The sign of coupling Hamiltonian in CQED

  • #1
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Hi all,

I've always regarded the coupling Hamiltonian for a bosonic cavity mode coupled to a two-level fermionic gain medium chromophore to be of the form,

$$H_{coupling}=\hbar g(\sigma_{10}+\sigma_{01})(b+b^{\dagger})$$,

where ##b## and ##b^{\dagger}## and annihilation and creation operators for the bosonic cavity mode and ##\sigma_{ij}## are the raising and lowering operators for a two level atom. ##g## is the coupling constant.

Using the rotating wave approximation, this sometimes is simplified to,

$$H_{coupling}=\hbar g(\sigma_{10}b+\sigma_{01}b^{\dagger})$$.

I've recently come across some texts that seems to use an alternate form(under the rotating wave approximation) to describe (what I perceive is) the exact same system,

$$H_{coupling}=\hbar g(\sigma_{10}b-\sigma_{01}b^{\dagger})$$,

the main difference being the negative sign. Would you be able to explain why this difference occurs and what the significance of the negative sign is?

Thanks!
 

Answers and Replies

  • #2
It depends on your convention for quantizing the EM field. You either get terms like [tex]\boldsymbol{E} \sim \boldsymbol{\epsilon_k}b_k+\boldsymbol{\epsilon_k}^*b_k^\dagger [/tex] or [tex]\boldsymbol{E} \sim i(\boldsymbol{\epsilon_k}b_k -\boldsymbol{\epsilon_k}^*b_k^\dagger).[/tex]

Then when you add in the dipole-field interaction you end up with extra minus signs.
 
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  • #3
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It depends on your convention for quantizing the EM field. You either get terms like [tex]\boldsymbol{E} \sim \boldsymbol{\epsilon_k}b_k+\boldsymbol{\epsilon_k}^*b_k^\dagger [/tex] or [tex]\boldsymbol{E} \sim i(\boldsymbol{\epsilon_k}b_k -\boldsymbol{\epsilon_k}^*b_k^\dagger).[/tex]

Then when you add in the dipole-field interaction you end up with extra minus signs.

Thank you very much for the reply! Do the conventions vary based on the redefinition of the field operators ##b_k^\dagger## and ##b_k##? If what I think is correct, in one convention, the ##b_k^\dagger,b_k## is proportional to the quantized vector potential, while in the other, it's the quantized generalized momentum. Am I right here?

Thanks.
 
  • #4
There is no redefinition of the field operators. Before the quantization it is the same as taking the E or A field as the real or imaginary part of a complex function.
 
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  • #5
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There is no redefinition of the field operators. Before the quantization it is the same as taking the E or A field as the real or imaginary part of a complex function.
Thanks for that. I wrote down the derivation and I see what you mean. It basically depends on whether the cosine or sine waveform is used to describe the oscillating field.
 

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