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.

Where It Enters

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.

Secular Lindblad Form

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

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

The matrix $\gamma_{\alpha\beta}(\omega)$ is built from bath spectra at the Bohr frequency $\omega$. 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.

The correct choice depends on what observables and time scales the model is meant to predict.

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. When the prediction target is only the population block, this leads to Pauli Rate Equations. 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,

$$H_S = \frac{\hbar\omega_0}{2}\sigma_z,$$

the lowering and raising operators 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

In a three-level system with two excited states decaying to a common ground state, the two transition frequencies may be nearly equal. If their difference is comparable to the decay rate, cross-damping terms can produce interference, bright and dark combinations, or long-lived coherences. Full secularization may incorrectly remove those effects.

Common Mistakes

Dropping exact degeneracies

Terms with the same Bohr frequency belong to the same secular block. Removing them is an additional modeling decision.

Assuming secular is always more accurate

Secularization can improve positivity while losing near-degenerate coherence physics. Accuracy depends on the observable and time scale.

Using the wrong frequency scale

The comparison is not simply “large frequency.” It is the separation $|\omega-\omega'|$ compared with dissipative rates, coarse-graining resolution, and relevant drive scales.

Confusing master-equation secularization with Hamiltonian rotating-wave approximation

Both are rotating-wave ideas, but they occur at different stages. A Hamiltonian rotating-wave approximation may already have removed counter-rotating interaction terms before the bath derivation begins. Master-equation secularization averages Bohr-frequency cross terms in the reduced generator.

Expecting secularization to fix bad Born or Markov assumptions

If the bath is strongly coupled, nonstationary, finite, or has long memory, secular averaging does not repair the earlier approximation.

Ignoring driven frames

For periodically driven systems, the relevant frequencies may be quasienergy differences plus drive harmonics. A secular approximation in the lab frame can differ from one in the Floquet or rotating frame.

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

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

away from the removable zero at $\Omega=0$, 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.

Near-degenerate test

Two transitions have frequencies $\omega_1$ and $\omega_2$, with $|\omega_1-\omega_2|=0.3\Gamma$. Should a full secular approximation drop their cross terms?

Solution

No. Their relative phase changes only on a time scale comparable to or longer than the dissipative time. Cross terms can affect observable coherence and steady-state behavior. A partial secular or nonsecular treatment is usually more appropriate.

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

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, 2nd ed., Springer, 2007.
• E. B. Davies, “Markovian master equations,” Communications in Mathematical Physics 39, 91–110 (1974).