Polarizaition and dipole moment?

[KFC]

I am learning a simple model for two-level system in which the dipole moment is written as

$$d = d_{12}|1\rangle\langle 2| + d_{21} |2\rangle\langle 1|$$

where d is the dipole moment element, "1" stands for ground state and "2" stands for excited state. For a system has the particle density be N, the polarization is

$$P = N \rangle d \langle = d_{12}\rho_{12} + d_{21} \rho_{21}$$

where $$\rho_{ij}$$ is the element of the density matrix. Now we apply an external field, with magnitue proportional to g, to the system, and suppose we'v already got the density matrix element is of the form (read it in a paper)

$$\rho_{12} = \rho_{12}^{(0)} + \rho_{12}^{(1)}ge^{-i\delta t} + \rho_{21}^{(1)*}g^*e^{i\delta t}$$

where $$\rho_{12}^{(0)}$$ is the density matrix element without external field, $$\rho_{12}^{(1)}$$ and $$\rho_{21}^{(1)*}$$ is the first-order of the corresponding matrix element when there is a very weak field of magnitude g. * stands for complex conjugate.

My question is:

1) In E&M, the linear susceptibility,$$\chi$$, should be related to polarization in the form

$$P = \chi \vec{E}$$

But in the text, it gives

$$\chi = \frac{N|d_{12}|^2}{\hbar}\rho_{12}$$

how to get this?

2) In the paper, when

$$\rho_{12} = \rho_{12}^{(0)} + \rho_{12}^{(1)}ge^{-i\delta t} + \rho_{21}^{(1)*}g^*e^{i\delta t}$$

$$\chi = \frac{N|d_{12}|^2}{\hbar}\rho_{12}^{(1)}$$

why only keep $$\rho_{12}^{(1)}$$? Where is $$\rho_{12}^{(0)}$$ and $$\rho_{21}^{(1)*}$$?

 

[eljose79]

Hello,

Thank you for your question. I can offer some clarification on the concepts presented in the forum post.

1) The linear susceptibility, \chi, is a measure of the response of a material to an external electric field. It is defined as the ratio of the induced polarization, P, to the applied electric field, \vec{E}. In the context of the two-level system, the induced polarization is given by P = N \rangle d \langle, where N is the particle density and d is the dipole moment element. Substituting this into the definition of \chi, we get:

\chi = \frac{P}{\vec{E}} = \frac{N \rangle d \langle}{\vec{E}}

Now, the electric field can be expressed as the gradient of the scalar potential, \vec{E} = -\nabla \phi. In the case of a weak field, the potential can be approximated as \phi \approx g e^{-i\delta t}, where g is the magnitude of the field and \delta is the frequency of the field. Substituting this into the expression for \vec{E}, we get:

\vec{E} = -\nabla \phi = -\nabla (ge^{-i\delta t}) = -g\nabla e^{-i\delta t}

Using the fact that \nabla e^{-i\delta t} = -i\delta e^{-i\delta t}, we can write:

\vec{E} = -g(-i\delta e^{-i\delta t}) = g i\delta e^{-i\delta t}

Substituting this into the expression for \chi, we get:

\chi = \frac{N \rangle d \langle}{g i\delta e^{-i\delta t}} = \frac{N \rangle d \langle}{g} \frac{1}{i\delta} e^{i\delta t}

Now, the density matrix element, \rho_{12}, can be written as:

\rho_{12} = \rho_{12}^{(0)} + \rho_{12}^{(1)}ge^{-i\delta t} + \rho_{21}^{(1)*}g^*e^{i\delta t}

Substituting this into the expression for \chi, we get:

\chi =

 

[Entropia]

I would first like to clarify that polarization and dipole moment are closely related concepts in the field of electromagnetism. Polarization refers to the alignment of electric dipoles within a material, while dipole moment is a measure of the strength of an electric dipole. In the context of a two-level system, dipole moment is written as a combination of ground and excited state elements, as shown in the provided formula.

To address your questions:

1) The relationship between polarization and linear susceptibility is typically given by P = \chi \vec{E}, where \vec{E} is the applied electric field and \chi is the linear susceptibility. In the provided text, the formula for \chi is derived from the dipole moment element and the particle density. This is a specific representation for a two-level system and may differ from the general formula in E&M. It is important to note that the linear susceptibility can vary depending on the system being studied.

2) The paper mentions the first-order contribution of the matrix element \rho_{12}. This means that only the terms involving the first power of the applied field are considered. The other terms, \rho_{12}^{(0)} and \rho_{21}^{(1)*}, are either constant or higher-order terms and are therefore neglected in the calculation of \chi. This is a simplification used to better understand the behavior of the system under a weak external field.

Overall, the provided text is discussing a simplified model for a two-level system and the formulas presented may not directly correspond to those in E&M. It is important to understand the assumptions and limitations of any model or theory being used in scientific research.