CAPITULO IX DISPOSICIONES FINALES
CALIDAD, DERECHOS Y DEBERES DE LOS ASOCIADOS
A source of photons is a prerequisite for performing quantum optics experiments. Lasers and thermal sources such as filament emission lamps provide light generated through non-deterministic processes comparable to the way that particles of radiation are emitted randomly by atoms in a sample of radioactive material. The number of photons n emitted by a source in a certain time frame will follow a statistical distribution related to the Poisson distribution which describes the expected frequency of discrete random events. A Poissonian photon source emitting an average of ¯n = 1 photons per unit time will have a statistical variance of (—n)2 = 1 and standard deviation of —n = 1 meaning at
many times the source will emit 0 or 2, 3, etc. photons simultaneously with the average over a long time tending towards ¯n = 1 (see Figure 2.3). Measurements of the emission from sources such as lasers confirms the Poissonian distribution of photons whereas thermal light sources tend to fluctuate over longer time scales creating broader so called super-Poissonian distributions for n.
Linear optical quantum computing (LOQC) as described in Section 2.3.2 can involve encoding a qubit on a single photon with the choice of two optical modes (e.g. two waveguide paths). To reliably prepare qubit states for an optical quantum logic gate it must be possible to determine when such a single photon state has been created [48], however measurement of a quit state necessarily involves loss of information [49, 50]. The reliable outcomes of quantum logic gates are dependent on receiving the correct incoming photon state, therefore variance in photon number from the source will directly increase the error rate further downstream in a quantum information processor. This creates a requirement for SPSs capable of reliably generating n = 1 photon states with minimal occurrences of n ”= 1 states [51, 52].
Correlated photon pairs can be generated through the process of spontaneous parametric down-conversion (SPDC) which converts 1 pump photon with energy
EP and momentum kP into two photons with conserved energy and momentum
such that
EP = Es+ Ei and
kP = ks+ ki
(2.12) where indices s, i represent the created photons referred to as signal and idler. The idler photon from a correlated pair source can be routed to an SPD to herald the generation of a pair leaving the signal photon free for use in the photonic logic gate. Down-converted photon pairs are generated at
Photon St. dev. Second order Example statistics and mean Classification correlation source
Super-Poissonian —n >Ô¯n Bunched g(2)(0) > 1 Thermal
Poissonian —n =Ô¯n Coherent (random) g(2)(0) = 1 Laser
Sub-Poissonian —n <Ô¯n Anti-bunched g(2)(0) < 1 Quantum dot
Table 2.1 Classification of photon emission by variance and correlation
random time intervals and the photon number follows a Poissonian distribution, therefore increasing the pump power necessarily casues the generation of more undesirable multi-photon states. SPDC sources must therefore be run at a very low pair generation rate and massive multiplexing is required to generate the large number of single photon states required for high data rate QIP [53]. An appealing feature of semiconductor QDs as SPSs is that they are able to reliably emit n = 1 photon states at high repetition rates without producing multi-photon states.
Fig. 2.3 Photon streams from various light sources. Dividing the streams into equal time bins (”t) illustrates that mean photon number ¯n can be the same for each source whereas variance (—n)2 can vary based on the emission pattern. [54]
SPSs are typically characterised by the first and second order correlation functions g(1)(⌧) and g(2)(⌧). First order correlation can be measured from
the self-interference pattern of a photon in a Michelson interferometer. The coherence time (or length) of a SPS describes how long the light emitted has a predictable phase. Constant phase allows for the visibility of interference fringes in a Michelson interferometer. Any shift in frequency or discontinuity in phase will cause the visibility of those fringes to wash out. The 1/e decay time
of the visibility of interference fringes is known as the first order correlation function (or the coherence time) of a light source.
A Hanbury-Brown Twiss (HBT) interferometer can be used to measure the second order correlation of a photon source. In this case g(2)(⌧) indicates
the statistical distribution of photons emitted from the source (see table 2.1) [55–57]. For an ideal SPS g(2)(0) = 0 implying that no multi-photon states
are ever emitted. Light emitted with these characteristics at steady time intervals is known as anti-bunched. By routing the stream of photons from an SPS through a 50:50 beam-splitter to two similar SPDs so called coincidence counts can be recorded when both SPDs trigger simultaneously. A variable time delay is introduced to the optical path leading to one detector and the number of coincidence counts measured as a function of the time difference ⌧ between the two paths. The absence of multi-photon states means that at ⌧= 0 the number of coincidence counts will drop to zero as shown in Figure 2.4. Recorded data from correlation measurements on SPSs are shown in Figure 2.4. In both cases semiconductor QDs are used as a source of single photons and Photo-luminescence (PL) is stimulated by pumping the QD with laser light. For pulsed pump light QDs ideally emit 1 photon per cycle and therefore the correlation data consists of a comb of peaks spaced by 1/f where f is the pump frequency. The peak at ⌧ = 0 is missing or at least suppressed to < 0.5 times the average peak height dependent on DCR and non-zero multi-photon emission probability. QDs pumped by a continuous wave (CW) laser will display a dip in the otherwise continuous coincidence rate at ⌧ = 0.
The indistinguishability of two photons can also be tested with a 50:50 beam-splitter and two SPDs. For two photons Ï1,2 arriving simultaneously at
the two input ports of the beam-splitter the four possible outcomes are shown in Figure 2.5. If the two photons are identical the probability amplitudes of outcomes (i) and (ii) sum to zero meaning that photons will always appear as pairs at one or other output ports of the beam splitter. A Hong–Ou–Mandel (HOM) measurement involves recording coincidence counts with SPDs placed at the outputs of the beam-splitter. By varying the arrival time of photons introduced to one input port of the beam-splitter the distinguishability of the photons will be tuned. If both photons are otherwise indistinguishable in terms of wavelength and phase the measurement will reveal a coincidence rate of zero for ⌧ = 0, a so called HOM dip as shown in Figure 2.5 (a) and (b).
Although the development of SPSs and SPDs are closely linked it can be necessary to benchmark one against the other. The current state of the art
Fig. 2.4 (a) An HBT measurement of a semiconductor QD under pulsed excitation will display a comb of coincidence peaks at the frequency of the pump laser. At time difference ⌧ = 0 there are zero coincidence counts if the QD emits only 1 photon per pulse. Detector dark counts or non-ideal QDs can create finite coincidences at ⌧ = 0. (b) Performing the same measurement with a CW pump laser creates a continuous
coincidence rate with a characteristic dip at ⌧ = 0. Data measured at the University of Sheffield by Dr. Nicola Prtljaga, adapted from reference 58.
in photonic technology makes it often easier to obtain a robust single photon detector than an efficient source of anti-bunched indistinguishable photons.
Fig. 2.5 (i–iv) Two photons incident simultaneously at a 50:50 beam-splitter can leave in one of four configurations. For indistinguishable photons the probabilities of configurations (i) and (ii) cancel out meaning the photons will always leave the beam-splitter as a pair on the same side. (a) A HOM measurement shows a dip in coincidence counts measured from the beam-splitter outputs as the incident photons are tuned into and out of distinguishability by moving the beam-splitter position. The solid line represents the ideal case whereas the experimental data are fitted to the dashed line which has a dip 0.9 times the depth. Data adapted from reference 59.