$T_{2}$ CPMG with delay sweep

This experiment determines the dephasing time, $T_{2}$, of a qubit by measuring the decay rate following a Carr-Purcell-Meiboom-Gill (CPMG) sequence¹. The number of $\pi$-pulses is fixed and the interpulse delay is swept.

Description

The CPMG sequence is a modified Echo sequence,² where the block [$\tau$ - $\pi$ - $\tau$] is repeated $N$ times between the two $\frac{\pi}{2}$-pulses and $\tau$ is the qubit free evolution time. Building on the coherence protection provided by the Echo sequence against low-frequency dephasing noise, the CPMG sequence is effective against dephasing at relatively higher frequencies ($0 \leq f < \frac{1}{4\tau}$).

Experiment steps

1. A $\frac{\pi}{2}$-pulse ($R_y(\frac{\pi}{2})$) is applied, which prepares the qubit in the superposition state $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$.

2. A [$\tau$ - $\pi$ - $\tau$] block is repeated $N$ times, where the $\pi$ pulse rotates the qubit around the $x$-axis ($R_x(\pi)$) and $N=1$ corresponds to an Echo sequence.

3. Another $\frac{\pi}{2}$-pulse ($R_y(\frac{\pi}{2})$) is applied.

4. The resonator transmission is measured.

5. Steps 1 to 4 are repeated for different values of $\tau$.

Analysis steps

1. The amplitude of the readout resonator's signal is calculated as $\sqrt{I^2 + Q^2}$, where $I$ and $Q$ are the in-phase and quadrature components of the transmitted signal, respectively.

2. The delay is converted into an effective time $t_{eff}= 2 N \tau + (N + 1) t_{\pi}$, where $t_{\pi}$ is the duration of the $\pi$-pulse.

3. The amplitude of the readout resonator's signal is plotted against $t_{eff}$ and $T_{2}$ is determined by fitting a multi-exponential function to the experimental trace. The fit function includes the effect of relaxation and coherence decay during the $\pi$-pulses.

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1. H. Y. Carr and E. M. Purcell. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev., 94:630–638, May 1954. doi:10.1103/PhysRev.94.630. ↩

2. Leigh M. Norris, Gerardo A. Paz-Silva, and Lorenza Viola. Qubit noise spectroscopy for non-gaussian dephasing environments. Phys. Rev. Lett., 116:150503, Apr 2016. doi:10.1103/PhysRevLett.116.150503. ↩