Table Of ContentReversible dynamics of single quantum emitters near metal-dielectric interfaces
A. Gonz´alez-Tudela,1,2,∗ P. A. Huidobro,1,† L. Mart´ın-Moreno,3 C. Tejedor,1 and F.J. Garc´ıa-Vidal1
1Departamento de F´ısica Teo´rica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),
Universidad Aut´onoma de Madrid, E-28049, Spain.
2Max-Planck-Institut fu¨r Quantenoptik Hans-Kopfermann-Str. 1. D-85748 Garching, Germany.
3Instituto de Ciencia de Materiales de Arago´n and Departamento de F´ısica de la Materia Condensada,
CSIC-Universidad de Zaragoza, E-50009, Zaragoza, Spain
(Dated: January 27, 2014)
Herewepresentasystematicstudyofthedynamicsofasinglequantumemitternearaflatmetal-
dielectric interface. We identify the key elements that determine the onset of reversibility in these
4
systems by using a formalism suited for absorbing media and through an exact integration of the
1
dynamics. Moreover, when the quantum emitter separation from the surface is small, we are able
0
todescribethedynamicswithinapseudomodedescriptionthatyieldsanalyticalunderstandingand
2
allows more powerful calculations.
n
a PACSnumbers: 42.50.Nn,73.20.Mf,71.36.+c
J
3
2 Metal-dielectric interfaces strongly modify the density effectivecavitythatisabletoexchangeenergycoherently
of electromagnetic (EM) modes in their surroundings. with the QE. This analogy allows us to map the prob-
] ThisisduetotheexistenceofsurfaceEMmodes,known lem into the dissipative Jaynes–Cummings (JC) model
l
l as surface plasmon polaritons (SPPs), which propagate [24],whichresultsinbothanalyticalformulasforthekey
a
h along the metal surface. This modified density of EM parameters determining the onset of reversibility and a
- modes reduces the lifetime of quantum emitters (QEs) powerful formalism to explore new physics.
s
when these are placed close to a metal surface [1, 2]. In the inset of Fig 1(a) we render the scheme of the
e
m Moreover, when the distance between the QE and the general configuration: a QE is embedded into a dielec-
metal surface is extremely small (less than around 10 tric host with permittivity (cid:15) , and placed in front of a
. d
t nm), the radiative emission is said to be quenched, be- metal layer of thickness h, which stands above another
a
m cause the QE decay is dominated by extremely fast non- dielectric matrix with the same (cid:15)d. We use a Drude di-
radiativelossychannelswithinthemetal. Althoughthere electric function for describing the EM response of the
-
d have been some theoretical studies dealing with the pos- metal (silver), (cid:15)m(ω) = (cid:15)m,∞ − ωp2/[ω(ω + iγp)], char-
n sibility of a coherent exchange of energy between a sin- acterized by its plasma frequency ((cid:126)ω = 9 eV), Ohmic
p
o gle QE and surface EM modes in metal nanostructures losses ((cid:126)γ = 0.07 eV) and high-frequency component
p
c [3–11], only the perturbative or weak-coupling regime in ((cid:15) = 5.7) [25]. The QE is described within the two-
[ m,∞
whichthequantumdynamicsisirreversiblehasbeenob- level-systemapproximation,{|g(cid:105),|e(cid:105)},andcharacterized
1 served in metal surfaces [12–14], plasmonic waveguides by its transition frequency ω and dipole moment, µ(cid:126),
0
v [15, 16] or metal nanoparticles [17, 18]. which we assume to be normally oriented to the surface
3
Inthisworkwepresentatheoreticalstudyonthepop- because this is the optimal direction to couple with the
9
ulation dynamics of a single QE coupled electromagneti- metal surface [26]. Nevertheless, similar results could be
1
callytoatwo-dimensional(2D)metal-dielectricinterface, obtained for the case of a dipole moment pointing par-
6
. including the situations where the emission is quenched. allel to the metal surface. Its Hamiltonian is given by
1 WeuseaquantummechanicalformalismfortheEMfield H =(cid:126)ω σ†σ, where σ† (σ) is the raising (lowering) op-
0 0 0
excitations that fully takes into account their lossy char- eratorofthetwo-levelsystemdescribingtheQE.Instead
4
acter[19–22]. Additionally,wegobeyondFermi’sgolden of the dipole moment, we use the intrinsic lifetime of the
1
: rule and integrate exactly the dynamics relying only on QE, given by
v
the rotating-wave approximation [21–23]. This is in con-
i 3π(cid:15) (cid:126)c3
X trast with previous works [11], in which a perturbative τ =γ−1 = 0 ,
r method was used to capture reversibility only up to low- 0 0 |µ(cid:126)|2ω03
a est order [23] in the coupling. Through the appearance
to measure the strength of the coupling to the EM field.
of oscillations in the dynamics of the QE population, we From here on we take (cid:126)=1.
determine the conditions under which the perturbative
Our first goal is to describe the coupling between the
regime breaks and reversible dynamics can be observed.
QEandtheEMfieldatthe2Dmetal-dielectricinterface,
Furthermore, in the limit of small separations the inter-
even in the region where quenching is expected. There-
ference between the QE dipole and its image creates an
fore, canonical quantization approaches are not suitable
forourproblembecausetheyarebasedeitheronneglect-
ing losses [27, 28] or considering them as a perturbation
∗ Correspondingauthor: [email protected] [29]. In order to properly take into account the lossy
† Correspondingauthor: [email protected] character of the EM field, we recourse to a macroscopic
2
(a) dependentcharactermakecalculationsgenerallyveryde-
12 3 12 3 0 4 manding. Here we assume that the QE is initially in the
Total
10 3 8 3 Quasistatic excitedstateandstudythedynamicsofitspopulationby
8 3 4 3 following the approach known as the Wigner-Weisskopf
problem. As H conserves the total number of excita-
6 3 0 x 10 tions, the most general state of the QE and EM field
3.0 3.2 3.4 3.6 3.8 4.0 x 500
4 3 x 5000 system reads [21]
2 3 |Ψ(t)(cid:105)=C (t)e−iω0t|e,0 (cid:105) (2)
e ω
0 (cid:90) (cid:90)
(b) + d3(cid:126)r dωC1((cid:126)r,ω,t)e−iωt|g,1(cid:126)r,ω(cid:105) ,
3 3
1
0 2 0 140 where |g,1(cid:126)r,ω(cid:105) = f(cid:126)†((cid:126)r,ω)|g,0ω(cid:105). The population of the
2 3 excited state is calculated as: nσ(t) = |Ce(t)|2. Apply-
10 ing the Sch¨odinger equation yields a set of differential
equations that can be formally integrated, resulting in
1 3 x 20 an integro-differential equation for C (t):
e
x 60
x 150 (cid:90) t
C˙ (t)=− K(t−t )C (t )dt , (3)
0 e 1 e 1 1
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0
with initial condition C (0) = 1. The kernel of this
e
equation is given by K(τ)=(cid:82)∞dωJ(ω)ei(ω0−ω)τ, where
0
FIG. 1. (a) Spectral density for a thick metal film embedded J(ω) is the so-called spectral density of the system. The
into a dielectric material with (cid:15)d = 1 for four different QE spectral density contains information on both the QE-
separations, z , from the metal surface. Inset: Total spectral
0 field coupling and the density of EM modes of the whole
density(black)forz =4nmanditsquasistatic(dashedred)
0 metal-dielectric system and reads
and plasmon pole (dotted blue) contribtions. (b) Same as in
panel (a) but for a thin metal film (h=5 nm). Inset: Con- ω2
figuration of a single QE placed in front of a metal-dielectric J(ω)= µ(cid:126) ·Im[Gˆ((cid:126)r ,(cid:126)r ,ω)]·µ(cid:126). (4)
π(cid:15) c2 0 0
interface. 0
InFig. 1(a),weplotthespectraldensitycorresponding
√
QEDformalismforabsorbingmedia[21,22]. Withinthis to a thick metal film embedded in vacuum (h (cid:29) (cid:15)dz0
framework, the EM field can be expanded in terms of with (cid:15)d = 1) for different QE-metal separations starting
bosonic field destruction (creation) operators, f(cid:126)(†)((cid:126)r,ω), at z0 =4 nm [30]. At very small distances (z0 =4 nm),
the spectral density is strongly peaked at the cut-off fre-
as follows: √
quency of the SPPs, ω = ω / (cid:15) +(cid:15) , anticipating
c p d m,∞
thattheEMmodesaroundthatfrequencywilldominate
(cid:115)
E(cid:126)((cid:126)r,ω)=i π(cid:15)ω4c4 (cid:90) d3(cid:126)r1(cid:112)Im[(cid:15)m(ω)]Gˆ((cid:126)r,(cid:126)r1,ω)·f(cid:126)((cid:126)r1,ω), tbheeloQw,ESdPyPnsapmlaicys.aNmointiocrerhoolewienvetrhethparto,caessswoefrweivlelrsshibowle
0 dynamics and that the key actors are the EM modes re-
(1)
sponsible for quenching of radiation. The same behavior
where the classical Green function of the system [26],
isobservedforthinmetalfilms(h=5nm);seeFig. 1(b)
Gˆ((cid:126)r,(cid:126)r ,ω), weights the contribution of the different EM
1 for z =4 nm. As the QE-interface separation grows the
modes. These f(cid:126)(†)((cid:126)r,ω)-modes are obtained [21, 22] by heigh0tofthepeakdecreasesanditswidthincreasesmak-
diagonalizing a Hamiltonian that includes both the EM ing the spectral density smoother for the case of a thick
excitations and a bath of harmonic oscillators describ- metal film. As a difference, for thin metal films in this
ingthemechanismsresponsibleforthedissipationinthe intermediate regime, the spectral density splits into two
metal. Thus, f(cid:126)(†)((cid:126)r,ω) represents the elementary ex- contributions corresponding to the short and long range
citations of the lossy EM field of the structure. The SPP modes of the thin film [31]. The poor confinement
free Hamiltonian of the EM excitations can be written of the long range mode makes its contribution pin at ω
as: HB = (cid:82) d3(cid:126)r(cid:82)0∞dωωf(cid:126)†((cid:126)r,ω)·f(cid:126)((cid:126)r,ω). The interac- independentlyofz0 whereasthecontributionoftheshorct
tionHamiltonianbetweentheEMfieldexcitationsanda rangemodeshiftstolowerfrequenciesasz increases. Fi-
0
QEplacedat(cid:126)r0,withintherotatingwaveapproximation nally, at large z0’s, the spectral densities corresponding
[21, 22], is given by: H =(cid:82) dω[−µ(cid:126) ·E(cid:126)((cid:126)r ,ω)σ†+h.c.]. to both thick and thin metal films become very smooth
int 0
The dynamics of the combined system is completely without signatures of resonant peaks.
determined by the total Hamiltonian: H = H +H + Inthefollowingwefocusourattentiononthedynamics
0 B
H . However, the infinite number of degrees of free- ofthesingleQEwhenitisplacedatveryshortdistances
int
domoftheEMmodesplustheirnonlocalandfrequency- from the metal surface. As the behaviors of both thick
3
andthinmetalfilmsareverysimilarforthisrangeofdis- 1.0 1.00
tances;fromnowonwestudythecaseofthickmetalfilms
withoutlosinggenerality. Inordertostudythedynamics 0.90
0.8
beyondtheperturbativeregime,wesolvenumericallyEq. 0.80
3 by using a grid-based method [32] without further ap-
proximations. This avoids the convergence problems of 0.6 0 100 200
perturbative methods [23] used in previous approaches
[11].
0.4
In Fig. 2, we render the excited state population dy-
namics for a QE placed at z = 5 nm for two different
0
0.2
energies ω . This distance corresponds to a case where (a)
0
the spectral density is strongly peaked at ω [see Fig.
c 0.0 0.5 1.0 1.5
1(a)]. The timescale of both panels is normalized to
the modified lifetime of the emitters, τ = 1/[2πJ(ω0)], 1.0 1.0
such that they appear in the same scale. In Fig. 2(a),
we show the dynamics of a QE with a transition fre- 0.8 0.8
quencylyingintheopticalregime,ω =2eV.Forτ =4
0 0
0.6
ns (dotted blue line), the population of the QE decays
0.6
exponentially. This trend can be described within the 0 100 200
so-called Markov approach: the spectral density in the
0.4
kernel can be approximated as J(ω) ≈ J(ω ), obtain-
0
ing K(τ) ≈ Γ(ω0)δ(τ) [33], with Γ(ω ) = 2πJ(ω ). The
2 0 0 0.2
integrationofEq. 3yieldsanexponentialirreversiblede-
cay at a rate Γ(ω ), then recovering the results of the
0 (b)
perturbative or weak-coupling regime where the dynam- 0.0
0 1 2 3 4 5
ics is solely determined by the value of J(ω) around ω
0
[26]. For τ =0.4 ns (dashed red line), however, the QE
0
dynamicsexhibitsacleardeviationfromtheexponential
FIG. 2. (a) Dynamics of the excited state population for a
decaywithfastoscillationsintheinitialtimes(seeinset).
QE, with ω =2 eV and z =5 nm, embedded in a dielectric
0
Finally,fortheshortestτ considered,0.04ns(blacksolid
0 with (cid:15)d =1 in front of a thick metal film for different τ0’s as
line), the QE dynamics shows strong oscillations, before specified in the legend. The time is normalized by the mod-
being attenuated by the metal losses. The emergence ified lifetime of the QE, τ = 1/Γ(ω ). Inset: Zoom of panel
0
of these oscillations as τ decreases is a manifestation of (a) showing the oscillations in the excited state population.
0
thecoherentexchangeofenergybetweentheQEandthe (b) Same as in panel (a), but for ω0 =3 eV.
EMfieldexcitationspresentatthemetalsurface. InFig.
2(b), we plot the same cases as in Fig. 2(a) but for a
QE with a transition frequency closer to the peak of the ric electron microscopy [37, 38].
spectral density, ω = 3 eV. The dynamics exhibits, as Through the numerical integration of Eq. 3, we have
0
in Fig. 2(a), a transition from irreversible to reversible explored the emergence of reversibility in all ranges of
dynamics with decreasing τ0, but showing oscillations of relevant parameters: z0,τ0,ω0, including the regions of
longer period and larger amplitude. The comparison be- bothsmalldistancesandω0 closetoωc thatwerenotac-
tweenthesetwopanelsprovesthatthedetuningbetween cessible with the approximations used in previous works
the peak of the spectral density and the energy of the [11]. The configurations that favor reversible dynamics
emitterisalsoaveryrelevantparameterfordetermining are: shorter τ0’s, ω0’s closer to ωc and the regions of
the visibility of the oscillations. very small separations. Interestingly, in this spatial re-
Animportantquestiontoaddressisthepossibleexper- gion [z0 (cid:28) ω0√c(cid:15)d], the main contribution to the spectral
imental verification of the reversible dynamics predicted densitydoesnotcomefromtheSPP(pole)contribution.
by our calculations. As shown in Fig. 2, the timescale Instead, it stems from the EM modes having an even
to experimentally observe these effects depends strongly larger parallel momentum than SPPs, i.e., EM modes
on the intrinsic lifetime τ of the chosen QE. There exist that are even more evanescent in the direction perpen-
0
several state-of-the-art QEs with transition frequencies dicular to the metal surface than SPPs. By only taking
lying within the optical regime such as nitrogen-vacancy into account these highly evanescent modes when eval-
centers, quantum dots and J-aggregates, with intrinsic uating Gˆ((cid:126)r ,(cid:126)r ,ω) in Eq. 4, we can find an analytical
0 0
lifetimes around τ ≈ 1 µs [34], 4 ns [15] and 70 ps [35], expression for J(ω):
0
respectively. In the fastest situation considered in Fig.
2, the experimental observation of the oscillations would
3 (cid:16) c (cid:17)3 (cid:16)(cid:15) (ω)−(cid:15) (cid:17)
require subpicosecond resolution, which can be achieved J(ω)= γ Im m d . (5)
with streak-camera experiments [36] or via interferomet- 016π ω0z0 (cid:15)m(ω)+(cid:15)d
4
We have checked numerically that this approximation the JC mapping can be extracted from the quasistatic
for the spectral density gives virtually the same results approximation. In this limit, the interference between
as those given by Eq. (2), provided that the distance of the QE dipole and its image results in a divergence of
the QE to the metal surface is very small as shown in the reflected field, expressed in the resonance condition
the inset of Fig. 1(a), where we compare the quasistatic (cid:15) (ω) + (cid:15) = 0, ultimately attenuated by the plasma
m d
contribution (dashed red) and the surface plasmon pole losses. This resonance can be seen as that supported
contribution [39] (dotted blue) to the total spectral den- by a dipole-image-induced effective cavity. The strongly
sity at z =4 nm. The formula of Eq. 5 for the spectral damped resonance interacts with the QE, being able to
0
densityisusuallyknownasthequasistaticapproximation coherently exchange energy with it. Note that in con-
andhasalreadybeenusedintheliteraturetoanalyzethe trast to previous studies [8–11], we identified that SPPs
strong modification of the QE’s lifetime in the presence playnoroleinreversibility. Thiscanbeconcludedmath-
ofmetalsurfaces,wiresornanoparticles[1,40–42]. How- ematicallyaswellfromtheabsenceofSPPcontributions
ever, we use it here to study the emergence of reversible to the spectral density of Eq. 5 that describes the EM
dynamics. Notice that the region of very short distances field when the QE is close to the surface.
was thought to only yield quenching [1, 2] and strong Within the JC model, it is well-established that re-
coherent effects were not expected. By using the Drude versible dynamics appears when the coupling strength is
expression for (cid:15) (ω), it is possible to further expand Eq. stronger than losses (g (cid:29) γ ). Thus, in Fig. 3 we
m eff eff
5: rendera(z ,τ )-spacediagramoftheparameterg /γ
0 0 eff eff
to get a better estimation of the optimal regions for ob-
servingreversibility[43]. Smallerτ ’sandz ’s[44]favour
3 (cid:16)ω (cid:17)3(cid:16) c (cid:17)3 γ /2 0 0
J(ω)= γ ω c p , reversibility as they increase g . Another strategy, not
0 p16π ω ω z (ω−ω )2+(γ /2)2 eff
p 0 0 c p exploredinthefigure, consistsofdecreasingtheeffective
(6)
losses, γ = γ , that control the timescale where the
eff p
oscillations get attenuated. This can be done by, e.g.,
showing a Lorentzian dependence on the frequency do-
lowering the temperature of the system [45]. Finally, as
main. This spectral shape allows us to find an analyt-
we showed in the numerical results of Fig. 2, there is
ical solution of the integro-differential Eq. 3 by using
another relevant magnitude for the characterization of
Laplacetransformtechniques[23]. Asaconsequence,for
reversible dynamics, namely, the amplitude of the oscil-
very short QE-metal distances, the dynamical evolution
lations. Basedonouranalyticalapproach,wecanextract
of the QE dictated by the general Hamiltonian H turns
ananalyticalformulaforthevisibilityoftheoscillations:
out to be mathematically equivalent to the solution of
the following master equation: max[n ]−min[n ] 2g2
V≡ σ σ = eff , (9)
max[n ]+min[n ] 2g2 +(ω −ω )2
γ σ σ eff 0 eff
ρ˙ =i[ρ,H +ω a†a+g (aσ†+σa†)]+ effL [ρ]. (7)
0 eff eff 2 a where we have neglected the amortiguation coming from
γ . This expression shows that there are two ways for
The above master equation appears in the dissipative p
increasing V: (i) by decreasing the effective detuning of
Jaynes-Cummings (JC) model [24], which is the corner-
the QE with the pseudomode as we showed in Figs. 2(a-
stone of cavity QED physics. The mapping of the dy-
b); (ii) by increasing the effective coupling, as done in
namics of a QE coupled to the continuum of EM modes
Fig. 2 by decreasing τ . In Fig. 3, we have plotted the
supported by a 2D metal surface to a simple JC model 0
frontiers in the phase-space diagram where V =1%, 5%
is one of the main results of this work. The first part of
and 10% in red, black and green, respectively. As ex-
the right term of Eq. 7 describes the coherent evolution
pected from Eq. (9), the higher the coupling and/or the
of the combined system: the QE and the “pseudomode”
smaller effective detuning between the modes the better
bosonic excitation of energy ω = ω (represented in
eff c the visibility.
Eq. 7 by the a-operator), which accounts for all the EM
Moreover,themappingofthephysicstoacavityQED
modes of the metal surface. The coherent coupling be-
problem of Eq. 7 has also important implications when
tween the QE and this pseudomode is given by:
thinking of incorporating new factors that could influ-
3 (cid:16)ω (cid:17)3(cid:16) c (cid:17)3 ence the QE dynamics. For example, the study of plas-
g2 =γ ω c . (8)
eff 0 p16 ω ω z mon nonlinear interaction can be done by adding sim-
p 0 0 ply a nonlinear term, H = U (a†a)2, in the coher-
nl a
The second part of the right term of Eq. 7 describes ent part of Eq. 7 whereas considering extra QE losses
irreversible mechanisms through the so-called Lindblad or pure dephasing can be done by including new Lind-
terms [23], with notation L [ρ] = (2OρO† − ρO†O − blad terms, (γ /2)L [ρ] [29] and (γφ/2)L [ρ] [46], re-
O σ σ σ σz
O†Oρ). Notice that the effective losses of the pseu- spectively. Noteworthily, the dynamics of a single QE
domode are solely determined by the metal properties, is dominated by the most evanescent EM modes of the
γ =γ . continuum, γ (cid:29)γ ,γφ, contrary to the situation of N
eff p eff σ σ
The physical picture that emerges from the mathe- QEs,wheretranslationalinvarianceforcesQEstocouple
matical equivalence between the initial Hamiltonian and to a single SPP mode whose associated propagation loss
5
10-7(a) (b) 20 fied the parameters that determine both the emergence
and the visibility of oscillations in the population of the
10-8
10 excitedstate. Contrarytointuition,theEMmodesoffre-
10-9 1 quenciesaroundthecut-offfrequencyoftheSPPmodes,
5
1 i.e.,themostdissipativeEMmodesofthesystem,arethe
10-10 2
most relevant for the observation of reversible dynamics.
10-11 10 1 Remarkably, in the region of very short distances, the
10
10-12 0 problem can be effectively treated within the dissipative
4 6 8 10 12 14 4 6 8 10 12 14 JCmodel,allowingforbothabetterunderstandingofthe
QE dynamics and for calculations otherwise intractable
due to their computational complexity.
FIG. 3. Contour plot of the real part of the ratio g /γ ,
eff eff
in z -τ for a QE placed in front of a thick metal film with
0 0
transitionfrequenciesω =2eVandω =3eVinpanels(a)
0 0
and(b)respectively,and(cid:15) =1. Inred,blackandgreen,itis
d ACKNOWLEDGEMENTS
drawn the line where the visibility of the oscillations reaches
1%, 5% and 10% respectively.
Work supported by the Spanish MINECO
(MAT2011-22997, MAT2011-28581-C02, CSD2007-046-
is much smaller than γφ [29]. NanoLight.es) and CAM (S-2009/ESP-1503). A.G.-T
σ
In conclusion, by using a suitable formalism to deal acknowledges funding by the EU integrated project
with absorbing media, we have studied the conditions SIQS. P.A.-H acknowledge FPU grant (AP2008-00021)
underwhichasingleQEinteractingwiththeEMfieldof from the Spanish Ministry of Education. This work has
a2Dmetal-dielectricinterfaceshowsreversibledynamics. beenpartiallyfundedbytheEuropeanResearchCouncil
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