Entangled Photon Resonance Energy Transfer in Arbitrary Media

Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 3181−3188

pubs.acs.org/JPCL

Entangled Photon Resonance Energy Transfer in Arbitrary Media K. Nasiri Avanaki†,‡ and George C. Schatz*,† †

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, United States

‡

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S Supporting Information *

ABSTRACT: Inspired by the unique nonclassical character of two-photon interactions induced by entangled photons, we develop a new comprehensive Förster-type formulation for entangled-two-photon resonance energy transfer (E2PRET) mediated by inhomogeneous, dispersive, and absorptive media with any spacedependent and frequency-dependent dielectric function and with any size of donor/ acceptor. In our theoretical framework, two uncoupled particles are jointly excited by the temporally entangled field associated with two virtual photons that are produced by three-level radiative cascade decay in a donor particle. The temporal entanglement leads to frequency anticorrelation in the virtual photon’s field, and vanishing of one of the time-ordered excitation pathways. The underlying mechanism leads to more than 3 orders of magnitude enhancement in the E2P-RET rate compared with the uncorrelated photon case. With the power of our new formulation, we propose a way to characterize E2P-RET through an effective rate coefficient KE2P, introduced here. This coefficient shows how energy transfer can be enhanced or suppressed depending on rate parameters in the radiative cascade, and by varying the donor−acceptor frequency differences.

E

states,25 and entangled-two-photon (E2P) states from spontaneous parametric down-conversion (SPDC)26,27 widely used in quantum information, data encryption28−30 and quantum communication.31,32 In recent experiments21,22,33,34 utilizing the SPDC technique, the phenomenon of entangled-two-photon absorption (E2PA) interestingly showed linear rather than quadratic dependence of the absorption rate on excitation intensity, which was dominant at low intensity. Indeed, the nonclassical approaches involving two or more entangled photons provide exceptional efficiency over conventional incoherent light sources. This motivates the present work. In this paper we seek to understand the underlying mechanism of RET for a system consisting of a single excited donor and a pair of uncoupled acceptors, with RET involving an entangled virtual pair of photons. The main goal here is to explore if the quantum state of light can give us better control/ enhancement for the RET rate. This work thus bridges between the two fields of quantum optics and resonance energy migration in photoactive materials, and a goal of our analysis is to develop the theory of RET in arbitrary media and going beyond the electric dipole approximation. There is a demanding list of expectations for an ideal entangled-photon source such as35−37 (i) entanglement fidelity in which the produced two photons should closely resemble the maximally entangled Bell state, (ii) deterministic generation

xcitation energy transfer including radiative and nonradiative mechanisms, is a universally important photophysical process in photoactive systems defined as the relocation of electronic excitation energy from an optically excited donor to a nearby acceptor. The originally formulated Förster theory1 can describe RET in various problems and with some changes it can be utilized in more efficient hybrid systems2−6 as well. With the design and synthesis of multichromophore macromolecules, a new theoretical framework has been developed for the general case of twin-donor RET7 in the vicinity of an acceptor,8−11 which is of interest for biomimetic energy conversion. These systems capture optical radiation with high efficiency due to the large number of antenna chromophores and efficient mechanisms for channeling energy to an acceptor core.12 In another direction, with advances in nonclassical light sources13−17 and their application in exploring new phenomena in multiphoton processes, there has been a rebirth of interest and extensive attention in nonlinear laser spectroscopy involving entangled photons, perhaps most prominently in two-photon absorption/emission as fundamental components of nonclassical light-matter interaction. The features of quantum light open up a new era for discovery of valuable information on relaxation, transport pathways, spectroscopy at extremely low input photon fluxes,18,19 entanglement-induced two-photon transparency,20 and entangled-photon virtual-state spectroscopy.17,21,22 These fascinating developments cannot be retrieved from the linear response of the system interacting with the classical form of the light. Today we have access to a variety of techniques for producing quantum light,23,24 entangled coherent © 2019 American Chemical Society

Received: March 29, 2019 Accepted: May 22, 2019 Published: May 22, 2019 3181

DOI: 10.1021/acs.jpclett.9b00902 J. Phys. Chem. Lett. 2019, 10, 3181−3188

Letter

The Journal of Physical Chemistry Letters

We assume that RET involves absorption of the two photons by a pair of uncoupled particles (see Figure 2) taken to be twolevel systems (ground and excited states |gi⟩ and |ei⟩, i = 1, 2). Generally, the acceptors are not identical with corresponding excitation energies ℏωi, and spontaneous emission rates γi. We assume that the mean excitation time for each particle is much shorter than the lifetimes of the two excited acceptors so that we can consider that the two excited states have infinite lifetimes (γ1,2 ≈ 0) and maintain their excitation forever. Under the rotating-wave approximation, the Hamiltonian of the system can be written as

(emission of one and only one pair of entangled photons), (iii) high collection efficiency of emission with very low loss, and (iv) indistinguishability (the emitted photons from different trials should be exactly identical in all degrees of freedom). However, fulfillment of all these features has been challenging so far. As for the SPDC photons, characteristics of (i) and (iii) simultaneously38 and very recently (i), (iii), and (iv)39 have been retrieved. However, in SPDC, the photon pairs are generated randomly and usually accompanied by undesirable multipair emissions. For solid-state artificial atomic systems with radiative cascades (single quantum emitters such as quantum dots), the quantum efficiency is near unity.40 Over the last decade, tremendous progress has been achieved in eliminating defects and improving the functionality of the cascade sources.41−46 Recently, entangled photons from a cascade source with high fidelity of 0.9 (i) were generated with a record-high pair efficiency of 0.37 per pulse, which is the product of a pair generation rate of 0.88 (ii) and a collection efficiency of 0.42 (iii), on the same device.47 This is an important step forward in the development of quantum light. However, since we are interested in energy transfer which might include the near field regime, we assume that the source of entangled photons is a biexciton cascade48−50 that takes place in a single-particle quantum dot (QD) with a perfect symmetric geometry (by optimizing the growth conditions of the quantum dots) (see the Supporting Information for more details). This provides us with a table-top source of triggered entangled-virtual-photon pairs, as it can produce no more than two photons per excitation cycle.48 Indeed, using pulsed excitation, the two emitted photons are “clocked” with one appearing shortly after the other.40,51−53 And the entangled state in this case is both correlated in time and anticorrelated in frequency. In general, for any frequencies ω1 and ω2, if the cross 54 the frequency correlation function satisfies g(2) × (r,ω) = 1, bipartite light beam is factorable; otherwise, the light beam has some frequency correlations between parts. In a bipartite twophoton state with total energy of ℏ(ω1+ω2) = E2p, frequency anticorrelation means that if there is one photon in the ω1frequency mode in the first partite, there is a higher probability to find the other photon in (E2p − ω1)-frequency mode in the second partite. We employ this concept in our method for the cascade emission from the three level QD source (see Figure 1) with a small width γα and a large width γβ. The entangled state produced by this emission process involves a single-particle donor. It should be noted that the excited QD is created by some prior event, and at time t = 0, it is ready for the emission such that the prior history is not important.

H = H0 + Vint = ℏω1b1b1† + ℏω2b2b2† + ℏ

∑ ∑ ω lalλalλ † + Vint λ = 1,2

l

(1)

where bi = |gi⟩⟨ei| and the annihilation operators al are timeindependent. Here [aλl , aλl′′†] = δλ,λ′δl,l′ and the interaction potential, Vint is defined as Vint(t ) = −∑ μi (bi + bi†) ·[Ei(+)(R i,t ) + Ei(+) †(R i,t )] i

(2)

Here μi is the electric dipole transition of acceptor i while E + E(+)† is the emitted induced electric field generally written as a mode expansion,12,55 arising from the donor at the position of the acceptor Ri (where the donor is placed at rD, the acceptor i is at rAi, and Ri = rAi − rD), (+)

E(+)(R i,t ) = i

∑ ∑ ϵl(λ)al(λ)ei(k ·R − ω t) l

λ = 1,2

ℏω l 2ϵ 0 V

In eq 3, ϵl(λ) =

i

l

(3)

l

e(l λ) and the two e(λ) l (polarization vectors)

are conventional unit vectors for left- and right-hand circularly polarized (LCP and RCP) waves, perpendicular to the wave vector, kl = ωl/c, and V is an arbitrary quantization volume. Then the field−matter interaction involves a coupling term of the form ,i(ωl) = −

i ℏ

= ,i le

∑ λ = 1,2

i(k l·R i − ω lt )

ωl 2ℏϵ0V

where ,i l = −i

⟨ei|μi ge |gi⟩·ϵl(λ)e i(k l·R i − ωlt ) (4)

× μige ·e(λ) is a slowly varying function of

the virtual photon frequency transferring energy from the donor to the acceptors. For the sake of simplicity, we assume a single polarization, so the creation/annihilation operators depend only on the frequency a(ωl) = al and the interaction potential is given by Vint = ℏb1† ∑ ,1(ω l)al + ℏb2† ∑ , 2(ω l)al + H.c. l

l

(5)

We proceed by defining the initial state of the donor as ρ0 = |ϕ⟩⟨ϕ| (ρ0;kk′,qq′ = ⟨1k, 1q|ρ0|1k′, 1q′⟩) and the initial state of the acceptors as |g1g2⟩⟨g1g2|. Note that the initial state of the donor can be defined as the pure two-photon state; |ϕ⟩ = ∑k,qηkq|1k, 1q⟩ or a mixed state in its spectral decomposition form. The characteristics of the whole system are then determined by |g1g2⟩⟨g1g2| ⊗ ρ0 and a unitary evolution super operator U(t), since we assume the two acceptors have infinite excited-state lifetimes. This informs us about the dynamics of the

Figure 1. Schematic generation of a two-photon state from the excited cascade of a three-level donor system. Initially, the system is in excited state |e⟩, and through the sequential emissions, a photon pair is generated with frequency anticorrelation due to energy conservation. 3182


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Figure 2. Schematic of the two-acceptor interacting with the E2P model. There is no interaction between the two acceptors, and the frequencies of the excited states are ω1 and ω2, respectively. The central frequencies of the two incident beams, ωα and ωβ, are far from resonance with the single particle, but their sum is almost equal to the sum of the excitation energy of the acceptors, i.e., δ = ωα + ωβ − ω1 − ω2 ≈ 0.

Note that eq 7 indicates that the state of the entangled photons cannot be factorized into two separable parts. The first term in the bracket represents the general single-photon emission process and the second term corresponds to the frequency anticorrelated (of the second) emission. The (anti)correlation term comes from energy conservation, since the total energy ℏ(ωk + ωq) of a photon pair should be close to ℏ(ωα + ωβ). Classically, there should be four possible ways of passing the two-photon energy from a single donor to two acceptors: two pathways corresponding to which photon is absorbed first, and two pairings corresponding to which acceptor absorbs which photon. Indeed, we have a temporally entangled field from the cascade state with four contributions from the joint-excitation amplitude. However, with the entangled input field, a time ordering is imposed at the emitter and two of the interfering pathways in each acceptor−field pairing have zero amplitude.58 Later we use the conclusion of this discussion to obtain the correct expression for the probability transition amplitude of the system with entangled photons. Given the above expression for the field states, the initial and final states of the system are described as

system over time using the density matrix of the entire system at any time t denoted by ρ̃(t) = U(t)|g1g2⟩⟨g1g2| ⊗ ρ0. Generally, a pure two-photon state in Hilbert space is represented by |II, donor⟩ =

∑ η(ωk ,ωq)|1 : ωk , α ; 1 : ωq , β⟩ (6)

k ,q

The symbol |1k, 1q⟩ represents the tensor product |1k⟩ ⊗ |1q⟩ of two single-photon states in frequency mode ωk(q) of subsystem α(β) with the normalized coefficient of η(ωk,ωq) ≡ ηkq. We then define the donor (emitter) as a three-level particle generating a photon pair through the cascade process (see Figure 2). In this process, the donor is initially excited at t = 0 to the top level |e⟩ with the energy ℏ(ωα + ωβ) and width γα. The first photon is radiated after the transition from |e⟩ to the intermediate state |m⟩ with the frequency ωα and a Lorentzian distribution in frequency in which its width is |γα − γβ|. For γα > γβ there will be some population accumulation, but if γα ≪ γβ, the state |m⟩ has a short lifetime and another photon is quickly emitted. At a given time t, the state after these emissions is given by56 |II, Cas⟩ =

∑ ηkqCas|1k , α ; 1q , β⟩

|i ⟩ ∝

l o 5 1 − e−γβt + i(ωq − ωβ)t o m o ω − ωβ + iγβ (ωk − ωα) + i(γα − γβ) o o n q | o 1 − e−i(ωα + ωβ − ωk − ωq)t − γαt o o − } o −(ωα + ωβ − ωk − ωq) + iγα o o ~ k ,q

ηkqCas =

k ,q

|f ⟩ ∝ |e1e2⟩|0⟩

In the above expression, 5 is the normalization of the two2c 3 γαγβ

spontaneous emission rate γα(β) =

56,57

U (t )|ϕ⟩ ∝

associated with the

V dα(β)2ωα(β)3 56 . Here dα = μem and 6πϵ0ℏc 3

∑ ηkq∫ k ,q

0

t

dt 2

∫0

t2

I I dt1 Vint (t 2) Vint (t1)|g1g 2⟩|1k , 1q ⟩

(9)

dβ = μ corresponds to the transition dipole between |e⟩ → |m⟩ and |m⟩ → |g⟩, respectively. mg

(8)

This means both acceptors are initially in the ground state, |g1g2⟩, and the field is initially in a pure two-photon state (or it can be a mixed state) ∑k,qηkq|1k, 1q⟩. From second-order perturbation theory, the state of the acceptor after two interaction events (at times t1 < t2) is obtained from

(7)

photon state defined as 5 =

∑ ηkq|g1g 2⟩|1k , 1q ⟩

VIint(t)

The donor−field coupling in the interaction picture, = eiH0tVint(t)e−iH0t can be simplified using the Baker−Campbell− 3183


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ωk and ωq, we can determine the total transition probability amplitude (more details in the Supporting Information). However, for the two-photon cascade state introduced in eq 7, if γα ≪ γβ, the decaying terms related to e−γβt vanish at a short time t ∼ γβ−1 and the only remaining term at longer time is

Hausdorff formula. The joint-excitation amplitude is then given by56,58,59 M(ω1 ,ω2 ,t ) = {⟨e1e2|⟨0|}U (t )|ϕ⟩ =

∑ ηkq⟨e1e2 , 0|U (t )|1k , 1q ; g1g 2⟩ k ,q

(10)

|II, Cas⟩ ≈

Assuming the interactions between entangled field and two acceptors are weak, the leading term from the evolution operator eq 9 (the second term of Dyson’s series) is presented as ⟨e1e2|U (2)(t )|g1g 2⟩ = e−iH0t / ℏ ∑ ak aqR kq k ,q

(11)

∑kq ↔ 2 ∫ 2 ∫

2

( )ω

k

2

ij yzl −(γ + iδ)t | o 1 1 o o1 − e α j zzo = −Λjjj + zzm } o o jj ωα1 + i(γ − γ ) z o o z + − − ω γ γ δ γ i( ) i α2 β α β α {n α ~ k | l ij y ( i ) t − γ + δ o zzo 1 1 1−e α o j zzo ≈ −Λjjj + } zzm o o jj ωα1 + iγ o o z ω + γ δ − γ i i α2 β β {n α (17) ~ k

ωq dωk dωq dΩk dΩq . Note the

⟨e1e2|U (2)(t )|g1g 2⟩ V2 4π 4c 6

∫ ∫ dωk dωq [ωk 2ωq2a(ωk) a(ωq)Rkq]

where ωij = ωi − ωj and Λ = 5

(12)

×

1 − e i(ω2 − ωq)t ωq − ω2

ω12ω2 2 ,1(ω1) , 2(ω2). We note that the distance dependence of the propagation length has been left out of the transition probability amplitude expression. If we had included it in our deviation, it would be hidden in Λ coefficient. We will come back to this later (eq 23 is the complete version). The detuning for two-entangled-virtual-photon absorption where none of the two photons are in resonance with the two acceptors is defined by δ = ωα + ωβ − ω1 − ω2. Note that in the resonance condition the sum of their two energies almost matches the sum of the two acceptor’s excitation energies; ωα + ωβ ∼ ω1 + ω2 (i.e., δ ≈ 0). It then follows from the above expression that the transition probability is given by

1 − e i(ω1− ωk)t ωk − ω1 (13)

7Cas(t ) = |4 Cas(ω1,ω2 ,t )|2 = |ℏΛ|2

4(ω1,ω2 ,t ) = Mαβ ,12 + Mαβ ,21

2

2

(18)

where Eδ = Eωα+ωβ − Eω1+ω2 = ℏδ and the E2P coefficient, KE2P, is defined as

where Mαβ,12 is defined as the probability amplitude that acceptor 1 interacts with virtual photon α first (at time t1) and that acceptor 2 absorbs virtual photon β after that (at time t2). This time-ordered excitation amplitude is closely related to the two-photon correlation amplitude.56,58 Furthermore, the time-dependent two-photon excitation probability due to quantum entanglement is defined as the projection (measurement) of the density matrix of the whole system at any time t; ρ̃ = U|g1g2⟩⟨g1g2| ⊗ ρ0 onto |e1e2⟩⟨e1e2|,

KE2P =

(ωα1 + ωα 2)2 + 4γβ 2 4ℏ2V 2 4 4 ω ω γ γ 1 2 α β (ωα12 + γβ 2)(ωα 2 2 + γβ 2) π 4c 6

(19)

In the case where γα ≪ δ, eq 18 is recast as 7Cas(t ) = KE2P |,1(ω1) , 2(ω2)|2

P = Tr ⟨e1e2|ρ ̃|e1e2⟩

sin 2(Eδt /2ℏ) (Eδ /2)2

γα ≪ δ (20)

= Tr ⟨e1e2|(U |g1g 2⟩⟨g1g 2|ρ0 )|e1e2⟩ = |4(ω1 ,ω2 ,t )|

(ωα12 + γβ 2)(ωα 2 2

1 − e−(γα + iδ)t 2 Eδ − iℏγα + γβ )

1 − e−(γα + iδ)t = KE2P |,1(ω1) , 2(ω2)| Eδ − iℏγα

2

(14)

2

(ωα1 + ωα 2)2 + 4γβ 2

2

∫ ∫ dωk dωq [ωk ωq a(ωk) a(ωq)Rkq](ηkq + ηqk) 2

, 2(ω2) =

π 2c 3

If we ignore the propagation length from donor to acceptor by setting z1 ∼ z2 ∼ 0, it then leads to the approximation ,i(ω l) ≈ ,i(ωi). (Later in this derivation we will include for the propagation length explicitly.) Imposing the time ordering t2 > t1, the resulting expression for the total joint-excitation amplitude becomes58 V2 = 4π 4c 6

V2 ω 2ω 2 ,1(ω1) π 2c 6 1 2

2V γαγβ

The response function of the uncoupled acceptors to the incoming field Rkq is defined as the product of two individual single-photon single-particle response functions R kq = ,1(ωk) , 2(ωq)e−i(k1·R1+ k 2·R 2)

(16)

4Cas(ω1,ω2 ,t )

2

factors of 2 includes for the polarization effects that are otherwise omitted in the derivation. Then we arrive at

=

l o o 1 − e−i(ωα + ωβ − ωk − ωq)t − γαt | o ×o m } o o o (ωα + ωβ − ωk − ωq) − iγα o o o n ~

With eq 13 and eq 14 and some rearrangement, the transition probability amplitude reads as follows

Suppose we have a continuous frequency distribution of the emitted field, so that we can make the replacement V (2πc)3

5 (ωk − ωα) + i(γα − γβ)

The above expression shows how the transition probability varies with line-shape parameter δ and time. At large enough time τ (τ ≥ 2πℏ/Eδ), the total probability of transition is a linear function of time and using Fermi’s Golden rule accordingly (more details in the Supporting Information), we can determine the rate as

(15)

“Tr” stands for the trace operation over the field variable. Substituting the response function eq 13 into eq 14 and employing the residue theorem to integrate over the frequencies 3184


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Figure 3. Contour plot of rate for E2P-RET (left) and S2P-RET (right) for identical acceptors in resonance condition. In this set of calculations γα = 0.005 eV, 0.05 < γβ < 0.35 eV, and ωα0 = δ − ωβ0.

W τCas = lim

τ →∞

The graphs in Figure 3 show us that the rate from E2P-RET is in general more than 3 orders of magnitude higher than the rate from S2P-RET. According to the E2P rate graph, the rate has a strong inverse dependence on both γβ and ωα0 when ωα0 is close to zero. However, when the detuning excitation energy ωα0 is increased, γβ can effectively control the rate. However, the S2PRET rate is weakly dependent on γβ for a wide range of ωα0 values. These differences arise from the quite different denominators in eq 18 and the corresponding equation for S2P-RET (see eq S17). Note that in the case of S2P-RET, the transition probability is not an explicit function of the two-photon two-acceptor detuning δ = ωα + ωβ − ω1 − ω2, while it is for E2P-RET. This plays an important role in determining the differences between E2P-RET and S2P-RET, and it reflects the crucial influence of the frequency anticorrelation that is built into entanglement. In eq 13 and thereafter, we ignored the propagation length in our derivation. However, in principle the distance dependence of RET in the dipole approximation regime is contained in matrix elements of the electric dipole−dipole coupling tensor, Θ⃡(ω ,rD,rA )55,65 that is hidden in the “Λ” coefficient of eq 17.

PCas(τ ) 2π = KE2P |,1(ω1) , 2(ω2)|2 δ(Eδ) τ ℏ (21)

Note that δ(E) stands for the Dirac delta function in this formula and is not to be confused with the line-shape parameter defined above. The above expression for the energy transfer rate is one of the main accomplishments of this work. This shows that the rate of the E2P-RET process functions in many respects like the usual one-photon RET rate (see also eq 21). In particular, we see that the rate is proportional to γα, which is the rate of emission of the first photon. Indeed, WCas τ /γα provides a measure of the quantum yield for RET as opposed to other processes that might happen. Also, in the limit where ωα1 and ωα2 are much greater than γβ, we see an inverse dependence on these frequency differences. This inverse dependence means inhibition of the rate as the transitions are detuned from resonance, as makes sense. Finally, note that the volume-dependent terms cancel in the overall rate expression, as is physically required. A discussion of what it means to take the time t long enough compared with the lifetime of the two transitions, t ≫ γβ−1, γα−1 can be found in the Supporting Information. RET rate calculations for separated two photons (S2P) may exemplify further details on the features and benefits of using E2P. If we follow the same procedure as above, we are able to derive the same type of expression for the RET rate with two separated photon wave packets of mean frequencies ωα and ωβ, and the corresponding spectral widths γα and γβ. For two identical acceptors (ω1,2 ≈ ω0), the resulting expression for S2Ptransition probability (see the Supporting Information for more details) is

i

Recalling that μem(rD), μmg(rD), and μge(rAi) are transition dipoles of the donor and acceptors, respectively, we revise the transition amplitude eq 17 to a more comprehensive expression for E2P-RET,

711(ω0 ,t ) = |411(ω0 ,t )|2 1 − e−γαt − iωα0t 1 − e−γβt − iωβ0t ≈ |Λ| × −ωα 0 + iγα −ωβ 0 + iγβ

V 2 2 2 em ω1 ω2 [μ (rD) Θ⃡(ωα ,rD,rA1) μge (rA1)] π 2c 6 × [μmg (rD) Θ⃡(ωβ ,rD,rA2) μge (rA2)]

2

4Cas(ω1,ω2 ,t ) = 5

2

yzl ij −(γ + iδ)t | o 1 1 o o1 − e α zzo j zzm × jjj + } o o z jj ω + iγ o δ − iγ o z ω + γ i 1 2 α α β β {n α ~ k

(22)

We numerically calculated the S2P-RET rate in the resonance condition and plotted the result in order to compare with the corresponding E2P-RET rate based on eq 18 (see Figure 3). We performed our calculations using parameters in the same range as polarization-entangled photon pairs from the biexciton cascade of a single InAs QD embedded in a GaAs/AlAs planar microcavity.60,61 In those experiments the pair-entangled photon emissions occur at 1.398 and 1.42 eV. Recent work on solid-state ensembles of highly entangled photon sources at rubidium atomic transitions by Keil et al.62 resembles the situation in our model system. References 4, 63, and 64 are also relevant.

(23)

Although the distance and orientation dependence of the energy transfer rate are nicely included in the above equation, the time-domain electrodynamics (TED)-RET formulation66,67 is more convenient for a wide spectrum of applications in inhomogeneous absorbing and dispersive media. Employing this scheme in the dipole approximation, the transition amplitude can be formulated on the basis of the induced field emerging from donor at the position of the two acceptors,68 3185


Letter

The Journal of Physical Chemistry Letters ÄÅ ÉÑ D Å ÑÑ e E ( r , ) · ω V 2 2 2ÅÅÅ em A A α ÑÑ 1 ÑÑ = 5 2 6 ω1 ω2 ÅÅÅμ (rD) μge (rA1) 1 ÅÅ pex (ωα) ÑÑÑ π c ÅÇ ÑÖ ÅÄÅ ÑÉÑ D ÅÅ e A . E (rA 2,ωβ ) ÑÑ ÑÑ × ÅÅÅÅμmg (rD) μge (rA 2) 2 ÑÑ ÅÅ ÑÑ pex (ωβ ) ÅÇ ÑÖ

effect of donor−acceptor separation, and the influence of inhomogeneous, dispersive, and absorptive materials with any space-dependent, frequency-dependent dielectric function and with any size of donor and acceptor. Since SPDC is a very common method for producing entangled photons in experimental studies, it will also be important to examine energy transfer associated with SPDC sources and compare it with the cascade source of the present study. The theoretical results of this work will lead the way to a new platform for exploring exciton and biexciton transport in coupled plasmonic−semiconductor nanostructures, with potential applications in spectroscopy, nanophotonics devices, biosensing, and quantum information. Details of theoretical derivations can be found in the Supporting Information.

4 Cas(ω1 ,ω2 ,t )

ij yzl −(γ + iδ)t | o 1 1 o o1 − e α zzo × jjjj + } zzm o o j ωα1 + iγ z o o i i ω + γ δ − γ 2 α β β α n ~ k {

(24)

Here, ED(rAi,ωα(β)) (i = 1, 2) is the induced electric field mode with angular frequency ωα(β) and unit polarization vector eAi and at the position of acceptor rAi. Note that pex(ωα/β) is the external polarization generated by the donor in a dielectric medium (more details can be found in the Supporting Information). We proceed by using the above concept to obtain a general expression for the E2P-RET rate in a manner similar to eq 21:

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b00902.

2

W τCas

e A1·ED(rA1,ωα) e A2 · ED(rA2,ωβ ) 2π = KE2P |Π|2 δ(Eδ ) ℏ pex (ωα) pex (ωβ )

General RET rate calculation, transition probability and rate, S2P-transition probability, Distance-dependence of E2P-RET process, QDs with strongly entangled photons, and schematic of radiative decay of the biexciton state (PDF)

(25)

where Π = μ (rD)μ (rA1)μ (rD)μ (rA2). The above expression is another major accomplishment in this work that shows a fourth-power dependence (compared with singlephoton energy transfer) of the rate on the induced electric fields by the donor. This expression allows us to calculate the enhanced E2P-RET rate in arbitrary media, which is more convenient for practical implementation. We also note that the dependence of the energy transfer rate on polarization properties of the donor and acceptors in included in this formula. In a regime where the size of the donor and acceptor are as big as the distance between them, we should go beyond the point dipole approximation by including the effect of higher order multipoles. In that case, the E2P-RET rate is calculated according to the total induced field in the system (more details in Supporting Information) using the following expression em

W τCas =

ge

mg

ge

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

K. Nasiri Avanaki: 0000-0001-9975-7991 George C. Schatz: 0000-0001-5837-4740 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. National Science Foundation under Grant No. CHE-1760537. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.

2π KE2P |, total(rA1,rD,ωα)|2 |, total(rA 2,rD,ωβ )|2 δ(Eδ) ℏ (26)

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The proposed theory provides a framework for simulation that has significant computational advantages compared to calculating the coupling factor utilizing dyadic Green’s functions. In conclusion, we developed a new theory for the resonance energy transfer between a donor and a pair of uncoupled acceptor particles via entangled photons. The underlying mechanism is uncovered through the joint excitation of acceptors using a temporally entangled field. The calculated result shows more than a 3 order of magnitude enhancement in the E2P-RET rate compared with the S2P case for parameters that are relevant to biexciton sources. Empowered with the quantum description of light, our theory provides a way to control the E2P-RET phenomena through the effective coefficient KE2P. This coefficient emphasizes the importance of the emission rate parameters of the donor, and there is also an important effect arising from detuning of the emitted photons energy relative to the excitation energies of the acceptors. Furthermore, we have extended our theory to include for the

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