What Determines the Lineshape in a Two-Level System with Spontaneous Emission?

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In summary, the lineshape of spontaneous emission for a two-level system is obtained by taking the Fourier transform of the trace of the operator rho d.d(t), where d is the dipole moment and rho is the density matrix. The diagonal terms of the density matrix contribute to the fluorescence, while the off-diagonal terms contribute to the line shape. The relative contribution to a specific line is proportional to the upper level population, which is determined by the diagonal matrix element of rho.
  • #1
KFC
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Take two level system as example, if I know the population (density matrix element) of excited and ground state, how do I get the lineshape of the spontaenous emission? Can I take the Fourier transformation on the diagonal density matrix element?
 
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  • #2
Lineshape is FT of
Tr[d(0).d(t) rho] (if you do not include the omega**4 power which is typically
a constant over the line profile)
 
  • #3
Thanks. What is d is your explanation? dipole moment? In my calculation, I have dipole moment to be constant and rho is time dependent
 
  • #4
d is dipole moment. rho is the density matrix. Why is d constant? You need matrix elements of d to do the trace. You can get a constant reduced matrix element out of d, but you still have the Clebsch-Gordan to sum
e.g. Ly-a:(sum over m)
<100|d|21m><21m|d(t)|100>
 
  • #5
Thanks. I am thinking the following question. For two-level system, if you take an average on dipole moment (trace), because of the parity, the diagonal terms vanish. So only the off-diagonal terms contribute to Tr[d(0).d(t) rho], namely, the line shape is from the off-diagonal terms? But as I read in other books, the fluorescence is same as the spontaneous spectrum (i.e. line shape obtained above), right? The fluorescence should be proportional to the diagonal terms (i.e. population), how come does this contradiction occurs?
 
  • #6
Look in my example for Ly-a: Assuming a diagonal density matrix,
the Trace is
sum_m <21m|rho|21m><21m|d|100><100|d(t)|21m>

no parity issue here.
The relative contribution to any particular line is proportional to the upper level population,
which (for a diagonal density matrix) is proportional to the diagonal matrix element of rho.
Even this is not that simple, because what counts are ALL states that contribute to the line in question, e.g. 210,211,21-1, but NOT 310 for example. When you talk about diagonal matrix elements, you are referring to the operator :rho d.d(t), not rho
 

FAQ: What Determines the Lineshape in a Two-Level System with Spontaneous Emission?

How does the lineshape of a spectrum affect the interpretation of data?

The lineshape of a spectrum is a measure of the shape of the peaks in the spectrum, and it can provide important information about the underlying physical processes that produced the data. For example, a broad lineshape may indicate a high degree of disorder or randomness in the system, while a narrow lineshape may suggest a more ordered or organized system. Therefore, understanding the lineshape is crucial for accurately interpreting the data and drawing meaningful conclusions.

What factors contribute to the lineshape of a spectrum?

The lineshape of a spectrum is affected by a variety of factors, including the energy levels and transitions of the system, the interactions between particles or molecules, the temperature and pressure of the system, and any external influences such as magnetic or electric fields. Each of these factors can alter the lineshape in different ways, so it is important to consider all of them when trying to understand the lineshape of a spectrum.

How is the lineshape of a spectrum measured?

The lineshape of a spectrum is typically measured by plotting the intensity of the signal (y-axis) against the energy or frequency of the signal (x-axis). This results in a curve that represents the shape of the peaks in the spectrum. This curve can then be analyzed and fitted with mathematical models to determine the line width, peak position, and other important parameters of the lineshape.

What are some common lineshape models used in spectroscopy?

There are several common lineshape models used in spectroscopy, including Gaussian, Lorentzian, and Voigt profiles. Each of these models has its own strengths and weaknesses, and the choice of model depends on the specific characteristics of the spectrum being analyzed. Often, a combination of models may be used to accurately describe the lineshape and extract the most information from the data.

How can the lineshape be manipulated or controlled in an experiment?

The lineshape of a spectrum can be manipulated or controlled in various ways depending on the system being studied. For example, the temperature and pressure can be adjusted to alter the interactions between particles and change the lineshape. External influences such as magnetic or electric fields can also be used to modify the lineshape. In addition, different experimental techniques and methods can be employed to enhance or suppress certain features of the lineshape, allowing for a more detailed understanding of the system.

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