Resonance

A driven quantum system responds dramatically when the drive frequency matches a natural transition frequency. This is resonance, and it is the mechanism behind spectroscopy, magnetic resonance imaging, and the control pulses that operate real qubits.

The resonance condition

For a harmonic perturbation the RWA transition probability is

P_{i\to f}(t) = \frac{|V_{fi}|^2}{4\hbar^2}\,\frac{\sin^2(\Delta t/2)}{(\Delta/2)^2}, \qquad \Delta = \omega_{fi}-\omega .

The factor $\sin^2(\Delta t/2)/(\Delta/2)^2$ is sharply peaked at $\Delta=0$ . The resonance condition is therefore

\boxed{\;\omega = \omega_{fi} = \frac{E_f - E_i}{\hbar}\;}\quad\Longleftrightarrow\quad \hbar\omega = E_f - E_i .

In words: the drive is resonant when one quantum of the drive, $\hbar\omega$ , exactly bridges the energy gap between initial and final states. This is the Bohr frequency condition, here recovered dynamically rather than postulated.

On resonance: quadratic growth

Setting $\Delta\to 0$ and using $\sin x \approx x$ gives $\sin^2(\Delta t/2)/(\Delta/2)^2 \to t^2$ , so

P_{\text{res}}(t) = \frac{|V_{fi}|^2}{4\hbar^2}\,t^2 .

The on-resonance probability grows quadratically in time. This is far faster than the small oscillations off resonance, which is exactly why resonant driving is the efficient way to move population between levels.

The lineshape: width and lifetime

As a function of detuning $\Delta$ at fixed $t$ , the response is a sinc-squared peak. Its central maximum has:

Height $\propto t^2$ — taller the longer you drive.
Width $\Delta_{\text{FWHM}} \sim 2\pi/t$ — narrower the longer you drive.

The first zeros of $\sin^2(\Delta t/2)$ occur at $\Delta = \pm 2\pi/t$ . So a long interaction time gives a sharp, frequency-selective response. Conversely, a short pulse has a broad spectral response: this is the time–energy uncertainty relation, $\Delta E\,\Delta t \gtrsim \hbar$ , expressed as a lineshape. Spectroscopic linewidths and the natural linewidth of atomic transitions both trace back to this.

Why resonance is universal

The same peaked-response-at-matching-frequency appears for a driven classical oscillator, a spin in NMR, and a superconducting qubit under a microwave tone. The quantum content is the quantization $\hbar\omega = E_f - E_i$ : energy is exchanged with the drive in discrete units set by the level spacing.

Try it

A system is driven exactly on resonance ( $\Delta=0$ ). With $V_{fi}=0.2$ , $t=3.0$ , and $\hbar=1$ , compute the resonant transition probability $P_{\text{res}}(t) = (|V_{fi}|^2/4\hbar^2)\,t^2$ and return it.

Run your code to see the quantum state.