Experiments sum up the noise on a qubit with two times. $T_{1}$, the relaxation time, is how long the excited state takes to decay: the populations relax as $e^{-t/T_{1}}$. $T_{2}$, the dephasing time, is how long a superposition keeps its phase: the coherences fade as $e^{-t/T_{2}}$. Decay alone already kills coherence at half its own rate, so extra dephasing only adds: $1/T_{2}=1/(2T_{1})+1/T_{\varphi}$, and $T_{2}$ can never exceed $2T_{1}$. Superconducting qubits reach a few hundred microseconds; trapped ions and nuclear spins keep their phase for seconds or more. A Ramsey measurement reads $T_{2}$ directly; a spin echo, which undoes slow drifts of the qubit’s frequency, recovers part of what a Ramsey measurement loses.

A spinning top eventually falls over; that is $T_{1}$. A crowd of tops set spinning together loses its common rhythm long before any falls, because each spins at a slightly different rate; that is $T_{2}$.

A qubit with $T_{1}=1$ and $T_{2}=0.7$, time in units of , each curve drawn relative to its value at the start: the excited population falls as , the coherence as . The dashed curve, , is the slowest a coherence can fall; extra dephasing makes it faster.

For a qubit with relaxation time $T_{1}$, pure-dephasing time $T_{\varphi}$ and dephasing time $T_{2}$:

• $T_{1}$ is set by energy exchanged with the environment; $T_{\varphi}$ by fluctuations of the qubit’s frequency that exchange no energy.
• $T_{2}\le2T_{1}$, with equality when there is no pure dephasing.

Decay, the jump $\sqrt{1/T_{1}}\ket{0}\bra{1}$, shrinks a coherence at half its rate; pure dephasing, , shrinks it at the rate . Both act at once, so their factors multiply and their rates add. Since the pure-dephasing rate cannot be negative, the total rate is at least .

The jump $\sqrt{1/T_{1}}\ket{0}\bra{1}$ shrinks coherences at half the decay rate.

The jump $\sqrt{1/2T_{\varphi}}\,\sigma_{z}$ shrinks them at the rate .

Both act together; the factors multiply.

The rates add, and $1/T_{\varphi}\ge0$.

With equality only without pure dephasing.

The excited population falls exponentially, by a factor $e$ in each relaxation time $T_{1}$.

$T_{2}$ is how long a superposition keeps its phase: the coherence of the density matrix fades by a factor $e$ in each $T_{2}$.

Decay alone already shrinks coherences at half its rate, and anything else only adds. So $T_{2}$ is at most $2T_{1}$.

The rates of the two causes of dephasing add up.

Great work! You now know how to:

• describe a qubit’s noise by $T_{1}$ and $T_{2}$
• prove that $T_{2}$ never exceeds