Photon correlations for colloidal nanocrystals and their clusters
O. A. Shcherbina, G. A. Shcherbina, M. Manceau, S. Vezzoli, L. Carbone, M. De Vittorio, A. Bramati, E. Giacobino, M. V. Chekhova, G. Leuchs
aa r X i v : . [ qu a n t - ph ] D ec Photon correlations for colloidal nanocrystals and their clusters
O. A. Shcherbina , G. A. Shcherbina , M. Manceau , S. Vezzoli , L. Carbone , M. De Vittorio ,A. Bramati , E. Giacobino , M. V. Chekhova , , , ∗ , G. Leuchs , M.V.Lomonosov Moscow State University, 119992 GSP-2 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow Reg., Russia Laboratoire Kastler Brossel, Universit´e Pierre et Marie Curie, Ecole Normale Sup´erieure, CNRS, 4 place Jussieu, 75252Paris Cedex 05, France National Nanotechnology Laboratory - Istituto Nanoscienze CNR, Via Arnesano 16 - 73100 Lecce, Italy Max-Planck Institute for the Science of Light, G.-Scharowsky Str 1/Bldg 24, 91058, Erlangen, Germany Universit¨at Erlangen-N¨urnberg, Staudtstraße 7/B2, 91058 Erlangen, Germany ∗ Corresponding author: [email protected]
Compiled March 20, 2018Images of semiconductor ‘dot in rods’ and their small clusters are studied by measuring the second-ordercorrelation function with a spatially resolving ICCD camera. This measurement allows one to distinguishbetween a single dot and a cluster and, to a certain extent, to estimate the number of dots in a cluster. A moreadvanced measurement is proposed, based on higher-order correlations, enabling more accurate determinationof the number of dots in a small cluster. Nonclassical features of the light emitted by such a cluster areanalyzed. c (cid:13)
OCIS codes:
Many quantum information protocols, such as quan-tum key distribution, linear optical quantum computa-tion, and quantum metrology, require on-demand single-photon sources [1, 2]. Among the sources available nowin the laboratory, there are atoms [3], molecules [4], ni-trogen vacancies in diamond [5], and epitaxial semicon-ductor quantum dots [6]. Colloidal quantum dots [7],despite their disadvantages such as blinking [8], bleach-ing [9], and a noticeable probability of two-photon emis-sion, are still one of the most promising types of single-photon emitters as they can be operated at room tem-perature and can be synthesized relatively easily. Espe-cially promising are dot-in-rods (DR) [10], which havea higher probability of single-photon emission, show re-duced blinking [11], and emit single photons with a highdegree of polarization [12].DRs can also merge into clusters. A cluster of DRs isnot a single-photon emitter but it can be used for quan-tum information purposes since it still emits nonclassicallight. In this work we perform measurements that allowone to distinguish between a single DR and a cluster.As a criterion, we use Glauber’s second-order correlationfunction (CF) and the brightness of emission. Anothermethod of distinguishing between a single DR and a clus-ter is defocused microscope technique [12], based on thedipole-like angular distribution of the DR emission; how-ever, this method only allows one to recognize clustersconsisting of differently oriented DRs. As an extensionof our correlation measurement, we propose the study ofhigher-order correlation functions. Such a measurementwill allow one to resolve the number of DRs in a clusterand also observe nonclassical behavior of its emission.For our experiment we used an intensified CCD(ICCD) camera. This device is equivalent to an array of single-photon detectors and can be used for spatiallyresolved measurements of Glauber’s CFs. Alternatively,the pixels can be used for photon-number, rather thanspatial, resolution, so that higher-order CFs can be meas-ured. The ICCD camera can be used in either analog orphoton-counting mode. In the photon-counting mode,used in our experiment, a certain level of the dimen-sionless readout signal S (proportional to the integralelectric charge acquired in a pixel) is chosen as a thresh-old, and any signal exceeding this value is interpretedas a single-photon event in a corresponding pixel. If thethreshold S th is taken too high, the resulting quantumefficiency becomes low, which fortunately does not affectthe normalized CF measured in our experiment [13]. Ifthe threshold is too low, the noise is increased. Laser405 nm OIODM ICCDIFLDRLaser405 nm OIODM ICCDIFLDRBS
Fig. 1. The experimental setup.The experimental setup is shown in Fig. 1. CdSe/CdSDRs with 4 . − mol/l and coated onto a glass substrate.After drying, the sample was placed over an NA1 . µ s. There was a possibility of excitation through thesame objective IO as well as through an NA0.75 objec-tive (O) placed on top of the sample. This way a largernumber of DRs could be excited. The excitation rate waschosen to be close to the saturation level.The emission from the DRs, centered at 630 nm, wascollected by the immersion objective and sent to theregistration part of the system by means of a dichroicmirror (DM). Afterwards, the beam was split in two bya beamsplitter (BS), and both beams were directed tothe Princeton Instruments PI-MAX3:1024i ICCD cam-era (ICCD) at a small angle. The path length differencebetween the beams was 20 cm, corresponding to a timedelay far below the smallest time scale in the experiment(10 ns). A lens (L) with the focal length 30 cm was placedin front of the camera so that the photocathode was inits focal plane, producing two images of a group of DRsand their clusters (parts A and B of the image shown inFig. 2, left). An interference filter with a width of 40 nmand a central wavelength of 650 nm was placed in frontof the camera to reduce the contribution of stray light.The ICCD camera was gated synchronously with thelaser pulses, and the gate width could be varied between10 ns and 40 ns. In order to optimize the single-photondetection, binning of pixels was performed (every groupof 4x4 pixels was joined into a single ‘superpixel’) and thereadout threshold was chosen. For this, an area arounda single DR was selected and the number of single-photon counts over this area (‘the signal’) was comparedwith the corresponding number for an identical emptyarea (‘the noise’). The threshold value S th = 685 corre-sponded to the largest signal-to noise ratio. However, inexperiment this value was slightly changed (within therange 685 < S th < frames was acquired. Because of the limited gaterate, the whole data acquisition took many hours. Theunavoidable small displacements of the images duringthis time (due to small temperature variations and themechanical vibrations) were taken into account by tak-ing a long frame after every 10 standard frames. These‘control’ frames allowed us to trace the displacement ofthe DR images and to make appropriate corrections tothe coordinates of pixels chosen in each frame.Five datasets were taken, with the gate times 10, 15,20, 30, and 40 ns. For each dataset, the mean number ofsingle-photon events per ‘superpixel’ per frame was lessthan 0 .
1. The probability of having two photons takenfor one was therefore negligible. Second-order CF at zero delay and displacement, g (2) (0 , g (2) (0 ,
0) = h N A N B ih N A ih N B i , (1)where the angular brackets denote averaging over allframes and N A,B are numbers of single-photon eventsfor an area associated with a given object, either a DRor a cluster, in the fields
A, B ( N A,B = 0 , B was calculated taking into accountthe average number of counts h N A,B i per gate, the gatetime T g , and the intensity I of the excitation beam atthe position of the object: B ≡ h N A i T g ˜ I . (2)Here, ˜ I = αI and α is a normalization factor, the samefor all measurements, which has been chosen in such away that most frequent objects have a value of B closeto 1. The distribution of the objects with respect to thebrightness value B is shown in Fig. 2, right. One can seethat, despite the low total number of objects in the im-age, several groups can be seen: roughly, with B < . . < B < .
5, and 2 . < B <
4. The comparison withthe CF measurements below will show that these groupsvery likely relate to a single DR and to clusters of 2 and3 DRs, respectively. Due to the mechanical vibrations in AB nu m be r o f ob j e c t s B Fig. 2. Left: an image of several DRs and their clusters.Right: brightness distribution for the objects registeredby the ICCD camera. The first three groups of bars areassociated with single DRs and 2-DR and 3-DR clusters,respectively.the system as well as to the bleaching and blinking ofthe DRs, the brightness of each object changed duringthe data acquisition time. This was taken into accountby normalizing the resulting CF g (2) (0 ,
0) of each objectto its value at a time delay of T = 30 ms, g (2) ( T, g (2) (0 ,
0) were also corrected taking into account thelevel of the noise, for which g (2) (0 ,
0) = 1.The results of g (2) measurement with the gate time T g = 10ns are shown in Fig. 3. The measurement er-ror is estimated from the number of registered single-2hoton events. For all objects, no anti-bunching is ob-served at low threshold values (Fig. 3a). This is becauseat low threshold, the noise is too high and its Poissonianstatistics masks the anti-bunching. At higher thresholdvalues, we see that an object that most probably is a sin-gle DR ( B = 0 .
74) manifests significant anti-bunching: g (2) = 0 . ± .
1. However, this value is relatively highbecause of the large gate width ( T g = 10 ns). The effect
650 660 670 680 690 7000.650.700.750.800.850.900.951.001.051.10 B=4.91 g ( ) ( ) S th abc
650 660 670 680 690 7000.00.20.40.60.81.01.2 B=0.74 g ( ) ( ) S th
650 660 670 680 690 7000.30.40.50.60.70.80.91.01.1 B=2.35 g ( ) ( ) S th Fig. 3. The bunching parameter measured for individualobjects in Fig. 2 versus the threshold value: (a) a ‘dimobject’, B = 0 .
74; (b) a ‘medium object’, B = 2 .
35; (c)a ‘bright object’, B = 4 .
91. The gate width is 10 ns.Dashed lines: theoretical predictions for 1,2, and 4 dots.of the averaging over T g can be described as follows. Forcw excitation [11, 12], the CF depends on the times ofthe first and second photons registration t , t as g (2) ( t , t ) = 1 − (1 − p ) exp[ − k | t − t | ] , (3)where k is the decay rate and p the probability of two-photon emission. In order to determine these parameters, g (2) ( t − t ) was measured in a different setup, with asingle DR excited at the saturation level and the emissionregistered by two avalanche photodiodes followed by acoincidence circuit. The results (Fig. 4, top) show that k = 0 . − and p = 0 .
22 if the noise contributioninto g (2) (0) is eliminated. Integration of Eq. (3) in t , t within the limits from 0 to T g yields g (2) int ( T g ) = 1 − (1 − p ) { kT g + 2( kT g ) (exp[ − kT g ] − } . (4) From this expression, the expected value of the bunch-ing parameter measured for a single DR with the gatetime 10 ns is 0 .
43 (dashed line in Fig. 3), which is inagreement with the measured value. For a brighter ob-
T , ns g g ( ) i n t g ( ) t -t , ns Fig. 4. Top: g (2) ( t − t ) dependence obtained by coin-cidence measurement with two avalanche photodiodes.Bottom: integrated bunching parameter g (2) int averagedover all ‘dim’ objects ( B < .
25) as a function of thegate (integration) time T g . Solid line: fit using (4).ject ( B = 2 . g (2) measured at optimal thresholdvalues is 0 . ± . g (2) in the presence of m independent contributions(modes) [14], g (2) m = 1 + g (2)1 − m , (5)where g (2)1 is the value for a single emitter. For m = 2and g (2)1 = 0 .
43, we get g (2)2 = 0 .
72, in agreement withthe data in Fig. 3b. Finally, Fig. 3c shows the bunch-ing parameter for an object with B = 4 .
91, for whichwe assume m = 4. This gives the theoretical prediction g (2)4 = 0 .
86, again in good agreement with the measure-ment (0 . ± . B < .
25 and which are therefore identified as singleDRs. The threshold was chosen to be 693. We see thatthe value of the bunching parameter grows with the gate3idth. The dependence is well fit with Eq. (4). The er-rors take into account both the number of single-photonevents and the spread of the data obtained for differ-ent DRs. In total, 10 DRs contributed into the plot, butonly 3 of them, for T g = 10ns, 2 for T g = 15ns, 2 for T g = 20ns, and 3 for T g = 40ns, as the statistical errorwas too large for the rest.It is also interesting to consider the measurement ofhigher-order CFs for a DR cluster. Indeed, for a cluster of N DRs, the N th-order CF will be nonzero, g ( N ) (0 , = 0while the next-order one will show the analog of anti-bunching, g ( N +1) (0 , ≈
0. This is because a clusterof N single-photon emitters cannot emit more than N photons within the lifetime. From such measurements,one can get more information about the number of DRsin a cluster. Note that here, it is not required that allDRs emit photons into the same radiation mode. Evenfor a product state of N photons simultaneously emittedinto different modes, the nonclassicality condition [15] g ( N − g ( N +1) < [ g ( N ) ] (6)will be satisfied.Moreover, if all DRs in the cluster emit photons intothe same mode, even with small (but equal) probabil-ities, another nonclassical feature can be observed. In-deed, the cluster emits then a Fock state | N i = 1 √ N ! ( a † ) N | vac i . (7)In the presence of losses leading to a finite detection effi-ciency η , one can account for the latter using the beam-splitter model, with the transmission coefficient t ≡ √ η and the reflection coefficient r ≡ √ − η . Then the lightstate becomes | Ψ i = N X i =0 √ N ! p i !( N − i )! t i r N − i | i i| N − i i , (8)and the probability to register k photons is p k = N ! k !( N − k )! η k (1 − η ) N − k . (9)Regardless of the detection efficiency, this probabil-ity distribution satisfies another nonclassicality condi-tion [15], k + 1 k p k +1 p k − p k < , (10)since for the state (8) k + 1 k p k +1 p k − p k = N − kN − k + 1 . (11)(Note that for a Poissonian state, both (11) and (6) willgive 1.) The nonclassical behavior will be especially no-ticeable for k close to N .In conclusion, we have observed spatially resolved im-ages of several single-photon emitters (‘dot-in-rods’) and their clusters. Using a single-photon photodetector array(ICCD camera), we were able to measure the bunch-ing parameter for each individual object and to distin-guish a single emitter from a cluster of such. By assum-ing that the brightness of an object scales as the num-ber of DRs in it, we identified two-dot, three-dot, andfour-dot clusters. This assumption is confirmed by theresults of the correlation function measurement. Finally,we propose the measurement of higher-order correlationsas a method for a better determination of the number ofsingle-photon emitters in a cluster.This work was supported by ERA-Net.RUS (projectNanoquint) and by the Russian Foundation for BasicResearch, grant 12-02-00965. O. A. S. acknowledges sup-port from the Dynasty Foundation. We are grateful toM. Sondermann and V. Salakhutdinov for helpful dis-cussions. References
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