Coherent and incoherent evolution of qubits in semiconductor systems Iain Chapman, Thornton...

Coherent and incoherent evolution of qubits in semiconductor systems

Iain Chapman, Thornton Greenland, Sev Savory and Andrew Fisher

Departments of Physics and Astronomy and Electrical Engineering

and

London Centre for Nanotechnology

UCL

Overview

• Some proposed and actual semiconductor qubits, and their principal sources of decoherence:– Spin states– Orbital states– Charge states

• “The standard model” of decoherence – a reminder• Some consequences for fluctuations, dissipation and

decoherence• Charge qubits in double quantum dots, surface acoustic

waves and double defects• Spin qubits at defects and the use of control spins

Electron spin qubits – quantum dots

Double quantum dot structures, e.g. Johnson et al. Nature 435 925 (2005)

Relaxation of spins in (1,1) charge state away from singlet state inhibits transfer to (0,2) (spin blockade)

Dominant decoherence mechanism is via nuclear spin bath, relaxation strongly suppressed by Bext. Spin-orbit dephasing will be important as an ultimate limit.

Spin echo and Rabi flopping on a single logical qubit

Petta et al. Science 309 2180 (2005)

T2*=9ns

Orbital qubits – excitons in quantum dots

1Y 1

Bonadeo et al. Science 282 1473 (1998)

Beats between x and y polarizations of excitons in a dot:

Large (band-gap) energy scales mean very rapid qubit evolution

0X

Monitor state of system by delayed probe pulse

0

or

Exciton decoherenceRapid initial decoherence (strongly pulse-area dependent) followed by long decay:

Time-domain

versus

Energy-domain

Borri et al. PRL 87157401 (2001) and PRB 66 081306(R) (2002)

Phonon dephasing

Long-time tail arises because phonon-induced processes cannot conserve energy in the long-time limit; rapid short-time decoherence comes from transient processes allowed by uncertainty principle (non-linear in optical field)

Dominant mechanism appears to be coupling to acoustic phonons

Förstner et al Phys. Stat. Sol. B 238 419 (2003); Phys. Rev. Lett. 91 127401 (2003)

Reproduction of weak-field lineshape:

Pulse-area dependent dephasing or Rabi oscillations:

Coherent oscillations of electron charge states in double quantum dots

Double quantum-dot structure in which dots can be gated separately

Gate-defined dot in GaAs with leads:

Hayashi et al. PRL 91 226804 (2003); period

~0.3ns, T2 ~ 2ns

Prepare electron in left dot, follow free evolution and detect probability it emerges on the right (“free-induction decay”)

SOI structure: Gorman et al. PRL 95 090502 (2005); period ~0.3 μs, T2~2 μs

Charge qubits in quantum dots (contd)

• Possible sources of decoherence:

– Co-tunnelling (in experiments with leads)

– Circuit noise

– Phonons

Petta et al.Phys. Rev. Lett. 93 186802 (2004)

Micro-wave-irradiation produces absorption peaks as the electron is excited into the upper state of the two-level system.

Relaxation of occupation gives T1=16ns; linewidths give T2* at least 0.4ns

Can be eliminated, at least in principle

Intrinsic

Overview





The Lindblad master equation

Lindblad (1966): most general form for Liouville equation in an open system that is Markovian (i.e. evolution depends only on current state) is

Hamiltonian (may be modified by environment)

Lindblad operators, cause incoherent

“jumps” in the state of the system

The Born-Markov limit

Evolution of full density matrix (interaction representation):

Find formal solution and trace over environment to get system evolution

In Markovian limit (i.e. can replace ρ(s) by ρ(t) and extend limit on integral to -∞), and also assuming

•First term is zero (achieve by redefining H0 if necessary)

•Factorization of density matrix into “system” and “environment” parts at all times

we get

Correlation functions

Explicit form for interaction Hamiltonian:

System operator Environment operator

Correlation functions of environment operators:

Write equation of motion in terms of these:

Not necessarily evaluated in thermal equilibrium

Good qubits: the Rotating Wave Approximation

For “good” qubits (systems which can evolve through many cycles before damped by environment), decompose system operators in terms of transition frequencies of system:

Neglecting rapidly-oscillating terms in ei(ω-ω’)t , equation of motion becomes

where we define the Fourier transforms

Hamiltonian (coherent) evolution (Lamb shift)

Quantum jumps

Causality (c.f. Kramers-Kronig):

Note non-zero S requires spectral structure in J

Overview





Coupling through the environment

A B

E

Environment in a typical solid-state implementation plays two roles:

(1) Generates interactions between qubits (essential for multi-qubit gates)

(2) Introduces decoherence (bad)

No direct coupling between spatially separate qubits, since physics is ultimately

local

Figure of merit: ~ # operations before decoherence sets in (c.f. Q-factor)

Good news, bad news

Good news: the two spins are coupled by an effective Hamiltonian

eff 00

( ')~ ', operating frequency of transition

'

JH d

Can generate time-evolution that entangles the spins, leads to a non-trivial 2-qubit gate

Bad news: there is inevitably a corresponding contribution to the decoherence (provided the wek-coupling limit is appropriate), scaling like

†0

ˆ ˆ ~ ( )L L J Destroys coherence of quantum evolution

Continuum environment

eff

2

( , ) ( , ) 0

ˆ ˆ ˆ

( )ˆ ˆ ˆ ˆ[ ]

2 ( )A B

H H H

Jde

Find standard Lindblad master equation for an open system (even though physics is partly non-Markovian):

†eff

1ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ[ , ] { , }2t S S S Si H L L L L

Effective Hamiltonian: Lindblad operators

Both determined by same spectral functions

†exp( ) ˆ ˆ( ) ( )n

n

C t n B t B nZ

( ) ( ) exp( )J C t i t dt

† 2, , 0

,

ˆ ˆ ˆ ˆ( )L L J

0† 2

, , 0,

ˆ ˆ ˆ ˆ( )L L e J

Coupling and decoherence

• Produces a link between coupling of qubits and decoherence introduced

• Inter-qubit coupling and decoherence linked by Hilbert transform

• Forms a generalisation of the fluctuation-dissipation theorem

†, ,( ') ( ')

'ˆ ( )2 '

L Ld

H

Coupling real part of susceptibility

Fluctuations (and hence decoherence) imaginary part of susceptibility

Enables us to bound the figure of merit (Q-factor) of system by knowing only the

spectrum of the correlations

†, ,( ') ( ')

'ˆ ( )2 '

L Ld

H

Discrete environment

' 0

0 0

' 0

0 '

( , ) ' ''exp[ ( ' ' '') ( '' ')];

( , ) ' ''exp[ ( ' ' '') ( '' ')];

t t

ss

t t

ss

t

t dt dt i st s t i t t

t dt dt i st s t i t t

2, *

eff , ' ', '( , ) ( , ) ( , ') ( , )

ˆ ˆ ˆ[ ( , ) ( , )]2 2

ns s n s s n s s

n A B s s

JitH t t

Effective Hamiltonian:

Effective Lindblad operators

† 2, , ' '

, '

ˆ ˆ ˆ ˆ( , )n s s n s sn ss

t L L J t

Suppose environment has discrete spectral response:

,( ) ( )n nn

J J

Both expressed in terms of :

Allows extra “engineering” possibilities:

(a) Choose operating frequencies far from environment frequencies Ωn

(b) Choose operating times to coincide with zeros of ψss’(t,Ωn)

c.f. Ion-trap “warm” entanglement (Mølmer,

Sørensen etc.)

Irreducible decoherence

• This decoherence is irreducible because– There may be other mechanisms, contributing additional

decoherence, which do not also contribute to the entanglement

– A decoherence-free subspace affords no protection:

effˆ ˆ0 0IH H

No decoherence No entanglement

Overview





Morivation

• Would like to – Understand processes operating in recent experiments– Assess the suitability of charge qubits over a wide

range of length and time scales

Charge qubits - minimal model

Form factor qˆ exp( )q q zL L iq a

L R

x=-a x=+a

σ σ

Two-state electronic system coupled to phonons:

†1,2

, ,

ˆˆ ˆ ˆ. .

ˆ ˆ ˆ

L R q q q q q qq q

e ph e ph

H L L R R t L R h c n M a a

H H H

Single electron in double dot coupled to phonons

ˆ exp( )q q zR R iq a

Coupling depends on interaction

Types of coupling

1/ 2

,acoustic, ,2q

q a s a

M q Dv

Two types of coupling to acoustic phonons important for low-frequency quantum-information processing:

1/ 2

i2q q

q s

M Cv

2

141 2 32 2

0 0

2qr

ee qC

q q

1 2 3, , polarization vector

ˆ( , , ) (i.e. direction cosines of )

q q

(1) Deformation potential:

(2) Piezoelectric coupling:

Direction-dependent coupling:

0 Debye optic( , )

7.5 8 8.5 9LogWavenumberm10.2

0.4

0.6

0.8

1

Ratio

Relative importance of couplings

10 0.5nmq

10 5nmq

10 20nmq

10q

Relative importance of piezoelectric term in GaAs very sensitive to assumptions about screening length q-1

0

Piezoelectric coupling dominates on distance scales above about 10nm provided do not exceed screening length

Two treatments of decohering processes

tot tot

ˆ ˆ ˆˆ ˆ ˆˆ ˆ( )t P K t P

(2)1 1

0

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )t

K t dt PL t L t P

0ˆ ˆL J

Two approaches:

†eff

1ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ[ , ] { , }2t AB AB AB ABi H L L L L

Rotating Wave Approximation: perturbative and Markovian (disregards memory of environment)

TCL (time-convolutionless) projection operator technique – perturbative but explicitly non-Markovian

“TCL kernel”. To 2nd order in coupling:

0

0

0

0

sin 2ˆ1 1 ;

2

sin 2ˆ1

2

q

q

q q

q q

qaL M n

qa

qaL M n

qa

Projector onto decoupled dots/phonons

Results similar in this case.

Is the RWA OK?

1 108

2 108

3 108

4 108

5 108

Dot size m 1 107

2 107

3 107

Separation am12108

6

4

Total LogT1s1 108

2 108

3 108

4 108

5 108

Dot size m

1 108

2 108

3 108

4 108

5 108

Dot size m 1 107

2 107

3 107

Separation am1210864

Deformation LogT1s1 108

2 108

3 108

4 108

5 108

Dot size m1 108

2 108

3 108

4 108

5 108

Dot size m 1 107

2 107

3 107

Separation am10

5

0Piezoelectric LogT1s

1 108

2 108

3 108

4 108

5 108

Dot size m

Results (RWA, GaAs parameters)

Deformation-potential

Piezo-electric (unscreened limit)

T=20 mK

Assumes electron transfer decays with distance like exp(- a/σ)

9.5 10 10.5 11 11.5 12LogOperating frequency s18

7

6

5

LogTotal T1sResults as a function of operating frequency

For the geometry of Hayashi et al:

Typical frequency range of experiments

Dot size = 30nm

Dot separation = 300nm

T=20mK

1 108

2 108

3 108

4 108

5 108

Dot size m 1 107

2 107

3 107

Separation am6

4Total LogT1s

1 108

2 108

3 108

4 108

5 108

Dot size m

Aside – charge decoherence in the GHz range and the behaviour of trapped charges

T=20 mK

Surface Acoustic Waves

• What is the relative magnitude of the surface and bulk contributions to relaxation and decoherence in a double dot?

a

b

Motivation: surface component readily easily controlled by surface cavities and transducers. Will this help?

Importance of anisotropy in piezoelectric terms when treating surface couplings

Anisotropy in piezoelectric matrix elements now becomes important:

SAW contribution to relaxation

Incomplete “burial” of dot charge density (unphysical)

Dot size = 30nm

Separation = 300nm

Splitting = 0.01 meV

Compare approximately 5ns lifetime from bulk terms, so SAW never dominates

Overview





Is there another way?Would like to

•Use (electron) spin qubits in order to avoid rapid phonon-induced decoherence as much as possible;

•Control coupling of qubits without presence of nearby electrodes and associated electromagnetic fluctuations;

•Avoid small excitation energies susceptible to decoherence at the lowest temperatures.

Our proposal (Stoneham et al., J Phys Conden Matt 15 L447 (2003)): use real optical transitions in a localized state to drive an atomic-scale gate:

Ground state

(no interaction)

Excited state

(interaction present)

Exploit properties of point defect systems conveniently occurring in Si, but concept also generalises to many other

systems

The basic idea• Qubits are S=1/2 electron spins which must be controlled by

one- and two-qubit gates

• The spins are associated with dopants (desirable impurities)– Chosen so they do not ionise thermally at the working

temperatures (“deep donors”)

• The dopants are spaced 7-10nm to have negligible interactions in the “off” state

Silicon

Dopants

Basic Ideas (Continued)…

• Uniquely, the distribution of dopant atoms is disordered– A disordered distribution is desirable for system reasons– Dopants do not have to be placed at precise sites

Silicon

Dopants

• The new concept is to control the spins producing the A-gates and J-gates using laser pulses

• Another new concept is separation of the storing of Quantum information from the control of Quantum interactions

Silicon

Source of Control Electron

Donors carrying Qubit Spins

Control gate by laser-induced electron transfer

ALL GATES OFF

Controlling Spins

Gate addressed by combination of position and

energy

ONE GATE ON

ALL GATES OFF

Silicon

Source of Control Electron

Donors carrying Qubit Spins

Control gate by laser-induced electron transfer

ALL GATES OFF

Controlling Spins

Gate addressed by combination of position and

energy

Many different charge transfer events possible

Different laser wavelengths

allow discrimination

ONE GATE ON

0 2 4 6 8 10 12 14 16 18 200

1

2

JT

0 2 4 6 8 10 12 14 16 18 200

1

2

JT

Measu

re o

f Enta

ngle

ment

(Eucl

idean n

orm

)

The dynamics of optically-controlled gates

Qubit 2

in

Qubit 0

Qubit 1

00

0

0

H

H

H

2

zR

SFGM=815N=904

SFGM=815N=904

4

P

4

P

2

zR

2

zR

2

zR

SFGM=1595N=2137

4

P

4

P

2

zR

2

zR

SFGM=1584N=2177

4

P

4

P

H

H

H

out zR

Identify which data manipulations can be efficiently produced using optical excitation without interfering with (decohering) the qubits, as a function of the controlling parameters:

Zero B-field

Finite field

Then analyse the chain of gates required to produce a demonstration quantum algorithm (3-qubit Deutsch-Jozsa), and estimate the overall accuracy (fidelity):

2Overall fidelity 0.999834out ideal out SFG

Laser onSafe to turn laser off without damaging quantum information

Conclusions

• Provided it may be safely applied, the weak-coupling approach to decoherence gives an attractively universal picture of incoherent processes and limits attainable figures of merit in many cases

• For charge qubits in semiconductors the model (with no free parameters) suggests recent experiments are close to the attainable limits

• Partly motivated by these considerations, we are investigating the optically-driven dynamics of defect spins in semiconductors

Acknowledgments

• Thanks to– EPSRC, IRC in Nanotechnology, UK Research

Councils Basic Technology Programme for support– Marshall Stoneham, Gabriel Aeppli, Wei Wu, Che

Gannarelli

Coherent and incoherent evolution of qubits in semiconductor systems Iain Chapman, Thornton...

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