Time evolution of one photon

In summary, the time evolution of a single photon with vacuum state and a Jaynes-Cummings Hamiltonian is given by cos(t/ℏ)|0,1⟩+isin(t/ℏ)|1,0⟩.
  • #1
deepalakshmi
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TL;DR Summary
How to evolve one photon using hamiltonian as beam splitter
During time evolution of one photon with vacuum state with hamiltonian as a^†b+b^†a, the answer is cos(t/ℏ)|0,1⟩+isin(t/ℏ)|1,0⟩. But i don't know how to do calculation to get this answer. Can someone please help me?
I tried to do this calculation:
|0⟩|1⟩(t)=e−iHtℏ|0⟩|1⟩
=(cos(tH/ℏ)−isin(tH/ℏ)) |0⟩|1⟩
=[cos(t/ℏ)−isin(t/ℏ)] H|0⟩|1⟩
=[cos(t/ℏ)−isin(t/ℏ)] [|0⟩|1⟩+i|1⟩|0⟩]

How to proceed?
 
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  • #2
A:The Hamiltonian you are using is the Jaynes-Cummings Hamiltonian, which can be written as $$H=\hbar \omega (a^{\dagger}a+b^{\dagger}b+1/2).$$For this Hamiltonian, the time evolution of a state $|\psi(0)\rangle$ is given by $$|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle.$$You are considering the initial state to be $|\psi(0)\rangle=|0\rangle|1\rangle$. Applying the time evolution operator to this state, we have\begin{align*}|\psi(t)\rangle&=e^{-iHt/\hbar}|0\rangle|1\rangle\\&=e^{-i\hbar \omega t/2}|0\rangle|1\rangle-ie^{-i\hbar \omega t/2}|1\rangle|0\rangle\\&=\cos(\omega t/2)|0\rangle|1\rangle+\sin(\omega t/2)|1\rangle|0\rangle\end{align*}This is the answer you are looking for. Note that I have used $\hbar \omega$ instead of $t/\hbar$ to make it consistent with the Hamiltonian.
 

1. What is the concept of time evolution in relation to one photon?

The concept of time evolution refers to the change in the state of a physical system over time. In the case of one photon, it refers to how the properties and behavior of the photon change as it moves through space and interacts with other particles.

2. How does the time evolution of one photon relate to the theory of relativity?

The time evolution of one photon is closely related to the theory of relativity, specifically the concept of time dilation. According to this theory, time moves slower for objects moving at high speeds, such as photons. This means that the time evolution of a photon can be different depending on the observer's frame of reference.

3. Can the time evolution of one photon be reversed?

No, the time evolution of one photon cannot be reversed. According to the laws of physics, the arrow of time only moves in one direction. This means that once a photon has moved through space and undergone certain interactions, it cannot go back to its previous state.

4. How does the time evolution of one photon differ from that of multiple photons?

The time evolution of one photon differs from that of multiple photons because of the concept of superposition. Multiple photons can exist in a superposition state, meaning they have multiple possible states at the same time. However, one photon can only exist in one state at a time, so its time evolution is more straightforward.

5. What are some real-world applications of understanding the time evolution of one photon?

Understanding the time evolution of one photon has many practical applications, such as in the development of technologies like lasers and fiber optics. It also plays a crucial role in fields like quantum computing and telecommunications. Additionally, studying the time evolution of one photon can help us better understand the fundamental laws of physics and the behavior of particles at the smallest scales.

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