin{array}[]{*{20}c}0&1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{array}}\right],\hskip
25.29494pt\lambda_{2}=\left[{\begin{array}[]{*{20}c}0&-i&0\\\ i&0&0\\\
0&0&0\\\ \end{array}}\right],\hskip
25.29494pt\lambda_{3}=\left[{\begin{array}[]{*{20}c}1&0&0\\\ 0&-1&0\\\
0&0&0\\\ \end{array}}\right],$

$\lambda_{4}=\left[{\begin{array}[]{*{20}c}0&0&1\\\ 0&0&0\\\ 1&0&0\\\
\end{array}}\right],\hskip
25.29494pt\lambda_{5}=\left[{\begin{array}[]{*{20}c}0&0&-i\\\ 0&0&0\\\
i&0&0\\\ \end{array}}\right],\hskip
25.29494pt\lambda_{6}=\left[{\begin{array}[]{*{20}c}0&0&0\\\ 0&0&1\\\ 0&1&0\\\
\end{array}}\right],$

$\lambda_{7}=\left[{\begin{array}[]{*{20}c}0&0&0\\\ 0&0&-i\\\ 0&i&0\\\
\end{array}}\right],\hskip
25.29494pt\lambda_{8}=\frac{1}{\sqrt{3}}\left[{\begin{array}[]{*{20}c}1&0&0\\\
0&1&0\\\ 0&0&-2\\\ \end{array}}\right].\hfill(2)$

These matrices follow the following commutation and anti-commutation relations

$[\lambda_{i},\lambda_{j}]=2if_{ijk}\lambda_{k}$,
$\\{\lambda_{i},\lambda_{j}\\}=\frac{4}{3}\delta_{ij}+2d_{ijk}\lambda_{k}$,
(3)

respectively, where $d_{ijk}$ and $f_{ijk}$ ($i,j=1,2,..8$) represent
completely symmetric and completely antisymmetric structure constants which
characterizes $SU(3)$ group [39]. It is customary to define the shift
operators $T$, $U$ and $V$ spin as

$T_{\pm}=\frac{1}{2}(\lambda_{1}\pm i\lambda_{2})$,
$U_{\pm}=\frac{1}{2}(\lambda_{6}\pm i\lambda_{7})$,
$V_{\pm}=\frac{1}{2}(\lambda_{4}\pm i\lambda_{5})$. (4)

They satisfy the closed algebra

$[U_{+},U_{-}]=U_{3}$, $[V_{+},V_{-}]=V_{3}$, $[T_{+},T_{-}]=T_{3}$, (5)

$[T_{3},T_{\pm}]=\pm 2T_{\pm}$, $[T_{3},U_{\pm}]=\mp U_{\pm}$,
$[T_{3},V_{\pm}]=\pm V_{\pm}$,

$[V_{3},T_{\pm}]=\pm T_{\pm}$, $[V_{3},U_{\pm}]=\pm U_{\pm}$,
$[V_{3},V_{\pm}]=\pm 2V_{\pm}$,

$[U_{3},T_{\pm}]=\mp T_{\pm}$, $[U_{3},U_{\pm}]=\pm 2U_{\pm}$,
$[U_{3},V_{\pm}]=\pm V_{\pm}$,

$[T_{+},V_{-}]=-U_{-}$, $[T_{+},U_{+}]=V_{+}$, $[U_{+},V_{-}]=T_{-}$,

$[T_{-},V_{+}]=U_{+}$, $[T_{-},U_{-}]=-V_{-}$, $[U_{-},V_{+}]=-T_{+}$,

where the diagonal terms are $T_{3}=\lambda_{3}$,
$U_{3}=(\sqrt{3}\lambda_{8}-\lambda_{3})/2$ and
$V_{3}=(\sqrt{3}\lambda_{8}+\lambda_{3})/2$, respectively.

The Hamiltonian of the semiclassical lambda model is given by

${\rm H^{\Lambda}}={\rm H_{I}^{\Lambda}}+{\rm H_{II}^{\Lambda}}$, (6a)

where the unperturbed and interaction parts including the detuning terms are
given by

${\rm
H_{I}^{\Lambda}}=\hbar(\Omega_{1}-\omega_{1}-\omega_{2})V_{3}+\hbar(\Omega_{2}-\omega_{1}-\omega_{2}){\rm
T}_{3}$, (6b)

and

${\rm
H_{II}^{\Lambda}}=\hbar(\Delta^{\Lambda}_{1}V_{3}+\Delta^{\Lambda}_{2}T_{3})+$

$\hbar\kappa_{1}(V_{+}\exp(-i\Omega_{1}t)+V_{-}\exp(i\Omega_{1}t))+\hbar\kappa_{2}(T_{+}\exp(-i\Omega_{2}t)+T_{-}\exp(i\Omega_{2}t))$,
(6c)

respectively. In Eq.(6), $\Omega_{i}$ ($i=1,2$) are the external frequencies
of the bi-chromatic field, $\kappa_{i}$ are the coupling parameters and
$\hbar\omega_{1}(=-{\rm E}_{1}),\hbar\omega_{2}(=-{\rm
E}_{2}),\hbar(\omega_{2}+\omega_{1})({=\rm E}_{3})$ be the respective energies
of the three levels.
$\Delta^{\Lambda}_{1}=(2\omega_{1}+\omega_{2}-\Omega_{1})$ and
$\Delta^{\Lambda}_{2}=(\omega_{1}+2\omega_{2}-\Omega_{2})$ represent the
respective detuning from the bi-chromatic external frequencies as shown in
Fig.1.

Proceeding in the same way, the semiclassical vee type three-level system can
be written as

${\rm H^{V}}={\rm H_{I}^{V}}+{\rm H_{II}^{V}},$ (7a)

where

${\rm
H_{I}^{V}}=\hbar(\Omega_{1}-\omega_{1}-\omega_{2})V_{3}+\hbar(\Omega_{2}-\omega_{1}-\omega_{2}){\rm
U}_{3},$ (7b)

and

${\rm H_{II}^{V}}=\hbar(\Delta^{V}_{1}V_{3}+\Delta^{V}_{2}U_{3})+$

$\hbar\kappa_{1}(V_{+}\exp(-i\Omega_{1}t)+V_{-}\exp(i\Omega_{1}t))+\hbar\kappa_{2}(U_{+}\exp(-i\Omega_{2}t)+U_{-}\exp(i\Omega_{2}t))$
(7c)

where $\Delta^{V}_{1}=(2\omega_{1}+\omega_{2}-\Omega_{1})$ and
$\Delta^{V}_{2}=(2\omega_{2}+\omega_{1}-\Omega_{2})$ be the detuning shown in
Fig.2.

Similarly the semiclassical cascade three-level model is given by

${\rm H^{\Xi}}={\rm H_{I}^{\Xi}}+{\rm H_{II}^{\Xi}}$, (8a)

where

${\rm H_{I}^{\Xi}}=\hbar(\Omega_{1}+\omega_{2}-\omega_{1}){\rm
U}_{3}+\hbar(\Omega_{2}+\omega_{1}-\omega_{2}){\rm T}_{3}$, (8b)

and

${\rm H_{II}^{\Xi}}=\hbar(\Delta_{1}^{\Xi}U_{3}+\Delta_{2}^{\Xi}T_{3})+$

$\hbar\kappa_{1}(U_{+}\exp(-i\Omega_{1}t)+U_{-}\exp(i\Omega_{1}t))+\hbar\kappa_{2}(T_{+}\exp(-i\Omega_{2}t)+T_{-}\exp(i\Omega_{2}t))$
(8c)

respectively with respective detuning
$\Delta^{\Xi}_{1}=(2\omega_{1}-\omega_{2}-\Omega_{1})$ and
$\Delta^{\Xi}_{2}=(2\omega_{2}-\omega_{1}-\Omega_{2})$.

Taking the fields to be the quantized cavity fields, in the rotating wave
approximation, the Hamiltonian of the quantized lambda configuration is given
by

${\it H^{\Lambda}}=H_{I}^{\Lambda}+H_{II}^{\Lambda}$, (9a)

where,

${\it H_{I}^{\Lambda}}=\hbar(\Omega_{2}-\omega_{1}-\omega_{2}){\rm
T}_{3}+\hbar(\Omega_{1}-\omega_{1}-\omega_{2})V_{3}+\sum\limits_{j=1}^{2}\Omega_{j}{a_{j}^{{\dagger}}a_{j}}$,
(9b)

${\it
H_{II}^{\Lambda}}=\hbar\Delta_{1}^{\Lambda}V_{3}+\hbar\Delta_{2}^{\Lambda}{\rm
T}_{3}+\hbar g_{1}(V_{+}a_{1}+V_{-}a_{1}^{\dagger})+\hbar
g_{2}(T_{+}a_{2}+T_{-}a_{2}^{\dagger}),$ (9c)

where $a^{\dagger}_{i}$ and $a_{i}$ ($i=1,2$) be the creation and annihilation
operators of the cavity modes, $g_{i}$ be the coupling constants and
$\Omega_{i}$ be the mode frequencies. Proceeding in the similar pattern, the
Hamiltonian of the quantized vee system is given by

${\it H^{V}}=H_{I}^{V}+H_{II}^{V}$, (10a)

where,

${\it H_{I}^{V}}=\hbar(\Omega_{2}-\omega_{1}-\omega_{2}){\rm
U}_{3}+\hbar(\Omega_{1}-\omega_{1}-\omega_{2})V_{3}+\sum\limits_{j=1}^{2}\Omega_{j}{a_{j}^{{\dagger}}a_{j}}$
(10b)

${\it H_{II}^{V}}=\hbar\Delta_{1}^{V}V_{3}+\hbar\Delta_{2}^{V}{\rm
U}_{3}+\hbar g_{1}(V_{+}a_{1}+V_{-}a_{1}^{\dagger})+\hbar
g_{2}(U_{+}a_{2}+U_{-}a_{2}^{\dagger}),$ (10c)

respectively. Similarly the Hamiltonian of the quantized cascade system reads

${\it H^{\Xi}}=H_{I}^{\Xi}+H_{II}^{\Xi}$, (11a)

where

${\it H_{I}^{\Xi}}=\hbar(\Omega_{2}-\omega_{1}-\omega_{2}){\rm
T}_{3}+\hbar(\Omega_{1}-\omega_{1}-\omega_{2})U_{3}+\sum\limits_{j=1}^{2}\Omega_{j}{a_{j}^{{\dagger}}a_{j}}$,
(11b)

${\it H_{II}^{\Xi}}=\hbar\Delta_{1}^{\Xi}U_{3}+\hbar\Delta_{2}^{\Xi}{\rm
T}_{3}+\hbar g_{1}(U_{+}a_{1}+U_{-}a_{1}^{\dagger})+\hbar
g_{2}(T_{+}a_{2}+T_{-}a_{2}^{\dagger}).$ (11c)

Using the algebra given in Eq.(5) and that of field operators, it is easy to
check that $[{\it H_{I}^{i}},{\it H_{II}^{i}}]=0$ for
$\Delta^{i}_{1}=-\Delta^{i}_{2}$ ($i=\Lambda$ and $V$) for the lambda and vee
model and $\Delta^{\Xi}_{1}=\Delta^{\Xi}_{2}$ for the cascade model which are
identified as the two photon resonance condition and equal detuning
conditions, respectively [18,21,22,24,26]. This ensures that each piece of the
Hamiltonian has the simultaneous eigen functions. Thus we note that, unlike
Ref.[18,21,22], precise formulation of the aforementioned three-level
configurations require the use of a subset of Gell-Mann $\lambda_{i}$ matrices
rather than the use of all matrices. We now proceed to solve the lambda and
vee configurations for the classical and the quantized field separately.

III.The semiclassical lambda system

At zero detuning the Hamiltonian of the lambda type three-level system is
given by

$H^{\Lambda}=\left[{\begin{array}[]{*{20}c}\hbar(\omega_{1}+\omega_{2})&\hbar\kappa_{2}\exp[-i\Omega_{2}t]&\hbar\kappa_{1}\exp[-i\Omega_{1}t]\\\
\hbar\kappa_{2}\exp[i\Omega_{2}t]&-\hbar\omega_{2}&0\\\
\hbar\kappa_{1}\exp[i\Omega_{1}t]&0&-\hbar\omega_{1}\\\
\end{array}}\right].\hfill(12)$

The solution of the Schrodinger equation corresponding to Hamiltonian (12) is
given by

$\Psi(t)=C_{1}(t)\left|1\right\rangle+C_{2}(t)\left|2\right\rangle+C_{3}(t)\left|3\right\rangle$
(13)

where $C_{1}(t)$, $C_{2}(t)$ and $C_{3}(t)$ be the time-dependent normalized
amplitudes of the lower, middle and upper levels with the respective basis
states,

$\left|1\right\rangle=\left[{\begin{array}[]{*{20}c}0\\\ 0\\\ 1\\\
\end{array}}\right],$ $\left|2\right\rangle=\left[{\begin{array}[]{*{20}c}0\\\
1\\\ 0\\\ \end{array}}\right],$
$\left|3\right\rangle=\left[{\begin{array}[]{*{20}c}1\\\ 0\\\ 0\\\
\end{array}}\right],$ (14)

respectively. We now proceed to calculate the probability amplitudes of the
three states. Substituting Eq.(13) in Schrödinger equation and equating the
coefficients of $\left|2\right\rangle$ , $\left|3\right\rangle$ and
$\left|1\right\rangle$ from both sides we obtain

$i\frac{{\partial C_{3}}}{{\partial
t}}=(\omega_{2}+\omega_{1})C_{3}+\kappa_{1}\exp(-i\Omega_{1}t)C_{1}+\kappa_{2}\exp(-i\Omega_{2}t)C_{2},$
(15a)

$i\frac{{\partial C_{2}}}{{\partial
t}}=-\omega_{2}C_{2}+\kappa_{2}\exp(i\Omega_{2}t)C_{3},$ (15b)

$i\frac{{\partial C_{1}}}{{\partial
t}}=-\omega_{1}C_{1}+\kappa_{1}\exp(i\Omega_{1}t)C_{3}.$ (15c)

Let the solutions of Eqs.(15a-c) are of the following form,

$C_{1}={\rm A}_{1}\exp(iS_{1}t),$ (16a)

$C_{2}={\rm A}_{2}\exp(iS_{2}t),$ (16b)

$C_{3}={\rm A}_{3}\exp(iS_{3}t),$ (16c)

where $A_{i}$s’ are the time independent constants to be determined. Putting
Eqs.(16a-c) in Eqs.(15a-c) we obtain

$(S_{3}+\omega_{2}+\omega_{1}){\rm A}_{3}+\kappa_{2}{\rm A}_{2}+\kappa_{1}{\rm
A}_{1}=0$, (17a)

$(S_{3}+\Omega_{2}-\omega_{2}){\rm A}_{2}+\kappa_{2}{\rm A}_{3}=0$, (17b)

$(S_{3}+\Omega_{1}-\omega_{1}){\rm A}_{1}+\kappa_{1}{\rm A}_{3}=0$. (17c)

In deriving Eqs.(17), the time independence of the amplitudes ${\rm A}_{3}$,
${\rm A}_{2}$ and ${\rm A}_{1}$ are ensured by invoking the conditions
$S_{2}=S_{3}+\Omega_{2}$ and $S_{1}=S_{3}+\Omega_{1}$. At resonance, we have
$\Delta^{\Lambda}_{1}=0=-\Delta^{\Lambda}_{2}$ i.e,
$(2\omega_{2}+\omega_{1})-\Omega_{2}=0=(\omega_{2}+2\omega_{1})-\Omega_{1}$
and the solution of Eq.(17) yields

$S_{3}=-(\omega_{2}+\omega_{1})\pm\Delta$, (18a)

$S_{3}=-(\omega_{2}+\omega_{1})$ (18b)

where $\Delta=\sqrt{\kappa_{1}^{2}+\kappa_{2}^{2}}$ and we have three values
of $S_{2}$ and $S_{1}$ namely

$S_{2}^{1}=\omega_{2},S_{2}^{2,3}=\omega_{2}\pm\Delta$, (19a)

$S_{1}^{1}=\omega_{1},S_{1}^{2,3}=\omega_{1}\pm\Delta$. (19b)

Using Eqs.(18) and (19), Eq.(16) can be written as

$C_{3}(t)={\rm A}_{3}^{1}\exp(-i(\omega_{2}+\omega_{1})t)$

$+{\rm A}_{3}^{2}\exp(i(-(\omega_{2}+\omega_{1})+\Delta)t)+{\rm
A}_{3}^{3}\exp(i(-(\omega_{2}+\omega_{1})-\Delta)t)$, (20a)

$C_{2}(t)={\rm A}_{2}^{1}\exp(i\omega_{2}t)+{\rm
A}_{2}^{2}\exp(i(\omega_{2}+\Delta)t)+{\rm A}_{2}^{3}(i(\omega_{2}-\Delta)t)$,
(20b)

$C_{1}(t)={\rm A}_{1}^{1}\exp(i\omega_{1}t)+{\rm
A}_{1}^{2}\exp(i(\omega_{1}+\Delta)t)+{\rm A}_{1}^{3}(i(\omega_{1}-\Delta)t)$,
(20c)

where $A_{i}$-s are the constants which can be calculated from the following
initial conditions:

Case-I: At $t=0$ let the atom is in level-1, i.e. $C_{1}(0)=1$, $C_{2}(0)=0$,
$C_{3}(0)=0$. Using Eqns (15) and (20), the corresponding time-dependent
probabilities of the three levels are

$\left|{C_{3}(t)}\right|^{2}=\frac{{\kappa_{1}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$, (21a)

$\left|{C_{2}(t)}\right|^{2}=4\frac{{\kappa_{1}^{2}\kappa_{2}^{2}}}{{\Delta^{4}}}\sin^{4}\Delta
t/2$, (21b)

$\left|{C_{1}(t)}\right|^{2}=\frac{1}{{\Delta^{4}}}(\kappa_{2}^{2}+\kappa_{1}^{2}\cos\Delta
t)^{2}$. (21c)

Case-II: If the atom is initially in level-2, i.e. $C_{1}(0)=0$, $C_{2}(0)=1$
and $C_{3}(0)=0$, the probabilities of the three states are

$\left|{C_{3}(t)}\right|^{2}=\frac{{\kappa_{2}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$, (22a)

$\left|{C_{2}(t)}\right|^{2}=\frac{1}{{\Delta^{4}}}(\kappa_{1}^{2}+\kappa_{2}^{2}\cos\Delta
t)^{2}$, (22b)

$\left|{C_{1}(t)}\right|^{2}=4\frac{{\kappa_{1}^{2}\kappa_{2}^{2}}}{{\Delta^{4}}}\sin^{4}\Delta
t/2$. (22c)

Case-III: When the atom is initially in level-3, i.e. $C_{1}(0)=0$,
$C_{2}(0)=0$ and $C_{3}(0)=1$, the time evolution of the probabilities of the
three states are

$\left|{C_{3}(t)}\right|^{2}=\cos^{2}\Delta t$, (23a)

$\left|{C_{2}(t)}\right|^{2}=\frac{{\kappa_{2}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$, (23b)

$\left|{C_{1}(t)}\right|^{2}=\frac{{\kappa_{1}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$. (23c)

We now proceed to solve the quantized version of the above model.

IV. The quantized lambda system

We now consider the three-level lambda system interacting with a bi-chromatic
quantized fields described by the Hamiltonian Eq.(9). At zero detuning the
solution of the Hamiltonian is given by

$\left|{\Psi_{\Lambda}(t)}\right\rangle=\sum\limits_{n,m=0}^{\infty}{[C_{1}^{n-1,m+1}(t)\left|{n-1,m+1,1}\right\rangle+C_{2}^{n,m}(t)\left|{n,m,2}\right\rangle}+C_{3}^{n-1,m}(t)\left|{n-1,m,3}\right\rangle],$

(24)
where $n$ and $m$ represent the photon number corresponding to two modes of
the bi-chromatic fields. This interaction Hamiltonian that couples the atom-
field states ${\left|{n-1,m,3}\right\rangle}$, ${\left|{n,m,2}\right\rangle}$
and ${\left|{n-1,m+1,1}\right\rangle}$ and forms the lambda configuration
shown in Fig.1 is given by

${\it
H_{II}^{\Lambda}}=\hbar\left[{\begin{array}[]{*{20}c}0&{g_{2}\sqrt{n}}&{g_{1}\sqrt{m+1}}\\\
{g_{2}\sqrt{n}}&0&0\\\ {g_{1}\sqrt{m+1}}&0&0\\\ \end{array}}\right].$ (25)

The eigenvalues of the Hamiltonian are given by
$\lambda_{\pm}=\pm\hbar\sqrt{ng_{2}^{2}+(m+1)g_{1}^{2}}\quad(=\pm\hbar\Omega_{nm})$
and $\lambda_{0}=0(=\Omega_{0})$, respectively with the corresponding dressed
eigenstates

$\left[{\begin{array}[]{*{20}c}{\left|{nm,3}\right\rangle}\\\
{\left|{nm,2}\right\rangle}\\\ {\left|{nm,1}\right\rangle}\\\
\end{array}}\right]=T_{n,m}(g_{1},g_{2})\left[{\begin{array}[]{*{20}c}{\left|{n-1,m,3}\right\rangle}\\\
{\left|{n,m,2}\right\rangle}\\\ {\left|{n-1,m+1,1}\right\rangle}\\\
\end{array}}\right].$ (26)

In Eq.(26), the dressed states are constructed by rotating the bare states
with the Euler matrix given by

$T_{n,m}(g_{1},g_{2})=\left[{\begin{array}[]{*{20}c}{c_{3}c_{2}-c_{1}s_{2}s_{3}}&{c_{3}s_{2}-c_{1}c_{2}s_{3}}&{s_{3}s_{1}}\\\
{-s_{3}c_{2}-c_{1}s_{2}c_{3}}&{-s_{3}s_{2}+c_{1}c_{2}c_{3}}&{c_{3}s_{1}}\\\
{s_{1}s_{2}}&{-s_{1}c_{2}}&{c_{1}}\\\ \end{array}}\right]$ (27)

where $s_{i}=\sin\theta_{i}$ and $c_{i}=\cos\theta_{i}$ ($i=1,2,3$). The
elements of the matrix are found to

$T_{n,m}(g_{1},g_{2})=\left[{\begin{array}[]{*{20}c}{\frac{1}{{\sqrt{2}}}}&{g_{2}\sqrt{\frac{{n}}{{2(ng_{2}^{2}+(m+1)g_{1}^{2})}}}}&{g_{1}\sqrt{\frac{{m+1}}{{2(ng_{2}^{2}+(m+1)g_{1}^{2})}}}}\\\
{0}&{g_{1}\sqrt{\frac{{m+1}}{{ng_{2}^{2}+(m+1)g_{1}^{2}}}}}&{-g_{2}\sqrt{\frac{{n}}{{ng_{2}^{2}+(m+1)g_{1}^{2}}}}}\\\
{-\frac{1}{{\sqrt{2}}}}&{g_{2}\sqrt{\frac{{n}}{{2(ng_{2}^{2}+(m+1)g_{1}^{2})}}}}&{g_{1}\sqrt{\frac{{m+1}}{{2(ng_{2}^{2}+(m+1)g_{1}^{2})}}}}\\\
\end{array}}\right],$ (28)

with corresponding Euler angles,

$\theta_{1}=\arccos[\frac{\sqrt{1+m}g_{1}}{\sqrt{2(1+m)g_{1}^{2}+2ng_{2}^{2}}}]$,
$\theta_{2}=-\arccos[-\frac{\sqrt{n}g_{2}}{\sqrt{(1+m)g_{1}^{2}+2ng_{2}^{2}}}]$,
$\theta_{3}=\arccos[-\frac{\sqrt{2n}g_{2}}{\sqrt{(1+m)g_{1}^{2}+2ng_{2}^{2}}}]$.

(29)
The time-dependent probability amplitudes of the three levels are given by

$\left[{\begin{array}[]{*{20}c}{C_{3}^{n-1,m}(t)}\\\ {C_{2}^{n,m}(t)}\\\
{C_{1}^{n-1,m+1}(t)}\\\
\end{array}}\right]=T_{n,m}^{-1}(g_{1},g_{2})\left[{\begin{array}[]{*{20}c}{e^{-i\Omega_{nm}t}}&0&0\\\
0&{e^{-i\Omega_{0}t}}&0\\\ 0&0&{e^{i\Omega_{nm}t}}\\\
\end{array}}\right]T_{n,m}(g_{1},g_{2})\left[{\begin{array}[]{*{20}c}{C_{3}^{n-1,m}(0)}\\\
{C_{2}^{n,m}(0)}\\\ {C_{1}^{n-1,m+1}(0)}\\\ \end{array}}\right].$

(30)
Now similar to the semiclassical model the probabilities corresponding to
different initial conditions are:

Case-IV: When the atom is initially in level-1, i.e, $C_{1}^{n-1,m+1}=1$,
$C_{2}^{n,m}=0$ and $C_{3}^{n-1,m}=0$, the time-dependent atomic populations
of the three states are given by

$\left|{C_{3}^{n-1,m}(t)}\right|^{2}=\frac{{(m+1)g_{1}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$,
(31a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=4\frac{{g_{1}^{2}g_{2}^{2}n(m+1)}}{{\Omega_{nm}^{4}}}\sin^{4}\Omega_{nm}t/2$,
(31b)

$\left|{C_{1}^{n-1,m+1}(t)}\right|^{2}=\frac{1}{{\Omega_{nm}^{4}}}[ng_{2}^{2}+(m+1)g_{1}^{2}\cos\Omega_{nm}t]^{2}$.
(31c)

Case-V: When the atom is initially in level-2, i.e, $C_{1}^{n-1,m+1}=0$,
$C_{2}^{n,m}=1$ and $C_{3}^{n-1,m}=0$, the probabilities of three states are

$\left|{C_{3}^{n-1,m}(t)}\right|^{2}=\frac{{ng_{2}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$,
(32a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=\frac{1}{{\Omega_{nm}^{4}}}[(m+1)g_{1}^{2}+ng_{2}^{2}\cos\Omega_{nm}t]^{2}$,
(32b)

$\left|{C_{1}^{n-1,m+1}(t)}\right|^{2}=4\frac{{g_{1}^{2}g_{2}^{2}n(m+1)}}{{\Omega_{nm}^{4}}}\sin^{4}\Omega_{mm}t/2$.
(32c)

Case-VI: If the atom is initially in level-3, then we have
$C_{1}^{n-1,m+1}=0$, $C_{2}^{n,m}=0$ and $C_{3}^{n-1,m+1}=1$ and the
corresponding probabilities are

$\left|{C_{3}^{n-1,m}(t)}\right|^{2}=\cos^{2}\Omega_{nm}t$, (33a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=\frac{{ng_{2}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$,
(33b)

$\left|{C_{1}^{n-1,m+1}(t)}\right|^{2}=\frac{{(m+1)g_{1}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$.
(33c)

We now proceed to evaluate the population oscillations of different levels of
the vee system with similar initial conditions.

V.The semiclassical vee system

At zero detuning, the Hamiltonian of the semiclassical three-level vee system
interacting with two-mode classical fields is given by

$H^{V}=\left[{\begin{array}[]{*{20}c}\hbar\omega_{1}&0&\hbar\kappa_{1}\exp[-i\Omega_{1}t]\\\
0&\hbar\omega_{2}&\hbar\kappa_{2}\exp[-i\Omega_{2}t]\\\
\hbar\kappa_{1}\exp[i\Omega_{1}t]&\hbar\kappa_{2}\exp[i\Omega_{2}t]&-\hbar(\omega_{1}+\omega_{2})\\\
\end{array}}\right].\hfill(34)$

Let the solution of the Schrodinger equation corresponding to Eq.(34) is given
by

$\Psi(t)=C_{1}(t)\left|1\right\rangle+C_{2}(t)\left|2\right\rangle+C_{3}(t)\left|3\right\rangle,$
(35)

where $C_{1}(t)$, $C_{2}(t)$ and $C_{3}(t)$ are the time-dependent normalized
amplitudes with the basis vectors defined in Eqs.(13). To calculate the
probability amplitudes of three states, substituting Eq.(35) into the
Schrödinger equation we obtain

$i\frac{{\partial C_{3}}}{{\partial
t}}=\omega_{1}C_{3}+\kappa_{1}\exp(-i\Omega_{1}t)C_{1}$, (36a)

$i\frac{{\partial C_{2}}}{{\partial
t}}=\omega_{2}C_{2}+\kappa_{2}\exp(-i\Omega_{2}t)C_{1}$, (36b)

$i\frac{{\partial C_{1}}}{{\partial
t}}=-(\omega_{1}+\omega_{2})C_{1}+\kappa_{2}\exp(i\Omega_{2}t)C_{2}+\kappa_{1}\exp(i\Omega_{1}t)C_{3}$.
(36c)

Let the solutions of Eqs.(36) are of the following form:

$C_{3}(t)={\rm A}_{3}\exp(iS_{3}t),$ (37a)

$C_{2}(t)={\rm A}_{2}\exp(iS_{2}t),$ (37b)

$C_{1}(t)={\rm A}_{1}\exp(iS_{1}t),$ (37c)

where $A_{i}$-s are the time independent constants to be determined from the
boundary conditions. From Eq.(36) and Eq.(37) we obtain

$(S_{1}-\Omega_{1}+\omega_{1}){\rm A}_{3}+\kappa_{1}{\rm A}_{1}=0$, (38a)

$(S_{1}-\Omega_{2}+\omega_{2}){\rm A}_{2}+\kappa_{2}{\rm A}_{1}=0$, (38b)

$(S_{1}-\omega_{2}-\omega_{1}){\rm A}_{1}+\kappa_{2}{\rm A}_{2}+\kappa_{1}{\rm
A}_{3}=0$. (38c)

In deriving Eqs.(38), the time independence of the amplitudes ${\rm A}_{3}$,
${\rm A}_{2}$ and ${\rm A}_{1}$ are ensured by invoking the conditions
$S_{2}=S_{1}-\Omega_{2}$ and $S_{3}=S_{1}-\Omega_{1}$. At resonance,
$\Delta^{V}_{1}=0=-\Delta^{V}_{2}$ i.e.
$(2\omega_{2}+\omega_{1})-\Omega_{2}=0=(\omega_{2}+2\omega_{1})-\Omega_{1}$
and the solutions of Eq.(38) are given by

$S_{1}=(\omega_{1}+\omega_{2})$ (39a)

$S_{1}=(\omega_{1}+\omega_{2})\pm\Delta$ (39b)

and we have three values of $S_{2}$ and $S_{3}$

$S_{2}^{1}=-\omega_{2},S_{2}^{2,3}=-\omega_{2}\pm\Delta$ (40a)

$S_{3}^{1}=-\omega_{1},S_{3}^{2,3}=-\omega_{1}\pm\Delta$. (40b)

Using Eqs.(39) and (40), Eqs. (37) can be written as

$C_{3}(t)={\rm A}_{3}^{1}\exp(-i\omega_{1}t)+{\rm
A}_{3}^{2}\exp(-i(\omega_{1}+\Delta)t)+{\rm
A}_{3}^{3}(-i(\omega_{1}-\Delta)t)$, (41a)

$C_{2}(t)={\rm A}_{2}^{1}\exp(-i\omega_{2}t)+{\rm
A}_{2}^{2}\exp(-i(\omega_{2}+\Delta)t)+{\rm
A}_{2}^{3}(-i(\omega_{2}-\Delta)t)$, (41b)

$C_{1}(t)={\rm A}_{1}^{1}\exp(i(\omega_{2}+\omega_{1})t)$

$+{\rm A}_{1}^{2}\exp(i((\omega_{2}+\omega_{1})+\Delta)t)+{\rm
A}_{1}^{3}\exp(i((\omega_{2}+\omega_{1})-\Delta)t)$, (41c),

where $A_{i}$-s are the constants which are calculated below from the various
initial conditions.

Case-I: Let us consider initially at $t=0$, the atom is in level-1, i.e,
$C_{1}(0)=1$, $C_{2}(0)=0$ and $C_{3}(0)=0$. Using Eqs. (36) and (41), the
time dependent probabilities of the three levels are given by

$\left|{C_{3}(t)}\right|^{2}=\frac{{\kappa_{1}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$, (42a)

$\left|{C_{2}(t)}\right|^{2}=\frac{{\kappa_{2}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$, (42b)

$\left|{C_{1}(t)}\right|^{2}=\cos^{2}\Delta t$. (42c)

Case-II: If the atom is initially in level-2, i.e, $C_{1}(0)=0$, $C_{2}(0)=1$
and $C_{3}(0)=0$, the corresponding probabilities of the states are given by

$\left|{C_{3}(t)}\right|^{2}=4\frac{{\kappa_{1}^{2}\kappa_{2}^{2}}}{{\Delta^{4}}}\sin^{4}\Delta
t/2$, (43a)

$\left|{C_{2}(t)}\right|^{2}=\frac{1}{{\Delta^{4}}}(\kappa_{1}^{2}+\kappa_{2}^{2}\cos\Delta
t)^{2}$, (43b)

$\left|{C_{1}(t)}\right|^{2}=\frac{{\kappa_{2}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$. (43c)

Case-III: When the atom is initially in level-3, i.e, $C_{1}(0)=0$,
$C_{2}(0)=0$ and $C_{3}(0)=1$, we obtain the the occupation probabilities of
the three states as follows:

$\left|{C_{3}(t)}\right|^{2}=\frac{1}{{\Delta^{4}}}(\kappa_{2}^{2}+\kappa_{1}^{2}\cos\Delta
t)^{2}$, (44a)

$\left|{C_{2}(t)}\right|^{2}=4\frac{{\kappa_{1}^{2}\kappa_{2}^{2}}}{{\Delta^{4}}}\sin^{4}\Delta
t/2$, (44b)

$\left|{C_{1}(t)}\right|^{2}=\frac{{\kappa_{1}^{2}}}{{\Delta^{2}}}\sin^{2}\Delta
t$. (44c)

VI.The quantized vee system

The eigenfunction of the quantized vee system described by the Hamiltonian in
Eq.(10) is given by

$\left|{\Psi_{V}(t)}\right\rangle=\sum\limits_{n,m=0}^{\infty}{[C_{1}^{n+1,m}(t)\left|{n+1,m,1}\right\rangle}+C_{2}^{n,m}(t)\left|{n,m,2}\right\rangle+C_{3}^{n+1,m-1}(t)\left|{n+1,m-1,3}\right\rangle].$

(45)
Once again we note that the Hamiltonian couples the atom-field states
${\left|{n+1,m,1}\right\rangle}$, ${\left|{n,m,2}\right\rangle}$ and
${\left|{n+1,m-1,3}\right\rangle}$ forming vee configuration depicted in
Fig.2. The interaction part of the Hamiltonian (45) can also be expressed in
the matrix form

${\it H_{II}^{V}}=\hbar\left[{\begin{array}[]{*{20}c}0&0&{g_{1}\sqrt{m}}\\\
0&0&{g_{2}\sqrt{n+1}}\\\ {g_{1}\sqrt{m}}&{g_{2}\sqrt{n+1}}&0\\\
\end{array}}\right],$ (46)

and the corresponding eigenvalues are
$\lambda_{\pm}=\pm\hbar\sqrt{mg_{1}^{2}+(n+1)g_{2}^{2}}\quad(=\pm\hbar\Omega_{nm})$
and $\lambda_{0}=0$ respectively. The dressed eigenstate is given by

$\left[{\begin{array}[]{*{20}c}{\left|{nm,3}\right\rangle}\\\
{\left|{nm,2}\right\rangle}\\\ {\left|{nm,1}\right\rangle}\\\
\end{array}}\right]=T_{n,m}\left[{\begin{array}[]{*{20}c}{\left|{n+1,m-1,3}\right\rangle}\\\
{\left|{n,m,2}\right\rangle}\\\ {\left|{n+1,m,1}\right\rangle}\\\
\end{array}}\right],$ (47)

the rotation matrix is found to be

$T_{n,m}=\left[{\begin{array}[]{*{20}c}{g_{1}\sqrt{\frac{{m}}{{2((n+1)g_{2}^{2}+mg_{1}^{2})}}}}&{g_{2}\sqrt{\frac{{n+1}}{{2((n+1)g_{2}^{2}+mg_{1}^{2})}}}}&{\frac{1}{{\sqrt{2}}}}\\\
{-g_{2}\sqrt{\frac{{n+1}}{{(n+1)g_{2}^{2}+mg_{1}^{2}}}}}&{g_{1}\sqrt{\frac{{m}}{{(n+1)g_{2}^{2}+mg_{1}^{2}}}}}&{0}\\\
{-g_{1}\sqrt{\frac{{m}}{{2((n+1)g_{2}^{2}+mg_{1}^{2})}}}}&{-g_{2}\sqrt{\frac{{n+1}}{{2((n+1)g_{2}^{2}+mg_{1}^{2})}}}}&{\frac{1}{{\sqrt{2}}}}\\\
\end{array}}\right].$ (48)

The straightforward evaluation yields the various Euler angles are

$\theta_{1}=-\frac{\pi}{4}$,
$\theta_{2}=\arccos[-\frac{\sqrt{n+1}g_{2}}{\sqrt{mg_{1}^{2}+(1+n)g_{2}^{2}}}]$,
$\theta_{3}=-\frac{\pi}{2}$. (49)

The time-dependent probability amplitudes of the three levels are given by

$\left[{\begin{array}[]{*{20}c}{C_{3}^{n+1,m-1}(t)}\\\ {C_{2}^{n,m}(t)}\\\
{C_{1}^{n+1,m}(t)}\\\
\end{array}}\right]={T^{-1}_{n,m}}\left[{\begin{array}[]{*{20}c}{e^{-i\Omega_{nm}t}}&0&0\\\
0&{e^{-i\Omega_{0}t}}&0\\\ 0&0&{e^{i\Omega_{nm}t}}\\\
\end{array}}\right]T_{n,m}\left[{\begin{array}[]{*{20}c}{C_{3}^{n+1,m-1}(0)}\\\
{C_{2}^{n,m}(0)}\\\ {C_{1}^{n+1,m}(0)}\\\ \end{array}}\right]$. (50)

Once again we proceed to calculate the probabilities for different initial
conditions.

Case-IV: Here we consider initially the atom is in level-1 i.e,
$C_{1}^{n+1,m}=1$, $C_{2}^{n,m}=0$ and $C_{3}^{n+1,m-1}=0$. Using Eqs.(49) and
(50) the time-dependent probabilities of the three levels are given by

$\left|{C_{3}^{n+1,m-1}(t)}\right|^{2}=\frac{{mg_{1}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$,
(51a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=\frac{{(n+1)g_{2}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$,
(51b)

$\left|{C_{1}^{n+1,m}(t)}\right|^{2}=\cos^{2}\Omega_{nm}t$. (51c)

Case-V: If the atom is initially in level-2 i.e, $C_{3}^{n+1,m-1}=0$,
$C_{2}^{n,m}=1$ and $C_{1}^{n+1,m}=0$, then corresponding probabilities are

$\left|{C_{3}^{n+1,m-1}(t)}\right|^{2}=4\frac{{g_{2}^{2}g_{1}^{2}(n+1)(m)}}{{\Omega_{mn}^{4}}}\sin^{4}\Omega_{mn}t/2$,
(52a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=\frac{1}{{\Omega_{nm}^{4}}}[mg_{1}^{2}+(n+1)g_{2}^{2}\cos\Omega_{nm}t]^{2}$,
(52b)

$\left|{C_{1}^{n+1,m}(t)}\right|^{2}=\frac{{g_{2}^{2}(n+1)}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$.
(52c)

Case-VI: Finally if the atom is initially in level-3 i.e, $C_{1}^{n+1,m}=0$,
$C_{2}^{n,m}=0$ and $C_{3}^{n+1,m-1}=1$, then

$\left|{C_{3}^{n+1,m-1}(t)}\right|^{2}=\frac{1}{{\Omega_{nm}^{4}}}[mg_{1}^{2}\cos\Omega_{nm}t+(n+1)g_{2}^{2}]^{2}$,
(53a)

$\left|{C_{2}^{n,m}(t)}\right|^{2}=4\frac{{g_{2}^{2}g_{1}^{2}(n+1)(m)}}{{\Omega_{nm}^{4}}}\sin^{4}\Omega_{nm}t/2$,
(53b)

$\left|{C_{1}^{n+1,m}(t)}\right|^{2}=\frac{{mg_{1}^{2}}}{{\Omega_{nm}^{2}}}\sin^{2}\Omega_{nm}t$.
(53c)

Finally we note that for large values of n and m, Case-IV, V and VI become
identical to Case-I, II and III, respectively. This precisely shows the
validity of the Bohr’s correspondence principle indicating the consistency of
our approach.

VII.Numerical results and discussion

Before going to show the numerical plots of the semiclassical and quantized
lambda and vee systems, we first consider their analytical results. If we
compare Case-I, II, III of both cases, we find that the probabilities in
Case-I (Case-III)) of lambda system is the same as in Case-III (Case-I) of vee
system except the populations of 1st and 3rd levels are interchanged. See
Eqs.(21 $\&$ 44) and Eqs.(23 $\&$ 42) for detailed comparison. Also Case-II
respective models are similar which is evident by comparing Eqs.(22 $\&$ 43).
In contrast, for the quantized model, Case-IV (Case-VI) of the lambda system
is no longer same as in Case-VI (Case-IV) of the vee system. This breaking of
symmetry is evident by comparing the analytical results, Eqs.(31 $\&$ 53),
Eqs.(32 $\&$ 52) and Eqs.(32 $\&$ 51) respectively. Unlike previous case, also
Case-V both the models are different which is evident from Eqs.(22 $\&$ 43).

In what follows, we compare the probabilities of the semiclassical and
quantized lambda and vee systems respectively. Fig.3 and 4 show the plots of
the probabilities $|C_{1}^{i}(t)|^{2}$ (blue line), $|C_{2}^{i}(t)|^{2}$
(green line) and $|C_{3}^{i}(t)|^{2}$ (red line) for the semiclassical lambda
and vee models when the atom is initially at level-1 (Case-I), level-2 (Case-
II) and level-3 (Case-III), respectively. The comparison of the plots shows
that the pattern of the probability oscillation of the lambda system for
Case-I shown in Fig.3a (Case-III in Fig.3c) is similar to that of Case-III
shown in Fig.4c (Case-I in Fig.4a) of the vee system. More particularly we
note that in all cases the oscillation of level-2 remains unchanged, while the
oscillation of level-3 (level-1) of the lambda system for Case-I is identical
to that of level-1 (level-3) of the vee system for Case-III. Furthermore,
comparison of Fig.3b and Fig.4b for Case-II shows that the time evolution of
the probabilities of level-2 of both systems also remains similar while those
of level-3 and level-1 are interchanged. From the behaviour of the probability
curve we can conclude that the lambda and vee configurations are essentially
identical to each other as we can obtain one configuration from another simply
by the inversion followed by the interchange of probabilities.

For the quantized field, we first consider the time evolution of the
probabilities taking the field is in a number state representation. In the
number state representation, the vacuum Rabi oscillation corresponding to
Case-IV, V and VI of the lambda and vee systems are shown in Fig.5 and Fig.6
respectively. We note that, unlike previous case, the Rabi oscillation for
Case-IV shown in Fig.5a (Case-VI shown in Fig.5c) for the lambda model is no
longer similar to Case-VI shown in Fig.6c (Case-IV shown in Fig.6a) for the
vee model. Furthermore, we note that for Case-V, the oscillation patterns of
Fig.5b is completely different from that of Fig.6b. In a word, for the
quantized field, in contrast to the semiclassical case, the symmetry in the
pattern of the vacuum Rabi oscillation in all cases is completely spoiled
irrespective of the fact whether the system stays initially in any one of the
three levels.

The quantum origin of the breaking of the symmetric pattern of the Rabi
oscillation is the following. We note that due to the appearance of the terms
like $(n+1)$ or $(m+1)$, several elements in the probabilities given by
Eqs.(31,32,33) for the lambda system and Eqs.(51,52,53) for the vee are non
zero even at $m=0$ and $n=0$. We argue that the vacuum Rabi oscillation
interferes with the probability oscillations of various levels and spoils
their symmetric structure. Thus as a consequence of the vacuum fluctuation,
the symmetry of probability amplitudes of the dressed states of both models
formed by the coherent superposition of the bare states is also lost. In the
other word, the invertibility between the lambda and vee models exhibited for
the classical field disappears as the direct consequence of the quantization
of the cavity modes.

Finally we consider the lambda and vee models interacting with the bi-
chromatic quantized fields which are in the coherent state. The coherently
averaged probabilities of level-3, level-2 and level-1 are given by

$\left\langle{P_{3}(t)}\right\rangle_{\Lambda}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{3}^{n-1,m}(t)}\right|^{2}$,
(54a)

$\left\langle{P_{2}(t)}\right\rangle_{\Lambda}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{2}^{n,m}(t)}\right|^{2}$,
(54b)

$\left\langle{P_{1}(t)}\right\rangle_{\Lambda}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{1}^{n-1,m+1}(t)}\right|^{2}$,
(54c)

for the lambda system and

$\left\langle{P_{3}(t)}\right\rangle_{V}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{3}^{n+1,m-1}(t)}\right|^{2}$,
(55a)

$\left\langle{P_{2}(t)}\right\rangle_{V}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{2}^{n,m}(t)}\right|^{2}$,
(55b)

$\left\langle{P_{1}(t)}\right\rangle_{V}=\sum\limits_{n,m}{W_{n}}W_{m}\left|{C_{1}^{n+1,m}(t)}\right|^{2}$,
(55c)

for the vee system, where $W_{n}=\frac{1}{n!}\exp[-\bar{n}]{\bar{n}}^{n}$ and
$W_{m}=\frac{1}{m!}\exp[-\bar{m}]{\bar{m}}^{m}$ with $\bar{n}$ and $\bar{m}$
be the mean photon numbers of the two quantized modes, respectively. Fig.7-9
display the numerical plots of Eq.(54) and Eq.(55) for Case-IV, V and VI
respectively where the collapse and revival of the Rabi oscillation is clearly
evident for large average photon numbers in both the fields. We note that in
all cases, the collapse and revival of level-2 of both the systems are
identical to each other. Furthermore, we note that the collapse and revival
for lambda system initially in level-1 shown in Fig.7a, Fig.7b and Fig.7c
(level-3 shown in Fig.9a, Fig.9b and Fig.9c) is the same as that of the vee
system if it is initially in level-3 shown in Fig.7f, Fig.7e and Fig.7d
(level-1 shown in Fig.9f, Fig.9e and Fig.9d) respectively. On the other hand,
if the system is initially in level-2, the collapse and revival of the lambda
systems shown in Fig.8a, Fig.8b and Fig.8c are identical to Fig.8f, Fig.8e and
Fig.8d respectively for the vee system. This is precisely the situation what
we obtained in case of the semiclassical model. Thus the symmetry broken in
the case of the quantized model is restored back again indicating that the
coherent state with large average photon number is very close to the classical
state where the effect of field population in the vacuum level is almost zero.
It is needless to say that the coherent state with very low average photon
number in the field modes can not show the symmetric dynamics in lambda and
vee systems.

VIII.Conclusion

This paper presents the explicit construction of the Hamiltonians of the
lambda, vee and cascade type of three-level configurations from the Gell-Mann
matrices of $SU(3)$ group and compares the exact solutions of the first two
models with different initial conditions. It is shown that the Hamiltonians of
different configurations of the three-level systems are different. We
emphasize that there is a conceptual difference between our treatment and the
existing approach by Hioe and Eberly [18,21,22]. These authors advocate the
existence of different energy conditions which effectively leads to same
cascade Hamiltonian ($h_{21}\neq 0$, $h_{32}\neq 0$ and $h_{31}=0$ in Eq.(1))
having similar spectral feature irrespective of the configuration. We justify
our approach by noting the fact that the two-photon condition and the equal
detuning condition is a natural outcome of our analysis. For the lambda and
vee models, the transition probabilities of the three levels for different
initial conditions are calculated while taking the atom interacting with the
bi-chromatic classical and quantized field respectively. It is shown that due
to the vacuum fluctuation, the inversion symmetry exhibited by the
semiclassical models is completely destroyed. In other words, the dynamics for
the semiclassical lambda system can be completely obtained from the knowledge
of the vee system and vice versa while such recovery is not possible if the
field modes are quantized. The symmetry is restored again when the field modes
are in the coherent state with large average photon number. Such breaking of
the symmetric pattern of the quantum Rabi oscillation is not observed in case
of the two-level Jaynes-Cummings model and therefore it is essentially a
nontrivial feature of the multi-level systems which is manifested if the
number of levels exceeds two. This investigation is a part of our sequel
studies of the symmetry breaking effect for the equidistant cascade three-
level and equidistant cascade four-level systems respectively [36,37].
Following the scheme of constructing of the model Hamiltonians, it is easy to
show that we have different eight dimensional Bloch equations and non-linear
constants for different configurations of the three-level systems and these
issues will be considered elsewhere [40]. The breaking of the inversion
symmetry of the lambda and vee models as a direct effect of the field
quantization is an intricate issue especially in context with future cavity
experiments with the multilevel systems.

Acknowledgement

MRN thanks University Grants Commission and SS thanks Department of Science
and Technology, New Delhi for partial financial support. We thank Dr T K Dey
for discussions. SS is also thankful to S N Bose National Centre for Basic
Sciences, Kolkata, for supporting his visit to the centre through the Visiting
Associateship program.

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(100,90)(200,90)(210,89)[]$\Delta_{2}$(260,90)[]$|3;m,n-1>$(155,85)(170,40)
(175,70)[]$\Omega_{2}$(100,40)(200,40) (250,40)[]$|2;m,n>$(100,80)(175,80)5
(150,85)(200,85)5 (110,100)(110,90) (110,70)(110,80) (195,100)(195,90)
(195,75)(195,85) (120,10)(150,80) (140,20)[]$\Omega_{1}$(100,10)(200,10)
(265,10)[]$|1;m+1,n-1>$(90,85)[]$\Delta_{1}$

Fig.1 : Lambda type transition

(100,90)(200,90) (250,90)[]$|3;m-1,n+1>$(120,85)(140,10)
(170,25)[]$\Omega_{2}$(100,40)(200,40) (250,40)[]$|2;m,n>$(100,85)(175,85)5
(140,35)(200,35)5 (145,10)(160,35) (120,60)[]$\Omega_{1}$(100,10)(200,10)
(260,10)[]$|1;m,n+1>$(110,100)(110,90) (110,78)(110,85) (195,47)(195,40)
(195,28)(195,35) (90,87)[]$\Delta_{1}$(210,39)[]$\Delta_{2}$

Fig.2 : Vee type transition

[Fig.3]: The time evolution of the probabilities $|C_{1}(t)|^{2}$ (blue line),
$|C_{2}(t)|^{2}$ (green line) and $|C_{3}(t)|^{2}$ (red line) of the
semiclassical lambda system for Case-I, II and III respectively with values
${\bf\kappa_{1}=.2}$, ${\bf\kappa_{2}=.1}$.

[Fig.4]: The time variation of the probabilities $|C_{1}(t)|^{2}$ (blue line),
$|C_{2}(t)|^{2}$ (green line) and $|C_{3}(t)|^{2}$ (red line) of the
semiclassical vee system for Case-I, II and III respectively with above values
of ${\bf\kappa_{1}}$, ${\bf\kappa_{2}}$.

[Fig.5]: The Rabi oscillation of the quantized lambda system when the fields
are in the number state for Case-I, II and III, respectively with ${\bf
g_{1}=.2}$, ${\bf g_{2}=.1}$, ${\bf n=1}$, ${\bf m=1}$.

[Fig.6]: The Rabi oscillation of the quantized vee system when the fields are
in the number states for Case-I, II and III, respectively for same values of
$\bf g_{1}$, $\bf g_{2}$, $\bf n$, $\bf m$.

[Fig.7]: Figs.7a-c display the time-dependent collapse and revival phenomenon
of level-3, level-2 and level-1 of the lambda system for Case-IV, while
Figs.7d-f show that of the level-3, level-2 and level-1 respectively for of
Case-VI of the vee system taking the field modes are in coherent states with
$\bar{n}=30$ and $\bar{m}=20$.

[Fig.8]: Figs.8a-c display the time-dependent of collapse and revival of
level-3, level-2 and level-1 of the lambda system for Case-V while Figs.8d-f
show that of level-3, level-2 and level-1 of the vee system for Case-V with
the same values of $\bar{n}$ and $\bar{m}$. as in Fig.7

[Fig.9]: Figs.9a-c display the time-dependent of collapse and revival of
level-3, level-2 and level-1 of the lambda system for Case-VI while Figs.9d-f
show that for level-3, level-2 and level-1 respectively for the vee system for
Case-IV with the same values of $\bar{n}$ and $\bar{m}$ as in Fig.7.