We show theoretically that nearly indistinguishable photons can be generated
with non-identical semiconductor-based sources. The use of virtual Raman
transitions and the optimization of the external driving fields increases the
tolerance to spectral inhomogeneity to the meV energy range. A trade-off
emerges between photon indistinguishability and efficiency in the
photon-generation process. Linear (quadratic) dependence of the coincidence
probability within the Hong-Ou-Mandel setup is found with respect to the
dephasing (relaxation) rate in the semiconductor sources

Atature

Boozer

Carson

E. Cancellieri

F. Troiani

G. Goldoni

Goldoni

Hayasaka

Jonsson

Laurent

Manin

Nakaoka

Nemoto

Solomon

Vuckovic

Optics Express

English

arXiv

Crossref

Optimal generation of indistinguishable photons from non-identical artificial molecules

We show theoretically that nearly indistinguishable photons can be generated with non-identical semiconductor-based sources. The use of virtual Raman transitions and the optimization of the external driving fields increases the tolerance to spectral inhomogeneity to the meV energy range. A trade-off emerges between photon indistinguishability and efficiency in the photon-generation process. Linear (quadratic) dependence of the coincidence probability within the Hong-Ou-Mandel setup is found with respect to the dephasing (relaxation) rate in the semiconductor sources

Cancellieri, E.

Troiani, F.

Goldoni, G.

White Rose Research Online

This is a repository copy of Optimal generation of indistinguishable photons from non-identical artificial molecules.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/84430/Version: Accepted VersionArticle:Cancellieri, E. orcid.org/0000-0002-4099-2716, Troiani, F. and Goldoni, G. (2009) Optimal generation of indistinguishable photons from non-identical artificial molecules. Optics Express, 17 (19). pp. 17156-17163. ISSN 1094-4087 https://doi.org/10.1364/OE.17.017156eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Optimal generation of indistinguishable photonsfrom non-identical artificial moleculesEmiliano Cancellieri,1, 2 Filippo Troiani,2 and Guido Goldoni2, 31NEST CNR-INFM and Scuola Normale Superiore, Pisa, Italy2S3 CNR-INFM, Modena, Italy3Dipartimento di Fisica, Università di Modena e Reggio Emilia(Dated: December 16, 2013)We show theoretically that nearly indistinguishable photons can be generated with non-identicalsemiconductor-based sources. The use of virtual Raman transitions and the optimization of theexternal driving fields increases the tolerance to spectral inhomogeneity to the meV energy range.A trade-off emerges between photon indistinguishability and efficiency in the photon-generationprocess. Linear (quadratic) dependence of the coincidence probability within the Hong-Ou-Mandelsetup is found with respect to the dephasing (relaxation) rate in the semiconductor sources.PACS numbers: 03.67.-a, 78.67.Hc, 42.50.-pSingle-photon sources (SPSs) are fundamental devicesin quantum communication [1] and linear-optics quantumcomputation [2]. Essentially, a SPS consists of an atomic-like system that can be deterministically excited and thustriggered to emit single-photon wavepackets into a prefer-ential mode. In addition, all photons have to be emittedin the same quantum state, in order to maximize thevisibility of two-photon interference.So far, single-photon generation has been demon-strated both with atomic [3, 4, 5, 6], molecular [7, 8, 9],and solid-state systems [10, 11, 12, 13, 14, 15]. Amongstthe latter, self-assembled semiconductor quantum dots(SAQDs) [16] are particularly promising, for they can becontrollably coupled to optical microcavities [17], so asto greatly enhance both the photon emission rate andFIG. 1: (Color online) Emission scheme of indistinguishablephotons from non-identical SPSs. Each source (A/B) consistsof an AM (gray), doped with an excess electron (red/blue),and coupled to an optical MC (black). The effect on thephoton wavepackets (green) of the differences between A andB, in terms of energies and oscillator strengths of the opticaltransitions, are compensated by properly tailoring the excit-ing laser pulses (straight arrows).the collection efficiency. As a major downside, SAQDscome with a finite dispersion in terms of size, shape,and composition. This typically gives rise to spectralinhomogeneities in the meV range, i.e., orders of magnti-tude larger than the homogeneous linewidths (few µeVs)of the lowest exciton transitions at cryogenic tempera-tures. Photons spontaneously emitted by two distinctSAQDs therefore tend to be completely distinguishable:this could impede scalability and ultimately limit the po-tential of semiconductor-based SPSs.In spite of its relevance to any solid-state approach,the problem of generating indistinguishable photons withnon-identical emitters has still received limited attention.In this Letter, we assess the possibility of compensatingthe effects of such differences between two SPSs by suit-ably tuning the exciting laser pulses. To this aim, wecombine a density-matrix approach [18] – to simulate thesystem dynamics and compute the coherence functions ofthe emitted radiation – with a genetic algorithm [25] – tooptimize the external driving fields. As a crucial point,the photon generation results from a (virtual) Ramantransition. Raman transitions have already been pro-posed as means to avoid the classical uncertainty on theinitial time of the emission process (the so-called time-jitter) [19, 20] and to tune both the temporal profile andthe central frequency of the emitted wavepacket [5]. Herewe show that such flexibility can be exploited to gener-ate two nearly indistinguishable photons, in spite of thespectral differences between their respective semiconduc-tor sources.We specifically consider the case where each source isrepresented by two coherently coupled SAQDs (often re-ferred to as artificial molecule, AM), embedded in an op-tical microcavity (MC) (Fig. 1). The AM is doped withan excess electron, whose levels 1 and 2 [Fig. 2(a)] definea pseudo-spin, corresponding to the hybridized – bond-ing and antibonding – states of the two dots [21, 22];together with the lowest charged exciton states (3 and4), these form a double Λ−scheme [23]. There, a RamanarXiv:0904.0146v1  [cond-mat.mes-hall]  1 Apr 20092FIG. 2: (Color online) (a) Relevant level scheme for eachnegatively-charged AM. The levels 1 and 2 correspond to thebonding and antibonding states of the excess electron; theseare optically coupled to the two lowest charged exciton states3 and 4. The off-resonant laser pulse (ωL) induces a virtualRaman transition from 1 to 2; this process results in the cre-ation and emission of a photon from the fundamental mode ofthe MC (ωc). (b) Schematics of the Hong-Ou-Mandel setup.The photons are emitted by the two sources A and B in theinput modes of a balanced beam-splitter (BS), whose outputmodes are coupled to the photodetectors C and D.transition can be induced by an off-resonant laser pulse(ωL), that drives the AM from 1 to 2, while generating acavity photon (ωc).We simulate the dynamics of the two sources (A and B)within a density-matrix approach [18]. The state of eachAM-MC system is denoted by |j, n〉, where j = 1, . . . , 4specifies the AM eigenstate, whereas n represents the oc-cupation number of the cavity mode. The time evolutionof the density matrix within such Hilbert space is givenby ρ̇ = i[ρ, HMC + HL] + Lρ (rotating-wave approxima-tion, ~ = 1). The coupling of the AM with the MC isaccounted byHMC =∑j=1,2∑k=3,4gjk(σjka† + aσ†jk), (1)where a is the annihilation operator of the cavity photonand σjk ≡ |j〉〈k| the ladder operator acting on the AMstate. The coupling of the AM with the exciting laser isgiven byHL =12NL∑p=1∑j=1,2∑k=3,4Ωpjk(t)(e−iδpjktσjk + H.c.), (2)where Ωpjk(t)/gjk = Ωp0 exp{(t− tp0)2/2σ2p} gives the timeenvelope of the p−th laser pulse, δpjk ≡ ǫk − ǫj −ωpL, andǫj the AM levels. The interaction of the AM-MC systemwith the environment is given by the superoperatorL =∑jL(√γjPj) +∑j,kL(√Γjkσjk) + L(√κa), (3)where L(A)ρ ≡ (2AρA† − A†Aρ − ρA†A)/2 and Pj ≡|j〉〈j|. The three terms in Eq. 3 account for pure de-phasing (with rate γ), radiative (jk = 13, 14, 23, 24) andnon-radiative (jk = 12, 34) relaxation of the AM, andcavity-photon emission, respectively. In the following,we assume for simplicity that Γjk ≡ Γr for all the radia-tive relaxation processes of the AM, and Γjk ≡ Γnr forthe non-radiative ones.The degree of indistinguishability between the photonsgenerated by the two sources can be measured withinthe Hong-Ou-Mandel setup [24] [Fig. 2(b)]. There, iftwo indistinguishable photons enter the input ports (Aand B) of a balanced beam splitter, two-photon inter-ference results in a vanishing probability of a coincidenceevent (PCD = 0) in the two photodetectors at the outputmodes (C and D). For distinguishable photons, instead,PCD = 1/2. The coincidence probability can thus be re-garded as a measure of the photon indistinguishability.Formally, PCD can be connected to the dynamics of theA and B sources through [19] PCD = F1 − F2, whereFp=1,2 = κ2∫dt∫dτ fp(t, τ) andf1(t, τ) = [nA(t + τ)nB(t) + nB(t)nA(t + τ)]/4,f2(t, τ) = Re{[G(1)A (t, t + τ)]∗ G(1)B (t, t + τ)}/2.Here, G(1)χ (t, t + τ) = 〈a†χ(t)aχ(t + τ)〉 and nχ(t) =〈a†χ(t)aχ(t)〉 are, respectively, the first-order coherencefunction and cavity-mode occupation corresponding tothe sources χ = A, B.In order to maximize the overlap between thewavepackets of the photons emitted by A and B, weoptimize the driving laser pulses and the frequency ofthe cavity mode, i.e., the vector X = (Ωn0 , tn0 , σn, ωc).For two given sources [each characterized by the vectorY = (ǫj , gjk, γj ,Γjk), with YA 6= YB ], the suitability ofeach set of laser pulses for the generation of two indis-tinguishable photons is quantified by a fitness functionF(XA,XB |YA,YB) ≥ 0, defined asF = (F2/F1) g(PAe , PBe ). (4)Here, 0 ≤ F2/F1 ≤ 1 measures the degree of indistin-guishability between the two photons, whereas g accountsfor the statistics of the two SPSs; more specifically, itimposes a penalty on the vectors Xχ corresponding to aphoton-emission probability Pχe =∫nχ(t)dt far from 1.We identify the best solution (X0χ) with the vectors thatcorrespond to the maximum value of the fitness func-tion, for the given values of the physical parameters thatcharacterize the SPSs (Yχ): FM = F(X0A,X0B |YA,YB).3FIG. 3: (Color online) Maximized fitness function FM asa function of the differences between the A and B sourcesin terms of frequencies (∆AB = ωA32 − ωB32) and oscilla-tor strengths (gAB ≡ gBjk/gAjk) of the optical transitions.The remaining parameters are: Γnr = ΓA/Br = 10−3 ps−1,γA/Bj = 10−2 ps−1, κA/B = 10−1 ps−1, gAjk = 0.120meV.Having found no evidence of an increased indistinguishabil-ity arising from non-gaussian laser-pulse profiles, we set hereNL = 1.Hereafter, we take: g = θ(PAe − P te) · θ(PBe − P te), withθ the Heaviside function. Therefore, provided that bothPAe and PBe are larger than the threshold Pte , 1 − F co-incides with the coincidence probability PCD, normal-ized to the joint emission probability of the two sourcesPAB = PAe · PBe = F1. To this aim, we combine thedensity matrix approach with a genetic algorithm [25];this allows to efficiently explore, within a large parame-ter space, the particularly complex landscape induced bythe penalty function g.The distinguishability between the photons emittedby the two sources mainly arises from the differencesbetween their transition energies (ωAjk 6= ωBjk, beingωjk ≡ ǫj − ǫk) and oscillator strengths (gAjk 6= gBjk). InFig. 3 we show the dependence of the maximum fitnessfor optimized laser pulses, FM , on gAB ≡ gAjk/gBjk and∆AB = ωA32 − ωB32 [26], for a threshold emission prob-ability P te = 0.9. The fact that FM < 1 even for thecase of identical sources (gAB = 1 and ∆AB = 0) isdue to the presence of dephasing and non-radiative re-laxation (see below). By adjusting the laser pulses, mod-erate differences between the oscillator strengths of thetwo sources (gAB & 0.9) can be efficiently compensated,whereas decreases of FM arising from mismatches be-tween the optical-transition energies of A and B in themeV range can be limited to a few percent.A couple of general comments are in order. On the onehand, stimulated Raman processes do increase the toler-ance to inhomogeneities between the SPSs from the µeV– as is the case with spontaneous emission – to the meVrange. On the other hand, such inhomogeneities cannotFIG. 4: (Color online) (a) Dependence of the maximum fit-ness function FM on the dephasing rate γ (upper axis, greensquares) and on the non-radiative relaxation rate Γ21 (loweraxis, red triangles), for gAB = 1 and ∆AB = 1.25 meV. Thegreen (red) dotted line represents the best fit of FM linear(quadratic) in γ (Γ21). (b) FM as a function of the thresh-old P te in the photon-emission probability for gAB = 1, with∆AB = 0 and 1.5 meV.be completely compensated by properly tuning the fre-quencies of the driving laser pulses, as could be naivelyexpected on the basis of a simple energy-conservation re-lation. These limitations are partially due to the effectsof decoherence; besides, the requirement that the twosources emit photons with a high probability seems toconflict with that of maximizing their indistinguishabil-ity. In the following we investigate separately these twoaspects.The coupling between the carriers confined in each AMand the phonons gives rise to energy relaxation and de-phasing. These two contributions are considered sepa-rately in Fig. 4(a), where we report FM as a function ofγ (with Γ21 = 0, squares), and of Γ21 (with γ = 0, trian-gles), within realistic ranges of values for these rates. Asin the case of two photons sequentially emitted by thesame source [19], the coincidence probability increaseslinearly with γ (dotted green curve). A stronger depen-dence is found with respect to the non-radiative relax-ation between the two lowest states in the Λ−scheme (afit by a quadratic function of Γ21 is given by the dottedred line). We note, however, that the value of Γ21 can bestrongly reduced by suitably engineering the AM geom-etry and by the application of external fields [27, 28].In order to investigate the relation between the emis-sion probability of the A and B photons and their in-4FIG. 5: (Color online) (a) Dependence of F2/F1 = 1 − PCDon the shift of the cavity and laser frequencies from their op-timized values, ∆ωA/Bα ≡ ωA/Bα − ωA/B,0α , for ∆AB = 0 (red)and ∆AB = 1.5meV (gray), with gAB = 0.88. Solid, dashedand dotted lines correspond respectively to: α = c; α = L,β = A; α = L, β = B. (b) Photon-emission probabilities,P Ae (solid lines) and PBe (dotted), as a function of ∆ωc, for∆AB = 0 (red) and ∆AB = 1.5meV (gray), with gAB = 0.88.distinguishability, we have computed FM as a functionof P te . In Fig. 4(b) we report two representative cases:∆AB = 0 (orange squares) and ∆AB = 1.5 meV (graytriangles), with gAB = 1. The photon emission proba-bility and the degree of indistinguishability are clearlyanti-correlated, thus demonstrating the existence of atrade-off between the two requirements. The dependenceof FM on P te , though evident even for identical sources(∆AB = 0), is more pronounced in the presence of a spec-tral mismatch (∆AB = 1.5). The above curves are rela-tively insensitive to differences in the oscillator strengths(gAB & 0.9, not shown here).We finally comment on the robustness of the solutionwith respect to small departures of the control parame-ters from their optimal values. As a representative ex-ample, we report the dependence of F2/F1 = 1 − PCDon the cavity frequency ωc ≡ ωAc = ωBc , referred to itsoptimized value ω0c [Fig. 5 (a)]. The robustness of thephoton indistinguishability with respect to non-optimalparameters decreases for incresing spectral mismatchesbetween the two sources. Besides, our simulations showa stronger dependence of PCD on the laser (solid and dot-ted lines) than on the cavity frequencies (dashed). Thisfeature can be traced to the fact that the uncertainty onthe cavity frequency (∼ 1/κ = 10 ps) is larger than thaton the laser frequency (typical duration of the optimizedlaser pulses are few tens of ps). As to PA/Be [Fig. 5 (b)],we note that the value corresponding to the optimizedparameter (∆ωc = 0) is close to the mimimum allowedvalue (P te): this provides a further indication on the ex-istence of a trade-off between photon indistinguishabilityand emission probability. Besides, the existence of a spec-tral mismatch (gray lines) makes also the fulfilment of therequirement PAe , PBe > Pte more sensitive to the tuningof the physical parameters. The required precision is ofthe order of a few µeV for ∆AB = 1.5 meV.In conclusion, we have investigated the degree of indis-tinguishability between single photons emitted by non-identical artificial molecules. We found that the use ofvirtual Raman transitions, combined with the optimaldesign of the driving laser pulses, increases the tolerancewith respect to the spectral inhomogeneities between thesources up to the meV energy range. Defined power-lawdependences of the coincidence probability on the de-phasing and (non-radiative) relaxation rates are identi-fied. Besides, unlike the case of identical sources, a trade-off emerges between the requirements of maximizing theemission efficiency and the photon indistinguishability. Itis finally worth mentioning that the use of a pseudo-spinwithin the Λ-level scheme offers promising possibilities[31]. These include the use of quantum-confined Starkeffect [29] for triggering the photon emission [30], andthe entangling of carrier spins localized in remote (andthus non-identical) semiconductor systems [32]. The lat-ter goal requires the combination of a spin-photon entan-gling [33] process with two-photon interference within theHong-Ou-Mandel setup.We acknowledge financial support from the ItalianMIUR under FIRB Contract No. RBIN01EY74.[1] J. I. Cirac et al., Phys. Rev. Lett. 78, 3221 (1997).[2] P. Kok et al., Rev. Mod. Phys. 79, 135 (2007).[3] M. Keller et al., Nature 431, 1075 (2004).[4] B. Darquié et al., Science 309, 454 (2005).[5] J. McKeever et al., Science 303, 1992 (2004).[6] J. Beugnon et al., Nature 440, 779 (2006).[7] F. D. Martini et al., Phys. Rev. Lett. 76, 900 (1996).[8] B. Lounis et al. Nature 407, 491 (2000).[9] C. Brunel et al., Phys. Rev. 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Lett. 59, 2044 (1987).[25] Z. Michalewicz, in Genetic Algorithms + Data Structures= Evolutionary Programming (Springer-Verlag, Berlin,1992).[26] For simplicity, we keep ωA34 = ωB34 = 5meV and ωA12 =ωB12 = 25 meV. Possible differences between the bonding-antibonding splittings of the two AMs can be triviallycompensated by adjusting ωAL − ωBL .5[27] P. Zanardi et al., Phys. Rev. Lett. 81, 4752 (1998).[28] A. Bertoni et al., Appl. Phys. Lett. 85, 4729 (2004).[29] H. J. Krenner et al., Phys. Rev. Lett. 97, 076403 (2006).[30] M. J. Fernée et al., Phys. Rev. A 75, 043815 (2007).[31] F. Troiani et al., Appl. Phys. Lett. 90, 144103 (2007).[32] C. Flindt et al., Phys. Rev. Lett. 98, 240501 (2007).[33] S. Cortez et al., Phys. Rev. Lett. 89, 207401 (2002).

GOLDONI, Guido

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Optimal generation of indistinguishable photons from non-identical
  artificial molecules

Optimal generation of indistinguishable photons from non-identical artificial molecules

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