Problem set 4

Problem set 4#

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#

  1. Understand the problem statement and requirements.

  2. Collaborate with your team to brainstorm ideas and approaches.

  3. Develop a solution using Python or Mathematica. You can use AI tools to assist you in coding and problem-solving.

  4. 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’)#

  1. Background:

    • What was the question?

    • How did you approach it?

  2. Solution development and implementation

  3. Implementation (mathematical and coding)

  4. 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?

  5. 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?

Problem 4. Coherent states#

A coherent state of a one-dimensional simple harmonic oscillator is defined to be an eigenstate of the non-Hermitian (i.e. \(a \neq a^{\dagger}\)) annihilation operator \(a\).

\[ \hat{a}|\alpha\rangle = \alpha |\alpha\rangle \]

where \(\alpha\), because of the non-Hermiticity of \(a\), is a complex number. Coherent states are specific linear combinations of harmonic oscillator eigenfunction that produce Gaussian wave packets that do not spread in time. Moreover, if the uncertainties in position and time are equal, then the resulting wave packet would be as close a representation of a classical particle as could be obtained whithin the bounds of the uncertainty principle.

  1. Show that a coherent state can also be obtained by appliying the translation (finite displacement) operator :

\[ e^{\alpha \hat{a}^\dagger - \alpha^{*} \hat{a}} | 0 \rangle = | \alpha \rangle \]

  1. Show that such a state satisifies the minimum uncertainty relation, i.e.

\[ \Delta x \Delta p = \hbar /2\]

where, \(\hbar =1\), and,

\[ x = \frac{a+a^{\dagger}}{\sqrt{2}}, \quad p = \frac{a-a^{\dagger}}{i\sqrt{2}} \]

  1. Jaynes-Cummings model.

Consider the two-level system interacting with a cavity. The Hamiltonian is given by:

\[ H = \frac{\omega}{2} \sigma_{z} + \omega_{c} a^{\dagger} a + g (\sigma^{+} a + \sigma^{-} a^{\dagger}) \]

The initial state of the system is given by

\[ | \psi_{in} \rangle = | 0 \rangle \otimes | \alpha \rangle \]

where the spin is in the ground state and the cavity in the coherent state.

Calculate \(\langle \sigma_{z}(t)\rangle\), and show that oscillations reappear some time after.

Hint: Use the case in which \(\omega = \omega_{c}\)

  1. Phase-space distribution

The Wigner function is a way to visualize and analyze quantum states in a manner similar to classical probability distributions and calculate observables as,

\[ \langle \vartheta (a,a^{\dagger}) \rangle = \int d\alpha^{2} \, \vartheta (\alpha,\alpha^{*}) W(\alpha) \]

Plot \(W( \alpha )\) dynamics at different times.

(Bonus)#

Try to repeat exercise 3 by using the Quantum Rabi model hamiltonian:

\[ H = \frac{\omega}{2} \sigma_{z} + \omega_{c} a^{\dagger} a + g \sigma_{x}( a + a^{\dagger}) \]

Explore the behavior by changing the parameters \(\omega\), \(\omega_{c}\), and \(g\).

For the curious readers:

Wigner function

Cat states

The quantum Rabi model

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