T2 Coherence Time

A qubit's power is not in its two levels but in the phase between them. Prepare an equal superposition of $|0\rangle$ and $|1\rangle$ and the physics that matters lives entirely in a single complex number — the off-diagonal element $\rho_{01}$ of the density matrix, whose magnitude is the length of the Bloch vector on the equator and whose argument is the relative phase that makes interference possible. Let the qubit sit in a real device for a while and that number shrinks toward zero. When it reaches zero the state is a lifeless classical mixture: the populations may be untouched, but every trace of "quantumness" is gone. The transverse coherence time $T_2$ is the timescale of that shrinking, and it is the single most quoted figure of merit for whether a piece of hardware can compute at all. Two mechanisms conspire to erase the coherence. Energy relaxation — the excited state decaying toward the ground state on a timescale $T_1$ — scrambles the phase as an unavoidable side effect and, through a factor-of-two that comes from an amplitude versus a probability, sets a hard ceiling $T_2 \le 2T_1$ . Pure dephasing — random wandering of the qubit's own transition frequency from charge, flux, and magnetic noise — scrambles the phase directly, with no energy ever changing hands, on a timescale $T_\phi$ . The two rates simply add: $1/T_2 = 1/(2T_1) + 1/T_\phi$ . What makes $T_2$ genuinely subtle is that its measured value depends on how you ask: a Ramsey experiment sees the slowest drifts and reports a short $T_2^*$ ; a Hahn echo refocuses those drifts and reports a longer $T_2^{\mathrm{echo}}$ ; dynamical-decoupling pulse trains push it toward the ceiling. $T_2$ is therefore not one number but a family, indexed by the pulse sequence that filters the noise.

Key takeaways

The transverse coherence time $T_2$ is the decay timescale of the off-diagonal density-matrix element $\rho_{01}$ — the relative phase between $|0\rangle$ and $|1\rangle$ that makes a superposition genuinely quantum.
Coherence is limited by relaxation through a hard ceiling $T_2 \le 2T_1$ , with equality only when pure dephasing is absent — the factor of two comes from coherence carrying the amplitude, not the probability, of survival.
Pure dephasing (rate $1/T_\phi$ ) adds to relaxation as independent rates: $1/T_2 = 1/(2T_1) + 1/T_\phi$ , cleanly separating energy loss from phase scrambling.
$T_2$ is protocol-dependent: Ramsey free induction gives $T_2^*$ (slowest noise), a Hahn echo gives a longer $T_2^{\mathrm{echo}}$ , and dynamical decoupling pushes toward the $2T_1$ ceiling — always quote the sequence.
The lineshape encodes the environment: white noise gives exponential decay, quasi-static or $1/f$ noise gives Gaussian decay, so $T_2$ makes the qubit a spectrometer of its own bath.
Key idea: $T_2$ measures how long the relative phase of a superposition survives. It combines unavoidable relaxation-induced dephasing ( $2T_1$ ) with pure dephasing ( $T_\phi$ ) via $1/T_2 = 1/(2T_1)+1/T_\phi$ , and its measured value depends on whether echo or dynamical decoupling refocuses low-frequency noise.

Where this lives today

$T_2$ is the daily currency of quantum-hardware engineering, and the last decade has been a race to extend it. Early superconducting charge qubits had coherence times of nanoseconds; the transmon design of 2007 — flattening the qubit's frequency against charge noise by operating deep in the $E_J/E_C\gg1$ regime, so $\partial\omega_0/\partial n_g$ is exponentially suppressed — lifted $T_2$ into the microseconds and then, with better materials, the hundreds of microseconds. Replacing niobium with tantalum in the capacitor pads (Place et al., 2021) pushed transmon $T_1$ past $0.3\ \mathrm{ms}$ , and echo and CPMG sequences now routinely bring $T_2^{\mathrm{echo}}$ close to the $2T_1$ ceiling. Fluxonium qubits, sitting at a flux sweet spot immune to first-order flux noise, have reached $T_2$ beyond a millisecond, while trapped-ion hyperfine qubits — whose "spin" is a pair of atomic clock states almost decoupled from the environment — hold coherence for seconds to minutes. The same physics runs in reverse as a tool: because $1/T_2$ is the overlap of the noise spectrum with a sequence-dependent filter, sweeping CPMG pulse spacing reconstructs $S_z(\omega)$ , turning the qubit into a spectrometer that maps charge, flux, and two-level-system noise in situ. The open frontier is the noise these clean pictures miss — non-Markovian, correlated $1/f$ fluctuations from surface two-level systems, whose microscopic origin remains the dominant unsolved limit on solid-state coherence.

The physical picture

Put the qubit on the equator of the Bloch sphere — an equal-weight superposition whose Bloch vector points outward, along some azimuthal direction fixed by the relative phase. Free evolution spins that vector around the vertical axis at the transition frequency $\omega_0$ ; this is harmless, a bookkeeping precession we track in a rotating frame. The damage comes from randomness in the precession rate. If the qubit frequency fluctuates, then across many repetitions of the experiment the vector spins forward by a different amount each time, and an ensemble of runs fans out into an arc. The measurable coherence is the vector average of that arc, and as it spreads the average shrinks. When the arc wraps the whole equator, the average is zero: coherence is dead, even though every individual run still has a unit-length Bloch vector. Coherence is not lost by any single trajectory — it is lost by our inability to know which trajectory we are on.

Figure

Dephasing on the Bloch sphere: an equatorial superposition (blue) precesses at a fluctuating rate, so repeated runs fan into an arc and the ensemble-averaged Bloch vector (red) shrinks — its equatorial length is the coherence, decaying as ρ₀₁(t) = ρ₀₁(0) e^(−t/T₂). The two erasure rates add: 1/T₂ = 1/(2T₁) + 1/T_φ

Pure dephasing is a mistuned clock. The qubit accumulates phase $\phi(t) = \int_0^t \delta\omega(t')\,\mathrm{d}t'$ from a fluctuating frequency offset $\delta\omega$ ; when the offset wanders, repeated runs accumulate different phases and the interference contrast washes out — without any energy ever leaving the qubit. Relaxation is a different erasure: an amplitude-damping event that drops $|1\rangle \to |0\rangle$ destroys coherence too, but at half the rate it destroys excited-state population, because coherence rides the amplitude (a square root) of the survival probability. And the whole game of extending $T_2$ is a game of refocusing: a Hahn echo places a $\pi$ -pulse at the midpoint that flips the accumulated phase, cancelling any offset that is constant over the sequence, so only fast noise survives. That is why the measured coherence time depends on the pulse sequence, and why the second figure — the decay envelopes for the same qubit under Ramsey, echo, and the relaxation ceiling — is the picture a device physicist actually reads.

Figure

The same qubit, three coherence times: for T₁ = 80 µs the relaxation ceiling is 2T₁ = 160 µs (grey, dashed). Ramsey free induction decays as a Gaussian e^(−(t/T₂)²) with T₂ = 40 µs (blue); a Hahn echo refocuses the slow noise to an exponential e^(−t/T₂) with T₂ = 120 µs (green). Each curve crosses the 1/e line at its own coherence time

The core equations

The state of a single qubit in contact with an environment is a $2\times2$ density matrix. Its diagonal holds the populations and its off-diagonal holds the coherences; everything about $T_2$ is the story of what happens to those off-diagonal entries.

\rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} \qquad (1)

where: $\rho_{00},\rho_{11}$ — populations of $|0\rangle,|1\rangle$ ; real, nonnegative, and summing to one; $\rho_{01}=\rho_{10}^*$ — the coherences, whose magnitude is the equatorial Bloch-vector length. Reading: a pure superposition has $|\rho_{01}| = \tfrac12$ (maximal); a classical mixture has $\rho_{01}=0$ . Under decoherence the coherence element decays exponentially while precessing at the transition frequency. The decay envelope defines $T_2$ ; the oscillation is the free precession we usually remove by working in a rotating frame.

\rho_{01}(t) = \rho_{01}(0)\, e^{-t/T_2}\, e^{-i\omega_0 t} \qquad (2)

where: $T_2$ — the transverse (coherence) time; $\omega_0$ — the qubit transition frequency; $t$ — elapsed idle time. Reading: only the magnitude $|\rho_{01}| = |\rho_{01}(0)|\,e^{-t/T_2}$ is physical decoherence — the phase factor $e^{-i\omega_0 t}$ carries no information loss. Two independent processes attack the coherence, and their rates simply add. Relaxation contributes half of its own rate $1/T_1$ ; pure dephasing contributes all of $1/T_\phi$ . This single relation is the master equation of the whole subject.

\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi} \qquad (3)

where: $T_1$ — the longitudinal (energy-relaxation) time; $T_\phi$ — the pure-dephasing time, phase noise with no energy exchange. Reading: because both terms are nonnegative, $1/T_2$ can never fall below $1/(2T_1)$ — the origin of the ceiling below. Setting the pure-dephasing rate to zero maximizes coherence. What remains is the relaxation-limited ceiling: no amount of engineering can push $T_2$ above twice the energy-relaxation time.

T_2 \le 2T_1 \qquad (4)

where: the bound is saturated exactly when $T_\phi \to \infty$ . Reading: a measured $T_2$ larger than $2T_1$ is not exotic physics — it is a fitting or calibration error. The pure-dephasing mechanism is captured by the phase-damping channel, a probabilistic mixture of doing nothing and applying a Pauli- $Z$ . It leaves populations exactly alone and shrinks coherence by a real factor.

\mathcal{E}(\rho) = (1-p)\,\rho + p\, Z\rho Z,\qquad p=\tfrac{1}{2}\!\left(1-e^{-t/T_\phi}\right) \qquad (5)

where: $Z=\mathrm{diag}(1,-1)$ — the Pauli- $Z$ operator; $p$ — the effective phase-flip probability in time $t$ ; $\mathcal{E}$ — the dephasing channel acting on $\rho$ . Reading: since $Z\rho Z$ flips the sign of the off-diagonals, the coherence is multiplied by $(1-2p)=e^{-t/T_\phi}$ while $\rho_{00},\rho_{11}$ are untouched. Relaxation is captured by the amplitude-damping channel. Its action on coherence carries a square root, and that square root is the entire origin of the factor of two.

\mathcal{A}(\rho):\;\rho_{01}\to \sqrt{1-\gamma}\,\rho_{01},\qquad \gamma = 1-e^{-t/T_1} \qquad (6)

where: $\gamma$ — the decay probability $|1\rangle\to|0\rangle$ in time $t$ ; $\mathcal{A}$ — the amplitude-damping (relaxation) channel. Reading: population scales by $1-\gamma=e^{-t/T_1}$ but coherence by $\sqrt{1-\gamma}=e^{-t/2T_1}$ , so relaxation alone gives $T_2=2T_1$ . The protocol dependence enters through the decay envelope. A Ramsey (free-induction) experiment left exposed to slow, quasi-static frequency noise decays as a Gaussian, not an exponential, and reports the inhomogeneous time $T_2^*$ .

S_{\mathrm{Ramsey}}(t) = \tfrac{1}{2}\left[1 + e^{-(t/T_2^*)^2}\cos(\delta\, t)\right] \qquad (7)

where: $T_2^*$ — the inhomogeneous coherence time from Ramsey fringe decay; $\delta$ — the detuning between drive and qubit. Reading: the Gaussian envelope is the fingerprint of low-frequency ( $1/f$ ) noise; a flat white spectrum would instead give a pure exponential. The unifying statement is the filter-function picture: the dephasing rate is an overlap integral between the noise spectrum and a sequence-dependent filter. Choosing the pulse sequence reshapes $|F(\omega)|^2$ and thereby the measured $T_2$ .

\frac{1}{T_2} = \frac{1}{2T_1} + \int_{-\infty}^{\infty} \frac{\mathrm{d}\omega}{2\pi}\, |F(\omega)|^2\, S_z(\omega) \qquad (8)

where: $S_z(\omega)$ — the power spectral density of the frequency noise; $F(\omega)$ — the filter function set by the sequence (Ramsey, echo, or CPMG). Reading: Ramsey's filter peaks at $\omega=0$ (maximal exposure to slow noise); an echo pushes its filter's peak away from zero, so it rejects the slow drifts and lengthens $T_2$ .

Derivation AThe factor of two — why T₂ equals 2T₁ from amplitude damping alone

The goal is to prove, with nothing but the two Kraus operators of the amplitude-damping channel, that pure energy relaxation degrades coherence at exactly half the rate it degrades excited-state population, so that relaxation on its own pins the coherence time at $T_2=2T_1$ . This is the origin of the ceiling in equation (4), and it is worth doing explicitly because the factor of two is the single most misremembered number in the subject — it is not a convention, it is a square root, and the derivation makes plain where that square root comes from.

Step 1. Write the channel in operator-sum form, $\mathcal{A}(\rho)=K_0\rho K_0^\dagger + K_1\rho K_1^\dagger$ , with the no-jump operator $K_0=\mathrm{diag}(1,\sqrt{1-\gamma})$ and the jump operator $K_1=\sqrt{\gamma}\,|0\rangle\langle1|$ , where $\gamma=1-e^{-t/T_1}$ . Why: $K_1$ is the literal emission event dropping the qubit from $|1\rangle$ to $|0\rangle$ , while $K_0$ is the "nothing was emitted" branch, which nonetheless reweights the excited amplitude — both branches must be summed because we never learn which occurred.

Step 2. Track the excited-state population by reading the lower-right entry of $\mathcal{A}(\rho)$ : only $K_0$ contributes on the diagonal there, giving $\rho_{11}\to(1-\gamma)\rho_{11}=e^{-t/T_1}\rho_{11}$ . Why: this is the definition of $T_1$ — the excited population decays with rate exactly $1/T_1$ , fixing the meaning of the relaxation time before we look at coherence.

Step 3. Track the coherence by reading the off-diagonal entry: the jump operator $K_1$ sends coherence to zero, and only the $K_0\rho K_0^\dagger$ term survives, contributing $\rho_{01}\to \sqrt{1-\gamma}\,\rho_{01}=e^{-t/2T_1}\rho_{01}$ . Why: the coherence sits between one factor of $K_0$ that carries a $1$ (the ground row) and one that carries $\sqrt{1-\gamma}$ (the excited column), so the coherence picks up a single square root where the population picked up a full factor — this asymmetry is the whole point.

Step 4. Compare the two exponents: population decays as $e^{-t/T_1}$ but coherence as $e^{-t/2T_1}$ , so matching $e^{-t/T_2}=e^{-t/2T_1}$ gives $T_2=2T_1$ . Why: with no pure dephasing this is the longest coherence time the qubit can have; any additional phase noise only adds a positive rate through equation (3), so $T_2\le2T_1$ is an inequality that relaxation alone saturates. *The amplitude-damping channel gives $T_2=2T_1$ exactly, with population factor $e^{-t/T_1}$ and coherence factor $e^{-t/2T_1}$ .*

Go deeper · Derivation B · 2 minCoherence decay from a fluctuating qubit frequency

Derivation BCoherence decay from a fluctuating qubit frequency

The goal is to derive the surviving coherence directly from a randomly wandering transition frequency, and to show that the shape of the decay — exponential versus Gaussian — is not universal but is dictated by the spectrum of the noise. This is what turns $T_2$ from a single number into a diagnostic instrument: measure the decay envelope and you have measured the character of the environment, distinguishing fast white noise from slow $1/f$ drift.

Step 1. Let the frequency fluctuate as $\omega_0+\delta\omega(t)$ with zero mean, so the coherence acquires a random phase $\phi(t)=\int_0^t \delta\omega(t')\,\mathrm{d}t'$ and $\rho_{01}(t)=\rho_{01}(0)\,e^{-i\omega_0 t}\,\langle e^{-i\phi(t)}\rangle$ , where the angle brackets average over noise realizations. Why: the measured coherence is an ensemble average over many identical runs, and it is precisely the spread of $\phi$ across those runs — not any single value of $\phi$ — that reduces the average and destroys coherence.

Step 2. For Gaussian-distributed noise the average of a phase factor closes exactly at second order, $\langle e^{-i\phi}\rangle = e^{-\frac{1}{2}\langle\phi^2\rangle}$ , so the entire problem reduces to computing the phase variance $\langle\phi^2(t)\rangle$ . Why: Gaussian statistics make the cumulant expansion truncate — higher moments vanish — so a single quantity, the variance, controls the whole decay, and this is an excellent approximation for the many-fluctuator baths of real devices.

Step 3. Express the variance through the noise power spectral density $S_z(\omega)$ : $\langle\phi^2(t)\rangle=\int \frac{\mathrm{d}\omega}{2\pi}\,S_z(\omega)\,\frac{\sin^2(\omega t/2)}{(\omega/2)^2}$ , where the $\mathrm{sinc}^2$ kernel is the Ramsey filter function $|F(\omega)|^2$ . Why: this is the filter-function picture of equation (8) made explicit — the qubit does not see the whole spectrum equally, it sees it weighted by a window set by how long, and with what pulses, it was allowed to evolve.

Step 4. Take the two limiting spectra. A flat (white) spectrum $S_z=S_0$ gives $\langle\phi^2\rangle=S_0 t$ , so $|\rho_{01}|\propto e^{-S_0 t/2}$ — a pure exponential with $1/T_\phi=S_0/2$ . A quasi-static or $1/f$ spectrum makes the variance grow as $t^2$ , giving $|\rho_{01}|\propto e^{-(t/T_2^*)^2}$ — the Gaussian Ramsey envelope of equation (7). Why: the decay shape is a direct readout of the spectrum, which is why the Ramsey lineshape on real superconducting qubits is Gaussian (dominated by slow $1/f$ charge and flux noise) rather than the textbook exponential. *This reproduces the two standard lineshapes: white noise gives an exponential decay with timescale $T_\phi$ , and quasi-static noise gives a Gaussian decay with timescale $T_2^*$ .*

Go deeper · Derivation C · 1 minHahn echo refocusing of static dephasing

Derivation CHahn echo refocusing of static dephasing

The goal is to show quantitatively how a single midpoint $\pi$ -pulse cancels any frequency offset that is constant over the sequence, so that the echo recovers coherence lost to slow noise and reports a $T_2^{\mathrm{echo}}$ longer than the Ramsey $T_2^*$ . This one trick — invented for nuclear spins in 1950 — is the seed of every dynamical-decoupling protocol used to protect qubits today, and its logic is pure geometry.

Step 1. Consider one noise realization with a fixed but unknown detuning $\delta$ , and let the qubit evolve freely for a time $t/2$ , accumulating phase $\phi_1=+\delta\,t/2$ . Why: on a single shot the slow noise looks like a constant offset, and this constant offset is exactly what a Ramsey experiment cannot distinguish from a real detuning — it is the dominant contribution to $T_2^*$ .

Step 2. Apply a $\pi$ -pulse, which swaps $|0\rangle\leftrightarrow|1\rangle$ and therefore negates the sign of all subsequently-referenced accumulated phase, $\phi\to-\phi$ . Why: flipping the two components of the superposition turns a "lead" into a matching "lag," so the phase clock that was running fast now effectively runs backward relative to the reference — this is the refocusing move.

Step 3. Let the qubit evolve for a second interval $t/2$ under the same detuning, accumulating $\phi_2=+\delta\,t/2$ , and add it to the negated first half: $\phi_{\mathrm{echo}}=-\phi_1+\phi_2=0$ . Why: because the detuning was constant across the whole sequence, the second half exactly undoes the negated first half, and the net phase error vanishes independent of the value of $\delta$ — the echo is blind to any static offset.

Step 4. Recognize what survives: only noise that changes between the two halves escapes cancellation, so the echo filters out everything below roughly $1/t$ in frequency and $T_2^{\mathrm{echo}}>T_2^*$ . Why: adding more pulses (Carr-Purcell, CPMG) narrows the filter window further and pushes coherence toward the $2T_1$ ceiling — dynamical decoupling is just this argument iterated. *This is the Hahn spin-echo argument, the basis of all dynamical decoupling: static-detuning phase cancels to zero, and $T_2^{\mathrm{echo}}>T_2^*$ follows.*

Limiting cases

Case	Result	Meaning
No pure dephasing	$T_\phi\to\infty \Rightarrow T_2=2T_1$	Coherence is limited solely by relaxation; the maximum achievable $T_2$ for a given $T_1$ .
Dephasing-dominated	$T_\phi\ll T_1 \Rightarrow T_2\approx T_\phi$	Phase noise, not energy loss, sets coherence — common when frequency fluctuations are large.
Inhomogeneous / static noise	$T_2^*\le T_2^{\mathrm{echo}}\le 2T_1$	Slow noise dominates the Ramsey decay; echo and decoupling progressively recover coherence.
White noise	$S_z=S_0 \Rightarrow$ exponential, $1/T_\phi=S_0/2$	A flat spectrum gives a pure exponential decay independent of the sequence.
Full dephasing channel	$p=\tfrac12 \Rightarrow \rho\to\mathrm{diag}(\rho_{00},\rho_{11})$	Coherence erased, populations intact — a classical bit remains.

Common trapA single quoted "T₂" is a complete specification of a qubit's coherence

It is not, and treating it as one is the most common error in reading device specs. Because real noise is correlated in time, $T_2$ depends on the pulse sequence that filters it: the Ramsey $T_2^*$ (sensitive to the slowest drifts) is generally the shortest, the Hahn-echo $T_2^{\mathrm{echo}}$ is longer, and multi-pulse dynamical decoupling can be longer still, approaching the ceiling $2T_1$ . A number quoted without its protocol is ambiguous by a factor that can exceed ten. Two further traps ride along. First, $T_2$ can never exceed $2T_1$ — a reported $T_2>2T_1$ signals a measurement or fitting error, not a record. Second, the Ramsey decay of a superconducting qubit is usually Gaussian, $e^{-(t/T_2^*)^2}$ , not the exponential the textbook two-level picture suggests, because slow $1/f$ noise dominates; fitting it to an exponential will misestimate the coherence time and hide the spectral character of the environment.

Worked examples

1 · Extracting the pure-dephasing time and the gate budget (fully worked)

A transmon is measured to have $T_1=80\ \mu\mathrm{s}$ and a Hahn-echo $T_2=120\ \mu\mathrm{s}$ . Single-qubit gates take $30\ \mathrm{ns}$ . Find the pure-dephasing time $T_\phi$ , classify the noise regime, and estimate how many gates fit inside a coherence time.

Step 1. Invert the rate relation (3) for the dephasing rate: $\dfrac{1}{T_\phi}=\dfrac{1}{T_2}-\dfrac{1}{2T_1}=\dfrac{1}{120}-\dfrac{1}{160}\ \mu\mathrm{s}^{-1}$ . Put over a common denominator: $\dfrac{1}{120}-\dfrac{1}{160}=\dfrac{4-3}{480}=\dfrac{1}{480}\ \mu\mathrm{s}^{-1}$ , so $T_\phi=480\ \mu\mathrm{s}$ .

Step 2. Compare the mechanisms. The relaxation contribution to the coherence rate is $1/(2T_1)=1/160=6.25\times10^{-3}\ \mu\mathrm{s}^{-1}$ , while the pure-dephasing contribution is $1/T_\phi=1/480=2.08\times10^{-3}\ \mu\mathrm{s}^{-1}$ . Relaxation supplies 75% of the decoherence rate and dephasing only 25% — this qubit is relaxation-dominated, sitting fairly close to its ceiling ( $T_2=120\ \mu\mathrm{s}$ against $2T_1=160\ \mu\mathrm{s}$ ).

Step 3. Estimate the circuit budget: the number of gates before coherence decays by $1/e$ is $T_2/\tau_{\mathrm{gate}}=120\ \mu\mathrm{s}/30\ \mathrm{ns}=\mathbf{4000}$ . This is the figure of merit against the third DiVincenzo criterion — thousands of operations per coherence time is what makes a transmon a serious computational element, and pushing $T_\phi$ higher (toward the relaxation ceiling) would gain only the last 25%, so the payoff now lies in improving $T_1$ itself. $T_\phi=480\ \mu\mathrm{s}$ , relaxation-dominated, roughly 4000 gates per $T_2$ .

2 · Coherence at a given idle time, Ramsey versus echo (fully worked)

The same qubit ( $T_1=80\ \mu\mathrm{s}$ ) has a Gaussian Ramsey decay with $T_2^*=40\ \mu\mathrm{s}$ and an exponential echo decay with $T_2^{\mathrm{echo}}=120\ \mu\mathrm{s}$ . After idling $t=60\ \mu\mathrm{s}$ , what fraction of the initial coherence survives in each protocol, and what is the relaxation-limited ceiling?

Step 1. Ramsey (Gaussian envelope, equation 7): $|\rho_{01}|/|\rho_{01}(0)|=e^{-(t/T_2^*)^2}=e^{-(60/40)^2}=e^{-2.25}=\mathbf{0.105}$ . Only about 10% of the coherence remains — the slow noise has fanned the ensemble almost all the way around.

Step 2. Hahn echo (exponential envelope): $|\rho_{01}|/|\rho_{01}(0)|=e^{-t/T_2^{\mathrm{echo}}}=e^{-60/120}=e^{-0.5}=\mathbf{0.607}$ . The midpoint $\pi$ -pulse refocused the static drift, so 61% of the coherence survives at the very same idle time — a sixfold improvement over Ramsey.

Step 3. The ceiling: pure relaxation would give $e^{-t/2T_1}=e^{-60/160}=e^{-0.375}=\mathbf{0.687}$ . The echo (0.607) sits just below this ceiling (0.687), confirming the qubit is close to relaxation-limited, while Ramsey (0.105) lies far below because it never refocuses the low-frequency noise. Echo recovers 61% versus Ramsey's 10%, against a 69% relaxation ceiling.

3 · Reading the environment off the lineshape

A Ramsey experiment on qubit A decays exponentially; on qubit B it decays as a Gaussian. What does each reveal about the dominant noise?

Step 1. From the derivation, an exponential Ramsey decay means the accumulated phase variance grows linearly in $t$ , the signature of a flat (white) spectrum at low frequency — qubit A's environment is effectively memoryless on the experiment timescale.

Step 2. A Gaussian decay means the variance grows as $t^2$ , the signature of quasi-static or $1/f$ noise — qubit B is dominated by slow drifts (charge or flux $1/f$ noise). Practically, qubit B stands to gain the most from a Hahn echo or dynamical decoupling, because those slow drifts are exactly what refocusing pulses cancel; qubit A's white noise is largely immune to echoing. Exponential → white noise (echo-resistant); Gaussian → 1/f noise (echo-recoverable).

Physics of the portrait

The two symbols $T_1$ and $T_2$ did not begin in a quantum computer. They were written down in 1946 for the magnetization of atomic nuclei in a magnet, in the phenomenological equations that founded nuclear magnetic resonance — and the same Bloch sphere we spin our qubits on carries the name of the man who wrote them.

Origins · 1 minFelix Bloch and how this was discovered

Felix Bloch

1905–1983 · nuclear magnetic resonance and the T₁ / T₂ relaxation times

Felix Bloch, a Swiss-American physicist and Stanford's first Nobel laureate, gave decoherence its enduring vocabulary. Trained under Heisenberg — his 1928 thesis produced the Bloch waves that underpin all of solid-state physics — he fled Europe in the 1930s and turned, at Stanford, to the magnetic behavior of nuclei. In his 1946 paper "Nuclear Induction" (Phys. Rev. 70, 460), independently of Purcell's group at Harvard, he wrote down the phenomenological equations of motion for a precessing nuclear magnetization and introduced two distinct relaxation times: the longitudinal $T_1$ for recovery of the magnetization along the field, and the transverse $T_2$ for decay of the magnetization perpendicular to it. That transverse time is exactly the coherence time of this article — the decay of the off-diagonal, phase-bearing part of the state — and the picture of a magnetization vector precessing and relaxing on a sphere is the direct ancestor of the Bloch sphere. Bloch shared the 1952 Nobel Prize in Physics with Edward Purcell for the discovery of nuclear magnetic resonance. When Hahn found the spin echo in 1950, and when superconducting-qubit experimentalists half a century later began quoting $T_1$ and $T_2$ for a single artificial atom, they were speaking Bloch's language: the qubit is a spin, its coherence is a transverse magnetization, and $T_2$ is the time that magnetization survives.

History

1946

— Bloch's "Nuclear Induction" equations define the relaxation times T₁ and T₂

1950

— Hahn discovers the spin echo, refocusing static dephasing

1950

— Ramsey introduces the method of separated oscillatory fields (Ramsey fringes)

1954

— Carr and Purcell extend the echo into a refocusing pulse train

1999

— Nakamura, Pashkin and Tsai show coherent oscillations of a superconducting qubit

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, section 8.3 (quantum operations; phase damping and amplitude damping) · Preskill, Lecture Notes for Physics 219, Chapter 3 (decoherence and master equations) · Wilde, Quantum Information Theory, Chapter 4 (dephasing and amplitude-damping channels) · Watrous, The Theory of Quantum Information, Chapter 2 (states and channels) · Krantz et al., A Quantum Engineer's Guide to Superconducting Qubits, Appl. Phys. Rev. 6, 021318 (2019). The key results throughout this article are: the amplitude-damping Kraus operators give coherence factor $\sqrt{1-\gamma}=e^{-t/2T_1}$ , hence $T_2=2T_1$ ; the dephasing channel $\mathcal{E}(\rho)=(1-p)\rho+pZ\rho Z$ attenuates coherence by $(1-2p)=e^{-t/T_\phi}$ ; the rate addition $1/T_2=1/(2T_1)+1/T_\phi$ holds with white noise giving an exponential lineshape and $1/f$ noise giving a Gaussian. Worked-example numbers: $1/T_\phi=1/120-1/160=1/480\ \mu\mathrm{s}^{-1}$ , so $T_\phi=480\ \mu\mathrm{s}$ ; $e^{-(60/40)^2}=0.105$ , $e^{-60/120}=0.607$ , $e^{-60/160}=0.687$ . Device figures ( $T_1$ past $0.3\ \mathrm{ms}$ in Ta transmons, fluxonium $T_2>1\ \mathrm{ms}$ , ion $T_2$ of seconds to minutes) are standard published ranges.

Prerequisites. DiVincenzo Criteria · Density Matrix and Bloch Sphere · Open Quantum Systems and the Master Equation. Leads to. Superconducting Qubits · The Transmon Qubit · Trapped-Ion Qubits · Dispersive Readout.

Ask the tutor: Ask me to run a superposition through relaxation and dephasing for a chosen $T_1$ , $T_\phi$ , and idle time, and watch the coherence, populations, and Bloch vector evolve step by step.

What is the difference between T₁, T₂, and T₂* in one sentence each?

$T_1$ is the energy-relaxation time — how long the excited state $|1\rangle$ survives before decaying to $|0\rangle$ , measured by preparing $|1\rangle$ and watching the population return to equilibrium. $T_2$ is the coherence (transverse) time — how long the relative phase of a superposition survives — and when measured with a refocusing Hahn echo it is written $T_2^{\mathrm{echo}}$ . $T_2^*$ is the inhomogeneous coherence time from a bare Ramsey experiment, which additionally suffers from slow, quasi-static frequency drifts that an echo would remove, so it is the shortest of the three: $T_2^*\le T_2^{\mathrm{echo}}\le 2T_1$ .

Why exactly can T₂ never exceed 2T₁?

Because relaxation alone already dephases the qubit at rate $1/(2T_1)$ , and nothing can subtract from that. When an excited amplitude decays, the amplitude-damping channel multiplies the coherence by $\sqrt{1-\gamma}=e^{-t/2T_1}$ — a square root of the population factor $e^{-t/T_1}$ — so even a perfectly noise-free environment gives $T_2=2T_1$ . Any real environment adds pure dephasing, a strictly nonnegative rate $1/T_\phi\ge0$ , through $1/T_2=1/(2T_1)+1/T_\phi$ . Adding a nonnegative number to $1/(2T_1)$ can only increase the total rate, hence only decrease $T_2$ . A measured $T_2>2T_1$ therefore always means the fit or the calibration is wrong, not that the bound was beaten.

How does a Hahn echo actually make coherence last longer if it does not touch the physical noise?

It does not slow the noise; it makes the qubit insensitive to the slow part of it. During the first half the qubit accumulates a phase from whatever frequency offset it currently feels. The midpoint $\pi$ -pulse swaps $|0\rangle$ and $|1\rangle$ , flipping the sign of all further accumulated phase, so during the second half the same offset unwinds exactly what the first half wound up. Any offset constant across the sequence cancels to zero. Only noise that changed between the two halves survives, so the echo acts as a high-pass filter rejecting the slow $1/f$ drifts that dominate $T_2^*$ . Adding more pulses (CPMG) narrows the surviving window further, which is why long pulse trains push $T_2$ almost to the $2T_1$ ceiling.

Why is the Ramsey decay Gaussian on real qubits instead of exponential?

The shape of the decay is set by how the accumulated-phase variance grows in time, and that in turn is set by the noise spectrum. Fast, memoryless (white) noise makes the variance grow linearly in $t$ , giving an exponential decay $e^{-t/T_\phi}$ — the textbook two-level answer. But solid-state qubits are dominated by slow $1/f$ charge and flux noise, which looks nearly static over a single Ramsey run; a static offset makes the phase variance grow as $t^2$ , so the coherence decays as $e^{-(t/T_2^*)^2}$ , a Gaussian. Seeing a Gaussian Ramsey envelope is thus a direct fingerprint that low-frequency noise controls the qubit, and it is why a Hahn echo — which cancels exactly that slow noise — buys such a large improvement on these devices.

If pure dephasing exchanges no energy, how can it destroy anything?

It destroys information, not energy. The coherence $\rho_{01}$ encodes a definite phase relationship between $|0\rangle$ and $|1\rangle$ — the resource that lets a superposition interfere. Pure dephasing lets the transition frequency wander randomly, so on each run the phase advances by a slightly different amount. No population ever moves — $\rho_{00}$ and $\rho_{11}$ are untouched, energy is exactly conserved — but the ensemble-averaged phase smears out and $|\rho_{01}|$ shrinks to zero. What is lost is our ability to predict the phase, and hence the interference contrast. This is why dephasing is often the dominant error even when $T_1$ is long: energy conservation offers no protection against a scrambled phase.

How is T₂ related to the DiVincenzo criteria and gate fidelity?

The third DiVincenzo criterion demands coherence times long compared to gate times, and $T_2$ is the coherence time that matters for phase-sensitive operations. The practical figure of merit is the ratio $T_2/\tau_{\mathrm{gate}}$ — the number of gates that fit inside a coherence time — and a coherence-limited gate error scales roughly as $\tau_{\mathrm{gate}}/T_2$ . A transmon with $T_2=120\ \mu\mathrm{s}$ and $30\ \mathrm{ns}$ gates therefore has a coherence-limited error near $30\ \mathrm{ns}/120\ \mu\mathrm{s}\approx2.5\times10^{-4}$ per gate, comfortably below the roughly $10^{-2}$ fault-tolerance threshold. This is exactly why $T_2$ is quoted first when a platform is assessed: it sets the depth of circuit that can run before decoherence overwhelms the signal.

Does measuring T₂ tell you anything about the environment itself?

Yes — this is one of its most useful features. The dephasing rate is an overlap integral, $1/T_2=1/(2T_1)+\int\frac{\mathrm{d}\omega}{2\pi}|F(\omega)|^2 S_z(\omega)$ , between the noise power spectrum $S_z(\omega)$ and a filter function $F(\omega)$ that you control through the pulse sequence. By changing the number and spacing of decoupling pulses (a CPMG sweep), you slide the filter across frequency and read off how $1/T_2$ responds, which inverts to give $S_z(\omega)$ itself. The qubit becomes a spectrometer of its own bath, and this "noise spectroscopy" has mapped charge, flux, and two-level-system noise in superconducting circuits and magnetic-field noise around trapped ions — diagnostics that then guide the material and design changes that raise $T_2$ .