1. Introduction

In recent years two-photon interference has been used to demonstrate violations of Bell’s inequalities [1], teleportation [2], multi-photon entanglement [3, 4], entanglement swapping [5] and entanglement purification [6]. All of these experiments have relied upon spontaneous parametric down-conversion to generate pairs of identical or entangled photons. Excitation with short optical pulses ensures that the photons are created at the same time and narrow-band filters can be used to make the photons time-bandwidth limited. However, the semi-classical nature of the down-conversion process means that these sources often generate more than one photon pair and these unwanted events limit the intensity of the source and lead to errors [6].

In contrast, single quantum dots are able to generate single photons with up to 50-fold suppression of multi-photon emission [7, 8]. This suppression is particularly important if linear optical quantum computing schemes based on two-photon interference are to be viable [9]. Inhomogeneous broadening and dephasing of the excitonic states witin the quantum dots are major obstacles to this application. Despite this, two-photon interference has been observed by using the Purcell effect to reduce the radiative lifetime of the exciton ground state and generate photons near the time bandwidth limit [10]. This has also allowed the generation of entangled photon pairs in the coincidence basis and teleportation of photon states [11, 12].

Here we describe a set of experiments on photons emitted from single quantum dots (QDs) in pillar microcavities. A series of single-photon intereference experiments enables us to study the decoherence processes occuring in the QDs under a range of excitation conditions. The longest coherence times are observed with weak, quasi-resonant excitation and it is under these conditions that we perform a two-photon interference experiment. We find that the magnitude of this interference effect is most strongly affected by the recombination dynamics within the dot itself. At high laser powers, we unavoidably populate the long-lived biexciton state of the QDs, thereby extending the time over which the exciton photon emission occurs and generating photons that are temporally distinguishable.

2. Description of samples

The sources consisted of QDs within 1.9 μm diameter circular pillar microcavities with quality factors, Q, of ~ 2000–3000. Details of the growth, processing and characterisation of these sources can be found elsewhere [8]. The pillars had larger mode volumes and higher Q factors than other pillars that have been previously reported [10, 13, 14] but display comparable photon collection efficiencies and Purcell factors [8]. Micro-photoluminescence (PL) was excited from the sources at 4 K using a tuneable pico-second pulsed laser operating at 80 MHz. A microscope objective with numerical aperture 0.5 was used to excite and collect emission from the sample along an axis perpendicular to the sample surface. The results presented here were taken from a pillar that displayed a bright narrow emission line within the HE ₁₁ cavity mode, the intensity of which varied linearly with excitation power below a saturation level. This emission was attributed to exciton emission (X) of a single QD. Photo-luminescence excitation measurements allowed us to identify a resonance at 908.15 nm which was used to excite the QD throughout this work. Several weaker resonances were identified in the 890–920 nm spectral range which may be due to either excited states of this or other QDs. At wavelengths between these absorption resonances the X emission intensity did not fall to zero suggesting that photo-excitation and carrier transfer to the QD is still possible [15].

 

Fig. 1. (a) Schematic diagram of the fibre-based interfereometer used in the experiment. (b) Histogram of time between coincidence counts in the two-photon interference experiment when overlap of the photon wave-packets from pillar B is maximised. (c) Measurement of opposite output probability, P_opp, as the time delay between the excitation of the two photons is varied. As a guide to the eye we have fitted this data with a function [0.5 - 0.33exp(- |∆τ|/τ_decay)].

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3. Two-photon interference

With this pillar we have performed a two-photon interference experiment by exciting pairs of photons with two laser pulses separated by 1.96 ns ± ∆τ. After spectral filtering, the photons were passed to a polarization-maintaining fibre Mach-Zehnder interferometer where one arm had a fixed delay of 1.96 ns. There was then a finite probability that the two photons were incident on the final 2×2 coupler of the fibre-interferometer, from opposite directions, at the same time. Fourth order interference between the photon wave-functions means that, if the photons were indistinguishable, two detectors placed at the output ports of this coupler would not detect a coincidence count [16].

The experimental data was recorded as a histogram showing the number of cross-counts occurring between the detectors versus the time delay (Fig. 1(a)). An opposite output probability, P_opp, describing the probability of the two coincident photons leaving the final coupler in opposite directions, is then given by the area of the peak at time zero divided by the sum of the areas of the peaks at + 2 ns and - 2 ns [10]. By adjusting the time delay between the median photon arrival times, ∆τ, we were then able to vary the probability of the two photons arriving simultaneously at the final coupler. In this case we see a characteristic dip in P_opp at ∆τ=0 (Fig. 1(b)). When the two photons are totally distinguishable, for instance if the difference between their arrival times at the final coupler is greater than their coherence times, there will be a 50% probability of them being detected at the two detectors simultaneously, in which case P_opp is 0.5.

4. Time-resolved and single-photon interference measurements

The data shown in Fig. 1 was recorded at a point where the intensity of the X state was half its saturated value. Obviously, it would be desirable to be able to generate a single photon from the X state for every laser pulse and therefore we have investigated the power dependence of this source. Shown in Fig. 2 as solid data points are the results of a series of measurements of P_opp taken with ∆τ = 0 at different excitation powers. As can be seen there is a strong variation in P_opp as a function of the laser excitation power. In order to understand these results direct measurements of the length of the photon wavepacket and the temporal jitter on the time of photon emission were made.

Firstly, we shall discuss the data in Fig. 3(a) relating to the measurement of the coherence times, τ_coh, of photons emitted by the source. The experimental apparatus for this single-photon interference experiment consisted of a non-polarizing coupler and two retro-reflectors arranged as a Michelson interferometer in free space. By mounting one retro-reflector on a variable delay line the difference between the two path lengths could be varied. The output of the interferometer was passed to a monochromator where the signal was measured with a CCD for each delay line position. Care was taken to integrate the intensity over a spectral range several times the width of each emission line. In this experiment a homogeneously broadened lorentzian emission line of width ∆E will give rise to an interferogram with a visibility decaying exponentially away from zero path length difference with a time-constant τ_coh = 2ħ/∆E[17, 18]. Data recorded at 50% of the saturated X intensity is shown in the insert to Fig. 3(a). An exponential fitted to this data gives a coherence time of τ_coh = 122 ps.

Repeating the single-photon interference mesurement at different excitation densities reveals that the τ_coh of the X photons falls with increasing excitation density. At the lowest excitation density investigated the probability of generating an X photon was 11% and we observe a τ_coh of 155 ± 8 ps. At the highest excitation density investigated, when the intensity of the X emission line has saturated, we observe a τ_coh of 118 ± 4 ps.

Some insight into this decoherence process can be gained through comparing these coherence times with those recorded under non-resonant excitation at 770 nm. Under conditions where the probability of generating an X photon is comparable we observe coherence times that are nearly 50% larger under quasi-resonant excitation than under non-resonant excitation. In this case the increase in τ_coh may be attributed to the elimination of carriers in the wetting-layer and GaAs band-edge states [17, 19]. But the decrease in τ_coh seen with progressively higher excitation densities under quasi-resonant excitation must be due to a reduction in the proportion of carriers in other QDs or in the background of states that exist at the same wavelength as the X state. These states, possibly defect states in the semiconductor, have been reported in photo-luminescence excitation experiments [15, 20]. In pillars that do not have QDs emitting near HE ₁₁ cavity mode this spectrally continuous emission allows us to measure the optical mode structure [8]. The background states that emit at 940 nm have a radiative lifetime of 600 ps so there will be carriers present, which can cause decoherence, when the X radiatively decays.

 

Fig. 2. Measured (solid symbols) opposite output probability, P_opp, as a function of excitation power. Also shown, as open symbols, is the predicted P_opp determined using independent measurements of the photon coherence times and the radiative decay time of the emission.

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Time-resolved measurements of the X photon emission are shown in the insert to Fig. 3(b). From these traces we have extracted a decay time, τ_decay, by fitting a single exponential decay to the data (Fig. 3(b), main panel). A significant increase in τ_decay is observed as we move to higher excitation densities. This is due to the fact that at higher excitation densities biexcitons are formed in the QD and it is the lifetime of the biexciton state (which is not efficiently coupled to the cavity) which dominates the jitter on the X photon emission time. Qualitaively, this variation in τ_decay with excitation power is the dominant factor in reducing the visibility of the two-photon interference effect (Fig. 2). At higher excitation denisties this increase in the jitter on the photon emission time greatly exceeds the extent of the photon wavepackets ensuring it is less likely that two photons arrive at the final coupler at the same time and interfere.

5. Comparison with theory

It is possible to predict P_opp for a given set of source parameters using equations similar to those quoted by Santori et al [10]. This analysis requires knowledge of a parameter g which is the probability of either pulse from the source containing more than one photon divided by the probability that each pulse delivers a single photon. This parameter, approximately 0.02 for the data in Fig. 1, increases P_opp. For simplicity, we have calculated a lower bound on P_opp for the case when g = 0 (Fig. 2, open symbols). Surprisingly, the values measured in the experiment are below this lower bound for the higher excitation powers. This suggests that a fluctuation in the X photoluminescence wavelength may be occurring on a timescale longer than 2 ns, possibly due to some charge fluctuation in the vicinity of the dot [10]. Indeed, the envelope of the single-photon interferogram of this exciton line displayed some Gaussian component (insert to Fig. 3(a)), as would be expected if the line were inhomogeneously broadened. At the lowest excitation power the measured and predicted P_opp values show a much closer agreement suggesting the process that causes this spectral shift is power dependent and less significant at lower excitation densities.

 

Fig. 3. (a) Measurements of the coherence time, τ_coh, as a function of laser excitation density. The insert shows the single-photon interferogram when the source is at 50% of its saturated intensity. (b) shows the best-fit decay time of the time-resolved photoluminscence measurement as a function of laser power. The insert to (b) shows the decay traces at three powers corresponding to 11, 50 and 100% of the saturated exciton intensity.

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This raises the question: what is the lowest P_opp we might expect to be able to observe in this system? If our couplers were perfectly balanced this would decrease the minimum P_opp we have measured from 0.169 to 0.165. More important is the fact that the finite jitter on the photon emission time is larger than the τ_coh of the photons. The parameter often cited to define how close the source is to generating time-bandwidth limited photons is R = 2τ_decay/τ_coh [10, 19, 21]. The value of R is equal to 1 when the uncertainty on the detection times of photons is entirely due to the quantum mechanical uncertainty in their wavepacket. For the pillar studied here R values down to 1.5 were observed. Decreasing the ratio R from 1.5 to 1.1 would give P_opp = 0.051. It may be feasible to reach such low R values if we are able to reduce the radiative lifetime further through optimization of the cavity or if we could increase τ_coh by reducing the number of carriers in the vicinity of the QD.

However, it may be possible to improve the visibility of the effect by improving the detection system we are using. The equations used to predict the peak areas in the auto-correlation histogram assume the system time-resolution is much greater than τ_coh/2. This is a valid assumption for our auto-correlation experiment which has a time resolution of 850 ps. In fact, as only photons arriving at the coupler within τ_coh/2 of each another can interfere the shape of the central peak will be changed. Figure 4 shows (dotted line) the predicted auto-correlation histogram that would be measured if the system time-resolution were infinitely fast and the photons were distinguishable. The solid line in Fig. 4 shows the histogram recorded when the photons are perfectly indistinguishable. If two photons can arrive at the coupler separated by a time ∆τ’ , then P_opp will be multiplied by a factor [1-exp(-2| ∆τ’|/∆τ_coh)] reducing the coincidence count rate near ∆τ’ = 0. A faster time-resolution detection system may enable us to post-select only those coincidences that occur within a time scale below τ_coh/2. A logical result of this is that it should be possible to observe a high visibility two-photon interference effect with a source delivering photons a long way from the time-bandwidth limit, if the time-response of the detection system is much less than τ_coh/2.

 

Fig. 4. Modelled auto-correlation histogram for distinguishable (dotted line) and indistinguishable (solid line) photons that would be recorded with an ideal detection system where τ_decay = 230 ps and τ_coh = 122 ps.

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6. Conclusion

In conclusion, we have presented experimental data on the temporal coherence, inhomogeneous broadening and interference of photons emitted from an on demand single-photon source. By varying the intensity of the quasi-resonant excitation it was possible to change how far the created photons were from the time bandwidth limit.