l.j.sham-restoring lost coherence of a two-level system in a mesoscopic spin bath

8/11/2019 L.J.sham-Restoring Lost Coherence of a Two-level System in a Mesoscopic Spin Bath

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04/11/06LJ Sham 1

Restoring lost coherenceRestoring lost coherenceof a two-level system in aof a two-level system in a mesoscopicmesoscopic spin bathspin bath

Lu J. ShamUniversity of California, San Diego

Research funded by NSA/ARO and NSF DMR

Wang Yao, graduate student, graduating this year Ren-Bao Liu, postdoc, now on faculty at

Chinese University of Hong Kong

Collaborators

W. Yao, Ren-Bao Liu, and L. J. Sham, cond-mat/0508441

KITP Spintronnics Program Seminar April 11, 2006


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04/11/06LJ Sham 2

Issues and ProblemIssues and Problem

1. Interaction between a micro object with a macrosystem: decoherence, control, measurement

2. Meso takes the place of macro:

a. Macro: N !" b. Meso: N large enough to assume the functions of

macro but small enough for its functions to be N dep

3. Formulation: a two-level system in contact with a

mesoscopic interacting spin bath.a. Coherence loss in a two-level state

b. Restoration of its coherence


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04/11/06LJ Sham 3

One electron in a III-V quantum dotOne electron in a III-V quantum dot

Focus on the decoherence problem (T 2 T 2*? How to measure T 2? Design of measures to restore electron spin coherence

Spin relaxation time (or longitudinal) measured T 1~ 50 s -- 20 ms

Spin decoherence time (or transverse) T 2 > T 2*~ 10 ns

GaAs gated dot(Kouwenhoven group,Marcus group), SAQD(Abstreiter group).

GaAs fluctuation dotensemble or time average of a single dot (Steel et al.),InAs SAQD (Braun et al.),and gated dot(Lukin/Marcus group).


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04/11/06LJ Sham 4

SpinSpin decoherencedecoherence of a localized electronof a localized electron

Theories antecedent Spectral diffusion:Stochastic model of frequency fluctuation

Born-Markov approxfor low energyexcitation mediation Electron-nuclear

hyperfine interaction Phonon mediatedlongitudinal spinfluctuations

Quantum treatment of electron-nuclei

Merkulov, Efros & Rosen, PRB 02 -

inhomogeneous broadening Khaetskii, Loss & Glazman, PRL 02

inhomogeneous broadeni

Semenov & Kim PRL 04, exp growth of T 2 with lowering of temperatureGolovach, Khaestskii and Loss PRL 04,T 2 ~ 2 T

Anderson et al. RMP 53, PR 59-phonon ex, PR 62 - ind. of interact.

De Sousa & Das Sarma, PRB 03 - n-ndipolar interaction, T 2 obtained fromtime dependence of spin echo

De Sousa, Shenvi & Whaley, PRB 72,045330 - GaAs n (I>1/2)

Witzel, De Sousa & Das Sarma, PRB (2005)

Shenvi, De Sousa & Whaley, PRB (2005),hyperfine, exact diagonalization of a small cluster


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04/11/06LJ Sham 5

DecoherenceDecoherence process in an open systemprocess in an open system

$ + % + &$' % Initial state

decoherence:

$+ % + &$' %)$ J % Reduced density matrix

spin bath $ + % $J + % + &$' %$J ' %Entanglement

Quantum system

t

Coherence ( +' decays with time

Hamiltonian

from ( ++ to ( '' negligible when T 2


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04/11/06LJ Sham 6

SpinSpin decoherencedecoherence of an electron in a III-V quantum dotof an electron in a III-V quantum dot

Mechanisms of decoherence (transversedephasing)

Electron-phonon: negligible below 1 K Decay by spontaneous emission: negligible Ultimate decoherence source, in general

terms of two-level system with a spin bath TLS - bath spin interaction bath spin-spin interaction


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Be '

Mesoscopic:N~ 10 5 - 10 6

second order perturbation

Single electron spin plus NSingle electron spin plus N interacting nuclear spinsinteracting nuclear spins

Nuclear-nuclear spin interaction via single electron spin A~a 2/(N 2 ) e)

Longitudinal coupling ~(a/N)S z J zn

Electron-nuclear hyperfine interaction: a

Neglect of electron spin-flip processes

Transverse coupling ~(a/N) (S + J ' n+ S ' J +n)contributes to n egligible electron spin flipsbecause of energy mismatch:) e>>T>> ) N

where H operate on the nuclear spins

cf Leggett

Nuclear-nuclear dipolar interaction B~b ) e~a>> ) N>>b~A gen


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Nuclear pair flip excitationsNuclear pair flip excitations

|m> 1 |m'> 2 |m-1> 1 |m'+1> 2 mapped to a pseudo-spin 1/2Reduced Hamiltonian on nuclear pairexcitation k for electron spin state $%

H-f mediatedIntrinsic n-n

Hyperfine energyDiag n-n energy

Pseudo-fields

Entanglement overlap

Independent nuclear spin pair correlation approximation


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04/11/06LJ Sham 9

Bath spin pair excitation statesBath spin pair excitation states


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Justification of independent pair correlation approximationJustification of independent pair correlation approximation

Short time behavior in agreement with perturbationtheory of nuclear pair flip interaction.

Error estimate of higher order correlations negligible if

Non-overlapping123

1234

Overlapping

at time t ~ T 2


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04/11/06LJ Sham 11

Solution for the electron spin coherenceSolution for the electron spin coherence

Spin coherence at time t

Inhomogeneous broadening

NB: the superoperator replaced by a multiplicative factor in t

Entanglement overlap of collective nuclear stateswhen el spin is - or +

Initial state ( e*( + ( (t)

( e+' = ,+$Tr N [( (t)] $'%

Origin of stat distribution:ensemble or repeated meas.


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04/11/06LJ Sham 12

Free induction decay for aFree induction decay for a single bath statesingle bath state

.

FID decoherence time - Spin echo decay time

GaAsGaAs dot (h=3 nm x r=15 nm)dot (h=3 nm x r=15 nm)


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04/11/06LJ Sham 13

Single particle levels Multi-particle states

. +

Adiabatic NR Raman Spin-flip Process

. +

Trion

Magnetic

field in x

Spin State to Trion

Pochung Chen, C. Piermarocchi, L.J. Sham, D. Gammon, and D.G. Steel, PRB 69 , 075320 (2004).

Electron spinElectron spin qubitqubit and singleand single qubitqubit gategate

Theo est. op time ~< 10 ps

Decoherence sources: optical & spin


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Physical picture of causes ofPhysical picture of causes of decoherencedecoherence

Intrinsic n-n interaction

Spin coherencedecreases as e-n

entanglement increases

exp[- (t/T 2,B)4]exp[- (t/T 2,A)2]

Hyperfine-mediated interaction

n-n interaction

hyperfine

FID

Spin echo: removes HMI but only reduces n-n i

Dot: d=6.2 nm, r=25 nm, B=12 T

|J %


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2 pulses on electron and effects on nuclear spin dynamics2 pulses on electron and effects on nuclear spin dynamics

Hyperfine-mediated interaction Intrinsic n-n interaction/ pulse at 0

CP: / pulse at 3 0

dis-entanglementat ( 12)0

echo at 2 0

echo at 4 0No N state prep!


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n =

4

6

8

1012

2

,

n nc l t T e !

/ -pulse sequences for concatenation level l =

( )0 1exp , l l l U iH U U U ! + + + " ++# " =

[001] 05.7 nm, 20 nm, 10 Text L r B= = =

Concatenated pulses for control of Concatenated pulses for control of entanglemententanglementConcatenated dynamical decoupling Khojasteh & Lidar, PRL 95, 180501 (2005)

Adopt concatenation idea but not to decouple the e-n interaction

t / ! = 0 1

0

1

2

3

3 4 5 6

Rot


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SpinSpin qubit decoherencequbit decoherence by nuclear spinsby nuclear spins

Quantum solution by method of the bath spincorrelation decoherence by entanglement

Meso bath + bath spin interaction a swap of evolution paths of bath when the qubit spin is flipped.

Spin echo, CPMG and concatenated pulses, alt. todynamical decoupling and complementary to decoherence free space

quantum error correction codes Mesoscopic spin bath model may be applied to other

systems

l.j.sham-restoring lost coherence of a two-level system in a mesoscopic spin bath

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