In analogy to the classical information science, which uses the binary bits 0 and 1 to store information, the basic entity of quantum information is the qubit. The basic states to form an orthogonal basis of the qubit space are called computational bases. Considering a two-state system, for example, if the computational bases are and , one possible pure qubit can be described as , where a and b are complex coefficients and they respect the normalization condition . In the classical information theory, a bit must be either 0 or 1, whereas in quantum theory, a qubit can be in both the and state at the same time. Such condition represents the coherent superposition of and . In the attempt of measuring the state of a qubit, one would obtain either or , with a probability of or , respectively. The coherent superposition and the probabilistic nature are the most distinct characters of qubits compared to classical binary bits. To date, qubits have been realized in several real physical systems. Figure 1 reports a list of some physical realizations of qubits, where the bases have been chosen by convention. Despite the great research efforts, most of these systems require rigorous operating conditions, such as ultra-low temperatures and almost adiabatic environments. At the present stage, photons are ideal candidates for many QIP tasks (e.g., quantum communication), given their weak interaction with the environment, the long coherent time, the fact that they travel at the speed of light and allow a multiple degree-of-freedom encoding.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Common physical realizations of qubits.[16]

The photon, whose modern concept was developed by Albert Einstein, is a fundamental particle exhibiting the wave-particle duality. In other words, photons are self-oscillating short electromagnetic waves and particles with spin 1 at the same time. Photons can be used as flying qubits in QIP, due to their unique properties mentioned above. In particular, a single photon possesses many degrees of freedom (e.g., spin angular momentum, orbital angular momentum, arrival time, frequency, and so on), which can be used to encode a coherent superposition of states. Figure 2 shows the Bloch sphere representations of three commonly used qubit encoding schemes.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Bloch sphere representations of (a) polarization qubit, (b) orbital angular momentum qubit, and (c) time-bin qubit.[17]

If we describe the photon as an electromagnetic wave, its polarization represents the oscillation direction of the electric field. In this viewpoint, the circular polarization is the spin angular momentum of the photon, with the left-handed and right-handed circular polarizations corresponding to +h and −h spin state, respectively. As shown in Fig. 2(a), each pair of opposite vertices on the Bloch sphere represents one pair of orthonormal bases. and denote the horizontal and vertical polarizations, respectively. Therefore, and denote the diagonal and circular polarization bases, respectively. The pure states are on the Bloch sphere, whereas the mixed states are in the interior of the Bloch sphere. The polarization encoding of photon qubits is one of the most mature techniques used in modern quantum optics.

The orbital angular momentum (OAM) of photons, which represents the spatial distribution of the electromagnetic field, is associated with the helical or twisted phase front. The schematic is shown in Fig. 3.^[18] Unlike the spin angular momentum of photons, which has only two computational bases, the OAM can assume any arbitrary integer value, and thus, offer a unique possibility for a high-dimensional encoding. One pair of orthonormal bases is shown in Fig. 2(b), where l denotes the angular quantum number. Ever since the first proposal,^[19] significant efforts have been made to use the OAM encoding in QIP.^[18,20] In recent years, one of the most striking results is the realization of a spin-OAM hyper-entanglement for single photons.^[21,22]

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Helical phase fronts of OAMs with different indexes.[18]

The time-bin encoding is based on the arrival time of a single photon. A detailed experimental configuration was firstly proposed by Brendel.^[23] A photon is sent into a two-path setup, then the quantum state of the photon can be described by or (also see Fig. 2(c)). Here, to ensure the unambiguous distinguishability of the two paths, the difference between the lengths of the two paths should be larger than the coherent length of the photon. The key point is to use a variable coupler and an unbalanced Mach–Zehnder interferometer (phase shifter) in one path to prepare and measure the qubits.^[24] The time-bin encoding greatly simplifies the realizations of two-photon quantum cryptography, quantum teleportation, dense coding, entanglement swapping, and GHZ-states sources.^[23] Furthermore, due to the robustness against the polarization mode dispersion, time-bin qubits are suitable for applications based on fiber optics. However, the environment insensitivity of time-bin qubits makes it difficult to perform quantum interferences. Building up an effective interference scheme is the goal to achieve.

The quantum entanglement describes a nonlocal property of quantum correlations between multiple systems. It can be described mathematically in an elegant way. Here, we refer to a two-body system for example. Let us consider two physical subsystems A and B, whose respective Hilbert spaces are and , and and are the bases of and , respectively. The quantum state of the composite system AB can be described as

Semiconductor QDs are zero-dimensional nanostructures, and their optical and electrical properties can be artificially tailored by shape, size and compositions. Due to the strong quantum confinement, excitons-related effects are dominating in QDs. Such feature makes semiconductor QDs eligible for application opportunities in quantum optics.

The various exciton states in a single III–V group semiconductor QD are shown in Fig. 8. The biexciton–exciton cascade process is used for the generation of polarization-entangled photon pairs, and the entanglement quality is definitely influenced by the QDs. In the next section, we will briefly review the optimization endeavors of QDs for entangled photon emitters.

As far as the III–V group semiconductor QDs concern, here we mainly focus on the InAs- and GaAs-based ones. In this context, the anisotropy of strain, composition and shape reduces the symmetry of QDs to C_2v or even lower C₁. Considering the exchange interaction, there will be an energetic splitting between the two bright excitons, which is called fine-structure splitting (FSS), as shown in Fig. 8. More specifically, when the FSS appears in QDs, the two-photon polarization state generated by the biexciton–exciton cascade process should be described by

Fig. 9.

	Figure Option ViewDownloadNew Window
	Fig. 9. (color online) Symmetry analysis for four different point groups. The single-particle levels for the highest occupied and lowest unoccupied states are indicated by solid black lines. The ensuing irreducible representation of bright and dark excitons are indicated by thick red and dashed black lines, respectively.[46]

More systematically, by constructing an effective two-band model based on the two bright excitons, Gong et al. claimed that the lower bound of FSS under uniaxial stress can be predicted by the polarization angle, and that both the FSS under zero stress and the critical stress can be determined by monitoring the changes in the polarization angle of the exciton.^[47] The FSS is related to the uniaxial stress according to a parabolic function; therefore, it is possible that the FSS cannot go down to zero under a mere uniaxial stress. This provides a feasible method to select QDs for entangled-photon sources.

Because semiconductor QDs are used as entangled-photon sources, the community has made great efforts to eliminate FSS. An ideal quantum light source, which can be excited either optically or electrically, should possess good properties of brightness, entanglement fidelity, indistinguishability, coherence and on-demand (tunable wavelength and repetition frequency) at the same time. For semiconductor QD-based polarization-entangled photon sources (PEPS), there are three main tasks to be carried out: the manipulation of FSS; the optimization of the excitation condition; and the rational design of optical structures. In recent years, we have witnessed great success in this field.

The FSS mainly originates from the structure asymmetry; therefore its manipulation via the growth control of QDs is straightforward. However, typical InAs QDs grown on GaAs (001) crystal plane using the Stranski–Krastanow method suffer from strain and alloy intermixing, as well as shape elongation.^[48,49] Hence, the symmetry of this type of QDs is lower than C_2v. In 2009, two theoretical articles^[50,51] predicted that InAs QDs on GaAs (111) crystal plane would have a symmetry higher than C_3v, hence a small FSS. The results of the calculation of the wave function symmetry are shown in Figs. 10(a) and 10(b) (see Ref. [50]). However, it has been proven that the direct growth of InAs QDs on the GaAs (111) crystal plane is difficult. In 2010, Mohan et al. realized a site-controlled pyramidal QDs grown on tetrahedral recesses patterned (111)B GaAs substrates, which exhibited a C_3v symmetry.^[52] In 2013, Juska et al. reported on the (111)-grown pyramidal site-controlled InGaAs_1−xN_x QDs and they demonstrated that the 15% of their QDs have an entanglement fidelity greater than 0.5.^[53] Figures 10(d) and 10(e) show the morphologies of these QDs.

Fig. 10.

Figure Option ViewDownloadNew Window

Fig. 10. (color online) (a) and (b) Orientation of electron (blue) and hole (yellow) wave functions for a lens-shaped QD on two substrates with different orientations.[50] (c) Atomic force microscope analysis of the sample surface in the work by Kuroda et al.[54] (d) Scheme of the pyramidal QD structure.[52] (e) Profile of the pyramidal InGaAs1−xNx QDs (the upper figure is the atomic force microscope results, the lower figure shows the SEM result of the sample after removing the substrate).[53]

GaAs/aluminum gallium arsenide (AlGaAs) QDs are an interesting alternative to the conventional InAs/GaAs QDs.^[49,55] In 2013, Kuroda et al. used a droplet epitaxy method to realize self-assembled GaAs QDs on (111)A substrates^[54] (see Fig. 10(c)). These QDs are highly symmetric in their shape. It was demonstrated that the 5% of QDs show no detectable FSS and an entanglement fidelity of up to 0.86. In 2004, the same group also reported on the growth of InAs/indium aluminum arsenide (InAlAs) QDs on a C_3v InP (111)A plane, with a vanishing-small FSS and emissions in the telecom band.^[56]

Another emerging possibility to obtain highly-symmetric growths of GaAs/AlGaAs QDs is to use the droplet etching and nanohole etching techniques.^[57–60] Recently, Huber et al. reported that, with preselected QDs in the sample, emissions of entangled photons can be obtained with high single-photon purity, high indistinguishability and high entanglement fidelity ( ).^[61] Meanwhile, Keil and Zopf et al. realized a large ensemble of QD-based PEPS on a wafer without any post-growth tuning.^[62] Thanks to the growth optimization, the QDs on the whole wafer present an average FSS of . Under two-photon excitations, approximately 100% of QDs can emit entangled photons with fidelity greater than 0.5; in this case, the QDs have an exciton radiative lifetime shorter than 220 ps. Moreover, a significant fraction of QDs is expected to exhibit a high fidelity (greater than 0.8) without any post-growth tuning^[62] (see Fig. 11).

Fig. 11.

	Figure Option ViewDownloadNew Window
	Fig. 11. (color online) (a) Droplet etching and nanohole infilling technique for the GaAs QDs growth. (b) Comparison of FSS between two kinds of QDs. (c) Entanglement fidelity as a function of FSS.[62]

Post-growth tuning knobs, for example, the electric field,^[63–67] magnetic field,^[68–70] strain field,^[47,71–85] thermal annealing,^[86–89] and their combinations, can effectively manipulate the FSS of QDs. For practical applications of QD-based sources, these techniques are indispensable.

4.4.1. Electric field and magnetic field tuning

The first breakthrough in electric field-controlled PEPS based on InAs QDs was reported by Bennett et al. in 2010.^[67] As shown in Fig. 12, the double AlGaAs high barriers prevent carriers from tunneling out the QDs when applying large electric fields, hence providing a large tuning scale of FSS. The variation of the exciton energy F with the external electric field can be described by , where is the permanent dipole moment in the z direction of the exciton states with H and V polarized emissions, respectively. The result is shown in Fig. 12(c).

Fig. 12.

	Figure Option ViewDownloadNew Window
	Fig. 12. (color online) (a) and (b) Device design and observed giant Stark shift of the excitonic transitions. (c) FSS of three QDs controlled by vertical electric field. (d) Entanglement fidelity of three QDs with different FSS.[67]

A similar work based on GaAs QDs was reported by Ghali et al.^[63] An n–i–Schottky diode was used to realize the electric field tuning of the QDs’ FSS. The device structure and energy band profile along the growth direction are shown in Fig. 13. It can be observed that the FSS shows a nonlinear behavior with respect to the applied electric field, unlike the linear behavior of the InAs QDs reported in Ref. [67]. The authors attributed this difference to the weak lateral confinement in the island-like GaAs QDs. Moreover, they obtained an entanglement fidelity of up to 0.72.

Fig. 13.

	Figure Option ViewDownloadNew Window
	Fig. 13. (color online) (a) Schottky-type device containing QDs. The polarization of QD exciton dipole moment is indicated by the dotted arrow. (b) z-direction energy band diagram. (c) Nonlinear dependence of FSS on external electric field. (d) Results of the entanglement fidelity.[63]

Another interesting result was reported by Muller et al.^[90] A continuous-wave laser was used to tune the QDs in the alternating current Stark limit to eliminate the FSS. It was observed that FSS decreases as the intensity of the tuning laser increases.

The effect of magnetic fields on QDs’ FSS was studied in detail by Bayer et al.^[70] The first PEPS based on magnetic field tuning was reported by Stevenson et al. in 2006.^[68] Four years later, they reported that the in-plane magnetic field can be used to tune the FSS more effectively, with a tuning coefficient of up to . In 2014, Pooley et al. reported the elimination of FSS by applying vertical electric and in-plane magnetic fields at the same time.^[91] They reported that the FSS can be reduced to its minimum value and the emission energy can be tuned over several meVs with a 5-T magnet.

4.4.2. FSS control by strain field

The electrical field tuning may introduce excessive carriers around the quantum dots and degrade the coherence of the emitting photons. Furthermore, large electrical fields can lead to quenching effects. Nevertheless, the magnetic field tuning is not suitable for practical applications due to the bulky setup. An emerging technique consists in the usage of strain fields. The early studies in this context were reported by Seidl et al.^[92] and Ding et al.^[71] In particular, they used piezoelectric lead zirconic titanate (PZT) and [Pb(Mg1/3Nb2/3)O3]0.72[PbTiO3]0.28(PMN-PT), respectively, to tune the exciton states of the self-assembled QDs. In the latter work, thin membranes containing QDs were used to significantly increase the strain tuning range. Plumhof et al. reported the influences of anisotropic strain fields on the FSS of InAs and GaAs QDs, for the first time.^[72] Trotta et al. realized the simultaneous application of large strain and electric fields, leading to the successful elimination of FSS for almost each QDs in the sample.^[75] In 2012, a strain-tunable quantum light-emitting diode (QLED) was also reported.^[74]

The first experimental demonstration of strain-tunable QLED emitting polarization-entangled photon pairs (also called ELED) was realized by Zhang et al.^[79] As shown in Fig. 14(a), an n–i–p structure, with embedded InGaAs QDs, was integrated into a PMN-PT substrate. The anisotropic in-plane strain fields provided by the PMN-PT can be used to effectively manipulate the FSS (see Fig. 14(c)). Figure 14(d) shows the real part of the density matrix of the two-photon state. The entanglement fidelity as a function of FSS is shown in Fig. 14(b). The ELED can emit entangled photons with a repetition rate of up to 400 MHz. Recently, there has been also one report on the strain-tunable single photon QLED based on GaAs/AlGaAs QDs.^[85]

Fig. 14.

	Figure Option ViewDownloadNew Window
	Fig. 14. (color online) (a) Sketch of the diode structure. The PMN-PT was cut in a way that it can exert large anisotropic strain fields on the bonded ELED. (b) Entanglement fidelity as a function of FSS. (c) Representative variation of FSS of the high-energy component of the exciton as a function of the strain tuning for five QDs. (d) Real part of the density matrix of the two-photon state.[79]

For most of the tuning techniques, it is impossible to realize the simultaneous tuning of FSS and the QD emission wavelength. As a consequence, it is difficult to realize an entanglement swapping between two separated QDs. In 2015, an in-plane strain tuning stage with three degrees of freedom, where the FSS and the exciton energy can be independently tuned, was theoretically investigated.^[82] The proposed device is shown in Fig. 15(a). Three pairs of piezoelectric strain arms are separated by 60 degrees. During the manipulation, the strains are controlled in the three different directions by the voltages V₁, V₂, and V₃, respectively. Another theoretical work by Wang et al. proposed that a three-dimensional stressor can be used for the same purpose (see Fig. 15(b)).^[81] The in-plane strain fields controlled by V_y and V_z are used to eliminate the FSS, while another PZT on top of the QDs is used to manipulate the exciton energy.

Fig. 15.

	Figure Option ViewDownloadNew Window
	Fig. 15. (color online) (a) Sketch of the strains stage providing anisotropic in-plane strain fields. Three independent voltages ( are applied across the three pairs of legs.[82] (b) Sketch of a three-dimensional strain stage to independently tune the FSS and exciton energy in QDs.[81]

In 2016, a six-legged piezoelectric device was realized experimentally by Trotta et al.^[80] Polarization-entangled photons from InGaAs QDs were interfaced with a Cs vapor cell. After slowing down the photons by the atoms, a decrease in the entanglement fidelity was observed. This work opens the door to the realization of a hybrid quantum system based on semiconductor QDs. The device structure and entanglement fidelity are shown in Figs. 16(a) and 16(b), respectively. In 2016, Chen et al. also realized an on-chip integrated wavelength tunable PEPS based on InGaAs QDs. The unique advantages of their device are the ultra-small footprint and ultra-low tuning voltage, which are important for the fabrication of advanced quantum photonic circuits for on-chip QIP applications.

Fig. 16.

Figure Option ViewDownloadNew Window

Fig. 16. (color online) (a) Sketch of the six-legged strain tuning device used to engineer a nanomembrane (grey region) containing QDs. (b) Entanglement fidelity with (blue data points) and without (red data points) the Cs cell in the exciton optical path.[80] (c) Scheme of the cross section of the four-legged strain-tuning device. A focused ion beam (FIB) and a wet-chemical undercut were used to fabricate the device. A thin GaAs nanomembrane containing In(Ga)As QDs was transferred to the suspended region between the four legs.[83] (d) Entanglement fidelity results after the FSS tuning, without any background subtraction.

To date, one of the most challenging tasks for QD-based PEPS is to enhance their brightness. Due to the high refractive index of the surrounding materials and isotropic emission of QDs, only a small portion of the photon pairs can be collected. The most common solution adopted to improve the extraction efficiency is to use the Purcell effect. When the QDs are coupled both spatially and spectrally with a cavity mode, their photon emissions are modified by

An elegant solution was proposed by Dousse et al.^[93] As shown in Fig. 17(a), two GaAs/AlGaAs micropillars, with Q factors equal to 4000 and 60000, respectively, are used to enhance the extraction efficiency of entangled photons from a single InAs QD. By precisely choosing the pillar diameters and distance, the exciton and biexciton photons can be coupled to the two cavity modes, respectively (see Fig. 17(c)). They reached a pair rate of 0.12 per excitation pulse into the first lens.

Fig. 17.

	Figure Option ViewDownloadNew Window
	Fig. 17. (color online) (a) Diagram of the device that contains two microcavities and one InAs QD. (b) Number of exciton and biexciton photons collected in each excitation pulse as a function of the excitation power. (c) The optical modes of the photonic molecule as a function of the intra-molecule distance.[93] (d) Structure of nanowire QDs.[95]

An alternative option, which also brings a broadband enhancement, is to use the nanowire QDs. Versteegh et al.^[94] reported that the nanowire waveguide has the advantages of a broad frequency bandwidth and a controlled direction, which suits for the QD-based PEPS. The light extraction efficiency of a single InAsP QD in InP nanowire could reach 18±3%. Besides, the entanglement fidelity to the maximum Bell state can be up to 0.765, without any time-gating technique. A similar work was reported by Huber et al.^[95] The device structure is shown in Fig. 17(d). It is worth noting that the nanowire QDs can also be strain tunable, as reported by Chen et al.^[96]

Semiconductor QDs can be excited either optically or electrically. The electrical injection is more promising in practical QIP, given that no sophisticated laser systems are required. The first ELED was realized by Salter et al.^[97] They showed that the device could emit polarization-entangled photon pairs under DC or AC injections, with the latter achieving an entanglement fidelity of up to 0.82. After this pioneering work, it took the community a long time to reproduce the result, simply due to the low probability of finding QDs with small enough FSS. Zhang et al. reported a strain-tunable ELED,^[79] as above mentioned. Another interesting recent work from Chung et al.^[98] is based on the realization of selectively carrier-injected and site-controlled arrays of ELED based on pyramidal QDs.

For optical excitations, QDs can be excited either resonantly or non-resonantly. For the non-resonant excitation, the carriers are excited to energy levels higher than the first excited states, and they decay to the first excited states of QDs. In this scheme, an inevitable time jitter of the emissions is created. Besides, charges in excess will form around the QDs, leading to spectral diffusions and a pure dephasing. The combination of these effects will significantly degrade both the indistinguishability and coherence of the QD emissions. Moreover, the biexciton population probability under the non-resonant excitation is quite low. In 2014, Müller et al. realized a two-photon excitation (TPE) to single QDs via a virtual state.^[99] The schematic is shown in Fig. 18(a).

Fig. 18.

Figure Option ViewDownloadNew Window

Fig. 18. (color online) (a) Energy diagram of TPE. (b) Photofluorescence spectra of the same QD, under non-resonant excitation (upper) and TPE (lower).[99] (c) Scheme of a pulse shaper to generate the chirp pulse (upper), and generation of a biexciton with broadband, chirped excitation (lower).[100] (d) Photofluorescence spectrum of a GaAs QD under phonon-assisted TPE for optimized detuning of the laser energy, pulse length ( ) and a pulse area of 6π. (e) Population of the exciton state as a function of the laser detuning for varying the excitation power.[101]

Figure 18(b) shows the photoluminescence spectra of a same QD, under either non-resonant excitation or TPE. Under TPE, the biexciton almost has the same intensity as that of the exciton. It is worth noting that the generation efficiency of the biexciton increases by a factor of 4 under resonant excitations.^[99] Their source also shows high entanglement fidelity and indistinguishability. Similar works were reported by several other groups.^[61,62]

However, TPE is sensitive to the excitation power and the QDs’ environment, and a slight perturbation may lead to dramatic variations in the population probability. Reindl et al. reported a scheme called phonon-assisted TPE.^[101] They claimed that, although the phonon-assisted TPE scheme is inherently non-resonant, the population inversion of exciton and biexciton states coupled to a quasi-continuum of vibrational modes is possible (see the inset of Fig. 18(d)). As a result, the population is robust against the detuning of the laser energy compared to traditional TPE, as shown in Fig. 18(e). Another alternative scheme is the adiabatic rapid passage (ARP), which is a resonant excitation technique. In the ARP, a broadband and chirped pulse is used to excite the QDs. According to several reports,^[100,102] the biexciton occupation is robust against the excitation power of the positive chirp. The experimental setup and excitation illustration are shown in Fig. 18(c).