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), G. de Lange, D. Riste, R. Hanson (TU Delft), G. D. Fuchs, D. Toyli, D. D. Awschalom (UCSB)

2. Quantum spins in the solid state settings NV center in diamond Quantum dots Fundamental questions How to manipulate quantum spins How to model spin dynamics Which dynamics is typical Which dynamics is interesting Which dynamics is useful Applications Nanoscale magnetic sensing High-precision magnetometry Quantum repeaters Quantum key distribution Quantum memory Magnetic molecules

4. General problem: decoherence Decoherence: nuclear spins, phonons, conduction electrons, … Quantum control of spin state in presence of decoherence

5. Spin control – important topic (>10,000 items on Amazon.com)

6. Preserving coherence: dynamical decoupling (DD) Employ time reversal, like in spin echo Spin echo: as if nothing happened Electron spin S Decohered by many nuclear spins I k Periodic DD (PDD): Central spin S is decoupled from the bath of spins I k ττ τ τ

7. 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) Viola, Knill, Lloyd, PRL 1999

8. Performance of DD and advanced protocols Assessing DD performance: Magnus expansion (asymptotic expansion for small delay τ, total experiment duration T ) Symmetrized XY PDD (XY SDD): XYXY-YXYX 2 nd order protocol, error O(τ 2 ) 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. Khodjasteh, Lidar, PRL 2005

9. 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 Traditional NMR and ESR: Only one spin component is preserved – others are often lost Only macroscopic systems Our focus: preserve complete quantum spin state for a single spin

10. The whole system (S+B) is isolated and is in pure quantum state Numerical simulations 1. Exact solution Very demanding: memory and time grow exponentially with N Special numerical techniques are needed to deal with d ~ 10 9 (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

11. Spectacular recent progress in DD on single spins Bluhm, Foletti, Neder, Rudner, Mahalu, Umansky, Yacoby: arXiv:1005.2995 de Lange, Wang, Riste, Dobrovitski, Hanson: Science 330, 60 (2010) Pulse imperfections start playing a major role Qualitatively change the spin dynamics Need to be carefully analyzed Ryan, Hodges, Cory: PRL 105, 200402 (2010) Naydenov, Dolde, Hall, Shin, Fedder, Hollenberg, Jelezko, Wrachtrup: arXiv:1008.1953

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

13. 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 Jelezko, Gaebel, Popa et al, PRL 2004 Gaebel, Jelezko, et al, Science 2006 Childress, Dutt, Taylor et al, Science 2006 Initialization: m = 0 state Readout (PL): population of m = 0

14. Theoretical picture: NV center and the bath of N atoms Most important baths: Single nitrogens (electron spins) 13 C 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 Hanson, Dobrovitski, Feiguin et al, Science 2008

15. Single central spin vs. Ensemble of similar spins Dilute dipolar-coupled baths Spectral line – Gaussian Spectral line – Lorentzian Rabi oscillations decay Dobrovitski, Feiguin, Awschalom et al, PRB 2008 Decoherence: Gaussian decay F ~ exp(-t 2 ) Decoherence: exponential decay F ~ exp(-t) Strong variation of local environment between different NV centers Prokof’ev, Stamp, PRL 1998

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

17. Strong driving of a single NV center Pulses 3-5 ns long → Driving field in the range of 0.1-1 GHz Standard NMR / ESR, weak driving x y Rotating frameSpinOscillating field co-rotating (resonant) counter-rotating (negligible) Rotating frame: static field B 1 /2 along X-axis

18. Strong driving of a single NV center “Square” pulses: ExperimentSimulation 29 MHz 109 MHz 223 MHz Gaussian pulses: 109 MHz 223 MHz Rotating-frame approximation invalid: counter-rotating field Role of pulse imperfections, especially at the pulse edges Time (ns) Fuchs, Dobrovitski, Toyli, et al, Science 2009

19. Characterizing / tuning DD pulses for NV center Known NMR tuning sequences: Long sequences (10-100 pulses) – our T 2 * is too short Some errors are negligible – for us, all errors are important Assume smoothly changing driving field – our pulses are too short Pulse error accumulation can be devastating at long times High-quality pulses are required for good DD Dobrovitski, de Lange, Riste et al, PRL 2010 Can reliably prepare only state Can reliably measure only S Z “Bootstrap” problem:

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

21. Bootstrap protocol: experiments Introduce known errors: - phase of π/2 Y pulse - frequency offset Self-consistency check: QPT with corrections Fidelity M2M2 - Prepare imperfect basis states - Apply corrections (errors are known) - Compare with uncorrected Ideal recovery: F = 1, M 2 = 0 - corrected - uncorrected

22. What to expect for DD? Bath dynamics Mean field: bath as a random field B(t) Confirmed by simulations simulation O-U fitting b – noise magnitude (spin-bath coupling) R = 1/τ C – rate of fluctuations (intra-bath coupling) Ramsey decay Experimental confirmation: pure dephasing by O-U noise T 2 * = 380 ns T 2 = 2.8 μs De Lange, Wang, Riste, et al, Science 2010 Dobrovitski, Feiguin, Hanson, et al, PRL 2009

23. CPMG (d/2)-X-d-X-(d/2) PDD d-X-d-X Short times (RT << 1):Long times (RT >> 1): PDD-based CDD Fast decaySlow decay All orders: fast decay at all times, rate W F (T) Slow decay at all times, rate W S (T) CPMG-based CDD All orders: slow decay at all times, rate W S (T) optimal choice Protocols for ideal pulses

24. 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:

25. Protocols for realistic imperfect pulses Pulses along X: CP and CPMG CPMG – performs like no errors CP – strongly affected by errors Pulses along X and Y: XY4 (d/2)-X-d-Y-d-X-d-Y-(d/2) (like XY PDD but CPMG timing) Very good agreement State fidelity ε X = ε Y = -0.02, m X = 0.005, m Z = n Z = 0.05·I Z, δB = -0.5 MHz

26. Quantum process tomography of DD t = 10 μs t = 24 μs t = 4.4 μs Only the elements ( I, I ) and (σ Z, σ Z ) change with time Pure dephasing No preferred spin component DD works for all states

27. DD on a single solid-state spin: scaling Master curve: for any number of pulses 136 pulses, coherence time increased by a factor 26 No limit is yet visible T coh = 90 μs at room temperature

28. What I will not show (for the lack of time) Single-spin magnetometry with DD Joint DD on central spin and the bath Quantum gates with DD … and much more to come in this field Ultimately – sensing a single magnetic molecule

29. Summary Dynamical decoupling – important for applications and for fundamental reasons DD on a single spin – challenging but possible Accumulation of pulse errors – careful design of DD protocols (Careful theoretical analysis) + (advanced experiments) = First implementation of DD on a single solid-state spin. Further advances: DD for control and study of the bath, DD with quantum gates, DD for improved magnetometry, etc.