Projection-Reduction Approach to Optical Conductivities for an Electron-Phonon System and Their Diagram Representation
Utilizing state-dependent projection operators and the Kang-Choi reduction identities, we derive the linear, first, and second-order nonlinear optical conductivities for an electron system interacting with phonons. The lineshape functions included in the conductivity tensors satisfy “the population criterion” saying that the Fermi distribution functions for electrons and Planck distribution functions for phonons should be combined in multiplicative forms. The results also contain energy denominator factors enforcing the energy conservation as well as interaction factors describing electron-phonon interaction properly. Therefore, the phonon absorption and emission processes as well as photon absorption and emission processes in all electron transition processes can be presented in an organized manner and the results can be represented in diagrams that can model the quantum dynamics of electrons in a solid.
Studies of the optical transitions for electron systems interacting with a background are useful for examining the electronic properties of solids because the absorption lineshapes are sensitive to the type of scattering mechanisms affecting the carrier behavior. One of the approaches to this problem is the projection method [1–17], which is generally based on the response theory. The projection techniques can be divided into two kinds: single-electron projection techniques [1–9] and many-electron projection techniques [10–17].
A proper many-body theory should satisfy the so-called “many-body criteria,” one of the most difficult being “the population criterion,” meaning that the Fermi distribution functions for electrons and the Planck distribution functions for phonons should be combined in multiplicative forms because the electrons and phonons belong to different categories in quantum-statistical physics. The population factors satisfying the population criterion should be combined in a physically acceptable manner with the energy denominator factor to maintain energy conservation and with the interaction factor to interpret the occurrence of local fluctuations due to electron-phonon interactions in such a way that the transition can take place via implicit states. Theories based on the single-electron projection technique starting from the Kubo formalism have failed to meet this criterion.
The many-electron projection techniques are classified into two categories; the first being state-independent or current-dependent [10–13] and the second being state-dependent [14–17]. The conductivity tensors [10, 11] produced using the isolation and projection operators, which are state-independent projection techniques, have limited applications for the following reasons. (a) They do not satisfy the population criterion because the two distribution functions are simply added (or subtracted), such as , where and are the Planck distribution function for a phonon with wave vector and the Fermi distribution function for an electron with energy , respectively. (b) They do not satisfy the “projection criterion,” so they are applicable only to cases where the spacing between the energy levels is constant, such as cyclotron resonance phenomena; that is, they are unsuitable for a system with nonuniform energy separation, such as a quantum square well (see the statements below (51)).
Since Franken et al.  pioneered the field of nonlinear optical phenomena by the experimental work on optical wave mixing, it has attracted considerable research attention [19, 20]. Most studies focused on the electronic structures and the electromagnetic properties of media with increasing the available laser power [20–33], but there were some problems. Shen  did not have explicit forms of the damping terms (relaxation rates) appearing in the conductivity. Suzuki and Ashikawa  reported the explicit forms of the damping term but did not satisfy the population criterion.
The present authors introduced an identity relation (the Kang-Choi reduction identity (KCRI) of the first kind) to derive the lineshape formula for a cyclotron transition . The population factors appearing in this formula satisfy the population criterion (see [12, equation (27)]). This shows that correct use of the KCRI can satisfy the population criterion. Later, the identity was generalized to cover the higher-order nonlinear optical conductivities [13–16]. The second kind (see [13, equation (4.18)]) was used along with current-dependent projectors, which is called the state-independent projectors, to derive the first-order nonlinear optical conductivity . This satisfies the population criterion but has limited applications because it does not satisfy the projection criterion. The corrected extended version of the nonlinear optical conductivity formalism, including the second-order nonlinear optical conductivity, was presented recently by applying the projection-reduction (PR) method by combining the generalized state-dependent projection operator (SDPO) and the KCRI of the 3rd kind (see [14, equation (2.5)]). The axes of the generalized projectors are the creation-annihilation operator pairs (see [14, equation (6.4)]), which makes the formalism applicable to general electron systems; that is, it satisfies the projection criterion.
The aim of this study is to generalize the KCRI to cover higher-order nonlinear optical conductivities and show how physically acceptable forms of the optical conductivities can be obtained by PR method. In addition, a method is introduced to represent the optical conductivity formulae using diagrams, through which a physical intuition to the quantum dynamics of the electrons in a solid can be achieved.
2. Hamiltonian and Density Operator
To derive the optical conductivities, the Hamiltonian is defined and the density operator is introduced.
We consider the total Hamiltonian, , for a system of electrons interacting with phonons as follows: where the time-independent part of the system in thermodynamic equilibrium, , is given by Here, , , and are the electron, phonon, and electron-phonon interaction Hamiltonians, respectively, is the creation (annihilation) operator for an electron in the state with energy , is that for a phonon in the state with energy , , is the phonon wave vector, is the polarization index, and is the electron-phonon interaction matrix element defined as where is the coupling factor that depends on the mode of phonons and is the electron position vector.
In (1), when a linearly polarized time-dependent electric field with angular frequency , , is applied, the following time-dependent interaction Hamiltonian is considered: where , , and denote the , , and components, respectively, of the electron position vector, for an arbitrary operator , and . means the complex conjugate.
2.2. Density Operator
To determine the general-order nonlinear optical conductivity, we assume that when the time-dependent electric field is applied to the system, the total density operator can be split into two parts as follows: where is the density operator for a system in thermal equilibrium and is the perturbed term by the time-dependent external field. Using the Liouville equation we can obtain Here, we have used the fact that and is the Liouville operator corresponding to the total Hamiltonian defined as for an arbitrary operator, . can be split into two parts for calculation convenience: where and , respectively, correspond to and .
To obtain , we define the density operator in the Dirac picture as By differentiating (8) and considering (7), where we have used . Integrating (9) from to subject to the initial condition , a solution to this equation can be obtained: By inserting (8) into (10) and replacing by , Finally, the following can be obtained by iteration: where involves ’s times.
Using (12), the ensemble average of the th component of the many-electron current density operator can be determined as follows: where , , and means the many-body trace. Equation (13) will be used in the following sections to calculate the linear and nonlinear optical conductivities.
3. Optical Conductivity
3.1. Linear Optical Conductivity
The following can be obtained by inserting in (12) into and considering (4): where we have used the cyclic property of trace, , and (). Using , (14) can be written as follows: Here, and the linear optical conductivity is defined as follows: where
3.2. First-Order Nonlinear Optical Conductivity
3.3. Second-Order Nonlinear Optical Conductivity
Inserting in (12) into , where , , and . Considering and inserting into (22), where . Therefore, inserting , , and into (23) results in Finally, considering (4), we obtain where and the second-order nonlinear optical conductivity is defined as
4. Kang-Choi Reduction Identity
The essential part of this paper is expansion of the optical conductivities (see (16), (21) and (26)) to easily solvable forms. For this purpose, this section introduces various forms of a useful identity, known as the Kang-Choi reduction identity (KCRI).
4.1. KCRI of First Kind (KCRI-I)
For arbitrary operators, and , we consider Considering and the trace property, , for the first and fourth terms, the KCRI-I of rank 1 can be obtained as follows: which is useful because and are separated. Similarly, for arbitrary operators, , , and , Considering and for the first and eighth terms and adding which is null, to the right-hand side of (29), the KCRI-I of rank 2 can be obtained as follows: Therefore, the KCRI-I of rank  can be inferred as follows: where . The present form given by (32) is the general case. The proof for is as follows. The left-hand side (lhs) of (32) for can be expanded as follows: which is the same as the right-hand side (rhs) of (32) for .
4.2. KCRI of Second Kind (KCRI-II)
The KCRI of the second kind (KCRI-II) is given by the following: which will be used for derivation of the first-order nonlinear optical conductivity formula. Equation (34) can be proven by applying the identity on ; that is, the lhs of (34) can be changed into Applying with and gives the following: Therefore, applying results in which is reduced to the rhs of (34) by applying (35) in a reverse manner.
4.3. A Lemma Useful for Derivation of the KC Expansion Relation
A lemma useful to derive the general form of KCRI is introduced. For an analytic function of , , for . It suffices to prove the case of , because can be expanded in a Taylor series. The case for is dealt with for illustration. Using , Applying , the rhs of (40) becomes Therefore, applying in a reverse manner results in completing the proof for . The cases for and can be proven by an iterative operation of (40). Using this lemma, the calculations needed to prove the KCRI for are reduced.
4.4. KCRI of Third Kind (KCRI-III)
The KCRI of the 3rd kind (KCRI-III) can be expressed as The proof can be obtained as follows. From (37), the lhs of (43) becomes where the KC expansion relation (see (39)) was applied to the last step. Therefore, applying (37) in a reverse manner results in the right-hand side of (43). KCRI-III (see (43)) is needed in deriving the second-order nonlinear optical conductivity formula.
5. State-Dependent Projection Operators
In this section, we calculate the linear, first-, and second-order optical conductivities using the PR method.
5.1. Linear Optical Conductivity
Using the many-electron current density operator , which is written in terms of the single-electron current operator, (), as the linear optical conductivity (16) becomes where
To calculate , the first state-dependent projection operators for an arbitrary operator are introduced as follows: where
Applying the identity, , to the right-hand side of the Liouville operator in (47) as and using (35) give which can be rearranged as Here, we have used and where , , and and are the Liouville operators corresponding to and , respectively.
Note that the projection operators given in (48) are state-dependent; that is, projects an arbitrary operator onto the operator , which depends on the states and . On the other hand, state-independent projection operators project into the state-independent current density operator, . The state-independent projection operator method is applicable only to the case in which or . Cyclotron phenomenon belongs to this category because for the current operator, , , where is the cyclotron frequency. This is called the “projection criterion.” Equation (48) satisfies this criterion because .
Taking the ensemble average of (51) gives Here which can be rearranged as where we have used (35) and in the second term. To calculate (54) further, we make use of the following relation in the second term of (54): where the first and second terms are canceled out. In this stage, is assumed and the terms up to the second order of are considered, assuming that the electron-phonon interaction is quite weak. Then, the first term in (54) and the fourth term in (55) are neglected because the ensemble averages of and are zeros. Therefore, where Equation (56) is an easily calculable form because it contains neither nor the projection operator in the denominator.
5.2. First-Order Nonlinear Optical Conductivity
By inserting (45) into (21), the first-order nonlinear optical conductivity becomes where which can be calculated by defining the second state-dependent projection operators for an arbitrary operator, , as follows: where
5.2.1. Calculation of
5.2.2. Calculation of
Applying KCRI-II to (65) results in Considering that we obtain, in the second-order approximation in , where we have used (37) in the first term of (69), Therefore, inserting (67) and (71) into (63) results in in the second-order approximation in , where Note that values in (74) are already approximated up to the second order of , so is approximated to .
5.3. Second-Order Nonlinear Optical Conductivity
By applying (45) to (26), the second-order nonlinear optical conductivity becomes where To calculate (77), the third state-dependent projection operators for an arbitrary operator, , are defined as follows: where
5.3.1. Calculation of
Considering (35) with and , (82) can be expanded as follows: where Applying (35) to again with and results in in the second-order approximation in , where we have used the . Equation (87) can be calculated as follows: where we have used and
5.3.2. Calculation of
Applying KCRI-III to (83) and using result in in the second-order approximation in , where