# Using graphene conductors to enhance the functionality of atom-chips

K. Wongcharoenbhorn, R. Crawford, N. Welch, F. Wang,
G. Sinuco-León1, P. Krüger2, F. Intravaia3, C. Koller4, T.M. Fromhold  School
of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
1Department of Chemistry, Durham University, South Road, Durham DH1 3LE, UK
2Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
UK  3 Humboldt-Universität zu Berlin, Institut für Physik, 12489 Berlin,
Germany  4University of Applied Sciences Wiener Neustadt, Johannes Gutenberg-
Straße 3, 2700 Wiener Neustadt, Austria

###### Abstract

We show that the performance and functionality of atom-chips can be
transformed by using graphene-based van der Waals heterostructures to overcome
present limitations on the lifetime of the trapped atom cloud and on its
proximity to the chip surface. Our analysis involves Green-function
calculations of the thermal (Johnson) noise and Casimir-Polder atom-surface
attraction produced by the atom-chip. This enables us to determine the
lifetime limitations produced by spin-flip, tunneling and three-body
collisional losses. Compared with atom-chips that use thick metallic
conductors and substrates, atom-chip structures based on two-dimensional
materials reduce the minimum attainable atom-surface separation to a few 100
nm and increase the lifetimes of the trapped atom clouds by orders of
magnitude so that they are limited only by the quality of the background
vacuum. We predict that atom-chips with two-dimensional conductors will also
reduce spatial fluctuations in the trapping potential originating from
imperfections in the conductor patterns. These advantages will enhance the
performance of atom-chips for quantum sensing applications and for fundamental
studies of complex quantum systems.

##  I Introduction

Cold atom systems play a key role in both fundamental and applied aspects of
quantum sensing as they provide a well isolated and controllable platform,
whilst still being sensitive to fundamentally interesting interactions such as
gravity or magnetic fields [1, 2]. The high level of uniformity and
homogeneity of cold atomic ensembles also provides a platform for high-
accuracy time standards [3, 4]. Recent experiments have demonstrated atomic
quantum sensors in precision accelerometers [5], clocks [6], and in measuring
magnetic fields with an unprecedented combination of high sensitivity (nT),
spatial resolution ($\text{\,}\mathrm{\SIUnitSymbolMicro m}$), and field of
view ($\sim$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) [7, 8, 9, 10, 11, 12,
13, 14]. As a result, there is now worldwide activity on the development of
cold-atom based quantum sensing and timing technologies [15, 16].

Miniaturizing and integrating cold-atom quantum systems for fundamental
experiments and technology development has advanced through the creation of
atom-chips, which use micro-fabricated current-carrying wires to trap and
control the atoms in an ultra-high vacuum and typically
$1$-$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the chip surface. Such
chips enable coherent manipulation of the atoms’ internal and external degrees
of freedom [17, 18] leading, for example, to on-chip formation of Bose-
Einstein Condensates (BECs) [19, 20], atom interferometers [21, 22], and
interfacing quantum gases with nanomechanical oscillators [23], carbon
nanotubes [24] and cryogenic surfaces [25, 26, 27, 28, 29, 30]. However,
commonly-used metal wires with a typical thickness of
$\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, mounted on bulk insulating
substrates, have adverse effects when trapped atom clouds approach the
surface. Spatial imperfections in the wires roughen the trapping potential,
Johnson noise currents induce spin-flip transitions that eject atoms from the
trap, and the strong Casimir-Polder (CP) attraction between the atoms and the
chip produce tunneling losses. Together, these loss mechanisms prevent the
formation of long-lived microtraps at distances closer than several microns
from the chip surface [31, 32].

Overcoming these limitations is needed to advance both the fundamental and
technological applications of micro- and nano-engineered environments for cold
atoms. Trapping atoms closer to the chip offers a number of advantages. Higher
magnetic field gradients and trap frequencies can be attained for a given
current, thereby facilitating fast initial cooling, i.e. before three-body
collisions become relevant, as required for creating BECs under less stringent
vacuum requirements. Higher trapping frequencies will also produce atomic
gases that are closer to the 1D limit and thus better for studying the
thermodynamics of low-dimensional gases. Sub-micron trapping was realized by
utilizing the balancing of attractive and repelling forces of light in nano-
fibres [33] [34], but has proven difficult for magnetic trapping [35].
Enabling this offers a pathway to creating hybrid quantum devices comprising
coherently-coupled atomic and solid-state elements [36, 37].

Achieving long lifetimes for atom clouds trapped within
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the chip surface will enable
quantum gases to be controlled, entangled, and addressed by potential
landscapes whose spatial features are finer than the intrinsic length scales
of atomic gases, for example the healing length. Sub-micron atom-surface
trapping distances will also advance chip-based sensors in which atoms are
used to measure external fields and forces. One such sensor, the BEC
microscope, can image current flow patterns in planar conductors with a
spatial resolution limited primarily by the distance of the BEC from the
conductor [7, 8, 9, 11, 12, 14].

Here, we show that atom-chips containing two-dimensional conductors can, in
principle, overcome the present limitations on the atom-surface separation and
lifetime of the trapped atom cloud. Such trapping structures can be fabricated
using, for example, graphene membranes that are either free standing or
enclosed by two-dimensional insulating layers so as to form a van der Waals
heterostructure [38, 39]. This opens a route to achieving sub-micron trapping
distances and, hence, fine features in the trapping potential landscape that
are not attainable when conventional metallic wires are used. We demonstrate
that van der Waals heterostructures can be used to form traps just a few
hundred nanometers from their surface whilst maintaining trap lifetimes of at
least ten seconds. This exceeds the duration of most experiments on the atom
clouds and of typical active operation cycles in cold-atom quantum sensors. In
previous work on the possible use of two-dimensional electron gases as
conductors in atom chips [40, 41], the lifetime of nearby atomic gases was
estimated by extrapolating from the rates of tunneling losses and Johnson
noise-induced spin flips near metallic conductors [41]. Here, we present
detailed calculations of the atom cloud lifetimes, in which the same Green
function formalism is used to determine both the CP potential and Johnson
noise lifetimes, thereby ensuring a fully consistent picture of atom loss
rates.

The paper is organized as follows: In Section II, we describe the layers of
two-dimensional materials that comprise the atom-chip and explain how the chip
affects the lifetime and minimum practical atom-surface separation of trapped
atom clouds. In Section III, we compare the limiting factors, specifically
atom cloud lifetime and spatial roughness, for traps formed less than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ away from the surface of chips
containing graphene or metallic trapping wires. Specifically, we present
detailed calculations that quantify how two-dimensional conductors such as
graphene can reduce both the spin-flip atom losses resulting from Johnson
noise in the conductor and the tunneling losses due to CP atom-surface
attraction sufficiently to enable stable sub-micron trapping with atom-cloud
lifetimes $>10$ s. In Section IV, we propose specific routes to the
realization of graphene-based atom-chips that operate under realistic
experimental conditions. Finally, in Section V we conclude with an overview of
possible further device geometries and experiments to demonstrate the
performance and versatility of graphene-based atom-chips.

##  II atom-chips: structure and influence on trapped atoms

Figure 1: Schematic diagram of the proposed graphene-based atom-chip showing
the Z-shaped graphene conducting channel (hexagonal graphene lattice pattern)
carrying current, $I$ (pink arrow), encased by thin, protective, hBN cladding
layers (upper and lower green slabs). The orientations of the applied magnetic
bias field and the offset field are shown by blue and orange arrows
respectively. These fields combine with that produced by the conducting
channel (current $I$) to trap a nearby atomic BEC (red).

We consider micro-fabricated atom-chip structures that produce magnetic traps
for clouds of rubidium-87 (87Rb) atoms, as this approach will facilitate
comparison to previous experiments. The implications for other species of
alkali atoms are straightforward to derive and do not differ qualitatively
from the 87Rb case. Our proposed graphene-based atom-chip is shown
schematically in Fig. 1. The chip comprises a Z-shaped graphene wire encased
by two cladding layers of $10\text{\,}\mathrm{nm}$-thick hexagonal boron
nitride (hBN). Electrical current, $I$, through the Z-shaped wire generates an
inhomogeneous magnetic field, which is supplemented by a constant applied bias
field, $\mathbf{B}_{b}$, and an offset (Ioffe) field, $\mathbf{B}_{0}$, to
create a magnetic field minimum at $\mathbf{r}_{0}=(x_{0},y_{0},z_{0})$. An
ultracold atom cloud is trapped near this magnetic field minimum, whose value
is non-zero due to the offset field, which suppresses atom losses due to
Majorana spin-flip transitions [17]. The potential energy, $U_{\mathrm{mag}}$,
of the trapped atoms equals the interaction energy between the atomic magnetic
moment $\bm{\mu}$ and the net magnetic field, $\mathbf{B}(\mathbf{r})$, where
$\mathbf{r}$ is the spatial position with respect to the coordinate origin,
i.e.

$U_{\mathrm{mag}}(\mathbf{r})=-\bm{\mu}\cdot\mathbf{B}(\mathbf{r})=m_{F}\mu_{B}g_{F}\absolutevalue{\mathbf{B}(\mathbf{r})}.$
(1)

Here, $\mu_{B}$ is the Bohr magneton and $g_{F}$ is the Landé factor of the
relevant hyperfine state. For the 87Rb atoms considered here, this is
typically the $\ket{F,m_{F}}=\ket{2,2}$ level of the $5^{2}S_{1/2}$ ground
state. Provided that the magnetic quantum number, $m_{F}$, is a good quantum
number, atoms in metastable low-field seeking states, whose magnetic moment is
aligned anti-parallel to the magnetic field orientation, will be trapped near
the magnetic field minimum, $\mathbf{r}_{0}$, where $U_{\mathrm{mag}}$ is also
minimal.

The interplay between three energy scales determines the lifetime of the
trapped atomic gas. The first energy scale is the trap depth given by
$V_{0}=\absolutevalue{\bm{\mu}\cdot\Delta\mathbf{B}}$, where
$\Delta\mathbf{B}$ is the difference between the maximum and the minimum
values of the magnetic field. The second scale is the thermal energy,
$k_{B}T_{\mathrm{cloud}}$, related to the temperature, $T_{\mathrm{cloud}}$,
of the trapped atoms, where $k_{B}$ is the Boltzmann constant. The last energy
scale is the ground-state energy of the trapped atoms, $E_{0}$. Provided that
a sufficiently deep magnetic trap, i.e. $V_{0}\gg k_{B}T_{\mathrm{cloud}}$,
$E_{0}$, is located far away from any surface the overall lifetime of the
trapped atoms is limited by collisions with the background gas. In typical
atom-chip experiments, pressures of $10^{-10}$ to $10^{-11}$ mbar or below can
be reached, for which the background pressure-limited lifetime of the trapped
atoms is of the order of tens of seconds or better [17]. As the atom cloud
approaches the chip surface, its lifetime is reduced by modification of the
trapping potential due to CP interactions with the surface, see Figs. 2 and 3,
and Johnson noise in the conductor, which can cause the atoms to undergo spin-
flip transitions into untrapped states. Lower-frequency Johnson noise,
comparable with the trap frequencies, can also potentially cause atom losses
due to parametric heating of the atom cloud. However, the rates of such
heating are orders of magnitude lower than the spin-flip loss rates [31]. Due
to the low Johnson noise and CP potential near graphene-based atom-chips, the
lifetime of atom clouds trapped near such chips will, beyond a certain
trapping distance, be limited only by the background pressure as we quantify
below.

As the atom-surface trapping distance decreases, the trap frequencies must be
increased in order to reduce depletion by the CP interaction. In turn, this
increases the density of the trapped atom cloud and, therefore, also increases
the rate of three-body collision losses discussed in Section III. In order to
determine the optimal trapping distance, the interplay of three-body losses,
the minimum detection density of the atom cloud, and the CP interaction all
have to be considered, as discussed in Section III.

###  II.1 CP potential and resulting atom tunneling towards the chip surface

The Casimir-Polder potential is essentially a position-dependent shift of the
atomic energy level structure, induced by the interaction of the atom with the
surrounding surface-modified electromagnetic radiation [42, 43]. In general,
the presence of an object modifies a system’s electromagnetic density of
states, due to the boundary conditions that the field has to satisfy on the
surface of the object. The extent of the modification, and therefore the
strength of the CP potential, depends on the object’s specific position in
space, on its form and on the material(s) from which it is made.

Generally, an atom experiences an attractive CP force towards metallic and
dielectric surfaces. In atom-chip systems this behavior effectively lowers the
barrier at the side of a magnetic trap that is nearest the surface, as shown
in Fig. 4. In turn, this enables atoms to tunnel out of the trap and be lost
from the atom cloud. tunneling losses induced by the CP potential affect key
atom-chip performance parameters such as the integration time for sensor
applications and the coherence time for quantum memories. In the present
generation of atom-chips, metal wires used to generate the magnetic field lead
to a large CP attraction on trapped atoms located within
$\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ of the surface. This
imposes a minimum trapping distance of
$10$-$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for typical atom-chip
experiments [17, 44, 45]. Our proposed 2D material-based atom-chips are
expected to exert very low CP attraction, due to their extremely small ($<$
100 nm) thickness and their specific material properties, thereby opening a
new route to entering the sub-micron atom-surface trapping regime.

Figure 2: Schematic diagram of a dipole (electric in the case of our CP
potential calculation but magnetic for our Johnson noise analysis) near an
$n$-layer system, where each layer is designated by index $l=1,2,...,n$. Each
layer is characterized by thickness, $t_{l}$, permeability, $\mu_{l}$, and
permittivity, $\epsilon_{l}$. The top surface of upper material layer 2
(yellow) coincides with the origin of the $y$ coordinate. The dipole (depicted
by the arrow labeled $\mathbf{d}$), is located at
$\mathbf{r}^{\prime}=(x^{\prime},y^{\prime},z^{\prime})$ in layer 1, above the
solid material layers, and acts as a point source. The arrow labeled
$\mathbf{E}$, represents the orientation of the electric field of frequency,
$\omega$, at point $\mathbf{r}$, which is related to the dipole $\mathbf{d}$
via a total Green’s tensor. Note also that $t_{1}$ and $t_{n}$ are infinite,
corresponding to semi-infinite top and bottom layers.

For a system in equilibrium at a temperature $T$ consisting of an atom located
at position $\mathbf{r}^{\prime}$ from a nearby material body (see Fig. 2),
both interacting with the electromagnetic field, the CP potential is given by
[46, 47]

$U_{CP}(\mathbf{r}^{\prime})=\mu_{0}k_{B}T\sum_{j=0}^{\infty}{}^{{}^{\prime}}\xi_{j}^{2}\alpha(\mathrm{i}\xi_{j})\,\mathrm{tr}\big{[}\mathbf{G}^{(1)}(\mathbf{r}^{\prime},\mathbf{r}^{\prime},\mathrm{i}\xi_{j})\big{]},$
(2)

where $\mu_{0}$ is the permeability of vacuum, $\hbar$ is the reduced Planck
constant $\alpha(\omega)$ is the atomic polarizability and $\xi_{j}=2\pi
k_{B}Tj/\hbar$, are commonly known as the Matsubara frequencies [48]. The
prime on the Matsubara sum in Eq. (2) indicates that the $j=0$ term carries
half weight [47]. In Eq. (2),
$\mathbf{G}^{(1)}(\mathbf{r}^{\prime},\mathbf{r}^{\prime},\omega)$ is the
scattering Green’s tensor, which contains the information about the material’s
optical properties and the geometry of the system.

For atom-surface separations shorter than the size of the components of an
actual atom-chip (typically of the order of a few tens of micrometers or
larger) we can consider that the atoms are interacting with a large layered
surface. In this case the expression for the Green tensor is known and we
present it explicitly in Appendix A. As shown there, in order to determine the
expression of the Green tensor we need to evaluate the reflection coefficients
of the electromagnetic field incident on the atom-chip structure. In our case,
the multi-layer configurations allow their determination using the scattering
or the transfer-matrix approach [49, 50] in combination with models describing
the optical properties of graphene, hBN and gold (see Appendix B for details).
In this work, we take the Fermi energy and electron relaxation rate of
graphene to be $E_{F}=$0.1\text{\,}\mathrm{e}\mathrm{V}$$ and
$\gamma=$4\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$, respectively,
corresponding to typical values found both theoretically [51, 52, 53, 54] and
in experiments [55].

In this paper, we consider a simple model of alkali atoms with atomic
polarisability of the form [56, 47]

$\alpha(\mathrm{i}\xi_{j})=\alpha_{0}\frac{\omega_{\mathrm{T}}^{2}}{\omega_{\mathrm{T}}^{2}+\xi_{j}^{2}},$
(3)

where $\alpha_{0}$ is the ground-state static polarizability and
$\omega_{\mathrm{T}}$ is the frequency of the dominant atomic transition. In
the case of 87Rb atoms,
$\alpha_{0}=$5.27\text{\times}{10}^{-39}\text{\,}\mathrm{F}\mathrm{m}^{2}$$
[57], and
$\omega_{\mathrm{T}}=2\pi\times$384\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$
is the D2 line transition frequency corresponding to a wavelength of
$780\text{\,}\mathrm{nm}$ [58].

Eq. (2) is sufficiently generic to enable the CP potential to be calculated
for our atom-chip system and compared consistently with other materials and
structures.

Figure 3: CP potential, $U_{CP}$, calculated versus the position,
$y^{\prime}$, of an 87Rb atom from the surface of: a graphene monolayer
(dashed red curve), a heterostructure comprising a graphene monolayer encased
by two $10$-$\text{\,}\mathrm{nm}$-thick hBN layers (solid green curve), a
$125$-$\text{\,}\mathrm{nm}$-thick gold slab (solid yellow curve) at
temperature $T=$300\text{\,}\mathrm{K}$$.

In Fig. 3, we compare the CP potential, $U_{CP}$, calculated versus the
separation, $y^{\prime}$, of an 87Rb atom from three different material
systems: a graphene monolayer (dashed red curve) for which CP potential
calculations have been reported previously [59, 60, 61, 56, 62, 63, 64], a
heterostructure comprising a graphene monolayer encased by two
$10$-$\text{\,}\mathrm{nm}$-thick hBN layers (solid green curve) and a
$125$-$\text{\,}\mathrm{nm}$-thick gold slab (solid yellow curve), all at
$T=$300\text{\,}\mathrm{K}$$. We choose the thickness of the gold slab to be
$125$-$\text{\,}\mathrm{nm}$ because this is among the smallest reported
thicknesses at which gold wires in atom-chip experiments [65] have a
conductivity that still behaves as bulk. Even in this limit of metallic
conductor thickness, at
$y^{\prime}\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the CP potential
for the heterostructure is approximately 40$\%$ of that for the thin gold
slab.

The effect of the CP potential on the total trapping potential,
$U_{\mathrm{tot}}$, can be illustrated by modelling the magnetic trapping
potential, $U_{H}$, as simple harmonic and adding the CP potential, giving

$U_{\mathrm{tot}}(y)=U_{\mathrm{H}}(y)+U_{\mathrm{CP}}(y).$ (4)

The simple harmonic potential takes the form

$U_{\mathrm{H}}(y)=\frac{1}{2}m\omega_{r}^{2}(y-y_{c})^{2},$ (5)

where $m$ is the mass of the trapped atom, $\omega_{r}$ is the radial trapping
frequency, and $y_{c}$ is the position of the centre of the simple harmonic
trap measured from the surface; note that $y_{c}$ is not necessarily equal to
the minimum of the total potential.

Figure 4: Total potential, $U_{\mathrm{tot}}$, calculated versus distance,
$y$, of an 87Rb atom from the surface of: a free-standing graphene monolayer
(dashed red curve); a graphene monolayer encased on each side by a
$10$-$\text{\,}\mathrm{nm}$-thick hBN sheet (solid green curve); a
$125$-$\text{\,}\mathrm{nm}$-thick gold sheet (solid yellow curve). The total
potential is the sum of the CP potential and the harmonic model trapping
potential, which is centred at $y=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
with radial trapping frequency,
$\omega_{r}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$. All curves
are for $T=$300\text{\,}\mathrm{K}$$.

Fig. 4 shows the resulting total trapping potential, $U_{\mathrm{tot}}$,
calculated versus distance, $y$, of an 87Rb atom from a graphene monolayer
(dashed red curve), an hBN-graphene monolayer-hBN heterostructure (solid green
curve) and a $125$-$\text{\,}\mathrm{nm}$-thick gold slab (solid yellow curve)
calculated taking (see Eq. (5))
$\omega_{r}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
$y_{c}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$T=$300\text{\,}\mathrm{K}$$, and the mass of an 87Rb atom,
$m=$1.44\text{\times}{10}^{-25}\text{\,}\mathrm{k}\mathrm{g}$$. It is apparent
that the CP potential distorts the simple harmonic trap: an energy barrier of
finite height and width appears near the surface for $y<y_{c}$. The height and
width effectively scale with distance of trap centre from the surface, thereby
giving rise to tunneling losses, which deplete the trapped atom cloud. Since
graphene creates a weaker CP attraction than even the thin gold conductor, the
tunneling loss rates for graphene-based atom-chips are lower than for
conventional atom-chips and we quantify this benefit below. Consequently,
graphene-based atom-chips offer a performance advantage over the present
generation of atom-chips, which use metallic conductors as current-carrying
wires.

####  II.1.1 Tunneling loss rate

In order to estimate the tunneling loss rate, $\Gamma_{\mathrm{tun}}$, of an
atom cloud trapped in the finite potential well shown schematically by the
solid black curve in Fig. 5, we employ Gamow’s theory of alpha decay [66]. In
this model, the atom is considered to oscillate inside the potential well and
can escape by tunneling through the finite barrier nearest the surface each
time it is incident on that barrier. The tunneling rate is determined by the
frequency at which the atom approaches the barrier, $f$, and the transmission
probability $\tilde{T}$ that the atom tunnels out at each attempt.
Mathematically, we have

$\Gamma_{\mathrm{tun}}=f\times\tilde{T}.$ (6)

Figure 5: Schematic diagram of the total trapping potential,
$U_{\mathrm{tot}}(y)$ (solid black curve) plotted versus distance, $y$, of an
87Rb atom from an atom-chip surface. The solid blue line indicates the ground-
state energy of the quantum harmonic oscillator,
$E=\hbar\omega_{\mathrm{eff}}/2$, where $\omega_{\mathrm{eff}}$ is the
effective characteristic frequency of the simple harmonic trap (dashed red
curve), which is perturbed by the CP potential, as described in the text, and
approximates $U_{\mathrm{tot}}(y)$ near the minimum. The positions $y_{0}$,
$y_{1}$, $y_{2}$, indicated by arrows, are, respectively, the actual trap
centre and the two classical turning points for the left-hand potential energy
barrier, where
$U(y_{1})-U(y_{0})=U(y_{2})-U(y_{0})=\hbar\omega_{\mathrm{eff}}/2$. The dash-
dotted yellow curve is the potential of the unperturbed simple harmonic trap.

Fig. 5 shows that the deformation of the unperturbed harmonic magnetic
potential (dot-dashed yellow curve) by the CP interaction also yields an
effective perturbed harmonic potential (dashed red curve) with a lower
trapping frequency, $\omega_{\rm eff}$, and whose minimum shifts from
$y=y_{c}$ to a new position, $y_{0}$. The perturbed harmonic potential
therefore takes the form

$U_{\mathrm{eff}}(y)=\frac{1}{2}m\omega_{\mathrm{eff}}^{2}(y-y_{0})^{2},$ (7)

where a Taylor expansion of $U_{\rm tot}(y)$ about $y=y_{0}$ yields

$y_{0}\approx y_{c}+\frac{U^{\prime}_{\rm
CP}(y_{c})}{m\omega_{r}^{2}},\quad\omega_{\rm
eff}^{2}\approx\omega_{r}^{2}+U^{\prime\prime}_{\rm CP}(y_{0}).$ (8)

Atoms in the ground-state of this effective potential have an energy
$E=\hbar\omega_{\mathrm{eff}}/2$ and approach the barrier at frequency
$f=\omega_{\mathrm{eff}}/2\pi$. Using the Wentzel–Kramers–Brillouin (WKB)
approximation, the transmission probability through the tunnel barrier is
given by [67, 68, 69]

$\tilde{T}=\mathrm{exp}\bigg{(}-2\int_{y_{1}}^{y_{2}}\kappa(y)\mathrm{d}y\bigg{)},$
(9)

where $y_{1}$, $y_{2}$, are the two classical turning points for the potential
barrier, $\kappa(y)=\sqrt{2m(U_{\mathrm{tot}}(y)-E)}/\hbar$, and
$U_{\mathrm{tot}}(y)$ is the form of the barrier in the total potential energy
curve.

The average tunneling-limited lifetime of a trapped atom is then defined by

$\tau_{\mathrm{tun}}(y_{0})=\frac{1}{\Gamma_{\mathrm{tun}}(y_{0})}.$ (10)

Figure 6: Tunneling lifetime, $\tau_{\mathrm{tun}}$, calculated versus the
position of the harmonic trap center, $y_{0}$, for an 87Rb atom trapped near a
graphene monolayer (dashed red curve) and a $125$-$\text{\,}\mathrm{nm}$-thick
gold slab (solid yellow curve). The weaker CP attraction for graphene gives
rise to a higher, wider, tunnel barrier and, consequently, a higher tunneling
lifetime. Parameters: $T=$300\text{\,}\mathrm{K}$$,
$\omega_{r}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

We calculated the tunneling loss rates for our model systems using the Gamow
formalism described above. Figure 6 shows the resulting lifetimes,
$\tau_{\mathrm{tun}}$, calculated versus the position of the trap center,
$y_{0}$ from a graphene monolayer (dashed red curve) and a thin gold slab
(solid yellow curve). The minimum distance that atoms can be trapped from the
surface is marked by the left-hand ends of the two curves, where the tunnel
barrier vanishes. Comparison of the curves shows that using a graphene
monolayer reduces this distance to
$\sim$0.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ compared to the value of
$\sim$0.45\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for the gold layer. For
$y_{0}>$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the tunneling lifetime
for the single layer of graphene is orders of magnitude higher than for the
gold slab due to the weaker CP potential.

###  II.2 Atom losses due to Johnson Noise

Magnetically trapped atoms only remain trapped when they are in a low field
seeking state with the magnetic moment aligned anti-parallel to the direction
of the magnetic field. In order to keep $m_{F}$ a good quantum number, an
offset magnetic field of a few Gauss is typically maintained at the trapping
position in atom-chip experiments [32]. Given a Zeeman splitting of, for
example, 0.7 MHz/G for the 87Rb ground state, this produces transition
frequencies of a few MHz between hyperfine states with different $m_{F}$
values, thus making the trapped atoms susceptible to magnetic fields in that
frequency range and therefore to noise in the radio frequency domain.

Johnson noise arises from electrical noise currents within a conductor, which
produce fluctuations of the magnetic field [70]. For near-surface traps formed
between $\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and 1 mm from a
metallic conductor on a conventional atom-chip, Johnson noise is usually the
main limitation on the lifetime of the atom cloud [32, 44, 45]. For example,
the measured lifetime of atoms trapped
$\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from thick metallic
conductors is limited to only $\approx 0.1$ s by the effects of Johnson noise
[32].

The key advantage of using trapping wires made from graphene, or other two-
dimensional conductors is their reduced level of Johnson noise [40, 41]. This
advantage originates from their orders-of-magnitude lower sheet electron
density compared with metals, which dominates over the tendency of higher
carrier mobility to increase current fluctuations. We now explore this
advantage using two models of increasing sophistication. Firstly, a crude
estimate based on previous models for metallic conductors and derived using
the fluctuation-dissipation theorem [32, 70, 71]. Secondly, we present a
rigorous quantum field theoretical calculation for 2D materials involving the
full Green’s function for the system, determined from the reflection
coefficients for graphene and 2D multilayers.

Comparing graphene and gold wires with a given top surface area, $A$, the
ratio of the number of free electrons in graphene, $N_{G}$, to that in gold,
$N_{Au}$, is $N_{G}/N_{Au}=n_{G}/n_{Au}t_{Au}$, where $t_{Au}$ is the
thickness of the gold wire and $n_{G}$, $n_{Au}$ are, respectively, the sheet
and volume electron densities of undoped graphene and gold. Taking
$n_{Au}=$5.9\text{\times}{10}^{28}\text{\,}\mathrm{m}^{-3}$$ for bulk gold and
typical values of $t_{Au}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$n_{G}=$9\text{\times}{10}^{14}\text{\,}\mathrm{m}^{-2}$$ [72] gives
$N_{G}/N_{Au}=1.5\times 10^{-8}$. The carrier mobility of gold is
$\mu_{Au}\approx 4.3\times 10^{-3}$ m2/Vs and, in graphene, mobilities up to
$\mu_{G}\approx 20$ m2/Vs have been reported in free-standing membranes [73,
74]. For graphene on a substrate, the electron mobility is typically at least
an order of magnitude lower, leading to the estimate $\mu_{Au}/\mu_{G}\gtrsim
2\times 10^{-4}$.

These results allow us to anticipate that the Johnson noise will be far
smaller in graphene than in gold. Indeed, to make a rough initial estimate of
this intuitive advantage, we now use the model presented in [32, 70, 71] to
evaluate the expected spin-flip lifetime enhancement. For an atom trapped at
distance, $d$, from a metal film of width $w\gg t$ and resistivity $\rho$ at
temperature $T$, the $|F,m\rangle\rightarrow|F,m-1\rangle$ spin-flip rate,
given in $\text{\,}\mathrm{s}^{-1}$, is
$\Gamma=C(T/\rho)\times[d(1+d/t)(1+2d/w)]^{-1}$, where $C$ is a constant that
depends on the Clebsch-Gordon coefficient for the transition and on the
transition frequency [32, 70, 71]. Assuming, as a crude initial approximation,
that this formula can also be used to estimate the rate of spin flips induced
by electrons in graphene, the ratio of the lifetimes, $\tau_{G}$ and
$\tau_{Au}$, of atom clouds trapped at a distance $d$ above graphene and gold
wires, respectively, is

$\frac{\tau_{G}}{\tau_{Au}}=\frac{\Gamma_{Au}}{\Gamma_{G}}=\frac{\rho_{G}}{\rho_{Au}}\frac{\left(1+\frac{d}{t_{G}}\right)}{\left(1+\frac{d}{t_{Au}}\right)},$
(11)

where $t_{G}=0.345$ nm is the thickness of a graphene monolayer and
$\rho_{Au}=1/(n_{Au}e\mu_{Au})$, $\rho_{G}=t_{G}/(n_{G}e\mu_{G})$ are the
resistivities of the gold and graphene and $e$ is the electron charge.

Since $d\gg t_{G}$, it follows that

$\begin{split}\frac{\tau_{G}}{\tau_{Au}}\approx\left(\frac{n_{Au}}{n_{G}}\right)\left(\frac{\mu_{Au}}{\mu_{G}}\right)\frac{t_{Au}d}{\left(d+t_{Au}\right)}\\\
=\left(\frac{N_{Au}}{N_{G}}\right)\left(\frac{\mu_{Au}}{\mu_{G}}\right)\frac{d}{\left(d+t_{Au}\right)}.\end{split}$
(12)

Unless $d\ll t_{Au}$, which is not the case for presently-attainable trapping
distances, the final term in the above equation is of order unity and so
$\tau_{G}/\tau_{Au}$ depends primarily on the relative number of free
electrons in gold and in graphene and on their mobility ratio.

Using the values for the carrier mobility and density given above we arrive at
$\tau_{G}/\tau_{Au}\gtrsim 1.3\times 10^{4}d/(d+t_{Au})$. We thus predict that
for atoms trapped $\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ away from
a conductor, the lifetime will increase from $\approx 0.1$ s for a
1-$\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick metallic wire, similar to
that reported in [32], to $\gtrsim 600$ s for graphene, i.e. an increase by a
factor of $\tau_{G}/\tau_{Au}\gtrsim 6.3\times 10^{3}$. The physical reason
for this is that although electrons in graphene have a higher mobility than in
a metal, and so produce more Johnson noise per carrier, this is more than
compensated by the far lower number of charge carriers in the graphene.

In the next section, we derive an expression for the Johnson noise produced by
van der Waals heterostructures. To quantify the advantages of using 2D
conductors, rather than metal wires, to reduce noise in atom-chips we consider
the particular case of graphene conduction channels. However, similar
advantages are expected from other 2D materials due to their low number of
electric current carriers.

####  II.2.1 Transition rates in terms of dyadic Green’s functions

The magnetic moment vector associated with the transition $\ket{i}\to\ket{f}$
of an atom is given by [75, 76]

$\bm{\mu}=-\bra{i}\frac{\mu_{B}}{\hbar}\Big{(}g_{S}\hat{\mathbf{S}}+g_{L}\hat{\mathbf{L}}-g_{I}\frac{m_{\mathrm{e}}}{m_{\mathrm{nuc}}}\hat{\mathbf{I}}\Big{)}\ket{f},$
(13)

where $\hat{\mathbf{S}}$, $\hat{\mathbf{L}}$, and $\hat{\mathbf{I}}$ are the
electron spin operator, the electron orbital angular momentum operator, and
the total nuclear angular momentum operator, respectively, with their
corresponding Landé g-factors $g_{S}$, $g_{L}$, and $g_{I}$, $m_{\mathrm{e}}$
is the electron mass and $m_{\mathrm{nuc}}$ is the nuclear mass. Here, the
magnitude of the angular momentum, for example, $\hat{\mathbf{S}}$, is
$\sqrt{S(S+1)}\hbar$ and the eigenvalue of the $z$-component of
$\hat{\mathbf{S}}$, i.e. $\hat{\mathbf{S}_{z}}$, is $m_{S}\hbar$, where $S$
and $m_{S}$ are the corresponding quantum numbers for $\hat{\mathbf{S}}$ and
$\hat{\mathbf{S}_{z}}$, respectively.

Taking $L=0$ for the electronic ground-state and neglecting the term
containing the total nuclear angular momentum operator $\hat{\mathbf{I}}$ in
equation (13) because $m_{\mathrm{e}}\ll m_{\mathrm{nuc}}$, the magnetic
moment vector becomes
$\bm{\mu}=-\mu_{B}g_{S}\bra{i}\hat{\mathbf{S}}\ket{f}/\hbar$, where $g_{S}=2$,
and the rate of magnetic spin-flip transitions from an initial hyperfine
magnetic state $\ket{i}$ to another state $\ket{f}$ is given by [76]

$\Gamma_{\mathrm{JN}}=\mu_{0}\frac{2(\mu_{B}g_{S})^{2}}{\hbar^{2}}\sum_{j,k}\Big{\\{}\bra{f}\hat{\mathbf{S}}_{j}\ket{i}\bra{i}\hat{\mathbf{S}}_{k}\ket{f}\\\
\times\mathrm{Im}[\curl{\curl{\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})}}]_{jk}(\bar{n}_{\mathrm{th}}+1)\Big{\\}},$
(14)

where $\hat{\mathbf{S}}_{j,k}$ denotes the $j$ and $k$ components of the
electron spin operator $\hat{\mathbf{S}}$, and
$\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})$ is the total dyadic
Green’s function describing the electromagnetic field of the transition
frequency $\omega_{if}$ at $\mathbf{r}_{0}$ due to a _magnetic_ dipole located
at $\mathbf{r}_{0}$ (see Appendix A). The mean thermal photon occupation
number is given by

$\bar{n}_{\mathrm{th}}=\frac{1}{\mathrm{e}^{\hbar\omega_{if}/k_{B}T}-1},$ (15)

where $T$ is the temperature of the electromagnetic field system that causes
the spin-flip transitions, rather than of the trapped atoms, and $\omega_{if}$
is the angular frequency of the radiation due to magnetic spin-flip
transitions. The Johnson noise lifetime of a single atom is defined as

$\tau=\frac{1}{\Gamma_{\mathrm{JN}}}.$ (16)

Note that, mathematically,
$\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})$ can be written as the
sum of a Green’s tensor, describing the field due to a dipole in an infinitely
extended homogeneous bulk medium, vacuum for example, and a scattering Green’s
tensor describing the reflected field in the presence of reflective bodies, so
that

$\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})=\mathbf{G}^{(0)}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})+\mathbf{{G}}^{(1)}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if}).$
(17)

The explicit forms of
$\mathbf{G}^{(0)}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{if})$ can be found in
[77, 78] and are summarized in Appendix A. Owing to the general properties of
a Green’s tensor, we have

$\mathrm{Im}[\curl{\curl{\mathbf{G}(\mathbf{r},\mathbf{r}_{0},\omega)}}]=\frac{\omega^{2}}{c^{2}}\mathrm{Im}[\epsilon(\omega)\mu(\omega)\mathbf{G}(\mathbf{r},\mathbf{r}_{0},\omega)],$
(18)

where $\epsilon(\omega)$ and $\mu(\omega)$ are, respectively, the permittivity
and permeability of the medium in which the field and source points are
located. The imaginary part of the Green’s tensor in vacuum has a simple form:

$\mathrm{Im}[\mathbf{G}^{(0)}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)]_{jk}=\frac{1}{6\pi}\frac{\omega}{c}\delta_{jk},$
(19)

where $\delta_{jk}$ is the Kronecker delta, which allows us to determine the
spin-flip rates in vacuum.

####  II.2.2 Johnson noise lifetime calculation

In this section, we use Eq. (1) to determine an appropriate value of the
atomic transition frequency for input into our Johnson noise lifetime
calculations in the presence of the magnetic trapping field. For the
$\ket{F,m_{F}}=\ket{2,2}$ $5^{2}S_{1/2}$ ground-state of the 87Rb atoms
considered here, $g_{F}=1/2$ [58, 75]. Here, we only consider the Zeeman
transition from $\ket{2,2}$ to $\ket{2,1}$ to facilitate direct comparison
with the results of Ref. [32]. The angular frequency of the radiation is then
given by

$\omega_{if}=\frac{\mu_{B}\absolutevalue{\mathbf{B(\mathbf{r}_{0})}}}{2\hbar}.$
(20)

Taking
$\absolutevalue{\mathbf{B(\mathbf{r}_{0})}}=$0.8\text{\times}{10}^{-4}\text{\,}\mathrm{T}$$
gives $\omega_{if}=2\pi\times$560\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.
Comparing with the hyperfine splitting frequency for the ground-state of the
87Rb atom, which is $2\pi\times$6.83\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$,
we now see that our assumption that $m_{F}$ is a good quantum number is
justified. The method for calculating the Clebsch-Gordon coefficients
associated with the spin-flip transition matrix elements,
$\bra{f}\hat{\mathbf{S}}_{j,k}\ket{i}$, can be found in [79]. For
completeness, we note that these matrix elements are

$\begin{split}\bra{2,2}\hat{\mathbf{S}}_{x}\ket{2,1}&=\phantom{-}\bra{2,1}\hat{\mathbf{S}}_{x}\ket{2,2}=\frac{1}{4},\\\
\bra{2,2}\hat{\mathbf{S}}_{y}\ket{2,1}&=-\bra{2,1}\hat{\mathbf{S}}_{y}\ket{2,2}=\frac{\mathrm{i}}{4},\\\
\bra{2,2}\hat{\mathbf{S}}_{z}\ket{2,1}&=\phantom{-}\bra{2,1}\hat{\mathbf{S}}_{z}\ket{2,2}=0.\end{split}$
(21)

To proceed with our calculations of the transition rates and comparisons for
different surface materials, we consider the typical thickness of the metallic
wires used to generate the magnetic field in atom-chips, which is
$\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ [32]. Figure 7 shows the
Johnson noise-limited lifetimes of the atom cloud, $\tau$, calculated versus
atom-surface distance, $y_{0}$, for a $1$-$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-thick gold slab (solid yellow curve), a $125$-$\text{\,}\mathrm{nm}$-thick
gold slab (solid green curve), a doped graphene monolayer with
$E_{F}=$0.1\text{\,}\mathrm{e}\mathrm{V}$$ (dashed red curve), an undoped
graphene monolayer (dashed blue curve) and a heterostructure consisting of a
graphene monolayer encased by two 10-nm-thick hBN layers (solid black curve).
As in [32], these Johnson noise lifetimes are calculated using Eqs. (14) and
(16). The graphene monolayers yield far longer lifetimes than the gold wires
do, even for a small wire thickness of $125\text{\,}\mathrm{nm}$. Making gold
wires thinner than $125$ $\text{\,}\mathrm{nm}$ is possible, but their
resistivities then become higher than for bulk gold [65]. At
$y_{0}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the lifetimes for the
undoped graphene monolayer and the $1$ $\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-thick gold slab are $\sim$2500\text{\,}\mathrm{s}$$ and
$0.34\text{\,}\mathrm{s}$, respectively, giving a lifetime ratio of
$\sim$7.4\text{\times}{10}^{3}\text{\,}$$, which is broadly consistent with
the estimate of $\sim$6.3\text{\times}{10}^{3}\text{\,}$$ obtained from Eq.
(11). The lifetime for the heterostructure is slightly longer than that for
the doped graphene layer.

Figure 7: Johnson noise lifetimes, $\tau$, calculated (using Eqs. (14) and
(16)) versus the position of the harmonic trap center, $y_{0}$, for an ultra-
cold gas of 87Rb atoms trapped above: an undoped graphene monolayer (dashed
blue curve); a doped graphene monolayer with Fermi energy
$E_{F}=$0.1\text{\,}\mathrm{e}\mathrm{V}$$ (dashed red curve); an hBN-encased
graphene-based heterostructure (solid black curve); a $1$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick gold slab (solid yellow curve);
a $125$ $\text{\,}\mathrm{nm}$-thick gold slab (solid green curve). The
graphene monolayers yield orders of magnitude longer lifetimes than the $1$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick gold slab because they have far
lower electromagnetic reflectance than gold (see text). Parameters:
$T=$300\text{\,}\mathrm{K}$$,
$\omega=2\pi\times$560\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

We conclude that Johnson noise in graphene conductors will produce negligible
spin-flip losses compared to the thick
($t_{Au}\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) metal wires typically
used in atom-chips, where it dominates the loss rate. Consequently, our
analysis of graphene atom-chips will, henceforth, focus on the effects of
tunneling and three-body losses and of spatial imperfections. We note,
however, from Eq. (11) that the lifetime above metallic conductors can be
increased by decreasing their thickness $t_{Au}$ and, hence, $N_{Au}$. For
wires with $t_{Au}=125$ nm, $\tau_{G}/\tau_{Au}\sim 1400$. Taking the limit of
the gold layer thickness to its lattice constant, 0.4 nm, gives
$\tau_{G}/\tau_{Au}\sim 5$. So the advantage of graphene over metallic
conductors persists even if the metal wire could be thinned close to the
theoretical limit of a monolayer. To our knowledge, though, graphene and other
exfoliated van der Waals materials are the only monolayers so far produced.
Moreover, their hexagonal crystal structure and resulting light-like linear
energy band dispersion relations ensure that they can carry high currents
despite their low thickness and carrier density. However, if high-quality
metallic monolayers could be produced, their low electron density may reduce
the Johnson noise and Casimir-Polder potential to levels comparable with
exfoliated 2D materials.

###  II.3 Negligible corrugation effects

Spatial meandering of the current stream lines can, in principle, be created
in four ways: deviation from strictly two-dimensional current flow (analogous
to surface roughness of 3D conductors); edge roughness resulting from
imperfect lithography; electrons scattering from one another or from phonons;
spatial variations in the electron potential energy created by impurities or
imperfections in, or near, the conducting channel [40, 41, 80, 81, 82]. We now
consider the importance of each potential source of roughness in turn.

When graphene is encapsulated in hBN, surface roughness and non-two-
dimensionality in graphene is only of order 12 pm [83] because the hBN
provides an ultraflat surface for the graphene and is closely lattice matched
to it [84, 85, 86]. Such low roughness is consistent with that of an
individual graphene layer in bulk graphite and will have a negligible effect
on the atom trapping potential landscape. Edge roughness will be determined by
the quality of the lithography used to define and create the conducting
channels. Since graphene is two dimensional, there will be negligible vertical
fluctuations in the channel wall. Edge fluctuations along the channel will be
determined by the lithographic process used and comparable to those in
existing atom-chips with metallic conductors. For electron beam lithography,
the edge fluctuations will be of order 35 nm [87], whereas for helium ion
beams, values below 5 nm are attainable [88].

In metallic conductors, grain boundaries give rise to local electron
scattering processes, which can be detected via their effect on the current
flow pattern and resulting modulation of the trapping potential and BEC atom
density [40, 41, 88, 89]. By contrast, graphene monolayers contain no grains
to induce position-specific scattering processes and resulting atom density
fluctuations. Electron-electron and electron-phonon scattering events do
occur, but these are spatio-temporally stochastic, rather than occurring at
particular fixed positions within the conducting channel and will therefore
not produce roughness in the trapping potential and BEC density profile
because of time averaging. Moreover, since their characteristic length scales
are shorter than the typical dimensions of atom-chip wires, ballistic
transport effects do not need to be considered. However, electron scattering
mechanisms do affect the diffusive electron mobility and, hence, the Johnson
noise-limited spin-flip lifetime of the trapped atom cloud.

Spatial fluctuations in the electronic potential energy created by
imperfections and impurities that are either within the graphene or accumulate
at interfaces in hBN-encased graphene structures have been studied
theoretically and measured in resonant-tunneling experiments [85, 90, 91, 92,
93, 94]. Self-consistent calculations [92, 93], which give excellent
quantitative agreement with measurements of graphene’s electron mobility,
$\mu_{G}$, versus impurity density and with scanning probe surface studies
[91], predict that the correlation length of these potential fluctuations is
$\approx 10$ nm. Recent experiments on graphene-boron nitride tunnel
transistors have shown that for graphene monolayers encased by several layers
of hexagonal boron nitride (hBN), the correlation length is $\approx 12$ nm
[90, 95]. Consequently, the associated small-angle current meander will have
negligible effect on the potential landscape of atoms trapped even as close as
150 nm from the graphene and will therefore not influence the minimum atom-
surface trapping distance.

When graphene is placed or grown epitaxially on hBN, the small lattice
mismatch between the two materials gives rise to a strain-induced moiré
pattern and superlattice potential, which can modify the electronic properties
of electrons within the graphene. Moiré periods up to 80 nm have been realized
[96] and further increases in period may modulate the current flow on a length
scale long enough to produce detectable variation in the density profile of a
BEC trapped nearby. Such variations could yield information about the
superlattice potential and the underlying strain mechanisms.

##  III Lifetime of A tRapped atomic BEC

In this section, we find an analytical expression for the total loss rate of
an elongated atomic BEC trapped in the vicinity of an atom-chip. We consider
contributions from atom tunneling towards the chip surface (Sec. II.1),
Johnson-noise induced losses (Sec. II.2) and the 3-body loss mechanism.

###  III.1 Methodology

First, we consider a harmonic magnetic trapping field,
$\mathbf{B}(\mathbf{r})$, formed near the surface of an atom-chip in the
coordinate system shown in Fig. 1, where $\mathbf{r}=(x,y,z)$, $\omega_{x}$,
$\omega_{y}$, and $\omega_{z}$ are the characteristic trapping frequencies in
the $x$-, $y$-, and $z$-axes, respectively, and the trap center is located at
$\mathbf{r}_{c}=(0,y_{c},0)$. The potential energy profile of an atom
interacting with this magnetic field is modelled by an anisotropic three-
dimensional harmonic-oscillator potential

$U(x,y,z)=\frac{1}{2}m\big{(}\omega_{x}^{2}x^{2}+\omega_{y}^{2}(y-y_{c})^{2}+\omega_{z}^{2}z^{2}\big{)}.\\\
$ (22)

Note that this magnetic potential originates from the interaction of the
magnetic moment of the trapped atom and the magnetic field given in Eq. (1)
and that the actual potential profile of an atom-chip trap is determined by
the wire and current configurations. Equation (22) gives a good approximation
for the potential landscape generated by the Z-shaped trapping wires often
used in atom-chip experiments.

We assume that such a magnetic trap has cylindrical symmetry and is elongated
along the $z$-axis, so that $\omega_{r}=\omega_{x,y}$, and
$\omega_{r}\gg\omega_{z}$, where $\omega_{r}$ denotes the trapping frequency
in the radial direction (i.e. in the $x$-$y$ plane). We also assume that
$\omega_{r}$ is so high that the trapped atoms only occupy the ground-state
energy in the radial direction. To include the perturbing effect of the CP
potential on the effective trapping frequency in the $y$-direction, henceforth
we approximate the radial trapping frequency as
$\omega_{r}=\sqrt{\omega_{x}\omega_{\mathrm{eff}}}$. An additional offset
magnetic field, $\mathbf{B}_{0}=(0,0,B_{z})$, of order
$\text{\,}\mathrm{m}\mathrm{T}$, is added in the $z$-direction to ensure that
the magnetic field is non-zero at the trap center. Together, these assumptions
enable us to treat the magnetic potential energy landscape as a highly
elongated, quasi one-dimensional, trap.

It follows from the above assumptions about the trapping frequencies that the
chemical potential, $\mu$, of the condensate must satisfy the following
constraints

$5\hbar\omega_{z}<\mu<\frac{3}{2}\hbar\omega_{r},$ (23)

which allows us to further assume that the mean atom density profile of the
condensate can be described by a one-dimensional Thomas-Fermi distribution in
the elongated ($z$) direction, and by the Gaussian ground-state wave function
of a quantum harmonic oscillator in the tightly-confining radial ($r$)
direction [97]. The atom density profile is then given by

$\rho_{0}(r,z)=\frac{1}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)}e^{-r^{2}/2a_{r}^{2}},$
(24)

where $U_{0}=4\pi\hbar^{2}a_{T}/m$, is the effective interaction strength for
a pair of slowly moving atoms of s-wave scattering length $a_{T}$ [98],
$\mu_{\mathrm{eff}}=\mu-\hbar\omega_{r}$,
$m=$1.44\text{\times}{10}^{-25}\text{\,}\mathrm{k}\mathrm{g}$$ is the mass of
an 87Rb atom, $r=\sqrt{x^{2}+(y-y_{c})^{2}}$ is the radial distance relative
to the center of the trap, and $a_{r}=\sqrt{\hbar/m\omega_{r}}$ is the
characteristic harmonic oscillator length.

Integrating Eq. (24) over the radial co-ordinate gives the mean line density
profile along the $z$-axis (see Appendix D)

$n_{0}(z)=\frac{2\pi
a_{r}^{2}}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)},$
(25)

where, the chemical potential $\mu$ of the trapped atom cloud is determined by
the peak mean line density, i.e. at $z=0$, as follows [99]:

$\mu=\big{(}2a_{T}n_{0}(0)+1\big{)}\hbar\omega_{r},$ (26)

where $a_{T}=$5.6\text{\,}\mathrm{nm}$$ is the scattering length for 87Rb
atoms in the $\ket{F,m_{F}}=\ket{2,2}$ state [100].

Figure 8: Color map of the atom volume density calculated for the trapped atom
cloud using the Thomas-Fermi distribution in Eq. (24). The color bar scale is
in units of $\text{\,}\mathrm{m}^{-3}$. Parameters:
$\omega_{r}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
$\omega_{z}=0.006\times\omega_{r}$, $N=750$.

We now define the total lifetime of the trapped atom cloud,
$\tau_{\mathrm{tot}}$, to be the time taken for the initial peak atom line
density, $n_{0}(z=0)$, to drop below the smallest experimentally-detectable
line density, which we take to be
$n_{\mathrm{min}}=$3\text{\times}{10}^{6}\text{\,}\mathrm{m}^{-1}$$ [101]. We
determine the upper limit of $\tau_{\mathrm{tot}}$ by taking $n_{0}(z=0)$ to
be the maximum possible value, $n_{\mathrm{max}}=14.8\times n_{\mathrm{min}}$,
satisfying inequality (23).

Let us now consider the density-dependent loss rates originating from three
distinct atom loss mechanisms; Johnson noise-induced spin flips, quantum
tunneling to the chip surface, and three-body processes. The Johnson noise-
induced loss rate is

$\frac{\mathrm{d}n_{0}(z)}{\mathrm{d}t}\biggr{\rvert}_{\mathrm{JN}}=-\Gamma_{\mathrm{JN}}n_{0}(z),$
(27)

where $\Gamma_{\mathrm{JN}}$ is the Johnson noise-induced spin-flip transition
rate given in Eq. (14).

As described above, the tunneling loss rate has a similar form

$\frac{\mathrm{d}n_{0}(z)}{\mathrm{d}t}\biggr{\rvert}_{\mathrm{tun}}=-\Gamma_{\mathrm{tun}}n_{0}(z),$
(28)

where $\Gamma_{\mathrm{tun}}=\omega_{\mathrm{eff}}\tilde{T}/2\pi$.

By contrast, the three-body loss rate is proportional to the cube of the mean
line density [102, 103],

$\frac{\mathrm{d}n_{0}(z)}{\mathrm{d}t}\biggr{\rvert}_{\mathrm{3b}}=-\Gamma_{\mathrm{3b}}n_{0}(z)^{3},$
(29)

where $\Gamma_{\mathrm{3b}}=\kappa_{\mathrm{Rb}}/12\pi^{2}a_{r}^{4}$ and
$\kappa_{\mathrm{Rb}}=$1.8\text{\times}{10}^{-41}\text{\,}\mathrm{m}^{6}\mathrm{s}^{-1}$$
is the three-body recombination rate for 87Rb in the $F=m_{F}=2$ state [104].

Combining all three distinct loss rates gives the total loss rate

$\frac{\mathrm{d}n_{0}(z)}{\mathrm{d}t}\biggr{\rvert}_{\mathrm{tot}}=-\Gamma_{\mathrm{3b}}n_{0}(z)^{3}-(\Gamma_{\mathrm{tun}}+\Gamma_{\mathrm{JN}})n_{0}(z).$
(30)

In this paper, we will only consider losses occurring at $z=0$, where the line
density peaks and so the total loss rate is maximal. Hence, we determine the
lower limit on the total lifetime given by the integral:

$\tau_{\mathrm{tot}}=\int_{n_{\mathrm{max}}}^{n_{\mathrm{min}}}\frac{\mathrm{d}n_{0}(z)}{-\Gamma_{\mathrm{3b}}n_{0}(z)^{3}-(\Gamma_{\mathrm{tun}}+\Gamma_{\mathrm{JN}})n_{0}(z)}\Big{\rvert}_{z=0},$
(31)

which can be integrated analytically to yield:

$\tau_{\mathrm{tot}}=\frac{\log{\Bigg{[}\frac{\Gamma_{\mathrm{3b}}n_{\mathrm{min}}^{2}+(\Gamma_{\mathrm{tun}}+\Gamma_{\mathrm{JN}})}{\alpha^{2}\Gamma_{\mathrm{3b}}n_{\mathrm{min}}^{2}+(\Gamma_{\mathrm{tun}}+\Gamma_{\mathrm{JN}})}\Bigg{]}}\\\
+2\log{(\alpha)}}{2(\Gamma_{\mathrm{tun}}+\Gamma_{\mathrm{JN}})},$ (32)

where $\alpha=n_{\mathrm{max}}/n_{\mathrm{min}}=14.8$.

###  III.2 Results

In this section, we calculate and compare atom-cloud lifetimes for three
different surface structures: a $1$ $\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-thick gold slab, a graphene monolayer, and a graphene monolayer encased by
two $10$ $\text{\,}\mathrm{nm}$-thick hBN layers. The first structure is
representative of the present generation of atom-chips, and provides typical
lifetimes for comparison with graphene-based devices. The second structure is
the theoretical limit for 2D materials and exemplifies the predicted
improvements in performance and functionality. The third atom-chip structure
is within the scope of existing fabrication techniques for vdW
heterostructures. The graphene conductor is encased within hBN multilayers,
which support it and shield it from adsorbates.

Fig. 9 shows the lifetimes resulting from each of the three loss mechanisms
considered in the previous section, together with the total lifetime,
calculated versus the position of the harmonic trap center, $y_{0}$, for 87Rb
atoms near (a) a graphene monolayer and (b) the $1$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick gold slab. Note that in this
figure and henceforth, all the lifetimes are calculated using Eqs. (27) to
(32), which account for the minimum experimentally-detectable atom density,
whereas those shown in Figs. 6 and 7 are calculated using Eqs. (10), (14) and
(16) to facilitate comparison with the corresponding lifetimes reported in
[32]. The unperturbed transverse trapping frequency,
$\omega_{y}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, is used in
all cases.

Considering Fig. 10, we firstly note that the 3-body loss lifetimes (dashed
red curves) are identical for the two structures, as expected from Eq. (29).
Secondly, as a consequence of weaker CP attraction, the tunneling loss
lifetime for the graphene monolayer is longer than for the gold slab (compare
dot-dashed green curves) and the minimum atom-surface trapping distance is
lower. Thirdly, significant improvement in the Johnson noise-limited lifetime
is apparent for the graphene monolayer. Whereas Johnson noise in the metal
wire limits the total atom lifetime, for graphene the 3-body lifetime of
$\sim$12.5\text{\,}\mathrm{s}$$ is the limiting factor and Johnson noise is
insignificant.

Figure 9: Lifetimes calculated (from Eqs. (27) to (32)) versus the position of
the harmonic trap center, $y_{0}$, for an 87Rb quasi-condensate trapped near
(a) a graphene monolayer and (b) a $1$ $\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-thick gold slab, for three different loss mechanisms: 3-body processes
(dashed red curves); tunneling losses (dot-dashed green curves); Johnson
noise-induced losses (solid blue curves). The total lifetime is shown by the
solid black curves. For the graphene monolayer, the total lifetime is limited
by 3-body losses to $\sim$12.5\text{\,}\mathrm{s}$$, whereas for the gold slab
the total lifetime is limited by Johnson noise. Parameter:
$\omega_{y}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

Figure 10 shows lifetimes calculated for 87Rb atoms near the hBN-graphene
heterostructure, taking the same trapping frequency as in Fig. 9. The only
noticeable difference in the lifetimes compared with those for a graphene
monolayer alone relates to tunneling loss (dashed red curve). For given
$y_{0}$, the hBN-graphene structure generates a higher CP potential and,
hence, a shorter tunneling lifetime and a higher minimum distance from the
trap center to the surface. The Johnson noise is insensitive to the addition
of the hBN cladding layers because such layers change neither the number nor
mobility of the free charge carriers in the graphene and the total lifetime is
still limited by the 3-body loss mechanism.

Figure 10: Lifetimes calculated (from Eqs. (27) to (32)) versus the position
of the harmonic trap center, $y_{0}$, for an 87Rb quasi-condensate trapped
near an hBN-encased graphene heterostructure for three different loss
mechanisms: 3-body processes (dashed red curve); tunneling losses (dot-dashed
green curve); Johnson noise-induced losses (solid blue curve). The total
lifetime is shown by the solid black curve. For
$y_{0}>$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, where the magnetic trap
has a well-defined barrier on the side near the surface, the lifetimes are
virtually identical to those for a single layer of graphene. Parameters:
$\omega_{y}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, hBN
thickness $=$10\text{\,}\mathrm{nm}$$.

Figure 11 shows color maps of the total lifetimes, calculated versus the
position of the trap center from the surface and the transverse trapping
frequency for (a) the hBN-graphene heterostructure and (b) the $1$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick gold slab. The color scale is
logarithmic and is common to (a) and (b). Whereas the total lifetime for the
thin gold slab is mainly below $1\text{\,}\mathrm{s}$ (yellow-green in color
scale), for the hBN-graphene structure it exceeds $100\text{\,}\mathrm{s}$
(red shading) for high $y_{0}$ and low $\omega_{y}$ values.

Figure 11: Color maps of total lifetimes, calculated (from Eqs. (27) to (32))
versus $y_{0}$ and $\omega_{y}/2\pi$ for (a) an hBN-encased graphene
heterostructure, (b) a $1$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick gold
slab with a common color scale (right). The lifetime is not defined in the
white region because the CP potential distorts the harmonic magnetic potential
(i.e. reduces the barrier nearest to the surface) so strongly that the trap
cannot be formed. For any given $y_{0}$ and $\omega_{y}/2\pi$ values, the
lifetime for the hBN-graphene structure is longer than for the thin gold slab.

##  IV Possible Implementations

In order to see whether a close surface magnetic trap can be realized using
graphene wires we assume a simple side guide configuration, consisting of
graphene wire carrying a current, $I$, and a bias magnetic field of magnitude
$\absolutevalue{\mathbf{B}_{b}}$. This will form a magnetic minimum at a
distance

$y_{0}=\frac{\mu_{0}}{2\pi}\frac{I}{\absolutevalue{\mathbf{B}_{b}}}$ (33)

from the graphene sheet. Using the derivation given in [17] and assuming that
the trapping distance is larger then the width of the graphene wire, the
trapping frequency is approximated by

$\omega_{y}=2\pi\sqrt{\frac{\mu_{B}g_{F}m_{F}}{m\absolutevalue{\mathbf{B}_{0}}}}\frac{\absolutevalue{\mathbf{B}_{b}}^{2}}{I\mu_{0}},$
(34)

where $\absolutevalue{\mathbf{B}_{0}}$ is the magnitude of the offset magnetic
field parallel to the direction of current flow used to avoid Majorana spin-
flips. A trap frequency of $\omega_{y}\approx
2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ is therefore realizable
at a distance of $400\text{\,}\mathrm{nm}$ with a total current of
$0.7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{A}$ in addition to a bias
field of $35\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$ and an offset
(Ioffe) field of $80\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{T}$. Since
exfoliated and epitaxial graphene on bulk substrates can support current
densities in excess of $\sim$ 1000 A/m even in an ultra-high vacuum [105,
106], and current densities as high as $\sim$ 700 A/m have been reported for
free-standing monolayer graphene [107], a graphene conducting channel only 50
nm wide would be sufficient to ensure trap operation with negligible heating.
Wires with larger widths could increase the possible trapping frequencies or
enable trapping further from the surface, which may assist with loading the
trap. We note, however, that such a trap could not be loaded directly but
would need to be mounted on a carrier chip featuring thick metal wires, which
generate the field used initially to cool and trap the atoms. This carrier
chip must be placed far enough from the graphene and the atoms to produce
negligible Johnson noise and CP attraction effects, but also close enough to
create a sufficiently compressed trap. Given a
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ separation between the atom cloud
and the carrier chip, gold wires carrying a current of 1A could produce the
transverse trap frequency of
$\omega_{r}=2\pi\times$20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ needed for
the traps shown in Figs. 4, 6, 9 and 10. Since thin van der Waals
heterostructures are almost transparent, laser light can pass through them and
be retro-reflected from a metal coating on the carrier chip in order to form a
mirror MOT.

In an alternative configuration, the potential for trapping the atoms could be
provided by optical fields, for example an electromagnetic standing wave
generated by on-chip mounted optics. In this case, the graphene wires could be
used to perturb strongly the optical potential or act as a source of magnetic
fields to enable, for example, tuning the scattering length via Feshbach
resonances.

Another issue that has been observed when trapping atoms close to a surface,
especially metal, is the effect of stray electric fields originating from the
polarization of adsorbed atoms [23]. Although this effect has not yet been
measured for graphene surfaces, covering the graphene layer with a dielectric
layer such as hBN is expected to suppress these effects by limiting
polarization of any adsorbates and keeping them away from the graphene
layer(s), so preventing them doping it.

Graphene-based atoms chips could be fabricated by MBE growth of graphene [108,
109, 110], or deposition of exfoliated graphene on hBN, followed by selective
etching of the graphene to define the conducting channel and, finally,
deposition of capping layers of hBN either by epitaxial growth or by placing
exfoliated hBN layers, as now widely done to create van der Waals
heterostructures [38, 39].

##  V Conclusions

In summary, we have presented a general formalism for calculating how the
lifetime of an atomic quantum gas is affected by the Johnson noise and atom-
surface CP attraction of van der Waals heterostructures comprising arbitrary
configurations of 2D materials such as graphene. The electromagnetic
reflection coefficients and corresponding electrical conductivities of the 2D
layers are of crucial importance in determining both the Johnson noise and CP
potential. Since both of these parameters are lower for graphene than for the
metal layers generally used in atom-chips, so too are the Johnson noise and CP
atom-surface attraction. Consequently, for given atom-surface separation, the
spin-flip and tunneling loss rates are both lower near graphene-based van der
Waals heterostructures than near metal wires, meaning that such
heterostructures can improve the performance of atom-chips. For example,
although high Johnson noise limits the lifetime of atom clouds trapped between
0.4 and $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the chip surface to
less than 1s, such noise is negligible for atoms trapped near graphene, whose
lifetime can, in principle, reach $\sim$ 100 s, limited only by 3-body losses
and background gas collisions. For atom-surface separations below
$0.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the lifetime of the atom cloud is
limited by tunneling losses for both metallic and 2D conductors. However, due
to the weak CP atom-surface attraction, such losses are lower near van der
Waals heterostructures; around 4 orders of magnitude lower for atoms held
$0.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the surface.

As a result of their favourable noise and CP characteristics, van der Waals
heterostructures offer a solid-state solution to the long-standing challenge
of holding ultracold atom clouds closer than
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from an atom-chip surface for long
enough (up to 100 s) to perform various experiments and measurements on the
atom cloud. Moreover, van der Waals heterostructures that enable robust
sub-$\text{\,}\mathrm{\SIUnitSymbolMicro m}$ atom trapping would control
atomic condensates on length scales that are smaller than presently achievable
optically and below the healing length, thereby providing access to new
regimes of quantum many-body physics. The ability to achieve long lifetimes
for ultracold atoms held as close as 400 nm to an electronic device also opens
a route to creating new hybrid atomic-solid state quantum systems, for example
a Rydberg atom coupled to a quantum dot formed within a 2D conductor [111,
112]. Since the micron-scale confinement length of electrons in the quantum
dot is similar to that of the excited electron in the Rydberg atom, new types
of electron orbital, shared between the atom and the condensed matter parts of
the system may be created. Such hybrid states may yield new regimes of quantum
control and information storage/processing, for example relating to side-band
cooling of graphene [113].

##  Appendix A Dyadic Green’s function

In this Appendix, we discuss Green’s tensors for planar multilayer systems,
like those considered in the main text. We will start by considering the
general characteristics of an electric field in a homogeneous space, in order
to provide an intuitive explanation of the Green’s tensors.

###  A.1 General definition

Let us recall the inhomogeneous Helmholtz equation describing the relationship
between an electric field, $\mathbf{E}$, and a current density, $\mathbf{j}$,
of angular frequency $\omega$ at position $\mathbf{r}$ with respect to a
linear, isotropic, inhomogeneous medium, with relative magnetic permeability,
$\mu(\omega)$, and relative permittivity $\epsilon(\omega)$ [114]:

$\curl{\curl{\mathbf{E}(\omega,\mathbf{r})}}-k^{2}\mathbf{E}(\omega,\mathbf{r})=\mathrm{i}\omega\mu_{0}\mu\mathbf{j}(\omega,\mathbf{r}).$
(35)

Here, $k=\sqrt{\epsilon(\omega)\mu(\omega)\omega^{2}/c^{2}}$ is the magnitude
of the wave vector associated with the electromagnetic wave, $\mu_{0}$ is the
permeability of free space, and

$\mathbf{j}(\omega,\mathbf{r})=j_{x}(\omega,\mathbf{r})\mathbf{e}_{x}+j_{y}(\omega,\mathbf{r})\mathbf{e}_{y}+j_{z}(\omega,\mathbf{r})\mathbf{e}_{z},$
(36)

where $\mathbf{e}_{x},\mathbf{e}_{y}$ and $\mathbf{e}_{z}$ are the unit
vectors in the $x$-, $y$\- and $z$-direction, respectively, with $j_{x}$,
$j_{y}$ and $j_{z}$ being the corresponding current density components.

Figure 12: Illustration of the dyadic Green’s function
$\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)$. The Green’s function
relates the local current source, $\mathbf{j}$, at point,
$\mathbf{r}^{\prime}$, and the associated electric field $\mathbf{E}$ at point
$\mathbf{r}$. The total electric field is the superposition of every field
corresponding to each point source in the source body of volume $V$.

The classical Green’s tensor
$\mathbf{G}=(\mathbf{G}_{x},\mathbf{G}_{y},\mathbf{G}_{z})$, as a function of
the field point position, $\mathbf{r}$, the source point position,
$\mathbf{r}^{\prime}$, and the wave angular frequency, $\omega$, is the unique
solution to the following differential equations:

$\curl{\curl{\mathbf{G}_{x}(\mathbf{r},\mathbf{r}^{\prime},\omega)}}-k^{2}\mathbf{G}_{x}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\delta(\mathbf{r}-\mathbf{r}^{\prime})\mathbf{e}_{x},$
(37)
$\curl{\curl{\mathbf{G}_{y}(\mathbf{r},\mathbf{r}^{\prime},\omega)}}-k^{2}\mathbf{G}_{y}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\delta(\mathbf{r}-\mathbf{r}^{\prime})\mathbf{e}_{y},$
(38)
$\curl{\curl{\mathbf{G}_{z}(\mathbf{r},\mathbf{r}^{\prime},\omega)}}-k^{2}\mathbf{G}_{z}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\delta(\mathbf{r}-\mathbf{r}^{\prime})\mathbf{e}_{z},$
(39)

where $\delta(\mathbf{r}-\mathbf{r}^{\prime})$ is the Dirac delta function.
These three Green’s functions, in a column vector form, can be combined into a
single tensor, giving the following general definition of the dyadic Green’s
function (Green’s tensor) for the electric field:

$\curl{\curl{\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)}}-k^{2}\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\mathbf{I}\delta(\mathbf{r}-\mathbf{r}^{\prime}),$
(40)

where $\mathbf{I}$ is the unit dyad (unit tensor).

We can see that each column of the tensor $\mathbf{G}$ can be mathematically
treated individually: the curl operators can be applied to any column of the
Green’s tensor as if they were to act on a single column vector. In addition,
each column can be interpreted separately: the first column of the Green’s
tensor describes the field due to a point source in the $x$-direction, the
second column the field due to a point source in the $y$-direction, and the
third column the field due to a point source in the $z$-direction.

Consequently, a particular solution of Eq. (35) defined by the dyadic Green’s
function is

$\mathbf{E}(\omega,\mathbf{r})=\mathrm{i}\omega\mu\mu_{0}\int_{V}\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)\mathbf{j}(\omega,\mathbf{r}^{\prime})\mathrm{d}^{3}r^{\prime},$
(41)

where the integral is evaluated over the volume $V$ of the current source body
(see Fig. 12).

###  A.2 Green’s tensor for planar multilayer systems

Let us consider the situation shown in Fig. 2, where a radiating electric
dipole is located above a layered substrate. We assume that the upper half-
space is vacuum while the lower half-space (atom-chip substrate) is optically
denser.

Since the electric field in layer 1 is the superposition of the field directly
radiated from the dipole and the field scattered by the material layers, the
Green’s function can, correspondingly, be decomposed into two contributions: a
Green’s function for homogeneous space and a scattering Green’s function
reflecting dielectric inhomogeneity. In order to find the primary dyadic
Green’s function, $\mathbf{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime},\omega)$,
(sometimes called the “free-space Green’s function”), we remove the interfaces
in Fig. 2 and assume that the electric dipole, $\mathbf{d}$, located at
$\mathbf{r}^{\prime}=(x^{\prime},y^{\prime},z^{\prime})$, is in a homogeneous,
linear and isotropic medium, characterized by permittivity and permeability
functions $\epsilon_{1}(\omega)$ and $\mu_{1}(\omega)$. The superscript $(0)$
here is to remind us that this Green’s function is the primary Green’s
function. The associated electric field at $\mathbf{r}=(x,y,z)$, and its
corresponding wave vectors, respectively, are

$\mathbf{E}(\mathbf{r},\omega)=\omega^{2}\mu_{0}\mu_{1}\mathbf{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime},\omega)\mathbf{d}(\mathbf{r}^{\prime},\omega),$
(42)
$\mathbf{k}_{1}(\omega)=k_{x}\mathbf{e}_{x}+k_{y1}\mathbf{e}_{y}+k_{z}\mathbf{e}_{z}=k^{\parallel}\mathbf{e}_{k^{\parallel}}+k_{y1}\mathbf{e}_{y},$
(43)

where $k_{x}$ and $k_{z}$ are the $x$\- and $z$-components of the wavevector
$k^{\parallel}\mathbf{e}_{k^{\parallel}}=k_{x}\mathbf{e}_{x}+k_{z}\mathbf{e}_{z}$
in the $x-z$-plane, which are the same in every layer, and $k_{y1}$ is the
$y$-component of the wave vector in layer 1.

The primary Green’s tensor can be written in the form [77, 53, 78]

$\mathbf{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\frac{\mathrm{i}}{8\pi^{2}}\iint_{-\infty}^{\infty}\frac{1}{k_{y1}}\mathbf{M}^{(0)}(k_{x},k_{z})\times\\\
\mathrm{e}^{\mathrm{i}[k_{x}(x-x^{\prime})+k_{z}(z-z^{\prime})+k_{y1}\absolutevalue{y-y^{\prime}}]}\mathrm{d}k_{x}\mathrm{d}k_{z},$
(44)

in which

$\mathbf{M}^{(0)}(k_{x},k_{z})=(\mathbf{e}_{s\pm}\otimes\mathbf{e}_{s\pm})+(\mathbf{e}_{p\pm}\otimes\mathbf{e}_{p\pm}),$
(45)

where $\otimes$ represents a tensor product. Here, the polarization unit
vectors for $s$\- and $p$-polarized waves in layer 1 are defined as (see Fig.
13)

$\displaystyle\mathbf{e}_{s\pm}$
$\displaystyle=\mathbf{e}_{k^{\parallel}}\times\mathbf{e}_{y},$ (46)
$\displaystyle\mathbf{e}_{p\pm}$
$\displaystyle=\frac{1}{k_{1}}(k^{\parallel}\mathbf{e}_{y}\mp
k_{y1}\mathbf{e}_{k^{\parallel}}),$ (47)

where
$k_{1}=\sqrt{\epsilon_{1}(\omega)\mu_{1}}(\omega)\omega/c=(k_{y1}^{2}+k^{\parallel
2})^{1/2}$ is the wave number and the upper (-) sign applies for
$y>y^{\prime}$ (i.e. waves propagating in the positive $y$-direction), whilst
the lower (+) sign applies for $y<y^{\prime}$ (i.e. waves propagating in the
negative $y$-direction).

Figure 13: Definition of the polarization unit vectors. The plane spanned by
the vector $\mathbf{e}_{y}$, pointing in the positive $y$-direction, and the
vector $\mathbf{e}_{k^{\parallel}}=(k_{x},0,k_{z})/k^{\parallel}$ defines the
plane of incidence. The $s$-polarization unit vectors, $\mathbf{e}_{s\pm}$,
are perpendicular to the plane of incidence, whilst the $p$-polarization unit
vectors, $\mathbf{e}_{p\pm}$, are parallel to the plane of incidence and
perpendicular to the directions of propagation (blue arrows).

Let us now consider what happens to the electric field described by the above
primary Green’s function when a reflective planar layered structure is added
to the system below the dipole, as depicted in Fig. 2. Physically, the role of
the layered structure is to reflect back the electromagnetic wave radiated
from the dipole. Therefore, mathematically, multiplying the individual
incident plane waves in $\mathbf{G}^{(0)}$ with the corresponding generalized
Fresnel reflection coefficients, $r_{s}^{(1)}$ and $r_{p}^{(1)}$, along with
changing the phase factor (the exponential term) accordingly, yields the
scattering Green’s tensor:

$\mathbf{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\frac{\mathrm{i}}{8\pi^{2}}\iint_{-\infty}^{\infty}\frac{1}{k_{y1}}\mathbf{M}^{(1)}(k_{x},k_{z})\\\
\times\mathrm{e}^{\mathrm{i}[k_{x}(x-x^{\prime})+k_{z}(z-z^{\prime})+k_{y1}(y+y^{\prime})]}\mathrm{d}k_{x}\mathrm{d}k_{z},$
(48)

where

$\mathbf{M}^{(1)}(k_{x},k_{z})=r_{s}^{(1)}(\mathbf{e}_{s+}\otimes\mathbf{e}_{s-})+r_{p}^{(1)}(\mathbf{e}_{p+}\otimes\mathbf{e}_{p-}).$
(49)

Here, the superscript $(1)$ is to remind us that the Green’s tensor is the
scattering Green’s tensor for layer 1.

The total electric field due to a radiating _electric_ dipole above a planar
structure can now be written as

$\mathbf{E}(\mathbf{r},\omega)=\omega^{2}\mu_{0}\mu_{1}\mathbf{G}_{E}(\mathbf{r},\mathbf{r}^{\prime},\omega)\mathbf{d}(\mathbf{r}^{\prime},\omega),$
(50)

where
$\mathbf{G}_{E}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\mathbf{G}^{(0)}(\mathbf{r},\mathbf{r}^{\prime},\omega)+\mathbf{G}^{(1)}(\mathbf{r},\mathbf{r}^{\prime},\omega)$
and we introduce a subscript $E$ to emphasize that this Green’s function is
for an electric dipole.

In order to obtain a scattering Green’s tensor for a _magnetic_ dipole, we
interchange the Fresnel reflection coefficients in (45).

The relevant Green’s tensors for calculating the CP potential and the Johnson
noise must be evaluated at $\mathbf{r}=\mathbf{r}^{\prime}=\mathbf{r}_{0}$,
where $\mathbf{r}_{0}=(x_{0},y_{0},z_{0})$ is the position of the center of
the magnetic trap. After straightforward manipulation of the polar
coordinates, the equal-position scattering Green’s tensor is, therefore, given
by [78, 54]

$\mathbf{G}^{(1)}(\mathbf{r}_{0},\mathbf{r}_{0},\omega)=\frac{\mathrm{i}}{8\pi}\int_{0}^{\infty}\mathrm{d}k^{\parallel}\frac{k^{\parallel}}{k_{y1}}\mathrm{e}^{2\mathrm{i}k_{y1}y_{0}}\\\
\times\Big{[}\mathbf{M}_{\alpha}r_{s}^{(1)}(k^{\parallel},\omega)+\frac{c^{2}}{\omega^{2}}\mathbf{M}_{\beta}r_{p}^{(1)}(k^{\parallel},\omega)\Big{]},$
(51)

where $y_{0}$ is the shortest distance between the surface and the center of
the magnetic trap and

$k_{y1}=\Big{(}\mu_{1}\epsilon_{1}\frac{\omega^{2}}{c^{2}}-k^{\parallel
2}\Big{)}^{1/2},$ (52)

with $k^{\parallel 2}=k_{x}^{2}+k_{z}^{2}$. The tensors $\mathbf{M}_{\alpha}$
and $\mathbf{M}_{\beta}$ in Eq. (51) are given by

$\displaystyle\mathbf{M}_{\alpha}=$ $\displaystyle\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix},$ (53) $\displaystyle\mathbf{M}_{\beta}=$
$\displaystyle\begin{pmatrix}-k_{y1}^{2}&0&0\\\ 0&2k^{\parallel 2}&0\\\
0&0&-k_{y1}^{2}\end{pmatrix}.$ (54)

Note that the forms of $\mathbf{M}_{\alpha}$ and $\mathbf{M}_{\beta}$ depend
on the coordinate system used and that Equation (51) only describes an
electromagnetic field with a _real_ frequency, created by a radiating
_electric_ dipole.

For a purely imaginary frequency, $\omega=\mathrm{i}\xi$, where $\xi$ is real,
e.g. the Matsubara frequencies that appear in the CP potential calculations,
the wave vector in the direction perpendicular to the surface is always purely
imaginary, $k_{y1}=\mathrm{i}\kappa_{1}^{\perp}$, with

$\kappa_{1}^{\perp}(\mu_{1},\epsilon_{1},\mathrm{i}\xi)=\sqrt{\mu_{1}(\mathrm{i}\xi)\epsilon_{1}(\mathrm{i}\xi)\frac{\xi^{2}}{c^{2}}+k^{\parallel
2},}$ (55)

and

$k^{\parallel}=\sqrt{\kappa_{1}^{\perp
2}-\mu_{1}\epsilon_{1}\frac{\xi^{2}}{c^{2}}}=\sqrt{\kappa_{l}^{\perp
2}-\mu_{l}\epsilon_{l}\frac{\xi^{2}}{c^{2}}},$ (56)

where the subscript $l$ denotes the layer index corresponding to each wave
vector. Equations (55) and (56) tell us that the wave numbers in the direction
perpendicular to the surface are functions of the optical properties of the
materials and the incident wave frequencies, whereas the wave numbers in the
direction parallel to the surface are constant for a given
$\kappa_{l}^{\perp}$. The equal-position Green’s tensor for _purely imaginary_
frequencies is then given by [77, 78, 114]

$\mathbf{G}^{(1)}(\mathbf{r}_{0},\mathbf{r}_{0},\mathrm{i}\xi)=\frac{1}{8\pi}\int_{\xi/c}^{\infty}\mathrm{d}\kappa_{1}^{\perp}\mathrm{e}^{-2\kappa_{1}^{\perp}y_{0}}\\\
\times\Big{[}\mathbf{M}_{\alpha}r_{s}^{(1)}(k^{\parallel},\mathrm{i}\xi)-\frac{c^{2}}{\xi^{2}}\mathbf{M}_{\beta}r_{p}^{(1)}(k^{\parallel},\mathrm{i}\xi)\Big{]}.$
(57)

Finally, we note that the Green’s tensor for a magnetic dipole, which is
needed for the Johnson noise calculations, can be readily obtained by
interchanging the reflection coefficients in (51).

##  Appendix B Optical properties of materials

In this section, we briefly introduce the optical properties of graphene, hBN
and bulk gold. They are used in order to determine the behavior of the
reflection coefficients of our system (see Appendix C).

Graphene’s optical conductivity can be split into two distinct contributions.
The first describes how the charge carriers respond to electromagnetic
radiation by transitioning to higher energy states within the same energy
bands without conserving momentum (intraband-transitions,
$\sigma_{\mathrm{intra}}(\omega)$). The second accounts for vertical momentum-
conserving transitions from the valence band to the conduction band, induced
by the electromagnetic radiation (interband- transitions,
$\sigma_{\mathrm{inter}}(\omega)$), where $\omega$ is the angular frequency of
the electromagnetic field to which the graphene is exposed.

The expression for graphene’s conductivity has been considered using multiple
approaches and limits (see for example [115, 116]) and the choice of a
specific description depends on the features of the system under analysis
and/or on the particular aspect under investigation (see for example [117,
118, 119, 120] and the references below). For our systems and its parameters
we can use the expression derived from the Kubo formula [53, 121, 122], which
gives:

$\sigma_{\mathrm{intra}}(\omega)=\frac{\sigma_{0}}{\pi}\frac{4}{\hbar\gamma-\mathrm{i}\hbar\omega}\bigg{[}E_{F}+2k_{B}T\ln\bigg(1+e^{-E_{F}/k_{B}T}\bigg{missing})\bigg{]},$
(58)

$\sigma_{\mathrm{inter}}(\omega)=\sigma_{0}\bigg{[}G(\hbar\omega/2)+\mathrm{i}\frac{4\hbar\omega}{\pi}\int_{0}^{\infty}dE\frac{G(E)-G(\hbar\omega/2)}{(\hbar\omega)^{2}-4E^{2}}\bigg{]},$
(59)

in which

$G(X)=\frac{\sinh\bigg{(}\frac{\displaystyle X}{\displaystyle
k_{B}T}\bigg{)}}{\cosh\bigg(\frac{\displaystyle E_{F}}{\displaystyle
k_{B}T}\bigg{missing})+\cosh\bigg(\frac{\displaystyle X}{\displaystyle
k_{B}T}\bigg{missing})},$ (60)

where $\sigma_{0}=e^{2}/(4\hbar)$ is the universal alternating-current
conductivity of graphene, $\gamma$ is the electron relaxation rate in
graphene, $E_{F}$ is the Fermi energy and $T$ is the temperature of the
graphene layer.

Within the four-parameter semi-quantum model [123, 124, 125, 126], the in-
plane optical conductivity of an ultra-thin hBN slab comprising a few
monolayers is

$\sigma_{\mathrm{hBN}}(\mathrm{i}\xi_{j})=\mathrm{i}\epsilon_{0}\xi_{j}t_{\mathrm{hBN}}[\epsilon_{z,\mathrm{hBN}}(\mathrm{i}\xi_{j})-\epsilon_{z}(\infty)].$
(61)

Here, the frequency-dependent permittivity of hBN is given by

$\epsilon_{f,\mathrm{hBN}}(\mathrm{i}\xi_{j})=\epsilon_{f}(\infty)+\frac{s_{\nu,f}\omega_{\nu,f}^{2}}{\omega_{\nu,f}^{2}+\gamma_{\nu,f}\xi_{j}+\xi_{j}^{2}},$
(62)

where $f=x,y,z$,
$\omega_{\nu,z}=$2.58\text{\times}{10}^{14}\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$,
$\gamma_{\nu,z}=$1.319\text{\times}{10}^{12}\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$,
$s_{\nu,z}=1.83$, $\epsilon_{z}(\infty)=4.87$ (see the supplementary
information of [127]) and $t_{\mathrm{hBN}}$ is the thickness of the hBN slab.

For metals, in particular gold, we use the Drude model for the permittivity

$\epsilon_{\mathrm{metal}}(\mathrm{i}\xi_{j})=1+\frac{\omega_{\mathrm{p}}^{2}}{\xi_{j}^{2}+\Gamma_{D}\xi_{j}},$
(63)

where, for gold,
$\omega_{\mathrm{p}}=$1.38\text{\times}{10}^{16}\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$
is the plasma frequency and
$\Gamma_{D}=$1.075\text{\times}{10}^{14}\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$
is the electron relaxation rate [128, 77].

##  Appendix C Reflection coefficients

Typical atom-chips can be modelled as planar multilayer structures (see Fig.
2) with generalized Fresnel reflection coefficients given by the following
recursive relations [78, 129]

$\begin{split}r_{s}^{(l)}&=r_{s}^{(l)}(k^{\parallel},\omega=\mathrm{i}\xi_{j})\\\
&=\frac{a+\\{b\cdot\mathrm{exp}(2\mathrm{i}k_{yl+1}^{\perp}t_{l+1})r_{s}^{(l+1)}\\}}{b+\\{a\cdot\mathrm{exp}(2\mathrm{i}k_{yl+1}^{\perp}t_{l+1})r_{s}^{(l+1)}\\}},\end{split}$
(64)

$\begin{split}r_{p}^{(l)}&=r_{p}^{(l)}(k^{\parallel},\omega=\mathrm{i}\xi_{j})\\\
&=\frac{c+\\{d\cdot\mathrm{exp}(2\mathrm{i}k_{yl+1}^{\perp}t_{l+1})r_{p}^{(l+1)}\\}}{d+\\{c\cdot\mathrm{exp}(2\mathrm{i}k_{yl+1}^{\perp}t_{l+1})r_{p}^{(l+1)}\\}},\end{split}$
(65)

where $a=(\mu_{l+1}k_{yl}-\mu_{l}k_{yl+1})$,
$b=(\mu_{l+1}k_{yl}+\mu_{l}k_{yl+1})$,
$c=(\epsilon_{l+1}k_{yl}-\epsilon_{l}k_{yl+1})$, and
$d=(\epsilon_{l+1}k_{yl}+\epsilon_{l}k_{yl+1})$ for $l=1,...,n-1$ with
$\mu_{l}=\mu_{l}(\mathrm{i}\xi),\epsilon_{l}=\epsilon_{l}(\mathrm{i}\xi)$ and
a termination condition $r_{s,p}^{n}=0$. Here, $k_{yl}$ is defined in the same
manner as Eqs. (55) and (56). Note that the superscripts, $(l)$, on $r_{s}$
and $r_{p}$ are indices denoting which layers the reflection coefficients
correspond to. We use this method to calculate the reflection coefficients of
the metallic conducting wires.

Another way to determine the reflection coefficients is to use a transfer-
matrix method (see, for example, [50]). We use this as a convenient method to
calculate the reflection coefficients of the structures that incorporate
graphene layers. We now present a concise description of this method. There
are two basic elements in this formalism, namely, transmission matrices and
propagation matrices: a transmission matrix describes the change of the wave
amplitudes when the wave crosses an interface between two media, whereas a
propagation matrix captures the phase change when the wave propagates through
a medium. Important physical quantities in this method are the conductivities
of the ultra-thin layers and the permittivities of thick media.

Figure 14: A schematic diagram of electromagnetic scattering in a structure
composed of a single graphene sheet (green) sandwiched by two semi-infinite
dielectric media of relative permittivities, $\epsilon_{1}$ (orange) and
$\epsilon_{2}$ (blue). The graphene sheet is located at the plane defined by
$y=0$ and its electromagnetic properties are encompassed by a conductivity,
$\sigma$. Arrows, all lying in a plane called the plane of incidence, which
coincides with the $y$-$z$ plane, indicate the propagation directions of the
electromagnetic waves (denoted by $A_{1}$ and $A_{2}$ for travelling towards
the negative $y$-direction; $B_{1}$ and $B_{2}$, the positive $y$-direction).
$\theta_{1}$ and $\theta_{2}$ are the incident and refracted angles,
respectively.

Let us consider the scattering of electromagnetic waves of frequency,
$\omega$, scatter from a planar structure consisting of a monolayer graphene,
cladded by two semi-infinite dielectric media of relative permittivities,
$\epsilon_{1}$ and $\epsilon_{2}$, as shown in Figure 14. The graphene sheet
is located in the $y=0$ plane and is assumed to be infinitesimally thin such
that medium 1 (characterized by a relative permittivity, $\epsilon_{1}$)
occupies the region defined by $y>0$ and medium 2 (characterized by a relative
permittivity, $\epsilon_{2}$) occupies the region defined by $y<0$. Let us
further assume that the waves are plane waves, that their $\mathbf{B}$-field
only has an $x$-component, and that their $\mathbf{E}$-field is parallel to
the plane of incidence, which coincides with the $y$-$z$ plane (commonly known
as transverse magnetic waves or p-polarized waves). Therefore, the
$\mathbf{B}$-fields can be written as

$\mathbf{B}_{x}^{(j)}(\mathbf{r,t})=(A_{j}e^{-\mathrm{i}k_{j,y}y}+B_{j}e^{\mathrm{i}k_{j,y}y})e^{\mathrm{i}(k_{j,z}z-\omega
t)}\mathbf{\hat{x}},$ (66)

where $j=1,2$ are for the waves in medium 1 and medium 2, respectively,
$A_{j}$ and $B_{j}$ are the amplitudes of the waves propagating in the
negative and positive $y$-directions, respectively, subscripts, $x,y,$ and $z$
denote the components associated with the coordinate axes, $t$ is time,
$\mathbf{\hat{x}}$ is a unit vector pointing in the positive $x$-direction
(out of the page) and the relation between the $y$\- and $z$\- components of
the wavevector is given by

$k_{j,y}^{2}=\epsilon_{j}\frac{\omega^{2}}{c^{2}}-k_{j,z}^{2}.$ (67)

.

In dielectric media, the relation between $\mathbf{B}$\- and $\mathbf{E}$\-
fields is given by the following Maxwell’s equation,

$\nabla\times\mathbf{B}=-\mathrm{i}\frac{\epsilon\omega}{c^{2}}\mathbf{E}.$
(68)

Using the equation given above, the $z$-component of the $\mathbf{E}$-fields
is found to be

$\mathbf{E}_{z}^{(j)}(\mathbf{r,t})=\frac{-k_{j,y}c^{2}}{\omega\epsilon_{j}}(A_{j}e^{-\mathrm{i}k_{j,y}y}-B_{j}e^{\mathrm{i}k_{j,y}y})e^{\mathrm{i}(k_{j,z}z-\omega
t)}\mathbf{\hat{z}},$ (69)

where $\mathbf{\hat{z}}$ is a unit vector pointing in the positive
$z$-direction.

In order to find the relations between the amplitudes of the waves in medium 1
and medium 2, we need to invoke the boundary conditions at the interface
between the two dielectric media for the parallel components of $\mathbf{E}$\-
and $\mathbf{B}$\- fields, which are

$\mathbf{\hat{n}}\times(\mathbf{E}^{(1)}-\mathbf{E}^{(2)})=0,$ (70)

$\mathbf{\hat{n}}\times(\mathbf{B}^{(1)}-\mathbf{B}^{(2)})=\mu_{0}\mathbf{J}_{s},$
(71)

where $\mathbf{\hat{n}}$ is a normal unit vector pointing from medium 2 into
medium 1 (equivalent to $\mathbf{\hat{y}}$), and $\mathbf{J}_{s}$ is the free
surface current density at the boundary, which is the graphene sheet in this
case. Writing out only the relevant components and using the generalized Ohm’s
law, $\mathbf{J}=\sigma\mathbf{E}$, we obtain

$E_{z}^{(1)}(y=0)=E_{z}^{(2)}(y=0),$ (72)

$B_{x}^{(1)}(y=0)-B_{x}^{(2)}(y=0)=\mu_{0}\sigma E_{z}^{(1)}(y=0),$ (73)

where $\sigma$ is the optical conductivity of the 2D material at the interface
(graphene). Substituting Eqs. (66) and (69) into the above boundary
conditions, yields

$A_{1}-B_{1}=\frac{\epsilon_{1}k_{2,y}}{\epsilon_{2}k_{1,y}}(A_{2}-B_{2}),$
(74)

$A_{1}+B_{1}=(A_{2}+B_{2})+\frac{\sigma
k_{2,y}}{\omega\epsilon_{0}\epsilon_{2}}(A_{2}-B_{2}).$ (75)

Converting (74) and (75) into a matrix equation yields

$\begin{pmatrix}1&-1\\\ 1&1\end{pmatrix}\begin{pmatrix}A_{1}\\\
B_{1}\end{pmatrix}=\begin{pmatrix}\eta_{p}&-\eta_{p}\\\
1+\xi_{p}&1-\xi_{p}\end{pmatrix}\begin{pmatrix}A_{2}\\\ B_{2}\end{pmatrix},$
(76)

where the following functions have been introduced to simplify our notation,

$\eta_{p}=\frac{\epsilon_{1}k_{2,y}}{\epsilon_{2}k_{1,y}}\quad\textrm{and}\quad\xi_{p}=\frac{\sigma
k_{2,y}}{\omega\epsilon_{0}\epsilon_{2}}.$ (77)

Multiplying (76) by the inverse of the leftmost matrix of (76), we obtain a
transfer-matrix equation

$\displaystyle\begin{pmatrix}A_{1}\\\ B_{1}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1+\eta_{p}+\xi_{p}&1-\eta_{p}-\xi_{p}\\\
1-\eta_{p}+\xi_{p}&1+\eta_{p}-\xi_{p}\end{pmatrix}\begin{pmatrix}A_{2}\\\
B_{2}\end{pmatrix},$ (78) $\displaystyle=\mathbf{T}_{p}\begin{pmatrix}A_{2}\\\
B_{2}\end{pmatrix},$ (79)

where

$\mathbf{T}_{p}=\frac{1}{2}\begin{pmatrix}1+\eta_{p}+\xi_{p}&1-\eta_{p}-\xi_{p}\\\
1-\eta_{p}+\xi_{p}&1+\eta_{p}-\xi_{p}\end{pmatrix}.$ (80)

Here, $\mathbf{T}_{p}$ is a transmission matrix for p-polarized
electromagnetic waves.

By following the same procedure, a transmission matrix for s-polarized waves
(transverse electric waves) is found to be

$\mathbf{T}_{s}=\frac{1}{2}\begin{pmatrix}1+\eta_{s}+\xi_{s}&1-\eta_{s}+\xi_{s}\\\
1-\eta_{s}-\xi_{s}&1+\eta_{s}-\xi_{s}\end{pmatrix},$ (81)

where

$\eta_{s}=\frac{k_{2,y}}{k_{1,y}}\quad\textrm{and}\quad\xi_{s}=\frac{\sigma\mu_{0}\omega}{k_{1,y}}.$
(82)

The propagation matrix can easily be derived by recalling the fact that an
electromagnetic wave with $y$-component of wavevector, $k_{y}$, propagating
through a distance, $d$, in the $y$-direction in a uniform medium only changes
its phase by $k_{y}d$. Hence, the propagation matrix is given by

$\mathbf{P}(d)=\begin{pmatrix}e^{-ik_{y}d}&0\\\ 0&e^{ik_{y}d}\end{pmatrix}.$
(83)

Now that we have obtained explicit forms for both transmission and propagation
matrices, a transfer-matrix for calculating the reflection coefficients can be
constructed from a series of matrix multiplications in reverse chronological
order of the scattering events that the matrices correspond to.

Writing the transfer-matrix in the form
$\mathbf{M}=\begin{pmatrix}M_{11}&M_{12}\\\ M_{21}&M_{22}\end{pmatrix}$, the
reflection coefficients can be straightforwardly obtained from two of the
matrix elements via

$r_{s,p}=\frac{M_{21}}{M_{11}}.$ (84)

For the atom-chip structure shown in Fig. 1, the associated transfer-matrix
can be written as

$\mathbf{M}=\mathbf{T}_{\mathrm{hBN}}\mathbf{P}(t_{\mathrm{hBN}})\mathbf{T}_{\mathrm{Gr}}\mathbf{P}(t_{\mathrm{hBN}})\mathbf{T}_{\mathrm{hBN}},$
(85)

where $\mathbf{T}_{\mathrm{hBN}}$ and $\mathbf{T}_{\mathrm{Gr}}$ are,
respectively, associated with transmission across the hBN interface and the
graphene interface, while $\mathbf{P}(t_{\mathrm{hBN}})$ corresponds to
propagation through the thickness, $t_{\mathrm{hBN}}$, of the hBN layer.

##  Appendix D Three-body loss rate in 1D BECs

In this section, we provide a more detailed derivation of the three-body loss
rate considered in the main text, which mainly follows [103, 102, 97, 130]. We
start by considering an 87Rb quasi-condensate with a mean atomic volume
density, $\rho_{0}(\mathbf{r},t)$, at time $t$, where $\mathbf{r}=(x,y,z)$
denotes the center of a cell of volume, $\Delta$, which is small enough for
the condensate to be considered homogeneous throughout the cell (i.e.
$\partial\rho_{0}(\mathbf{r},t)/\partial\Delta\approx 0$), but also large
enough to accommodate many atoms. Assuming that the condensate is subject to a
three-body loss process, its mean volume density evolves in time according to
[102, 103]

$\frac{\mathrm{d}\rho_{0}}{\mathrm{d}t}=-\kappa_{\mathrm{Rb}}\rho_{0}^{3},$
(86)

where
$\kappa_{\mathrm{Rb}}=$1.8\text{\times}{10}^{-41}\text{\,}\mathrm{m}^{6}\mathrm{s}^{-1}$$
is the three-body recombination rate for 87Rb atoms in the $F=m_{F}=2$ state
[104], and we have dropped the explicit dependence of $\rho_{0}$ on
$(\mathbf{r},t)$ for simplicity. Henceforth, we derive the three-body loss
rate at the center of the trap ($z=0$) for a 1D quasi-condensate by
integrating the loss rate for a 3D quasi-condensate given in Eq. (86) over the
radial coordinate.

But first, let us assume that this condensate is greatly elongated in one
dimension, i.e. that it is trapped in a smoothly-varying anisotropic harmonic
potential with radial trapping frequency $\omega_{r}=\omega_{x}=\omega_{y}$
and axial trapping frequency $\omega_{z}$, where $\omega_{r}>>\omega_{z}$.
Consequently, the density profile of the condensate can be described by a one-
dimensional Thomas-Fermi distribution in the $z$-direction, multiplied by the
Gaussian ground-state quantum harmonic oscillator wavefunctions in the $x$\-
and $y$\- directions [97]:

$\rho_{0}(r,z)=\frac{1}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)}e^{-r^{2}/2a_{r}^{2}},$
(87)

where $U_{0}=4\pi\hbar^{2}a_{T}/m$, $a_{T}=$5.6\text{\,}\mathrm{nm}$$ is the
s-wave scattering length [100], $\mu_{\mathrm{eff}}=\mu-\hbar\omega_{r}$,
$\mu$ is the chemical potential of the condensate,
$m=$1.44\text{\times}{10}^{-25}\text{\,}\mathrm{k}\mathrm{g}$$ is the mass of
an 87Rb atom, $a_{r}=\sqrt{\hbar/m\omega_{r}}$, and
$r=\sqrt{x^{2}+(y-y_{c})^{2}}$ is the radial distance measured from the mean
positions of the harmonic oscillator states at $x=0$ and $y=y_{c}$. It can be
seen that in the $z$-direction, the mean volume density peaks at the trap
center, where $z=0$.

We can obtain the mean line density, $n_{0}(z)$, by integrating Eq. (87)
radially from $r=0$ to $r=\infty$:

$\begin{split}\int_{0}^{\infty}\rho_{0}2\pi
r\mathrm{d}r&=\frac{1}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)}\int_{0}^{\infty}e^{-r^{2}/2a_{r}^{2}}2\pi
r\mathrm{d}r,\\\
n_{0}(z)&=\frac{1}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)}\Big{[}0+2\pi
a_{r}^{2}\Big{]},\\\ n_{0}(z)&=\frac{2\pi
a_{r}^{2}}{U_{0}}\Big{(}\mu_{\mathrm{eff}}-\frac{m\omega_{z}^{2}}{2}z^{2}\Big{)}.\end{split}$
(88)

Performing a radial integration on Eq. (86) in the same manner, we obtain the
time evolution of the mean line density as follows

$\begin{split}\int_{0}^{\infty}\frac{\mathrm{d}\rho_{0}}{\mathrm{d}t}2\pi
r\mathrm{d}r&=-\int_{0}^{\infty}\kappa_{\mathrm{Rb}}\rho_{0}^{3}2\pi
r\mathrm{d}r,\\\
\frac{\mathrm{d}n_{0}(z)}{\mathrm{d}t}&=-\frac{\kappa_{\mathrm{Rb}}}{12\pi^{2}a_{r}^{4}}n_{0}(z)^{3},\end{split}$
(89)

where we have used Eqs. (87) and (88) to convert from a volume to a line
atomic density.

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