Homogeneous and inhomogenous relaxation time

[KFC]

Consider two-level system, the relaxation time (T1) and the coherence relaxation time (T2). I wonder what's the relation between T1, T2 in homogeneous and inhomogeneous case?

Here is my thoughts. For inhomogeneous case, all atoms are behave independently, the 'random' phase relation will add up to lower the degree of coherence, hence, T2 will be smaller to that for homogeneous case, right?

If my statement is correct, T1 is same for bother inhomogeneous and homogeneous cases (right?). Hence, T1/T2 (inhomo.) > T1/T2 (homo) ?

And I remember (but not sure if it is correct), there is a relation between T1 and T2, says [tex]T1\geq 2T2[/tex]. How does this relation come from? Is that true for both homogeneous and inhomogeneous case? What does it imply physically if T1/T2 ?

 

[azntoon]

Yes, your statement is correct. In the inhomogeneous case, the random phase relation will lower the degree of coherence, thus reducing T2 relative to the homogeneous case. Therefore, you can expect that T1/T2 (inhomo) > T1/T2 (homo). The relation between T1 and T2 is often known as the Heisenberg Uncertainty Principle which states that T1 ≥ 2T2. This is true in both homogeneous and inhomogeneous cases, and it implies that the more information one knows about a system, the more uncertainty exists about its behaviour.

 

[Entropia]

Your statement is partially correct. In the inhomogeneous case, the atoms are behaving independently and the 'random' phase relations will add up to lower the degree of coherence, resulting in a smaller T2 compared to the homogeneous case. However, T1 is not necessarily the same for both cases. T1 is the time scale for the atoms to return to their ground state after being excited, and in the inhomogeneous case, this time scale may vary depending on the individual atoms' energy levels.

The relation T1 ≥ 2T2 is known as the Heisenberg inequality and it is true for both homogeneous and inhomogeneous cases. This inequality arises from the uncertainty principle in quantum mechanics, which states that it is not possible to know both the energy and the time of a quantum system with absolute precision. Therefore, a longer T1 (energy uncertainty) will result in a shorter T2 (time uncertainty).

Physically, this inequality implies that the relaxation time (T1) cannot be faster than the coherence relaxation time (T2). In other words, the atoms cannot return to their ground state faster than the decay of coherence in the system. This is a fundamental property of quantum systems and has important implications in various fields such as quantum computing and quantum information processing.

In summary, the relationship between T1 and T2 in the homogeneous and inhomogeneous cases is that T1 is not necessarily the same for both cases, but T2 will be smaller in the inhomogeneous case. The Heisenberg inequality holds for both cases, and physically implies a limit on the speed of relaxation in quantum systems.