# Rigorous definition of phenomenological master equation for decoherence

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 $$T_i$$ ?

I think you are mostly correct (although I am not an expert).

I think it makes it easier to understand these terms to remember that these terms are used by experimentalits. They refer to things that can be measured, and their connection to theoretical time-constants is not simple.

I agree with your definitions for T1 and T2, but I would give a different definition of T2* (which might be equivalent):

• T2: The time-scale on which a single copy of my system will dephase (loose coherence and become a mixture).

• T2*: The time-scale on which a large ensemble of "copies" of my system will fall out of phase with one another. IF every single copy is identical to every other this is equal to T2. However in practice it is typical in many systems for (say) Qubit 1 to have a energy difference between ground and excited states of $$E_1$$, and qubit two to have energy gap $$E_2$$, with some small difference - its an imperfectly engineered system. Consequently you would still expect the two qubits to fall out of phase with one another even without any dephasing on either of them individually. A single Qubit or spin cannot have a T2*, only an ensemble. (Just like the standard deviation in YOUR height doesn't make sense, you need the standard deviation of a population).

By this definition T2* is always < than T2.

Another point to consider is that the decay of the excited state (the T1 process) will also destroy coherence between the ground and excited states. So the experimentally measured T2 time includes something related to the decay in addition to other effects. I am not 100% about this but my understanding is that in the absence of any additional dephasing process (any σ_z process) the measured T2 is 2*T1. This means that the dephasing timescale you want to include in your model should be considerably less than 1/T2.

(You definitely don't want to take my word for this without checkking but I think:

$$\frac{1}{T_2} = \frac{1}{2 T_1} + \frac{1}{T_{\text{dephaisng}}}$$

$$(γ1,X1)=(\frac{1}{T_1},|g⟩⟨e|)$$
$$(γ2,X2)=( (\frac{1}{T_2} - \frac{1}{2 T_1}) ,σ_z)$$