Relaxation time and spatial width

This has significance in understanding the emission of light from a sample under different conditions.
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Consider a two-level system, let T1 be the relaxation time for the excited system and T2 is the relaxation time for coherence. If we use a Gaussian (with small constant waist w) [tex]exp(-x^2/w^2)[/tex] to shine on the sample, because the waist is small, propably only the atoms around x=0 will be excited. Note that the excited atoms will emit lights eventually. If we collect the lights from the sample, we will get a Gaussian profile with constant waist eventually. What interesting is if we do the same thing on two medium, for one T1/T2 is small but for the other, T1/T2 is big. You will find that the waist of the emitted Gaussian spatial profile for big T1/T2 is smaller than that for small T1/T2, why's that? What's the significance for big T1 and small T2? How does this affect the waist of the emitted light?
 
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The significance of having a large T1 and a small T2 is that it leads to a shorter lifetime of the excited state. This means that the atoms around x=0 will emit light faster than atoms in other regions, resulting in a narrower Gaussian profile. The smaller T2 means that the coherence between the excited and ground state will quickly diminish, which also contributes to a narrower Gaussian profile. In summary, having a large T1 and a small T2 leads to a faster relaxation from the excited state, resulting in a narrower Gaussian profile.
 
  • #3


The relaxation time and spatial width in a two-level system are important factors in understanding the behavior of excited atoms and the resulting emission of light. In this scenario, we are using a Gaussian beam to excite the atoms in the sample, with a small waist (w) indicating a focused and intense beam.

When considering the relaxation time T1, we are looking at the rate at which the excited atoms return to their ground state. A smaller T1 means that the atoms will quickly return to their ground state, resulting in a shorter lifetime for the excited state. This can be observed in the emitted light, as the Gaussian profile will have a wider waist due to a larger number of excited atoms emitting light.

On the other hand, the coherence relaxation time T2 is a measure of how long the atoms remain in a coherent superposition state before losing their phase information. A smaller T2 means that the atoms will lose their phase information quickly, resulting in a shorter coherence lifetime. In this case, the emitted light will also have a wider waist due to the larger number of atoms emitting light incoherently.

When comparing two mediums with different T1/T2 ratios, we can see that the waist of the emitted Gaussian profile is smaller for the medium with a bigger T1/T2 ratio. This is because in this medium, the atoms have a longer relaxation time (T1) and a shorter coherence time (T2). This leads to a smaller number of excited atoms and a shorter coherence lifetime, resulting in a narrower Gaussian profile.

The significance of a big T1 and a small T2 is that it indicates a longer lifetime for the excited state, but a shorter coherence lifetime. This can be useful in certain applications where a short coherence time is desired, such as in quantum information processing. It also affects the waist of the emitted light, as a shorter coherence time leads to a narrower Gaussian profile, which can have practical applications in imaging and sensing.

In conclusion, the relaxation time and spatial width in a two-level system are closely related and can provide valuable insights into the behavior of excited atoms and the resulting emission of light. The T1/T2 ratio plays a crucial role in determining the waist of the emitted Gaussian profile, highlighting the importance of understanding these parameters in various scientific and technological applications.
 

1. What is relaxation time?

Relaxation time is a measure of how quickly a system or particle returns to its equilibrium state after being disturbed.

2. How is relaxation time related to spatial width?

Relaxation time and spatial width are inversely related. As the relaxation time increases, the spatial width decreases, and vice versa.

3. What factors affect relaxation time?

Relaxation time is affected by various factors such as temperature, external fields, and interactions with other particles.

4. How is relaxation time measured?

Relaxation time can be measured through various techniques, such as spectroscopy, microscopy, or rheology.

5. Why is understanding relaxation time important in scientific research?

Relaxation time is crucial in understanding the dynamics of systems, from small particles to large-scale structures. It also provides insights into the stability and behavior of systems under different conditions.

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