# Room temperature single-photon superfluorescence from a single epitaxial
cuboid nano-heterostructure

John P. Philbin  Harvard John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA<EMAIL_ADDRESS>Joseph Kelly  Department of Chemistry, Stanford University, Stanford, CA
94305, USA  Lintao Peng  Center for Nanoscale Materials, Argonne National
Laboratory, Lemont, IL 60439, USA  Igor Coropceanu  Department of Chemistry
and James Franck Institute, University of Chicago, IL 60637, USA  Abhijit
Hazarika  Department of Chemistry and James Franck Institute, University of
Chicago, IL 60637, USA  Dmitri V. Talapin  Center for Nanoscale Materials,
Argonne National Laboratory, Lemont, IL 60439, USA  Eran Rabani  Department of
Chemistry, University of California, Berkeley, CA 94720, USA  Xuedan Ma
Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL 60439,
USA<EMAIL_ADDRESS>Prineha Narang  Harvard John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
<EMAIL_ADDRESS>
Department of Chemistry and James Franck Institute, University of Chicago, IL
60637, USA Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA The Sackler Center for Computational Molecular and
Materials Science, Tel Aviv University, Tel Aviv 69978, Israel

Single-photon superradiance can emerge when a collection of identical emitters
are spatially separated by distances much less than the wavelength of the
light they emit, and is characterized by the formation of a superradiant state
that spontaneously emits light with a rate that scales linearly with the
number of emitters.1, 2, 3, 4 This collective phenomena has only been
demonstrated in a few nanomaterial systems, all requiring temperatures below
$10$ K.5, 6, 7, 8 Here, we rationally design a single colloidal nanomaterial
that hosts multiple (nearly) identical emitters that are impervious to the
fluctuations which typically inhibit room temperature superradiance in other
systems such as molecular aggregates.9 Specifically, by combining molecular
dynamics, atomistic electronic structure calculations, and model Hamiltonian
methods, we show that the faces of a heterostructure “nanocuboid” mimic
individual quasi-2D nanoplatelets 10, 11 and can serve as the robust emitters
required to realize superradiant phenomena at room temperature. Leveraging
layer-by-layer colloidal growth techniques 12, 13 to synthesize a nanocuboid,
we demonstrate single-photon superfluorescence _via_ single-particle time-
resolved photoluminescence measurements at room temperature. This robust
observation of both superradiant and subradiant states in single nanocuboids
opens the door to ultrafast single-photon emitters and provides an avenue to
entangled multi-photon states14, 15 _via_ superradiant cascades.2, 16

Studies of superradiance provide many fundamental insights into the many-body
physics of photons, atoms, molecules, and nanomaterials,17, 18, 4, 19, 9, 8,
20, 21 and lay foundation for technological applications based on both
superradiance and subradiance.22, 23, 24 These applications range from
nanomaterial-based lasers 25 and optical mirrors 24 to quantum-optical
networks 22 and quantum information processing.26, 27, 28 This _Article_
presents how rational design combined with state-of-the-art synthetic
techniques can accelerate the emission rates of colloidal nanomaterials by
nearly an order of magnitude _via_ superradiance and, arguably more
importantly, introduces nanocuboids as a platform to investigate the unique
interplay of quantum optics, quantum confinement, and many-body physics at
room temperature.

The observation of superradiant phenomena from a single colloidal nanomaterial
is challenging even at low temperatures, requiring multiple identical quantum
emitters with large oscillator strengths to be spatially separated by
distances on the nanometer scale.20, 29 To address these requirements, we
specially-designed an epitaxial nano-heterostructure, termed a nanocuboid, to
contain multiple quasi-2D nanoplatelets, each with the electronic structure of
a quantum well, separated only by a few nanometers. We describe these
nanocuboids as a set of coupled nanoplatelets (similar to coupled quantum
wells), but they are actually much more complex objects. However, prior to
discussing the specifics of our nanocuboids, it is worthwhile to analyze the
superradiant and subradiant properties of an idealized nanocuboid, where each
face hosts two degenerate excited states localized to its center. We assume
that the excited states are optically active with transition electric dipoles
of equal magnitude polarized in the plane of each face 30 (e.g. the $xy$-faces
have $x$ and $y$ polarized excitations). This gives a total of twelve excited
states (four with $x$-polarizations, four with $y$-polarizations, and four
with $z$-polarizations) for a single nanocuboid, as shown in Fig. 1a.

Figure Fig. 1: (a) The schematic shows a nanocuboid along with arrows
displaying the emission polarization of the two lowest energy, optically
bright excited states located along each face of the nanocuboid. The colors of
the arrows indicate the direction of the emission polarization. The
superradiant and subradiant states inherent to a nanocuboid arise from the
constructive and destructive interference, respectively, of the transition
electric dipole moments between the excited states of the different faces of
the nanocuboid, as shown for the $x$-polarized transitions in the schematic.
(b) Low-magnification bright field TEM (left) and HAADF-STEM (middle) images
are shown alongside a schematic of our nanocuboid in which the active parts (4
monolayers of CdSe, 4CdSe) of the nanocuboid, including the initial 4CdSe
seed, are shown in red and the interior and exterior insulating, passivating
layers (CdS and ZnS) are shown in gray. The labels are used to identify the
$7$ embedded 4CdSe nanoplatelets in the nanocuboid (c) Low-magnification
bright field TEM (left) and HAADF-STEM (middle) images are shown alongside a
schematic of our single core/shell nanoplatelet in which the active seed
(4CdSe) is shown in red and the insulating, passivating layers are shown in
gray. The label is used to identify the $1$ embedded 4CdSe nanoplatelet in the
core/shell nanoplatelet. The color codes for the HAADF-STEM images in (b) are:
Red = Se, Yellow = S, Cyan = Cd/Zn and in (c) are: Red = Se, Yellow = S, Cyan
= Cd, Purple = Zn.

In order to understand the superradiant and subradiant properties of
nanocuboids, we invoke a simple and general model Hamiltonian ($H$) to
describe its excited states:2, 9

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i\alpha}E_{i\alpha}a_{i\alpha}^{\dagger}a_{i\alpha}+\sum_{i\alpha,j\beta}J_{i\alpha,j\beta}\left(a_{i\alpha}^{\dagger}a_{j\beta}+a_{j\beta}^{\dagger}a_{i\alpha}\right).$
(1)

In equation (1), $i,j,...$ are indices going over all faces of the nanocuboid,
$\alpha,\beta,...$ are indices indicating the direction of the transition
electric dipole of the excited state, $E_{i\alpha}$ is the energy of the
excited state, $a_{i\alpha}^{\dagger}$ ($a_{i\alpha}$) is a creation
(annihilation) operator for the excitonic state $i\alpha$, and
$J_{i\alpha,j\beta}$ is the dipole-coupling interaction energy:

$\displaystyle J_{i\alpha,j\beta}$ $\displaystyle=$
$\displaystyle\frac{\left|\vec{\mu}_{i\alpha}\right|\left|\vec{\mu}_{j\beta}\right|}{4\pi\epsilon_{0}\epsilon_{r}\left|\vec{r}_{i}-\vec{r}_{j}\right|^{3}}\left[\hat{\mu}_{i\alpha}\cdot\hat{\mu}_{j\beta}-3\left(\hat{\mu}_{i\alpha}\cdot\hat{r}_{ij}\right)\left(\hat{\mu}_{j\beta}\cdot\hat{r}_{ij}\right)\right].$
(2)

In equation (2), $\epsilon_{r}$ is the relative permittivity of the host
material, $\vec{\mu}_{i\alpha}$ is the transition electric dipole moment,
$\vec{r}_{i}$ is the position of the excited state, $\vec{r}_{ij}$ is the
vector connecting sites $i$ and $j$, and $\hat{\mu}_{i\alpha}$ and
$\hat{r}_{ij}$ are unit vectors. Refer to the Supplementary Information for a
discussion of the approximations involved in this model Hamiltonian.

Next we compare the eigenstates predicted by equation (1) for the ideal
nanocube described above. Upon diagonalization of equation (1) within the
singly excited subspace for this idealized nanocube, three superradiant states
are observed, corresponding to the fully symmetric combination of each set of
four identical transitions.31 These three superradiant states have radiative
decay rates that are $4$ times faster than that of a single emitter (i.e.
solving equation (1) with just a single site such as a single core/shell
nanoplatelet). Accompanying these three superradiant states are $9$ subradiant
states that cannot radiatively decay. Fig. 1a contains visual representations
of the one superradiant state and three subradiant states built from linear
combinations of the $x$-polarized transitions. Having established the general
superradiant capabilities of the surface of a nanocuboid, we will now turn to
the specific heterostructure nanocuboid for which we realized single-photon
superfluorescence at room temperature.

We designed a nanocuboid by epitaxially embedding multiple quasi-2D CdSe
nanoplatelets (NPLs) along the surface of a nanocuboid of insulating CdS and
ZnS layers. The nanocuboids shown in Fig. 1b overall dimensions ($L_{x}\times
L_{y}\times L_{z}$) of approximately $23\times 10\times 7$ nm3. Embedded NPLs
are a prudent choice for the emitting material for a few reasons. First, NPLs
are the fastest fluorescent emitter of any colloidal nanomaterial11 due to the
giant oscillator strength inherent to two-dimensional materials.32 Second, the
emission energy and polarization can be precisely tuned by controlling the
thickness and lateral dimensions, respectively.10, 33 Specifically, the
atomically precise control of the NPL thickness results in very strong
confinement of a single dimension which causes the emission energy to increase
upon increasing the confinement by synthesizing thinner NPLs. NPLs with
thicknesses ranging from three to eleven metal layers (MLs) separated by
chalcogen atoms can be directly synthesized,34, 35 and many more
nanostructures can be prepared by the methods of colloidal Atomic Layer
Deposition (c-ALD).12, 13 For the case of equal (i.e. square) lateral
dimensions, NPLs emit plane-polarized light.36, 37 However, synthetic
protocols can yield rectangular NPLs that can behave similar to quasi-1D
nanorods and preferentially emit linearly polarized light.36, 38 Therefore,
NPLs are a unique material because both the energies and polarizations of
their excited states can be precisely controlled. This control is vital to the
success of realizing superradiant phenomena in multi-NPL colloidal
nanomaterials such as the nanocuboids described herein, because superradiance
relies on independent and identical emitters interacting with the same photon
mode as defined by the energy and polarization of the photon mode.2 The final
reason is that NPLs embedded within an epitaxial structure are resilient to
the fluctuations that typically inhibit room temperature superradiance in
molecular systems. In particular, the electron-phonon coupling is more than an
order of magnitude weaker in NPLs than in molecular systems, as evidenced by
the Stokes shift being on the order of $10$ meV in NPLs and greater than $100$
meV in molecules.9 Additionally, epitaxially embedding the NPLs removes the
possibility of misorientation of the transition electric dipoles between
neighboring emitters that is known to hinder superradiance in molecular
aggregates.9

For the aforementioned reasons, embedded NPLs are prime candidates for
realizing superradiant phenomena at room temperature. The challenge was, thus,
to design a synthetic protocol to yield a single nanostructure containing
multiple NPLs. We addressed this challenge by utilizing a recently developed
layer by layer colloidal growth technique, colloidal Atomic Layer Deposition
(c-ALD),12, 13 to synthesize a nanocuboid for which each face of the
nanocuboid acts as an independent NPL, mimicking the nanocuboid shown in Fig.
1a. The synthesis of the core/shell NPLs and nanocuboids shown in Fig. 1
starts with the synthesis of a colloidal NPL seed (a $4$ ML thick CdSe NPL
with lateral dimensions of $17\text{ x }4\text{ nm${}^{2}$}$). Next, layer by
layer growth was performed via c-ALD to grow passivating shell layers in the
case of the core/shell NPL and, in the case of the nanocuboid, passivating
layers followed by $4$ MLs of CdSe followed by more passivating layers. The
insulating, passivating layers are important for improving the optical
efficiencies of colloidal nanomaterials39 but also introduce strain that can
influence the optical properties,40, 41 as further discussed below. The final
core/shell NPLs and nanocuboids are denoted 4CdSe-2CdS-2ZnS-2CdS and
4CdSe-1CdS-3CdZnS-1CdS-4CdSe-1CdS, respectively, where the number prior to the
II-VI pair indicates the number of layers grown and the layers are ordered
from inner to outer layers (e.g. both nanostructures began with a $4$ ML CdSe
core NPL and ended with CdS shell layers). Fig. 1 shows the atomistic
structures and the structural characterizations of the synthesized core/shell
NPLs and nanocuboids. The set of embedded NPLs in the nanocuboids resemble a
vertical stack of three quantum wells when considering just the top ($T$),
bottom ($B$), and center ($C$) embedded NPLs. However, there is lateral
confinement and additional embedded NPLs, labelled in ($I$), out ($O$), left
($L$), and right ($R$) in Fig. 1b, present in our synthesized nanocuboids that
are not present in stacks of three quantum wells. The dimensions of each
embedded NPL is given in Supplementary Table S3.

Importantly, the parameters ($E_{i\alpha},\leavevmode\nobreak\
\vec{\mu}_{i\alpha},\leavevmode\nobreak\ \vec{r}_{i}$) in equations (1) and
(2) for the specific core/shell NPL ($i\in\\{C\\}$) and nanocuboid
($i\in\\{C,T,B,I,O,L,R\\}$) shown in Fig. 1 can be obtained from atomistic
electronic structure calculations, structural characterization techniques, and
known scaling laws for NPLs. To this end, we utilized molecular dynamics based
structural minimization of the final heterostructure nanomaterials to study
the strain and atomistic structure of these materials.42, 43 Additionally, a
semi-empirical pseudopotential Hamiltonian44, 45 and the Bethe–Salpeter
equation46 with a static dielectric constant of $\epsilon=5$ were solved to
understand the excited states in the core/shell NPL and nanocuboids. These
computational techniques have recently been successfully applied to
understanding both radiative 30, 13 and nonradiative 47, 48 processes of
excitons in colloidal nanomaterials. (See the Supplementary Information for
more details on the synthetic protocol and computational methods.)

The low energy excitonic states relevant to emission from the core/shell NPLs
(4CdSe-2CdS-2ZnS-2CdS) can be modelled and understood using a simple few-level
system, ignoring the exciton fine structure and associated dark states. In
Supplementary Table S4, we explicitly show how almost all of the oscillator
strength within the low energy, thermally accessible excitonic states of a
single NPL is concentrated within a few ($<10$) excitonic states. Furthermore,
there are often only $2$ or $4$ states that contain a majority of this
oscillator strength depending on whether the NPL is rectangular or square,
respectively. In both cases, all of the transition electric dipole moments of
these few low energy bright excited states are orthogonal to the confined
axis, just as in each face of the nanocuboid shown in Fig. 1a. The magnitude
of the transition electric dipole moments ($\left|\vec{\mu}_{\alpha}\right|$)
and the energies ($E_{\alpha}$) of the few important excited states depend on
the lateral dimensions of the NPL. The longer lateral dimension will be
slightly lower in energy and have a larger transition electric dipole moment
(as shown in detail in the Supplementary Information). The 4CdSe NPLs within
the core/shell NPL (4CdSe-2CdS-2ZnS-2CdS) have lateral dimensions of
$L_{x}=17\text{ nm}$ and $L_{y}=4\text{ nm}$. Thus, the $x$-polarized excited
states will be lower in energy and contain more of the oscillator strength
than the $y$-polarized excited states (Supplementary Table S4). Fig. 2a
contains a schematic of the simplest three-level system that can be used to
model the core/shell NPL in this work, and these three-level systems and their
extension to few-level ($<10$) systems will be used to model the excited
states of the faces of the nanocuboids which will then couple via equation (2)
to form superradiant states.

Figure Fig. 2: Strain profiles in the 4CdSe parts of the nanoplatelet (a,
4CdSe-2CdS-2ZnS-2CdS) and each face of the nanocuboid (b-d,
4CdSe-1CdS-3CdZnS-1CdS-4CdSe-1CdS) are displayed. The set of three plots for
each face correspond to the percent change of the CdSe lattice constant ($a$)
in the three cardinal directions relative to the ideal projected bond lengths.
The percent change of the CdSe lattice constant along the $x$-dimension
($\Delta a_{x}$) is shown on top, $y$-dimension ($\Delta a_{y}$) in the
middle, and $z$-dimension ($\Delta a_{z}$) on the bottom. In all cases, the
lattice compresses along the large, weakly confined dimensions and expands in
the small, confined dimension, demonstrating the biaxial strain in these
heterostructure nanomaterials. (e,f) Wavefunctions for the lowest energy hole
states are shown for (e) a core/shell nanoplatelet (4CdSe-2CdS-2ZnS-2CdS) with
4CdSe lateral dimensions of $17\times 4$ nm2 and (f) a nanocuboid
(4CdSe-1CdS-3ZnS-1CdS-4CdSe-1CdS) with an initial 4CdSe seed with lateral
dimensions $4\times 4$ nm2. These wavefunctions are shown to exemplify how the
low energy hole states are localized to the 4CdSe parts of the nanocuboids and
resemble the states in a single nanoplatelet due to quantum confinement,
strain, and band alignment effects.

The connection between the 4CdSe NPL in the core/shell NPL and the faces of
the nanocuboid is given credence by noting the similar strain profiles and
hole wavefunctions between the embedded 4CdSe NPL in the core/shell NPL and
the 4CdSe faces of the nanocuboid shown in Fig. 2. In particular, biaxial
strain, in which the confined dimension expands and the lateral dimensions
shrink due to the smaller lattice constants of the CdS and ZnS being grown
around the active 4CdSe NPLs,13 is seen in both the core/shell NPL and each
face of the nanocuboid. The mapping of each face of the nanocuboid to NPL is
further supported by noting how the low energy excited states are localized to
the 4CdSe faces of the nanocuboid. This effect is demonstrated by the hole
wavefunctions in Fig. 2e,f. It is worth noting that in this atomistic
electronic structure calculation, the initial center 4CdSe seed in the
nanocuboid (4CdSe-1CdS-3ZnS-1CdS-4CdSe-1CdS) has lateral dimensions of only
$4\times 4$ nm2 in order to make the calculation for the entire nanocuboid
computationally tractable. The similar strain profiles and band alignment
(i.e. confinement) induced excited state localization to the 4CdSe parts of a
nanocuboid is key to the mapping of excited states of the nanocuboids to
equations (1) and (2) as it allows us to reasonably define the $\vec{r}_{i}$
as the centers of each 4CdSe face of the nanocuboid and the 4CdSe initial
center seed.

Another important finding is that within a single face of the nanocuboid, the
excitons are delocalized, which is significant for the radiative decay of
excitons in NPLs as it, together with the small average electron-hole
separations in CdSe NPLs,30 is responsible for the giant oscillator strengths
of NPLs. Mathematically, the radiative decay rate ($k_{\text{rad}}$) of an NPL
is proportional to the ratio of the exciton coherence area ($A_{\text{exc}}$)
to the square of the exciton Bohr radius ($a_{\text{B,exc}}$,
$k_{\text{rad}}\propto A_{\text{exc}}/a_{\text{B,exc}}^{2}$).32, 49, 50
Because the lateral dimensions of all NPLs (i.e. the faces of the nanocuboid)
are much smaller than the exciton mean free path in these systems even at room
temperature,51 the exciton coherence area within an NPL is simply proportional
to the area of the NPL ($A_{\text{exc}}\propto A_{\text{NPL}}$). Furthermore,
the linear proportionality of the radiative decay rate of an excited state to
the lateral area of the face of the nanocuboid is also consistent with a
transition electric dipole moment that scales as
$k_{\text{rad}}\propto\left|\mu\right|^{2}$.49 Our electronic structure
calculations of 4CdSe NPLs reported in the Supplementary Information also
support this scaling.

In terms of equations (1) and (2), the aforementioned results from the
molecular dynamics and electronic structure calculations can be summarized in
three main points. First, strain and confinement induce exciton localization
to 4CdSe of the nanocuboid such that we can reasonably define the positions
($\vec{r}_{i}$) as the center of the six faces of the nanocuboid and the
starting center 4CdSe NPL seed to give seven total sites
($i\in\\{C,T,B,I,O,L,R\\}$). This demonstrates the connection to the idealized
nanocube for which we already detailed its superradiant potential. Second,
there are a few ($<10$) low energy excitonic states with large transition
electric dipole moments ($\vec{\mu}_{i\alpha}$) in each embedded NPL and the
magnitude of the transition dipoles are proportional to the lateral dimensions
of the NPL (Supplementary Table S4).32 And, third, the rigid structure imposed
by embedding the 4CdSe NPLs in CdS and ZnS layers results in the geometrical
relationship between the transition dipoles of differing faces of the
nanocuboid to be constant (i.e. insensitive to thermal fluctuations unlike in
neighboring molecules in molecular aggregates). Thus, there is no need to
account for fluctuations in the orientation of the transition dipoles when
modelling the superradiant properties of the nanocuboids at room temperature.
However, disorder in the onsite energies (${\delta}E_{i\alpha}$ where
$E_{i\alpha}=E+{\delta}E_{i\alpha}$) can play a central role, as further
discussed below.

Figure Fig. 3: (a) The ratio of the fastest radiative lifetime for a
nanocuboid (${\tau}_{\text{rad}}$) to that of a single nanoplatelet
(${\tau}_{\text{rad}}^{(0)}$) is shown as a function of the level degeneracy
and the ratio of the reorganization energy ($\lambda$) to the magnitude of the
dipole-dipole coupling strength ($\left|J\right|$). Each pixel represents a
different system and $100$ samples were averaged over for each pixel (see the
Supplementary Information for more details). The radiative lifetimes of the
nanocuboids are shortest for small reorganization energies, defining a
“superradiant regime.” As the reorganization energy increases, a continuous
transition to Anderson-type localization occurs, and the radiative lifetimes
of the nanocuboids begin to resemble those of single nanoplatelets. Increasing
the number of levels per site (i.e. level degeneracy) makes the superradiant
states more resilient to larger energetic fluctuations. Lastly, the location
of our synthesized nanocuboids is shown by the gray circle, clearly falling
within the “superradiant regime.” (b) A visual representation of the
superradiant state of our synthesized nanocuboid is shown. The superradiant
state is a delocalized excited state in which the $x$-polarized transition
electric dipoles of the $C,T,B,I,O$ embedded 4CdSe NPLs constructively
interfere.

To this end, we calculated the radiative decay rates of the excitonic states
of nanocubes with various degrees of energetic disorder
(${\delta}E_{i\alpha}$) and number of excited state levels per face of the
nanocube. The results of these calculations are shown in Fig. 3a. The color of
each pixel in Fig. 3a represents the ratio of the shortest radiative lifetime
(averaged over $100$ samples) for a nanocube to that of a NPL with their
excited state energies ($E_{i\alpha}$) sampled from a Gaussian distribution
with an average energy of $2\text{ eV}$ and a standard deviation equal to the
reorganization energy ($\lambda$) and with a fixed dipole-dipole coupling
magnitude ($\left|J\right|$) in the nanocubes. The diagram in Fig. 3a shows
that increasing the disorder in the onsite energies (modelled by increasing
$\lambda$) leads to longer radiative lifetimes for a nanocube as a result of
Anderson-type localization of the excited states to a single face of the
nanocuboid and thus, the radiative lifetimes just resemble those of the
individual NPLs (recall that each face of the nanocuboid is similar to an
individual NPL). In other words, the delocalized, superradiant states are
destroyed by the presence of large energetic disorder (Fig. 3a). That being
said, increasing the number of excited state levels per face of the nanocuboid
stabilizes the superradiant regime, as demonstrated in Fig. 3a by the decrease
in radiative lifetimes as a function of the number of levels per face.

Next, we diagonalize the Hamiltonian given by equation (1) using the
parameters ($E_{i\alpha},\leavevmode\nobreak\
\vec{\mu}_{i\alpha},\leavevmode\nobreak\ \vec{r}_{i}$) for our specific
nanocuboids (4CdSe-1CdS-3CdZnS-1CdS-4CdSe-1CdS) and specific core/shell NPLs
(4CdSe-2CdS-2ZnS-2CdS) in order to make predictions on the superradiant
properties of these specific structures. Focusing on the single excitation
subspace of the Hamiltonian, clear signatures of both superradiant (Fig. 3b)
and subradiant states are observed (Supplementary Table S10). Specifically,
singly excited states are found that have radiative decay rates that are
faster (i.e. superradiant) and slower (i.e. subradiant) in the nanocuboid
relative to the core/shell NPL. Quantitatively, there is always a superradiant
eigenstate that has a radiative decay that is approximately $5$ times faster
than that of the core/shell NPL even when disorder in the onsite energies is
taken into account. This signature of single-photon superradiance is
highlighted by the gray circle in Fig. 3a, where enhancement of the radiative
decay rate is primarily due to the superradiant effects that are not destroyed
by the onsite disorder and are further promoted by the large number of
excitonic states per facet. Fig. 3b shows a visual representation of the
superradiant eigenstate of our nanocuboid that will be responsible for the
short radiative lifetime of our nanocuboid compared to our core/shell NPL.
There are also subradiant eigenstates that are either completely or nearly
completely dark with radiative decay rates that are $>5$ times slower than
that of the core/shell NPL (Supplementary Table S10). The ideal separation
between superradiant and subradiant states discussed above for the idealized
nanocube is broken by the disorder (${\delta}E_{i\alpha}$) and the
inequivalent positions ($\vec{r}_{i}$) that results in inhomogeneous dipole-
dipole coupling ($J_{i\alpha,j\beta}$) between the different 4CdSe NPL faces
(Supplementary Table S3 and Supplementary Fig. S11).

In order to test the predictions of superradiant and subradiant eigenstates in
nanocuboids, we performed time-resolved photoluminescence experiments on the
single-particle level as shown schematically in Fig. 4a. Fig. 4 also shows
ensemble absorption and emission spectra (Fig. 4b) along with representative
single-particle blinking (Fig. 4c) and time-resolved photoluminescence (Fig.
4d) spectra for the core/shell NPLs (4CdSe-2CdS-2ZnS-2CdS) and the nanocuboids
(4CdSe-1CdS-3CdZnS-1CdS-4CdSe-1CdS) in blue and red, respectively. We
extracted the reorganization energy ($\lambda$) from the Stokes shift of
approximately $20$ meV seen in Fig. 4b, which was used as a metric when
accounting for the magnitude of the energetic disorder in the final model of
our nanocuboids shown in Fig. 3. The single-particle blinking spectra show the
typical blinking behavior of colloidal nanomaterials.52 Because time-resolved
photoluminescence spectra have contributions from both radiative and
nonradiative components, we extract radiative lifetimes from just the highly
emissive part of the blinking spectrum,53, 54, 51 as depicted in Fig. 4c,d.

Figure Fig. 4: (a) A schematic representation of superfluorescence from a
nanocuboid. The non-resonant optical excitation that initiates
superfluorescence is captured by the blue light shining on a nanocuboid in the
top left. The bottom right then shows the symmetric, superradiant excitonic
states emitting red light at a future time. The superradiant states are
represented by the red ovals along each face of the nanocuboid. (b) Ensemble
absorption and emission spectra for the core/shell nanoplatelets (blue lines)
and nanocuboids (red lines). (c) Representative emission time trajectory of a
core/shell nanoplatelet (top in blue) and nanocuboid (bottom in red) showing
the typical blinking pattern of colloidal nanomaterials. (d) Single-particle
time-resolved photoluminescence spectrum extracted from the highly luminescent
part of the emission time trajectory. The thick lines are best fits to a
single exponential for the nanoplatelet and a bi-exponential for the
nanocuboid. The single exponential (thick blue line) has a lifetime of
$11.6\text{ ns}$ and the bi-exponential (thick red line) has a fast component
with a lifetime of $1.8\text{ ns}$ and a slow component with a lifetime of
$67.2\text{ ns}$.

We find that a single exponential with a radiative decay lifetime of
$11.6\text{ ns}$ fits the photoluminescence decay for the core/shell NPL in
Fig. 4d very well, in agreement with previous room temperature radiative
lifetime measurements of core/shell NPLs.55 For the nanocuboid, a bi-
exponential function is needed to fit the photoluminescence decay, and there
is a clear separation of time scales in the two exponentials. The fast,
superradiant component in Fig. 4d has a lifetime of $1.8\text{ ns}$ whereas
the slow, subradiant component in Fig. 4d has a lifetime of $67.2\text{ ns}$.
Recall that the diagonalization of equation (1) using the parameters derived
from calculations of these exact nanocuboids predicted both superradiant and
subradiant excited states with radiative decay rates approximately $5$ times
faster (Fig. 3b) and $5$ times slower than that of the radiative decay rates
of the core/shell NPL. These predictions are in excellent quantitative
agreement with the measured lifetimes. Importantly, these radiative lifetimes
are reproducible between samples; the Supplementary Information contains the
sample to sample variations and demonstrates that the traces presented in Fig.
4d are an accurate representation of all the single-particle fluorescent data
obtained (Supplementary Fig. S5 and Fig. S6).

An aspect of these measurements that is worth highlighting is that they were
performed at room temperature, making nanocuboids one of the first, if not the
first, superfluorescent single nanomaterial at room temperature ($298\text{
K}$). Two more subtle, but very interesting, findings from these experiments
is that the observation of both superradiant and subradiant states in the
emission dynamics of the nanocuboids suggests that both dynamical dephasing
between the faces of the nanocuboid (which was not accounted for in this work)
and phonon-mediated transitions between the superradiant and subradiant states
are slow, occurring on the nanosecond timescale. Future theoretical and
experimental studies plan to address the mechanism(s) behind these findings.

A brief summary of the properties of the nanocuboids reported herein that were
critical to the realization of room temperature superfluorescence from a
single nanostructure is warranted. The multi-level nature of the excited
states of each emitter (i.e. 4CdSe face) greatly aided in satisfying the
energetic resonance conditions between faces required to observe superradiant
phenomena. The combination of these many near resonances with the large
transition electric dipoles inherent to NPLs and the rigidity of the
nanocuboids which allowed us to neglect polarization fluctuations resulted in
the delocalized, superradiant excited states in the nanocuboids being stable
to the fluctuations inherent to room temperature experiments.

To conclude, herein we introduced nanocuboids, outlined their superradiant
properties, and realized single-photon superfluorescence _via_ single-particle
time-resolved photoluminescence. Nanocuboids are a promising colloidal
nanomaterial due to the state-of-the-art atomic level precision of their
synthesis afforded by colloidal atomic layer deposition and the unique
interplay of strain and quantum confinement induced exciton localization
causing each face of the nanocuboid to act as a nearly identical emitter with
large transition electric dipoles of fixed polarization because of the
quasi-2D nature of the NPL faces. This localization combined with the non-
negligible dipole-dipole coupling between the emitting sites that are spaced
only a few nanometers apart results in both superradiant and subradiant states
that can be observed _via_ single-particle time-resolved photoluminescence
experiments at room temperature.

We envisage future investigations of nanocuboids under higher optical
excitation densities might be able to realize entangled photon-pair
generation14, 15 _via_ emission cascades.2, 16 Furthermore, the energy of the
emitted photons should be easily tunable by changing the thickness of the
emitting NPLs,11, 34, 35 which is a clear advantage of choosing a quasi-2D
material as the emitter. This demonstration of room temperature
superfluorescence from a single colloidal nanomaterial is a powerful example
of how colloidal synthesis and atomistic modelling can be combined to
rationally design new nanomaterials with entirely new quantum optoelectronic
functionality, and as a platform to explore previously inaccessible regimes of
quantum many-body physics.

##  1 Data Availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

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J.P.P., E.R., and P.N. acknowledge helpful feedback from Eli Yablonovitch and
Wenjie Dou. This work was primarily supported by the Department of Energy,
Photonics at Thermodynamic Limits Energy Frontier Research Center, under Grant
No. DE-SC0019140. This work was partially supported by the Army Research
Office MURI (Ab-Initio Solid-State Quantum Materials) Grant No.
W911NF-18-1-0431 that supported development of new computational methods. The
atomistic calculations used resources supported by the Center for
Computational Study of Excited State Phenomena in Energy Materials (C2SEPEM),
which is funded by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division via Contract No.
DE-AC02-05CH11231, as part of the Computational Materials Sciences Program and
the National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility operated under Contract
No. DE-AC02-05CH11231. The work on nanocuboid synthesis was supported by the
Department of Defense (DOD) Air Force Office of Scientific Research under
grant number FA9550-18-1-0099 and by NSF under award number CHE-1905290. This
work was performed, in part, at the Center for Nanoscale Materials, a U.S.
Department of Energy Office of Science User Facility, and supported by the
U.S. Department of Energy, Office of Science, under Contract No. DE-
AC02-06CH11357. X.M. acknowledges support from the Center for Molecular
Quantum Transduction (CMQT), an Energy Frontier Research Center funded by the
U.S. Department of Energy (DOE), Office of Science, Basic Energy Science
(BES). J.P.P. is a Ziff Fellow at the Harvard University Center for the
Environment. P.N. is a Moore Inventor Fellow through Grant GBMF8048 from the
Gordon and Betty Moore Foundation.

##  2 Author Contributions

The project was jointly conceived by J.P.P., D.V.T., E.R., X.M., and P.N.
J.P.P. led the theory and calculations, under the guidance of E.R. and P.N.
J.K. performed the molecular dynamics calculations of the core/shell
nanoplatelet and nanocuboid, under the guidance of E.R. I.C. and A.H. carried
out the experiments on materials synthesis, basic optical and structural
characterization and corresponding data analysis, under the guidance of D.V.T.
L.P. performed the optical measurements of the single nanoparticles and
analyzed the experimental data, under the guidance of X.M. All authors
participated in the data analysis and jointly wrote the manuscript.

##  3 Competing Interests

The Authors declare no Competing Financial or Non-Financial Interests.