# Controlling Single-Photon Emission with Ultrathin Transdimensional Plasmonic
Films

Igor V. Bondarev<EMAIL_ADDRESS>[

###### Abstract

We study theoretically the properties of a two-level quantum dipole emitter
near an ultrathin transdimensional plasmonic film. Our model system mimics a
solid-state single-photon source device. Using realistic experimental
parameters, we compute the spontaneous and stimulated emission intensity
profiles as functions of the excitation frequency and film thickness, followed
by the analysis of the second-order photon correlations to explore the photon
antibunching effect. We show that ultrathin transdimensional plasmonic films
can greatly improve photon antibunching with thickness reduction, which allows
one to control quantum properties of light and make them more pronounced.
Knowledge of these features is advantageous for solid-state single-photon
source device engineering and overall for the development of the new
integrated quantum photonics material platform based on the transdimensional
plasmonic films.

###### keywords:

Single-Photon Source, Transdimensional Plasmonic Materials, Photon Emission,
Photon Antibunching

NCCU] Department of Mathematics and Physics, North Carolina Central
University,
Durham, NC 27707, USA TD, CN, QED

##  0.0.1 Introduction

Transdimensional (TD) quantum materials are atomically-thin films of precisely
controlled thickness, films made of precisely controlled finite number of
monolayers 1, 2, 3. The term ’transdimensional’ refers to the transitional
range of thicknesses — a regime that is neither three (3D) nor two (2D)
dimensional but rather something in between, turning into 2D as thickness
tends to zero, challenging to study what the 3D-to-2D continuous transition
has to offer to improve material functionalities. Currently available due to
the rapid progress in nanofabrication techniques 4, 5, 6, 7, 8, 9, 10, 11,
such materials offer high tailorability of their electronic and optical
properties not only by altering their chemical and/or electronic composition
(stoichiometry, doping) but also by merely varying their thickness (number of
monolayers) 12, 13, 14, 15. Materials like these are indispensable for studies
of fundamental properties of the light-matter interaction as it evolves from a
single 2D atomic layer to a larger number of layers approaching the 3D bulk
material properties. With thickness of only a few atomic layers, ultrathin TD
films of metals, doped semiconductors, or polar materials can support
plasmon-, exciton-, magnon-, and phonon-polariton eigenmodes 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. Plasmonic TD
materials (ultrathin metallic films) offer controlled light confinement, large
tailorability and dynamic tunability of their optical properties due to their
thickness-dependent localized surface plasmon (SP) modes 14, 15, 16, 17, 18,
19, 20, 21, which are distinctly different from those of conventional thin
films commonly described by either purely 2D or by 3D material properties with
boundary conditions imposed on their top and bottom interfaces 33, 34, 35, 36,
37, 38, 39, 40, 41. In such systems, the vertical quantum confinement enables
a variety of new quantum phenomena, including the thickness-controlled plasma
frequency red shift 2, 11, the SP mode degeneracy lifting 14, 18, a series of
magneto-optical effects 13, and even atomic transitions that are normally
forbidden 1, 20, 21, to mention a few.

Previously, we have implemented the confinement-induced nonlocal Drude
response model based on the Keldysh-Rytova (KR) pairwise electron interaction
potential 2 to study theoretically the electronic properties of ultrathin TD
plasmonic films 13, 14 and demonstrate their major manifestations
experimentally 11, 42. The KR interaction potential takes into account the
vertical electron confinement 43, 44, which makes it much stronger than the
electron-electron Coulomb potential 43, offering also the film thickness as a
parameter to control the electronic properties of the TD plasmonic films.
Here, we use the KR model to further explore the capabilities of the TD
plasmonic films, focusing on the light-scattering properties of a quantum
dipole emitter (DE) placed at a distance near the film surface; see the inset
in Fig. 1 (b). The model system we study mimics a quasi-2D solid-state single-
photon source device, 45 for which our goal is to show the advantages of using
the ultrathin TD plasmonic films. So far room-temperature solid-state quantum
emitters have been observed in wide-bandgap semiconductors such as diamond 46
and silicon carbide 47, nanocrystalline quantum dots 48, 49, 50, and in carbon
nanotubes 51. Single-photon emission from localized color-center defects in 2D
materials has been reported both at cryogenic 52, 53, 54, 55 and at room
temperatures 56, 57. An aspiring goal, however, is to use these materials in
the quantum regime where the photon emission is of robust and controllable
single-photon character, to enable applications in quantum information
processing 45, 58. We show that the ultrathin TD plasmonic films have this
outstanding potential due to their unique tailorability by means of their
thickness adjustment.

Using the nanodiamond nitrogen-vacancy (NV) center 59, 60 near the TiN film
surface as a prototype coupled ’DE-TD film’ system 14, we compute the
spontaneous and stimulated emission intensity profiles as functions of the
excitation frequency and film thickness, followed by the intensity correlation
function analysis to explore the photon antibunching effect. We show that the
film thickness can be used to tune from weak to strong the evanescent coupling
of the DE to the plasma modes of the ultrathin TD films. The strong evanescent
coupling hybridizes the transition energy levels of the DE whereby its
stimulated emission is quenched. Increased resonance optical pumping reduces
this quenching effect. The overall film thickness effect on the intensity
correlation function is to increase its positive slope with thickness
reduction and thus to improve the photon antibunching and related
(nonclassical) sub-Poissonian photon counting statistics 61. Knowledge of
these features is advantageous for solid-state single-photon source device
engineering 45 and overall for the development of the new integrated quantum
photonics material platform 62 based on the ultrathin TD materials for quantum
information processing applications.

Figure 1: (a) The plasma mode distribution derived from the $p$-evanescent
wave reflection coefficient of the ultrathin TD plasmonic film as a function
of the film thickness. (b) The spontaneous emission rate (averaged over the
three dipole orientations and normalized by the spontaneous emission rate in
vacuum) for the DE coupled to the plasma modes in (a), as a function of the
film thickness $d$ and the DE position $z_{A}$ near the TD film surface where
the evanescent contribution to the spontaneous emission is dominant. Inset
shows the geometry of the problem. See Ref. 14 for details.

##  0.0.2 a. Spontaneous Emission Spectrum

Figure 1 (a) shows the earlier reported plasma mode distribution derived from
the $p$-evanescent wave reflection coefficient of the ultrathin TD plasmonic
slab as a function of the thickness $d$ of the slab 14. The associated
spontaneous emission rate (averaged over the three dipole orientations and
normalized by the spontaneous emission rate in vacuum) is shown in Fig. 1 (b)
as a function of $d$ and the DE position $z_{A}$ near the surface (see inset)
where the evanescent contribution to the spontaneous emission is dominant. The
DE transition frequency is taken to be $1.81$ eV consistent with what was
reported recently from the density functional theory simulations for the zero-
phonon line transition of the NV-center in diamond 60, with all other
parameters being essentially representative of the ultrathin TiN film as
reported earlier 14. Comparing the (a) and (b) panels, it can be clearly seen
that the fast increase followed by the drop-off of the DE spontaneous emission
rate at $d\\!<\\!10$ nm originates from the DE coupling to the split-off
plasma mode of the lowest out-of-plane momentum which is present in there for
$10\gtrsim d\gtrsim 5$ nm. The presence itself of this split-off mode is a
remarkable feature of the confinement-induced nonlocality captured by the KR
model we use 14, 11.

To see how the DE coupling to the plasma modes in Fig. 1 (a) affects its
spontaneous emission intensity profile, we use the analytical approach
previously developed by one of us for an analogous problem of the DE coupled
to a resonance of the local density of photonic states (LDOS) close to the
nanotube surface 63. In our notations here, the spectral lineshape profile of
the spontaneous emission intensity is given by

$I(d,z_{A},\omega)=\frac{\Gamma(d,z_{A})}{2\pi}\frac{(\omega-\omega_{A})^{2}+\delta\omega_{r}^{2}}{\left[(\omega-\omega_{A})^{2}-X^{2}/4\right]^{2}+\delta\omega_{r}^{2}(\omega-\omega_{A})^{2}}\,,$
(1) $X(d,z_{A})=\sqrt{2\delta\omega_{r}\Gamma(d,z_{A})}\,.$

Here, the DE transition frequency $\omega_{A}$ includes the red shift due to
the DE interaction with the surface 64 and is assumed to be dimensionless
(divided by $\Gamma_{0}$, the vacuum spontaneous emission rate 65), and so too
are all quantities in the equation including the half-width-at-half maximum
$\delta\omega_{r}$ of the LDOS resonance. The thickness-dependent spontaneous
emission rate $\Gamma(d,z_{A})\\!=\Gamma(d,z_{A},\omega_{A})/\Gamma_{0}$ is
the one presented in Fig. 1 (b). The parameter $X$ is the Rabi-splitting to
represent the DE transition level hybridization due to the coupling to the
medium-assisted excitations of the material subsystem (plasma modes of the TD
film herein). The coupling is termed weak if
$(X/\delta\omega_{r})^{2}\\!\ll\\!1$ and strong if
$(X/\delta\omega_{r})^{2}\\!\gg\\!1$, in which cases Eq. (1) generates the
single-peaked $\omega_{A}$-centered Lorentzian profile and the double-picked
profile of two symmetrically split peaks at $\omega_{A}\pm X/2$, respectively
63.

##  0.0.3 b. Stimulated Emission Spectrum

In resonance-fluorescence experiments, the DE is irradiated by laser light
tuned to the DE transition frequency, so that the DE can be excited into an
upper quantum state. The excitation is followed by two competing processes.
They are the (coherent) driving of the DE transition by the laser field,
whereby stimulated (coherent) photon emission and absorption of a laser photon
take turns, and spontaneous (incoherent) emission of the DE excited state. An
interplay of the two, of which one is strongly $d$-dependent in our case (Fig.
1), is expected to give rise to new quantum-statistical features of the
scattered radiation.

It was first recognized by R.Feynmann et al. 66 that the resonance-
fluorescence process can be described by optical Bloch equations, a
generalization of the Bloch equations of the magnetic resonance 67, whereby
the DE stimulated emission spectrum under resonance excitation can be shown to
take the form as follows 65

$S(\omega,\Gamma_{f})=\frac{1}{\pi}\,\mbox{Re}\Big{\\{}\tilde{S}_{12}\Big{[}i\big{(}\omega-\omega_{L}\big{)}+\frac{\Gamma_{f}}{2}\Big{]}\Big{\\}}\,.$
(2)

Here, $\omega_{L}$ is the laser field frequency, $\Gamma_{f}$ is the spectral
apparatus passband width, and

$\tilde{S}_{12}(s)=\frac{\sigma_{22}(\infty)}{s+\Gamma_{2}}\Big{\\{}1+\frac{\Gamma_{1}}{2s}\Big{[}1-\frac{\Omega/\Gamma_{1}}{\Omega/(s+\Gamma_{1})+(s+\Gamma_{2})/\Omega}\Big{]}\Big{\\}}$
(3)

is the Laplace transform of the spectral distribution function under the
resonance ($\omega_{A}\\!=\omega_{L}$) single-photon transition
$|1\rangle|n+1\rangle\\!\leftrightarrow|2\rangle|n\rangle$ ($n$ is the photon
occupation number) between the lower $|1\rangle$ and upper $|2\rangle$ DE
states. In this equation,
$\Omega\\!=|\textbf{d}_{21}\\!\cdot\textbf{E}_{L}|/2\hbar$ is the Rabi
frequency to represent the DE transition dipole coupling strength to an
external laser field, $\Gamma_{1}$ and $\Gamma_{2}$ are the longitudinal and
transverse relaxation rates, respectively, analogous to those of the Bloch
magnetic resonance theory 67. In our case here, we have
$\Gamma_{1}\\!=2\Gamma_{2}\\!=\Gamma(d,z_{A})$ since major relaxation comes
due to spontaneous emission from the upper DE state 65. The prefactor

$\sigma_{22}(\infty)=\frac{1}{2}\frac{\Omega^{2}}{\Gamma_{1}\Gamma_{2}+\Omega^{2}}$
(4)

is the time-dependent upper-state occupation probability taken at
$t\\!=\\!\infty$. All quantities in Eqs. (2)–(4) are assumed to be
dimensionless as per convention introduced above for Eq. (1).

Equations (2)–(4) allow for two distinct limiting cases 65. They are (1) the
weak-driving-field limit, where $\Omega\\!\ll\\!\Gamma_{1},\Gamma_{2}$ to
result in the $\omega_{L}$-centered ($\omega_{L}\\!=\omega_{A}$) single-peak
Rayleigh scattering spectral profile, and (2) the high-driving-field limit,
where $\Omega\\!\gg\\!\Gamma_{1},\Gamma_{2}$ and the spectrum is reconstructed
due to the ”dressed” state formation to yield the Mollow triplet profile
consisting of the $\omega_{L}$-centered main peak and the two symmetric
satellites at $\omega_{L}\pm\Omega$.

##  0.0.4 c. Intensity Correlation and Photon Antibunching

Having emitted a photon, the DE undergoes a quantum jump from the upper to the
lower (ground) quantum state, whereby it can no longer emit a second photon.
Only after performing a laser-induced transition back to the upper quantum
state, the DE is ready to emit a second photon. The probability of emitting
the first photon at an instant of time $t$ is proportional to the upper-state
photon occupation probability $\sigma_{22}(t)$, which also controls the
intensity $S$ of the scattered light as per Eqs. (2)–(4), and which can be
found from the optical Bloch equations to have the form as follows 65

$\sigma_{22}(t)=\sigma_{22}(\infty)\Big{(}1+\frac{\lambda_{2}\,e^{\lambda_{1}t}}{\lambda_{1}-\lambda_{2}}+\frac{\lambda_{1}\,e^{\lambda_{2}t}}{\lambda_{2}-\lambda_{1}}\Big{)}\,.$
(5)

Here

$\lambda_{1,2}=-\frac{1}{2}\big{(}\Gamma_{1}+\Gamma_{2}\big{)}\pm\sqrt{\frac{1}{4}\big{(}\Gamma_{1}-\Gamma_{2}\big{)}^{2}-\Omega^{2}}\,,$
(6)

and the DE is assumed to be resonantly ($\omega_{L}\\!=\omega_{A}$) excited
out of its initial lower (ground) quantum state. Similarly, the probability of
emitting a second photon at a later instant of time $t+\tau$ is proportional
to the DE upper-state occupation probability $\sigma_{22}(t+\tau)$ under the
condition that at time $t$ the DE is in its lower (ground) quantum state.
Hence, the probability of emitting a first photon at time $t$ and a second
photon at time $t+\tau$ is proportional to
$\sigma_{22}(t)\,\sigma_{22}(t+\tau)$. Under the steady-state observation
conditions, this turns into the two-photon scattered-intensity correlation
function $\sigma_{22}(\infty)\,\sigma_{22}(t+\tau)|_{t\rightarrow\infty}$, a
quantity directly related to the normalized second-order autocorrelation
function

$g^{(2)}(\tau)=\frac{\sigma_{22}(|\tau|)}{\sigma_{22}(\infty)}=1+\frac{\lambda_{2}\,e^{\lambda_{1}|\tau|}}{\lambda_{1}-\lambda_{2}}+\frac{\lambda_{1}\,e^{\lambda_{2}|\tau|}}{\lambda_{2}-\lambda_{1}}=1-(1+a)\,e^{\lambda_{1}|\tau|}+a\,e^{\lambda_{2}|\tau|}$
(7)

with $a\\!=\\!\lambda_{1}/(\lambda_{2}-\lambda_{1})$, commonly used in single-
photon source characterization experiments 56.

The function $g^{(2)}(\tau)$ can be seen to have a positive initial slope,
which reflects the quantum optics effect of photon antibunching. Clearly, the
greater this slope is, the more pronounced quantum features of light are. In
addition, it can be shown that since $g^{(2)}(\tau)<1$, the scattered
fluorescence light gives rise to a (nonclassical) sub-Poissonian photocounting
statistics 61. Both photon antibunching nature and the sub-Poissonian
statistics of the resonance fluorescence from a two-level DE system may be
regarded as proofs of the quantum nature of light which cannot be understood
and explained in terms of classical optics.

Figure 2: (a),(b) Spontaneous emission spectra calculated from Eq. (1) with
two representative $\delta\omega_{r}$ and $\Gamma(d,z_{A})$ shown in Fig. 1
(b) for the DE positioned at $z_{A}\\!=20$ nm and coupled to the TD film
plasma modes shown in Fig. 1 (a). (c),(d) Stimulated emission spectra
calculated from Eqs. (2)–(4) with two representative $\Omega$ and
$\Gamma(d,z_{A})$ shown in Fig. 1 (b) for the DE positioned at $z_{A}\\!=20$
nm and coupled to the TD film plasma modes shown in Fig. 1 (a). (e),(f) Same
as in (c),(d) for the DE positioned at $z_{A}\\!=10$ nm.

##  0.0.5 d. Discussion of Numerical Results

Figure 2 (a) presents the spontaneous emission spectrum calculated from Eq.
(1) with $\delta\omega_{r}\\!=5$ (taken as a model parameter) and
$\Gamma(d,z_{A})$ shown in Fig. 1 (b) for the DE positioned at $z_{A}\\!=20$
nm and coupled to the TD film plasma modes shown in Fig. 1 (a). Figure 2 (b)
presents the same for $\delta\omega_{r}\\!=10$ to compare. The most
interesting feature there is that as $d$ decreases, in the domain $20\gtrsim
d\gtrsim 5$ nm, the DE-plasmon coupling strengthens to result in the fast
spontaneous emission rate increase, as can be seen in Fig. 1 (b), whereby the
coupled system is driven in the strong coupling regime
$(X/\delta\omega_{r})^{2}\\!\gg\\!1$ where the new set of (hybridized) eigen
levels forms 63, $\omega_{A}\\!\pm X/2$. This turns the single-peaked
$\omega_{A}$-centered Lorentzian spontaneous emission profile into the double-
picked profile of the two symmetrically split resonances at $\omega_{A}\\!\pm
X/2$. Increased $\delta\omega_{r}\\!=10$ does not eliminate but just smoothes
out this effect. In this domain of film thicknesses, $\omega_{A}$ is no longer
the DE eigen transition frequency, and so the intensity of the stimulated
emission Rayleigh (same frequency) scattering profile drops down, as can be
seen in Fig. 2 (c) for the same $z_{A}$ and $\Omega=1$, the DE radiative
coupling to the external laser field. Instead, it can be shown that the
plasmon enhanced Raman (shifted frequency) scattering is very efficient in
this regime 68. Figure 2 (d) shows that the increased DE-laser field coupling,
$\Omega=10$, can compete with the DE-plasmon coupling to restore the
stimulated Rayleigh scattering intensity by driving the system in the strong
radiative coupling regime with ”dressed” states formed where the
representative Mollow triplet spectral profile appears. Figure 2 (e), where
$z_{A}$ is decreased down to $10$ nm, demonstrates that bringing the DE closer
to the film surface does drop down the intensity of the stimulated emission
Rayleigh scattering profile for $\Omega=1$. The obvious reason is that the DE-
plasmon coupling increases as well (not shown) so that even increased
$\Omega=10$ can barely restore it completely as can be seen from Fig. 2 (f).
Finally, decreasing $d$ below $5$ nm restores the sharp central peaks for both
spontaneous and stimulated emission profiles in Fig. 2 as only one plasma mode
is found in this domain in Fig. 1 (a), which dies with $d$ decreasing to
reduce $\Gamma(d,z_{A})$ as can be seen in Fig. 1 (b). This brings our system
back in the weak DE-plasmon coupling regime,
$(X/\delta\omega_{r})^{2}\\!\ll\\!1$, followed by making it transparent in the
$d\\!\rightarrow 0$ limit.

Figure 3: (a),(b) Normalized second-order autocorrelation function
$g^{(2)}(\tau)$ calculated from Eq. (7) with two representative $\Omega$ and
$\Gamma(d,z_{A})$ shown in Fig. 1 (b) for the DE positioned at $z_{A}\\!=10$
nm and coupled to the TD film plasma modes shown in Fig. 1 (a). (c),(d) Same
as in (a),(b) for the DE positioned at $z_{A}\\!=20$ nm. Left panels show the
3D graphs of the functions $g^{(2)}(\tau\\!>\\!0)$; right panels are their
crosscuts by planes of constant $d=50$ nm (line 1), $d=30$ nm (line 2), and
$d=10$ nm (line 3).

Figure 3 presents our calculations of the normalized second-order
autocorrelation function $g^{(2)}(\tau)$ of Eq. (7) for two representative
$\Omega=1$ and $10$ with $\Gamma(d,z_{A})$ shown in Fig. 1 (b), for the DE
placed at $z_{A}\\!=10$ nm and $20$ nm and coupled to the TD film plasma modes
shown in Fig. 1 (a). In all four graphs, the left panels show
$g^{(2)}(\tau\\!>\\!0)$ as functions of $d$ and $\Gamma_{0}\tau$, while the
right panels are their crosscuts by planes of constant $d=50$ nm (line 1),
$d=30$ nm (line 2), and $d=10$ nm (line 3). There are two regimes that can be
clearly seen there. They are the weak-driving-field regime and the strong-
driving field regime. The former occurs for $\Omega=1$ in the entire domain of
$d$ presented. Here $g^{(2)}(\tau)$ (and the two-photon scattered-intensity
correlation function, accordingly) is seen to grow up monotonically from the
value $g^{(2)}(0)=0$ to approach the stationary value of $g^{(2)}(\infty)=1$,
with characteristic time scale controlled by the DE spontaneous relaxation
times $\Gamma_{1}=\Gamma(d,z_{A})$ and $\Gamma_{2}=\Gamma(d,z_{A})/2$. The
latter takes place for $\Omega=10$, where the function $g^{(2)}(\tau)$
oscillates at Rabi frequency in the domain of moderate $d$, with Rabi
oscillations being damped due to the DE spontaneous relaxation processes. With
decrease of $d$, however, in both cases the sharp photon antibunching can be
seen in the ultrathin film thickness regime. The positive initial slope of the
function $g^{(2)}(\tau)$ increases with the film thickness decrease to make
the quantum single-photon features of light more pronounced for ultrathin TD
films.

##  0.0.6 Conclusion

In this work we use the parameters of the nanodiamond NV center near the TiN
film surface to compute the spontaneous and stimulated emission intensity
profiles as functions of the excitation frequency and film thickness, followed
by the intensity correlation function analysis to explore the photon
antibunching effect for the quantum DE coupled the ultrathin TD plasmonic
film. We show that the film thickness can be used to tune the evanescent
coupling of the DE to the plasma modes of the film. The strong evanescent
coupling hybridizes the transition energy levels of the DE, whereby its
stimulated emission Rayleigh scattering profile is quenched. Increased
resonance optical pumping drives the system in the strong radiative coupling
regime, which competes with the DE-plasmon coupling, to restore the stimulated
Rayleigh scattering intensity. The overall film thickness effect on the
second-order photon autocorrelation function is to increase its positive slope
with thickness reduction and thus to improve the photon antibunching
properties and related (nonclassical) sub-Poissonian photon counting
statistics. Knowledge of these features is advantageous for solid-state
single-photon source device engineering and overall for the development of the
new integrated quantum photonics material platform based on the ultrathin TD
materials for quantum information processing applications.

This research is supported by the U.S. National Science Foundation under
Condensed Matter Theory Program Award No. DMR-1830874 (I.V.B.). I.V.B.
acknowledges hospitality of the Kavli Institute for Theoretical Physics
(KITP), UC Santa Barbara, where this work was completed during his invited
visit as a KITP Fellow 2022–23 (the research program supported by the U.S.
National Science Foundation under Grant No. PHY-1748958).

## References

  * 1 N.Rivera, I.Kaminer, B.Zhen, J.D.Joannopoulos, and M.Solja$\check{\mbox{c}}$ić, Shrinking light to allow forbidden transitions on the atomic scale, Science 353, 263 (2016).
  * 2 I.V.Bondarev and V.M.Shalaev, Universal features of the optical properties of ultrathin plasmonic films, Optical Mater. Express 7, 3731 (2017).
  * 3 A.Boltasseva and V.M.Shalaev, Transdimensional photonics, ACS Photon. 6, 1 (2019).
  * 4 A.Kossoy, V.Merk, D.Simakov, K.Leosson, S.Kéna-Cohen, and S.A.Maier, Optical and structural properties of ultra-thin gold films, Adv. Opt. Mater. 3, 71 (2015).
  * 5 J.Halim, S.Kota, M.R.Lukatskaya, M.Naguib, M.-Q.Zhao, E.J.Moon, J.Pitock, J.Nanda, S.J.May, Y.Gogotsi, M.W.Barsoum, Synthesis and characterization of 2D molybdenum carbide (MXene), Adv. Funct. Mater. 26, 3118 (2016).
  * 6 D.Shah, H.Reddy, N.Kinsey, V.M.Shalaev, and A.Boltasseva, Optical properties of plasmonic ultrathin TiN films, Adv. Optical Mater. 5, 1700065 (2017).
  * 7 D.Shah, A.Catellani, H.Reddy, N.Kinsey, V.M.Shalaev, A. Boltasseva, and A. Calzolari, Controlling the plasmonic properties of ultrathin tin films at the atomic level, ACS Photon. 5, 2816 (2018).
  * 8 A.Rodríguez Echarri, J.D.Cox, and F.J.García de Abajo, Quantum effects in the acoustic plasmons of atomically thin heterostructures, Optica 6, 630 (2019).
  * 9 Z.M.Abd El-Fattah, V.Mkhitaryan, J.Brede, L.Fern?ndez, C.Li, Q.Guo, A.Ghosh, A.Rodríguez Echarri, D.Naveh, F.Xia, J.Enrique Ortega, and F.J.García de Abajo, Plasmonics in atomically thin crystalline silver films, ACS Nano 13, 7771 (2019).
  * 10 R.A.Maniyara, D.Rodrigo, R.Yu, J.Canet-Ferrer, D.S.Ghosh, R.Yongsunthon, D.E.Baker, A.Rezikyan, F.J.García de Abajo, and V.Pruneri, Tunable plasmons in ultrathin metal films, Nature Photonics 13, 328 (2019).
  * 11 D.Shah, M.Yang, Z.Kudyshev, X.Xu, V.M.Shalaev, I.V.Bondarev, and A.Boltasseva, Thickness-dependent Drude plasma frequency in transdimensional plasmonic TiN, Nano Lett. 22, 4622 (2022).
  * 12 X.Liu, J.-H.Kang, H.Yuan, J.Park, S.J.Kim, Y.Cui, H.Y.Hwang, and M.L.Brongersma, Electrical tuning of a quantum plasmonic resonance, Nature Nanotechnology 12, 866 (2017).
  * 13 I.V.Bondarev, H.Mousavi, and V.M.Shalaev, Optical response of finite-thickness ultrathin plasmonic films, MRS Commun. 8, 1092 (2018).
  * 14 I.V.Bondarev, H.Mousavi, and V.M.Shalaev, Transdimensional epsilon-near-zero modes in planar plasmonic nanostructures, Phys. Rev. Research 2, 013070 (2020).
  * 15 L.Zundel, P.Gieri, S.Sanders, and A.Manjavacas, Comparative analysis of the near- and far-field optical response of thin plasmonic nanostructures, Adv. Optical Mater. 10, 2102550 (2022).
  * 16 A.Manjavacas and F.J.Garcia de Abajo, Tunable plasmons in atomically thin gold nanodisks, Nature Commun. 5, 3548 (2014).
  * 17 F.J.Garcia de Abajo and A.Manjavacas, Plasmonics in atomically thin materials, Faraday Discuss. 178, 87 (2015).
  * 18 S.Campione, I.Brener, and F.Marquier, Theory of epsilon-near-zero modes in ultrathin films, Phys. Rev. B 91, 121408(R) (2015).
  * 19 I.V.Bondarev, Finite-thickness effects in plasmonic films with periodic cylindrical anisotropy [Invited], Optical Mater. Express 9, 285 (2019).
  * 20 I.V.Bondarev and C.M.Adhikari, Collective excitations and optical response of ultrathin carbon-nanotube films, Phys. Rev. Applied 15, 034001 (2021).
  * 21 C.M.Adhikari and I.V.Bondarev, Controlled exciton-plasmon coupling in a mixture of ultrathin periodically aligned single-wall carbon nanotube arrays, J. Appl. Phys. 129, 015301 (2021).
  * 22 B.Miller, A.Steinhoff, B.Pano, J.Klein, F.Jahnke, A.Holleitner, and U.Wurstbauer, Long-lived direct and indirect interlayer excitons in van der Waals heterostructures, Nano Lett. 17, 5229 (2017).
  * 23 I.V.Bondarev and M.R.Vladimirova, Complexes of dipolar excitons in layered quasi-two-dimensional nanostructures, Phys. Rev. B 97, 165419 (2018).
  * 24 T.Deilmann and K.S.Thygesen, Interlayer trions in the MoS2/WS2 van der Waals heterostructure, Nano Lett. 18, 1460 (2018).
  * 25 L.A.Jauregui, A.Y.Joe, K.Pistunova, D.S.Wild, A.A. High, Y.Zhou, G.Scuri, K.De Greve, A.Sushko, C.-H.Yu, T.Taniguchi, K.Watanabe, D.J.Needleman, M.D.Lukin, H.Park, and P.Kim, Electrical control of interlayer exciton dynamics in atomically thin heterostructures, Science 366, 870 (2019).
  * 26 E.V.Calman, L.H.Fowler-Gerace, D.J.Choksy, L.V.Butov, D.E.Nikonov, I.A.Young, S.Hu, A.Mishchenko, and A.K.Geim, Indirect excitons and trions in MoSe2/WSe2 van der Waals heterostructures, Nano Lett. 20, 1869 (2020).
  * 27 I.V.Bondarev, O.L.Berman, R.Ya.Kezerashvili, and Y.E.Lozovik, Crystal phases of charged interlayer excitons in van der Waals heterostructures, Communications Physics 4, 134 (2021).
  * 28 X.Wang, J.Zhu, K.L.Seyler, P.Rivera, H.Zheng, Y.Wang, M.He, T.Taniguchi, K.Watanabe, J.Yan, D.G.Mandrus, D.R.Gamelin, W.Yao, and X.Xu, Moiré trions in MoSe2/WSe2 heterobilayers, Nature Nanotechn. 16, 1208 (2021).
  * 29 Z.Sun, J.Beaumariage, Q.Wan, H.Alnatah, N.Hougland, J.Chisholm, Q.Cao, K.Watanabe, T.Taniguchi, B.M.Hunt, I.V.Bondarev, and D.Snoke, Charged bosons made of fermions in bilayer structures with strong metallic screening, Nano Lett. 21, 7669 (2021).
  * 30 J.Pelliciari, S.Lee, K.Gilmore, J.Li, Y.Gu, A.Barbour, I.Jarrige, C.H.Ahn, F.J.Walker, and V.Bisogni, Tuning spin excitations in magnetic films by confinement, Nature Materials 20, 188 (2021).
  * 31 A.M.Dubrovkin, B.Qiang, H.N.S.Krishnamoorthy, N.I.Zheludev, and Q.J.Wang, Ultra-confined surface phonon polaritons in molecular layers of van der Waals dielectrics, Nature Commun. 9, 1762 (2018).
  * 32 K.P.Kelley, T.G.Folland, J.Ryan Nolen, N.Engheta, J.D.Caldwell, and J.-P.Maria, Polaritonic hybrid-epsilon-near-zero modes: Beating the plasmonic confinement vs propagation-length trade-off with doped cadmium oxide bilayers, Nano Lett. 19, 948 (2019).
  * 33 R.H.Ritchie, Plasma losses by fast electrons in thin films, Phys. Rev. 106, 874 (1957).
  * 34 E.N.Economou, Surface plasmons in thin films, Phys. Rev. 182, 539 (1969).
  * 35 D.A.Dahl and L.J.Sham, Electrodynamics of quasi-two-dimensional electrons, Phys. Rev. B 16, 651 (1977).
  * 36 T.N.Theis, Plasmons in inversion layers, Surf. Sci. 98, 515 (1980).
  * 37 T.Ando, A.B.Fowler, and F.Stern, Electronic properties of two-dimensional systems, Rev. Mod. Phys. 54, 437 (1982).
  * 38 A.V.Chaplik, Absorption and emission of electromagnetic waves by two-dimensional plasmons, Surf. Sci. Rep. 5, 289 (1985).
  * 39 Z.L.Wang, Valence electron excitations and plasmon oscillations in thin films, surfaces, interfaces and small particles, Micron 27, 265 (1996).
  * 40 J.M.Pitarke, V.M.Silkin, E.V.Chulkov, and P.M.Echenique, Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys. 70, 1 (2007).
  * 41 A.Politano and G.Chiarello, Plasmonic modes in thin films: quo vadis?, Front. Mater. 1, 9 (2014).
  * 42 L.Vertchenko, L.Leandro, E.Shkondin, O.Takayama, I.V.Bondarev, N.Akopian, and A.V.Lavrinenko, Cryogenic characterization of titanium nitride thin films, Optical Mater. Express 9, 2117 (2019).
  * 43 L.V.Keldysh, Coulomb interaction in thin semiconductor and semimetal films, Pis’ma Zh. Eksp. Teor. Fiz. 29, 716 (1979) [Engl. translation: JETP Lett. 29, 658 (1980)].
  * 44 N.S.Rytova, Screened potential of a point charge in a thin film, Moscow University Physics Bulletin 3, 30 (1967).
  * 45 M.Kianinia, Z.-Q.Xu, Milos Toth, and I.Aharonovich, Quantum emitters in 2D materials: Emitter engineering, photophysics, and integration in photonic nanostructures, Appl. Phys. Rev. 9, 011306 (2022).
  * 46 I.Aharonovich and E.Neu, Diamond nanophotonics, Adv. Opt. Mater. 2, 911 (2014).
  * 47 S.Castelletto, B.C.Johnson, V.Ivády, N.Stavrias, T.Umeda, A.Gali, and T.Ohshima, A silicon carbide room-temperature single-photon source, Nature Mater. 13, 151 (2014).
  * 48 B.D.Mangum, F.Wang, A.M.Dennis, Y.Gao, X.Ma, J.A.Hollingsworth, and H.Htoon, Competition between Auger recombination and hot-carrier trapping in PL intensity fluctuations of type II nanocrystals, Small 10, 2892 (2014).
  * 49 F.Pisanello, G.Leménager, L.Martiradonna, L.Carbone, S.Vezzoli, P.Desfonds, P.D.Cozzoli, J.-P.Hermier, E.Giacobino, R.Cingolani, M.De Vittorio, and A.Bramati, Non-blinking single-photon generation with anisotropic colloidal nanocrystals: Towards room-temperature, efficient, colloidal quantum sources. Adv. Mater. 25, 1974 (2013).
  * 50 P.Michler, A.Imamo$\breve{\mbox{g}}$lu, M.D.Mason, P.J.Carson, G.F.Strouse, and S.K.Buratto, Quantum correlation among photons from a single quantum dot at room temperature, Nature 406, 968 (2000).
  * 51 X.Ma, N.F.Hartmann, J.K.Baldwin, S.K.Doorn, and H.Htoon, Room-temperature single-photon generation from solitary dopants of carbon nanotubes. Nature Nanotech. 10, 671 (2015).
  * 52 C.Chakraborty, L.Kinnischtzke, K.M.Goodfellow, R.Beams, and A.N.Vamivakas, Voltage-controlled quantum light from an atomically thin semiconductor. Nature Nanotechnology 10, 507 (2015).
  * 53 Y.-M.He, G.Clark, J.R.Schaibley, Y.He, M.-C.Chen, Y.-J.Wei, X.Ding, Q.Zhang, W.Yao, X.Xu, C.-Y.Lu, and J.-W.Pan, Single quantum emitters in monolayer semiconductors, Nature Nanotechnology 10, 497 (2015).
  * 54 M.Koperski, K.Nogajewski, A.Arora, V.Cherkez, P.Mallet, J.-Y.Veuillen, J.Marcus, P.Kossacki, and M.Potemski, Single photon emitters in exfoliated WSe2 structures, Nature Nanotechnology 10, 503 (2015).
  * 55 A.Srivastava, M.Sidler, A.V.Allain, D.S.Lembke, A.Kis, and A.Imamo$\breve{\mbox{g}}$lu, Optically active quantum dots in monolayer WSe2, Nature Nanotechnology 10, 491 (2015).
  * 56 T.T.Tran, K.Bray, M.J.Ford, M.Toth, and I.Aharonovich, Quantum emission from hexagonal boron nitride monolayers, Nature Nanotechnology 11, 37 (2016).
  * 57 A.Dietrich, M.W.Doherty, I.Aharonovich, and A.Kubanek, Solid-state single photon source with Fourier transform limited lines at room temperature, Phys. Rev. B 101, 081401(R) (2020).
  * 58 J.L.O${}^{{}^{\prime}}$Brien, A.Furusawa, and J.Vu$\check{\mbox{c}}$ković, Photonic quantum technologies. Nature Photon. 3, 687 (2009).
  * 59 S.I.Bogdanov, M.Y.Shalaginov, A.S.Lagutchev, C.-C Chiang, D.Shah, A.S.Baburin, I.A.Ryzhikov, I.A.Rodionov, A.V.Kildishev, A.Boltasseva, and V.M.Shalaev, Ultrabright room-temperature sub-nanosecond emission from single NV-centers coupled to nanopatch antennas, Nano Lett. 18, 4837 (2018).
  * 60 R.Löfgren, S.Öberg, and J.A.Larsson, The diamond NV-center transition energies in the vicinity of an intrinsic stacking fault, AIP Advances 12, 035009 (2022).
  * 61 R.J.Cook, Photon number statistics in resonance fluorescence, Phys. Rev. A 23, 1243 (1981).
  * 62 S.Bogdanov, M.Y.Shalaginov, A.Boltasseva, and V.M.Shalaev, Material platforms for integrated quantum photonics, Optical Mater. Express 7, 111 (2017).
  * 63 I.V.Bondarev and B.Vlahovich, Optical absorption by atomically doped carbon nanotubes, Phys. Rev. B 74, 073401 (2006).
  * 64 I.V.Bondarev and Ph.Lambin, van der Waals coupling in atomically doped carbon nanotubes, Phys. Rev. B 72, 035451 (2005).
  * 65 W.Vogel and D.-G.Welsch, Quantum Optics (Wiley-VCH, New York, 2006).
  * 66 R.P.Feynman, F.L.Vernon, and R.W.Hellwarth, Geometrical representation of the Schrödinger equation for solving maser problems, J. Appl. Phys. 28, 49 (1957).
  * 67 K.Bloom, Density Matrix Theory and Applications (Springer-Verlag, Berlin, 2012).
  * 68 I.V.Bondarev, Plasmon enhanced Raman scattering effect for an atom near a carbon nanotube, Optics Express 23, 3971 (2015).