Neutrino oscillations

Demonstrate that neutrinos have mass.

Objectives

In the following we will go through the theory that leads to the conclusions above and see how the time evolution of a neutrino can be modeled by changing the ratio of the distance between source and detector and the energy of the neutrino. This will translate into running several circuits on the quantum computer for different values of this ratio and measuring the corresponding probability of finding the neutrino in one of the three different species.

Understand why the metamorphosis of neutrinos imply that they must have mass.
Experimentally verify that neutrino`'s species changes over timesteps on the quantum computer.

Background

Neutrinos are elementary particles constantly created through a lot of different processes: in space, we can find them in an exploding supernova or as a product of nuclear reactions inside the Sun; on Earth, they are released in reactions in nuclear power plants or even in our bodies, due to the decay of potassium-40 (fun fact here). They interact exclusively via the weak interaction and gravity, and their name reveals two of their most important characteristics: they are not electrically charged and they have a very small mass with respect to the majority of the other known elementary particles. In fact, for many years neutrinos were thought to be massless, until two experiments revealed that they must have a mass, even if tiny! This discovery had an impact on well established models in particle physics, like the Standard Model, which is now known to be incomplete because it does not provide a mass generation mechanism for neutrinos.

Theory

Neutrinos come in three flavors (or species): electron neutrino, muon neutrino and tau neutrino. More precisely, when they are produced in a weak interaction, they always come together with a charged lepton (an electron, a muon or a tauon). The fact that a neutrino is created with a definite flavor implies, from the quantum mechanical point of view, that their state coincides with one of the three flavor eigenstates. However, flavor eigenstates are not the same as the mass ones, or stated in other words, masses of neutrinos are not diagonal in the flavor basis. This means that when a neutrino $|\nu_{\alpha}\rangle$ is created with flavor $\alpha$, it is in a coherent supersposition of the mass eigenstates $|\nu_{1}\rangle,|\nu_{2}\rangle,|\nu_{2}\rangle$. The two different bases are related by a unitary transformation called the Pontecorvo-Maki-Nakagawa-Sakata matrix, which we indicate with $U_{PMNS}$ in the following, also known as neutrino mixing matrix:$$\begin{pmatrix} \nu_e \\ \nu_\mu \\ \nu_\tau \\ \end{pmatrix} = U_{PMNS} \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \\ \end{pmatrix}. $$To be able to encode this system in a two-qubit one, we will introduce fictitious $|\nu_{X}\rangle$and $|\nu_{4}\rangle$ which have no physical significance (the explicit expression of $U_{PMNS}$ is given below in the Appendix).

Imagine now that a neutrino $|\nu_{\mu}\rangle$ is generated at $t=0$. At time $t>0$ the probability of detecting a different flavor neutrino $|\nu_{\alpha}\rangle$ is given by $$p_{\alpha}(t)=|\langle \nu_{\alpha}|e^{-\frac{iHt}{\hbar}}|\nu_{\mu} \rangle|^2$$and we can express the time evolution operator $V(t)=e^{-\frac{iHt}{\hbar}}$ as a diagonal matrix in the mass basis (explicit expression in the appendix). The entries of this matrix only depends on the energy difference $E_{k1}$ that can be approximated as: $$E_{k1}\simeq\frac{\Delta m_{k1}^2c^3}{2p} \text{ for } k={2,3}$$where we used the Taylor expansion with respect to $m$ in $E_k=\sqrt{p^2c^2 + m_k^2c^4}$.

We finally arrived to the big revelation moment: for the neutrino oscillation to occur, at least one of the masses in $\Delta m_{k1}$ must be non-zero! This also means that neutrino oscillation experiments can only inform us on the difference between the mass values, and not on the absolute masses, which still remain unknown to date.

Last but not least, if we use the approximations $E\simeq pc$ and $L \simeq ct$, we can rewrite:$$e^{-iE_{k1}t/\hbar} \simeq e^{-i\frac{\Delta m^2_{k1}c^3}{2\hbar}\frac{L}{E}}$$The parameter $\frac{L}{E}$ is in fact what experimentalists can control to detect neutrino oscillations. $L$ is the distance from the neutrinos source to the detector, while $E$ is the energy of the neutrino. When a neutrino travels for a distance $L$, since the masses are different, the phase between the mass states changes with distance from the source; once the neutrino arrives to the detector it can happen that we find a different flavor state than at the beginning (i.e. a different linear combination of mass states). At fixed $\Delta m_{k1}^2$, the probability of oscillation changes as we move away from the detector, or if we consider different neutrino energies.

Appendix

The values below for the neutrino mixing matrix correspond to the experimentally measured ones in November 2022 :$$U_{PMNS}=\begin{pmatrix} 0.8255 & 0.5445 & -0.142+0.0434 i & 0 \\ - 0.2709+0.02739 i & 0.6057, +0.0181 i & 0.7475 & 0 \\ 0.4938, +0.0237 i & -0.5798+0.0157 i & 0.6475 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$The explicit expression of the time evolution operator in the mass basis is $V(t)=S_1(t)\otimes S_2(t)$ with:$$S_1(t) = \begin{pmatrix} 1 & 0 \\ 0 & e^{-iE_{31}t/\hbar} \\ \end{pmatrix},~~ S_2(t) = \begin{pmatrix} 1 & 0 \\ 0 & e^{-iE_{21}t/\hbar} \\ \end{pmatrix}$$The numerical values for the mass differences are:$$\Delta m^2_{21}=7.39\times10^{-5}eV^2 , \Delta m^2_{31}=2.45\times10^{-3} eV^2$$NB: the division by $c^2$ is very often omitted in particle physics!

Assignment

You will use a system of $2$ qubits to model neutrino oscillations. Since the bases of flavor and mass eigenstates consist of only three elements each, the last computational basis state will not be identified with a physical quantity. Mapping the first three computational basis states to the neutrino mass eigenstates and applying time evolution on the $2$-qubit system, we will show how the probability of measuring a certain flavor eigenstate oscillates.

Express the state of the neutrino $|\nu_{\mu}\rangle = |01\rangle$ in terms of the mass eigenstates using the conjugate transpose of the Pontecorvo-Maki-Nakagawa-Sakata matrix. Use the gate representation of this matrix to run it as a circuit on the quantum computer.
Apply the time evolution operator (as defined in the Appendix) on the state above. Transform the state back to the flavor basis and measure.
Run the circuit constructed with the steps above for different values of $\frac{L}{E}$.
Plot the experimentally obtained probabilities and compare them with the theoretical values provided.

Questions for Preparation

Do neutrino oscillations imply that all neutrinos must have mass?
Do charged leptons (electrons, muons, tauons) oscillate?
Why studying solar neutrinos was important for the discovery of neutrinos oscillation?

Literature

The chameleons of space (The Nobel Prize in Physics 2015).

On-premises superconducting quantum computer for education and research (EPJ Quantum Technol. 11, 32, 2024).

Neutrino oscillations (Lecture Notes University of Warwick 2020).