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)

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.

Quantum spins in solid state NV center in diamond Localized electron spin S=1 Quantum dots Localized electron S=1/2 P donor in silicon 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

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

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

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

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

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)

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 2 nd order protocol, error O(T 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.

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

The whole system (S+B) is isolated and is in pure quantum state Numerical approaches 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

Dynamical decoupling for a single-electron quantum dot

Single electron spin in a quantum dot Single electron QD electron spin (delocalized) nuclear spins (Ga, As nuclei) Hyperfine spin coupling Fermi contact interaction control Hamiltonian Hahn echo : from T 2 * ~ 10 ns to T 2 ~ 1 μs Universal DD: protect all three components of the spin

PDD SDD CDD2 ME valid Magnus expansion is valid only for  ≤ 10 ps 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 Is Magnus expansion sufficient ?

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 Preserving unknown state of the spin Worst-case scenario: minimum fidelity Decoherence:

Long times: fidelity saturation τ = 1 τ = 0.1 τ = 0.01 S X (t)S Z (t) τ = 0.01 τ = 0.1 τ = 1 XY PDD – commutes with S z S z is a “quasi-conserved” quantity Quantum tomography is a must to confirm decoupled qubit

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

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

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

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

Simplified analytics (leading order in pulse errors) XZ PDDXY PDD All rotation axes close to Y Rotation angle – 1st order in ε X, ε Y S Y – frozen, S X and S Z decay fast All rotation axes close to Z Rotation angle – 2nd order in ε X, ε Y S Z – frozen, S X and S Y decay slow In agreement with experiment

Quantitative treatment: numerics vs. experiment SYSY SZSZ SXSX Hollow squares – experiment, dots – theory SYSY SXSX SZSZ XZ PDDXY PDD Rotation angle errors (ε X, ε Y ) – distribution width 0.3 (~15º) Rotation axis errors (n Z, m Z ) – distribution width 0.12 (~7º)

Concatenation: single-cycle fidelity XZ CDDsXY CDDs Nothing to show All fidelities are 1 (within 2%) SYSY SZSZ SXSX Analytical result: CDDs of all levels have the same error, in spite of exponentially increasing number of cycles

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) SYSY SZSZ SXSX Hollow circles – PDD XY Solid circles – SDD (XY-8) Less freezing Overall better fidelity

Aperiodic sequences: Uhrig’s DD All errors n Z errors only N p = 20: UDD is not robust wrt pulse errors Very susceptible to the rotation angle errors SXSX SYSY SXSX SYSY SZSZ SZSZ Optimization of the inter-pulse intervals: UDD

Aperiodic sequences: Quadratic DD 3 rd order QDD: U 4 (Y)-X-U 4 (Y)-X-U 4 (Y)-X-U 4 (Y)-X U 4 (Y) = Uhrig’s DD with 4 pulses All errors ε X only ε Y only N p = 20 SXSX SYSY SZSZ

1.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 Lessons learned so far:

DD for spins in diamond Nitrogen-vacancy centers

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

Individual NV centers can be initialized and read out: access to a single spin dynamics 2.87 GHz m = 0 m = ±1 ISC (m = ±1 only) 532 nm Ground state triplet: 3E3E 3A3A 1A1A Initialization: m = 0 state Readout (PL level): population of m = 0 NV center – solid-state version of trapped atom m = 0 – always emits light m = ±1 – not

NV center and bath spins 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

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

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 x y Rotating frameSpinOscillating field co-rotating (resonant) counter-rotating (negligible) Rotating frame: static field B 1 /2 along X-axis

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)

Characterizing / tuning DD pulses for NV center We want to determine and/or reduce the pulse errors Pulse errors - important: see Si:P DD - unavoidable: counter-rotating field, pulse edges - all errors (n X, n Y, n Z, ε X ) Known NMR tuning sequences: Long sequences ( 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 Can not be directly applied to strong driving

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

“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

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

What to expect for DD? Bath dynamics Mean field: bath as a random field B(t) Gaussian, stationary, Markovian noise: Ornstein-Uhlenbeck process simulation O-U fitting b – noise magnitude (spin-bath coupling) R = 1/τ C – rate of fluctuations (intra-bath coupling) Ramsey decay Agrees with experiments: pure dephasing by O-U noise T 2 * = 380 ns T 2 = 2.8 μs

Protocols for ideal pulses … τττ XXX T = N τ … +1 –1 … B(t)B(t) τττ XXXX Pulses Total accumulated phase:

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 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

Aperiodic sequences: UDD and QDD Are expected to be sub-optimal: no hard cut-off in the bath spectrum State fidelity Total time (  s) UDD CPMG N p = 6 sim. 1/e decay time ( μ s) NpNp 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

Visibility issue QDD, S X QDD, S Y XY4, S X XY4, S Y Small times: QDD: F = XY4: F = Sensitive to different kind of errors QDD: XY4: Solution: symmetrization XY8 No 1st-order errors. Initial F = but decays slowly as XY4 XY8, S X XY8, S Y

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

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

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