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Explore bang-bang decoupling, dynamical decoupling, and pulse sequences for qubits in experimental quantum tests. Learn about decoupling methods, concatenated sequences, and noise reduction strategies.
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Pulse techniques for decoupling qubits from noise: experimental tests Steve Lyon, Princeton EE • Bang-bang decoupling 31P nuclear spins • Low-decoherence electron-spin qubits and global 1/f noise • Dynamical decoupling of the qubits • Periodic pulse sequences • Concatenated pulse sequences • Summary Alexei Tyryshkin, Shyam Shankar, Forrest Bradbury, Jianhua He, John Morton
Experiments /2 • 2-pulse Hahn echo • Decoupling Pulses (|0 + |1) FID – T2* Signal Echo T (|0 + |1) /2 Pulses (|0 + |1) Signal Echo T (|0 + |1)
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-(pn-1-X-pn-1-Z-pn-1-X-pn-1-Z)--X--echo with Z=XY • Yao, Liu, Sham, PRL 98, 077602 (2007). – concatenated CPMG • x/2-(pn-1-Y-pn-1-pn-1-Y-pn-1)--Y--echo • Experimental pulses ~ 1s (for -pulse) • Power ~ 1/(pulse length)2 Energy/pulse ~ power1/2
The Qubits: 31P donors in Si 31P donor: Electron spin (S) = ½ and Nuclear spin (I) = ½ • Blue (microwave) transitions are usual ESR • All transitions can be selectively addressed ↓e,↓n |3 X-band: magnetic field = 0.35 T w1 ~ 9.7 GHz ≠ w2 ~ 9.8 GHz rf1 ~ 52 MHz ≠ rf2 ~ 65 MHz rf1 ↓e,↑n |2 w1 w2 ↑e,↑n |1 rf2 ↑e,↓n |0
Bang-Bang control Fast nuclear refocusing 31P donor: S = ½ and I = ½ ↓e,↓n |3 ↓e,↑n |2 2p rotation w1 ↑e,↑n |1 rf1 ↑e,↓n |0 2p • i = a|0 + b|1f = a|0 - b|1 Nuclear refocusing pulse would be ~10 s but electron pulse ~30 ns
Electron spin qubits • Doping ~1015/cm3 • Isotopically purified 28Si:P • 7K electron T1 ~ 100’s milliseconds • 7K electron T2 ~ 60 milliseconds (extrapolating to ~single donor) 3 10 (Feher et al, 1958) T1 28Si:P 9.767 GHz 2 10 T2 28Si:P 9.767 GHz 1 10 T T2 Si:P 16.44 GHz 0 10 1 -1 10 “real” T2 x T1, T2 (sec) -2 10 (Gordon, 28Si, 1958) -3 10 T -4 10 2 -5 10 -6 10 -7 10 0 2 4 6 8 10 12 14 16 18 20 22 Temperature (K)
Noise in electron spin echo signals Decoherence In-phase Single-pulse T2 = 2 msec Out-of-phase averaged Signal transferred: in-phase out-of-phase • Must use single pulses to measure decoherence • About 100x sensitivity penalty
B-field noise Measure noise voltage induced in coil • Origin of noise unclear • Background field in lab? • Domains in the iron? • Essentially 1/f
Microwave Field Inhomogeneity Vertical B-field Sapphire cylinder Metal Wall Sapphire • Carr-Purcell (CP) sequence • x/2--X-2-X-2-…-X--echo
Periodic (standard) CPMG x/2--Y-2-Y-2-…-Y--echo Self correcting sequence
Concatenated and periodic CPMG Concatenated CPMG 42 pulses Periodic CPMG 32 pulses
Fault-Tolerant Dynamical Decoupling • x/2-(pn-1-X-pn-1-X-Y-pn-1-X-pn-1-X-Y)--X--echo • Not obvious that it self-corrects
Sanity check: collapse adjacent pulses • Effect of combining pairs of adjacent pulses • Ex. Z-Z I • nth level concatenation without combining 2*4n – 2 = 510 for n=4 • nth level concatenation with combined pulses = 306 for n=4
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
