Dynamical decoupling with imperfect pulses Viatcheslav Dobrovitski Ames Laboratory US DOE, Iowa State University

1. Dynamical decoupling with imperfect pulses Viatcheslav Dobrovitski Ames Laboratory US DOE, Iowa State University Works done in collaboration with Z.H. Wang, B. N. Harmon (Ames Lab), G. de Lange, D. Riste, R. Hanson (TU Delft), G. D. Fuchs, D. D. Awschalom (UCSB), L. Santos (Yeshiva U.), K. Khodjasteh, L. Viola (Dartmouth College) S. Lyon, A. Tyryshkin (Princeton) W. Zhang (Fudan), N. Konstantinidis (Fribourg)

2. Outline 1. Introduction – what are we doing and why. 2. Quantum dots – some lessons and caveats. 3. P donors in Si – how pulse errors qualitatively change the spin dynamics. 4. Dynamical decoupling of a single spin – decoupling protocols for a NV center in diamond.

3. Quantum spins in solid state P donor in silicon Localized electron S=1/2 NV center in diamond Localized electron spin S=1 Quantum dots Localized electron S=1/2 Fundamental questions: How to reliably manipulate quantum spins How to accurately model dynamics of driven spins Which dynamics is typical Which dynamics is interesting Which dynamics is useful

4. Possible applications Magnetometry with nanoscale resolution STM ODMR nanoprobe: quantum dot, NV center, … Quantum computation NV centers in a waveguide Array of quantum dots Quantum repeater 2-qubit quantum computer NV center with an electron and a nuclear spin (15N or 13C)

5. General problem: decoherence Influence of environment: nuclear spins, phonons, conduction electrons, … Decoherence: phase is forgotten Dynamical decoupling: applying a sequence of pulses to negate the effect of environment

6. Spectacular recent progress in DD on single spins Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby, 2010: 16-pulse CPMG sequence on quantum dot arXiv:1005.2995 de Lange, Wang, Riste, Dobrovitski, Hanson, 2010: DD on a single solid-state spin (NV center in diamond) 136 pulses, ideal scaling with Np Coherence time increased by a factor of 26 arXiv:1008.2119 Pulse imperfections start playing a major role Qualitatively change the spin dynamics Need to be carefully analyzed

7. Talking about dirt Studying dirt can be useful Antoni van Leeuwenhoek Delft, 17th century Studied dirt – discovered germs Ames Lab + TU Delft, 21st century Studied dirt, achieved DD on a single solid-state spin

8. Dynamical decoupling protocols General approach – e.g., group-theoretic methods Examples: Periodic DD (CPMG, pulses along X): Period d-X-d-X (d – free evolution) Universal DD (2-axis, e.g. X and Y): Period d-X-d-Y-d-X-d-Y Can also choose XZ PDD, or YZ PDD – ideally, all the same (in reality, different – see further)

9. Performance of DD and advanced protocols Assessing DD performance: Magnus expansion (asymptotic expansion for small period duration T ) Symmetrized XY PDD (XY SDD): XYXY-YXYX 2nd order protocol, error O(T2) Concatenated XY PDD (CDD) level l=1 (CDD1 = PDD): d-X-d-Y-d-X-d-Y level l=2 (CDD2): PDD-X-PDD-Y-PDD-X-PDD-Y etc.

10. Why we need something else? • Deficiencies of Magnus expansion: • Norm of H(0), H(1),… – grows with the size of the bath • Validity conditions are often not satisfied in reality • (but DD works) • Behavior at long times – unclear • Role of experimental errors and imperfections – unknown • Possible accumulation of errors and imperfections with time Numerical simulations: realistic treatment and independent validity check

11. Numerical approaches 1. Exact solution The whole system (S+B) is isolated and is in pure quantum state Very demanding: memory and time grow exponentially with N Special numerical techniques are needed to deal with d ~ 109 (Chebyshev polynomial expansion, Suzuki-Trotter decomposition) Still, N up to 30 can be treated 2. Some special cases – bath as a classical noise Random time-varying magnetic field acting on the spin

12. Dynamical decoupling for a single-electron quantum dot

13. Single electron spin in a quantum dot Single electron QD Hyperfine spin coupling Fermi contact interaction electron spin (delocalized) nuclear spins (Ga, As nuclei) Hahn echo : from T2* ~ 10 ns to T2 ~ 1 μs Universal DD: protect all three components of the spin control Hamiltonian

14. ME valid PDD SDD CDD2 Is Magnus expansion sufficient ? Periodic DD (PDD) d-X-d-Y-d-X-d-Y Symmetrized DD (SDD) XYXY-YXYX Concatenated, level 2 (CDD2) PDD-X-PDD-Y-PDD-X-PDD-Y Magnus expansion is valid only for  ≤ 10 ps

15. Preserving unknown state of the spin Decoherence: Worst-case scenario: minimum fidelity • 8 different protocols • Large τ(up to 5 ns) • Long times • Imperfections considered • Finite-width pulses • Intra-bath interactions DD works very well – but ME is not valid

16. Long times: fidelity saturation SX (t) SZ (t) XY PDD τ= 0.01 τ= 0.1 τ= 0.01 τ= 0.1 τ= 1 τ= 1 – commutes with Sz Sz is a “quasi-conserved” quantity Quantum tomography is a must to confirm decoupled qubit

17. DD for P donors in silicon: pulse errors and fidelity saturation

18. DD for P donors in silicon, fidelity for different states (S. Lyon and A. Tyryshkin) XZ PDD SY quasi-conserved Initial state along Y Initial state along X Initial state along Y XY PDD SZ quasi-conserved Initial state along X

19. P donors in Si: key features 1. Ensemble experiments: ESR on a large number of P spins 2. 29Si – depleted sample: f = 800 ppm (naturally, f=4.67%) 3. Inhomogeneous broadening: cw ESR linewidth 50 mG 4. However, T2 = 6 ms – plenty of room for DD Dephasing by almost static bath – decoupling should be perfect Model: pulse field inhomogeneity Bpulse (x) • Rotation angle is not exactly π • everywhere • Rotation axis is not exactly X • (or Y) everywhere Sample x

20. Freezing in Si:P, qualitative picture Consider some spin PDD, after 1/2-cycle: (composition of rotations = rotation) After N cycles: Each spin rotates around its own axis, by its own angle But all axes are close to Y (for PDD XZ) Total spin component along Y – conserved, other components average to zero

21. Simplified analytics (leading order in pulse errors) XZ PDD XY PDD All rotation axes close to Y Rotation angle – 1st order inεX , εY SY – frozen, SX and SZ decay fast All rotation axes close to Z Rotation angle – 2nd order inεX , εY SZ – frozen, SX and SY decay slow In agreement with experiment

22. Quantitative treatment: numerics vs. experiment XZ PDD XY PDD SZ SY SX SY SX SZ Hollow squares – experiment, dots – theory Rotation angle errors (εX , εY) – distribution width 0.3 (~15º) Rotation axis errors (nZ, mZ) – distribution width 0.12 (~7º)

23. Concatenation: single-cycle fidelity XZ CDDs XY CDDs Nothing to show All fidelities are 1 (within 2%) SY SZ SX Analytical result: CDDs of all levels have the same error, in spite of exponentially increasing number of cycles

24. Symmetrization: XY-8 sequence Periodic DD (PDD) d-X-d-Y-d-X-d-Y Symmetrized DD XYXY-YXYX (called XY-8 in the original paper) Hollow circles – PDD XY Solid circles – SDD (XY-8) SX SZ SY • Less freezing • Overall better fidelity

25. Aperiodic sequences: Uhrig’s DD Optimization of the inter-pulse intervals: UDD Np = 20: SX SX SY SZ SY SZ All errors nZ errors only UDD is not robust wrt pulse errors Very susceptible to the rotation angle errors

26. Aperiodic sequences: Quadratic DD 3rd order QDD: U4(Y)-X-U4(Y)-X-U4(Y)-X-U4(Y)-X U4(Y) = Uhrig’s DD with 4 pulses Np = 20 εX only All errors SX SY εY only SZ

27. Lessons learned so far: • Pulse errors are important • 2. Pulse errors can accumulate pretty fast • 3. Concatenated design is very good: errors stay the same • in spite of exponentially growing number of pulses • 4. Fidelity of different initial states must be measured. • 5. Freezing is a sign of low fidelity • 6. UDD and QDD require very precise pulses

28. DD for spins in diamond Nitrogen-vacancy centers

29. Studying a single solid-state spin: NV center in diamond Diamond – solid-state version of vacuum: no conduction electrons, few phonons, few impurity spins, … Simplest impurity: substitutional N Nitrogen meets vacancy: NV center Bath spins S = 1/2 Distance between spins ~ 10 nm Ground state spin 1 Easy-plane anisotropy Distance between centers: ~ 2 μm

30. NV center – solid-state version of trapped atom 3E ISC (m = ±1 only) 1A 532 nm 3A m = 0 – always emits light m = ±1 – not Initialization: m = 0 state Readout (PL level): population of m = 0 Ground state triplet: Individual NV centers can be initialized and read out: access to a single spin dynamics m = ±1 2.87 GHz m = 0

31. NV center and bath spins • Most important baths: • Single nitrogens (electron spins) • 13C nuclear spins • Long-range dipolar coupling DD on a single NV center • Absence of inhomogeneous broadening • Pulses can be fine-tuned: small errors achievable • Very strong driving is possible • (MW driving field can be concentrated in small volume) • NV bonus: adjustable baths – good testbed for DD and • quantum control protocols

32. C C C N V C C C C 0.5 -0.5 0 0.2 0.4 0.6 0.8 t(µs) NV center in a spin bath NV spin Bath spin – N atom ms=+1 m=+1/2 ms=-1 ms=-1/2 MW MW ms=0 Electron spin: pseudospin 1/2 14N nuclear spin: I = 1 B B No flip-flops between NV and the bath Decoherence of NV – pure dephasing Ramsey decay Decay of envelope: T2* = 380 ns A = 2.3 MHz Need fast pulses Slow modulation: hf coupling to 14N

33. Strong driving of a single NV center Pulses 3-5 ns long → Driving field in the range close to GHz Standard NMR / ESR, weak driving Rotating frame Spin Oscillating field x co-rotating (resonant) y counter-rotating (negligible) Rotating frame: static field B1/2 along X-axis

34. Strong driving of a single NV center Experiment Simulation “Square” pulses: 29 MHz 109 MHz 223 MHz Time (ns) Time (ns) Gaussian pulses: 109 MHz 223 MHz • Rotating-frame approximation invalid: counter-rotating field • Role of pulse imperfections, especially at the pulse edges

35. Characterizing / tuning DD pulses for NV center Pulse errors - important: see Si:P DD - unavoidable: counter-rotating field, pulse edges - all errors (nX, nY, nZ, εX) We want to determine and/or reduce the pulse errors • Known NMR tuning sequences: • Long sequences (10-100 pulses) – our T2* is too short • Some errors are negligible – for us, all errors are important • Assume smoothly changing driving field – our pulses are too short Can not be directly applied to strong driving

36. – linear relation between “in” and “out” 1. Prepare full set of basis states 2. Apply process L[ρ] to each of them 3. Measure in the same basis: determine χ Our situation: • Can reliably prepare only state • Can reliably measure only SZ Quantum process tomography Describes most of experimental situations – QM is linear ! a’s and b’s are linearly related – matrix χ– complete description of L “Bootstrap” problem

37. “Bootstrap” protocol Assume: errors are small, decoherence during pulse negligible Series 0: π/2X and π/2Y Find φ' and χ' (angle errors) Series 1: πX – π/2X, πY – π/2Y Find φ and χ (for π pulses) Series 2: π/2X – πY, π/2Y – πX Find εZ and vZ (axis errors, π pulses) Series 3: π/2X – π/2Y, π/2Y – π/2X π/2X – πX – π/2Y, π/2Y – πX – π/2X π/2X – πY – π/2Y, π/2Y – πY – π/2X Gives 5 independent equations for 5 independent parameters All errors are determined from scratch, with imperfect pulses • Bonuses: • Signal is proportional to error (not to its square) • Signal is zero for no errors (better sensitivity)

38. - corrected - uncorrected Bootstrap protocol: experiments Introduce known errors: - phase of π/2Y pulse - frequency offset Self-consistency check: QPT with corrections - Prepare imperfect basis states - Apply corrections (errors are known) - Compare with uncorrected Ideal recovery: F = 1, M2 = 0 M2 Fidelity

39. 0.5 0.5 Spin echo 0 -0.5 1 10 0 0.2 0.4 0.6 0.8 free evolution time (ms) t(µs) What to expect for DD? Bath dynamics Mean field: bath as a random field B(t) simulation O-U fitting Gaussian, stationary, Markovian noise: Ornstein-Uhlenbeck process b – noise magnitude(spin-bath coupling) R = 1/τC – rate of fluctuations (intra-bath coupling) Agrees with experiments: pure dephasing by O-U noise Ramsey decay T2* = 380 ns T2 = 2.8 μs

40. … B(t) Protocols for ideal pulses X X X X X X X τ τ τ τ τ τ Pulses … +1 … –1 T = N τ Total accumulated phase:

41. Short times (RT << 1): Long times (RT >> 1): PDD d-X-d-X Fast decay Slow decay PDD-based CDD All orders: fast decay at all times, rate WF (T) optimal choice CPMG (d/2)-X-d-X-(d/2) Slow decay at all times, rate WS (T) CPMG-based CDD All orders: slow decay at all times, rate WS (T)

42. 1.0 x y 0.6 simulation 0 5 10 15 total time (ms) 1.0 x y simulation 0.6 0 5 10 15 total time (ms) Protocols for realistic imperfect pulses εX = εY = -0.02, mX = 0.005, mZ= nZ= 0.05·IZ, δB = -0.5 MHz Pulses along X: CP and CPMG CPMG – performs like no errors CP – strongly affected by errors State fidelity Pulses along X and Y: XY4 (d/2)-X-d-Y-d-X-d-Y-(d/2) (like XY PDD but CPMG timing) State fidelity Very good agreement

43. 20 CPMG CPMG 1 UDD UDD 1/e decay time (μs) State fidelity exp. Np = 6 sim. 5 0.5 5 10 15 5 0 10 15 Np Total time (ms) Aperiodic sequences: UDD and QDD Are expected to be sub-optimal: no hard cut-off in the bath spectrum Robustness to errors: UDD vs XY4 QDD6 vs XY4 UDD,SX QDD,SX UDD,SY QDD,SY Np= 48 XY4,SX XY4, SY Np= 48 XY4,SX XY4, SY

44. Visibility issue Small times: QDD: F = 0.992 XY4: F = 0.947 QDD,SX QDD,SY XY4,SX XY4, SY XY4: QDD: Sensitive to different kind of errors Solution: symmetrization XY8 No 1st-order errors. Initial F = 0.9999 but decays slowly as XY4 XY8,SX XY8, SY

45. Master curve: for any number of pulses 100 1 NV2 SE N = 4 State fidelity 1/e decay time (μs) N = 8 10 N = 16 N = 36 NV1 0.5 N = 72 N = 136 0.1 1 10 1 10 100 Normalized time (t / T2 N 2/3) number of pulses Np DD on a single solid-state spin: scaling 136 pulses, coherence time increased by a factor 26 No limit is yet visible Tcoh = 90 μs at room temperature

46. 1 0 Re(χ) Im(χ) -1 1 0 -1 1 0 -1 1 0 -1 1 1 0 0 -1 -1 Quantum process tomography of DD t = 4.4 μs Pure dephasing t = 10 μs Only the elements (I, I) and (σZ , σZ ) change with time t = 24 μs

47. Summary • Standard analytics (Magnus expansion) is often insufficient • Numerical simulations are useful and often needed for • realistic assessment of DD protocols • In-out fidelity for a single state is not enough (freezing happens) • Tomography is needed, at least partial • Pulse errors are more than a little nuisance: can seriously plague • advanced DD sequences • Pulse errors need to be seriously addressed, theoretically and • experimentally • All taken into account, DD on a single solid-state spin achieved