I need to have a resource in which phenomenological decoherence model are well defined and explained.
Indeed, from what I read there are three usual timescales considered to model decoherence of a qubit: $T_1,T_2,T_2^*$
$T_1$ from what I understood is the typical time for which the qubit will fall in the ground state
$T_2$ is the typical time for which the qubit will lose its coherence. More precisely that the quantum superposition between excited and ground state will become a classical mixture
$T_2^*$ is the one that confuses me the most but represents the following as far as I understood.
I take two qubits $A$ and $B$ in some superposition of excited and ground states. I call $\phi_1$ and $\phi_2$ their respective angle to the $x$ axis of the Bloch sphere.
If $T_2^*$ is infinite: $\forall t, \phi_1(t)-\phi_2(t)=\phi_1(0)-\phi_2(0)$
$T_2^*$ thus represent the typical time for which $\phi_1(t)-\phi_2(t)$ is a uniform random variable that can take any value in $[0,\pi]$ (I don't know at all what is the relative phase between the two).
I don't know if there is a way to make this definition more precise.
Now, in practice, a reasonable model for some physical situations in which we assume the noise is markovian is a master equation like:
$$ \frac{\partial \rho}{\partial t} = \frac{1}{i \hbar} [H,\rho]+\sum_i \gamma_i D[X_i](\rho) $$
Where:
$$D[X_i](\rho)=X_i \rho X_i^{\dagger} - \frac{1}{2} X_i^{\dagger} X_i \rho - \frac{1}{2} \rho X_i^{\dagger} X_i $$
And $H$ is the hamiltonian of the system that may include the drive to manipulate our qubit (if we want to perform a gate for example).
My questions are:
- How to relate this master equation to those phenomenological times ? Would I be correct to do the following:
$$(\gamma_1, X_1)=(\frac{1}{T_1}, |g\rangle \langle e|)$$ $$(\gamma_2, X_2)=(Max(\frac{1}{T_2^*},\frac{1}{T_2}) , \sigma_z)$$
Would you agree that it can be considered as a phenomenological model to describe two level system evolution ? For example if in some experiment they give values of $T_1$, $T_2$, $T_2^*$, as a first approximation the dynamic can be modelled like this
Are there good resources that state clearly what I said. I would like to have a reference clearly stating master equation description relating the $T_1$ $T_2$ $T_2^*$ like I tried to do here.
I have heard somewhere that "in general" we have: $T_2^* < T_2 < T_1$, do you confirm that it is indeed the case in most physical situations ?
Do you agree with my understanding of $T_2^*$ ? Am I wrong in my formulation of its meaning ?
Is the master equation written in the rotating frame at the frequency of the qubit (interacting picture) ? So the noises rate are indeed time independent and correspond to the inverse of the $T_i$ ?
