# Observation of dressed states of distant atoms with delocalized photons in
coupled-cavities quantum electrodynamics

Shinya Kato  Department of Applied Physics, Waseda University, 3-4-1 Okubo,
Shinjuku, Tokyo 169-8555, Japan  JST, PRESTO, 4-1-8 Honcho, Kawaguchi,
Saitama, 332-0012, Japan  Nikolett Német  The Dodd-Walls Centre for Photonic
and Quantum Technologies, New Zealand  Department of Physics, University of
Auckland, Auckland 1010, New Zealand  Kohei Senga  Department of Applied
Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
Shota Mizukami  Department of Applied Physics, Waseda University, 3-4-1 Okubo,
Shinjuku, Tokyo 169-8555, Japan  Xinhe Huang  Department of Applied Physics,
Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan  Scott Parkins
The Dodd-Walls Centre for Photonic and Quantum Technologies, New Zealand
Department of Physics, University of Auckland, Auckland 1010, New Zealand
Takao Aoki<EMAIL_ADDRESS>Department of Applied Physics, Waseda University,
3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

###### Abstract

In a cavity quantum electrodynamics (QED) system, where atoms coherently
interact with photons in a cavity, the eigenstates of the system are the
superposition states of atoms and cavity photons, the so-called dressed states
of atoms. When two cavities are connected by an optical fiber with negligible
loss, the coherent coupling between the cavities gives rise to photonic normal
modes. One of these normal modes is the fiber-dark mode, in which photons are
delocalized in the two distant cavities. Here we demonstrate the setting of
coupled-cavities QED, where two nanofiber cavity-QED systems are coherently
connected by a meter-long low-loss channel in an all-fiber fashion.
Specifically, we observe dressed states of distant atoms with delocalized
photons of the fiber-dark normal mode. Our system will provide a platform for
the study of delocalized atomic and photonic states, photonic many-body
physics, and distributed quantum computation.

When atoms are coherently coupled to each other via their interaction with a
common mode of the electromagnetic field, they form collective states, the
dynamics of which can be drastically different from that of independent atoms.
For the case of atoms in free space, collective effects become observable, as
a change in radiative decay rates, when inter-atomic separations are smaller
than a few wavelengths Dicke_PR_1954 ; DeVoe_PRL_1996 ; Scully_Science_2009 ;
Guerin_PRL_2016 . Recently, such super-radiance and sub-radiance phenomena
have been observed for atoms interacting with a common guided mode of a
photonic crystal waveguide Goban_PRL_2015  and an optical
nanofiberSolano_Nature_Commun_2017 , where atoms are separated by a
macroscopic distance up to several hundred microns, much larger than the
wavelength. On the other hand, in the setting of cavity quantum
electrodynamics (QED), where coherent atom-atom coupling is mediated by the
confined mode of the cavity, the collective effects can be observed as
coherent, reversible dynamics of the system, which is evident as a structural
change in the energy spectra, in contrast to changes in atomic dissipation
rates as for the above cases. For example, the vacuum Rabi splitting for the
dressed states of an atom in the Jaynes-Cummings model Jaynes_Proc_IEEE_1963 ;
Thompson_PRL_1992 ; Boca_PRL_2004 ; Maunz_PRL_2004  is enhanced by a factor of
$\sqrt{N}$ for the collective dressed states in the $N$-atom Tavis-Cummings
model Tavis_PR_1968 ; Kaluzny_PRL_1983 ; Raizen_PRL_1989 ; Ruddell_Optica_2017
. These coherent collective effects may be extended to a configuration of
coupled, but distant, cavity QED systems. In particular, when two cavities are
connected via a channel whose loss is negligible compared to the coupling
between each cavity and the channel, photons can deterministically propagate
back and forth between the cavities many times before being lost. Such
coherent coupling between two cavities gives rise to normal modes, or
superpositions in certain combinations of the two cavities and the connecting
fiber, each of which extends to the whole system nonlocally. Notably, one of
these normal modes is a superposition of the two cavity modes but has no
contribution from the connecting fiber (fiber-dark mode) Serafini_PRL_2006 .

Here, we demonstrate an all-fiber, coupled-cavities QED system in which either
a single ensemble or two distantly separated ensembles of several tens of
atoms coherently interact with the delocalized normal modes of coherently
coupled, distant cavities, and we observe collective dressed states of atoms
with the fiber-dark mode. This is the demonstration of coherent, reversible
coupling between distant atoms, with a separation of the order of a meter,
which is made possible by the all-fiber nature of the connection between the
two cavities, with a loss as low as 2%. Our achievement is an important step
towards the physical implementation of cavity QED-based distributed quantum
computation Serafini_PRL_2006 ; Yang_NJP_2013 ; Koshino_PRAppl_2017 ;
Cao_SciRep_2018  and a quantum network Kimble_Nature_2008 ; Reiserer_RMP_2015
, where a large number of cavity QED systems are coherently connected by low-
loss fiber channels. In such systems, quantum entanglement over the whole
network can be created deterministically Pellizzari_PRL_1995 ; Cirac_PRL_1997
, instead of probabilistically Ritter_Nature_2012 . Our achievement also paves
the way for the study of many-body physics with atoms and photons in a
coupled-cavities QED system, such as quantum phase transitions of light
Torma_PRL_1998 ; Hartmann_NaturePhys_2006 ; Greentree_NaturePhys_2006 ;
Rossini_PRL_2007 ; Angelakis_PRA_2007 ; Cho_PRL_2008 ; Irish_PRA_2008 ;
Hartmann_LPR_2008 .

Figure 1: Schematic of the coupled-cavity QED system. Cavity 1 of length
$L_{1}$ and Cavity 2 of length $L_{2}$ are coupled to a connecting fiber of
length $L_{\rm f}$ with coupling rates of $v_{1}$ and $v_{2}$, respectively.
Atomic ensembles are coupled to Cavities 1 and 2 with atom-cavity coupling
rates of $g_{1}$ and $g_{2}$, respectively. Probe beam is input from the left
mirror of Cavity 1 (Mirror 1), and its output from the right mirror of Cavity
2 (Mirror 4) is measured.

As illustrated in Fig. 1, our system consists of two nanofiber cavity QED
systems Kato_PRL_2015  connected in an all-fiber fashion. In each cavity, an
ensemble of several tens of atoms interacts with the cavity field through the
evanescent field of a nanofiber, both ends of which are connected to standard
optical fibers through tapered regions and sandwiched by a pair of fiber-
Bragg-grating mirrors. A probe laser at the frequency $\omega_{\rm p}$ is
input from mirror 1 (left mirror of cavity 1), and the output from mirror 4
(right mirror of cavity 2) is measured.

For simplicity, we first consider the system with one atom for each cavity,
which is modeled by the following Hamiltonian ($\hbar=1$) in a frame rotating
at $\omega_{\rm p}$:

$\displaystyle H$ $\displaystyle=$ $\displaystyle\Delta_{\rm
c}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+b^{\dagger}b\right)+\sum_{i=1,2}v_{i}\left(a_{i}^{\dagger}b+b^{\dagger}a_{i}\right)$
$\displaystyle+\Delta_{\rm
a}\left(\sigma_{1}^{+}\sigma_{1}^{-}+\sigma_{2}^{+}\sigma_{2}^{-}\right)+\sum_{i=1,2}g_{i}\left(a_{i}^{\dagger}\sigma_{i}^{-}+\sigma_{i}^{+}a_{i}\right),$

where we assume, for simplicity, that the cavity and connecting-fiber modes
$(a_{1},a_{2},b)$ have the same frequency $\omega_{\rm c}$ so that
$\Delta_{\rm c}=\omega_{\rm c}-\omega_{\rm p}$ (see Appendix). The atom-probe
detuning is given by $\Delta_{\rm a}=\omega_{\rm a}-\omega_{\rm p}$, where
$\omega_{\rm a}$ is the atomic transition frequency. The coupling rates of
cavities 1 and 2 with the connecting fiber are given by

$\displaystyle v_{1,2}=\frac{c}{2}\sqrt{\frac{T_{2,3}}{L_{\rm f}L_{1,2}}},$
(2)

where $c$ is the speed of light in the fiber and $T_{i}$, $L_{i}$, and $L_{\rm
f}$ are the transmittance of the mirror $i$, length of the cavity $i$, and
length of the connecting fiber, respectively. The atoms are coupled to their
respective cavity modes with strengths $g_{1}$ and $g_{2}$.

Considering just the cavities and connecting fiber, we can move to a normal-
mode picture, with normal-mode operators given by

$\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\tilde{v}}(v_{2}a_{1}+v_{1}a_{2}),$ (3)
$\displaystyle c_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2\tilde{v}}(v_{1}a_{1}+v_{2}a_{2})\pm\frac{1}{\sqrt{2}}b,$
(4)

where

$\displaystyle\tilde{v}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}}{2}}.$ (5)

Note that the normal mode $d$ has no contribution from the connecting fiber
mode $b$. Rather, it has only contributions from the two cavity modes $a_{1}$
and $a_{2}$, and it has the frequency $\omega_{\rm c}$. On the other hand, the
normal modes $c_{\pm}$ have contributions from the two cavities and the
connecting fiber, and they are shifted in frequency by $\pm\sqrt{2}\tilde{v}$
from $\omega_{\rm c}$. If $\sqrt{2}\tilde{v}$ is sufficiently large, the
atomic transition frequency is close to the bare cavity resonance,
$\omega_{\rm a}\simeq\omega_{\rm c}$, and the frequency of the probe laser is
scanned only in this vicinity, then it is possible to focus on the system
dynamics involving only the mode $d$. That is, we can focus on the reduced
Hamiltonian

$\displaystyle H_{d}$ $\displaystyle=$ $\displaystyle\Delta_{\rm
c}d^{\dagger}d+\sum_{i=1,2}\left[\Delta_{\rm
a}\sigma_{i}^{+}\sigma_{i}^{-}+g_{di}\left(d^{\dagger}\sigma_{i}^{-}+\sigma_{i}^{+}d\right)\right],$

where

$\displaystyle g_{d1,d2}=\frac{v_{2,1}}{\sqrt{2}\tilde{v}}g_{1,2}.$ (7)

This Hamiltonian is identical to the Hamiltonian for a standard single-cavity
QED system with a cavity mode $d$ and two atoms (Tavis-Cummings
HamiltonianTavis_PR_1968 ) having single-atom coupling strengths of $g_{d1}$
and $g_{d2}$. The eigenstates of this system are the collective dressed states
of atoms in distant cavities and of photons in the delocalized normal mode
$d$. In other words, each atom interacts with both cavities simultaneously and
collectively, but not with the connecting fiber. Indeed, the atom-field
coupling strengths, $g_{d1}$ and $g_{d2}$, do not depend on the length of the
connecting fiber.

For a general case of the system with many atoms in each cavity, the linear
optical response in the weak-driving limit is identical to the single-atom
model discussed above, with replacements of single-atom coupling strengths
$g_{i}$ with the collective coupling strengths $g_{i,{\rm
eff}}=g_{i,(0)}\sqrt{N_{i,{\rm eff}}}$, where $g_{i,(0)}$ is the single-atom
coupling strength for an atom located at a potential minimum of the atomic
trap in cavity $i$ and $N_{i,{\rm eff}}$ is the effective number of atoms in
cavity $i$ (see Appendix).

In order to investigate the interaction between atoms and the normal modes of
the coupled-cavities system, we measure transmission spectra with different
atom-loading conditions and various lengths of the connecting fiber.

Each cavity QED system is similar to the previous setup described in
Kato_PRL_2015 . The transmittances of the mirrors are
$(T_{1},T_{2},T_{3},T_{4})=(0.13,0.39,0.33,0.06)$. The single-pass losses
inside the cavities and that for the connecting fiber are all 0.02, which are
dominated by the splicing losses (two splices for each cavity and the
connecting fiber). The cavity lengths are $(L_{1},L_{2})=(0.92,1.38)$ m.

Each experimental sequence starts with cooling and collecting Cs atoms in
standard magneto-optical traps (MOTs). We use the D2-line F=4$\rightarrow$F′=5
transition for cooling and the F=3$\rightarrow$F′=4 transition for repumping
in the MOT. The numbers of atoms in the MOTs are 7$\times 10^{6}$ and
3.3$\times 10^{7}$ for cavities 1 and 2, respectively. The positions of the
MOTs are intentionally shifted from the cavities, and atoms are not coupled to
the cavities at this stage. After the numbers of atoms in the MOTs are
saturated, we scan the lengths of the cavities and the connecting fiber with
different frequencies and monitor the transmission of a laser with the
frequency $\omega_{\rm a}$ (atomic F=4$\rightarrow$F′=5 transition). The
transmission varies with the lengths of the cavities and the connecting fiber,
i.e., the detunings of the modes $a_{1},a_{2},$ and $b$ from $\omega_{\rm a}$,
and takes the maximum value when all the three modes are resonant to
$\omega_{\rm a}$. When the transmission reaches a certain threshold value, we
assume that this resonance condition is satisfied and proceed to the next
step, in which we switch off the monitor laser and change the detuning and
intensity of the cooling laser for 32 ms to further cool the atoms down to 20
$\mu$K. Subsequently, we move the MOTs to overlap with the cavities to load
atoms in the optical traps in the evanescent fields of the nanofibers.

Atoms are trapped in a state-insensitive, two-color evanescent-field optical
trap Kien2005 ; Vetsch2010 ; Lacroute2012 ; Goban2012 ; Kato_PRL_2015 . We use
counterpropagating red-detuned beams ($\lambda_{\rm red}=$937 nm), which form
a 1-dimensional optical lattice, and a blue-detuned beam ($\lambda_{\rm
blue}=$688 nm). The trap depth of the lattice well is about 260 $\mu$K.

After loading atoms in the optical traps, we measure the transmission spectrum
of the system by sending a probe pulse with the atom-probe detuning
$\Delta_{\rm a}$ scanned over $\pm$30 MHz within 4 ms and by detecting the
transmitted probe beam from the system by an avalanche photodiode after
removing unwanted stray light with filters. The power of the input probe is
210–310 pW. Next, we optically pump the atoms into the F=3 state, which is
sufficiently off-resonant from the cavity modes, by irradiating a pumping
laser resonant to the atomic F=4$\rightarrow$F′=3 transition from the sides of
the cavities for 1 ms, and we send a frequency-scanned probe pulse again to
measure the transmission spectrum of the coupled empty cavities, i.e., the
system without atoms. We then switch on the cooling and repumping lasers for 4
ms to cool and load the atoms into the optical trap again. We repeat the above
procedure for spectroscopy and re-cooling five times per MOT loading sequence,
and we use the average of the second, third, and fourth data for each sequence
(see Appendix for details).

Figure 2: Transmission spectra with different atom-loading conditions. a, No
atoms are loaded. b, Atoms are loaded in cavity 1 only. c, Atoms are loaded in
cavity 2 only. d, Atoms are loaded both cavities. Blue solid lines represent
experimental data, while red dashed lines and pink shaded bands are
theoretical curves for $(g_{1,{\rm eff}},g_{2,{\rm eff}})=2\pi\times(7.2,7.3)$
MHz and $(g_{1,{\rm eff}},g_{2,{\rm eff}})=2\pi\times(7.2\pm 1.0,7.3\pm 1.0)$
MHz, respectively. When no atoms are loaded (a), formation of normal modes for
three coupled empty cavities results in the clear triplet spectrum. The
central peak corresponds to the fiber-dark mode $d$, while the two side peaks
correspond to the other two normal modes $c_{\pm}$. When atoms are loaded in
one of the two (b and c), or both cavities (d), coupling between atoms and the
fiber-dark mode results in the splitting of the central peak observed in a. In
particular, the splitting observed in d is the signature of the dressed states
of distant atoms with delocalized photons in the fiber-dark mode.

Firstly, we fix the length of the connecting fiber as $L_{\rm f}=1.23$ m and
measure transmission spectra of the coupled-cavities QED system with different
atom-loading conditions, in which the atoms are not loaded, loaded in either
cavity, or loaded in both cavities. In Fig. 2a, the blue solid line shows the
measured transmission spectrum normalized to the maximum transmission for the
case without atoms. Three peaks for the normal modes $d$ and $c_{\pm}$ are
clearly observed. The red dashed line shows the corresponding theoretical
curve (see Appendix) with no free fitting parameter, which agrees well with
the experimental data. In particular, the splittings of the side peaks from
the center peak match with the theoretical value of the frequency difference
of the modes $c_{\pm}$ from the fiber-dark mode $d$ given by
$\sqrt{2}\tilde{v}=2\pi\times 12.1$ MHz. The slight broadening in linewidths
and the asymmetry of the spectral shape in the experiments are, we believe,
due to the instability of the detunings of the cavity and fiber modes
($a_{1},a_{2},$ and $b$) from the atomic frequency $\omega_{\rm a}$ during the
measurement. Figures 2b and c show the spectra for the cases of atoms loaded
only in cavities 1 and 2, respectively. It can be clearly seen that the
interaction of atoms with the fiber-dark mode $d$ causes a splitting of the
center peak in Fig. 2a, which is the signature of the dressed states of atoms
with delocalized photons of the fiber-dark mode. On the other hand, the (off-
resonant) interaction of atoms with the modes $c_{\pm}$ causes the frequency
shifts of the side peaks. It can be seen that the spectra agree reasonably
well with the theoretical curves of a linearized model with the atom-cavity
coupling strengths $(g_{1,{\rm eff}},g_{2,{\rm eff}})=2\pi\times(7.2\pm
1.0,7.3\pm 1.0)$ MHz ($(g_{d1},g_{d2})=2\pi\times(4.3\pm 0.6,5.8\pm 0.8)$ MHz)
as the only free parameters. Figure 2d shows the spectrum for the case of
atoms loaded in both cavities. The measured spectra clearly also agree quite
well with the theoretical curves with no additional free parameters. Note that
the splitting of the center peak is larger than that observed in Figs. 2b and
c and agrees with the theoretical value of
$\sqrt{g_{d1}^{2}+g_{d2}^{2}}=2\pi\times 7.3$ MHz, which is the signature of
the collective dressed states of distant atoms with delocalized photons of the
fiber-dark mode.

Figure 3: Transmission spectra with different connecting-fiber lengths. Blue
solid lines represent experimental data, while red dashed lines and pink
shaded bands are theoretical curves for $(g_{1,{\rm eff}},g_{2,{\rm
eff}})=2\pi\times(7.2,7.3)$ MHz and $(g_{1,{\rm eff}},g_{2,{\rm
eff}})=2\pi\times(7.2\pm 1.0,7.3\pm 1.0)$ MHz, respectively. The small peaks
at $\Delta_{\rm a}\approx 2\pi\times 25$ MHz in a and c correspond to a mode
with different polarization, which appears because of imperfect polarization
compensation in the connecting fiber (see Appendix). The dependence of the
cavity-fiber coupling rates $v_{1}$ and $v_{2}$ on the connecting-fiber length
is clearly observed as the change of the splitting in the triplet structure in
a and b. In contrast, the splitting of the central peak in c and d remains
unchanged. This is because the coupling rate between atoms and the delocalized
photons in the fiber-dark mode does not depend on the connecting-fiber length.

Secondly, we measure transmission spectra of the coupled-cavities QED system
with different lengths of the connecting fiber. Figures 3a and b show the
spectra without atoms for the cases of $L_{\rm f}=0.83$ m and 2.27 m,
respectively. The measured spectra (blue solid lines) reasonably agree with
the theoretical curves (red dashed lines). A change of splittings between the
center and side peaks from that in Fig. 2a ($L_{\rm f}=1.23$ m) is clearly
observed. Specifically, the observed splittings roughly match the theoretical
values of $\sqrt{2}\tilde{v}=2\pi\times 14.7$ MHz and $2\pi\times$ 8.9 MHz for
$L_{\rm f}=0.83$ m and 2.27 m, respectively. Figures 3c and d show the spectra
with atoms for the cases of $L_{\rm f}=0.83$ m and 2.27 m, respectively.
Again, the measured spectra (blue solid lines) reasonably agree with the
theoretical curves (red dashed lines). Note that the splittings of the center
peak associated with the coupling of atoms to the fiber-dark mode do not
change for different lengths of the connecting fiber, in agreement with the
above theory.

Figure 4: Saturation behavior of the on-resonance transmission. Normalized
transmission is plotted as a function of input probe power for atoms loaded in
a, cavity 1 and b, cavity 2. Red curves are semiclassical state equations with
$(g_{1,(0)},g_{2,(0)})=2\pi\times(0.75,1.2)$ MHz and $(N_{1,{\rm
eff}},N_{2,{\rm eff}})=(92,37)$. The corresponding saturation photon numbers
are $(n_{1,{\rm sat}},n_{2,{\rm sat}})=(6.9,2.7)$, and thus, our system shows
strong nonlinearity at the few-photon level.

Lastly, we investigate saturation behaviors of the coupled-cavities QED system
to confirm that the above measurements are conducted in the weak-driving
regime. Specifically, we load atoms in either of the cavities and measure
transmission as a function of the input probe power. The length of the
connecting fiber is fixed at $L_{\rm f}=1.23$ m. Figures 4a and b show the
normalized transmission at zero detuning $\Delta_{\rm a}=0$ for atoms loaded
in cavities 1 and 2, respectively. It can be clearly seen that the system is
in the weak-driving regime at the input power of 210–310 pW for the above
measurement and that the transmission starts to saturate as the input power
exceeds $\sim 10^{3}$ pW.

We further compare the measured saturation behaviors with semiclassical state
equations for the many-atom coupled-cavities QED system (see Appendix). The
red solid lines in Figs. 4a and b are the theoretical curves with
$(g_{1,(0)},g_{2,(0)})=2\pi\times(0.75,1.2)$ MHz, from which we obtain
$(N_{1,{\rm eff}},N_{2,{\rm eff}})\equiv((g_{1,{\rm
eff}}/g_{1,(0)})^{2},(g_{2,{\rm eff}}/g_{2,(0)})^{2})=(92,37)$. It can also be
seen that these curves reasonably agree with the experimental results.
Saturation behavior of a cavity QED system can be characterized with the so-
called saturation photon number, which is a measure of the system
nonlinearity. Note that, the corresponding saturation photon numbers are
$(n_{1,{\rm sat}},n_{2,{\rm sat}})=(6.9,2.7)$ (see Appendix), and thus, our
system shows strong nonlinearity at the few-photon level.

In summary, we have demonstrated the setting of a coupled-cavities QED system
in which two nanofiber cavity-QED systems are coherently connected via a
meter-long, low-loss fiber channel. Specifically, we have observed the
collective dressed states of distant atoms with delocalized photons of a
fiber-dark normal mode. We note that the delocalization of photons for the
fiber-dark mode would be explicitly demonstrated by the inability to drive or
detect this mode through the connecting fiber, which could be facilitated by
inserting a fiber beam splitter in the connecting fiber. Also, while in the
present study the coherent coupling between distant atoms has been observed in
the steady-state spectra, it will be possible, and very interesting, to also
investigate transient dynamics of this system, where deterministic, reversible
exchange of excitation between distant atoms would be observable.

It is straightforward to increase the number of coupled cavities in our
system, and by driving the atoms from the side of the cavities with classical
fields, it will be possible to realize a system of strongly interacting
polaritons Hartmann_NaturePhys_2006 . Since the cavities are coupled via
optical fibers, it is also possible to design a system with arbitrary geometry
of connections. Note that ensembles of atoms $\gtrsim 10^{3}$ are routinely
loaded into the optical traps around nanofibers Vetsch2010 ; Goban2012 ;
Beguin2014 . By reducing the number of atoms in each cavity to one
Kato_PRL_2015 , on the other hand, it will be possible to construct a fiber
network of single-atom cavity QED systems Kimble_Nature_2008 ;
Reiserer_RMP_2015 . With such a system, deterministic creation of quantum
entanglement over the whole network Pellizzari_PRL_1995 ; Cirac_PRL_1997 ,
instead of probabilistic creation Ritter_Nature_2012 , will be possible.
Furthermore, provided that the normal mode description is valid, i.e., if the
coherent coupling rate between each cavity and the fiber is larger than the
loss rates, the interaction of atoms with the resulting fiber-dark mode can be
utilized in quantum gates for distributed quantum computation
Serafini_PRL_2006 . In order to extend our work to the construction of a fiber
network of coherently coupled single-atom cavity QED systems, we are currently
making technical improvements to the setup, for example, further reduction of
the internal losses in the cavities, active stabilization of the cavities, and
extending the lifetime of the atomic traps Meng_PRX_2018 .

##  Appendix A Theory

###  A.1 Coupled-cavities system

Figure 5: Schematic of the coupled-cavities system (not to scale). The
transmittance of mirror $i$ is $T_{i}$. Other parameters are defined in the
text.

As a simple model of our system (Fig. 5), we can use single modes for the
fields in the cavities and in the connecting fiber, as well as single, two-
level atoms in each cavity, and a master equation for the density operator
$\rho$ of the composite system (atoms plus fields) that takes the form (in a
frame rotating at the probe laser frequency $\omega_{\rm p}$, and setting
$\hbar=1$)

$\displaystyle\dot{\rho}$ $\displaystyle=-i[H,\rho]+\kappa_{1}{\cal
D}[a_{1}]\rho+\kappa_{2}{\cal D}[a_{2}]\rho+\kappa_{b,{\rm loss}}{\cal
D}[b]\rho+\frac{\gamma_{\parallel}}{2}\left({\cal D}[\sigma_{1}^{-}]\rho+{\cal
D}[\sigma_{2}^{-}]\rho\right)$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}+\gamma_{\rm
las}\left({\cal D}[a_{1}^{\dagger}a_{1}]\rho+{\cal
D}[a_{2}^{\dagger}a_{2}]\rho+{\cal D}[b^{\dagger}b]\rho+{\cal
D}[\sigma_{z,1}]\rho+{\cal D}[\sigma_{z,2}]\rho\right).$ (8)

where ${\cal D}[O]\rho=2O\rho O^{\dagger}-O^{\dagger}O\rho-\rho O^{\dagger}O$.
The Hamiltonian is

$\displaystyle H$ $\displaystyle=\Delta_{\rm
c}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+b^{\dagger}b\right)+\left(v_{1}^{\ast}a_{1}^{\dagger}b+v_{1}b^{\dagger}a_{1}\right)+\left(v_{2}^{\ast}a_{2}^{\dagger}b+v_{2}b^{\dagger}a_{2}\right)+\left({\cal
E}_{\rm p}^{\ast}a_{1}+{\cal E}_{\rm p}a_{1}^{\dagger}\right)$
$\displaystyle~{}~{}~{}+\Delta_{\rm
a}\left(\sigma_{1}^{+}\sigma_{1}^{-}+\sigma_{2}^{+}\sigma_{2}^{-}\right)+\left(g_{1}a_{1}^{\dagger}\sigma_{1}^{-}+g_{1}^{\ast}\sigma_{1}^{+}a_{1}\right)+\left(g_{2}a_{2}^{\dagger}\sigma_{2}^{-}+g_{2}^{\ast}\sigma_{2}^{+}a_{2}\right),$
(9)

where we assume, for simplicity, that the cavity and connecting-fiber modes
($a_{1}$, $a_{2}$, $b$) have the same frequency $\omega_{\rm c}$, so that
$\Delta_{\rm c}=\omega_{\rm c}-\omega_{\rm p}$. The atom-probe detuning is
$\Delta_{\rm a}=\omega_{\rm a}-\omega_{\rm p}$, where $\omega_{\rm a}$ is the
atomic transition frequency, and ${\cal E}_{\rm p}$ is the probe driving
strength. The atoms couple with strengths $g_{1,2}$ to their respective cavity
modes, while the coupling rates between the cavity modes (of lengths
$L_{1,2}$) and the fiber mode (of length $L_{\rm f}$) are given by

$\displaystyle v_{1}=\sqrt{\frac{\kappa_{1r}}{\pi}\omega_{\rm
FSR,f}}\equiv\frac{c}{2}\sqrt{\frac{T_{2}}{L_{1}L_{\rm
f}}}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}v_{2}=\sqrt{\frac{\kappa_{2l}}{\pi}\omega_{\rm
FSR,f}}\equiv\frac{c}{2}\sqrt{\frac{T_{3}}{L_{2}L_{\rm f}}}.$ (10)

Here, $\omega_{\rm FSR,f}=\pi c/L_{\rm f}$ is the free spectral range of the
coupling fiber mode, where $c$ is the speed of light in the fiber, and
$\kappa_{1r}=cT_{2}/(4L_{1})$ and $\kappa_{2l}=cT_{3}/(4L_{2})$ correspond to
the decay rates of the respective cavity fields through mirrors 2 and 3 in the
case that the outputs from these mirrors couple to a continuum of modes (for
example, in the limit that $L_{\rm f}\rightarrow\infty$).

The remaining terms in the master equation describe losses and dephasing
effects in the system. The fiber mode $b$ has an intrinsic loss rate
$\kappa_{b,{\rm loss}}$, while the field decay rates of cavities 1 and 2 are
given by

$\displaystyle\kappa_{1}=\kappa_{1l}+\kappa_{\rm
1,loss}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}\kappa_{2}=\kappa_{2r}+\kappa_{\rm
2,loss},$ (11)

where $\kappa_{1l}=cT_{1}/(4L_{1})$ and $\kappa_{2r}=cT_{4}/(4L_{2})$. The
intrinsic loss rates are determined from the (intensity) transmission
coefficients of the fiber segments that support the various modes as

$\displaystyle\kappa_{\rm
1,loss}=-\frac{1}{2}\frac{c}{L_{1}}\ln(1-\alpha_{1}),~{}~{}~{}~{}\kappa_{b,{\rm
loss}}=-\frac{1}{2}\frac{c}{L_{\rm f}}\ln(1-\alpha_{\rm
f}),~{}~{}~{}~{}\kappa_{\rm
2,loss}=-\frac{1}{2}\frac{c}{L_{2}}\ln(1-\alpha_{2}),$ (12)

where $\alpha_{1}$, $\alpha_{\rm f}$, $\alpha_{2}$ are single-pass losses for
the segments in cavity 1, the connecting fiber, and cavity 2, respectively.
The term proportional to $\gamma_{\rm las}$ – the laser linewidth (HWHM) – is
included so as to incorporate the effect of laser frequency fluctuations,
which appears as phase damping of the field and atomic amplitudes. The atoms
decay into free space with rate $\gamma_{\parallel}$.

Parameter |  $2\pi\cdot$MHz
---|---
$\kappa_{1l}$ | 1.16
$\kappa_{\rm 1,loss}$ | 0.36
$\kappa_{1r}$ | 3.48
$\kappa_{2l}$ | 1.97
$\kappa_{\rm 2,loss}$ | 0.24
$\kappa_{2r}$ | 0.357
$\kappa_{b,{\rm loss}}$ ($L_{\rm f}=0.83\,{\rm m}$) | 0.40
$\kappa_{b,{\rm loss}}$ ($L_{\rm f}=1.23\,{\rm m}$) | 0.27
$\kappa_{b,{\rm loss}}$ ($L_{\rm f}=2.27\,{\rm m}$) | 0.15
$v_{1}$ ($L_{\rm f}=0.83\,{\rm m}$) | 11.7
$v_{1}$ ($L_{\rm f}=1.23\,{\rm m}$) | 9.65
$v_{1}$ ($L_{\rm f}=2.27\,{\rm m}$) | 7.10
$v_{2}$ ($L_{\rm f}=0.83\,{\rm m}$) | 8.82
$v_{2}$ ($L_{\rm f}=1.23\,{\rm m}$) | 7.25
$v_{2}$ ($L_{\rm f}=2.27\,{\rm m}$) | 5.33
$\gamma_{\parallel}$ | 5.2
$\gamma_{\rm las}$ | 0.365
Table 1: List of parameter values for modeling of the experiment.
$\kappa_{1l}$, $\kappa_{1r}$, $\kappa_{2l}$, $\kappa_{2r}$ correspond to the
decay rates of the respective cavity fields through mirrors 1, 2, 3, and 4 (in
the case that the outputs from these mirrors couple to a continuum of modes).
$\kappa_{1,\rm loss}$, $\kappa_{b,\rm loss}$, $\kappa_{2,\rm loss}$ are the
intrinsic loss rates for cavity 1, connecting fiber, and cavity 2. $v_{1}$ and
$v_{2}$ are the coupling rates between the cavity modes and the fiber mode.
$\gamma_{\parallel}$ and $\gamma_{\rm las}$ are the atomic decay rate into
free space and the laser linewidth (HWHM), respectively.

###  A.2 Weak probe driving: linearized equations of motion

If we assume weak driving and, hence, weak excitation of the atoms, then we
may derive the following linear equations of motion for the field and atomic
amplitudes:

$\displaystyle\dot{\left\langle a_{1}\right\rangle}$
$\displaystyle=-(\kappa_{1}^{\prime}+i\Delta_{\rm c})\left\langle
a_{1}\right\rangle-iv_{1}\left\langle b\right\rangle-
ig_{1}\left\langle\sigma_{1}^{-}\right\rangle-i{\cal E}_{1},$ (13)
$\displaystyle\dot{\left\langle a_{2}\right\rangle}$
$\displaystyle=-(\kappa_{2}^{\prime}+i\Delta_{\rm c})\left\langle
a_{2}\right\rangle-iv_{2}\left\langle b\right\rangle-
ig_{2}\left\langle\sigma_{2}^{-}\right\rangle,$ (14)
$\displaystyle\dot{\left\langle b\right\rangle}$
$\displaystyle=-(\kappa_{b}+i\Delta_{\rm c})\left\langle b\right\rangle-
iv_{1}^{\ast}\left\langle a_{1}\right\rangle-iv_{2}^{\ast}\left\langle
a_{2}\right\rangle,$ (15)
$\displaystyle\dot{\left\langle\sigma_{1}^{-}\right\rangle}$
$\displaystyle=-(\gamma_{\perp}+i\Delta_{\rm
a})\left\langle\sigma_{1}^{-}\right\rangle-ig_{1}^{\ast}\left\langle
a_{1}\right\rangle,$ (16)
$\displaystyle\dot{\left\langle\sigma_{2}^{-}\right\rangle}$
$\displaystyle=-(\gamma_{\perp}+i\Delta_{\rm
a})\left\langle\sigma_{2}^{-}\right\rangle-ig_{2}^{\ast}\left\langle
a_{2}\right\rangle,$ (17)

where $\kappa_{1,2}^{\prime}=\kappa_{1,2}+\gamma_{\rm las}$,
$\kappa_{b}=\kappa_{b,{\rm loss}}+\gamma_{\rm las}$, and
$\gamma_{\perp}=\gamma_{\parallel}/2+\gamma_{\rm las}$. Setting the time
derivatives to zero, we find the general steady state solution for the
amplitude of cavity 2 as

$\displaystyle\left\langle a_{2}\right\rangle_{\rm ss}=\frac{A}{B}\,,$ (18)

where

$\displaystyle A=-i{\cal E}_{1}\left(\frac{v_{2}}{\kappa_{b}+i\Delta_{\rm
c}}\right)\frac{v_{1}^{\ast}}{\kappa_{1}^{\prime}+i\Delta_{\rm
c}+\dfrac{|g_{1}|^{2}}{\gamma_{\perp}+i\Delta_{\rm
a}}+\dfrac{|v_{1}|^{2}}{\kappa_{b}+i\Delta_{\rm c}}},$ (19)

and

$\displaystyle B=-(\kappa_{2}^{\prime}+i\Delta_{\rm
c})-\frac{|v_{2}|^{2}}{\kappa_{b}+i\Delta_{\rm
c}}-\frac{|g_{2}|^{2}}{\gamma_{\perp}+i\Delta_{\rm
a}}+\frac{|v_{1}v_{2}|^{2}}{(\kappa_{b}+i\Delta_{\rm
c})^{2}}\,\frac{1}{\kappa_{1}^{\prime}+i\Delta_{\rm
c}+\dfrac{|g_{1}|^{2}}{\gamma_{\perp}+i\Delta_{\rm
a}}+\dfrac{|v_{1}|^{2}}{\kappa_{b}+i\Delta_{\rm c}}}.$ (20)

Solutions for the steady state amplitudes of cavity 1 and fiber mode $b$ then
follow from

$\displaystyle\left\langle a_{1}\right\rangle_{\rm ss}=-\frac{i{\cal
E}_{1}+\dfrac{v_{1}v_{2}^{\ast}}{\kappa_{b}+i\Delta_{\rm c}}\,\left\langle
a_{2}\right\rangle_{\rm ss}}{\kappa_{1}^{\prime}+i\Delta_{\rm
c}+\dfrac{|g_{1}|^{2}}{\gamma_{\perp}+i\Delta_{\rm
a}}+\dfrac{|v_{1}|^{2}}{\kappa_{b}+i\Delta_{\rm c}}},$ (21)

and

$\displaystyle\left\langle b\right\rangle_{\rm
ss}=-\frac{iv_{1}^{\ast}}{\kappa_{b}+i\Delta_{\rm c}}\,\left\langle
a_{1}\right\rangle_{\rm ss}-\frac{iv_{2}^{\ast}}{\kappa_{b}+i\Delta_{\rm
c}}\,\left\langle a_{2}\right\rangle_{\rm ss}.$ (22)

The output photon flux from cavity 2, in the linear regime, is then

$\displaystyle\left|\left\langle a_{\rm
out,2}\right\rangle\right|^{2}=2\kappa_{2r}\left|\left\langle
a_{2}\right\rangle\right|^{2}.$ (23)

When plotting this quantity, we normalize by the on-resonance ($\Delta_{\rm
c}=0$) flux with no atoms ($g_{1,2}=0$).

###  A.3 Normal mode description

We assume, for simplicity, that all of the parameters in the Hamiltonian are
real. Considering just the cavities and coupling fiber, we can move to a
normal mode picture, with normal mode operators given by

$\displaystyle
d=\frac{1}{\sqrt{2}\tilde{v}}\left(v_{2}a_{1}+v_{1}a_{2}\right),~{}~{}~{}~{}c_{\pm}=\frac{1}{2\tilde{v}}\left(v_{1}a_{1}+v_{2}a_{2}\right)\pm\frac{1}{\sqrt{2}}b,$
(24)

where

$\displaystyle\tilde{v}=\sqrt{\frac{\left(v_{1}^{2}+v_{2}^{2}\right)}{2}}.$
(25)

Expressed in terms of these normal mode operators, the Hamiltonian becomes

$\displaystyle H$ $\displaystyle=\Delta_{\rm c}d^{\dagger}d+\left(\Delta_{\rm
c}+\sqrt{2}\tilde{v}\right)c_{+}^{\dagger}c_{+}+\left(\Delta_{\rm
c}-\sqrt{2}\tilde{v}\right)c_{-}^{\dagger}c_{-}+\Delta_{\rm
a}\left(\sigma_{1}^{+}\sigma_{1}^{-}+\sigma_{2}^{+}\sigma_{2}^{-}\right)$
$\displaystyle~{}~{}~{}+{\cal E}_{\rm
p}\frac{v_{1}}{2\tilde{v}}\left(c_{+}^{\dagger}+c_{+}\right)+{\cal E}_{\rm
p}\frac{v_{1}}{2\tilde{v}}\left(c_{-}^{\dagger}+c_{-}\right)+{\cal E}_{\rm
p}\frac{v_{2}}{\sqrt{2}\tilde{v}}\left(d^{\dagger}+d\right)$
$\displaystyle~{}~{}~{}+\frac{1}{2\tilde{v}}\left[\left(c_{+}^{\dagger}+c_{-}^{\dagger}\right)\left(v_{1}g_{1}\sigma_{1}^{-}+v_{2}g_{2}\sigma_{2}^{-}\right)+\textrm{H.c.}\right]+\frac{1}{\sqrt{2}\tilde{v}}\left[d^{\dagger}\left(v_{2}g_{1}\sigma_{1}^{-}+v_{1}g_{2}\sigma_{2}^{-}\right)+\textrm{H.c.}\right].$
(26)

Let us now assume that the normal mode splitting $\sqrt{2}\tilde{v}$ is large
compared to all other parameters. If the atomic transition frequency is close
to the bare cavity resonance ($\omega_{\rm a}\simeq\omega_{\rm c}$), and the
probe laser frequency is scanned just in this vicinity as well, then it is
possible to focus on the system dynamics involving only the resonant mode $d$.
That is, we can focus on the reduced Hamiltonian

$\displaystyle H_{d}$ $\displaystyle=\Delta_{\rm c}d^{\dagger}d+\Delta_{\rm
a}\left(\sigma_{1}^{+}\sigma_{1}^{-}+\sigma_{2}^{+}\sigma_{2}^{-}\right)+{\cal
E}_{\rm
p}\frac{v_{2}}{\sqrt{2}\tilde{v}}\left(d^{\dagger}+d\right)+\frac{1}{\sqrt{2}\tilde{v}}\left[d^{\dagger}\left(v_{2}g_{1}\sigma_{1}^{-}+v_{1}g_{2}\sigma_{2}^{-}\right)+\textrm{H.c.}\right]$
$\displaystyle\equiv\Delta_{\rm c}d^{\dagger}d+\sum_{i=1,2}\left[\Delta_{\rm
a}\sigma_{i}^{+}\sigma_{i}^{-}+g_{di}\left(d^{\dagger}\sigma_{i}^{-}+\sigma_{i}^{+}d\right)\right]+{\cal
E}_{d}\left(d^{\dagger}+d\right),$ (27)

where

$\displaystyle
g_{d1}=\frac{v_{2}}{\sqrt{2}\tilde{v}}g_{1},~{}~{}~{}g_{d2}=\frac{v_{1}}{\sqrt{2}\tilde{v}}g_{2},~{}~{}~{}\textrm{and}~{}~{}~{}{\cal
E}_{d}={\cal E}_{\rm p}\frac{v_{2}}{\sqrt{2}\tilde{v}}.$ (28)

Now considering the dissipative terms in the master equation and noting the
inverse relations

$\displaystyle
b=\frac{1}{\sqrt{2}}\left(c_{+}-c_{-}\right),~{}~{}~{}a_{1}=\frac{1}{2\tilde{v}}\left[v_{1}\left(c_{+}+c_{-}\right)+\sqrt{2}v_{2}d\right],~{}~{}~{}a_{2}=\frac{1}{2\tilde{v}}\left[v_{2}\left(c_{+}+c_{-}\right)-\sqrt{2}v_{1}d\right],$
(29)

one can see that, in general, the decay of each normal mode cannot be
decoupled from the other normal modes, i.e., there are cross-terms between the
normal mode operators arising from the Lindblad forms $\kappa_{1}{\cal
D}[a_{1}]\rho+\kappa_{2}{\cal D}[a_{2}]\rho+\kappa_{b,{\rm loss}}{\cal
D}[b]\rho$ (for simplicity, we set $\gamma_{\rm las}=0$ and ignore phase
damping in this discussion). Once again, however, if $\sqrt{2}\tilde{v}$ is
sufficiently large, then the cross-terms can be neglected (as in a rotating-
wave approximation) and the normal modes can be regarded as decaying according
to

$\displaystyle\kappa_{d}{\cal D}[d]\rho+\kappa_{+}{\cal
D}[c_{+}]\rho+\kappa_{-}{\cal D}[c_{-}]\rho,$ (30)

with

$\displaystyle\kappa_{d}=\frac{v_{2}^{2}\kappa_{1}+v_{1}^{2}\kappa_{2}}{2\tilde{v}^{2}}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}\kappa_{+}=\kappa_{-}=\frac{1}{2}\left(\kappa_{b,{\rm
loss}}+\frac{v_{1}^{2}\kappa_{1}+v_{2}^{2}\kappa_{2}}{2\tilde{v}^{2}}\right).$
(31)

So, we can in turn focus on the reduced master equation,

$\displaystyle\dot{\rho}$ $\displaystyle=-i[H_{d},\rho]+\kappa_{d}{\cal
D}[d]\rho+\gamma_{\perp}\left({\cal D}[\sigma_{1}^{-}]\rho+{\cal
D}[\sigma_{2}^{-}]\rho\right),$ (32)

for probe driving around the atom-cavity resonance $\omega_{\rm
c}\simeq\omega_{\rm a}$ and far from the normal modes at $\omega_{\rm
c}\pm\sqrt{2}\tilde{v}$.

###  A.4 Vacuum Rabi splitting of the fiber-dark mode $d$

The photon flux transmitted through the coupled-cavities system is determined
by the output photon flux from cavity 2, i.e., by $2\kappa_{2r}\langle
a_{2}^{\dagger}a_{2}\rangle$. Given weak probe driving strength in equation
(32), and the relationship between $a_{2}$ and $d$ in equation (29), the
transmitted spectrum will show, for sufficiently large atom-cavity coupling
strengths, vacuum Rabi splitting of the fiber-dark mode $d$ given by

$\displaystyle\pm\sqrt{\left(g_{d1}^{2}+g_{d2}^{2}\right)-\frac{1}{4}(\kappa_{d}-\gamma_{\perp})^{2}},$
(33)

which is approximately $\pm\sqrt{g_{d1}^{2}+g_{d2}^{2}}$ for the parameters of
the experiment.

###  A.5 Saturation

From the semiclassical (nonlinear) equations of motion in steady state, we can
eliminate the atomic variables, as well as the connecting-fiber mode
amplitude, to obtain the following coupled equations for the amplitudes of the
two cavity modes,

$\displaystyle-i\frac{{\cal E}_{\rm p}}{\kappa_{1}^{\prime}}$
$\displaystyle=\left\langle
a_{1}\right\rangle\left\\{\left(1+i\frac{\Delta_{\rm
c}}{\kappa_{1}^{\prime}}+\frac{v_{1}^{2}/\kappa_{1}^{\prime}}{\kappa_{b}+i\Delta_{\rm
c}}\right)+\left(1-i\frac{\Delta_{\rm
a}}{\gamma_{\perp}}\right)\sum_{j_{1}}\frac{C_{1j_{1}}}{1+\dfrac{\Delta_{\rm
a}^{2}}{\gamma_{\perp}^{2}}+\dfrac{\left|\left\langle
a_{1}\right\rangle\right|^{2}}{n_{1j_{1}}}}\right\\}+\frac{v_{1}v_{2}/\kappa_{1}^{\prime}}{\kappa_{b}+i\Delta_{\rm
c}}\left\langle a_{2}\right\rangle,$ (34) $\displaystyle 0$
$\displaystyle=\left\langle
a_{2}\right\rangle\left\\{\left(1+i\frac{\Delta_{\rm
c}}{\kappa_{2}^{\prime}}+\frac{v_{2}^{2}/\kappa_{2}^{\prime}}{\kappa_{b}+i\Delta_{\rm
c}}\right)+\left(1-i\frac{\Delta_{\rm
a}}{\gamma_{\perp}}\right)\sum_{j_{2}}\frac{C_{2j_{2}}}{1+\dfrac{\Delta_{\rm
a}^{2}}{\gamma_{\perp}^{2}}+\dfrac{\left|\left\langle
a_{2}\right\rangle\right|^{2}}{n_{2j_{2}}}}\right\\}+\frac{v_{1}v_{2}/\kappa_{2}^{\prime}}{\kappa_{b}+i\Delta_{\rm
c}}\left\langle a_{1}\right\rangle,$ (35)

where

$\displaystyle
C_{lj_{l}}=\frac{g_{lj_{l}}^{2}}{\kappa_{l}^{\prime}\gamma_{\perp}}~{}~{}~{}\textrm{and}~{}~{}~{}n_{lj_{l}}=\frac{\gamma_{\perp}\gamma_{\parallel}}{4g_{lj_{l}}^{2}}~{}~{}~{}~{}~{}~{}~{}(l=1,2).$
(36)

Here, $g_{lj_{l}}$ denotes the coupling strength of atom $j_{l}$ to cavity
mode $l$. Note that we will assume for simplicity that the parameters
$\\{g_{1j_{1}},g_{2j_{2}}\\}$, $\\{v_{1},v_{2}\\}$, and ${\cal E}_{\rm p}$ are
all real. In what follows, we will also assume resonance between atoms, field
modes and driving laser field, i.e., $\Delta_{\rm c}=\Delta_{\rm a}=0$.

####  A.5.1 Atoms in cavity 1 only

Setting $C_{2j_{2}}=0$ (no atoms in cavity 2), we can eliminate the amplitude
$\left\langle a_{2}\right\rangle$ to obtain an equation for the amplitude
$\left\langle a_{1}\right\rangle$ alone, which in scaled variables takes the
form

$\displaystyle-iy_{1}$
$\displaystyle=X_{1}\left\\{1+\frac{v_{1}^{2}/(\kappa_{b}\kappa_{1}^{\prime})}{1+v_{2}^{2}/(\kappa_{b}\kappa_{2}^{\prime})}+C_{\rm
1,(0)}\sum_{j_{1}}\frac{C_{1j_{1}}/C_{\rm
1,(0)}}{1+\left|X_{1}\right|^{2}\left(g_{1j_{1}}/g_{\rm
1,(0)}\right)^{2}}\right\\},$ (37)

where

$\displaystyle y_{1}=\frac{{\cal E}_{\rm p}}{\kappa_{1}^{\prime}\sqrt{n_{\rm
1,sat}}},~{}~{}~{}X_{1}=\frac{\left\langle a_{1}\right\rangle}{\sqrt{n_{\rm
1,sat}}},~{}~{}~{}n_{\rm
1,sat}=\frac{\gamma_{\perp}\gamma_{\parallel}}{4g_{\rm
1,(0)}^{2}},~{}~{}~{}C_{\rm 1,(0)}=\frac{g_{\rm
1,(0)}^{2}}{\kappa_{1}^{\prime}\gamma_{\perp}}.$ (38)

Here, $g_{\rm 1,(0)}$ is the single-atom coupling strength for an atom located
at a potential minimum of the dipole trap in cavity 1. The normalized probe
transmission through the system (i.e., the normalized output intensity from
cavity 2) in the semiclassical model is given by

$\displaystyle T=\left|\frac{\left\langle a_{2}\right\rangle}{\left\langle
a_{2}\right\rangle_{0}}\right|^{2}=\frac{1}{y_{1}^{2}}\left(1+\frac{v_{1}^{2}}{\kappa_{b}\kappa_{1}^{\prime}}+\frac{v_{2}^{2}}{\kappa_{b}\kappa_{2}^{\prime}}\right)^{2}\frac{\left|X_{1}\right|^{2}}{\left(1+\dfrac{v_{2}^{2}}{\kappa_{b}\kappa_{2}^{\prime}}\right)^{2}},$
(39)

where $\left\langle a_{2}\right\rangle_{0}$ is the mode amplitude of cavity 2
for resonant driving and no atoms in either cavity. Finally, the input probe
power can be related to $y_{1}$ and $n_{\rm 1,sat}$ through the relation

$\displaystyle P_{\rm in}=\frac{{\cal E}_{\rm
p}^{2}}{2\kappa_{1l}}\hbar\omega_{\rm p}=\frac{{\cal E}_{\rm
p}^{2}}{2\kappa_{1l}}\frac{2\pi\hbar
c}{\lambda}=y_{1}^{2}\left(\frac{2\pi\hbar
c}{\lambda}\right)\left(\frac{{\kappa_{1}^{\prime}}^{2}}{2\kappa_{1l}}\right)n_{\rm
1,sat}.$ (40)

To compare theory and experiment, we plot $T$ versus $P_{\rm in}$ as given in
equation (40), with $n_{\rm 1,sat}$ as a scaling factor.

####  A.5.2 Atoms in cavity 2 only

Now setting $C_{1j_{2}}=1$ (no atoms in cavity 1), we can similarly eliminate
the amplitude $\left\langle a_{1}\right\rangle$ to obtain an equation for the
amplitude $\left\langle a_{2}\right\rangle$ alone, which in scaled variables
takes the form

$\displaystyle iy_{2}$
$\displaystyle=X_{2}\frac{v_{1}\kappa_{2}^{\prime}}{v_{2}\kappa_{1}^{\prime}}\left(1+\frac{\kappa_{b}\kappa_{1}^{\prime}}{v_{1}^{2}}\right)\left\\{1+\frac{v_{2}^{2}\kappa_{1}^{\prime}}{v_{1}^{2}\kappa_{2}^{\prime}}\frac{1}{1+\kappa_{b}\kappa_{1}^{\prime}/v_{1}^{2}}+C_{\rm
2,(0)}\sum_{j_{2}}\frac{C_{2j_{2}}/C_{\rm
2,(0)}}{1+\left|X_{2}\right|^{2}(g_{2j_{2}}/g_{\rm 2,(0)})^{2}}\right\\},$
(41)

where

$\displaystyle y_{2}=\frac{{\cal E}_{\rm p}}{\kappa_{1}^{\prime}\sqrt{n_{\rm
2,sat}}},~{}~{}~{}X_{2}=\frac{\left\langle a_{2}\right\rangle}{\sqrt{n_{\rm
2,sat}}},~{}~{}~{}n_{\rm
2,sat}=\frac{\gamma_{\perp}\gamma_{\parallel}}{4g_{\rm
2,(0)}^{2}},~{}~{}~{}C_{\rm 2,(0)}=\frac{g_{\rm
2,(0)}^{2}}{\kappa_{2}^{\prime}\gamma_{\perp}}.$ (42)

Here, $g_{\rm 2,(0)}$ is the single-atom coupling strength for an atom located
at a potential minimum of the dipole trap in cavity 2. The normalized probe
transmission through the system in this case is given by

$\displaystyle T=\left|\frac{\left\langle a_{2}\right\rangle}{\left\langle
a_{2}\right\rangle_{0}}\right|^{2}=\frac{1}{y_{2}^{2}}\left(\frac{1+v_{1}^{2}/(\kappa_{b}\kappa_{1}^{\prime})+v_{2}^{2}/(\kappa_{b}\kappa_{2}^{\prime})}{v_{1}v_{2}/(\kappa_{b}\kappa_{2}^{\prime})}\right)^{2}\left|X_{2}\right|^{2},$
(43)

and the relationship to the input power becomes

$\displaystyle P_{\rm in}=\frac{{\cal E}_{\rm
p}^{2}}{2\kappa_{1l}}\hbar\omega_{\rm p}=y_{2}^{2}\left(\frac{2\pi\hbar
c}{\lambda}\right)\left(\frac{{\kappa_{1}^{\prime}}^{2}}{2\kappa_{1l}}\right)n_{\rm
2,sat}.$ (44)

####  A.5.3 Position dependence of the atom-cavity coupling

From Lacroute2012 , we take the (square of the) atom-cavity mode coupling
strength, in both cavities, to have the form ($l=1,2$)

$\displaystyle g_{l}(r,\phi,z)^{2}$ $\displaystyle={\cal
C}\left(\frac{\beta}{2q}\right)^{2}\left\\{\left[(1-s)K_{0}(qr)+(1+s)K_{2}(qr)\cos(2\phi)\right]^{2}+(1+s)^{2}K_{2}^{2}(qr)\sin^{2}(2\phi)\right\\}\cos^{2}(\beta
z)$ $\displaystyle+{\cal
C}\left\\{K_{1}^{2}(qr)\cos^{2}(\phi)\right\\}\sin^{2}(\beta z),$ (45)

where ${\cal C}$ is a constant and we use cylindrical coordinates, with the
$z$-direction along the fiber axis and $r$ ($>a$) measured from the center of
the fiber. The propagation constant $\beta=7.87925\times 10^{6}~{}{\rm
m}^{-1}$, and $q=\sqrt{\beta^{2}-n_{2}^{2}k^{2}}$ with $n_{2}=1$ and
$k=2\pi/\lambda$ ($\lambda=852~{}{\rm nm}$). The parameter $s=-0.828$ depends
on $q$, on $h=\sqrt{k^{2}n_{1}^{2}-\beta^{2}}$ ($n_{1}=1.4525$), and on the
radius ($a$) of the nanofiber Lacroute2012 . With respect to the position of a
minimum of the dipole trapping potential located at $(r=r_{l,0},\phi=0,z=0)$,
we find that the above function is in fact very well approximated by the
simplified form

$\displaystyle g_{l}(r,\phi,z)^{2}$
$\displaystyle=g_{l,(0)}^{2}\frac{1}{2}\left[1+A+B\cos(2\beta
z)\right]\frac{e^{-2q^{\prime}(r-r_{l,0})}}{r/r_{l,0}}\cos^{2}(\phi),$ (46)

with $A+B=1$ and $q^{\prime}\simeq 1.3q$. We shall use this approximate form
in the calculations that follow.

####  A.5.4 Integral approximation to the summation over atoms

Assuming sufficiently many atoms in each cavity, we replace the summations in
(37) and (41) as follows:

$\displaystyle
C_{l,(0)}\sum_{j_{l}}\frac{C_{lj_{l}}/C_{l,(0)}}{1+\left|X_{l}\right|^{2}(g_{lj_{l}}/g_{l,(0)})^{2}}\longrightarrow
C_{l,(0)}\int_{0}^{L}\int_{-\pi}^{\pi}\int_{a}^{\infty}\rho_{l}(r,\phi,z)\frac{(g_{l}(r,\phi,z)/g_{l,(0)})^{2}}{1+\left|X_{l}\right|^{2}(g_{l}(r,\phi,z)/g_{l,(0)})^{2}}\,rdrd\phi
dz,$ (47)

where $\rho_{l}(r,\phi,z)$ is the atom density distribution in cavity $l$.
With respect to the $z$-direction, atoms are tightly confined by a standing
wave optical potential. However, this standing wave is incommensurate with the
cavity mode standing wave and so, on average, the atomic distribution along
the $z$-direction, with regards to the cavity mode, can be regarded as
uniform. That is, we take $\rho_{l}(r,\phi,z)=\rho_{l}(r,\phi)$, and then the
integration over $z$ can be carried out exactly, reducing the above expression
to

$\displaystyle
C_{l,(0)}L\frac{1}{\left|X_{l}\right|^{2}}\int_{-\pi}^{\pi}\int_{a}^{\infty}\rho_{l}(r,\phi)f_{l}(r,\phi)rdrd\phi,$
(48)

where

$\displaystyle
f_{l}(r,\phi)=1-\left[\left(1+A\left|X_{l}\right|^{2}\dfrac{e^{-2q^{\prime}(r-r_{0})}}{r/r_{0}}\cos^{2}(\phi)\right)\left(1+\left|X_{l}\right|^{2}\dfrac{e^{-2q^{\prime}(r-r_{0})}}{r/r_{0}}\cos^{2}(\phi)\right)\right]^{-1/2}.$
(49)

Making a harmonic approximation to the atom trapping potential, the atomic
density distribution can be written, in Cartesian coordinates, as

$\displaystyle\rho_{l}(x,y)=\rho_{l,0}e^{-(x-x_{l,0})^{2}/\sigma_{l,x}^{2}}e^{-y^{2}/\sigma_{l,y}^{2}},$
(50)

where $x_{l,0}=r_{l,0}$ and $\sigma_{x,y}=\sqrt{2k_{\rm
B}T/m\omega_{x,y}^{2}}$ for a gas at temperature $T$ and with trapping
frequencies $\omega_{x,y}$.

The double integral (48) can be rewritten in Cartesian coordinates as

$\displaystyle
C_{l,(0)}L\rho_{l,0}\frac{1}{\left|X_{l}\right|^{2}}\iint_{x^{2}+y^{2}\geq
a^{2}}e^{-(x-x_{l,0})^{2}/\sigma_{l,x}^{2}}e^{-y^{2}/\sigma_{l,y}^{2}}f_{l}(x,y)\,dxdy.$
(51)

In the $x$-direction, the function $f_{l}(x,y)$ varies slowly (i.e.,
approximately linearly) in comparison to the Gaussian density distribution for
the characteristic parameters of the experiment. Given this, and assuming
$\sigma_{l,x}\ll x_{l,0}-a$ ($x_{l,0}-a$ is the distance from the trap center
to the surface of the nanofiber), the integral is well approximated by

$\displaystyle
C_{l,(0)}L\rho_{l,0}\frac{1}{\left|X_{l}\right|^{2}}\sqrt{\pi}\sigma_{l,x}\sigma_{l,y}\int_{-\infty}^{\infty}e^{-u^{2}}f_{l}(x_{l,0},u)\,du,$
(52)

where

$\displaystyle
f_{l}(x_{l,0},u)=1-\frac{1}{\sqrt{\left(1+A\left|X_{l}\right|^{2}s(x_{l,0},u)\right)\left(1+\left|X_{l}\right|^{2}s(x_{l,0},u)\right)}},$
(53)

with

$\displaystyle
s(x_{l,0},u)=\frac{\exp\left[-2q^{\prime}x_{l,0}\left(\sqrt{1+(\sigma_{l,y}^{2}/x_{l,0}^{2})u^{2}}-1\right)\right]}{\left[1+(\sigma_{l,y}^{2}/x_{l,0}^{2})u^{2}\right]^{3/2}}.$
(54)

The trapping along the $y$-direction is typically much weaker than in the
other directions (i.e., $\sigma_{l,y}\gtrsim 5\sigma_{l,x}$), but, given
uncertainty in the exact trapping parameters, and making the (not
unreasonable) assumption that $\sigma_{l,y}^{2}/x_{l,0}^{2}\ll 1$, we can set
$s(x_{l,0},u)=1$ in the integral, and then (52) reduces to

$\displaystyle C_{l,(0)}N_{l,{\rm
eff}}\frac{2}{(1+A)}\frac{1}{\left|X_{l}\right|^{2}}\left[1-\frac{1}{\sqrt{(1+A\left|X_{l}\right|^{2})(1+\left|X_{l}\right|^{2})}}\right],$
(55)

where $N_{l,{\rm eff}}=\rho_{l,0}(L/2)(1+A)\pi\sigma_{l,x}\sigma_{l,y}$. In
our comparison between theory and experiment, we use (55) to approximate the
summations in (37) and (41). We use a value $A=0.17$, deduced from the
experimental parameters and (A.5.3). As mentioned earlier, matching the theory
curves for $T$ versus $P_{\rm in}$ to the experimental data yields estimates
of $n_{l,{\rm sat}}$, from which we can deduce values for $g_{l,(0)}$. From
these, and the estimates of $g_{l,{\rm eff}}$ obtained from the weak-field
transmission spectra, we then deduce estimates for $N_{l,{\rm eff}}$.

##  Appendix B Experimental Setup

The schematic of the experimental setup is shown in Fig. 6. Each cavity
consists of a nanofiber, both ends of which are connected to standard single-
mode fibers through tapered regions, and sandwiched by a pair of fiber-Bragg-
grating (FBG) mirrorsKato_PRL_2015 . The single-pass loss inside each cavity
is 2%, which is dominated by the losses in two fiber splices. The diameter and
length of the nanofibers are 400 nm and 3 mm, respectively. The nanofibers are
located in two separate UHV chambers.

We put three-paddle polarization controllers in the connecting fiber and
cavity 2 to compensate polarization rotation in the optical fiber. For the
case of $L_{\rm f}=0.83$ m (Figs. 3a and c in the main text), we replace the
polarization controller in the connecting fiber with a two-paddle one because
of the limitation of the fiber length, which cause the imperfect polarization
compensation. The lasers for the optical trap and probe are mixed by using
dichroic mirrors and coupled to the optical fiber. In contrast, the optical
pumping pulses are irradiated on the nanofiber from its side. The
polarizations of the trapping fields are linearly polarized and parallel to
each other at the nanofiber regions. The transmitted probe beam passes the
filtering system to block unwanted stray light, and is detected by an
avalanche photodetector. The detection efficiency including the transmittance
of the filtering system and the quantum efficiency of the detector is 0.26.

Figure 6: Schematic of the setup. Cavity 1 of length $L_{1}=0.92$ m and Cavity
2 of length $L_{2}=1.38$ m are connected by a fiber of length $L_{\rm f}$. DM:
Dichroic mirror, F: filters, APD: avalanche photodiode.

##  Appendix C Measurement sequence and data analysis

Figure 7: Sequence diagram of the spectroscopy. After the preparation of atoms
in the MOT and additional cooling, the spectroscopy procedure repeats five
times.

The schematic diagram of the pulse sequence for the transmission spectroscopy
is shown in Fig. 7. Each sequence starts from cooling and trapping cesium
atoms in the two UHV chambers by using standard six-beam magneto-optical traps
(MOT). We use the D2 line F=4$\rightarrow$F’=5 transition for cooling and the
F=3$\rightarrow$F’=4 transition for repumping in the MOTs. In the loading
stage of the MOTs, the total intensity and the detuning of the cooling beams
are, respectively, 16$I_{s}$ and -1.25$\Gamma$, where $I_{s}$ and $\Gamma$ are
the saturation intensity and natural linewidth of the cooling transition.

The number of the atoms in the MOTs saturates within the loading time of about
500 ms. After the preparation of the atoms in the MOT, we adjust the resonance
frequency of the cavity array close to the F=4$\rightarrow$F’=5 transition. We
input a monitor beam with the power of 130 pW and the frequency locked to the
F=4$\rightarrow$F’=5 transition, while changing the cavity lengths of the
cavity array (cavity 1, 2, and the connecting fiber), and the transmission of
the monitor beam is used as a start trigger for the following sequence. The
MOTs are kept during the monitoring stage, and the positions have a small
offset from the nanofibers to keep the cavities without atoms. When the
transmission surpass a threshold, we switch off the beam for the monitor, and
change the detuning and intensity of the cooling laser to 0.5$I_{s}$ and
-3.45$\Gamma$, respectively, and hold for 32 ms for further cooling and
loading the atoms into the optical trap. During the cooling stage, the
positions of the atoms are overlapped onto the nanofibers to load them in the
traps. After the loading atoms into the traps, we send a probe pulse and sweep
the frequency detuning to the F=4$\rightarrow$F’=5 over $\pm$30 MHz within 4
ms. To observe the transmission spectrum without atoms (empty cavity), we
optically pump the atoms into the F=3 state, which is off-resonant to the
cavity field, by using the optical pumping pulse irradiated from the side of
nanofibers. We use the F=4$\rightarrow$F’=3 transition for the pumping, and
the pulse duration is 1 ms. After the optical pumping, we send a probe pulse
again, and sweep the frequency detuning in the same manner as the previous
probe pulse. We then switch on the cooling and repumping lasers to cool and
load the atoms into the optical trap again. The cooling duration is 4 ms. The
spectroscopy procedure repeats five times per single MOT loading event. Figure
8 shows typical transmission spectra for a sequence of five spectroscopy
procedures, each of which is an average of about 15 sets of data. It can be
seen that the splittings for the center peak (fiber-dark mode) reduces as the
spectroscopy procedure is repeated, which is due to the decrease of the number
of the atoms in the optical trap. We use the average of the second, third, and
forth spectra (after taking an average of about 15 sets of data for each of
the five spectroscopy procedures in the same way as the spectra in Fig. 8) for
the data analysis in the main text.

Figure 8: Typical transmission spectra for the five consecutive spectroscopy
procedures per single MOT loading event. The experimental condition is the
same as that of Fig. 2d in the main text. The spectra include vertical offsets
for clarity.

In the data analysis, the observed spectrum without the atom-cavity
interaction (empty cavity), where the atoms are optically pumped into the F=3
state, is used to judge whether the cavity array is resonant to the atomic
transition or not. When all cavities (Cavity1, 2 and the channel part) are
resonant to the atom, the transmission spectrum has a symmetric triplet peaks
centered at the zero atom-probe detuning (see Figs 2a, 3a, 3c in the main
text). In order to choose the data in the relevant condition, we put the
following criteria to filter the data; the spectrum of the empty cavity has a
largest transmission peak around the atomic resonance frequency within
$\pm$0.9 MHz, and has symmetrical sideband peaks at $\pm\sqrt{2}\tilde{v}$
within $\pm$1.8 MHz.

## Acknowledgments

This work was supported by JST CREST Grant Number JPMJCR1771, JSPS KAKENHI
Grant Numbers 16H01055, 18H04293, JST PRESTO Grant Number JPMJPR1662, Japan,
and Institute for Advanced Theoretical and Experimental Physics, Waseda
University. S. P. acknowledges support from the Marsden Fund of the Royal
Society of New Zealand (Contract No. UOA1328).

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