1. Pulse techniques for decoupling qubits from noise: experimental tests Bang-bang decoupling 31 P nuclear spins Low-decoherence electron-spin qubits and global 1/f noise Dynamical decoupling of the qubits – Periodic pulse sequences – Concatenated pulse sequences Summary Steve Lyon, Princeton EE Alexei Tyryshkin, Shyam Shankar, Forrest Bradbury, Jianhua He, John Morton

2. Experiments 2-pulse Hahn echo Decoupling  /2   FID – T 2 * Echo Signal Pulses T  /2  Echo Signal Pulses T  (|0  + |1  )

3. Dynamical Decoupling Replace single  -pulse with sequence of pulses – Refocus spins rapidly (< noise correlation time) – “Bang-bang” – fast strong pulses (or 2 different spins) – CP (Carr-Purcell) – periodic  -pulses  x /2-  -X-2  -X-2  -…-X-  -echo – CPMG (Carr-Purcell-Meiboom-Gill) – periodic  -pulses  x /2-  -Y-2  -Y-2  -…-Y-  -echo – Aperiodic pulse sequences – concatenated sequences Khodjasteh, Lidar, PRL 95, 180501 (2005); PRA 75, 062310 (2007). –  x /2-(p n-1 -X-p n-1 -Z-p n-1 -X-p n-1 -Z)-  -X-  -echo with Z=XY Yao, Liu, Sham, PRL 98, 077602 (2007). – concatenated CPMG –  x /2-(p n-1 -Y-p n-1 -p n-1 -Y-p n-1 )-  -Y-  -echo Experimental pulses ~ 1  s (for  -pulse) – Power ~ 1/(pulse length) 2  Energy/pulse ~ power 1/2

4. The Qubits: 31 P donors in Si Blue (microwave) transitions are usual ESR All transitions can be selectively addressed 31 P donor: Electron spin (S) = ½ and Nuclear spin (I) = ½ ↑e,↓n ↑e,↑n ↓e,↑n ↓e,↓n rf1  w1  w2 rf2 |3  |2  |1  |0  X-band: magnetic field = 0.35 T  w1 ~ 9.7 GHz ≠  w2 ~ 9.8 GHz rf1 ~ 52 MHz ≠ rf2 ~ 65 MHz

5. Bang-Bang control ↑e,↓n ↑e,↑n ↓e,↑n ↓e,↓n rf 1     2  rotation w1w1 Fast nuclear refocusing  i = a|0  + b|1   f = a|0  - b|1  22 31 P donor: S = ½ and I = ½ Nuclear refocusing pulse would be ~10  s but electron pulse ~30 ns

6. Electron spin qubits Doping ~10 15 /cm 3 Isotopically purified 28 Si:P 7K  electron T 1 ~ 100’s milliseconds 7K  electron T 2 ~ 60 milliseconds (extrapolating to ~single donor) x “real” T 2

7. Noise in electron spin echo signals averaged Single-pulse T 2 = 2 msec Decoherence In-phase Out-of-phase Signal transferred: in-phase  out-of-phase Must use single pulses to measure decoherence  About 100x sensitivity penalty

8. B-field noise Origin of noise unclear Background field in lab? Domains in the iron?  Essentially 1/f Measure noise voltage induced in coil

9. Microwave Field Inhomogeneity Sapphire Vertical B-field Sapphire cylinder Metal Wall  x /2-  -X-2  -X-2  -…-X-  -echo Carr-Purcell (CP) sequence

10. Periodic (standard) CPMG  x /2-  -Y-2  -Y-2  -…-Y-  -echo Self correcting sequence

11. Coherence after N pulses

12. Concatenated CPMG

13. Coherence vs. concatenation level

14. Concatenated and periodic CPMG Periodic CPMG 32 pulses Concatenated CPMG 42 pulses

15. Fault-Tolerant Dynamical Decoupling  x /2-(p n-1 -X-p n-1 -X-Y-p n-1 -X-p n-1 -X-Y)-  -X-  -echo Not obvious that it self-corrects

16. Coherence vs. concatenation level

17. Sanity check: collapse adjacent pulses Effect of combining pairs of adjacent pulses – Ex. Z-Z  I – n th level concatenation without combining  2*4 n – 2 = 510 for n=4 – n th level concatenation with combined pulses = 306 for n=4

18. Sanity check: white noise

19. Summary Dynamical decoupling can work for electron spins Through the hyperfine interaction with the electron can generate very fast bang-bang control of nucleus CPMG preserves initial  x /2 with fewest pulses – But does not deal with pulse errors for  y /2 – CPMG cannot protect arbitrary state Concatenated CPMG does no better Can utilize concatenated XZXZ sequence out to at least 1000 pulses – Situation with  y /2 initial states is more complex Not clear fidelity improves monotonically with level But much better than CP May need to combine XZXZ with composite pulses