Plasmonic bio-sensing for the Fenna-Matthews-Olson complex Guang-Yin Chen1,†, Neill Lambert2, Yen-An Shih3, Meng-Han Liu3, Yueh-Nan Chen3,4⋆, and Franco Nori2,5 7 1 0 2 1DepartmentofPhysics,NationalChungHsingUniversity,Taichung402,Taiwan n a 2CenterforEmergentMatterScience,RIKEN,Wako-shi,Saitama351-0198,Japan J 9 1 3DepartmentofPhysics,NationalCheng-KungUniversity,Tainan701,Taiwan ] h 4PhysicsDivision,NationalCenterforTheoreticalSciences,Hsinchu,Taiwan p - t n 5PhysicsDepartment,UniversityofMichigan,AnnArbor,Michigan48109-1040,USA a u q ⋆e-mail:[email protected] [ 1 ⋆e-mail:[email protected] v 5 0 4 Westudy theoretically thebio-sensing capabilitiesofmetalnanowire surfaceplasmons. Asa 5 0 . 1 specific example, wecouple the nanowire to specific sites (bacteriochlorophyll) of the Fenna- 0 7 1 Matthews-Olson (FMO) photosynthetic pigment protein complex. In this hybrid system, we : v i findthatwhencertainsitesoftheFMOcomplexaresubjecttoeitherthesuppressionofinter- X r a sitetransitions orare entirely disconnected from the complex, the resulting variations inthe excitation transfer rates through the complex can be monitored through the corresponding changesinthescatteringspectraoftheincidentnanowiresurfaceplasmons. Wealsofindthat these changes can be further enhanced by changing the ratio of plasmon-sitecouplings. The changeoftheFanolineshapeinthescatteringspectrafurtherrevealsthat“site5”intheFMO 1 complexplaysadistinctrolefromothersites. Ourresultsprovideafeasibleway,usingsingle photons, to detect mutation-induced, or bleaching-induced, local defects or modifications of the FMOcomplex, and allowsaccess to both the localand globalproperties ofthe excitation transfer insuch systems. Photosynthesis, the transformation of light into chemical energy, is one of the most crucial bio-chemical processes for life on earth 1. When a light-harvesting antenna absorbs photons, the resulting electronic excitation is transferred to a reaction center where it is transformed into other types of energy. With its relatively small size, homogenous structure, and solubility, the Fenna- Matthew-Olson (FMO) complex in green sulfur bacteria has attracted much research attention and has been widely studied 2 as a prototypical example of a photosynthetic complex. It consists of eight sites (chromophores), each of which can be regarded as an effective two-level system withflourescentresonantenergytransportcouplingbetweenoneanother3–8. TheFMOcomplexis surroundedbyaproteinenvironment,whichnormallyleadstodecoherenceandnoise,butiswidely thought to play a role in assisting the excitation transfer in the complex 3,9–13. The excitation transfer in the FMO complex was first demonstrated 14 to exhibit signatures of non-negligible quantumcoherenceat 77 K,and then morerecently advanced toroomtemperature15,16. Recently,hybridquantumsystems,whichcombinetwoormorephysicalsystems,enableone to combine the strengths and advantages of individual systems. Work in this area has led to new phenomenaandpotentiallynewquantumtechnologies17. Inspiredbythisapproach,andmotivated by recent developments on achieving interactions between surface plasmons (SPs) and organic 2 molecules18–20,here weinvestigateahybridsystemwhichintegratesnanowireSPs withtheFMO complex, with the goal of probing properties of the complex via the surface plasmon scattering spectra. SuchSPsareelectromagneticexcitationsexistingonthesurfaceofmetals21,whichcanbe excited by external fields. Due to their strong interactions with emitters, significant enhancement of the atomic or excitonic decay rates have been observed 21–28. With strong analogies to light propagation in conventional dielectric components 29,30, nanowire SP have been used to achieve subwavelengthwaveguidingbelowthe diffractionlimit,bipartitequantumentanglement31,32, and the miniaturization of existing photonic circuits 33. The strong coupling between the emitters and SP fields also enables the system to act like a lossy optical cavity; namely, the interaction can be coherent 24,26. These advantages of strong coupling between SPs and emitters and the relatively low-power propagation loss of SP in the nanowire make nanowire SPs an attractive system to combine with the environment-assisted transport in the FMO complex, for the goal of enhanced bio-sensing. Whilechallenging,therecentbreakthroughsoncouplingSPstoJ-aggregates19,20and Photosystem I trimer complexes 18 suggests the hybrid devices we study here may be feasible in thefuture. Like all other species, the in vivo FMO complex can experience both mutation-induced and bleaching-induced local defects or non-functional sites, leading to blocked excitation transferring pathways. This has been demonstratedin recent experiments34,35. These effects may be observed in the changes in excitation dynamics, the efficiency of excitation arrival at the reaction center 36, or thespectraof thephotoluminescence18,19. Still lackinghoweverisboth a means to prepare the complexwithasinglelocalizedexcitation,andameanstomeasurepopulationsoflocalsitesofthe 3 complex. HereweexaminehowtheSPcanbeusedasbothasinglephotonsource22andadetector. ThroughthescatteringoftheincidentSPduetothecouplingsbetweentheSPsandsites1and6of the FMO complex, the changes in the transmission spectra can indicate the presence of pathway- inhibitionormissingsites. Thisprovidesanalternativemethodtodetectmutation-induceddefects ornon-functionalsites,and suggestsfurtherapplicationsforsuch hybridsystems. Results WemodelasingleFMOmonomerasanetworkofN = 7sites(seeFig.1),whichcanbedescribed by ageneral Hamiltonian N HFMO = ǫn|nihn|+ Jn,n′(|nihn′|+|n′ihn|) (1) Xn=1 nX<n′ where the state |ni represents an excitation at site n (n ∈ 1,...,7), ǫ is the site energy of chro- n mophore n, and Jn,n′ is the excitonic coupling between the n-th and n′-th sites. For simplicity, heretherecentlydiscovered37 eighthsitehasbeenomittedbecauseresultsofmoleculardynamics simulations37 suggestthatthissiteplaysaminimalrolein theprocesses weare interestedin. In the bacterial photosynthesis, the excitation from the light-harvesting antenna enters the FMO complex at sites 1 or 6 and then transfers from one site to another. When the excitation gets to the site 3, it hops irreversibly to the reaction center. In the regime that the excitonic cou- pling Jn,n′ is large compared with the reorganization energy, the electron-nuclear coupling can be treatedperturbatively3,andthedynamicsofthesystemcanbegovernedbythequantumLiouville equation. ThestrongcouplingbetweentheexcitonicdynamicsandtheFMOenvironment,aswell 4 as the structure of that environment, will quantitatively affect the excitation transfer 38,39. Here, however, we wish to focus on the interplay between the excitation transfer and the SP scattering; sowefocusonthesimplestpossiblemodelofsuchenvironmentaleffects. Thisallowsustoclearly identify in what way defects in the FMO complex affect the SP scattering. A full investigationof theenvironmentinfluencewillbeconsideredinfuturework. Also,itshouldbenotedthatexcitonic fluorescencerelaxationisnotincludedintheLiouvilleequation. Thisisbecauseitstimescale(∼ 1 ns) is much longer compared with that of the excitation transfer from site 3 to the reaction center (∼ 1 ps), the typical excitation transfer timeacross the complex, and the dephasing 40 (∼ 100 fs), such thatthisrelaxationprocess can thenbeomittedforsimplicity. The Surface plasmon and FMO Hybrid system. As shown in Fig. 1, we considera metal nanowirethat is placed close to theFMO complex,as a substitutefor thelight-harvestingantenna found in vivo. A SP with energy E = ~v k incident from theleft end of the wire can bestrongly k g coupled 18,41,42 to both sites 1 and 6. Here, v and k are the group velocity and wave vectorof the g incidentSP,respectively,andv issettobeunitythroughoutthispaper. TheincidentSPcanthenbe g scattered by the two sites, due to these strong SP-site couplings. Alternatively, it can be absorbed bythesetwositesandbedissipatedduetolossintheFMOcomplex. ThetotalHamiltonianH of T thisSP-FMO hybridsystemcan bewrittenas 31,43,44 H = H +H +H , with T sp sp-site FMO H = dk ~ω a†a sp k k k Z H = dk ~ (g σ+a +g σ+a eikd)+H.c. , sp-site 1 1 k 6 6 k Z (cid:2) (cid:3) (2) 5 where H stands for the energy of the SPs with a† being the creation operator of the k-mode sp k SP, H denotes the interaction between the SPs and the sites of the FMO complex. In H , sp-site sp-site d is the separation between site 1 and 6, while σ+ denotes the raising operator for site 1 (6). 1(6) Here, the coupling strength g between the SPs and site 1 (6) is assumed to be independent of 1(6) k under the Markovian approximation 43,45. The strong decay rate into SPs then takes the form 24, γ = 4πg2 /(dω /dk), and the values we choose for g and g are consistent with a recent sp 1(6) k 1 6 analysisofnanowireSPs coupled toJ-aggregates18–20. When the SP is propagating on the surface of the metal nanowire, it inevitably suffers from dissipation,such as Ohmicloss, that can be described by a Markovian channel. Thus, here we in- cludetheMarkovianchannels forirreversibleexcitationtransferfrom site3 to thereaction center, aswellastheOhmicloss,byintroducingnon-HermitiantermsinthetotalHamiltonian. Moreover, since we are interested in time-independent solutions of the photon state, the effect of quantum jumps 46 on the dynamics is neglected, such that the phonon dephasing leads to population reduc- tion in our model. We therefore simply include the phonon dephasing as a non-Hermitian term. Whilethisintroducesextrapopulationloss,itallowsusto includeina simpleway thebroadening effect of dephasing on the SP spectra. The site energy of the FMO complex in Eq. (1) is then modifiedas N=7 γ n ǫ −i |nihn|, (3) n 2 Xn=1(cid:16) (cid:17) where γ are mostly zero, except for the rates referring to the Markovian channels, γ = γ = n 1 6 γ + γ and γ = γ . Here, the Ohmic loss rate γ is set to be γ = 20−1γ 21,24,47. Note that dp ol 3 s ol ol sp theseparationbetweensite1and6 isabout1-2nm48,whichismuchsmallerthanthewavelength 6 of the incident SP. Therefore, one can set the separation d = 0. Similarly, since the separation is small compared with the distance (∼ 30 nm) that SPs propagate during the dephasing time of scattering49,50,weneglectplasmondephasing. Notethatweonlyincludeγ onsites1and6,and dp neglect dephasing for other sites, because the broadening of these sites, which couple to the SPs, typicallydominatesthescatteringspectra51. The energy eigenstate of the hybrid system with an energy matching the incident SP, E = k v k, can bewrittenas 24,43: g |E i = dx φ(x)† C†(x)+φ† (x)C†(x) |g,0 i k Z h k,R R k,L L i sp 7 + ξ |n,0 i (4) n sp Xn=1 where |g,0 i describes that the FMO complex is in the ground state with no SPs, and |n,0 i sp sp stands for that the excitation is in the site n, while ξ is the probability amplitude that the site n n absorbstheexcitation. WealsoassumethattheSP fieldisincidentfromtheleft ofthewaveguide, thescatteringamplitudesφ† (x) and φ† (x) thereforetaketheform, k,R k,L φ† (x) ≡ [exp(ikx)θ(−x)+texp(ikx)θ(x)], k,R φ† (z) ≡ r exp(−ikx)θ(−x). (5) k,L Here, tand r are thetransmissionand reflection amplitudes,respectively,and θ(x) istheunit step function. ThetotalHamiltonianH can befurthertransformed 43 by Fouriertransformationintoa T 7 real-space representation, ∂ ∂ H˜ = ~ dx −iv c† (x) c (x)+iv c†(x) c (x) T Z (cid:26) g R ∂x R g L ∂x L +~g δ(x) c (x)σ+ +c (x)σ+ +H.c. 1 R 1 L 1 (cid:2) (cid:3) +~g δ(x−d) c (x)σ+ +c (x)σ+ +H.c. 6 R 6 L 6 (cid:2) (cid:3)(cid:9) N γ + ǫn −i n |nihn|+ Jn,n′(|nihn′|+H.c.) (6) 2 Xn=1(cid:16) (cid:17) nX<n′ where c (x) [c (x)] is a bosonic operator annihilating a right-going (left-going) photon at x, and R L δ(x) is the Dirac delta function. This real-space total Hamiltonian can be applied to the energy eigenstate [Eq. (4)]. The transmission spectrum T = |t|2 and the probability amplitudes ξ can n thenbeobtainedbysolvingtheeigenvalueequationH˜ |E i = E |E i. Notethatinthefollowing T k k k calculations, we employ the energies and excitoniccouplings from Ref. [50] for the FMO Hamil- tonian. Therateofexcitationtransferfromsite3tothereactioncenterissettobeγ−1 = 1ps,and s thedephasingrateγ , proportionalto thetemperature40,is chosento be77cm−1. dp Detectingdefectsfromchangesinthescatteringspectra. Ithasbeendemonstratedexper- imentally34,35 that each chromophorein the FMO complexcan be decoupled from its nearest site due to rotations of the chromophore or mutation-induced local defects. This can lead to damage of the chromophores, such as blocked energy pathways or entirely non-functional sites. Here we first calculate the transmission spectra of the incident SP field in the presence of defects in the FMO complex, and compare them to the case without any defects. As can be seen in Fig. 2, the black-solid curve shows the transmission spectrum of the incident SP field for the normal FMO complex. The dips in the spectrum correspond to the the eigenenergies of the hybrid system, i.e., 8 thedipsoccurwhen thesitesresonant withtheSPreflect theincidentfield 24. Thereason whythe dips do not reach zero is because here we have included dephasing and the irreversible excitation transferfromsite3tothereactioncenter. Thesedissipativechannelsdecreasetheamplitudeofthe dips. AsseeninFig.2,thered-dashedcurvesshowthetransmissionspectrawhencertainexcitonic couplings are inhibited. The differences between the red-dashed and black-solid curves can indi- cate the occurrence of a blocked transfer pathway. However, the spectral differences in Fig. 2(a) and (d)are larger than thosein Fig. 2(b)and (c). Thisis because inthenormal casethesesitesare strongly coupled, and thus when suppressed the inhibition strongly affects not only the eigenen- ergies, but also the quantum coherence 53 in the site basis. On the other hand, the transmission spectrainFig.2 showtypicalFanolineshapes54 stemmingfromtheinterference betweendiscrete (the sites) and continuous channels (the SPs). As the inhibition of the relatively strong coupling J , between sites 5 and 6 leads to the enhancement of the coherence between other the sites of 5,6 theFMOcomplex,theFanoresonanceisstronglyaltered[Fig.2(d)]. Thisresultisconsistentwith ourpreviouswork 55 on thedynamicsoftheexcitationtransferin FMO complex. Similarly, mutation-induced defects or local environment modification 34,35 can also lead to the disconnection of the chromophores from the FMO complex entirely, rather than just suppres- sion of certain couplings. In Fig. 3, we plot the transmission spectra for the cases where certain sites with relatively large excitonic couplings are removed entirely. The differences between the black-solid (the normal situation) and red-dashed curves can again indicate this drastic alteration of the complex. One interesting question can be raised here, can one distinguish the following two situations: (i) inhibitinga strong excitoniccoupling, and (ii) totally removing one of the sites 9 which contains the very excitoniccoupling entirely? The answer is yes. By comparing Fig. 2 and 3,theinhibitionofthetransferpathwayonlyresultsinchangesofthetransmissionspectra,butthe missingsitecase also leads to vanishingpeaks/dips,indicatingtheremovalof an eigenstateofthe system. Note that in plotting Fig. 2 and 3, we assume the SP-site couplings to be symmetric, i.e., g = g = 10cm−1. Inthefollowingsection,wewilldiscusstheeffects ofasymmetriccouplings. 1 6 Discussion So far we have shown that local defects or missing sites can be detected via changes in the trans- mission spectra of the incident SP. However, if these changes in the transmission spectra can be enhanced,thentheobservationsofsuchaneffectinexperimentalrealizationswouldbecomemore feasible. We have observed that the transmission spectra contain Fano lineshapes, thus we can predictthatadiscrepancybetweenthediscrete(theFMO sites)andcontinuous(theSPs)channels willstronglyaffectthebehaviorofsuchFanoresonances56–58. Largerchangesinthetransmission spectracanthenbeachievedbymakingthesetwochannelsmoredisparate. InFig.4,weshowthe transmission spectra when inhibiting J [Fig. 4(a)] and entirely removing the site 2 [Fig. 4(b)] 1,2 with asymmetric SP-site couplings g /g = 100 (g = 10 cm−1, g = 0.1 cm−1). The changes in 1 6 1 6 the transmission spectra are enhanced compared to those in Fig. 2(a) and Fig. 3(a), respectively. This is because when the damage (pathway-inhibition or missing site) occurs to site 2, the coher- ence between other sites in the FMO complex increases 55, and the large ratio g /g reduces the 1 6 communication between discrete and continuous channels. As a result, the two channels become moredisparate,leadingto an enhancementin thetransmissionspectra. 10