Superradiance and Symmetry in Resonant Energy Transfer
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Superradiance and symmetry in resonant energy transfer Severin Bang, Stefan Yoshi Buhmann, and Robert Bennett Physikalisches Institut, Albert-Ludwigs-Universit¨atFreiburg, Hermann-Herder-Str. 3, D-79104 Freiburg i. Br., Germany (Dated: December 13, 2019) Closely-spaced quantum emitters coherently sharing excitation can release their energy faster than suggested by a simple sum over their individual emission rates | a phenomenon known as superradiance. Here, we show that the assumption of closely-spaced emitters can be relaxed in the context of resonant energy transfer, instead finding that certain symmetrical arrangements of donors are just as effective. We derive exact expressions for the superradiant fidelity in such situations, finding some surprising results such as complete suppression of the rate for a single acceptor within a homogeneous spherically symmetric distribution of coherent donors. When quantum emitters share an excitation and in- to spontaneous decay, achieving a superradiant FRET teract coherently, the stored energy can be released more rate does not require closely-spaced donors or acceptors. quickly than it would be if the emitters were taken as in- We will do this by building up a general expression for dependent. This phenomenon is known as superradiance, the superradiant fidelity, taking into account arbitrary and is one of the most prominent examples of collective arrangements of donors and acceptors as well as both effects in quantum theory. First predicted in 1954 [1], its near- and far-field behaviour. first experimental verification was carried out in sodium We work within the general formalism of macroscopic vapour [2] and has been numerously observed since. Its QED, making our work amenable to inclusion of dielec- most convenient theoretical description is based on the tric bodies near the decaying cloud. Within this frame- Dicke states, where the ensemble of two-level emitters is work, the transfer rate Γ from a single donor at rD to a mapped to the algebra of angular momentum (see [3] for single acceptor at rA is given by [20]; a review). The criterion for superradiant behaviour in 2 2 2πµ0! 2 this context is that the atom{atom separation should be Γ = dA G(rA; rD;!) dD (1) significantly smaller than the wavelength of the involved ~ j · · j transitions. where dD and dA are the transition dipole moments of The energy released from a decaying system can also be donor and acceptor, respectively. The spectral overlap captured by another system, causing the latter to reach of donor and acceptor is taken here to be dominated by an excited state. This is known as (F¨orster)resonant en- a single frequency, denoted as !. The matrix G(r; r0;!) ergy transfer (FRET/RET) [4], and is a pervasive mech- is the dyadic Green's tensor describing propagation of anism by which energy can be transported from donor to electromagnetic radiation of frequency ! = k=p" from r0 acceptor (for recent reviews, see [5{7]). Its characteris- to r in a given environment, being defined as the solution tic inverse sixth power distance-dependence in the near to field is a widely-used `spectroscopic ruler' in the anal- 2 G(r; r0;!) k G(r; r0;!) = Iδ(r r0) : (2) ysis of macromolecular structure, while analogous pro- r × r × − − cesses such as interatomic Coulombic decay [8] are the supplemented by appropriate boundary conditions. The subject of intense interest due to their relevance in radi- derivation of (1) was based on Fermi's golden rule ation biology [9]. A description beyond the electrostatic dipole{dipole regime was developed on the basis of molec- 2π 2 Γ = Mfi δ(Ei Ef ) (3) ular quantum electrodynamics (QED), revealing that the ~ j j − arXiv:1912.05892v1 [quant-ph] 12 Dec 2019 resonance energy transfer rate beyond the nonretarded with the transition matrix element F¨orsterregime falls off with an inverse quadratic distance M = f f H i i (4) law [10]. Many-particle effects in the form of passive me- fi h jA ⊗ h jD int j iD ⊗ j iA diator atoms have been shown to modify the transfer of the dipole coupling interaction Hamiltonian [11{13]. On a macroscopic level, multilayer structures can have a similar effect [14, 15]. H = d^ E^(r ) d^ E^(r ) (5) int − D · D − A · A Generalised F¨orstertheory is a many-particle version written in terms of the macroscopic QED electric field of FRET where the excitations mediating the process [21, 22] can be spread over a collection of donors or acceptors making up a molecular aggregate [16{19]. Superradi- 2 ^ 1 ! 3 ~ ant behaviour is a signature of this arrangement, usually E(r) = i d! 2 d r0 Im"(r0;!) c π0 expressed as delocalisation of a quasiparticle excitation Z0 Z r G(r; r0;!) ^f(r0;!) + H.c. (6) known as an exciton. Here we will show that in contrast × · 2 where ^f y and ^f are the creation and annihilation opera- tors for elementary polariton-like excitations of the com- posite field-matter system, and "(r;!) is the relative per- mittivity at position r and frequency !. The important assumption here was that the donor and acceptor ini- tial states i D and i A were simply the eigenstates e and g correspondingj i j i to the upper and lower levelsj ini the involvedj i transition, respectively. If the excitation is instead coherently spread over N sites, we have; 1 i = ( e + e + ::: + e ) (7) j iD pN j i1 j i2 j iN one can follow the same steps as the previous calculation to find N N 2πµ2!2 Γ = Γ = 0 [d G(r ; r ;!) d ] tot ij A · A Di · Dj i;j=1 i;j=1 ~ X X [d G∗(r ; r ;!) d ] (8) × Di · Dj A · A which reduces to Eq. (1) if N = 1. Equation (8) gives the energy transfer rate for an excitation spread over N FIG. 1. Superradiant fidelity F for a two-donor system. The donors arranged in an arbitrary way, and is the main acceptor (red) is placed at the origin, while the first donor (blue) is fixed 1:8µm away. The second donor is free to move, expression we will be using for the rest of this work. with the colour bar representing the superradiant fidelity for As a point of comparison, we will consider a convenient a donor placed at that point. Both donors have transition representation for a cloud of atoms which can be consid- wavelengths of 19µm, meaning this example is in the non- ered as identical. These are the Dicke states, based on the retarded (electrostatic) regime. algebra of angular momentum. The general Dicke state for an N-particle system with M excitations can be found For the case of ideal (Dicke) superradiance have F = 1, from the completely excited state e i e 2 e N E via j i ⊗j i · · · j i ≡ j i and for no superradiance (no coherence) we have F = 1=N. (J + M)! J M We can now calculate the fidelity in a variety of differ- J; M = J^ − E (9) j i sN!(J M)! − j i ent situations. The simplest and most instructive is to − consider a setup with two donors. We will assume that N where J = , M runs from J to J in integer steps and both donors and the acceptor are randomly oriented, un- 2 − J^ is the collective angular momentum lowering operator der which conditions we can take d d d 2I=3, mean- ^− N N ⊗ ! j j J = σ^i− = g e If the system has one ing Eq. (8) simplifies to; − i=1 i=1 j ii h ji excitation, we have M = J + 1. Using this in (9), we 2 2 P P − N 1 2πµ0! 2 2 get; J; J + 1 = 1=N!(N 1)!J^ − E ; from which Γij = dA dD − 9 it followsj − that thei matrix element− of thej collectivei dipole ~ j j j j p Tr [G(rA; rDi;!) G∗(rDj; rA;!)] (13) operator D^ = d^i = di(J^ + J^+) for decay to the i − × · ground state M = J is given by; J; J D^ J; J + 1 = We will use the vacuum Green's tensor, given by P d pN Taking d D in (1), weh then− j havej − i i D ! c2ei!ρ/c !ρ !ρ 2 G(r; r0;!) = f I g eρ eρ ΓSR = N Γ (10) − 4π!2ρ3 c − c ⊗ h (14)i which is also what we find when using Eq. (8) with Γ = ij where ρ = ρ = r r , e = ρ/ρ, f(x) = 1 ix x2, Γ. If we instead carry out an incoherent sum (Γ = 0 0 ρ ij g(x) = 3 3ij xj xj2 −andj we have dropped a term− propor-− for i = j) over the N donor-acceptor pairs, we get − − 6 tional to δ(r r0) since the donor and acceptor points N are never coinciding.− Using this in Eq. (13), we find Γ0 = Γi = NΓ (11) the results shown Fig.1, where it is clear that optimal i=1 X placement of the second donor relative to the first is not This leads to a natural definition of the superradiant fi- limited to simply being near it as is usually assumed in delity F : Dicke physics, rather there is another region on precisely N the other side of the acceptor for which the fidelity is Γ Γij F = SR = i;j=1 (12) maximised. Placing the second donor there, then sub- NΓ N 0 NP i=1 Γii sequently placing a third donor at the new position of P 33 1.0 0.8 0.6 F 0.4 0.2 0.0 2 4 6 8 10 z0/R0 FIG.