Relaxation Times T1 and T2

Give a two-level atom a name for its two ways of forgetting. From quantum mechanics we take the $2\times2$ density matrix $\rho$ : its diagonal entries $\rho_{ee},\rho_{gg}$ are the populations — the odds of finding the atom up or down — and its off-diagonal entry $\rho_{eg}$ is the coherence, the locked phase relation between the two states that is proportional to the induced electric dipole. These two kinds of quantity are physically different, and Felix Bloch's 1946 insight, carried unchanged from nuclear magnetic resonance into the optical Bloch equations, is that they die on two independent clocks. $T_1$ , the longitudinal relaxation time, is how fast energy leaves: spontaneous emission, inelastic collisions, and coupling to a thermal bath drive the population back toward equilibrium. In a purely radiative atom $1/T_1$ is exactly the Einstein $A$ coefficient, so $T_1$ equals the radiative lifetime $\tau$ . $T_2$ , the transverse relaxation time, is how fast the atomic dipoles lose their mutual phase. Every population-changing event scrambles the phase too, but elastic, phase-interrupting collisions scramble it further while leaving the populations untouched. That asymmetry is the whole story: $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ , so $T_2 \le 2T_1$ always, and $T_2$ can be made arbitrarily short by dephasing alone. It is why the homogeneous linewidth $\Delta\nu = 1/(\pi T_2)$ of a gas is so often broader than the natural width, why pressure broadens spectral lines, and why in a qubit the phase memory $T_2$ — not the energy memory $T_1$ — is nearly always the harder-won number.

Key takeaways

$T_1$ (longitudinal, energy, or spin-lattice relaxation) is the time in which the population difference of a two-level system returns to thermal equilibrium; it governs the decay of the diagonal density-matrix elements.
$T_2$ (transverse, or phase relaxation) is the time in which the coherence — the off-diagonal element carrying the phase of the induced dipole — decays; it sets the homogeneous linewidth $\Delta\nu = 1/(\pi T_2)$ .
The two are tied by $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ , so $T_2 \le 2T_1$ : energy decay must destroy coherence, and pure dephasing $1/T_2^{*}$ can shorten $T_2$ further without touching $T_1$ .
In the purely radiative limit $T_2^{*}\to\infty$ gives $T_2 = 2T_1 = 2\tau$ , recovering the natural linewidth $\Delta\nu = A/(2\pi)$ .
These constants enter the optical Bloch equations phenomenologically, generalizing Bloch's original nuclear-induction equations; their product $T_1 T_2$ sets the saturation of a driven line.
Key idea: populations and coherences are different objects on different clocks — $T_1$ for energy (diagonal $\rho$ ), $T_2$ for phase (off-diagonal $\rho$ ) — constrained by $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ so that $T_2 \le 2T_1$ .

The core insight

Populations and coherences are physically different objects that decay on independent timescales: $T_1$ for energy (diagonal $\rho$ ), $T_2$ for phase (off-diagonal $\rho$ ), constrained by $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ so that $T_2 \le 2T_1$ .

Where this lives today

Every magnetic-resonance image is a $T_1$ – $T_2$ map. Radiologists pick pulse sequences that weight the contrast toward one clock or the other: a $T_1$ -weighted scan brightens fat and shows anatomy, a $T_2$ -weighted scan brightens fluid and reveals oedema, and the instrumental cousin $T_2^{*}$ — static field inhomogeneity exactly like the inhomogeneous ensemble of this article — underlies functional MRI's BOLD signal. The same two numbers now govern the quantum-computing roadmap: a superconducting or trapped-ion qubit is characterized first by its $T_1$ (energy-relaxation) and $T_2$ (dephasing) times, and because $T_2\le 2T_1$ with pure dephasing usually dominant, raising $T_2$ toward its $2T_1$ ceiling — through better materials, filtered noise, and dynamical-decoupling sequences that echo away low-frequency fluctuations — is where much of the engineering effort goes. And in precision spectroscopy, separating the homogeneous $T_2$ -limited width from Doppler broadening via saturation spectroscopy is what lets an optical-clock transition be split to a part in $10^{18}$ .

The physical picture

Put the atom's state on the Bloch sphere. The vertical, north–south component records the population — how far the atom is toward excited ( $|e\rangle$ ) or ground ( $|g\rangle$ ) — while the length of the horizontal, equatorial component records the coherence, the strength of the oscillating dipole. Two different things can go wrong with this arrow, and each has its own clock. $T_1$ governs the vertical. When energy leaves the atom — a photon spontaneously emitted, an inelastic collision, a quantum handed to a cold bath — the arrow slides toward its equilibrium latitude near the pole — longitudinal relaxation changes which state the atom is in. $T_2$ governs the horizontal. Imagine not one arrow but a whole ensemble of them, all launched pointing the same way around the equator. Elastic collisions give each atom's precession a random little shove in phase without moving it off the equator; the arrows fan out, and their vector sum — the macroscopic dipole that radiates and absorbs — collapses to zero even though every single atom keeps its energy. That fanning-out is pure dephasing, and it can kill the dipole long before any energy is lost.

Figure

Two relaxations on the Bloch sphere — T₁ shortens the vertical population arrow toward equilibrium, while T₂ collapses the equatorial fan of dipole phases even when every atom keeps its energy

Why the factor of two between the two clocks? Because probability decays twice as fast as amplitude. When the population $\rho_{ee}\propto e^{-t/T_1}$ falls off, it is the probability that follows that rate; the underlying amplitude $c_e$ falls off half as fast, at $e^{-t/2T_1}$ . The coherence is built from amplitudes, $\rho_{eg}=c_e c_g^{*}$ , so energy relaxation drags it down at only half its own rate — hence a floor of $1/(2T_1)$ on the dephasing rate and the ceiling $T_2\le 2T_1$ . Any elastic dephasing sits on top of that floor. The next figure makes the consequence concrete: adding collisions crushes the coherence while leaving the population decay where it was.

Figure

Population and coherence decay on independent clocks for an atom with T₁ = 16 ns — the radiative coherence e^(−t/T₂) with T₂ = 2T₁ = 32 ns decays at half the population rate giving Δν = 9.9 MHz, while elastic collisions cut T₂ to 4 ns and broaden the line to 79.6 MHz with T₁ untouched

The core equations

The two relaxation times enter as the damping constants of the density matrix. Longitudinal relaxation drives the diagonal elements toward their thermal values; write it for the excited-state population, with all coherent drive folded into the unwritten terms.

\dot{\rho}_{ee} = -\frac{\rho_{ee}-\rho_{ee}^{\mathrm{eq}}}{T_1} + (\text{drive terms}) \qquad (1)

where: $\rho_{ee}$ — excited-state population, a diagonal density-matrix element; $\rho_{ee}^{\mathrm{eq}}$ — its thermal-equilibrium value; $T_1$ — longitudinal (energy) relaxation time. Physical reading: switch the drive off and the population relaxes to equilibrium as a single exponential with time constant $T_1$ ; this is exactly what a fluorescence-decay measurement records. Transverse relaxation damps the off-diagonal element, which also precesses at the transition frequency. The coherence carries both a fast phase rotation and a slow decay, and it is the decay rate that we name.

\dot{\rho}_{eg} = -\left(i\omega_0 + \frac{1}{T_2}\right)\rho_{eg} + (\text{drive terms}) \qquad (2)

where: $\rho_{eg}$ — coherence, the off-diagonal element, proportional to the induced dipole; $\omega_0$ — atomic transition angular frequency; $T_2$ — transverse (phase) relaxation time. Physical reading: left alone, the dipole oscillates at $\omega_0$ inside an envelope $e^{-t/T_2}$ ; the shorter $T_2$ , the faster the macroscopic polarization fades. The link between the two clocks is a sum of rates: the unavoidable contribution from energy decay plus a pure-dephasing rate from elastic events that shuffle phase without moving population. This single line contains the ceiling $T_2\le 2T_1$ .

\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_2^{*}} \qquad (3)

where: $1/(2T_1)$ — unavoidable dephasing forced by population decay; $1/T_2^{*}$ — pure elastic dephasing rate, leaves populations untouched. Physical reading: rates add, so $1/T_2$ is never smaller than $1/(2T_1)$ ; measuring both $T_1$ and $T_2$ isolates $1/T_2^{*}$ and hence the collisional or fluctuation physics. A coherence that oscillates and decays has a Lorentzian spectrum, and the homogeneous line's full width at half maximum is fixed entirely by $T_2$ . This is the number a spectroscopist reads off a scan.

\Delta\nu_{\mathrm{FWHM}} = \frac{1}{\pi T_2} \qquad (4)

where: $\Delta\nu_{\mathrm{FWHM}}$ — homogeneous full width at half maximum in Hz; $T_2$ — transverse relaxation time. Physical reading: line width and phase memory are reciprocal — a 32 ns coherence gives a 9.9 MHz line, a 4 ns coherence a 79.6 MHz line, independent of how long the atom stores its energy. Writing out that spectrum explicitly, the normalized homogeneous line shape is the Fourier transform of the damped coherence, a Lorentzian centred on $\omega_0$ .

S(\omega) = \frac{1}{\pi}\,\frac{1/T_2}{(\omega-\omega_0)^2 + (1/T_2)^2} \qquad (5)

where: $S(\omega)$ — normalized homogeneous spectral line shape; $1/T_2$ — the half-width at half maximum in angular frequency. Physical reading: the peak sits at the transition frequency, its half-width is $1/T_2$ , and its full width $2/T_2$ in angular units is the $\Delta\nu = 1/(\pi T_2)$ of equation (4).

Derivation ASplitting the dephasing rate into energy decay and pure dephasing

The goal is to explain, from the structure of the density matrix alone, why energy relaxation feeds the coherence-decay rate with a factor of one-half and not one, and why every additional elastic process can only make $T_2$ shorter. The result is the master relation $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ together with its immediate corollary $T_2\le 2T_1$ . The trick is to follow amplitudes rather than probabilities, because the coherence is bilinear in amplitudes while the population is quadratic, and that difference in degree is exactly where the factor of two is born.

Step 1. Start with the population. With energy relaxation acting, the excited-state population decays as $\rho_{ee}(t)=\rho_{ee}(0)\,e^{-t/T_1}$ , a single exponential at rate $1/T_1$ . Why: this is the definition of $T_1$ — it is read directly off a fluorescence-decay curve, and it is a statement about a probability, $\rho_{ee}=|c_e|^2$ .

Step 2. Descend to amplitudes. Since $\rho_{ee}=|c_e|^2\propto e^{-t/T_1}$ , the excited-state amplitude itself must fall off as $c_e\propto e^{-t/2T_1}$ — half the exponent, because squaring an amplitude doubles its decay rate. Why: probability is amplitude squared, so whatever kills probability at rate $1/T_1$ kills amplitude at only $1/(2T_1)$ ; this is the hinge of the whole argument.

Step 3. Build the coherence. The off-diagonal element is $\rho_{eg}=c_e\,c_g^{*}$ , so its magnitude inherits the amplitude decay, $|\rho_{eg}|\propto e^{-t/2T_1}$ . Energy relaxation therefore contributes a rate $1/(2T_1)$ to the coherence decay, exactly half the population rate. Why: the coherence is a product of amplitudes, so it dies at the amplitude rate, not the probability rate — the factor of two is structural, not a modelling choice.

Step 4. Add the elastic events. Phase-interrupting collisions randomize the phase of $\rho_{eg}$ without changing $|c_e|^2$ or $|c_g|^2$ ; being an independent Poisson process, their rate $1/T_2^{*}$ simply adds. The total transverse rate is $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ , and since $1/T_2^{*}\ge0$ we get $1/T_2\ge 1/(2T_1)$ , i.e. $T_2\le 2T_1$ . Why: independent random-phase-kicking processes contribute additive rates, so the radiative floor $1/(2T_1)$ can only be exceeded, never undercut — hence the hard ceiling on $T_2$ .

Go deeper · Derivation B · 1 minThe Lorentzian line shape as the Fourier transform of the coherence

Derivation BThe Lorentzian line shape as the Fourier transform of the coherence

The goal is to show that the homogeneous spectral line is not an independent postulate but a direct consequence of exponential coherence decay: an oscillation at $\omega_0$ inside a decaying envelope $e^{-t/T_2}$ has, by the Wiener–Khinchin route, a Lorentzian power spectrum whose half-width is precisely the decay rate $1/T_2$ . Carrying the transform through delivers both the shape of equation (5) and the width of equation (4), and pins the peak to the transition frequency. This is why phase memory and line width are two views of one number.

Step 1. Write the free coherence. After the drive is removed the dipole radiates as $\rho_{eg}(t)=\rho_{eg}(0)\,e^{-i\omega_0 t}\,e^{-t/T_2}$ for $t\ge 0$ , a one-sided damped oscillation. Why: this is the source term for the emitted field, so its frequency content is the spectrum an instrument records.

Step 2. Take the spectrum. The line shape is the squared magnitude of the Fourier transform, $S(\omega)\propto\left|\int_0^{\infty} e^{-i\omega_0 t}\,e^{-t/T_2}\,e^{i\omega t}\,\mathrm{d}t\right|^2$ , an elementary integral of a decaying exponential. Why: a stationary decaying signal has a power spectrum given by the transform of its autocorrelation, which for a single exponential reduces to this one integral.

Step 3. Evaluate and normalize. The integral gives $1/[\,1/T_2 - i(\omega-\omega_0)\,]$ ; its squared magnitude, normalized to unit area, is $S(\omega)=\frac{1}{\pi}\frac{1/T_2}{(\omega-\omega_0)^2+(1/T_2)^2}$ . The half-width at half maximum is $1/T_2$ , so the full width is $2/T_2$ in angular frequency, i.e. $\Delta\nu = 1/(\pi T_2)$ . Why: a shorter coherence time means a faster-decaying signal, and a faster decay must contain a broader spread of frequencies — the reciprocity of $T_2$ and $\Delta\nu$ is Fourier's, not the atom's.

Go deeper · Derivation C · 1 minTying T1 to the Einstein A coefficient and the natural linewidth

Derivation CTying T1 to the Einstein A coefficient and the natural linewidth

The goal is to nail $T_1$ to atomic structure in the cleanest possible case — an isolated atom in vacuum with no drive and no collisions — where the only way for energy to leave is spontaneous emission. There the longitudinal rate is nothing but the Einstein $A$ coefficient, so $T_1$ equals the radiative lifetime $\tau$ ; with no elastic dephasing the transverse time saturates its ceiling $T_2 = 2T_1$ , reproducing the natural linewidth. This anchor fixes the absolute scale of both clocks from the transition dipole moment.

Step 1. Write the decay. With no drive and no collisions the population obeys $\dot{\rho}_{ee}=-A_{eg}\,\rho_{ee}$ , spontaneous emission being the sole channel. Why: the Einstein $A$ coefficient is by definition the vacuum spontaneous-decay rate, so it is the entire longitudinal relaxation here.

Step 2. Identify $T_1$ . Integrating gives $\rho_{ee}(t)=\rho_{ee}(0)\,e^{-A_{eg}t}\equiv\rho_{ee}(0)\,e^{-t/T_1}$ , so $T_1 = 1/A_{eg} = \tau$ , the radiative lifetime. Why: matching the exponential to the definition of $T_1$ turns an abstract relaxation constant into the measurable lifetime of the excited state.

Step 3. Insert atomic structure. The $A$ coefficient is set by the transition dipole moment, $A_{eg}=\omega_0^{3}|\mathbf{d}_{eg}|^{2}/(3\pi\varepsilon_0\hbar c^{3})$ , so $1/T_1$ is fixed by the atom's own wavefunctions. Why: this closes the loop to the dipole-moment article — the timescale of energy relaxation is computed, not fitted, once the oscillator strength is known.

Step 4. Reach the radiative $T_2$ . With $T_2^{*}\to\infty$ , equation (3) gives $T_2 = 2T_1 = 2\tau$ , and the line width becomes $\Delta\nu = 1/(\pi T_2) = A/(2\pi)$ — the natural linewidth. Why: in the collision-free limit the only broadening is the finite lifetime itself, so the homogeneous width collapses onto the natural width set by $A$ .

Limiting cases

Case	Result	Meaning
Purely radiative	$T_1=\tau=1/A,\ \ T_2=2\tau$	No elastic dephasing ( $T_2^{*}\to\infty$ ). Homogeneous width equals the natural linewidth $\Delta\nu=A/(2\pi)$ .
Collision-dominated gas	$\dfrac{1}{T_2}\approx\dfrac{1}{T_2^{*}}\gg\dfrac{1}{2T_1}$	Pressure broadening: elastic collisions dominate, $T_2\ll 2T_1$ , and $\Delta\nu$ grows linearly with pressure while $T_1$ barely moves.
Cold bath ( $T\to0$ )	$1/T_1\to A,\ \ \rho_{ee}^{\mathrm{eq}}\to0$	No thermal excitation; the system relaxes to the ground state by spontaneous emission alone.
Static inhomogeneous ensemble	$1/T_2^{\mathrm{obs}}=1/T_2+\text{(inhomog.)}$	Free-induction decay is faster than $1/T_2$ ; a photon or spin echo refocuses the static spread to recover the intrinsic $T_2$ .
Strong drive (saturation)	$\Delta\nu_{\mathrm{sat}}=\Delta\nu\sqrt{1+s_0}$	Power broadening: both clocks shape the driven line, $s_0\propto T_1T_2$ .

Common trapA long T₁ does not guarantee a long T₂, and T₂ is not simply equal to T₁.

The single most common error is to treat the two clocks as one. They are not. Elastic dephasing can drive $T_2$ far below its $2T_1$ ceiling while leaving $T_1$ completely untouched — a superposition can lose its phase long before the excitation decays, which is precisely why coherence, not energy storage, is usually the binding constraint on a qubit. Equally treacherous is confusing the intrinsic homogeneous $T_2$ with the inhomogeneous $T_2^{*}$ seen in a free-induction signal from an ensemble whose atoms have slightly different transition frequencies (Doppler shifts, strain, field gradients). That ensemble dephasing is static and reversible: only a spin echo or photon echo refocuses it to reveal the true $T_2$ . Read a raw free-induction decay as $T_2$ and you will report a coherence time that is too short, sometimes by orders of magnitude.

Worked examples

1 · The sodium D₂ line, from lifetime to linewidth (confidence builder)

The sodium D₂ transition has a radiative lifetime $\tau = 16.2\ \mathrm{ns}$ and, in a dilute collision-free vapour, no elastic dephasing. Find the Einstein $A$ coefficient, $T_1$ , $T_2$ , and the natural linewidth.

Step 1. Energy clock. With spontaneous emission the only decay channel, $1/T_1 = A = 1/\tau = 1/(16.2\times10^{-9}\,\mathrm{s}) = \mathbf{6.17\times10^{7}\ s^{-1}}$ , so $T_1 = \tau = \mathbf{16.2\ ns}$ . The longitudinal time is just the radiative lifetime.

Step 2. Phase clock. No elastic dephasing means $T_2^{*}\to\infty$ , so equation (3) collapses to $1/T_2 = 1/(2T_1)$ and $T_2 = 2T_1 = \mathbf{32.4\ ns}$ . The coherence lives twice as long as the population takes to decay, because amplitude decays half as fast as probability.

Step 3. Linewidth two ways. From $T_2$ : $\Delta\nu = 1/(\pi T_2) = 1/(\pi\cdot32.4\times10^{-9}\,\mathrm{s}) = \mathbf{9.8\ MHz}$ . From $A$ : $\Delta\nu = A/(2\pi) = 6.17\times10^{7}/(2\pi) = 9.8\ \mathrm{MHz}$ — the two agree exactly, as they must in the radiative limit. A = 6.17×10⁷ s⁻¹, T₁ = 16.2 ns, T₂ = 32.4 ns, Δν = 9.8 MHz.

2 · Pressure broadening: collisions crush T₂ but spare T₁

The same sodium vapour now carries a buffer gas that produces elastic, phase-interrupting collisions at a rate $1/T_2^{*} = 5.0\times10^{8}\ \mathrm{s^{-1}}$ . Find the new $T_2$ and homogeneous linewidth, and compare with the natural line.

Step 1. Keep $T_1$ . Elastic collisions do not move population between levels, so $T_1 = 16.2\ \mathrm{ns}$ is unchanged and the radiative floor is still $1/(2T_1) = 3.09\times10^{7}\ \mathrm{s^{-1}}$ . A fluorescence-decay measurement would see no difference.

Step 2. Add the rates. Equation (3): $1/T_2 = 3.09\times10^{7} + 5.0\times10^{8} = 5.31\times10^{8}\ \mathrm{s^{-1}}$ , dominated by the collisional term. Inverting, $T_2 = \mathbf{1.88\ ns}$ — seventeen times shorter than the radiative $32.4\ \mathrm{ns}$ , and far below the $2T_1$ ceiling.

Step 3. New width. $\Delta\nu = 1/(\pi T_2) = 1/(\pi\cdot1.88\times10^{-9}\,\mathrm{s}) = \mathbf{169\ MHz}$ , a seventeen-fold broadening of the 9.8 MHz natural line — with the atom's energy storage untouched. This is exactly the fingerprint of pressure broadening: the line widens linearly with buffer-gas density while the excited-state lifetime holds firm. T₂ = 1.88 ns, Δν = 169 MHz, T₁ unchanged at 16.2 ns.

3 · A superconducting qubit: extracting the pure-dephasing time

A transmon qubit is measured to have energy-relaxation time $T_1 = 100\ \mu\mathrm{s}$ and coherence time $T_2 = 120\ \mu\mathrm{s}$ . Find the pure-dephasing time $T_\varphi\equiv T_2^{*}$ and check the fundamental inequality.

Step 1. Isolate pure dephasing. Rearranging equation (3), $1/T_\varphi = 1/T_2 - 1/(2T_1)$ . Numerically $1/T_2 = 8.33\times10^{3}\ \mathrm{s^{-1}}$ and $1/(2T_1) = 5.00\times10^{3}\ \mathrm{s^{-1}}$ , so $1/T_\varphi = 3.33\times10^{3}\ \mathrm{s^{-1}}$ .

Step 2. Invert. $T_\varphi = 1/(3.33\times10^{3}\,\mathrm{s^{-1}}) = \mathbf{300\ \mu s}$ . Low-frequency flux and charge noise, not energy loss, set this number.

Step 3. Check the ceiling. The bound is $T_2\le 2T_1 = 200\ \mu\mathrm{s}$ , and the measured $T_2 = 120\ \mu\mathrm{s}$ sits safely below it, with pure dephasing accounting for the shortfall. Reach $T_\varphi\to\infty$ and $T_2$ would climb to the $200\ \mu\mathrm{s}$ ceiling. T_φ = 300 μs, and T₂ = 120 μs < 2T₁ = 200 μs as required.

Physics of the portrait

The names $T_1$ and $T_2$ are one physicist's shorthand that quietly organized three fields — magnetic resonance, laser spectroscopy, and quantum computing. They belong to the man who had the taste to give energy and phase two separate symbols.

Origins · 1 minFelix Bloch and how this was discovered

Felix Bloch

1905–1983 · nuclear induction and the two relaxation times

Felix Bloch, a Swiss-born theorist who had already given solid-state physics its Bloch waves, turned in the 1940s to nuclear magnetism. In his 1946 paper Nuclear Induction (Phys. Rev. 70, 460) he wrote a phenomenological equation of motion for the nuclear magnetization and — crucially — damped it not with one relaxation time but with two. The longitudinal time $T_1$ set how fast the component along the field recovered toward equilibrium, exchanging energy with the lattice; the transverse time $T_2$ set how fast the perpendicular, precessing component decayed as spins lost their mutual phase. That the two need not be equal, and that $T_2$ can be much the shorter, was the physical content Purcell, Torrey, and Pound were detecting the same year. The distinction proved astonishingly portable: rewrite the magnetization as a Bloch vector for a two-level atom driven by light and Bloch's equations become the optical Bloch equations, with $T_1$ now the population time and $T_2$ the optical coherence time. Bloch shared the 1952 Nobel Prize with Purcell for the discovery of nuclear magnetic resonance; the two subscripts he chose are now spoken daily by people who image tissue, cool atoms, and build qubits.

History

1946

— Felix Bloch formulates the nuclear induction equations, defining T1 and T2

1948

— Bloembergen, Purcell and Pound relate the relaxation times to molecular motion

1950

— Erwin Hahn discovers the spin echo, separating the true T2 from inhomogeneous decay

1953

— Wangsness and Bloch derive spin relaxation from a microscopic master equation

1975

— Allen and Eberly recast T1 and T2 into the optical Bloch equations for two-level atoms

Going further

Texts. Foot, Atomic Physics, §7 (the optical Bloch equations and relaxation) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 (density matrix and damping) · Allen & Eberly, Optical Resonance and Two-Level Atoms (Bloch-vector formalism) · Saleh & Teich, Fundamentals of Photonics, 2nd ed., Ch. 13 (laser amplifiers, line shape, and relaxation) · Demtröder, Laser Spectroscopy, Vol. 1 (homogeneous versus inhomogeneous broadening).

Provenance. The definitions of $T_1$ , $T_2$ and the two-time damping structure trace to Bloch, Phys. Rev. 70, 460 (1946); their transcription into the optical Bloch equations follows Foot §7, Loudon Ch. 2, and Allen & Eberly. The relation $1/T_2 = 1/(2T_1) + 1/T_2^{*}$ and the ceiling $T_2\le 2T_1$ are standard (Bloch 1946; Demtröder). A verifier should reconfirm: (i) $\Delta\nu = 1/(\pi T_2)$ as the Fourier transform of a damped exponential; (ii) $A = \omega_0^{3}|\mathbf{d}_{eg}|^{2}/(3\pi\varepsilon_0\hbar c^{3})$ with $T_1 = 1/A$ ; (iii) the sodium D₂ numbers $\tau = 16.2$ ns, $A = 6.17\times10^{7}\ \mathrm{s^{-1}}$ , natural width $\approx 9.8$ MHz; (iv) the pressure-broadening arithmetic $T_2 = 1.88$ ns, $\Delta\nu = 169$ MHz; (v) the qubit extraction $T_\varphi = 300\ \mu$ s from $T_1 = 100\ \mu$ s, $T_2 = 120\ \mu$ s. All numerical examples are textbook-standard; no constants are fabricated.

Builds on: Spontaneous Emission · Einstein A and B Coefficients · Transition Dipole Moment · Radiative Lifetime and Natural Linewidth Leads to: Lorentzian Line Shape · Resonant Absorption · Detailed Balance and the Planck Spectrum

Ask the tutor: Want me to walk through why a spin echo refocuses $T_2^{*}$ but cannot recover the true $T_2$ , step by step on the Bloch sphere?

Are T₁ and T₂ ever equal?

Only by coincidence, and never with $T_2$ larger. In the purely radiative limit $T_2 = 2T_1$ , so the coherence clock is the longer one; as elastic dephasing is added $T_2$ falls, passing through $T_2 = T_1$ at one particular dephasing rate and continuing toward $T_2\ll T_1$ in a dense gas. So $T_2 = T_1$ is not a natural resting point, just a value $T_2$ crosses on its way down. The safe statement is the inequality $T_2\le 2T_1$ , which always holds, rather than any equality, which almost never does.

Where does the factor of two in 1/(2T₁) actually come from?

From the difference between a probability and an amplitude. The population $\rho_{ee}=|c_e|^2$ is an amplitude squared, so when it decays at rate $1/T_1$ , the amplitude $c_e$ underneath it decays at only half that rate, $1/(2T_1)$ . The coherence $\rho_{eg}=c_e c_g^{*}$ is built from amplitudes, not probabilities, so it inherits the slower amplitude rate. That is the entire origin of the one-half: energy relaxation drains the coherence at half the speed it drains the population. Any elastic dephasing then adds on top, which is why the coherence-decay rate is $1/(2T_1)$ plus $1/T_2^{*}$ , never less than the first term.

What is the difference between T₂ and T₂-star?

$T_2$ is the intrinsic homogeneous coherence time of a single atom or an ensemble of identical atoms — the real phase memory. $T_2^{*}$ (T-two-star) is the faster decay you actually see in a free-induction signal from an ensemble whose members have slightly different transition frequencies: Doppler shifts in a gas, strain in a crystal, field gradients in a magnet. Those static differences make the individual dipoles drift out of step and their sum collapse, even though each one is perfectly coherent. Because the spread is static, it is reversible: a spin echo or photon echo flips the evolution and refocuses the dipoles, cancelling the inhomogeneous part and revealing the true $T_2$ underneath. Always $1/T_2^{*}\ge 1/T_2$ .

Does a short T₂ mean the atom loses its energy quickly?

No — that is the crux of the whole topic. $T_2$ measures how fast the phase of a superposition is lost, not how fast energy leaves. A collision that merely interrupts the precession phase, leaving the atom in exactly the same energy level it started in, shortens $T_2$ while contributing nothing to $T_1$ . So you can have an atom whose excited state lives for microseconds ( $T_1$ ) while its coherence lasts only nanoseconds ( $T_2$ ). A fluorescence-decay experiment measures the energy clock $T_1$ ; a linewidth or echo experiment measures the phase clock $T_2$ ; the two answers are routinely very different, and confusing them is the classic mistake.

How do T₁ and T₂ enter the width of a driven line?

Separately, and through their product. The unsaturated homogeneous width is set by $T_2$ alone, $\Delta\nu = 1/(\pi T_2)$ . Drive the transition hard and the steady-state saturation parameter scales as $s_0\propto T_1 T_2$ — $T_1$ governing how easily the population inverts, $T_2$ the coherence — while the line power-broadens as $\Delta\nu_{\mathrm{sat}} = \Delta\nu\sqrt{1+s_0}$ . So a weak probe reads $T_2$ off the width directly, but on a strong drive both clocks shape the line, and separating them means working at low intensity or measuring the saturation explicitly.

Why is T₂ usually the harder figure of merit for a qubit?

Because storing energy and storing phase are different tasks, and phase is more fragile. A qubit's $T_1$ is limited by processes that actually move the excitation — spontaneous emission, coupling to lossy modes — which good engineering can make slow. But $T_2$ is limited by any fluctuation that jitters the transition frequency, including slow flux, charge, or field noise that never exchanges energy at all. These low-frequency fluctuations dephase the qubit while leaving $T_1$ intact, so pure dephasing usually dominates and $T_2$ sits below its $2T_1$ ceiling. Since a quantum algorithm needs a definite phase, not just a surviving excitation, $T_2$ is what caps the number of coherent gates — and why dynamical decoupling is worth the trouble.