Downloaded by guest on September 26, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1814178116 Sch Christian chemistry in quantum-electrodynamical transfer cavity charge and excitation of Modification E chemistry QED reactions. chemical accep- of control or the toward donor a approach chemistry promising either polaritonic rendering with processes, environment chemical photonic resonance redefine the can changing to that highlight close enhance results Our is should tor. that cavity mode the times), a weak for coherence while acceptor, (long and decoherence donor the of excitations between isolated in the be should frequency cavity decoherence the strong times), For coherence (short times. coherence distance- the effi- on highest essentially depending of ciency efficient, configurations the different more that further and transfer find independent, we the transfer, renders excitation selectively cavity For even efficiency. alter and the drastically behavior can improve cavity charge-transfer the characteristic to of coupling the the degrees that photonic large and over distances the correlations electron–electron that extra introduce find freedom We interaction. self- and polarization Coulomb often-disregarded the account results into taking quantum-chemical tions, well-known F and charge-transfer to Dexter mode, as connect such photon effective easily an can by we environment sim- and the explicit in model all changes the in keeps ulates freedom that of degrees principles electronic first the from of formulation By cavity. real-space as a a in well field using modifying electromagnetic as by the processes of altered fluctuations drastically these vacuum chem- the be that quantum can highlight properties modern chemical the we of other work, of challenges this major one In the is istry. of charge Sandoghdar) one Vahid controlling or is and and Narang, understanding them excitation Prineha and nature, Batista, of in S. processes Victor basic terms by most reviewed in 2018; 16, transfer August review Energy for (sent 2018 12, December Rubio, Angel by Contributed Germany Hamburg, a unu-pia oes hs oesueavr etitdrep- restricted very a use models with These usually models. considered, quantum-optical is photons reactions and individual laser structures under of a chemical system with interplay as the of the such heating source, Theoretically, or energy ionization investigation. to external lead an could that need field, and not conditions, does ambient it and that temperature be room to control for seems it even and that robust, features investigate strong appealing to the a has way properties have not chemical alternative can are This an mechanics that (3–7). by quantum influence modifications e.g., traditional field, tiny by photon seemingly captured the these this cavity, with optical and chang- chemical By environment and (2). the fluctu- research structures intense ing vacuum of on focus of the field become form has electromagnetic reactions the the in of photons ations individual of influence lead then can which 1), changes. For (Fig. structural (A) excitation large to subsystems. molecule or acceptor charge a between an transfer to to can transfer energy (D) determined molecule charge donor are or a instance, energy irradiation, solvents, by and of extent heat modification and surfaces, or investigated reac- catalytic pressure, traditionally chemical e.g., are further by, which controlled undergo reactions, can Such structures constituents, tions. these individual the how from and distinct properties have which O interaction light–matter a lnkIsiuefrteSrcueadDnmc fMte,271Hmug emn;and Germany; Hamburg, 22761 Matter, of Dynamics and Structure the for Institute Planck Max nyrcnl,mil rvnb xeietlrsls() the (1), results experimental by driven mainly recently, Only ttet uha lcrn n tm omnwstructures, new form con- atoms different and how electrons is as chemistry such of stituents questions basic the of ne | afer orltdchemistry correlated ¨ a,b,1 | ihe Ruggenthaler Michael , aiyQED cavity | ogrneeeg transfer energy long-range rtrectto-rnfrreac- excitation-transfer orster ¨ a,b ek Appel Heiko , | strong a,b n ne Rubio Angel and , 1073/pnas.1814178116/-/DCSupplemental. y at online information supporting contains article This 1 BY-NC-ND) (CC 4.0 NoDerivatives License distributed under RouTe is QuantERA article access the open of This members collaborating.y both currently are not V.S. are and They consortium. A.R. statement: interest of Conflict Institute Planck Light.y Max of V.S., Science and the University; for Harvard P.N., University; Yale V.S.B., Reviewers: omdrsac;CS,MR,HA,adAR nlzddt;adCS,MR,HA,adA.R. and per- H.A., M.R., A.R. C.S., and and H.A., data; analyzed M.R., A.R. paper. C.S., the and wrote research; H.A., M.R., designed C.S., A.R. research; formed and C.S. contributions: Author yamnmllna obnto faoi rias(LCAO). orbitals atomic of combination described linear well are minimal D and a A by where onset limit, earlier an static-correlation others, the among of to, leads This them distances. entangles large and over subsystems matter between also simplified correlations by captured be cou- cannot strong that a matter to models. also few-level due and reactions, effects light chemical find of we of pling Furthermore, knob space. control real promising in photons a that fluctuations) for strongly vacuum highlights quantum allow from be This originating manner. those can only controlled (even probabilities, a charge-transfer in modified Dexter effects, quantum-chemical as a well-known of in such that principles vacuum show first We from the space. subsystem to matter real the coupling treat we to when due cavity reactivities and properties for responsible are space reactions. real and in structures between freedom detailed interplay of complex degrees the matter in case, the systems this of In treatment in chemistry. first-principle is quantum usual the challenging, to very contrast observables stark determination correlation effective the and an makes optical sys- real-space in which an of few-level subsumed treatment, of often the simplified then mode of This is bath. the environment mode one to complex to it The coupled couple 9). tem of 8, then molecules) (5, and and cavity levels (atoms few subsystem a matter only the of resentation owo orsodnemyb drse.Eal ne.ui@pdmgd or [email protected] Email: addressed. be may [email protected]. correspondence whom To essdpn rcal nteCuobadself-polarization and Coulomb the interactions. pro- on the that crucially highlight situa- depend We cesses possible. free-space be corresponding transfer should transfer the no enhanced tion in to when even leads reactions, interaction where experiments, light–matter recent with corre- strong line electron–electron in of results impact find We the lations. on where emphasis models, an donor–acceptor put we real-space trans- with can in processes processes changes these these fer cavity, consider a We different. e.g., dramatically to, be under- due of electromagnetic changed well the goal is when are case, vacuum major free-space usual processes a the for these is stood While processes chemistry. these in quantum processes controlling fundamental and are nature, transfer charge and Excitation Significance safis xml,w osdrhwapoo oeinduces mode photon a how consider we example, first a As chemical in changes fundamental identify we work, this In y b h etrfrFe-lcrnLsrSine 22761 Science, Laser Free-Electron for Center The a,b,1 y . y raieCmosAttribution-NonCommercial- Commons Creative www.pnas.org/lookup/suppl/doi:10. NSLts Articles Latest PNAS | f10 of 1

PHYSICS in Photon-Induced Correlations. Finally, we present the conclu- sions of our work and provide a perspective outlook in Summary and Conclusion. Theoretical Framework We focus on changes of the electronic properties (which drive the aforementioned energy-transfer processes) due to coupling Fig. 1. Schematic illustration of typical excitation transfer in free space to the photon vacuum and, hence, keep photonic as well as all between donor and acceptor, which consists of transversal (radiative) and electronic degrees of freedom explicit. We include the effects longitudinal (nonradiative, Forster)¨ contributions. Here, R is the distance of the nuclei in the Born–Oppenheimer approximation—i.e., we between donor and acceptor. If the electromagnetic vacuum is changed due consider the electronic wave function as a conditional wave func- to, e.g., an optical cavity, especially the transversal contribution is expected tion of the nuclear positions.∗ Furthermore, following the highly to differ from its free-space form, therefore deviating from the geometric successful approach of quantum chemistry, we take the photon dilution 1/R2 behavior in three dimensions. bath of the bare electromagnetic vacuum into account by renor- malizing the bare masses of the charged particles and use their respective physical values (13). Instead of performing a renor- Together with the following investigations, this illustrates pos- malization of the masses to new values that take into account sibilities to use cavities and their photons to investigate directly the changes in the vacuum, we simulate these changes by explic- matter–matter correlations. itly keeping one of the enhanced modes due to a cavity. This Next, to contrast to well-known results from quantum mechan- allows us to recover the well-known matter Hamiltonian when ics in real space, we then investigate changes in charge-transfer we let the coupling to this mode go to zero. While this sim- ¨ (Dexter) and excitation-transfer (Forster) reactions for a donor– plified treatment of dissipation is expected to be accurate for acceptor system. In quantum mechanics, charge transfer is static (eigenfunction) calculations, in the time-dependent case, understood perturbatively by considering the overlap of expo- this simplified treatment will lead to wrong long-time dynamics. nentially decaying wave functions. In this case, indeed, elec- For long-time dynamics, the influence of the photon and phonon tronic charge density moves from D to A. As a consequence, bath (that we disregarded due to the Born–Oppenheimer and with increasing distance between the two components, the zero-temperature approximation) will become essential. The exponential decaying overlap leads to an exponentially decay- latter one will become increasingly important, already in equi- Γ ∼ exp(−(I + I )|R − R |) ing charge-transfer probability A D A D , librium at finite temperature, leading to a thermal occupation of where IA/D and RA/D are the corresponding ionization poten- phononic eigenstates on the energy-scale kB T . We take this into tials and (mean) positions, respectively. This perturbative limit account effectively by introducing relevant coherence times— of Dexter charge transfer dominates typically length scales of i.e., we consider coherent dynamics up to a finite time T , after a few to tens of Angstroms depending on its participants (10). which we assume that the bath will damp the dynamics. The Here, we show how the coupling to a cavity can change this well- coherence time is usually determined by the coupling to the known behavior and allows, by increasing the distance between A phonon modes, which is typically between a few tens up to hun- and D, to even invert the charge transfer—i.e., charge flows from dreds of femtoseconds in, e.g., light-harvesting complexes (14, A to D. 15). For such short-time dynamics, the decoherence due to pho- Excitation energy transfer, on the other hand, does not ton loss (typically on the order of picoseconds) is indeed a minor demand a transfer of charge, but is mediated by transversal contribution. (observable/real) and longitudinal (Coulomb) photons as illus- As a minimal example of the above description, we consider trated in Fig. 1. In free space, the transfer rate decreases with 2 a one-dimensional dimer model coupled to one effective pho- 1/|RA − RD | due to geometric dilution, dominating the far- ton mode. The polarization of the effective mode is therefore field rate. The Coulombic participation, typically referred to in the direction of the one-dimensional model. We assume the as Forster¨ excitation energy transfer, can be approximated as validity of the long-wavelength approximation—i.e., since the a dipole–dipole interaction after a certain spatial separation wavelength of the photonic mode is much larger (hundreds of 6 and is decaying as Γ ∼ 1/|RA − RD | , dominating typically the nanometers) than the extension of the molecular system (few near-field beyond the Dexter domain up to 30 nm (11). Angstroms), we can disregard the inhomogeneity of the electro- If we couple A and D to a cavity, the characteristic transver- magnetic field to determine the electronic properties (2, 16). In sal contribution changes. Its efficiency depends strongly on the this case, the Hamiltonian in SI units is given by intrinsic coherence time of the coupled system, and we iden- 2 2 tify two major domains. For strongly decoherent systems (short X2 2 e X2 Zj Hˆ (t) = − ~ ∇ − coherence times), a cavity that has a frequency in between the xn p n=1 2me 4πε0 n,j =1 (ˆx − R )2 + 1 isolated resonances of A and D shows the highest excitation- n j 2 transfer efficiency. For long-time coherent systems, we find that ξ(t) e † + + ωaˆ aˆ p 2 ~ the highest efficiency is provided if the cavity is in resonance 4πε0 (ˆx1 − xˆ2) + 1 with the isolated D or A resonances. Furthermore, we find that r 1 ω   the usually discarded dipole self-polarization term has a large + (λ(t)Xˆ )2 − ~ λ(t)Xˆ aˆ† +a ˆ , influence on the dynamics of the combined light–matter system, 2 2 especially for strong-coupling situations. Finally, we highlight that, even for the coupled system, it is the electron–electron with the electronic dipole operator Xˆ = −|e| [ˆx1 +x ˆ2]. Here, the correlation that dominates the excitation transfer. physical mass me already contains the effect of the continuum This paper is structured as follows: First, the theoretical set- of all modes other than the one enhanced by the cavity. To ting is explained in Theoretical Framework. We then consider find the proper physical transfer behavior, it is essential that the the influence of matter–photon correlations on the equilibrium longitudinal interaction and nuclear potentials (at the positions structure in Equilibrium Long-Range Correlation, before we investigate charge transfer in Charge Transfer. Next, we dis- cuss excitation energy transfer in Excitation Energy Transfer *The here-applied description can be extended to account for the full quantum behavior and then highlight the influence of matter–matter correlations of nuclei, electrons, and photons following, e.g., refs. 2 and 12.

2 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1814178116 Schafer¨ et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 Sch on decoherence for freedom accounts of system degrees electron–photon of the amount in enormous the Consequently, {φ by states single-electron corresponding the denote charges nuclear of of with values charge (5), experimental to nuclear close energies higher 2 excitation has slightly 2) a Fig. in has hydrogen- a D is 2) Z where Fig. in dimer, (described 1 like Setup model. dimer above structure is electronic transition, the splitting electronic that Rabi the large. mean an also is resonance, in we with if and, strong, resonance considerably, changes By in identi- small. is almost (λ very cavity is case the structure free-space if electronic the situations. the to coupling that strong cal mean or we weak different investigate weak, however, to time, want This we off. nonzero and interac- for on transversal the switched values also exchange be 17), the can 13, to Simi- tion due (2, 17). is photons (16, which longitudinal interaction, excitations of longitudinal matter the the to to lar relation in interaction between that relation such unitless the via given coupling-strength volume is mode coupling effective the with lowing, cavity and electronic system between interaction light–matter transversal the mines coordinates displacement q the with rdo aibelnt,adtepoo oei xaddin electron– expanded combined of is the consists mode space that photon Hilbert means photon the This and states. length, Fock-number variable of grid of states many-body system. the light–matter and between coupled A relations the and the D discuss of to polaritons us individual allows cou- the This problem cavity. A the or cavity—i.e., to D single-electron pled the the to solve one we only coupled differently, with Put individually independently still solved are but specifi- A electron and we D Transfer), that Energy mean cally Excitation (in constituents lated” of (16) Hamiltonian as reads the and 17) of 13, term. limit (2, (QED) frequency nonrelativistic interaction electrodynamics with quantum the Coulombic mode from a the for deduced on coupling switching electron–photon The by coefficient system a charge the effective have soft-Coulomb the applied have we widely nuclei the The use approximation. therefore We dimension. nuclei the of † ee efcso oa hne ftemlclrssesadcmeto possible on comment and systems molecular the of molecules). changes in of modifications local individ- ensemble collective on the to to focus (coupling (coupling we field enhancement Here, electromagnetic collective the a and of rela- increase molecules) photons) are an ual (16): rates of contributions loss (number two the has far, excitations molecules so of of work, numbers this as the single-molecule in such Increasing presented for high. molecules values tively allows to of nanoplasmonics close number scales e.g., the on While, couplings increases chemistry. or QED), polaritonic mirrors circuit in the or done of nanoplasmonics of reflectivity which domain smaller, the cavity the increases the into of (leading volume interactions the light-matter mode-length makes effective really an either demand One would so. do to ways several hl eew utaotteculn teghb ad na xeietteeare there experiment an in hand, by strength coupling the adopt just we here While D − ntefloig ecnie w ifrn elztoso the of realizations different two consider we following, the In h ueia acltosaepromdwt real-space a with performed are calculations numerical The fre al. et afer 0 D ~ω ¨ 2 1 = 3 , i φ eV (ˆ 1 D .2 a utemr,i iutosweew ee o“iso- to refer we where situations in Furthermore, }. † nrlto oAwith A to relation in iikn h nrei tutr fcaiedyes cyanine of structure energetic the mimicking , − g /~ω a ˆ swl stecoupling the as well as ), R H ˆ 1/2 ep hrceie h tegho h light–matter the of strength the characterizes g eebetecorrect the resemble ) = umr n Conclusion. and Summary = λ(t 2 1 ea " ) ξ 0 p ˆ (t q ilb sddpnigo whether on depending used be will 2 Z ) + A ~ω ∈ 2 Z 2 = ω = 0 1] [0, λ L A 2 ewe 7ad5.7 and 67 between g  1 = /9 n eeatenergy-scale relevant and 101 0 = q ˆ − eu ilsrtda well as (illustrated 2 Setup . q and ˆ htalw st quench to us allows that 2 ,adteRb splitting, Rabi the and ), = · ω λ λ N q = · Z pt 1/r X ˆ D p 2ω ~ to  We 1/5)9. + (2 = 1/ε 2 (ˆ om loi one in also form, # a 301 orahteapplied the reach to A ˚ , ˆ + V 0 V {φ 2 a ntefol- the In . · † 0 A htdeter- that N ) Z , pt and φ 1/2 1 A states. g } and , † 0 6 = ω and p ˆ ~ω By = is , . lcrncoebd eue-est arx(1RDM) matrix reduced-density one-body electronic nteCuobccs,w antmk h ttccreainany correlation static the make cannot we is case, However, which Coulombic (20). correlation, the concurrence in static generalized called nonzero a often to context equivalent present the in is simple—i.e., specifically becomes then (NOs) orbitals natural so-called ϕ the by diagonalized be can which unger- and gerade of LCAO states local minimal two ade of a combinations orbitals—i.e., antisymmetric atomic well and are symmetric is that to determinants function two due of wave combination fully linear the a remaining by consequence, represented while become a components electrons As the both antisymmetrized. limit), on atomistic distributed simple (the equally and lengths bond Hartree–Fock very large For domain, adequately. perform this approximations decreases density-functional In (correlation) influ- determinants fast. the Slater exponentially and accu- additional determinant, be Slater of can single ence system a electronic by the reproduced lengths, rately bond small very For nsrcue thsi hscs tl uiysalrta one— than smaller purity den- a pure still simple a case specifically this of is in notion allows i.e., 1RDM has the the This it though to structure, one. case, in even connection to usual since simple 1RDM matrix, the a the sity to draw normalize contrast to to in us chosen that, have Note we 3). (Fig. suppressed ϕ can we that problem Born–Oppenheimer or different length investigate. bond a each get for dis- we where the on dimer, distance freedom model interatomic of our degrees the of photonic the consider sociation harness extra will To the we electrons. of formulation, the influence real-space of our mass usu- of physical is possibilities as the by subsumed, in is account done, bath into photon ally the take in of change we rest The a which (see information). mode by 19), ground photon influenced 18, The effective be (13, one dimer. can bath model systems, photonic the open the photonic for of changed also the equilibrium of state, the influence the on analyzing vacuum by start us Let Correlation Long-Range Equilibrium figures the below given the are in details and More scales. time attosecond respectively. 2, is A of single-electron excitation lowest electron The is interactions. D of cavity excitation and Coulomb, length, bond 2. Fig. Setup 12 k g ∗ For (x (x tr 1 0 ) + ) (γ n aua occupations natural and g eu n ftetoeeto ie oe ihvariable with model dimer two-electron the of 2 and 1 Setup e 2 0 = ϕ ) aeil n Methods. and Materials u ≈ Donor (D) {ϕ hspolmhstowl-nw iiigcases: limiting well-known two has problem this , (x γ P γ e 1 g ... ~ω (x e )ϕ , k ∞ (x ϕ =1 1 D u ∗ , 1 u = x (x , o hs(pnsnlt rudsaethe state ground (spin-singlet) this For }. length Bond n 1 x 0 12.62 k 1 |R = ) 1 2 0 0 ~ω adalohrNsaeexponentially are NOs other all ))—and = ) < A A Z n cuainbyn igeNO single a beyond occupation Any 1. − eV X = dx R 10.75 or k ∞ D 2 =1 ~ω n Ψ(x | eto 1 section Appendix, SI k normdlsse Fg 2), (Fig. 1 system model our in eV D n as = k 1 ϕ , or 2.617 x k 2 ~ω (x NSLts Articles Latest PNAS )Ψ 1 A )ϕ eV ∗ = γ (x e k ∗ n h oetsingle- lowest the and , 2.351 (x Acceptor (A) 1 (x 0 , 1 1 x 0 , ), 2 x ... eV ), 1 0 ) o eu or 1 setup for ≈ 1 2 o more for (ϕ | Setup 12 f10 of 3 g (x 1 )

PHYSICS local polarizations. As a consequence, bound molecules tend to reduce their bond length (also observed in ref. 18) due to an accumulation of charge between the molecules, while separated charges accumulate at their local molecules [also observed in theoretical calculations for realistic molecules (21)]. This effect, however, is rather small in relation to effects discussed in the following sections. The major difference is that in equilibrium without external perturbations, all of the effects are purely due to quantum fluctuations, and the (expectation values of the) elec- tromagnetic fields are zero. In the following sections, where the dynamics of the electrons also generate nonzero electromagnetic fields, the classical part of the photon field can enhance effects. But we will see in Excitation Energy Transfer that photon- assisted electron–electron correlations still play a major role in Fig. 3. Natural occupations of γe,p with and without photonic coupling molecular dynamical processes. for frequency ~ω = 12.62 eV. While for the uncoupled systems, the higher- lying NOs, e.g., 3 (purple) and 4 (yellow), are exponentially suppressed with Charge Transfer bond length, and the dark cavity introduces explicit electron–electron cor- relation that become distance-independent. In the atomistic limit, the cavity Let us next move on to dynamical processes where energy is therefore introduces static correlations beyond the usual Coulomb case. The transferred between two subsystems. We first investigate the simulation box is 26.5 A,˚ with a spacing of 0.0529 A˚ and 10 photon number influence of the photon field on charge-transfer processes. We states. consider setup 1, presented in Fig. 2, and take as the initial state an eigenstate of the coupled Hamiltonian that has most of its charge density on D. Then, we perturb this eigenstate weakly larger. Even if we increased the longitudinal interaction, in the with an external pulse |e|xEˆ 0δ(t − t0), t0 = 0.12 fs, where the atomistic limit, we would still only have two occupied NOs. delta peak was numerically represented by a sharp Lorentzian Next, we couple this system to a photonic mode (details are with width σ = 10−4 fs, triggering a broad spectral evolution as given in Fig. 3). Again, we want to consider the static correlation, commonly done for response calculations, and the origin of the but since we now also have the photon mode, we consider a slight coordinate system is located between A and D. From the dynam- extension of the 1RDM to the photonic case by ics of the system, we investigate the induced charge-transfer Z process (see SI Appendix, section 2 for details on an alternative 0 0 ∗ 0 0 eV γe,p(x1q, x q ) = dx2Ψ(x1, x2, q)Ψ (x , x2, q ). E0 = 0.144 1 1 approach). We apply here a positive field e·A˚ , which leads to an almost pure charge transfer from D to A.§ We then If we accordingly extend also the definition of the NOs measure the efficiency by dividing the system into two parts—one 0 0 to include the photon coordinate, we find γe,p(x1q, x1q ) = to the left of x = 0 (associated with D) and the other to the right P∞ ∗ 0 0 (associated with A)—and then define the total charge transfer k=1 nk ϕk (x1q)ϕk (x1q ). If we would integrate out also the photon coordinate, we could get static correlation, even for sit- (leaving D toward A) uations where there is no true electronic correlation.‡ If we Z t Z ∞ find occupation numbers that are different from the above 0 0 c(t) = dt dx[n(x, t ) − n(x, 0)], LCAO case, then we have influenced the genuine static electron– 0 0 electron correlations of the system. Indeed, in Fig. 3 we see that, already for small bond lengths, and the maximal time-resolved transfer the coupled and uncoupled cases are different. The photon mode makes the single-Slater-determinant ansatz less accurate—i.e., max c (T ) = max t∈T [|c(t)|], the higher-lying NOs have a larger occupation. These differences become more pronounced as we approach the atomistic limit— for a fixed coherence time T = 10 fs, while keeping track of the i.e., for large bond lengths. While both cases have dominant sign of c(t ).¶ We then repeat this for different interatomic occupations of the two first NOs, in the coupled case, the higher- max distances RAD = |RA − RD |. lying NOs are no longer exponentially suppressed, but saturate The pure Coulombic system (ξ = 1 and g = 0) has two dif- and become distance-independent. Thus, in the atomistic limit, ferent domains. The first one is the molecular domain (below we have more than only two NOs occupied and, hence, have ˚ larger static correlations. Physically, this is to be expected, since the interatomic distance of 5 A; see also Fig. 3), where we as long as the dipole approximation is valid, the photon field have significant electronic delocalization and charge is trans- interacts with both electrons simultaneously, independently of ferred quickly, oscillating multiple times forward and backward their distance. That also means that there is a tiny bond energy, with small amplitudes such that the effective transfer averages even over very large distances, due to the photons shared by the to almost zero. More interesting is the second domain shown in two electrons. Fig. 4A starting at ∼5 A.˚ For this interatomic distance, the effi- Furthermore, the cavity-induced correlation results in a ciency to transfer charge from D to A is maximal, and beyond slightly earlier onset of the static-correlation limit. This happens this point, we observe decay with exponential character over because the photonic interaction tends to localize charge densi- the interatomic distance (note especially the blue curve with ties stronger—i.e., for realistic light–matter couplings, it reduces

§The results are invariant under inversion of the kick, i.e., whether we consider the transfer from D to A or the inverse process from A to D by inverting the kick E → −E . ‡For instance, if we have only one electron coupled to a mode then while the extended 0 0 ¶ 1RDM (γe,p) would have only one nonvanishing occupation, the reduced (γe) would While our results depend quantitatively on the integration time T, qualitatively, also have more than one, simply as a consequence of that the correlated state cannot be indicated by higher amplitudes at the resonances, our results are consistent for dif- factorized. Furthermore, since the electrons and photons were uncoupled in the pre- ferent T. For the energies present in this setup, 10 fs relates to a medium to high vious case, it is easy to extend it to the present situations by just multiplying with the coherence. The maximum time-resolved transfer observable is less dependent on the bare photonic vacuum. This does not affect the occupation numbers. chosen integration time.

4 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1814178116 Schafer¨ et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 oadsalritrtmcdsacs hrfr,b dutn fre- reso- adjusting by the Therefore, 9) distances. Fig. interatomic to smaller shifted e.g., toward is close transfer (see, Coulomb-mediated efficiency competing increases the while the polariton nance, middle amplifies the A consequentially to or and contribution A A, the and to interaction, D D light–matter from of increasing transfer With (resonant) D. states, efficient these to an of character find the can on one depending in and, decrease dipole, sudden total but the small a undergoes crossing polariton eigenstate, other) avoided also many-body an principle in shows (and then middle spin-singlet D the on with charge all initial almost the energetic distances, with specific its state At change states. other state to initial relation in the position lets and dipole the molecule increases the Stretching the 4B). of (Fig. energies system matter–photon many-body coupled the from behavior physically Dexter understood usual be the can of violation This distances. for suppressed larger exponentially even is and from distances transfer increasing for efficient A very the to a D than smaller into distances switches interatomic suddenly for it A) resonance, to D (A of transfer instead the D inverts to now interaction light–matter characteristic— the transfer while the i.e., of inversion resonance sudden the a around features which behavior exponential an observe system. again Coulomb We pure the to relation in efficiency of amplification 4D Fig. (see 8 around find trans- Dexter we light– optimal fer, and strong previously the distance the bleaches While interatomic interaction behavior. the matter different change drastically again a We observe 4). Fig. the in are (details between given respectively A, in and D frequency of a excitation first select single-electron and coupling light–matter the 4C Fig. see transfer; g many-body polariton with eigenstate many-body initial the of states. crossings coupling. avoided increasing to with distances connected smaller are density-difference to they decay and exponential arise, in Dexter-type eigenstates transfer of (MB) charge domain of the maxima shifts coupling (inverse) increasing New with charge of localization increasing 4. Fig. Sch # led xetta h hnei h Oocptoswl aesm nuneo the on influence therefore some have can will we occupations before, NO the From dynamics. in transfer. charge-transfer change Dexter the the that form expect of interatomic that already picture orbitals large A perturbative For and 3. the D Fig. single-electron mine the in only NOs is higher-order it of distances, suppression exponential the from ent eeta h xoeta upeso fcag rnfrcnas einferred be also can transfer charge of suppression exponential the that here note We /~ω et esic nteculn otepoo ed evary We field. photon the to coupling the on switch we Next, fre al. et afer ¨ 0 = h nertdcag rnfra ucino h neaoi itnefrdfeetlgtmte opig ( couplings light–matter different for distance interatomic the of function a as transfer charge integrated The (A) .Ti xoeta ea orsod oDxe charge Dexter to corresponds decay exponential This ). C A o h lcrncdnmc)wihfaueadrastic a feature which dynamics) electronic the for n(x In B. , t ) C o ersnaieexample. representative a for − eoacswt pia hretransfer charge optimal with resonances A ˚ and n(x eso h hrednmc fDxe-yeadrsnn ih–atroiiae rnfri em ftetime-dependent the of terms in transfer light–matter-originated resonant and Dexter-type of dynamics charge the show we D, )frteidctditrtmcdsacs h iuainbxi 31.84 is box simulation The distances. interatomic indicated the for 0) , {ϕ # g , ϕ u } htdeter- that D rtectdsaeo ,a ela h aepooi vacuum, photonic bare the single-electron as the well and as A D, of of state state first-excited ground product single-electron singlet) (spin the symmetric typical spatially of the the be to beyond chosen F far is in state going nanometers of 6), tens distances (5, of over limit nanometers transfer of energy excitation hundreds allow of cav- to a proven where we has studies, experimental ity Here, recent to energy. comparable are excitation that of 2 transfer setup the investigate consider we Next, Transfer Energy Excitation observables. common and in reactions, changes chemical effects—e.g., states, transition light–matter-induced and chemistry treatment consistent of beyond fully a extension enables further The descriptions by few-level vacuum. dynamics electromagnetic quantum electronic the high- steering controlling this of 4), feasibility ref. the e.g., lights (see, reactions of chemical demonstrations of cavity-controlled experimental be position with can the together transfer control Taken charge the controlled. can that we such resonances cavity, many-body the these of coupling or quency n denote and first-excited the on to state evolved starts we state target the and process, transfer of system excitation-energy-transfer projection coupled the the the evaluate monitor of state To eigenstate initial an evolve. change—the longer conditions no the is than slower sys- reacts the that tem assumes this Hamiltonian—i.e., the quenching After k iewt h olwn eut.Nt htdfeeteeg clsrsl codnl in accordingly result scales energy frames. different time that and in Note couplings qualitatively results. adjusted are following Those the 1. setup with previous line our using calculations reference performed oaodta u bevtosaelmtdt pca ofiuain eselectively we configuration, special a to limited are observations our that avoid To i.e., |ψ 1 |ψ (0)i (0)i = B = p ,wt pcn f0.1058 of spacing a with A, p k ˚ 1/2(|φ e speetdi i.2 hc a energies has which 2, Fig. in presented as 1 1/2(|φ (t = ) 1 A |hψ 0 A | ⊗ i | ⊗ i 1 rtreeg rnfr h initial The transfer. energy orster ¨ (0)|Ψ(t φ 0 D φ 1 D i + i ξ + |φ = NSLts Articles Latest PNAS )i| | 0 D φ and 1 2 n 0poo number photon 30 and A 1 D | ⊗ i ˚ , | ⊗ i ~ω φ 1 A φ = 0 A i) i) 11.97 ⊗ |0i ⊗ 1, ˆ | eV f10 of 5 p .The ). . [1]

PHYSICS the first-order or linear excitation energy transfer. Here, first- order/linear corresponds to the fact that we exchange the lowest- order excitation from D to A and that this excitation energy transfer would be dominant in the linear-response regime. From this, we can define the integrated first-order excitation energy transfer Z T E1(T ) = dt e1(t), [2] 0 where the upper limit T is the chosen coherence time—i.e., small T (of the order of a few femtoseconds) indicates that the dis- carded bath leads to a fast decay of coherence. By exchanging A the first-excited state with the second-excited state—i.e., φ1 → A φ2 —we can accordingly define a second-order or nonlinear Fig. 6. First- and second-order excitation energy transfer with and with- 2 excitation transfer e2(t) and integrated second-order excitation out the self-polarization contribution R with frequency ~ω = 2.340 eV and transfer E2(T ). interatomic distance 42.3 A.˚ For weak light–matter coupling g/~ω = 0.0058 (main plot), small differences are visible. For longer times, these small Distance Dependence of Excitation Energy Transfer. Let us focus changes accumulate and lead to substantial differences, even for weak first on the distance dependence of the excitation energy trans- coupling (see also Fig. 10). For strong coupling (Inset; g/~ω = 0.0579) the fer. If we choose a long coherence time of T = 60.5 fs and then differences are substantial already for short propagation times. This effect plot the resulting integrated first-order excitation energy transfer is not restricted to a resonant frequency but persists also off resonance. We further note that also the mode occupation changes drastically, e.g., for for different interatomic distances in Fig. 5, we find for the purely g/~ω = 0.0579 from ≈ 45 without to ≈ 1 with self-polarization. This origi- Coulombic case that the longitudinal transfer decays as expected nates from a strong charge displacement that appears for short times only 6 and resembles the usual 1/|R1 − R2| Forster¨ behavior. In the without the self-polarization contribution, and, in contrast to other calcula- coupled case (see Fig. 5 for details), the transversal light–matter tions in this section, the charge transfer dominates then the energy transfer. coupling is strongly enhancing the excitation energy transfer (note The simulation box is 79.75 A,˚ with a spacing of 0.397 A˚ and 100 photon the factor of 400 in Fig. 5), and for larger interatomic distances, number states, except for g/~ω = 0.0579 without self-polarization, where the efficiency is even slightly increasing after an initial decay, with we used 250 photon number states. a tendency to saturate once the Coulombic near-field effects van- ish. This is in stark contrast to the usual Forster¨ behavior. Thus, increasing the coupling strength amplifies the transfer drastically, for a long time. Put differently, if the coherence times are dominating the longitudinal interaction, even for small distances, large. For strong coupling, however, this term leads to sub- and leading to an almost distance-independent transfer efficiency stantial differences already for very short times. This shows for large interatomic distances. This finding is in agreement with that the self-polarization term, which has been proven to be recent experiments (5). essential in the equilibrium, especially the ground state (16, 21, However, it is important to note that with increasing light– 22), also can have a strong influence on dynamical processes matter coupling, the excitation-transfer process strongly depends such as excitation-transfer reactions, especially for higher-order on the self-polarization dipole–dipole interaction (16, 21, 22) excitations. 2 2 P2 2 2 (λ · Xˆ ) = e (λxˆn ) + 2e (λxˆ1)(λxˆ2), which is often dis- n=1 Resonances and Efficiency. While we feature that light–matter regarded in quantum-optical models of excitation transfer. For coupling can lead to an efficient long-range excitation energy weak coupling, this contribution, which for free-space situa- transfer, which is very distinct to the usual longitudinal Forster¨ tions approximately cancels with the longitudinal interaction transfer, we would also expect that the efficiency can be after a certain distance (23), only slightly influences the exci- enhanced resonantly for specific frequencies and couplings. The tation energy transfer (Fig. 6). Hence, in such cases, the self- isolated energies of D and A coupled independently to the cavity polarization term would only become visible if we propagate (see discussion in Theoretical Framework) show the well-known Rabi splitting (see Fig. 7 for details). As we couple D and A by the Coulomb interaction—i.e., ξ = 1—and also together to the cavity, the independent electron– photon states (lower and upper polariton for D and A, respec- tively) interact and build lower, upper, and middle polariton states. These states consist dominantly of a single excitation on D or A and the many-body ground-state plus a photonic excitation. The creation of such new light–matter-correlated states was not only observed in dilute gases (5), but also in extended systems (24) and circuit structures (25). The assumption of a resonance that emerges from crossings of isolated polaritons—that is, shifting the lower D polariton into resonance with the upper A polariton—is in this case a too- drastic simplification, as can be seen from Fig. 7. This can only Fig. 5. Integrated first-order excitation energy transfer E1(T) for T = 60.5 hold if A and D have drastically different numbers of particles, fs for different interatomic distances. The Coulombic case (blue) decays as such that the effective coupling is different and the effective col- expected and is multiplied by a factor of 400 here to present the otherwise lective bright polariton state of one constituent is shifted stronger vanishingly small purely longitudinal transfer. The coupled case (green) with (7). However, the lack of such a simplified crossing picture does g/~ω = 0.0055 and ~ω = 2.612 eV shows a drastic enhancement of the exci- tation energy transfer and is mostly distance-independent. The simulation not exclude a resonance or most-favorable setup. Indeed, we will box is 79.75 A,˚ with a spacing of 0.397 A˚ up to a interatomic distance of see in the following that the definition of such an optimal setup 44.45 A,˚ and 106.3 A,˚ with a spacing of 0.529 A˚ for interatomic distances depends on the time scales that we are interested in and with this >44.45 A.˚ We use six photon number states. the given coherence times.

6 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1814178116 Schafer¨ et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 Sch At D. and middle trans- A energy the subsystems excitation matter Consequently, for both state A. between essential and fer most most D the is is of that polariton state mixture many-body a correlated dominantly the is which polariton, many-body the to states isolated the states. of contribution relative the vis- (see is coefficients it polariton—are difference upper where energetic and middle 10, identical.∗∗ the and middle Fig. phases—i.e., and (see lower relative between oscillation both ( fast when setup very of ible) favorable a excitations isolated most shows the The that between A. middle and the D in lie which cies with transfer short-time efficient to efficiency maxima. time high short-time with and no connected resonances, directly many-body almost is of assume space has the system explore we electronic where femtoseconds—the regime the Configuration. Short-Time polariton A is isolated it the calculations. still with many-body/isolated for resonance, cross states of number 40.07 not photon is out does box far simulation but The is cavity, energies. D the isolated by the affected Although systems. lated interatomic an difference chosen energy have 21.2 we of and A, distance of excitation single-electron strengths. cavity lowest coupling the different of for frequency upper calculated The and lines) middle, lower, (thick system energies many-body polariton interacting fully the and lines) 7. Fig. hw nFg itgae rtodrectto nrytrans- energy excitation first-order identify [integrated 8 To Fig. situation weak-coupling in different. differ- the shown In three very limits. strong-coupling following be and the weak- can times in coherence investigate scales ent we trans- time conditions, energy such excitation these femtosec- efficient on of for hundreds conditions fer over The (14). dynamics onds coherent to times potentially coherence long Configuration. Long-Time with line in Master-equation is two-level 9). (8, dephasive observation treatment a This in field. strong calculations a photon few-level bypass the to transfer contribution of the photon excitation allows the which time, small, same very the becomes coupling, contributions at D while strong and A larger, For the become effi- since energy. efficient an more excitation allows even becomes which of this equal, transfer become direct contributions cient A and D the ** ewe oe A n pe D oaio.A hsmxmmpit h relation middle-upper the and point, lower-middle maximum energies this competing maximum. At the its reaches polariton. and polariton (D) energy upper this and between (A) lower between h rqec once oti aiu orsod oteeegtcdifference energetic the to corresponds maximum this to connected frequency The nFg ,w hwteHpedcefiinsfrtemiddle the for coefficients Hopfield the show we 9, Fig. In very a details), for 8 Fig. (see coupling weak consider we If fre al. et afer ¨ h sltdAadDlwraduprplrtneege (dotted energies polariton upper and lower D and A isolated The hsrsl sspotdb osdrn h Hopfield the considering by supported is result This .W hfe h sltdeege yteground-state the by energies isolated the shifted We A. ˚ E 0 g=0 eto 3 section Appendix, SI − T ε 0 A = 0f r osbe(4 5,leading 15), (14, possible are fs >40 oee,i ih-avsigcomplexes, light-harvesting in However, ~ω − o hr oeec ie—.. in times—i.e., coherence short For 5 0 135 60, {15, ε = 0 D = 2.340 0.367 ,wt pcn f0.397 of spacing a with A, ˚ T T eV 15 = eV } scoet eoac ihthe with resonance to close is fteodro few a of order the of ~ω ewe orltdadiso- and correlated between s n ecnie the consider we and fs, o oedetails)—i.e., more for shpesfrfrequen- for happens fs ≈ 2.5 eV ~ω steone the is ) ' n 6/40 and A ˚ 2.5 eV , transfer a rcse togydpn nteCuobitrcinand interaction nonlin- Coulomb These the in on fast. transfer), depend very coherence strongly first-order happens long processes the this ear a beyond case over far strong-coupling increase up the can build (and processes weak-coupling time the nonlinear nonlinear in for the While indication transfer. case an excitation the as strong- in use as processes we well which as weak-coupling situation, the coupling in transfer excitation order weak-coupling processes for 10). observed nonlinear (Fig. be situations highly pho- already can and and as matter complex important of become very energies is approximations, different subsystems few-level the ton of and interplay single-mode the ques- of since results validity strong-coupling the these tion However, strongly figure. depend dedicated situation, time coupling max- coherence weak specific the the on The to broadened. contrast is in transfer ima, the energy D have in effective to and peaked range very frequency A a was possible efficiency the of now transfer energies regime, max- weak-coupling the excitation a while find isolated Also, not the anymore. completely do at we become efficiency spectrum, systems imal many-body isolated the regime in this the “dissolved” in of Since eigenenergies enhanced. the strongly is subsystems different frequency reference the is (for 10 of factor a many-body the in frequencies, exist those longer spectrum. no around energies D efficiency isolated and the transfer A although maximal isolated a the favor to close excitations. are lowest peaks These peaks. distinct fer 79.75 is states. box shift number simulation detun- photon energetic The slight many-body–induced 7. the the Fig. with that of coincides Notice resonance minimum. from a ing short- (equal the becomes maximum are short-time coefficients) long the lines For Hopfield and excitations. vertical maxima, isolated become by D latter and Indicated the A coherences, 100. the and of frequency factor cavity optimal a time by amplified is fs times integration ferent 2.340 42.3 distance 8. Fig. ‡‡ †† rvisas fw ees h rcs,ie,ta ecnie xiaineeg transfer energy excitation consider we that D. i.e., to process, A the from reverse we if also prevails configuration between this result, amplitude a scales. As the time larger. long is and on state dominates minimal, A/D and becomes function wave polariton time-propagated middle and lower/upper ent htasihl ihritgae rtodrefiinya h resonance D the at efficiency first-order integrated higher slightly a that note We between difference energetic the with associated phase corresponding the case, this In ti motn ohglgtterlvneo h second- the of relevance the highlight to important is It by coupling the increase we where limit, strong-coupling the In g /~ω E eV 1 (T nertdfis-re xiaineeg rnfrwt interatomic with transfer energy excitation first-order Integrated ref ehave we e n 0[rt n eododrectto energy excitation second-order and [first- 10 and )] 1 ‡‡ 0 = (t n ekculn frterfrnefrequency reference the (for coupling weak and A ˚ ) ,teectto nrytase ewe the between transfer energy excitation the .058), and g/~ω †† e 2 ref T (t hsidctsta ogtr coherences long-term that indicates This = = ,w bev o ogtm propagation long-time for observe we )], T 5 0 135} 60, {15, 08 o ifrn rqece n he dif- three and frequencies different for 0.0058) n ec erfanfo hwn a showing from refrain we hence and , ,wt pcn f0.397 of spacing a with A, ˚ s oeta h eutfor result the that Note fs. NSLts Articles Latest PNAS ~ω ref 2 = .340 eV n six and A ˚ | ~ω T f10 of 7 this , = ref 15 =

PHYSICS self-polarization term, which are both usually neglected for sim- ple few-level systems. Hence, when the coherence time is large, as in light-harvesting complexes (14), the influence of these terms can become apparent. To conclude this section about excitation energy transfer, we highlight the difference in short-time vs. long-time behavior. If due to strong system–bath interactions, we only have relatively short coherent dynamics, then a cavity frequency between the isolated D and A resonances leads to a strong energy transfer. For longer times, a cavity with a frequency near the isolated A or D resonances is beneficial. Especially in this latter domain, the influence of the self-polarization and Coulomb interaction as well as higher excited states becomes obvious.

Photon-Induced Correlations Let us finally return to photon-assisted electron–electron corre- lations. That photons induce such correlations is not surprising, Fig. 10. First-order (Upper) and second-order (Lower) excitation energy since it is the (longitudinal) photons that induce the Coulomb transfer for weak coupling (for the reference frequency ~ωref = 2.340 eV, ˚ interaction among charged particles. A little more interesting is this is g/~ωref = 0.0058) with interatomic distance 42.3 A. Notice the differ- ence in scales. The relative strength of the second-order excitation energy the finding of Equilibrium Long-Range Correlation, where we transfer is very sensitive to the self-polarization and Coulomb interaction. highlight that also the transversal photons can induce electron– For short times (t < 15 fs), first-order excitation energy transfer is domi- electron correlations. These correlations, however, are very nant, with an optimal transfer for a cavity frequency in between D and weak compared with the longitudinal correlations. In the time- A isolated excitations. Between 15 and 60 fs, long-time coherences build dependent case, we expect a stronger influence of the transversal up, which lead to the long-time peak structure of Fig. 8 and allow a dras- photons. Here, we want to quantify their contribution in the tic amplification of second-order excitations. For t > 60 fs, the long-time afore introduced excitation energy transfer setup 2 (Fig. 2). coherences determine the excitation energy transfer, and the second-order To investigate and quantify these photon-induced correla- transfer becomes dominant. The simulation box is 79.75 A,˚ with a spacing ˚ tions, we consider besides the electronic 1RDM two further types of 0.397 A and six photon number states. of reduced density matrices Z Z single-particle state and is consequently idempotent—i.e., their γ (q, q 0, t) = dx dx Ψ(x , x , q, t)Ψ∗(x , x , q 0, t), [3] 2 P 1 2 1 2 1 2 purity tre/p(γe/p) should be equal to one. If we find a purity Z that is <1, we have a linear combination of several single- 0 0 ∗ 0 0 Γe(x1x2, x1x2, t) = dqΨ(x1, x2, q, t)Ψ (x1, x2, q, t). [4] particle states. In the specific case that we start from an equi- librium configuration with essentially only two occupied Slater determinants—i.e., the minimal basis LCAO limit of Equilib- Again, we choose a normalization of these density matrices to rium Long-Range Correlation—then the electronic 1RDM only one, such that the following holds contains two single-particle states such that the purity is exactly 1/2. If the purity drops <1/2, then we know that we have tr(|ΨihΨ|) = trp(γP)tree(Γe) = trp(γP)tre(γe)tre(γe) = 1, more than a two-determinant wave function and more single- where the traces run over different (sub)spaces as indicated. particle states contribute, we can talk about nontrivial dynamic Of course, this equality does not hold on the level of den- correlation. sity matrices, unless they are completely uncorrelated. In this If we plot the purities for weak light–matter coupling (see case, each 1RDM individually should correspond to a pure Fig. 11 for details), we find for ~ω = 2.340 eV that the photonic state is almost the pure vacuum and the dynamical correla- tion (beyond LCAO correlation) in the electronic coordinates remains small. If we increase the photon occupation by increas- ing the frequency to ~ω = 2.612 eV (close to resonance with the isolated excitation of D), the photon purity is reduced, as now two states (|0ip, |1ip) are present. We also see that this reduces the electronic purity. Besides the purity, the light–matter interaction also affects the photonic fluctuations since the interaction with the elec- trons induces anharmonicities in the photonic system (16, 26). Consequently, the photonic vacuum state is no longer a simple Gaussian, but is elongated along the displacement coordinate q 2 ~ †
h∆ˆq i with ∆ˆq =q ˆ − hqˆi,q ˆ = 2ω aˆ +a ˆ . This elongation h∆ˆq 2i (Fig. 11) is therefore a measure for the nonlinearity of the photonic system due to the presence of matter—i.e., the amount to which photons can interact with each other via a polarizable Fig. 9. Hopfield coefficients for the middle polariton with interatomic dis- §§ tance 21.2 A˚ for different frequencies. The coupling is given with respect medium. The fluctuations increase substantially, even for weak to the reference frequency ~ωref = 2.340 eV. For ~ω = 2.50 eV, we observe equal weights of D and A. For weak coupling, the photonic (P) contribution is dominant, while for strong coupling, it is A and D. The equal weight in §§In reality, as a medium consists of an ensemble of molecules, the induced anhar- the middle polariton appears at the short-time excitation-energy-transfer monicities are influenced also by collective effects of the full ensemble. Furthermore, maximum of Fig. 8. The simulation box is 40.07 A,˚ with a spacing of 0.397 A˚ higher-order perturbative effects can become essential in the thermodynamic limit, and six photon number states. which is indeed achievable by modern experimental investigations (27).

8 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1814178116 Schafer¨ et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 ecnfcoietesi-ige w-oyrdcddniymatrix density reduced Γ two-body spin-singlet the factorize can we classical. purely is photons and electrons between γ facvt nisvcu tt losu ocreaeteelectronic the correlate to us allows state vacuum its in cavity a of process. excitation–transfer the define electron–electron correlations photon-assisted therefore and We Coulombic wrong. that completely conclude is single- exact) Eq. the are (still quantities of mean-field electron on approximation correlation the electron–electron the the of accurate, explicit all photons) very keeps the but to is level due correla- (also mean-field electronic correlations classical electron–electron a the on on photons Eq. of find and strongly approximation we first the depends details), While tions. process for 12 the Fig. that (see approximations transfer uncorrelated (WF) wave-function–based the compare where photonic and electronic uncorrelated—i.e., that are assumed freedom we of step, degrees last the In that space. used we where h ierodrectto rnfrsol hnapproximately then by should given be transfer excitation contribution. linear-order mean-field The or uncorrelated the to corresponds This the rewrite we this, For the influenced. also be transfer fluctuations, excitation photon can first-order the itself as processes such quantities damental fluctuations factor. momentum similar the a by also increase that note We coupling. states. number 79.87 photon for is six substantially and box increases simulation axis) The frequency. right higher The (red, occupation. variance mode displacement the dominantly photonic follow purities Both frequencies. ent xs o ekculn frrfrnefrequency reference (for g/~ω coupling weak for axis) 11. Fig. Sch ¶¶ p e oi n lcrncsse ee ece h mato ietelectron–electron direct of Eq. of impact deviation the the reaches Eq. calculations—i.e., never our in system correlation electronic and tonic ent,ta vnfrsrn ih–atrculn,tecreainbtenpho- between correlation the coupling, light–matter strong for even that note, We et loasmn htteeetoi ytmi uncorrelated, is system electronic the that assuming also Next, h htnasse orlto hne u otepresence the to due channel correlation photon-assisted The eie h nuneo htnasse orltoso fun- on correlations photon-assisted of influence the Besides (t fre al. et afer ¨ by e 6. hscly eteeoeasm htteinteraction the that assume therefore we Physically, ). ref 1 (t = γ = ) htnc(h;sld n lcrnc(l;dse)prte (left purities dashed) (El.; electronic and solid) (Ph.; Photonic e 1 08 fstp2wt neaoi itne42.3 distance interatomic with 2 setup of 0.0058) e ≈ (0) Γ 1 MF e tr tr (x ee ee (t steeetoi RMof 1RDM electronic the is 1 (Γ = )  x 2 X e , |ψ x tr 1 0 k N 1 ˆ 1 e x ,n (γ p (0)ihψ 2 0 = =0 , e t P (t ) hψ ≈ )γ n N t e 1 =0 p e γ e 1 (0)|hk (0)|) 1 e (0)) (t (x |n ) 1 ihn , 2 fEq. of X x |Ψ(t X 1 0 | , k N norrsrce photon restricted our in t ,wt pcn f0.529 of spacing a with A, ,n k N p ˚ )γ )ihΨ(t ,n p =0 1 e 5 =0 Ψ(t ~ω (x ntefloigway following the in hk httet electrons treats that 5 ψ hk 2 ref 6 smc mle hnof than smaller much is |γ 1 , )Ψ )|n fw then we If (0). | x = p γ htas treats also that 2 0 |n p ∗ , 2.340 i|ψ e ¶¶ | (t t n 1 i, ). (t ) n differ- and A i, ˚ t e ≈ ) (0)i eV Γ n the and h∆ˆ e htis that  (t p ) [5] [6] 2 ⊗ A i ˚ ela h hsnchrnetms u netgtosstrongly investigations as Our times. interactions coherence self-polarization chosen very the and are as Coulomb effects well observed the the to of free-space sensitive Many the distance. suppressed to exponentially increasing contrast is and with in efficiency donor is the the where This transfer, when apart. free- Dexter even far of very processes, degrees are crossings efficient photon acceptor The very avoided the to to efficiency. where lead due transfer transfer, dom spectrum enhanced charge many-body the for drastically in true a holds as same trans- well energy as excitation usual distance-independent fer the almost to an to contrast interaction lead In many-body nuclei. iso- and the F exact electrons free-space of A the and consequence with from D (long a resonance between shift system is in energetic acceptor energies frequency slight or lated a donor A has the times). excita- of that coherence the excitations cavity between isolated a in the coherence by is (short that system or acceptor frequency and times) donor a isolated the has of that tions cavity efficiently a most enhanced by espe- coherence be can the important, transfer on energy become Depending excitation times, transfer. and energy degrees distances excitation electronic in large cially of over entanglement freedom an of to lead which relations, a within dynamics interval. only effects coherence-time considering specific Decoherence by situa- mechanics. effectively included quantum-chemical quantum were standard free-space the to for with access tions that compare easy efficiencies could transfer us for- we as allowed real-space such it first-principles quantities since distance-dependent a advantageous exci- change and Here was drastically charge a can mulation transfer. as cavity how such energy a process, of tation chemical to examples fundamental due a presented vacuum alter have electromagnetic the we of work, this In Conclusion and Summary transfer. on excitation impact linear direct the with as effect such nonnegligible observables a correlation is multiple this in that seen measures have We distances. large over system ml o hr ie,snetecasclapoiaino Eq. of approximation classical remains the transfer since excitation times, linear short the on for system small photonic and electronic Eq. 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Fig. n lcrnccreainrslsi rmnosdvain.Tesimulation The deviations. 79.87 tremendous is in box results correlation Disregard- electronic transfer. ing the overestimates (~ω drastically vanishingly occupation approximation mode is mean-field substantial field of mean situation the the For and small. transfer, (~ω the occupations dominates photonic correlation tronic small for Especially accurate. efudta akcvt a nueeeto–lcrncor- electron–electron induce can cavity dark a that found We ~ω is-re xiaintase o eu ihitrtmcdis- interatomic with 2 setup for transfer excitation First-order ref o o(oetefco 0)adwa opig(g/~ω coupling weak and 100) factor the (note no for A ˚ ,wt pcn f0.529 of spacing a with A, 5 ˚ = and rtrcs,tetaseslpoosdet cavity a to due photons transversal the case, orster ¨ 2.340 o ekculn,teipc fcreainbetween correlation of impact the coupling, weak For 6. eV n w ifrn rqece.W opr the compare We frequencies. different two and ) n i htnnme states. number photon six and A ˚ NSLts Articles Latest PNAS 1 ihteapproxima- the with = = 2.340 2.612 eV 5 | eV ,elec- ), svery is f10 of 9 ,the ), ref =

PHYSICS suggest that the theoretical description of chemistry under the Materials and Methods influence of a strongly coupled quantized mode demands a The electronic structure is calculated on a 2D grid as indicated in consistent first-principles description. Novel techniques such as the figures, where derivatives are performed by fourth-order finite dif- quantum-electrodynamic density-functional theory (17) and the ferences. The photonic contribution is included through a converged cavity-Born–Oppenheimer approximation (18) could realize this truncated expression for the creation and destruction operators with milestone in the near future, although our results, especially pre- a dimension as indicated. Special caution is demanded for ultrastrong sented in Photon-Induced Correlations, set demanding require- couplings and large dipole moments, as the photonic occupation then rises ∝ (λX)2. Further research in this direction could profit from an ments for those techniques. Many well-established results of adjusted basis according to ref. 16. Time propagations use the Lanc- molecular physics change under those novel conditions, and con- zos method. We ensured that the results do not change more than a trol of chemical properties by adopting the photon field seems few percent, especially the physical conclusion, for increasing the sim- possible. The present findings could be extended to the cavity- ulation box or decreasing the spacing. The interatomic distances are mediated interaction between 2D materials and nanostructures. selected such that charge and excitation energy transfer are significantly By selectively addressing single-molecular dimers in specific fixed distinct. configurations as, e.g., possible on surfaces (28) or in strongly ACKNOWLEDGMENTS. We thank Arunangshu Debnath and Johannes Flick confined fields (29), the gathered insights could be directly val- for insightful discussions. This work was supported by European Research idated in experiment, paving the way to novel technological Council Grant ERC-2015-AdG-694097 and partially supported by Federal advances. Ministry of Education and Research Grant RouTe-13N14839.

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