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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

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

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  • $\begingroup$ Thanks a lot for this! It does make sense now. However, I realized I am also a bit confused about the way you actually realize the pi/2 pulse in practice. First of all you need the Rabi freq /Omega (which tells you how long the pulse should be), which is a dipole matrix element between the 2 levels. I assume that is quite hard to calculate in practice, and from what I saw, the relative errors in multi electron systems wavefunctions are about 10%-15% how do they get that value, such that they can tune the time precisely? $\endgroup$
    – BillKet
    Dec 31 '20 at 22:05
  • $\begingroup$ Also, when you don't know \omega_0, what value do you use for t, as you can't calculate W. Do you just use \Omega again i.e. you keep the pulse length as t=pi/(2*Omega) all the time and just vary the frequency? $\endgroup$
    – BillKet
    Dec 31 '20 at 22:06
  • $\begingroup$ 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.) $\endgroup$ Dec 31 '20 at 22:19
  • $\begingroup$ Oh, so you measure the Rabi freq directly. I assume you send pulses of different lengths and see where you get a maximum (for a pi pulse)? However, how do you measure the frequency in the first place (sorry i got quite confused)? Initially I thought that in an experiment, if you want to find an unknown transition you would scan the frequency over a given range until you see some (for example) fluorescence. But now I am not so sure. Don't you still need to send pulses for this freq measurement, too? $\endgroup$
    – BillKet
    Jan 1 at 0:02
  • $\begingroup$ And if you are unlucky, can't it be that the length of your pulse is 2pi, such that even if you are on resonance, the system will go back to the ground state and you see no fluorescence at all? Do they scan both the frequency and pulse length at the same time? Or am I missing sometime? $\endgroup$
    – BillKet
    Jan 1 at 0:02

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