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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 P donor in silicon Localized electron S=1/2 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 Possible applications Magnetometry with nanoscale resolution STM ODMR nanoprobe: quantum dot, NV center, … Quantum computation Array of quantum dots NV centers in a waveguide Quantum repeater 2-qubit quantum computer NV center with an electron and a nuclear spin (15N or 13C) General problem: decoherence a be i a2 i ab e ab e i 2 b Influence of environment: nuclear spins, phonons, conduction electrons, … a2 0 0 2 b 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: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 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 Dynamical decoupling protocols General approach – e.g., group-theoretic methods Examples: H SZ Ak I kZ Periodic DD (CPMG, pulses along X): Period d-X-d-X (d – free evolution) H SX Ak IkX SY Bk IkY SZ Ck IkZ 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 ) Uper exp[i T ( H (0) H (1) H ( 2) ...)] O (1) Symmetrized XY PDD (XY SDD): O (T ) O(T 2 ) 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): etc. PDD-X-PDD-Y-PDD-X-PDD-Y 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 Numerical approaches 1. Exact solution The whole system (S+B) is isolated and is in pure quantum state H H S H B H SB (t ) T exp(iHt) (0) U (t ) (0) (t ) Ckm (t ) Sk Bm Sk , Bm - basis statesof thesystem and thebath 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 Dynamical decoupling for a single-electron quantum dot Single electron spin in a quantum dot Hyperfine spin coupling Fermi contact interaction Single electron QD H Ak S I k electron spin (delocalized) N k 1 nuclear spins (Ga, As nuclei) Hahn echo : from T2* ~ 10 ns to T2 ~ 1 μs Universal DD: protect all three components of the spin H H C (t ) Ak S I k H B N k 1 control Hamiltonian 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 PDD Magnus expansion is valid only for ≤ 10 ps SDD CDD2 ME valid Preserving unknown state of the spin t Decoherence: 0 S (t ) Fmin (t ) min 0 S (t ) 0 Worst-case scenario: minimum fidelity • 8 different protocols 1.0 0.9 SRPD Minimum fidelity • Imperfections considered CDD2 SHD • Large τ (up to 5 ns) • Long times 0 0.8 SDD RPD PDD NRD 0.7 • Finite-width pulses FID 0.6 FID PDD SDD CDD2 • Intra-bath interactions 0.5 0.1 1 10 100 Time DD works very well – but ME is not valid NRD RPD SRPD SHD 1000 Long times: fidelity saturation H 0 Ak S I k N k 1 SX (t) 1.0 XY PDD 0.8 0.6 τ = 0.01 0.4 100 τ = 0.01 0.8 τ = 0.1 0.6 200 300 τ=1 0.2 τ=1 0 B B B 1.0 0.4 τ = 0.1 0.2 0.0 SZ (t) 400 0.0 0 Time 50 100 150 Time 0 : U p / 2 X U ( ) Y U ( ) exp(i S z ) – commutes with Sz Sz 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 DD for P donors in silicon, fidelity for different states (S. Lyon and A. Tyryshkin) XZ PDD SY quasi-conserved Fidelity 1.0 Initial state along Y 0.5 Initial state along X 0.0 0 5 10 15 Number of Repeats XY PDD SZ quasi-conserved Fidelity 1.0 Initial state along Y 0.5 Initial state along X 0.0 0 5 10 15 20 25 Number of Repeats 30 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) Sample x • Rotation angle is not exactly π everywhere • Rotation axis is not exactly X (or Y) everywhere Freezing in Si:P, qualitative picture Consider some spin U X exp[i ( X )(Sn )] UY exp[i ( Y )(Sm )] PDD, after 1/2-cycle: U1/ 2 Ud exp[i B SZ ] exp[i( ) ( a )] (composition of rotations = rotation) After N cycles: U N exp[i 2 N ( a)] 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 PDD XY PDD O ( X , Y ) O ( X2 , Y2 ) aX 0 aX 0 aY 1 aY 0 aZ 0 a Z 1 All rotation axes close to Y Rotation angle – 1st order in εX , εY All rotation axes close to Z Rotation angle – 2nd order in εX , εY SY – frozen, SX and SZ decay fast SZ – frozen, SX and SY decay slow In agreement with experiment Quantitative treatment: numerics vs. experiment XZ PDD XY PDD 1.0 1.0 Fidelity 0.8 SY 0.8 0.6 0.6 0.4 0.4 0.2 SX SZ 0.0 5 10 NC (the number of cycles) 15 SZ SX SY 0.2 0.0 0 B C D B C 0 5 10 15 20 25 30 NC (the number of cycles) 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) 35 Concatenation: single-cycle fidelity XZ CDDs XY CDDs 1.0 B Nothing to show C D All fidelities are 1 (within 2%) Fidelity 0.8 0.6 SY SZ SX 0.4 0.2 0.0 0 1 2 3 4 Concatenation level 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) 1.0 Hollow circles – PDD XY Fidelity 0.8 Solid circles 0.6 SX SZ 0.4 0.2 0.0 SY 0 5 10 15 20 25 NC (number of cycles) 30 35 dXdYdXdY-YdXdYdXd, (cycle PDD (<Sx>) –<Sy> SDD (XY-8) B C D • Less freezing • Overall better fidelity Aperiodic sequences: Uhrig’s DD Optimization of the inter-pulse intervals: UDD j t j T sin 2( N p 1) 2 Np = 20: 1.0 SX 0.5 Fidelity Fidelity 1.0 Y Axis Title B SZ SY 0.0 X Axis Title SX 10000 (pulse) SY 0.5 number (2<Sy>) Np= (2<Sz>) SZ 0.0 0 100 Total time (s) All errors 200 0 50 100 150 Total time (s) nZ errors only UDD is not robust wrt pulse errors Very susceptible to the rotation angle errors 200 Aperiodic sequences: Quadratic DD 3rd order QDD: U4(Y)-X-U4(Y)-X-U4(Y)-X-U4(Y)-X Np = 20 U4(Y) = Uhrig’s DD with 4 pulses Fidelity 1.0 0.5 All errors 0.0 0 50 100 B C D 0.5 εX only 0.0 150 200 0 Total time (s) 50 100 150 200 Total time (s) 1.0 SX SY SZ Fidelity Fidelity 1.0 0.5 εY only B C D 0.0 0 50 100 150 Total time (s) 200 Lessons learned so far: 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 DD for spins in diamond Nitrogen-vacancy centers 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 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 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: m = ±1 2.87 GHz m=0 Individual NV centers can be initialized and read out: access to a single spin dynamics 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 NV center in a spin bath C C N V C C Bath spin – N atom NV spin ms = +1 ms = -1 C C MW Electron spin: pseudospin 1/2 14N nuclear spin: I = 1 ms = 0 m = +1/2 1 MW ms = -1/2 0 B B No flip-flops between NV and the bath Decoherence of NV – pure dephasing 0.5 Ramsey decay Decay of envelope: exp[ t T -0.5 0 0.2 0.4 0.6 t (µs) 0.8 ] * 2 2 Slow modulation: hf coupling to 14N T2* = 380 ns A = 2.3 MHz Need fast pulses 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 BL B1 cosLt L Rotating frame y L Spin x Oscillating field L S L co-rotating (resonant) counter-rotating (negligible) Rotating frame: static field B1/2 along X-axis Strong driving of a single NV center Experiment “Square” pulses: Simulation 29 MHz 109 MHz 223 MHz Time (ns) Gaussian pulses: Time (ns) 109 MHz 223 MHz • Rotating-frame approximation invalid: counter-rotating field • Role of pulse imperfections, especially at the pulse edges Characterizing / tuning DD pulses for NV center U X exp[i( X )(Sn)] n (nX , nY , nZ ) 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 Quantum process tomography out L[ in ] – linear relation between “in” and “out” Describes most of experimental situations – QM is linear ! in a0 a1 X a2 Y a3 Z out b0 b1 X b2 Y b3 Z a’s and b’s are linearly related – matrix χ – complete description of L 1. Prepare full set of basis states , , X , Y i 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 “Bootstrap” problem “Bootstrap” protocol Assume: errors are small, decoherence during pulse negligible U X exp[i( )( n) / 2] i( X Y Y Z Z ) Series 0: Series 1: Series 2: π/2X and π/2Y πX – π/2X, πY – π/2Y π/2X – πY, π/2Y – πX Find φ' and χ' (angle errors) Find φ and χ (for π pulses) 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) Bootstrap protocol: experiments Introduce known errors: - phase of π/2Y pulse - frequency offset Self-consistency check: QPT with corrections - Prepare imperfect basis states 1 , 0 , 1 0 , 1 i 0 - corrected - uncorrected - Apply corrections (errors are known) - Compare with uncorrected Ideal recovery: F = 1, M2 = 0 F T r[ 0 ] M 2 M 2 , M 0 M2 Fidelity What to expect for DD? Bath dynamics B(0) B(t ) Mean field: bath as a random field B(t) 1.0 Gaussian, stationary, Markovian noise: Ornstein-Uhlenbeck process 0.8 B(0) B(t ) b2 exp(Rt) 0.4 simulation O-U fitting 0.6 b – noise magnitude (spin-bath coupling) R = 1/τC – rate of fluctuations (intra-bath coupling) 0.2 0.0 0 10 20 30 Time Agrees with experiments: pure dephasing by O-U noise 0.5 0.5 Ramsey decay Spin echo * T2 = 380 ns exp[ t T -0.5 0 0.2 0.4 t (µs) ] * 2 2 0.6 0.8 T2 = 2.8 μs exp[t T2 ] 3 0 1 10 free evolution time (s) 40 Protocols for ideal pulses … B(t) X Pulses (t ) τ X τ X τ X X τ … +1 … –1 T=Nτ T Total accumulated phase: (t ) B (t )dt 0 X τ X τ T T s 0 0 Signal(T ) exp i exp ib W (T ) , W (T ) ds e Rs (t ) (t s)dt 2 Short times (RT << 1): PDD d-X-d-X 4 WF (T ) NR 3 3 Fast decay PDD-based CDD CPMG (d/2)-X-d-X-(d/2) CPMG-based CDD Long times (RT >> 1): 1 WS (T ) NR 3 3 Slow decay All orders: fast decay at all times, rate WF (T) Slow decay at all times, rate WS (T) optimal choice All orders: slow decay at all times, rate WS (T) Protocols for realistic imperfect pulses Pulses along X: CP and CPMG CPMG – performs like no errors CP – strongly affected by errors State fidelity εX = εY = -0.02, mX = 0.005, mZ = nZ = 0.05·IZ, δB = -0.5 MHz 1.0 x y 0.6 simulation 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 0 5 10 total time (s) 15 1.0 x y simulation 0.6 0 5 10 total time (s) 15 Aperiodic sequences: UDD and QDD Are expected to be sub-optimal: no hard cut-off in the bath spectrum 20 State fidelity 1 0.5 UDD Np = 6 0 CPMG 1/e decay time (μs) CPMG UDD exp. sim. 5 5 10 15 Total time (s) 5 10 Np 15 Robustness to errors: QDD6 vs XY4 1.0 B C QDD, SX QDD, SY 0.5 Np= 48 0.0 0 10 20 30 Total time (s) 40 XY4, SX XY4, SY 1.0 UDD vs XY4 B C B CX 0.5 UDD, S UDD, SY 0.0 Np= 48 -0.5 0 10 20 30 Total time (s) 40 XY4, SX XY4, SY Visibility issue Small times: 1.0 QDD, SX QDD, SY XY4, SX XY4, SY 0.5 0.0 0 10 20 30 40 Total time (s) QDD: F = 0.992 XY4: F = 0.947 B C XY4: U 1 i 24(mX nY ) Z QDD: U 1 i3 X X 3 Y Y Sensitive to different kind of errors 1.0 Solution: symmetrization XY8, SX B XY8, SY C XY8 B C B C No 1st-order errors. Initial F = 0.9999 0.5 0 but decays slowly as XY4 10 20 Total time (s) 30 DD on a single solid-state spin: scaling Master curve: for any number of pulses 3 S (T ) exp t 3 / Tcoh Tcoh T2 N p2 / 3 100 1/e decay time (μs) State fidelity 1 SE N=4 N=8 N = 16 N = 36 0.5 N = 72 NV2 10 NV1 N = 136 0.1 1 Normalized time (t / T2 N 2/3) 10 1 10 100 number of pulses Np 136 pulses, coherence time increased by a factor 26 No limit is yet visible Tcoh = 90 μs at room temperature Quantum process tomography of DD Re(χ) Im(χ) 1 1 0 0 -1 Iˆ ˆ x ˆ y ˆ z Iˆ ˆ z ˆ ˆ y -1 x 1 1 0 0 -1 Iˆ ˆ x ˆ y ˆ z ˆ ˆ z ˆ Iˆ x -1 y 1 1 0 0 -1 Iˆ ˆ x ˆ y ˆ z ˆ ˆ z ˆ y Iˆ x -1 t = 4.4 μs Iˆ ˆ x ˆ y ˆ z ˆ yˆ z ˆ Iˆ x Pure dephasing t = 10 μs Iˆ ˆ x ˆ y ˆ z ˆ z Only the elements ( I, I ) and (σZ , σZ ) change with time ˆ ˆ y Iˆ x t = 24 μs Iˆ ˆ x ˆ y ˆ z ˆ yˆ z ˆ Iˆ x 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