Quantum of electrodynamics theory of high-efficiency excitation energy transfer inlaser-driven nanostructure systems∗

Dilusha Weeraddana, Malin Premaratne†

Advanced Computing and Simulation Laboratory (AχL),Department of Electrical and Computer Systems Engineering,

Monash University, Clayton, Victoria, 3800, Australia

David L. Andrews‡

School of Chemistry, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, United Kingdom(Dated: April 23, 2016)

A fundamental theory is developed for describing laser driven resonance energy transfer (RET) indimensionally constrained nanostrcutures within the framework of quantum electrodynamics. Theprocess of RET communicates electronic excitation between suitably disposed emitter and detectorparticles in close proximity, activated on excitation of the emitter. Here, we demonstrate that thetransfer rate can be significantly increased by propagation of an auxiliary laser beam through a pairof nanostructure particles. This is due to the higher order perturbative contribution to the Forstertype RET, in which laser field is applied to stimulate the energy transfer process. We constructa detailed picture of how excitation energy transfer is affected by an off-resonant radiation field,which includes the derivation of second and fourth order quantum amplitudes, and the derivationof transfer rates, orientational, distance and laser intensity dependences, providing a comprehensivefundamental understanding of laser driven RET in nanostructures. The results of the derivationsdemonstrate that the geometry of the system exercises considerable control over the laser assistedRET mechanism. Thus, under favourable conformational conditions and relative spacing of donor-acceptor nanostructures, auxiliary laser beam is shown to produce an enhancement in the energymigration rate up to 70 %.

PACS numbers: 31.30.J-,78.70.-g,33.50.Hv,78.67.De,78.67.Uh,78.67.Hc

I. INTRODUCTION

In the past few years, semiconductor nanowires andquantum dots have developed into advanced, promisingclasses of materials for numerous next-generation appli-cations in photonics [1–4]. The quantum confinement inthese materials make them capable of providing uniqueand superior optical properties. Thus, nanowires andquantum dots have been extensively exploited in lighting,light harvesting and electronic devices due to strong con-finement effects [5–7]. In particular, exciton-exciton in-teractions through nonradiative resonance energy trans-fer (RET) of these dimensionally constrained nanostruc-tures have enabled exciting opportunities in light gener-ation and energy harvesting [8–10].

RET gains control across a chemically diverse andextensive range of material systems [11–15]. Energyfrom the sunlight is captured by photo-biological an-tenna chromophores, subsequently transferred to a re-action center by a series of hops between other chro-mophore units with great efficiency by RET [16, 17].The phenomenon also has an important function in theoperation of organic light-emitting diodes, solar light har-vesting and luminescence detectors [11, 18, 19].

∗† dilusha.weeraddana@monash.edu,malin.premaratne@monash.edu‡ d.l.andrews@uea.ac.uk

The process of resonance energy transfer was first cor-rectly described by Theodor Forster [20], and is alsoknown as Forster resonance energy transfer (FRET).FRET can be described as the transfer of the excitationenergy from an excited donor to a ground-state detec-tor without the process of photon emission/reabsorption.This mechanism of transfer of energy in near-field occursdue to the Coulombic interaction between the resonanttransition dipoles of donor and detector species. Fur-thermore, RET was first discussed in terms of quantumelectrodynamics (QED) in pioneering studies by Avery[21], Gomberoff and Power [22]. These studies showedthat far-field retardation effects embellish a near-fieldCoulombic interaction and, radiative and radiationlessmechanisms are both necessarily incorporated within acommon theory as asymptotic limits. In fact, RET isfully quantum mechanical in nature, and can be com-prehensively analysed considering a closed quantum me-chanical system where both matter and radiation arequantized [23–26].

Moreover, it has been shown that, electronically ex-cited nanostructures interact with their neighbors differ-ently from their ground-state counterparts [27, 28]. Italso emerges from the use of QED, that a throughput off-resonant radiation can also produce significant additionaleffects on the process of RET between two molecules(Laser Assisted Resonance Energy Transfer) [29]. Thisis due to coupled absorption and stimulated emission ofphotons from and into the applied beam. Therefore, fol-lowing conventional excitation of the donor particle, the

2

α β

0

Donor (D)

Acceptor (A)

Energy transfer

Laser beam

Decay of D

Excitation of A

FIG. 1. Schematic depiction of energy transfer from donor toacceptor. Green arrows represent emission and absorption ofenergy in donor and acceptor particles. Horizontal red dasharrow illustrates the interaction of the laser beam with thedonor and acceptor. Horizontal orange wavy arrow representsthe energy transfer from the donor to the acceptor. The Greekletters indicate the relevant electronic excited states and 0 theground state

transfer rate can be appreciably increased by the prop-agation of an auxiliary laser beam through the donor-acceptor system [30]. A schematic illustration of laserdriven RET process is shown in Fig.1.

However, the origin of the complications associatedwith laser driven RET in nanostructures is deeply rootedin the nature of real and virtual photons which interactwith confined geometries. Although the basic principlesof RET are well established, there still exist open ques-tions regarding the mechanisms of RET phenomenon inmany aspects.

Therefore, the main objective of this paper is to studythe effects on the energy transfer process due directlyto a laser field impinging on nanowire and quantum dotsystems. We propose and model a system consisted ofnanowires or quantum dots irradiated by an auxiliarylaser beam, by developing a comprehensive quantum elec-trodynamical analysis. We show that the rate of reso-nance energy transfer can be greatly increased throughan applied off-resonant radiation field under favourablephysical configurations of nanostrcutures.

II. QED PERSPECTIVE OF LASER ASSISTEDRESONANCE ENERGY TRANSFER (LARET)

We review and sharpen the concept of RET based onthe quantum theory of light-matter interaction. Thekey feature of quantum electrodynamics is that boththe radiation and matter are subject to quantum me-chanical rules. The system Hamiltonian comprises un-

perturbed operators for the two matter components, asource, HD

mat and detector, HAmat, two corresponding

light-matter interaction terms, HDint, H

Aint, and also the

second-quantized radiation field Hrad;

Htotal = HDmat +HA

mat +HDint +HA

int +Hrad. (1)

Within the electric dipole approximation, the matter-

field coupling Hamiltonian Hξint (ξ = D,A) is explicitly

given by,

Hξint = −µ(ξ) ·E(Rξ) (2)

where the interaction Hamiltonian compromises contri-butions for each species ξ located at Rξ, the µ(ξ) is theelectric-dipole moment operator and E(Rξ) is the oper-ator for the electric displacement field at the specifiedlocation Rξ.

The throughput radiation (with wave vector, k and po-larization, λ) emerges in the final state that is unchangedfrom its initial state, while the matter system experiencesa transfer of energy from D to A. Thus, for the initial(|ΨI〉) and final (|ΨF 〉) matter-radiation state vectors arechosen to be the following eigenvectors

|ΨI〉 = |DαA0;n(k, λ)〉 (3)

|ΨF 〉 = |D0Aβ ;n(k, λ)〉 (4)

Here, superscripts denote donor and acceptor states α, β.The quantum probability amplitude or matrix element,

M , which is expressed through the time-dependent per-turbation expansion:

M =

∞∑m=1

M (m) (5)

where m is the number of photonic interactions. Forthe conventional direct energy transfer, leading contri-butions to the matrix element are associated with m=2,expressive of the two interactions shown in Fig. 2 (a),(b)[23, 30], and given by

M (2) =∑R

〈F |Hint|R〉〈R|Hint|I〉EI − ER

. (6)

The effects on the energy-transfer process manifestthrough interaction with an auxiliary beam, the lowest-order contribution to effect a rate modification will bedue to two extra laser-particle interactions. As depictedin Figs. 2 (c), (d), (e), (f), depending on how thethroughput radiation interacts with the emitter-detectorsystem, a number of possible laser modified energy trans-fer mechanisms emerge. Each entails real photon absorp-tion and emission, coupled by a virtual photon mediator.Therefore, the quantum amplitude which accounts for

3

β0

α 0k

p

s

D A

β0

α 0

p

t

D A

β0

α 0

p

D A

β0

α 0

p

D A

β0

α 0

p

t

D A

β0

α 0

p

t

D A

t

k k

k

k

k

k

k

ss

(a) (b)

(c) (d) (e) (f )

s

FIG. 2. Time-ordered Feynman diagrams representing resonance energy transfer between a donor D and acceptor A. On eachmatter vertical line, Greek symbols identify particle electronic excited states, with 0 the corresponding ground state and r, sare intermediate matter states. The transfer is mediated by the virtual photon with wave vector p. (a),(b) illustrate Feynmandiagrams for direct RET. (c) one of 24 time orderings representing one type of laser modified process where both donor andacceptor interact with the auxiliary beam (with wave vector k). (d) the mirror case of (c). (e) one of 24 time orderingsrepresentative of LARET interactions where only the donor D interacts with the laser beam. (f) photons are absorbed fromand emitted back into the laser beam at the acceptor A.

these corrections is a fourth-order perturbational theorysumming over three intermediate states R, S, and T .

M (4) =∑R,S,T

〈F |Hint|T 〉〈T |Hint|S〉〈S|Hint|R〉〈R|Hint|I〉(EI − ER)(EI − ES)(EI − ET )

(7)

Finally, the energy-transfer process modified by thelaser field impinging on a donor-acceptor system, can bededuced using Fermi’s golden rule [31].

Γtotlaser =2π

~

∣∣∣ ∞∑m=1

M2m∣∣∣2ρ (8)

where ρ is the density of final molecular states of theacceptor particle. The even constraint on the value ofm is a result of the nature of LARET. This is because,every energy transfer entails at least two photonic inter-actions and, in order for the auxiliary laser field to remainunperturbed overall, each matter-field photonic annihila-tion needs to be twinned with a creation and vice versa.Here, only the second and fourth orders of the pertur-bation are sufficient as the series rapidly converges form ≥ 3.

III. ENHANCEMENT OF RESONANCEENERGY TRANSFER BY AN AUXILIARY

LASER BEAM

We explore the effects on the process of energy trans-fer due directly to a laser beam impinging on a donor-acceptor nanostructure system consisting of nanowiresand quantum dots. We shall concentrate exclusively onthe dominant near-field regime, where the compellingphoto-physical mechanism, RET gains control.

A. Nanowire System

Nanowires are specific type of nanostructures withlarge aspect ratios and small diameters, and have beenthe focus of extensive research during the last few decades[32, 33]. Their length is sufficiently large for easy manip-ulation as building blocks in fabricating superstructures.Here, we consider the process of resonance energy trans-fer in a system consisted of NWs illuminated by an aux-iliary laser field as illustrated in Fig. (3). Owing to thecylindrical symmetry of NWs, it is convenient to modelEM waves using Hankel function of order n [34–36]. Di-rectly substituting into Eq. (6), performing contour in-

4

ħk

Donor (D)

Acceptor (A)Laser beam

R

Donor transition

dipole plane

Acceptor transition

dipole plane

(a) (b)

θD

θA

θDA

A

D

D A

R

FIG. 3. Schematics for the (a) direct resonance energy transfer in NW to NW with the orientational factors in Eq. (12), R isthe distance between two NWs; (b) laser assisted resonance energy transfer in a NW pair.

tegration and by the residue theorem yields

MdFI =

µ0αi (D)µβ0j (A)

8π2Lε0(−∇2δij +∇i∇j) (9)∫ ∞

0

∫ 2π

0

H(1)0 (pR)

k − p− H

(2)0 (pR)

k + pdφdp

M(2)NW =

µ0αi (D)µβ0j (A)

4Lε0

[pδij

{− Y2(pR) +

Y1(pR)

pR

}−

p{Y1(pR)

(δij − RiRjR

)+

pRiRj

(− Y2(pR) +

Y1(pR)

pR

)}− ipδij

{− J2(pR) +

J1(pR)

pR

}+ ip

{J1(pR)

(δij − RiRjR

)+

pRiRj

(− J2(pR) +

J1(pR)

pR

)}](10)

where Yz(pR), Jz(pR) are second and first kind of Besselfunctions respectively, z is the order number.

We now deploy the asymptotic series for 0 < pR �√n+ 1 and n 6= 0 [37]By applying short-range limits 0 < pR � 1 on Eq.

(10) yields

M(2)NW =

µ0αi (D)µβ0j (A)

2πLε0R2(δij − 2RiRj)

=κNW |µ0α

i (D)||µβ0j (A)|2πLε0R2

(11)

The above equation is dominant in the short-range regionand indicates a highly ‘virtual’ character of the electro-magnetic mediator. Here, |κNW |2 is an orientation fac-tor, which describes the influence of the relative orienta-tions of the transition dipole moments of the donor andacceptor NWs, as given by

κNW = µ0αi (D)(δij − 2RiRj)µ

β0j (A)

= cos(θDA)− 2 cos(θD) cos(θA)(12)

where θD is the angle between donor and separation vec-tor (R), and θA is the angle between acceptor and R.θDA is the angle between donor and acceptor NWs (seeFig.3). Due to the cylindrical symmetry and the physicalnature of the exchanged photon virtue in the 2D geom-etry, the orientation factor varies from 0 ≤ κ2NW ≤ 1[28].

In a similar manner, the quantum amplitude arisingfrom the input auxiliary beam is given by

M(4)NW = −n~ck

2ε0V

{eλi (k)α

0α(D)ij (k)

(δjk − 2RjRk)

2πLε0R2

×αβ0(A)kl (k)eλl (k) + eλi (k)α

0α(D)ij (k)

(δjk − 2RjRk)

2πLε0R2

×αβ0(A)kl (k)eλl (k) + eλi (k)eλj (k)β

0α(D)ijk (k)

× (δkl − 2RkRl)

2πLε0R2× µ0β(A)

l + µ0α(D)i

× (δij − 2RiRj)

2πLε0R2× eλk(k)eλl (k)β

0α(D)jkl (k)

}(13)

where n is the number of photons (proportional tolaser intensity) in the quantization volume V , e and ~ck

5

represent the polarization (e denoting its complex con-jugation) and energy of the input photon, respectively.

Further, αfiij (k) and βfiijk(k) are generalized and hyper-

polarizability defined as [38, 39]

αfiij (k) =∑s

{ µfsi µsij

Esf − ~ck+

µfsj µsii

Esi + ~ck

}(14)

βfiijk(k) =∑s,t

{ µfti µtsj µ

sik

(Esi − ~ck)(Eti − ~ck)+

µfti µtsk µ

sij

(Esi − ~ck)(Eti − ~ck)+

µftj µtsi µ

sik

(Esi − ~ck)(Eti − ~ck)+

µftj µtsk µ

sii

(Esi − ~ck)(Eti − ~ck)+

µftk µtsi µ

sij

(Esi − ~ck)(Eti − ~ck)+

µftk µtsj µ

sii

(Esi − ~ck)(Eti − ~ck)

}(15)

The Eq. (13) represents the optically nonlinear influ-ence of the input beam, involves four components: (i)laser photon absorption at D, stimulated emission at A;(ii) the converse; (iii) absorption and emission at D; and(iv) both photon events at A.

According to Fermi’s Golden Rule given in Eq. (8), thematrix elements of the interaction operator between theinitial and final states of the field determine the transitionprobability

ΓlaserNW =2πρ

~

∣∣∣M (2)NW

∣∣∣2 +2πρ

~

∣∣∣M (4)NW

∣∣∣2+

4πρ

~Re{M (2)

NWM(4)

NW } (16)

The first term represents the normal Forster energytransfer rate

Γ(2−2)NW =

|µ0αi (D)|2|µβ0j (A)|2|κNW |2ρ

2πL2R4ε02~(17)

The middle term depends on the laser intensity, andexplicitly given by

Γ(4−4)NW =

I2(k)ρ

8L2~c2πε40R4|eλi (k)α

0α(D)ij (k)

×(δjk − 2RjRk)αβ0(A)kl (k)eλl (k) + eλi (k)α

0α(D)ij (k)

×(δjk − 2RjRk)αβ0(A)kl (k)eλl (k) + eλi (k)eλj (k)β

0α(D)ijk (k)

×(δkl − 2RkRl)µ0β(A)l + µ

0α(D)i (δij − 2RiRj)

×β0α(D)jkl (k)eλk(k)eλl (k)|2

(18)

I(k) ≡ n~c2k/V is the auxiliary laser intensity, de-scribed as comprising n photons with wave-vector k. The

third term, linear in I, signifies a quantum interferenceof these two concurrent processes

ΓinterNW = −I(k)|µ0α

i (D)||µβ0j (A)|κNW ρ2L2~cπε30R4

×Re{eλi (k)α0α(D)ij (k)(δjk − 2RjRk)α

β0(A)kl (k)eλl (k)

+eλi (k)α0α(D)ij (k)(δjk − 2RjRk)α

β0(A)kl (k)eλl (k)

+eλi (k)eλj (k)β0α(D)ijk (k)(δkl − 2RkRl)µ

0β(A)l

+µ0α(D)i (δij − 2RiRj)β

0α(D)jkl (k)eλk(k)eλl (k)}

(19)

By implementing conservative estimates for parame-ters, µ0α(D) ≈ µβ0(A) ≈ 5 × 10−30 C m for tran-sition dipole moment magnitudes and α(D)α(A) ≈β(D)µ(A) ≈ µ(D)β(A) ≈ 1×10−78 C4 m2 J−2 for tensorproducts, we can estimate the relative magnitude of theeffect with increasing laser intensity. We use above pa-rameters for the development of the plots in Sec. III A,III B.

1. Laser irradiance and orientational dependence ontransfer rate

We envisage three vital orientational factors and thelaser irradiance dependence on each case to enhance thetransfer rate.

1. κNW = 1: When cos(θDA) = π, cos(θD) = π (or0) and cos(θA) = 0 (or π), orientation factor κNWbecomes 1. Fig. 5 (a) illustrates the total energytransfer rate and the magnitudes of the individualcontributions to the transfer rate with increasinglaser intensity. The direct RET rate is constantwith respect to I as non-mediated RET is inde-pendent of the radiation intensity. The LARETgradually increases with I, showing the quadraticnature of Γ4−4

NW . Nevertheless, quantum interfer-ence (ΓinterNW ), leanerly decreases with I, owing tothe quantum amplitudes of direct and laser medi-ated RET (see Fig. 4 (a) and Eq. (19)). Theseresults exhibit a gradual decline of the total en-ergy transfer efficiency until it reaches its minimumpoint when I becomes 1.33 × 1017 Wm−2. Afterthis point, the total energy transfer rate steadilyelevates its value and finally starts dominating thedirect transfer rate when I > 2.66 × 1017 Wm−2

(see point X in Fig. 5 (a)).

2. κNW = 0: This is when both transition dipolemoments are perpendicular to each other and per-pendicular to the donor-acceptor separation vector.This configuration prohibits the direct transfer ofenergy, hence it excludes the quantum interference.Therefore, only laser driven RET term survives, ex-hibiting a steady increment of total energy transfer

6

rate with respect to I. Transfer rates and corre-sponding quantum amplitudes are depicted in Fig.5 (b) and Fig. 4 (b) respectively.

3. κNW = −1: When cos(θDA) = cos(θD) =cos(θA) = 0, orientation factor κNW becomes −1.In this case, the coupling matrix element of directRET acquires a negative value as illustrated in Fig.4 (c), delivering a positive quantum interference.Analogous to the case of κNW = 1, laser mediatedRET gradually increases as a function of I and di-rect RET is independent of I. Thus, the total rateenhances steadily with increasing I, displaying ahigher total transfer rate compared to the Forstertype direct RET for the whole spectrum of laserintensities. These results are illustrated in Fig. 5(c).

B. Quantum dot System

In contrast to molecular fluorophores, individual QDshave discrete atom-like discrete energy transitions, whicharise from band splitting due to the quantum confine-ment effect. In particular, owing to its unique opticalproperties, QDs are being exploited as highly efficient flu-orophores, typically possessing excellent quantum yieldsand photostability. Furthermore, QDs exhibit excellentphotophysical properties that are highly desired in a RETsystem, including: (1) broad absorption spectra; (2) largeabsorption cross-sections; (3) narrow, size-tunable emis-sion spectra to name a few [40–42]. In this section, apair of QDs separated by distance R interacting with alaser field is considered. The direct interaction betweenthe donor and acceptor can be calculated from the secondorder quantum amplitude [27, 43]

M(2)QD =

µ0αi (D)µβ0j (A)

16π3ε0(−∇2δij +∇i∇j)

∫ ∞0

∫ 2π

0

∫ 1

−1

×{eip·Rpk − p

− e−ip·Rp

k + p

}d(cos θ)dφdp

(20)

Performing contour integration with a suitable contourand by the residue theorem yields

M(2)QD = µ0α

i (D)δij − 3RiRj

4πε0R3µβ0j (A) (21)

The application of the Fermis golden rule gives rise to thefollowing expression for the direct energy transfer ratebetween two QDs:

Γ(2−2)QD =

|µ0αi (D)|2|µβ0j (A)|2|κQD|2ρ

8πR6ε02~(22)

where |κQD|2 is an orientation factor

κQD = µ0αi (D)(δij − 3RiRj)µ

β0j (A)

= cos(θDA)− 3 cos(θD) cos(θA)(23)

where θD is the angle between donor and separation vec-tor (R), and θA is the angle between acceptor and R.θDA is the angle between donor and acceptor QDs. Ow-ing to the spherical symmetry of the QDs, the orientationfactor varies from 0 ≤ κ2QD ≤ 4.

The indirect transfer rate Γ(4−4)QD , represents the opti-

cally nonlinear influence of the input beam, is deliveredby previous work [30], [44]

Γ(4−4)QD =

I2(k)ρ

32~c2πε40R6|eλi (k)α

0α(D)ij (k)

×(δjk − 3RjRk)αβ0(A)kl (k)eλl (k) + eλi (k)α

0α(D)ij (k)

×(δjk − 3RjRk)αβ0(A)kl (k)eλl (k) + eλi (k)eλj (k)β

0α(D)ijk (k)

×(δkl − 3RkRl)µ0β(A)l + µ

0α(D)i (δij − 3RiRj)

β0α(D)jkl (k)|2 × eλk(k)eλl (k)

(24)

Thus, the quantum interference due to the laser field isgiven by

ΓinterQD = −I(k)|µ0α

i (D)||µβ0j (A)|κQDρ8~cπε30R6

×Re{eλi (k)α0α(D)ij (k)(δjk − 3RjRk)α

β0(A)kl (k)eλl (k)

+eλi (k)α0α(D)ij (k)(δjk − 3RjRk)α

β0(A)kl (k)eλl (k)

+eλi (k)eλj (k)β0α(D)ijk (k)(δkl − 3RkRl)µ

0β(A)l

+µ0α(D)i (δij − 3RiRj)β

0α(D)jkl (k)eλk(k)eλl (k)}

(25)

1. Laser irradiance and orientational dependence on RETrate

Similar to the NW systems, we study three non-trivialorientational factors and the laser irradiance dependenceon each case to enhance the transfer rate.

1. κQD = 2: Similar to the NW case, laser modified

coupling matrix element (M(4−4)QD ) decreases with

intensity while direct coupling remains static forall the laser intensities, producing a negative quan-tum interference value which steadily declines as afunction of I. This is shown in Fig. 7 (a). Asdepicted in Fig. 8 (a), the LARET in QDs grad-ually increases with I. On the otherhand quan-tum interference experienced a steady drop whenI increases, displaying a gradual decline in the to-tal energy transfer rate until I reaches its valueof 2.66 × 1017 Wm−2. It is immediately apparent

7

-2.5

0x 10

−4

0 0.5 1 1.5 2 2.5 3

x 1017

−3

0

1.5x 10

−4

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

x 1017

−3

−1

0x 10

−4

x 1017

0

R

R

R

θD=πD

A

D

A

D A

θA=0

θDA=π

θD=0

θA=0

θDA=0

θD= θA=π/2θDA=

(a) (b)

(c)

K=1

K= -1

K= 0

direct RET

quantum interferenceLARET

RD

DI (

eV

)

RD

DI (

eV

)

RD

DI (

eV

)

−3

0

4

x 105

qu

an

tum

inte

rfe

ren

ce (

ns

)

0

3

7 x 105

Laser intensity (Wm )-2 Laser intensity (Wm )

-2

Laser intensity (Wm )-2

-1

qu

an

tum

inte

rfe

ren

ce (

ns

)-1

qu

an

tum

inte

rfe

ren

ce (

ns

)-1

FIG. 4. RDDI strengths of the direct RET (ash colour line), laser driven RET (pink colour line) and the quantum interference(blue colour line) when κ = 1, 0,−1 are shown in (a)-(c) respectively as a function of the laser intensity

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3

4

x 105

K=1

0 5 10 15 20 25 30

0

1

2

3

4

5

6

7

x 105

K= -1

0 5 10 15 20 25 30

0

0 .5

1

1..5

2

2. 5

3

3..5

x 105

K= 0

(a) (b) (c)

direct RET rate

quantum interference

LARET rate

total rate

x 1016

x 1016

x 1016

RE

T r

ate

s (n

s )-1

RE

T r

ate

s (n

s )-1

RE

T r

ate

s (n

s )-1

Laser intensity (Wm )-2

Laser intensity (Wm )-2

Laser intensity (Wm )-2

X

FIG. 5. Transfer rates of the direct RET (ash colour line), laser driven RET (red colour line), quantum interference (bluecolour line) and the total LARET rate (green colour line) when κ = 1, 0,−1 are shown in (a)-(c) respectively as a function ofthe laser intensity

8

Donor Acceptorħk

Laser beam

R

Donor transition

dipole plane

Acceptor transition

dipole plane

(a)

θD

θA

θDA

A

D

(b)

D A

R

FIG. 6. Schematics for the (a) direct resonance energy transfer in QD to QD with the orientational factors in Eq. (23), R isthe distance between two QDs; (b) laser assisted resonance energy transfer in a QD pair.

0 0.5 1 1.5 2 2.5 3

x 1017

−1

0

x 10−4

0

0 0.5 1 1.5 2 2.5 3

x 1017

-1

0x 10

−4

0

3

0 0.5 1 1.5 2 2.5 3

x 1017

−1.5

0

1.5x 10

−4

−1.5

1.5

R

θD=πD

A θA=0

θDA=π

K=2

R

θD=0D

A θA=0

θDA=0

K=-2

R

D A

θD= θA=π/2θDA=

K= 0

(a) (b)

(c)

direct RET

quantum interferenceLARET

RD

DI (

eV

)

RD

DI (

eV

)

RD

DI (

eV

)

qu

an

tum

inte

rfe

ren

ce(n

s )

-1

x 106

x 106

Laser intensity (Wm )-2

Laser intensity (Wm )-2

Laser intensity (Wm )-2

-1

qu

an

tum

inte

rfe

ren

ce (

ns

)-1

qu

an

tum

inte

rfe

ren

ce (

ns

)-1

FIG. 7. RDDI strengths of the direct RET (ash colour line), laser driven RET (pink colour line) and the quantum interference(blue colour line) when κ = 2, 0,−2 are shown in (a)-(c) respectively as a function of the laser intensity

that, at intensities less than 2.66 × 1017 Wm−2,the transfer is Forster dominated. Due to the ef-fect of LARET, total energy transfer elevates withincreasing I and finally achieves higher transfer ef-ficiencies when I > 5.32×1017 Wm−2 (see point Y

in Fig. 8 (a)).

2. κQD = 0: This case is analogous to the caseof κNW = 0. Here the direct coupling has been“switched off”between QD particles by arranging

9

0 10 20 30 40 50 60

−1.5

0

1.5x 10

6

K=2

0 10 20 30 40 50 60

1.5

3x 10

6

0 10 20 30 40 50 60

1.4x 10

6

K=-2

K=0

direct RET rate

quantum interference

LARET rate

total rate

(a) (b) (c)

x 1016

x 1016

x 1016

RE

T r

ate

s (n

s )-1

RE

T r

ate

s (n

s )-1

RE

T r

ate

s (n

s )-1

Laser intensity (Wm )-2

Laser intensity (Wm )-2

Laser intensity (Wm )-2

Y

FIG. 8. Transfer rates of the direct RET (ash colour line), laser driven RET (red colour line), quantum interference (bluecolour line) and the total LARET rate (green colour line) when κ = 2, 0,−2 are shown in (a)-(c) respectively as a function ofthe laser intensity

them such that the transition dipole moments areperpendicular to each other and donor-acceptorseparation vector, hence excluding the quantum in-terference. Therefore, the total energy transfer onlycontains the contribution of laser dependent RET,which gradually increases as a function of I. Theseresults are shown in Fig. 7 (b), Fig. 8 (b).

3. κQD = −2: This is when both transition dipolemoments are parallel to each other and parallel tothe donor-acceptor separation vector. The quan-tum amplitude of direct RET acquires a negativevalue as depicted in Fig. 7 (c), resulting in a posi-tive quantum interference. Analogous to the case ofκQD = 2, laser mediated RET gradually increasesas a function of I and direct RET is independent ofI. Therefore, the total rate enhances steadily withI. Here, the direct RET between a pair of QDs isdominated by the LARET for the whole spectrumof laser intensities. These results are shown in Fig.8 (c). However, unfocused light does not perturbRET to a significant extent.

C. Design guidelines for high efficiency laser drivenRET

The RET rate can be enhanced efficiently, when theoptimal configuration of the nanostructure system is im-plemented. Here, for fast transfer between nearby sitesare those where the particle transition dipole momentsand the separation vector are collinear (θDA = θD =

θA = 0). The transfer rate enhancement with increas-ing laser intensity for this scenario is depicted in Fig. 9(a), comparing both QD and NW cases. Then we per-formed calculation to perceive RET rate enhancement asa percentage (Fig. 9 (b)). The results demonstrate thateven for moderate laser intensities (I ≈ 1016), up to 70%and 30% rate enhancements can be achieved for QDs andNWs respectively.

Moreover, we investigated the distance dependence inlaser driven RET in NWs and QDs. Owing to the (R)−4

and (R)−6 distance dependence on the total transfer ratein NW and QD respectively, lower relative spacing be-tween donor and acceptor particles displays higher trans-fer efficiency. This is shown in Fig. 9 (c). Further, it canalso be observed from the plots, that QDs produce higherLARET efficiency in relatively lower intensities. How-ever, when increasing the laser beam intensity, LARETin NWs starts to outperform QDs.

IV. DISCUSSION

In Sec. (III), laser driven resonance energy trans-fer rate equations, corresponding quantum amplitudesand quantum interferences for nanowire and quantumdot systems have been derived and the results are dis-cussed with figures. The results have demonstratedthat the transfer of energy can be enhanced to a sig-nificant and measurable degree when the laser inten-sity is pulsed and highly focused or through deployingfavourable configurations of the nanostructure particles(κNW = −1, κQD = −2). It is also important that the

10

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10

16

10

30

50

70

1 2 3 4 5 6

x 1017

0

2

4

6x 10

7

34 35 36 37 38 39 40 41

51

60

(a) (b)

(c)

QD,NW, K=-1

QDNW

QD-3nm

QD-5nm

QD-7nm

NW-3nm

NW-5nm

NW-7nm

RE

T r

ate

s (n

s )-1

RE

T r

ate

en

ha

nce

me

nt

%

Laser intensity (Wm )-2

Laser intensity (Wm )-2

log (I /Wm )10

-2

log

(

rate

/n

s )-1

10

K=-2

FIG. 9. (a) The brown colour and purple colour solid lines represent the LARET rates when κNW = −1 and κQD = −2;(b) RET rate enhancement factor in QD and NW systems; (c) Log-Log plot of transfer rate as a function of log(I), for threedifferent relative distances (3 nm, 5 nm, 7 nm) between D and A, dash lines represent the transfer rates for the case of NWpair.

laser radiation is off-resonant, in order to prevent directexcitation of the acceptor.

Exploration of the quantum amplitudes in laser as-sistant RET is the core contribution of all derivations.Our calculations for quantum amplitude are based onSchrodinger state vector representation of quantum dy-namics, where matrix element for RET is represented asa sum of differently time-ordered contributions [23]. Fig.4(a)-(c) indicate the variation of RDDI strengths in a NWpair against the laser irradiance for different orientationfactors. Interestingly, laser driven RDDIs are identicalfor all three cases. This is because quantum amplitudeis independent of the orientational factor. Nevertheless,direct RET and quantum interference vary with κ, andvanish when NWs are perpendicular to each other and tothe distance vector. The variations of RDDIs of QDs withlaser irradiation and orientational factor exhibit similarpatterns observed for NWs (Figs. 7 (a)-(c)), with one dif-ference. Due to the physical nature and geometry of thequantum dots, the κQD value is higher than the κNW .

Furthermore, rates for unmediated RET, laser drivenRET and the quantum interference for a pair of NWsare illustrated in Fig. 5 (a)-(c) for three non trivial ori-entational factors. In Fig. 5 (a), where the Γtot is plot-ted against laser irradiation when κNW = 1, a gradualdecrement and then an increment of total transfer ratewith the laser intensity can be observed. Here, when thelaser intensity is low, quantum interference dominatesthe transfer rate. However, when the intensity increases,laser assisted RET dominates the transfer process andfinally for high intensities, an enhancement in RET ratecan be observed. Moreover, when κNW = 0 (Fig. 5 (b)),direct RET and hence the quantum interference are ex-plicitly forbidden, allowing full control to the laser driventransfer. In this case, optical switching can be producedby the throughput of a single off-resonant beam (or, withmore control options, by two coincident beams) [44–46].Further, for angles θDA = θD = θA = 0, a gradual in-crease of the total transfer rate against laser intensitycan be observed as depicted in Fig. 5 (c). Thus the RET

11

rate can be enhanced to a greater extent under this condi-tion. Additionally, the results for QDs show similar pat-terns (Fig. 8 (a)-(c)) and hold same explanations, withone significant difference. Due to the nature of the realand virtual photon virtue in spherical geometry, whenκQD = 2 (θDA = θD = π, θA = 0), higher laser inten-sities are needed to suppress the quantum interferencewhich arises from the applied laser beam.

Moreover, by suitably configuring an arrangement oftransition dipoles, it proved possible to design opticalswitches [44, 47]. Here, the optical switching actionscan be implemented by exploiting RET mechanism ac-tivated by throughput laser radiation of an appropriateintensity, frequency and polarization. Therefore, imple-mented in a configuration with nanowires (or quantumdots) arrayed in thin films, the process may offer designof logic gates, optical transistor and ultrafast parallel-processing capabilities etc. This represents scope for po-tential development of the theory.

V. CONCLUSIONS AND OUTLOOK

We calculated the rates of laser driven resonance en-ergy transfer in optically excited systems composed ofNWs and QDs. A full quantum electrodynamical treat-ment for laser assisted energy migration is developed andformulated with the aid of Feynman diagram methods[48]. In a nutshell, The results expounded here indicatethat at relatively higher laser intensities, the higher-ordereffects are significant even in a non-favourable configura-

tion of the particles (when donor-acceptor particles makean angle of π with each other and one particle is paral-lel to the donor-acceptor displacement vector). Besides,the results show that, at reasonable levels of laser in-tensity, quantum interference arises from the auxiliarylaser beam drops to insignificant levels, enhancing thetotal RET rate between nanostuctures. Thus, at suit-ably higher laser intensities, for example those readilyavailable from a focused, Q-switched laser or a spaser,energy-transfer rate explicitly enhanced up to 70%.

In addition to the calculations presented in this ar-ticle, it is viable to envisage high-intensity nanolasersbased on spasers to enhance and control the process ofRET. In this case, the emitted photons from and into thelaser beam are replaced by surface plasmons. The elec-tric field operator for the surface plasmons would modifythe quantum amplitudes and the transfer rate, provide ahigh efficiency controlling mechanism for resonance en-ergy transfer.

The ensuing results open up a new avenue for appli-cations of high-efficiency artificial nano-antenna systems.Furthermore, the proposed laser driven RET mechanismholds potential for optical transistors, optical logic gates,providing building blocks for more complex nanophotoniccircuitry.

ACKNOWLEDGMENTS

The work of D.W. is supported by the Monash Univer-sity Institute of Graduate Research. The work of M.P. issupported by the Australian Research Council, throughits Discovery Grant No. DP140100883.

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