Understanding the diagonal elements of the transition dipole moment

In summary, the conversation discusses the difficulty in interpreting the results of a numerical calculation of matrix elements for a superconducting qubit. The main diagonal of the matrix contains large, non-zero elements which can be interpreted as indicating allowed transitions between quantum states. However, there is uncertainty about the meaning of these values, with one possible interpretation being that the system can be polarized. Further opinions on this matter are welcome.
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TL;DR Summary
How am I supposed to interpret the diagonal elements of the transition dipole matrix. We learned that the matrix elements ##\neq 0## tells us that a certain transition is possible but what if that matrix element is on the diagonal of my matrix. A transition from ##|2\rangle## to ##|2\rangle## for example doesn't make much sense to me.
Hey guys, I just numerically calculated the matrix elements for a superconducting qubit and I'm having some trouble to interpret the result.
I will include a plot of the matrix I got below:

Unbenannt.JPG

I basically have "large" non zero elements on the main diagonal of this matrix. In previous courses I was told that for example the matrix element ##\langle 2|d|3\rangle## would indicate me that transitions for this quantum system from ##2## to ##3## and vice versa are allowed. Therefore I have a little bit of a hard time understanding the non zero values on the main diagonal.

One interpretation I have is that the system can therefore be polarized but I'm not quite sure about that.

Would be really awesome if someone could share his opinion on this, thank you!
 
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  • #2
The transition matrix can always have non-zero diagonal elements. There is some probability that no transition occurs, and that's reflected by the diagonal elements.
 
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FAQ: Understanding the diagonal elements of the transition dipole moment

What is the transition dipole moment?

The transition dipole moment is a vector quantity that describes the probability of a transition between two quantum states of a molecule, typically between an initial state and an excited state. It is defined as the integral of the product of the wavefunctions of the initial and final states and the dipole operator. It provides insight into the electronic transitions and is essential for understanding spectroscopy and photophysical processes.

Why are diagonal elements of the transition dipole moment important?

The diagonal elements of the transition dipole moment are important because they represent the transitions within the same electronic state, which can provide information about the symmetry and properties of the molecular system. These elements are crucial for understanding the intensity of spectral lines and the likelihood of certain transitions occurring, thereby influencing the overall spectroscopic behavior of the molecule.

How do you calculate the diagonal elements of the transition dipole moment?

To calculate the diagonal elements of the transition dipole moment, one typically evaluates the integral of the product of the wavefunction of the state with itself and the dipole operator. Mathematically, this is expressed as ⟨ψ|μ|ψ⟩, where ψ is the wavefunction of the state and μ is the dipole moment operator. This calculation often involves using quantum mechanical principles and may require numerical methods for complex systems.

What factors influence the diagonal elements of the transition dipole moment?

Several factors influence the diagonal elements of the transition dipole moment, including the molecular geometry, the electronic structure of the molecule, and the nature of the interactions between electrons. Additionally, the presence of external fields, such as electric or magnetic fields, can alter these elements by affecting the distribution of electronic charge and the overall symmetry of the molecule.

How do diagonal elements relate to spectroscopic transitions?

Diagonal elements of the transition dipole moment directly relate to the intensity and likelihood of spectroscopic transitions. A larger diagonal element indicates a stronger transition probability, which typically results in more intense absorption or emission lines in spectroscopic measurements. Understanding these elements helps in predicting and interpreting the spectral features observed in techniques such as UV-Vis spectroscopy and fluorescence spectroscopy.

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