Coherence Time Hierarchy

Every qubit is characterized by a hierarchy of timescales that quantify how long quantum information survives. Understanding the relationships between $T_1$, $T_2$, $T_2^{\ast }$, and $T_{\phi }$ is essential for diagnosing noise sources, comparing qubit technologies, and predicting gate fidelities.

The Timescales

$T_1$ — Energy relaxation time

$T_1$ characterizes the rate at which an excited qubit decays to the ground state, releasing energy to the environment. It is measured by preparing $|1⟩$ and monitoring the probability of finding $|0⟩$ as a function of delay time:

$P_1(t)=e^{-t/T_1}$

$T_1$ is set by the coupling between the qubit and dissipative channels (dielectric loss, quasiparticle tunneling, radiative decay, phonon emission, etc.). It represents an irreversible loss of both energy and information.

$T_{\phi }$ — Pure dephasing time

$T_{\phi }$ characterizes the rate at which the relative phase between $|0⟩$ and $|1⟩$ randomizes due to fluctuations in the qubit transition frequency, without energy exchange. If the qubit frequency fluctuates as $\omega (t)={\omega }_0+\delta \omega (t)$, then the accumulated phase noise is:

$⟨e^{i{\int }_0^t\delta \omega (t^{\prime })d{t}^{\prime }}⟩=e^{-t/T_{\phi }}$

(for Markovian noise; the decay envelope can be Gaussian for $1/f$ noise).

$T_2^{\ast }$ — Free induction decay time (Ramsey)

$T_2^{\ast }$ is measured in a Ramsey experiment: $\pi /2$ pulse — free evolution for time $t$ — $\pi /2$ pulse. The fringe visibility decays as:

$\mathcal{V}(t)=e^{-t/T_2^{\ast }}$

$T_2^{\ast }$ includes contributions from both relaxation and dephasing, and is also degraded by slow, quasi-static frequency fluctuations (e.g., $1/f$ flux noise, magnetic field drift).

$T_2$ — Echo (Hahn or CPMG) coherence time

$T_2$ is measured by inserting a refocusing $\pi$ pulse (Hahn echo) or a train of $\pi$ pulses (CPMG, XY-$n$ dynamical decoupling) between the Ramsey pulses. The echo refocuses quasi-static phase accumulation, making $T_2$ sensitive only to noise at frequencies $\gtrsim 1/t$.

Fundamental Relations

The total dephasing rate is the sum of the relaxation-induced dephasing rate and the pure dephasing rate:

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

This gives the fundamental bound:

$\boxed{T_2\le 2T_1}$

The factor of 2 arises because relaxation destroys coherence (the off-diagonal density matrix element ${\rho }_{01}$) at half the rate it destroys population (${\rho }_{11}$).

The hierarchy in practice is:

$T_2^{\ast }\le T_2^{\text{Hahn}}\le T_2^{\text{CPMG}}\le 2T_1$

with each successive measurement filtering out more low-frequency noise.

Noise Spectroscopy Connection

The relationship between coherence times and the noise power spectral density $S(\omega )$ is:

• $T_1$: Set by $S({\omega }_0)$, the noise spectral density at the qubit frequency:

$\frac{1}{T_1}=\frac{1}{{\hslash }^2}{\left|\frac{\partial H}{\partial \lambda }\right|}^2{S}_{\lambda }({\omega }_0)$

• $T_{\phi }$: Set by $S(\omega \to 0)$, the low-frequency noise:

$\frac{1}{T_{\phi }}=\frac{1}{2}{\left|\frac{\partial {\omega }_{01}}{\partial \lambda }\right|}^2{S}_{\lambda }(\omega \to 0)$

• $T_2^{\ast }$ vs. $T_2$: $T_2^{\ast }$ is sensitive to noise at all frequencies below $\sim 1/t$; echo sequences act as bandpass filters centered at $f_{\pi }=1/(2\tau )$ where $\tau$ is the inter-pulse spacing.

For $1/f$ noise ($S(\omega )=A/\omega$), the Ramsey decay is Gaussian ($e^{-t^2/T_2^{\ast 2}}$) while the Hahn echo decay is typically $e^{-(t/T_2{)}^3}$ or exponential, depending on the noise spectrum details.

Dynamical Decoupling

Echo and dynamical decoupling sequences extend coherence by filtering out low-frequency noise. A CPMG sequence with $N$ equally spaced $\pi$ pulses over total time $t$ creates a filter function peaked at $f=N/(2t)$:

$W_N(f)=\frac{8{\sin }^4(\pi ft/2N)}{{\cos }^2(\pi ft/N)}$

This suppresses noise at frequencies $f\ll N/(2t)$, effectively moving the sensitivity window to higher frequencies where $S(\omega )$ is typically smaller. More pulses → higher effective $T_2$, up to the $2T_1$ limit.

Platform Comparison

Platform	$T_1$	$T_2^{\ast }$	$T_2$ (echo)	$T_2/2T_1$	Dominant noise
Transmon	100–300 μs	50–150 μs	100–500 μs	0.5–0.9	Dielectric TLS, QP
Fluxonium	100 μs–1 ms	10–100 μs	100 μs–1 ms	0.3–0.9	Flux noise (away from sweet spot)
${}^{171}{\text{Yb}}^{+}$ ion	>10 s	1–10 s	>10 min	~1.0	Magnetic field fluctuations
NV center	1–10 ms	1–100 μs	0.1–1 ms	0.05–0.1	${}^{13}$C spin bath
Si spin qubit	1–10 s	1–100 μs	1–10 ms	<0.01	Charge noise, nuclear spins

Historical Context

• Hahn (1950) introduced the spin echo in NMR, the precursor to $T_2$ measurement.
• Carr & Purcell (1954) and Meiboom & Gill (1958) developed multi-pulse decoupling (CPMG).
• Ithier et al. (2005) performed the first comprehensive noise spectroscopy of a superconducting qubit using $T_1$, $T_2^{\ast }$, and $T_2^{\text{echo}}$.
• Bylander et al. (2011) demonstrated CPMG noise spectroscopy in transmons.

References

• transmon
• fluxonium
• trapped-ion-qubit
• ytterbium-hyperfine-qubit
• charge-noise-in-superconducting-qubits