We provide an intuitive platform for engineering exciton transfer dynamics.
We show that careful consideration of the spectral density, which describes the
system-bath interaction, leads to opportunities to engineer the transfer of an
exciton. Since excitons in nanostructures are proposed for use in quantum
information processing and artificial photosynthetic designs, our approach
paves the way for engineering a wide range of desired exciton dynamics. We
carefully describe the validity of the model and use experimentally relevant
material parameters to show counter-intuitive examples of a directed exciton
transfer in a linear chain of quantum dots

Alan Aspuru-Guzik

Alejandro Perdomo

Ali Najmaie

Leslie Vogt

May V.

Nitzan A.

English

arXiv

We provide an intuitive platform for engineering exciton transfer dynamics. We show that careful consideration of the spectral density, which describes the system-bath interaction, leads to opportunities to engineer exciton transfer. Since excitons in nanostructures are proposed for use in quantum information processing and artificial photosynthetic designs, our approach paves the way for engineering a wide range of desired exciton dynamics. We carefully describe the validity of the model and use experimentally relevant material parameters to show counter-intuitive examples of directed exciton transfer in a linear chain of quantum dots.Chemistry and Chemical BiologyAccepted Manuscrip

Perdomo, Alejandro

Vogt, Leslie

Najmaie, Ali

Aspuru-Guzik, Alan

Harvard University - DASH

 Engineering Directed Excitonic Energy Transfer  (Article begins on next page)The Harvard community has made this article openly available.Please share how this access benefits you. Your story matters.Citation Perdomo, Alejandro, Leslie Vogt, Ali Najmaie, and Alán Aspuru-Guzik. 2010. Engineering directed excitonic energy transfer.Applied Physics Letters 96(9): 093114.Published Version doi: 10.1063/1.3323108Accessed February 19, 2015 3:21:54 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:4686408Terms of Use This article was downloaded from Harvard University's DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth athttp://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAParXiv:1001.2602v1  [quant-ph]  15 Jan 2010Engineering directed excitonic energy transferAlejandro Perdomo, Leslie Vogt, Ali Najmaie, and Alan Aspuru-GuzikDepartment of Chemistry and Chemical Biology,Harvard University, 12 Oxford Street, 02138, Cambridge, MAWe provide an intuitive platform for engineering exciton transfer dynamics. We show that carefulconsideration of the spectral density, which describes the system-bath interaction, leads to opportu-nities to engineer the transfer of an exciton. Since excitons in nanostructures are proposed for use inquantum information processing and artiﬁcial photosynthetic designs, our approach paves the wayfor engineering a wide range of desired exciton dynamics. We carefully describe the validity of themodel and use experimentally relevant material parameters to show counter-intuitive examples of adirected exciton transfer in a linear chain of quantum dots.The widely-applied F¨ orster theory for energy transferlinks experimental results to estimates of system infor-mation, particularly in biological and nanoscale appli-cations [1, 2]. The usefulness of this theory is partlydue to the simple expression of the kinetic rate constantsas a product of electronic coupling and a spectral over-lap factor which captures the complexity of the environ-ment. F¨ orster theory describes transport in the inco-herent limit, but a complementary and more elaborateapproach, such as Redﬁeld theory, is often required todescribe energy transfer. However, the information es-sential to understanding the dynamics is buried withinthe structure of the equations. In this letter, we employa quantum kinetic rate approach to distill the informa-tion contained in equations into a simple, yet instructive,formula. We use this approach to design directed excitontransfer mediated by an environment.Excitonic energy transfer (EET) has been studied insystems as varied as quantum dot (QD) nanostructures[3, 4], polymer chains [5], and photosynthetic complexes[6, 7]. Many applications of EET would beneﬁt fromcontrolling exciton dynamics. Perfect state transfer, asstudied in the quantum computing community, is achiev-able in certain engineered systems, but only at particulartimes during coherent evolution [8]. Recent works haveshown that environment-induced decoherence can alterexciton dynamics [9–11], although controlling the trans-fer direction has only been achieved using external poten-tials [12]. Our paper builds upon the idea of engineeringexciton transfer by designing appropriate system-bath in-teractions [13, 14]. We show that it is possible to designexperimentally realizable systems where the environmentcan be used to direct the ﬂow of energy.The Hamiltonian used in our simulation aims to cap-ture dynamics in a single-exciton manifold [15] interact-ing with an environment,ˆ H = ˆ Hs + ˆ Hb + ˆ Hsb (1)withˆ Hs =N Xn=1En |sn  sn| +Xn =mJmn |sm  sn|. (2)This representation is in the site basis {|sn } of local-ized excitations on each of N sites, (e.g. QDs or chro-mophores), with excitation energy En for each site andinter-site coupling Jmn. The environment is described bya phonon bath,ˆ Hb =Xq~ωq(b†qbq + 1/2), (3)where b†q (bq) is the creation (destruction) operator for aphonon with wavevector q. The system-bath interactionis assumed to be linear,ˆ Hsb =N Xn=1|sn  sn|Xq~ωq￿gnqb†q + (gnq)∗bq￿. (4)where gnq describes the site-speciﬁc coupling of electronicand vibrational degrees of freedom. We ignore the oﬀ-diagonal terms in the above equation, which correspondto phonon-induced modulations of the inter-site coupling,Jmn [15]. While ﬂuctuations of the gap give rise to the di-agonal electron-phonon coupling considered here [16], theinter-site couplings are usually one or two orders of mag-nitude smaller than the excitation gap and are thereforekept constant. In general, the validity of this approxima-tion is still an open question and its applicability variesfrom system to system.To describe the excitonic quantum dynamics we useRedﬁeld theory [17–19], which is a reduced density ma-trix approach in the regime of weak system-bath cou-pling. The formalism involves second-order perturbationtheory in the system-bath interaction, ˆ Hsb. This methodassumes the Markov approximation, no initial correla-tions between system and bath degrees of freedom, anda thermalized bath. We avoid the frequently employedsecular approximation (Bloch equations) [15, 19]; it isimportant to note that for systems where the time scale|ωab − ωcd|−1 is comparable or larger than the charac-teristic decoherence time, the coherence to populationtransfers contribute considerably to the dynamics of thesystem and must therefore be included.The equation of motion for the density operator ˆ ρ(t)in the excitonic energy basis representation, ˆ Hs |ea  =ǫa |ea , is given by [17–19],dρab(t)dt= −iωabρab(t) +XcdRab,cdρcd(t), (5)2with ρab(t) ≡  ea| ˆ ρ(t)|eb  and ωab = (ǫa − ǫb)/~. Theﬁrst term on the right hand side of Eq. 5 describes thefully coherent dynamics in the absence of ˆ Hsb and thesecond term describes the irreversible dynamics from theinteraction with the phonon bath.Correlations in bath-ﬂuctuations on diﬀerent sites [20]are taken into account by using the relation gnqgm∗q =gmq gn∗q = g2qe−Rmn/Rcorr, where Rmn ≡ |Rmn| is the dis-tance between the sites and Rcorr is the phonon correla-tion length [21].Using this relation for the electron-phonon couplings,the cross-correlation Cmn(ω) can now be written as,Cmn(ω) = e−Rmn/RcorrC(ω), (6)where the frequency correlation function C(ω) =2π[n(ω)+1](J(ω)−J(−ω)), with n(ω) the Bose-Einsteindistribution and the spectral density of the bath J(ω) = Pq|gq|2ω2qδ(ω−ωq), where δ(ω) is the Dirac delta func-tion.For the Hamiltonian speciﬁed in Eq. 1, the Redﬁeldtensor elements are given byRab,cd = Γdb,ac(ωca) + Γ∗ca,bd(ωdb)− δbdXeΓae,ec(ωce) − δacXeΓ∗be,ed(ωde), (7)where δij is the Kronecker delta andΓab,cd(ωdc) =12ζab,cdC(ωdc)+i2πζab,cdP￿Z ∞−∞C(ω)ωdc − ωdω￿,(8)ζab,cd =Xn,m(U−1)anUnb(U−1)cmUmde−Rmn/Rcorr, (9)with P denoting the Cauchy principal value of the in-tegral and Una =  sn|ea  the transformation matrix el-ements relating the site basis {|sn } and the excitonicbasis {|ea }. Since |ea  =Pn Una |sn , then |Una|2 canbe interpreted as the contribution of the n-th site to thea-th eigenstate of ˆ Hs.Hereafter, we will focus on QDs at low temperature(10 K) as our prototypical experimental realization. Weneglect possible inversion asymmetry of the crystal, andtherefore the contribution of the piezoelectric coupling,and focus instead on the deformation potential coupling.As shown by Calarco et al. [22], the spectral densitydescribing this coupling (in the absence of an externalelectric ﬁeld) is given byJ(ω) = Θ(ω)ηω3e−ω2/ω2c, (10)where, Θ(ω) is the Heaviside step function. We usetypical values for GaAs QDs for our numerical simula-tions [23, 24], giving η =(De−Dh)24π2ρu5~ = 0.035 ps2 andωc =p2u2/l2 = 1.41 ps−1, where De (Dh) is the de-formation coupling potential for electron (hole), u thespeed of sound within the quantum dot, ρ its mass den-sity and l the ground state localization length, assumedto be the same for electron and holes. We also assume acorrelation length, Rcorr, of 3 nm.Using Eqns. 7-9, the population transfer rates, kab, be-tween the eigenstates a → b is given by,kab = Rbb,aa = ζab,baC(ωab). (11)This equation is central to our insight into designing exci-tonic transfer. Though derived for a diﬀerent regime, theform of Eq. 11 is similar to the widely used rate equationin F¨ orster theory (incoherent limit), kForsterab ∝ J2abIab,where J is the electronic coupling, and I is the spectraloverlap integral [25]. Both equations are a product of twoterms: one dealing primarily with the description of thesystem (ζ in this paper or J2 for F¨ orster theory), and an-other largely depending on and arising from the system-bath interaction (C(ω) in this paper or I for F¨ orster the-ory). F¨ orster theory has been applied in ﬂuorescence res-onance energy transfer (FRET) to design chromophoresfor biosensing assays [2, 26, 27] and was recently veriﬁedexperimentally for semiconductor QDs [28]. We use thesimple structure of Eq. 11 to gain microscopic and exper-imentally relevant insight into engineering directed andoptimized EET.While C(ωab) depends on the overlap of systemeigenenergies with the spectral properties of the phononbath (e.g., lattice vibrations, solvent, protein environ-ment, etc.), ζ depends on the transformation matrix U inEq. 9, determined by the relative magnitude of electroniccouplings with respect to site energies, site connectivity,and spatial correlation between sites. The aim is then tomaximize (minimize) the product of these two factors inEq. 11 to favor (suppress) the desired rates.To illustrate the applicability of Eq. 11, we choose twothree-site examples to highlight the importance of thephonon bath interaction to achieve directed EET. Sincemultiplying a system Hamiltonian by a scalar does notchange the maximum exciton transfer probability in thecoherent limit, the distances and site energies for the twocases are chosen so that the Hamiltonians are relatedby a multiplicative factor (3.5 ˆ H1s = ˆ H2s ). As a conse-quence, the ζ values are roughly the same for cases 1and 2 (Table I); slight diﬀerences are introduced by thebath correlation term in Eq. 9. In the cases chosen, thefully coherent evolution gives a maximum probability ofﬁnding an excitation of 5% (1%) for site 1(2). Therefore,any diﬀerence between the two examples is a result ofinteraction with the environment.In contrast to the ζ factors, the C(ωab) values for cases1 and 2 at 10 K diﬀer by at least one order of magnitudedue the position of the transition frequencies with respectto the spectral density (Table I). Any changes to thesystem Hamiltonian can aﬀect C(ωab). In our examples,the scalar factor mentioned above changes the C(ωab)such that the largest population transfer rate is switched3from site 1 to site 2 since ω131 ≈ ω232 and C(ω131) ≈ C(ω232)(Fig. 1).TABLE I: Contributions of the system factor, ζab,ba, and ofthe overlap between transition frequency and phonon bathspectral properties, C(ωab), to the calculation of the quantumkinetic rates kab from energy eigenstate |ea  to |eb . The twocases considered are described in Fig. 1.Case 1 Case 2|ea  → |eb  logζ logC logk logζ logC logk3 → 1 -1.5 11.6 10.0 -1.5 7.9 6.33 → 2 -2.4 10.7 8.3 -2.4 11.6 9.12 → 1 -4.3 11.4 7.1 -4.3 10.2 5.8From the values of ζ and C(ωab), it is clear that case 1is designed such that an excitation starting on site 3 willtend to transfer to site 1, but in case 2 the population willgo to site 2, albeit at diﬀerent rates. Simulations of thequantum dynamics according to Redﬁeld theory conﬁrmsthis result (Fig. 2). Over the typical exciton lifetime of 1ns in QDs, we not only achieve directed transfer, but alsopopulation enhancement compared to both the maximumsite population during fully coherent dynamics and thepopulation expected at thermal equilibrium.Moreover, while there are always experimental limi-tations in tuning parameters, the structure of Eq. 11 isvaluable since it partitions the eﬀects due to the systemand bath. Using the calculated rates and/or a visual in-spection of C(ω) (Fig. 1B), it is easy to determine theimpact of varying a system parameter on exciton trans-fer. Future work will address exciton and electron trans-fer between sites with varying spectral density functions,as well as the role of aligning dipole moment orienta-tions in engineering EET. We are also working to identifyregimes in which preserved coherences enhance or reducethe eﬃciency of excitonic transfer. Of course, in situa-tions where multiple excitons are present in the systemdue to incident light intensity, frequency range, and/oroptical spectral density of the quantum dots, the Hamil-tonian used to describe the system needs to be expandedaccordingly [29]. To the extent that each exciton cou-ples to the environment through a spectral density asdescribed in this letter, some of the intuition developedhere should be transferable to these systems. However,there are a number of interesting complications, includingmany-body interactions among the excitons and correc-tions to the rates due to multi-phonon processes. A morecareful analysis is needed when the oﬀset in QD excita-tion energies is large compared with the characteristicfrequency expanded by the phonon spectral densities orfor systems at much higher temperature, e.g., room tem-perature, where multi-phonon processes are expected tobe relevant, or even dominant. These eﬀects are cur-rently under investigation and are beyond the scope ofthe current communication.In summary, we develop a framework for engineeringenvironment-assisted and directed excitonic transfer ina network of coupled QDs based on a quantum kineticrate approach. We emphasize the importance of howcharacteristic frequencies of a system ﬁt within the spec-tral bath structure. Our examples utilize the factoredand intuitive form of the population transfer rates equa-tion, which separates the contributions from the system(electronic) and bath (vibrational) degrees of freedom.This equation is similar in spirit to the rate equation forFRET, making it convenient to design interesting scenar-ios for environment-assisted transfer. Although we focuson QD examples, the principles presented here form thebasis for engineering a wide range of desired EET in avariety of nanostructures or artiﬁcial molecular photo-synthetic units.The authors thank Semion Saikin for helpful discus-sions. A. A.-G. and A. P. were supported as part ofthe Center for Excitonics, an Energy Frontier ResearchCenter funded by the U.S. Department of Energy, Oﬃceof Science, Oﬃce of Basic Energy Sciences under AwardNumber de-sc0001088. L. V. acknowledges support fromthe NSF Graduate Research Fellowship. A.N. acknowl-edges support from NSERC (Canada)[1] G. D. Scholes, Annual Review of Physical Chemistry 54,57 (2003).[2] J. R. Lakowicz, Principles of Fluorescence Spectroscopy(New York: Springer, 2006), 3rd ed.[3] G. D. Scholes and G. Rumbles, Nat Mater 5, 683 (2006).[4] S. A. Crooker, J. A. Hollingsworth, S. Tretiak, and V. I.Klimov, Physical Review Letters 89, 186802 (2002).[5] E. Collini and G. D. Scholes, Science 323, 369373 (2009).[6] H. Lee, Y. Cheng, and G. R. Fleming, Science 316, 1462(2007).[7] Y. Cheng and G. R. Fleming, Annual Review of PhysicalChemistry 60, 241 (2009).[8] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay,and A. J. Landahl, Phys. Rev. A 71, 032312 (2005).[9] E. Rozbicki and P. Machnikowski, Physical Review Let-ters 100, 027401 (2008).[10] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, andA. Aspuru-Guzik, New Journal of Physics 11, 033003(2009).[11] M. B. Plenio and S. F. Huelga, New Journal of Physics10, 113019 (14pp) (2008).[12] A. A. High, E. E. Novitskaya, L. V. Butov, M. Hanson,and A. C. Gossard, Science 321, 229 (2008).[13] K. M. Gaab and C. J. Bardeen, The Journal of ChemicalPhysics 121, 7813 (2004).[14] J. Cao and R. J. Silbey, The Journal of Physical Chem-istry A 113, 13825 (2009).[15] V. May and O. Kuhn, Charge and Energy Transfer Dy-namics in Molecular Systems (Wiley-VCH, 2004), 2nded.4[16] J. A. Leegwater, The Journal of Physical Chemistry 100,14403 (1996).[17] A. G. Redﬁeld, IBM Journal of Research and Develop-ment 1, 19 (1957).[18] W. T. Pollard and R. A. Friesner, The Journal of Chem-ical Physics 100, 5054 (1994).[19] A. Nitzan, Chemical Dynamics in Condensed Phases:Relaxation, Transfer, and Reactions in Condensed Molec-ular Systems (Oxford University Press, USA, 2006).[20] M. Richter, K. J. Ahn, A. Knorr, A. Schliwa, D. Bimberg,M. E. Madjet, and T. Renger, physica status solidi (b)243, 2302 (2006).[21] T. Renger and R. A. Marcus, The Journal of ChemicalPhysics 116, 999710019 (2002).[22] T. Calarco, A. Datta, P. Fedichev, E. Pazy, and P. Zoller,Physical Review A 68, 012310 (2003).[23] We consider a typical GaAs QD with deformation po-tentials De = −14.6 eV, Dh = −4.8 eV, mass densityof ρ = 5.4 g/cm3, speed of sound u = 5000 cm/s, andradius of l = 5 nm.[24] O. Madelung, ed., Semiconductors: data handbook(Springer, Berlin, 2004), 3rd ed.[25] G. D. Scholes, Annual Review of Physical Chemistry 54,57 (2003), ISSN 0066-426X.[26] A. L. Rogach, T. A. Klar, J. M. Lupton, A. Meijerink,and J. Feldmann, Journal of Materials Chemistry 19,1208 (2009).[27] I. L. Medintz and H. Mattoussi, Physical ChemistryChemical Physics 11, 17 (2009).[28] D. Kim, S. Okahara, M. Nakayama, and Y. Shim, Phys-ical Review B 78, 153301 (2008).[29] V. I. Klimov, Annual Review of Physical Chemistry 58,635 (2007).[30] A. Nazir, B. W. Lovett, S. D. Barrett, J. H. Reina, andG. A. D. Briggs, Physical Review B 71, 045334 (2005).5(a) Case 13 E  = 0.9 meVE  = 0.7 meV 2E  = 0 11 2 3R    = 20.7 nm 23 R    =  139.6 nmJ    = 0.004 meV 12J    = 0.11 meV 13J    = 0.01 meV 23Case 2E  = 0 1E  = 2.4 meV23 E  = 3.2 meV1 2 3R    = 13.6 nm 23 R    =  136.3 nmJ    = 0.40 meV 13J    = 0.01 meV 12J    = 0.04 meV 23312 212321211311322 (b)FIG. 1: Two cases of a three-site system are considered. a) Scaled schematic of system spacing and energy levels with details of variable site energies (En) and intersiteF¨ orster coupling strengths (Jmn). For QDs with transition dipole moments aligned perpendicular to Rmn, Jmn = 100meV/R3mn, with Rmn in nanometers [30]. b)Frequency correlation function for a super-ohmic spectral density. Energy basis transition frequencies for case 1 (2) are indicated by solid (dashed) vertical lines.6(a) (b)(c) (d)FIG. 2: Site basis population probabilities for an excition starting on site 3 for case 1 (a) and case 2 (b) demonstrate the change in transfer dynamics obtained byscaling the Hamiltonian (simulation at 10 K). Dashed lines indicate site populations at thermal equilibrium. Energy basis coherences for case 1 (c) and case 2 (d)display characteristic oscillations and damping over the course of 1 ns, a typical recombination time in QD systems.

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Engineering Directed Excitonic Energy Transfer

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Engineering directed excitonic energy transfer

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