††thanks: M. Tian and F. Sun contributed equally to this work.††thanks: M.
Tian and F. Sun contributed equally to this work.

#  Nonreciprocal Amplification Transition in a Driven-Dissipative Quantum
Network

Mingsheng Tian  State Key Laboratory for Mesoscopic Physics, School of
Physics, Frontiers Science Center for Nano-optoelectronics, $\&$ Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
Fengxiao Sun  State Key Laboratory for Mesoscopic Physics, School of Physics,
Frontiers Science Center for Nano-optoelectronics, $\&$ Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
Kaiye Shi  Department of Physics, Renmin University of China, Beijing 100872,
China  Haitan Xu<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science
and Engineering, Southern University of Science and Technology, Shenzhen
518055, China  School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Qiongyi He<EMAIL_ADDRESS>State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers
Science Center for Nano-optoelectronics, $\&$ Collaborative Innovation Center
of Quantum Matter, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi
030006, China  Wei Zhang<EMAIL_ADDRESS>Department of Physics, Renmin
University of China, Beijing 100872, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China

###### Abstract

We study the transport properties of a driven-dissipative quantum network,
where multiple bosonic cavities such as photonic microcavities are coupled
through a nonreciprocal bus with unidirectional transmission. For short-range
coupling between the cavities, the occurrence of nonreciprocal amplification
can be linked to a topological phase transition of the underlying dynamic
Hamiltonian. However, for long-range coupling, we find that the nonreciprocal
amplification transition deviates drastically from the topological phase
transition. Nonetheless, we show that the nonreciprocal amplification
transition can be connected to the emergence of zero-energy edge states of an
auxiliary Hamiltonian with chiral symmetry even in the long-range coupling
limit. We also investigate the stability, the crossover from short to long-
range coupling, and the bandwidth of the nonreciprocal amplification. Our work
has potential application in signal transmission and amplification, and also
opens a window to non-Hermitian systems with long-range coupling and
nontrivial boundary effects.

Introduction.$-$ Nonreciprocity breaks the invariance of transmission
amplitudes under the exchange of source and detector, and is of great value in
numerous circumstances Deák and Fülöp (2012). For instance, it offers new
functionalities to photonic networks Lodahl  _et al._ (2017); Ranzani and
Aumentado (2015); Pichler  _et al._ (2015); Metelmann and Türeci (2018),
enhances the information processing capacity Miller (2010); Jalas  _et al._
(2013); Guimond  _et al._ (2020), and acts as a resource for quantum metrology
Lau and Clerk (2018). A number of strategies have been developed to realize
nonreciprocity, such as magneto-optical effects Dötsch  _et al._ (2005),
nonreciprocal phase response in Josephson circuits Abdo  _et al._ (2013);
Sliwa  _et al._ (2015), spatial-temporal refractive index modulations Yu and
Fan (2009); Lira  _et al._ (2012), and optomechanically induced nonreciprocity
Manipatruni  _et al._ (2009); Verhagen and Alù (2017); Hafezi and Rabl (2012);
Ruesink  _et al._ (2016); Xu  _et al._ (2016, 2019). Of particular interest is
an amplifier with nonreciprocity which protects weak signals against noises
from the read-out electronics Clerk  _et al._ (2010), and can be achieved by
reservoir engineering with interfering coherent and dissipative processes
Metelmann and Clerk (2015, 2017, 2014); Malz  _et al._ (2018); Shen  _et al._
(2016); Fang  _et al._ (2017); Bernier  _et al._ (2017); Peterson  _et al._
(2017); Barzanjeh  _et al._ (2017); Mercier de Lépinay  _et al._ (2019).

Recent works show that a correspondence between nonreciprocal amplification
and topological phase can be established Porras and Fernández-Lorenzo (2019);
Wanjura  _et al._ (2020). The transport properties of a driven-dissipative
system can be characterized by a topological invariant Rudner and Levitov
(2009); Gong  _et al._ (2018). Since the topological characterization of a
system is mostly formulated in the context of condensed matter physics where
the interaction is usually short-ranged, its application in understanding
nonreciprocal amplification was focused on systems with short-range coupling
Wanjura  _et al._ (2021); Porras and Fernández-Lorenzo (2019). However, for a
driven-dissipative system possessing long-range coupling, the discussion from
a topological perspective is still lacking.

In this paper, we study the transport properties of a driven-dissipative
system where a chain of cavities supporting bosonic modes are coupled to a
common nonreciprocal bus with unidirectional transmission. The range of the
effective coupling between different cavities is determined by the coherence
length of the bus, which can be much longer than the spacing between nearest
neighbors. We find that a nonreciprocal amplification can be achieved within a
wide range of parameters. In the short-range coupling limit, the amplification
transition can be linked to the topological phase transition either in the
analysis of non-Hermitian dynamic matrix Wanjura  _et al._ (2020) or within
the framework of topological band theory with auxiliary chiral symmetry Porras
and Fernández-Lorenzo (2019). However, this connection fails in the long-range
coupling limit, in which case the amplification transition deviates
drastically from the topological phase transition associated with the dynamic
matrix. This discrepancy originates from the nontrivial boundary effect
induced by the long-range coupling, which breaks the conventional bulk-
boundary correspondence in systems with short-range coupling. By correctly
addressing the boundary effect, we analytically establish a connection between
the amplification transition and the emergence of symmetry-protected zero-
energy edge states for general cases with arbitrary coupling range. We further
investigate the stability of the system, the crossover from short to long-
range coupling, and the bandwidth of amplification.

Figure 1:  A chain of $N$ cavities with identical bosonic modes coupled to a
nonreciprocal bus. $\Gamma$ denotes the coupling between each cavity mode and
the bus, $\kappa$ is the input/output coupling rate, $\gamma$ is the net
pumping rate (including internal damping), and $d$ is the spacing between
neighboring cavities.

Nonreciprocal Amplification Transition.$-$ We consider a one-dimensional array
of $N$ equidistantly separated cavities such as photonic microcavities with
local driving and dissipation as shown in Fig. 1. We focus on a single bosonic
mode $a_{j}$ of the $j^{\rm th}$ cavity, and the bosonic modes of different
cavities are assumed to have an identical frequency $\omega_{0}$. The cavities
are coupled to a nonreciprocal bus with a continuum of right-propagating
bosonic modes. The Hamiltonian reads $H_{\rm tot}=H_{S}+H_{R}+H_{SR}$, where
$H_{S}=\hbar\omega_{0}\sum_{j=1}^{N}a_{j}^{\dagger}a_{j}$ describes the $N$
cavities, and $H_{R}=\int
d\omega_{b}\,\hbar\omega_{b}\,b^{\dagger}(\omega_{b})b(\omega_{b})$ is the
free Hamiltonian of the bus modes. The cavity-bus coupling can be modeled as
$H_{SR}=i\hbar\sum_{j}\int
d\omega_{b}\sqrt{\Gamma/2\pi}b^{\dagger}(\omega_{b})a_{j}e^{i\left(k_{b}x_{j}-\omega_{b}t\right)}+{\rm
H.C.}$, where $|\omega_{b}-\omega_{0}|\ll\omega_{0}$, $x_{j}$ denotes the
position of the $j^{\rm th}$ cavity, and $\Gamma$ is the coupling rate between
the cavity mode $a_{j}$ and the bus modes $b(\omega_{b})$. As the bus modes
transmit unidirectionally from the left ($j=1$) to the right ($j=N$), the
wavevector $k_{b}$ satisfies Re[$k_{b}]>0$, and can be expressed as
$k_{b}=\omega_{b}/v+ik_{b}^{\prime\prime}$ where $v$ is the group velocity of
the mode and $k_{b}^{\prime\prime}$ is related to the loss rate.

We focus on the characteristics of the input fields $\langle a_{j,{\rm
in}}(t)\rangle$ and the output fields $\langle a_{j,{\rm out}}(t)\rangle$
through the driven-dissipative chain. Under Born-Markov approximation, the
equations of motion for the amplitudes $\langle a_{j}(t)\rangle$ are given by

$\displaystyle\langle\dot{a}_{j}(t)\rangle=$
$\displaystyle\frac{\gamma_{j}-\kappa_{j}-\Gamma-2i\omega_{0}}{2}\langle
a_{j}(t)\rangle-\sqrt{\kappa_{j}}\langle a_{j,{\rm in}}(t)\rangle$
$\displaystyle-\sum_{j>l}^{N}\Gamma_{jl}e^{ik_{\omega}x_{jl}}\langle
a_{l}(t)\rangle$ $\displaystyle=$ $\displaystyle\sum_{l}{H_{jl}}\langle
a_{l}(t)\rangle-\sqrt{\kappa_{j}}\langle a_{j,{\rm in}}(t)\rangle,$ (1)

where $k_{\omega}=\omega/v$, $\omega$ is the signal frequency,
$x_{jl}=x_{j}-x_{l}$, and $\Gamma_{jl}=\Gamma e^{-x_{jl}/\zeta}$ with
$\zeta=1/k_{b}^{\prime\prime}$ being the coherence length of the bus mode. For
simplicity, we assume that the input/output coupling rates and pumping rates
are identical for all cavities, i.e. $\kappa_{j}=\kappa$ and
$\gamma_{j}=\gamma$, and we take the natural unit of $d=1$. As $\zeta$ varies
from order $1$ to the order of system size $N$, the effective coupling between
different cavities changes from short-range to long-range. Especially, in the
infinitely long-range limit of $\zeta\gg N$, the $l^{\rm th}$ cavity is
coupled to all cavities on the right with $j>l$.

The stability of the system is determined by the local parameters. A stable
system requires that the real part of the eigenvalues $q_{n}$ of the non-
Hermitian matrix $H$ to be negative, since otherwise any fluctuation gets
exponentially amplified. In our case, the eigenvalues are degenerate and
$Re(q_{n})=(\gamma-\kappa-\Gamma)/2,$ thus we conclude that the system is
stable only when $\gamma<\kappa+\Gamma$.

By Fourier transforming $a(t)$ and considering the input-output boundary
conditions $a_{j,{\rm out}}(\omega)=a_{j,{\rm
in}}(\omega)+\sqrt{\kappa}a_{j}(\omega)$ Gardiner and Collett (1985), we
obtain the input-output relation in the frequency domain

$\displaystyle\mathbf{a}_{\mathrm{out}}(\omega)$ $\displaystyle=$
$\displaystyle\left[\mathbb{I}+\kappa(H+i(\omega-\omega_{0})\mathbb{I})^{-1}\right]\mathbf{a}_{\mathrm{in}}(\omega)$
(2) $\displaystyle=$ $\displaystyle
S(\omega)\mathbf{a}_{\mathrm{in}}(\omega),$

where $\mathbf{a}_{\rm in/out}=(\langle
a_{1,\mathrm{in/out}}\rangle,\cdots,\langle a_{N,\mathrm{in/out}}\rangle)^{T}$
are the input and output amplitude vectors. The nonreciprocal amplification
can be captured by the scattering matrix $S(\omega)$. In particular, a large
component in the bottom left corner of $S(\omega)$ as shown in Fig. 2(a)
corresponds to a strong amplification of the input signal from the first
cavity ($j=1$) to the last one ($j=N$).

From Eq. (2), we can define a non-Hermitian dynamic matrix $M\equiv
H+i(\omega-\omega_{0})\mathbb{I}$ which plays an important role in
characterizing the transport properties. Using the singular value
decomposition (SVD) $M=U\Lambda V^{\dagger}$ Ramos  _et al._ (2021), where $U$
and $V$ are unitary matrices and $\Lambda$ is a diagonal matrix
$\Lambda_{nm}=\lambda_{n}\delta_{nm}$ with $\lambda_{n}>0$, we obtain an
analytical solution for the scattering matrix

$\displaystyle|S_{jl}|$ $\displaystyle=$
$\displaystyle|\delta_{jl}+\kappa\sum_{n}V_{jn}\lambda_{n}^{-1}U_{ln}^{*}|$
(3) $\displaystyle=$
$\displaystyle\begin{cases}\left|\frac{\gamma+\kappa-\Gamma+2i\Delta\omega}{\gamma-\kappa-\Gamma+2i\Delta\omega}\right|,&\text{
for }j=l;\\\ 0,&\text{ for }j<l;\\\ Be^{(j-l)(\xi^{-1}-\zeta^{-1})},&\text{
for }j>l.\end{cases}$

The coefficients are

$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{4\kappa\Gamma}{|\gamma+\Gamma-\kappa+2i\Delta\omega||\gamma-\Gamma-\kappa+2i\Delta\omega|},$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\left[\log\left|\frac{\gamma+\Gamma-\kappa+2i\Delta\omega}{\gamma-\Gamma-\kappa+2i\Delta\omega}\right|\right]^{-1},$
(4)

where $\Delta\omega=\omega-\omega_{0}$ is the frequency detuning. An
amplification from the $l^{\rm th}$ cavity to the $j^{\rm th}$ cavity occurs
when $|S_{jl}|>1$. Especially, if the prefactor $B=1$, we have $|S_{jl}|=1$
for all $j>l$ when $\zeta=\xi$ (i.e.,
$\gamma=\kappa+\Gamma-\frac{2\Gamma}{e^{1/\zeta}+1}$ with zero detuning), and
the amplification occurs for $\zeta>\xi$, while the attenuation occurs if
$\zeta<\xi$. For general cases with an arbitrary $B$, the same criterion of
amplification holds if the separation $j-l$ is large enough.

Of particular interest is the case of infinitely long-range coupling, i.e.,
$\zeta\gg N$, as it is nearly impossible in solid state systems and thus
considered to be irrelevant in the context of condensed matter physics. In
such case, the condition of amplification translates to $\gamma>\kappa$. This
result suggests that any small cavity-bus coupling would suffice to sustain a
nonreciprocal amplification provided that the pumping overcomes the
dissipation to the input-output ports. As shown in Fig. 2(b), the amplitudes
of all scattering matrix elements are less than unity for $\gamma<\kappa$.
When $\gamma$ goes beyond the transition point [red dot in Fig. 2(c)],
$|S_{N1}|$ becomes larger than 1 and increases rapidly with $N$, as depicted
in Fig. 2(c). When $\gamma$ is further increased beyond $\kappa+\Gamma$, the
system enters an unstable regime labeled by shaded area in Fig. 2(c).

Figure 2: Scattering matrices for (a) $\gamma=0.2$ and (b) $\gamma=0.5$ with
$N=20$. (c) The gain $|S_{N1}|^{2}$ for different system sizes as a function
of pumping rate $\gamma$. In all plots, $\Gamma=1$, $\kappa=0.25$,
$\Delta\omega=0$ and $\zeta\gg N$.

Topological invariants of dynamic matrix.$-$ The nonreciprocal amplification
exhibited in the scattering matrix $S$ leaves evident signature in the non-
Hermitian dynamic matrix $M$, which satisfies $S=\mathbb{I}+\kappa M^{-1}$. If
the system has only nearest-neighbor coupling between the cavities, a direct
correspondence can be established between the nonreciprocal amplification and
a nontrivial topological invariant defined on the spectrum of the dynamic
matrix under a periodic boundary condition (PBC) Wanjura  _et al._ (2020).
However, as we will show, this correspondence does not always hold in our
system.

Under PBC, the dynamic matrix
$M_{\rm{p}}=\sum_{k}h_{\rm{p}}(k)|k\rangle\langle k|$ is diagonal in the basis
$|k\rangle=({1}/{\sqrt{N}})\sum_{j}e^{ikj}|j\rangle$, where $k=2\pi n/N$ with
$n=0,1,\cdots,N-1$, and $|j\rangle$ represents the real-space wave function.
The spectrum of $M_{\rm{p}}$ reads

$\displaystyle h_{\rm{p}}(k)=\sum_{m=0}^{N-1}\mu_{m}e^{-ikm},$ (5)

where $\mu_{0}=(\gamma-\kappa-\Gamma)/2+i\Delta\omega$ and $\mu_{m>0}=-\Gamma
e^{-m/\zeta}{\color[rgb]{0,0,0}e^{ik_{\omega}m}}$ (see details in
Supplementary Material sup ). Since $M_{\rm{p}}$ is non-Hermitian,
$h_{\rm{p}}(k)=h_{{\rm p},x}(k)+ih_{{\rm p},y}(k)$ is in general complex with
$h_{{\rm p},x},h_{{\rm p},y}\in\mathbb{R}$. Furthermore, since $h_{\rm{p}}(k)$
is periodic in $k$ with period $2\pi$, it describes a closed curve in the
complex plane. Thus we can define a winding number from the argument principle
Gong  _et al._ (2018); Kawabata  _et al._ (2019),

$\displaystyle\nu_{\rm{p}}=\frac{1}{2\pi
i}\int_{0}^{2\pi}dk\frac{h_{\rm{p}}^{\prime}(k)}{h_{\rm{p}}(k)}=\frac{1}{2\pi
i}\int_{0}^{2\pi}dk\frac{d}{dk}\log h_{\rm{p}}(k).$ (6)

The winding number is an integer by counting the number of times
$h_{\rm{p}}(k)$ wrapping around the origin as $k$ varies from $0$ to $2\pi$.
The topological phase transition occurs at the location where the real and the
imaginary parts of $h_{\rm{p}}(k)$ are both zero for some $k$. If the coupling
range is much smaller than the system size ($\zeta\ll N$) and the detuning
$\Delta\omega$ is zero, the transition point is determined by the condition
$\kappa=\gamma+\Gamma-\frac{2\Gamma}{e^{1/\zeta}+1}$, which is consistent with
the amplification transition at $\zeta=\xi$ obtained from Eq. (3). However, in
the case of long-range coupling $\zeta\gg N$ and zero detuning, the
topological phase transition occurs at $\gamma={\kappa-\Gamma}$, which is
clearly different from the nonreciprocal amplification transition at
$\gamma=\kappa$ sup .

The breakdown of the correspondence between the topological phase transition
and the amplification transition is rooted in the long-range interaction. In
the limit of $\zeta\gg N$, as the bus is unidirectional, there are in total
$N(N-1)/2$ coupling terms in the dynamic matrix when discussing the input-
output relation under an open boundary condition (OBC). However, in the
analysis of topological properties of the spectra $h_{\rm p}(k)$ under PBC,
all pairs of sites are coupled and the number of coupling terms in
$M_{\rm{p}}$ is doubled. Thus the difference between OBC and PBC can no longer
be ignored in the thermodynamic limit. As a comparison, if the coupling is
short-ranged with $\zeta\ll N$, by changing from PBC to OBC one only needs to
disconnect an order of $\zeta$ couplings and induce negligible effect with
large enough $N$.

Figure 3: (a) Energy spectrum of the auxiliary Hamiltonian. A pair of zero-
energy edge states (orange solid line) emerge as $\gamma$ inreases beyond a
critical point (red dashed line). The unstable region is labeled by shaded
area. (b) The amplitude of the wavefunctions $|\psi_{0}|$ for the zero-energy
edge states with $\gamma=0.5$. In all plots, $N=40$, $\Gamma=1$,
$\kappa=0.25$, $\Delta\omega=0$ and $\zeta\gg N$.

From zero-energy edge states to nonreciprocal amplification.$-$ Due to the
drastic difference between models under OBC and PBC, we turn to the edge
properties of an open boundary system to seek further understanding of the
nonreciprocal amplification transition. To this aim, we define an auxiliary
Hamiltonian of the system by introducing a spin and extending the dynamic
matrix $M_{\rm{o}}$ to acknowledge the chiral symmetry Porras and Fernández-
Lorenzo (2019) as
$\mathcal{M}_{\rm{o}}=M_{\rm{o}}\otimes\sigma_{+}+M_{\rm{o}}^{\dagger}\otimes\sigma_{-},$
where $\sigma_{\pm}$ are the spin ladder operators.

The symmetry-protected zero-energy edge states of $\mathcal{M}_{\rm{o}}$
emerge when $\zeta>\xi$ [as shown in Fig. 3(a)], which coincides with the
nonreciprocal amplification transition. In fact, by directly diagonalizing
$\mathcal{M}_{\rm{o}}$ in the thermodynamic limit $N\to\infty$, one can obtain
an analytic result for the edge state wavefunctions
$|U_{lN}|=N_{0}e^{-(l-1)/{\xi^{\prime}}}$ and
$|V_{jN}|=N_{0}e^{-(N-j)/{\xi^{\prime}}}$ [see Fig. 3(b)], where $N_{0}$ is
the normalization coefficient and $\xi^{\prime}=1/(\xi^{-1}-\zeta^{-1})$ is
the localization length of edge states. The corresponding eigenvalues are
$\lambda_{\pm}=\pm\frac{e^{-1/\zeta}(\gamma+\Gamma-\kappa)^{2}}{4\Gamma}N_{0}^{2}e^{-N/\xi^{\prime}}$,
which go to 0 in the thermodynamic limit (see Supplementary Material sup ).
The scattering matrix can therefore be approximated as

$\displaystyle|S_{jl}|$ $\displaystyle\approx$
$\displaystyle\kappa|V_{jN}|\lambda_{+}^{-1}|U_{lN}^{*}|$ (7) $\displaystyle=$
$\displaystyle\frac{4\kappa\Gamma}{|\gamma-\Gamma-\kappa||\gamma+\Gamma-\kappa|}e^{(j-l)/\xi^{\prime}},$

which is consistent with that obtained from the input-output theory as given
in Eq. (3) for $j>l$. We emphasize that such correspondence between the
emergence of symmetry-protected zero-energy edge states and the nonreciprocal
amplification transition is valid for the general case with an arbitrary
coupling range.

In Fig. 4(a), we present the gain $|S_{N1}|^{2}$ from the first cavity to the
last by changing the pumping rate $\gamma$ and the coupling range $\zeta$.
Three regions can be clearly identified, i.e., attenuation (left),
amplification (middle), and unstable (right) regions. The transition between
attenuation and amplification (black dashed line) coincides exactly with the
emergence of symmetry-protected zero-energy edge states (blue circles) for all
parameters. As a comparison, the topological phase transition of the dynamic
matrix $M_{\rm p}$ under PBC (red dashed line) is very close to the
amplification transition only for short-range coupling ($\zeta\ll N$), where
the boundary effect is negligible Porras and Fernández-Lorenzo (2019); Wanjura
_et al._ (2020); Schnyder  _et al._ (2008); Ryu  _et al._ (2010); Asbóth  _et
al._ (2016); Chiu  _et al._ (2016). With increasing $\zeta$, the system
gradually becomes a long-range coupling model and the correspondence between
topological phase transition and nonreciprocal amplification transition breaks
down. As shown in Fig. 4(a), the topological phase transition starts to
deviate from the amplification transition when $\zeta$ approaches a
significant fraction of $N$, and becomes far off in the long-range coupling
limit.

Figure 4: (a) The gain $|S_{N1}|^{2}$ as a function of the pumping rate
$\gamma$ and the localization length $\zeta$. The red dashed curve labeled by
‘TPT’ corresponds to the topological phase transition under PBC, which agrees
well with the nonreciprocal amplification transition (black dashed curve) when
$\zeta\ll N$. When $\zeta$ increases, the TPT starts to deviate from
amplification transition. The emergence of zero-energy edge states (blue
circles) always coincides with the amplification transition for all the range
of $\zeta$. The white dashed curve indicates the critical pumping beyond which
the system becomes unstable. (b) The gain $|S_{N1}|^{2}$ as a function of
$\gamma$ and the frequency detuning $\Delta\omega$ for the long-range coupling
limit $\zeta\gg N$. The black dashed curve corresponds to nonreciprocal
amplification transition. In all plots, we take $N=40$, $\Gamma=1$, and
$\kappa=e^{-1/\zeta}/(1+e^{-1/\zeta})^{2}$.

Furthermore, we calculate the gain $|S_{N1}|^{2}$ for nonzero detuning to
manifest the bandwidth of amplification. We plot in Fig. 4(b) the gain
$|S_{N1}|^{2}$ as a function of frequency detuning $\Delta\omega$ and the
pumping rate $\gamma$, and conclude that the amplification bandwidth decreases
with increasing $\gamma$, which is determined by Eq. (3). Finally, we discuss
possible experimental realizations of the proposed model. Recently,
nonreciprocal transport and amplification have been observed by optomechanical
interactions Shen  _et al._ (2016); Fang  _et al._ (2017). The whispering
gallery modes in microresonators can be chirally coupled with forward
propagating wave in tapered fibre, which is also a feasible platform to
simulate the model described by Eq. (Nonreciprocal Amplification Transition in
a Driven-Dissipative Quantum Network). Besides, the nonreciprocal
amplification can also be realized in superconducting systems O’Brien  _et
al._ (2014); Macklin  _et al._ (2015).

Conclusion.$-$ We have studied the amplification transition in a quantum
network of cavities coupled to a nonreciprocal bus with unidirectional
transmission. When the coherence length of the bus is comparable to or even
exceeds the system size, an effective long-range coupling between the cavities
can be established. We have derived the input-output relation for this system,
and obtained an analytical condition for the nonreciprocal amplification
transition, which, however, cannot be directly linked to the topological phase
transition associated with the spectrum of the dynamic matrix under a periodic
boundary condition. In the long-range coupling limit, the topological phase
transition drastically deviates from the amplification transition, owing to
the significant boundary effect. To fill this gap, we work on the system with
open boundary condition and establish an exact connection between the
nonreciprocal amplification transition and the emergence of symmetry-protected
zero-energy edge states for arbitrary coupling range. Our work can be applied
to signal transmission and amplification, and also paves the way to study the
long-range coupling and nontrivial boundary effects in non-Hermitian systems.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grants No. 11974031, 11975026, 12125402, 12147148, and 12074428), Beijing
Natural Science Foundation (Grant No. Z190005 and No. Z180013), the Key R$\&$D
Program of Guangdong Province (Grant No. 2018B030329001 and No.
2018YFA0306501). F. S. acknowledges the China Postdoctoral Science Foundation
(Grant No. 2020M680186).

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# Augmented Reality Remote Operation of Dual
Arm Manipulators in Hot Boxes

Frank Regal1, Young Soo Park2, Jerry Nolan2, and Mitch Pryor1 1The University
of Texas at Austin, Austin, TX 78712, USA<EMAIL_ADDRESS>National
Laboratory, Lemont, IL 60439, USA<EMAIL_ADDRESS>
##  I INTRODUCTION

In nuclear isotope and chemistry laboratories, hot cells and gloveboxes
provide scientists with a controlled and safe environment to perform
experiments [1]. Working on experiments in these isolated containment cells
requires scientists to be physically present. For hot cell work, scientists
manipulate equipment and radioactive material inside through a bilateral
mechanical control mechanism. Motions produced outside the cell with the
master control levers are mechanically transferred to the internal grippers
inside the shielded containment cell. Similar to hot cells, gloveboxes provide
scientists with the capability to manipulate equipment inside, but instead,
the users can perform work with their hands by using sealed glove inserts on
the side of the chamber.

There is a growing need to have the capability to conduct experiments within
these cells remotely. During past pandemics, scientists often could not enter
facilities and had to restart their time-sensitive work. In other situations,
scientists’ home sites often do not have all equipment needed for the
experiment that other sites might have. Therefore, scientists need a solution
to perform physical experiments outside the laboratory.

Remote robotic control has long been a significant area of research. Robotic
teleoperated systems have assisted in natural disaster clean-up, hazardous
environment inspections, and nuclear facility decommissioning with success
[2], [3]. A simple method to enable remote manipulations within hot cell and
glovebox cells is to mount two robotic arms inside a box to mimic the motions
of human hands. An initial prototype, coined the hot box, has been built and
is shown in Figure 1. Capable manipulators can perform structured pre-defined
experiments autonomously without issue with the robotic arms programmed to fit
the experiment. Problems arise when the experimental workspace within the cell
changes. Enabling a robot to understand, plan, and act in an unstructured
ever-changing experimental environment in a fully autonomous manner is
difficult to repeat and not robust enough to be deployed for laboratory
applications where the task is not known before hand or may only be undertaken
a single time. Therefore, teleoperation of the robotic arms is required even
if automation is possible. Teleoperation will always be a necessary capability
for safety purposes. Scientists need to operate the dual arm, robotic
manipulators, in the hot box with intuitive and natural controls and be fully
situationally aware of their experiments.

Current solutions to perform remote teleoperation of robotic systems use
2-dimensional computer monitors with multiple windows, viewports, buttons,
joysticks, and keyboards that often overload untrained operators [4], [5].

Augmented Reality (AR) headsets, such as the Microsoft HoloLens 2, have been
used to connect human operators with robotic agents to communicate and command
mobile robotic platforms [6], [7]. [8] paired a HoloLens with an industrial
manipulator to allow users to visualize planned joint states and provide the
arm with goal poses via manipulation of a virtual cube. Manring et al. [9]
used a HoloLens to allow users to manipulate a virtual representation of a
robotic arm, visualize the planned motion in the AR environment, and then send
execution commands to the manipulator to proceed.

Building upon work from [8] and [9], our team at Argonne National Laboratory
developed an AR application for a HoloLens 2 headset that enables users to
teleoperate hot box dual arm manipulators by grasping robotic end-effectors of
digital replicas in AR from a remote location. In addition to the real-time
replica of the physical robotic arms in AR, the application also enabled users
to view a live video stream attached to the robotic arms and a parsed 3D point
cloud of 3D objects in their remote AR environment for better situational
awareness. It also provides users with virtual fixtures to assist in
manipulation and other teleoperation tasks.

Figure 1: [Left] New hot box design with two robotic arms mounted to the top
manipulating chemistry tools. [Right] First-person-view of a HoloLens 2 user
seeing both a virtual replica of the hot box atop the table and the real
physical hotbox on the right. Figure 2: Images captured from the perspective
of a HoloLens 2 user. [Left] User pinching the virtual end effector commanding
the physical arm to move. [Middle] Robotic arm interacting with a virtual
plane in AR used to assist in teleoperation. [Right] Virtual point cloud
visualization of the physical tube. Figure 3: System overview of how the
HoloLens 2 integrates with the robotic manipulators inside of the hot box.

##  II SYSTEM OVERVIEW

A Microsoft HoloLens 2, ROS packages, and Kinova Robotic’s Gen3 robotic arms
were combined to create a framework that allows remote operation via natural
controls. Unity game engine and ROS package scripts created the communication
pipeline required to communicate back and forth between the headset and
robotic arms. With the augmented reality application scene setup, a user can
manipulate the robotic arms with a pinch, using their thumb and index finger
on the virtual arm grippers. The HoloLens 2 tracks the pose of the hand
positioning. Those hand poses convert to serialized messages and are sent to
an Ubuntu machine over a wireless network acting as a control station server.
This server deserializes the messages and broadcasts them to specific scripts
running on the server and robotic arms that interpret the messages and
converts them to twist commands. When the robots move their position, the
HoloLens 2 receives those location messages of the robotic arms through the
reverse route, and the virtual arms within the HoloLens 2 send updates
accordingly. Through this configuration, a user has an experience that allows
them to see precisely the position of the robotic arms in the hot box from
remote locations and the ability to control them effortlessly.

###  II-A HoloLens 2

As shown in Figure 3, the Unity game engine was used to build the AR
application. The game engine application provides HoloLens 2 users with
virtual representations of the hot box and read the hand tracking data from
the HoloLens. The Unity application on the HoloLens 2 publishes the hand
positions and subscribes to the robotic arms’ joint positions on ROS topics
published on a RosBridge server. These ROS topic messages are standard
PoseStamped and JointState ROS messages for the hands and robot joint
positions, respectively. Before the Unity application sends the messages over
the wireless network to the server, the Unity application serializes the
messages into a standard JSON format. This JSON format allows for rapid
translation of data across a local router.

###  II-B Linux Machine

When the serialized ROS messages leave the HoloLens 2, a ROS package called
RosBridge, running on a Linux server, begins to receive these messages,
deserializes them, and broadcasts them locally to the ROS nodes on the Linux
machine. One specific node is the Kinova AR Teleop Package. This custom ROS
package takes the PoseStamped positions of the hands that the HoloLens
published and converts them to a linear and angular velocity that the robots
understand, called a Twist message. After converting to a Twist message, the
Kinova AR Teleop Package publishes the new Twist message to the Kinova ROS
Kortex package. This package is the low-level ROS package that sends the move
commands to the robots. The ROS Kortex takes the Twist messages, calls the
internal inverse kinematic solver, and sends a move command to the arms. Once
the arms move, the ROS Kortex package takes the continuously published
JointState information and publishes this back to the Kinova AR Teleop
Package. The messages are then packaged, serialized, and sent back to the
HoloLens 2 device across the network. The HoloLens 2 finally interprets the
message, deserializes, and then updates the virtual representation position of
the holographic Kinova arms in the Hololens 2 so that the user can see its new
position from a remote location.

##  III TEST IMPLEMENTATION

The application has been used to control both the Kinova dual arm and a
ReThink Robotics Baxter dual arm. Users can move the manipulators with a pinch
and grab, from a remote location. A live video stream, virtual fixtures, and
point cloud visualizations can be toggled within the interface. The virtual
robots can also scale in their scene for precise manipulations or confined
space arrangements.

##  IV CONCLUSION

Within this work, an AR application was created to allow scientists to
remotely operate robotic manipulators via hand gestures inside a hot box to
enable scientists to work remotely in a hot cell or glovebox environment. A
live video stream, 3D point cloud visualization, and virtual fixtures have
been tested within the AR environment as well. Future research needs to be
done to understand the best hand gesturing method to control the robotic arms’
position and orientation. Currently, users are using pinch and drag, but with
the HoloLens 2, much more sensor data about the human user is captured. There
could be more research done to understand if inputs like voice, eye gaze, and
head position could be used to improve the users’ remote operation experience.

Demonstration Video
$\centering\href https://youtu.be/aPauLc2_o00\@add@centering$

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