# How is interference between photon wavefunctions within the Hong-Ou-Mandel effect different from interference between coherent laser beams?

Does it have anything to do with the difference between Fock states and coherent states? Do photons also exit together on the same output port of a beamsplitter in the case of interference between laser beams like in the case of Hong-Ou-Mandel effect or is it a different situation?

The interference that one can find in a coherent laser beam can be described purely classically. On the other hand, the type of interference described by the Hong-Ou-Mandel (HOM) effect is a quantum effect. The latter involves multiple particles, whereas the former can be decribed with only one particle, one photon.

With the HOM effect one has two photons entering the two input ports of a beam splitter. Let's call them $a$ and $b$. Let's assume the two photons are identical (same polarization, same frequency, same spatial modes). Then we have the input state given by $|1\rangle_a |1\rangle_b$. The beamsplitter is described by a unitary process. One way to represent this process is as follows (assuming we can also label the two output ports as $a$ and $b$): $$|1\rangle_a \rightarrow \frac{1}{\sqrt{2}}(|1\rangle_a - |1\rangle_b)$$

$$|1\rangle_b \rightarrow \frac{1}{\sqrt{2}}(|1\rangle_a + |1\rangle_b)$$

So, now it is clear to see that for the given input states (two identical photons) one would have $$|1\rangle_a|1\rangle_b \rightarrow \frac{1}{2}(|1\rangle_a |1\rangle_a - |1\rangle_b |1\rangle_b) = \frac{1}{\sqrt{2}}(|2\rangle_a + |2\rangle_b .$$ This means that either both photons leave through port $a$ or both leave through port $b$. The case where one leaves through $a$ and the other through $b$ is canceled by this process of quantum interference. Those terms appear with opposite signs and therefore cancel. In a sense the interfere destructively. This is different from the interference found in laser beams.

It does not have anything to do with being described by coherent states, as opposed to Fock states, because for optical interference the result is the same regardless whether one uses Fock states or coherent states. For instance, considering a one-particle Fock state (single photon) going through an interferometer, one would have $$|\psi\rangle= \frac{1}{\sqrt{2}}[|1\rangle_a + \exp(i\phi) |1\rangle_b] ,$$ where the subscripts again represent the two input port to beam splitter that would combine the two paths of the interferometer and $\phi$ is a relative phase due to a difference in pathlength. Now we apply the operation of the beam splitter $$|\psi\rangle= \frac{1}{2}[(|1\rangle_a - |1\rangle_b) + \exp(i\phi) (|1\rangle_a + |1\rangle_b)] .$$ To see the interference we measure the probability to detect a single photon after one of the output ports. For this we use a projection operator $P_a$, which the selects the part after port $a$ $$|\psi_a\rangle = P_a|\psi\rangle = \frac{1}{2} |1\rangle_a [1+ \exp(i\phi) ] .$$ The probability to detect a single photon is then given by $$|\langle 1_a|\psi_a\rangle|^2 = \frac{1}{4} |1+ \exp(i\phi) |^2 = \frac{1}{2} [1+ \cos(\phi)] .$$ Here we see the interference as a function of the relative phase. One can see that this is quite different form the situation with the HOM effect.

• I agree, but what is the quantum state for purely classical interference? Do they exit in different output ports unlike the Hong ou mandel effect? – Luparu Dorian Nov 9 '16 at 5:03
• I understand how a single photon interferes with itself, but I do not understand what happens when many photons interfere with each other. I read that a laser is composed of many photons in the same quantum state (macroscopic quantum coherence), so it should be easy for you to write the interference between two laser beams in a equation similar to the ones you wrote since they are macroscopically populated single quantum states. – Luparu Dorian Nov 9 '16 at 8:59
• I also found this quote. What does it tell you? I do not understand it since it is not reffered to in terms of photons, but waves. <quote>A classical analog to the HOM effect occurs when two coherent states interfere at the beamsplitter. If the states have a rapidly varying phase difference then a dip will be observed in the coincidence rate equal to one half the average coincidence count at long delays . Consequently, to prove that destructive interference is two-photon quantum interference rather than a classical effect, the HOM dip must be lower than one half.</quote> – Luparu Dorian Nov 9 '16 at 9:09
• arxiv.org/pdf/1309.2265.pdfhttps://arxiv.org/pdf/1309.2265.pdf This tells us that the Hong ou Mandel effect is observed for two lasers with random relative phase: The U-shape can be also observed for classical beams with the fluctuations in the relative phase increased artificially.Whenever the phase of a classical oscillator is distributed uniformly, the quadrature will have the arcsine probability distribution, imitating the U-shape. – Luparu Dorian Nov 9 '16 at 9:36
• In order to observe the classical analogue of the HOM effect, a random phase φ should be introduced into one of the beams to ‘erase’ the usual first-order interference. (For the case of twin beams, there is no first-order interference due to their uncertain relative phase.) IT SEEMS that interference between Fock states is a second order interference, not the normal one! – Luparu Dorian Nov 9 '16 at 9:46