Problem set 1

Problem set 1#

Short guidelines#

Description#

Today you will be working on a specific challenge related to quantum physics or quantum computing. In groups, your task is to develop a solution using the concepts and tools you’ve learned throughout the course. You can use any modules or libraries you find useful, such as NumPy, SciPy or QuTiP.

We will be using AI copilots to assist you in coding and problem-solving. The focus will be on collaboration, creativity, and critical thinking.

The final part of the session will involve a short presentation (15’), where each group will share their approach, solution, and reflections on the use of AI tools in their process.

Objectives#

Understand the problem statement and requirements.
Collaborate with your team to brainstorm ideas and approaches.
Develop a solution using Python or Mathematica. You can use AI tools to assist you in coding and problem-solving.
Prepare a presentation to share your findings and reflect on the use of AI in your process.

Resources you can use#

Access to AI copilots/tools for coding assistance. Some examples include GenAI, ChatGPT, or other similar tools.
Documentation and resources from previous weeks.

Short presentation outline (15’)#

Background:
- What was the question?
- How did you approach it?
Solution development and implementation
Implementation (mathematical and coding)
Critical reflection on the use of AI copilots. Eg:
- Which AI tool/s did you use?
- Was it helpful, misleading, or counterproductive?
- What resources did you need other than AI?
- How confident are you in your solution?
- Which parts are less clear or hard to verify?
Coding skills evaluation
- To what extent do you understand the code generated with AI assistance?
- Were there functions or code constructs used that you didn’t know before? Do you understand them now?

Section 1: Many-body quantum systems#

1.1 Identical particle time-evolution.#

Explain why an initially completely symmetric or anti-symmetric wave function describing a system of identical particles remains symmetric or anti-symmetric at later times.

1.2 Polarization states.#

We model the phase of a photon as a two-level system (q-bit) whose Hilbert space is spanned by a orthonormal basis of two states: |1〉 and |2〉. They correspond, for instance, to two orthogonal linear polarizations with observable polarization operators:

\[ P_{1} = |1\rangle\langle1|, P_{2} = |2\rangle\langle2| \]

Provide an example of a normalized pure state for which the expectation value of both $P_{1}$ and $P_{2}$ equals 1/2, and a two-by-two matrix representation of the corresponding density-mmatrix operator in the given basis.
Find the density matrix for the most general normalized pure state with the properties specified in the previous task.
The state with partial polarization has the general representation:

\[ \rho = \frac{1}{2}(\mathbb{1} + \xi \cdot \sigma ) \]

where 1 is the 2x2 identity matrix, $\sigma$ is the vector of Pauli matrices, $\xi \in \mathbb{R}^{3}$. For which $\xi$ is this state a pure state?

For which $\xi$ is $\langle P_{1} \rangle = 1$? For which $\xi$ is $\langle P_{2} \rangle = 1$?
Define the states with circular polarizations: $$ | \pm \rangle = \frac{1}{2} ( |1\rangle \pm i |2\rangle ) $$

For which $\xi$ is $\langle P_{+} \rangle = 1$? For which $\xi$ is $\langle P_{-} \rangle = 1$?

Section 2: Ensembles#

2.1 Ensembles of $\frac{1}{2}$ spin systems#

Consider a pure ensemble of identically prepared spin $\frac{1}{2}$ systems. Suppose the expectation values $\langle S_{x} \rangle$ and $\langle S_{z} \rangle$ and the sign of $\langle S_{y} \rangle$ are known. Show how we may determine the state vector. Why is it unnecesary to know the magnitude of $\langle S_{y} \rangle$?
Consider now a mixed ensemble of spin $\frac{1}{2}$ systems. Suppose the ensemble averages $\langle S_{x} \rangle$, $\langle S_{y} \rangle$, and $\langle S_{z} \rangle$ are known. Show how we may determine the 2x2 density matrix of the ensemble.

2.2 Mixed ensembles#

The density matrix is defined as:

\[ \rho = \sum_{i} w_{i} | \alpha^{i} \rangle \langle \alpha^{i} | \]

If $w_{i} = 1$ for a given $i$ and 0 elsewhere, then the ensemble is said to be a pure ensemble. Consider a mixed ensemble of spin $\frac{1}{2}$ systems, containing 75% of $S_{z}^{+}$ systems and 25% of $S_{x}^{+}$ systems. Calculate the density matrix and the ensemble average expectation value [S]. Consider now that the $S_{z}$ systems are replaced by systems with $S\cdot \hat{n}(+)$, where $\hat{n}=\cos \theta \hat{y} + \sin \theta \hat{z}$. Calculate the ensemble average values now.

Section 3: Galuber states#

By using the commutstion relation $[a, a^{\dagger}] =1$, show that

\[ e^{-\beta a^{\dagger}} ae^{\beta a^{\dagger}} = \beta + a \]

Using this result, show that $\beta = N e^{\beta a^{\dagger}} |0\rangle$ is a coherent state, i.e. $a|\beta\rangle = \beta |\beta\rangle$. Finally show that the normalization, $N=e^{- |\beta|^{2}/2}$.

Hint: To prove it, consider the $\beta$ derivative of this expression.

Calculate the expectation values, $x_{0}=\langle \hat{x} \rangle$ and $p_{0}=\langle \hat{p} \rangle$, with respect to $|\beta \rangle$ and, by considering $\langle \hat{x}^{2} \rangle$, and $\langle \hat{p}^{2} \rangle$, show that,

\[ (\Delta p)^{2}(\Delta x)^{2} = \frac{\hbar^{2}}{4} \]

where $\Delta p = \hat{p} - \langle \hat{p} \rangle$ (similarly $x$).

Hint: Remember how creation and anhiliation operators are related to the phase space operators $\hat{x}$ and $\hat{p}$. Also, note that the Hermitian conjugate of the eigenvalue equation $a|\beta \rangle = \beta |\beta \rangle$ leads to the relation $\langle \beta | a^{\dagger} = \beta | \beta^{*} \rangle$.

To determine the coordinate representation to the coherent state, $\psi(x) = \langle x|\beta\rangle$, it is helpful to revert back to the expression for $a$ as a differential operator. Show that the eigenvalue equation $a|\beta \rangle = \beta|\beta \rangle$ translates to the equation:

\[ (x+ \frac{\hbar}{m \omega} \partial_{x}) \psi(x) = \beta \psi(x)\]

Show that this equation has the solution:

\[ \psi(x) = N exp [-\frac{(x-x_{0}^{2})}{4 (\Delta x)^{2}} + i \frac{p_{0}x}{\hbar}] \]

where $x_{0}$ and $p_{0}$ are defined in part (2).

(Bonus) Maths problem#

Asymptotic and Convergent series#

A sequence of function $\{ \psi_{n} \}$ with $ \psi_{n} : C/z_{0} \rightarrow C , n=0,1,2,... $ is called an asymptotic sequence as $z \rightarrow z_{0}$ if for each $n=1,2,...$

\[ \psi_{n-1}(z) = o(\psi_{n}(z)) , z \rightarrow z_{0} \]

Let $f(z)$ be a continuous function such that $f(x):z \in C/z_{0} \rightarrow f(z) \in C$. We say that $f(z)$ allows an asymptotic series expansion for $z \rightarrow z_{0}$ if there exist an asymptotic sequence $\{ \psi_{n} \}$, such that for each $N=1,2,...$

\[ |f(z) - \sum_{n=0}^{N-1} a_{n}\psi_{n}(z) | = O(\psi_{N}(z)), \quad z \rightarrow z_{0}\]

or equivalently,

\[ |f(z) - \sum_{n=0}^{N-1} a_{n}\psi_{n}(z) | = o(\psi_{N}(z)), \quad z \rightarrow z_{0}\]

Asymptotic series often appear in many branches of physics when performing perturbative expansions. Note that asymptotic series need not to converge. Similarly, a convergent series need not to be asymptotic.

Consider the (uniformly convergent) Taylos series for the exponential function $f(z) = e^{z}$. Prove that it does not define an asymptotic series for $e^{z}$. Prove that it does not define an asymptotic series for $e^{z}$ when $|z| \rightarrow \infty$.
Consider now the function,

\[ -E_{i}(-x) = E_{i}(x) = \int_{x}^{\infty} \frac{exp(-t)}{t} dt \]

and prove that $E_{1}(x)$ does not converge in any standard sense, but admits an asymmptotic expansion for $x \rightarrow \infty$.