Harmonic Perturbations

A harmonic perturbation resolves a drive into positive- and negative-frequency components. Their phases distinguish absorption from stimulated emission, while their matrix elements decide which transitions are allowed. The finite-time calculation below retains both components before making a near-resonance approximation.

Required background. First-Order Transition Probability supplies the amplitude integral. Selection Rules explains symmetry restrictions; Fermi’s Golden Rule treats the continuum limit.

Write the perturbation as

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

so that $V(t)$ is Hermitian. Take $\omega>0$. The operator $W$ multiplies the negative exponential and need not be Hermitian. In particular, $W^\dagger_{fi}=W_{if}^*$, which is generally different from $W_{fi}^*$. For a real cosine drive $A\cos\omega t$ with Hermitian $A$, $W=A/2$.

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

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

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

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

where

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

For a given nonzero matrix element, a nearly stationary phase permits coherent accumulation. For short pulses, both frequency components can contribute appreciably and must be added as complex amplitudes before squaring.

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. Parity and Selection Rules in Transition Rates derive these zeros. A forbidden first-order matrix element need not forbid a higher-order multiphoton transition.

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. For linear polarization with peak field $E_0$, $W=-E_0\mathbf d\cdot\boldsymbol\epsilon/2$. Thus the resonant two-level Rabi angular frequency is $2|W_{fi}|/\hbar=E_0|\mathbf d_{fi}\cdot\boldsymbol\epsilon|/\hbar$. The factor of two comes from resolving a real cosine, not from a change of physical field strength.

Exercises

Circular versus linear driving

For $H_0=\hbar\omega_0\sigma_z/2$, compare $W=g\sigma_+$ with $W=g\sigma_x$, where $g$ is real. Which first-order frequency components can excite the lower state?

Solution

Since $\langle+|\sigma_+|-\rangle=1$ and $\langle+|\sigma_-|-\rangle=0$, the circular drive has only the excitation amplitude proportional to $g\int e^{i(\omega_0-\omega)t}dt$. For $W=g\sigma_x$, both $W_{+-}$ and $W^\dagger_{+-}$ equal $g$, so both difference- and sum-frequency integrals occur. The latter Hamiltonian is $2g\cos\omega t\,\sigma_x$; its peak transverse coefficient is twice $g$.

Interference between frequency components

For the linear drive above, take $T\ll1/(\omega_0+\omega)$. Compare the full first-order excitation probability with the result obtained by dropping the counter-rotating term.

Solution

Both integrals approach $T$, so $c_+^{(1)}\simeq-2igT/\hbar$ and $P\simeq4g^2T^2/\hbar^2$. Keeping only one component gives $g^2T^2/\hbar^2$. There has been no time to average the rapid component; “weak drive” alone does not justify the rotating-wave approximation on this time scale.

References

• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.
• 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.