Hong–Ou–Mandel effect
The Hong–Ou–Mandel effect is a two-photon interference effect in quantum optics that was demonstrated in 1987 by Chung Ki Hong (Korean: 홍정기), Zheyu Jeff Ou (Chinese: 区泽宇; pinyin: Oū Zéyǔ) and Leonard Mandel at the University of Rochester:.
[1] The effect occurs when two identical single photons enter a 1:1 beam splitter, one in each input port.
In this way, the interferometer coincidence signal can accurately measure bandwidth, path lengths, and timing.
Since this effect relies on the existence of photons and the second quantization it can not be fully explained by classical optics.
When a photon enters a beam splitter, there are two possibilities: it will either be reflected or transmitted.
Here, we assume a 1:1 beam splitter, in which a photon has equal probability of being reflected and transmitted.
In addition, reflection from the bottom side of the beam splitter introduces a relative phase shift of π, corresponding to a factor of −1 in the associated term in the superposition.
This sign is required by the reversibility (or unitarity of the quantum evolution) of the beam splitter.
Since the two photons are identical, we cannot distinguish between the output states of possibilities 2 and 3, and their relative minus sign ensures that these two terms cancel.
Consider two optical input modes a and b that carry annihilation and creation operators
Therefore The relative minus sign appears because the classical lossless beam splitter produces a unitary transformation.
Physically, this beam splitter transformation means that reflection from one surface induces a relative phase shift of π, corresponding to a factor of −1, with respect to reflection from the other side of the beam splitter (see the Physical description above).
When two photons enter the beam splitter, one on each side, the state of the two modes becomes where we used
The result is non-classical: a classical light wave entering a classical beam splitter with the same transfer matrix would always exit in arm c due to destructive interference in arm d, whereas the quantum result is random.
For a more general treatment of the beam splitter with arbitrary reflection/transmission coefficients, and arbitrary numbers of input photons, see the general quantum mechanical treatment of a beamsplitter for the resulting output Fock state.
Customarily the Hong–Ou–Mandel effect is observed using two photodetectors monitoring the output modes of the beam splitter.
The coincidence rate of the detectors will drop to zero when the identical input photons overlap perfectly in time.
The precise shape of the dip is directly related to the power spectrum of the single-photon wave packet and is therefore determined by the physical process of the source.
Common shapes of the HOM dip are Gaussian and Lorentzian.
A classical analogue to the HOM effect occurs when two coherent states (e.g. laser beams) interfere at the beamsplitter.
If the states have a rapidly varying phase difference (i.e. faster than the integration time of the detectors) then a dip will be observed in the coincidence rate equal to one half the average coincidence count at long delays.
(Nevertheless, it can be further reduced with a proper discriminating trigger level applied to the signal.)
The Hong–Ou–Mandel effect can be directly observed using single-photon-sensitive intensified cameras.
Such cameras have the ability to register single photons as bright spots clearly distinguished from the low-noise background.
[3] In most cases, they appear grouped in pairs either on the left or right side, corresponding to two output ports of a beam splitter.
Occasionally a coincidence event occurs, manifesting a residual distinguishability between the photons.
[5] In 2006, an experiment was performed in which two atoms independently emitted a single photon each.
In 2015 the Hong–Ou–Mandel effect for photons was directly observed with spatial resolution using an sCMOS camera with an image intensifier.
[8] The HOM effect can be used to measure the biphoton wave function from a spontaneous four-wave mixing process.
[11] Topological photonics have intrinsically high-coherence, and unlike other quantum processor approaches, do not require strong magnetic fields and operate at room temperature.
