Assume we have a 2 level system with the frequency between the 2 levels $$\omega_0$$ and ignore the lifetime of the upper state. In order to measure the transition frequency using Ramsey technique, you apply 2 $$\pi/2$$ pulses separated by a time $$T$$. We have that the probability of finding an atom in the excited sate under the interaction with light (assuming strong field i.e. not using perturbation theory) is given by: $$\frac{\Omega^2}{W^2}sin^2(Wt/2)$$ where $$\Omega$$ is the Rabi frequency and $$W^2=\Omega^2+(\omega-\omega_0)^2$$ with $$\omega$$ being the laser frequency (which we scan in order to find $$\omega_0$$). If we are on resonance, i.e. $$W=\Omega$$, a $$\pi/2$$ pulse is a pulse of duration $$t=\frac{\pi}{2\Omega}$$ such that, starting in the ground state, the atom becomes an equal superposition of upper and lower states. However in the Ramsey technique, we don't know $$\omega_0$$ beforehand (this is what we want to measure). So I am not sure I understand how we can still create an equal superposition of upper and lower level using a $$\pi/2$$ pulse. Assuming we use a pulse of duration $$t=\frac{\pi}{2W}$$, from the equation above we get that the population of the upper state is $$\frac{\Omega^2}{2W^2}$$ which is not $$1/2$$ as in the resonance case. How do we still get equal superposition of upper and lower case when we are not on resonance? Thank you

• You vary the frequency, to find the frequency at which you see the effects of a $\pi/2$ pulse.
– Buzz
Dec 31 '20 at 20:40

You are correct that the Ramsey sequence should be applied close to resonance, but it does not need to be perfectly on resonance -- it only needs to be close to resonance, relative to the Rabi frequency used for the $$\pi/2$$ pulses.
The Ramsey technique gives a measurement signal that oscillates as a function of the gap time $$T$$ between $$\pi/2$$ pulses, where the oscillation frequency is given by the detuning $$\omega-\omega_0$$ and the contrast is determined by the weights of the superposition produced by the $$\pi/2$$ pulse. In the ideal case (maximal contrast), the $$\pi/2$$ pulse would transfer 50% of the population to the second state; however, the signal is still usable when the $$\pi/2$$ is imperfect. In practice, as long as $$|\omega - \omega_0| \lesssim \Omega$$, then you find from the expression that you wrote $$\frac{\Omega^2}{W^2} \sin^2(Wt/2)$$ that close to 50% of the population will be transferred.
If you know approximately where the resonance is to within a bandwidth of $$\sim \Omega$$, then Ramsey can be used to more precisely find the resonance (ultimately limited by the longest gap time $$T$$ that can be used). If you don't have any idea where the resonance is, then you can first find it by applying an approximate $$\pi$$ pulse over a wider range of frequencies and finding which frequency gives the largest population transfer.
• Usually you would measure the Rabi frequency in practice by scanning to find the approximate resonance, and then driving Rabi oscillations to measure $\Omega$. (You would know in advance what order of magnitude to expect for $\Omega$, but it's hard to know all details including precise laser/MW intensity at the atoms, etc.). And yes, for Ramsey you could use a fixed time of $t=\pi/2\Omega$. (Note that the more general $\pi/2W$ is first-order insensitive to detuning from resonance.) Dec 31 '20 at 22:19