Fundamentals of Quantum Engineering
Term 3, 2022
Assignment 2
Due date: Friday 25 November 2022, 17:00
Assignments submitted after the due date will attract a penalty of 10% per day (including weekends).
Cutoff date: Monday 28 November 2022, 17:00
The submission portal will close on the cutoff date. No submissions will be accepted beyond that date
and time.
The assignment report should be typed in a word processor of your choice. It should include electronic
graphs and figures to illustrate the results of your calculations, with appropriate axis labels and figure
captions. We require that your figures are easily readable, which must be done by setting axis font sizes
to 18, and plot line widths to 2.
When using Jupyter Notebook and MS Word, the figures can be included in the document using any
method you prefer. One suggested method is to use the Plotly tool (for information on installing and
using this tool please see the Lab 1 code). Once a plot has been generated using Plotly then you can
simply select the ‘Download plot as png’ icon in the top right of the plot, before copy and pasting this
image into the word processor.
At your discretion, you may include in the assignment report a printout of the important parts of the
Jupyter Notebook code you wrote to answer the questions. Do not include the whole code in your
report (not even as an appendix) – this must be submitted in separate files.
Upload to Moodle the following files:
1) The electronic version of your assignment report, as a single pdf file;
2) The Jupyter Notebook code used to generate the answers. The lecturer should be able to press the
“run” button on your Jupyter Notebook file and reproduce the results you have shown in the report.
Please add clear headings and comments to the Jupyter Notebook file to indicate which question the
code below it refers to i.e. if you are answering question 1_1 then the heading above the code for this
question should be ‘Question 1.1’.
3) Once you are finished, combine your Jupyter notebook code and pdf report into a single zipped folder
with your full name and zID.
The assignment contributes a total of 20 marks to the final course score.
Part 1 – Semiconductor “charge qubits” (50 marks)
We have seen in the lectures that quantum dots can be fabricated starting from a 2-
dimensional electron gas (2DEG), locally depleted using metallic electrodes to which we apply
negative voltages. In this exercise, we consider two double quantum dots, each one of them
containing a single electron. A crude schematic of such a system would look like this:
The blue rectangles represent metallic gates fabricated on top of a GaAs/AlGaAs
heterostructure, so that a 2DEG is present ~100 nm below the surface. A central gate, running
from top to bottom, separates the left double dot (LDD) from the right double dot (RDD) and
creates a very high potential energy barrier, so that no electron can move from left to right.
On each side, five metallic gates define a quantum dot. The gate in the middle is used to tune
the tunnel barrier between the top dot and the bottom dot. Therefore, the voltage on this gate
controls the tunnel rate of an electron between the top and the bottom dot.
We will work in a configuration where each double dot contains only one electron.
As usual, the starting point for a quantum mechanical description is to define the basis of the
Hilbert space. We will use a natural basis, defined as the “up” or “down” position of the
electron in each dot. For example:
This system is called a charge qubit. It has two basis states, that we denote with |0? and |1?,
and be used to store a bit of quantum information, encoded in the position of the electron
charge (indicated by the red dot in the drawings above). Note that, in this problem, the spin of
the electron does not play any role.
The Hamiltonian of a single electron in a double quantum dot looks exactly the same as that of
the ammonia molecule. For the left quantum dot (denoted by the subscript L) we would write:
= Δ +
and similarly for the right double dot:
= Δ +
Here and in the rest of this question, we will assume that the Hamiltonians are written in actual
units of energy (in Joule.)
The term Δ represents the tunnel coupling between upper and lower dot, and is controlled by
the voltage on the gate T. We will assume that each of the tunnel couplings depends on
voltage as:
Δ = Δ0 exp(/0)
with
Δ0/? = 1 GHz
0 = 1 V
(? = 6.626 × 10?34 J ? s is the Planck constant).
The term represents the difference in energy between the basis states |↑?, |↓? and is
controlled by the difference between the voltages , applied to the “plunger gates” PU and
PD. Having different voltages on those two gates induces an electric field oriented along the
axis of the double quantum dot, thus pulling the electron preferably towards one side. Again,
this is exactly the same situation as in the ammonia molecule.
Here, the detunings are simply given by:
= ( ? ) ≡ ?
where = ?1.602 × 10?19 C is the electron charge and we have defined = ? , i.e.
the voltage difference between the upper and the lower plunger gate.
Now let’s consider the two double dots side-by-side. Given two electrons, we have four basis
states:
We assume that the big electrode running vertically between the two sides is biased with a
large negative voltage, such that it is impossible for an electron to cross from left to right. Since
there is no tunnel coupling between left and right side, there cannot be any exchange
interaction. Therefore, again, the spin states of the electrons do not play any role.
However, the charge positions do matter, because electrons repel each other via the Coulomb
force. You can intuitively see that the |01? and |10? states will have lower energy than the |00?
and |11?, because the electrons are further apart from each other. Therefore, the Coulomb
Hamiltonian can be written in the following form:
where . If > 0, the above Hamiltonian describes two qubits that
have an energy advantage when they are in the opposite states.
This Hamiltonian looks very similar to the exchange coupling we saw with spins, but here it’s
only the z-components that are coupled, since it is the operator that describes the position
of the charge. This type of interaction is often called “Ising interaction”.
In the case of spins, > 0 indicates that the spins prefer to point in opposite directions
(antiferromagnetic coupling). Therefore, here we see the charge equivalent of an
antiferromagnetic interaction.
1.1 [5 marks]
Using the rules of tensor product, write a Python code that describes the full Hamiltonian of
the system:
Write the individual parameters in this Hamiltonian as per their definitions given on page 3,
where Δ and are functions of the applied voltages, namely:
ΔL = Δ0 exp(/0)
ΔL = Δ0 exp(/0)
with Δ0/= 1 GHz and 0 = 1 V.
The interaction is set simply by the geometry of the dots (how far from each other the
electrons are) and cannot, in this case, be tuned. Everywhere in this exercise we will use:
= 1 GHz
1.2 [7 marks]
In the Hamiltonian written at 1.1, assume the following numerical values:
= = 0 V
= = 0 V
Calculate the eigenvalues and eigenvectors of this Hamiltonian. (1 mark)
In two separate graphs (one for the left DQD and one for the right DQD) plot the expectation
values of , , , in each of the four eigenstates. (3 marks)
Recall the meaning of the operator in this context: for example, if you had a state |? such
that ?||? = ?1, this would mean that the state |? describes an electron fully localized
in the bottom left dot.
What is the meaning of ? If you had, for example, a state such that ?||? = +1, where
would the electron in the left dot be? (3 marks)
1.3 [8 marks]
Next, we want to use the metric of entanglement entropy to distinguish whether or not the
eigenvectors obtained at point 1.2 represent entangled states.
In the Moodle page you fill find a Jupyter notebook file pt_function.ipynb.This file
contains a Python function that calculates the partial trace over a density matrix. In order to
calculate the partial trace you can copy this function into your notebook and run the following
code:
rho1 = partial_trace(rho_total,[2,2],axis=0)
The arguments for this function are the density matrix of your total system, the dimensions of
every component subsystem, in this case we have a 2-qubit systems, each of dimension 2 and
the axis you want to extract (i.e. axis=0 extracts the density matrix of the first qubit and axis=1
the second qubit).
You are also free to create your own partial trace code if you wish.
Once you have calculated the partial trace you can calculate the entropy using the following
formula:
= Tr( log())
Where, in the notation we used here, = Tr(
) is obtained by taking the partial trace
over qubit L of the density matrix of the combined system.
Having calculated the values of the entanglement entropy, discuss whether or not the four
eigenstates calculated at 1.3 represent an entangled state of the two charge qubits.
1.4 [15 marks]
The key feature of quantum dots is that they allow tuning of the parameters of the Hamiltonian
by simply changing some voltages. In the device considered here, you can change the four
parameters Δ , Δ , , by tuning the voltages , , , .
In this exercise, you are allowed to imagine that all voltages can be tuned between -5 V and +5
V. They can be changed from one value to another in instantaneous steps, but you cannot
apply oscillating voltages.
The only parameter that cannot be changed dynamically is the electrostatic coupling ,
because it depends on geometry, not on voltages.
Find three different sets of voltages that yield the following configurations:
a) All four eigenstates are maximally entangled; (5 marks)
b) The lowest-energy state is one where both electrons are in the upper dots;
(5 marks)
c) The lowest-energy state is one where the left electron is in the upper dot, while the
right electron is in the lower dot; (5 marks)
To be clear: these three demands are mutually exclusive! You must come up with three
different sets of voltages. For each configuration, use an appropriate measure (expectation
values, entanglement entropy, etc.) to demonstrate that you have achieved the requested
result.
1.5 [15 marks]
Now assume you have applied the set of voltages that give the configuration c) in point 1.4,
that is, the left electron is in the upper dot, while the right electron is in the lower dot when
the system is in the ground state. You may imagine that the system will find itself in such
lowest-energy state if you wait long enough.
Next, suddenly switch to a different set of voltages, chosen such that the electrons begin to
oscillate between the upper and the lower dot with opposite phase (that is, the system
oscillates between “upper-left / lower-right” and “lower-left / upper right”.
Plot (in the same graph) the time evolution of the expectation values of and , to prove
that you have made the right choice. (5 marks)
Also plot (in a separate graph) the entanglement entropy as a function of time. (5 marks)
Comment on the results. In particular, discuss why you think that a certain set of parameters
gives the required results. You are always welcome to help yourself using the analogy with a
spin, or with the ammonia molecule. (5 marks)
Note: if you had a lot of experience with coupled quantum systems (and quantum dots in
particular), you could immediately guess what set of parameters gives you the desired results.
Here, we do not expect you to be able to make such a guess (it’s great if you do, of course!),
but the purpose of this exercise is to allow you to “experiment” with a simulated quantum
system. The Python code is simple to write and runs instantly, so you will be able to work by
“trial and error”, checking different combinations and see what happens, until you find the
parameters that work.
We strongly recommend that, as you try different combinations of parameters, you do so in a
systematic way, and take careful notes of the relation between the parameter choices and the
behavior of the system. This will help you finding the correct choices, and understand why they
work.
Part 2 – Magnetic Frustration [50 marks]
Three electrons experiencing mutual exchange coupling demonstrates a phenomenon known as
magnetic frustration. Specifically, this only occurs when we have an anti-ferromagnetic exchange
interaction between the spins, i.e. > 0. Imagine 3 spins at the vertices of a triangle as shown in figure
1.
Under exchange, the first and second spin must be antiparallel, but this leaves an impossible problem
for the third spin trying to minimise its energy, by being antiparallel to both spins, 1 and 2. Hence, the
third spin is said to be frustrated. We will work in the following basis:
Ultimately, we will be exploring the idea of entanglement for these coupled spins. To do this, we will
again be making use of the partial trace to calculate the Von Neumann entropy of entanglement. In
this task you will again be making use of the partial trace function in the Jupyter notebook file
pt_function.ipynb, which calculates the partial trace for a system of arbitrary dimensions.
Figure 1: 3 electron spins under mutual exchange shown in the -plane. The first spin (blue)
is up, the second spin (red) must therefore be down, this leaves the third spin (green) in a
frustrated state of being unable to be anti-parallel to both spins 1 and 2.
2.1 [3 marks]
Construct all the necessary spin operators for our 3 spin system.
Sx1 = np.kron(sigma_x,np. kron(np.identity(2), np.identity (2)))
You can make this programmatic to the number of electron spins you are considering.
2.2 [10 marks]
First, we will consider the exchange interaction to be of Ising form. This kind of interaction only couples
the z-components of the spins. It arises in complex spin systems or “artificial atoms”. The Ising
Hamiltonian (in unit of frequency, in hertz) is:
= ∑ ,,
,, ?≠
where = 1 MHz,
Find the eigenstates and their associated eigenenergies of this Hamiltonian. Give a physical
description of each eigenstate and elaborate on the relationship with their eigenenergies.
2.3 [10 marks]
We will now introduce an external magnetic field which creates the familiar Zeeman interaction,
along the -axis,
=
for each electron; 1, 2 and 3. As usual, = 28 GHz/T. Therefore, the full Hamiltonian is
= ∑ (,,)
,, ?≠
+ ∑ ,
Plot the eigenenergies as a function of magnetic field, varying from = 0 T to = 100 μT. How
many energy levels appear, how many would you expect to appear?
2.4 [10 marks]
In reality, it is most likely that each spin will experience a slightly different magnetic field due to the
practical impossibility of making a magnetic field that is perfectly uniform in space. We will model this
as a gradient magnetic field that decays linearly along the line 1? + 1?, i.e.
Calculate the magnetic field at each electron position, noting that the first spin is at (0,0) in the -
plane. Repeat the calculation of 1.3 with this added magnetic field gradient and comment on the
differences you observe. Simulate the same values of as in the previous question.
2.5 [5 marks]
Next, we will consider the exchange interaction to be of Heisenberg form. This is the “normal” form we
have seen in the lectures.
Where the and are defined similarly to above.
Repeat questions 1.2 and 1.3 for this different exchange form, and again comment on the nature of the
eigenstates, the eigenvalues, and how they differ from the Ising form.
2.6 [12 marks]
The ground state, i.e. the lowest energy state, of a frustrated magnetic system has interesting
properties when in the presence of an external magnetic field. Repeat questions 1.3 and 1.4, ensuring
that you store all of the lowest energy eigenstates for each magnetic field you simulate.
For every value of , take the lowest energy eigenstate and calculate the entanglement entropy of
the first spin to the others. For any specific points of interest, note down what the lowest energy
eigenstate was before and after the point. Do this using the Heisenberg form of the exchange
interaction.