# Excitons guided by polaritons

K. Mukherjee1 and S. Wüster1 1Department of Physics, Indian Institute of
Science Education and Research, Bhopal, Madhya Pradesh 462 066, India
<EMAIL_ADDRESS>
###### Abstract

We show that an exciton on a discrete chain of sites can be guided by
effective measurements induced by an ambient, non-equilibrium medium that is
synchronised to the exciton transport. For experimental verification, we
propose a hybrid cold atom platform, carrying the exciton as electronic
excitation on a chain of atoms, which are surrounded by a slow light medium
supporting polaritons. The chain is coupled to the medium through long-range
Rydberg interactions. Despite the guiding mechanism being incoherent, the
exciton pulse can be coherently transported with high fidelity. The
implementation requires careful alignment of chain and medium but then no
further time-dependent control. Our concept can be ported to other exciton and
polariton carrying media or devices, and will enable switches and waveguides
operating with the two quasi particles involved, as we demonstrate.

##  1 Introduction

While computational devices of the 20th century mainly utilized electrons and
holes as operational entities, quantum devices of the present century will
likely exploit a significantly wider pool, functionalising spinons, polaritons
[1, 2], plasmons [3] and excitons [4, 5], among many other quasi-particles.
Future technologies will also leverage quantum coherence, essential for
quantum information and quantum computing. However, as we progress towards
larger quantum devices, decoherence will become significant [6, 7], as will
transport of quantum information from one part of the device to another.
Although decoherence is typically considered detrimental [8], it can also be
turned into a resource [9, 10, 11].

Along that line, we develop a design principle for hybrid platforms in which a
polariton in a bulk medium coherently guides an exciton on a discrete chain or
waveguide embedded in that medium. For this, chain and medium are linked by
localized interaction sites, which affect the exciton only when they are
reached by the polariton and then guide its transport. We show that guiding
enables high fidelity exciton transport and provides a control interface
linking the polariton carrying component and exciton carrying component. This
hybrid quantum technology functionalises both types of quasi particles and
turns decoherence by the interactions between the components into an asset.

We find highly coherent and controlled exciton transport, despite the
underlying guiding mechanism relying on an effective measurement and hence
decoherence. Our concept can be leveraged for a variety of quantum
technologies, since the guided transport remains coherent and could thus carry
quantum information, while the guiding mechanism is decoherence, for which
sources are broadly available. For that reason, we expect that the present
scheme can find applications for example in future electromagnetically induced
transparency (EIT) in semi-conductor optics [12], Rydberg-excitons in
semiconductors [13, 14], artificial photosynthetic devices [15, 16, 17, 18],
quantum computing [19], quantum batteries [20], quantum switches [21] and
other solid state based devices.

As a specific example here, that also can be demonstrated with state of the
art experiments, we consider a regular chain of Rydberg atoms, a Rydberg
aggregate [22], embedded in a cold gas medium, while the two are interfaced
through suitably distributed detector atoms. These detector atoms can control
the exciton transport through measurement induced guiding (MIG), when they are
triggered by a suitably designed polariton pulse.

Figure 1:  Setup for guiding excitons with polaritons. (a) An equidistant
chain of Rydberg atoms (large blue-balls), that can be in two internal states
as shown on the right, is embedded in background gas of carrier atoms (green-
balls), that also contains detector atoms as interface (magenta-balls within
square). The latter are realized with 87Rb and the former with 85Rb, to allow
separate control and trapping of the components. (b) In each isotope, we
require an effective three level system to realize EIT, shown separately for
detector atoms ($\Xi$ \- medium) and carrier atoms ($\Lambda$ \- medium). Red
boxes near detector atoms in (a) represent focussed coupling beams, with Rabi
frequency $\Omega_{c}$.

The article is structured as follows: we first establish the exemplary
physical system in section 2, discussing the polariton and exciton carrying
media. In section 3, we demonstrate high-fidelity exciton transport guided by
polaritons, and in section 4 incoherent switching of exciton transport, before
we conclude in section 5. Detailed level schemes for the proposed
implementation, and options for further fidelity enhancement are provided in
appendices.

##  2 The system

We consider an atomic gas composed of the two Rubidium isotopes, 85Rb and
87Rb. Each is used for a different purpose, with 87Rb contributing two types
of atoms: (i) aggregate atoms, shown as blue balls in figure 1 (a), that will
carry the exciton and (ii) detector atoms (magenta balls). Both are trapped at
specified locations, providing the interface between excitons and polaritons.
The third type of atoms, carrier atoms (green balls), are distributed as an
untrapped bulk 85Rb gas surrounding the first two types of trapped atoms, and
will carry the polaritons.

The $N$ aggregate atoms are excited into a Rydberg state, forming a Rydberg
aggregate [22], evenly spaced with separation $R$. Each is flanked by a
detector atom as shown. For aggregate atoms, we consider only two Rydberg
states: $|\,{s}\,\rangle\equiv|\,{n=43;l=0}\,\rangle$ and
$|\,{p}\,\rangle\equiv|\,{n=43;l=1}\,\rangle$, and then further restrict the
many-body Hilbert-space to contain just a single $p$-excitation on the chain.
The resulting many-body basis is
$|\,{\pi_{n}}\,\rangle=|\,{ss..p..s}\,\rangle$, where the $n$’th atom is in
the Rydberg-$p$ state and the remaining aggregate atoms are in the Rydberg-$s$
state.

The heart of the proposed platform is the separate implementation of ladder-
type EIT [23, 24, 25, 26, 27, 28, 29, 30, 31] ($\Xi$-medium) for the detector
atoms and lambda-type EIT [23] ($\Lambda$-medium) for the carrier atoms, as
illustrated in figure 1 (b). To describe these, we use the electronic basis
$\\{g_{r},e_{r},u_{r}\\}$ for the detector atoms (EIT-$\Xi$) and
$\\{g_{1},e_{1},g_{2}\\}$ for the carrier atoms (EIT-$\Lambda$). We require
$|\,{u_{r}}\,\rangle$ to be a strongly interacting state that will couple
excitons and polaritons. In our example, where we consider two atomic species
forming the $\Xi$ and $\Lambda$ media, respectively, this is realized when the
upper level of EIT-$\Xi$ is a Rydberg state. Specific hyperfine states
proposed for those respective atomic species are indicated in figure 1 (b).
The exemplary states are selected such that
$|\,{g_{r}}\,\rangle\rightarrow|\,{e_{r}}\,\rangle$ and
$|\,{g_{1}}\,\rangle\rightarrow|\,{e_{1}}\,\rangle$ can be coupled using the
same optical field, which is the case here when both transition frequencies
are made equal with an additional Zeeman shift as discussed in A. Subsequent
transitions in the two level schemes however have to be species selective.

The specific proposal of an atomic system above is for concreteness and direct
experimental demonstration, while the design principle introduced is more
general. For example EIT is generic to three level systems, occurring also in
optomechanics [32] or semi-conductor quantum wells [33], while the chain could
be a quantum dot array [34] instead of atoms.

The setup hybridizes two components: Firstly the aggregate, which allows a
(Frenkel) exciton in the form of the Rydberg $p$-excitation to be transported
coherently in absence of environmental interactions [22, 35]. This excitation
transport can be significantly affected by controlled decoherence through an
effective measurement made by the detector atoms [36, 37] as discussed in 2.1.
Secondly, the background gas, which can carry polaritons formed through the
light-matter interaction within EIT-$\Lambda$ involving the carrier atoms.
This polariton transport is tunable using EIT-$\Lambda$ parameters, as
reviewed and applied in section 2.2.

###  2.1 Exciton dynamics

To model the exciton carrying component, the Hamiltonian of a Rydberg
aggregate describing dipole-dipole interactions is given by

$\displaystyle{\hat{H}}_{\mbox{\scriptsize agg}}=\sum_{n,m\neq
n}^{N}\frac{C_{3}}{|\mathbf{r}_{n}-\mathbf{r}_{m}|^{3}}|\,{\pi_{n}}\,\rangle\langle\,{\pi_{m}}\,|,$
(1)

where $C_{3}$ is the dipole-dipole dispersion coefficient and $\mathbf{r}_{n}$
the position of the $n$th Rydberg atom in the aggregate. The Hamiltonians to
describe EIT of the detector and carrier atoms respectively, after the dipole
and rotating wave approximation, are

$\displaystyle\hat{H}_{\Xi}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}^{{N}_{\mbox{\scriptsize
det}}}\frac{\Omega_{p}(\mathbf{x}_{\alpha},t)}{2}\hat{\sigma}^{(\alpha)}_{g_{r}e_{r}}+\frac{\Omega_{c}}{2}\hat{\sigma}^{(\alpha)}_{e_{r}u_{r}}+h.c.,$
(2) $\displaystyle\hat{H}_{\Lambda}$ $\displaystyle=$
$\displaystyle\sum_{\beta}^{{N}_{\mbox{\scriptsize
car}}}\frac{\Omega_{p^{\prime}}(\mathbf{x}_{\beta},t)}{2}\hat{\sigma}^{(\beta)}_{g_{1}e_{1}}+\frac{\Omega_{c^{\prime}}(\mathbf{x})}{2}\hat{\sigma}^{(\beta)}_{e_{1}g_{2}}+h.c.,$
(3)

where $\hat{\sigma}^{(\zeta)}_{kl}=[|\,{k}\,\rangle\langle\,{l}\,|]_{\zeta}$
for $\zeta\in\\{\alpha,\beta\\}$ acts on the $\alpha$th detector atom (or
$\beta$th carrier atom) only, the former is at position $\mathbf{x}_{\alpha}$,
and $\Omega_{p}$ ($\Omega_{c}$) are the probe (coupling) Rabi frequency for
detector atoms (ladder EIT, $\Xi$-medium), while $\Omega_{p^{\prime}}$
($\Omega_{c^{\prime}}$) are those for carrier atoms (Lambda EIT,
$\Lambda$-medium). $\Omega_{p}$ and $\Omega_{p^{\prime}}$ should involve the
same transition frequency, such that the same field can drive
$|\,{g_{r}}\,\rangle\rightarrow|\,{e_{r}}\,\rangle$ and
$|\,{g_{1}}\,\rangle\rightarrow|\,{e_{1}}\,\rangle$ simultaneously. This will
enable a polariton propagating in the $\Lambda$-medium to trigger a detector
atom. The coupling beam $\Omega_{c}$ is tightly focused onto the detector
atoms only, in order to prevent any conflict between the two EIT schemes, as
indicated by pink shades in figure 1(b). In addition, our exemplary choice of
states also makes sure that the transitions
$|\,{e_{1}}\,\rangle\rightarrow|\,{g_{2}}\,\rangle$ and
$|\,{e_{r}}\,\rangle\rightarrow|\,{u_{r}}\,\rangle$ are only driven for the
respective target species, see A.

Spontaneous decay of states involved in EIT is modelled using decay operators
$\hat{L}^{\zeta}_{kl}=\sqrt{\Gamma_{kl}}\hat{\sigma}_{kl}^{(\zeta)}$ with
$\zeta\in\\{\alpha,\beta\\}$ and $\Gamma_{kl}$ the spontaneous decay rate of
level $|\,{l}\,\rangle$ to $|\,{k}\,\rangle$. Here, we only explicitly require
the decay operator relevant for EIT-$\Xi$, which is
$\hat{L}_{\alpha}=\sqrt{\Gamma_{p}}\hat{\sigma}_{g_{r}e_{r}}^{(\alpha)}$ with
$\Gamma_{p}\equiv\Gamma_{g_{r}e_{r}}$.

Since the upper level of EIT-$\Xi$ is a Rydberg state
$|\,{u_{r}}\,\rangle\equiv|\,{38;l=0}\,\rangle$, the detector atoms realize
Rydberg-EIT [23, 24, 25, 26, 27, 28, 29, 30] and experience long-range van der
Waals interactions with Rydberg aggregate atoms. The interaction Hamiltonian
${H}_{\mbox{\scriptsize int}}$ accounting for these is

$\displaystyle{\hat{H}}_{\mbox{\scriptsize
int}}=\sum_{n,\alpha}\bar{V}_{\alpha
n}[|\,{u_{r}}\,\rangle\langle\,{u_{r}}\,|]_{\alpha}|\,{\pi_{n}}\,\rangle\langle\,{\pi_{n}}\,|,$
(4)

where $\bar{V}_{\alpha n}=V^{\alpha n}_{u_{r}p}+\sum_{m\neq n}V^{\alpha
m}_{u_{r}s}$ based on $V^{\alpha
n}_{u_{r}a}=C_{\eta(a),u_{r}a}/|\mathbf{r}_{\alpha}-\mathbf{r}_{n}|^{\eta(a)}$
with $\eta(a)=6,4$ for $a=s,p$, respectively, is the net interaction of the
$|\,{u_{r}}\,\rangle$ state with the aggregate in $|\,{\pi_{n}}\,\rangle$
[36]. Here, $\mathbf{r}_{\alpha}$ and $\mathbf{r}_{n}$ are the position of the
$\alpha$th detector atom and the $n$th Rydberg atom. Since the probability of
two detector atoms being simultaneously excited to $|\,{u_{r}}\,\rangle$ will
remain small, interactions between detector atoms are negligible [36, 38].
Carrier atoms (EIT-$\Lambda$) are only present in low lying states, hence
interactions between Rydberg atoms and carrier atoms (EIT-$\Lambda$) can be
ignored in comparison to those between Rydberg states.

We describe the dynamics of the system incorporating decoherence effects from
EIT-$\Xi$ with the Lindblad master equation ($\hbar=1$)

$\displaystyle\dot{\hat{\rho}}=-i[\hat{H},\hat{\rho}]+\sum_{\alpha}\mathcal{L}_{\hat{L}_{\alpha}}[\hat{\rho}],$
(5)

where $\hat{H}$ is the total Hamiltonian, $\hat{\rho}$ is the density operator
and $\mathcal{L}_{\hat{L}_{\alpha}}$ is a superoperator, defined in the form
$\mathcal{L}_{\hat{O}}=\hat{O}\hat{\rho}\hat{O}^{\dagger}-(\hat{O}^{\dagger}\hat{O}\hat{\rho}+\hat{\rho}\hat{O}^{\dagger}\hat{O})/2$.

To reduce the dimension of the Hilbert-space and reach a simpler master
equation, we use an effective model described in [36], derived using the
effective operator formalism of Ref. [39]. It allows a description of dynamics
in the space of the aggregate alone, using a reduced density matrix
${\hat{\rho}}^{\mbox{\scriptsize
agg}}=\sum_{nm}\rho_{nm}|\,{\pi_{n}}\,\rangle\langle\,{\pi_{m}}\,|$, evolving
according to

$\displaystyle{\dot{\hat{\rho}}}^{\mbox{\scriptsize agg}}$
$\displaystyle=-i[{\hat{H}}_{\mbox{\scriptsize
agg}}+{\hat{H}}_{\mbox{\scriptsize eff}},{\hat{\rho}}^{\mbox{\scriptsize
agg}}]+\sum_{\alpha}\mathcal{L}_{{\hat{L}}_{\mbox{\scriptsize
eff}}^{(\alpha)}}[{\hat{\rho}}^{\mbox{\scriptsize agg}}],$ (6)

where ${\hat{H}}_{\mbox{\scriptsize agg}}$ is defined in (1), and the
effective Hamiltonian and the Lindblad operator incorporating
${\hat{H}}_{\mbox{\scriptsize int}}$, $\hat{H}_{\Xi}$ and $\hat{L}_{\alpha}$
[36] are written as

$\displaystyle{\hat{H}}_{\mbox{\scriptsize eff}}$ $\displaystyle=$
$\displaystyle\sum_{n}\bigg{[}\sum_{\alpha}\frac{\Omega_{p}^{2}(\mathbf{x}_{\alpha},t)}{\Omega_{c}^{2}}\frac{\bar{V}_{\alpha
n}}{1+(\bar{V}_{\alpha
n}/V_{c})^{2}}\bigg{]}|\,{\pi_{n}}\,\rangle\langle\,{\pi_{n}}\,|,$
$\displaystyle{\hat{L}}_{\mbox{\scriptsize eff}}^{(\alpha)}$ $\displaystyle=$
$\displaystyle-\sum_{n}\bigg{[}\frac{\Omega_{p}(\mathbf{x}_{\alpha},t)}{\sqrt{\Gamma_{p}}}\frac{1}{i+V_{c}/\bar{V}_{\alpha
n}}\bigg{]}|\,{\pi_{n}}\,\rangle\langle\,{\pi_{n}}\,|,$ (7)

where $V_{c}=\Omega_{c}^{2}/(2\Gamma_{p})$. The operators
${\hat{L}}_{\mbox{\scriptsize eff}}^{(\alpha)}$ describe decoherence that
depends on the aggregate state $|\,{\pi_{n}}\,\rangle$ and atomic positions,
caused by the fact that van der Waals interactions can break EIT-$\Xi$ by
shifting the state $|\,{u_{r}}\,\rangle$ out of resonance, as first proposed
in [40] and demonstrated in [41].

Away from the aggregate Rydberg atoms, EIT-$\Xi$ turns the background gas
transparent for the probe beam. The presence of embedded Rydberg atoms, due to
the interaction energy shift discussed above, then give rise to two main
features: (i) van der Waals interactions between Rydberg atoms and detector
atoms disrupt EIT-$\Xi$ at close proximity, generating an absorption shadow of
radius $R_{c,s}$ around a Rydberg atom in state $|\,{s}\,\rangle$ and of
radius $R_{c,p}$ around a Rydberg atom in state $|\,{p}\,\rangle$. The radii
are $R_{c,a}=(2C_{\eta(a),ra}\Gamma_{p}/\Omega_{c}^{2})^{1/\eta(a)}$ where
$\eta(a)=6,4$ for $a=s,p$, and can be tuned to significantly differ [36]. The
shadows indicate the presence of Rydberg atoms and indirectly measure their
location (and state) via the centre (and size) of the shadow, implementing a
quantum non-demolition (QND) measurement of both properties. (ii) Controllable
decoherence can thus be induced in the excitation transport by positioning
detector atoms between the shadow radii $R_{c,s}$ and $R_{c,p}$. Then they can
measure the location of the Rydberg-$p$ excitation on the aggregate, and thus
the state $|\,{\pi_{n}}\,\rangle$. The resultant effect on the quantum state
of the aggregate can give rise to tunable measurement-induced guiding (MIG),
presented in section 3.

###  2.2 Tuned polariton propagation

In the second component of the hybrid platform, we now discuss how
EIT-$\Lambda$ on the carrier atoms can control the motion of the probe field
polaritons [42], to synchronize MIG with dipolar interactions between the
Rydberg atoms. For this purpose, writing $\mathbf{x}=[x,y,z]^{T}$, we consider
a spatially dependent coupling field for EIT-$\Lambda$

$\displaystyle\Omega_{c^{\prime}}(\mathbf{x})$ $\displaystyle=$
$\displaystyle\Omega_{c^{\prime},min}+\frac{(\Omega_{c^{\prime},max}-\Omega_{c^{\prime},min})}{2}\sum_{n=-2}^{N+1}\bigg{(}\tanh\left[\frac{x-(n-1/2)R+w}{\sigma_{c}}\right]$
(8) $\displaystyle-\tanh\left[\frac{x-(n-1/2)R-w}{\sigma_{c}}\right]\bigg{)},$

shown in figure 2 (b). In (8), $\Omega_{c^{\prime},min}$ and
$\Omega_{c^{\prime},max}$ are the minimum and maximum Rabi frequency,
$\sigma_{c}$ controls the spatial sharpness of the increasing and decreasing
flanks of the profile, $N$ is the number of aggregate atoms and $R$ is the
separation between them. We have taken the half width of square waveforms as
$w=3R/8$. The coupling field $\Omega_{c^{\prime}}(\mathbf{x})$ varies along
the $x$-component of $\mathbf{x}$ only, and for subsequent equations in one
dimension we shall thus use the notation
$\Omega_{c^{\prime}}(\mathbf{x})\rightarrow\tilde{\Omega}_{c^{\prime}}(x)$,
for simplicity. Complex beam shapes such as (8) can be created using e.g.
spatial light modulators [43].

Figure 2: (a) Array of $5$ Rydberg aggregate atoms (blue) spaced by
$R=20\mu$m, flanked by detector atoms (magenta), all trapped in individual
harmonic traps with ground-state width ${\sigma}_{\mbox{\scriptsize
trap}}=0.5\mu$m. All are immersed in a background gas of 85Rb carrier atoms at
uniform density $\varrho_{85}=3.75\times 10^{13}$m-2. (b) The coupling field
profile $\tilde{\Omega}_{c^{\prime}}(x)$ as in (8), using a rise-length
$\sigma_{c}=1.0\mu$m and $\Omega_{c^{\prime},min}/2\pi=0.5$ kHz,
$\Omega_{c^{\prime},max}/2\pi=5.28$ kHz. (c) Resultant propagation of two
probe pulses in $\Omega_{p}(\mathbf{x},t)$ with Gaussian envelope of width
$\sigma_{p}=3\mu m$. White vertical lines in (c-d) indicate the position of
aggregate atoms marked with the atom index $n$. (d) Modelled using (6), an
excitation initialised at site 1, at $x=0$ $\mu$m, is transported to site 5,
at $x=80$ $\mu$m, via site 2,3 and 4 with high fidelity, enabled by
synchronized activation of detector atoms and polariton transport shown in
(c). We indicate the excitation population $\rho_{nn}$ of each atom by the
width and amplitude of the colour shading surrounding the white vertical
lines. The right inset in (d) again shows the atomic populations $\rho_{nn}$
for the guided case, and the left inset for the isolated aggregate that has no
surrounding medium. Parameters used are $C_{3}/2\pi=1619$ MHz $\mu$m3,
$C_{6,s}/2\pi=-87$ MHz $\mu$m6, $C_{4,p}/2\pi=-1032$ MHz $\mu$m4,
$\Omega_{p,0}/2\pi=2.5\Omega_{p^{\prime},0}/2\pi=18$ MHz, $\Omega_{c}/2\pi=90$
MHz and $\Gamma_{p}/2\pi=6.1$ MHz.

The polaritons formed in EIT-$\Lambda$, can be written as [42],

$\displaystyle\hat{\Psi}(x,t)$ $\displaystyle=$
$\displaystyle\cos\theta(x)\hat{E}(x,t)-\sin\theta(x)\sqrt{\varrho_{85}}\hat{\sigma}_{gg_{1}}(x,t),$
(9) $\displaystyle\cos\theta(x)$ $\displaystyle=$
$\displaystyle\frac{\tilde{\Omega}_{c^{\prime}}(x)}{\sqrt{\tilde{\Omega}_{c^{\prime}}^{2}(x)+g^{2}\varrho_{85}}}.$
(10)

Here, $\hat{E}$ is a quantum field resonantly coupling the states
$|\,{g_{1}}\,\rangle\leftrightarrow|\,{e_{1}}\,\rangle$ with coupling constant
$g=\mu\sqrt{\omega_{0}/2\pi\epsilon_{0}}$, where $\mu$ and $\omega_{0}$ are
the transition dipole matrix element and transition frequency, respectively.
$\varrho_{85}$ is the background gas density of 85Rb carrier atoms. As is well
known, $\hat{\Psi}$ obeys the equation of motion

$\displaystyle\bigg{[}\frac{\partial}{\partial
t}+c\cos^{2}[\theta(x)]\frac{\partial}{\partial x}\bigg{]}\hat{\Psi}(x,t)=0,$
(11)

which describes a shape-preserving wave propagation with group velocity
$v_{g}(x)=c\cos^{2}[\theta(x)]$, where $c$ is the vacuum speed of light. Since
$\cos\theta(x)$ decreases with $\tilde{\Omega}_{c^{\prime}}(x)$ in (10), the
group velocity of the polariton decreases with the strength of
$\tilde{\Omega}_{c^{\prime}}(x)$, giving rise to slow light [23]. Note, that
the functionality of the detector atoms in EIT-$\Xi$ requires
$\Omega_{c}\gg\Omega_{p}$, while slowing down polaritons in EIT-$\Lambda$
requires small $\Omega_{c^{\prime}}$, thus necessitating two different atomic
species for EIT-$\Xi$ and EIT-$\Lambda$.

In the classical limit, the field part of the propagating polariton gives rise
to a probe Rabi frequency
$\Omega_{p^{\prime}}(\mathbf{x},t)=g\sqrt{\varrho_{85}}\langle\hat{E}(x,t)\rangle$,
which we can use in (3). Importantly, since we have designed the probe
transition frequencies for EIT-$\Lambda$ and EIT-$\Xi$ to be equal, these
polaritons will trigger the detector atoms and we take
$\Omega_{p}(\mathbf{x},t)\approx 2.5\Omega_{p^{\prime}}(\mathbf{x},t)$, where
the proportionality factor arises due to different transition dipole matrix
elements.

Incorporating the mean field polariton propagation dictated by (11), we thus
assume for $\Omega_{p}(\mathbf{x},t)$ a train of Gaussian pulses that
propagate with group-velocities locally controlled by
$\tilde{\Omega}_{c^{\prime}}(x)$ according to

$\displaystyle\Omega_{p}(\mathbf{x},t)=\sum_{n}\Omega_{p,0}\exp\bigg{(}-\frac{(x-x_{n}(t))^{2}}{2\sigma_{p}^{2}}\bigg{)},$
(12)

with $\mathbf{x}=[x,y,z]$,
$x_{n}(t)=x_{n}(0)+\int_{0}^{t}v_{g}(x_{n}(t^{\prime}))dt^{\prime}$ the
instantaneous x-position of the $n$th probe pulse (polariton), $\Omega_{p,0}$
its peak Rabi frequency and $\sigma_{p}$ its spatial width. The corresponding
profile $\Omega_{p}(\mathbf{x},t)$ along the $x$-axis is shown in figure 2(c).
Since the profile $\tilde{\Omega}_{c^{\prime}}(x)$ in (8) shall later vary on
scales $\sigma_{c}\approx\sigma_{p}/3$ the pulses will in reality be distorted
during their fast transits from detector atom to detector atom, but should
remain Gaussian shaped while remaining at one detector atom, which is where
they matter for our simulations.

##  3 Polaritons guide excitons

To demonstrate how the combination of effective measurement through
interactions and polariton transport discussed in the previous sections gives
rise to measurement induced guiding (MIG) of the exciton, we consider a
Rydberg aggregate of $N=5$ atoms, equidistantly separated by $R=20\mu$m and
immersed in a background gas, as shown in figure 2 (a). The life-time of the
aggregate is $17.8$ $\mu$s [44]. Each Rydberg atom is accompanied by a
detector atom, offset by $\delta_{y}=1$ $\mu$m in the y-direction only. The
carrier atoms (green balls) are uniformly distributed in the $x-y$ plane shown
in figure 2 (a).

We assume the time-independent spatial coupling profile
$\tilde{\Omega}_{c^{\prime}}(x)$ in (8), shown in figure 2 (b), and insert two
strongly occupied polariton pulses at $x_{1}=-21.5$ $\mu$m and $x_{2}=38.5$
$\mu$m into the probe field, as shown in figure 2 (c). The precise location of
polaritons is important to ensure synchronization with exciton transport and
thus a high fidelity of the latter. The aggregate is initialised with exciton
localised on the first site:
$\hat{\rho}=|\,{\pi_{1}}\,\rangle\langle\,{\pi_{1}}\,|$, and the system is
then evolved following the effective master equation in (6), assuming the
propagating profile for the polaritons given in (12), with results shown in
figure 2 (d).

We see in panel (d) and its top right inset, that the exciton is guided to the
final site with high fidelity, kept localized on at most two sites, in
contrast to the free propagation on the isolated aggregate chain, shown in the
bottom left inset. To understand this, let us consider the process step by
step: In the first stage, from $t=0$ to $t\approx 1.23$ $\mu$s ($\approx\pi
R^{3}/2C_{3})$, the polaritons are slowed down by the weak coupling amplitude
$\tilde{\Omega}_{c^{\prime}}$ near their initial positions at
$x_{1}\approx-20\mu$m and $x_{2}\approx 40\mu$m. During this period they only
activate the detector atom flanking the third site. This results in continuous
strong measurement of the third aggregate atom through the nearby detector
atom. That atom thus strongly “measures” whether the adjacent aggregate atom
carries the exciton, freezing its state in the $|\,{s}\,\rangle$ state though
a quantum Zeno effect [45, 46]. The energy shift in
${\hat{H}}_{\mbox{\scriptsize eff}}$ linked to the detector atom is not
significant here, which we verify by setting it to zero, finding nearly
unchanged dynamics. Having thus effectively removed $|\,{\pi_{3}}\,\rangle$
from the Hilbert space through the Zeno effect, the exciton is transferred by
a coherent oscillation to site $|\,{\pi_{2}}\,\rangle$. This is the essence of
measurement induced guiding. The EIT-$\Lambda$ coupling profile is designed
such that exactly after this transfer, the polariton pulses leave their
respective slow zone, make a quick transit to the next site within $t\approx
2ns$, and now block site 1 and 4, thus nextly allowing a complete exciton
transfer from site 2 to site 3, and so forth.

Repeating these steps, we demonstrate a transport fidelity $\rho_{55}$ of up
to $87\%$ in figure 2 (d). The simulation did include position uncertainties
of aggregate atoms and detector atoms, assuming each to be harmonically
trapped with width ${\sigma}_{\mbox{\scriptsize trap}}=0.5\mu$m.111Our
proposal assumes a uniform distribution of carrier atoms, although these do
not explicitly enter simulations. We then averaged over $10^{3}$ realisations
of aggregate and detector atom positions. The fidelity can be enhanced further
to over $95\%$ with four polaritons, which can also suppress next-to-nearest
neighbour dipole interactions between aggregate atoms, see B. Furthermore, the
design can be extended to a 2D lattice with $N\times N$ aggregate atoms, where
the exciton can be guided using a two-dimensional spatially dependent coupling
field $\tilde{\Omega}_{c}(x,y)$.

Note, although the QND measurement that guides transport implies a decohering
process, the transport itself remains coherent with purity
$P=\mathrm{Tr}[{\hat{\rho}}_{\mbox{\scriptsize agg}}(t)^{2}]$ dropping only
marginally to $P\approx 0.8$, similar to the drop in fidelity. For the case
with four polaritons, the purity remains near $P>0.96$ resulting in
effectively coherent transport, despite the process relying on decoherence.

##  4 Incoherent switching of coherent exciton transport

Figure 3: (a) Probe field $\Omega_{p}(\mathbf{x},t)$ with set of instructions
for controlled excitation switching in the system. (b) Excitation probability
$\rho_{nn}$ of aggregate atom $n$. The exciton is initialized on atom $3$ and
evolved using (6), with the exciton transport controlled by polaritons using
switch operations. Other parameters are the same as in figure 2. The vertical
lines indicate the timing of implementation of each operation, labelled at the
top of (a). The inset in (b) shows the purity
$P=\mathrm{Tr}[{\hat{\rho}}_{\mbox{\scriptsize agg}}(t)^{2}]$

Even without the feature of synchronised propagating polaritons, the MIG
through detector atoms can be used to orchestrate complex transport sequences
on the Rydberg aggregate. As an example, we show in figure 3 how it can be
used to move the exciton one site to the left, and then two sites to the
right.

For this, we consider a set of time-dependent pulses for the probe field
$\Omega_{p}$, shown in figure 3 (a). The probe pulses can be manually switched
“on” and “off”, to carry out a set of instructions that enable or disable
exciton transport in a certain direction, thus implementing bi-directional
guiding of the exciton within the aggregate. For the demonstration, we
initialise the array of $N=5$ Rydberg atoms in
$\hat{\rho}=|\,{\pi_{3}}\,\rangle\langle\,{\pi_{3}}\,|$, which would result in
its spreading in both directions $-x$ / $+x$ in the unguided aggregate.
Instead we implement three operations: (i) switch OFF, to hold the excitation
on a single atom, (ii) switch ONL, only allowing coherent excitation transport
by one site to the left (in the $-x$ direction), and (iii) switch ONR, similar
to ONL but transporting to the right (in the $+x$ direction).

We again see in figure 3, that the purity remains close to $0.95$ after these
operations. The switch off feature can have a flexible duration, only limited
by the life-time of the system, but the timing of switch OFF and switch ON
needs to be precise and depends on the Rabi period during which the excitation
transfers coherently from one site to a neighboring site, given by
${t}_{\mbox{\scriptsize period}}=\pi R^{3}/2C_{3}$.

In more complex scenarios, such as considering two dimensions, the switching
can again be induced by propagating and then slowed down polariton pulses as
in section 3, providing a hybrid interface between excitonic and polaritonic
platforms [2], with possible applications in measurement based quantum
computing [47].

##  5 Conclusion

We have shown that guided exciton transport is possible in hybrid platforms
that carry excitons and polaritons. The guiding is based on continuous Zeno-
like measurements of the exciton-carrying component, that are synchronised
with transport by propagating polaritons in a guidance medium. We demonstrate
the proposed concept explicitly, by modelling a platform compatible with
state-of-the-art experiments, in which Rydberg atoms, as exciton carrying
component, are embedded in a cold gas medium involving two separate
implementations of EIT, the latter providing the polariton-carrying component.

It is known that an environment can be beneficial for transport, despite
decohering it, for example in exciton migration through molecular aggregates
relying on intra-molecular vibrations [48, 49, 50, 51], which was recently
more broadly discussed [52, 53, 54]. In stark contrast to these, our proposal
uses an environment out of equilibrium, and then does not decohere the
transport, while enhancing it.

Beyond the cold atom demonstration here, the design principle proposed will
apply to Rydberg-excitons in semiconductors [13, 14], artificial
photosynthetic devices [15, 16, 17, 18], quantum computing [19], quantum
batteries [20], quantum switches [21] and other solid state based devices. The
key feature to guiding excitons by polaritons that then has to be ported to
these systems, is that the guiding medium accesses a strongly interacting
state only in the presence of the polariton, which then triggers decoherence
in the discrete component to guide the exciton.

We acknowledge helpful discussions with Sourav Dutta, Ashok Mohapatra,
Swayangdipta Bera, Abhijit Pendse and Aniruddha Mitra. We gladly thank Shannon
Whitlock for discussions and reading the manuscript. We also thank the Max-
Planck society for financial support under the MPG-IISER partner group program
as well as the Indo-French Centre for the Promotion of Advanced Research -
CEFIPRA for funding. K.M. acknowledges the Ministry of Education for the Prime
Minister’s Research Fellowship (PMRF).

##  Appendix A Level scheme

Figure 4:  (a) Energy levels for each EIT scheme as depicted in figure 1 (c).
(b) More detailed energy level diagram for 87Rb (left) and 85Rb (right)
showing hyperfine levels and their corresponding Zeeman shift for the levels
in consideration here [55, 56].

A more detailed energy level diagram for 87Rb [56] and 85Rb [55] than the one
sketched in figure 1 is shown in figure 4 (b), including hyperfine levels and
Zeeman shifts. As discussed in the main text, the probe field (blue arrows)
drives the transitions $|\,{g_{r}}\,\rangle\rightarrow|\,{e_{r}}\,\rangle$ in
87Rb and $|\,{g_{1}}\,\rangle\rightarrow|\,{e_{1}}\,\rangle$ in 85Rb. In the
absence of a magnetic field, these transition are detuned by
$\Delta_{p}=\epsilon_{e_{r}g_{r}}-\epsilon_{e_{1}g_{1}}\approx 1.3$ GHz.
However, the difference in the transition energies ($\Delta_{p}$) can be tuned
to zero by Zeeman shifting the levels using a magnetic field of $B_{z}=20.12$
mT, making $\Delta_{p}=0$. This is a strong magnetic field, but within range
of typical Feshbach coils used in similar experiments. In addition, we also
make sure the coupling field of EIT-$\Xi$ (red arrow) does not interact with
EIT-$\Lambda$ by making sure there is no state at the energy of
$|\,{u_{r}}\,\rangle$ in EIT-$\Lambda$ and only focussing that coupling beam
onto detector atoms. Also the coupling field of EIT-$\Lambda$ (green arrow)
does not interfere with EIT-$\Xi$ due to the non-existence of a
$|5S_{1/2}F=1,m_{F}=2\rangle$ state, thereby making the $\sigma_{+}$
transition forbidden for any other state of 87Rb.

##  Appendix B Enhancing fidelity using four polaritons

Figure 5: (a) The probe field $\Omega_{p}(x,t)$ with two Gaussian pulses as
defined in (12) for $(n=2)$, with peak Rabi frequency $\Omega_{p}$
corresponding to EIT-$\Xi$. Black solid (dashed) line indicates the initial
(final) position of the polariton, induced by the spatially dependent coupling
field $\tilde{\Omega}_{c^{\prime}}(x)$ shown in figure 2 (b). (b) Excitation
$\rho_{nn}$ on each atom, with index $n$ indicated near each line, for $N=5$
Rydberg atoms shown in figure 2 (a). The excitation is initialised on atom 1
with $\hat{\rho}=|\,{\pi_{1}}\,\rangle\langle\,{\pi_{1}}\,|$ (black solid) and
transported to atom 5 ($|\,{\pi_{5}}\,\rangle$, red solid) via atom 2 (blue
dashed), 3 (green dot dashed) and 4 (purple dotted) following (6). (c-d) are
similar to (a-b) but with four polaritons ((12) with $n=4$). The parameters
used are the same as in figure 2

Here we compare the fidelity in two cases: first with two polaritons
initialized at position $x_{1}=-21.5$ $\mu$m and $x_{2}=38.5$ $\mu$m as
discussed in figure 2 (a) of the main text, while in the second case we
consider a set of four polaritons initialized at $x_{1}=-41.5$ $\mu$m,
$x_{2}=-21.5$ $\mu$m, $x_{3}=38.5$ $\mu$m and $x_{4}=58.5$ $\mu$m, as
presented in figure 5 (c). In either case, we initialize the aggregate at site
1 $\hat{\rho}=|\,{\pi_{1}}\,\rangle\langle\,{\pi_{1}}\,|$, and the system is
then evolved with the effective master equation (6). We see that in either
case, synchronization of the MIG by detector atoms with exciton transport can
guide the exciton with high fidelity, but since four polaritons can suppresses
long-range dipole-dipole interactions beyond the nearest neighbor, they yield
a higher fidelity of up to 95%.

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