2. General Description of Operators
2.1. Atom-field Interaction Hamiltonian in the Electric Dipole ApproximationThe Hamiltonian 
of a bare atom (with no radiation present) can be written as 2, 4
 | (2.1.) |
where 
and 
are, respectively, the eigenvectors and eigenvalues of 
[2.4] satisfying the orthonormality (
) and closure (
) relations, whereas 
are the atomic operators satisfying the commutation relation 
in which 
stands for the Kronecker delta.
Like the bare atom Hamiltonian 
, any linear operator 
can be expressed in terms of the atomic operators 
by applying the closure relation twice as
 | (2.2) |
Here, the expansion coefficients 
are the matrix elements of 
in the basis 
. If 
is a Hermitian operators (as any observable should be), it is represented by a Hermitian matrix 5, one in which all the diagonal elements 
are real, and any two elements symmetric with respect to the diagonal are complex conjugates of each other (
).
The interaction of an atom with a radiation field is represented by an interaction Hamiltonian 2 
given by Eq.2.3. In this equation, 
is the atomic dipole moment vector which is expanded in terms of the atomic operators 
in accordance with Eq.2.2, 
are the matrix elements of 
, 
is the magnitude of 
, and 
is the direction (unit vector) of 
. The matrix elements 
are in general complex, and (since 
is Hermitian) 
. Moreover, since 
is a vector operator, its diagonal matrix elements 
are all zero 5.
 | (2.3) |
In the full quantum mechanical treatment of atom-field interaction, the atom is treated quantum mechanically, and the radiation field is taken to be a quantized field 7, 8, 9. In this work, we employ the semiclassical approach in which the atom is described quantum mechanically whereas the radiation field is taken to be a purely classical field (by assuming the field to be sufficiently strong). Moreover, we assume the radiation field to be a monochromatic (or quasi-monochromatic) laser light with central frequency 
which is assumed to be in the visible region. We also assume the light to be linearly polarized in the direction of a unit vector 
, and propagating in the direction of the wave vector 
. Taking the nucleus of the atom as the origin of coordinates, the electric field 
of the applied laser light can be expressed as
 | (2.4) |
Here, 
is a real amplitude of the electric field, 
is the central frequency of the field, and 
is the corresponding wave number. If 
is truly monochromatic, 
will be constant in time, whereas, more generally, 
varies with time.
When the applied light is in the optical (visible) region so that the wavelength of the light is much greater than the size of the atom, the spatial variation of the electric field of the light across the atom can be ignored to a good approximation, and this is what is meant by electric dipole approximation 5. Invoking this approximation in Eq.2.4, we ignore 
in the exponent, and 
in 
to write the applied electric field as
 | (2.5) |
Using Eq.2.5 in Eq.2.3 and expanding the result according to Eq.2.2, we obtain
 | (2.6) |
The quantity 
has the dimensions of frequency and is called the Rabi frequency of the external laser radiation. It quantifies the interaction of the external laser radiation with the relevant atomic transition 
and is zero when 
(because 
, in accordance Eq. 2.3)
 | (2.7) |
In general, the laser field envelope 
defined by Eq. 2.4 depends on time, and so do the quantities 
and 
which are dependent on 
as defined by Eq.2.6. However, if the external laser field is purely monochromatic, 
is a constant independent of time so that 
and 
are also constants independent of time. For most applications, if 
varies sufficiently slowly within an optical period, it (and, therefore, both 
and 
) can be taken to be constants to a good approximation.
Combining Eq.2.1 and Eq. 2.6, we finally obtain
 | (2.8) |
for the total Hamiltonian describing atom-field interaction in the electric dipole approximation. In connection with the coefficients 
of the interaction Hamiltonian 
, we introduce the atomic transition frequency 
, and the detuning frequency 
by
 | (2.9) |
The atomic transition frequency 
is the frequency difference between any two atomic levels 
and 
as shown in Fig.1, whereas the detuning frequency 
is the frequency difference between 
and the radiation frequency 
.
2.2. State Vectors and Density Operators in GeneralIn this paper, we choose to work in the Schrödinger picture where the basis vectors 
are stationary but the state vectors 
vary in time according to the time dependent Schrödinger equation. Each time dependent state vector 
can be expanded on the time independent basis 
uniquely as in Eq. 2.10.
Here, the expansion coefficient 
gives the probability amplitude for finding the atom in the eigenstate 
and is, in general, complex. We assume that 
is normalized (
) so that the coefficients 
satisfy the probability conservation condition 
.
 | (2.10) |
The density operator 
of a system is such that, characterizing the system by a state vector 
is completely equivalent to characterizing it by the corresponding 
acting in that state space, in the sense that all physical information that can be obtained from 
can also be obtained from 
5. For each state vector 
, the corresponding density operator 
and its time evolution are given by,
 | (2.11) |
where 
is the total Hamiltonian of the atom-radiation system given by Eq. 2.8. The density operator 
can be expanded in terms of the atomic operators 
in accordance with Eq. 2.2 to obtain,
 | (2.12a) |
 | (2.12b) |
 | (2.12c) |
 | (2.12d) |
Here 
are elements of the density matrix, and 
are the expansion coefficients of the state vector 
given by Eq. 2.10. The diagonal elements of the density matrix 
are clearly real, and the trace of the density matrix is equal to unity which follows directly from the probability condition of Eq. 2.10. The off-diagonal elements of the density matrix are in general complex, and satisfy 
, since 
is a Hermitian operator, as can be easily deduced from its definition by Eq. 2.11. From the definition of the density operator (Eq. 2.11), the orthonormality condition (
), and Eq. 2.12c for the trace of the density matrix, it readily follows that,
 | (2.13) |
The diagonal elements the density matrix 
represent the probability of finding the atom in its eigenstate state 
and, therefore, are known as the populations. The non-diagonal elements
express the interference effects between the states 
and 
when the atom is in a coherent linear superposition of these states and, therefore, are known as the coherences. According to Eq. 2.12b, 
is non-zero only when both 
and 
are non-zero. Therefore, the density matrix can have coherences (non-diagonal elements) only between populated states.
Figure 1. (a) Two-level atom, (b) three-level atom in the V-configuration, (c) three-level atom in the Ʌ-configuration, and (d) three-level atom in the cascade configuration. For a three-level atom in the V-configuration, the upper levels |3> and |2> are close to each other but far from the ground level |1>, and the transition |3> ßà |2> is not dipole allowed. On the other hand, for a three-level atom in the Λ-configuration the lower levels |2> and |1> are close to each other but far from the third (highest) level |3>, and the transition |3> ßà |2> is not dipole allowed. For a three-level atom in the cascade configuration, the transition |3> ßà |1> is not dipole allowed
3. Optical Bloch Equations for a General Three-Level Atom
3.1. Parametric Equations of the Density Matrix Elements of a General Three-Level AtomFor a three-level atom, the state space is a 3- dimensional space spanned by the eigenvectors 
, 
and 
, Figure. 1. The most general (normalized) state vector of this 3-dimensional state space is given by the linear superposition of these eigenvectors as
 | (3.1b) |
In the three state basis 
arranged in this order, any operator 
is represented by a 
matrix in accordance with Eq. 2.2. In particular, the density operator 
of Eq. 2.11 corresponding to the state vector 
of Eq. 3.1 is represented by a 
density matrix whose elements are, according to Eq. 2.12b, 
.
 | (3.2) |
Noting that the diagonal elements of the density matrix are real, and add up to unity (Eq. 2.12), whereas the non-diagonal elements are related by complex conjugation, the nine elements of density matrix of Eq. 3.2 can be parameterized by a new set of eight variables as
 | (3.3) |
Comparing Eqs. 3.2 and 3.3, we obtain the following expressions for 
in terms of 
 | (3.4a) |
 | (3.4b) |
 | (3.4c) |
 | (3.4d) |
 | (3.4e) |
 | (3.4f) |
 | (3.4g) |
 | (3.4h) |
where 
and 
denote real and imaginary parts, respectively.
From Eq. 3.4, we see that all eight parameters of the density matrix are real. The parameter 
and 
are related to the populations of the upper levels 
and 
. When the atom is certainly in level 
so that 
, we obtain 
and 
. On the other hand, when the atom is certainly not in level 
so that 
, we obtain 
and 
. Therefore, the population parameters 
and 
vary between 2 and -1.
 | (3.5a) |
 | (3.5b) |
Apart from the population parameters 
and 
, all the other six parameters of the density matrix are coherence parameters related to interference effects between the atomic levels 
, 
and 
. The quantity
 | (3.6) |
is related to the population inversion between the upper levels 
, and is relevant for three-level atoms in the 
configuration, as will be seen in section (4). Taking the square of the density matrix in parameter form (Eq. 3.3), and then applying Eq. 2.13 to equate the trace of this square matrix to 1, we obtain,
 | (3.7) |
which can also be obtained directly from Eq. 3.4.
For a general three-level atom, the density matrix given by Eq. 3.3 is expanded in terms of the atomic operators 
of Eq. 2.1, in accordance with Eq. 2.12a as
 | (3.8) |
Moreover, the total Hamiltonian of a general three-level atom is obtained from the general Hamiltonian of an n-level atom (Eq. 2.8) to be,
 | (3.9) |
where we have used Eq. 2.7 to set 
for 
. Using Eq. 3.8 and Eq. 3.9 in Eq. 2.11, we obtain, after considerable careful accounting of terms, the following eight coupled equations for time evolutions of the eight parameters of the density matrix of a general three-level atom defined by Eq. 3.3
 | (3.10a) |
 | (3.10b) |
 | (3.10c) |
 | (3.10d) |
 | (3.10e) |
 | (3.10f) |
 | (3.10g) |
 | (3.10h) |
Here, a dot over a symbol represents time derivative, and 
is the frequency difference between the atomic levels 
and 
, as defined by Eq. 2.9.
Eqs. 3.10 are general equations applicable for any three-level atom interacting with a radiation field in the optical region under the electric dipole approximation. In these equations, all coefficients containing 
and 
are of the form,
 | (3.11) |
and, therefore, are real, indicating that Eqs.3.10 are equations for real variables with real coefficients. These equations are simplified a great deal if we assume the atomic dipole matrix elements 
, 
, and 
associated with the atomic transitions 
, 
, and 
, and (as defined by Eq. 2.3) are all real so that
 | (3.12) |
where 
, 
and 
(defined by Eq. 2.6) are now real. Using Eq. 3.12 in Eq. 3.10, we obtain
 | (3.13a) |
 | (3.13b) |
 | (3.13c) |
 | (3.13d) |
 | (3.13e) |
 | (3.13f) |
 | (3.13g) |
 | (3.13h) |
These equations govern the time evolutions of the matrix elements of the density matrix of a general three-level atom (Eq. 3.3) when the matrix elements of the dipole moment of the atom are assumed to be real. They can be cast in matrix form as
 | (3.14) |
where 
is a vector in an 8-dimensional space given in column form as
 | (3.15) |
and 
is an 
coefficient matrix given by
 | (3.16) |
According to Eq. 3.7, the norm of vector 
is given by,
 | (3.17) |
Therefore,
is not constant but depends on the population parameters 
and 
. However, if we introduce the normalized population parameters 
and 
by
 | (3.18a) |
so that
 | (3.18b) |
the norm of the new vector 
defined by
 | (3.19) |
will be a constant and is given by,
Therefore, there is a conservation law associated with Eq. 3.13 when the population parameters 
and 
are replaced by their normalized versions 
and 
. This constraint on 
is nothing but another way of expressing the probability conservation of Eq.2.10 in terms of the parameters of the density matrix.
 | (3.20) |
3.2. Rotating Wave Approximation for a General Three-Level AtomEq. 3.13 contain terms which vary rapidly at the optical frequencies 
, 
, 
and 
. Switching to not so rapidly varying variables will render the dynamics of the system more transparent. This can be achieved by an orthogonal transformation - a linear transformation in which the sum of the squares of the new variables is equal to the sum of the squares of the old variables 10. The necessary and sufficient condition for a linear transformation to be orthogonal is that the transformation matrix 
be orthogonal (that is, the transpose 
of 
be equal to its inverse 
). If 
is orthogonal, then its determinant is necessarily 
. If 
, the transformation is called proper rotation because it involves only rotation, and no reflection. On the other hand, if 
the transformation involves reflection on all or some of the axes in addition to rotation 10.
An orthogonal matrix representing pure (proper) rotation in an 8-dimensional state space is given by,
 | (3.21) |
For the matrix M given by Eq. 3.21, direct multiplication shows that 
(where 1 is the 
identity matrix) so that 
. Moreover, 
, proving our assertion that 
is orthogonal, and, therefore, represents proper rotation. Under 
, the 8-dimensional vector 
of Eq. 3.15 orthogonally transforms to another 8-dimensional vector 
which, by the very definition of orthogonal transformation, has the same norm as 
 | (3.22a) |
 | (3.22b) |
 | (3.22c) |
 | (3.22d) |
Differentiating both sides of the transformation equation 
with respect to time, rearranging results, and then using 
from Eq. 3.14, we obtain
 | (3.23) |
Multiplying both sides of Eq. 3.23 from the right by 
, we finally obtain
 | (3.24) |
This is the equation of motion for the components of the transformed vector 
obtained from the original vector
by the orthogonal transformation 
. It is a general equation valid for any orthogonal transformation of the form 
, independent of the dimension of vector 
, and the corresponding sizes of the coefficient matrix 
and the transformation matrix
.
Using Eq. 3.16 for the coefficient matrix 
, and Eq. 3.21 for the transformation matrix 
in Eq. 3.24, we obtain (after considerable algebra) the following eight equations for the time evolution of the eight components of vector 
.
 | (3.25a) |
 | (3.25b) |
 | (3.25c) |
 | (3.25d) |
 | (3.25e) |
 | (3.25f) |
 | (3.25g) |
 | (3.25h) |
In these equations, 
is the detuning of the atomic transition frequency 
from the applied laser field frequency 
, as defined by Eq. 2.9
In Eqs. 3.25, the terms containing 
, 
, and 
represent high frequency evolutions, and, therefore, average out to zero over the evolution of the slowly varying terms containing the detuning frequencies 
. For this reason, these rapidly varying terms can be ignored to a good approximation, and this is what is meant by the rotating wave approximation (RWA) 1. Therefore, when the RWA is invoked in Eqs. 3.25, we obtain,
 | (3.26a) |
 | (3.26b) |
 | (3.26c) |
 | (3.26d) |
 | (3.26e) |
 | (3.26f) |
 | (3.26g) |
 | (3.26h) |
Eqs. 3.26h-h are the principal equations of this paper. They are the optical Bloch equations for a general three-level atom interacting with a linearly polarized light in the visible region under the electric dipole and rotating wave approximations, and under the assumption that the matrix elements of the atomic dipole are all real. They reduce to the Bloch equations for all possible special cases of a general three-level atom as well as to the Bloch equations for a two-level atom as shown in the next section.
4. Optical Bloch Equations for Special Cases of a General Three-Level Atoms
4.1. Optical Bloch Equations for Two-Level AtomA general three-level atom with three atomic transitions 
, 
, and 
reduces to a two-level atom [Fig. 1(a)] when only one of these three transitions is dipole allowed. The density matrix of such a two-level atom can be written in terms of the parameters of the density matrix of a three-level atom (Eq. 3.3) as follows.
 | (4.1a) |
 | (4.1b) |
 | (4.1c) |
From Eq. 4.1c we see that, when a three-level atom is considered as a two-level atom consisting of levels 
and 
, the parameters 
and 
must be equal and opposite so that the trace of the density matrix is equal to unity, in accordance with Eq. 2.12d.
 | (4.2) |
For a two-level atom consisting of levels 
and 
, the only relevant quantities in Eqs. 3.26a-h are those related to the transition 
, namely 
, 
, 
, 
and 
. Similarly, for a two-level atom consisting of levels 
and 
, the only relevant quantities in Eqs. 3.26a-h are those related to the transition 
, namely 
, 
, 
, 
and 
. Likewise, for a two-level atom consisting of levels 
and 
, the only relevant quantities in Eqs. 3.26a-h are those related to the transition 
, namely 
, 
, 
, 
, 
and 
where, according to Eq. 4.2, 
. The general Bloch equations for a general three-level atom (Eqs. 3.26a-h) reduce to those of a two-level atom when all quantities non-relevant to the atom are set to zero. As a result, we obtain,
 | (4.3a) |
 | (4.3b) |
 | (4.3c) |
These optical Bloch equations are identical to the optical Bloch equations derived in Allen and Eberly 1 specifically for two-level atoms, showing that our general Bloch equations for three-level atoms contain those for two-level atoms as special cases.
4.2. Optical Bloch Equations for Three-Level Atoms in the 
, 
, and Cascade ConfigurationsA three-level atom is said to be in the so called 
configuration, if the upper levels 
and 
are close to each other but far from the ground level 
, and if these upper levels are of the same symmetry so that the atomic transition 
is not dipole allowed, Figure 1(b). Therefore, for a three-level atom in the 
configuration, the only relevant quantities in Eqs. 3.26a-h are 
, 
, 
, 
and 
(associated with the atomic transition 
) as well as 
, 
, 
, 
and 
(associated with the atomic transition 
). As a result, Eqs. 3.26a-h for a general three-level atom reduce (as they should) to those of a three-level atom in the 
configuration when all quantities non-relevant to the configuration are set to zero.
 | (4.4) |
In the case of a three-level atom in the 
configuration, the lower levels 
and 
are close to each other but far from the third (highest) level 
, and these lower levels are of the same symmetry so that the atomic transition 
is not dipole allowed, Figure 1(c). Therefore, in such a case, the only relevant quantities in Eqs. 3.26a-h are 
, 
, 
, 
and 
(associated with the atomic transition 
) as well as 
, 
, 
, 
and 
(associated with the atomic transition 
). As a result, Eqs. 3.26a-h for a general three-level atom reduce to those of a three-level atom in the 
configuration when all quantities non-relevant to the configuration are set to zero.
 | (4.5) |
The main difference between the Bloch equations for 
and 
configurations is that, in addition to the individual populations 
and 
of the upper levels 
and 
, the 
configuration (because of its allowed 
transition) involves the population difference 
between these upper levels, as indicated by the second term on the RHS of Eq. 4.5e.
For a three-level atom in the cascade configuration, there is no direct transition from
to 
, Figure 1(d). Therefore, in such a case, the only relevant quantities in Eqs. 3.26a-h are 
, 
, 
, 
and 
(associated with the atomic transition 
) as well as 
, 
, 
, 
and 
(associated with the atomic transition 
). As a result, Eqs. 3.26a-h for a general three-level atom reduce to those of a three-level atom in the cascade configuration when all quantities non-relevant to the configuration are set to zero.
Eqs. 4.4, 4.5, and 4.6 are identical to the optical Bloch equations derived separately for 
-configuration,
-configuration, and cascade configuration 7, 8, showing once again that our general Bloch equations for three-level atoms contain those for 
, 
, and cascade configurations as special cases.
 | (4.6) |
5. Conclusions
We derived a single set of optical Bloch equations for a general three-level atom from which the optical Bloch equations for three-level atoms in the 
, 
, and cascade configurations as well as those for two-level atoms can be easily obtained as special cases. These general optical Bloch equations govern the interaction of a general three-level atom (and all its special cases) with linearly polarized light under the electric dipole and rotating wave approximations and under the assumption that all relevant electric dipoles are real.
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density matrix of a general three-level atom interacting with a classically described linearly polarized light are parameterized by eight (8) variables related to the populations of the atomic levels, and the coherences (interference effects) between the levels. The dynamical equations of these eight parameters (that is, the optical Bloch equations for a general three-level atom) are derived under the electric dipole as well as the rotating wave approximations, and under the assumption that the matrix elements of the atomic electric dipole are all real. The general optical Bloch equations so derived are shown to contain the well-known optical Bloch equations for three-level atoms in the 
, and cascade configurations as well as those for two-level atoms as special cases. Expressing the interaction of light with a three-level atom in all possible configurations as well as with a two-level atom by a single set of generalized optical Bloch equations is the main result of this work.
orthogonal transformation matrix representing “pure rotation” in an 8-dimensional state space, and then ignoring the rapidly varying terms. The matrix elements of the electric dipoles associated with the atomic transitions can be made real by adjusting the arbitrary phases of the wave functions of the relevant atomic levels, provided the transitions between the levels obey the ∆m = 0 selection rule 1. The optical Bloch equations for the special cases of three-level atoms in the V, Λ, and cascade configurations as well as those for two-level atoms are easily derived from the general Bloch equations by setting to zero all quantities non-relevant for each of those special cases.
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