1. Quantum dynamics and quantum control of spins in diamond Viatcheslav Dobrovitski Ames Laboratory US DOE, Iowa State University Works done in collaboration with Z.H. Wang (Ames Lab – now USC), T. der Sar, G. de Lange, T. Taminiau, R. Hanson (TU Delft), G. D. Fuchs, D. Toyli, D. D. Awschalom (UCSB), D. Lidar (USC)

2. Individual quantum spins in solid state  Quantum information processing  Single-spin coherent spintronics and photonics  High-precision metrology and magnetic sensing at nanoscale Quantum spin coherence: valuable resource NV center in diamond Quantum dots Donors in silicon

3. Fundamental problems: 1.Understand dynamics of individual quantum spins 2. Control individual quantum spins 3. Preserve coherence of quantum spins 4. Generate and preserve entanglement between quantum spins Grand challenge – controlling single quantum spins in solids Spins in diamond – excellent testbed for quantum studies Long coherence time Individually addressable Controllable optically and magnetically Jelezko et al, PRL 2004; Gaebel et al, Nat.Phys. 2006; Childress et al, Science 2006

4. Dynamical decoupling protocols Symmetrized protocol: τ-X-τ-X-X- τ -X- τ = τ -X- τ - τ -X- τ 2 nd order protocol, error O(τ 2 ) Concatenated protocols (CDD) level l=1 (CDD1 = PDD): τ -X- τ -X level l=2 (CDD2): PDD-X-PDD-X etc. Simplest – Periodic DD : Period τ -X- τ -X CPMG sequence Traditional analysis and classification: Magnus expansion

5. 2. Approximate – but very accurate – numerics: coherent spin states Assessing the quality of coherence protection Up to 32 spins (Hilbert space d = 4×10 9 ) on 128 processors Parallel code, 80 % efficiency 3. Analytical mean-field techniques 1. Exact numerical modeling 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 (the UV cutoff is too large) but DD works Behavior at long times – unclear Accumulation of pulse errors and imperfections – unknown

6. Outline de Lange, Wang, Riste, Dobrovitski, Hanson: Science 2010 Ryan, Hodges, Cory: PRL 2010 Naydenov, Dolde, Hall, Fedder, Hollenberg, Jelezko, Wrachtrup: PRB 2010 Spectacular recent progress: DD on a single NV spin 1.Quantum control and dynamical decoupling of NV center: protecting coherence 2. Decoherence-protected quantum gates 3. Decoherence-protected quantum algorithm: first 2-qubit computation with invidivual solid-state spins

7. Simplest impurity: substitutional N (P1 center) Environment (spin bath) S = 1/2 Long-range dipolar coupling Nitrogen meets vacancy: NV center Central spin S = 1, I = 1 HF coupling onsite Dipolar coupling to the bath NV center in diamond Single NV spin can be initialized, manipulated and read out

8. ISC (m = ±1 only) 532 nm Excited state: Spin 1 orbital doublet Ground state: Spin 1 Orbital singlet 1A1A Single NV center – optical manipulation and readout m = 0 – always emits light m = ±1 – not m = +1 m = –1 m = 0 m = +1 m = –1 m = 0 MW Initialization: m = 0 state Readout (PL): population of m = 0

9. Decoherence: NV center in a spin bath NV electron spin: pseudospin S = 1/2 (qubit) NV spin ms = 0ms = 0 B m s = –1 m s = +1 Bath spin – N atom B m = +1/2 m s = -1/2 No flip-flops between NV and the bath: energy mismatch – field created by the bath spins Time dependence governed by H B C C C C C C N V C

10. Mean field picture: bath as a random field Gaussian, stationary, Markovian noise b – noise magnitude (spin-bath coupling) τ C – correlation time (intra-bath coupling) Direct many-spin modeling: confirms mean field simulation O-U fitting Dobrovitski et al, PRL 2009Hanson et al, Science 2008

11. Free decoherence T 2 * = 380 ns Spin echo: probing the bath dynamics Decay due to field inhomogeneity from run to run τ C = 25 μs Modulation: HF coupling to 14 N of NV

12. Quantum control and Dynamical decoupling: Extending coherence time of a single NV center

13. CPMG τ-X- 2τ -X-τ PDD τ -X- τ -X Short times (T << τ C ): Long times (T >> τ C ): Fast decaySlow decay Slow decay at all times, rate W S (T) optimal choice Choice of the DD protocol: theory Concatenated PDD Fast decay at all times, makes things worse Concatenated CPMG Slow decay at all times, no improvement and many other protocols have been analyzed…

14. Qualitative features Coherence time can be extended well beyond τ C as long as the inter-pulse interval is small enough: τ/τ C << 1 Magnus expansion (also similar cumulant expansions) predict: W(T) ~ O(N τ 4 ) for PDD but we have W(T) ~ O(N τ 3 ) Symmetrization or concatenation give no improvement Source of disagreement: Magnus expansion is inapplicable Ornstein-Uhlenbeck noise: Second moment is (formally) infinite – corresponds to Cutoff of the Lorentzian:

15. DD “as usual” Pulses only along X: τ-X-2τ-X- τ X component – preserved well Y component – not so well State fidelity What is wrong? Control pulses are not perfect

16. Fast rotation of a single NV center Example pulse shape: ExperimentSimulation 29 MHz 109 MHz 223 MHz Rotating-frame approximation invalid: counter-rotating field Pulse imperfections important Time (ns) Fuchs et al, Science 2009 1.Bootstrap protocol - characterize all pulse errors from scratch Dobrovitski et al, PRL 2010 2. Understand well the accumulation of the pulse errors Wang et al, arXiv:1011.6417; Khodjasteh et al PRA 2011

17. Protecting all initial states Pulses only along X: τ -X-2 τ -X- τ X component – preserved well Y component – not so well Pulses along X and Y: τ -X-2 τ -Y-2 τ -X-2 τ -Y- τ Both components are preserved Coherence extended far beyond echo time State fidelity Solution: two-axis control

18. Aperiodic sequences: UDD and QDD Are expected to be sub-optimal: no hard cut-off in the bath spectrum State fidelity 0 5 1015 0.5 1 Total time (  s) UDD CPMG N p = 6 sim. 1/e decay time ( μ s) NpNp 5 1015 5 20 exp. CPMG UDD Robustness to errors: QDD, S X QDD, S Y XY4, S X XY4, S Y QDD6 vs XY4 N p = 48 UDD, S X UDD, S Y XY4, S X XY4, S Y UDD vs XY4 N p = 48

19. Extending coherence time with DD Master curve: for any number of pulses 136 pulses, coherence time increased by a factor 26 T coh = 90 μs at room temperature, and no limit in sight De Lange, Wang, Riste, et al, Science 2010

20. Using DD for other good deeds Single-spin magnetometry with DD Detailed probe of the mesoscopic spin bath de Lange, Riste, Dobrovitski et al, PRL 2011 Taylor, Cappellaro, Childress et al. Nat Phys 2008 Naydenov, Dolde, Hall et al. PRB 2011 de Lange, van der Sar, Blok et al, arXiv 2011

21. Combining DD and quantum operation Gates with resonant decoupling

22. Coupling NVs to each other – hybrid systems Hybrid systems: different types of qubits for different functions NV centers – qubits Nanomechanical oscillators – data bus Rabl et al, Nat Phys 2010 NV centers – qubits Spin chain (other spins) – data bus Cappellaro et al PRL 2010; Yao et al. PNAS 2011 Electron spins – processors Nuclear spins – memory Many works since Kane 1998, maybe before

23. Bath Unprotected quantum gate Bath Protected storage: decoupling “Standard” quantum operation Contradiction: DD efficiently preserves the qubit state but quantum computation must change it

24. Bath Unprotected quantum gate Bath Protected storage: decoupling Gates with integrated decouplind Bath Protected gate DD TgTg

25. Nuclear 14 N spin: memory, Electronic NV spin: processing (quantum memory, quantum repeater, magnetic sensing, etc.) But control of nuclear spin takes much longer than T 2 * Poor choice: either decouple the electron – no gates possible or gating without DD – no gates possible Gate with resonant decoupling (GARD) for hybrid systems Childress, Taylor, Sorensen et al. PRL 2006 Taylor, Marcus, Lukin PRL 2003 Jiang, Hodges, Maze et al. Science 2009 Neumann, Beck, Steiner et al. Science 2011 C C C C C C N V C A way out: use internal resonance in the system Different qubits have different coherence and control timescales One qubits decoheres before another starts to move

26. How the GARD works - 1 Rotating frame (ω N << A ) A ωNωN Nuclear rotation around XNuclear rotation around Z Rotating frame: A = 2π ∙ 2.16 MHz ω N = 2π ∙ 18 kHz 100 times smaller

27. How the GARD works - 2 Main problem: electron switches very frequently between 0 and 1 and slow nuclear spin should keep track of this Contradiction with the very idea of DD? 0-X-1-1-Y-0 1-X-0-0-Y-1 XY4 unit: τ -X- τ - τ -Y- τ Motion of the nuclear spin: conditional single-spin rotation Axes n 0 and n 1 are both close to z (A >> ω 1 ): small Resonance: smth 2 also becomes small when

28. How the GARD works - 3 RZ(π)RZ(π) R X (2α)RZ(π)RZ(π) RX(α)RX(α)R Z (2π)RX(α)RX(α) IN: OUT: XY-4 unit:

29. Experimental implementation of GARD Nuclear spin rotation conditioned on the electron: Unconditional nuclear spin rotation: All nuclear gates are produced only by changing τ Resonances are very narrow, ~ (ω N /A) 2 Timing jitter < 1 ns over 100 μs time span Error by 10 ns – fidelity drops by 10%

30. Protected C-Rot gate T G = 60 μs >> T 2 * Experimental implementation: proof of concept IZIZ electron nucleus 0.5 -0.5 IZIZ Fidelity 97% mostly, T1 decay CNOT gate

31. How good is GARD: protected CNOT gate Controllable decoherence: inject a noise into the system Decoherence time: T 2 = 50 μs; Gate time T G = 120 μs

32. GARD implementation of Grover’s algorithm 2 qubits – Grover’s algorithm converges in one iteration Total time: 330 μs, T 2 time only 250 μs First quantum computation on two individual solid-state spins

33. 123 456 Fidelity: 95% for GARD implementation of Grover’s algorithm For other states: 0.93, 0.92, 0.91 High fidelity beyond coherence time

34. Conclusions 1.Diamond-based QIP becomes truly competitive 2.Coherence time can be extended, 25-fold demonstrated 3.DD can be efficiently combined with gates 4.GARD algorithms demonstrated, 50% longer than T 2 Fidelity above 90% First 2-qubit computation on individual solid state spins