# The sign of coupling Hamiltonian in CQED

• I
• thariya
In summary: In the first case, the terms are cosine, while in the second case, they are sine. Is that what you're saying?Yes, that's correct.
thariya
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!

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

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

thariya
vancouver_water said:
It depends on your convention for quantizing the EM field. You either get terms like $$\boldsymbol{E} \sim \boldsymbol{\epsilon_k}b_k+\boldsymbol{\epsilon_k}^*b_k^\dagger$$ or $$\boldsymbol{E} \sim i(\boldsymbol{\epsilon_k}b_k -\boldsymbol{\epsilon_k}^*b_k^\dagger).$$

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.

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.

thariya
vancouver_water said:
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.

## 1. What is the significance of the coupling Hamiltonian in CQED?

The coupling Hamiltonian is an important component in the field of cavity quantum electrodynamics (CQED). It describes the interaction between a quantum system, such as an atom or quantum dot, and an electromagnetic field within a cavity. This interaction allows for the study and manipulation of quantum states and enables applications such as quantum computing and quantum communication.

## 2. How is the coupling Hamiltonian calculated?

The coupling Hamiltonian is typically calculated using perturbation theory, where the interaction between the quantum system and the cavity field is treated as a small perturbation. This allows for the calculation of the energy levels and transitions of the coupled system, which can then be used to study its behavior and make predictions about its dynamics.

## 3. Can the coupling Hamiltonian be controlled or manipulated?

Yes, the coupling Hamiltonian can be controlled and manipulated through various techniques such as adjusting the coupling strength between the quantum system and the cavity, changing the frequency of the cavity field, or altering the properties of the quantum system itself. These manipulations can be used to tailor the behavior of the coupled system for specific applications.

## 4. What are some real-world applications of the coupling Hamiltonian in CQED?

The coupling Hamiltonian has numerous applications in the field of CQED, including quantum information processing, quantum simulation, and quantum sensing. It is also being explored for use in technologies such as quantum computers, quantum sensors, and quantum communication devices.

## 5. Are there any limitations or challenges associated with the coupling Hamiltonian in CQED?

While the coupling Hamiltonian has many potential applications, there are also challenges associated with its use. One limitation is the potential for decoherence, where the quantum system loses its coherence due to interactions with the environment. This can impact the accuracy and stability of the system and is an ongoing challenge for researchers in the field of CQED.

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