Hong-Ou-Mandel effect

The Hong-Ou-Mandel effect is a two-photon interference effect in quantum optics. The effect was demonstrated experimentally by Hong, Ou, and Mandel [ [http://prola.aps.org/abstract/PRL/v59/i18/p2044_1 C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).] ] in 1987. The effect occurs when two identical photons enter a 50:50 beam splitter. When the temporal overlap of the photons on the beam splitter is perfect, the two photons will always exit together in the same output mode, which is chosen randomly. Since this effect relies on the existence of photons it can not be fully explained by classical optics.

The effect provides one of the underlying physical mechanisms for logic gates in linear optical quantum computation [ [http://www.nature.com/nature/journal/v409/n6816/abs/409046a0.html E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001).] ] (the other mechanism being the action of measurement).

Quantum-mechanical description

Physical description

When a photon enters a beam splitter, there are two possibilities: it will either be reflected or transmitted. The relative probabilities of transmission and reflection are determined by the reflectivity of the beam splitter. Here, we assume a 50:50 beam splitter, in which a photon has equal probability of being reflected and transmitted.

Next, consider two photons, one in each input mode of a 50:50 beam splitter (see figure 1). There are four possibilities for the photons to behave: 1) The photon coming in from above is reflected and the photon coming in from below is transmitted; 2) Both photons are transmitted; 3) Both photons are reflected; 4) The photon coming in from above is transmitted and the photon coming in from below is reflected. We assume now that the two photons are identical in their physical properties (i.e., polarization, spatio-temporal mode structure, and frequency). Since the state of the beam splitter does not "record" which of the four possibilities actually happens, Feynman's rule dictates that we have to add all four possibilities at the amplitude level. In addition, reflection off the bottom side of the beam splitter introduces a relative phase shift of -1 in the associated term in the superposition. Since the two photons are identical, we cannot distinguish between the output states of possibilities 2 and 3 in figure 1, and their relative minus sign ensures that these two terms cancel. This can be interpreted as destructive interference.

Mathematical description

Consider two optical modes "a" and "b" that carry creation and annihilation operators $hat\{a\}$, $hat\{a\}^\{dagger\}$, and $hat\{b\}$, $hat\{b\}^\{dagger\}$. Two identical photons in different modes can be described by the Fock states

$hat\{a\}^\{dagger\}\; hat\{b\}^\{dagger\}\; |0,\; 0\; angle\_\{ab\}\; =\; |\; 1,\; 1\; angle\_\{ab\}$,

where $|1\; angle$ is a single-photon state. When the two modes "a" and "b" are mixed in a 50:50 beam splitter, they turn into new modes "c" and "d", and the creation and annihilation operators transform accordingly:

$hat\{a\}^\{dagger\}\; ightarrow\; frac\{hat\{c\}^\{dagger\}\; +\; hat\{d\}^\{dagger\{sqrt\{2\; quad\; ext\{and\}quad\; hat\{b\}^\{dagger\}\; ightarrow\; frac\{hat\{c\}^\{dagger\}\; -\; hat\{d\}^\{dagger\{sqrt\{2$.

The relative minus sign appears because the beam splitter is a unitary transformation. This can be seen most clearly when we write the two-mode beam splitter transformation in matrix form:

Unitarity of the transformation now means unitarity of the matrix, which requires that the two column vectors in the matrix are orthogonal.

Physically, this beam splitter transformation means that reflection off one surface induces a relative phase shift of -1 with respect to reflection off the other side of the beam splitter. Similar transformations hold for the annihilation operators.

When two photons enter the beam splitter, one on each side, the state of the two modes becomes

$|\; 1,\; 1\; angle\_\{ab\}\; =\; hat\{a\}^\{dagger\}\; hat\{b\}^\{dagger\}\; |0,\; 0\; angle\_\{ab\}\; ightarrow\; frac\{1\}\{2\}\; left(\; hat\{c\}^\{dagger\}\; +\; hat\{d\}^\{dagger\}\; ight)\; left(\; hat\{c\}^\{dagger\}\; -\; hat\{d\}^\{dagger\}\; ight)\; |0,0\; angle\_\{cd\}\; =\; frac\{1\}\{2\}\; left(\; hat\{c\}^\{dagger\; 2\}\; -\; hat\{d\}^\{dagger\; 2\}\; ight)\; |0,0\; angle\_\{cd\}\; =\; frac\{|2,0\; angle\_\{cd\}\; -\; |0,2\; angle\_\{cd\{sqrt\{2$.

Therefore, when two identical photons enter a 50:50 beam splitter, they will always exit the beam splitter in the same, randomly chosen output mode.

Experimental signature

When two photodetectors monitor 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. This is called the "Hong-Ou-Mandel dip", or HOM dip, shown in figure 2. The coincidence count reaches a minimum, indicated by the dotted line in figure 2. The minimum drops to zero when the two photons are perfectly identical in all properties. When the two photons are perfectly distinguishable, the dip completely disappears. The precise shape of the dip is directly related to 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 analog 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. Consequently, to prove this is two-photon quantum interference the HOM dip must be lower than one half.

Applications and experiments

The Hong-Ou-Mandel effect can be used to test the degree of indistinguishability of the two photons involved. When the HOM dip in figure 2 reaches all the way down to zero coincidence counts, the incoming photons are prefectly indistinguishable, whereas if there is no dip the photons are distinguishable. In 2002, the Hong-Ou-Mandel effect was used to demonstrate the purity of a solid-state single-photon source by feeding two successive photons from the source into a 50:50 beam splitter. [ [http://www.nature.com/nature/journal/v419/n6907/abs/nature01086.html C. Santori et al. Nature 419, 594 (2002)] ] The interference visibility "V" of the dip is related to the states of the two photons $ho\_a$ and $ho\_b$ by:

$V\; =\; Trleft(\; ho\_\{a\}\; ho\_\{b\}\; ight)\; .$

If $ho\_a=\; ho\_b=\; ho$ then the visibility is equal to the purity $P=Tr(\; ho^2)$ of the photons. In 2006, an experiment was performed in which two atoms independently emitted a single photon. These photons subsequently produced the Hong-Ou-Mandel effect. [ [http://www.nature.com/nature/journal/v440/n7085/abs/nature04628.html J. Beugnon et al., Nature 440, 779 (2006)] ]

The Hong-Ou-Mandel effect also underlies the basic entangling mechanism in linear optical quantum computing, and the two-photon quantum state $|2,0\; angle\; +\; |0,2\; angle$ that leads to the HOM dip is the simplest non-trivial state in a class called NOON states.

References