Qubit dephasing times

Question

The dephasing time (also known as decoherence time in the literature), $T_2$, speaks about the timescale in which qubits experience dephasing effects. This relates with the decay of the off-diagonal components of the density matrix that describe such qubit. However, usually two different dephasing times are reported (see Fault-tolerant operation of a logical qubit in a diamond quantum processor, for example): the Ramsey dephasing time ($T_2^*$) and the Hahn-echo dephasing time ($T_2^{(echo)}$, or sometimes just $T_2$). I understand both the experiments in order to measure such parameters, but I am confused about their actual meaning when relating them with the dephasing rate experienced by such qubits. Specifically, I understand that $T_2^*$ speaks about the dephasing effect suffered by the qubit when it is left to evolve freely, while the echo sequences are used in order to compensate the low frequency decoherence. This results in $T_2^*<T_2^{(echo)}$. My question is, which of those describes the actual dephasing rate that a qubit will suffer, that is, the rate at which the off-diagonal elements will decay. This question comes from the fact that if one uses the Hahn-echo technique, he will actually be "manipulating" the dephasing of such qubit. In this way, my reasoning is that the Ramsey time will be more precise in describing such effect (even if the experiment is more delicate). A bit of clarity and/or references about this topic would be helpful for me.

Answer 1

As you described, $T_2^*$ defines the decay time of coherence in a Ramsey experiment, while $T_2^{(echo)}$ and more generally $T_2^{(DD)}$ run a similar experiment but with intermediate refocusing pulses that generally serve to increase this time.

For a concrete example, consider nitrogen-vacancy (NV) centers in diamond. NVs typically have $T_2^{(DD)}$ 1-2 orders of magnitude longer than $T_2^*$ because carbon-13 and nitrogen electron spins near the NV tend to introduce low frequency magnetic field noise that's effectively DC compared to the frequency of the NV (See this for much more detail if you're interested about diamond in particular). This means that during the free precession window, the NV qubit will pick up some noisy amount of phase due to the DC noise. However, if we introduce a $\pi$-pulse in the middle of the free precession time, then the qubit will accrue this extra phase for the first half, but then this phase will be inverted, and then the phase accumulated in the 2nd half will cancel with the original noise, making the qubit insensitive to this kind of DC noise.

Now as for why everyone doesn't always run dynamical decoupling, there are two main reasons: First, dynamical decoupling makes you less sensitive to lower frequency noise, but increases your sensitivity to higher frequency noise. For example, in a standard Hahn echo experiment, you become insensitive to DC noise as mentioned above. However, if you imagine sinusoidal noise which peaks halfway through the first half of the precession time and then troughs halfway through the second half, the phase accumulated from it would cancel out over the whole window during a Ramsey experiment, but the $\pi$-pulse from a Hahn echo experiment causes the accumulated phases from the two halves to add, giving a worse result. Every dynamical decoupling sequence introduces a frequency-dependent filter function on the noise, and more complicated schemes tend to help shift the frequencies you're sensitive to to be higher and higher, where there's generally less noise for most experiments.

Second, implementing DD sequences requires more complicated experiments which introduce practical concerns. The faster you wish to $\pi$-pulse your qubit in a CPMG DD sequence to minimize your sensitivity to lower frequency noise, the more you need to blast your system with microwaves, lasers, or whatever you drive it with. Any sort of timing imprecision, power or phase instability, etc in this drive signal will be amplified as you use it more, which can actually hurt your qubit coherence more than you help it. This is one of the reasons why there are many different DD schemes (e.g. I believe XY8,16,etc schemes generally introduce less error due to drive power instability compared to similar CPMG sequences).