Secular Approximation

The secular approximation is the rotating-wave step in weak-coupling master-equation derivations. After system operators are decomposed into Bohr-frequency components, the master equation contains terms oscillating as $e^{i(\omega'-\omega)t}$. The secular approximation drops cross terms whose oscillations are fast compared with dissipative evolution.

In its simplest form, keep terms with

$$\omega=\omega'$$

and drop terms with

$$|\omega-\omega'|\gg \Gamma,$$

where $\Gamma$ is a characteristic relaxation or dephasing rate. Exact degeneracies and near-degeneracies require more care: they are not supposed to be erased by a blind algebraic rule.

Helpful background. Bohr-frequency components identify the rotating terms; the Markov approximation supplies the short-memory generator.

Bohr-frequency blocks in the master equation

The Born Approximation handles weak system–bath correlations. The Markov Approximation handles short bath memory. The secular approximation handles fast system phases.

For a system Hamiltonian

$$H_S = \sum_\epsilon \epsilon\,\Pi(\epsilon),$$

decompose a coupling operator into Bohr-frequency components:

$$A_\alpha(\omega) = \sum_{\epsilon'-\epsilon=\hbar\omega} \Pi(\epsilon)A_\alpha\Pi(\epsilon').$$

These components satisfy

$$[H_S,A_\alpha(\omega)] = -\hbar\omega A_\alpha(\omega),$$

and therefore

$$A_\alpha(t) = \sum_\omega e^{-i\omega t} A_\alpha(\omega)$$

in the interaction picture.

After Born and Markov approximations, nonsecular terms often carry factors of the form

$$e^{i(\omega'-\omega)t} \mathcal L_{\omega\omega'}(\rho).$$

When $\omega\ne\omega'$ and the phase rotates many times before dissipation appreciably changes $\rho$, the time-averaged influence is small.

Time-Averaging View

The basic averaging estimate is

$$\frac{1}{T} \int_0^T dt\, e^{i\Omega t} = e^{i\Omega T/2} \frac{\sin(\Omega T/2)}{\Omega T/2}.$$

This is small when

$$|\Omega|T\gg1.$$

For open systems, the averaging window must also be short compared with dissipative evolution. A useful hierarchy is

$$\tau_E \ll T \ll \Gamma^{-1}, \qquad |\omega-\omega'|T\gg1,$$

where $\tau_E$ is the bath memory time. The first inequality belongs to the Markov/coarse-graining logic; the last inequality is the secular averaging condition. For example, $\tau_E=10^{-9}\,\mathrm s$, $\Gamma=10^3\,\mathrm{s}^{-1}$, and a splitting $10^7\,\mathrm{rad\,s}^{-1}$ admit $T=10^{-5}\,\mathrm s$: $\Gamma T=0.01$ while $|\omega-\omega'|T=100$, so the cross-term average has magnitude at most $0.02$. A splitting $300\,\mathrm{rad\,s}^{-1}$ does not admit a window long enough to average the beat while remaining much shorter than $\Gamma^{-1}$.

Secular Lindblad Form

After full secularization, a common weak-coupling generator has the form

$$\frac{d\rho}{dt} = -\frac{i}{\hbar} [H_S+H_{\text{LS}},\rho] + \sum_{\omega,\alpha,\beta} \gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho A_\alpha^\dagger(\omega) - \frac12 \{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\} \right).$$

Here $A_\alpha$ is dimensionless and $\gamma_{\alpha\beta}(\omega)$ includes the coupling prefactor $\lambda^2/\hbar^2$ multiplying the two-sided bath spectrum, so it has rate units. In the standard construction it is positive semidefinite for each fixed $\omega$:

$$\sum_{\alpha,\beta} v_\alpha^* \gamma_{\alpha\beta}(\omega) v_\beta \ge0.$$

This blockwise positivity is why secularization often leads to Lindblad–GKSL form. Diagonalize the rate matrix:

$$\gamma_{\alpha\beta}(\omega) = \sum_r \kappa_r(\omega) u_{\alpha r}(\omega) u_{\beta r}^*(\omega), \qquad \kappa_r(\omega)\ge0.$$

Then the Lindblad operators may be chosen as

$$L_{r\omega} = \sqrt{\kappa_r(\omega)} \sum_\alpha u_{\alpha r}^*(\omega) A_\alpha(\omega).$$

The nonsecular equation can be accurate over a perturbative time window but fail to display complete positivity in a simple semigroup form. Secularization trades some coherent near-resonant physics for a cleaner completely positive structure.

What Is Kept

The secular approximation does not keep only one transition. It keeps all terms within the same Bohr-frequency block.

If several transitions have the same energy gap, their shared $\omega$ terms must remain coupled. This matters in:

• degenerate excited-state manifolds;
• harmonic oscillators, where many transitions share the same frequency;
• angular-momentum multiplets;
• multilevel atoms with parallel transition dipoles;
• systems with symmetry-protected degeneracies.

Dropping cross terms inside an exactly degenerate block is an extra approximation, not the secular approximation itself.

Near Degeneracies

The dangerous case is

$$|\omega-\omega'| \lesssim \Gamma.$$

Then cross terms rotate slowly on the dissipative time scale and may affect line shapes, dark states, quantum beats, transport coherences, or steady states.

A full secular approximation can be too aggressive near degeneracy. Alternatives include:

• partial secular approximation, grouping frequencies whose splittings are small;
• coarse-grained master equations with an explicit averaging time;
• retaining a Redfield Equation in its perturbative regime;
• enlarging the system to include near-resonant modes or levels;
• using a driven or Floquet basis when the Hamiltonian is time periodic.

Keeping selected nonsecular terms by hand does not automatically preserve complete positivity. A partial-secular or coarse-grained construction must retain a positive coefficient matrix and a justified frequency grouping. The choice depends on the observables and time scales to be predicted.

One controlled construction groups transition splittings of the same perturbative order as the decay rates, evaluates the dissipative spectrum at each group’s central frequency, and retains the small splittings in the Hamiltonian (Trushechkin, 2021, Sec. III).

Physical Interpretation

Density-matrix elements in the interaction picture can rotate at different Bohr-frequency differences. The secular approximation says that terms coupling widely separated rotating sectors average out before dissipation builds up.

In Schrödinger picture, the approximation tends to decouple populations from rapidly rotating coherences and separates dissipative channels by transition frequency. For nondegenerate energy levels with a closed diagonal block, this leads to Pauli Rate Equations. Degenerate energy sectors can retain coupled populations and coherences even after full secularization. This is why a secular master equation often has a transparent interpretation in terms of emission, absorption, and dephasing at definite frequencies.

But transparency is not the same as universal accuracy. If coherence between nearby transitions is the phenomenon of interest, full secularization may remove the effect being studied.

Examples

Two-level atom

For a two-level system,

$$H_S = \hbar\omega_0|e\rangle\langle e|,$$

define $\sigma_-=|g\rangle\langle e|$ and $\sigma_+=\sigma_-^\dagger$. They have frequencies $\omega_0$ and $-\omega_0$:

$$\sigma_-(t)=e^{-i\omega_0 t}\sigma_-, \qquad \sigma_+(t)=e^{i\omega_0 t}\sigma_+.$$

Cross terms between raising and lowering rotate at roughly $2\omega_0$. The secular approximation is well justified when

$$\omega_0\gg\Gamma,$$

and the bath spectrum is smooth on the line-width scale. The resulting dissipators resemble amplitude damping and thermal excitation terms.

Pure dephasing

If the coupling operator commutes with $H_S$, only the $\omega=0$ component is present. There are no distinct Bohr-frequency sectors to average away. The issue is then not secularization but the validity of the weak-coupling and memory approximations behind the dephasing channel.

Harmonic oscillator

For an oscillator with equally spaced levels, many transitions have the same Bohr frequency. Secularization keeps the whole frequency block. The annihilation operator $a$ is a single lowering-frequency operator, so the usual damping dissipator can remain compact:

$$\mathcal D[a]\rho = a\rho a^\dagger - \frac12\{a^\dagger a,\rho\}.$$

Nearly degenerate V system

Take two exactly degenerate excited states $|e_1\rangle,|e_2\rangle$ decaying to $|g\rangle$ through the same bath coupling, with $A_1=|g\rangle\langle e_1|$, $A_2=|g\rangle\langle e_2|$ and rate matrix

$$\gamma_{\alpha\beta}=\gamma\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad \gamma>0.$$

Its eigenvalues are $2\gamma$ and zero, so one jump suffices: $L=\sqrt\gamma(A_1+A_2)$. The bright state $(|e_1\rangle+|e_2\rangle)/\sqrt2$ decays with population rate $2\gamma$, while the dark state $(|e_1\rangle-|e_2\rangle)/\sqrt2$ is annihilated by $L$.

If one incorrectly drops the cross terms and uses separate jumps $\sqrt\gamma A_1$ and $\sqrt\gamma A_2$, even the dark state decays with rate $\gamma$. This is a measurable change, not a notation choice. The common-bath correlations specified by this matrix are essential; two independent baths would instead justify the diagonal matrix. A small nonzero level splitting also mixes the bright and dark combinations, so perfect darkness is not generic near degeneracy.

A different rotating-wave approximation

Master-equation secularization averages cross terms in an already reduced generator. A Hamiltonian rotating-wave approximation removes terms from the microscopic interaction before the bath is traced out. They use related oscillation arguments but have different error scales and need separate justification. Neither repairs an uncontrolled Born or Markov approximation. For driven systems, the relevant frequencies may include quasienergy differences and drive harmonics rather than only undriven energy gaps.

Exercises

Averaging a cross term

Show that

$$\frac{1}{T} \int_0^T dt\, e^{i(\omega'-\omega)t}$$

is small when $|\omega'-\omega|T\gg1$.

Solution

Let $\Omega=\omega'-\omega$. Then

$$\frac{1}{T} \int_0^T dt\, e^{i\Omega t} = \frac{e^{i\Omega T}-1}{i\Omega T} = e^{i\Omega T/2} \frac{\sin(\Omega T/2)}{\Omega T/2}.$$

The magnitude is bounded by

$$\frac{1}{|\Omega|T/2}$$

for $\Omega\ne0$; the exact average tends to one as $\Omega\to0$, so it is small when $|\Omega|T\gg1$.

Equal spacing

A three-level ladder has energies $0$, $\hbar\omega_0$, and $2\hbar\omega_0$. Which downward transitions share a Bohr frequency?

Solution

The transitions $2\hbar\omega_0\to\hbar\omega_0$ and $\hbar\omega_0\to0$ both have energy difference $\hbar\omega_0$. They belong to the same Bohr-frequency block. Secularization should not discard their shared-frequency cross terms merely because they connect different pairs of levels.

Lifetime of an almost dark state

In the V-system example, retain the common-bath jump $L=\sqrt\gamma(A_1+A_2)$ and add, in a frame rotating at the mean transition frequency,

$$H_\delta=\frac{\hbar\delta}{2} (|e_1\rangle\langle e_1|-|e_2\rangle\langle e_2|), \qquad 0<|\delta|\ll\gamma.$$

Take this as a specified effective model of an unresolved doublet with equal couplings to a broad vacuum bath, no excitation or extra dephasing, and the displayed Hamiltonian including the retained shifts. Start in the dark state $|D\rangle=(|e_1\rangle-|e_2\rangle)/\sqrt2$. Find the slow decay rate of the total excited population after the bright transient. Compare with separate jumps $\sqrt\gamma A_1$ and $\sqrt\gamma A_2$.

Solution

Let $|B\rangle=(|e_1\rangle+|e_2\rangle)/\sqrt2$. Then $H_\delta$ mixes $B$ and $D$ with matrix element $\hbar\delta/2$, while $L^\dagger L=2\gamma|B\rangle\langle B|$. A jump deposits population only in $|g\rangle$. The excited block consequently evolves as $\rho_e(t)=|\psi_e(t)\rangle\langle\psi_e(t)|$, where the unnormalized amplitudes obey

$$\frac{d}{dt}\begin{pmatrix}a_B\\a_D\end{pmatrix} =\begin{pmatrix}-\gamma&-i\delta/2\\-i\delta/2&0\end{pmatrix} \begin{pmatrix}a_B\\a_D\end{pmatrix}, \qquad (a_B(0),a_D(0))=(0,1).$$

Their squared norm is the surviving excited population, not a normalized conditional state. The amplitude exponents are

$$r_\pm=\frac{-\gamma\pm\sqrt{\gamma^2-\delta^2}}{2}.$$

For $0<|\delta|<\gamma$, the initial dark state has a nonzero slow component. Once the fast component has decayed, $p_e(t)\propto e^{-\Gamma_D t}$, with

$$\Gamma_D=-2r_+ =\gamma-\sqrt{\gamma^2-\delta^2} =\frac{\delta^2}{2\gamma} +O\!\left(\frac{\delta^4}{\gamma^3}\right).$$

The factor of two converts amplitude decay to population decay. For $|\delta|=0.1\gamma$, the exact rate is $0.0050126\gamma$ and the tail lifetime is about $199.5/\gamma$. Perfect darkness returns at $\delta=0$.

With separate jumps, the total excited population instead obeys $\dot p_e=-\gamma p_e$, including for this initial state. The distinction is observable as a long population tail. These rates solve the stated generator; establishing that generator for a particular bath still requires the weak-coupling, bandwidth, and frequency-grouping checks above.

Diagonalizing the rate matrix

Suppose $\gamma_{\alpha\beta}(\omega)$ is positive semidefinite for fixed $\omega$. Explain why the corresponding secular dissipator can be written with Lindblad operators.

Solution

A positive semidefinite matrix can be diagonalized as

$$\gamma_{\alpha\beta}(\omega) = \sum_r \kappa_r(\omega) u_{\alpha r}(\omega) u_{\beta r}^*(\omega), \qquad \kappa_r(\omega)\ge0.$$

Substituting this decomposition into the dissipator and defining

$$L_{r\omega} = \sqrt{\kappa_r(\omega)} \sum_\alpha u_{\alpha r}^*(\omega) A_\alpha(\omega)$$

puts the generator into Lindblad form for that frequency block.

References

• R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, 2nd ed., Springer (2007).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• E. B. Davies, “Markovian master equations,” Communications in Mathematical Physics 39, 91–110 (1974). DOI.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004). Publisher.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• A. Trushechkin, “Unified Gorini–Kossakowski–Lindblad–Sudarshan quantum master equation beyond the secular approximation,” Physical Review A 103, 062226 (2021). Author manuscript.