Hong-ou-almond effect

The Hong-Ou-Mandel effect (often HOM effect for short ) is the quantum mechanical interference between two photons . It is named after its discoverers CK Hong, Zhe-Yu Ou and Leonard Mandel (1987).

The effect occurs when two indistinguishable photons each hit an input of a 50:50 beam splitter . It can be observed experimentally that the photons are always found in pairs at one of the two outputs. The case that a photon can be found in each of the two outputs does not occur.

Quantum mechanical description

The four possibilities for two photons on a beam splitter.

The HOM effect is a non-classical effect and can only be understood in the context of quantum mechanics. The state of the system after the interference results from the superposition of all possible paths that the photons can take through the beam splitter. These four possibilities are shown in the figure. In cases 1 and 4, one photon is transmitted while the other is reflected. In cases 2 and 3, both photons are transmitted or reflected. In the case of identical particles, processes 2 and 3 cannot be distinguished from one another (only the final state can be observed and it is the same in both cases). In addition, the probability amplitudes of the two paths are the same except for the sign . They therefore interfere destructively and only cases 1 and 4 can occur. The origin of the signs can be clearly explained with the help of the phase jump of 180 ° known from classical physics when reflecting on the denser medium.

Mathematical description

In quantum mechanics, light is no longer described by the classical electric field amplitudes, but by creation and annihilation operators. The coupling of the modes at the two inputs and to the output modes and through the beam splitter is described by the unitary time evolution operator : ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a'}$ ${\displaystyle b'}$

${\displaystyle U=\operatorname {e} ^{{\frac {i}{2}}\theta (ab^{\dagger }+a^{\dagger }b)}}$

The "angle of rotation" describes the splitting ratio of the beam splitter. For the 50:50 beam splitter assumed below, is . ${\displaystyle \theta }$ ${\displaystyle \theta =\pi /2}$

In the Schrödinger picture , the initial state is then obtained by multiplying the expansion operator with the initial state. The input state is represented by the ket , which describes that one photon is present at each input of the beam splitter. ${\displaystyle |1,1\rangle }$

${\displaystyle U|1,1\rangle ={\frac {1}{2}}\left(a^{\dagger }a^{\dagger }+a^{\dagger }b^{\dagger }-b^{\dagger }a^{\dagger }-b^{\dagger }b^{\dagger }\right)|0,0\rangle ={\frac {1}{\sqrt {2}}}\left(|2,0\rangle -|0,2\rangle \right)}$

The four terms in the equation correspond exactly to the four possibilities shown in the figure for the photons to fly through the beam splitter. Since, according to the assumption, the photons are indistinguishable, the operators and swap , so that the terms are proportional to and cancel each other and no longer appear in the final state. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a^{\dagger }b^{\dagger }}$ ${\displaystyle b^{\dagger }a^{\dagger }}$

Experimental observation

In order to demonstrate the Hong-Ou-Mandel effect, the outputs of the beam splitter are observed with photo multipliers and a coincidence measurement is carried out. The non-occurrence of temporal coincidences between the detectors shows that the photons always leave the beam splitter together at the same exit.

Web links

• QuantumLab Interactive experiment on Hong-Ou-Almond interference

Individual evidence

1. ↑ CK Hong, ZY Ou, L. Mandel: Measurement of subpicosecond time intervals between two photons by interference . In: Physical Review Letters . tape 59 , no. 18 , October 2, 1987, pp. 2044 , doi : 10.1103 / PhysRevLett.59.2044 .