Adiabatic elimination in the presence of multiphoton transitions in atoms
inside a cavity

Prosenjit Maity 222Corresponding author<EMAIL_ADDRESS>orcid id:
0000-0001-5417-8654.

Department of Physics, Ramakrishna Mission Residential College,
Narendrapur, Kolkata-700103, India.

###### Abstract

Various approaches have been used in the literature for eliminating
nonresonant levels in atomic systems and deriving effective Hamiltonians.
Important among these are elimination techniques at the level of probability
amplitudes, operator techniques to project the dynamics on to the subspace of
resonant levels, Green’s function techniques, the James’ effective Hamiltonian
approach, etc. None of the previous approaches is suitable for deriving
effective Hamiltonians in intracavity situations. However, the James’ approach
does work in the case of only two-photon transitions in a cavity. A
generalization of the James’ approach works in the case of three-photon
transitions in a cavity, but only under Raman-like resonant conditions.
Another important approach for adiabatic elimination is based on an adaptation
of the Markov approximation well-known in the theory of system-bath
interactions. However, this approach has not been shown to work in intracavity
situations. In this paper, we present a method of adiabatic elimination for
atoms inside cavities in the presence of multiphoton transitions. We work in
the Heisenberg picture, and our approach has the advantage that it allows one
to derive effective Hamiltonians even when Raman-like resonance conditions do
not hold.

##  I Introduction

Various approaches for the adiabatic elimination of nonresonant levels in
atomic systems to derive effective Hamiltonians have been studied in the
literature. Several of these approaches are inspired by existing approaches in
the theory of system-bath interactions [[1, 2, 3, 4, 5, 6, 7, 8, 9]]. Such
effective Hamiltonians have played an important role in the study of atom-
field interactions, both in free space as well as intracavity situations
[[1]]. A well-known example is that a three-level lambda system, where an
effective Hamiltonian for the Raman transition is derived by adiabatically
eliminating the intermediate nonresonant level.

We shall briefly review here some of these approaches. The simplest ones are
based on adiabatic elimination of nonresonant levels in terms of the relevant
probability amplitudes. More sophisticated approaches employ a combination of
projection operator and Green’s function techniques. While these approaches
work very well in the case of few-level systems, they become cumbersome when a
large number of levels are involved. In particular, in intracavity situations,
where the cavity modes necessarily span an infinite dimensional Hilbert space
the above-mentioned methods are difficult to adopt. An important alternative
to the above-mentioned approaches was given James and Jerke [[10]]. They have
given a remarkably simple derivation of an effective Hamiltonian for a
multilevel system in the presence of two-photon processes, using second-order
perturbation theory. Although James and Jerke did not design their approach to
work in intracavity situations, it turns out that their approach as has been
shown by P. Z. Zhao et al [[11]], does allow one to derive an effective
Hamiltonian in intracavity situations as well.

W. Shao et al [[12]] have given a generalization of the James’ approach to the
case of three-photon transitions by using third-order perturbation theory.
Again, W. Shao et al did not indicate whether their approach would work in
intracavity situations. In the spirit of the analysis of Zhao et al, we have
shown in this paper, that their approach indeed works in intracavity
situations as well. However, it is possible to derive an effective Hamiltonian
only in the case of a Raman-like resonance condition. The generalized James’
approach fails in situations where such a resonant condition is not satisfied.
Recently, V. Paulisch et al [[8]] have presented a method of adiabatically
eliminating irrelevant nonresonantly coupled levels by employing a Markov
approximation on the integro-differential equation for the probability
amplitudes of the relevant levels. They have also discussed a way of going
beyond adiabatic elimination in terms of a hierarchy of approximations.

The aim of the present paper is to present an adaptation of the Markovian
approach to derive effective Hamiltonians for atoms interacting with fields in
the presence of multiphoton transitions and in cavities. An important
difference between the approach of Paulisch et al and ours is that we work in
the Heisenberg picture. This allows us to treat the dynamics of the atoms as
well as cavity modes in a similar manner. The effective Hamiltonians that can
be derived using our approach do not require any conditions such as Raman-like
resonance to be satisfied.

This paper is organized as follows. In Sec. II, we briefly review some of the
approaches for deriving effective Hamiltonians which are relevant in our
context. At first, we discuss the simplest one among them based on the
analysis of Brion et al [[7]] in the context of Raman transition in a three-
level lambda system interacting with classical laser fields. Thereafter, we
review more sophisticated approach, a combination of projection operator and
Green’s function techniques. Subsequently, we revisit the James’ approach and
its generalization [[10, 12]] and discuss their merits and demerits in the
case of multiphoton transitions in intracavity situations. Furthermore, we
review the integro-differential equation approach of Paulisch et al [[8]] and
discuss its adaptability in the context of intracavity situations. In Sec.
III, we introduce our approach i.e. adiabatic elimination in the Heisenberg
picture using Markov approximation for multiphoton transitions inside a
cavity. We demonstrate it with a few examples and compare the effective
Hamiltonians which are derived using James’ and generalized James’ techniques
in the subsequent sections. Section IV concludes the paper with a brief
summary and some discussion.

##  II A brief review of adiabatic elimination methods

###  II.A Simplest technique of adiabatic elimination using Brion et al
approach

To begin with, we adopt a simple example to describe the adiabatic elimination
technique. We look at an atomic lambda system [[7]] with two lower states
$\ket{g}$, $\ket{2}$ and an excited state $\ket{1}$ connected by two off-
resonance lasers. The transition $\ket{g}\leftrightarrow\ket{1}$ is governed
by the classical laser pulse having Rabi frequency $\Omega_{1}(t)$ and the
detuning $\Delta_{1}$, whereas the transition $\ket{1}\leftrightarrow\ket{2}$
is facilitated by the Rabi frequency $\Omega_{2}(t)$ and the detuning
$\Delta_{2}$, as shown in the Fig. (1). The detunings are read as:
$\Delta_{1}=(\omega_{1}-\omega_{g})-\omega_{g}^{(l)};$ and
$\Delta_{2}=(\omega_{1}-\omega_{2})-\omega_{1}^{(l)}$, where
$\omega_{g}^{(l)}$ and $\omega_{1}^{(l)}$ are frequencies of the laser pulses.
The Hamiltonian describing the system can be written as
$(\hbar=1\;\text{herein})$

Figure 1: Energy-level diagram of the three-level atom in lambda
configuration. The detuning reads:
$\Delta_{1}=\omega_{1g}-\omega_{g}^{(1)};\;\Delta_{2}=\omega_{12}-\omega_{1}^{(l)}$.
The transition between $\ket{g}$ and $\ket{1}$ is governed by Rabi frequency
$\Omega_{1}(t)$ whereas the transition between $\ket{1}$ and $\ket{2}$ is
controlled by the Rabi frequency $\Omega_{2}(t)$.

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\omega_{g}\sigma_{gg}+\omega_{1}\sigma_{11}+\omega_{2}\sigma_{22}+\Omega_{1}\exp{(-i\omega_{g}^{(l)}t)}\;\sigma_{1g}+\Omega_{2}\exp{(-i\omega_{1}^{(l)}t)}\;\sigma_{12}+\text{H.c}.,$
(1)

where the atomic transition operators $\sigma_{ij}\equiv\ket{i}\bra{j}$. In
the rotating frame defined by the transformation
$\ket{\psi}\rightarrow\ket{\chi}=\exp{(i\zeta t)}\ket{\psi}$, where

$\zeta=\begin{pmatrix}\frac{\Delta}{2}&0&0\\\
0&\omega_{g}-\omega_{1}-\frac{\Delta}{2}&0\\\
0&0&\omega_{g}-\bar{\Delta}\end{pmatrix},$ (2)

and carrying out the rotating wave approximation, the Hamiltonian (1) in the
unperturbed energy basis $\\{\ket{g},\ket{2},\ket{1}\\}$ reads as

$\mathcal{H}^{{}^{\prime}}=\begin{pmatrix}-\frac{\Delta}{2}&0&\frac{\Omega_{1}^{*}}{2}\\\
0&\frac{\Delta}{2}&\frac{\Omega^{*}_{2}}{2}\\\
\frac{\Omega_{1}}{2}&\frac{\Omega_{2}}{2}&\bar{\Delta}\end{pmatrix},$ (3)

where the abbreviations: $\Delta=\Delta_{1}-\Delta_{2}$ and
$\bar{\Delta}=\frac{1}{2}(\Delta_{1}+\Delta_{2})$.

To establish a two-photon Raman transition between the states $\ket{g}$ and
$\ket{2}$, we adiabatically eliminate the nonresonant level $\ket{1}$ under
the assumption of large detuning condition i.e.
$|\Delta|,|\bar{\Delta}|\gg\Omega_{1},\Omega_{2}$. The most common method for
removing the excited state by invoking the Schrödinger equation written for
the state $\ket{\psi}=C_{g}\ket{g}+C_{1}\ket{1}+C_{2}\ket{2}$,

$\displaystyle\begin{cases}i\dot{C}_{g}(t)\;\;=\;\;-\frac{\Delta}{2}C_{g}+\frac{\Omega^{*}_{1}}{2}C_{1}\\\
i\dot{C}_{1}(t)\;\;=\;\;\frac{\Omega_{2}}{2}C_{g}+\frac{\Omega_{1}}{2}C_{2}+\bar{\Delta}C_{1},\\\
i\dot{C}_{2}(t)\;\;=\;\;\frac{\Delta}{2}C_{2}+\frac{\Omega^{*}_{2}}{2}C_{1}.\end{cases}$
(4)

Since under the large detuning condition, the excited state $\ket{1}$ is
nonresonantly coupled to the states $\ket{g}$ and $\ket{2}$, we can claim
$\dot{C}_{1}(t)=0$, which implies

$C_{1}=-\frac{\Omega_{1}}{2\bar{\Delta}}C_{g}-\frac{\Omega_{2}}{2\bar{\Delta}}C_{2}.$
(5)

Inserting the result in (5) into the dynamical equations of $C_{g}$ and
$C_{2}$ in (4), we have

$i\begin{pmatrix}\dot{C}_{g}\\\
\dot{C}_{2}\end{pmatrix}=\mathcal{H}_{\text{eff}}\begin{pmatrix}C_{g}\\\
C_{2}\end{pmatrix},$ (6)

where the effective Hamiltonian describing the two-photon Raman transition
reads,

$\mathcal{H}_{\text{eff}}\;\;=\;\;-\begin{pmatrix}\frac{\Delta}{2}+\frac{|\Omega_{1}|^{2}}{4\bar{\Delta}}&\frac{\Omega_{1}^{*}\Omega_{2}}{4\bar{\Delta}}\\\
\frac{\Omega_{1}\Omega_{2}^{*}}{4\bar{\Delta}}&-\frac{\Delta}{2}+\frac{|\Omega_{2}|^{2}}{4\bar{\Delta}}\end{pmatrix}.$
(7)

###  II.B  Green’s function and projection operator formalism

The Green’s function and projection operator formalism was first introduced in
the Refs. [[13, 14]] and was extensively presented in the Ref. [[15]]. For a
given system of Hamiltonian $H$ which is consisting of two parts, the
unperturbed part $H_{0}$ and a small perturbed part $H^{{}^{\prime}}$, one can
define the Green’s function $\mathcal{G}(\xi)$, $\xi\in\mathbb{C}$ as follows:

$\mathcal{G}(\xi)=\frac{1}{\xi-H}.$ (8)

Let $\mathcal{P}$ be projector onto a subspace $\mathscr{E}_{0}$ subtended by
an ensemble unperturbed states
$\\{\ket{\phi_{1},\ket{\phi_{2}},....,\ket{\phi_{N}}}\\}$ of $H_{0}$. Assuming
$\ket{\phi_{1}},\ket{\phi_{2}},....,\ket{\phi_{N}}$ are orthonormal, the
projector onto the subspace $\mathscr{E}_{0}$ can then be written as,

$\mathcal{P}=\ket{\phi_{1}}\bra{\phi_{1}}+\ket{\phi_{2}}\bra{\phi_{2}}+...+\ket{\phi_{N}}\bra{\phi_{N}},$
(9)

which satisfies the relations:
$\mathcal{P}=\mathcal{P}^{\dagger},\;\mathcal{P}^{2}=\mathcal{P}$. Let the
supplementary subspace of $\mathscr{E}_{0}$ is $\mathscr{C}_{0}$ and the
corresponding projector $Q$ is defined as $\mathcal{Q}=1-\mathcal{P}$, which
also obeys the relations:
$\mathcal{Q}=\mathcal{Q}^{\dagger},\;\mathcal{Q}^{2}=\mathcal{Q}$. Following
this, one can derive the expression for projection of $G(\xi)$ in (8) as

$\mathcal{P}\mathcal{G}(\xi)\mathcal{P}=\frac{\mathcal{P}}{\xi-\mathcal{P}H_{0}\mathcal{P}-\mathcal{P}\mathcal{R}(\xi)\mathcal{P}}\;,$
(10)

where the displacement operator $\mathcal{R}(\xi)$ is defined by

$\mathcal{R}(\xi)=H^{{}^{\prime}}+H^{{}^{\prime}}\frac{\mathcal{Q}}{\xi-\mathcal{Q}H_{0}\mathcal{Q}-\mathcal{Q}H^{{}^{\prime}}\mathcal{Q}}H^{{}^{\prime}}.$
(11)

Based on this construction, the authors in the Ref. [[7]] have shown the
effective Hamiltonian for the lambda system discussed above (Fig. 1). In this
case,

$H_{0}=\begin{pmatrix}-\frac{\Delta}{2}&0&0\\\ 0&\frac{\Delta}{2}&0\\\
0&0&\bar{\Delta}\end{pmatrix},\quad
H^{{}^{\prime}}=\frac{1}{2}\begin{pmatrix}0&0&\Omega^{\ast}_{1}\\\
0&0&\Omega^{\ast}_{2}\\\ \Omega_{1}&\Omega_{2}&0\end{pmatrix}.$ (12)

Here, the projectors are read: $\mathcal{P}=\ket{g}\bra{g}+\ket{2}\bra{2}$ and
$\mathcal{Q}=\ket{1}\bra{1}$. After little algebra, they have shown the
effective Hamiltonian
$H_{\text{eff}}=\mathcal{P}H_{0}\mathcal{P}+\mathcal{P}\mathcal{R}(0)\mathcal{P}$,
which describes the reduced dynamics of the system in terms of two-photon
Raman transition.

Although, this method has been utilized to study an effective operator
formalism for open quantum systems [[16, 17]], however, to our knowledge, this
approach does not include the cavity mode where the applied laser fields are
treated classically in a more general situation. The pole analysis will be
cumbersome task when the cavity modes will be incorporated in this situation.
As a result, extending this approach to eliminate nonresonant levels in the
case of a multiphoton transition in an atom will be a difficult undertaking.

###  II.C Integro-differential equation approach of Paulisch et al

In this approach, the authors in the Ref. [[8]] have considered a multi-level
system where the atomic transition between the successive levels happens under
the application of classical laser fields. They have used an integro-
differential equation approach for the probability amplitudes for the atomic
levels which are resonantly coupled to each other. The integro-differential
equation is then solved under Markov approximation to adiabatically eliminate
the nonresonantly coupled states. For instance, in case of aforesaid lambda
system discussed in subsection II.B, the effective Hamiltonian in the
unperturbed basis $\\{\ket{g},\ket{2}\\}$ is shown to be

$\mathcal{H}_{\text{eff}}=-\dfrac{1}{2}\begin{pmatrix}\Delta+\frac{|\Omega_{1}|^{2}}{2\bar{\Delta}}&\frac{\Omega_{1}^{\ast}\Omega_{2}^{\ast}}{2\Delta}\\\
\frac{\Omega_{1}\Omega_{2}}{2\Delta}&-\Delta+\frac{|\Omega_{1}|^{2}}{2\bar{\Delta}}\end{pmatrix}.$
(13)

Despite the fact that this method has been used to investigate the effective
dynamics in squeezed-light-enhanced atom interferometry [[18]], adiabatic
elimination for strong field light matter coupling [[19]], third-order
diffraction in Raman scattering [[20]], etc., it does not include the cavity
modes. In this approach, they have worked in interaction picture and to make
the Hamiltonian time independent, a suitable transformation has been utilized,
which will be difficult to obtain when the cavity interactions are present.
Furthermore, in the presence of cavity modes, the Schrödinger equation for
probability amplitudes for the states becomes complicated. However, we
demonstrate how to construct an effective Hamiltonian in this general scenario
in our technique.

###  II.D  James’ and generalized James’ effective Hamiltonian approaches

In this subsection, we discuss the merits and demerits of the James’ and
generalized James’ effective Hamiltonian methods [[10, 12]] briefly. The
authors in the Ref. [[10]] have considered the time-averaged dynamics of the
system to discard the various highly oscillating terms present in the
effective Hamiltonian. Keeping the terms up to second order of the interacting
Hamiltonian in the time-ordered expansion of the evolution operator, they have
shown the effective Hamiltonian as

$\mathcal{H}_{\text{eff}}^{(2)}(t)\;\;=\;\;\sum\limits_{\alpha,\beta=1}^{N}\frac{1}{\bar{\omega}_{{}_{\alpha\beta}}}\big{[}\Lambda_{\alpha}^{\dagger},\Lambda_{\beta}\big{]}\exp(i(\omega_{\alpha}-\omega_{\beta})t),$
(14)

where $\Lambda_{\alpha}$ is the relevant atomic transition operators, $N$ is
the total number of different harmonic term and
$\bar{\omega}_{{}_{\alpha\beta}}$ is harmonic average of $\omega_{\alpha}$ and
$\omega_{\beta}$ i.e.
$\frac{1}{\bar{\omega}_{{}_{\alpha\beta}}}\;=\;\frac{1}{2}\left(\frac{1}{\omega_{\alpha}}+\frac{1}{\omega_{\beta}}\right)$.
Although this approach explains two-photon Raman transition in a three-level
system, quantum AC Stark shits [[21]], single-qubit and two-qubit holonomic
gates [[11]] etc., it does not demonstrate the phenomena related to three-
photon or higher order transition processes.

Later on, adopting James’ idea, W. Shao et al [[12]] have extended the James’
effective Hamiltonian method up to third-order in time-ordered expansion of
the evolution operator. By neglecting the rapidly oscillating terms under
rotating wave approximation, they have derived the effective Hamiltonian which
reads

$\displaystyle\mathcal{H}_{\text{eff}}^{(3)}(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{\alpha,\beta,\gamma}\big{\lgroup}\frac{1}{\omega_{\gamma}(\omega_{\gamma}-\omega_{\beta})}\big{[}\Lambda_{\alpha}\Lambda_{\beta}\Lambda_{\gamma}\exp{(i(\omega_{\alpha}-\omega_{\beta}+\omega_{\gamma})t)}+\Lambda_{\alpha}^{\dagger}\Lambda_{\beta}\Lambda_{\gamma}^{\dagger}\exp{(i(\omega_{\alpha}-\omega_{\beta}+\omega_{\gamma})t)}$
(15) $\displaystyle+$
$\displaystyle\Lambda_{\alpha}\Lambda_{\beta}\Lambda_{\gamma}^{\dagger}\exp{(i(\omega_{\alpha}+\omega_{\beta}-\omega_{\gamma})t)}+\Lambda_{\alpha}^{\dagger}\Lambda_{\beta}^{\dagger}\Lambda_{\gamma}\exp{(i(-\omega_{\alpha}-\omega_{\beta}+\omega_{\gamma})t)}\big{]}+\frac{1}{\omega_{\gamma}(\omega_{\gamma}+\omega_{\beta})}\times$
$\displaystyle\times$
$\displaystyle\big{[}\Lambda_{\alpha}^{\dagger}\Lambda_{\beta}\Lambda_{\gamma}\exp{(i(-\omega_{\alpha}+\omega_{\beta}+\omega_{\gamma})t)}+\Lambda_{\alpha}\Lambda_{\beta}^{\dagger}\Lambda_{\gamma}^{\dagger}\exp{(i(\omega_{\alpha}-\omega_{\beta}-\omega_{\gamma})t)}\big{]}\big{\rgroup}.$

It is worth noting that to prove the Hermiticity of
$\mathcal{H}_{\text{eff}}^{(3)}(t)$ (15), the authors in the Ref. [[12]] have
introduced the condition that all of the frequencies $\omega_{m}$ are
distinct, and the algebraic sum of any three frequencies is zero i.e.

$\omega_{\alpha}+\omega_{\beta}-\omega_{\gamma}=0.$ (16)

We observe that this condition of Hermiticity actually leads to the
transitions at resonance which we shall demonstrate in Sec. III.B in detail.
Because of this condition of Hermiticity, the generalized James’ effective
Hamiltonian method does not explain off-resonant transitions. However, we
propose an approach based on the adiabatic elimination under Markov
approximation which includes the transitions both at off-resonance and as well
as at resonance.

##  III  Adiabatic elimination in the Heisenberg picture using Markov
approximation

As we have pointed out in the previous sections, the adiabatic elimination
techniques already existing in the literature work very well in the case of
few level systems interacting with classical laser fields. However, these are
either not suitable or are too cumbersome to deal with multiphoton resonances
in multilevel systems in the presence of interactions with cavity modes.

We propose here an alternative approach to the problem of deriving effective
Hamiltonians in the case of atomic systems interacting with both the classical
laser fields as well as quantized cavity fields, by adiabatically eliminating
nonresonantly coupled n-photon intermediate transitions. We use an adaptation
of the Markovian approach as discussed earlier by Paulisch et al [[8]]. In
contrast to the previous approaches, we work entirely in the Heisenberg
picture. While the cavity modes are represented by the corresponding creation
and annihilation operators ($a^{\dagger}$, $a$), the atomic systems are
represented by appropriate transition operators such as
$\sigma_{\imath\jmath}\equiv\ket{\imath}\bra{\jmath}$, where $\ket{\imath}$
and $\ket{\jmath}$ are some unperturbed energy states. For the sake of
illustration, let us consider the simple case of a transition from level
$\ket{g}$ to level $\ket{N-1}$ by absorption of photons from laser fields with
frequencies, $\omega_{\jmath}(\jmath=1,2,....,N-1)$. The levels $\ket{N-1}$
and $\ket{N}$ are connected by a one-photon transition induced by single
cavity mode of frequency $\omega_{c}$ and Rabi frequency $\eta$. The
Hamiltonian describing this system is given by

Figure 2:  Energy-level diagram for n-photon transitions in atoms inside a
microwave cavity under the application of classical laser pulses and the
cavity mode.

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H^{{}^{\prime}},$ (17)

where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N}E_{\jmath}\sigma_{\jmath\jmath}+E_{g}\sigma_{gg}+\omega_{c}a^{\dagger}a,$
(18)

is the free atom plus free field Hamiltonian and

$\displaystyle H^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\big{\\{}\Omega_{\jmath+1}(t)\exp{(-i\omega_{\jmath+1}t)}\sigma_{\jmath+1,\jmath}+\Omega_{\jmath+1}^{\ast}(t)\exp{(i\omega_{\jmath+1}t)}\sigma_{\jmath,\jmath+1}\big{\\}}+\big{(}\Omega_{{}_{1}}(t)\exp{(-i\omega_{1}t)}\sigma_{{}_{1g}}$
(19) $\displaystyle+$
$\displaystyle\Omega_{{}_{1}}^{\ast}(t)\exp{(i\omega_{{}_{1}}t)}\sigma_{{}_{g1}}\big{)}+\left(\eta
a\sigma_{{}_{N,N-1}}+\eta a^{\dagger}\sigma_{{}_{N-1,N}}\right)$

is the interaction Hamiltonian. For simplicity, we assume the energy of the
level $\ket{g}$, $E_{g}=0$, which implies $E_{\jmath}=\hbar\omega_{\jmath g}$
and $E_{{}_{N}}=\hbar\omega_{{}_{Ng}}$. We further assume that
$E_{\jmath}<E_{\jmath+1}$ and $E_{{}_{N-1}}<E_{{}_{N}}$. In the interaction
picture defined with $H_{0}$ (18), the reduced form of the Hamiltonian (17)
can be written as ($\hbar=1$ herein)

$\displaystyle H_{{}_{I}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\big{\\{}\Omega_{\jmath+1}(t)\exp{(i\Delta_{\jmath+1}t)}\sigma_{\jmath+1,\jmath}+\Omega_{\jmath+1}^{\ast}(t)\exp{(-i\Delta_{\jmath+1}t)}\sigma_{\jmath,\jmath+1}\big{\\}}+\big{(}\Omega_{1}(t)\exp{(i\Delta_{1}t)}\sigma_{{}_{1g}}$
(20) $\displaystyle+$
$\displaystyle\Omega_{1}^{\ast}(t)\exp{(-i\Delta_{1}t)}\sigma_{{}_{g1}}\big{)}+\left(\eta\exp{(i\Delta_{{}_{N}}t)}a\sigma_{{}_{N,N-1}}+\eta\exp{(-i\Delta_{{}_{N}}t)}a^{\dagger}\sigma_{{}_{N-1,N}}\right),$

where the detunings,
$\Delta_{\jmath+1}=\left(\omega_{\jmath+1,g}-\omega_{\jmath
g}\right)-\omega_{\jmath+1}$; $\Delta_{1}=\omega_{1g}-\omega_{1}$ and
$\Delta_{{}_{N}}=\omega_{{}_{Ng}}-\omega_{c}$.

Using the Hamiltonian (20), we obtain the Heisenberg equations for the
operators $\sigma_{\jmath,\jmath+1}$, $\sigma_{{}_{g1}}$ and
$\sigma_{{}_{N-1,N}}$

$\displaystyle i\dot{\sigma}_{{\jmath,\jmath+1}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\imath=1}^{N-2}\big{\\{}\Omega_{\imath+1}(t)\exp{(i\Delta_{\imath+1}t)}\left(\sigma_{\jmath\imath}\delta_{\jmath+1,\imath+1}-\sigma_{\imath+1,\jmath+1}\delta_{\imath\jmath}\right)+\Omega_{\imath+1}^{\ast}(t)\exp{(-i\Delta_{\imath+1}t)}\qquad\quad$
$\displaystyle\times\left(\sigma_{\jmath,\imath+1}\delta_{\jmath+1,\imath}-\sigma_{\imath,\jmath+1}\delta_{\imath+1,\jmath}\right)\big{\\}}-\Omega_{1}^{\ast}(t)\exp{(-i\Delta_{1}t)}\sigma_{g,\jmath+1}\delta_{1\jmath}\qquad\quad$
(21)
$\displaystyle+\;\;\eta\exp{(-i\Delta_{{}_{N}}t)}a^{\dagger}\sigma_{{}_{\jmath
N}}\delta_{\jmath+1,N-1},$ $\displaystyle i\dot{\sigma}_{{}_{g1}}$
$\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\Omega_{\jmath+1}^{\ast}(t)\exp{(-i\Delta_{\jmath+1}t)}\sigma_{g,\jmath+1}\delta_{1\jmath}+\Omega_{1}(t)\exp{(i\Delta_{1}t)}(\sigma_{gg}-\sigma_{11}),\qquad\qquad\qquad\;\;\;$
(22) $\displaystyle i\dot{\sigma}_{{}_{N-1,N}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}-\;\Omega_{\jmath+1}^{\ast}(t)\exp{(-i\Delta_{\jmath+1}t)}\sigma_{\jmath
N}\delta_{\jmath+1,N-1}+\eta\exp{(-i\Delta_{{}_{N}}t)}a(\sigma_{{}_{N-1,N-1}}-\sigma_{{}_{NN}}).\qquad\;\;\;\;$
(23)

Now further assuming the detunings $\Delta_{\jmath}\;(\jmath=1,2,...,N)$ to be
large compare to $|\Omega_{\jmath}(t)|$ and $\eta$, levels
$\ket{1},\ket{2},.....,\ket{N-1}$ may be adiabatically eliminated. To this
end, for solving the Eqs. (21-23), we make use of Markov approximation which
allows us to consider
$\Omega_{\jmath}(t^{{}^{\prime}})\approx\Omega_{\jmath}(t)$. Therefore from
Eq.(21), we have

$\displaystyle\sigma_{\jmath,\jmath+1}$ $\displaystyle=$
$\displaystyle\sum\limits_{\imath=1}^{N-2}\big{\\{}-\frac{\Omega_{\imath+1}(t)}{\Delta_{\imath+1}}\exp{(i\Delta_{\imath+1}t)}\left(\sigma_{\jmath\imath}\delta_{\jmath+1,\imath+1}-\sigma_{\imath+1,\jmath+1}\delta_{\imath\jmath}\right)+\frac{\Omega_{\imath+1}^{\ast}(t)}{\Delta_{\imath+1}}\exp{(-i\Delta_{\imath+1}t)}$
(24)
$\displaystyle\times\;\left(\sigma_{\jmath,\imath+1}\delta_{\jmath+1,\imath}-\sigma_{\imath,\jmath+1}\delta_{\imath+1,\jmath}\right)\big{\\}}-\frac{\Omega_{1}^{\ast}(t)}{\Delta_{1}}\exp{(-i\Delta_{1}t)}\sigma_{g,\jmath+1}\delta_{1\jmath}$
$\displaystyle+\;\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}a^{\dagger}\sigma_{{}_{\jmath
N}}\delta_{\jmath+1,N-1},$

Following the above approximation, from the Eqs.(22) and (23) we obtain,

$\displaystyle\sigma_{{}_{g1}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\;\frac{\Omega_{\jmath+1}^{\ast}(t)}{\Delta_{\jmath+1}}\exp{(-i\Delta_{\jmath+1}t)}\sigma_{g,\jmath+1}\delta_{1\jmath}-\frac{\Omega_{1}(t)}{\Delta_{1}}\exp{(i\Delta_{1}t)}(\sigma_{gg}-\sigma_{11}),\qquad\qquad\qquad\;\;\;$
(25) $\displaystyle\sigma_{{}_{N-1,N}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}-\;\frac{\Omega_{\jmath+1}^{\ast}(t)}{\Delta_{\jmath+1}}\exp{(-i\Delta_{\jmath+1}t)}\sigma_{{}_{\jmath
N}}\delta_{\jmath+1,N-1}+\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}a(\sigma_{{}_{N-1,N-1}}-\sigma_{{}_{NN}}),\quad\quad\;$
(26)

respectively. Now assuming that Stark shifts resulting from the diagonal terms
in (24-26) are small we have

$\displaystyle\sigma_{\jmath,\jmath+1}$ $\displaystyle\simeq$
$\displaystyle\sum\limits_{\imath=1}^{N-2}\bigg{\\{}\frac{\Omega_{\imath+1}^{\ast}(t)}{\Delta_{\imath+1}}\exp{(-i\Delta_{\imath+1}t)}\left(\sigma_{\jmath,\imath+1}\delta_{\jmath+1,\imath}-\sigma_{\imath,\jmath+1}\delta_{\imath+1,\jmath}\right)\bigg{\\}}$
(27) $\displaystyle-$
$\displaystyle\frac{\Omega_{1}^{\ast}(t)}{\Delta_{1}}\exp{(-i\Delta_{1}t)}\sigma_{g,\jmath+1}\delta_{1\jmath}+\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}a^{\dagger}\sigma_{{}_{\jmath
N}}\delta_{\jmath+1,N-1},\qquad\qquad\qquad\qquad$
$\displaystyle\sigma_{{}_{g1}}$ $\displaystyle\simeq$
$\displaystyle\frac{\Omega_{2}^{\ast}(t)}{\Delta_{2}}\exp{(-i\Delta_{2}t)}\sigma_{{}_{g2}},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;$
(28) $\displaystyle\sigma_{{}_{N-1,N}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\Omega_{N}^{\ast}(t)}{\Delta_{N}}\exp{(-i\Delta_{N}t)}\sigma_{{}_{N-1,N}}.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;$
(29)

Inserting these results into the equation (20), we have

$\displaystyle H_{{}_{I}}^{(1)}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-1}\big{\\{}\Omega_{\jmath+1}^{\ast}(t)\Omega_{\jmath+2}^{\ast}(t)\left(\frac{1}{\Delta_{\jmath+2}}-\frac{1}{\Delta_{\jmath+1}}\right)\exp{\left(-i(\Delta_{\jmath+1}+\Delta_{\jmath+2})t\right)}\sigma_{\jmath,\jmath+2}+H.c.\big{\\}}$
(30) $\displaystyle+$
$\displaystyle\Omega_{1}^{\ast}(t)\Omega_{2}^{\ast}(t)\left(\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}}\right)\exp{\left(-i(\Delta_{1}+\Delta_{2})t\right)}\sigma_{g2}$
$\displaystyle+$
$\displaystyle\eta\;\Omega_{{}_{N-1}}^{\ast}(t)\left(\frac{1}{\Delta_{{}_{N}}}-\frac{1}{\Delta_{{}_{N-1}}}\right)\exp{\left(-i(\Delta_{{}_{N-1}}+\Delta_{{}_{N}})t\right)}\;a^{\dagger}\sigma_{{}_{N-2,N}}+H.c.$

At this point the Hamiltonian has three parts, the first and second parts
correspond to the effective two-photon transitions between
$\ket{\jmath}\leftrightarrow\ket{\jmath+2}$ and
$\ket{g}\leftrightarrow\ket{2}$ respectively while the third one gives the
usual Raman transitions between transitions between the states $\ket{N-1}$ and
$\ket{N}$. To pursue the adiabatic elimination of the intermediate levels, we
further need the equations for $\sigma_{\jmath,\jmath+2}$, $\sigma_{g2}$ and
$\sigma_{{}_{N-1,N}}$.

Following the above procedure we find

$\displaystyle\sigma_{\jmath,\jmath+2}$ $\displaystyle=$
$\displaystyle\sum\limits_{\imath=1}^{N-2}\bigg{\\{}-\frac{\Omega_{\imath+1}(t)}{\Delta_{\imath+1}}\exp{(i\Delta_{\imath+1}t)}\left(\sigma_{\jmath\imath}\delta_{\jmath+2,\imath+1}-\sigma_{\imath+1,\jmath+2}\;\delta_{\imath\jmath}\right)+\frac{\Omega_{\imath+1}^{\ast}(t)}{\Delta_{\imath+1}}\exp{(-i\Delta_{\imath+1}t)}$
(31)
$\displaystyle\times\;\left(\sigma_{\jmath,\imath+2}\;\delta_{\jmath+2,\imath}-\sigma_{\imath,\jmath+2}\;\delta_{\imath+1,\jmath}\right)\bigg{\\}}-\frac{\Omega_{1}^{\ast}(t)}{\Delta_{1}}\exp{(-i\Delta_{1}t)}\sigma_{g,\jmath+2}\;\delta_{1\jmath}$
$\displaystyle+\;\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}\;a^{\dagger}\sigma_{{}_{\jmath
N}}\delta_{\jmath+2,N-1},$ $\displaystyle\sigma_{{}_{g2}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}-\frac{\Omega_{\jmath+1}(t)}{\Delta_{\jmath+1}}\exp{(i\Delta_{\jmath+1}t)}\sigma_{g\jmath}\;\delta_{\jmath+1,2}+\frac{\Omega_{\jmath+1}^{\ast}(t)}{\Delta_{\jmath+1}}\exp{(i\Delta_{\jmath+1}t)}\sigma_{g,\jmath+1}\delta_{2\jmath}\qquad\qquad\;\;\;\;$
(32) $\displaystyle+$
$\displaystyle\frac{\Omega_{1}(t)}{\Delta_{1}}\exp(i\Delta_{1}t)\sigma_{{}_{12}},$
$\displaystyle\sigma_{{}_{N-2,N}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\bigg{\\{}-\;\frac{\Omega_{\jmath+1}(t)}{\Delta_{\jmath+1}}\exp{(i\Delta_{\jmath+1}t)}\left(\sigma_{{}_{N-2,\jmath}}\;\delta_{N,\jmath+1}-\sigma_{\jmath+1,N}\;\delta_{\jmath,N-2}\right)\qquad\qquad\qquad\qquad\quad$
(33)
$\displaystyle+\;\frac{\Omega_{\jmath+1}^{\ast}(t)}{\Delta_{\jmath+1}}\exp{(-i\Delta_{\jmath+1}t)}\left(\sigma_{{}_{N-2,\jmath+1}}\;\delta_{N\jmath}-\sigma_{\jmath
N}\;\delta_{\jmath+1,N-2}\right)\bigg{\\}}$
$\displaystyle+\;\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}\;a^{\dagger}\sigma_{{}_{N-2,N-1}}.\qquad\qquad\qquad\qquad\qquad\qquad$

As we have already assumed that the detunings $\Delta_{\jmath}$ are large,
therefore neglecting the terms on the order of
$(\Delta_{\imath}\Delta_{\jmath})^{-1}\;(\imath,\jmath=1,2,....,N)$ in (31
-33) we have

$\displaystyle\sigma_{\jmath,\jmath+2}$ $\displaystyle\simeq$
$\displaystyle\sum\limits_{\imath=1}^{N-1}\bigg{\\{}\frac{\Omega_{\imath+1}^{\ast}(t)}{\Delta_{\imath+1}}\exp{(-i\Delta_{\imath+1}t)}\left(\sigma_{\jmath,\imath+1}\delta_{\jmath+2,\imath}-\sigma_{\imath,\jmath+2}\delta_{\imath+1,\jmath}\right)\bigg{\\}}$
(34) $\displaystyle-$
$\displaystyle\frac{\Omega_{1}^{\ast}(t)}{\Delta_{1}}\exp{(-i\Delta_{1}t)}\sigma_{g,\jmath+2}\;\delta_{1\jmath}+\frac{\eta}{\Delta_{{}_{N}}}\exp{(-i\Delta_{{}_{N}}t)}\;a^{\dagger}\sigma_{{}_{\jmath
N}}\delta_{\jmath+2,N-1},\qquad\qquad\qquad\qquad\qquad\qquad\;\ $
$\displaystyle\sigma_{{}_{g2}}$ $\displaystyle\simeq$
$\displaystyle\frac{\Omega_{3}^{\ast}(t)}{\Delta_{3}}\exp{(-i\Delta_{3}t)}\sigma_{{}_{g3}},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;$
(35) $\displaystyle\sigma_{{}_{N-2,N}}$ $\displaystyle\simeq$
$\displaystyle-\frac{\Omega_{N-2}^{\ast}(t)}{\Delta_{{}_{N-2}}}\exp{(-i\Delta_{{}_{N-2}}t)}\sigma_{{}_{N-3,N}}.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;\;$
(36)

Inserting these results into the Eq. (30), we obtain

$\displaystyle H_{{}_{I}}^{(2)}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\bigg{\\{}\Omega_{\jmath+1}^{\ast}(t)\Omega_{\jmath+2}^{\ast}(t)\Omega_{\jmath+3}^{\ast}(t)\left(\frac{1}{\Delta_{\jmath+3}}\left(\frac{1}{\Delta_{\jmath+2}}-\frac{1}{\Delta_{\jmath+1}}\right)-\frac{1}{\Delta_{\jmath+1}}\left(\frac{1}{\Delta_{\jmath+3}}-\frac{1}{\Delta_{\jmath+2}}\right)\right)$
(37) $\displaystyle\times$
$\displaystyle\exp{\left(-i\left(\Delta_{\jmath+1}+\Delta_{\jmath+2}+\Delta_{\jmath+3}\right)t\right)}\sigma_{\jmath,\jmath+3}+\text{H.c.}\bigg{\\}}$
$\displaystyle+$
$\displaystyle\Omega_{1}^{\ast}(t)\Omega_{2}^{\ast}(t)\Omega_{3}^{\ast}(t)\left(\frac{1}{\Delta_{3}}\left(\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}}\right)-\frac{1}{\Delta_{1}}\left(\frac{1}{\Delta_{3}}-\frac{1}{\Delta_{2}}\right)\right)\times\exp{\left(-i\left(\Delta_{1}+\Delta_{2}+\Delta_{3}\right)t\right)}\sigma_{{}_{g3}}$
$\displaystyle+$
$\displaystyle\eta\;\Omega_{N-1}^{\ast}(t)\Omega_{N-2}^{\ast}(t)\left(\frac{1}{\Delta_{{}_{N}}}\left(\frac{1}{\Delta_{{}_{N-1}}}-\frac{1}{\Delta_{{}_{N-2}}}\right)-\frac{1}{\Delta_{{}_{N-2}}}\left(\frac{1}{\Delta_{{}_{N}}}-\frac{1}{\Delta_{{}_{N-1}}}\right)\right)$
$\displaystyle\times$
$\displaystyle\exp{\left(-i\left(\Delta_{{}_{N-2}}+\Delta_{{}_{N-1}}+\Delta_{{}_{N}}\right)t\right)}\;a^{\dagger}\sigma_{{}_{N-3,N}}+\text{H.c.}$

As above, the first two terms in the Hamiltonian represent an effective three-
photon transitions between $\ket{\jmath}\leftrightarrow\ket{\jmath+3}$ and
$\ket{g}\leftrightarrow\ket{3}$ while the last one describes the three-photon
hyper Raman transitions between $\ket{N-3}$ and $\ket{N}$. From this point, it
is clear that to eliminate next levels we need to calculate the equations of
motion for the operators $\sigma_{\jmath,\jmath+3}$, $\sigma_{{}_{g3}}$ and
$\sigma_{{}_{N-3,N}}$.

Therefore, following the above procedure and based on the assumption adopted
in case of Eqs. (34-36), one can obtain the approximate solutions for
$\sigma_{\jmath,\jmath+3}$, $\sigma_{g3}$ and $\sigma_{{}_{N-3,N}}$. Inserting
these results into (37) we have

$\displaystyle
H_{{}_{I}}^{(3)}\;\;=\;\;\sum\limits_{\jmath=1}^{N-2}\bigg{\\{}\bar{\Omega}_{\jmath}\Bigg{(}\frac{1}{\Delta_{\jmath+4}}\left(\frac{1}{\Delta_{\jmath+3}}\left(\frac{1}{\Delta_{\jmath+2}}-\frac{1}{\Delta_{\jmath+1}}\right)-\frac{1}{\Delta_{\jmath+1}}\left(\frac{1}{\Delta_{\jmath+3}}-\frac{1}{\Delta_{\jmath+2}}\right)\right)-\frac{1}{\Delta_{\jmath+1}}\qquad\qquad\;\;\;$
$\displaystyle\times\left(\frac{1}{\Delta_{\jmath+4}}\left(\frac{1}{\Delta_{\jmath+3}}-\frac{1}{\Delta_{\jmath+2}}\right)-\frac{1}{\Delta_{\jmath+2}}\left(\frac{1}{\Delta_{\jmath+4}}-\frac{1}{\Delta_{\jmath+3}}\right)\right)\Bigg{)}\exp{\left(-i\bar{\Delta}_{\jmath}t\right)}\;\sigma_{\jmath,\jmath+4}+\text{H.c}\Bigg{\\}}\;\;\;\;$
$\displaystyle+\;\;\bar{\Omega}_{0}\Bigg{(}\frac{1}{\Delta_{4}}\left(\frac{1}{\Delta_{3}}\left(\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}}\right)-\frac{1}{\Delta_{1}}\left(\frac{1}{\Delta_{3}}-\frac{1}{\Delta_{2}}\right)\right)-\frac{1}{\Delta_{1}}\Bigg{(}\frac{1}{\Delta_{4}}\left(\frac{1}{\Delta_{3}}-\frac{1}{\Delta_{2}}\right)-\qquad\qquad\;\;$
$\displaystyle-\frac{1}{\Delta_{2}}\left(\frac{1}{\Delta_{4}}-\frac{1}{\Delta_{3}}\right)\Bigg{)}\Bigg{)}\exp{\left(-i\bar{\Delta}_{0}t\right)}\;\sigma_{{}_{g4}}+\text{H.c.}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\;\;\;$
$\displaystyle+\;\bar{\eta}\;\Bigg{(}\frac{1}{\Delta_{{}_{N}}}\Bigg{(}\frac{1}{\Delta_{{}_{N-1}}}\left(\frac{1}{\Delta_{{}_{N-2}}}+\frac{1}{\Delta_{{}_{N-3}}}\right)-\frac{1}{\Delta_{{}_{N-3}}}\left(\frac{1}{\Delta_{{}_{N-1}}}-\frac{1}{\Delta_{{}_{N-2}}}\right)\Bigg{)}-\frac{1}{\Delta_{{}_{N-3}}}\times\quad\quad\quad\qquad$
$\displaystyle\times\;\Bigg{(}\frac{1}{\Delta_{{}_{N}}}\left(\frac{1}{\Delta_{{}_{N-1}}}-\frac{1}{\Delta_{{}_{N-2}}}\right)-\frac{1}{\Delta_{{}_{N-2}}}\left(\frac{1}{\Delta_{{}_{N}}}-\frac{1}{\Delta_{{}_{N-1}}}\right)\Bigg{)}\Bigg{)}\exp{\left(\
-i\bar{\Delta}_{{}_{N-4}}t\right)}\;a^{\dagger}\sigma_{{}_{N-4,N}}+\text{H.c.},$
(38)

where the abbreviations:
$\bar{\Omega}_{\jmath}=\Omega_{\jmath+1}^{\ast}(t)\Omega_{\jmath+2}^{\ast}(t)\Omega_{\jmath+3}^{\ast}(t)\Omega_{\jmath+4}^{\ast}(t)$,
$\bar{\Delta}_{\jmath}=\Delta_{\jmath+1}+\Delta_{\jmath+2}+\Delta_{\jmath+3}+\Delta_{\jmath+4}$
and
$\bar{\eta}=\eta\;\Omega_{N-3}^{\ast}(t)\;\Omega_{N-2}^{\ast}(t)\;\Omega_{N-1}^{\ast}(t)$.
Now, by observing the Eqs. (30), (37) and (38), one can find that there exists
recurrence relations between the coefficients of successive transition
operators. For instance, let us first consider the coefficients of the
transition operators $\sigma_{\jmath,\jmath+2}$, $\sigma_{\jmath,\jmath+3}$
and $\sigma_{\jmath,\jmath+4}$. Let’s define
$\tilde{\Omega}_{\jmath}^{(1)}=\Omega_{\jmath+1}^{\ast}(t)\Omega_{\jmath+2}^{\ast}(t)$
and $\tilde{\Delta}_{\jmath}^{(1)}=\Delta_{\jmath+1}+\Delta_{\jmath+2}$ in the
Eq. (30). With these, from the Eqs.(37) and (38), we have
$\tilde{\Omega}_{\jmath}^{(2)}=\Omega_{\jmath+3}^{\ast}(t)\tilde{\Omega}_{\jmath}^{(1)}$,
$\tilde{\Delta}_{\jmath}^{(2)}=\Delta_{\jmath+3}+\tilde{\Delta}_{\jmath}^{(1)}$
and
$\tilde{\Omega}_{\jmath}^{(3)}=\Omega_{\jmath+4}^{\ast}(t)\tilde{\Omega}_{\jmath}^{(2)}$,
$\tilde{\Delta}_{\jmath}^{(3)}=\Delta_{\jmath+4}+\tilde{\Delta}_{\jmath}^{(2)}$
respectively. Moreover, let us define
$\mathcal{C}_{\jmath}^{(1)}=\frac{1}{\Delta_{\jmath+2}}-\frac{1}{\Delta_{\jmath+1}}$
in the Eq. (30). Thus,
$\mathcal{C}_{\jmath}^{(2)}=\frac{1}{\Delta_{\jmath+3}}\mathcal{C}_{\jmath}^{(1)}-\frac{1}{\Delta_{\jmath+1}}\mathcal{C}_{\jmath+1}^{(1)}$
and
$\mathcal{C}_{\jmath}^{(3)}=\frac{1}{\Delta_{\jmath+4}}\mathcal{C}_{\jmath}^{(2)}-\frac{1}{\Delta_{\jmath+1}}\mathcal{C}_{\jmath+1}^{(2)}$.
It is noted that $\jmath=0$ gives the coefficients of $\sigma_{g2}$,
$\sigma_{g3}$ and $\sigma_{g4}$.

Furthermore, by looking at the last term in the Eqs. (30), (37) and (38),
recurrence relations between the coefficients of successive transition
operators can be found. To this end, let us define,
$\tilde{\eta}_{{}_{N}}^{(1)}=\eta\;\Omega_{N-1}^{\ast}(t)$ and using the above
definition we have
$\mathcal{C}_{{}_{N-2}}^{(1)}=\frac{1}{\Delta_{{}_{N}}}-\frac{1}{\Delta_{{}_{N-1}}}$
and $\tilde{\Delta}_{{}_{N-2}}^{(1)}=\Delta_{{}_{N}}+\Delta_{{}_{N-1}}$ in the
Eq. (30). Thus, from the Eqs. (37) and (38), we have
$\tilde{\eta}_{{}_{N}}^{(2)}=\Omega_{N-2}^{\ast}(t)\tilde{\eta}_{{}_{N}}^{(1)}$,
$\tilde{\Delta}_{{}_{N-2}}^{(2)}=\Delta_{{}_{N-2}}+\tilde{\Delta}_{{}_{N-2}}^{(1)}$
and
$\tilde{\eta}_{{}_{N}}^{(3)}=\Omega_{N-3}^{\ast}(t)\tilde{\eta}_{{}_{N}}^{(2)}$,
$\tilde{\Delta}_{{}_{N-2}}^{(3)}=\Delta_{{}_{N-3}}+\tilde{\Delta}_{{}_{N-2}}^{(2)}$
respectively. Therefore, after performing $m$ iteration $(m=2,3,...,N-2)$ of
substitution of the solutions of transition operators obtained under Markov
approximation to the Hamiltonian in the $(m-1)$ step, one can obtain

$\displaystyle H_{{}_{I}}^{(m)}$ $\displaystyle=$
$\displaystyle\sum\limits_{\jmath=1}^{N-2}\Big{\\{}\tilde{\Omega}_{\jmath}^{(m)}\mathcal{C}_{\jmath}^{(m)}\exp{(-i\tilde{\Delta}_{\jmath}^{(m)}t)}\sigma_{\jmath,\jmath+m+1}+\text{H.c.}\Big{\\}}+\Big{(}\tilde{\Omega}_{0}^{(m)}\mathcal{C}_{0}^{(m)}\exp{(-i\tilde{\Delta}_{0}^{(m)}t)}\sigma_{g,m+1}$
(39) $\displaystyle+$
$\displaystyle\tilde{\eta}_{{}_{N}}^{(m)}\mathcal{C}_{{}_{N-2}}^{(m)}\exp{(-i\tilde{\Delta}_{{}_{N-2}}^{(m)}t)}\;a^{\dagger}\sigma_{{}_{N-m-1,N}}+\text{H.c.}\Big{)},$

where
$\tilde{\Omega}_{\jmath}^{(m)}=\Omega_{\jmath+m+1}^{\ast}(t)\tilde{\Omega}_{\jmath}^{(m-1)}$,
$\tilde{\Delta}_{\jmath}^{(m)}=\Delta_{\jmath+m+1}+\tilde{\Delta}_{\jmath}^{(m-1)}$,
$\mathcal{C}_{\jmath}^{(m)}=\frac{1}{\Delta_{\jmath+m+1}}\mathcal{C}_{\jmath}^{(m-1)}-\frac{1}{\Delta_{\jmath+1}}\mathcal{C}_{\jmath+1}^{(m-1)}$,
$\tilde{\eta}_{{}_{N}}^{(m)}=\tilde{\Omega}_{N-m}^{\ast}(t)\tilde{\eta}_{{}_{N}}^{(m-1)}$,
$\tilde{\Delta}_{{}_{N-2}}^{(m)}=\Delta_{{}_{N-m}}+\tilde{\Delta}_{{}_{N-2}}^{(m-1)}$,
and
$\mathcal{C}_{{}_{N-2}}^{(m)}=\frac{1}{\Delta_{{}_{N}}}\mathcal{C}_{{}_{N-3}}^{(m-1)}-\frac{1}{\Delta_{{}_{N-m}}}\mathcal{C}_{{}_{N-2}}^{(m-1)}$.
Since $\jmath$ runs from $1$ to $N-2$, the first term in (39) is ruled out
when $m=N-2$. Therefore, we have

$\displaystyle H_{{}_{I}}^{(N-2)}$ $\displaystyle=$
$\displaystyle\tilde{\Omega}_{0}^{(N-2)}\mathcal{C}_{0}^{(N-2)}\exp{(-i\tilde{\Delta}_{0}^{(N-2)}t)}\sigma_{{}_{g,N-1}}+\tilde{\eta}_{{}_{N}}^{(N-2)}\mathcal{C}_{{}_{N-2}}^{(N-2)}\exp{(-i\tilde{\Delta}_{{}_{N-2}}^{(N-2)}t)}\;a^{\dagger}\sigma_{{}_{1N}}+\text{H.c.}.$
(40)

At this stage, the first term in the Hamiltonian (40) represents the
transition between the levels $g$ and $N-1$ while the second term corresponds
to the transition between the levels $1$ and $N$. Finally, using (20), we
calculate the equations of motion for $\sigma_{{}_{g,N-1}}$ and
$\sigma_{{}_{1N}}$ and substituting their solutions into (40), we obtain the
effective Hamiltonian

$\displaystyle H_{\text{eff}}(t)$ $\displaystyle=$
$\displaystyle\eta\;\tilde{\Omega}_{0}^{(N-2)}\left(\frac{1}{\Delta_{{}_{N}}}\mathcal{C}_{0}^{N-2}-\frac{1}{\Delta_{{}_{1}}}\mathcal{C}_{{}_{N-2}}^{(N-2)}\right)\;\exp{\left(-i\tilde{\Delta}_{0}^{(N-1)}t\right)}\;a^{\dagger}\sigma_{{}_{gN}}+\text{H.c.}$
(41) $\displaystyle=$
$\displaystyle\eta\;\tilde{\Omega}_{0}^{(N-2)}\mathcal{C}_{0}^{(N-1)}\exp{\left(-i\tilde{\Delta}_{0}^{(N-1)}t\right)}\;a^{\dagger}\sigma_{{}_{gN}}+\text{H.c.},$

which describes the effective dynamics between the states $\ket{g}$ and
$\ket{N}$.

###  III.A Revisit the two-photon transition in a three-level atom inside a
cavity using our approach

As an example, we consider a three-level atomic system in the so-called lambda
configuration inside a cavity as shown in the Fig. 3.

Figure 3:  A typical lambda system inside a cavity is shown. The transition
$\ket{g}\leftrightarrow\ket{1}$ is controlled by the Rabi frequency
$\Omega(t)$ and the corresponding detuning
$\Delta_{1}\;(\omega_{21}-\omega_{l})$. The transition
$\ket{1}\leftrightarrow\ket{2}$ happens under cavity the mode $\omega_{c}$ and
governed by the detuning $\Delta_{2}\;(\omega_{12}-\omega_{c})$ and the
coupling constant $\eta$ .

The atomic system interacts with a pump mode of frequency $\omega_{l}$ and a
Stokes mode of frequency $\omega_{c}$ which is the frequency of cavity mode.
The Hamiltonian describing the system can be written as

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\omega_{g}\sigma_{gg}+\omega_{1}\sigma_{11}+\omega_{2}\sigma_{22}+\omega_{c}a^{\dagger}a+\Omega(t)\exp{(-i\omega_{l}t)}\sigma_{1g}$
(42) $\displaystyle+$
$\displaystyle\eta\exp{(-i\omega_{c}t)}\;a\sigma_{12}+\mathrm{H.c.}$

By using the rotating frame and the rotating wave approximation the
Hamiltonian of the system reads

$\displaystyle\mathcal{H}_{I}(t)$ $\displaystyle=$
$\displaystyle\Omega(t)\exp{(i\Delta_{1}t)}\sigma_{1g}+\eta\exp{(i\Delta_{2}t)}a\sigma_{12}+\mathrm{H.c.}$
(43)

To establish an effective two-level Raman transition between
$\ket{g}\leftrightarrow\ket{2}$, we need to eliminate the state $\ket{1}$ from
the effective dynamics. Since in this case $N=2$, therefore, in the Eq. (41),
we have
$\tilde{\Omega}_{0}^{(N-2)}(t)=\Omega(t);\;\mathcal{C}_{0}^{(1)}=-\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}};\;\tilde{\Delta}_{0}^{(1)}=-\Delta_{2}+\Delta_{1}$.
By inserting these values into the Eq. (41), we get the effective Hamiltonian

$\displaystyle\mathcal{H}_{\text{eff}}(t)$ $\displaystyle=$
$\displaystyle-\eta\Omega(t)\bigg{(}\frac{1}{\Delta_{1}}+\frac{1}{\Delta_{2}}\bigg{)}\exp{(i(\Delta_{1}-\Delta_{2})t)}a^{\dagger}\sigma_{2g}+\mathrm{H.c.},$
(44)

which can be obtained following the Jame’s effective Hamiltonian theory of
two-photon transition [[10]].

###  III.B The effective Hamiltonian for a three-photon transition using
generalized James’ approach

To realize the three-photon transition process, we consider a four-level atom
inside a cavity as shown in the Fig. 4. We assume the energies of the states
$\ket{g},\;\ket{1},\;\ket{2},\;\ket{3}$ are $E_{g},\;E_{1},\;E_{2}$ and
$E_{3}$ respectively. To establish an effective three-photon transition
between $\ket{g}\leftrightarrow\ket{3}$, firstly we employ the generalized
James’ effective Hamiltonian method [[12]]. In the interaction picture, the
Hamiltonian of the system reads

Figure 4:  A typical three-photon hyper Raman transition inside a single mode
cavity of frequency $\omega_{c}$ is displayed. The detuning
$\Delta_{1}(\Delta_{2})$ and the Rabi frequency $\Omega_{1}(t)(\Omega_{2}(t))$
facilitate the transition
$\ket{{g}}\leftrightarrow\ket{{1}}(\ket{{1}}\leftrightarrow\ket{2})$. The
coupling constant $\eta$ and detuning $\Delta_{3}$ control the transition
$\ket{{2}}\leftrightarrow\ket{{3}}$.

$\displaystyle\mathcal{H}_{I}(t)$ $\displaystyle=$
$\displaystyle\Omega_{1}(t)\exp{(i\Delta_{1}t)}\sigma_{\text{1g}}+\Omega_{2}(t)\exp{(i\Delta_{2}t)}\sigma_{\text{21}}+\eta\exp{(i\Delta_{3}t)}a\sigma_{23}+\mathrm{H.c.}$
(45)

From the Eqs.(15) and (45), we can consider
$\omega_{1}=\Delta_{1},\;\omega_{2}=\Delta_{2},\;\omega_{3}=\Delta_{3}$ and
the corresponding $\Lambda_{\alpha}\;(\alpha=1,2,3)$ are of the form as
$\Lambda_{1}=\Omega_{1}\sigma_{1g}$, $\Lambda_{2}=\Omega_{2}\sigma_{21}$ and
$\Lambda_{3}=\eta a\sigma_{23}$. By utilizing the third-order James’ effective
Hamiltonian method (15) and the Hermiticity condition used in (16), we arrive
at

$\displaystyle\mathcal{H}_{\text{eff}}^{(3)}(t)=\frac{\eta\Omega_{1}(t)\Omega_{2}(t)}{\Delta_{2}\big{(}\Delta_{1}+\Delta_{2}\big{)}}\;a^{\dagger}\sigma_{3g}+\mathrm{H.c.}+\text{Stark
shifts.}$ (46)

This establishes the effective dynamics between the states $\ket{g}$ and
$\ket{3}$ at resonance. However, we show that our approach based on the Markov
approximation approach produces effective dynamics at off-resonance as well as
at the resonance.

###  III.C The effective Hamiltonian using our Markovian approximation
approach

To this end, we revisit to the aforesaid effective dynamics for three-photon
hyper Raman transition process using our Markovian approximation approach. In
this case, $N=3$, and in the Eq. (41), we have
$\tilde{\Omega}_{0}^{(1)}=\Omega_{1}^{*}\Omega_{2}^{*}$,
$\mathcal{C}_{0}^{(2)}=\frac{1}{\Delta_{3}}\mathcal{C}_{0}^{(1)}-\frac{1}{\Delta_{1}}\mathcal{C}_{1}^{(1)}=\frac{1}{\Delta_{3}}\left(\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}}\right)-\frac{1}{\Delta_{1}}\left(\frac{1}{\Delta_{3}}-\frac{1}{\Delta_{2}}\right)$,
and $\tilde{\Delta}_{0}^{(2)}=\Delta_{3}+\Delta_{2}+\Delta_{1}$. Therefore,
the effective Hamiltonian (41) will take the following form

$\displaystyle\mathcal{H}_{\text{eff}}(t)=\eta\Omega_{1}^{*}\Omega_{2}^{*}\left(\frac{1}{\Delta_{3}}\left(\frac{1}{\Delta_{2}}-\frac{1}{\Delta_{1}}\right)-\frac{1}{\Delta_{1}}\left(\frac{1}{\Delta_{3}}-\frac{1}{\Delta_{2}}\right)\right)\exp{\left(-i\Delta_{0}^{(2)}t\right)}a^{\dagger}\sigma_{g3}+\text{H.c}.$
(47)

Now, at the resonance we can put $\Delta_{1}+\Delta_{2}+\Delta_{3}=0$. As the
detunings $\Delta_{1}$ and $\Delta_{2}$ are large enough, we can neglect the
first term within the first bracket in the Eq. (47). Therefore, we have the
effective Hamiltonian at resonance as

$\mathcal{H}_{\text{eff}}(t)=\frac{\eta\Omega_{1}^{*}(t)\Omega_{2}^{*}(t)}{\Delta_{2}(\Delta_{1}+\Delta_{2})}a^{\dagger}\sigma_{g3}+\text{H.c}.,$
(48)

which is obtained above (46) employing the generalized James’ effective
Hamiltonian approach.

##  IV Summary and discussion

We have proposed an approach to derive the effective Hamiltonian under the
adiabatic elimination process for multiphoton transition in atom in the
presence of classical laser fields and the cavity mode interaction. By
calculating the equations of motion for the transition operators in the
Heisenberg picture and solving it under Markov approximation, we have obtained
the effective dynamics for the system. We have compared our procedure with the
James’ and generalized James’ effective Hamiltonian approaches with a few
examples. From the above analysis it is evident that our approach works well
at off-resonance and as well as at resonance. Unlike the generalized James’
approximation approach, our approach can be utilized to get rid of cavity
modes from the effective dynamics. For instance, the implementation of
holonomic quantum gates demands that the exclusion of cavity modes from the
effective Hamiltonian in order to make the gate operations resilient to the
cavity decay [[11]]. Our approach could be convincingly adopted to implement
the multi-qubit holonomic quantum gates in quantum computation.

## Acknowledgement

I would like to thank M. Sanjay Kumar for his helpful comments and
encouragement. I acknowledge the financial support from DST (India) through
the INSPIRE Fellowship Programme.

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