Single Electron Spin in Interacting Nuclear Spin Bath Coherence Loss and Restoration

Transcription

1 Asilomar, CA, June 6 th, 2007 Single Electron Spin in Interacting Nuclear Spin Bath Coherence Loss and Restoration Wang Yao Department of Physics, University of Texas, Austin Collaborated with: L. J. Sham (UC San Diego) Renbao Liu (CUHK) S. K. Saiin (UC San Diego) Supported by NSF DMR, ARDA/ARO, DARPA/AFOSR

2 Issues and Problems Single spin in solids: decoherence and coherence protection

3 Single electrons localized in solids By quantum dot By impurities By electrical gate

4 Issues and Problems Single spin in solids: decoherence and coherence protection A 2-level system + many interacting bath spins

5

6 Theories antecedent on spin bath Spectral diffusion theory Herzog & Hahn, PR 56, Klauder & Anderson, PR 61, semi-classical theory Electron nuclear hyperfine interation Schulten & Wolynes, J. Chem. Phys., 78 frozen nuclear configuration Merulov, Efros & Rosen, PRB 02 Khaetsii, Loss & Glazman, PRL 02, Coish & Loss, PRB 04 Schliemann, Khaetsii & Loss, PRB 02 sys-bath entanglement Shenvi, De Sousa & Whaley, PRB 05 small environment ~10 nuclei Nuclear dipolar interaction plus diagonal e-n hyperfine coupling De Sousa & Das Sarma, PRB 03 semiclassical stochastic solution De Sousa, Shenvi & Whaley, PRB 05 semiclassical theory Witzel, De Sousa & Das Sarma, PRB 05 - ensemble study Other low energy modes mapped to spin bath Proov ef & Stamp, Rep. Prog. Phys. 00 magnets and superconducting systems Sorry, not an exhaustive list.

7 Issues and Problems Single spin in solids: decoherence and coherence protection A 2-level system + many interacting bath spins Single electron in III-V semiconductor quantum dot

8 Spin decoherence in QD: Phonon Vs. Nuclei Decoherence by phonon scattering Spin relaxation T 1 ~ ms s, at ~ K in QD theories (Khaetsii and Nazarov; Woods, Reinece and Lyanda-Gella); experiments (Tarucha group; Kouwenhoven group; Finley group; Steel group) Pure dephasing suppressed at low-temp theoretical estimation: T 2 = 2T ~ K and below (Golovach, Khaestsii and Loss, PRL 04 ; Semenov & Kim, PRL 04 ) Spin transverse decoherence time from exp Single spin T 2? not measurable with current capability T 2 *~1-10 ns >> T 1 gated dot in GaAs, SAD T H ~ 1.2 µs >> T 1 gated dot in GaAs (Petta ea., Koppens ea.) Nuclear spin is the dominant cause for transverse decoherence at low temp.

9 Electron spin in a nuclear spin bath E-N coupled through contact hyperfine interaction Mesoscopic size N ~ 10 6, posterior justification

10 e-n interaction off-diagonal! + amse Im Electron nuclear dynamics Intrinsic n-n interaction, e.g. dipolar b I I +! m, n m n Dynamical fluctuation of zeeman energy a! a " 0 +! anse In n e-n interaction diagonal m a S I z z n e n aman S z e I + m I! n " e! = gµ B Inhomogeneous broadening Extrinsic n-n interaction* by nuclear field! T 2 e B Zeeman energy gµ B B ~ 10 GHz (1 T) V.S. e-n off-diagonal energy A ~ 1 THz A / N ~ 0.1! 1 GHz Polarized Unpolarized 100 mt 1 T 10 T

11 Issues and Problems Single spin in solids: decoherence and coherence protection A 2-level system + many interacting bath spins Single electron in III-V semiconductor quantum dot Low T and high field Pure transverse decoherence ( T 1! ") Mescoscopic nuclear bath, N ~ 10 6, unpolarized 1 + N well isolated from the rest universe Quantum mechanical origin of decoherence: entanglement

12 Decoherence of single quantum system Pure system state initially factorized Sys-bath entanglement from bath 2 $ *!!" % e # (0) ( Tr[ # tot ] = & ' 2 2 & * * $ + %! )! " " '!" J J* & e ' ( * + $ 2 ')! " J J " (* # ( t) = Zure, RMP 03 ; Schlosshauer, RMP 05 Bath dynamics conditioned on system states Bath state bifurcates in the Hilbert space System (which-state) information measured by the bath Decoherence: decays of off-diagonal DM element (T 2 ) Nuclear correlation func: L ( t) # J ( t) J ( t) ;! ( t) = L ( t)! (0) s " + e s e +," +," +," +,"

13 Issues and Problems Single spin in solids: decoherence and coherence protection A 2-level system + many interacting bath spins Single electron in III-V semiconductor quantum dot Low T and high field Pure transverse decoherence ( T 1! ") Mescoscopic nuclear spin bath, N ~ 10 6, unpolarized 1 + N well isolated from the rest universe Quantum mechanical origin of decoherence: entanglement Coherence restoration by disentanglement

14 Recoherence by disentanglement Flip of system states redirects bath evolution Bath states intersect in Hilbert space (which-state info erased) System disentangled from bath Coherence restored Possible in mesoscopic bath: sys+bath isolate from rest universe

15 Nuclear spin dynamics: pair wise flip-flop Elementary excitation: th pair-flip """"""# j1 L jn L jm L j j N 1 L j + 1 L j! 1 L j Transition amplitudes: Energy cost: j + 1 j! 1 j j n m m n ± A + B D ± E ( if e-spin state is ± ) n m N A Transition amplitude by hyperfine mediated n-n coupling infinite-range, dominates non-local pair flip-flop B Transition amplitude by intrinsic nuclear coupling finite-range, dominates local pair flip-flop D Energy cost by diagonal nuclear couplings E Energy cost by hyperfine interaction E = a! a ( ) 2 n m B

16 Nuclear spin dynamics: pair wise flip-flop Hierarchy of dynamics: 1 excitation M J, M 1 3 J, 1, 2 1 J, 1, 2, 3 J " # j n,! J, 1 vacuum In time scale 2 of interest, 2 J, 1, 3the number n,! 3 M Lof J, 1, 2, nuclear pair-flip 3 3 M excitations 1 is small! J, 2, 3 J, 2 M 3 J,,, State at time t 2 excitations M M J,, 1 5 M M J, 3, 4 M M J,,, M M 2 3 4!, 1 1!, 1, L J ( t) = C ( t) J + C ( t) J, + C ( t) J,, + ± ± ± ± J J J, Size of relevant Hilbert space / 4 / 4 / 4 ~ C N N N N C3 N / 4 CN / 2

17 Nuclear spin dynamics: pair wise flip-flop Unconnected pair-flips are independent Unconnected Connected Probability of having connected pair-flips con J! j1 L j L jl LL jm L jn L jn 2 ( flip ) P! 1" exp " N N 1 j L j! 1 l j l j m! 2 L 2 j + 1 Number of nuclear available for flip large ~ N (~10 6 ) for typical J Number of flips much smaller than N 2 2 2! J 2 1! J 3 1 2! J 1 2 L 3 N = C + C + C + N flip,,,,,,,,, m 2! 1 L j + 1 n

18 Pair-Correlation-Method The two states connected by n-n pair-flip interactions is mapped into one two-state system ( th pseudo-spin ). Flip-pair to pseudo spin: Initial state: j j! ", j + 1 j # 1! $ n m n m flip-flop of nuclear pair mapped to single pseudo spin flip J! " # The dynamics of the pseudo-spins (pair-flips) is treated independent of each other. Flip-flop dynamics to pseudo spin rotations: ± ± J ( t ) " #!,! ± ± " ih t # e $ s Nuclear correlation func: L ( t) J " ( t) J + ( t)! " ( t)! + +,- = = # ( t) WY, Liu and Sham, PRB 74, , 2006

19 Pair-Correlation-Method Verified from a lined cluster expansion approach. ( ) 2!! 0 t V ( t) + V ( t) + L J T exp " i# V ( t ) dt J = e V ( t ) = 2 4 Lined diagrams V ( t ) = 4 V 4 = + 1/2 + 1/2 + Independent pseudospin rotations: Saiin, WY and Sham, PRB 75, , 2007

20 Geometrical picture for decoherence h ± = ( B ± A,0, D ± E ) " = ( h # óˆ ) Rotation of Bloch vector ± =! ± ( t) óˆ! ± ( t) in effective h ± field. Distance! = S + # S # = 1 # " # ( t) " + ( t) is a direct measure of electron spin coherence. S i! ± ( t) ± 2! ± ( t) t 2

21 Two mechanisms for pair-flip Step 1: sort out all pseudo-spins (flip-pairs) from a given configuration s L ( t) = J " ( t) J + ( t) = #! " ( t)! + +,- ( t) Step 2: determine the pseudo-field from 1 st -principle interactions ( B A,0, D E ) h ± = ± ± ( h óˆ ) i"! ± ( t) = ± # 2! ± ( t) t D by diagonal nuclear couplings 2 1 Non-local pairs: Extrinsic 10 s! n-n interaction dominant E h ± = by ( ± diagonal,0, ) ~ N 2 A ± hyperfine interaction E pairs magnetic field dependent s! A Local by extrinsic hyperfine mediated nuclear coupling pairs: Intrinsic n-n interaction! 1 dominant infinite-range, " B!1, 1! 10 s (for B " 40! 1 T) h ± = ( B,0, ± E ) B ~ N pairs no field dependence by intrinsic nuclear coupling 2 1 finite-range, 10 s! for near neighbors

22 Two mechanisms for pair-flip Extrinsic n-n interaction ( A,0, E ) h ± = ± ± Intrinsic n-n interaction ( B,0, E ) h ± = ±! "! + y! "!! + y s L +,- T 2, A! " t x ( t) = exp(! t / T ) " N# N! e A 2 2 2, A! 2 : QD size e : electron zeeman energy s L +,- 2, B! " t , B 1 4! 1 2! 1 2 x ( t) = exp(! t / T ) T " N A b A : hyperfine constant b: intrinsic n-n coupling strength

23 Single spin FID: intrinsic vs extrinsic Nuclear bath begins on a pure state of random configuration GaAs,, d=6 nm, r 0 =25 nm, B = 12T r 0 [001] B ext [110] all extrinsic intrinsic 2 exp[!( t / T2, A) ] d Extrinsic interaction dominates small dot Intrinsic interaction dominates large dot WY, Liu and Sham, PRB 74, , 2006

24 Single System FID: field dependence Nuclear bath begins on a pure state of random configuration Strong field dependence shows extrinsic n-n interaction important GaAs d=3 nm r 0 =15 nm

25 Ensemble dynamics Single system (t=0):! e (0) " J Ensemble (t=0): Ensemble correlation function: e!(0) =! (0) "# PJ J J E J nuclear Overhauser field =! a j J n n n by the spectrum of flip-pairs of J! ( t) = L ( t)! (0) e en e +," +," +," en! i J L +! ( t) = " P e E! +, J J ( t) J ( t) Number of flip-pairs M ~ N (or N 2 ) for most Identical excitation spectrum up to an error of Identical temporal behavior for J! + J ( t) J ( t) t J 1 M L +! = " E i J ( t) J ( t) J ( t) P e! en! +, 0 0 System-bath entanglement J J t Inhomogeneous broadening * 2! i J t!( / 2 ) * P e = e t T, T ~ 10 ns " E J J 2

26 Single System FID and Ensemble Echo Nuclear bath begins on a pure state of random configuration GaAs d=3 nm r 0 =15 nm Spin echo (T H ) does NOT measure single spin FID (T 2 ). WY, Liu and Sham, PRB 74, , 2006

27 Ensemble Spin Echo Bext = 10 T, d = 3 nm, r0 = 7 nm 10 nm 15 nm 30 nm 50 nm Decoherence due to hyperfine mediated nuclear interactions is removed in spin echo! (also in simulation with ~10 bath spin, Shenvi, desousa and Whaley, PRB 05 )

28 Pulses on electron and effects on nuclear spin! pulse! pulse τ 2τ e spin: + " pseudo spin:! ± (2" ) = exp( $ ih m " ) exp ( $ ih ± " ) # Extrinsic n-n interactions (Nonlocal pairs) Spin echo measures the single spin decoherence induced h ± = ± A,0, ± E h ± = ± ( ) π pulse at τ 2τ Intrinsic n-n interactions (Local pairs) ( B,0, E ) by intrinsic nuclear interactions only! T 2 H = T 2, B

29 Control of Bath Evolution for Disentanglement! pulse τ x Flip of electron redirects evolution of bath Pseudo-spin paths re-intersects Universally at 2! for all pseudospins y System disentangled from bath (! + + " $ )% J ( 2 # ) WY, Liu and Sham, PRL 98, , 2007 Liu, WY and Sham, cond-mat/

30 Coherence Echo in Single System Dynamics d = 6 nm, r = 40 nm, B = 10 T, Coherence Nuclear bath begins on a pure state of random configuration t ( µ S ) coherence n( n + 1)! intrinsic n-n interaction dominant Different from dynamical decoupling, sys-bath coupling does not vanish " ih ( 2 1) eff 2 ih! + "! " " ih! e # e e $ H % eff 0

31 Disentanglement Concealed in Ensemble Dynamics d = 6 nm, r = 40 nm, B = 10 T, 0 intrinsic n-n interaction dominant L +! = "# E J ( t) P e en! +! i, J 0 ( t ) J 0 ( t ) J J t coherence t = 2!! t = " t = 2! T 2 * T 2

32 Disentanglement Concealed in Ensemble Dynamics To observe quantum disentanglement in ensemble dynamics a. Filter inhomogeneous broadening by measurement projection Schemes to be realized in near future: Klauser, Coish & Loss, PRB 2006 Giede, Taylor, D Alessandro, Luin & Imamoglu, PRA 2006 Stepaneno, Burard, Giede & Imamoglu, PRL 2006 or b. Arrange disentanglement time to coincide ensemble echo time WY, Liu and Sham, PRL 98, , 2007 Liu, WY and Sham, cond-mat/

33 Two-pulse control: Disentanglement at echo time! " y! 3" Absences of spin echo x! Irreversible loss of coherence 1 st 2! 2 nd 4! t = 4!! pulse t = 2!! pulse 2 nd echo time

34 Concatenated Dynamical Disentanglement Concept of concatenation on dynamical decoupling, Khodjasteh and Lidar, PRL 05 U U U U U ± 0 # e " ih ±! ± ± 1! U m 0U0 ± ± 2! U m 1 U1 ± ± 3! U m 2U 2 ± ± 4! U m 3U3! =! Disentanglement occur at 2 l l ( ) 2 b! l Decoherence reduced by with each level of concatenation Controlling small parameter: bath spin interaction 2 1 b 10 s!

35 Concatenated Dynamical Disentanglement n 2 4 = 2l WY, Liu and Sham, PRL 98, , 2007 Liu, WY and Sham, cond-mat/ Stabilization with repeated # of Concatenated units

36 Summary 1+N isolated from rest universe in relevant timescale Quantum theory of decoherence: bath evolution conditioned on system state Decoherence by sys-bath entanglement Operations on system affects bath evolution Spin echo removes more than inhomogenesous dephasing Echo decay time T H NOT equivalent to single spin T 2 Slow bath dynamics: pair-correlations dominates Sys-bath disentanglement by pulse sequences on electron. Concatenation design more efficient then periodic sequence.

37 Canonical transformation: Effective Hamiltonian: Rotation of wavefunction: Visibility loss: for B > 1T Effective dynamics:

38 Description of 2 level quantum system Single quantum system! = " + + # $ Ensemble of identical quantum systems 2 * + Pi! i + P N i! i" i ' i i ( = + Pi i i = * 2 i= 1 ' ( ' + Pi! i " i + Pi " i ( i i # $ $ + 2 & ) = $ * %) ( * ' * * 2 % & ) * Reduced density matrix for open quantum system %! = " $ # tot i i sys i en )( ( #! "! = Tr [" " ] sys en tot tot

39 Decoherence: : relaxation and pure dephasing Geometric representation! = 1 ( ), 2 I + a r #" r Bloch vector : 1 2 a r = Tr "! S ur # $ % + + ax = 2 Re! a = 2 Im! a 21 y 21 z =! "! Longitudinal relaxation!! T rl = 2T 2 1 Decay of coherence as a result of decay of population Pure dephasing deph T 2 No decay of population. Usually much faster than T 1

40 Inhomogeneous broadening & spin echo + L + + L +! 1!! i N 1 echo (t = 2! ) = N " i= 1 P! i i * T 2! (t flip =! )