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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
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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
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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
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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
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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)
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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:
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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.)
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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
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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
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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
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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
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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( , )
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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
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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.
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Is the RWA OK?
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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)
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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
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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
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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?
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Importance of anisotropy in piezoelectric terms when treating surface couplings
Anisotropy in piezoelectric matrix elements now becomes important:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Acknowledgments
• Thanks to– EPSRC, IRC in Nanotechnology, UK Research
Councils Basic Technology Programme for support– Marshall Stoneham, Gabriel Aeppli, Wei Wu, Che
Gannarelli