# Facilitation Induced Transparency and Single-Photon Switch with Dual-Channel
Rydberg Interactions

Yao Ding1 Zhengyang Bai1<EMAIL_ADDRESS>Guoxiang Huang1 Weibin Li2
1State Key Laboratory of Precision Spectroscopy, East China Normal University,
Shanghai 200062, China
2School of Physics and Astronomy, and Centre for the Mathematics and
Theoretical Physics of Quantum Non-equilibrium Systems, University of
Nottingham, Nottingham, NG7 2RD, UK

###### Abstract

We investigate facilitation induced transparency (FIT) enabled by strong and
long-range Rydberg atom interactions between two spatially separated optical
channels. In this setting, the resonant two-photon excitation of Rydberg
states in a target channel is conditioned by a single Rydberg excitation in a
control channel. Through the contactless coupling enabled by the Rydberg
interaction, the optical transparency of the target channel can be actively
manipulated by steering the optical detuning in the control channel. By
adopting a dressed-state picture, we identify two different interference
pathways, in which one corresponds to Rydberg blockade and an emergent one
results from facilitation. We show that the FIT is originated from the Rydberg
interaction and the quantum interference effect between the two pathways,
which is different from conventional electromagnetically induced transparency
realized by single-body laser-atom coupling. We find that the FIT in such a
dual-channel setting is rather robust, insensitive to changes of systemic
parameters, and can be generalized to multi-channel settings. Moreover, we
demonstrate that such a FIT permits to realize controllable single-photon
switches, which also paves a route to detect Rydberg facilitation by using
optical absorption spectra. Our study contributes to current efforts in
probing correlated many-body dynamics and developing single-photon quantum
devices based on Rydberg atom ensembles.

##  I Introduction

Cold gases of Rydberg atoms have emerged as a versatile platform for studying
quantum nonlinear optics, non-equilibrium statistical physics, and quantum
simulation of strongly interacting many-body systems Saffman  _et al._ (2010);
Murray and Pohl (2016); Adams  _et al._ (2019); Browaeys and Lahaye (2020),
including antiferromagnetic phase transition Lee  _et al._ (2011); Bernien
_et al._ (2017), quantum many-body scars Lesanovsky and Katsura (2012); Turner
_et al._ (2018); Bluvstein  _et al._ (2021), and Heisenberg XYZ spin model
Geier  _et al._ (2021). At the same time, Rydberg atoms are of technological
importance, which allow to create collectively encoded qubit Lukin  _et al._
(2001); Spong  _et al._ (2021), entanglement Carr and Saffman (2013); Jau  _et
al._ (2016), photonic or neutral-atom gates for quantum information processing
Saffman  _et al._ (2010); Khazali  _et al._ (2015); Liu  _et al._ (2021,
2022a), precision measurements Sedlacek  _et al._ (2012); Jing  _et al._
(2020), and so on. The enabled fundamental research and practical applications
are rooted largely by the fact that Rydberg atoms offer strong and long-ranged
interactions. In addition, their internal and external states can be actively
manipulated under current experimental conditions Browaeys and Lahaye (2020);
Vasilyev  _et al._ (2020).

Rydberg many-body physics is largely based on two mechanisms: Rydberg blockade
and facilitation (antiblockade), which can be realized by tuning the
excitation laser on resonance with atomic transitions, and for matching the
Rydberg interaction induced energy shifts Su  _et al._ (2020), respectively.
The former prevents multiple Rydberg excitations in the vicinity of a Rydberg
atom, which benefits to the creation of quantum correlation and entanglement
between atoms Carr and Saffman (2013); Li  _et al._ (2014). The facilitation,
opposite to the blockade, allows the formation of Rydberg clusters Ates  _et
al._ (2007); Malossi  _et al._ (2014); Schempp  _et al._ (2014); Khazali  _et
al._ (2015); Marcuzzi  _et al._ (2017); Liu  _et al._ (2022b); it has been a
central building block of non-equilibrium phase transitions, such as
aggregation Schempp  _et al._ (2014); Lesanovsky and Garrahan (2014), epidemic
spreading Pérez-Espigares  _et al._ (2017); Wintermantel  _et al._ (2021), and
self-organized criticality Helmrich  _et al._ (2020); Ding  _et al._ (2020),
etc.. Experimentally, Rydberg facilitation can be detected via absorption
imaging Helmrich  _et al._ (2020) or direct ionization Amthor  _et al._
(2010); Gutiérrez  _et al._ (2017); Bai  _et al._ (2020a). However, although
these methods have high efficiency, they typically damage many-body coherence
after experimental operations Mohapatra  _et al._ (2007).

In this work, we theoretically investigate the optical property of a dual-
channel Rydberg ensemble He  _et al._ (2014); Viscor  _et al._ (2015); Busche
_et al._ (2017); Thompson  _et al._ (2017); Khazali  _et al._ (2019); Chen
_et al._ (2021), and illustrate that the facilitation mechanism can lead to an
effective interaction to probe photons, which can be directly measured by
using atomic absorption spectrum. The system we consider consists of two
optical channels that are remotely separated in space, in which target channel
$A$ is resonantly excited to a Rydberg state via electromagnetically induced
transparency (EIT) Pritchard  _et al._ (2010) and affected by control channel
$B$ via long-ranged Rydberg interactions. Such a setting is available to
realize with present-day experimental techniques Busche  _et al._ (2017);
Thompson  _et al._ (2017); Chen  _et al._ (2021).

We show that a transparency window for probe photons opens through adjusting
the detuning of the channel $B$, and this transparency window is sensitive to
the Rydberg interaction. By using numerical and dressed-state calculations, we
reveal many-body quantum pathways and characterize an interaction-dependent
scaling function. A key finding is that the two resonant transition pathways
are respectively originated from the Rydberg blockade and facilitation; it is
the quantum interference effect between the two pathways that leads to the
facilitation induced transparency (FIT) of the probe photons. The relevant
model can moreover be generalized to Rydberg atomic gases with multiple
optical channels.

The results on the FIT described above may have potential applications in all-
optical quantum information processing. To realize optical quantum devices at
single photon level, up to now various quantum systems have been suggested to
implement optical switches or transistors, such as single molecules Hwang  _et
al._ (2009), quantum dots Bose  _et al._ (2012); Englund  _et al._ (2012); Loo
_et al._ (2012); Volz  _et al._ (2012) and neutral atoms Bajcsy  _et al._
(2009); O’Shea  _et al._ (2013). However, due to the saturable absorption of
single quantum emitters, it is difficult to realize a sizable optical non-
linearity at single-photon level. To enhance light-matter interactions, one
way is to use high-finesse optical resonators to confine photons to small mode
volumes Chen  _et al._ (2013); Chang  _et al._ (2014). At variance with
single-body quantum systems, strong and long-range interactions between
Rydberg atoms can be effectively mapped onto photon-photon interactions via
two-photon scheme. This provides an alternative avenue to design single-photon
switching in free space Baur  _et al._ (2014); Gorniaczyk  _et al._ (2014);
Tiarks  _et al._ (2014). We demonstrate that the system proposed here can be
used to design contactless single-photon switches, which is different from
early studies, where Rydberg single-photon switches were realized in the
blockade regime Baur  _et al._ (2014); Gorniaczyk  _et al._ (2014); Tiarks
_et al._ (2014); Li and Lesanovsky (2015); Murray  _et al._ (2016); Yu  _et
al._ (2020); Chen  _et al._ (2021). By employing the presented dual-channel
system, one can easily launch both facilitated and blockade single-photon
switch by manipulating the control channel.

Moreover, different from that of stationary Rydberg single-photon switches
reported before, the efficiency of the FIT-based photon switch suggested here
can be controlled dynamically. The use of the dual-channel setting brings more
potential benefits. Here Rydberg atoms in the control and target channels are
spatially separated. For example, complicated atomic and molecular processes
could be avoided, which play roles in the Rydberg blockade in atomic ensembles
Derevianko  _et al._ (2015). As we will show later, Rydberg populations in the
FIT is relatively weak. This could mitigate the formation of ultralong-range
Rydberg molecules Bendkowsky  _et al._ (2009); Li  _et al._ (2011) and Rydberg
polarons Camargo  _et al._ (2018) due to collisions between ground state and
Rydberg atoms Bai  _et al._ (2020b). These features mean that coherence of the
atom could be increasingly protected in the FIT regime.

The remainder of the article is arranged as follows. In Sec. II, we introduce
the Rydberg dual-channel model based on a master equation description. The
level scheme and typically parameters are given. In Sec. III, we reveal the
FIT mechanism through analyzing a two-atom model both analytically and
numerically. In Sec. III.1, coherence of the probe field is obtained
numerically. By comparing with the approximate result, we reveal the
importance of two-channel correlations in the emergence of FIT. In Sec. III.2,
we present an analytical dressed-state approach on the model and show the
scaling of FIT as a function of the inter-channel interaction. In Sec. IV,
based on the FIT we discuss controllable single-photon switch protocol.
Finally, Sec. V gives a summary of the main results obtained in this work.

##  II Dual-channel Rydberg model

The system we consider consists of two ensembles of cold atoms driven by
channel-dependent laser fields. The two ensembles are prepared in parallel,
elongated traps along $z$ direction, and separated by distance $d$; see Fig.
1(a).

Figure 1: Scheme of dual-channel Rydberg atomic gas for realizing FIT. Two
atomic ensembles, i.e. target channel $A$ and control channel $B$, are
prepared in parallel and separated by distance $d$. (a) A weak probe field
$\hat{{\cal E}}_{p}$ propagates in the channel $A$, whose dynamics is
facilitated by the Rydberg excitation in the channel $B$. Long-range inter-
channel Rydberg interaction permits a contactless control for the two channels
and hence the propagation of the probe photon. (b) Level diagram and
excitation scheme. Atoms in the channel $A$ have three quantum states (i.e.
$|1\rangle$, $|2\rangle$, $|3\rangle$; $|3\rangle$ is a Rydberg state) forming
a Rydberg-EIT, where resonant transitions $|1\rangle\leftrightarrow|2\rangle$
and $|2\rangle\leftrightarrow|3\rangle$ are respectively provided by weak
probe field $\hat{\cal E}_{p}$ and strong coupling field (Rabi frequency
$\Omega$). In the channel $B$, a control field with Rabi frequency
$\Omega_{c}$ drives the transition between the ground state $|1\rangle$ to the
Rydberg state $|3\rangle$ with detuning $\Delta_{c}$. $V_{AB}$ is the van der
Waals interaction between the atoms in the two channels.

They form two optical channels, i.e. a target channel $A$ (containing $N_{A}$
atoms) and a control channel $B$ ($N_{B}$ atoms). The atomic level structure
and the optical excitation scheme are illustrated schematically in Fig. 1(b).
We assume that, in the target channel $A$, a weak probe laser field $\hat{\cal
E}_{p}(z)$ [Rabi frequency $\hat{\Omega}_{p}(z)=g\hat{\cal E}_{p}(z)$]
resonantly couples the ground state $|1\rangle$ and the excited state
$|2\rangle$, and a coupling laser field (Rabi frequency $\Omega$) resonantly
couples the state $|2\rangle$ and the Rydberg state $|3\rangle$, which forms a
scheme of conventional Rydberg-EIT Mohapatra  _et al._ (2007). Here
$g=\mu_{21}\sqrt{\omega_{p}/(2\hbar\epsilon_{0}V_{p})}$ is the atom-field
coupling constant, with $\omega_{p}$ the central angular frequency of the
probe field, $V_{p}$ the quantization volume of the probe field, and
$\mu_{21}$ the electric-dipole matrix element associated with the transition
between $|1\rangle$ and $|2\rangle$. In the control channel $B$, atomic levels
$|1\rangle$ and $|3\rangle$ are coupled off-resonantly by another control
laser field (Rabi frequency $\Omega_{c}$) with detuning $\Delta_{c}$, which
plays a critical role for realizing the FIT. We assume that both $\Omega$ and
$\Omega_{c}$ are strong enough, thus they can be taken as classical fields and
their depletion are negligible.

Under the rotating wave approximation, by setting $\hbar=1$ the Hamiltonian
for the channels $A$ and $B$ are respectively given by

$\displaystyle\hat{H}_{A}=-\frac{1}{2}\sum_{j\in
A}\left[\hat{\Omega}_{p}\hat{\sigma}_{21}(z_{j})+\Omega\hat{\sigma}_{32}(z_{j})+{\rm
H.c.}\right],$ (1a) $\displaystyle\hat{H}_{B}=-\sum_{j\in
B}\left[\frac{\Omega_{c}}{2}\hat{\sigma}_{31}(z_{j})-\Delta_{c}\hat{\sigma}_{33}(z_{j})+{\rm
H.c.}\right],$ (1b)

with $\hat{\sigma}_{ab}(z_{j})\equiv|a_{j}\rangle\langle b_{j}|$ ($z_{j}$ is
the position of $j$th atom in the respective ensemble) and H.c. representing
Hermitian conjugate of the preceding terms. We assume that the spatial
extension of each atomic ensemble is small, so that multiple Rydberg
excitations within each ensemble is prohibited by the strong and long-ranged
interaction between Rydberg states. However, there exists an inter-channel
Rydberg interaction between the target channel $A$ and the control channel $B$
(with the distance $d$ typically Rydberg blockade radius $\sim 5-10$ $\mu$m),
which leads to the following inter-channel interaction Hamiltonian

$H_{\rm int}=\sum_{j\in A,l\in
B}V(z_{jl})\hat{\sigma}_{33}(z_{j})\hat{\sigma}_{33}(z_{l}),$ (2)

Here we have assumed that the target and control channels are both excited to
$|nS\rangle$ states (or $|nD\rangle$ states). Two Rydberg channels interact
via the van der Waals (vdW) interaction
$V(z_{jl})=-C_{6}/[(z_{j}-z_{l})^{2}+d^{2}]^{3}$, with $C_{6}$ the dispersion
coefficient. Note that, except for the vdW interaction, a resonant dipole-
dipole interaction can also be engineered by coupling the ensembles to Rydberg
states $|S\rangle$ and $|P\rangle$, which may lead to spin-exchange between
remote Rydberg channels Li  _et al._ (2014); Viscor  _et al._ (2015); Thompson
_et al._ (2017); Khazali  _et al._ (2019). This is a topic deserving to be
explored, but here we focus our consideration only on the case of the vdW
interaction.

The dynamics of the system including the two channels is governed by the
master equation,

$\displaystyle\frac{d\rho}{dt}=-i\left[H,\rho\right]+D(\rho)+D_{d}(\rho),$ (3)

where $\rho(t)$ is density matrix, and $\hat{H}=\hat{H}_{A}+\hat{H}_{B}+H_{\rm
int}$ is the total Hamiltonian of the system. In Eq. (3)

$\displaystyle D(\rho)=$
$\displaystyle\sum_{\mu=A,B}\sum_{j\in\mu}\Gamma_{ba}^{\mu}\left\\{\hat{\sigma}_{ba}(z_{j})\rho\hat{\sigma}_{ab}(z_{j})\right.$
(4)
$\displaystyle\left.-\frac{1}{2}\left[\rho\hat{\sigma}_{ab}(z_{j})\hat{\sigma}_{ba}(z_{j})+\hat{\sigma}_{ab}(z_{j})\hat{\sigma}_{ba}(z_{j})\rho\right]\right\\},$

describes the respective dissipation processes from $|a\rangle\to|b\rangle$
with the rates $\Gamma_{ba}^{\mu}=\Gamma_{12}^{A}$, $\Gamma_{23}^{A}$, and
$\Gamma_{13}^{B}$;

$D_{d}(\rho)=\sum_{\mu=A,B}\sum_{j\in\mu}\gamma_{a}^{\mu}\left\\{2\hat{\sigma}_{aa}(z_{j})\rho\hat{\sigma}_{aa}(z_{j})-\left[\rho\hat{\sigma}_{aa}(z_{j})+\hat{\sigma}_{aa}(z_{j})\rho\right]\right\\},$
(5)

gives the dephasing of the atomic coherence (originated from atomic
collisions, residue Doppler effect, dipole-dipole interaction between the
Rydberg atoms, finite laser linewidth, etc.), with rates $\gamma_{a}^{\mu}$
($\mu=A,B$; $a=2,3$). In the following, for the convenience of numerical
calculations, we measure all atomic parameters (e.g. $\Omega_{c}$,
$\Omega_{p}$, $\Delta_{c}$) in unit of $\Gamma_{12}^{A}$ (typically,
$\Gamma_{12}^{A}\simeq 2\pi\times 6.06$ MHz for $|5P_{3/2}\rangle$ state in Rb
atom; $\Gamma_{12}^{A}\simeq 2\pi\times 5.22$ MHz for $|6P_{3/2}\rangle$ state
in Cs atom Steck (2019)).

In this setting, the inter-channel interaction is the central element in the
Hamiltonian, with which the dynamics of the target channel $A$ can be actively
manipulated by varying the relevant atomic parameters in the control channel
$B$ (i.e., $\Omega_{c}$, $\Delta_{c}$). Note that the control channel $B$ is
adopted with a two-photon Raman scheme. Due to the large single-photon
detuning, the channel $B$ can be described by an effective two-level system.
Though this is a simplified approach, it can demonstrate the essential physics
inherent in the model. Dynamics of the two-photon scheme in channel $B$ is
presented in Appendix A and will be briefly discussed below in Fig. 3(c).

##  III Facilitation Induced Transparency

In order to demonstrate the FIT and to reveal its basic physical mechanism, we
first consider a simple two-atom scenario [i.e. one atom (atom $A$) is in the
channel $A$ and another one (the atom $B$) is in the channel $B$] that
represents the minimal model of the dual-channel setting Li  _et al._ (2014);
Viscor  _et al._ (2015).

With such a consideration, the effective Hamiltonian is reduced into the form
of $\hat{H}=\hat{H}_{A}+\hat{H}_{B}+\hat{H}_{\rm int}$, where
$\hat{H}_{A}=-(\hat{\Omega}_{p}\hat{\sigma}_{21}^{A}/2+\Omega\hat{\sigma}_{32}^{A}/2)+{\rm
H.c.}$,
$\hat{H}_{B}=-\Omega_{c}\hat{\sigma}_{31}^{B}/2+\Delta_{c}\hat{\sigma}_{33}^{B}+{\rm
H.c.}$, and $H_{\rm int}=V_{AB}\hat{\sigma}_{33}^{A}\hat{\sigma}_{33}^{B}$.
Here $\hat{\sigma}_{ab}^{A}=\hat{\sigma}_{ab}(z_{A})$,
$\hat{\sigma}_{ab}^{B}=\hat{\sigma}_{ab}(z_{B})$, the two-body interaction is
given by $V_{AB}=-C_{6}/d^{6}$. Note that under the EIT condition the probe
field has negligible attenuation and can be approximately treated as a
classical one Gorshkov  _et al._ (2007), hence for simplicity we replace
$\hat{\Omega}_{p}$ by $\Omega_{p}$ in the approach of this section. The
spatial-temporal dynamics of the quantized probe field at a level of single
photon Fleischhauer and Lukin (2002); Li and Lesanovsky (2015); Zhang and
Huang (2021) in high density atom gases will be analyzed in section IV.

###  III.1 Optical coherence and inter-channel correlation

The response character of the system to the probe field is characterized by
the atomic coherence between the states $|1\rangle$ and $|2\rangle$ in the
channel $A$, i.e. $\rho_{21}^{A}$, where $\rho_{ab}^{\mu}\equiv{\rm
Tr}(\rho\sigma_{ba}^{\mu})$ ($\mu=A,B$) are one-body reduced density matrix
(DM) elements. The absorption (refraction) of the probe field is determined by
the imaginary (real) part Im($\rho_{21}^{A}$) [Re($\rho_{21}^{A}$) ] of
$\rho_{21}^{A}$. The corresponding optical susceptibility will be given in
section IV; see Eq. (14) below. In the following calculations, we shall
examine in particular the behavior of $\rho_{21}^{A}$ when adjusting the
detuning $\Delta_{c}$ in the channel $B$, which is one of the system
parameters easy to control experimentally.

Figure 2: Optical coherence and inter-channel correlation in the two-atom
system. (a) Level diagram and excitation scheme between the Rydberg blockade
state $|1_{A}3_{B}\rangle$ (occurring at $\Delta_{c}=0$) to the facilitated
state $|3_{A}3_{B}\rangle$ (occurring at $\Delta_{c}=-V_{AB}$) of the two-atom
model. For more details, see the text. (b) FIT transparency window induced by
the inter-channel interaction $V_{AB}$, as seen in the absorption spectrum
Im($\rho_{21}^{A}$) as a function of $\Delta_{c}$. $\Delta E$ is the energy
separation between centers of the two peaks of the FIT spectrum. (c1)
Im($\rho_{21}^{A}$) as a function of $\Delta_{c}$ and $V_{AB}$.
Im($\rho_{21}^{A}$) exhibits a single peak for smaller $V_{AB}$; it however
splits into two peaks for larger $V_{AB}$. The dip between the two peaks
becomes wider and deeper as $V_{AB}$ increases, similar to the behavior of EIT
in three-level systems. Corresponding Rydberg excitations are shown in (b)
(dashed red line for $\rho_{33}^{A}$; dot-dashed blue line for
$\rho_{33}^{B}$), and also in (c2) which shows that the channel $A$ is fully
blockaded independent of $V_{AB}$ (solid blue line) for $\Delta_{c}=0$ but
$\rho_{33}^{A}$ is facilitated around $\Delta_{c}=-V_{AB}$ (dashed red line).
(d) Analytical result (dotted red line) given by Eq. (6), which captures well
with the one by solving the master equation (ME) numerically ( solid black
line). The quantum correlation $O_{AB}$ (dot-dashed blue line) is significant
for the FIT effect. The parameters used are
$\Omega=\Omega_{c}=5\Gamma_{12}^{A}$, $\Omega_{p}=0.5\Gamma_{12}^{A}$,
$\Gamma_{23}^{A}=10^{-3}\Gamma_{12}^{A}$, and
$\Gamma_{13}^{B}=\Gamma_{12}^{A}$; in panels (b) and (d),
$V_{AB}=15\Gamma_{12}^{A}$.

Fig. 2(b) shows the result on the absorption spectrum $\rm{Im}(\rho_{21}^{A})$
as a function of $\Delta_{c}$ through solving the master equation (3)
numerically. We see that $\rm{Im}(\rho_{21}^{A})$ has an EIT-like pattern,
where a transparent window (called FIT transparent window hereafter) in the
absorption spectrum opens. For weak inter-channel interaction (i.e. small
$V_{AB}$), Im($\rho_{21}^{A}$) displays only a single absorption peak, while
for stronger interaction (i.e. larger $V_{AB}$) two absorption peaks occur
[see Fig. 2(c1)]. Concretely, in the region of large $V_{AB}$, one peak (the
lower one) locates at $\Delta_{c}=0$ and the other (the higher one) locates at
$\Delta_{c}=-V_{AB}$. The width of the two peaks is $\Delta E$ [see Fig.
2(b)]. Our calculating result also shows that $\rho_{21}^{A}$ strongly
correlates to Rydberg populations $\rho_{33}^{A}$ and $\rho_{33}^{B}$.

In Fig. 2(c1), one can observe clearly that there is a crossover from one peak
to two ones for $\rm{Im}(\rho_{21}^{A})$ through the adjustment of $V_{AB}$,
and the transparent window becomes wider and deeper as inter-channel
interaction $V_{AB}$ increases. This fact tells us that the splitting of the
probe-field absorption spectrum (i.e. FIT) is facilitated by the contribution
of the inter-channel interaction $V_{AB}$. For illustrating this point
further, the left part of Fig. 2(a) [which corresponds to the point indicated
by the small green circle in Fig. 2(b)] gives the level diagram and excitation
scheme between the Ryder blockade state $|1_{A}3_{B}\rangle$ (occurring for
$\Delta_{c}=0$) to the facilitated state $|3_{A}3_{B}\rangle$ (occurring for
$\Delta_{c}=-V_{AB}$) of the two-atom model; the solid gray circles (solid
blue circles) denote ground-state (Rydberg state) atoms. The right part of the
figure is a simplified representation of the excitation scheme showing in the
left part. Although the transition of the target atom $A$ to the Rydberg state
acquires an energy shift because of the inter-channel interaction $V_{AB}$, by
sweeping the detuning $\Delta_{c}$ in the channel $B$ the system can be driven
from the Rydberg blockade state $|1_{A}3_{B}\rangle$ to the facilitated state
$|3_{A}3_{B}\rangle$, where the channels $A$ and $B$ are simultaneously
excited to the Rydberg state near at $\Delta_{c}=-V_{AB}$ [see Fig. 2(b) and
(c2)]. Since the positions of the excitation blockade and the facilitation
overlaps well with the peaks of $\rm{Im}(\rho_{21}^{A})$, the FIT transparency
window between the two strong absorption peaks is thus induced by the both
effects of the excitation blockade and the facilitation.

To verify the behavior of the atomic coherence and the FIT phenomenon in a
direct way, we have solved the Heisenberg equations of motion of the system.
The calculation shows that $\rho_{21}^{A}$ is given by the following form

$\displaystyle\rho_{21}^{A}\simeq-\frac{2V_{AB}\,\rho_{31,33}^{AB}}{\Omega},$
(6)

where
$\rho_{31,33}^{AB}\equiv\langle\hat{\sigma}_{13}^{A}\hat{\sigma}_{33}^{B}\rangle$
is two-body DM element (two-body correlator) [see Appendix B for details].

The result predicted by using this approach is plotted in Fig. 2(d) by a
dotted red line. One sees that the analytical (6) captures well with the one
by solving the master equation (ME) numerically (given by the solid black
line). One notes that the variance
$O_{AB}=\langle\hat{\sigma}_{13}^{A}\hat{\sigma}_{33}^{B}\rangle-\langle\hat{\sigma}_{13}^{A}\rangle\langle\hat{\sigma}_{33}^{B}\rangle=\rho_{31,33}^{AB}-\rho_{31}^{A}\rho_{33}^{B}$
exhibits the similar double peak structure of the FIT. In the above
calculations, system parameters used are $\Omega=\Omega_{c}=5\Gamma_{12}^{A}$,
$\Omega_{p}=0.5\Gamma_{12}^{A}$, $\Gamma_{23}^{A}=10^{-3}\Gamma_{12}^{A}$,
$\Gamma_{13}^{B}=\Gamma_{12}^{A}$, and $V_{AB}=15\Gamma_{12}^{A}$ in panels
(b) and (d). Based on the results described in Fig. 2, we conclude that FIT is
mainly contributed by the inter-channel quantum correlation, contributed by
the Rydberg interaction between the atom $A$ and the atom $B$. It should be
stressed that such results cannot be predicted by using the approach of mean
field theory Schempp  _et al._ (2010); Pritchard  _et al._ (2010). This
indicates that the two-body correlation could be effectively mapped to the
atomic coherence $\rho_{21}^{A}$, providing a method to detect Rydberg
facilitation directly from the optical absorption spectrum, which is the key
finding of this work.

###  III.2 Excitation pathways identified through dressed-state approach and
generation to multiple target channels

Even when the system involves only two atoms, it is difficult to solve the
master equation analytically (e.g. $36$ coupled differential equations are
involved). Motivated by the dressed-state theory of conventional EIT
Fleischhauer  _et al._ (2005), here we develop an analytical approach of
dressed state to identify the excitation pathways of the FIT through the
facilitated many-body states.

We consider the interaction-dressed subspace spanned by the tensor product of
two single-atom basis, i.e.
$\\{|2_{A}\rangle,|3_{A}\rangle\\}\otimes\\{|1_{B}\rangle,|3_{B}\rangle\\}$,
indicated by the dashed gray box in Fig. 3(a). The subspace is coupled by the
ground state $|1_{A}\rangle$ of the channel $A$ through the probe field
$\Omega_{p}$.

Figure 3: Dressed-state picture and the scaling of the FIT peak. (a)
Interaction-dressed subspace spanned by the basis
$\\{|2_{A}\rangle,|3_{A}\rangle\\}\otimes\\{|1_{B}\rangle,|3_{B}\rangle\\}$,
indicated by the dashed gray box. (b) Dressed eigen values $\epsilon_{j}$ as
functions of $\Delta_{c}$ for $V_{AB}=15\Gamma_{12}^{A}$; corresponding
eigenstates $\\{|\psi_{j}\rangle;j=1,4\\}$ (dashed gray lines) and
$\\{|\psi_{j}\rangle;j=2,3\\}$ (solid blue lines) are also shown. $\Omega_{p}$
can resonantly drive the transition $|1_{A}\rangle\rightarrow|\Psi_{F}\rangle$
($|1_{A}\rangle\rightarrow|\Psi_{E}\rangle$) at $\Delta_{c}=\Delta_{c}^{-}$
($\Delta_{c}=\Delta_{c}^{+}$), where $|\Psi_{E}\rangle$ ($|\Psi_{F}\rangle$)
is the blockade (facilitation) state (see the Appendix C). (c) Energy
separation $\Delta E$ [described also in Fig. 2(a)] as a function $V_{AB}$ for
$\Omega_{c}=3\Gamma_{12}^{A}$ (yellow circles) and
$\Omega_{c}=5\Gamma_{12}^{A}$ (green crosses) with fixed
$\Omega=3\Gamma_{12}^{A}$ and $\Gamma_{13}^{B}=0.1\Gamma_{12}^{A}$, obtained
by numerically solving the master equation (3). Triangles: the result for the
channel $B$ is driven by a three level scheme (3LS). Inset: details of $\Delta
E$ in the different parameter regimes for small $V_{AB}$. Dashed red line: the
line $\Delta E=V_{AB}$, which matches well with the numerical data for larger
$V_{AB}$. (d) Fidelities ${\cal F}_{B}$ (dashed red line) and ${\cal F}_{F}$
(dash-dotted blue line) as functions of $\Delta_{c}$. Maximal fidelities
overlap with the doublet peaks of the absorption spectrum
$\rm{Im}(\rho_{21}^{A})$ (solid black line). (e) Mandel $Q$ factor as a
function of $\Delta_{c}$ for different $V_{AB}$, which is enhanced drastically
at $\Delta_{c}=\Delta_{c}^{\pm}$ for large $V_{AB}$. (f) Suggested model with
multiple target channels, in which the control channel (solid red circle)
locates at the center of a ring (with radius $R_{\rm fac}$) where the target
channels (solid blue circles) locate. Such a setting can facilitates the
excitation of $N$-target channels (solid blue circles) within a shell (gray
region). Shown in the box are lines of Im($\rho_{21}$) for $N=2,3,4$,
indicating that FIT is supported for such a system.

For simplicity and for obtaining explicit analytical results, we neglect the
small damping (i.e. the spontaneous emission and dephasing). The eigenspectra
$\epsilon_{j}$ of the subspace can be obtained via the diagonalization of the
system Hamiltonian (see Appendix C) expanded in this subspace, with
corresponding set of eigen basis given by $\\{\,|\psi_{j}\rangle$;
$j=1,2,3,4\\}$.

To highlight the roles played by the control channel $B$, we make a
calculation especially on the eigenspectra and eigenfunctions. Shown in Fig.
3(b) is the result for the eigenvalues $\epsilon_{j}$ as functions of
$\Delta_{c}$ for $V_{AB}=15\Gamma_{12}^{A}$. Corresponding eigenstates
$\\{|\psi_{j}\rangle;j=1,4\\}$ and $\\{|\psi_{j}\rangle;j=2,3\\}$ are given by
the dashed gray and solid blue lines, respectively. We see that the prove
field $\Omega_{p}$ can resonantly drive the transition
$|1_{A}\rangle\rightarrow|\Psi_{F}\rangle$
($|1_{A}\rangle\rightarrow|\Psi_{E}\rangle$) at $\Delta_{c}=\Delta_{c}^{-}$
($\Delta_{c}=\Delta_{c}^{+}$), where $|\Psi_{E}\rangle$ ($|\Psi_{F}\rangle$)
is the blockade (facilitation) state. For more details, see the discussion
given in the Appendix C.

From the figure, we see that, when $\epsilon_{j}=0$, the probe field
$\Omega_{p}$ can resonantly drive the transition from $|1_{A}\rangle$ to the
dressed subspace at the particular detuning
$\Delta_{c}^{\pm}=-V_{AB}/2\pm\sqrt{V_{AB}^{2}+\left(\Omega-\Omega_{c}^{2}/\Omega\right)^{2}}/2$,
which locates the double peaks of the absorption spectrum Im($\rho_{21}^{A}$)
[predicted also in Fig. 2(b)], with energy separation $\Delta
E=\Delta_{c}^{+}-\Delta_{c}^{-}$. In the strong interaction dominant regime
(i.e. $|V_{AB}|>|\Omega|,\,|\Omega_{c}|\gg|\Omega-\Omega_{c}^{2}/\Omega|$),
one can obtain $\Delta E\simeq V_{AB}$. This linear dependence on $V_{AB}$ for
the distance between the two absorption peaks can also be seen in the
numerical data given by Fig. 3(c). Deviations for smaller $V_{AB}$ attribute
to spontaneous emission, dephasing of the atomic coherence and light induced
shifts, which are included in the numerical simulation. In Fig. 3(c), the
result for the case when the control channel described by three-level (3L)
model (e.g. two-photon Raman scheme) is also shown. One sees that the energy
separation $\Delta E$ as a function $V_{AB}$ also have a linear dependence for
large $V_{AB}$.

Now lets turn to investigating the eigenstates of the system, which, in the
interaction dominant region, are given by

$\displaystyle|\Psi_{E}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|3_{A}1_{B}\rangle-|2_{A}3_{B}\rangle),\,\,\,\,{\rm
for}\,\,\Delta_{c}=\Delta_{c}^{+},$ (7) $\displaystyle|\Psi_{F}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|3_{A}3_{B}\rangle-|2_{A}1_{B}\rangle),\,\,\,\,{\rm
for}\,\,\Delta_{c}=\Delta_{c}^{-},$ (8)

where $|\Psi_{E(F)}\rangle$ are Rydberg entangled and facilitated Bell states.
The excitation of these states provides two pathways. When the inter-channel
interaction is strong, they interfere destructively and lead to the FIT.
However, due to the weak probe field and the large spontaneous emission at the
intermediate level $|2_{A}\rangle$, the entangled state $|\Psi_{E}\rangle$
decays to the blockade state $|\Psi_{B}\rangle=|1_{A}3_{B}\rangle$, which
partially destroys the transparency between the two absorption peaks,
consisted with the result given in Fig. 2.

To verify the validity of the dressed-state approach in the presence of
dissipation, we quantify the difference between states
$\hat{\rho}_{B(F)}=|\Psi_{B(F)}\rangle\langle\Psi_{B(F)}|$ and the steady
state $\hat{\rho}_{s}$ from master equation simulations with fidelity ${\cal
F}_{B(F)}=({\rm Tr}|\sqrt{\hat{\rho}_{B(F)}}\sqrt{\hat{\rho}_{s}}|)^{2}$. When
sweeping $\Delta_{c}$, ${\cal F}_{B(F)}$ indeed displays maxima at
$\Delta_{c}^{\pm}$ [see Fig. 3(d)]. This indicates that the dressed-states can
be used to characterize the Rydberg blockade and facilitation in the mixed-
state system.

We emphasize that two-body quantum correlations are important to the FIT. We
further quantify the inter-channel correlation by the Mandel $Q$ factor
defined as
$Q=\langle(\Delta\hat{n}_{33})^{2}\rangle/\langle\hat{n}_{33}\rangle-1$ with
$\hat{n}_{33}=\hat{\sigma}_{33}^{A}+\hat{\sigma}_{33}^{B}$. As illustrated in
Fig. 3(e), Mandel $Q$ factor is enhanced drastically at
$\Delta_{c}=\Delta_{c}^{\pm}$, and exhibits sub-Poissonian counting statistics
with $Q<0$ around $\Delta_{c}=\Delta_{c}^{+}$, and the super-Poissonian
processes characterized by $Q>0$ around $\Delta_{c}=\Delta_{c}^{-}$. This is
an unique property for such a system, where exotic statistics characteristics
can be probed via tuning laser parameters Schempp  _et al._ (2014); Valado
_et al._ (2016).

The dual channel model discussed above can be extended to the situations with
multiple target channels. One possible generation is sketched in Fig. 3(f), in
which the control channel (denoted by the solid red circle) locates at the
center of a ring (with radius $R_{\rm fac}$) where the target channels
(denoted by solid blue circles) locate. Such a setting can facilitates the
excitation of $N$-target channels (represented by solid blue circles) within a
shell (represented by the gray region) Lesanovsky and Garrahan (2014). The
lower part of the figure (i.e. the box) shows the lines of the probe-field
absorption spectrum Im($\rho_{21}$) for the target-channel number $N=2,3,4$,
which indicates that FIT character still persists with multiple target
channels.

In our simulation, interactions between different target channels have been
taken into account. Due to the strong interaction contributed from the control
channel, the target channels are fully blockade or facilitated, and thus the
target-target (TT) interaction plays a minor role on the FIT Marcuzzi  _et
al._ (2017); Helmrich  _et al._ (2020). To demonstrate this point further, we
have constructed a three-atom FIT model (in which one control atom and two
target atoms are considered) and investigated its FIT behavior. The results
shows that the TT interaction have a small effect on the FIT in the system;
for more detail, see Appendix D.

##  IV Facilitated single-photon switch protocol

The above analysis shows that the target channel $A$ can be changed from
opaque to transparent by tuning $\Delta_{c}$. Such tunability allows us to
design optical switch via FIT by using contactless atom-atom interactions. In
the rest of the work, we consider the propagation dynamics of quantized probe
fields in high density atom gases. We will show that a facilitated single-
photon switch can be realized in the dual-channel system suggested above.

###  IV.1 Optical susceptibility of the probe field

Before examining quantum dynamics of the single-photon switch, we first
analyze the polarization of the probe field ${\cal E}_{p}$, defined by
${P}_{p}\equiv\epsilon_{0}\chi_{p}{{\cal E}}_{p}$. The susceptibility
$\chi_{p}$ is, in general, of a nonlinear function of ${\cal E}_{p}$ if the
probe-field intensity is high. The relation between the optical susceptibility
of the probe field and the one-body DM elements can be built with the formula

$\chi_{p}=\frac{\hbar\varepsilon_{0}}{{\cal
N}_{A}\mu_{21}^{2}\Omega_{p}}\,\rho_{21}^{A},$ (9)

where ${\cal N}_{A}$ is the atomic density of the ensemble $A$ (assumed to be
homogeneous). In our setting, $\chi_{p}$ depends not only on the parameters of
the ensemble $A$ (e.g, $\Omega_{p}$, $\Omega$), but also on those of the
ensemble $B$ (e.g., $\Omega_{c}$, $\Delta_{c}$).

Figure 4: Optical susceptibility. (a) Im($\chi_{p}$) as a function of
$\Delta_{c}$ (for $V_{AB}=15\Gamma_{12}^{A}$) when the Rabi frequency of the
probe field takes values of
$0.01\Gamma_{12}^{A},0.1\Gamma_{12}^{A},0.5\Gamma_{12}^{A},$ and
$\Gamma_{12}^{A}$, respectively. (b1) Im($\chi_{p}$) as a function of
$\Omega_{p}$ and $V_{AB}$ for $\Delta_{c}=-V_{AB}$ [corresponding to the left
peak of panel (a)]. (b2) The same as (b1) but for small values of
$\Omega_{p}$. (c1) and (c2): The same as (b1) and (b2) but with $\Delta_{c}=0$
[corresponding to the right peak of panel (a)]. The result shows that when
decreasing $\Omega_{p}$, Im($\chi_{p}$) saturates for
$\Omega_{p}<0.1\Gamma_{12}^{A}$, such that $\chi_{p}$ behaves as a linear
susceptibility, independent of $\Omega_{p}$.

Figure 4(a) shows the imaginary part of the susceptibility Im($\chi_{p}$) as a
function of Rabi frequency of the probe field, $\Omega_{p}$, with fixed
$V_{AB}=15\Gamma_{12}^{A}$. The dashed blue, dotted yellow, dot-dashed red,
and solid green lines are for
$\Omega_{p}=0.01\Gamma_{12}^{A},0.1\Gamma_{12}^{A},0.5\Gamma_{12}^{A},$ and
$\Gamma_{12}^{A}$, respectively. Illustrated in panels (b1) is Im($\chi_{p}$)
as a function of $\Omega_{p}$ and $V_{AB}$ for $\Delta_{c}=-V_{AB}$, which
corresponds to the left peak of panel (a); panel (b2) is same as panel (b1)
but for small values of $\Omega_{p}$. Panels (c1) and (c2) are the same as
(b1) and (b2) but with $\Delta_{c}=0$ [corresponding to the right peak of
panel (a)]. The results are obtained by solving the Heisenberg equations of
the system. From the figure we see that, as the probe Rabi frequency is
decreased, there is a dramatic enhancement of Im($\chi_{p}$) around
$\Delta_{c}=0$ and $-V_{AB}$, manifested by the two FIT peaks in the figure.
Moreover, there is no broadening or detuning of the FIT, and the
susceptibility saturates when $\Omega_{p}<0.1\Gamma_{12}^{A}$, for which
Im($\chi_{p}$) does not change as $\Omega_{p}$ is decreased. The absence of
the broadening or shift is vital for the detection of Rydberg facilitation
from the optical absorption spectrum. When the probe field is weak (e.g. at
the level of single photons), $\chi_{p}$ becomes linear, i.e. its value is
largely independent of ${\cal E}_{p}$ in both the facilitation and blockade
region [see Fig. 4(b2) and (c2)]. Outside the linear regime, one can achieve a
strong Rydberg nonlinearity in such a system, which is a topic deserving
explorations but beyond the scope of the present work.

###  IV.2 Facilitated single-photon switch

In the following, we will focus on the linear regime and demonstrate how to
realize a single-photon switch with the FIT predicted above. To this end, we
consider an elongated atomic gas and study propagation of the probe field
along the $z$ direction, while neglecting diffraction effect, which is valid
for the case of short propagation distance or the atomic gas is prepared in a
trap with a small transverse cross section Gorniaczyk  _et al._ (2014); Busche
_et al._ (2017); Chen  _et al._ (2021). Treating the probe field quantum
mechanically, the dynamics of its propagation is governed by the Hamiltonian
$\hat{H}_{p}=-(c/L)\int{dz\hat{{\cal
E}}_{p}^{\dagger}(z,t)i\partial_{z}\hat{{\cal E}}_{p}(z,t)}$. Here,
$\hat{{\cal E}}_{p}(z,t)$ is the slowly varying annihilation operator of probe
photons, satisfying the equal-time commutation relation $[\hat{{\cal
E}}_{p}(z,t),\hat{{\cal
E}}_{p}^{\dagger}(z^{\prime},t)]=L\delta(z-z^{\prime})$, with $L$ the
quantization length along the $z$ axis. By applying the continuous density
approximation Fleischhauer and Lukin (2002) the Hamiltonian of the system
reads

$\displaystyle\hat{H}_{A}$ $\displaystyle=$
$\displaystyle-\frac{N_{A}}{L}\int{dz}\left[\frac{g\hat{{\cal
E}}_{p}(z,t)}{2}\hat{\sigma}_{21}^{A}(z,t)+\frac{\Omega}{2}\hat{\sigma}_{32}^{A}(z,t)+{\rm
H.c.}\right],$ $\displaystyle\hat{H}_{B}$ $\displaystyle=$
$\displaystyle-\frac{N_{B}}{L}\int{dz}\left[\frac{\Omega_{c}}{2}\hat{\sigma}_{31}^{B}(z,t)-\Delta_{c}\hat{\sigma}_{33}^{B}(z,t)+{\rm
H.c.}\right],$ (10b)

here $N_{A}$ and $N_{A}$ are respectively atomic numbers in the channels $A$
and $B$, $\hat{\sigma}_{ab}^{\mu}(z)$ satisfies the canonical commutation
relation
$[\hat{\sigma}_{ab}^{\mu}(z),\hat{\sigma}_{cd}^{\mu}(z^{\prime})]=\delta(z-z^{\prime})[\delta_{bc}\hat{\sigma}_{ad}^{\mu}(z)-\delta_{da}\hat{\sigma}_{cb}^{\mu}(z)]L/N_{\mu}$.
The Hamiltonian describing the inter-channel interaction is given by

$\displaystyle H_{\rm
int}=\frac{N_{A}}{L}\sum_{j}\int{dz}V(z-z_{j})\hat{\sigma}_{33}^{A}(z,t)\hat{\sigma}_{33}^{B}(z_{j},t),$
(11)

where the index $j$ labels the respective Rydberg excitation in the channel
$B$ Li and Lesanovsky (2015). We assume that the atoms within each channel are
trapped within a small spatial interval $\delta r\ll d$; the atom-atom
interaction is described by the potential $V(z-z_{j})\simeq-
C_{6}/[(z-z_{j})^{2}+d^{2}]^{3}$ with $V_{AB}=-C_{6}/d^{6}$ Busche  _et al._
(2017).

Suppose that the quantum state of the single-photon input probe field has the
form $|\Psi_{P}\rangle=(1/L)\int{dz}E_{p}(z,t)\hat{{\cal
E}}_{p}^{\dagger}(z,t)|0\rangle$, where $E_{p}(z,t)$ is the single-photon
probability amplitude. Based on the Maxwell equation for $\hat{\cal
E}_{p}(z,t)$, one can derive the evolution equation for $E_{p}(z,t)$ Gorshkov
_et al._ (2007); Shou  _et al._ (2021), which reads

$\displaystyle\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
z}\right)E_{p}(z,t)=ig\,N_{A}\rho_{21}^{A}(z,t)/2.$ (12)

It can be solved numerically in conjunction with the Bloch equations for the
atomic DM elements $\rho_{ab}^{A}$ and $\rho_{ab}^{B}$. The transmission $T$
of the single-photon probe field, used to quantify efficiency of the photon
switch, can be obtained at $z=L$, i.e. $T=\mathcal{I}_{s}(L)$, where
$\mathcal{I}_{s}(z)=|E_{p}(z)|^{2}/|E_{p}(0)|^{2}$ is the relative intensity
of the photon field at position $z$.

To understand the propagation dynamics of the probe photon, lets first
consider a simple scenario in which the Rydberg excitation in the control
channel $B$ is localized at $z=0$. Faraway from $z=0$, the photon field can
propagate under perfectly EIT condition in the channel $A$ [where the
excitation scheme is the one of an exact ladder-type EIT; see Fig. 1(a)], as
illustrated in Fig. 5(a1).

Figure 5: Facilitation enabled single-photon switch. (a) The control channel
$B$ is driven with the excitation localized at $z=0$ (a1). In the vicinity of
$z=0$, $\rho_{33}^{A}$ is blockaded for $\Delta_{c}=0$ or facilitated for
$\Delta_{c}=-V_{AB}$ ($\Delta_{c}=-V_{AB}/2$ is an intermediate case). They
both result in a strong photon scattering, as can be seen from the decreasing
relative intensity $\mathcal{I}_{s}$ (a2). Shown in (a3) and (a4) are
$\rho_{33}^{A}$ and $\rho_{33}^{B}$ as functions of $z$ for $\Delta_{c}=0$
(red dashed line), $-V_{AB}/2$ (green dash-dotted line), and $-V_{AB}$ (blue
solid line), respectively. (b) When the channel $B$ is dynamically driven
(b1), one can change the control detuning $\Delta_{c}$ from a far-detuned
value to $-V_{AB}$ (or $0$) to launch facilitated (blockade) single-photon
switch, where the relative intensity of the probe field quickly decreases to
zero (b2). Shown in (b3) and (b4) are $\rho_{33}^{A}$ and $\rho_{33}^{B}$ as
functions of $z$ for $\Delta_{cF}=0$ (red dashed line), and $-V_{AB}$ (blue
solid line), respectively. In the simulation, system parameters
$\Omega=\Omega_{c}=3\Gamma_{12}^{A}$, $V_{AB}=15\Gamma_{12}^{A}$, and
$\mathcal{N}_{A}=3\times 10^{12}{\rm cm}^{-3}$ are used. See text for more
details.

However, in close proximity to $z=0$ (noticing that $V_{AB}$ varies with $z$
and reaches its maximal value $V_{AB}$ at $z=0$), the channel $B$ inhibits
($\Delta_{c}=0$) or facilitates ($\Delta_{c}=-V_{AB}$) the Rydberg excitation
in the channel $A$, and causes strong photon loss from the medium; see Fig.
5(a2), where $\mathcal{I}_{s}$ as a function of $z$ is plotted for
$\Delta_{c}=0$, $\Delta_{c}=-V_{AB}/2$, and $\Delta_{c}=-V_{AB}$,
respectively. The former (i.e. the result for $\Delta_{c}=0$) is the
phenomenon of Rydberg blockade, which has been studied in Ref. Baur  _et al._
(2014); Gorniaczyk  _et al._ (2014); Li and Lesanovsky (2015); the latter
(i.e. the result for $\Delta_{c}=-V_{AB}$) behaves as a facilitated photon
switch where the probability of the Rydberg excitation in the both channels is
relatively large. This can be seen clearly in Fig. 5(a3) and (a4), where
$\rho_{33}^{A}$ and $\rho_{33}^{B}$ as functions of $z$. When passing from the
interaction region, the photon field propagates freely again under the EIT
condition where the photon scattering is negligible. Therefore, the reduced
transmission is caused by the facilitation induced photon scattering in the
vicinity of $z=0$.

For an ideal photon switch, the photon field should be completely blocked from
transmission (i.e. $T=0$). To achieve this, we consider the light field in the
control channel co-propagate with the probe field in the target channel,
depicted in Fig. 5(b1). To demonstrate the controllability of the photon
switch, the detuning $\Delta_{c}$ is chosen to be time-dependent but varies
slowly, which has the from of
$\Delta_{c}=\Delta_{c0}+(\Delta_{c0}-\Delta_{cF})[{\rm
tanh}(1-(z-z_{q})/z_{s})-1]/2$ [see the inset of Fig. 5(b2)]. Initially,
$\Omega_{c}$ is far detuned before the photon field travels to $z=z_{q}$. Then
it is rapidly changed to $\Delta_{cF}\simeq-V_{AB}$ (or $0$). In response to
such changes, the system immediately evolves from a non-interacting dark
states into an absorbing facilitated state (or blockade state) [see the
behavior of $\rho_{33}^{A(B)}$ shown in Fig. 5(b3)-(b4)]. The transmission of
the photon field quickly decreases from almost $100\%$ to zero within a short
propagation distance. Consequently, for a medium of less than 100 $\mu$m, the
photon field transmission can be made completely vanish, which is promising
for designing highly efficient and controllable single-photon switches.

###  IV.3 Discussion on delocalized control-atom excitation

In the above consideration, we have assumed the spatial distribution of each
atom ensemble is small, the excitation of the control atom is treated to be
localized at a fixed position in the channel $B$, such that the interaction is
described by the simple form of $V_{AB}=-C_{6}/d^{6}$. In reality, atoms in
the ensemble spread in space Busche  _et al._ (2017). For practical
realization of the FIT and photon switch predicted here, the spatial
dependence of the atom-atom interaction needs to be taken into account.

When there is a delocalization of the control-atom excitation over the channel
$B$, the inter-channel interaction takes the form
$V_{AB}=-C_{6}/|\mathbf{d}+\mathbf{r}_{B}|^{6}$. Here $d=|\mathbf{d}|$ is the
center-of-mass (COM) separation between the channels $A$ and $B$, and
$r_{B}=|\mathbf{r}_{B}|$ is the distance of the control excitation relative to
the COM of the channel $B$; see Fig. 6(a). Under such a consideration, we can
obtain $\mathbf{r}_{B}$-dependent $\rho_{21}^{A}$, which reads

$\displaystyle\rho_{21}^{A}(\mathbf{r}_{B})\simeq-\frac{2V_{AB}(\mathbf{r}_{B})\,\rho_{31,33}^{AB}(\mathbf{r}_{B})}{\Omega}.$
(13)

Then optical susceptibility of the probe field is given by

$\chi_{p}=\int
d\mathbf{r}_{B}f(\mathbf{r}_{B})\frac{\hbar\varepsilon_{0}}{{\cal
N}_{A}\mu_{21}^{2}\Omega_{p}}\,\rho_{21}^{A}(\mathbf{r}_{B}),$ (14)

where $f(\mathbf{r}_{B})$ is a normalized distribution function describing the
spatial delocalization of the control-atom excitation in the channel $B$. For
simplicity, we assume $f(\mathbf{r}_{B})$ takes the form of the Gaussian
distribution, i.e. $f(\mathbf{r}_{B})=1/(\sqrt{\pi}\sigma){\rm
exp}[-(\mathbf{r}_{B}/\sigma)^{2}]$), with $\sigma$ being the distribution
width. The delocalization degree can be characterized by using parameter ${\rm
max}(r_{B})/d$.

Figure 6: FIT with delocalized control-atom excitation. (a) Schematic of the
delocalized control-atom excitation in the channel $B$. The delocalization
modifies the atom-atom interaction form from $V_{AB}=-C_{6}/d^{6}$ to
$V_{AB}=-C_{6}/|\mathbf{d}+\mathbf{r}_{B}|^{6}$, here $d=|\mathbf{d}|$ is the
center-of-mass separation between the $A$ and $B$ channels and
$r_{B}=|\mathbf{r}_{B}|$ is the relative variation of the control excitation.
(b) Photon transmission $T$ as a function of $\Delta_{c}$ for different $d$,
but with fixed $V_{AB}(r_{B}=0)=15\Gamma_{12}^{A}$, when the photon propagates
to distance $L=20\,{\rm\mu m}$. Solid black line: the result of the ideal FIT
(i.e. the case for localized control-atom excitation). (c) is similar to Fig.
5(b). To launch facilitated (blockade) switch, here $\Delta_{c}$ is changed
from the far-detuned value to $-V_{AB}$ (or $0$). (c1)-(c3) show the relative
intensity of the probe field $I_{s}$ and the Rydberg population
$\rho_{33}^{A(B)}$ in the facilitated regime; (d1)-(d3) are similar to
(c1)-(c3) but for the blockade regime. Blue curves in these figures are
results of single trajectory simulation in the delocalized regime. Their mean
values are calculated by casting 300 trajectories for different initial
delocalized distribution (see green-dotted line). The coherence of the control
atom is modified during the operation of the photon switch, manifested by the
real Re($\rho_{31}^{B}$) of the atomic coherence [see panel (c4) and (d4)].
System parameters used in the simulation are
$\Omega=\Omega_{c}=3\Gamma_{12}^{A}$ and $\mathcal{N}_{A}=3\times 10^{12}{\rm
cm}^{-3}$.

Fig. 6(a) shows the schematic of the delocalized control-atom excitation in
the channel $B$ with a three-dimensional Gaussian-shaped distribution. To get
a quantitative result of the behavior of single-photon switch in the presence
of the delocalized control-atom excitation, a numerical simulation is carried
out by using a combination of the four-order Runge-Kutta and finite difference
methods to solve the corresponding coupled Maxwell-Bloch equations. In the
simulation, we average the transmission $T$ over different sites until the
result converges.

Shown in Fig. 6(b) is the transmission $T$ as a function of $\Delta_{c}$ for
different separation $d$, but with $V_{AB}(r_{B}=0)=15\Gamma_{12}^{A}$. To
obtain such an interaction strength, it requires $C_{6}=2.68\times 10^{4}{\rm
GHz\cdot\mu m}^{6}$ (with state $|93S\rangle$) for $d=6{\rm\mu m}$ and
$C_{6}=5.77\times 10^{5}{\rm GHz\cdot\mu m}^{6}$ (with state $|122S\rangle$)
for $d=10{\rm\mu m}$ in Rb atom. We can see that, in the presence of the
delocalized control-atom excitation, the FIT can still be observed. However,
in comparison with the case of the localization regime (i.e., the regime with
no position fluctuation of the control-atom excitation), there has a slight
shift on the absorption peak located at $\Delta_{c}=-V_{AB}$ (shown by the
black solid line). In addition, in the delocalization regime, an inhomogeneous
broadening for the FIT peak occurs; as $d$ decreases, the delocalization
degree of the control-atom excitation increases, and the FIT peak becomes
smaller. Consequently, to realize a robust FIT, the system must have strong
Rydberg interactions, larger separation between the two channels, and tightly-
focused control field.

To test the protocol of single-photon switch, we have conducted a simulation
similar to that shown in Fig. 5(b), but with delocalized control-atom
excitation here. Fig. 6(c1) shows the results of the relative probe-field
intensity ${\cal I}_{s}$ as a function of $z$, for regimes of the
delocalization (blue solid line) and localization (red dashed line) are given.
We see that ${\cal I}_{s}$ decreases slower in the delocalization regime but
still approaches to zero within a short propagation distance. Drawn in Fig.
6(c2) [Fig. 6(c3)] is the behavior of the Rydberg population $\rho_{33}^{A}$
($\rho_{33}^{B}$). One sees that, due to the delocalized excitation, both
$\rho_{33}^{A}$ and $\rho_{33}^{B}$ exhibit large fluctuations; however, their
envelopes are still facilitated, and captured well with the corresponding
localized curves, respectively. To see the average behaviors of ${\cal I}_{s}$
and $\rho_{33}^{A(B)}$ in the presence of the fluctuations, in the simulation
300 trajectories for different initial delocalization distributions are
considered, and these quantities are evaluated by calculating their mean
values. The results of such an average are given by the green-dotted lines in
the figure.

Shown in Fig. 6(d1)-(d3) are also for the quantities ${\cal I}_{s}$ and
$\rho_{33}^{A(B)}$, but for the case of the blockade regime. Blue curves in
these figures are results of single trajectory simulation in the
delocalization regime. Their mean values are also calculated by casting 300
trajectories for different initial delocalized distributions (give by green-
dotted lines). We see that the quality of the blockade switch is not to be
significantly affected by the delocalized distribution of the control-atom
excitation.

In order to know the coherence behavior of the Rydberg spin wave in the
control channel $B$ during the operation of the photon switch, we have
calculated Re$(\rho_{31}^{B})$ as a function of the propagation distance $z$,
with the results being presented in Fig. 6(c4) and Fig. 6(d4). We see that in
the case of the facilitation regime Re$(\rho_{31}^{B})$ becomes increasingly
protected during the operation of the switch [Fig. 6(c4)]; however, in the
case of the blockade regime Re$(\rho_{31}^{B})$ first increases due to
resonance driving, and then rapidly decays to zero [Fig. 6(d4)] as $z$
increases. The reason for the protected coherence during the switch operation
in the facilitation regime is due to the fact that the control atom is driven
away from the resonance with the choice of $\Delta_{c}=-V_{AB}$. Thereby, the
coherence of the control atom is not diminished even in the case of the photon
scattering.

##  V Summary and discussion

We have investigated the FIT effect in a dual-channel Rydberg atom setting. It
is found that optical properties of the probe photon in the target channel can
be actively controlled by varying laser detuning in the control channel. The
FIT window emerges, and becomes wider and deeper as the inter-channel Rydberg
interaction increases. Using a dressed state approach, we have shown that the
quantum interference pathways of the FIT are induced from the blockade and
facilitated states. The FIT and its scaling on the interaction are examined
both numerically and analytically. We have demonstrated that a contactless
single-photon switch can be realized with FIT in the dual-channel system. Due
to weak excitation under facilitated condition, coherence of the control
channel is well protected during the operation of the switch. This work
moreover opens new opportunities to directly detect Rydberg facilitation using
the optical transmission of a neighboring atom ensemble.

Beyond the single-photon situation considered here, multiple Rydberg
excitations can form strongly correlated states in spatially large atomic
gases (e.g. Rydberg excitation crystals Pohl  _et al._ (2010) and quantum
scarring states Turner  _et al._ (2018)). An interesting question is whether
one could use this contactless multi-channel setting to probe Rydberg
excitations and phases through Rydberg interaction induced optical responses
Olmos  _et al._ (2011); Günter  _et al._ (2013). This might provide new
contactless detection methods without demolishing many-body coherence.

###### Acknowledgements.

We thank useful discussions with Jianming Zhao, Lin Li, Yuechun Jiao, and Qi
Zhang. ZB and GH acknowledge National Science Foundation of China (NSF)
(11904104, 11975098), and the Shanghai Pujiang Program under grant No.
21PJ1402500. W. L. acknowledges support from the Engineering and Physical
Sciences Research Council [grant no. EP/W015641/1]. The data that support this
work are available in the Appendix of this article and at
http://doi.org/10.17639/nott.7200.

##  Appendix A Multi-photon excitations in the control channel B

One can also adopt an two- and multi-photon excitation scheme in the control
channel $B$. Here we gives a simple description on a two-photon excitation
scheme, shown in Fig. A1.

Figure A1: Two-photon excitation scheme for the control channel $B$. (a) The
channel $A$ is the same as that of Fig. 1. The control channel $B$ is however
driven by a two-photon process, where the transition
$|1\rangle\leftrightarrow|2\rangle$ ($|2\rangle\leftrightarrow|3\rangle$) is
coupled by the control field of Rabi frequency $\Omega_{c1}$ ($\Omega_{c2}$),
with detuning $\Delta$ ($\Delta_{c}$). (b) Level diagram of the system.

In this scheme, the channel $A$ is the same as that in Fig. 1 in the main
text, but the control channel $B$ is a three-level system, where the first
(second) control field with Rabi frequency $\Omega_{c1}$ ($\Omega_{c2}$)
drives the transition $|1\rangle\leftrightarrow|2\rangle$
($|2\rangle\leftrightarrow|3\rangle$) with detuning $\Delta$ ($\Delta_{c}$).
Under rotating wave approximation, the two-atom Hamiltonian is given by
$\hat{H}=\hat{H}_{A}+\hat{H}_{B}+\hat{H}_{\rm int}$, with

$\displaystyle\hat{H}_{A}$ $\displaystyle=$
$\displaystyle-\frac{\Omega_{p}}{2}\hat{\sigma}_{21}^{A}-\frac{\Omega}{2}\hat{\sigma}_{32}^{A}+{\rm
H.c.},$ $\displaystyle\hat{H}_{B}$ $\displaystyle=$
$\displaystyle-\left(\frac{\Omega_{c1}}{2}\hat{\sigma}_{21}^{B}+\frac{\Omega_{c2}}{2}\hat{\sigma}_{32}^{B}-\Delta\hat{\sigma}_{22}^{B}-\Delta_{c}\hat{\sigma}_{33}^{B}+{\rm
H.c.}\right),$ $\displaystyle\hat{H}_{\rm int}$ $\displaystyle=$
$\displaystyle V_{AB}\hat{\sigma}_{33}^{A}\hat{\sigma}_{33}^{B}.$

The dynamical behavior of such a system can be obtained by numerically solving
the corresponding master equation. Shown in Fig. A2(a)

Figure A2: Atomic coherence and Rydberg excitations in the dual-channel model
when the control channel $B$ is driven by a two-photon process. (a) FIT effect
emerges in the absorption spectrum of the probe field, manifested by the
imaginary part Im($\rho_{21}^{A}$) of the atomic coherence $\rho_{21}^{A}$ as
a function of $\Delta_{c}$ (solid black line). Corresponding Rydberg
excitations $\rho_{33}^{A}$ (dot-dashed blue line) and $\rho_{33}^{B}$ (dashed
red line) are also plotted, where the channel $A$ is blockaded at
$\Delta_{c}=0$, but facilitated around $\Delta_{c}=-V_{AB}$. (b) As $V_{AB}$
increases, the FIT transparency window becomes wider and deeper. Two peaks are
visible even for weak interaction strength (e.g, $V_{AB}=5\Gamma_{12}^{A}$).
System parameters used are $\Omega_{c1}=\Omega_{p}=0.8\Gamma_{12}^{A}$,
$\Omega_{c2}=\Omega=4\Gamma_{12}^{A}$, and $\Delta=10\Gamma_{12}^{A}$.

is the result of Im($\rho_{21}^{A}$) (the imaginary part of the atomic
coherence $\rho_{21}^{A}$) as a function of $\Delta_{c}$, plotted by the solid
black line. The corresponding Rydberg excitations $\rho_{33}^{A}$ and
$\rho_{33}^{B}$ are also illustrated by the dot-dashed blue line and dashed
red line, respectively. We see that the channel $A$ is blockaded at
$\Delta_{c}=0$, but facilitated near $\Delta_{c}=-V_{AB}$. Consequently, the
FIT phenomenon can emerge in such a system. i.e. two absorption peaks emerge
in the profile of Im($\rho_{21}^{A}$) [Fig. A2(b)]. Moreover, the FIT
transparency window becomes wider and deeper as $V_{AB}$ increases. Notice
that, due to the large single-photon detuning $\Delta$ in the channel $B$
(which makes the effective Rabi frequency $\Omega_{\rm
eff}=\Omega_{c1}\Omega_{c2}/(2\Delta)$ small), the linewidth of the absorption
peak (proportional to $\Omega_{\rm eff}$) is much narrower than that obtained
by using the one-photon excitation of the channel $B$ given in the main text.
Thus by using such a scheme one can obtain a visible FIT doublet even for weak
atomic interaction (e.g., $V_{AB}=5\Gamma_{12}^{A}$). This result is also
shown in Fig. 3(c) (the blue triangle line) in the main text, and agrees well
with the dress-state theory described there.

##  Appendix B Atomic coherences in stationary states

The Heisenberg equations for operators $\hat{\sigma}_{13}^{A}$ and
$\hat{\sigma}_{12}^{A}$ read

$\displaystyle i\left(\frac{\partial}{\partial
t}+\gamma_{12}^{A}\right)\hat{\sigma}_{12}^{A}-\frac{\hat{\Omega}_{p}}{2}(\hat{\sigma}_{22}^{A}-\hat{\sigma}_{11}^{A})+\frac{\Omega^{\ast}}{2}\hat{\sigma}_{13}^{A}=0,$
(15a) $\displaystyle i\left(\frac{\partial}{\partial
t}+\gamma_{13}^{A}\right)\hat{\sigma}_{13}^{A}-\frac{\hat{\Omega}_{p}}{2}\hat{\sigma}_{23}^{A}+\frac{\Omega}{2}\hat{\sigma}_{12}^{A}+V_{AB}\hat{\sigma}_{13}^{A}\hat{\sigma}_{33}^{B}=0.$

Taking the average (trace) on the above equations (i.e.
$\rho_{ab}^{A(B)}\equiv\langle\sigma_{ba}^{A(B)}\rangle={\rm
Tr}[\rho\sigma_{ba}^{A(B)}]$) and solving them for stationary states (i.e.,
$\partial/\partial t=0$), we obtain the atomic coherence between the states
$|1\rangle$ and $|2\rangle$:

$\displaystyle\rho_{21}^{A}=\frac{-2V_{AB}\,\rho_{31,33}^{AB}\Omega^{\ast}+2i(\rho_{11}^{A}-\rho_{22}^{A})\gamma_{13}^{A}\Omega_{p}+\rho_{32}^{A}\Omega^{\ast}\Omega_{p}}{\Omega^{2}+4\gamma_{12}^{A}\gamma_{13}^{A}}.$

where
$\rho_{31,33}^{AB}\equiv\langle\hat{\sigma}_{13}^{A}\hat{\sigma}_{33}^{B}\rangle$
is two-body DM element (or called two-body correlator). The terms related to
the damping rate $\gamma_{13}^{A}$ are very small (e.g., 100${\rm\mu s}$) and
hence can be neglected. Moreover, since the probe field in the channel $A$ is
weak, $\rho_{32}^{A}$ is negligible. Under these considerations,
$\rho_{21}^{A}$ can be reduced to

$\displaystyle\rho_{21}^{A}\simeq-\frac{2V_{AB}\,\rho_{31,33}^{AB}}{\Omega}.$
(17)

We see that $\rho_{21}^{A}$ depends mainly on the two-body density matrix
element $\rho_{31,33}^{AB}$. From the result illustrated in Fig. 2(d) of the
main text, one can see that Eq. (17) can capture well with that obtained by
solving the master equation numerically.

To get $\rho_{21}^{A}$, one must solve the equations of motion for two-body
correlators $\rho_{ab,cd}^{AB}$ ($a,b,c,d=1,2,3$). For saving space, we do not
list these equation here. For more detail, see Bai  _et al._ (2019).

##  Appendix C Dressed-state picture for the FIT

As shown in Fig. 3(a) in the main text, the interaction-dressed subspace is
considered with the basis
$\\{|2_{A}\rangle,|3_{A}\rangle,|1_{B}\rangle,|3_{B}\rangle\\}$. The
Hamiltonian of this subspace reads

$\displaystyle
H_{0}=\begin{bmatrix}0&-\Omega_{c}^{\ast}/2&-\Omega^{\ast}/2&0\\\
-\Omega_{c}/2&\Delta_{c}&0&-\Omega^{\ast}/2\\\
-\Omega/2&0&0&-\Omega_{c}^{\ast}/2\\\
0&-\Omega/2&-\Omega_{c}/2&\Delta_{c}+V_{AB}\end{bmatrix}\quad.$

After neglecting the small damping (i.e. spontaneous emission and dephasing)
in the system, the eigenvalue (eigenstate) $\epsilon_{j}$ $(|\psi_{j}\rangle)$
of the subspace can be obtained by diagonalizing the above Hamiltonian, with
the result sketched in Fig. 3(b) of the main text.

In particular, for the zero eigenvalue ($\epsilon_{j}=0$), the probe field
$\Omega_{p}$ can resonantly drive the transition from $|1_{A}\rangle$ to
dressed subspace at the detuning defined by

$\displaystyle\Delta_{c}^{\pm}=-\frac{V_{AB}}{2}\pm\frac{\sqrt{V_{AB}^{2}+[(\Omega^{2}-\Omega_{c}^{2})/\Omega]^{2}}}{2}.$
(18)

In the strong interaction regime (i.e. $V_{AB}\gg\Omega,\Omega_{c}$), we have
$\Delta_{c}^{+}\simeq 0$ and $\Delta_{c}^{-}\simeq-V_{AB}$. By adiabatically
sweeping $\Delta_{c}$, the system can respectively excited to the Rydberg
blockade and facilitation states, with the eigen state functions given by

$\displaystyle\mid\Psi_{\rm E}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mid 3_{A}1_{B}\rangle-\mid
2_{A}3_{B}\rangle),\,\,\,\,{\rm for}\,\,\Delta_{c}=\Delta_{c}^{+},$ (19a)
$\displaystyle\mid\Psi_{\rm F}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mid 3_{A}3_{B}\rangle-\mid
2_{A}1_{B}\rangle),\,\,\,\,{\rm for}\,\,\Delta_{c}=\Delta_{c}^{-}.$ (19b)

However, due to the large decay of the level $|2_{A}\rangle$ (which is not
included in the calculation on the eigenstates), entangled state
$|\Psi_{E}\rangle$ rapidly evolves to trivial blockade state
$|\Psi_{B}\rangle=|1_{A}3_{B}\rangle$. The energy gap between
$|\Psi_{E}\rangle$ and $|\Psi_{B}\rangle$ is

$\displaystyle\Delta
E=\sqrt{V_{AB}^{2}+(\Omega-\Omega_{c}^{2}/\Omega)^{2}}\simeq V_{AB},$ (20)

which is plotted (with dashed red line) in Fig. 3(c) of the main text. We find
that the result of this dressed-state approach can capture the numerical data
very well in the interaction dominant regime.

Figure A3: Three-atom model. (a1) FIT spectrum Im($\rho_{21}$) for the target
channel as functions of $\Delta_{c}$ and $V_{TT}$, which exhibits double peaks
even for a strong $V_{TT}$, with $V_{CT}=15\Gamma_{12}^{A}$. The inset shows
the geometry of the system, in which the control channel (red circle) locates
at the center of a ring (with radius $R_{\rm fac}$), and the target channels
(two blue circles) locate on the shell. (a2) The corresponding contour map as
functions of $\Delta_{c}$ and $V_{TT}$ is sketched with
Im$[\rho_{21}-\rho_{12}(V_{TT}=0)]$. It is clear to see that the results reach
saturation as $V_{TT}$ increases. (b1) and (b2) are similar to (a1) and (a2)
but with $V_{CT}=10\Gamma_{12}^{A}$.

##  Appendix D FIT in the three-atom model

To explore the influence of the target-target (TT) interaction (described by
$V_{TT}$) in systems with multi-channels, here we consider a simple three-atom
scenario with one control channel and two target channels; see the inset of
Fig. A3(a1) and the corresponding caption. For simplicity, we assume that the
control-target (CT) interaction $V_{CT}$ is fixed, but $V_{TT}$ can be changed
by varying position of target channels on the shell.

The FIT behavior in such a system is investigated by solving the corresponding
Maxwell-Bloch equations of the model, with results given by in Fig. A3. Panel
(a1) shows the FIT spectrum Im($\rho_{21}$) for the target-channel as a
function of $\Delta_{c}$ and $V_{TT}$, with $V_{CT}=15\Gamma_{12}^{A}$. We see
that it exhibits two peaks even $V_{TT}$ is strong. Illustrated in panel (a2)
is the contour map of $V_{TT}$ as a function of $\Delta_{c}$, sketched with
Im$[\rho_{21}-\rho_{12}(V_{TT}=0)]$. One sees that it reaches a saturation as
$V_{TT}$ increases. For comparison, similar result for case of
$V_{CT}=10\Gamma_{12}^{A}$ is also provided; one see that FIT can also
saturate but need larger $V_{TT}$ [see panels (b1) and (b2)]. From these
results, we conclude that the TT interaction has a minor influence on the
occurrence of the FIT. In fact, under the condition of $\Delta\simeq-V_{CT}$,
the control atom can facilitate the all atoms located on the shell of the
blockade sphere, and it can lead to a self-organized criticality, which has
been observed in a recent study Helmrich  _et al._ (2020). This example shows
that the number of target channels is largely not limited in the FIT model as
long as the conditions are met. In practice, of course, the number of target
channels will be limited by physical dimensions of the channel. Hence future
experiments can explore such practical limitations.

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