Simulating Charge Stability Diagrams for Double and Triple Quantum Dot Systems Ian E. Powell (Dated: September 2, 2014) Abstract We recreate Sandia National Laboratories’ double quantum dot charge stability diagram simu- lation using their rate equation approach and compare our simulation’s results to some touchstone situations as well as their results. We find qualitative agreement between charge stability diagrams and extend the code to construct the charge stability diagram for a triple dot structure. We note how the triple dot charge stability diagrams change as the third gate voltage is increased. 1
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Simulating Charge Stability Diagrams for Double and Triple
Quantum Dot Systems
Ian E. Powell
(Dated: September 2, 2014)
Abstract
We recreate Sandia National Laboratories’ double quantum dot charge stability diagram simu-
lation using their rate equation approach and compare our simulation’s results to some touchstone
situations as well as their results. We find qualitative agreement between charge stability diagrams
and extend the code to construct the charge stability diagram for a triple dot structure. We note
how the triple dot charge stability diagrams change as the third gate voltage is increased.
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I. INTRODUCTION
A. Quantum Computing in Solid State Quantum Dots
The field of quantum computing has received a great deal of attention in the physics com-
munity due to its exciting prospective applications and recent progress made in constructing
systems with characteristics that allow for the formation of so called “qubits.” A qubit, short
for quantum bit, represents a two-level, quantum mechanical system. Canonical examples
of a qubit are a photon’s polarization–be it linearly horizontal or linearly vertical–or an
electron’s spin state–where down and up, or horizontal and vertical correspond to 0 and 1
in terms of binary. The difference, between a qubit and a classical bit is the qubit’s ability
to be in both 0 and 1 simultaneously–a superposition of up and down for the example of the
electron’s spin. It is this property of superposition, along with something called quantum
entanglement, that differentiates quantum computing from its classical counterpart.
David Divincezo of IBM famously constructed a list of requirements to construct quantum
computer. The “DiVincenzo Criteria” are: (1) the system must have a large number of well
defined qubits, (2) initialization to a known state for the system must be easily performed,
(3) a universal set of quantum logic gates must be developed, (4) qubit-specific measurements
must be able to be performed, (5) and long coherence times must exist for the qubits in
the system1–where coherence time describes how long a quantum state can exist on average
before being perturbed, and, in turn, altered by the environment.
Many different types of systems show promise in fulfilling said criteria–for example, super-
conducting systems have shown promise in quantum information processing due to the ease
of fabricating larger systems. Cold neutral atom systems’ qubits have some of the longest co-
herence times–offering promise for storing information for longer periods of time. Solid-state
systems in which spin states of electrons or nuclei represent the qubits have serious promise
in their scalability, due to extensive research done on the miniaturizing of semiconductors
for computing. This summer I worked on researching a solid state system–namely quantum
dots–for quantum computing purposes.
A quantum dot is generally any 3-D potential well, but in the context of solid-state
physics it is a structure made of semiconducting material that is small enough to exhibit
quantum mechanical phenomena. The dot’s bound states of electron and electron-hole, or
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excitons, are confined in all three spatial dimensions. The electronic properties of a single
dot, its discrete energy levels etc., are akin to that of an atom–hence the nickname for a
quantum dot is “artificial atom.” When another dot is introduced into the system and there
exists a strong, coherent, in-phase tunneling of electrons between the two dots, the system
behaves similar to a covalently bonded molecule.
We utilize double and triple quantum dots which have been fashioned onto a Si-SiO2
heterostructure as our means for quantum computing; for a schematic of a GaAs-AlGaAs
heterostructure see Fig. I.
FIG. 1. Schematic of a GaAs-AlGaAs heterostructure taken from [2].
At the interface between the two different materials a two dimensional electron gas (2DEG)
forms. To further confine the other two dimensions of motion we apply voltages to the
gates to create potential wells and trap some target number of electrons. We utilize a
quantum point contact (QPC) to determine the current through the dots and, in turn, the
occupancy of each dot. This is due to the fact that the QPC has a systematically varying
conductance as the occupation of the dots varies. The QPC, as opposed to some ammeter,
has the advantage of measuring the current of electrons through the dots indirectly–thereby
not perturbing our system of qubits, and thus not destroying any information stored in our
system.
There are two different types of qubits that are viable for double dot systems–spin qubits
and charge qubits. The bases for a double dot system’s spin qubits are the singlet and triplet
states–i.e. (S = 0, Sz = 0) = |0〉, (S = 1, Sz = 0) = |1〉. The bases for the charge qubits are
based on electron occupation of the dots–for example (0,1) occupancy could be |0〉 whereas
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(1,0) would be |1〉–for a helpful visualization of the wave function of our system written in
terms of our qubit bases see Fig. II. For the triple dot system we utilize the Heisenberg
FIG. 2. The “Bloch Sphere” visualization of a qubit’s wave function.
interaction and the “singlet” and “triplet-like” states as our bases–where, for example, |0〉