# Quantum dot single-photon source

A quantum dot single-photon source is based on a single quantum dot placed in an optical microcavity. It is an on-demand single photon source. A laser pulse can excite a pair of carriers known as an exciton in the quantum dot. The decay of a single exciton due to spontaneous emission leads to the emission of a single photon. Due to interactions between excitons, the emission when the quantum dot contains a single exciton is energetically distinct from that when the quantum dot contains more than one exciton. Therefore, by suitable energy filtering, the quantum dot becomes a nonclassical light source that emits photons one by one and thus shows photon antibunching. The emission of single photons can be proven by measuring the second order intensity correlation function. The linewidth of the emitted photons can be reduced by using distributed Bragg reflectors (DBR’s). Additionally, DBR's lead to an emission in a well-defined direction.

## History

With the growing interest in quantum information science since the beginning of the 21st century, research in different kinds of single photon sources was growing. Early single-photon sources such as heralded photon sources[1] that were first reported in 1985 are based on non-deterministic processes. Quantum dot single-photon sources are on-demand. A single photon source based on a quantum dot in a microdisk structure was reported on in 2000.[2] Sources were subsequently embedded in different structures such as photonic crystals[3] or pillars.[4] The addition of DBR's allowed emission in a well defined direction and increased emission efficiency.[5] Quantum dot single photon sources need to work at cryogenic temperatures, which is still a technical challenge.[5]

## Theory of realizing a single-photon source

Figure 1: Schematic structure of an optical microcavity with a single quantum dot placed between two layers of DBR's. This structure works as a single photon source.

Single photons are extracted out of a semiconductor by spontaneous emission from the decay of a single excitation. Inside the cavity spontaneous emission is increased due to the Purcell effect.[5] The challenge in making in a single photon source is to make sure that there is only one excited state in the system at a time. To do that, a quantum dot is placed in a microcavity (Fig. 1). A quantum dot has discrete energy levels. An excitation from its ground state to an excited state will create an exciton. The eventual decay of this exciton due to spontaneous emission will result in the emission of a single photon. DBR’s are placed in the cavity to achieve a well-defined spatial mode and to reduce linewidth broadening due to the lifetime ${\displaystyle \Delta t}$ of the excited state (see Fig. 2).

Figure 2: The decay of a linewidth broadened excited state results in the emission of a photon of frequency ħω. The linewidth broadening is a result of the finite lifetime of the excited state.

The system can then be approximated by the Jaynes-Cummings model. In this model, the quantum dot only interacts with one single mode of the optical cavity. The frequency of the optical mode is well defined. This makes the photons indistinguishable if their polarization is aligned by a polarizer. The solution of the Jaynes-Cummings Hamiltonian is a vacuum Rabi oscillation. A vacuum Rabi oscillation of a photon interacting with an exciton is known as a exciton-polariton.

To eliminate the probability of the simultaneous emission of two photons it has to be made sure that there can only be one exciton in the cavity at one time. The discrete energy states in a quantum dot allow only one excitation. Additionally, the Rydberg blockade prevents the excitation of two excitons at the same space...[6] The electromagnetic interaction with the already existing exciton changes the energy for creating another exciton at the same space sightly. If the energy of the pump laser is turned on resonance, the second exciton cannot be created. Still, there is a small probability of having two excitations in the quantum dot at the same time. Two excitons confined in a small volume are called biexcitons. They interact with each other and thus slightly change their energy. Photons resulting from the decay of biexcitons have a different energy than photons resulting from the decay of excitons. They can be filtered out by letting the outgoing beam pass an optical filter.[7] The quantum dots can be excited both electrically and optically.[5] For optical pumping, a pulsed laser can be used for excitation of the quantum dots. In order to have the highest probability of creating an exciton, the pump laser is tuned on resonance.[8] This resembles a ${\displaystyle \pi }$-pulse on the Bloch sphere. However, this way the emitted photons have the same frequency as the pump laser. A polarizer is needed to distinguish between them.[8] As the direction of polarization of the photons from the cavity is random, half of the emitted photons are blocked by this filter.

## Experimental realization

A microcavity with only a single quantum dot in it is built. The DBR’s can be grown by molecular beam epitaxy (MBE). For the mirrors two materials with different indices of refraction are grown in alternate order. Their lattice parameters should match to prevent strain. A possible combination is a combination of aluminum arsenide and gallium arsenide-layers.[8] A material with smaller band gap is used to grow the quantum dot. In the first few atomic layers of growing this material, the lattice constant will match that of the DBR. A tensile strain appears. At a certain thickness, the energy of the strain becomes too big and the layer contracts to grow with its own lattice constant. At this point, quantum dots have formed naturally. The second layer of DBR’s can now be grown on top of the layer with the quantum dots.

The diameter of the pillar is only a few microns wide. To prevent the optical mode from exiting the cavity the micropillar must act as a waveguide. Semiconductors usually have relatively high indices of refraction about n≅3.[9] Therefore, their extraction cone is small. On a smooth surface the micropillar works as an almost perfect waveguide. However losses increase with roughness of the walls and decreasing diameter of the micropillar.[10]

The edges thus must be as smooth as possible to minimize losses. This can be achieved by structuring the sample with Electron beam lithography and processing the pillars with reactive ion etching.[7]

### Verification of emission of single photons

Single photon sources exhibit antibunching. As photons are emitted one at a time, the probability of seeing two photons at the same time for an ideal source is 0. To verify the antibunching of a light source, one can measure the autocorrelation function ${\displaystyle g^{(2)}(\tau )}$. A photon source is antibunched if ${\displaystyle g^{(2)}(0)}$${\displaystyle g^{(2)}(\tau )}$.[11] For an ideal single photon source, ${\displaystyle g^{(2)}(0)=0}$. Experimentally, ${\displaystyle g^{(2)}(\tau )}$ is measured using the Hanbury Brown and Twiss effect. Devices experimentally exhibit values between ${\displaystyle g^{(2)}(0)=0.009}$[8] and ${\displaystyle g^{(2)}(0)=0.024}$[12] at cryogenic temperatures.

### Indistinguishability of the emitted photons

For applications the photons emitted by a single photon source must be indistinguishable. The theoretical solution of the Jaynes-Cummings Hamiltonian is a well-defined mode in which only the polarization is random. After aligning the polarization of the photons, their indistinguishability can be measured. For that, the Hong-Ou-Mandel effect is used. Two photons of the source are prepared so that they enter a 50:50 beam splitter at the same time from the two different input channels. A detector is placed on both exits of the beam splitter. Coincidences between the two detectors are measured. If the photons are indistinguishable, no coincidences should occur.[13] Experimentally, almost perfect indistinguishability is found.[12][8]

## Applications

Single-photon sources are of great importance in quantum communication science. They can be used for truly random number generators.[5] Single photons entering a beam splitter exhibit inherent quantum indeterminacy. Random numbers are used extensively in simulations using the Monte Carlo method.

Furthermore, single photon sources are essential in quantum cryptography. The BB84[14] scheme is a provable secure quantum key distribution scheme. It works with a light source that perfectly emits only one photon at a time. Due to the no-cloning theorem,[15] no eavesdropping can happen without being noticed. The use of quantum randomness while writing the key prevents any patterns in the key that can be used to decipher the code.

Apart from that, single photon sources can be used to test some fundamental properties of quantum field theory.[1]

## References

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