What is the meaning of coherent decay rate and population decay rate? In quantum optics, especially the part when talking about atom-light interaction, there are two different kinds of decay rate, coherent decay rate $\gamma_{ij}$ where $i$ and $j$ are atomic energy states, and population decay rate $\Gamma_{ij}$.
I don't clearly understand these two. Could you explain in term of both physical and mathematical meaning?
Thank you!
 A: Density Matrices
You can model a quantum system with a density operator $\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$ , which represents a mixed state: a probabilistic mixture of pure states $|\psi_i\rangle$. If you choose a basis to represent the the state vectors $|\psi_i\rangle$, then the density operator can be written as a density matrix with elements $\rho_{ij}$ such that $\rho=\sum_{ij} \rho_{ij} |i\rangle\langle j |$.
The elements $\rho_{ii}$ of the diagonal of the density matrix are known as populations and represent the probability the system will be measured to be in a particular state. Note that populations, like probabilities, must sum to one: $\sum \rho_{ii} = 1$.
The off-diagonal elements $\rho_{ij}$ (where $i\neq j$) are known as coherences, and represent the quantum statistics of your system: the relative phases and other information is stored here. (For example, you need the coherences to tell whether the state is mixed or pure.)
Coherence and Population Decays
With the above in mind: a coherence decay $\gamma_{ij}$ likely means the coherence $\rho_{ij}$ simply has an exponential decay rate $\gamma_{ij}$ such that $\frac{\partial}{\partial t} \rho_{ij} = -\gamma_{ij} \rho_{ij}$.
In this vein, a population decay $\Gamma_{i\rightarrow j}$ would simultaneously mean $\frac{\partial}{\partial t} \rho_{ii} = -\Gamma_{i\rightarrow j} \rho_{ii}$ and
$\frac{\partial}{\partial t} \rho_{jj} = +\Gamma_{i\rightarrow j} \rho_{ii}$, so population is transferred from state $i$ to state $j$.
Physically, population and coherence decay models can be used to help describe several different phenomena. Spontaneous emission, for example, consists of both a population decay $\Gamma_{i\rightarrow j}$ to a lower state, and coherence decays $\gamma_{ik}$ between upper state $i$ and every other state $k\neq i$. Different population and coherence decays can sum together when effects are simultaneously included.
Using a Master Equation
I would recommend looking at the Lindblad equation and the Liouvillian superoperator for a better idea of how this works in practice.
For example, spontaneous decay I described above corresponds to a Lindblad jump operator $|j\rangle\langle i|$, which once plugged into the Lindblad equation generates the decay terms described above. For simple cases like this, the jump operator interpretation is quite intuitive: we stochastically, irreversibly go from state $i$ to state $j$ with jump operator $|j\rangle\langle i|$.
Note that certain combinations of coherence and population decays can lead to unphysical density matrices, and sticking to master equations like the Lindblad equation when possible helps the construction of physically sensible dynamical models.
A: I am not a specialist of quantum optics but I am familiar with laser and classical optics.
I would say that coherent decay is stimulated emission and population decay rate is simply spontaneous emission. 
When atoms/electrons are interacting with light they have a certain probability of absorbing photons to get into higher energy level states. Once they are in the higher energy states they have 2 different possibilities to relax back to original state : 
They can loose their energy by spontaneous radiation which only depends on the nature of the transition. The spontaneous emission probability can be assessed from quantum vacuum fluctuation / virtual photon. The most important is that this radiation does only depend on the transition and is random in phase and direction, hence referred to as incoherent . 
The other way to radiate back into original state is through stimulated emission. It actually consists in a radiation that is excited by another photon. The newly created photon is this time in phase with the exciting photon and has the same direction hence said as being coherent . The likelihood of stimulated emission depends on the transition and the amount of exciting photons.
Mathematically : 
The coherent decay is given by :
$\gamma_{ij}=B_{ij}u(\nu)$
The population/incoherent decay is given by : 
$\Gamma_{ij}=A_{ij}=\frac{1}{\tau_{ij}}$
$A_{ij}$ and $B_{ij}$ are the Einstein coefficients and their ratio can be established with thermodynamic considerations(photons being bosons whereas atoms follows Boltzmann statistics) : 
$\frac{A_{ij}}{B_{ij}}=\frac{8\pi h\nu^{3}}{c^{3}}$
