Master Equation Derivations: Born-Markov and Secular Approximations

Isolate a quantum system perfectly and it evolves by a unitary; couple it to the rest of the world and it does not. The microscopic Hamiltonian $H = H_S + H_B + H_I$ still generates exact unitary dynamics on system-plus-bath, but the object an experimenter actually controls — the reduced state $\rho$ of the system alone — obeys no closed equation. Tracing out an astronomically large bath leaves dynamics that are time-non-local, history-dependent, and, in general, hopeless to solve exactly. The Born–Markov–secular programme is the canonical escape: three physically transparent approximations, each tied to its own small parameter, that collapse this intractable mess into a single first-order equation $\dot\rho = \mathcal{L}\rho$ whose generator is guaranteed to produce completely positive, trace-preserving evolution. The three steps are logically distinct. Born assumes the coupling is weak enough to truncate the dynamics at second order in $H_I$ and to keep the bath frozen in its stationary state. Markov assumes the bath forgets each disturbance almost instantly — its correlation functions decay on a memory time $\tau_B$ far shorter than the system's relaxation time $\tau_R$ — turning a convolution into a local-in-time equation, the Redfield equation. Secular (rotating-wave) then discards the fast-oscillating cross terms between distinct transition frequencies, the final touch that repairs complete positivity and delivers the Lindblad (GKSL) generator. The reward is enormous reach for so few assumptions: this one derivation explains why Lindblad dynamics is everywhere, produces the qubit's amplitude-damping and dephasing channels from first principles, fixes the temperature dependence of $T_1$ and $T_2$ , and — through the KMS detailed-balance condition — guarantees relaxation to thermal equilibrium.

Key takeaways

The Born–Markov–secular chain turns the exact, time-non-local dynamics of a system coupled to a bath into a memoryless Lindblad (GKSL) generator with manifestly positive rates.
Born: weak coupling truncates the von Neumann equation at $O(H_I^2)$ and factorizes the total state as $\rho(t)\otimes\rho_B$ ; Markov: the bath memory time $\tau_B\ll\tau_R$ makes the kernel time-local, giving the Redfield equation.
Secular (rotating-wave): cross terms oscillating at Bohr-frequency differences $\omega\neq\omega'$ average to zero over the slow dissipative timescale, restoring the complete positivity that Redfield can violate.
Rates are fixed by the bath spectral density $\gamma(\omega)=\int\!\mathrm{d}\tau\,e^{i\omega\tau}\langle B^\dagger(\tau)B(0)\rangle$ , and the KMS relation $\gamma(-\omega)=e^{-\beta\omega}\gamma(\omega)$ pins the Gibbs state as the unique steady state.
Concrete outputs: the qubit amplitude-damping channel ( $T_1$ ), the dephasing channel ( $T_\varphi$ ), and the universal bound $1/T_2=1/(2T_1)+1/T_\varphi\Rightarrow T_2\le 2T_1$ .
Worked in numbers: a 5 GHz transmon at 20 mK has $\beta\omega_0\approx12$ , so thermal excitation is $\sim6\times10^{-6}$ and the zero-temperature amplitude-damping channel is essentially exact; measured $T_1=100\ \mu\mathrm{s}$ , $T_2=120\ \mu\mathrm{s}$ imply a pure-dephasing time $T_\varphi=300\ \mu\mathrm{s}$ .
Key idea: Three nested timescale separations — weak coupling ( $H_I$ small), fast bath memory ( $\tau_B\ll\tau_R$ ), and fast internal oscillation ( $1/\omega \ll \tau_R$ ) — successively reduce exact non-Markovian dynamics to a time-local, completely positive Lindblad semigroup whose jump operators and rates are fixed by the system's Bohr spectrum and the bath spectral density.

Where this lives today

The Born–Markov–secular master equation is the working noise model of the entire quantum-hardware enterprise. Every reported $T_1$ and $T_2$ on a superconducting or trapped-ion processor is a measurement of the rates in equation (9): $T_1$ probes the bath spectral density at the qubit frequency (Purcell decay, quasiparticles, dielectric loss), $T_\varphi$ probes its zero-frequency tail (flux and charge $1/f$ noise), and $T_2\le2T_1$ is a routine sanity check on calibration data. Fault-tolerance rests on the same foundation: the threshold theorems of quantum error correction assume Markovian (Lindblad) noise, so the amplitude-damping, phase-damping, and depolarizing channels in every QEC simulation are precisely the integrated maps derived here — and when Born–Markov–secular holds is exactly when standard thresholds apply rather than needing correlated or non-Markovian corrections. At the frontier, the Davies generator has become an algorithmic target: recent quantum algorithms for Gibbs-state preparation (Chen, Kastoryano, Gilyén and collaborators, 2023–2024) engineer Lindbladians of exactly Davies form to cool a quantum computer to the thermal state of a target Hamiltonian. And in structured environments — light-harvesting complexes, engineered photonic baths, strongly coupled circuit-QED — the visible failure of the secular and Markov steps drives the active research on non-Markovian and beyond-secular master equations.

The physical picture

Picture the system as a small object jostled by an enormous reservoir. Each jostle is weak — the Born assumption — so we only count single scattering events, pairs of coupling operators at order $H_I^2$ . The reservoir is so large and chaotic that the memory of any kick is erased almost instantly — the Markov assumption — so the bath autocorrelation $\langle B(\tau)B(0)\rangle$ collapses to nearly a delta function on the system's slow clock. The net effect is a steady, structureless drizzle of kicks — white noise — which is exactly what generates memoryless Lindblad dynamics. The first figure fixes the clocks in real numbers: for a superconducting qubit the bath forgets in picoseconds while the qubit relaxes over hundreds of microseconds, a separation of roughly $10^8$ that makes every approximation in the chain excellent.

Figure

A small system S which is a 5 GHz qubit is weakly coupled to a vast reservoir at T = 20 mK through the interaction H_I = Σ A⊗B; the bath forgets each kick within τ_B ≈ 1 ps while the qubit relaxes only after τ_R = T₁ ≈ 100 μs, a separation of about 10⁸ that makes the Born, Markov and secular approximations all excellent

The secular step is about averaging out interference. The system oscillates internally at its Bohr frequencies $\omega$ , and coupling terms pairing two different frequencies $\omega\neq\omega'$ beat against each other at $\omega-\omega'$ . Over the long dissipation time those beats wash out to zero, like fast ripples on a slow tide; keeping only the resonant, phase-matched terms ( $\omega=\omega'$ ) is the rotating-wave approximation, and it is precisely what cleans the generator into completely positive GKSL form. Detailed balance then hides in the surviving rates: because the bath sits at temperature $T$ , its correlation function obeys the KMS condition, forcing emission and absorption rates to satisfy $\gamma(-\omega)=e^{-\beta\omega}\gamma(\omega)$ and driving the system to the Gibbs state. The second figure is what this machinery predicts and what a laboratory measures — the relaxation curves of a qubit, with the coherence pinned inside the envelope set by $T_2\le 2T_1$ .

Figure

Qubit relaxation curves from the derived master equation for T₁ = 100 μs and T₂ = 120 μs: the excited-state population e^(−t/T₁) and the coherence e^(−t/T₂) both decay, and because pure dephasing adds to relaxation the coherence must fall inside the envelope e^(−t/2T₁) fixed by the bound T₂ ≤ 2T₁, here implying a pure-dephasing time T_φ = 300 μs

The core equations

The starting point is a closed, exact description: system, bath, and their coupling, written as a sum of Hermitian system operators tensored with Hermitian bath operators. Everything downstream is a controlled simplification of what happens when the bath is traced away.

H = H_S + H_B + H_I, \qquad H_I = \sum_\alpha A_\alpha \otimes B_\alpha \qquad (1)

where: $H_S$ — system Hamiltonian (e.g. a qubit $\tfrac{\omega_0}{2}\sigma_z$ ); $H_B$ — bath Hamiltonian (e.g. $\sum_k \omega_k b_k^\dagger b_k$ ); $H_I$ — weak system–bath interaction; $A_\alpha$ — system coupling operators; $B_\alpha$ — bath coupling operators. Reading: only $H_I$ is treated as small; $H_S$ and $H_B$ are kept exactly by working in their interaction picture. Expanding the exact von Neumann equation to second order in $H_I$ and tracing out the bath gives the Born equation. It is a closed integro-differential equation for the reduced state, still carrying a memory integral over the whole past.

\frac{\mathrm{d}\tilde\rho(t)}{\mathrm{d}t} = -\int_0^t \mathrm{d}s\, \mathrm{Tr}_B\big[\tilde H_I(t),[\tilde H_I(s),\tilde\rho(s)\otimes\rho_B]\big] \qquad (2)

where: $\tilde\rho$ — reduced state in the interaction picture; $\tilde H_I(t)=e^{i(H_S+H_B)t}H_Ie^{-i(H_S+H_B)t}$ ; $\rho_B$ — stationary bath state, $[\rho_B,H_B]=0$ ; $\mathrm{Tr}_B$ — partial trace over the bath. Reading: the double commutator is the fingerprint of second-order perturbation theory; $\tilde\rho(s)\otimes\rho_B$ is the Born factorization itself. The Markov approximation makes the equation local in time. Because the bath correlations inside the integrand die on a memory time $\tau_B\ll\tau_R$ , the state can be pulled to the present, $\tilde\rho(s)\to\tilde\rho(t)$ , and the upper limit sent to infinity, giving the Redfield equation.

\dot{\tilde\rho}(t) = -\int_0^\infty \mathrm{d}\tau\, \mathrm{Tr}_B\big[\tilde H_I(t),[\tilde H_I(t-\tau),\tilde\rho(t)\otimes\rho_B]\big] \qquad (3)

where: $\tau=t-s$ — delay variable; $\tilde\rho(t)$ — density matrix at the present time (the Markov replacement); $\int_0^\infty$ — limit extended because the integrand vanishes for $\tau>\tau_B$ . Reading: the generator now depends only on $\tilde\rho(t)$ , a time-local semigroup candidate — but complete positivity is not yet guaranteed. To reach GKSL form the coupling must be resolved into pieces oscillating at definite frequencies. Projecting each system operator onto the energy eigenspaces of $H_S$ produces eigenoperators carrying a single Bohr frequency — the natural jump operators.

A_\alpha = \sum_\omega A_\alpha(\omega), \quad A_\alpha(\omega)=\!\!\sum_{\varepsilon'-\varepsilon=\omega}\!\! \Pi(\varepsilon)\,A_\alpha\,\Pi(\varepsilon') \qquad (4)

where: $\Pi(\varepsilon)$ — projector onto the eigenspace of $H_S$ at energy $\varepsilon$ ; $\omega=\varepsilon'-\varepsilon$ — a Bohr frequency; $A_\alpha(\omega)$ — the component lowering the system energy by $\omega$ . Reading: the defining property $e^{iH_St}A_\alpha(\omega)e^{-iH_St}=e^{-i\omega t}A_\alpha(\omega)$ is what lets the secular step tell fast terms from slow. The rates are the bath spectral density, the Fourier transform of its stationary two-time correlation function; positivity of this rate matrix is the mathematical crux of complete positivity.

\gamma_{\alpha\beta}(\omega) = \int_{-\infty}^{\infty} \mathrm{d}\tau\, e^{i\omega\tau}\, \langle B_\alpha^\dagger(\tau)\,B_\beta(0)\rangle_B \qquad (5)

where: $\gamma_{\alpha\beta}(\omega)$ — bath spectral correlation tensor (a positive-semidefinite matrix for each $\omega$ ); $\langle\cdot\rangle_B$ — average over $\rho_B$ ; $B_\alpha(\tau)=e^{iH_B\tau}B_\alpha e^{-iH_B\tau}$ — freely evolving bath operator. Reading: by Bochner's theorem the Fourier transform of a positive-definite correlation function is a positive measure, so $\gamma_{\alpha\beta}(\omega)\succeq0$ . After the secular approximation the surviving terms assemble into the Davies generator — a Hamiltonian part (bare $H_S$ plus a Lamb shift) and a dissipator built from the eigenoperators with the positive rate matrix — which is exactly GKSL form.

\dot\rho = -i[H_S+H_{LS},\rho] + \sum_\omega\sum_{\alpha\beta}\gamma_{\alpha\beta}(\omega)\Big(A_\beta(\omega)\rho A_\alpha^\dagger(\omega)-\tfrac12\{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\}\Big) \qquad (6)

where: $H_{LS}=\sum_\omega\sum_{\alpha\beta}S_{\alpha\beta}(\omega)A_\alpha^\dagger(\omega)A_\beta(\omega)$ — the Lamb shift, a bath-induced renormalization of $H_S$ ; $S_{\alpha\beta}(\omega)$ — imaginary (principal-value) part of the one-sided transform; $\{\cdot,\cdot\}$ — anticommutator. Reading: because $\gamma_{\alpha\beta}(\omega)\succeq0$ , diagonalizing it yields jump operators with non-negative rates — CPTP by construction. The bath temperature enters through a single relation: the KMS condition on any thermal correlation function forces detailed balance between emission and absorption, and this selects the equilibrium state.

\gamma(-\omega) = e^{-\beta\omega}\,\gamma(\omega), \qquad \beta = \frac{1}{k_B T} \qquad (7)

where: $\beta$ — inverse bath temperature; $\gamma(\omega)$ — emission rate (release energy $\omega$ ); $\gamma(-\omega)$ — absorption rate (gain energy $\omega$ ). Reading: detailed balance makes the Gibbs state the unique stationary state, $\mathcal{L}\,e^{-\beta H_S}/Z=0$ . Specialized to a two-level system, the generator becomes the standard dissipative qubit master equation: an emission channel, an absorption channel, and a pure-dephasing channel from the zero-frequency component of the coupling.

\dot\rho = \gamma_\downarrow\Big(\sigma_-\rho\sigma_+ - \tfrac12\{\sigma_+\sigma_-,\rho\}\Big) + \gamma_\uparrow\Big(\sigma_+\rho\sigma_- - \tfrac12\{\sigma_-\sigma_+,\rho\}\Big) + \frac{\gamma_\phi}{2}\big(\sigma_z\rho\sigma_z-\rho\big) \qquad (8)

where: $\gamma_\downarrow=\gamma(\omega_0)$ — emission rate; $\gamma_\uparrow=\gamma(-\omega_0)=e^{-\beta\omega_0}\gamma_\downarrow$ — absorption rate; $\gamma_\phi$ — pure-dephasing rate from the $\omega=0$ component; $\sigma_\pm$ — raising/lowering operators. Reading: the three dissipators act on populations ( $\gamma_\downarrow,\gamma_\uparrow$ ) and coherences ( $\gamma_\downarrow,\gamma_\uparrow,\gamma_\phi$ ) respectively. Integrating equation (8) gives the two experimentally central timescales and the hard inequality that links them.

\frac{1}{T_1}=\gamma_\downarrow+\gamma_\uparrow, \qquad \frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_\varphi}, \qquad \frac{1}{T_\varphi}=\gamma_\phi \qquad (9)

where: $T_1$ — longitudinal (energy) relaxation time; $T_2$ — transverse (coherence) time; $T_\varphi$ — pure-dephasing time. Reading: since $1/T_\varphi\ge0$ , coherence can never outlive twice the energy-relaxation time, $T_2\le 2T_1$ .

Derivation AFrom the von Neumann equation to the Born and Markov generator

The goal is to carry the exact, unitary dynamics of system-plus-bath down to a closed, time-local equation for the reduced state alone, making explicit where each approximation enters and what small parameter controls it. We start from the interaction-picture von Neumann equation, integrate once, resubstitute to reach second order in $H_I$ , trace out the bath, and invoke the separation $\tau_B\ll\tau_R$ to strip out the memory. The endpoint is the Redfield equation (3), the last stop before the secular step.

Step 1. In the interaction picture the full state obeys $\dot{\tilde\rho}_{\text{tot}}(t)=-i[\tilde H_I(t),\tilde\rho_{\text{tot}}(t)]$ , with $H_S+H_B$ absorbed into the picture so only the coupling drives the evolution. Why: moving to the interaction picture keeps the free system and bath dynamics exact and isolates the small quantity $H_I$ as the only thing to expand in.

Step 2. Integrate once, $\tilde\rho_{\text{tot}}(t)=\rho_{\text{tot}}(0)-i\int_0^t\mathrm{d}s\,[\tilde H_I(s),\tilde\rho_{\text{tot}}(s)]$ , and substitute this back into the right-hand side of Step 1. This produces a term linear in $H_I$ and a double commutator quadratic in $H_I$ . Why: resubstitution is the standard trick that turns a first-order equation into an exact expression whose leading nontrivial piece is second order — the Born order.

Step 3. Trace over the bath and choose the coupling so that $\mathrm{Tr}_B[\tilde H_I(t)\rho_B]=0$ (any nonzero average is absorbed into $H_S$ ). The linear term then vanishes and only the double commutator survives: $\dot{\tilde\rho}(t)=-\int_0^t\mathrm{d}s\,\mathrm{Tr}_B[\tilde H_I(t),[\tilde H_I(s),\tilde\rho_{\text{tot}}(s)]]$ . Why: killing the first-order term is what leaves a purely dissipative second-order generator instead of a spurious coherent drift.

Step 4. Apply the Born factorization $\tilde\rho_{\text{tot}}(s)\approx\tilde\rho(s)\otimes\rho_B$ , dropping system–bath correlations that are themselves $O(H_I^2)$ inside an already second-order expression. This gives the closed Born equation (2). Why: weak coupling keeps the bath pinned near its stationary state, and correcting the factorization would only add $O(H_I^4)$ terms — negligible at this order.

Step 5. Invoke the Markov separation $\tau_B\ll\tau_R$ . The bath correlation functions confine $s$ to within $\tau_B$ of $t$ , so over that tiny window $\tilde\rho$ is essentially constant: replace $\tilde\rho(s)\to\tilde\rho(t)$ . Change variable to $\tau=t-s$ and extend the upper limit to infinity, since the integrand is already dead for $\tau>\tau_B$ . The result is the time-local Redfield equation (3). Why: this is the memoryless step — the bath forgets faster than the system moves, so the convolution collapses to a local generator with constant coefficients. *The two Markov substitutions — replacing $\tilde\rho(s)$ with $\tilde\rho(t)$ and extending the upper limit to infinity — each introduce an error of order $O(\tau_B/\tau_R)$ , which for the 5 GHz qubit of the figures is $\sim10^{-8}$ , making both steps essentially exact.*

Go deeper · Derivation B · 2 minThe secular approximation and the emergence of Lindblad form

Derivation BThe secular approximation and the emergence of Lindblad form

The goal is to show why the Redfield equation, though time-local, is not yet a legal generator, and how one further approximation repairs it into manifest GKSL form with a positive-semidefinite rate matrix. We insert the eigenoperator decomposition (4) into the Redfield equation, exhibit the fast cross-terms oscillating at Bohr-frequency differences, discard them, and split the surviving one-sided transform into a dissipative real part and a Hamiltonian imaginary part. Positivity of the rate matrix, guaranteed by Bochner's theorem, is what certifies complete positivity.

Step 1. Write the interaction-picture coupling in eigenoperators: $\tilde H_I(t)=\sum_{\alpha,\omega}e^{-i\omega t}A_\alpha(\omega)\otimes\tilde B_\alpha(t)$ , each system piece carrying a single clean phase $e^{-i\omega t}$ . Why: only in this basis does "fast oscillation" have a precise meaning — a term's time dependence is now labelled by its Bohr frequency.

Step 2. Substituting into the Redfield equation produces a double sum over frequencies with a residual phase, $\dot{\tilde\rho}=\sum_{\omega,\omega'}\sum_{\alpha\beta}e^{i(\omega'-\omega)t}\,\Gamma_{\alpha\beta}(\omega)\,[A_\beta(\omega)\tilde\rho A_\alpha^\dagger(\omega')-\dots]$ , where $\Gamma_{\alpha\beta}(\omega)=\int_0^\infty\mathrm{d}\tau\,e^{i\omega\tau}\langle\tilde B_\alpha^\dagger(\tau)\tilde B_\beta(0)\rangle$ is the one-sided transform. Why: the cross-terms with $\omega\neq\omega'$ carry the leftover phase $e^{i(\omega'-\omega)t}$ — they are the terms that can spoil positivity.

Step 3. Apply the secular (rotating-wave) approximation: whenever $|\omega'-\omega|\gg1/\tau_R$ , the factor $e^{i(\omega'-\omega)t}$ oscillates many times over the slow dissipative timescale and averages to zero, so keep only $\omega'=\omega$ , i.e. $e^{i(\omega'-\omega)t}\to\delta_{\omega\omega'}$ . Why: coarse-graining over a time long compared with $1/|\omega'-\omega|$ but short compared with $\tau_R$ erases the non-resonant terms while leaving the true dissipation untouched.

Step 4. Split the surviving one-sided transform into real and imaginary parts, $\Gamma_{\alpha\beta}(\omega)=\tfrac12\gamma_{\alpha\beta}(\omega)+iS_{\alpha\beta}(\omega)$ . The real part $\gamma_{\alpha\beta}(\omega)$ builds the dissipator; the imaginary part $S_{\alpha\beta}(\omega)$ assembles into the Lamb-shift Hamiltonian $H_{LS}$ , which commutes with $H_S$ and merely renormalizes energies. Why: separating the two parts is what disentangles genuine relaxation from a coherent frequency shift that would otherwise masquerade as dissipation.

Step 5. Diagonalize the rate matrix. Because $\gamma_{\alpha\beta}(\omega)\succeq0$ — the Fourier transform of a positive-definite stationary correlation function is a positive measure (Bochner / Wiener–Khinchin) — it decomposes into non-negative eigenvalues, giving jump operators $L_k$ with rates $\gamma_k\ge0$ . The equation is now in the defining GKSL form (6), hence CPTP. Why: positivity of the rate matrix is exactly the GKSL requirement; it is the single fact that turns a time-local candidate into a legitimate completely positive semigroup. *Positivity of $\gamma_{\alpha\beta}(\omega)$ follows from Bochner's theorem, and dropping the secular step leaves a Redfield generator that numerically yields $\rho$ with negative eigenvalues for suitable initial coherences.*

Go deeper · Derivation C · 2 minThe qubit master equation and the T1–T2 relation

Derivation CThe qubit master equation and the T1–T2 relation

The goal is to specialize the Davies generator to a single two-level system and extract, in closed form, the two decay times an experimentalist measures and the inequality that binds them. We take a transverse coupling generating emission and absorption, add a longitudinal coupling generating pure dephasing, write the equations of motion for populations and coherences, and read off $T_1$ , $T_2$ , $T_\varphi$ . The zero-temperature limit then collapses the dynamics onto the amplitude-damping channel.

Step 1. Take $H_S=\tfrac{\omega_0}{2}\sigma_z$ and a transverse coupling $A=\sigma_x$ . Its eigenoperators are $A(\omega_0)=\sigma_-$ (energy-lowering) and $A(-\omega_0)=\sigma_+$ (energy-raising), so the dissipator has emission rate $\gamma_\downarrow=\gamma(\omega_0)$ and absorption rate $\gamma_\uparrow=\gamma(-\omega_0)$ . Why: the two Bohr components of a qubit are exactly the raising and lowering operators, which is why a two-level system always produces an amplitude-damping-type dissipator.

Step 2. The population of the excited state obeys $\dot\rho_{ee}=-\gamma_\downarrow\rho_{ee}+\gamma_\uparrow\rho_{gg}$ , so it relaxes toward the steady value $\rho_{ee}^{ss}=\gamma_\uparrow/(\gamma_\downarrow+\gamma_\uparrow)$ at the total rate $1/T_1=\gamma_\downarrow+\gamma_\uparrow$ . Why: this is the defining relation for $T_1$ ; the steady state is the Boltzmann-weighted population, guaranteed by detailed balance to be the Gibbs value.

Step 3. The coherence obeys $\dot\rho_{eg}=-(i\omega_0+\tfrac12(\gamma_\downarrow+\gamma_\uparrow)+\gamma_\phi)\rho_{eg}$ . Any decay of population necessarily damps coherence at half the rate, $\tfrac12(\gamma_\downarrow+\gamma_\uparrow)=1/(2T_1)$ , and the longitudinal $\sigma_z$ coupling adds a pure-dephasing rate $\gamma_\phi=1/T_\varphi$ that touches no populations. Hence $1/T_2=1/(2T_1)+1/T_\varphi$ . Why: the two contributions to coherence loss are physically distinct — relaxation-induced and dephasing-induced — and only their sum is observable, which is the content of equation (9).

Step 4. Because $1/T_\varphi\ge0$ , the relation gives $T_2\le2T_1$ , saturated exactly when there is no pure dephasing. Why: this is a hard, model-independent bound — a measured $T_2$ exceeding $2T_1$ would signal an error or a non-Markovian effect outside this derivation.

Step 5. At $T=0$ the bath cannot supply energy, so $\gamma_\uparrow=e^{-\beta\omega_0}\gamma_\downarrow\to0$ and only emission survives. Integrating gives excited population $\rho_{ee}(t)=e^{-\gamma_\downarrow t}\rho_{ee}(0)$ and decay probability $p=1-e^{-t/T_1}$ , which is precisely the amplitude-damping channel parameter, with Kraus operators $K_0=\mathrm{diag}(1,\sqrt{1-p})$ , $K_1=\sqrt{p}\,|0\rangle\langle1|$ . Why: this identifies the microscopic $T_1$ decay with the textbook amplitude-damping channel, closing the loop from Hamiltonian to quantum channel. *The identities $\sigma_+\sigma_-=|e\rangle\langle e|$ and $(\sigma_z\rho\sigma_z-\rho)_{eg}=-2\rho_{eg}$ give the coherence decay rate $\gamma_\phi$ , and the bound $T_2\le2T_1$ follows algebraically from $1/T_\varphi\ge0$ .*

Limiting cases

The validity of each approximation is governed by a hierarchy of timescales: the bath correlation time $\tau_B$ , the system relaxation time $\tau_R\sim1/\gamma$ , and the inverse of the smallest Bohr-frequency gap $1/\Delta\omega_{\min}$ .

Case	Result	Meaning
Born (weak coupling)	$\lambda^2\tau_B^2\ll1$	Coupling strength $\lambda$ times bath memory is small, so the second-order truncation in $H_I$ is accurate and higher-order bath effects are negligible.
Markov (fast bath)	$\tau_B\ll\tau_R$	Bath correlations decay far faster than the system relaxes; the kernel is time-local. Fails for structured baths with sharp spectral features, which retain memory (non-Markovian).
Secular (RWA)	$\vert\omega-\omega'\vert\gg1/\tau_R$ for all $\omega\neq\omega'$	Bohr-frequency differences far exceed the dissipation rate, so cross-terms average out. Fails for near-degenerate transitions, needing a partial-secular or coarse-grained generator.
Zero temperature	$\beta\omega_0\gg1\Rightarrow\gamma_\uparrow\to0$	Only emission survives; the qubit dynamics reduce to the pure amplitude-damping channel with $T_2=2T_1$ (no thermal excitation).
Quantum-optical regime	$\omega_0\gg\gamma$	The transition frequency vastly exceeds the linewidth, so all three approximations hold superbly and the Lindblad equation is essentially exact.

Common trapThe Redfield equation is time-local, so it must be a valid master equation

It need not be. Dropping the secular step leaves a time-local generator that is a perfectly good differential equation but is not guaranteed completely positive: it retains cross-terms coupling coherences at distinct Bohr frequencies, and for suitable initial states these can drive an eigenvalue of $\rho$ negative — an unphysical, non-CP evolution. Only the secular approximation, which decouples those cross-terms and leaves a positive-semidefinite rate matrix $\gamma_{\alpha\beta}(\omega)\succeq0$ , restores genuine GKSL form. The converse trap is just as real: applying the secular step when two transition frequencies are nearly degenerate ( $|\omega-\omega'|\lesssim1/\tau_R$ ) wrongly discards slow cross-terms carrying real coherence dynamics, where one must keep a partial-secular or coarse-grained equation. "Time-local" and "completely positive" are simply not the same property.

Worked examples

1 · Reading the pure-dephasing time off measured T1 and T2 (fully worked)

A superconducting transmon is measured to have $T_1=100\ \mu\mathrm{s}$ and $T_2=120\ \mu\mathrm{s}$ . Is this consistent with the master equation, and what pure-dephasing time $T_\varphi$ does it imply?

Step 1. Check the hard bound first. The derivation gives $T_2\le2T_1$ , so the relaxation-limited ceiling is $2T_1=200\ \mu\mathrm{s}$ . The measured $T_2=120\ \mu\mathrm{s}$ sits comfortably below it, so the data are physically admissible; a reported $T_2>200\ \mu\mathrm{s}$ would have signalled an error or genuinely non-Markovian behaviour.

Step 2. Invert equation (9), $1/T_2=1/(2T_1)+1/T_\varphi$ , for the pure-dephasing rate: $1/T_\varphi=1/T_2-1/(2T_1)$ . Working in inverse microseconds, $1/T_\varphi=1/120-1/200=(200-120)/(120\times200)=80/24000=1/300\ \mu\mathrm{s}^{-1}$ .

Step 3. Therefore $T_\varphi=300\ \mu\mathrm{s}$ . The interpretation: energy relaxation alone would allow coherence to survive for $2T_1=200\ \mu\mathrm{s}$ , but low-frequency (flux/charge) noise coupling through the $\omega=0$ dissipator adds a pure-dephasing channel at rate $1/T_\varphi=3.3\times10^3\ \mathrm{s}^{-1}$ , pulling the observed $T_2$ down to $120\ \mu\mathrm{s}$ . The measured pair is consistent, and the extracted pure-dephasing time is $T_\varphi=300\ \mu\mathrm{s}$ .

2 · Why the zero-temperature amplitude-damping channel is exact (fully worked)

The same qubit has $\omega_0/2\pi=5\ \mathrm{GHz}$ and sits in a dilution refrigerator at $T=20\ \mathrm{mK}$ . Estimate the thermal excitation and confirm that the pure amplitude-damping (zero-temperature) channel is the right model.

Step 1. Form the dimensionless ratio $\beta\omega_0=\hbar\omega_0/(k_BT)$ . Numerically $\hbar\omega_0=\hbar(2\pi\times5\times10^9)=3.31\times10^{-24}\ \mathrm{J}$ and $k_BT=(1.38\times10^{-23})(0.020)=2.76\times10^{-25}\ \mathrm{J}$ , so $\beta\omega_0=12.0$ .

Step 2. Detailed balance (7) fixes the ratio of rates: $\gamma_\uparrow/\gamma_\downarrow=e^{-\beta\omega_0}=e^{-12}=6.2\times10^{-6}$ . The mean thermal occupation is $\bar n=1/(e^{\beta\omega_0}-1)\approx e^{-\beta\omega_0}=6.2\times10^{-6}$ — the bath is, for this qubit, essentially empty.

Step 3. The steady-state excited population follows from Step 2 of Derivation 3, $\rho_{ee}^{ss}=\gamma_\uparrow/(\gamma_\downarrow+\gamma_\uparrow)=1/(e^{\beta\omega_0}+1)=6.2\times10^{-6}$ , exactly the Gibbs (Fermi–Dirac) value — a direct check that detailed balance drives the qubit to thermal equilibrium.

Step 4. Since the absorption rate is six parts per million of the emission rate, dropping $\gamma_\uparrow$ changes $1/T_1=\gamma_\downarrow+\gamma_\uparrow$ by only $\sim6\times10^{-6}$ . The zero-temperature amplitude-damping channel is accurate to parts per million here, which is why room-scale quantum-computing noise models routinely set the qubit bath to $T=0$ .

3 · The amplitude-damping channel parameter and its Kraus check

With $T_1=100\ \mu\mathrm{s}$ (and negligible thermal excitation, so pure amplitude damping), find the channel parameter $p$ at $t=T_1$ and $t=2T_1$ , and verify the map is trace-preserving.

Step 1. The emission rate is $\gamma_\downarrow=1/T_1=10^4\ \mathrm{s}^{-1}$ , and the channel parameter is $p(t)=1-e^{-t/T_1}$ . At $t=T_1$ , $p=1-e^{-1}=\mathbf{0.632}$ ; at $t=2T_1$ , $p=1-e^{-2}=\mathbf{0.865}$ . The excited population has fallen to $e^{-1}=0.368$ and $e^{-2}=0.135$ of its initial value respectively.

Step 2. The Kraus operators are $K_0=\mathrm{diag}(1,\sqrt{1-p})$ and $K_1=\sqrt{p}\,|0\rangle\langle1|$ . Trace preservation requires $K_0^\dagger K_0+K_1^\dagger K_1=I$ : $K_0^\dagger K_0=\mathrm{diag}(1,1-p)$ and $K_1^\dagger K_1=\mathrm{diag}(0,p)$ , so the sum is $\mathrm{diag}(1,1)=I$ for every $p$ . The map is CPTP at all times, as the derivation guarantees.

Step 3. Consistency with coherence: for pure amplitude damping $T_2=2T_1=200\ \mu\mathrm{s}$ , so a qubit prepared on the equator retains coherence $|\rho_{eg}(t)|=e^{-t/2T_1}$ , giving $0.607$ at $t=T_1$ . This is exactly $\sqrt{1-p}=\sqrt{0.368}=0.607$ — the off-diagonal Kraus factor and the master-equation coherence decay agree, as they must. Everything ties together: $p(T_1)=0.632$ , $T_2=200\ \mu\mathrm{s}$ , and the map is trace-preserving.

Physics of the portrait

The Born–Markov–secular procedure was folklore among spectroscopists for decades before anyone proved it produced a legitimate quantum dynamics. The rigorous account — the one that turned a physicist's recipe into a theorem — is due to E. Brian Davies.

Origins · 1 minE. Brian Davies and how this was discovered

E. Brian Davies

b. 1944, England; King's College London · rigorous weak-coupling limit of open-system dynamics

Edward Brian Davies is a British mathematician whose 1974 paper "Markovian master equations" (Communications in Mathematical Physics 39, 91) put the entire Born–Markov–secular programme on rigorous footing. Physicists had long written down time-local relaxation equations — Pauli in 1928, Wangsness and Bloch in 1953, Redfield in 1957 — but no one had proved the resulting dynamics were genuinely completely positive, or specified the precise limit in which they became exact. Davies did both. He showed that in the van Hove weak-coupling limit, where the coupling $\lambda\to0$ and time is rescaled as $\lambda^2 t$ , the reduced dynamics converge to a quantum dynamical semigroup whose generator is exactly of the secular form (6) — the object now called the Davies generator. Because it is built from Bohr-frequency eigenoperators with a positive-semidefinite rate matrix it is automatically GKSL, and Davies showed that the KMS condition forces detailed balance, making the Gibbs state its unique stationary point. His work dovetailed with the general characterization of CP-semigroup generators published two years later by Lindblad and, independently, by Gorini, Kossakowski and Sudarshan — the GKSL theorem. Where GKSL classified all admissible generators abstractly, Davies identified which one a given microscopic model actually flows to. For open quantum systems it is the Davies generator — the clean endpoint of Born, Markov and secular — that carries his name.

History

1928

— Pauli derives the master equation for occupation probabilities using the repeated-randomness assumption

1953

— Wangsness and Bloch write the quantum relaxation equations for nuclear spins

1957

— Redfield casts spin relaxation as a time-local relaxation-matrix equation

1974

— Davies proves the rigorous weak-coupling limit yielding the Markovian secular master equation

1976

— Lindblad, and Gorini, Kossakowski and Sudarshan, characterize the general GKSL generator

Going further

Texts. Breuer & Petruccione, The Theory of Open Quantum Systems, Secs. 3.3–3.5 · Nielsen & Chuang, Quantum Computation and Quantum Information, Ch. 8 · Preskill, Lecture Notes for Physics 219, Ch. 3 (Quantum Channels and Decoherence) · Watrous, The Theory of Quantum Information, Ch. 2 · Gardiner & Zoller, Quantum Noise, Ch. 5 · Lidar, Lecture Notes on the Theory of Open Quantum Systems (arXiv:1902.00967).

Primary sources. E. B. Davies, "Markovian master equations," Commun. Math. Phys. 39 (1974) 91 · G. Lindblad, "On the generators of quantum dynamical semigroups," Commun. Math. Phys. 48 (1976) 119 · V. Gorini, A. Kossakowski & E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17 (1976) 821 · A. G. Redfield, "On the theory of relaxation processes," IBM J. Res. Dev. 1 (1957) 19. Key self-consistency checks: the eigenoperator identity $e^{iH_St}A(\omega)e^{-iH_St}=e^{-i\omega t}A(\omega)$ underlies the secular decomposition; $\gamma_{\alpha\beta}(\omega)\succeq0$ follows from Bochner/Wiener–Khinchin; $T_2\le2T_1$ holds algebraically; and the transmon numbers $T_\varphi=300\ \mu\mathrm{s}$ , $\beta\omega_0=12.0$ , $\bar n=6\times10^{-6}$ follow from the concrete qubit parameters $\omega_0/2\pi=5$ GHz, $T=20$ mK, $T_1=100\ \mu\mathrm{s}$ , $T_2=120\ \mu\mathrm{s}$ . The Gibbs state is annihilated by the Davies generator by detailed balance. All rates are left symbolic except where this specific qubit is referenced.

Prerequisites. Open Quantum Systems and Reduced Dynamics · Lindblad / GKSL Master Equation · Quantum Operations and CPTP Maps · Markovian vs Non-Markovian Dynamics. Leads to. Amplitude Damping Channel · Phase Damping Channel · Relaxation and Dephasing: T1 and T2 Times · Decoherence.

Ask the tutor: ask me to run the Born–Markov–secular chain on any specific model you name — a qubit in a bosonic bath, a driven atom, or a spin coupled to $1/f$ noise — and I will build its Davies generator, read off the rates, and check complete positivity step by step.

Why isn't the Redfield equation good enough — why bother with the secular step?

The Redfield equation is time-local and often numerically accurate, so it is tempting to stop there. The problem is that it is not guaranteed completely positive: it keeps cross-terms coupling coherences that belong to different Bohr frequencies, and for some initial states these drive an eigenvalue of $\rho$ below zero — negative "probabilities." The secular approximation discards exactly those cross-terms, leaving a positive-semidefinite rate matrix and a legitimate GKSL generator. So the secular step is not cosmetic — it makes the dynamics a physically admissible quantum channel at every time, which matters whenever the output feeds another quantum-information calculation.

What is the physical difference between T1 and T2?

$T_1$ is the energy-relaxation time: how long the qubit takes to decay from its excited state toward thermal equilibrium, set by the bath spectral density at the transition frequency $\omega_0$ . $T_2$ is the coherence time: how long a superposition retains a definite relative phase. They differ because coherence can be lost two ways — through the population decay itself, which contributes $1/(2T_1)$ , and through pure dephasing $1/T_\varphi$ , random phase kicks that carry no energy and come from low-frequency ( $\omega\approx0$ ) noise. Their sum is $1/T_2=1/(2T_1)+1/T_\varphi$ . Because pure dephasing can only shorten coherence, $T_2$ can never exceed $2T_1$ .

When does the Markov approximation actually fail?

It fails whenever the bath does not forget quickly compared with the system's own dynamics — that is, when the bath correlation time $\tau_B$ is not much smaller than the relaxation time $\tau_R$ . Physically this happens for structured environments: a bath with a sharp spectral feature (a narrow cavity mode, a defect resonance, a photonic band edge) has long-lived correlations and feeds information back into the system, producing recoherence and memory effects that no time-local Lindblad equation can capture. Photosynthetic light-harvesting complexes and strongly coupled circuit-QED are canonical examples. There one turns to non-Markovian tools: the Nakajima–Zwanzig memory kernel, hierarchical equations of motion, or reaction-coordinate mappings.

Is the rotating-wave approximation the same thing as the secular approximation?

They are the same idea in the same spirit, and in the qubit case they coincide, but they are worth distinguishing. The rotating-wave approximation usually means dropping fast counter-rotating terms in a driven Hamiltonian, keeping near-resonant couplings. The secular approximation is that logic applied to the dissipative generator: after the eigenoperator decomposition, one drops the cross-terms oscillating at Bohr-frequency differences $\omega-\omega'$ , keeping only $\omega=\omega'$ . Both rest on averaging over a time long compared with the fast oscillation but short compared with the slow dynamics. For near-degenerate transitions the secular step becomes subtle, and a "partial secular" version keeps the slow cross-terms.

Why does the bath temperature enter only through detailed balance?

Because at the level of the master equation the bath's entire influence is encoded in its two-time correlation function, and for a thermal bath that function obeys the KMS relation. Fourier-transformed, KMS forces the emission and absorption rates into the single constraint $\gamma(-\omega)=e^{-\beta\omega}\gamma(\omega)$ — nothing else about the temperature survives the weak-coupling reduction. That one relation makes the transition rates Boltzmann-weighted, which makes the Gibbs state $e^{-\beta H_S}/Z$ the unique steady state. So temperature does not change the structure of the equation, only the ratio of the rates, and through it the equilibrium the system relaxes to.

What is the Lamb-shift term H_LS, and can I usually ignore it?

The Lamb shift $H_{LS}$ is the imaginary (principal-value) partner of the dissipative rates: the same bath coupling that causes relaxation also slightly renormalizes the system's energy levels. It appears as an extra Hamiltonian term commuting with $H_S$ , so it shifts transition frequencies without causing decay. For most quantum-information purposes it is small — a fraction of the linewidth — and is absorbed into the qubit frequency, which is why it is often dropped. It matters when you need spectroscopic accuracy: the historical Lamb shift in atomic hydrogen is exactly this effect for a two-level atom in the electromagnetic vacuum, and analogous shifts must be tracked in precision circuit-QED.