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\(T_{2}\) CPMG with pulse number sweep

This experiment determines the dephasing time, \(T_{2}\), of a qubit by measuring the decay rate following a Carr-Purcell-Meiboom-Gill (CPMG) sequence12. The number of \(\pi\)-pulses is swept and the interpulse delay is fixed.

Description

The CPMG sequence is a modified Echo sequence,3 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 \(N\).

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 pulse number 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.

  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. S. Meiboom and D. Gill. Modified Spin‐Echo Method for Measuring Nuclear Relaxation Times. Review of Scientific Instruments, 29(8):688–691, 08 1958. doi:10.1063/1.1716296. ↩

  3. 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. ↩