Quantum bomb detection: Tasks Part 1 & 2 with solutions

Solution

• Without the beam splitter 2, the light wave reaches both detector 1 and detector 2 – the light intensity registered at each detector amounts to 50% of the light intensity of the incoming wave.
• When inserting beam splitter 2, the light wave is detected by detector 2 only (100% of the incoming wave’s intensity).

Explanation

• At beam splitter 1, the incoming wave splits up into two waves, each one having 50% of the incoming wave’s intensity. One wave is reflected at mirror 1, the other one at mirror 2.
• Once these two waves have passed beam splitter 2, they interfere with each other.
• When being reflected at the beam splitters, the waves are phase-shifted. Depending on the phase difference between the two waves after the second beam splitter, one can observe destructive or constructive interference.
• In the direction of detector 1, one observes a destructive interference: the two waves cancel each other, the light intensity is zero, thus nothing is detected.
• In the direction of detector 2, the interference is constructive: the two waves “add up” and the light reaches detector 2. The light intensity detected at detector 2 corresponds to the light intensity of the incoming wave.

Solution

• At the beam splitters, a classical particle takes either one path or the other.
• The number of particles reaching detector 1 is approximately equal to the number of particles reaching detector 2. The probability that a particle is detected by either one of the detectors is 50%.
• A wave can be “split up” whereas a classical particle cannot.

Explanation using states

• Physicists describe the behaviour of a system by tracking its state(s). To describe the state of a system (e. g. a wave, a particle), they use the so-called (bra)-ket notation. Example: a particle taking the upper path in the interferometer would be described by |up⟩.
• An incoming classical particle that has passed beam splitter 2 can be in two possible states: a particle taking the upper path is described by |up⟩ and is detected by detector 1. Likewise, a particle taking the lower path is described by |right⟩ and is detected by detector 2.
• The following table resumes how the state of the incoming classical particle changes when going through the interferometer.

Unlike in the case of light being considered as an electromagnetic wave, a classical particle can be detected by both detectors – with an equal probability of 50%.

Conclusion: Light does not behave like a classical particle.

Solution

• The single photons are detected by detector 2 only (100% probability), whereas classical particles were detected in both detector 1 (50% probability) and detector 2 (50% probability).

Explanation

• Although a photon has properties of a particle, it can simultaneously be in both states, ${\displaystyle |up⟩}$ or ${\displaystyle |right⟩}$, taking the upper or the right path, after having passed beam splitter 1.

• The single photon is said to be in a superposition of the two states ${\displaystyle |up⟩}$ and ${\displaystyle |right⟩}$. The superposition state is written as: ${\displaystyle \frac{1}{\sqrt{2}}|up⟩+\frac{1}{\sqrt{2}}|right⟩}$.
  In the animation, the superposition of the two states is symbolised by the two wave packets (representing the photon) being connected by a dotted line.
   

  

• In order to find out which path the photon takes, you have to perform a measurement. If you square the factors before the states ${\displaystyle |up⟩}$ and ${\displaystyle |right⟩}$, in this case ${\displaystyle (\frac{1}{\sqrt{2}}{)}^2=(\frac{1}{2})}$, you get the probability that the photon will take the upper or the right path. The probability is ${\displaystyle \frac{1}{2}}$ in each case, i. e. 50%. Without measuring, the path of the individual photon is not determined.
• After beam splitter 2, the two possible states of the single photon (${\displaystyle |up⟩}$ and ${\displaystyle |right⟩}$) interfere – just like the electromagnetic wave. In the direction of detector 1, the interference is destructive, and in the direction of detector 2 it is constructive. Thus, the photon can only be detected by detector 2.

Conclusion

If the photon can be in several possible states, this is called a “superposition state”, like here the state ${\displaystyle \frac{1}{\sqrt{2}}|up⟩+\frac{1}{\sqrt{2}}|right⟩}$ describing the photon going both the upper and the right path. The two states can interfere – just like waves.

The following table resumes what happens:

Conclusion

A measurement changes the state of the photon. In the experiment above, it destroys the superposition state ${\displaystyle \frac{1}{\sqrt{2}}|down⟩+\frac{1}{\sqrt{2}}|right⟩}$. After the measurement, the photon is in either state ${\displaystyle |down⟩}$ or ${\displaystyle |right⟩}$ .