T2* (dephasing time)

Three times describe how a two-level system loses the state it was prepared in. $T_1$, the longitudinal or population lifetime, is how long an excited population lasts. $T_2$, the transverse or coherence time, is how long a superposition keeps a definite phase against the irreversible noise of its environment. $T_2^{\ast }$, read "T-two-star", is what an ensemble measurement of $T_2$ actually returns: it contains $T_2$ and, on top of it, the reversible dephasing that occurs because members of the ensemble oscillate at slightly different frequencies and drift out of step with one another. The ordering $T_2^{\ast }\le T_2\le 2T_1$ always holds. The star is inherited from nuclear magnetic resonance, where the decay of the free-induction signal in an imperfectly uniform magnet was marked $T_2^{\ast }$ to distinguish it from the intrinsic $T_2$ of the spins.

When both contributions are exponential the rates add, $1\mathrm{/}{T}_2^{\ast }=1\mathrm{/}{T}_2+1\mathrm{/}{T}_2^{\mathrm{\prime }}$, where $T_2^{\mathrm{\prime }}$ is the inhomogeneous dephasing time. In the frequency domain the same statement is that the measured line is the intrinsic (homogeneous) line broadened by the spread of transition frequencies. For a Lorentzian line the conversions are $\mathrm{\Delta }{\nu }_{\mathrm{h}\mathrm{o}\mathrm{m}}=1\mathrm{/}(\pi T_2)$ and $\mathrm{\Delta }{\nu }_{\mathrm{i}\mathrm{n}\mathrm{h}}=1\mathrm{/}(\pi T_2^{\ast })$ with widths as FWHM in hertz; for a Gaussian distribution of frequencies the free-induction envelope is itself Gaussian and its $1\mathrm{/}e$ time is $0.53\mathrm{/}\mathrm{\Delta }\nu$ rather than $0.32\mathrm{/}\mathrm{\Delta }\nu$. The numerical factor is a convention, so a quoted $T_2^{\ast }$ should travel with its definition. In the Lorentzian convention a $T_2^{\ast }$ of 1 µs corresponds to an inhomogeneous linewidth of 320 kHz.

Ramsey interferometry measures $T_2^{\ast }$: two $\pi \mathrm{/}2$ pulses separated by a free-evolution time $t$, with the fringe contrast decaying as the ensemble dephases. A Hahn echo measures $T_2$: a $\pi$ pulse inserted a time $\tau$ after the first $\pi \mathrm{/}2$ pulse reverses the phase each member has accumulated, so the static frequency spread refocuses at $2\tau$ and only the irreversible part of the decay survives. The echo is what makes the distinction operational. A static field gradient, a strain distribution, or a Doppler spread of velocities can be undone by a refocusing pulse; coupling to a fluctuating bath cannot. Dynamical-decoupling sequences (CPMG and its descendants) extend the idea with trains of $\pi$ pulses and push the measured coherence toward the ceiling of $2T_1$.

The distinction is everywhere in photonics once you look for it. A room-temperature rubidium vapor has an optical transition with a natural linewidth of 6.1 MHz ($T_2=52$ ns, radiatively limited) inside a Doppler profile 520 MHz wide, so the free-induction decay of the vapor lasts about 1 ns while each atom stays coherent fifty times longer; saturated-absorption and Ramsey-Bordé spectroscopy exist to see through the inhomogeneous envelope. A self-assembled quantum-dot ensemble has an inhomogeneous width of tens of meV (20 meV is 4.8 THz, a dephasing time of 66 fs) while a single dot's optical coherence runs to about a nanosecond, limited by its radiative lifetime, a ratio of ${10}^4$ that is the reason single-dot spectroscopy was developed. Rare-earth ions in crystals are the extreme case: europium in yttrium orthosilicate at 2 K has an optical homogeneous linewidth of 122 Hz ($T_2=2.6$ ms) inside an inhomogeneous line of order a gigahertz, seven orders of magnitude apart, which is the resource that spectral hole burning and atomic-frequency-comb quantum memories exploit.

For optically addressed spins the number sets the sensitivity of a sensor. The nitrogen-vacancy center in diamond, read out through its photoluminescence, has a spin $T_2^{\ast }$ of order a microsecond in natural-abundance diamond, limited by the bath of 1.1% carbon-13 nuclear spins and by strain; its Hahn-echo $T_2$ is hundreds of microseconds, and 1.8 ms has been reached in isotopically purified material. A DC magnetometer based on Ramsey interrogation has a shot-noise-limited sensitivity that improves as $\sqrt{T_2^{\ast }}$, whereas an AC magnetometer built on echo sequences improves as $\sqrt{T_2}$. That is why isotopic purification (to lengthen $T_2^{\ast }$) and pulsed AC protocols (to reach $T_2$) are the two routes to a better sensor, and why a data sheet that quotes $T_2$ for a DC sensing application is quoting the wrong number.

The same physics wears different clothes in laser gain media. A homogeneously broadened medium, such as a semiconductor at room temperature where carrier scattering on a 100 fs timescale sets a homogeneous width of order 10 meV, shares its gain among all modes and tends toward single-mode operation once one mode wins; an inhomogeneously broadened one, a Doppler-broadened gas or a glass host with site-to-site variation, can sustain several modes at once because different atoms feed different frequencies. Linewidth as used for lasers, the width of the emitted field's spectrum, is a different quantity from either; coherence length is its time-domain partner, where the coherence time of the field plays the role $T_2$ plays for an emitter.

References: F. Bloch, "Nuclear induction," Phys. Rev. 70, 460 (1946); E. L. Hahn, "Spin echoes," Phys. Rev. 80, 580 (1950); N. F. Ramsey, "A molecular beam resonance method with separated oscillating fields," Phys. Rev. 78, 695 (1950); G. Balasubramanian et al., "Ultralong spin coherence time in isotopically engineered diamond," Nature Materials 8, 383 (2009); R. W. Equall, Y. Sun, R. L. Cone and R. M. Macfarlane, "Ultraslow optical dephasing in Eu³⁺:Y₂SiO₅," Phys. Rev. Lett. 72, 2179 (1994); C. L. Degen, F. Reinhard and P. Cappellaro, "Quantum sensing," Rev. Mod. Phys. 89, 035002 (2017). The rubidium numbers are for the ⁸⁵Rb D2 line at 300 K (natural linewidth from D. A. Steck, Rubidium 85 D Line Data).