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Radiative decay of an emitter due to non-Markovian interactions with dissipating

matter

Kritika Jain and Murugesan Venkatapathi∗

Computational and Statistical Physics Laboratory, Indian Institute of Science, Bangalore, 560012

It is known that the more tractable Markovian models of coupling suited for weak interactions

may overestimate the Rabi frequency signiﬁcantly, and alter the total decay rate marginally, when

applied to the strong-coupling regime. Here we describe a more signiﬁcant consequence of the non-

Markovian interaction between a photon emitter and dissipating matter such as resonant plasmonic

nanoparticles. We show a large increase of radiative decay and a diminished non-radiative loss, which

unravels the mechanism behind the large enhancements of surface-enhanced-Raman-spectroscopy

(SERS), as well as the anomalous enhancements of emission due to extremely small fully absorbing

metal nanoparticles less than 10 nm in dimensions. We construct the mixture of pure states of

the coupled emitter-nanoparticle system, unlike conventional methods that rely on the orthogonal

modes of the nanoparticle alone.

In the absence of an enclosing cavity, in the weak-

coupling regime of a photon emitter and vacuum, Rabi

oscillations between the emitter and a proximal resonant

object indicates a strong coupling between them [1–3].

The increase in decay rates due to the strongly coupled

proximal object, is evaluated using the number of addi-

tional optical modes available for the spontaneous emis-

sion [4]. Regardless of the quantization of the emitter

and the object, the theoretical partition of the additional

optical states into the radiative and non-radiative parts,

reﬂected the classical scattering and absorption eﬃcien-

cies of this object [5, 6].

We can recast the coupling of an emitter and a

nanoparticle as a quantum interference due to the many

paths of a photon from the emitter to a point in vacuum

[7–10]. The paths of a photon from the emitter in the

presence of the nanoparticle are described in Figure 1,

where two Markovian paths of decay are marked as Aand

A0. These two processes represent the memory-less expo-

nentially decaying probability of emission into vacuum,

one directly from the emitter, and one from the nanopar-

ticle excited by the photon from emitter. The sum of am-

plitudes over the multiple paths of the photon to a point

P, both through the particle and directly from the emit-

ter, represents the weak-coupling approximation. The

multiple paths through the particle result from probable

absorption and re-emission by constituents of the par-

ticle. Numerous experiments have conﬁrmed the weak-

coupling predictions of large gains in the radiative decay

of the emitter compared to the non-radiative losses, when

the larger strongly scattering plasmonic particles (&50

nm in dimension) are placed at optimal distances from

the emitter [11–13]. In typical evaluations, this sum of

amplitudes over all the paths is substituted by a more

convenient sum of the electric ﬁelds due to the emitter

and particle, over each orthogonal mode of the particle.

These semi-classical approaches used for determining the

increase in local density of optical states (LDOS) due to

∗murugesh@iisc.ac.in

e

P

B

’

A’

B

A

A’

X

Z

FIG. 1. Coupling of emitter ewith the polarizable matter

of a spherical particle and resulting paths of the photon to a

far-ﬁeld point P. Markovian paths of decay from the emitter

and the nanoparticle respectively, are shown by Aand A0.

Non-Markovian decay Bis due to B0(in blue) that returns a

photon to the emitter.

a weak coupling, can be extended to stronger couplings

where non-Markovian interactions between the particle

and the emitter can not be ignored. Figure 1 also shows

an additional loop that may return the photon from the

particle to the emitter, which makes the two decays from

the particle and the emitter as dependent processes. This

can result in exponentially damped oscillatory decays

characterizing the non-Markovian process [14, 15]. The

eﬀect of such interactions on the total decay rates and

Rabi frequencies have been elucidated earlier [16, 17].

We show that the non-Markovian interaction enhances

the radiative decay by large factors, along with a di-

minished non-radiative decay in the dissipating parti-

cle. Earlier, we proposed a ﬁrst order (one loop) cor-

rection to the conventional decomposition into radia-

tive and non-radiative parts to account for the non-

Markovian eﬀects, using a phenomenological extension

[18]. This ﬁrst-order correction is shown here to be

useful when separations are not very small. The non-

Markovian process results in counter-intuitive and large

divergences with the conventional predictions, for the

case of proximal nanostructures that are either strongly

absorbing or do not scatter light at all. It includes the

arXiv:2107.01484v1 [physics.optics] 3 Jul 2021

2

unexpected giant enhancements of emission in surface-

enhanced-Raman-spectroscopy (SERS) where a strongly

absorbing rough metal structure increases the radiation

exciting a molecule in the near-ﬁeld, by a few orders of

magnitude, but surprisingly with no apparent absorp-

tion of the photons emitted by the excited molecule.

This divergence of SERS from ﬁrst-principle theoreti-

cal predictions has been widening for decades, as the

reported SERS enhancements grew from 104to 1014 [19–

21]. Meanwhile, anomalous enhancements of sponta-

neous emission near fully absorbing metal nanoparticles

less than 10 nm in dimensions, have also been reported

[22–26].

We begin with a single two-level emitter and n−1

spatially distributed two-level systems to represent the

proximal nanoparticle. A superposition of paths based

on this distributed interacting system is an explicit de-

scription of the non-local process, both in time as well as

in space. An initial state of the excited system can be

described by the ninteracting subsystems as

ψinit. ={c1|ˆe1i+c2|ˆe2i+. . . cn|ˆeni} (1)

where c∗

jcjrepresents the unknown probability of exci-

tation of the two-level system j. Here ˆejrepresents the

canonical basis vectors in ndimensions, and Pjc∗

jcj= 1

for a known initial state. Using these individual two-level

systems as the basis for representing the collective system

more concisely, ψinit. = [c1, c2,...cn], where the emitter

is given the index 1. Radiative decay of an initial state is

given by the superposition of the radiative decays of the

basis states |ˆeji.

I. WEAK COUPLING WITH THE

NANOPARTICLE

In the regime of weak coupling, the initial state of the

excited coupled system is given by

ψinit. = [ψe

init., ψp

init.] (2)

where the initial state of the particle, ψp

init., is determined

using the nature of its excitation due to the given initial

state of the emitter, ψe

init., under the Markovian approx-

imation.

Without any signiﬁcant loss of accuracy, the initial

state of the excited particle can also be constructed in

the form of npolarizable point dipoles using a balance of

forces. The weak coupling with vacuum allows an evalua-

tion of the interactions using classical ﬁelds in the dipole

approximation [27–29]. Note that the absolute values of

self-energy of these representative oscillators provides the

probability of excitation of the mutually interacting two-

level systems. It is implicit that each two-level system

has a transition energy of ~ω0. Given the polarization of

the dipole emitter, we balance the forces at each dipole

using

−1

αj

Pj+

n

X

k=2,k6=j

G(rj,rk)Pk=Einit. =G(rj,r1)P1

(3)

where the polarizability αjof a grain in the particle can

be determined from its size and the dispersive permittiv-

ity of the material, using the Clausius-Mosotti relation

[30] and its extensions to include the lattice dispersion

[31]. Einit. is the ﬁeld incident on the dipoles due to the

emitter. Solving the above coupled system of equations

gives us the polarization Pjfor j= 2,3. . . n representing

the particle. The required Green dyad Gare evaluated

for a given frequency ω, and is deﬁned in the appendix.

The maximum size of the dipole grain is determined by

the wavelength of emission and material properties of the

nanoparticle, and the number of oscillators nis further

increased to meet the error tolerance allowed in the so-

lution of the system of linear equations [30, 31]. The so-

lutions can be weighted and integrated with a line-shape

around the resonance frequency of the emitter, ω0, if re-

quired.

After solving the above, the required superposition of

Markovian paths in Figure 1 is obtained using the sum of

electric ﬁelds at points rdue to all oscillators jincluding

the emitter, and the total decay rate due to the particle

is evaluated by the imaginary part of the self-interaction

of the emitting dipole.

Γr=0c2

2IE(r)2dr=0c2

2I{

n

X

j=1

G(r,rj)Pj}2dr(4)

Γtotal =4πq2ω

mc2·Im{

n

X

j=2

P1G(r1,rj)Pj}+ Γ0(5)

=4πq2ω

mc2·Im{P1[ˆ

G1jˆ

G−1

jj ˆ

Gj1]P1}+ Γ0(6)

The explicit solution of the coupled system of equa-

tions introduced in equation (3) containing all the 3 ×3

dyads G, can be rearranged concisely in the form of ma-

trices ˆ

Gwhere the subscripts jand 1 refer to the inter-

acting elements of the particle and the emitter respec-

tively [32]; see appendix for the full description. The

non-radiative decay rate Γnr = Γtotal −Γr.

Since this solution from equation (3) uses the coupling

of n−1 oscillators only in the nanoparticle, it can be

always rewritten as a weighted sum of the n−1 orthog-

onal modes of the particle if required. For particles of

regular shapes like spheres, a more compact set of or-

thogonal modes can also be analytically constructed us-

ing the vector harmonics of the Helmholtz equation. In

the weak-coupling regime, the noscillators and the an-

alytical vector harmonics provide identical results. But

an analytical decomposition into orthogonal modes of the

emitter-particle system is not tractable for retarded in-

teractions, and the possible quasi-static solutions become

3

inaccurate for stronger couplings. Whereas this descrip-

tion using a large set of noscillators allows us to numer-

ically construct the eigenstates for the coupled system

even with retarded interactions, and this becomes essen-

tial in the next section. Higher the refractive index and

size of the particle, larger is the number of optical paths

(states) in the particle, and so is the number of orthog-

onal modes or the number of oscillators n, required in

the summation. A nanoparticle can also produce a large

density of paths when a mode is resonant. Note that this

weak-coupling approximation does not include the loop

that returns the photon to the emitter as shown in Fig-

ure 1. This possible re-absorption of the photon renders

the initial state of system as a mixture.

II. MODERATE AND STRONG COUPLINGS

WITH THE NANOPARTICLE

Proceeding from the previous section on the Marko-

vian approximation for a weak coupling, we introduce

two signiﬁcant reﬁnements to include eﬀects of the non-

Markovian interaction on the radiative decay from the

system. The former converges to the latter when separa-

tions between the emitter and the dissipating nanopar-

ticle increase, and when the size of the particle is suﬃ-

ciently large. Firstly, we construct a mixture of initial

states to represent the coupled system where the pho-

ton can also be re-absorbed by the emitter. Secondly, to

obtain the superposition of radiative decays over all the

oscillators and the ensemble of initial states, we decom-

pose this mixture into a set of orthogonal pure states.

The solution of the coupled classical system using a

balance of forces, in equation (3), is used here to fur-

ther construct a non-local system that shares a photon

[33, 34]. Solutions Pjof the noscillators and their self

energy components now represent a single photon. The

self energy components normalized by Planck’s constant

[35] are given by:

Σjk =∆Ej k

~−iΓjk

2=Pj·G(rj,rk)·Pkfor j6=k(7)

Σjj = Ωj j −i·Im{Pj·Pj

αj}(8)

Ωjj determines the strength of coupling of the isolated

oscillators with vacuum, and the conclusions presented

here are agnostic to it for weak vacuum-coupling; see [36]

for more details. The real parts of the above symmetric

matrix provide the rates of exchange of the photon from

the dipole jto another dipole k. In the rotating wave

approximation valid here as |∆Ejk | ~ω0, the Rabi fre-

quencies are given by the absolute value of shifts in the

energy (Ωjk =|∆Ej k|/~). The imaginary parts Γjk rep-

resent the rates of decay of the photon from dipole jdue

to the other dipole k, and the diagonal entries represents

the self-interaction of the excited oscillators due to vac-

uum. The ncollective modes (eigenstates) of the excited

system provide us a complete set of initial states, ψi, and

012345678

h/R

0

0.5

1

1.5

2

2.5

Coupling strength (g)

non Markovian

Markovian (weak)

2 nm

25 nm

FIG. 2. Coupling strength g= Ω/Γ, varying with the relative

separation of the emitter from the surface of gold particles of

radii R= 2 nm and R= 25 nm in a medium of refractive

index 1.5; ~ω0= 2.21 eV. Ω can be notably diﬀerent for the

two models [36]. Results presented in the main paper repre-

sent X or Y polarized initial state of emitter as in Figure 1;

corresponding results of a Z-polarized emitter are in SI [36].

corresponding sets of phases and amplitudes of these os-

cillators. The mixture of these collective eigenstates, that

are not necessarily orthogonal, determines the Hermitian

density matrix ρof the system.

Σ|ψii=λi|ψii(9)

ρ=1

Pn

i=1 |λi|

n

X

i=1 |λi||ψiihψi|(10)

The self-energy of an initial state in the form of its

eigenvalue provides its relative weight in the mixture.

The decay rates and Rabi frequencies of the initial state

appear in the imaginary and real parts of the eigenval-

ues. The ensemble averaged total decay rate and the ex-

pected Rabi frequency of oscillation are given by, Γtotal =

-2Im{tr(Σ)}and hΩi=Pn

i=1 |Re{λi}|. The predictions

of Rabi frequencies can be notably diﬀerent in the non-

Markovian model, as shown by the coupling strengths in

Figure 2.

We decompose the mixture of initial states into a set

of orthogonal pure states, and this allows us to sum over

the superpositions of oscillators in each orthogonal state,

to evaluate Γr. Solving Hermitian eigenvalue problems

ρ|φii=pi|φii, we have probabilities piand the pure

states |φiiin the mixture. The amplitude and the relative

phase of an oscillator in the pure state φi, is used with the

normalized polarization set by the solution of the coupled

system in equation (3). The polarization of dipoles jfor

state φiare given by

Pi

j=φi

jPj/kPjk(11)

The radiative decay rates Γr

iof a pure state φiis eval-

4

0 1 2 3 4 5 6 7 8

h/R

2

4

6

8

10

12

14

r

Markovian (weak) : R = 2 nm

non Markovian : R = 2 nm

Markovian (weak) : R = 25 nm

non Markovian : R = 25 nm

First order correction

FIG. 3. Radiative decay rates Γrnormalized by free-space

decay rates Γr

0for emission energy ~ω0= 2.21 eV at varying

separations from gold nanospheres. The ﬁrst order correction

of the weak-coupling values for non-Markovian eﬀects [18] is

given by the dashed lines. This correction for eﬀective values

was given by Γr

eff . = Γr+ Γnr e−1/g.

uated using the polarization given by the above equa-

tion and the conventional integral for the superposition

of the radiated ﬁeld from all oscillators jas in equa-

tion (4). The radiative decay rate of the system is given

by Γr=Pn

i=1 piΓr

i. The plots of the the radiative de-

cay in Figure 3 show an unconventional large peak for

the 2 nm radii fully absorbing metal particles even at

large relative separations. The earlier proposed ﬁrst or-

der correction of the Markovian evaluations suited for

weak coupling, captures the large increase of radiative

decays, but it diverges for smaller separations and larger

coupling strengths. These corrections of the Markovian

model are nevertheless useful, and the limiting case of

large particles representing a plane surface can also be

used to predict the enhancements in SERS [18].

Similarly, the quantum eﬃciencies shown in Figure 4

for X/Y polarized initial state of emitter contrast the pre-

dictions of the Markovian model with large enhancements

for the smaller particles; values for Z-polarization show

suppressed quenching in these cases [36]. The movement

of the peak eﬃciencies towards the smaller separations,

and the possible enhancement due to smaller particles,

are signiﬁcant for unraveling the mechanism of SERS.

The rough metal surface and its smaller features while

enhancing the incident radiation in its near-ﬁeld, are also

shown here to enhance the emission from the molecules

at such separations less than 10 nm. Further, the larger

Γrpredicted in the non-Markovian model can play a sig-

niﬁcant role in the SERS gains, due to the repeated ex-

citations possible, even when the predicted quantum eﬃ-

ciencies of the two models may be similar due to a large

non-radiative component [18].

The Markovian models of coupling considered only the

uncertainty of the path of the photon from the emit-

1234567

0

0.5

1

1.5

2

12345678

h/R

0

1

2

Emission enhancement (Q/Qo)

R = 2 nm

R = 10 nm

R = 25 nm

R = 50 nm

Markovian (weak)

non Markovian

FIG. 4. Normalized quantum eﬃciencies show enhancement

of Q=Γr/Γtotal for even the smaller 2 nm gold nanoparticles,

in the case of the non-Markovian model. Q0=1/3 was as-

sumed; the enhancements have an inverse relationship with

Q0.

012345678

h/R

0

0.2

0.4

0.6

0.8

Normalized entropy

R = 2 nm

R = 10 nm

R = 25 nm

R = 50 nm

FIG. 5. The normalized von Neumann entropy of the mixture,

1

log nPn

1pilog pi, varying with relative separations for gold

particles of diﬀerent radii R. Number of oscillators nused

was 1419.

ter, and the corresponding interference. In the case of

strongly absorbing or non-scattering nanostructures, the

large eﬀect of the possible re-absorption of photons from

the excited nanostructure by the proximal emitter at

ground-state, represented by a non-Markovian loop in

Figure 1, has to be accounted by a mixture of initial

states of the system; see Figure 5 for entropy of the mix-

tures. From a classical perspective, one may relate the

origin of this eﬀect to the evanescent ﬁelds [37–39] of the

excited dissipating nanostructure, coupling back to the

emitter at ground-state.

5

ACKNOWLEDGMENTS

M.V. thanks Girish S. Agarwal for illuminating dis-

cussions of the literature on self-interactions. K.J. and

M.V. thank the department of Computational & Data

Sciences, Indian Institute of Science for its generous sup-

port.

Appendix A: Green Dyads

The required dyads Gfor interaction among the point-

dipoles, are solutions for a point source in a homogeneous

background:

5 × 5 ×G(r,rj;ω)−k2G(r,rj;ω) = Iδ(r−rj).(A1)

giving us

G(ri,rj;ω)=(I+55

k2)g(kri−rjk) (A2)

where g(r) = eikr

4πr .Iis a unit dyad and the wave number

k=√ω

c, and δ(r−rj) represents the point source at rj.

The global Green tensors ˆ

Gcan be written in terms of

3×3 blocks using the above dyads as

ˆ

G3i→3i+2,3j→3j+2 =G(ri,rj;ω) (A3)

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6

SUPPLEMENTARY INFORMATION

7

012345678

h/R

0

1

2

3

4

Coupling strength (g)

non Markovian

Markovian (weak)

2 nm

25 nm

FIG. 6. Coupling strength gfor Z-polarization (initial polarization of emitter) given by Ω/Γ, varying with the relative separation

of the emitter from the surface of gold particles of radii R= 2 nm and R= 25 nm.

012345678

h/R

2

4

6

8

10

12

14

r

Markovian (weak) : R = 2 nm

non Markovian : R = 2 nm

Markovian (weak) : R = 25 nm

non Markovian : R = 25 nm

First order correction

FIG. 7. Radiative decay rates Γrfor Z-polarization (initial polarization of emitter) normalized by free-space decay rates Γr

0

at emission energy ~ω0= 2.21 eV when gold nanoparticles are introduced with varying separations in a medium of refractive

index 1.5. The ﬁrst order correction of the weak-coupling values for non-Markovian eﬀects [18] is given by the dashed lines.

This correction for eﬀective values was given by Γr

eff . = Γr+ Γnr e−1/g.

8

1234567

0

0.5

1

1.5

2

1234567

h/R

0

1

Emission enhancement (Q/Qo)

R = 2 nm

R = 10 nm

R = 25 nm

R = 50 nm

Markovian (weak)

non Markovian

FIG. 8. Normalized quantum eﬃciencies for Z-polarization (initial state of emitter) show less quenching of Qfor the smallest

and largest gold nanoparticle, in the case of the non-Markovian model. Q0=1/3 was assumed, and the enhancements have an

inverse relationship with Q0.

012345678

h/R

0

0.2

0.4

0.6

0.8

1

Normalized entropy

R = 2 nm

R = 10 nm

R = 25 nm

R = 50 nm

FIG. 9. The normalized von Neumann entropy of the mixture for Z-polarization (initial state of emitter), 1

log nPn

1pilog pi,

varying with relative separations for particles of diﬀerent radii R.nis 553 in the evaluations presented.

9

012345678

h/R

2

4

6

8

10

12

14

r

Markovian (weak) : R = 10 nm

non Markovian : R = 10 nm

Markovian (weak) : R = 50 nm

non Markovian : R = 50 nm

First order

correction

FIG. 10. Radiative decay rates Γrfor X/Y-polarization (initial state of emitter) normalized by free-space decay rates Γr

0at

emission energy ~ω0= 2.21 eV when gold nanoparticles are introduced with varying separations in a medium of refractive

index 1.5. The ﬁrst order correction of the weak-coupling values for non-Markovian eﬀects [18] is given by the dashed lines.

This correction for eﬀective values was given by Γr

eff . = Γr+ Γnr e−1/g.

012345678

h/R

2

4

6

8

10

12

r

Markovian (weak) : R = 10 nm

non Markovian : R = 10 nm

Markovian (weak) : R = 50 nm

non Markovian : R = 50 nm

First order

correction

FIG. 11. Radiative decay rates Γrfor Z-polarization (initial state of emitter) normalized by free-space decay rates Γr

0at

emission energy ~ω0= 2.21 eV when gold nanoparticles are introduced with varying separations in a medium of refractive

index 1.5. The ﬁrst order correction of the weak-coupling values for non-Markovian eﬀects [18] is given by the dashed lines.

This correction for eﬀective values was given by Γr

eff . = Γr+ Γnr e−1/g.

10

012345678

h/R

100

102

104

106

Decay rate ( )

non Markovian

Markovian (weak)

2 nm

25 nm

FIG. 12. Total decay rates Γ for X/Y-polarization (initial state of emitter) normalized by free-space decay rates Γr

0at emission

energy ~ω0= 2.21 eV when gold nanoparticles of radii 2 nm and 25 nm are introduced with varying separations in a medium

of refractive index 1.5.

012345678

h/R

10-4

10-2

100

102

104

106

Rabi frequency ( )

non Markovian

Markovian (weak)

2 nm

25 nm

FIG. 13. Rabi frequency Ω for X/Y-polarization (initial state of emitter) normalized by free-space decay rates Γr

0at emission

energy ~ω0= 2.21 eV when gold nanoparticles of radii 2 nm and 25 nm are introduced with varying separations in a medium

of refractive index 1.5.

11

012345678

h/R

100

102

104

106

Decay rate ( )

non Markovian

Markovian (weak)

25 nm

2 nm

FIG. 14. Total decay rates Γ for Z-polarization (initial state of emitter) normalized by free-space decay rates Γr

0at emission

energy ~ω0= 2.21 eV when gold nanoparticles of radii 2 nm and 25 nm are introduced with varying separations in a medium

of refractive index 1.5.

012345678

h/R

10-5

100

105

Rabi frequency ( )

non Markovian

Markovian (weak)

25 nm

2 nm

FIG. 15. Rabi frequency Ω for Z-polarization (initial polarization of emitter) normalized by free-space decay rates Γr

0at emission

of refractive index 1.5.

12

FIG. 16. Normalized quantum eﬃciencies for X/Y-polarization (initial state of emitter) show enhancement of Qfor the smallest

and large gold nanoparticle and no enhancement for 10 nm particle, in the case of the ﬁrst order non-Markovian model. Q0=1/3

was assumed, and the enhancements have an inverse relationship with Q0.

13

FIG. 17. Normalized quantum eﬃciencies for Z-polarization (initial polarization of emitter) show more enhancement of Qfor

the smallest and largest gold nanoparticle, in the case of the ﬁrst order non-Markovian model. Q0=1/3 was assumed, and the

enhancements have an inverse relationship with Q0.

14

FIG. 18. Change in the coupling strength of emitter with the particle, due to diﬀerent coupling strengths (0.05 and 0.1) of the

isolated oscillators and vacuum