Harmonic Perturbations

A harmonic perturbation is a weak drive with a single angular frequency. It is the basic model for absorption, stimulated emission, driven transitions, spectroscopy, and the weak-drive limit of two-level dynamics. Resonant Driving analyzes detuning, finite-pulse linewidth, and the long-time breakdown of first order. Linear Response Preview reorganizes the same transition frequencies and matrix elements into a susceptibility. The coherent two-level version is introduced in Rabi Oscillations: First Encounter, with the explicit perturbative match derived in Rabi Formula in the Weak-Drive Limit.

Write the perturbation as

$$V(t) = W e^{-i\omega t} + W^\dagger e^{i\omega t},$$

so that $V(t)$ is Hermitian. The operator $W$ carries the part of the drive that oscillates with frequency $\omega$.

First-Order Amplitude

Let

$$\omega_{fi} = \frac{E_f-E_i}{\hbar}.$$

The first-order transition amplitude is

$$c_f^{(1)}(T) = - \frac{i}{\hbar} \int_0^T dt\, e^{i\omega_{fi}t} \langle f\vert V(t)\vert i\rangle.$$

Substituting the harmonic perturbation gives

$$\begin{aligned} c_f^{(1)}(T) = - \frac{i}{\hbar} \bigg[ &W_{fi} \int_0^T dt\,e^{i(\omega_{fi}-\omega)t} \\ &+ W^\dagger_{fi} \int_0^T dt\,e^{i(\omega_{fi}+\omega)t} \bigg], \end{aligned}$$

where

$$W_{fi}=\langle f\vert W\vert i\rangle, \qquad W^\dagger_{fi}=\langle f\vert W^\dagger\vert i\rangle.$$

Resonance Denominator

For any detuning $\Delta$,

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

Thus

$$c_f^{(1)}(T) = - \frac{iT}{\hbar} \left[ W_{fi} e^{i(\omega_{fi}-\omega)T/2} \operatorname{sinc} \left( \frac{(\omega_{fi}-\omega)T}{2} \right) + W^\dagger_{fi} e^{i(\omega_{fi}+\omega)T/2} \operatorname{sinc} \left( \frac{(\omega_{fi}+\omega)T}{2} \right) \right],$$

where

$$\operatorname{sinc}x=\frac{\sin x}{x}.$$

The amplitude is largest when one of the oscillatory phases is nearly stationary.

Absorption and Emission Terms

If $E_f>E_i$, then $\omega_{fi}>0$. The term with denominator $\omega_{fi}-\omega$ is resonant when

$$\omega\approx\omega_{fi}.$$

This is the absorption condition: the drive supplies energy $\hbar\omega$.

If $E_f<E_i$, then $\omega_{fi}<0$. The term with denominator $\omega_{fi}+\omega$ is resonant when

$$\omega\approx-\omega_{fi}.$$

This is the stimulated-emission condition: the system loses energy $\hbar\omega$ to the drive.

The labels “absorption” and “emission” depend on which transition is being considered and how the external field is modeled. The mathematical test is always the phase-matching condition in the time integral.

Rotating and Counter-Rotating Contributions

Near a positive-frequency transition $\omega_{fi}\approx\omega$, the $W e^{-i\omega t}$ term varies slowly in the interaction picture. The $W^\dagger e^{i\omega t}$ term varies with frequency approximately $2\omega$ and often averages out.

Keeping only the slowly rotating term is the rotating-wave approximation. It is not part of first-order perturbation theory itself; it is an additional approximation based on near resonance and weak drive. It fails for strong drives, very short pulses, or regimes where counter-rotating effects accumulate.

Transition Probability

The first-order probability is

$$P_{i\to f}(T) \approx \lvert c_f^{(1)}(T)\rvert^2.$$

Near a single resonance, this becomes approximately

$$P_{i\to f}(T) \approx \frac{\lvert W_{fi}\rvert^2T^2}{\hbar^2} \operatorname{sinc}^2 \left( \frac{(\omega_{fi}-\omega)T}{2} \right).$$

The finite-time line shape has width of order $1/T$. In the long-time continuum limit, this finite-time peak becomes the delta-function structure behind Fermi’s golden rule.

Selection Rules

The drive can be perfectly resonant and still produce no transition if the matrix element vanishes:

$$W_{fi}=0.$$

Such zeros may come from parity, angular momentum, spin, translation symmetry, or other quantum numbers. The symmetry logic belongs in Why Symmetry Matters and Parity; this page uses those zeros inside the transition amplitude.

Light-Matter Preview

In the electric dipole approximation, a classical monochromatic electric field gives an interaction of the schematic form

$$V(t) = - \mathbf d\cdot\mathbf E_0\cos\omega t,$$

where $\mathbf d$ is the dipole operator. Writing the cosine as positive- and negative-frequency parts puts the perturbation in harmonic form. Transition strengths are controlled by dipole matrix elements such as

$$\langle f\vert\mathbf d\cdot\boldsymbol\epsilon\vert i\rangle,$$

where $\boldsymbol\epsilon$ is the polarization direction.

Common Mistakes

• Calling a transition resonant while its matrix element is zero by symmetry.
• Forgetting the counter-rotating term before justifying a rotating-wave approximation.
• Treating the finite-time sinc peak as an exact delta function at finite $T$.
• Letting first-order probabilities grow beyond the perturbative regime.
• Confusing angular frequency $\omega$ with ordinary frequency.

Exercises

1. Derive the finite-time integral for a detuning $\Delta$.

Solution

Compute

$$\int_0^T dt\,e^{i\Delta t} = \frac{e^{i\Delta T}-1}{i\Delta}.$$

Factor out $e^{i\Delta T/2}$:

$$e^{i\Delta T}-1 = e^{i\Delta T/2} \left( e^{i\Delta T/2}-e^{-i\Delta T/2} \right) = 2i e^{i\Delta T/2}\sin\frac{\Delta T}{2}.$$

Therefore

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

2. For $E_f>E_i$, which term is resonant when $\omega>0$?

Solution

If $E_f>E_i$, then $\omega_{fi}>0$. The denominator $\omega_{fi}-\omega$ can vanish when $\omega=\omega_{fi}$. The denominator $\omega_{fi}+\omega$ is then far from zero for positive $\omega$. Thus the term proportional to $W_{fi}$ is resonant.

3. Why does a zero matrix element suppress a resonant transition?

Solution

The resonant enhancement multiplies the matrix element. If $W_{fi}=0$ by symmetry, the resonant term in the amplitude vanishes. Resonance can enhance an allowed transition, but it cannot make a forbidden matrix element nonzero unless the symmetry assumptions are broken.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.