# Ramsey interferometry
**Ramsey interferometry** is a technique for measuring the transition frequency of a quantum system with very high precision using two separated pulses. It was developed by Norman Ramsey in 1950 as an improvement on Rabi's continuous-drive spectroscopy, and it forms the basis of modern atomic clocks and qubit frequency calibration.

The Ramsey protocol applies two $\pi/2$ pulses separated by a free-precession time $T$. The first $\pi/2$ pulse creates an equal superposition $\lvert +\rangle$ from $\lvert 0\rangle$. During the free evolution time $T$, the qubit accumulates a phase $\delta T$ relative to the rotating frame, where $\delta = \omega - \omega_0$ is the detuning between the drive frequency $\omega$ and the qubit frequency $\omega_0$. The second $\pi/2$ pulse converts this accumulated phase into a population difference, which is then measured.

$$P_1(T) = \frac{1}{2}(1 - \cos(\delta T))$$

## Advantages over Rabi spectroscopy
In a Rabi experiment, the qubit is driven continuously and the spectral resolution is set by the Rabi frequency $\Omega$, which must be large enough to drive the transition. In the Ramsey experiment, the qubit evolves freely for time $T$ and the spectral fringes oscillate at the small detuning $\delta \ll \Omega$. By increasing $T$, the fringe spacing $1/T$ decreases, giving much higher frequency resolution.

## Coherence time measurement
Ramsey interferometry is also used to measure the dephasing time $T_2^*$. Because different parts of the qubit ensemble (or repeated measurements of the same qubit) experience slightly different values of $\omega_0$ due to environmental fluctuations, the Ramsey fringes lose contrast exponentially.

$$P_1(T) = \frac{1}{2}\left(1 - e^{-T/T_2^*}\cos(\delta T)\right)$$

Fitting this decay envelope gives $T_2^*$. The related Hahn echo sequence adds a refocusing $\pi$ pulse between the two $\pi/2$ pulses to cancel out low-frequency dephasing noise, yielding the longer $T_2$ coherence time.