Photons and Interference

Abstract

In this chapter, we will give the mathematical description of the previous chapter’s experiments. This will show us how we can calculate the probabilities of measurement outcomes in quantum mechanics. We need to introduce simple matrices and basic complex numbers.

Exercises

• 1. Multiply the following matrices:

  $$ A = \begin{pmatrix} 12 &{}\quad -3 \\ 6 &{}\quad 9 \end{pmatrix} \qquad \text {and}\qquad B = \begin{pmatrix} 2 &{}\quad 5 \\ -1 &{}\quad 0 \end{pmatrix} . $$

  Is AB the same as BA?

• 2. Express the following complex numbers in polar representation $r e^{i\phi }$:

  $$ z_1 = 3+4i\, , \qquad z_2 = 12 - 8 i\, , \qquad \text {and}\qquad z_3 = z_1 + z_2^*\, . $$

• 3. Find the complex solutions to the quadratic equation

  $$ x^2 - 2x + 10 = 0\, . $$

• 4. Every matrix has two special properties: the trace and the determinant. The trace is the sum over the diagonal elements (top left to bottom right), while the determinant of a $2\times 2$ matrix is given by

  $$ \det \begin{pmatrix} a &{}\quad b \\ c &{}\quad d \end{pmatrix} = ad - bc\, . $$

  Calculate the trace and the determinant of the following matrices:

  $$ \begin{pmatrix} 3 &{}\quad 1 \\ 2 &{}\quad 6 \end{pmatrix} , \qquad \begin{pmatrix} 0 &{}\quad -i \\ i &{}\quad 0 \end{pmatrix} , \qquad \text {and}\qquad \begin{pmatrix} 1 &{}\quad 1 \\ -1 &{}\quad -1 \end{pmatrix} . $$

• 5. Normalise the vector $|\psi \rangle = \begin{pmatrix} 3 \\ 4i \end{pmatrix}$.

• 6. Calculate the scalar product between

  $$ |\psi \rangle = \begin{pmatrix} 3i \\ -3 \end{pmatrix} \qquad \text {and}\qquad |\phi \rangle = \begin{pmatrix} 5 \\ 7 \end{pmatrix} . $$

  Show that $\langle \psi |\phi \rangle = \langle \phi |\psi \rangle ^*$.

• 7. Every month a lottery jackpot is one million pounds. A lottery ticket costs £5, and there are four million people playing the lottery each month. Calculate your average gain at any given month. How long do you have to play the lottery in order to have a 50% chance of winning at least once?

• 8. A car comes at a fork in the road, and the driver can turn left or right. If we describe the car quantum mechanically (not very realistic!) and the car is twice as likely to turn left as it is to turn right, what is the quantum state of the car after the fork?

• 9. The state of a photon in an interferometer is given by

  $$ |\psi \rangle = \frac{3i}{5} |\text {right}\rangle + \frac{4}{5} |\text {down}\rangle \, . $$

  What is the probability of getting the measurement outcome “right”?

• 10. Construct the matrix for a beam splitter with a 70:30 ratio between reflection and transmission. Show how we can achieve perfect destructive interference in a Mach–Zehnder interferometer using two of these beam splitters (see also Exercise 3 of Chap. 1).

• 11. The unitary transformation of a Mach–Zehnder interferometer is described by

  $$ U_\mathrm{MZ} = \frac{1}{4} \begin{pmatrix} 3+\sqrt{3}\, i &{} 1-\sqrt{3}\, i \\ 1-\sqrt{3}\, i &{} 3+\sqrt{3}\, i \end{pmatrix} . $$

  Calculate the relative phase difference between the arms in the interferometer. You may use Eq. (2.46).

• 12. A photon in the state

  $$ |\psi \rangle = \frac{1}{\sqrt{5}} |\text {right}\rangle + \frac{2}{\sqrt{5}} |\text {down}\rangle $$

  is sent into a beam splitter. What is the probability of finding the photon in the “down” path?

• 13. We replace the lower-left mirror of the Mach–Zehnder interferometer in Fig. 1.4 with another 50:50 beam splitter. There are now three input beams and three output beams of the resulting interferometer. Assuming there are no relative path differences, construct the $3\times 3$ transformation matrix describing the interferometer. Calculate the nine probabilities of a photon in any of the three inputs going to any of the three outputs. Compare your results with the $3\times 3$ transformation matrix.

• 14. Consider again the Mach–Zehnder interferometer in Fig. 1.4. Instead of a single phase shift $\phi$ in the upper arm, we place two identical phase shifts $\phi$, one in the upper arm and one in the lower arm. Calculate the matrix transformation of this interferometer. Sending a photon into the left input, what is the quantum state vector of the photon at the output? Can we measure the phase shift $\phi$ in both arms based on the probabilities of finding the photon in detectors $D_1$ and $D_2$?

• 15. Consider a quantum state $|\psi \rangle$ that accumulates a global phase $e^{i\phi }$. Show that any probability calculated using this state does not depend on the phase $\phi$. The phase is unobservable.

• 16. We record 1000 photons in a gravity wave detector (see Sect. 2.5), and we find $N_1 = 35$ in detector $D_1$ and $N_2 = 965$ in detector $D_2$. Calculate the phase shift $\phi$.