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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
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
Petta et al.
Science 309
2180 (2005)
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)
Spin echo and Rabi flopping on
a single logical qubit
T2*=9ns
Dominant decoherence mechanism is via nuclear spin
bath, relaxation strongly suppressed by Bext. Spin-orbit
dephasing will be important as an ultimate limit.
Orbital qubits – excitons in quantum dots
Beats between x and y polarizations of X  0
excitons in a dot:
1
Y 1
or
0
Large (band-gap) energy
scales mean very rapid qubit
evolution
Monitor state of system by delayed
probe pulse
Bonadeo et al. Science 282 1473 (1998)
Exciton decoherence
Rapid initial decoherence (strongly
pulse-area dependent) followed by long
decay:
Borri et al. PRL 87157401 (2001) and
PRB 66 081306(R) (2002)
Time-domain
versus
Energy-domain
Phonon dephasing
Förstner et al Phys. Stat. Sol. B
238 419 (2003); Phys. Rev. Lett.
91 127401 (2003)
Dominant mechanism appears to be coupling to acoustic phonons
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)
Pulse-area dependent dephasing
Reproduction of weak-field
or Rabi oscillations:
lineshape:
Coherent oscillations of electron charge states in
double quantum dots
Prepare electron in
left dot, follow free
evolution and detect
probability it emerges
on the right (“freeinduction decay”)
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
SOI structure: Gorman et
al. PRL 95 090502 (2005);
period ~0.3 μs, T2~2 μs
Charge qubits in quantum dots (contd)
Micro-wave-irradiation produces
Relaxation of occupation gives
absorption peaks as the electron is
T1=16ns; linewidths give T2* at least
excited into the upper state of the two- 0.4ns
level system.
• Possible sources of
decoherence:
Petta et al.Phys. Rev. Lett. 93 186802
– Co-tunnelling (in
Can be
(2004)
experiments with leads) eliminated, at
least in principle
– Circuit noise
Intrinsic
– Phonons
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
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
Hamiltonian (coherent) evolution (Lamb shift)
motion becomes
Quantum
jumps
where we define the Fourier transforms
Note non-zero S
requires spectral
structure in J
Causality (c.f. Kramers-Kronig):
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
Coupling through the environment
Environment in a typical solid-state implementation plays two roles:
(1) Generates interactions between
qubits (essential for multi-qubit gates)
(2) Introduces decoherence (bad)
Figure of merit: ~ # operations before
decoherence sets in (c.f. Q-factor)
No direct coupling between
spatially separate qubits,
since physics is ultimately
local


A
B


E
Good news, bad news
Good news: the two spins are coupled by an effective Hamiltonian
H eff ~ 
J ( ')
d ', 0  operating frequency of transition
0   '
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
Lˆ† Lˆ ~ J (0 )
Destroys coherence of quantum evolution
Continuum environment
Find standard Lindblad master equation for an open system (even
though physics is partly non-Markovian):
1


it ˆ S  [ Hˆ eff , ˆ S ]    Lˆ ˆ S Lˆ  {Lˆ† Lˆ , ˆ S }
2

 
Effective Hamiltonian:
Hˆ eff  Hˆ   Hˆ 
J ( )
d  
   
ˆ
ˆ
ˆ
ˆ

[



e


]
 
 
 
(  0 )
( ,  )  ( A, B )  2

2
Lindblad operators
†
2
 
ˆ
ˆ
ˆ
L
L


J
(

)

  ,  ,   0  ˆ 

 ,
 0 2
†
 
ˆ
ˆ
ˆ
L
L

e

J
(

)

  ,  ,
  0  ˆ 

Both determined by same
spectral functions
 ,
C (t )  
n
exp(  n )
n Bˆ † (t ) Bˆ  n
Z
J ( ) 

 C (t ) exp(it )dt

Coupling and decoherence
Coupling  real part of susceptibility
• 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
Enables us to bound the figure of merit (Qfactor) of system by knowing only the
spectrum of the correlations
Fluctuations (and hence decoherence) 
imaginary part of susceptibility


†
  L , ( ') L , ( ') 

d '  

Hˆ  ( )  
2
 ' 



†
L
(

')
L
(

')
  ,

 ,

d

'


Hˆ  ( )  
2
 ' 

Discrete environment
Suppose environment has discrete spectral
response:
J ( )   J ,n (  n )
Effective Hamiltonian:
itHˆ eff  
2
2
  

n ( , )  ( A, B ) ( s , s ') (  ,  )
J ,n
2
n
[s , s ' (t , n )  *s ', s (t , n )]ˆ s' ˆ s
Effective Lindblad operators
Both expressed in terms of :
t
t
0
0
 ss ' (t , )   dt '  dt ''exp[i0 ( st ' s ' t '')  i(t '' t ')];
t

n  ,  ss '
t
ss ' (t , )   dt '  dt ''exp[i0 ( st ' s ' t '')  i(t '' t ')];
0
t  Lˆ† Lˆ   2  J ,n s ,s ' (t , n )ˆ s'ˆ s
t'
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:
Hˆ I   0  Hˆ eff   0
No decoherence
No entanglement
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
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
Single electron in
double dot coupled
to phonons
L
σ
σ
x=-a
Form factor  q
R
x=+a
L ˆ q L  q exp(iqz a)
R ˆ q R  q exp(iqz a)
Two-state electronic system coupled to phonons:
H   L L L   R R R  t  L R  h.c.    q  nˆ q  12    M  q ˆ q  aˆ q  aˆ† , q 
q ,
 Hˆ e  Hˆ ph  Hˆ e  ph
q ,
Coupling depends on
interaction
Types of coupling
Two types of coupling to acoustic phonons important for lowDebye , optic )
frequency quantum-information processing: (0
1/ 2
(1) Deformation potential:

Mq  
 2  v
q ,a s ,a





(2) Piezoelectric coupling:

 i
 2  v
 q s





Direction-dependent
coupling:
M q
q D  ,acoustic
1/ 2
C q
q2
C q  2
    2  3 
 r  0 q 2  q02 1
ee14
 1 , 2 , 3   polarization vector
( ,  ,  )  qˆ (i.e. direction cosines of q)
Relative importance of couplings
Piezoelectric coupling dominates on
distance scales above about 10nm
provided do not exceed screening
length
Ratio
1
0.8
0.6
0.4
0.2
7.5
8
8.5
9
Log Wavenumber
Relative importance of piezoelectric term in
GaAs very sensitive to assumptions about
screening length q-10
m
1
q0 1  0.5nm
q0 1  5nm
q01  20nm
q01  
Two treatments of decohering processes
Two approaches:
Rotating Wave
Approximation: perturbative
and Markovian (disregards
memory of environment)
1 ˆ† ˆ

ˆ , ˆ ]    L
ˆ
ˆ
i  t ˆ AB  [ H
eff
AB
  ˆ AB L  {L L , ˆ AB } 
2

 
ˆ 
L

L  M qq
TCL (time-convolutionless)
projection operator technique –
perturbative but explicitly nonMarkovian
Projector onto decoupled
dots/phonons
L  M qq
q 0
q 0
J  0 ˆ 

sin 2qa 
1 

2qa 


sin 2qa 
1 

2qa 

n 0 ˆ 
ˆˆ ˆ
ˆ
ˆ
t P
 tot  Kˆ (t ) Pˆ ˆ tot
“TCL kernel”. To 2nd order in
coupling:
t
Results similar in this case.
n 0   1ˆ  ;
ˆˆ ˆˆ ˆˆ
ˆ
K (2) (t )   dt1 PL
(t ) L(t1 ) Pˆ
0
Is the RWA OK?
Results (RWA, GaAs parameters)
Piezo electric Log T1 s
0
3 10
7
5
Deformation Log T1 s
4
6
8
10
12
3 10
2 10
10
7
1 10
2 10
1 10
Dot size
1 10
8
3 10
m
Separation a m
8
2 10
Dot size
Separation a m
8
2 10
7
7
1 10
8
3 10
m
8
4 10
5 10
8
8
5 10
8
Deformationpotential
Total Log T1 s
Assumes electron transfer
decays with distance like
exp(- a/σ)
4
6
8
10
12
3 10
2 10
1 10
7
7
Separation a m
8
2 10
Dot size
1 10
8
3 10
m
7
8
4 10
8
5 10
8
7
8
8
4 10
7
T=20 mK
Piezo-electric
(unscreened
limit)
Results as a function of operating frequency
For the geometry of Hayashi et al:
Log Total T1 s
Dot size = 30nm
5
Dot separation = 300nm
6
T=20mK
7
8
9.5
10
10.5
11
11.5
Typical frequency range
of experiments
12
Log Operating frequency
s
1
Aside – charge decoherence in the GHz range
and the behaviour of trapped charges
Total Log T1 s
4
3 10
7
6
2 10
1 10
7
Separation a m
8
2 10
Dot size
1 10
8
3 10
m
8
7
T=20 mK
4 10
8
5 10
8
Surface Acoustic Waves
• What is the relative magnitude of the surface and
bulk contributions to relaxation and decoherence
in a double dot?
b
a
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
Compare approximately 5ns
lifetime from bulk terms, so SAW
never dominates
Incomplete “burial” of dot
charge density
(unphysical)
Dot size = 30nm
Separation = 300nm
Splitting = 0.01 meV
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
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:
Excited state

(interaction present)
Ground state

(no interaction)
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”)
Dopants
Silicon
•
The dopants are spaced 7-10nm to have negligible interactions in the
“off” state
Basic Ideas (Continued)…
•
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
Dopants
Silicon
•
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
Controlling Spins
Control gate by laserinduced electron transfer
ALL
ONEGATES
GATE
OFF
ON
Gate addressed
by combination
of position and
energy
Silicon
Source of
Control Electron
Donors carrying
Qubit Spins
Controlling Spins
Control gate by laserinduced electron transfer
ALL
ONEGATES
GATE
OFF
ON
Many different
charge transfer
events possible
Gate addressed
by combination
of position and
energy
Different laser
wavelengths
allow
discrimination
Silicon
Source of
Control Electron
Donors carrying
Qubit Spins
The dynamics of optically-controlled gates
Identify which data manipulations can be efficiently produced using optical excitation without
interfering with (decohering) the qubits, as a function of the controlling parameters:
2
Zero B-field
Measure of Entanglement
(Euclidean norm)
1
0
0
2
4
6
8
10
12
14
16
18
20
JT
2
1
Finite field
0
0
2
4
6
8
10
12
14
16
18
20
JT
Then analyse the chain of gates required to produce a demonstration quantum algorithm (3qubit Deutsch-Jozsa), and estimate the overall accuracy (fidelity):
0 Qubit
0
 in
0 Qubit 1
0 Qubit
2
H R   2 
Laser on
H
z
H
 
Rz   
 2
SFG
M=815
N=904
 
 
P 
Rz  
2
 4 Safe to
turnSFG
laser
M=1595
off without
 
N=2137
R 
damaging
2
quantum
 
P information
4
z
SFG
M=815
N=904
 
P  
 4
 
P  
 4
H
 
Rz  
2
 
Rz  
2
SFG
M=1584
N=2177
 
P  
 4
Rz  
 
P  
 4
Overall fidelity   out ideal  out SFG
H
 out
H
2
 0.999834
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