# What is the explicit form of $T_1$ decay channel?

I see $$T_1$$ error mentioned in many experimental papers (for example, page $$10$$ of this paper mentions such a decay.)

How do I model $$T_1$$ decay theoretically? More concretely, here's my question. Say I have a single qubit density matrix $$\rho$$, and a noise channel $$\mathcal{N}$$ that models $$T_1$$ decay. How does $$\mathcal{N}(\rho)$$ look like explicitly?

Relaxation time $$T_1$$ describes the strength of amplitude damping$$^1$$ by specifying the mean lifetime$$^2$$ of the $$|1\rangle$$ state. More precisely, $$\mathcal{N}$$ has a Kraus representation $$E_0=\begin{bmatrix}1&0\\0&\sqrt{1-\gamma}\end{bmatrix}\quad E_1=\begin{bmatrix}0&\sqrt{\gamma}\\0&0\end{bmatrix}$$
where $$\gamma=1-e^{-\frac{t}{T_1}}$$. Explicitly$$^3$$, \begin{align} \mathcal{N}\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)&=\begin{bmatrix}a+d\gamma&b\sqrt{1-\gamma}\\c\sqrt{1-\gamma}&d-d\gamma\end{bmatrix}\\ &=\begin{bmatrix}a+d(1-e^{-\frac{t}{T_1}})&be^{-\frac{t}{2T_1}}\\ce^{-\frac{t}{2T_1}}&de^{-\frac{t}{T_1}}\end{bmatrix}. \end{align} In the geometric picture of the Bloch sphere, the channel appears as a contraction pulling the sphere towards $$|0\rangle$$. In fact $$|0\rangle$$ is the channel's sole fixed point.
$$^1$$ We assume here that the bath absorbing energy from the qubit is so cold that at thermal equilibrium the qubit is in the ground state $$|0\rangle$$. If bath's temperature is higher and hence the equilibrium state has non-zero population in the $$|1\rangle$$ state, then the appropriate model is the generalized amplitude damping channel. See section $$8.3.5$$ in Nielsen & Chuang for details.
$$^2$$ The inverse $$\frac{1}{T_1}$$ is the decay rate and $$T_1\ln 2$$ is the half-life, see exponential decay for more details.
$$^3$$ Note that amplitude damping does not only reduce the occupation of the $$|1\rangle$$ state, but also acts like a dephasing channel on the off-diagonal elements, albeit at half the rate. This is a mathematical necessity since otherwise the output would not necessarily be positive semi-definite. This is why in more general noise models with both amplitude and phase damping we have $$T_2\leqslant 2T_1$$ where $$T_2$$ describes the exponential decay of the off-diagonal elements.