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As Mad Scientist said in the comments, measurement of the transverse signal in NMR occurs in both the $x$- and $y$-directions. This is known as quadrature detection. In practice this is not achieved with two detector coils, but rather with a "trick" which is described more fully in Keeler's Understanding NMR Spectroscopy, 1st ed., section 5.6.

However, a rf pulse centred at $400.002~\mathrm{MHz}$ would be less effective at exciting the spins (as it is further away from being on-resonance), and essentially you are throwing away half of the rf pulse (any components of the rf pulse with frequency greater than $400.002~\mathrm{MHz}$ are wasted). So, it's preferable to put $\omega_\mathrm{rf}$ right in the middle of the spectrum; but as said before, that would necessitate quadrature detection so that you can obtain information about the sign of the offset.

As Mad Scientist said in the comments, measurement of the transverse signal in NMR occurs in both the $x$- and $y$-directions. This is known as quadrature detection. In practice this is not achieved with two detector coils, but rather with a "trick" which is described more fully in Keeler's Understanding NMR Spectroscopy, 1st ed., section 5.6. For a more mathematical description, see also Appendix A.5 of Levitt's Spin Dynamics, 2nd ed.

However, a rf pulse centred at $400.002~\mathrm{MHz}$ would be less effective at exciting the spins (as it is further away from being on-resonance), and essentially you are throwing away half of the rf pulse (any components of the rf pulse with frequency greater than $400.002~\mathrm{MHz}$ are wasted). So, it's preferable to put $\omega_\mathrm{rf}$ right in the middle of the spectrum; but as said before, that would necessitate quadrature detection so that you can obtain information about the sign of the offset.

As a final comment, the fact you note about how $\hat{I}_{\!x} + \mathrm{i}\hat{I}_{\!y}$ is the raising operator $\hat{I}_{\!+}$ is reflected in many books when they discuss coherence transfer pathways: often it is said that the detected signal has coherence order $-1$. This corresponds precisely to detecting the $\hat{I}_+$ operator, or specifically, one off-diagonal entry of the density matrix.

As Mad Scientist said in the comments, measurement of the transverse signal in NMR occurs in both the $x$- and $y$-directions. This is known as quadrature detection.

The benefit of quadrature recording (which isn't mentioned in Keeler's 2nd ed - I don't know if it was in the 1st) is that it allows us to find out the sign of the offset $\Omega$. Let's say that instead of having quadrature recording we only measure along the $x$-axis. Our signal is

and if you take the full expression for this and replace $\Omega$ by $-\Omega$, you will find that $S(\omega)$ is the same regardless of the sign of the offset $\Omega$! This means that our spin of interest could have either a positive or a negative offset, relative to the spectrometer reference frequency $\omega_\mathrm{rf}$ (which is equal to the frequency of the rf pulse $\omega_1$) but we would not be able to determine it.

Here, the authors have Fourier transformed $M_x$ and $M_y$ separately before adding them up. Since the Fourier transform is linear, this has the same effect as adding $M_x$ and $M_y$ and Fourier transforming the sum. In any case, it shows that if the magnetisation vector starts off along the $y$-axis, you could, in theory, measure only the $x$-magnetisation to determine whether the offset is negative (case a) or positive (case b).

Such a spectrum would firstly be very difficult to interpret, especially if positive and negative peaks overlap. But perhaps more fundamentally, that would require you to know the exact phase difference between the initial signal and your single detection coil. This is (due to several reasons) not experimentally possible, so unfortunately, there's no way to extract this information using single detection. (This is also the reason why NMR spectra have to be phase corrected - for the average NMR user the computer does this automatically.)

However, a rf pulse centred at $400.002~\mathrm{MHz}$ would be less effective at exciting the spins, and essentially you are throwing away half of the rf pulse (any components of the rf pulse with frequency greater than $400.002~\mathrm{MHz}$ are wasted). So, it's preferable to put $\omega_\mathrm{rf}$ right in the middle of the spectrum; but as said before, that would necessitate quadrature detection so that you can obtain information about the sign of the offset.

As Mad Scientist said in the comments, measurement of the transverse signal in NMR occurs in both the $x$- and $y$-directions. This is known as quadrature detection. In practice this is not achieved with two detector coils, but rather with a "trick" which is described more fully in Keeler's Understanding NMR Spectroscopy, 1st ed., section 5.6.

The benefit of quadrature recording (which seems to have been cut from Keeler's 2nd ed., but remains in the 1st ed.) is that it allows us to find out the sign of the offset $\Omega$. Let's say that instead of having quadrature recording we only measure along the $x$-axis. Our signal is

and if you take the full expression for this and replace $\Omega$ by $-\Omega$, you will find that $S(\omega)$ is the same regardless of the sign of the offset $\Omega$. This means that our spin of interest could have either a positive or a negative offset, relative to the spectrometer reference frequency $\omega_\mathrm{rf}$, but we would not be able to determine it.

Here, the author Fourier transformed $M_x$ and $M_y$ separately before adding them up. Since the Fourier transform is linear, this has the same effect as adding $M_x$ and $M_y$ and Fourier transforming the sum. In any case, it shows that if the magnetisation vector starts off along the $y$-axis, you could, in theory, measure only the $x$-magnetisation to determine whether the offset is negative (case a) or positive (case b).

Such a spectrum would be very difficult to interpret, and would probably be impossible to analyse if positive and negative peaks overlapped. On top of that, it would also require you to know the exact phase difference between the initial signal and your single detection coil. I believe this is (due to several reasons) not experimentally possible, so unfortunately, there's no way to extract this information using single detection. (This is also part of the reason why NMR spectra have to be phase corrected – for the average NMR user the computer does this automatically.)

However, a rf pulse centred at $400.002~\mathrm{MHz}$ would be less effective at exciting the spins (as it is further away from being on-resonance), and essentially you are throwing away half of the rf pulse (any components of the rf pulse with frequency greater than $400.002~\mathrm{MHz}$ are wasted). So, it's preferable to put $\omega_\mathrm{rf}$ right in the middle of the spectrum; but as said before, that would necessitate quadrature detection so that you can obtain information about the sign of the offset.

The benefit of quadrature recording (which as far as I know, is not mentioned in Keeler's book) is that it allows us to find out the sign of the offset $\Omega$. Let's say that instead of having quadrature recording we only measure along the $x$-axis. Our signal is

The benefit of quadrature recording (which isn't mentioned in Keeler's 2nd ed - I don't know if it was in the 1st) is that it allows us to find out the sign of the offset $\Omega$. Let's say that instead of having quadrature recording we only measure along the $x$-axis. Our signal is