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Quantum Dots
By
Timothy Paik
Marcus Dahlstrom
Michael Nip
Implementing Quantum
Computers
• Many implementations for quantum
computing
• Why solid state?
– Scalability
– Decoherence is less of a problem
What is a quantum dot?
• In two words, a
semiconductor
nanocrystal.
• Easily tunable by
changing the size and
composition of the
nanocrystal
Gallium Arsenide Quantum Dots
• Gallium arsenide is a III-V semiconductor
– Higher saturated electron velocity and higher
electron mobility than silicon
– Gallium arsenide can emit and absorb light,
unlike silicon
• No silicon laser is possible (or has been made yet)
Energy Band Levels
• Electrons exist in discrete
energy levels in bulk
semiconductor material.
– There exists a forbidden
range of energy levels in
any material called the
band gap.
Energy Band Levels
• By absorbing some sort
of stimulus (in light or
heat form), an electron
can rise to the conduction
band from the valence
band.
– This action leaves behind a
“hole” in the valence band.
The hole and the electron
together are called an
exciton.
Energy Band Levels
• The average distance
between an electron and
a hole in a exciton is
called the Excited Bohr
Radius.
• When the size of the
semiconductor falls below
the Bohr Radius, the
semiconductor is called a
quantum dot.
Tuning Quantum Dots
• By changing size,
shape, and
composition,
quantum dots can
change their
absorptive and
emissive properties
dramatically
Manufacturing methods
• Electron beam lithography
• Molecular beam epitaxy
Electron Beam Lithography
• Electrons are accelerated
out of an electron gun
and sent through
condenser lens optics
directly onto a wafer
• λ = (12.3 Å / √V)
• Advantages:
– generation of micron and
submicron resist geometries
– greater depth of focus than
optical lithography
– masks are unnecessary
– Optical diffraction limit is not a
real concern
Electron Beam Lithography
• Disadvantage(s):
– The lithography is serial
(masks aren’t used; instead
the beam itself sweeps
across the wafer) =>
Comparatively low
throughput ~5 wafers per
hour at less than 1
micrometer resolution
– The proximity effect:
Electrons scatter because
they are relatively low in
mass, reducing the
resolution.
• Heavy ion lithography has
been proposed, but still is
in development stages
Molecular Beam Epitaxy
• Molecular beam epitaxy (MBE) is the deposition
of one or more pure materials onto a single
crystal wafer one layer of atoms at a time in
order to form a perfect crystal
– This is done by evaporating each of the elements to
combine, then condensing them on top of the wafer.
– The word “beam” means that the evaporated atoms
only meet each other on the wafer
Spin Quantum Computing
Qubit information is stored in the spin state of an
electron in an artificial atom
Advantages:
Long decoherence time
Future Scalabilty
Artifical atoms are bigger than regular atoms therefore
easier to manipulate
Decoherence time ~ 100ns
• Time before the quantum mechanical system starts
acting in a classical way with it's complex
environment
• The state of the system has not yet collapsed due
to (unwanted) environmental effects
• Spin - DT are 100 as long as for the Exciton
• Need to SWITCH 104 during DT for reliable error
correction. This requirement is met.
Artificial Atom
• Double Barrier
Heterostructure
• Dot: In0.05Ga0.95As
• Source &Drain : GaAs
• 2D Electron Gas
• Confine with gate bias
• D ~ Fermi wavelength
→ Discrete energy
levels
Adding Electrons, changing Vgate
• 2D-Harmonic
Oscillator
• Shell structure as in
atoms
• Magic Numbers: 2, 6,
12...
• To add “even”
electron requires only
additional Coulomb
energy
Comparison with Hydrogen
• Artificial Atom:
• Hydrogen:
Energy levels ~
1meV
Energy levels ~ 1eV
Size ~ 10μm
Only strong magnetic
fields can perturb
energy levels
Weak magnetic
fields can affect
energy levels
Size ~ 1Å
Factor 1000...
Tuning the Quantum Dot
• Tune so we have one
valence electron
• Initial state can be set
by applying
homogeneous
magnetic field → |0>
• Low temperature:
kT < ΔE (state gap)
• Now we have defined
our single qubit
Energy
Unoccupied state
Gate bias
Spin up - electron
position
Single Qubit manipulation
• Unitary operations can
be made by applying
a local magnetic field:
HZE = -μ·B = g μB S·B
• MF microscope
• AF microscope
• Sub grid of current
• Magnetic dots
• Etc...
(Magnetic force microscope tip)
Two Qubit Manipulation
• Complete set of logic
requires a CNOT
• Dots are placed so
close that they overlap
and interact:
• Hspin = J(t)S1·S2
Exchange coupling:
J(t,E,B) = Etriplet -Esinglet
(4:th order Harmonic Oscillator)
Ground State Splitting (J = Et –
Es)
• 2 coupled fermions must have an total antisymmetric wave function
• Lowest coupled state is the singlet. It has a
symmetric spatial wave function and an anti
symmetric spin (Coulomb dominates):
|ψs> ~ (|12> + |21>) (|↓↑> - |↑↓>)
• The triplet states are:
(|↓↓>)
|ψt> ~ (|12> - |21>) (|↓↑> + |↑↓>)
(|↑↑>)
• <1|2> ≠ 0, |i> is spatial w.f. Coulomb dominates
Solving J(B(t)): Exchange
Coupling
• Different solutions:
* Heitler-London
* Hund-Mulliken
* Hubbard
• Important conclusion:
We can control
coupling from zero to
non-zero by changing
the magnetic field →
We can perform two
qubit operations.
SWAP - gate
• Assume J can be pulsed:
J(t) = {0, J0}
Formula 1
Formula 2
• Now we can put many qubits on a line
and move them so that they all can
interact [not all at once though]
XOR ~ CNOT
• Formula 3
• Requirements:
* Spin rotations about the z-axis
* Squareroot of Uswap
Read out / Memory
• Assume dot with an electron with some
information stored in spin-state
• Connect two leads to dot
• Apply a small bias (ΔV) → Current (i)?!
Energy
Unoccupied state
i?
Gate bias
Spin up - electron
position
Another Spin up electron enters
dot
• Pauli principle forces electrons with spin
up to occupy the higher energy state
• Negligible chance of tunneling
i=0
E
Higher energy level
(forbidden classically)
Gate bias
Spin up - electron
position
Spin down electron enters dot
• Pauli principle allows the new electron to join the
same energy level as the original electron
• Coulomb interaction perturbs the ground-state so
that it is raised above the right bias and current
will flow
E
Unoccupied state
i≠0
Gate bias
Spin up - electron
position
Read out / Memory
• We have a way of measuring the spin state of an
electron in a quantum dot
• The first electron that passes though measures
the spin-state in the dot and other electrons that
follow will all have the same spin properties
• To be able to predict the original state of the dot,
the state has to be prepared again and then
measured using the same technique
• The electron current can be specialized (we can
aim it's spin to make measurement efficient)
5 DiVincenzo QC Criteria
1.
2.
3.
4.
5.
A scalable physical system with well-characterized
qubits.
The ability to initialize the state of the qubits to a
simple fiducial state.
Relatively long decoherence times compared to gateoperation times.
A universal set of quantum gates.
Qubit-specific measurement capability.
The Physical System: Excitons
Trapped in GaAs Quantum Dots
•
•
•
•
•
Exciton - a Coulomb correlated
electron-hole pair in a semiconductor,
a quasiparticle of a solid.
Often formed when photons excite
electrons from the valence band into
the conduction band.
Wavefunctions are “hydrogen-like” i.e.
an “exotic atom” though the binding
energy is much smaller and the extent
much larger than hydrogen because of
screening effects and the smaller
effective masses
Decay by radiating photons. Decay
time ~50ps-1ns
Hence can define the computational
basis as absence of an exciton |0>, or
existence of an exciton |1>
Initialization
• Register relaxes to the |00…0> state within 50ps-1ns
due to radiative decay
– Experimental systems are cooled to liquid helium temps ~4K to
prevent thermal excitations
• Hence initialization with such a system is relatively easy
• Other states can be initialized by applying gates to the
register
Relatively Long Decoherence
Times
• Mechanisms:
– Radiative Decay ~10ps-1ns
• Can be lengthened by electron-hole separation
– Background Electromagnetic fluctuations
• Less of a problem than in other systems since the
exciton and III-V heterostructure is on average
electrically neutral.
• Gate times are determined by energy band spacing, i.e.
creation and annihilation energies.
– Gate operations for GaAs QDs are estimated at ~1ps
or less
A Universal set of Quantum Gates
• Single Qubit Rotations through laser induced
Rabi Oscillations
• CNOT operations through dipole interactions
and laser excitation
Single Qubit Gates: Rabi Flopping
• Light-particle interaction is
characterized by the product of
the dipole moment and the
electric field:
μ•E(t)= ħR(t)
Where R(t) is the Rabi frequency
and the pulse area is given by:
Θ(t)=∫R(t)dt
and the state at time t is then
given by:
Cos(Θ/2)|0>+Sin(Θ/2)|1>
Stufler et al.
Large wafer containing InGaAs
QD was placed between a bias
voltage and exposed to
ultrafast laser pulses.
Cos(Θ/2)|0>+Sin(Θ/2)|1>
|1> => electric charge
=>Photocurrent (PC)
PC~Sin2(Θ/2)
π-pulse corresponds to a
population inversion
CNOT: Dipole Coupling
Nearest neighbor interactions alter the energy states:
Effective energy:
E’i = Ei + ∑j≠i ∆Eij nj
Hence, a coherent π-pulse with energy E’t(nc) results in a
state flop iff the control state is occupied.
Overcoming Short Interaction
Distances
•
•
•
Electrostatic Dipole fields fall off
as 1/R^3 hence the CNOT gate
can only be used for closely
neighboring QDs.
Solution: Use a sequence of
CNOTs on nearest neighbors to
swap the desired qubits until they
are contained in adjacent dots.
Optical Cavity coupling and fiber
optical interconnects have also
been proposed.
Read Out of Specified Qubit States
• Optical readout:
Excitons decay spontaneously and the resulting radiation can be
measured.
Alternatively, an excitation/probe beam spot can be physically
positioned in the region of the desired QD.
Due to the statistical distribution of QD shape and size variations,
individual QDs can be more accurately identified and addressed
through frequency discrimination.
In either case, repeated measurements have to be made. A single shot
readout still needs to be developed.
5 DiVincenzo QC Criteria
1.
2.
3.
4.
5.
A scalable physical system with well-characterized
qubits.
The ability to initialize the state of the qubits to a
simple fiducial state.
Relatively long decoherence times compared to gateoperation times.
A universal set of quantum gates.
Qubit-specific measurement capability.