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

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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.
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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’
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 n1
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
G1
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 n1 oscillators only in the nanoparticle, it can be
always rewritten as a weighted sum of the n1 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)
jj 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 hi=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 e1/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 Qr/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(rrj).(A1)
giving us
G(ri,rj;ω)=(I+55
k2)g(krirjk) (A2)
where g(r) = eikr
4πr .Iis a unit dyad and the wave number
k=ω
c, and δ(rrj) 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
G3i3i+2,3j3j+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 e1/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 e1/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 e1/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
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.
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
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