# Quantum size effects, multiple Dirac cones and edge states in ultrathin
Bi(110) films

Asish K. Kundu<EMAIL_ADDRESS>Genda Gu  Tonica Valla<EMAIL_ADDRESS>Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA

###### Abstract

The presence of inherently strong spin-orbit coupling in bismuth, its unique
layer-dependent band topology and high carrier mobility make it an interesting
system for both fundamental studies and applications. Theoretically, it has
been suggested that strong quantum size effects should be present in the
Bi(110) films, with the possibility of Dirac fermion states in the odd-bilayer
(BL) films, originating from dangling $p_{z}$ orbitals and quantum-spin hall
(QSH) states in the even-bilayer films. However, the experimental verification
of these claims has been lacking. Here, we study the electronic structure of
Bi(110) films grown on a high-$T_{c}$ superconductor, Bi2Sr2CaCu2O8+δ (Bi2212)
using angle-resolved photoemission spectroscopy (ARPES). We observe an
oscillatory behavior of electronic structure with the film thickness and
identify the Dirac-states in the odd-bilayer films, consistent with the
theoretical predictions. In the even-bilayer films, we find another Dirac
state that was predicted to play a crucial role in the QSH effect. In the low
thickness limit, we observe several extremely one-dimensional states, probably
originating from the edge-states of Bi(110) islands. Our results provide a
much needed experimental insight into the electronic and structural properties
of Bi(110) films.

keywords: Ultrathin films, Electronic structure, Dirac cone, Edge states,
Spin-orbit coupling, Quantum size effects, Angle-resolved photoemission
spectroscopy

††preprint: APS/123-QED

##  I Introduction

In recent years, low-dimensional systems have attracted considerable
scientific interest as they often show anomalous atomic and electronic
structure and can display many exotic properties such as layer-dependent
nontrivial band topology, the appearance of Dirac fermions, QSH effects, etc.
which are often absent in their bulk form [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In
particular, group V elements (e.g., Bi and Sb) are known to show rich
allotropic transformation when grown in the form of thin-films due to their
semimetallic bonding character [11, 12, 13, 7]. Both Bi and Sb host spin-
polarized surface states, potentially applicable in spin-sources and/or
filters in the field of spintronics [3].

Depending on the substrate and the film thickness, the film surface and the
associated electronic properties can be very different [12, 6, 13, 14, 15,
16]. For example, bulk bismuth crystalizes in a rhombohedral A7 structure
(space group R-3m) with two atoms per unit cell. In bismuth films, the
hexagonal (HEX) surface, corresponding to Bi(111) is the preferred structure
for thicker films, while the pseudocubic (PC) phase, corresponding to the
(110) surface is more stable for thinner films [17, 18, 16]. According to
theoretical studies, both PC and HEX phases should show exotic surface
electronic structure with Dirac states in the interior and edge-states at step
edges, that could be topological in some circumstances [6, 2, 8, 14, 19, 20].
The electronic character of odd-bilayer Bi(110) films is expected to be very
different from Bi(111) films since one nearest neighbor bond of Bi is broken,
leaving a dangling bond [21, 6]. In the even-bilayer Bi(110) films, the
dangling bonds are removed due to the inter-bilayer covalent bonding [6].
Theoretical studies have predicted ample of interesting properties in Bi(110)
films. For example, the Dirac states originating from the surface dangling
bonds at $\bar{M}$ point of surface Brillouin zone (SBZ) are predicted in odd-
bilayer films [6] while QSH states should exist in the even-bilayer films
below a critical film thickness of $\leq$ 4 BL. The Dirac-states near $E_{F}$
could lead to unconventional optical and transport properties, such as
enhanced spin-polarized transport, while the QSH could be used in
dissipationless transport devices.

Topological edge-states, were predicted [22] and experimentally detected at
Bi(111) surfaces using scanning tunneling microscopy/spectroscopy (STM/STS)
[19, 23, 20]. Later, an ARPES study questioned the topological nature of these
states [14]. Topological edge-states have also been observed in Bi(110) films
with a black-phosphorus (BP)-like structure (no buckling within a bilayer)
grown on highly-oriented pyrolytic graphite (HOPG), and Nb(110) substrates
[24, 25, 26]. Such topological states were not observed in Bi(110) films with
a distorted BP-like structure (buckled bilayers) grown on Si(111) [13, 12] and
4H-SiC(0001) [27]. Theoretical studies suggest that atomic buckling plays a
crucial role in determining the topological character of these films [24, 2].
Below the critical value of buckling, $h=0.1$ Å and the film thickness ($\leq$
4 BL), the Bi(110) films should be nontrivial and above that, they should be
trivial [24]. However, the dispersion of the Bi(110) edge-states has never
been experimentally observed. Furthermore, the surface structure of Bi(110)
films, a crucial parameter that determines the outcome of theoretical studies,
is still debated.

Figure 1: Truncated A7-bulk structure of Bi(110) and the LEED pattern of
substrate (Bi2212) as well as Bi films. (a) Top view of Bi(110) where pseudo-
square unit cell is marked. Dotted line represent the mirror plane. (b) Side
view (parallel to the mirror plane). Within 1 BL (3.28 Å) two consecutive
atoms (different colors) share different atomic position, resulting in buckled
structure. (c) Schematic of the Bi(110) surface Brillouin zone (SBZ). (d) LEED
pattern of substrate (crystallographic $\vec{b_{s}}$ -axis is marked), (e) 3
BL, and (f) 5 BL Bi films on Bi2212, respectively. Dotted square in (d) and
(e, f) represent the reciprocal unit cell of Bi2212 and Bi(110) films,
respectively.

The Bi(110) films have been synthesized on various substrates [13, 17, 28, 29,
12, 6, 30, 31, 27, 18, 16]. It has been found that on HOPG, only the even
bilayer islands nucleate, whereas on Si(111)$(\surd 3\times\surd 3)$-B and
Bi2212 both the even and odd bilayer Bi(110) islands coexist. The experimental
studies were mostly performed by using STM, with some ARPES results, typically
lacking a systematic thickness dependence [17, 6, 8, 30, 16]. Therefore, very
important questions such as the existence of Dirac-bands in the odd-bilayer
Bi(110) films, or gapped Dirac states necessary for the edge-states, as well
as the dispersion of the latter in the even-bilayer films remain open.

To answer some of these questions, we have grown Bi(110) films in-situ on
optimally doped Bi2212 substrate and studied their layer-dependent structural
and electronic properties using low-energy electron diffraction (LEED) and
ARPES. Our results show that both the even- and odd bilayer islands grow on
Bi2212. For the first time, we provide direct evidence for the Dirac cone at
the $\bar{M}$ point of SBZ in the odd-bilayer Bi(110) films. We also observe
another Dirac state of different origins in the odd-bilayer Bi(110) films.
Finally, in the low-thickness limit, we observe a highly 1D-like state with
the dispersion that resembles the theoretically predicted dispersion of the
edge-states. Our results also provide evidence for different surface
structures of even and odd bilayer films.

##  II Results and discussions

###  II.1 Surface structure of Bi(110) films

The top and side views of Bi(110) surface with an A7 crystal structure are
shown in figure 1(a) and (b), respectively. The pseudocubic surface unit cell
is marked by a rectangle in figure 1(a), with the corresponding SBZ shown in
figure 1(c). The cleaved Bi2212 surface shows a typical diffraction pattern
with the long-range supermodulation along the $b_{S}$ direction, figure 1(d)
[32]. The LEED pattern of a film with the nominal thickness of 3 BL, figure
1(e), displays both the substrate and overlayer-Bi related spots, an
indication that films do not grow in a layer-by-layer mode, but follow a
three-dimensional (3D) island growth. The similar growth mode was also
observed for other substrates [31, 12, 18]. The observed pseudo-square LEED
pattern indicates the formation of Bi(110) films. As the lattice parameters of
Bi(110) surface ($a=4.54$ Å, $b=4.75$ Å) are smaller than the lattice
parameter of Bi2212 ($a_{S}\approx b_{S}\approx 5.4$ Å), the Bi(110)
reciprocal unit-cell is bigger than the Bi2212 one. The arc-like feature of
the second-order diffraction spots is the signature of coexisting orthogonal
domains originating from different lattice parameters, $a$ and $b$. Similar
LEED pattern was reported in previous studies of Bi/Bi2212 system [18]. The
LEED patterns indicate that the Bi-O-plane of the Bi2212 substrate and Bi(110)
films are aligned so that $a\parallel b_{S}$ and $b\parallel b_{S}$.

At a nominal coverage of $\sim$ 5 BL, the substrate spots are no longer
visible, figure 1(f). Considering the atomic structure of Bi(110) surface, the
islands are expected to propagate along the zigzag chain as Bi atoms are only
bonded along that direction, whereas the atoms in adjacent chains are not
bonded directly in plane. Therefore, during the growth, migrating atoms tend
to attach to the ends of the zigzag chains. STM studies on Bi(110) indicate
that the longer side of islands is the zigzag direction (i.e., the shorter
side of the unit-cell) [12]. If this is also true for the Bi/Bi2212 system,
the obtained LEED pattern (figure 1(e)) implies that the Bi islands grow so
that the $\bar{\Gamma}-\bar{X_{1}}$ and $\bar{\Gamma}-\bar{X_{2}}$ directions
are along the $a_{S}$ and $b_{S}$ crystallographic axes of Bi2212. Indeed, in
the previous STM study, it was apparent that islands nucleated along the two
specific orthogonal directions [18]. However, these studies could not resolve
whether the islands had BP or A7 structures, as both structures have very
similar unit cells.

###  II.2 Electronic structure of Bi(110) films

The experimental band dispersions along the $\bar{\Gamma}-\bar{M}$ direction
of Bi(110) SBZ for the nominal film coverage of $\sim 2$ BL, $\sim 3$ BL,
$\sim 3.5$ BL , $\sim 4$ BL and $\sim 5$ BL are presented in figure 2(a)-(e).
With the increasing Bi coverage, a complex evolution of electronic bands
indicates that multiple film thicknesses are present simultaneously, in
accordance with our LEED results. To understand the origin of individual
bands, the theoretical layer-dependent dispersions are reproduced from Ref.
[24] (even-bilayer) and Ref. [8] (odd-bilayer), figure 2(f)-(j). The
comparison enables the identification of spectroscopic features corresponding
to the specific film thickness. For example, in the case of even-bilayer
films, going from 4 BL to higher coverage, the electronic states marked by 1,
2, and 4 in figure 2(f)-(i) show very similar dispersions. On the other hand,
the calculations predict quite a different behavior for the odd-bilayer films,
with a linear dispersive band and Dirac cone at $\bar{M}$ (figure 2(g)-(j))
[8, 6]. Variation of spectral intensity of a particular band with the nominal
Bi coverage is related to the change in the relative area covered by the
corresponding thickness.

Figure 2: Measured (top panels) and theoretical (bottom panels) electronic
structure of Bi(110) films along the $\bar{\Gamma}$-$\bar{M}$ for various film
thicknesses. (a)-(e) Experimental band dispersions for the nominal film
thickness of 2 BL, 3 BL, 3.5 BL, 4 BL, and 5 BL, respectively. (f)-(j)
Calculated layer-dependent bands of various bilayers as indicated. Bands are
reproduced from Ref. [24] (even-bilayer) and Ref. [8] (odd-bilayer),
respectively. To match the experimental data, a rigid downward shift of 130
(90) meV was applied to the calculated bands of 4 (8) BL films. Experimental
bands marked by numbers and arrows in (a)-(e) can be identified in theoretical
calculations (f)-(j). Bands indicated by arrows are the Dirac states
originating from the dangling bonds. The bands 6 and 7 probably originate from
the 10 BL islands.

According to theory, in the 2 BL and 4 BL films (figure 2(f)) there should be
essentially no electronic states crossing the $E_{F}$ and a spectral gap
should be observed [24, 8]. Our results for the nominal film coverage of
$\sim$ 2 BL, figure 2(a), show prominent signatures corresponding to 4 BL
Bi(110) films, with some contribution from 3 BL and/or 6 BL thickness forming
a hole-like band at $\bar{\Gamma}$. The states originating from 4 BL islands
show a spectral gap along $\bar{\Gamma}-\bar{M}$, in agreement with the
calculations.

The film with a total nominal thickness of $\sim$ 3 BL, shows a few more
states appearing (figure 2(b)). Comparison with the theoretical result (figure
2(g)) indicates that the additional states are mainly originating from the 6
BL islands. The presence of a linear dispersive band, marked by an arrow in
figure 2(b), suggests an additional contribution from odd-bilayer islands,
most likely from the 3 BL and/or 5 BL islands. The exact discrimination
between them is difficult due to its extremely low intensity. Thus, the 3 BL
nominal thickness film consists of 4 BL, 5 BL (and/or 3 BL), and 6 BL Bi(110)
islands and small regions of uncovered Bi2212, as inferred from LEED (figure
1(e)). We note that there are some additional states which are visible near
$E_{F}$ (0 to $-$0.3 eV) in between $\bar{\Gamma}$ and $\bar{M}$. These states
are absent in the layer-dependent band calculations and their intensity is
strongly suppressed with further Bi deposition.

Figure 2(c) shows the electronic states for the nominal coverage of 3.5 BL
film. Now, the electronic states corresponding to 4 BL are suppressed, and the
film is dominated by 6 BL Bi(110) islands. The electronic states 2 and 3 are
well reproduced by theoretical calculations for 6 BL Bi(110) film (figure
2(h)). In addition, the presence of a less intense band 4 indicates a small
contribution from 8 BL islands. Upon further Bi deposition with a nominal
coverage $\sim$ 4 BL, the bands 4 and 5, originating from 8 BL Bi(110) islands
get sharper and well-resolved as can be seen in figure 2(d). The linearly
dispersing band near the $\bar{M}$ point also gained in intensity compared to
the 3.5 BL film. This band shows a local minimum at $\sim$ k∥=0.6 Å-1 and its
corresponding energy position of $\sim$ $-$0.57 eV matches well with the
theoretical values of 5 BL Bi(110) film (figure 2(i)). Upon further Bi
deposition, the local minimum further shifts to $\sim$ $-$0.47 eV (figure
2(e)) corresponding to the 7 BL Bi(110) film (figure 2(j)). The observed trend
would suggest that the bands marked by 6 and 7 in figure 2(e) originate from
the 10 BL islands.

###  II.3 Fermi surface of Bi(110)

Figure 3: Fermi surface and band dispersions along the high symmetry
directions of Bi(110) SBZ for bare substrate and various film thickness of Bi
films. (a)-(d) substrate (Bi2212), (e)-(h) 2 BL, (i)-(l) 3 BL, (m)-(p) 3.5 BL,
and (q)-(t) 5 BL films, respectively. For the FS plots, the photoemission
intensities are integrated within $\pm$8 meV around Fermi level. The dotted
square represents the one quadrant of the Bi(110) SBZ. Dirac-cone (DC) and its
corresponding linear dispersive band at $\bar{M}$ are marked by the arrows
(red). Electronic states corresponding to the CuO plane of Bi2212 are
indicated in (a). White color arrows in (i) and (j) indicating the 1D FS. The
dispersions of these states are indicated by arrows in (k). Calculated band
dispersions of 4 BL, 6 BL, and 8 BL Bi(110) films are reproduced from Ref.
[24] and superimposed on (g), (o) and (s), respectively. Where the red and
black dots represent calculated bands along $\bar{\Gamma}$-$\bar{X_{1}}$ and
$\bar{\Gamma}$-$\bar{X_{2}}$ directions of SBZ (figure 1(c)), respectively.

The Fermi surfaces and constant-energy contours at $-50$ meV, along with the
band dispersions for bare substrate and various film thicknesses are shown in
figure 3: (a-d) bare Bi2212, (e-h) 2 BL, (i-l) 3 BL, (m-p) 3.5 BL and (q-t) 5
BL, respectively. In figure 3(a), the states originating from the Cu-O plane
of the bare Bi2212 substrate are marked. The Fermi surface, dispersion and
superconducting gap are consistent with our previous results on optimally
doped Bi2212 [33, 34]. After the deposition of 2 BL film, the intensity of
these substrate related states is strongly suppressed whereas an intense spot
from the Bi film appears near the $\bar{X}$ point along the
$\bar{\Gamma}$-$\bar{X}$ (figure 3(e) and (f)). The $k_{F}$ position of this
spot agrees with the expected $k_{F}$ from the 4 BL Bi(110) islands [35, 24].

According to calculations, this band hosts a massive Dirac cone with a gap
size of $\sim$ 70 meV, formed by the bonding and antibonding states of
Bi-$p_{z}$ orbitals [24, 2]. In figure 3(g), the theoretical dispersions are
shown on top of the experimental data. The bands for both
$\bar{\Gamma}$-$\bar{X_{1}}$ (red dots) and $\bar{\Gamma}$-$\bar{X_{2}}$
(black dots) directions are shown, as they should coexist due to the formation
of orthogonal domains, as discussed earlier. The red dots represent the Dirac
cone. The more detailed picture of the experimental electronic structure of
that Dirac state is given in figure 4. The Dirac point is located $\sim$ 150
meV below $E_{F}$. Such Dirac cone does not exist on the surface of bulk
Bi(110) crystal [3].

Figure 4: Fermi surface and band dispersions of 2 BL Bi(110) film (nominal
thickness). (a) Fermi surface with the zoomed-in views of features inside blue
(b) and red (c) boxes, as indicated. (d) Variation of constant energy contours
for the state inside the red box from (a, c). (e)-(g) Energy-momentum
dispersions of the Dirac-like features along the momentum cuts indicated in
(a). Arrows in (d)-(e) indicate the Dirac point (DP). The photoemission
intensity is given by the gray-scale, with high-intensity corresponding to
black in (a)-(d) and white in (e)-(g), respectively

In the case of 3 BL coverage, the FS (figure 3(i)) looks more complex and in
addition to the 4 BL, the features from 3 BL and/or 5 BL and 6 BL are also
visible. In figure 3(k) and (l), the hole-like band at $\bar{\Gamma}$,
predominantly originating from 6 BL islands, creates a circular feature in the
FS (figure 3(i)) and in the $-50$ meV constant-energy contour (figure 3(j)).
The Dirac-cone at $\bar{M}$, marked by the red arrows in figure 3(i), (j) and
(l) originates from the 3 BL and/or 5 BL islands. Overall, these features are
in good agreement with the calculated FS of 3 BL/5 BL [21] and 6 BL [35]
Bi(110) films. Interestingly, we also observe a set of new features that form
extremely one-dimensional (1D) FS, running orthogonal to each other, as
indicated by the white arrows in figure 3(i). These features are better
visualized in the second derivative of spectral intensity, as shown in the
supplementary information, figure S1. They were not observed on the Bi(110)
surface of a bulk crystal [36] and do not appear in the layer-dependent FS
calculations [21, 35]. We also note that they quickly fade away for thicker
films (figure 3(q) and (r)). One feasible interpretation would be that these
states are the edge-states of Bi(110) islands. Their 1D-like FSs are aligned
with the edges of the Bi(110) islands, $\bar{\Gamma}$-$\bar{X}$. Their energy-
momentum dispersions as indicated by arrows in figure 3(k) are also in good
agreement with the calculated edge-states of Bi(110) ribbons [24, 2]. A direct
side-by-side comparison of experimental and theoretical dispersion of these
states is further presented in figure 5. The theoretical edge-state
calculations for 6 nm wide Bi(110) nano-ribbon with BP-like (panel (a)) and
the distorted BP-like structure (panel (c)) [19] are compared to the
experimentally observed edge-state dispersion (panel (b)) of 3 BL films. The
measured states resemble the edge-states from calculations. However, their
relatively low-intensity and the presence of other bands make it difficult to
discriminate between the BP-like and distorted BP-like structure of the film.
We note that the Bi(110) edge-states were previously observed in the STM
measurements [24, 25, 26], but their full energy-momentum dispersion has never
been reported before.

The 3.5 BL film (figure 3(m)-(p)), does not show any 4 BL-related features,
while the 1D-like features are now suppressed. The spectra show mainly 6 BL
(indicated in figure 3(m)) and 5 BL-related features (Dirac cone at
$\bar{M}$).

In the case of 5 BL films, the elongated FS segments, marked by “8 BL” in
figure 3(q) are not originating from the edge-states, but from the electronic
states of 8 BL films, as shown in figure 3(s). The main differences between
3.5 BL and 5 BL films are that the circular feature (figure 3(q) and (r)) and
the linearly dispersing bands (figure 3(s) and (t)) around $\bar{\Gamma}$
originate predominantly from the 7 BL islands, not from 6 BL. Furthermore, the
presence of intense spots at $\bar{M}$ in figure 3(m) and (q), clearly suggest
that the Dirac point in these films are very close to $E_{F}$, in good
agreement with the theoretical results (figure 2(j)). The Dirac cone at
$\bar{M}$ was also observed at the surface of bulk Bi(110) due to the presence
of surface dangling bonds [3]. We also note the 3 BL, 3.5 BL and 5 BL films
all show an arc-like feature along the $\bar{X_{1}}-\bar{M}-\bar{X_{2}}$ in
the FS (figure 3(i), (m) and (q)) and in the constant-energy contour (figure
3(j), (n) and (r)). This feature is due to the presence of a shallow electron
pocket that is also present in the theoretical calculations for 3 BL and 5 BL
Bi(110) films [21].

Further, we want to point out that, although the ARPES is generally not
sensitive to the edge-state, as the probed volume fraction of the edges is
usually extremely small, in the case of Bi/Bi2212, the edge density might just
be large enough to generate a detectable photoemission signal. In thinner
films, the islands are relatively small, with widths much smaller than the
island lengths [18], while in thicker films the islands generally grow bigger
and the effective edge density within the ARPES probing region decreases.

Figure 5: Comparison of theoretical [24] and experimental edge-state
dispersions in thin 3 BL Bi(110) films. (a) Theoretical edge-state dispersion
for 6 nm Bi(110) nano-ribbon with BP-like (topological) and (c) the distorted
BP-like structure (trivial) [24]. (b) Experimentally observed edge-state
dispersion, marked with the white arrows. Arrows in (a) and (c) represents the
edge-states those are identified experimentally.

As already noted, the theoretical results [24, 2] show that the 2 BL and 4 BL
Bi(110) films should host gapped Dirac states along $\bar{\Gamma}-\bar{X}$ $-$
the prerequisite conditions to support QSH states [2]. Our results on 4 BL
islands (figure 3(g) and figure 4) suggest that the films are metallic and
without a significant gap at the Dirac point. The absence of a gap would
suggest that the observed 1D-like states in our experiment might be the edge
resonances, rather than in-gap edge-localized states. This would further
suggest that it is unlikely that the Bi/Bi2212 system hosts QSH states.
However, as the magnitude, the position relative to the Fermi level and the
character of the gap are determined by the atomic buckling, lattice parameter,
and charge transfer at the film-substrate interface, it is expected that by
growing Bi films on appropriate substrates, the Dirac state could be tuned so
that it supports a QSH effect [24, 2]. We also expect that an application of
hydrostatic or uniaxial pressure could be used for fine tuning of a given
system into a QHS state.

It is important that, unlike in Bi films on HOPG, where only even bilayer
islands are formed, both even and odd bilayer films can be stabilized on
Bi2212. This suggests that the substrate plays a crucial role in stabilizing
the nucleation of islands that might be otherwise energetically unfavorable.
The electronic fingerprint of the odd-bilayer islands is a Dirac-like band at
$\bar{M}$ [6]. Theoretical study shows that this cone is a consequence of the
dangling $p_{z}$ orbitals on the surface, the strong spin-orbit coupling and
mirror symmetry of the lattice [6]. Thus, experimental observation of this
feature confirms the existence of surface dangling bonds which in turn would
imply the A7-like (bulk-like) surface structure of the odd-bilayer films.

A direct comparison of experimental data with the BP-like and A7-like
structure can be seen in figure S2, suggesting that the surface structure of
even-bilayer films is most likely very close to the BP-like. Further,
considering the theoretical prediction that the Bi(110) films ($>$ 4 BL) with
BP-like structure are topologically trivial and metallic [24, 2] would suggest
that the edge-like states detected in our study are likely topologically
trivial. The spin-polarised ARPES and STM studies would be helpful in
addressing this issue.

##  III Summary

In summary, we have studied the electronic structure of Bi(110) films for
various thicknesses and discovered the multiple Dirac states in the electronic
structure. The possibility of a Dirac cone for odd-bilayer films was predicted
theoretically, but our study offers the first experimental confirmation. The
observed Dirac bands are a consequence of mirror symmetry of the lattice and a
strong spin-orbit coupling [6]. Our results also suggest the A7-like surface
structure of the odd-bilayer and the BP-like of the even-bilayer films. We
also observe a highly 1D-like FS in thinner films, suggestive of the edge-
states localized to the edges of small but dense (110) islands. Further
studies, including the STM and spin-resolved ARPES, would be necessary to
resolve whether these 1D states are topologically trivial or nontrivial. We
also show that for even bilayer films, the required conditions for QSH effects
are not fulfilled in the Bi/Bi2212 system and indicate that such states could
be achieved by tuning film-substrate interaction. Our results provide a solid
basis for a complete understanding of the electronic structure of this system
and indicate a possibility of using the chiral surface states (Dirac state) of
the Bi(110) films in spintronics. However, in accordance with our previous
studies, an electron transfer from the Bi films renders the interface region
of the Bi2212 substrate to become strongly hole-underdoped and non-
superconducting [37, 38]. This effectively prohibits the superconducting
proximity effect and prevents this interesting system from becoming a platform
for topological superconductivity.

##  IV Experimental

Bi films were grown on in-situ cleaved optimally doped Bi2212 single crystals.
The Bi2212 substrates were synthesized by the traveling floating zone method,
clamped to the sample holder and cleaved with Kapton tape in the ARPES
preparation chamber. The high purity bismuth (99.99%) was evaporated from a
resistively heated Al2O3 crucible. During the film growth, substrate was kept
at room temperature (RT). The photoemission experiments were carried out on a
Scienta SES-R4000 electron spectrometer with the monochromatized He Iα(21.2
eV) and He IIα(40.8 eV) radiation (VUV-5k) [39]. The total instrumental energy
resolution was $\sim$ 20 meV and 30 meV for the ARPES and core-level
measurements, respectively. The angular resolution was better than 0.15∘ and
0.3∘ along and perpendicular to the slit of the analyzer, respectively. Most
of the data were taken at $\sim$30 K. The nominal film thickness was
determined by measuring the attenuation of Bi 5$d_{5/2}$ core-levels of Bi2212
substrate [37].

## Acknowledgements

This work was supported by the US Department of Energy, Office of Basic Energy
Sciences, contract no. DE-SC0012704.

## References

  * Novoselov  _et al._ [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science 306, 666 (2004).
  * Li  _et al._ [2017a] S.-s. Li, W.-x. Ji, P. Li, S.-j. Hu, L. Cai, C.-w. Zhang, and S.-s. Yan, Tunability of the quantum spin hall effect in bi(110) films: Effects of electric field and strain engineering, ACS Applied Materials & Interfaces 9, 21515 (2017a).
  * Hofmann [2006] P. Hofmann, The surfaces of bismuth: Structural and electronic properties, Progress in surface science 81, 191 (2006).
  * Matusalem  _et al._ [2017] F. Matusalem, D. S. Koda, F. Bechstedt, M. Marques, and L. K. Teles, Deposition of topological silicene, germanene and stanene on graphene-covered sic substrates, Scientific reports 7, 1 (2017).
  * Galeotti  _et al._ [2020] G. Galeotti, F. De Marchi, E. Hamzehpoor, O. MacLean, M. R. Rao, Y. Chen, L. Besteiro, D. Dettmann, L. Ferrari, F. Frezza, _et al._ , Synthesis of mesoscale ordered two-dimensional $\pi$-conjugated polymers with semiconducting properties, Nature Materials , 1 (2020).
  * Bian  _et al._ [2014] G. Bian, X. Wang, T. Miller, T.-C. Chiang, P. J. Kowalczyk, O. Mahapatra, and S. A. Brown, First-principles and spectroscopic studies of bi(110) films: Thickness-dependent dirac modes and property oscillations, Phys. Rev. B 90, 195409 (2014).
  * Holtgrewe  _et al._ [2020] K. Holtgrewe, S. Mahatha, P. Sheverdyaeva, P. Moras, R. Flammini, S. Colonna, F. Ronci, M. Papagno, A. Barla, L. Petaccia, _et al._ , Topologization of $\beta$-antimonene on bi 2 se 3 via proximity effects, Scientific reports 10, 1 (2020).
  * Bian  _et al._ [2019] G. Bian, X. Wang, P. J. Kowalczyk, T. Maerkl, S. A. Brown, and T.-C. Chiang, Survey of electronic structure of bi and sb thin films by first-principles calculations and photoemission measurements, Journal of Physics and Chemistry of Solids 128, 109 (2019).
  * Wang  _et al._ [2017] Y.-p. Wang, W.-x. Ji, C.-w. Zhang, P. Li, S.-f. Zhang, P.-j. Wang, S.-s. Li, and S.-s. Yan, Two-dimensional arsenene oxide: A realistic large-gap quantum spin hall insulator, Applied Physics Letters 110, 213101 (2017).
  * Li  _et al._ [2017b] S.-s. Li, W.-x. Ji, S.-j. Hu, C.-w. Zhang, and S.-s. Yan, Effect of amidogen functionalization on quantum spin hall effect in bi/sb (111) films, ACS applied materials & interfaces 9, 41443 (2017b).
  * Hirayama [2020] H. Hirayama, Nucleation and growth of ultrathin bi films, Advances in Physics: X 6, 1845975 (2020).
  * Nagase  _et al._ [2018] K. Nagase, I. Kokubo, S. Yamazaki, K. Nakatsuji, and H. Hirayama, Structure and growth of bi(110) islands on si(111) 3$\times$ 3-b substrates, Physical Review B 97, 195418 (2018).
  * Nagao  _et al._ [2004] T. Nagao, J. Sadowski, M. Saito, S. Yaginuma, Y. Fujikawa, T. Kogure, T. Ohno, Y. Hasegawa, S. Hasegawa, and T. Sakurai, Nanofilm allotrope and phase transformation of ultrathin bi film on si(111)-7$\times$ 7, Phys. Rev. Lett. 93, 105501 (2004).
  * Takayama  _et al._ [2015] A. Takayama, T. Sato, S. Souma, T. Oguchi, and T. Takahashi, One-dimensional edge states with giant spin splitting in a bismuth thin film, Phys. Rev. Lett. 114, 066402 (2015).
  * Peng  _et al._ [2019] L. Peng, J. Qiao, J.-J. Xian, Y. Pan, W. Ji, W. Zhang, and Y.-S. Fu, Unusual electronic states and superconducting proximity effect of bi films modulated by a nbse2 substrate, ACS nano 13, 1885 (2019).
  * Yamada  _et al._ [2018] K. Yamada, S. Souma, K. Yamauchi, N. Shimamura, K. Sugawara, C. X. Trang, T. Oguchi, K. Ueno, T. Takahashi, and T. Sato, Ultrathin bismuth film on 1t-tas2: Structural transition and charge-density-wave proximity effect, Nano letters 18, 3235 (2018).
  * Bian  _et al._ [2009] G. Bian, T. Miller, and T.-C. Chiang, Electronic structure and surface-mediated metastability of bi films on si(111)-$7\times{}7$ studied by angle-resolved photoemission spectroscopy, Phys. Rev. B 80, 245407 (2009).
  * Shimamura  _et al._ [2018] N. Shimamura, K. Sugawara, S. Sucharitakul, S. Souma, K. Iwaya, K. Nakayama, C. X. Trang, K. Yamauchi, T. Oguchi, K. Kudo, _et al._ , Ultrathin bismuth film on high-temperature cuprate superconductor ${\mathrm{bi}}_{2}{\mathrm{sr}}_{2}{\mathrm{cacu}}_{2}{\mathrm{o}}_{8+\delta}$ as a candidate of a topological superconductor, ACS nano 12, 10977 (2018).
  * Drozdov  _et al._ [2014] I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. Cava, B. A. Bernevig, and A. Yazdani, One-dimensional topological edge states of bismuth bilayers, Nature Physics 10, 664 (2014).
  * Kim  _et al._ [2014] S. H. Kim, K.-H. Jin, J. Park, J. S. Kim, S.-H. Jhi, T.-H. Kim, and H. W. Yeom, Edge and interfacial states in a two-dimensional topological insulator: Bi(111) bilayer on bi2te2se, Physical Review B 89, 155436 (2014).
  * Koroteev  _et al._ [2008] Y. M. Koroteev, G. Bihlmayer, E. V. Chulkov, and S. Blügel, First-principles investigation of structural and electronic properties of ultrathin bi films, Phys. Rev. B 77, 045428 (2008).
  * Murakami [2006] S. Murakami, Quantum spin hall effect and enhanced magnetic response by spin-orbit coupling, Phys. Rev. Lett. 97, 236805 (2006).
  * Peng  _et al._ [2018] L. Peng, J.-J. Xian, P. Tang, A. Rubio, S.-C. Zhang, W. Zhang, and Y.-S. Fu, Visualizing topological edge states of single and double bilayer bi supported on multibilayer bi(111) films, Phys. Rev. B 98, 245108 (2018).
  * Lu  _et al._ [2015] Y. Lu, W. Xu, M. Zeng, G. Yao, L. Shen, M. Yang, Z. Luo, F. Pan, K. Wu, T. Das, _et al._ , Topological properties determined by atomic buckling in self-assembled ultrathin bi(110), Nano letters 15, 80 (2015).
  * Yang  _et al._ [2017] F. Yang, J. Jandke, T. Storbeck, T. Balashov, A. Aishwarya, and W. Wulfhekel, Edge states in mesoscopic bi islands on superconducting nb(110), Physical Review B 96, 235413 (2017).
  * Gou  _et al._ [2020] J. Gou, L. Kong, X. He, Y. L. Huang, J. Sun, S. Meng, K. Wu, L. Chen, and A. T. S. Wee, The effect of moiré superstructures on topological edge states in twisted bismuthene homojunctions, Science Advances 6, eaba2773 (2020).
  * Sun  _et al._ [2012] J.-T. Sun, H. Huang, S. L. Wong, H.-J. Gao, Y. P. Feng, and A. T. S. Wee, Energy-gap opening in a bi(110) nanoribbon induced by edge reconstruction, Physical review letters 109, 246804 (2012).
  * Yaginuma  _et al._ [2007] S. Yaginuma, K. Nagaoka, T. Nagao, G. Bihlmayer, Y. M. Koroteev, E. V. Chulkov, and T. Nakayama, Electronic structure of ultrathin bismuth films with a7 and black-phosphorus-like structures, Journal of the Physical Society of Japan 77, 014701 (2007).
  * Kokubo  _et al._ [2015] I. Kokubo, Y. Yoshiike, K. Nakatsuji, and H. Hirayama, Ultrathin bi(110) films on si(111)3$\times$ 3-b substrates, Phys. Rev. B 91, 075429 (2015).
  * Kowalczyk  _et al._ [2020] P. J. Kowalczyk, S. A. Brown, T. Maerkl, Q. Lu, C.-K. Chiu, Y. Liu, S. A. Yang, X. Wang, I. Zasada, F. Genuzio, _et al._ , Realization of symmetry-enforced two-dimensional dirac fermions in nonsymmorphic $\alpha$-bismuthene, ACS nano 14, 1888 (2020).
  * Hatta  _et al._ [2009] S. Hatta, Y. Ohtsubo, S. Miyamoto, H. Okuyama, and T. Aruga, Epitaxial growth of bi thin films on ge (111), Applied Surface Science 256, 1252 (2009).
  * Lindberg  _et al._ [1988] P. Lindberg, Z.-X. Shen, B. Wells, D. Mitzi, I. Lindau, W. Spicer, and A. Kapitulnik, Surface structure of ${\mathrm{bi}}_{2}{\mathrm{sr}}_{2}{\mathrm{cacu}}_{2}{\mathrm{o}}_{8+\delta}$ high-temperature superconductors studied using low-energy electron diffraction, Applied physics letters 53, 2563 (1988).
  * Drozdov  _et al._ [2018] I. K. Drozdov, I. Pletikosić, C.-K. Kim, K. Fujita, G. Gu, J. S. Davis, P. Johnson, I. Božović, and T. Valla, Phase diagram of bi 2 sr 2 cacu 2 o 8+ $\delta$ revisited, Nature communications 9, 1 (2018).
  * Valla  _et al._ [2020] T. Valla, I. K. Drozdov, and G. D. Gu, Disappearance of Superconductivity Due to Vanishing Coupling in the Overdoped ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\delta}$, Nature Communications 11, 569 (2020).
  * Kowalczyk  _et al._ [2013] P. J. Kowalczyk, O. Mahapatra, S. Brown, G. Bian, X. Wang, and T.-C. Chiang, Electronic size effects in three-dimensional nanostructures, Nano letters 13, 43 (2013).
  * Agergaard  _et al._ [2001] S. Agergaard, C. Søndergaard, H. Li, M. B. Nielsen, S. Hoffmann, Z. Li, and P. Hofmann, The effect of reduced dimensionality on a semimetal: the electronic structure of the bi(110) surface, New journal of physics 3, 15 (2001).
  * Kundu  _et al._ [2020] A. K. Kundu, Z.-B. Wu, I. Drozdov, G. Gu, and T. Valla, Origin of suppression of proximity-induced superconductivity in bi/${\mathrm{bi}}_{2}{\mathrm{sr}}_{2}{\mathrm{cacu}}_{2}{\mathrm{o}}_{8+\delta}$ heterostructures, Advanced Quantum Technologies 3, 2000038 (2020).
  * Yilmaz  _et al._ [2014] T. Yilmaz, I. Pletikosić, A. P. Weber, J. T. Sadowski, G. D. Gu, A. N. Caruso, B. Sinkovic, and T. Valla, Absence of a proximity effect for a thin-films of a ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ topological insulator grown on top of a ${\mathrm{bi}}_{2}{\mathrm{sr}}_{2}{\mathrm{cacu}}_{2}{\mathrm{o}}_{8+\delta}$ cuprate superconductor, Phys. Rev. Lett. 113, 067003 (2014).
  * Kim  _et al._ [2018] C. K. Kim, I. K. Drozdov, K. Fujita, J. S. Davis, I. Božović, and T. Valla, In-situ angle-resolved photoemission spectroscopy of copper-oxide thin films synthesized by molecular beam epitaxy, Journal of Electron Spectroscopy and Related Phenomena https://doi.org/10.1016/j.elspec.2018.07.003 (2018).


# Sideband ground-state cooling of graphene with Rydberg atoms via vacuum
forces

M. Miskeen Khan  Instituto Superior Técnico, Universidade de Lisboa, Portugal
Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade
de Lisboa, Portugal<EMAIL_ADDRESS>S. Ribeiro  Joint Quantum
Centre (JQC) Durham-Newcastle, Department of Physics, Durham University,
United Kingdom  J. T. Mendonça  Instituto Superior Técnico, Universidade de
Lisboa, Portugal  Instituto de Plasmas e Fusão Nuclear, Instituto Superior
Técnico, Universidade de Lisboa, Portugal  H. Terças  Instituto Superior
Técnico, Universidade de Lisboa, Portugal  Instituto de Plasmas e Fusão
Nuclear, Instituto Superior Técnico, Universidade de Lisboa, Portugal
<EMAIL_ADDRESS>
###### Abstract

We present a scheme leading to ground-state cooling of the fundamental out-of-
plane (flexural) mode of a suspended graphene sheet. Our proposal exploits the
coupling between a driven Rydberg atom and the graphene resonator, which is
enabled by vacuum forces. Thanks to the large atomic polarizability of the
Rydberg states, the Casimir-Polder force is several orders of magnitude larger
than the corresponding force achieved for atoms in the ground state. By
playing with the distance between the atom and the graphene membrane, we show
that resolved sideband cooling is possible, bringing the occupation number of
the fundamental flexural mode down to its quantum limit. Our findings are
expected to motivate physical applications of graphene at extremely low
temperatures.

Introduction.$\displaystyle-$ The interest around atomically thin mechanical
resonators, such as graphene and other two-dimensional (2D) materials Bunch
_et al._ (2007); Steele  _et al._ (2009), has grown in the last years thanks
to their low masses and large stiffnesses, leading to large oscillation
frequencies and high-quality factors Chen  _et al._ (2009); Will  _et al._
(2017); Moser  _et al._ (2014); Castellanos-Gomez  _et al._ (2015). Those
features put graphene in the run for competitive nanomechanical solutions,
with a variety of important applications in quantum technology, such as
storing and processing quantum information by invoking coherent coupling Luo
_et al._ (2018), high-precision quantum sensing Chen  _et al._ (2009); Singh
_et al._ (2014); Muschik  _et al._ (2014a); de Bonis  _et al._ (2018), and
quantum interferometry Khosla  _et al._ (2018). From the condensed matter
physics perspective, the ability to control the out-of-plane vibration of a 2D
material (the so-called flexural phonon) is also very challenging, as it
allows for tunable control of pseudo-magnetic fields Guinea  _et al._ (2008),
electron-phonon coupling Benyamini  _et al._ (2014) and selective band-gap
engineering Neto and Novoselov (2011). However, the quantum control of the
flexural modes occurs in the near-zero thermal noise limit, where the
mechanical motion is mostly due to quantum fluctuations LaHaye  _et al._
(2004). As such, ground-state cooling of graphene nanoresonators still
preludes the applications mentioned above, and therefore an efficient, active
cooling protocol is urgent. Since laser (or radiation pressure) and photo-
thermal cooling schemes are extremely ineffective in graphene - as a result of
its broad-band absorption spectrum Barton  _et al._ (2012); Thongrattanasiri
_et al._ (2012); Yan  _et al._ (2012); Song  _et al._ (2014); Alba  _et al._
(2016) \- a possible solution could be exploiting the electromagnetic vacuum
fluctuations (EVF), emerging as a powerful resource to couple graphene
resonators with a quantum emitter. Such hybrid setups offer all-optical
quantum control of the membrane motion, enabling force sensing Muschik  _et
al._ (2014b) and mechanical squeezing Li  _et al._ (2017). More recently, EVF
has also been used for ground-state cooling of an h-BN monolayer based on
electromagnetically induced transparency (EIT) Abdi and Plenio (2019).
Additionally, dispersion forces have a measurable effect on the levels and
lifetimes of Rydberg atoms when brought close to surfaces Carter and Martin
(2011); Hogan  _et al._ (2012). However, solutions combining the features of
both vacuum forces and high-lying Rydberg states in graphene electro-
mechanical setups are still at their infancy Ribeiro and Scheel (2013), and
the possibilities offered by Rydberg states are far from being exhausted.

Figure 1: (Color online) a) Schematic representation of the experimental
setup. A Rydberg atom is placed at a distance $\displaystyle d$ from a
suspended graphene membrane. The out-of-plane (flexural) motion results in a
modulation of the Rydberg transition frequency $\displaystyle\omega_{a}$. b)
(not in scale) The states $\displaystyle|s\rangle$ contributing to the
calculation of ground-state energy shift $\displaystyle\delta\omega_{g}$,
lying above the energy level. For Rydberg states, the energy shift
$\displaystyle\delta\omega_{r}$ accounts for contributions from both the
above- and below-lying transitions $\displaystyle|k\rangle$. In addition, the
graphene resonator induces a modification in the free-space Rydberg decay rate
$\displaystyle\Gamma_{0}$.

In this Letter, we describe an experimentally feasible scheme to achieve
ground-state cooling of a graphene resonator by coupling it to a Rydberg atom
via vacuum forces. We exploit the large atomic polarizability of the Rydberg
states, and the fact that the back-action of the Casimir-Polder (CP) force
depends on the atom-surface distance, to construct a resolved sideband cooling
protocol bringing the out-of-plane (flexural) mode down to its quantum ground
state. Our findings pave the way for both quantum technological and condensed
matter investigations for which ground-state cooling of flexural modes in
suspended graphene (and eventually other 2D materials) is required.

Modification of the atomic transitions due to vacuum
fluctuations.$\displaystyle-$ Our setup is depicted in Fig. 1 a), where we
consider an atom placed close to a graphene membrane at a distance
$\displaystyle d$ along the $\displaystyle z$-direction. The interaction
potential is calculated assuming the graphene sheet to be infinitely extended,
thereby neglecting possible finite-size effects. Within the formalism of
macroscopic quantum electrodynamics, we can write the energy shift in the
ground-state $\displaystyle\delta\omega_{g}$, and total shift in the Rydberg
state $\displaystyle\delta\omega_{r}$ as the sum of the non-resonant
$\displaystyle\delta\omega_{r}^{\rm NR}$ and the resonant
$\displaystyle\delta\omega_{r}^{\rm R}$ parts Buhmann (2013); Scheel and
Buhmann (2008)

$\displaystyle\displaystyle\delta\omega_{g}$
$\displaystyle\displaystyle=\frac{\hbar\mu_{0}}{2\pi}\int_{0}^{\infty}d\xi\xi^{2}{\alpha}_{g}\left(i\xi\right)\operatorname{Tr}\left[\mathbf{G}\left(\mathbf{r},\mathbf{r},i\xi\right)\right],$
(1) $\displaystyle\displaystyle\delta\omega_{r}^{\rm NR}$
$\displaystyle\displaystyle=\frac{\hbar\mu_{0}}{2\pi}\int_{0}^{\infty}d\xi\xi^{2}{\alpha}_{r}\left(i\xi\right)\operatorname{Tr}\left[\mathbf{G}\left(\mathbf{r},\mathbf{r},i\xi\right)\right],$
(2) $\displaystyle\displaystyle\delta\omega_{r}^{\rm R}$
$\displaystyle\displaystyle=-\frac{\mu_{0}}{3}\sum_{\ell<r}\omega^{2}_{r\ell}|\mathbf{d}_{r\ell}|^{2}\mathrm{Tr}\left[\text{Re}\left[\mathbf{G}\left(\mathbf{r},\mathbf{r},\omega_{r\ell}\right)\right]\right].$
(3)

The free-space spontaneous decay rate of the Rydberg state
$\displaystyle\Gamma_{0}$ is also modified due to the presence of the graphene
sheet as

$\displaystyle\displaystyle\Gamma=\Gamma_{0}+\frac{2\mu_{0}}{3\hbar}\sum_{\ell<r}\omega^{2}_{r\ell}|\mathbf{d}_{r\ell}|^{2}\mathrm{Tr}\left[\text{Im}\left[\mathbf{G}\left(\mathbf{r},\mathbf{r},\omega_{r\ell}\right)\right]\right],$
(4)

where $\displaystyle\mathbf{G}\left(\mathbf{r},\mathbf{r},\omega\right)$ is
the vacuum field Green tensor accounting for the relevant electromagnetic
properties of graphene (refer to the Supplementary Material located at sup
for further details). In particular, we take the values $\displaystyle E_{\rm
F}\simeq 0.8~{}\text{eV}$ and $\displaystyle\gamma_{g}=10^{12}~{}{\rm s^{-1}}$
for the graphene Fermi energy and relaxation rate, respectively Chen  _et al._
(2011); Efetov and Kim (2010); Koppens  _et al._ (2011); Tielrooij  _et al._
(2015). The properties of the atom are introduced in this formalism via the
atomic polarizability function $\displaystyle{\alpha}_{x}\;(\omega)$, with
$\displaystyle x=g\;(r)$ labeling the ground (Rydberg) state, and is given by

$\displaystyle\displaystyle\alpha_{x}(\omega)=\lim_{\epsilon\to
0}\frac{2}{3\hbar}\sum_{\ell}\frac{\omega_{\ell
x}|\mathbf{d}_{x\ell}|^{2}}{\omega^{2}_{\ell x}-\omega^{2}-i\omega\epsilon}.$
(5)

Here, the sum is extended to all possible atomic transitions, as depicted in
Fig. 1 b).

Figure 2: (Color online) a) Casimir-Polder vacuum coupling $\displaystyle
g_{0}$ and b) spontaneous decay rate of the Rydberg states
$\displaystyle\left|80S_{1/2}\right\rangle$ (solid line),
$\displaystyle\left|60S_{1/2}\right\rangle$ (dashed line) and
$\displaystyle\left|50S_{1/2}\right\rangle$ (dotted line). The intersection of
grey lines fixes our chosen working point.

It is evident from Eqs. (3), (4) and (5) that, due to their large principal
quantum numbers, the energy shift and relaxation rate are much larger for
Rydberg atoms than in the case of regular atoms. As we show below, this fact
is crucial for the achievement of resolved sidebands cooling protocol.

Coupling of Rydberg state and phonon modes.$\displaystyle-$ Since the CP
induced energy shifts are extremely sensitive to small distance fluctuations
(it scales as $\displaystyle\sim(d+\delta z)^{-3}$ in the nonretarded limit,
with $\displaystyle\delta z$ denoting the departure from equilibrium), the
out-of-plane motion of graphene introduces a modulation in the local
transition frequencies. For small amplitudes, $\displaystyle\delta z\ll d$,
one can write the total Hamiltonian of the system as Muschik  _et al._
(2014a),

$\displaystyle\displaystyle\hat{H}=\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\frac{\Omega}{2}\sigma_{x}-\hbar\frac{\Delta}{2}\sigma_{z}+\hbar
g(\hat{b}+\hat{b}^{\dagger})\sigma_{z}.$ (6)

The first term is the oscillator Hamiltonian, with
$\displaystyle\hat{b}\left(\hat{b}^{\dagger}\right)$ standing for the phonon
annihilation (creation) operator; the second term describes the Rydberg laser
driving of effective Rabi frequency $\displaystyle\Omega$ and detuning
$\displaystyle\Delta$ with respect to Rydberg transition frequency
$\displaystyle\omega_{a}=\omega_{0}+\Delta\omega(d)$ Muschik  _et al._
(2014a); Bernien  _et al._ (2017); Zhang  _et al._ (2015), and the surface-
induced shift which can be calculated as $\displaystyle\Delta\omega(\delta
z+d)=\delta\omega_{r}-\delta\omega_{g}$. Finally, the last term in Eq. (6) is
the (electro-mechanical) interaction between the atom and the oscillating
surface with effective coupling $\displaystyle g=g_{0}z_{\rm zpf}$. Here,
$\displaystyle g_{0}=\partial_{z}\Delta\omega|_{z=d}$ is the CP coupling and
$\displaystyle z_{\rm zpf}=\sqrt{\hbar/2m\omega_{m}}$ is the zero-point
fluctuation of the oscillator, characterized by its effective mass
$\displaystyle m$ and frequency $\displaystyle\omega_{m}$; the position
operator is given by $\displaystyle\hat{z}=z_{\rm
zpf}(\hat{b}+\hat{b}^{\dagger})$. A similar situation involving spin-
mechanical coupling has been investigated in a variety of setups, aiming at
phonon control Cirac  _et al._ (1992); Wilson-Rae  _et al._ (2004); Martin
_et al._ (2004); Jaehne  _et al._ (2008); Zippilli  _et al._ (2009); Romero-
Isart  _et al._ (2012); Kepesidis  _et al._ (2013); Abdi  _et al._ (2017) or
at efficient mechanical readout Didier  _et al._ (2015). In the remainder of
this manuscript, we show that the Hamiltonian in Eq. (6) allows for an
unprecedented resolved sideband cooling of the graphene flexural mode.

In Fig. 2, we present the numerical results for the vacuum coupling
$\displaystyle g_{0}$ and the surface-assisted decay rate
$\displaystyle\Gamma$ as a function of the atom-surface distance
$\displaystyle d$ for different Rydberg states $\displaystyle n$ of 87Rb. As
one can immediately see, it is possible to achieve couplings of the same order
of magnitude as in the case of a two-level emitter, where only a single
transition $\displaystyle|g\rangle\leftrightarrow|e\rangle$ is taken into
account for the calculations of energy shifts ($\displaystyle g_{0}\simeq
10^{11}$ Hz/nm for $\displaystyle d\simeq 20~{}\rm nm$ in Ref. Muschik  _et
al._ (2014b)). However, for the Rydberg case presented here, we incorporate
all the allowed transitions
$\displaystyle\textstyle\sum_{k}(|r\rangle\leftrightarrow|k\rangle)$ (with
$\displaystyle|k\rangle$ denoting the states lying below and above the Rydberg
state $\displaystyle|r\rangle$ as a consequence of the surface-induced
degeneracy lifting - see Fig. 1). As a result, equally large couplings are
obtained for distances at least one order of magnitude larger. Moreover, while
the free-space spontaneous decay rate of Rydberg state is of the order of a
few kHz, the presence of a macroscopic surface significantly increase these
values up to few MHz for such large distances due to the changes in the vacuum
density of states. As we are about to see, similar to cavity-enhanced decay
that assists cooling of polar molecules André  _et al._ (2006), these enhanced
decay rates make sideband cooling feasible for graphene flexural modes.

The advantage of exploiting the Rydberg states for graphene cooling is then
twofold: i) strong enough couplings achieved for sufficiently large atom-
surface distances ($\displaystyle d\sim\mu$m) is a crucial element in real-
life experiments, as it allows to place atoms near surfaces with standard
quantum optics techniques without heating graphene Chang  _et al._ (2009); Yu
_et al._ (2018); Delić  _et al._ (2019); ii) the resulting surface-assisted
decay $\displaystyle\Gamma$ falls below the mechanical frequency range
($\displaystyle\omega_{m}/2\pi\simeq 1-100~{}\rm MHz$), a central ingredient
for the establishment of the resolved sideband condition. The latter is
otherwise unavailable in similar protocols involving two-level emitters (see
for instance Ref. Abdi and Plenio (2019)). As such, we have tuned the atom-
surface distance and chosen the Rydberg state number to obtain appropriate
values of $\displaystyle g_{0}$ and $\displaystyle\Gamma$. In the remainder of
our calculations, we have set $\displaystyle d=0.7\mu\text{m}$ for the state
$\displaystyle n=80$ of 87Rb, as depicted in Fig. 2.

Figure 3: (Color online) a) Real part of the atomic spectrum for
$\displaystyle\Omega=0.85~{}\omega_{m}$ (dot-dashed line),
$\displaystyle\Omega=\omega_{m}$ (solid line) and
$\displaystyle\Omega=0.4~{}\omega_{m}$ (dotted line), with peaks
$\displaystyle\omega_{s}=\pm\sqrt{\Omega^{2}+\Delta^{2}}$. Optimal driving is
achieved for $\displaystyle\Omega=0.85~{}\omega_{m}$. For
$\displaystyle\Omega=\omega_{m}$, the heating and cooling peak balance each
other, therefore, the coupled atomic system does not work as cold bath. b) Net
cooling rate $\displaystyle\Gamma_{c}$ (in units of kHz) of the graphene
resonator. c) Steady-state phonon number $\displaystyle n_{\rm ss}$. d) The
same as in c) but considering a dephasing rate
$\displaystyle\tilde{\Gamma}=0.015~{}\omega_{m}$. For b), c) and d), we have
set $\displaystyle\omega_{m}=2\pi\times 20$ MHz and $\displaystyle m=1.6\times
10^{-18}$ kg.

Sideband cooling of the graphene flexural mode.$\displaystyle-$ Our cooling
scheme is based on a reservoir engineering protocol, where phonons are
scattered away thanks to a sufficiently fast relaxation of the atomic degrees
of freedom (DOF), to which the graphene flexural mode is coupled with via the
CP force. Similar reservoir-engineering schemes are at the basis of different
ground-state cooling solutions, achieved for single ions Diedrich  _et al._
(1989), optomechanical setups Marquardt  _et al._ (2007); Schliesser  _et al._
(2008); Teufel  _et al._ (2011), single atoms Thompson  _et al._ (2013) and
nitrogen-vacancy centres in diamond MacQuarrie  _et al._ (2017). These have
also been recently proposed for carbon nanotubes Stadler  _et al._ (2014);
Zippilli  _et al._ (2009); Wang  _et al._ (2017).

Taking advantage of the features of Rydberg atoms discussed above, we
construct a resolved sideband cooling scheme for the fundamental flexural mode
of a graphene resonator. We consider that the mechanical mode is initially in
equilibrium with a (phonon) bath of temperature $\displaystyle T$, and thus
the mean phonon number in the mode is given by $\displaystyle\bar{n}_{\rm
th}=(\exp({\hbar\omega_{m}/k_{B}T})-1)^{-1}$. For our purpose, we will
consider a graphene resonator of total mass $\displaystyle m=1.6\times
10^{-18}$ kg and quality factor $\displaystyle Q=2\times 10^{5}$ Weber  _et
al._ (2016); Chen  _et al._ (2009); Castellanos-Gomez  _et al._ (2015). The
graphene resonator of frequency $\displaystyle\omega_{m}=2\pi\times 20$ MHz is
kept at a cryogenic temperature $\displaystyle T=30$ mK (see e.g Ref. Weber
_et al._ (2016); Castellanos-Gomez  _et al._ (2015)), for which we find
$\displaystyle\bar{n}_{\rm th}\simeq 30$. The chosen values for the coupling
$\displaystyle g_{0}=1.12$ GHz/nm (correspondingly, $\displaystyle g=0.57$
MHz) and decay $\displaystyle\Gamma=3.24$ MHz allow us to operate within
resolved sideband cooling regime, since the conditions $\displaystyle
g/\Gamma\simeq 0.18<1$ and $\displaystyle\Gamma/\omega_{m}\simeq 0.03<1$ are
simultaneously satisfied, while keeping the strong cooperativity condition,
$\displaystyle g^{2}/\Gamma(\gamma_{m}n_{\rm th})\simeq 5.2>1$ Jaehne  _et
al._ (2008); Abdi  _et al._ (2017); MacQuarrie  _et al._ (2017); sup . The
dynamics of the open system is governed by Markov master equation

$\dot{\hat{\rho}}=-(i/\hbar)\left[\hat{H},\hat{\rho}\right]+\mathcal{L}(\hat{\rho}),$
(7)

where the Liouville operator accounting for irreversible dynamics due to the
coupling with the various dissipative channels are given by

$\displaystyle\displaystyle\mathcal{L}(\hat{\rho})$
$\displaystyle\displaystyle=\frac{\gamma_{m}}{2}(\bar{n}_{\rm
th}+1)\mathcal{D}_{\hat{b}}\left[\hat{\rho}\right]+\frac{\gamma_{m}}{2}\bar{n}_{\rm
th}\mathcal{D}_{\hat{b}^{\dagger}}\left[\hat{\rho}\right]$
$\displaystyle\displaystyle\quad+\Gamma\mathcal{D}_{\hat{\sigma}_{-}}\left[\hat{\rho}\right]+\frac{\tilde{\Gamma}}{4}\mathcal{D}_{\hat{\sigma}_{z}}\left[\hat{\rho}\right],$
(8)

with
$\displaystyle\mathcal{D}_{\hat{o}}\left[\hat{\rho}\right]=2\hat{o}\hat{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}\hat{\rho}-\hat{\rho}\hat{o}\hat{o}^{\dagger}$.
The first and the second terms in Eq. (8) describe the thermalisation process
with the thermal bath, respectively accounting for the (stimulated and
spontaneous) thermal emission and thermal excitation of the mechanical mode of
vacuum decay rate $\displaystyle\gamma_{m}=\omega_{m}/Q$. The third one
describes the atomic spontaneous emission in the presence of a graphene
resonator $\displaystyle\Gamma$; the fourth term accounts for the dephasing
occurring at the rate $\displaystyle\tilde{\Gamma}$. Note that for the
cryogenic temperatures considered, we can safely neglect the atomic thermal
excitation.

In the Lamb-Dicke regime, $\displaystyle g\sqrt{\bar{n}_{\rm
th}+1}<\omega_{m}$ Wilson-Rae  _et al._ (2004), where the mode-atom
interaction is only a perturbative effect, the system will relax into the
product state $\displaystyle\hat{\rho}(t)\simeq\hat{\rho}_{\rm
ss}\otimes\hat{\rho}_{m}(t)$, which will be composed of atomic steady-state
density matrix $\displaystyle\hat{\rho}_{ss}$ and the reduced mechanical
density matrix
$\displaystyle\hat{\rho}_{m}=\operatorname{Tr}_{a}\left[\rho\right]$, with the
trace performed over the atomic degrees. This allows us to adiabatically
eliminate the atomic degrees and re-write the master equation for the
mechanical modes alone Jaehne  _et al._ (2008),

$\displaystyle\displaystyle\dot{\hat{\rho}}_{m}=-i\omega_{m}\left[\hat{b}^{\dagger}\hat{b},\hat{\rho}_{m}\right]+A_{-}\mathcal{D}_{\hat{b}}\left[\hat{\rho}_{m}\right]+A_{+}\mathcal{D}_{\hat{b}^{\dagger}}\left[\hat{\rho}_{m}\right],$
(9)

where $\displaystyle A_{-}=g^{2}{S(\omega_{m})}+(\gamma_{m}/2)(\bar{n}_{\rm
th}+1)$ and $\displaystyle
A_{+}=g^{2}{S(-\omega_{m})}+(\gamma_{m}/2)\bar{n}_{\rm th}$ describe the
effective flexural cooling and heating rates, respectively. These processes
are a result of the coupling to the dissipative atomic bath, characterized by
its steady-state spectral density Breuer  _et al._ (2002)

$S(\omega)={\rm Re}\left[\int_{0}^{\infty}d\tau
e^{i\omega\tau}\langle\delta\sigma_{z}(\tau)\delta\sigma_{z}\rangle_{\rm
ss}\right],$ (10)

evaluated at the mechanical frequency $\displaystyle\omega_{m}$. The spectrum
is determined by the correlation of the atomic steady-state fluctuation
operator
$\displaystyle\delta\sigma_{z}=\sigma_{z}-\langle\sigma_{z}\rangle_{\rm ss}$
and obtained by solving Bloch’s equations for the Rydberg atom alone (cf. Eq.
8) and by applying the quantum regression theorem Breuer  _et al._ (2002). The
cooling and heating rates are evaluated at positive $\displaystyle\omega_{m}$
and negative $\displaystyle-\omega_{m}$ frequencies. This characterizes the
ability of the atom to respectively absorb and emit phonons at that very
frequency Clerk  _et al._ (2010). As one can see from Eq. (9), one must
enhance the phonon absorption in detriment of its emission, in order to
increase the effective cooling rate. This means that we need to look at the
parameters that enhance the positive frequency component of the (asymmetric)
spectrum. In Fig. 3 a), we have plotted the steady-state spectrum for
different Rabi frequencies. As we can see, $\displaystyle S(\omega)$ exhibits
three well resolved peaks at $\displaystyle\omega=0$ and
$\displaystyle\omega=\pm\omega_{s}$. Here
$\displaystyle\omega_{s}=\sqrt{\Omega^{2}+\Delta^{2}}$ is the energy splitting
of the dressed atomic states
$\displaystyle\left|+\right\rangle=\sin({\alpha/2})\left|g\right\rangle+\cos({\alpha/2})\left|r\right\rangle$
and
$\displaystyle\left|-\right\rangle=\cos({\alpha/2})\left|g\right\rangle-\sin({\alpha/2})\left|r\right\rangle$,
with $\displaystyle\tan{\alpha}=(\Omega/|\Delta|)$, as evaluated in the
interaction picture (cf. Eq. 6 and Ref. Hauss  _et al._ (2008); sup ). In the
resolved sideband regime, $\displaystyle\Gamma\ll\omega_{s}$, the optimal
cooling conditions (i.e. when the phonon absorption capacity by the atom is
maximized) is achieved at resonance $\displaystyle\omega_{s}=\omega_{m}$, for
a driving strength of $\displaystyle\Omega\simeq 0.85~{}\omega_{m}$ Rabl
(2010); Jaehne  _et al._ (2008); sup . In the interaction picture, the process
$\displaystyle\left|-,n\right\rangle\mapsto\left|+,n-1\right\rangle$
corresponds to cooling, which is manifested by the spectrum peak at
$\displaystyle\omega=\omega_{s}$. Such cooling is obtained for red detuned
driving, $\displaystyle\omega_{l}-\omega_{a}=-\Delta$. From Eq. (9), we obtain
the mean flexural number,

$\langle\dot{n}\rangle\equiv\operatorname{Tr}\left[\dot{\hat{\rho}}_{m}\hat{b}^{\dagger}\hat{b}\right]=-\Gamma_{c}\langle
n\rangle+A_{+},$ (11)

where the net cooling rate is given by $\displaystyle\Gamma_{c}=A_{-}-A_{+}$.
Ground-state cooling is achieved if the steady-state flexuron number,
$\displaystyle n_{\rm ss}=A_{+}/\Gamma_{c}$, satisfies the condition
$\displaystyle n_{\rm ss}<1$. Using Eq. 9 leads us the optical cooling rate
$\displaystyle\gamma_{\text{opt}}$ and the minimal occupation number
$\displaystyle n_{0}$ (see Ref. sup ), reading

$\displaystyle\displaystyle\gamma_{\text{opt}}=2g^{2}[S(\omega_{m})-S(-\omega_{m})],~{}n_{0}=\frac{S(-\omega_{m})}{S(\omega_{m})-S(-\omega_{m})}.$
(12)

The net cooling rate $\displaystyle\Gamma_{c}$ and steady-state phonon number
$\displaystyle n_{\text{ss}}$ then take the form,

$\displaystyle\displaystyle\Gamma_{c}=\frac{1}{2}\left(\gamma_{\text{opt}}+\gamma_{m}\right),\quad
n_{\text{ss}}=\frac{\gamma_{\text{opt}}n_{0}+\gamma_{m}\bar{n}_{\rm
th}}{\gamma_{\text{opt}}+\gamma_{m}}.$ (13)

Since $\displaystyle\gamma_{\text{opt}}\sim(g^{2}/\Gamma)\gg\gamma_{m}$ sup ,
one can write $\displaystyle n_{\text{ss}}\simeq n_{0}+(\gamma_{m}\bar{n}_{\rm
th}/{\gamma_{\text{opt}}})$. Given that the heating process (related to the
peak $\displaystyle S(-\omega_{m})$) is highly suppressed, the final
occupation number is limited by the smallness of the ratio
$\displaystyle\gamma_{m}\bar{n}_{\rm th}/{\gamma_{\text{opt}}}$, which is
guaranteed for large values of $\displaystyle g$. The high polarizability of
the Rydberg state provides such a requirement. In Fig. 3 b), we plot
$\displaystyle\Gamma_{c}$, displaying a maximum for the parameters contained
by the dotted quarter circle. Moreover, the steady-state phonon number is
shown in Fig. 3 c). For the optimal cooling conditions, we get a record value
$\displaystyle n_{\rm ss}\lesssim 0.7$.

Finally, we investigate how robust the ground-state cooling protocol can be in
the presence of dephasing, which may enter the cooling dynamics as a result of
the atomic trap fluctuations and surface-induced noise. In fact, provided that
the condition $\displaystyle\tilde{\Gamma}<\omega_{m}$ is satisfied, ground-
state cooling is still attained. This fact is patent in Fig. 3 d), revealing
that $\displaystyle n_{\rm ss}\lesssim 1$ for
$\displaystyle\tilde{\Gamma}=0.015~{}\omega_{m}$, which is estimated to be met
in typical conditions Li  _et al._ (2013); Contreras-Reyes  _et al._ (2010).
These dephasing effects will set a cooling threshold, clearly shortening the
region of parameters for which ground-state cooling is achievable. This is
therefore desirable to understand at which extent dephasing effects can be
mitigated or controlled in state-of-the-art experimental conditions, although
our calculations seem to suggest that our protocol is quite robust.

Conclusions.$\displaystyle-$ We have constructed a protocol enabling ground-
state sideband cooling of out-of-plane (flexural) mode in suspended graphene
based on vacuum forces. Our setup consists of a driven Rydberg atom placed a
few micrometres away from a graphene nanoresonator, which is coupled with the
help of CP forces. Given the high-atomic polarizability of Rydberg atoms, the
conditions for resolved sideband cooling are achievable, enabling to bring the
fundamental flexural mode down to the quantum limit. Our findings overcome the
difficulty associated to cooling schemes based on optomechanical laser cooling
(a fact inherited from the large light absorbency of graphene in usual
optomechanical interfaces), thus paving the way towards practical applications
in quantum technology for which quantum motion is necessary. Moreover, this
may also contribute to the enrichment of condensed matter platforms, as
flexural modes are known to strongly affect the electronic mobility is
suspended graphene Castro  _et al._ (2010). For example, we anticipate that
our scheme could be used to control the emission of graphene plasmons
Tielrooij  _et al._ (2015), eventually potentiating the generation of plasmon
lasers and control for the emission properties of atomic Ryberg states Chang
_et al._ (2017); Yang  _et al._ (2018).

Acknowledgments.$\displaystyle-$ MK and HT acknowledge support from Fundação
para a Ciência e Tecnologia (FCT-Portugal) through Grant No PD/BD/114345/2016
and through Contract No IF/00433/2015 respectively. SR thanks the EPSRC grant
EP/R002061/1.

## References

  * Bunch  _et al._ (2007) J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead, and P. L. McEuen, Science 315, 490 (2007).
  * Steele  _et al._ (2009) G. A. Steele, A. K. Huttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant, Science 325, 1103 (2009).
  * Chen  _et al._ (2009) C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, Nature nanotechnology 4, 861 (2009).
  * Will  _et al._ (2017) M. Will, M. Hamer, M. Müller, A. Noury, P. Weber, A. Bachtold, R. V. Gorbachev, C. Stampfer, and J. Güttinger, Nano Letters 17, 5950 (2017).
  * Moser  _et al._ (2014) J. Moser, A. Eichler, J. Güttinger, M. I. Dykman, and A. Bachtold, Nature nanotechnology 9, 1007 (2014).
  * Castellanos-Gomez  _et al._ (2015) A. Castellanos-Gomez, V. Singh, H. S. J. van der Zant, and G. A. Steele, Annalen der Physik 527, 27 (2015).
  * Luo  _et al._ (2018) G. Luo, Z.-Z. Zhang, G.-W. Deng, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, L. Tian, and G.-P. Guo, Nature Communications 9, 383 (2018).
  * Singh  _et al._ (2014) V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, Nature Nanotechnology 9, 820 (2014).
  * Muschik  _et al._ (2014a) C. A. Muschik, S. Moulieras, A. Bachtold, F. H. L. Koppens, M. Lewenstein, and D. E. Chang, Phys. Rev. Lett. 112, 223601 (2014a).
  * de Bonis  _et al._ (2018) S. L. de Bonis, C. Urgell, W. Yang, C. Samanta, A. Noury, J. Vergara-Cruz, Q. Dong, Y. Jin, and A. Bachtold, Nano Letters 18, 5324 (2018).
  * Khosla  _et al._ (2018) K. E. Khosla, M. R. Vanner, N. Ares, and E. A. Laird, Phys. Rev. X 8, 021052 (2018).
  * Guinea  _et al._ (2008) F. Guinea, M. I. Katsnelson, and M. A. H. Vozmediano, Phys. Rev. B 77, 075422 (2008).
  * Benyamini  _et al._ (2014) A. Benyamini, A. Hamo, S. V. Kusminskiy, F. von Oppen, and S. Ilani, Nature Physics 10, 151 (2014).
  * Neto and Novoselov (2011) A. H. C. Neto and K. Novoselov, Reports on Progress in Physics 74, 082501 (2011).
  * LaHaye  _et al._ (2004) M. LaHaye, O. Buu, B. Camarota, and K. Schwab, Science 304, 74 (2004).
  * Barton  _et al._ (2012) R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S. P. Wang, P. Ong, P. L. McEuen, J. M. Parpia, and H. G. Craighead, Nano Letters 12, 4681 (2012).
  * Thongrattanasiri  _et al._ (2012) S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, Phys. Rev. Lett. 108, 047401 (2012).
  * Yan  _et al._ (2012) H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, Nature Nanotechnology 7, 330 EP (2012).
  * Song  _et al._ (2014) X. Song, M. Oksanen, J. Li, P. J. Hakonen, and M. A. Sillanpää, Phys. Rev. Lett. 113, 027404 (2014).
  * Alba  _et al._ (2016) R. D. Alba, F. Massel, I. R. Storch, T. S. Abhilash, A. Hui, P. L. McEuen, H. G. Craighead, and J. M. Parpia, Nature Nanotechnology 11, 741 (2016).
  * Muschik  _et al._ (2014b) C. A. Muschik, S. Moulieras, A. Bachtold, F. H. L. Koppens, M. Lewenstein, and D. E. Chang, Phys. Rev. Lett. 112, 223601 (2014b).
  * Li  _et al._ (2017) X. Li, W. Nie, A. Chen, and Y. Lan, Phys. Rev. A 96, 063819 (2017).
  * Abdi and Plenio (2019) M. Abdi and M. B. Plenio, Phys. Rev. Lett. 122, 023602 (2019).
  * Carter and Martin (2011) J. D. Carter and J. D. D. Martin, Phys. Rev. A 83, 032902 (2011).
  * Hogan  _et al._ (2012) S. D. Hogan, J. A. Agner, F. Merkt, T. Thiele, S. Filipp, and A. Wallraff, Phys. Rev. Lett. 108, 063004 (2012).
  * Ribeiro and Scheel (2013) S. Ribeiro and S. Scheel, Phys. Rev. A 88, 052521 (2013).
  * Buhmann (2013) S. Buhmann,  _Dispersion Forces II: Many-Body Effects, Excited Atoms, Finite Temperature and Quantum Friction_ , Vol. 248 (Springer, 2013).
  * Scheel and Buhmann (2008) S. Scheel and S. Buhmann, Acta Physica Slovaca. Reviews and Tutorials 58, 675 (2008).
  * (29) Check the Supplemental Material. .
  * Chen  _et al._ (2011) C.-F. Chen, C.-H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, _et al._ , Nature 471, 617 (2011).
  * Efetov and Kim (2010) D. K. Efetov and P. Kim, Phys. Rev. Lett. 105, 256805 (2010).
  * Koppens  _et al._ (2011) F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, Nano Letters 11, 3370 (2011).
  * Tielrooij  _et al._ (2015) K. Tielrooij, L. Orona, A. Ferrier, M. Badioli, G. Navickaite, S. Coop, S. Nanot, B. Kalinic, T. Cesca, L. Gaudreau, _et al._ , Nature Physics 11, 281 (2015).
  * Bernien  _et al._ (2017) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, _et al._ , Nature 551, 579 (2017).
  * Zhang  _et al._ (2015) Y. Zhang, J. Fan, J.-Q. Liang, J. Ma, G. Chen, S. Jia, and F. Nori, Scientific Reports 5 (2015), 10.1038/srep11510.
  * Cirac  _et al._ (1992) J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, Phys. Rev. A 46, 2668 (1992).
  * Wilson-Rae  _et al._ (2004) I. Wilson-Rae, P. Zoller, and A. Imamoḡlu, Phys. Rev. Lett. 92, 075507 (2004).
  * Martin  _et al._ (2004) I. Martin, A. Shnirman, L. Tian, and P. Zoller, Phys. Rev. B 69, 125339 (2004).
  * Jaehne  _et al._ (2008) K. Jaehne, K. Hammerer, and M. Wallquist, New Journal of Physics 10, 095019 (2008).
  * Zippilli  _et al._ (2009) S. Zippilli, G. Morigi, and A. Bachtold, Phys. Rev. Lett. 102, 096804 (2009).
  * Romero-Isart  _et al._ (2012) O. Romero-Isart, L. Clemente, C. Navau, A. Sanchez, and J. I. Cirac, Phys. Rev. Lett. 109, 147205 (2012).
  * Kepesidis  _et al._ (2013) K. V. Kepesidis, S. D. Bennett, S. Portolan, M. D. Lukin, and P. Rabl, Phys. Rev. B 88, 064105 (2013).
  * Abdi  _et al._ (2017) M. Abdi, M.-J. Hwang, M. Aghtar, and M. B. Plenio, Phys. Rev. Lett. 119, 233602 (2017).
  * Didier  _et al._ (2015) N. Didier, J. Bourassa, and A. Blais, Phys. Rev. Lett. 115, 203601 (2015).
  * André  _et al._ (2006) A. André, D. DeMille, J. M. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, Nature Physics 2, 636 (2006).
  * Chang  _et al._ (2009) D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 103, 123004 (2009).
  * Yu  _et al._ (2018) Y. Yu, N. R. Hutzler, J. T. Zhang, L. R. Liu, J. D. Hood, T. Rosenband, and K.-K. Ni, Phys. Rev. A 97, 063423 (2018).
  * Delić  _et al._ (2019) U. c. v. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, and M. Aspelmeyer, Phys. Rev. Lett. 122, 123602 (2019).
  * Diedrich  _et al._ (1989) F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989).
  * Marquardt  _et al._ (2007) F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007).
  * Schliesser  _et al._ (2008) A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, Nature Physics 4, 415 (2008).
  * Teufel  _et al._ (2011) J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Nature 475, 359 (2011).
  * Thompson  _et al._ (2013) J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, Phys. Rev. Lett. 110, 133001 (2013).
  * MacQuarrie  _et al._ (2017) E. MacQuarrie, M. Otten, S. Gray, and G. Fuchs, Nature communications 8, 14358 (2017).
  * Stadler  _et al._ (2014) P. Stadler, W. Belzig, and G. Rastelli, Phys. Rev. Lett. 113, 047201 (2014).
  * Wang  _et al._ (2017) X. Wang, A. Miranowicz, H.-R. Li, and F. Nori, Phys. Rev. B 95, 205415 (2017).
  * Weber  _et al._ (2016) P. Weber, J. Güttinger, A. Noury, J. Vergara-Cruz, and A. Bachtold, Nature communications 7, 12496 (2016).
  * Breuer  _et al._ (2002) H.-P. Breuer, F. Petruccione, _et al._ ,  _The theory of open quantum systems_ (Oxford University Press on Demand, 2002).
  * Clerk  _et al._ (2010) A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
  * Hauss  _et al._ (2008) J. Hauss, A. Fedorov, S. André, V. Brosco, C. Hutter, R. Kothari, S. Yeshwanth, A. Shnirman, and G. Schön, New Journal of Physics 10, 095018 (2008).
  * Rabl (2010) P. Rabl, Phys. Rev. B 82, 165320 (2010).
  * Li  _et al._ (2013) L. Li, Y. O. Dudin, and A. Kuzmich, Nature 498, 466 (2013).
  * Contreras-Reyes  _et al._ (2010) A. M. Contreras-Reyes, R. Guérout, P. A. M. Neto, D. A. R. Dalvit, A. Lambrecht, and S. Reynaud, Phys. Rev. A 82, 052517 (2010).
  * Castro  _et al._ (2010) E. V. Castro, H. Ochoa, M. I. Katsnelson, R. V. Gorbachev, D. C. Elias, K. S. Novoselov, A. K. Geim, and F. Guinea, Phys. Rev. Lett. 105, 266601 (2010).
  * Chang  _et al._ (2017) C.-H. Chang, N. Rivera, J. D. Joannopoulos, M. Soljacic, and I. Kaminer, ACS Photonics 4, 3098 (2017).
  * Yang  _et al._ (2018) Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, Nature Physics 14, 894 (2018).

Supplemental Material: Sideband ground-state cooling of graphene with Rydberg
atoms via vacuum forces

##  .1 Resonant and non-resonant part contribution in the energy shifts

We briefly state about the origin of resonant and non-resonant contributions
to the energy shifts described in the main text. The presence of a nearby
surface changes the electromagnetic environment of the atom and results in
substantial energy shifts (in comparison with the Lamb shift) and transition
broadenings. The distance-dependent energy shift (Casimir-Polder shift) in the
ground-state, $\displaystyle\delta\omega_{g}$, is due to non-resonant terms
($\displaystyle\hat{\sigma}_{gk}\hat{a}_{l}^{\dagger}$ and
$\displaystyle\hat{\sigma}_{kg}\hat{a}_{l}$) of the interaction Hamiltonian
$\displaystyle H_{\rm int}=-\mathbf{d}\cdot\mathbf{E}$. Here,
$\displaystyle\hat{a}_{l}$ denotes the annihilation operator of the scattered
electromagnetic field $\displaystyle\mathbf{E}$. In addition, the atomic
operator $\displaystyle\sigma_{gk}=\left|g\right\rangle\left\langle k\right|$
describes the transitions from the ground-state
$\displaystyle\left|g\right\rangle$ to the intermediate states
$\displaystyle\left|s\right\rangle$ (see Fig. 1). In the single-photon Hilbert
space, $\displaystyle H_{\rm int}$ couples the polaritonic ground
($\displaystyle\left|g,0_{l}\right\rangle$) and the intermediate
($\displaystyle\left|k,1_{l}\right\rangle$) states. These excitations are
followed by the re-absorption process of a photon by the atom, bringing the
atom back to its ground-state (the field mode to its vacuum state). As these
processes violate energy conservation, they are purely virtual ensuring that
the ground-state energy shift is due to non-resonant part of the interaction
as stated in the main text.

For the Rydberg state, however, the transition manifold contains both above-
and low-lying states, denoted by $\displaystyle|k\rangle$ in Fig. 1. As a
result, the corresponding shift $\displaystyle\delta\omega_{r}$ contains both
non-resonant and resonant terms (thus coupling the polaritonic
$\displaystyle\left|r,0_{l}\right\rangle$ and
$\displaystyle\left|k,1_{l}\right\rangle$ states). While former accounts for
virtual processes, the latter allow real photon transitions. The total shift
$\displaystyle\delta\omega_{r}$ is then the sum of non-resonant
($\displaystyle\delta\omega_{r}^{\rm NR}$) and resonant
($\displaystyle\delta\omega_{r}^{\rm R}$) parts and the expression are given
in main text.

##  .2 Surface-scattered Green’s function

The scattered part the electromagnetic field is characterized by the dyadic
Green’s function $\displaystyle\mathbf{G}$, which is the solution of Helmholtz
equation
$\displaystyle[(\mathbf{\nabla}\times\mathbf{\nabla}\times)-(\omega^{2}/c^{2})\epsilon(\textbf{r},\omega)]\mathbf{G}(\textbf{r},\textbf{r}^{\prime},\omega)=\delta(\textbf{r}-\textbf{r}^{\prime})\otimes\mathbb{I}$,
solved for the planar geometry. Evaluating it at a transverse distance
$\displaystyle z$, it encompass the effects of the graphene surface via its
optical conductivity $\displaystyle\sigma$ through the transverse electric
$\displaystyle R_{\mathrm{\rm TE}}$ ($\displaystyle s$-polarized) and
transverse magnetic $\displaystyle R_{\mathrm{\rm TM}}$ (p-polarized)
reflection coefficients Dispersion Forces II ,

$\mathbf{G}\left(z,z,\omega\right)=\frac{i}{8\pi}\int_{0}^{\infty}dk_{\parallel}\frac{k_{\parallel}}{k_{\perp}}e^{2ik_{\perp}z}\text{Diag}\left[R_{\mathrm{TE}}-\left(\frac{c^{2}k_{\perp}^{2}}{\omega^{2}}\right)R_{\mathrm{TM}},R_{\mathrm{TE}}-\left(\frac{c^{2}k_{\perp}^{2}}{\omega^{2}}\right)R_{\mathrm{TM}},\left(\frac{2c^{2}k_{\parallel}^{2}}{\omega^{2}}\right)R_{\mathrm{TM}}\right].$
(S1)

Here, $\displaystyle\rm{Diag}\left[.,.,.\right]$ is a $\displaystyle 3\times
3$ diagonal matrix. The free-space wavevector satisfies $\displaystyle
k^{2}=k_{\parallel}^{2}+k_{\perp}^{2}=(\omega^{2}/c^{2})$, while
$\displaystyle k_{\perp}^{\prime}=\sqrt{\epsilon k^{2}-k_{\parallel}^{2}}$
would be the wavevector in the graphene plane (for suspended graphene,
$\displaystyle\epsilon=1$). The Fresnel refection coefficients $\displaystyle
R_{\mathrm{TE}}=R_{\mathrm{\rm TE}}(\omega,k_{\parallel})$, $\displaystyle
R_{\mathrm{TM}}=R_{\mathrm{TM}}(\omega,k_{\parallel})$ can be found by
applying appropriate boundary conditions, yielding D. E. Chang ,

$\displaystyle\displaystyle R_{\mathrm{TM}}=\frac{\epsilon
k_{\perp}-k_{\perp}^{\prime}+4\pi\sigma
k_{\perp}k_{\perp}^{\prime}/\omega}{\epsilon
k_{\perp}+k_{\perp}^{\prime}+4\pi\sigma
k_{\perp}k_{\perp}^{\prime}/\omega},\quad
R_{\mathrm{TE}}=\frac{k_{\perp}-k_{\perp}^{\prime}+4\pi\sigma
k/c}{k_{\perp}+k_{\perp}^{\prime}+4\pi\sigma k/c}.$ (S2)

To calculate the non-resonant part of the energy shift, we have to introduced
the imaginary frequency $\displaystyle\omega=i\xi$ allowing for a better
evaluation of the poles Dispersion Forces II ). For the real-frequency
dependence, the two regions $\displaystyle 0<k_{\parallel}<\omega/c$
(propagating waves) and $\displaystyle k_{\parallel}>\omega/c$ (evanescent
waves) are considered in the numerical computation of (S1).

Figure S1: (Color online) (a) Photonic conductivity behaviour of graphene. (b)
Dimensionless function $\displaystyle f(\alpha)$.

##  .3 Graphene properties

The graphene conductivity $\displaystyle\sigma=\sigma(\omega)$ is given within
the Random Phase Approximation (RPA), whose real
$\displaystyle\sigma_{R}(\omega)$ and imaginary
$\displaystyle\sigma_{I}(\omega)$ parts takes the form D. E. Chang ,

$\sigma_{R}(\omega)=\frac{e^{2}E_{\text{F}}}{\pi\hbar^{2}}\frac{1/\tau}{\omega^{2}+1/\tau^{2}}+\frac{e^{2}}{4\hbar}\theta(\hbar\omega-2E_{\text{F}}),$
(S3)
$\sigma_{I}(\omega)=\frac{e^{2}E_{\text{F}}}{\pi\hbar^{2}}\frac{\omega}{\omega^{2}+1/\tau^{2}}-\frac{e^{2}}{4\pi\hbar}\text{log}\left|\frac{2E_{\text{F}}+\hbar\omega}{2E_{\text{F}}-\hbar\omega}\right|,$
(S4)

where $\displaystyle\tau=1/\gamma_{g}$ denotes the electron scattering time.
The typical behavior of both the real and imaginary parts of the optical
conductivity is shown in Fig. S1 a). For $\displaystyle E_{\rm F}\lesssim
0.5~{}\text{max}[\hbar\omega_{rk}]$, the conductivity is mostly real and the
atomic transition decay dumps energy into the particle-hole (inter-band)
continuum. In the region, $\displaystyle
0.5~{}\text{max}[\hbar\omega_{rk}]\lesssim
E_{\text{F}}\lesssim~{}0.7~{}\text{max}[\hbar\omega_{rk}]$, the atom decays
into the free space, while for $\displaystyle E_{\text{F}}\gtrsim
0.7~{}\text{max}[\hbar\omega_{rk}]$ the decay process results in a plasmon
excitation (intra-band), as witnessed by a positive imaginary part in the
conductivity L. Orona .

It is well known, and also verified by our numerics, that the shift in the
state $\displaystyle n$ is mainly due to the transition
$\displaystyle|nS_{1/2}\rangle\mapsto|nP_{1/2,3/2}\rangle$, i.e. the
transition of lowest frequency (i.e.
$\displaystyle\text{min}[\hbar\omega_{rk}]\simeq 0.031~{}\text{meV}$) and
largest dipole moment S.A.Ellingsen . For $\displaystyle n=80$, the choice of
the Fermi energy in the main text ($\displaystyle
E_{\text{F}}=0.8~{}\text{eV}$) suggests that emissions in these frequency
range result in plasmon excitations for almost $\displaystyle\simeq 97\%$ of
whole the transition manifold, as the condition $\displaystyle
E_{\text{F}}\gtrsim 0.7[\hbar\omega_{rk}]$ is fulfilled. Therefore, only a
small percentage of the transitions (and thus with a negligible contribution
to the atomic energy shifts) will emit in the particle-hole continuum. Both
intra- and inter-band emissions could potentially heat graphene, via the
sequential excitation of in-plane phonons. However, their effect in the
heating via the production of out-of-plane phonons (flexurons) are unknown.
Although interesting, the latter are out of the scope of present work.

##  .4 Atomic dressed states

We start by considering the Hamiltonian appearing in Eq. (6) of the manuscript
($\displaystyle\hbar=1$),

$\displaystyle\displaystyle\hat{H}=\omega_{m}\hat{b}^{\dagger}\hat{b}+\frac{\Omega}{2}\sigma_{x}-\frac{\Delta}{2}\sigma_{z}+g(\hat{b}+\hat{b}^{\dagger})\sigma_{z},$
(S5)

and transforming it into the basis in which the two-level atom is diagonal
(Dicke basis),

$\displaystyle\displaystyle\hat{H}=-\frac{\omega_{s}}{2}\sigma_{z}+\omega_{m}\hat{b}^{\dagger}\hat{b}+[g\cos(\alpha)\sigma_{z}-g\sin({\alpha})\sigma_{x}](\hat{b}+\hat{b}^{\dagger}),$
(S6)

where $\displaystyle\omega_{s}=\sqrt{\Delta^{2}+\Omega^{2}}$ is the energy
splitting of two dressed state
$\displaystyle\left|-\right\rangle,\left|+\right\rangle$ discussed in the
text. Here, $\displaystyle\sin{(\alpha)}=\Omega/\omega_{s}$ and
$\displaystyle\cos{(\alpha)}=-\Delta/\omega_{s}$. Then, we move to the
interaction picture via the unitary transformation
$\displaystyle\hat{U}=\exp[i(\omega_{m}\hat{b}^{\dagger}\hat{b}+(\omega_{s}/2)\sigma_{z})t]$.
After performing a rotating wave approximation (RWA) (valid for $\displaystyle
g\ll\omega_{s}=\omega_{m}$), we recast Eq. (S6) as

$\displaystyle\displaystyle\hat{H}_{I}=\hat{U}^{\dagger}\hat{H}\hat{U}=-\tilde{g}(\sigma_{-}\hat{b}^{\dagger}+\sigma_{+}\hat{b}),$
(S7)

with $\displaystyle\tilde{g}=g\sin({\alpha})$. In the single-phonon sector,
the Hamiltonian in Eq. (S7) drives the transition
$\displaystyle\left|-,n\right\rangle\leftrightarrow\left|+,n-1\right\rangle$.
Since the atomic decays are fast, $\displaystyle\Gamma>\tilde{g}$, the latter
is over damped, which guarantees that phonons can be damped away thanks to the
dressing with the atomic degrees of freedom.

##  .5 Atomic power spectrum

Coming back to Eq. (7) of main text, we can obtain dynamics of Bloch vector
$\displaystyle\vec{\sigma}=\left[\sigma_{x},\sigma_{y},\sigma_{z}\right]^{T}$
from the isolated Bloch equation, i.e. considering only the atomic part of the
Hamiltonian, thus taking $\displaystyle\gamma_{m}=0$. In the limit of pure
atomic decay (i.e. $\displaystyle\tilde{\Gamma}\mapsto 0$), the Bloch vector
evolves as

$\displaystyle\displaystyle\partial_{t}\langle\vec{\sigma}\rangle=A\langle\vec{\sigma}\rangle-\vec{\Gamma},$
(S8)

where

$A=\left(\begin{array}[]{cccc}{-\Gamma}&{-\omega_{s}\cos(\alpha)}&{0}\\\
{\omega_{s}\cos(\alpha)}&{-\Gamma}&{-\omega_{s}\sin(\alpha)}\\\
{0}&{\omega_{s}\sin(\alpha)}&{2\Gamma}\\\
\end{array}\right),\quad\vec{\Gamma}=\left(\begin{array}[]{c}{0}\\\ {0}\\\
{2\Gamma}\\\ \end{array}\right).$ (S9)

Using the quantum regression theorem K. Hammerer , one can write the spectrum
$\displaystyle\mathcal{S}(\omega)=\langle\delta\sigma_{z}(-i\omega)\delta\sigma_{z}(0)\rangle_{\rm
ss}$ for the steady-state correlation of the fluctuation operator
$\displaystyle\delta\sigma_{z}$ (defined in the main text) as

$\mathcal{S}(\omega)=-(0,0,1)\cdot\left[i\omega\textbf{1}+A\right]^{-1}\vec{B},$
(S10)

where
$\displaystyle\vec{B}=\langle\delta\vec{\sigma}\delta\sigma_{z}\rangle_{\rm
ss}$. Given the inverse of the matrix
$\displaystyle\left[i\omega\textbf{1}+A\right]^{-1}=\text{Adj}\left[i\omega\textbf{1}+A\right]/\text{Det}\left[i\omega\textbf{1}+A\right]$
(with $\displaystyle\text{Adj}\left[.\right]$ and
$\displaystyle\text{Det}\left[.\right]$ denoting the adjugate and determinant,
respectively), one can write

$\mathcal{S}(\omega)=\frac{h(\omega)}{(i\omega+\epsilon_{0})(i\omega+\epsilon_{+})(i\omega+\epsilon_{-})},\quad\text{where}\quad
h(\omega)=-(0,0,1)\cdot\text{Adj}\left[i\omega\textbf{1}+A\right]^{-1}\vec{B}.$
(S11)

Here, $\displaystyle\epsilon_{i}$ are the eigenvalues of matrix $\displaystyle
A$. The spectrum generally exhibits three peaks, whose frequencies and width
are respectively given by the imaginary and real parts of
$\displaystyle\epsilon_{i}$. In the resolved sideband regime,
$\displaystyle\Gamma/\omega_{s}\ll 1$, the eigenvalues values are given by

$\epsilon_{0}\simeq-\Gamma_{\epsilon_{0}},\quad\epsilon_{\pm}\simeq\pm
i\omega_{s}-\Gamma_{\epsilon_{+}},$ (S12)

where,

$\Gamma_{\epsilon_{0}}=\frac{\Gamma}{2}(\cos(2\alpha)+3),\quad\Gamma_{\epsilon_{+}}=\frac{\Gamma}{4}(\cos(2\alpha)-5).$
(S13)

It is therefore clear that the spectrum exhibits well-resolved peaks at
$\displaystyle\omega=(0,\pm\omega_{s})$, as illustrated in Fig. 3 of the
manuscript. In the same regime, the real part of the spectrum in the vicinity
of $\displaystyle\pm\omega_{s}$ reads,

$S(\omega)\equiv\text{Re}\left[{\mathcal{S}(\omega\simeq\pm\omega_{s})}\right]=\frac{\Gamma_{\epsilon_{+}}}{\Gamma_{\epsilon_{+}}^{2}+(\omega\mp\omega_{s})}(P_{\pm}),\quad\text{with}\quad
P_{\pm}=\text{Re}\left[\frac{-h(\pm\omega_{s})}{(\pm
i\omega_{s}-\Gamma_{\epsilon_{0}})(\pm 2i\omega_{s}-\Gamma_{+})}\right],$
(S14)

To the lowest order in $\displaystyle\Gamma/\omega_{s}$, $\displaystyle
P_{\pm}$ is explicitly given by

$P_{+}=\frac{\sin^{6}(\alpha)\csc^{4}\left(\frac{\alpha}{2}\right)}{4(\cos(2\alpha)+3)},\quad
P_{-}=\frac{4\sin^{4}\left(\frac{\alpha}{2}\right)\sin^{2}(\alpha)}{\cos(2\alpha)+3}.$
(S15)

For convenience, we finally define a dimensionless function $\displaystyle
f(\alpha)=\Gamma[S(\omega)-S(-\omega)]$ capturing the dependence of the net
cooling rate on the relevant parameters of the problem. In the limit
$\displaystyle\omega\rightarrow\omega_{s}$, the function turns out to be

$f(\alpha)=-\frac{8\sin(\alpha)\sin(2\alpha)}{(\cos(2\alpha)-5)(\cos(2\alpha)+3)}.$
(S16)

As it can be seen in Fig. S1, the latter is peaked at
$\displaystyle\Omega\simeq 0.85\omega_{m}$, revealing the optimal
configuration for resolved sideband cooling to take place.

##  .6 Contribution by coupled dissipative atomic bath and net cooling rate

From Eq. (8) of the main text, one can isolate the effects of the thermal bath
(the uncontrolled reservoir) and that of the “dissipative atomic bath”
(controlled reservoir). This allows us to write the equation in the form

$\displaystyle\displaystyle\dot{\hat{\rho}}_{m}=-i\omega_{m}\left[\hat{b}^{\dagger}\hat{b},\hat{\rho}_{m}\right]+\frac{\gamma_{\text{opt}}}{2}(n_{0}+1)\mathcal{D}_{\hat{b}}\left[\hat{\rho}_{m}\right]+\frac{\gamma_{\text{opt}}}{2}n_{0}\mathcal{D}_{\hat{b}^{\dagger}}\left[\hat{\rho}_{m}\right]+\
\frac{\gamma_{m}}{2}(\bar{n}_{\rm
th}+1)\mathcal{D}_{\hat{b}}\left[\hat{\rho}\right]+\frac{\gamma_{m}}{2}\bar{n}_{\rm
th}\mathcal{D}_{\hat{b}^{\dagger}}\left[\hat{\rho}\right],$ (S17)

where $\displaystyle\gamma_{\text{opt}}=2g^{2}[S(\omega_{m})-S(-\omega_{m})]$
is the optical cooling rate and $\displaystyle
n_{0}=S(-\omega_{m})/[S(\omega_{m})-S(-\omega_{m})]$ is the minimal (or
residual) flexural occupation number. Both quantities are a result of the
coupling to the dissipative atomic bath alone. Conversely, the net cooling
rate $\displaystyle\Gamma_{c}$ and the steady-state phonon number
$\displaystyle n_{\text{\rm ss}}$ described in the manuscript take the form

$\displaystyle\displaystyle\Gamma_{c}=\frac{1}{2}\left(\gamma_{\text{opt}}+\gamma_{m}\right),\quad
n_{\text{ss}}=\frac{\gamma_{\text{opt}}n_{0}+\gamma_{m}\bar{n}_{\rm
th}}{\gamma_{\text{opt}}+\gamma_{m}}.$ (S18)

Using Eq. S16, and taking $\displaystyle\omega_{s}=\omega_{m}$, the total
optical cooling rate reads,

$\displaystyle\displaystyle\gamma_{\text{opt}}\simeq(2g^{2}/\Gamma)f(\alpha)=\frac{8\sin^{2}(\alpha)\sqrt{1-\sin^{2}(\alpha)}}{4-\sin^{4}(\alpha)}\frac{g^{2}}{\Gamma}.$
(S19)

The net optical cooling rate turns out to be on the order of
$\displaystyle\mathcal{O}(g^{2}/\Gamma)$. To achieve ground-state cooling,
$\displaystyle\gamma_{\text{opt}}$ must exceed the flexural thermal heating
rate $\displaystyle\gamma_{m}\bar{n}_{\rm th}$. As a consequence, a large
value of the ratio $\displaystyle(\gamma_{\text{opt}}/\gamma_{m}\bar{n}_{\rm
th})=g^{2}/\Gamma\gamma_{m}\bar{n}_{\rm th}$ (dubbed the cooperativity
parameter in the literature) is required to achieve ground-state cooling
Rabl2010 .

## References

  * (1) S. Buhmann, Dispersion Forces II: Many-Body Effects, Excited Atoms, Finite Temperature and Quantum Friction, Vol. 248 (Springer, 2013).
  * (2) F. H. L. Koppens, D. E. Chang, and F. J. G. de Abajo, Nano Letters 11, 3370 (2011).
  * (3) S. A . Ellingsen, S. Y. Buhmann, and S. Scheel, Phys. Rev. Lett. 104, 223003 (2010).
  * (4) K. Tielrooij, L. Orona, A. Ferrier, M. Badioli, G. Navickaite, S. Coop, S. Nanot, B. Kalinic, T. Cesca, L. Gaudreau, et al., Nature Physics 11, 281 (2015)
  * (5) J.A.Crosse, S.A.Ellingsen, K.Clements, S.Y.Buhmann, and S. Scheel, Phys. Rev. A 82, 010901 (2010).
  * (6) K. Jaehne, K. Hammerer, and M. Wallquist, New Journal of Physics 10, 095019 (2008).
  * (7) P. Rabl, Phys. Rev. B 82, 165320 (2010).