Interference of dissimilar photon sources
A. J. Bennett, R. B. Patel, C. A. Nicoll, D. A. Ritchie, A. J. Shields
aa r X i v : . [ qu a n t - ph ] J un Interference of dissimilar photon sources
A. J. Bennett, ∗ R. B. Patel,
1, 2
C. A. Nicoll, D. A. Ritchie, and A. J. Shields Toshiba Research Europe Limited, Cambridge Research Laboratory,208 Science Park, Milton Road, Cambridge, CB4 OGZ, U. K. Cavendish Laboratory, Cambridge University,JJ Thomson Avenue, Cambridge, CB3 0HE, U. K. (Dated: October 31, 2018)
Abstract
If identical photons meet at a semi-transparent mirror they appear to leave in the same direc-tion, an effect called “two-photon interference”. It has been known for some time that this effectshould occur for photons generated by dissimilar sources with no common history, provided themeasurement cannot distinguish between the photons[1]. Here we report a technique to observesuch interference with isolated, unsynchronized sources whose coherence times differ by severalorders of magnitude. In an experiment we interfere photons generated via different physical pro-cesses, with different photon statistics. One source is stimulated emission from a tuneable laser,which has Poissonian statistics and a neV bandwidth. The other is spontaneous emission from aquantum dot in a p-i-n diode [2, 16] with a µeV linewidth. We develop a theory to explain thevisibility of interference, which is primarily limited by the timing resolution of our detectors.
PACS numbers: 78.67.-n, 85.35.Ds ∗ Electronic address: [email protected] h ψ a | ψ a i = exp ( − α ) P n a α na n a ! h n a | n a i and for the quantum light source h ψ b | ψ b i = (1 − η ) h b | b i + η h b | b i . In our notation η and α are proportional to the proba-2 ource Asource B D1D2 g ( ) τ ( τ ) b g (2) (cid:65) (0) g (2)// (0) (cid:3)(cid:75)(cid:18)(cid:68) Source bSource a a FIG. 1: Interference of a single-photon source with a weak Poissonian source. (a) Shows a schematicof the experiment. (b) Shows the predicted correlation function at zero time delay, g (2) (0) forparallel (black) and orthogonal (red) photons as a function of the intensity ratio of the sources, η/α . Shown in blue is the resulting visibility of two-photon interference, ( g (2) ⊥ (0) − g (2) k (0)) /g (2) ⊥ (0). bilities of detecting either a photon from the anti-bunched source or the laser, respectively,at the output of the experiment. For simplicity we only expand the coherent state up to n a = 2, which is valid for a strongly attenuated laser α ≪
1. If the sources do not interferethere are a number of ways the detectors can collect one photon each: either they collectboth photons from the laser (with probability 2
RT α /
4) or they collect one photon fromthe laser and one from the single photon source. This second possibility can occur if bothphotons are reflected (with probability proportional to R ηα /
2) or if they are both trans-mitted ( T ηα / R = T . These joint-detection probabil-ities g (2) (0) are normalized to the probability of detecting two photons at different times,( η + α ) . Equation 1a gives the probability of a detecting a photon in both outputs atthe same time when the sources have parallel polarisations and are indistinguishable. Auseful control measurement is to measure data with the sources orthogonally polarised, so3he photons are entirely distinguishable and no interference occurs (1b). g (2) k (0) = (cid:16) ηα (cid:17) − (1a) g (2) ⊥ (0) = (cid:18) ηα (cid:19) (cid:16) ηα (cid:17) − (1b)Fig. 1 (b) plots g (2) ⊥ (0) and g (2) k (0) versus the ratio of source intensities, η/α . For g (2) k (0), allcoincidence counts at time delay zero are due to multi-photon emission from the laser, whichfalls as η/α increases. Using this simple analysis we can make the surprising predictionthat the visibility of two-photon interference, ( g (2) ⊥ (0) − g (2) k (0)) /g (2) ⊥ (0), can approach unityas η/α increases.We now determine the coherence properties of the two sources used in our experiment.Single-photon interference measurements were carried out using a free space Michelson inter-ferometer with a variable time delay [17] (Fig. 2 (a)). The interference pattern as a functionof delay is measured using an avalanche photodiode (D1). Looking at emission from the X − state of the quantum dot source [16] on its own we see that the interference has a fringecontrast which varies as A exp ( −| t | /τ coh ) where A is the fringe contrast at zero delay, t thedelay time and τ coh the coherence time. For the dot studied here τ coh = 285 ps at 100 µA .This characteristic exponential variation in contrast is an indication that the state has aLorentzian line-shape in energy of width 4.4 µeV . Thus we can be sure that homogeneousprocesses dominate the line broadening mechanisms [18]. We note that the maximum fringecontrast observed at zero time delay, A , is below unity due to the finite spatial overlap oflight that travels along the two arms of the interferometer. Separately, we have shown thecoherence time of this source is sufficient to post-select interference events between successivephotons emitted by the source. The other source we employ is an external cavity solid-statelaser diode which can be tuned several hundred µeV using a piezo-electric actuator. Thissource has a coherence time of 1 µs , which is three orders of magnitude longer than we areable to probe with our Michelson interferometer. Thus, the fringe contrast is constant at A over the range of delays we probe.In our experiment, to obtain a finite visibility of two-photon interference these sourcesmust have the same energy to within the sum of their linewidths [10]. However, our spec-trometer and CCD system only have a spectral resolution ∼ µeV . Hence, to ensurespectral indistinguishability we employ a scheme based on single-photon interference, thelayout of which is shown in Fig. 2(a). Clearly the two photons have orthogonal polarisation4 C on t r a s t/ A Piezo offset (V)-1000 0 1000-1000100 Time delay (ps) E ne r g y s p li tt i ng ( (cid:80) e V ) b c SpectralFilterPBSH VV 1nF50 (cid:58)
Laserdiode D1Variabledelay a FIG. 2: Measurements of single-photon interference. (a) Layout for the experiment (b) Predictedfringe contrast as a function of the energy difference between the sources and the delay in theMichelson interferometer. (c) Experimental fringe contrast for the dot and laser signals combinedat 380 ps delay as the bias applied to the piezo-electric stack tunes the laser wavelength (black).Also, shown are the fringe contrasts of the laser (red) and dot (blue) at the same delay, measuredseparately. Error bars represent standard deviations determined from least-squares fits to the data. at the detectors and so will not give rise to two-photon interference. However, the single-photon interference patterns of the separate sources have a period given by their wavelength.Thus, when both sources are detected at the same time we observe “beating” in the intensityat the detector. The period of the beats is inversely proportional to the energy differencebetween the states. We have developed a simple model to illustrate how the single-photoninterference fringe contrast, normalized to A , varies with both the energy difference be-tween the sources and interferometer delay (Fig. 2 (b)). We consider only the case wherethe sources appear to have the same intensity on the detectors.In practice, we set the interferometer delay to a fixed value and measure the fringecontrast as a function of the piezo-voltage applied to the laser, which results in a near-linear variation in laser energy. As can be seen in Fig. 2(c), the fringe contrast variescosinusoidally as a function of the energy splitting between the two sources. For a delay of380ps the period of the cosine variation is 34 µ eV. With a least squares fit to the experimentaldata we can ensure degeneracy with an error estimated below 1 µ eV, less than the line-widthof the broader source. Using this method we have experimentally verified that the sources’wavelengths remain stable within the accuracy of this measurement over 24 hours.5 (cid:73)(cid:3)(cid:32)(cid:3)(cid:19) g ( ) H B T ( (cid:87) ) c Time (ns) g ( ) (cid:73) ( (cid:87) ) b ed (cid:73)(cid:3)(cid:32)(cid:3)(cid:83)(cid:18)(cid:21) a HVH V SpectralFilter PBSPBS HWPV 1nF50 (cid:58)
Laser diode D D C B C A FIG. 3: Measurements of photon statistics. (a) experimental layout for two-photon interferencebetween the sources. Intensity correlation functions, recorded for (b) the quantum light sourceonly, (c) the laser only, with both sources having (d) orthogonal and (e) parallel polarisations.These plots show the measured data (black), predicted correlations for infinitely fast detectors(blue) and for the measured detectors’ response function (red).
We are now able to perform the two-photon interference experiment using the apparatusin Fig. 3a. The co-linear and oppositely polarised photons are passed to an interferometermade of polarisation-maintaining optic fibre. The first, polarising, coupler C A ensures everyphoton from the dot takes the upper path to the final non-polarising, 50 /
50 coupler C B andevery photon from the laser takes the lower path. This design increases the probability ofthe two photons reaching the final coupler from opposite sides by a factor of four, relative toprevious experiments [9]. Correlations at the outputs of C B are measured with two siliconavalanche photodiodes (APDs). A half-wave plate in the path taken by the laser photonswitches its polarisation between being parallel and orthogonal to the quantum dot’s photonevery few minutes. This allows us to build up the correlations for the case where the photonsare and are not interfering within the same integration time. Thus any slow drift in the6osition of the sources, or the fibres, which might change the ratio of their intensities atthe detectors, is averaged out between pairs of measurements. During the course of eachmeasurement the ratio of intensities is stable to within 5 %.Fig. 3(d) and (e) shows experimental data recorded for equal intensity sources. Themeasurement of g (2) ⊥ ( τ ) shows a dip at time-zero due to the anti-bunched nature of thequantum light source, as expected. More strikingly, we can see a clear difference betweenthe measurements for parallel and orthogonal polarisations, which is a result of two-photoninterference. This finite visibility constitutes the main result of our experiment and occursdue to interference between photons from the weak laser and the anti-bunched source despitetheir different linewidths and lack of common history.To further quantify this result a full analysis, including non-ideal source parameters,allows us to calculate the correlation as a function of time (equation 2). g (2) φ ( τ ) = R f ( τ ) ⊗ " ηα (1 − γ cos ( φ ) exp( −| τ | τ coh )) + ( η g (2) HBT ( τ ) + α )( η + α ) (2)Where φ is the angle between the polarisations of the two photons, γ = h ψ a | ψ b i a measure ofthe overlap of the two photon’s wave-functions, R f the detection system response functionand R = T. We note that in the case where g (2) HBT (0) = 0, γ =1 and τ = 0 this reverts to theform given in equations 1a and 1b, as expected. In this experiment η and α are of the orderof 10 − . Separately we measure the photon statistics of our sources using a Hanbury-Brownand Twiss (HBT) arrangement (equivalent to Fig. 3 (a) with only one source operationalat a time). For the laser the auto-correlation function g (2) HBT ( τ ) =1 (Fig. 3 (c)) , as wouldbe expected for a photon source with Poissonian statistics. For the QD source we expectanti-bunched emission with a dip at time zero. We can predict the precise shape of thisauto-correlation [18, 19] using the independently measured radiative lifetime (985ps), thecontribution from background and dark counts (which sum to 0.04 of the signal from thequantum state) and the resolution of our detection system (a Gaussian with width 428ps).This model suggests g (2) HBT (0) = 0 .
19, consistent with our experimental measurement (Fig. 3(b)). From these parameters we calculate g (2) ⊥ ( τ ) and g (2) k ( τ ) for equal intensity sources beingmixed. Shown as blue lines in Fig. 3 (d) and (e) are the correlations that would be observedfor infinitely fast detectors. However, when the response function of the detection system isincluded we obtain the curves shown in red. The only free parameter is the wave-functionoverlap γ = 0.91. 7 V i s i b ili t y (cid:3) (cid:75) (cid:18) (cid:68) FIG. 4: Measurement of the visibility of two-photon interference as a function of η/α . Includedas a solid line is a fit using the known detector response time and γ = 91 %. Error bars representstandard deviations determined from least-squares fits to the data. Finally, a series of measurements were made of the visibility of interference as a functionof the intensity ratio of the two sources, η/α . Fig. 4 shows this data and our prediction for γ = 0.91. Our theory predicts a maximum visibility will be observed for η/α ∼
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