Non-Markovian dynamics in plasmon-induced spontaneous emission interference

0.57 MB·

by  I. Thanopulos

Non-Markovian dynamics in plasmon-induced spontaneous emission interference
I. Thanopulos1,∗ V. Yannopapas2,† and E. Paspalakis3‡
1 Department of Optics and Optometry, T.E.I. of Western Greece, Aigio 251 00, Greece
2 Department of Physics, National Technical University of Athens, Athens 157 80, Greece and
3 Department of Materials Science, School of Natural Sciences, University of Patras, Patras 265 04, Greece
Weinvestigatetheoreticallythenon-MarkoviandynamicsofadegenerateV-typequantumemitter
inthevicinityofametallicnanosphere,asystemthatexhibitsquantuminterferenceinspontaneous
emissionduetotheanisotropicPurcelleffect. WecalculatenumericallytheelectromagneticGreen’s
tensor and employ the effective modes differential equation method for calculating the quantum
dynamics of the emitter population, with respect to the resonance frequency and the initial state
of the emitter, as well as its distance from the nanosphere. We find that the emitter population
7
evolution varies between a gradually total decay and a partial decay combined with oscillatory
1
population dynamics, depending strongly on the specific values of the above three parameters.
0
2 Understrongcouplingconditions,coherentpopulationtrappingcanbeobservedinthissystem. We
compare our exact results with results when the flat continuum approximation for themodified by
n
the metallic nanosphere vacuum is applied. We conclude that the flat continuum approximation is
a
anexcellentapproximationonlywhenthespectraldensityofthesystemunderstudyischaracterized
J
bynon-overlapping plasmonic resonances.
6
1
PACSnumbers: 42.50.Nn,42.50.Ct,73.20.Mf,78.67.Bf

]
h
I. INTRODUCTION tive refractive index metamaterials25–27, metasurfaces28,
p
- hyperbolic metamaterials29, as well as plasmonic
nt Metallic nanoparticles (MNPs) can confine light in- nanostructures30–32.
a
side deep subwavelength volumes in electromagnetic
u
(EM)modes,whicharecalledlocalizedsurfaceplasmons
q
[ (LSPs). This strong field localization leads to an en-
hancement of the light-matter interaction in the vicinity |1 > |2 >
1
of the MNP, featuring some very interesting physics in
v w w
a broad range of contexts such as optical antennas1,2, x 0 0
0
surface-enhancedRamanscattering3, photovoltaics4, en-
4 |0 >
4 ergy transfer in light-harvesting5 and biosensing6.
4 The presence of LSPs in metal-dielectric interfaces
0 strongly modifies the density of the EM modes in the D z
.
1 surroundings, leading to strong modification of the life-
0 time of quantum emitters (QE) placed close to the
7 MNP. Moreover, when the distance between the QE Q E
y
1
and the MNP surface is very small, theory predicts
:
v that coherent energy exchange between the QE and the M N P
i modified EM modes accompanied with non-Markovian
X
dynamics may take place7–12. The above phenom-
r FIG.1: (coloronline) TheconfigurationofthedegenerateV-
ena have been demonstrated mostly experimentally in
a typeQEwithresonancefrequencyω0,placedatadistanceD
the perturbative (weak-coupling) regime wherein quan-
from the surface of a spherical MNP.
tum dynamics is irreversible13–19. Nevertheless, recently
strongcouplingatroomtemperaturebetweenanorganic
molecule and a plasmonic nanocavity has been demon-
strated experimentally20, followed by a detailed theoret- Here,incontrasttopreviouswork22–32,theinteraction
ical analysis21. of the QE with the MNP is not restricted to the weak-
In this work we study theoretically the popu- coupling regime. We use a quantum-mechanical formal-
lation dynamics of a V-type QE coupled electro- ism for the EM field excitations9–11, which goes beyond
magnetically to a MNP. The spontaneous emis- the Markov approximation and incorporates the exact
sion of such a QE may exhibit interference ef- populationdynamicsbyemployingtheeffectivemodedif-
fects due the fact that the MNP affects differently ferential equation (EMDE) method33. The anisotropic
the EM modes along mutually perpendicular direc- enhancement of the spontaneous decay rates of a QE
tions, known as the anisotropic Purcell effect22,23. in the proximity of a MNP is calculated using an EM
This phenomenon has been studied in various pho- Green’s tensor technique34. The purpose of the present
tonic structures, including periodic dielectrics24, nega- work is to study in detail the crossover regime in which
2
theQEpopulationdynamicschangesfromirreversibleto The state of the system is given by
reversible for a system shown in Fig. 1, manifested by
the emergence of non-decaying oscillations in the evolu- Ψ(t) = c1(t)e−iω0t 1;0ω +c2(t)e−iω0t 2;0ω +
| i | i | i
tion of the QE population. Such oscillations signify the
d~r dω C(~r,ω,t)e−iωt 0;1 , (4)
coherent exchange of energy between the QE and modi- Z Z | ~r,ωi
fied reservoir of EM modes due to the nearby MNP.
This paper is organized as follows: In Section II we where 0;1 fˆ†(~r,ω)0;0 . Here, n;a = n
presentourmethodologyforcalculatingthe spontaneous a , wh|ere~r,nωi(≡n =λ0,1,2)|denωoites the q|uantium s|taite⊗s
emissiondynamicsofaQEnearaMNP.Wethenpresent o|fithe V-ty|pei system (see Fig. 1) and a denotes the
anddiscusstheresultsforaV-typesysteminSectionIII. photonicstates(statesofthemodifiedEM|viacuum),with
We alsopresentrelevantresults for atwo-levelsystemin 0 meaning zero and 1 one photon states. The
ω ~r,ω
Section IV. Finally, we conclude our findings in Section |appilication of the time-d|epenident Schr¨odinger equation
V. iΨ˙(t) = H Ψ(t) yields a set of differential equations
| i | i
thatcanbe formallyintegrated,resultingintwointegro-
differential equations for c (t) and c (t), given by
1 2
II. CALCULATION OF THE SPONTANEOUS
t
EMISSION DYNAMICS OF A V-TYPE QE NEAR c˙ (t)=i dt′ K11(t t′)c (t′)+K12(t t′)c (t′) ,
A MNP 1 Z0 (cid:16) − 1 − 2 (cid:17)
(5)
We consider a degenerate V-type QE at distance D
t
fromthe surface of a sphericalMNP, as shownin Fig. 1. c˙ (t)=i dt′ K21(t t′)c (t′)+K22(t t′)c (t′) ,
2 1 2
The origin of the coordinate system coincides with the Z0 (cid:16) − − (cid:17)
center of the sphere and the QE lies on the z axis of the (6)
coordinate system. The Hamiltonian of the system9,11, where
is given by (we use ¯h=1 throughout this work)
eiω0τ ∞
Kij(τ) = i ij(ω)e−iωτdω (7)
∞ π Z G
H = ω σˆ + d~r dω ω ˆf †(~r,ω) ˆf(~r,ω) 0
iX=1,2 0 ii ∞Z Z0 · Gij(ω) ≡ ǫω0c22hµ~†i0·Im[Gˆ(~r0,~r0,ω)]·~µj0i, (8)
σˆ dω~µ Eˆ(~r,ω)+H.c. . (1)
− (cid:20) i0Z i0· (cid:21) with τ t t′ and i,j =1,2.
iX=1,2 0 ≡ −
In general, the MNP affects differently the EM modes
along the tangential ( ) and the radial ( ) directions,
In Eq. (1), ˆf(~r,ω), ˆf †(~r,ω) stand for the bosonic vec- k ⊥
corresponding to a QE with a transition dipole oriented
tor field operators for the elementary excitations of the
parallel and perpendicular to the surface of the MNP,
system, σˆ denotes the Pauli operator, ~µ = µεˆ and
ij 10 − respectively. In our case, the radial direction coincides
~µ = µεˆ are the dipole moments of the QE, where
20 + with the z axis, and the tangential direction is parallel
εˆ (εˆ iεˆ )/√2 describe the right-rotating (εˆ )
± z x(y) + to the x and y axes, according to Fig. 1. In such a case,
≡ ±
andleft-rotating(εˆ )unitvectors(µistakentobereal).
− we have (i,j =1,2)
Also, ω stands for the resonance frequency between the
0
two degenerate upper levels and the lower level of the ∞
V-type QE, with the energy of the lower level taken as Kii(τ)=K+(τ)=ieiω0τ J+(ω)e−iωτdω, (9)
Z
zero. 0
The electric field vector operator Eˆ(~r,ω) is given by
∞
Kij(τ)=K−(τ)=ieiω0τ J−(ω)e−iωτdω,i=j
Eˆ(~r,ω)= iω2 d~s Im[ǫ(~s,ω)]Gˆ(~r,~s,ω) ˆf(~s,ω), Z0 6
√πǫ c2 Z · (10)
0 p
(2) with
with Gˆ(~r,~s,ω) being the dyadic EM Green’s tensor de-
1 ω2µ2
fined as J±(ω) = Im[Gˆ (~r ,~r ,ω) Gˆ (~r ,~r ,ω)]
2π ǫ c2 ⊥ 0 0 ± k 0 0
0
Gˆ(~r,~s,ω) ǫ(~r,ω)ω2Gˆ(~r,~s,ω)=ˆIδ(~r ~s). (3) ≡ J⊥(ω)±Jk(ω), (11)
∇×∇× − c2 −
where Gˆ Gˆ and Gˆ Gˆ . The decay rate
⊥ zz k xx(yy)
In Eqs. (2) and (3), ˆI is the unit dyad (unit tensor), Γ(ω) in the≡presence of a MN≡P is given by35
ǫ(~r,ω)is the spatially and frequency dependent complex
dielectricfunctionoftheMNP,andcisthespeedoflight 2µ2ω2
Γk(ω)= Im[Gˆ (~r ,~r ,ω)], k = , , (12)
in the vacuum. ǫ c2 k 0 0 ⊥ k
0
3
t
and we thus obtain 2(t)=i W−c (t′)+W+c (t′) e−i(ω0−ωi)t′dt′,
Jk(ω)= 1 Γk(ω) = 1 λk(ω,D)Γ0(ω), (13) Ji Z0 (cid:16) i 1 i 2 (cid:17) (23)
2π 2 2π 2 for which holds
with Γ0(ω) being the decay rate of the QE in free- J˙i1(t) = i(Wi+c1(t)+Wi−c2(t))e−i(ω0−ωi)t, (24)
space,andλk(ω,D)isthedirectional(k = , )enhance- ˙2(t) = i(W−c (t)+W+c (t))e−i(ω0−ωi)t. (25)
ment factor of the free-space decay rate du⊥e tko the pres- Ji i 1 i 2
In this way, we have transformed the two integro-
ence of the MNP at distance D from the QE. Due to
the anisotropy, the enhancement factors λk(ω,D) and differentialequations,Eqs. (5),(6),intoasetof2(M+1)
λ⊥(ω,D) are not equal, in general. We further write effectivemodes(ordinary)differentialequations(EMDE)
with constant coefficients, given by Eqs. (21), (24) and
Γ0(ω)= 3ωπ03ǫµ0c23(ωω0)3 ≡Γ0(ω0)(ωω0)3, (14) (e2x5p)(.−iDωeitn)oJtiiνn(gt),C˜νw(itt)h≡ν e=xp1(,−2iωa0ntd)cνi(t=) a1n,.d..J,˜iMν(t,)w≡e
obtain Eqs. (21), (24) , and (25) in a matrix form
since in free-space Im[Gˆ0(~r0,~r0,ω)] = 6ωπcˆI is valid12,35, C˜˙(t)=iH˜ C˜(t), (26)
with Gˆ (~r ,~r ,ω) being the free-space Green’s tensor. ·
We now0ob0tai0n whereC˜(t)=(C˜ (t),C˜ (t), ˜1(t),..., ˜1 (t), ˜2(t),..., ˜2(t))T
1 2 J1 JMT J1 JM
(C˜ (t),C˜ (t), ˜(1)(t),..., ˜(2M)(t)) , with all
∞ 1 2
K±(τ) = i J±(ω)e−i(ω−ω0)τdω, (15) ≡˜(i)(0) = 0. HerJe the supJerscript T is the trans-
Z0 Jpose operation and H˜ is a time-independent complex
Γ (ω )λ⊥(ω,D) λk(ω,D) ω
J±(ω) 0 0 ± ( )3 matrix, given by
≡ 2π 2 ω
0 W(1)
≡ Jrad(ω)±Jtan(ω), (16) H˜ =(cid:18)S Y (cid:19), Z =... , Y =(cid:18)ΩΩ((12)) (cid:19) .
Z R W(2M)
with Γ0(ω0) = 1/τ0, and τ0 standing for the free-  
(27)
space decay time of the QE. Taking into account that
wexrpit(e−(i(ωωω0)−3 ≈ω01),τ)inctorondtruicbiuntgesthmuasitnhlye flaraotucnodntωin0u,uwme caapn- tahInneddERiaqig.,io′n(=a2l7R)b,lio+wcMkes,i.h′+aFvMuert=Shke,δrkim′,i′o=ωrei,,δt(khi,k,e′iω′20=M, (1rk,o,.wk.s.′,o=Mf )1,,af2or)er,
proximation (FCA), and obtaining Z
definedasW(η) =(W+,W−)andW(M+η) =(W−,W+),
η η η η
K±(τ) K± (τ) (17) for η = 1,...,M. Lastly, the two rows of are defined
≈ FCA as [Ω(1)] = [Ω(2)] = 1, for 3 m Y2+M and
∞ 1,m 2,n
K± = i J± (ω)e−i(ω−ω0)τdω, (18) 3+M n 2(M +1), with [Ω(1)]≤ =≤[Ω(2)] = 0,
FCA Z0 FCA for m a≤nd n≤otherwise. Given these1d,mefinitions w1,enwrite
J± (ω) Γ0(ω0)λ⊥(ω,D)±λk(ω,D) the solution of Eq. (26) as,
FCA ≡ 2π 2
M′
≡ JFraCdA(ω)±JFtaCnA(ω). (19) C˜(t)= ΛjeiλjtC˜j0, (28)
Xj=1
We cannowexpand(to anygivenaccuracy)K±(τ) as
a sumofM exponentialterms ateffective mode frequen- whereM′ =2(M+1)isthenumberofλj eigenvaluesand
cies ωi (i=1,...,M)33, Λj eigenvectorsof H˜. The vectorof complex coefficients
C˜0 is given as C˜0 = L−1 C˜(0). The columns of the
K±(τ)=ieiω0τ M Wi±e−iωiτ. (20) nLoj≡te[tΛha1,t..s.i,nΛceMH˜′] misantoritxHaer·remtihteiaeni,gLen−v1ec=toLrs†.of H˜. We
Xi=1 6
The EM response of the 5 nm silver sphere used in
Then, the equations for the two upper states of the QE this study is described by a Drude dielectric function,
are given by ǫm(ω) = ǫm,∞ − ωp2/(ω2 + iωγ), characterized by its
plasma frequency ω = 9.176 eV, high-frequency com-
p
M ponent ǫ = 3.718 eV, and Ohmic losses γ = 0.021
m,∞
c˙ν(t)=i ei(ω0−ωi)tJiν(t), ν =1,2 (21) eV. The radial and tangential enhancement factors of
Xi=1 the free-space decay rate of the QE due to the presence
of the MNP as a function of frequency ω at distance D,
where we have introduced new variables given by
calculated using an EM Green’s tensor technique34, are
presented in Fig. 2.
t
1(t)=i W+c (t′)+W−c (t′) e−i(ω0−ωi)t′dt′,
Ji Z0 (cid:16) i 1 i 2 (cid:17)
(22)
4
7 large enhancement factors λk(ω ,D) and λ⊥(ω ,D), re-
0 0
6 spectively, shown in Fig. 2. We also define the effective
50 DD == 11 .n5m5 nm decaytimeofaQEwithresonancefrequencyω0andfree-
1 5 D = 2 nm space decayrate τ atdistance D fromthe MNP surface
/ 0
) 04 as τeff ≡ (τerfafd +τetfafn)/2, given by the average of the
Γ corresponding radial τrad τ /λ (ω ,D) and tangen-
/ ad3 tial τetfafn ≡τ0/λk(ω0,De)ffeff≡ectiv0e d⊥ecay0 times.
Γr2

(
1
A. QD dynamics at ω0 =3.84 eV
3.7 3.8 3.9 4 4.1 4.2
ω / eV
3 1
510 DD == 11 .n5m5 nm 0.8 |c(t)|2
2 D = 2 nm 1
Γ ) /0 2t)|0.6 |c2(t)|2
(
/ an1 |ci0.4
Γt
( 0.2
3.7 3.8 3.9 4 4.1 4.2 00 0.05 0.1 0.15 0.2 0.25
ω / eV t / ps
FIG.2: (coloronline)Radialλ⊥(ω,D) Γrad/Γ0(toppanel)
≡
and tangential λk(ω,D) Γtan/Γ0 (bottom panel) enhance-
≡
mentfactorsasfunctionofthedistanceDoftheQEfromthe
MNP and EM mode frequency ω. c(0)=0.7071; c(0)=0.7071
0.4 c1(0)=0.7071; c2(0)=-0.7071
2 1 2
)|
t
(
2
c
=|
2
We note that our results below are convergedwith re- )|0.2
t
(
specttothenumberofeffectivemodesM includedinthe 1
c
|
dynamics; we use M = 1001 in the energy range 3.5-4.5
eV.WealsonotethatourresultsonV-typeandtwo-level
systems presented below are obtained using the K±(τ) 0
0 0.05 0.1 0.15 0.2 0.25
kernel as defined in Eq. (15) unless otherwise stated. t / ps
FIG.3: (coloronline)Top: Populationevolutionof 1 (black
| i
III. DYNAMICS OF A V-TYPE QE NEAR A solid curve) and 2 (red dashed curve) of a QD with ω0 =
| i
3.84 eV at D = 1 nm. Bottom: Population evolution of 1
MNP
| i
of the QD with ω0 = 3.84 eV at D = 1 nm for the initial
states, Ψ(0) = √0.51 +√0.52 (black solid curve) and
In this work, we investigate the c1(t)2 and c2(t)2 Ψ(0) =| √0.i51 √0|.5i2 (red d|asihed curve).
| | | | | i | i− | i
time evolution for state-of-the-art QEs with transition
frequencies in the optical regime, such as quantum dots
(QD) and J-aggregates (J-AGR), with free-space decay
timeτ 4ns36 and 70ps37,respectively;correspond- The population dynamics of a QD, i.e. a QE with
0
≈ ≈
ingly,weoftenusethetermsQDandJ-AGRforreferring τ = 4 ns, with ω = 3.84 eV at D = 1 nm for different
0 0
to QEs with such free-space decay times below. In each initial states is shown in Fig. 3. In the top panel of this
case, we study the dynamics for different initial states figure, the time-evolution of c (t)2 (black solid curve)
1
of the QE. Moreover, for each QE, we consider two res- and c (t)2 (red dashed curv|e) is|shown for the initial
2
| |
onance frequencies, ω = 3.84 eV and ω = 4.16 eV, state Ψ(0) = 1 . We find that the initial population
0 0
| i | i
the frequencies corresponding to (relatively) small and in state 1 decays rapidly into the modified EM modes,
| i
5
while thereisapartialpopulationtransferfromstate 1 and state 2 of such a system is presented in Fig. 4.
| i | i
into state 2 within about 50 fs. Partial revivals of the
| i
populations in 1 and 2 at 125 fs and 175 fs, respec-
| i | i
tively, are further observed, before all population finally
endsupinthemodifiedEMcontinuumafterabout250fs.
TheeffectivedecaytimeforaQEwithτ =4ns,ashere,
0
is τeff ≈ 89 fs (τerfafd ≈35 fs and τetfafn ≈143 fs) consider- 0.8 |c(t)|2
ingthatλ (3.84,1) 1.13 105andλ (3.84,1) 2.8 104 1
asshowni⊥nFig. 2. T≈hetim·e-evolutioknofstate≈1 sh·own 2 0.6 |c2(t)|2
| i )|
in the top panelof Fig. 3, at earlytimes, is practically a (t
decay,characterisedprimarilybythedecaytimeτrad,in- |ci0.4
eff
stead of τ , indicating that the clearly non-Markovian
eff
dynamics are dominated by the radial part (along the z 0.2
axis)ofthe modified by the MNP spectraldensity ofthe
EM modes. Moreover, small oscillations on the popu- 0
0 0.05 0.1 0.15 0.2 0.25
lation evolution dynamics of 1 and 2 , which are dis- t / ps
| i | i
cernible also in the top panel of Fig. 3, further point
to a rather moderate non-Markovian character. There
is also a partial oscillatory exchange of population be-
tweenstate 1 and state 2 via the modified EM modes
| i | i c(0)=0.7071; c(0)=0.7071
as indicated by a phase difference, which we define as π, 0.4 c1(0)=0.7071; c2(0)=-0.7071
since, when one is at a (local) maximum the other is at 2 1 2
)|
a (local) minimum, and vice versa. (t
2
c
In the bottom panel of Fig. 3 we present the popu- =|
2
lation dynamics for 1 and 2 for the symmetric initial )|0.2
state Ψ(0) =√0.5|1i+√0.|5i2 (black solid curve) and (t1
c
| i | i | i |
theantisymmetricinitialstate Ψ(0) =√0.51 √0.52
| i | i− | i
(reddashedcurve),denotedbySISandAIS,respectively.
WinecaosbeseorfvSeItShaantdthAeIpSoepquulaaltsiotnoecvo(ltu)ti2o,nde|cc1a(yts)|f2a,swthinictho 00 0.05 t 0/. 1ps 0.15 0.2
2
| |
the modified EMmodes for both initial conditions; how-
ever, the explicit dynamics depends on the initial state. FIG.4: (coloronline)Top: Populationevolutionof 1 (black
| i
In case of a SIS, the population decay of 1 is slower solidcurve)and 2 (reddashedcurve)ofaJ-AGRwithω0=
| i | i
than in case of an AIS; in addition, in the latter case, 3.84 eV at D = 2 nm. Bottom: Population evolution of 1
| i
thenon-Markoviancharacteroftheunderlyingdynamics of the J-AGR with ω0 =3.84 eV at D=2 nm for the initial
is apparentlystronger. The differencesinthe population states, Ψ(0) = √0.51 +√0.52 (black solid curve) and
Ψ(0) =| √0.i51 √0|.5i2 (red d|asihed curve).
evolution of state 1 and state 2 in case of a SIS and
| i | i− | i
| i | i
anAISclearlyindicate thatquantuminterferenceeffects
affect the underlying dynamics, as well as the strength
of the non-Markovian character on it. Furthermore, we In the top panel of Fig. 4, the evolution of c (t)2
1
| |
observethat the populationdynamics with a SIS and an (black solid curve) and c (t)2 (red dashed curve) for
2
| |
AIS, at early times, can be understood as decays, with initial state Ψ(0) = 1 are shown. We find that both
| i | i
decay times τtan 145 fs and τrad 35 fs, respectively; c (t)2and c (t)2decaytotallywithin200-250fs. Inthe
eff ≈ eff ≈ | 1 | | 2 |
apparently,thedynamicswithaSISisdominatedbythe meanwhile, though, there is partial oscillatory exchange
tangential Purcell effect and the dynamics with an AIS ofpopulationbetweenstate 1 andstate 2 viathemod-
| i | i
is dominated by the radial Purcell effect. ified EM modes as indicated by the phase difference of
π. Moreover, the evolution of c (t)2 and c (t)2 point
1 2
| | | |
to a rather strong non-Markovian character of the un-
derlying dynamics, as there are very distinct oscillations
B. J-AGR dynamics at ω0=3.84 eV on the top of the overall gradual decay of the popula-
tion of these states. The population evolution of c (t)2
1
and c (t)2 for a SIS (black solid curve) and an AI|S (re|d
2
We now focus on the dynamics of a J-AGR, i.e. a QE |
dashed curve) are shown in the bottom panel of Fig. 4.
with τ =70 ps, with resonance frequency ω =3.84 eV
0 0 We observe that, in both cases, the population of state
at D = 2 nm, and corresponding τ 4 fs (τrad
eff ≈ eff ≈ 1 and 2 decay totally in the modified EM mode con-
1.6 fs and τetfanf ≈ 6.5 fs), after taking into account that |tiniuum w| iithin about200 fs. Furthermore,in both cases,
λ (3.84,2) 4.40 104 and λ (3.84,2) 1.07 104 as the overall population decay occurs with strong, gradu-
⊥ k
≈ · ≈ ·
shown in Fig. 2. The population evolution of states 1 ally decreasing in magnitude, oscillations of population
| i
6
transfer between states 1 and 2 and the modified EM curve) and c (t)2 (red dashed curve) with initial state
2
| i | i | |
continuum simultaneously. However, the density of os- Ψ(0) = 1 areshownduringthefirst100fs. Atthevery
| i | i
cillations within a given time interval is lower in case early stage of the population evolution of state 1 (not
| i
of a AIS than in case of a SIS, which points to destruc- shownhere),thedynamicsispracticallyafastdecaywith
tivequantuminterferenceeffectsintheunderlyingstrong τ ; however, very quickly the strongly non-Markovian
eff
non-Markovian population dynamics in case of an AIS. character of the underlying dynamics manifests itself as
Wenotethatacloserlookatthepopulationevolutionup an oscillatory transfer of population between the states
to 5 fs in both panels of Fig. 4 (not shown here) shows 1 and 2 ,withthemodifiedEMmodecontinuumasan
| i | i
that the decay time in each case is close to the corre- intermediate, as clearly indicated by the π-phase differ-
sponding τrad, instead of τ , which indicates, that the ence between c (t)2 and c (t)2. The initial population
radialparteofffspectraldensietfyfisdominatingtheunderly- of the V-type|sy1ste|m is d|ec2reased to about half within
ing non-Markovian population dynamics here, as in the the first 100 fs.
case of a QD with the same ω shown Fig. 3.
0
c(0)=0.7071; c(0)=0.7071
0.6 1 2
c(0)=0.7071; c(0)=-0.7071
|c(t)|2 2)| 1 2
0.8 1 (t
|c2(t)|2 =|c20.4
2|c(t)|i00..46 2|c(t)|10.2
0.2
0
0 0.01 0.02 0.03 0.04 0.05
t / ps
0
0 0.02 0.04 0.06 0.08 0.1
t / ps
c(0)=0.7071; c(0)=0.7071
0.4 0.6 1 2
c(0)=0.7071; c(0)=-0.7071
0.3 ||cc12((tt))||22 2=|c(t)|20.4 1 2
2
2(t)|0.2 c(t)|1
ci |0.2
|
0.1
0
0.15 0.16 0.17 0.18 0.19 0.2
t / ps
0
0.4 0.42 0.44 0.46 0.48 0.5
t / ps
FIG.6: (coloronline)Top: Populationevolutionof 1 ofaJ-
| i
AGRwithω0 =3.84eVatD=1.55nmfortheinitialstates,
FIG.5: (coloronline)Top: Populationevolutionof 1 (black Ψ(0) =√0.51 +√0.52 (black solid curve) and Ψ(0) =
solidcurve)and 2 (reddashedcurve)ofaJ-AGRw|itihω0 = √| 0.5i1 √0.5| 2i (red d|aished curve) up to 50 fs. |Bottoim:
3.84 eV at D =|1.i55 nm up to 100 fs. Bottom: Population Popu|laiti−on evol|utiion of 1 of aJ-AGR with ω0 =3.84 eV at
evolutionof 1 (blacksolidcurve)and 2 (reddashedcurve) D=1.55nmfortheinit|iailstates, Ψ(0) =√0.51 +√0.52
ofaJ-AGRw|iithω0 =3.84eVatD=1|.5i5nm,atlatertimes (blacksolidcurve)and Ψ(0) =√0|.51 i √0.52|(ireddash|edi
(400-500 fs). curve) at later times (1|50-20i0 fs). | i− | i
In Fig. 5, we study the population evolution of states In the bottom panel of Fig. 5 we present the popu-
n|1mi,awndith|2iτfor a J2-.3AGfsR(τwraitdh ω00.=9 f3s.8a4ndeVτtaant D3=.81f.s5)5, lation evolution |c1(t)|2 (black solid curve) and |c2(t)|2
eff ≈ eff ≈ eff ≈ (red dashed curve) at later times, in the time interval
taking into account that λ⊥(3.84,1.55) 7.50 104 and 400 - 500 fs. We now find that the time-evolution of
≈ ·
λk(3.84,1.55) 1.85 104 as shown in Fig. 2. In the top the population has reached a steady state, indicating a
≈ ·
panelofFig. 5,the time evolutionof c (t)2 (blacksolid populationexchangebetweenthe V-type systemandthe
1
| |
7
modified EM mode continuum, which amounts in total (τrad 6 fs and τtan 14 fs), taking into account that
eff ≈ eff ≈
to 50% of the initial population. Interestingly, the π- λ (4.16,1) 6.54 105 and λ (3.84,1) 2.83 105, as
pha≈seshiftbetween c1(t)2 and c2(t)2,whichexistedat sh⊥own in Fi≈g. 2. I·n Fig. 7, kwe presen≈t the ti·me evo-
| | | |
earlier times of the dynamics, shown in the top panel of lution of c (t)2 (=c (t)2 here) in case of a SIS (black
1 2
thisfigure,isnowlost. Thepopulationexchangenowhas solid curv|e) an|d an|AIS (|red dashed curve). We observe
nowbecome practicallya transferofpopulationbetween that the population evolutionis different for the two ini-
the states of the V-type system and the modified EM tial states, which again points to quantum interference
continuum, which occurs simultaneously for both states. effects in the underlying dynamics. As in case of a QD
In Fig. 6, we investigate the influence of the initial with ω = 3.84 eV, shown in Fig. 3, the population dy-
0
stateonthe populationevolutionofstates 1 and 2 for namicsshowsamoderatenon-Markoviancharacter,and,
| i | i
a J-AGR with ω0 = 3.84 eV at D = 1.55 nm. In the atearlytimes,ispracticallyadecaywithτeff,whileafter
top panel of this figure, we present the time evolution 200 fs, the population of the upper states is (almost)
of c1(t)2 and c2(t)2 for a SIS (black solid curve) and ≈completely lost. We note that the population dynamics
| | | |
an AIS (red dashed curve) for the first 50 fs. We find of 1 and 2 in case of the initial state Ψ(0) = 1
| i | i | i | i
that the population evolution of states 1 and 2 are (not shown here) looks analogous to the corresponding
| i | i
strongly dependent on the initial state, which indicates dynamics in case of a QD with ω = 3.84 eV at D = 1
0
significantquantuminterferenceeffectsintheunderlying nm, presented in the top panel of Fig. 3 above.
non-Markovian dynamics. In case of a SIS, the c (t)2
1
| |
(= c (t)2)evolutionshowsasteadyoscillatoryexchange
2
of p|opula|tion between the V-type system and the modi- D. J-AGR dynamics at ω0 =4.16 eV
fiedEMmodecontinuum. IncaseofanAIS,thesituation
is strikingly different; the c (t)2 evolution indicates an
| 1 | We now study the population dynamics of a J-AGR,
oscillatorygradually decreasingin magnitude transfer of
i.e. aQEwithτ =70ps,withω =4.16eVatD =1.55
theinitialpopulationoftheV-typesystemintothemod- nm, with τ 0 0.35 fs (τrad 0 0.2 fs and τtan 0.5
ified EM mode continuum. At later times, as shown in eff ≈ eff ≈ eff ≈
fs), taking into account that λ (4.16,1) 3.21 105 and
the bottom panel of Fig. 6, a marginal, only about 5% ⊥ ≈ ·
λ (4.16,1.55) 1.39 105, as shown in Fig. 2. Here, the
of the initial population in total, oscillatory population k ≈ ·
averaged effective decay rate, as well as the radial and
transfer between states 1 and 2 and the modified EM
| i | i tangential effective rates, are in attosecond time range,
modes, is still observable.
which is much smaller than the duration of the dynam-
ics observed; we thus conclude that they are practically
of minor importance for understanding the underlying
C. QD dynamics at ω0=4.16 eV strongly non-Markoviandynamics in this case.
InthetoppanelofFig. 8,thetimeevolutionof c (t)2
1
| |
(black solid curve) and c (t)2 (red dashed curve) are
2
| |
shown for the initial state Ψ(0) = 1 during the first
c(0)=0.7071; c(0)=0.7071 150fs. The stronglynon-Ma|rkoviianc|hairacterofthe un-
2 0.4 c11(0)=0.7071; c22(0)=-0.7071 derlying dynamics manifests itself as an oscillatory ex-
(t)|2 change of population between states |1i and |2i with
c the modified EM mode continuum as an intermediate,
=|
2 as clearly indicated by the phase difference of π between
|c(t)|10.2 o|cf1(tth)e|2Va-ntydp|ec2s(yt)s2t.emFuirsthdeercmreoarsee,dthtoe ianbitoiuatl p4o0p%uloaftioitns
initial value within the first 150 fs.
0
0 0.05 0.1 0.15 0.2
t / ps
In the bottom panel of Fig. 8 we show c (t)2 (black
1
solidcurve)and c (t)2(reddashedcurve)|atlat|ertimes,
FIG. 7: (color online) Population evolution of 1 and 2 2
of a QD with ω0 = 4.16 eV at D = 1 nm for|tihe init|iail 400 - 500 fs. We|now|observe that the population evolu-
states, Ψ(0) = √0.51 +√0.52 (black solid curve) and tion has reached a steady state; now, the non-decaying
Ψ(0) =| √0.i51 √0|.5i2 (red d|aished curve). exchange of population between the V-type system and
| i | i− | i
themodifiedEMmodecontinuumamountsintotalto
≈
25% of the initial population. Interestingly, contrary to
the findings when studying the analogous configuration
We now study the population evolution for a QD at ω = 3.84 eV shown in Fig. 5, the π-phase differ-
0
with ω = 4.16 eV at D = 1 nm, with τ 10 fs ence between c (t)2 and c (t)2, which existed at ear-
0 eff 1 2
≈ | | | |
8
c(0)=0.7071; c(0)=0.7071
|c(t)|2 0.6 1 2
0.8 |c12(t)|2 2(t)| c1(0)=0.7071; c2(0)=-0.7071
2
2|c(t)|i00..46 2(t)|=|c10.4
c
|0.2
0.2
0
0 0 0.01 0.02 0.03 0.04 0.05
0 0.05 0.1 t / ps
t / ps
0.4
0.4
2
|c(t)|
1 c(0)=0.7071; c(0)=0.7071
2 1 2
|c(t)| 0.3 c(0)=0.7071; c(0)=-0.7071
0.3 2 2)| 1 2
(t
2
2 c
)| =|0.2
(t0.2 2
ci )|
| (t
1
c
|0.1
0.1
0
0 0.15 0.16 0.17 0.18 0.19 0.2
0.4 0.42 0.44 0.46 0.48 0.5 t / ps
t / ps
FIG. 9: (color online) Top: Population evolution of 1
FIG.8: (coloronline)Top: Populationevolutionof 1 (black | i
solidcurve)and 2 (reddashedcurve)ofaJ-AGRw|itihω0 = of the J-AGR with ω0 = 4.16 eV at D = 1.55 nm for
| i the initial states, Ψ(0) = √0.51 + √0.52 (black solid
4.16 eV at D = 1.55 nm up to 100 fs. Bottom: Population curve) and Ψ(0) |= √0i.51 √|0.i52 (red| diashed curve)
evolutionof 1 (blacksolidcurve)and 2 (reddashedcurve) | i | i − | i
| i | i up to 50 fs. Bottom: Population evolution of 1 of the J-
ofaJ-AGRwithω0 =4.16eVatD=1.55nm,atlatertimes | i
(400-500 fs). AGR with ω0 = 4.16 eV at D = 1.55 nm for the initial
states, Ψ(0) = √0.51 +√0.52 (black solid curve) and
Ψ(0) =| √0.i51 √0|.5i2 (red d|aished curve) at later times
| i | i− | i
(150-200 fs).
lier times of the dynamics, is still present; here, clearly
the population exchange between the two states of the
V-type system with the modified EM continuum as an
intermediate, suffers no dephasing, even over long time
periods, in this case.
In Fig. 9, we investigate the influence of the initial
state on the population evolution of states 1 and 2
| i | i
for a J-AGR with ω = 4.16 eV at D = 1.55 nm. In
0
the top panel of this figure, we present the time evolu- erate oscillatorypopulation exchange,of about 20% and
tion of c (t)2 and c (t)2 for a SIS (black solid curve) 10%oftheinitialpopulation,incaseofaSISandanAIS,
1 2
| | | |
and an AIS (red dashed curve) for the first 50 fs. We respectively, between states 1 and 2 and the modified
| i | i
find that the population evolution of states 1 and 2 EM modes occurs simultaneously. These minor qualita-
| i | i
are strongly dependent on the initial state, which indi- tive differences on the population evolution of 1 and
| i
cates strong quantum interference effects in the under- 2 are safely attributed to quantum interference effects
| i
lying non-Markovian dynamics. In case of a SIS, the in the underlying non-Markovian dynamics. Most inter-
c (t)2 (= c (t)2 here) evolution shows a slowly de- estingly, however, for both initial states, at later times,
1 2
| | | |
creasing in magnitude oscillatory transfer of population thepopulationevolutionshowsclearlythatabout20%of
between the V-type system and the modified EM mode the initial population is “trapped” in the V-type system
continuum. IncaseofanAIS,thesituationissimilar,al- at all times, which indicates that coherent population
thoughthe rate of magnitude decreaseis faster. At later trapping38 under conditions of strong coupling between
times, as shown in the bottom panel of Fig. 9, a mod- a QE and a MNP, as studied here, is observable.
9
1
E. The influence of the FCA on the dynamics of
the V-type QE: K±(τ) vs KF±CA(τ) with K±(τ)
0.8
with K± (τ)
FCA
We now investigate the influence of the FCA on the
non-Markovian spontaneous emission dynamics of a V- 2)|0.6
(t
typeQE.InthetoppanelofFig. 10,wepresentthepop- |c10.4
ulation evolution of state 1 with initial state Ψ(0) =
| i | i
1 , for a J-AGR at D =1.55 nm with ω =3.84 eV, ob-
|taiinedusingtheK±(τ)kernel(blacksolid0curve),defined 0.2
by Eq. (15), and using the K± (τ) kernel (red dashed
FCA 0
curve), defined by Eq. (17). We observe that the FCA 0 0.05 0.1
t / ps
introduces a phase shift between the two c (t)2 curves.
1
More specifically, the c (t)2 obtained w|ith |K± (τ)
| 1 | FCA
precedes the corresponding population evolution when
the FCA is not invoked; we denote such a phase shift 1
between the two curves as positive. In the bottom panel
with K±(τ)
of Fig. 10, we present the population dynamics of state 0.8
with K± (τ)
1 for a J-AGR, again at D = 1.55 nm, but now with FCA
| i
ω0 = 4.16 eV, obtained using the K±(τ) kernel (black 2 0.6
solid curve) and with the K± (τ) kernel (red dashed t)|
FCA (
curve). We once more find that the two |c1(t)|2 curves |c10.4
are phase-shifted to each other. However, interestingly,
in this case, the phase shift is negative.
0.2
0
0 0.05 0.1
t / ps
In order to better understand the phase shift in the
population dynamics introduced by invoking the FCA,
as discussed above, we further focus on the radial spec- FIG. 10: (color online) Population evolution of 1 of a QE
| i
traldensity Jrad(ω) andJrad (ω), withandwithout the with τ0 = 70 ps at D = 1.55 nm obtained using the kernel
FCA K±(τ) (defined by Eq. (15)) (black solid curve) and using
FCA, respectively, to be used in the corresponding ker-
thekernelK± (τ)(definedbyEq. (17))(reddashedcurve),
nels. We do not elaborate on the case of the tangential FCA
withω0 =3.84eV(top)andω0=4.16eV(bottom). Seetext
spectral density here, since one comes to analogousfind-
for discussion.
ings as for the radial spectral case, to be presented now.
In Fig. 11, we show the Jrad(ω) (black solid curve)
and Jrad (ω) (red dashed curve) in case of a QE with
FCA
ω = 3.84 eV (top panel) and ω = 4.16 eV (bottom
0 0
a MNP, for which the spectral density is dominated by
panel). We observe that in both cases, around the reso-
nancefrequencyω ,Jrad(ω)andJrad (ω)arepractically overlapping plasmonic resonances, introduces a positive
0 FCA
(negative) phase shift in the population evolution, when
thesame,whichimpliesthatinthecommoncaseofa(ra-
theQEresonancefrequencyliesinthelower(higher)end
dial) spectral density dominated by only one plasmonic
of its non-vanishing spectral density.
resonance,or more generally,by severalnon-overlapping
resonances, calculating the non-Markovian dynamics of
a QE in the proximity of a MNP, with or without the
FCA,leadstopracticallyindistinguishableresults. How-
ever, when the (radial) spectral density is dominated by
overlapping plasmonic resonances, as in this work (see
IV. DYNAMICS OF A TWO-LEVEL QE NEAR
Fig. 2), invoking the FCA leads to observable differ-
A MNP
ences between the two spectral densities as one moves
awayfrom the resonancefrequency of the QE.In partic-
ular, in presence of overlapping resonances in the spec- In this section, we consider a V-type QE, for which,
tral density, JFraCdA(ω) < Jrad(ω) is valid when ω > ω0, however, the transition dipole moments ~µ10 and ~µ20 are
withreversedrelationbetweenthetwospectraldensities, along the z and the x axis, i.e. we use ~µ10 = µǫˆz(x) and
when ω <ω . Furthermore,with helpof Fig. 11, we can ~µ =µǫˆ in Eq. (7). In sucha case, since Kij(τ)=0
0 20 x(z)
rationalisequalitativelythephasedifferenceintheevolu- (i = j), the population dynamics of one excited state of
tion of c (t)2, when obtained with K± (τ), insteadof the6 V-type system does not depend on the dynamics of
K±(τ).| W1 e|conclude that the introduFcCtAion of the FCA the other excited state; therefore one has effectively two
for computing the population dynamics of a QE near independent two-level systems, instead of a three-level
10
10 1
9 →µ along z axis
8 0.8 →µ10 along x axis
7 10
14-10 sec 465 JJrrFaaCdd(Aω(ω)) 2|c(t)|100..46
1
3
2 0.2
1
0 00 0.05 0.1 0.15 0.2
3.7 3.8 3.9 ω /4 eV 4.1 4.2 t / ps
FIG. 12: (color online) Population evolution of 1 of a two-
| i
level system with ω0 = 4.16 eV and τ0 = 4 ns at D = 1 nm
8 withµ~10 =µǫˆz (blacksolidcurve)andµ~10 =µǫˆx (reddashed
7 curve). See text for discussion.
Jrad (ω)
6 FCA
Jrad(ω)
1
-c5
e
s4
14 are 6 fs and 14 fs, respectively, taking into account that
03
1 λ⊥(4.16,1) 6.54 105 and λk(3.84,1) 2.83 105 as
≈ · ≈ ·
2 shown in Fig. 2. The time evolution of c (t)2 at early
1
| |
timescanbeunderstoodasadecaywiththecorrespond-
1
ing effectivedecaytime. However,quickly,anoscillatory
0
3.7 3.8 3.9 4 4.1 4.2 decayofthepopulationof 1 intothemodifiedEMmode
ω / eV | i
continuumwithdistinctlynon-Markoviancharacterdom-
inatesthedynamics. Moreover,aninterestingconjecture
FIG. 11: (color online) The radial spectral density Jrad(ω) regarding the population dynamics shown in this figure
(blacksolid curve)andJFraCdA(ω)(reddashedcurve)foraQE can be made, when a comparison with the population
withτ0 =70psatD=1.55nmwithω0 =3.84eV(top)and dynamics for the V-type system with ω = 4.16 eV and
ω0 =4.16 eV (bottom). Seetext for discussion. 0
τ = 4 ns at D = 1 nm presented in Fig. 7 above is
0
made. More specifically, we observe that the curves of
c (t)2 in Fig. 7 and in Fig. 12 look very similar, when
1
| |
taking into account the different amount of initial pop-
V-type system. We also note that the two-level transi- ulation, although they correspond to distinctly different
tion dipole polarisation ~µ along the z (along the x axis) initialconditions. The comparisonofthe results ofthese
implies that the populationdynamics of sucha two-level two figures suggests strongly that the different popula-
systemisaffectedbytheradial(tangential)spectralden- tion evolution of c1(t)2 for the SIS and the AIS shown
| |
sity only. in Fig. 7 is due to the fact that for a SIS state, due to
quantuminterference, the dynamics is dominated by the
spectral density along the z axis, i.e. the radial spectral
density. IncaseofanAIS,the c (t)2 evolutioninFig. 7
1
We now study the dynamics for such a two-level sys- | |
isdominatedbythetangentialspectraldensity,i.e. along
tem in order to compare its population dynamics to the the x axis31. We can thus understand better, with help
dynamics of a V-type system presented above. For the
of the two-level system studied here, the direct result of
sake of concreteness, we focus on a two-level QE, com- the quantum interference on the dynamics of the V-type
posed of states 0 and 1 (see Fig. 1), with resonance
system with ω = 4.16 eV and τ = 4 ns at D = 1 nm,
| i | i 0 0
frequency ω = 4.16 eV; the results at a lower ω (not
0 0 discussed above.
shown here) are similar to the results at ω = 4.16 eV,
0
withonlyminorquantitativedifferencesduetothemuch
smaller enhancement factors λ⊥(ω ,D) and λk(ω ,D),
0 0
when ω0 <4.16 eV. In Fig. 13, we consider a two-level QE with ω0 =4.16
In Fig. 12, we present the population dynamics of eVandτ =70pswithatransitiondipolemomentalong
0
state 1 of a two-state QE with τ =4 ns at D =1 nm, the z axis, which implies that the dynamics is affected
0
| i
with a transition dipole moment along the z axis (black only by the radial spectral density. In this figure we
solid curve) and along the x axis (red dashed curve); presentthe populationevolution c (t)2 atD =1.85nm
1
the corresponding effective decay times τrad and τtan (black solid curve) and at D =2|nm (|red dashed curve)
eff eff