Method of images applied to driven solid-state emitters

2.9 MB·

by  Dale Scerri

Method of images applied to driven solid-state emitters
Dale Scerri,∗ Ted S. Santana, Brian D. Gerardot, and Erik M. Gauger†
SUPA, Institute of Photonics and Quantum Sciences,
Heriot-Watt University, EH14 4AS, United Kingdom.
(Dated: January 18, 2017)
Increasingthecollectionefficiencyfromsolid-stateemittersisanimportantsteptowardsachieving
robust single photon sources, as well as optically connecting different nodes of quantum hardware.
A metallic substrate may be the most basic method of improving the collection of photons from
quantumdots,withpredictedcollectionefficiencyincreasesofupto50%. Theestablished‘method-
of-images’ approach models the effects of a reflective surface for atomic and molecular emitters
by replacing the metal surface with a second fictitious emitter which ensures appropriate electro-
magnetic boundary conditions. Here, we extend the approach to the case of driven solid-state
emitters,whereexciton-phononinteractionsplayakeyroleindeterminingtheopticalpropertiesof
7 thesystem. Wederiveanintuitivepolaronmasterequationanddemonstrateitsagreementwiththe
1
complementary half-sided cavity formulation of the same problem. Our extended image approach
0
offersastraightforwardroutetowardsstudyingthedynamicsofmultiplesolid-stateemittersneara
2
metallic surface.

n
a
J I. Introduction

6
1
Theproblemofadipoleemitterplacedclosetoareflec-

tive surface has received much interest over the last few
]
h decades: seminalwork1 byDrexhagein1970firstdemon-
p strated that a reflective interface modifies the intrinsic
-
properties of the emitter, influencing both the emission
t
n frequency2,3 and the emitter’s excited lifetime3–7. Re-
a
cently, a sound analogue of Drexhage’s experiment has
u
been performed to study the acoustic frequency shifts of
q
[ a gong struck near a hard wall8.

Mirrors have widespread use for directing light from
1
sources that emit across a extended solid angle, for ex-
v
2 ample in the form parabolic reflectors in everyday light
3 sources. On the nanoscale, precise guiding of photons
4 intoparticularopticalmodesisofparamountimportance
4 forquantuminformationprocessingandcommunication,
0 where on demand single photons are required9–12. Al-
.
1 though micron-sized spherical mirrors for open access FIG. 1: Artistic rendition of a driven quantum dot
0 microcavities13 have recently enabled the investigation (QD), depicted as a cyan spheroid, in the proximity of a
7 of quantum dot–cavity systems in the strong coupling golden metallic surface. The corresponding ‘image dot’
1
regime14,15, the use of sophisticated mirrors remains a is shown blurred on the other side ‘below’ of the
:
v challenge for solid-state quantum emitters that are often semiconductor-gold interface. The optical dipoles are
Xi embeddedinheterogenouslayersofsubstrateswithvary- depicted as ‘dumbbells’ within the QDs. The vertical
ing refractive indices. This motivates the more straight- red beam represents the laser driving, and the magenta
r
a forwardalternativeofincreasingthephotoncollectionef- spiralling arrows indicate scattered photons.
ficiency by placing the emitter above a planar mirroring
interface16–18. Interestingly, the presence of even such a
simple mirror also affects the physical properties of the
emitter, as discussed above. interactionwiththemirror. Withimprovedatom-mirror
In recent years, progress in the synthesis and con- coupling, Hoi et al. managed to collect over 99% of the
trol of solid-state emitters has enabled experimental in- radiation by coupling a transmon microwave emitter to
vestigation of these modified properties of condensed- a 1D superconducting waveguide23.
state emitters including quantum dots (QDs)19,20 as Several theoretical investigations2–4,7 have shown that
well as perovskite21 and transition metal dichalcogenide an atomic two-level system (TLS) near a reflective sur-
monolayers22 deposited on reflective surfaces. Circuit face can be modelled as a pair of emitters: the real one
QED analogues of an atom and a variable mirror have as well as an identical emitter that is placed equidistant
also been successfully implemented23,24; these offer the from,butontheoppositesideof,theinterface(seeFigs.1
advantage of increased control over the artificial atom’s and 2). The basic idea follows that of the electrostat-
2
ics concept of an image charge to capture the surface Half-sided cavity Dot-Image
chargedistributionthatensuresmeetingtheelectricfield
boundary conditions25. In the optical case, the ‘method
γ γ
of images’ relies on considering the emission from the
combined dipole-image system. This yields the same
expression for the modified spontaneous emission (SE)
rate which one obtains from a full QED treatment (em-
ploying surface-dependent response functions to arrive
at the modifications to the emitter’s lifetime and transi-
tion frequency)26. The image dipole treatment has also
been applied to model the surface-induced modifications
of more complex structures such as molecules27,28, mul-
tiple dipole emitters29–31 and solid state-emitters19,21.
To date, however, the latter have largely ignored the
FIG.2: Twoequivalentdescriptionsofanemitterneara
vibrational solid state environment and the continuous
perfect metallic mirror. Left: schematic of the Green’s
wave (cw) laser driving typical of a resonance fluores-
function and half-sided cavity approaches. Right: the
cence (RF) setting.
emitter supplemented with a fictitious image dipole.
Motivated by these successes, we here present a full The solid (dashed) red arrows indicate emitted
image dipole polaron master equation (ME) treatment (reflected) photons whereas the solid (dashed) red curve
of a driven TLS (such as, e.g., a quantum dot) in the indicates the incident (reflected) driving beam.
proximity of a metal surface (see Fig. 1). Our calcula-
tionsextendpreviousimagedipolestudiesasfollows: (i)
weconsiderdrivensystems,showinghowtoincorporatea
II. Green’s function approach: Brief summary
laserdrivingtermintothedipoleandimageHamiltonian;
(ii)wediscusstheneedforintroducinganadditional‘se-
lectionrule’topreventunphysicaldoubleexcitation;(iii) We begin by summarising the main results of the
we demonstrate how a solid-state phonon environment Green’s function approach for modelling the optical en-
can be accounted for – via a single bosonic bath that is vironment of a dipole emitter. This can be applied to
perfectlycorrelatedacrosstherealemitteranditsimage. obtain the SE rate of an emitter in free space32 as well
as in the presence of a metallic surface5,32,33. Whilst
Wewillshowthattheresultingmasterequationmodel
this approach gives a closed analytical solution for the
remainshighlyintuitiveandpossessesappealingsimplic-
case of a single dipole, a numerical route has to be
ity. We establish the correctness of this model by com-
taken to model a system comprised of a larger number
paring its results to those obtained from an alternative
of emitters31,32, even in the absence of a driving field
calculation which does not involve fictitious entities or
and phonon-environments. Therefore, we here limit the
rely on ad-hoc assumptions: the half-sided cavity model.
discussion to a single ‘bare’ emitter as an independent
This agreement gives us confidencethat the modelcould
reference point for the SE rate (and energy shift) in that
also be extended to the case of multiple solid-state emit-
idealised configuration.
ters near a reflective surface, laying the groundwork for
Let the dipole be situated at position r , where r is
theinvestigationofcollectiveeffectsinthissetting,where d d
perpendicular to a metal surface containing the origin of
we believe that an image approach will be easier to de-
thecoordinatesystem. IntheGreen’sfunctionapproach,
ploy than both the Green’s function and the half-sided
the emitter is usually modelled as a classical dipole os-
cavity approach.
cillating harmonically with amplitude x at frequency ω
0
This Article is organised as follows: We will start about r 31. In vacuum, the SE rate can be calculated as
d
by briefly summarising the results from the established
Green’s function method for calculating the SE rate of γpt(ω )= 4ω02 (cid:104)dˆ·Im{G(r ,r ;ω )}·dˆ(cid:105) , (1)
a ‘bare’ dipole emitter. Next, we shall derive a ME for 0 0 π(cid:15) (cid:126)c2 d d 0
0
the emitter by treating the metal surface as a half-sided
Fabry–P´erot cavity, providing the benchmark model for where (cid:15)0 is the electric permittivity of vacuum, c is the
a single TLS near the metal surface (see Fig. 2a). Fi- speed of light, dˆ is a unit vector indicating the direction
nally, we formulate the ME using the method of images of the emitter’s dipole moment, and G(rd,rd;ω0) is the
(see Fig. 2b). We show that, with suitable alterations, Fourier transform of the dyadic Green’s function at the
the two-body ME reduces to an effective two level sys- emitter’s position32. In Ref.31, Choquette et al. studied
temwithratesandenergyshiftsagreeingwiththecavity the the collective decay rate of N such classical emitters
model. Finally, we put our model to use to obtain the near a planar interface, arriving at a diagonal Green’s
RFspectrumofthemodifiedsystem,featuringaphonon function matrix, so that Eq. (1) allows one to find the
sideband, the Mollow triplet, and the ratio of coherently SE rate for arbitrary dipole orientations.
to incoherently scattered light. To obtain the SE rate in a dielectric environment, we
3
consider the following expression for the normalised dis-
sipated power:
q q
− +
P =1+ 6π(cid:15)0(cid:15)rIm{d∗·E (r )} , (2) r =0
P |d|2k3 s d
0
where P is rate of energy dissipation in free space, (cid:15)
0 r
z
andk aretherelativepermittivityandwavevectormag- r =−1
nitude in the dielectric surrounding the emitter, respec-
tively, and E (r ) is the scattered electric field at the
s d
dipole’s position (which, for a single dipole near the sur-
face,correspondstothereflectedfield)32. Theconnection FIG. 3: The limiting case of the Fabry–P´erot cavity,
effectively reducing to a single perfectly reflecting
between the Green’s function and the decay rate of the
surface. The arrows indicate the wavevectors in (5) and
dipole emitter is established via the relationship
(10), and r denotes the surface reflection coefficient7,36.
P γpt(ω )
= 0 . (3)
P0 γ0pt(ω0)
Rearrangingtheabovethenyieldsanintegralexpression
Ω∗
for the desired SE rate γpt(ω0). HS =δ|X(cid:105)(cid:104)X|+ c2av |0(cid:105)(cid:104)X|+H.c. , (4)
We note that the Green’s function method is not lim-
ited to ideal metallic interfaces but can also be applied whereH.c.denotestheHermitianconjugateandδ =ω −
0
straightforwardly to reflective dielectric interfaces, sim- ω is the detuning between the TLS transition frequency
l
ply by substituting appropriate dielectric constants into ω and the laser frequency ω . Ω is the effective Rabi
0 l cav
theaboverelevantexpressions32. Inthiscase,oneobtains frequency in the presence of the metal surface, given by
qualitatively very similar results for a dielectric mirror,
(cid:114)
eimspaegceisalfluynadtalmaregnetralsleypraerlaietsioonns3t2h.eWashsiulsmtpthtieonmoefthaopdeor-f Ωcav =2 2ω(cid:15)Vl d·(cid:0)el−e−iqlr−el+eiqlr(cid:1) , (5)
fectly conducting interface, it is fair to assume its quali-
tativepredictionswillbyanalogyalsocarryacrosstothe where ql is the laser field wavevector, with polarisation
case of dielectric mirrors. el− (el+ after reflection), as shown in Fig. 3 for the case
of the laser beam being perpendicular to the surface.
Photon and phonon environments are modelled by the
Hamiltonians
III. Half-sided Cavity Model
(cid:88)
Hpt = ν a† a , (6)
E q qλ qλ
In the previous section, we discussed how to deter- q,λ
minetheSErateforanundrivenemitterinteractingonly Hpn =(cid:88)ω b†b , (7)
with a photonic environment. However, in order to fully E k k k
k
model a solid-state emitter such as a QD, we need to
include interactions between the emitter and its phonon whereb† anda† (b anda )arethek-phononandqλ-
environment34,35. Now we shall derive the polaron ME k qλ k qλ
photoncreation(annihilation)operators,respectively. In
foraTLSnearametalsurface,bymodellingthelatteras
the dipole approximation, the photon interaction Hamil-
ahalf-sidedFabry–P´erotcavitypositionedatz =0lying
tonian is of the form
in the xy plane, and the QD positioned at z = r ≥ 0,
d
wityhemreodrdel=fro|rmd|R. eOfsu.r7,3c6a,lctualkaitnigonthfeolalopwpsrotphreiagteenleimrailtscafovr- HIpt =−d·E(rd)(|0(cid:105)(cid:104)X|+|X(cid:105)(cid:104)0|) (8)
the reflectivity and transmittivity of the two mirrors to with E(r) being the Schr¨odinger picture electric field for
obtain, effectively, only a single perfectly reflecting sur- the half-sided cavity7,36,
face (see Fig. 3).
(cid:88)
E(r)=i [u (r)a −H.c.] . (9)
qλ qλ
q,λ
A. Hamiltonian
Thespatialmodefunctionsu (r)foranidealhalf-sided
qλ
cavity (of perfect reflectivity) are given by
WeconsideradrivenTLSwithgroundstate|0(cid:105)andex-
citedstate|X(cid:105),whichisgovernedbythefollowingHamil-
(cid:114)
twoanvieanapinpraoxrimotaattiionng(f(cid:126)ra=me1)and after the usual rotating uqλ(r)= 2ω(cid:15)qVλ (cid:0)eq−λeiq−r−eq+λeiq+r(cid:1) . (10)
4
Here,q (q )istheincident(reflected)wavevector,with phonon bath degrees of freedom, we thus obtain
− +
corresponding polarisation e (e ). For simplicity,
we have assumed that the diqp−oλle mqo+mλent d of the TLS Trpn(cid:20)(cid:18)Ω∗cav |0(cid:105)(cid:104)X|B + Ωcav |X(cid:105)(cid:104)0|B (cid:19)ρpn(cid:21)
is real. E 2 − 2 + E
Ω∗ Ω
= cav(cid:104)B(cid:105)|0(cid:105)(cid:104)X|+ cav(cid:104)B(cid:105)|X(cid:105)(cid:104)0| , (16)
2 2
The interaction with the phonon bath can be generi-
cally represented by the Hamiltonian37 where
Hpn =|X(cid:105)(cid:104)X|(cid:88)g (b† +b ) , (11) (cid:104)B(cid:105)=exp(cid:20)−1(cid:90) ∞dωJpn(ω)coth(βω/2)(cid:21) . (17)
I k k k 2 ω2
k 0
In order to expand perturbatively, we therefore define
where g is the coupling strength of the TLS’s ex-
k
the system-bath interaction with respect to this value.
cited electronic configuration with phonon mode k. We
To this end, we add the expectation value by defining
move to the polaron frame by employing the stan-
dard Lang–Firsov-type transformation U = eS, S = B± =B±−(cid:104)B(cid:105) and Ωpcanv =(cid:104)B(cid:105)Ωcav and regrouping our
|X(cid:105)(cid:104)X|(cid:80) (g /ω )(b† − b ), obtaining the following system and interaction Hamiltonian terms, obtaining:
k k k k k
transformed system Hamiltonian: Ωpn∗ Ωpn
H =δ(cid:48)|X(cid:105)(cid:104)X|+ cav |0(cid:105)(cid:104)X|+ cav |X(cid:105)(cid:104)0| , (18)
Ω∗ SP 2 2
HSP =δ(cid:48)|X(cid:105)(cid:104)X|+ Ωc2av |0(cid:105)(cid:104)X|B− (12) HIpPn = Ω∗c2av |0(cid:105)(cid:104)X|B−+ Ωc2av |X(cid:105)(cid:104)0|B+ , (19)
+ cav |X(cid:105)(cid:104)0|B ,
2 + As for Eq. (15), we introduce operator labels Bpn =
1/2
where δ(cid:48) = δ −(cid:80)kgk2/ωk (becoming δ −(cid:82)0∞Jpn(ω)/ω B∓, Ap1n =Ω∗cav/2 |0(cid:105)(cid:104)X| and Ap2n =Ap1n† to recast the
in the continuum limit), and the phonon bath operators above interaction Hamiltonian into the compact form
B are defined as B =Π D (g /ω ), with D (±α)=
± ± k k k k k
exp[±(αb†k−α∗bk)]beingthekthmodedisplacementop- Hpn =(cid:88)2 Apn⊗Bpn (20)
erator. For numerical results we shall later use a super- IP i i
i=1
ohmic exciton-phonon spectral density J (ω) with ex-
pn
ponential cut-off at frequency ωc that is appropriate for which will prove useful for the derivation of the master
self-assembled III-V quantum dots38,39: equation.
Jpn(ω)=αω3e−ωωc22 . (13)
B. Master Equation
In the polaron frame the light-mattter interaction
Hamiltonian Eq. (8) becomes
(cid:88)
Hpt =i|0(cid:105)(cid:104)X|B d·u∗ (r )a†
IP − qλ d qλ HavingobtainedourHamiltonianinthepolaronframe
q,λ (14) and partitioned it into system, interaction and environ-
(cid:88)
−i|X(cid:105)(cid:104)0|B d·u (r )a . ment parts, we can make use of the generically derived
+ qλ d qλ
microscopic second-order Born-Markov master equation
q,λ
of Ref.41 (Eqn. 3.118). The interaction terms Eqs. (15)
With the definitions Apt = |0(cid:105)(cid:104)X|, Apt = Apt†, Bpt ≡ and (20) are of the required form underlying this deriva-
1 2 1 1/2
B , C = i(cid:80) d·u∗ (r )a† , and C = C†, we can tion, and the resultant ME (in the interaction picture)
∓ 1 q,λ qλ d qλ 2 1 reads:
compactly write the above Hamiltonian as
d
Hpt =(cid:88)2 Apt⊗Bpt⊗C , (15) dtρSP(t)= (21)
IP i i i (cid:90) ∞
i=1 − dτ Tr [H (t),[H (t−τ),ρ (t)⊗ρ (0)]] ,
E IP IP SP E
0
SincethesecondterminEq.(12)containssystemanden-
vironmentoperators,weidentifythisasournewexciton- where H (t) = Hpn(t)+Hpt(t), and Tr denotes the
IP IP IP E
phonon interaction term40. This new interaction term traceoverbothenvironments41. Itcanbeeasilyshown40
possesses a non-zero expectation value with respect to thattheright-handside(RHS)oftheaboveequationcan
the thermal equilibrium bath state ρpn; tracing out the be split into two parts:
E
5
2. Electromagnetic bath correlations
d
ρ (t)= (22)
dt SP
(cid:90) ∞
− dτTrpn[Hpn(t),[Hpn(t−τ),ρ (t)⊗ρpn(0)]]
E IP IP SP E
0 Having arrived at a ‘Lindblad-like’ phonon
(cid:90) ∞
− dτTr [Hpt(t),[Hpt(t−τ),ρ (t)⊗ρ (0)]] . dissipator46, we now turn our attention to the sec-
E IP IP SP E ond term of the RHS of Eq. (22). This term will yield
0
the modified SE rate of the TLS near the cavity, as
Since we assume that the (initial) environmental state is well as account for the frequency shift via a unitary
thermal, ρE(0) factorises: ρE(0)=ρpEn(0)⊗ρpEt(0). renormalisation term. As in the previous section, we
begin by explicitly printing the correlation functions
obtained from Eq. (22):
1. Phonon bath correlations Cpt(τ) (25)
ij
=Tr (cid:104)(cid:16)Bpt†(τ)⊗C†(τ)(cid:17)(cid:0)Bpt(0)⊗C (0)(cid:1)ρ (0)(cid:105) ,
E i i j j E
(cid:104) (cid:105) (cid:104) (cid:105)
We proceed by analysing the first term on the RHS =Trpn Bpt†(τ)Bpt(0)ρpn(0) Trpt C†(τ)C (0)ρpt(0) ,
of Eq. (22) which captures the influence of phonons on E i j E E i j E
the TLS dynamcis with scattering rates determined by
where i,j ∈{1,2}. After substituting for the bath oper-
phonon correlation functions42–44. In the ME formal-
ators, we make use of the following relations41
ism, the rate γ(ω) of a dissipative process is given by
γ(ω) = 2Re(cid:2)(cid:82)0∞dsK(s)(cid:3), where K(s) is the relevant Trpt(cid:2)a a ρpt(0)(cid:3)=Trpt(cid:104)a† a† ρpt(0)(cid:105) = 0 ,
correlation function [c.f. Eq. (3.137) in Ref.41]. For our E qλ q(cid:48)λ(cid:48) E E qλ q(cid:48)λ(cid:48) E
phonon dissipator, these functions are given by Trpt(cid:104)a a† ρpt(0)(cid:105)=δ δ (1+N(ν )) ≈ δ δ ,
E qλ q(cid:48)λ(cid:48) E qq(cid:48) λλ(cid:48) q qq(cid:48) λλ(cid:48)
(cid:104) (cid:105)
Cpn(τ)=Trpn B†(τ)B (0)ρpn(0)
ii E ± ± E (cid:104) (cid:105)
Trpt a† a ρpt(0) =δ δ N(ν ) ≈ 0 ,
=(cid:104)B(cid:105)2(eφ(τ)−1) , (23) E qλ q(cid:48)λ(cid:48) E qq(cid:48) λλ(cid:48) q
(cid:104) (cid:105)
Cpn(τ)=Trpn B†(τ)B (0)ρpn(0) where we have assumed that ∀ω > 0, the Planck distri-
ij E ± ∓ E bution N(ω) ≈ 047. This means that we only have a
=(cid:104)B(cid:105)2(e−φ(τ)−1) , (24) singlenon-vanishingcorrelationfunctionCpt(τ). Follow-
11
ingRef.48,weconsiderwell-separatedphotonandphonon
where i,j ∈{1,2}, i(cid:54)=j. After some algebra, we obtain
correlation times (appropriate for an unstructured pho-
a phonon dissipator of the form tonic environment), so that Cpt(τ) reduces to the pho-
11
ton bath correlation function in the absence of a phonon
γpn(ω(cid:48))L[σ ]+γpn(−ω(cid:48))L[σ ]
− + bath. The latter is given by
−γcpdn(ω(cid:48))Lcd[σ−]−γcpdn(−ω(cid:48))Lcd[σ+] , Cpt(τ)= |d|2 (cid:90) ∞dν ν3[1+F (qr )] , (26)
11 6π2(cid:15)c3 q q cav d
where L[C] = Cρ C† − 1{C†C,ρ } and L [C] = 0
SP 2 SP cd
CρSPC− 12{C2,ρSP}. The rates γpn(±ω(cid:48)) and γcpdn are where the term
(cid:18) (cid:19)
γpn(±ω(cid:48))= |Ωpcanv|2 (cid:90) ∞ dτ e±iω(cid:48)τ(cid:16)eφ(τ)−1(cid:17) , Fcav(x)= 23 −sin2(x2x) − co(2sx(2)x2) + s(in2(x2)x3) , (27)
4
−∞
(Ωpn∗)2 (cid:90) ∞ (cid:16) (cid:17) describestheinfluenceofthemetalsurface. TheSErate
γpn(ω(cid:48))= cav dτ cos(ω(cid:48)τ) 1−e−φ(τ) ,
cd 4 then evaluates to
−∞
γpn(−ω(cid:48))= (Ωpcanv)2 (cid:90) ∞ dτ cos(ω(cid:48)τ)(cid:16)1−e−φ(τ)(cid:17) , γcpatv(ω(cid:48))=(1+Fcav(q0rd))γ0pt(ω(cid:48)) , (28)
cd 4
−∞
where γpt(ω(cid:48)) is the bare SE rate for an isolated TLS,
0
where φ(τ) = (cid:82)∞dωJpn(ω)[coth(βω/2)cos(ωτ) − and is given by γpt(ω(cid:48)) = |d|2ω(cid:48)3/3π(cid:15)c3. The imag-
0 ω2 0
isin(ωτ)]. Our rates match the ones obtained by Roy- inary part of the correlation tensor has two compo-
Choudhuryetal.43inpreviouswork45. Theratesγpn(ω(cid:48)) nents: the first term is the usual Lamb shift (whose ex-
and γpn(−ω(cid:48)) correspond to enhanced radiative decay pression is divergent unless one adopts a full QED ap-
andincoherentexcitationoftheTLS,respectively,whilst proachbasedonarelativisticHamiltonianandappropri-
γpn(±ω(cid:48)) is associated with cross-dephasing42. aterenormalisation49). Thesecondtermistheadditional
cd
6
������ �����������
���
���
��� ���
��� ���
����γ()/γ����������� ��()/γ������-������
���
��� -���
��� -���
��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ���
�=��/λ� �=��/λ�
FIG. 4: Spontaneous emission rate (left) and energy shift (right) for the half-sided cavity model (red), where we
divided expressions (28) and (29) by the bare SE rate in order to avoid dependence on its value. The blue energy
shift curve denotes the energy shift obtained using a full QED approach26, showing a distinctively different
behaviour at smaller separations ((cid:46)0.05λ ) when compared to the half-sided cavity and image approaches. The
0
oscillations persist even at larger separations, of the order of the emission wavelength λ for the SE rate. As x→∞,
0
the SE rate tends to that of a bare emitter and the energy shift vanishes, as expected.
energy shift term and takes the form7,50,51 multiples of 1/8n (where n is the refractive index of the
hostmaterial, takentobeGaAsinourcase), takenfrom
1
V = G (q r )γpt(ω(cid:48)) , (29) Eqns.(28)and(29),serveasaguidetotheeyefortheap-
cav 2 cav 0 d 0 proximatefrequencyofoscillation,anddemonstratethat
multiple periods occur within a wavelength’s separation
where the function G is given by
cav of emitter to surface. In the limiting case r → ∞, we
d
3(cid:18) sin(2x) cos(2x) cos(2x)(cid:19) have Vcav → 0 and γcpatv(ω(cid:48)) → γ0pt(ω(cid:48)), i.e. we recover
Gcav(x)= 2 − (2x)2 − (2x)3 + 2x . (30) the case of an isolated QD as required.
Overall, the transition frequency for the TLS in the po-
laron frame is now given by IV. Image Emitter Approach
ω˜(cid:48) =ω(cid:48)+V (31)
cav Models involving emission from a combination of two
identical TLS have been used extensively to study the
and the final polaron frame ME takes the following form
modificationstotheSErateofanemitterintheproxim-
in the Schr¨odinger picture:
ity of a dielectric or metal surface. After setting up the
d appropriateHamiltonian,weshalloncemorederiveapo-
ρ =
dt SP laron frame ME. We then show that this ME is identical
(32)
i to the one derived using the half-sided cavity approach,
− (cid:126)[HS(cid:48)P,ρSP(t)]+Dpn(ρSP)+Dpt(ρSP) , providedwedisregardcertaintermsinordertoconstrain
thedynamicsofourtwoemittermodeltothe‘right’sub-
where D (ρ ) = γpn(ω(cid:48))L[σ ] + γpn(−ω(cid:48))L[σ ] − space.
pn SP − +
γpn(ω(cid:48))L [σ ] − γpn(−ω(cid:48))L [σ ] and D (ρ ) =
cd cd − cd cd + pt SP
γpt (ω(cid:48))L[σ ]. H(cid:48) is the system Hamiltonian in the
cav − SP
polaron frame including the energy shift from Eq. (29). A. Setup
In summary, Eqs. (28) and (29) capture how the pres-
ence of a metal surface (here treated as a perfect reflec- We focus on the case where the dipole is oriented par-
tor) alters the SE rate and the transition frequency of allel to the surface52 (as is appropriate for a typical self-
the TLS, respectively. Considering our results in the ab- assembled QD emitter), implying that the image dipole
sence of phonons, we find full analytical agreement with will be antiparallel4,26,50,51. In what follows, we shall
thepriorliteratureontheimagedipoleapproach7,26,and once again take the real emitter to be situated at a dis-
except for very small separations, we also have excellent tance r > 0 along the positive z-axis, with the dipole
d
numerical agreement with the full QED approach26. We vector oriented in the positive x-direction. Hence, the
showthisagreementinFig.4asafunctionofthedistance corresponding image dipole is positioned at z = −r ,
d
oftheemittertothesurface. Thedashedverticallinesat with its dipole vector being parallel to the negative x-
7
axis. our relevant Hamiltonian reads
Hpn =Hpn,1+Hpn,2
I I I
(cid:88)2 (cid:88) (38)
B. Hamiltonian = |X (cid:105)(cid:104)X |g (b† +b ) .
j j k k k
j=1 k
Next, we move into the polaron frame with the trans-
The Hamiltonian of the two driven TLS in a frame
formation eS1+S2 = eS1eS2, obtaining the transformed
rotating with frequency ω is given by
l Hamiltonians
HS =(cid:88)j=21δ|Xj(cid:105)(cid:104)Xj|+Ω2∗j |0j(cid:105)(cid:104)Xj|+Ω2j |Xj(cid:105)(cid:104)0j| , (33) HSP =(cid:88)2 δ(cid:48)|Xj(cid:105)(cid:104)Xj|+ Ωp2n∗ |0j(cid:105)(cid:104)Xj|+H.c. , (39)
j=1
(cid:88)
where the subscript j = 1,2 denotes the real and im- Hpt,j =i|0 (cid:105)(cid:104)X |B d ·u∗ (r )a†
IP j j − j qλ j qλ
age TLS, respectively. In order to match the boundary
q,λ
conditions required for reflection, we model the classical (cid:88)
−i|X (cid:105)(cid:104)0 |B d ·u (r )a ,
drivingfieldastwocounter-propagatingbeams,withthe j j + j qλ j qλ
secondary ‘reflected’ beam having a π phase shift with q,λ
respect to the original beam. For simplicity, we model Hpn,j =Ω∗ |0 (cid:105)(cid:104)X |B + Ω|X (cid:105)(cid:104)0 |B . (40)
these as plane waves propagating along the z-axis and IP 2 j j − 2 j j +
polarised in the x-direction. In phasor notation, these
AsinSec.III,thelattertwocaneasilybeseentobeofthe
two waves can be written as
following generic form (with appropriate identifications
E1(r)=Eincident(r)=E0eiql·rxˆ , for the A,B,C operators) which will enable straightfor-
(34) ward use of the ME (3.118) from Ref.41:
E2(r)=Ereflected(r)=−E0e−iql·rxˆ ,
2
giving rise to the following Rabi frequencies at the posi- Hpn,j =(cid:88)Apn,j ⊗Bpn,j , (41)
tions r of the two emitters: IP i i
1,2 i=1
2
Ω1 =2d1·(E1(r1)+E2(r1)) , (35) HIpPt,j =(cid:88)Apit,j ⊗Bipt,j ⊗Cij . (42)
Ω =2d ·(E (r )+E (r )) .
2 2 1 2 2 2 i=1
Since r =−r and d =−d , we have Ω:=Ω =Ω .
2 1 2 1 1 2
C. Master equation
Wenowturntothewiderelectromagneticenvironment
(excluding the coherent driving field discussed above).
The electric field operator can be written as in Eq. (9)
The ME for our system can, once again, be written as
but with the spatial mode functions now being replaced
by the free-space functions d
ρ (t)= (43)
(cid:114) dt SP
uqλ(r)= 2ω(cid:15)qVλeqλeiqr . (36) −(cid:90) ∞dτTrpEn[HIpPn(t),[HIpPn(t−τ),ρSP(t)⊗ρpEn(0)]]
0
(cid:90) ∞
The interaction Hamiltonian of the TLS with the pho- − dτTr [Hpt(t),[Hpt(t−τ),ρ (t)⊗ρ (0)]] ,
tonic environment is then given by E IP IP SP E
0
Hpt =Hpt,1+Hpt,2 however, it now features a larger number of correlation
I I I
functions due to the presence of the image emitter. Fol-
(cid:88)2 (37) lowing the general procedure in Sec. IIIB, we shall anal-
=− d ·E(r )(|0 (cid:105)(cid:104)X |+|X (cid:105)(cid:104)0 |) .
j j j j j j
yse different contributions in turn to arrive at our final
j=1
ME of the image emitter model.
Fortheinteractionwithvibrationalmodes,weassume
thatbothrealandimageTLSseethesamephononbath
1. Phonon dissipator
and possess perfectly correlated coupling constants g .
k
Thisensurestheimagesystemexactlyfollowsthedynam-
ics of real dipole, as is required for matching the bound- The correlation functions (including cross correlation
ary condition of a perfectly reflecting interface. Thus, terms between bath operators of the real and image sys-
8
tem) result in the following phonon dissipator geticshiftscanbeabsorbedintothebareTLStransition
frequency. We thus focus on the off-diagonal element
Dpn(ρSP)= (44) which is of the form:
2 (cid:18) (cid:19)
(cid:88) 1 1
γpn(ω(cid:48)) σj ρ (t)σi − {σi σj ,ρ (t)} V = G (q∆r)γpt(ω(cid:48)) , (47)
ji − SP + 2 + − SP 12 2 12 0
i,j=1
(cid:88)2 (cid:18) 1 (cid:19) where the function G12 is
+ γpn(−ω(cid:48)) σjρ (t)σi − {σi σj,ρ (t)}
ji + SP − 2 − + SP 3(cid:18) sin(x) cos(x) cos(x)(cid:19)
i,j=1 G (x)= − − + . (48)
12 2 x2 x3 x
2 (cid:18) (cid:19)
(cid:88) 1
− γpn (ω(cid:48)) σj ρ (t)σi − {σi σj ,ρ (t)}
cd,ji − SP − 2 − − SP Again,thiscorrespondstothesameenergyshifttermwe
i,j=1
have previously encountered in Sec. IIIB2. After diago-
2 (cid:18) (cid:19)
(cid:88) 1 nalisingtheHamiltonian,thefrequencyofthesymmetric
− γpn (−ω(cid:48)) σjρ (t)σi − {σi σj,ρ (t)} ,
cd,ji + SP + 2 + + SP excited to ground state transition (in the polaron frame)
i,j=1 is then given by
where the rates γpn(±ω(cid:48)) and γpn are given by
ji cd,j ω˜(cid:48) =ω(cid:48)+V , (49)
12
|Ωpn|2 (cid:90) ∞ (cid:16) (cid:17)
γpn(±ω(cid:48))= dτ e±iω(cid:48)τ eφ(τ)−1 , exactlymatchingthetransitionfrequencyEq.(31)ofthe
ji 4 half-sided cavity model.
−∞
(Ωpn∗)2 (cid:90) ∞ (cid:16) (cid:17)
γpn (ω(cid:48))= dτ cos(ω(cid:48)t) 1−e−φ(τ) ,
cd,ji 4
−∞
(Ωpn)2 (cid:90) ∞ (cid:16) (cid:17)
γpn (−ω(cid:48))= dτ cos(ω(cid:48)t) 1−e−φ(τ) .
cd,ji 4
−∞
|e(cid:105)
We shall return back to the phonon dissipator when dis-
cussingtheMEequationinthesymmetric-antisymmetric
basis,whichallowsustoderiveamodelagreeingwiththe
half-sided cavity approach.
|s(cid:105)
2. Photon dissipator
2V
12
We now turn our attention to the photon dissipator |a(cid:105)
term from Eq.(43). After evaluating thecorrelation and
cross-correlation functions, we obtain the usual expres-
sion for two emitters7 in a shared electromagnetic envi-
ronment,
Dpt(ρSP)= |g(cid:105)
(cid:88)2 (cid:18) 1 (cid:19) (45)
γpt σj ρ (t)σi − {σi σj ,ρ (t)} ,
ji − SP + 2 + − SP FIG. 5: Energy level diagram for the two emitter
i,j=1 system. The symmetric (|s(cid:105)) and antisymmetric (|a(cid:105))
levels are shifted up and down by V , respectively. The
where the diagonal terms γpt(ω(cid:48)) = γpt(ω(cid:48)) = γpt(ω(cid:48)), 12
22 11 0 black arrows indicate the laser driving; the
whilst the off diagonal terms are given by γpt(ω(cid:48)) =
12 antisymmetric state is decoupled. Blue and red wavy
γpt(ω(cid:48)) = F (q ∆r)γpt(ω(cid:48)) with ∆r = r −r = 2r ,
21 12 0 0 1 2 d lines indicate photon emission from the antisymmetric
and where
and symmetric channel, respectively. As discussed in
(cid:18) (cid:19) the text, it is necessary to disable driving on the
3 sin(x) cos(x) sin(x)
F (x)= − − + . (46) |s(cid:105)↔|e(cid:105) transition (black dashed) to recover the
12 2 x x2 x3
effective two level-system |g(cid:105)↔|s(cid:105). For environments
permitting photon absorption, the dashed wavy
Thisisthesamefunctionobtainedforthehalf-sidedcav-
transitions also need to be explicitly disabled.
ity approach [c.f. Eq. (27)]. The imaginary part of the
correlation function yields the ‘correction’ term to the
unitary part of the ME7,41,50: its diagonal contribution
represents diagonal Lamb shift terms. Their small ener-
9
Phonons:No No Yes Yes
Mirror: No Yes No Yes
γ0 =γ0pt(ω0) γ0 =(1+F12(q0∆r))γ0pt(ω0) γ0 =γ0pt(ω(cid:48)) γ0 =(1+F12(q0∆r))γ0pt(ω(cid:48))
ω =ω ω =ω +V ω =ω(cid:48) ω =ω(cid:48)+V
0 0 12 12
FIG. 6: Overview of the four scenarios for an optical dipole considered in this work. All cases have a schematic
depiction accompanied by the corresponding SE rates γ and transition frequencies ω. Here, ∆r is the separation
0
between the real and image dipole, F (q ∆r) and V are given by Eqns. (B1) and (47), respectively, and ω and ω(cid:48)
12 0 12 0
are the bare and polaron shifted frequencies, respectively. The blue ‘masses on springs’ (blue circles) denote the
phonon bath. Note that the driving field is not shown here, as its presence or absence does not influencing the
relevant properties.
D. Effective TLS in the energy eigenbasis space only served to let us calculate the correct proper-
ties of this single transition. Fully decoupling the anti-
symmetric singly and the doubly excited states from the
Asstatedintheintroduction,previousliteraturetreat-
dynamics is achieved by disabling the laser driving on
ing spontaneous emission from initially excited emitters
the |s(cid:105) ↔ |e(cid:105) transition. For finite temperature photon
consideredthetransitionfromthesymmetricallyexcited
environmentswithN(ω)(cid:54)=0,wealsoneedtoremovedis-
to the ground state, as this choice yields matching re-
sultswithothermethods4,7. Wefollowthisapproachand sipativephotonabsorptionchannels,bydroppingthean-
tisymmetric dissipator term Da (ρ ) from the ME and
adopt the b√asis {|e(cid:105),|s(cid:105),|a(cid:105),|g(cid:105)} with |s(cid:105)=(|01(cid:105)√|X2(cid:105)+ pt SP
explicitly removing the dissipative |s(cid:105)↔|e(cid:105) operator.
|X (cid:105)|0 (cid:105))/ 2 and |a(cid:105) = (|0 (cid:105)|X (cid:105)−|X (cid:105)|0 (cid:105))/ 2, see
1 2 1 2 1 2
The image approach can thus be reduced to an effec-
Fig. 5. In this basis, our full polaron ME reads:
tive TLS model featuring the same Rabi frequency, SE
d i rate,andtransitionfrequencyasthehalf-sidedcavityap-
ρ (t)=− [H(cid:48) ,ρ (t)]
dt SP (cid:126) SP SP (50) proach–i.e. displayingfullequivalencebetweenthetwo
+Ds (ρ )+Da (ρ )+Ds (ρ ) , representations.
pn SP pt SP pt SP
In Fig. 6, we summarise the key results from the pre-
where the dissipator terms are explicitly given in Ap- vioussections: WeshowthetransitionfrequencyandSE
pendix A. Here, H(cid:48) denotes the system diagonalised ratefortheallfourcasesconsideredinthisArticlealong-
SP
Hamiltonian [including the energy shift term Eq. (47)]. side their schematic depictions. The driving term is not
The ME photonic dissipator separates into a symmetric includedasithasnodirectinfluenceonthepropertiesof
channel (|g(cid:105) ↔ |s(cid:105) ↔ |e(cid:105)) and an antisymmetric one the optical dipole transition.
(|g(cid:105) ↔ |a(cid:105) ↔ |e(cid:105)). Courtesy of the fully correlated
phonon bath, phonons also only act in the symmetric
channel. V. Resonance Fluorescence Spectrum
Since Ω =Ω , the symmetric channel Rabi frequency
1 2 √ √
becomes Ω :=(Ω +Ω )/ 2= 2Ω=Ω and hence Having included the possibility of laser driving in our
sg 1 2 cav
weobtainthesamephononratesasinthehalf-sidedcav- model, a natural application is to study the resonance
ity approach53. Furthermore, the a√ntisymmetric channel fluorescence (RF) spectrum of a condensed matter TLS
Rabi frequency Ω := (Ω −Ω )/ 2 = 0, meaning that near a mirroring surface. We use the ME (50) (after
a 1 2
the laser field is completely decoupled from the antisym- discarding the antisymmetric channel, as argued above)
metric state. to calculate the spectral function, which is given by the
Consistency with the Green’s function and half-sided Fourier transform of the (steady-state) first order corre-
cavity approach demands that we restrict the dynam- lation function lim (cid:104)E(−)(R,t)E(+)(R,t+τ)(cid:105), where
t→∞
icsofourfour-dimensionalHilbertspacetothesubspace E(−)(R,t) and E(+)(R,t) are, respectively, the negative
spanned by the states {|g(cid:105),|s(cid:105)}, i.e. the larger Hilbert and positive components of the electric field operator
10
�
���
������ �����
������ �����
� ������ ����� ���
)����� ������������ ���������� ��������
(ω)(������ ������ -� -� -� � � � � ����-���� -��� ��� ��� ��� ����������� wwiitthhoouuttmmiirrrroorr,,wwiitthhopuhtopnhoonnsons
�
��� withmirror,withphonons
withmirror,withoutphonons
���
�
-���� -���� -���� ���� ���� ���� ���� ����� ���� � ���
ω-ω�(��-�) Ω(��-�)
FIG. 7: Left: Incoherent component of the RF spectrum for a single TLS (blue) and the effective TLS
incorporating surface-induced modifications (red). Right: Ratio of coherent emission for all four cases
(with/without mirror, with/without the phonon environment) as a function of the (normalised) effective Rabi
frequency. Ω denotes the saturation Rabi frequency for γpt =0.001 ps−1. See text for a discussion.
s 0
evaluated at the position R of the detector7. These op- the left panel of Fig. 7 show the much broader phonon
erators are related to the system operators σ =|0(cid:105)(cid:104)X| sideband, which receives ∼16% of the scattered photons
−
and σ =|X(cid:105)(cid:104)0|, and hence, after applying the polaron for the chosen spectral density at a phonon temperature
+
transformation, the RF spectral function can be written of T=10 K.
as
In the right panel of Fig. 7, we plot the fraction of
(cid:90) ∞ coherently scattered photons as a function of the renor-
S(ω)∝ dτe−i(ω−ω(cid:48))τ× malised effective Rabi frequency. This ratio is obtained
(51)
−∞ numerically as the (integrated) coherent spectrum di-
(cid:104)σ+(τ)B+(τ)σ−(0)B−(0)(cid:105)s , vided by the total integrated spectrum. There are two
pairs of curves: one with and one without phonons. For
where we have exploited the temporal homogeneity of
the former, the finite area under the phonon sideband
the stationary correlation function, and where the sub-
means that the coherent fraction does not go to unity
script‘s’denotesthetracetakenwithrespectthesteady-
even when driving far below saturation. The level at
state density matrix41. The correlation function appear-
which this fraction plateaus is phonon coupling strength
ing in Eq. (51) involves two timescales, the nanosecond and temperature dependent54. By contrast, in the ab-
timescale associated with the exciton lifetime, and the
sence of phonons, almost all light is coherently scattered
shorter picosecond phonon bath relaxation timescale, al-
at weak enough driving. The close agreement between
lowing us to separate the correlation function into the
the two curves in each pair bears testament to the fact
product (cid:104)σ (τ)σ (0)(cid:105) (cid:104)B (τ)B (0)(cid:105) 54. Substituting
+ − s + − s that the surface-modified emitter largely behaves like a
the expression for the phonon bath correlation function,
bare emitter once the effective Rabi frequency has been
we obtain the spectral function
corrected for (with the slight remaining discrepancy due
(cid:90) ∞ to modifications of the natural lifetime). Indeed, plot-
S(ω)∝(cid:104)B(cid:105)2 dτe−i(ω−ω(cid:48))τ× ting this ratio directly as a function of the laser driving
−∞ (52) field amplitude reveals sizeable horizontal shifts between
eφ(τ)(cid:104)σ (τ)σ (0)(cid:105) . these two curves in each pair (not shown).
+ − s
In the left panel of Fig. 7, we show the incoherent part
of the emission spectrum of our surface-modified system
VI. Summary and Discussion
as well as that of a reference TLS (also subject to the
same phonon environment). Following Ref.20, we take
the TLS’s position relative to the surface as r ∼ 177 Wehaveextendedthemethodofimages–traditionally
d
nm. The reference TLS is driven with ‘free space’ Rabi developed for capturing spontaneous emission in atomic
frequency given by Ωpn = 2(cid:104)B(cid:105)d·E . As expected, the ensembles near reflective interfaces – to the case of a
0
curvesdifferinthepositionoftheMollowsidebandsand driven solid-state emitter near a metal surface. We have
the width of the three peaks, since the former is deter- developedtwoapproaches: ahalf-sidedcavityandimage
mined by the effective Rabi frequency and the later de- dipole,andshownthatthelatteragreeswiththeformer,
pendsontheemissionrate,whichbothundergoachange but only when additional ‘selection rules’ are introduced
in the presence of a reflective surface. The two insets in toconstrainthedynamicstotherelevantsubspace. Both
our approaches agree with a Green’s function treatment