Quadrature distribution for an optical mode in the coherent state

In summary, the task is to find the quadrature distribution ρ(q) for an optical mode in a coherent state |α> by using the formula ρ(q)=<q|\hat{\rho}|q>, where q is a generalized coordinate and n is the number of particles in the system. This formula involves calculating the sum of <q| \hat{\rho} {{\psi}_n}^* |n> for all values of n, and then using the result to find the value of ρ(q). However, the concept may be difficult to understand without further explanations or examples.
  • #1
proton4ik
15
0
Hey there, the task I'm working on is written below.

Find the quadrature distribution ρ(q), for an optical mode being in the coherent state |α>.
Hint: use ∑Hn
(x)*(t^n)/(n!)

I really am struggling with this type of tasks :D
I tried to follow a solved example that I found in my workbook, but there are no explanations and I'm really not sure if I do anything correctly.

[tex] \rho(q)=<q|\hat{\rho}|q>=\sum_{n=0}^{\infty} <q| \hat{\rho} {{\psi}_n}^* |n> = \frac {1} {1+n_{th}} \sum_{n=0}^{\infty} (\frac{n_{th}}{1+n_{th}})^n {{\psi}_n}^* |{\psi}_n (q)|^2=\frac{e^{\frac{-q^2}{1+2n_{th}}}}{{\pi (1+2n_{th})}^{1/2}} [/tex]

I have five more exercises like that, but I can't understand the concept. Any help is much appreciated!
 
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  • #2
You'll have to give more details. What is q here?
 
  • #3
DrClaude said:
You'll have to give more details. What is q here?
I think that q is a generalized coordinate
 

FAQ: Quadrature distribution for an optical mode in the coherent state

1. What is quadrature distribution for an optical mode in the coherent state?

The quadrature distribution for an optical mode in the coherent state is a probability distribution that describes the fluctuations of the electric field in a coherent laser beam. It is a measure of the amplitude and phase of the electric field at a specific point in space and time.

2. How is the quadrature distribution related to the coherent state?

The quadrature distribution is closely related to the coherent state, which is a quantum state of light that exhibits classical properties such as a well-defined amplitude and phase. The quadrature distribution is a mathematical representation of the coherent state and provides information about its properties.

3. What is the significance of the quadrature distribution in quantum optics?

The quadrature distribution is important in quantum optics because it allows us to fully characterize the properties of a coherent laser beam. It provides information about the amplitude and phase of the electric field, which are crucial for understanding the behavior of light in various optical systems.

4. How is the quadrature distribution measured experimentally?

The quadrature distribution can be measured experimentally using homodyne detection, which involves mixing the coherent laser beam with a local oscillator beam. The resulting interference pattern can be measured and used to reconstruct the quadrature distribution.

5. Are there any applications of the quadrature distribution in real-world systems?

Yes, the quadrature distribution has many practical applications in fields such as quantum communication, quantum cryptography, and quantum computing. It is also used in the design and optimization of optical systems, such as laser interferometers used in gravitational wave detection.

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