What's the Difference between T2 and T2*?

Question

$T_2$ generally refers to the measurement of the coherence of the qubit with respect to its dephasing (that's a rotation through the $|0\rangle$ - $|1\rangle$ axis of the Bloch sphere for those of us visualizing). But sometime in the literature, it's called $T_2$ and other times it's referred to as $T_2^*$. The fact that it is never explained leads me to believe the distinction is very simple. What's the distinction between these two concepts?

I assert that this nomenclature is very common in the literature (at least regarding solid-state QC). Here is one example: Ultralong spin coherence time in isotopically engineered diamond.

My internet search for an explanation has come up dry. Please help.

Answer 1

The naming started in NMR and it has become the difference between the following two experiments.

Experiment one: Prepare the qubit in a superposition state (apply a H gate) and vary the wait time and then measure in the superposition basis (apply another H gate). The decay time of this experiment is $T_2^*$. We commonly call this a Ramsey experiment.

Experiment two: Prepare the qubit in a superposition state and apply half the wait time and then apply a pi-pulse (X operator) and then the remainder of the wait time and measure in the superposition basis. The decay time of this experiment is $T_2$. We commonly call this a Hahn Echo.

In the second experiment, the pi-pulse refocuses slow noise which depending on the system can be due to many reasons. There are higher order experiments that refocus the noise better and this is an active research area.

To see this imagine the simplest case where the noise can be explained by a Hamiltonian $H = \Delta |1\rangle\langle 1|$ that is constant and unknown.

In the first experiment:

Step 1. Apply H makes $(|0\rangle+|1\rangle)/\sqrt(2)$

Step 2. Wait makes $(|0\rangle+\exp(-i\Delta t)|1\rangle)/\sqrt{2}$

Step 3. Apply H $[(1+e^{-i\Delta t})|0\rangle+(1-e^{-i\Delta t})|1\rangle]/2$.

The probability of getting outcome 0 is $1/2+\cos(\Delta t)/2$.

In the second experiment:

Step 1. Apply H gives $(|0\rangle+|1\rangle)/\sqrt(2)$

Step 2. Half Wait gives $(|0\rangle+\exp(-i\Delta t/2)|1\rangle)/\sqrt{2}$

Step 3. Apply X gives $(|1\rangle+\exp(-i\Delta t/2)|0\rangle)/\sqrt{2}$.

Step 3. Half Wait gives $\exp(-i\Delta t/2)(|1\rangle+|0\rangle)/\sqrt{2}$.

Step 4. Apply H gives $\exp(-i\Delta t/2)|0\rangle$.

The probability of getting outcome $0$ is always $1$ (there is no decay). So the $T_2$ of this experiment would be infinite.

In the more general case $H = \Delta(t) |1\rangle\langle 1|$ the first experiment gives $\mathrm{Pr}(0) = 1/2+\langle\cos(\int_0^t \Delta(t))\rangle/2$ where we have averaged over different shots (runs of the experiment). I similar but different expression can be derived for the second experiment and I will leave as an exercise what happens but depending on the assumptions you are willing to make about the noise correlations you can simplify this expression in terms of the noise spectrum.

If you want to go check it out on a real experiment you can try the notebook https://github.com/QISKit/qiskit-tutorial/blob/master/reference/qcvv/relaxation_and_decoherence.ipynb but you need an account on the IBM Q experience.

Answer 2

From Chapter 15 of NII's quantum information lecture series on "Fundamentals of Noise processes" (link here):

An applied DC field $H_0$ is not completely uniform in all space points. If many spin qubits are placed in such an inhomogeneous DC field, they have different Larmor frequencies. This leads to the dephasing effect if we compare the phase difference between different qubits. A time constant for this dephasing process is determined by the spatial (not temporal) inhomogeneous broadening of the DC field and distinguished from $T_2$ process. A new time constant is often referred to as $T_2^*$.

So, if I am understanding this correctly (and I am not an expert in this), then $T_2^*$ is the combined dephasing from the standard dephasing mechanism (described by $T_2$) and any inhomogeneities in the DC field. Presumably, the reason it is often not distinguished from $T_2$ in papers is that it is assumed that such inhomogeneities always exist and so $T^*_2$ is almost always the value being measured and the most relevant in practise.