# Breaking reciprocity by designed loss

I. Peshko  B. I. Stepanov Institute of Physics, National Academy of Sciences
of Belarus, Nezavisimosti Ave. 68, Minsk 220072, Belarus  D. Pustakhod
Eindhoven Hendrik Casimir Institute (EHCI), Eindhoven University of
Technology, 5600 MB, Eindhoven, the Netherlands  D. Mogilevtsev  B. I.
Stepanov Institute of Physics, National Academy of Sciences of Belarus,
Nezavisimosti Ave. 68, Minsk 220072, Belarus

(April 2022)

###### Abstract

In this paper we show how designed loss in open quantum systems can break
reciprocity of the state propagation, and how the non-reciprocal and even
unidirectional propagation can be achieved for different kinds of designed
loss, both linear and nonlinear ones. In particular, we show how a
unidirectional propagation can be achieved for states of certain symmetry in
linear schemes, demonstrate possibility of building a single-mode optical
insulator by combining two kinds of nonlinear designed losses, and the way to
build non-reciprocal circulator with a planar structure of dissipatively
coupled waveguides. We discuss feasibility of the considered schemes and
suggest possible realizations.

##  I Introduction

Recently, non-reciprocal propagation of electromagnetic field became a popular
research theme. Non-reciprocity is important for quite a wide of practical
tasks, from field distributors and circulators to lasing and high-precision
sensing [1, 2, 3, 4, 5, 6]. Current interest to realization of non-reciprocal
systems, and especially isolators, is born of necessity to extent methods
commonly applied for longer wavelength (radio, microwaves, etc.) toward
optical region and possibility to integrate non-reciprocal systems into
photonic circuitry. Traditional realizations implementing magneto-sensitive
media are either hard to realize on the basis of existing integration
platforms, or difficult even to realize for optical wavelength, or both [4,
5]. So, a plethora of novel schemes for breaking reciprocity has appeared in
recent years. There are schemes exploiting time-modulation [7, 8],
nonlinearities [9, 10, 11] and interaction with few-level systems, such as
atom-like structures [12, 4, 13, 14], optomechanics [15], topological
properties of structured media [5, 16, 17].

Some quite interesting results were also obtained by considering loss.
Recently it was shown that even common single-photon energy losses in
conjunction with the usual (i.e., unitary) coupling can also be a tool for
devising non-reciprocal multi-mode structures [18, 19]. Even more interesting
results are obtained with designed loss, i.e., for example, when different
systems are coupled to the same loss reservoir inducing dissipative coupling
between them, or when loss is made to be nonlinear. For instance, designed
collective linear loss can lead to unidirectional propagation and
amplification [20, 21].

Here we present another interesting feature of designed loss: it is ability to
break reciprocity without conjoining with the simultaneous unitary coupling.
We discuss a general recipe for doing so with the designed loss allowing for
several stationary states, provide several examples and outline ways for
possible practical realizations, in particular, with integrable planar
waveguide systems. Notice that schemes relying only on the designed loss might
be easier to realize in practice that schemes combining unitary and
dissipative coupling, since one does not need adjusting different types of
coupling. We show how the designed nonlinear loss can break reciprocity even
for a single-mode system, and how to realize this system using conventional
Kerr nonlinearity and dissipative coupling. We also demonstrate the way to
achieve non-reciprocal optical circulation using the dissipative coupling.

The outline of the paper is as follows. Firstly, in the second Section we
recall a non-reciprocity criterion for linear systems and suggest a simple
generalization of the reciprocity concept suitable for quantum
state/correlation transfer, since nonlinear designed loss can produce non-
classical states from the classical input. Then, in the third Section we
describe the mechanics of the reciprocity breaking for the dissipatively
coupled systems. In the fourth and fifth Sections we discuss examples of
single-mode reciprocity breaking and unidirectional propagation with nonlinear
designed loss in asymptotic and non-asymptotic regimes, and address practical
realization of such schemes. We argue that such schemes are quite feasible
with existing photonic integration platforms and with such common modeling and
simulation tools as systems of laser-written waveguides in balk dielectric. In
sixth Section we show how such common type of loss as dephasing can also break
reciprocity, and demonstrate reciprocity breaking in correlations propagation
for a simple device of a beam-splitter and dephaser. In seventh Section we
discuss a non-reciprocal optical circulator built only with dissipatively
coupled waveguides. In the Appendix we discuss several designs of integrable
dissipatively coupled structures.

##  II Reciprocity

First of all, let us clarify the concept of reciprocity and its breaking as
applicable for the designed loss structures. We are considering here both
linear and nonlinear structures, and the latter ones are known to produce non-
classical states from classical ones [22, 23, 24, 25]. So, it is necessary to
provide for a suitable generalization of the reciprocity concept for the open
quantum systems. Here we will not pursue the task of discussing the general
reciprocity concept for the open quantum systems in the manner as it was done
for unitary systems, for example, in the works [26, 27]. Here we introduce a
simple generalization of the classical definition [4, 28, 29, 26, 27]). We
formulate the classical definition using quantum operators in the following
way [30, 31]. Let us define a set of $N$ input modes described by the bosonic
annihilation operators $\vec{a}=[a_{1},a_{2},\ldots a_{N}]$ (and the
corresponding creation operators $a_{n}^{\dagger}$), and a set of $N$ output
modes described by the bosonic annihilation operators
$\vec{b}=[b_{1},b_{2},\ldots b_{N}]$ (and the corresponding creation operators
$b_{n}^{\dagger}$). For the linear device transforming input into the output,
one can write [30]:

$\vec{b}=\mathbf{S}\vec{a}+\vec{N},$ (1)

where the vector of operators $\vec{N}$ describes noise. For passive systems
with $\langle\vec{N}\rangle=0$, one gets

$\langle\vec{b}\rangle=\mathbf{S}\langle\vec{a}\rangle,$ (2)

where the expression $\langle x\rangle=Tr\\{x\rho\\}$ denotes the quantum
mechanical average of the operator $x$ over the density matrix $\rho$
describing the quantum state of the whole system. For the coherent modal
states, $\langle\vec{a}\rangle$, $\langle\vec{b}\rangle$ are the modal
amplitudes. Thus one arrives to the well-know classical description of
reciprocity for the system of electromagnetic field modes. Reciprocity is
directly associated with the features of the scattering matrix $\mathbf{S}$.
Non-symmetrical scattering matrix, $\mathbf{S}\neq\mathbf{S}^{T}$ means that
the system breaks reciprocity [4, 28, 29, 26, 27]. Notice that symmetry should
be considered taking into account arbitrariness of the modal phase, i.e.
possibility to make all the elements of the scattering matrix real by the
local gauge transformation

$x\rightarrow x\exp\\{i\phi_{x}\\},$ (3)

$x=a,b$ with some phases $\phi_{x}$ [18].

Generally, Eq.(1) allows extending the description of reciprocity for the
quantum case. For example, one can derive the transfer equation similar to
Eq.(2) also for the correlation function of different orders. For nonlinear
transformations Eq.(1) can hardly be applied. However, one can return to
linearity just by considering transformation of the density matrix elements
instead of modal operators. Let us proceed discussing reciprocity in open
quantum systems in such a way as it was done for unitary systems in Refs. [26,
27].

Let us assume that our system of interest is described by the density matrix
$\rho_{in}$, and we have two propagators describing propagation in different
directions, $\overleftarrow{\mathcal{D}}$ and $\overrightarrow{\mathcal{D}}$.
The state propagation corresponding to these directions is described by the
mapping

$\rho_{in}\rightarrow\overrightarrow{\rho}_{out}=\overrightarrow{\mathcal{D}}\rho_{in},\quad\rho_{in}\rightarrow\overleftarrow{\rho}_{out}=\overleftarrow{\mathcal{D}}\rho_{in}.$
(4)

Here we are not interested in the complete description of the transferred
density matrix but rather in dynamics of certain observable quantities, for
example, energy. If for the same input state of our passive system the output
energy for different propagation directions is different, it is a sufficient
condition of the reciprocity breaking. So, let us introduce a set of Hermitian
operators $P_{j}$ describing the observables of interest, so that quantities
${p}_{j}=Tr\\{{P}_{j}\rho_{in}\\}$,
$\overrightarrow{p}_{j}=Tr\\{{P}_{j}\overrightarrow{\rho}_{out}\\}$
($\overleftarrow{p}_{j}=Tr\\{{P}_{j}\overleftarrow{\rho}_{out}\\}$) can be
measured at input and output of our device described by the map
$\overrightarrow{\mathcal{D}}$ ($\overleftarrow{\mathcal{D}}$). For the
reciprocal transmission one would have

$\overleftarrow{p}_{j}=\overrightarrow{p}_{j},\quad\forall j.$ (5)

Thus, breaking of the equality (5) would mean non-reciprocity. Notice that
this simple equation implies a number of important observations. Firstly, the
condition (5) is state dependent. So, one needs defining a class of initial
density matrices, $\\{\rho_{in}\\}$ for discussing reciprocity. Secondly,
reciprocity is defined for a particular set of $\\{P_{j}\\}$. This set might
be complete and sufficient for description/inference of the density matrix, or
not. Different sets of $\\{P_{j}\\}$ might demonstrate both reciprocity and
breaking of it for the same system. Only proving reciprocity for arbitrary
initial states and observables allows calling the system completely
reciprocal.

Figure 1: (a) A scheme of the dissipative coupling of two waveguides A and B
through the common loss reservoir represented as a lossy waveguide C. (b) An
illustration of the modal amplitude dynamics for the direct and reverse
propagation through the two waveguides with different dissipative coupling in
the first part (denoted as $L_{1}$ in the panel (c)) and in the second part
(denoted as $L_{2}$ in the panel (c)). Solid and dotted curves correspond to
the field propagation in the waveguides A and B in the direction $1\rightarrow
2$, the dashed and dash-dotted lines correspond to the field propagation in
the waveguides A and B in the direction $2\rightarrow 1$. For the first device
part the Lindblad operator is $L_{1}=\gamma(a-0.5b)$, for the second part the
Lindblad operator is $L_{2}=\gamma(a-b)$. The initial amplitude for the mode
$a$ is 1, for the mode $b$ is -1. Panel (d) depicts a scheme of the integrable
device combining two different kinds of dissipative coupling by variable
placing of a central lossy waveguide between two low-loss waveguides (see
discussion in Appendix).

##  III Breaking reciprocity by linear dissipative coupling

First of all, let us demonstrate how a purely dissipative device breaks
reciprocity. We consider a set of dissipatively coupled waveguides without an
additional unitary coupling between them. Such a coupling can be realized by
usual unitary coupling of several waveguides to the common dissipative
reservoir, say, the waveguide with strong loss (as depicted in Fig. 1(a)).
Such an arrangement can be easily realized in practice, for example, by laser-
writing a systems of single-mode waveguides in a balk glass [32, 24, 33, 25,
34], or growing planar semiconductor structures on some platform, for example,
InGaAsP material on InP [6, 35] (see the Appendix for discussion of several
designs of dissipative waveguide couplers on the integrated photonic
platform).

To demonstrate a breaking of reciprocity, let us consider a bipartite two-
waveguides system as schematically shown in Fig. 1(c). During propagation
through the $j$-th part of the system, the density matrix $\rho$ dynamics is
described by the following master equation

$\frac{d}{dt}\rho=\gamma_{j}\left(2L_{j}\rho L_{j}^{\dagger}-\rho
L_{j}^{\dagger}L_{j}-L_{j}^{\dagger}L_{j}\rho\right),$ (6)

where the Lindblad operators for the $j$-th part are taken as
$L_{j}=a+\delta_{j}b$, $a$ and $b$ being the annihilation operators for the
modes A and B, $\delta_{j}$ are asymmetry parameters describing the respective
strength of waveguides coupling to the common reservoir; $\gamma_{j}$ are
decay rates corresponding to each part. Taking into account a possibility of
the local gauge transformation (3), one can assume real $\delta_{j}$ (this we
also assume in the further considerations). In Fig. 1(b) one can see an
illustration of the reciprocity breaking for the bipartite systems with
$L_{1}=\gamma(a-0.5b)$ and $L_{2}=\gamma(a-b)$. For the asymmetric input state
of both modes (i.e. when the sum of the amplitudes is zero), one has a
unidirectional energy flow through the system. For such an input, the action
of the second part of the device just exponentially reduces the state to the
vacuum.

Such behavior can be easily understood from the character of dynamics dictated
by the master equation (6). The dissipative coupling between modes in $j$-th
part of the system drives their state to the one satisfying $L_{j}\rho=0$ [24,
34]. Asymptotically (i.e. for the interaction time tending to infinity), the
modal amplitude transformation can be described by the following scattering
matrix

$\begin{bmatrix}\langle a_{out}\rangle\\\ \langle
b_{out}\rangle\end{bmatrix}=\mathbf{S}_{j}\begin{bmatrix}\langle
a_{in}\rangle\\\ \langle
b_{in}\rangle\end{bmatrix},\quad\mathbf{S}_{j}=\frac{1}{1+\delta_{j}^{2}}\begin{bmatrix}\delta_{j}^{2}&-\delta_{j}\\\
-\delta_{j}&1\end{bmatrix}.$ (7)

Generally, for $\delta_{1}\neq\delta_{2}$ and $\delta_{1}\delta_{2}\neq-1$,
the product of the matrices $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ is not
symmetric

$\mathbf{S}_{1}\mathbf{S}_{2}=\frac{1+\delta_{1}\delta_{2}}{(1+\delta_{1}^{2})(1+\delta_{2}^{2})}\begin{bmatrix}\delta_{1}\delta_{2}&-\delta_{1}\\\
-\delta_{2}&1\end{bmatrix}.$ (8)

So, our bipartite device is indeed not reciprocal. Moreover, it does not
allows the states satisfying $\delta_{1}\langle a_{in}\rangle-\langle
b_{in}\rangle=0$ to propagate in the direction $1\rightarrow 2$, and it does
not allows the states satisfying $\delta_{2}\langle a_{in}\rangle-\langle
b_{in}\rangle=0$ to propagate in the opposite direction. Notice that different
dissipative coupling in the realization schematically depicted in Fig. 1(a)
can be simply realized by adjusting coupling with the common reservoirs, for
example, by adjusting distances between the waveguides A and B and the central
dissipative waveguide as shown in Fig. 1(d) [25, 34].

Interestingly, that a simple linear device discussed here can offer quite
considerable asymmetry in energy exchange between waveguides for different
propagation directions. Indeed, in is easy to get from Eq.(8) that the ratio
of the off-diagonal elements of the matrix products

$\left|\frac{[\mathbf{S}_{1}\mathbf{S}_{2}]_{12}}{[\mathbf{S}_{2}\mathbf{S}_{1}]_{12}}\right|=\left|\frac{\delta_{1}}{\delta_{2}}\right|$
(9)

is proportional to the ratio of the asymmetry parameters. As long as the value
of $\delta_{j}\gamma$ remains much larger than the rate of uncorrelated
single-photon loss inevitably present in a realistic waveguide system, one can
get $\left|\frac{\delta_{1}}{\delta_{2}}\right|^{2}$ difference in energy
transmission between waveguides in different directions. For example, for
planar systems described in Appendix it seems realistic to reach the ratio of
0.1 between waveguide coupling rates in the scheme depicted in Fig. 1(d), thus
to have $\left|\frac{\delta_{1}}{\delta_{2}}\right|^{2}=0.01$ and 20 dB in
energy exchange asymmetry.

Naturally, our discussion can be easily extended for more involved
dissipatively coupled system of modes. The main message would be the same:
conjunction of parts with different dissipative coupling leads to breaking of
reciprocity.

Also, it is useful to note that breaking reciprocity can be achieved with only
one of two parts being a dissipative coupler. Indeed, replacing, for example,
the first part with the beam-splitter transforming some set of input states to
the ones satisfying $\delta_{2}\langle a_{in}\rangle-\langle b_{in}\rangle=0$,
one gets unidirectional propagation of energy for this set of states (only for
$2\rightarrow 1$). Also, to break reciprocity one can simply introduce
different uncorrelated loss in both modes of the first part. Indeed, for the
uncorrelated loss in both waveguides $A$ and $B$ (taken to be finite-length)
the scattering matrix is $\mathbf{S}_{1}=\mathrm{diag}\\{T_{A},T_{B}\\}$,
where $T_{A}$, $T_{B}$ are transmission coefficients for the modes $A$ and
$B$. We assume the asymptotic regime for the second, dissipative coupler part.
So, for the transmission through the bipartite compound device one has the
following scattering matrix

$\mathbf{S}_{2}\mathbf{S}_{1}=\frac{1}{1+\delta_{2}^{2}}\begin{bmatrix}T_{A}\delta_{2}&-T_{B}\delta_{2}\\\
-\delta_{2}T_{A}&T_{B}\end{bmatrix}.$ (10)

The product (10) is asymmetric. The asymmetry in energy exchange between
waveguides then is defined by the ratio $(T_{A}/T_{B})^{2}$.

Figure 2: Dynamics of the population for the single-mode bipartite system with
parts described by the Lindblad operators (13) with $m_{1}=4$ and $m_{2}=2$ as
given by the master equation (6) for the initial coherent state with 5 photons
and the same rate $\gamma$. Solid lines both in the main panel and in the
upper inset correspond to the propagation $1\rightarrow 2$; dashed lines both
in the main panel and the inset correspond to the propagation $2\rightarrow
1$. For the main panel propagation through each part occurs during the
interval $3\gamma t$; for the upper panel propagation through each part occurs
during the interval $0.5\gamma t$. Lower panel shows the photon number
distribution of the output state for the propagation $1\rightarrow 2$ depicted
in the main panel.

##  IV Breaking reciprocity with nonlinear dissipation

The obvious idea of breaking commutativity of dynamics in different parts of a
compound system to break reciprocity can be easily extended for the general
dissipative dynamics. Generally, the state transformation of the open system
can be described in a standard way with help of the Kraus operators $D_{j}$ as

$\rho_{out}=\sum\limits_{\forall j}D_{j}\rho_{in}D^{\dagger}_{j},$ (11)

where the Kraus operators satisfy $\sum\limits_{\forall
j}D_{j}^{\dagger}D_{j}=\mathds{1}$ [36].

It is easy to surmise that for our bipartite system as depicted in Fig. 1(c),
non-commutativity of the Kraus operators $D_{j}^{(1)}$ and $D_{k}^{(2)}$
corresponding to the first and the second parts of the system, can also lead
to breaking of reciprocity. Now let us show that the dissipative systems
possessing several stationary states (similarly to the linear case described
above) can also break reciprocity. Moreover, in stark difference with the
linear case, such system can provide for a single-mode non-reciprocity.

Indeed, let us consider a simple example of single-mode subsystems 1 and 2
with just two stationary states each. These systems are described by the
following Kraus operators

$\displaystyle D_{m}^{(1,2)}=\begin{cases}|m_{1,2}\rangle\langle m|&\forall
m\geq m_{1,2}\\\ |0\rangle\langle m|&\forall m<m_{1,2},\end{cases}$ (12)

where the states $|m\rangle$ are Fock states with $m$ photons of the single
considered mode, and $m_{1,2}\neq 0$. For $m_{1}\neq m_{2}$ some Kraus
operators corresponding to different parts are obviously not commuting, so,
the state propagation is not reciprocal. Moreover, combination of the
subsystems 1 and 2 realizes a single-mode optical insulator. Indeed, for
$m_{1}>m_{2}$, any Fock-state input with $m>m_{1}$ produces Fock-state with
$m_{2}$ photons when propagating $1\rightarrow 2$. Whereas any state input
from the opposite side (i.e., when $2\rightarrow 1$) produces just the vacuum
state.

Despite its seemingly idealized character, the scheme with the Kraus operators
(12) can be realized, for example, with nonlinear coherent loss described by
the Lindblad operators

$L_{j}=a(a^{\dagger}a-m_{j}),\quad j=1,2,$ (13)

$m$ being a positive integer number [37].

It is interesting that the regime of optical isolation with the device
described by the Lindblad operators (13) functions also with classical inputs.
An example of such functioning is shown in Fig. 2. The main panel of Fig. 2
shows how the average number of photons behaves for the input at the part with
$m_{1}=4$ (solid line) and output at the part with $m_{2}=2$. Dashed lines
shows dynamics of the average number of photons for propagation in the
opposite direction. Low inset shows the output state for the propagation
$1\rightarrow 2$ is non-classical being a mixture of the vacuum with the two-
photon Fock state.

Another interesting feature with the scheme (13) is that non-reciprocity can
be quite pronounced even far from asymptotic regime, i.e., when the state of
the mode is far form the one given by the condition $L_{j}\rho=0$. This
situation is illustrated in the upper panel in Fig. 2. Actually, this feature
makes it feasible looking for practical realization of non-reciprocity with
designed nonlinear loss of the considered kind. Indeed, close to the
stationary state satisfying $L_{j}\rho=0$, the dynamics resulted from non-
linear loss is rather slow and can be easily disrupted by common single-photon
loss unavoidable in realistic systems [24, 25].

Figure 3: (a) The scheme of the single-mode bipartite non-reciprocal device
described by Eq. (14). The panel (b) shows dynamics of the average number of
photons for the structure schematically depicted in the panel (a). The solid
line corresponds to propagation $1\rightarrow 2$, the dashed line corresponds
to the propagation in the opposite direction. For the Part 1 the dynamics is
described by the master equation (14) with decay rates
$\gamma_{0}=\gamma_{1}=\gamma$, $\gamma_{2}=0$. For the Part 2 the decay rates
are $\gamma_{0,1,2}=\gamma$. The unitary transformation between the two parts
is supposed to leave the collective mode state unchanged. The initial state at
each input is taken as the coherent state of the corresponding ”stationary”
collective mode with the amplitude $\alpha=20$. The inset shows the output
state photon number distribution, $\rho_{nn}$, for the propagation
$1\rightarrow 2$ (gray thick bars) and for the propagation in the opposite
direction, $2\rightarrow 1$ (thin black bars). (c) An example of the bipartite
nonlinear loss scheme for realizing non-asymptotic breaking of reciprocity by
single-mode nonlinear loss. Two self-Kerr nonlinear waveguides are coupled to
the third waveguide, but not to each other. The third waveguide is designed to
have excess loss. In the part 1 the third waveguide is coupled symmetrically
to the waveguides A and B realizing the two-photon loss. In the part 2 the
coupling is asymmetric realizing both the two-photon loss and the nonlinear
coherent loss (NCL). We assume a unitary coupling between the parts 1 and 2
allowing for lossless transformation of a collective mode $\tilde{A_{1}}$ to
the collective mode $\tilde{A_{2}}$ and vice versa.

##  V Toward practical non-reciprocity with nonlinear dissipation

As it was already mentioned here, practical realization of the dissipative
coupling between waveguides is rather simple to achieve, for example, by
laser-writing single-mode waveguides in glass [32, 24, 33, 25, 34], or
fabricating planar structures as described in the Appendix (and one might
mention that waveguide-writing in borosilicate glass has become a very
versatile photonic instrument for modeling a plethora of exotic physical
effects (see, for example, Refs.[34, 38])). Designing nonlinear loss is more
difficult and involved. Some schemes were suggested and experimentally
demonstrated with trapped ions and atoms, nanomechanical resonators,
superconducting microwave circuits, nonlinear Kerr resonators, etc. [39, 40,
41, 42, 43, 44]. To demonstrate the way of experimental realization of
reciprocity breaking purely by designed nonlinear loss, here we consider a
recently suggested scheme with self-Kerr nonlinear waveguides that implements
the dissipative coupling scheme depicted in Fig. 1(a) [24, 25, 45]. This
scheme allows to produce two-photon loss or nonlinear coherent loss of the
kind given by Eq.(13) for $m_{j}=1$ combined with two-photon loss for the
essentially single-mode device.

Dynamics of the single-mode system considered in Refs. [24, 25] is described
by the following master equation

$\displaystyle\frac{d}{dt}\rho=\sum\limits_{j=0}^{2}\gamma_{j}\Bigl{(}2L_{j}\rho
L_{j}^{\dagger}-\rho L_{j}^{\dagger}L_{j}-L_{j}^{\dagger}L_{j}\rho\Bigr{)},$
(14)

where $L_{0}=a$ describes common single-photon loss; $L_{1}=a^{2}$ describes
two-photon loss and $L_{2}=a(a^{\dagger}a-1)$ describes so called nonlinear
coherent loss (NCL) [37]; $\gamma_{0,1,2}$ are the corresponding loss rates.

Firstly, let us show that the bipartite system composed of the parts described
by the master equation (14) with different sets of decay rates $\gamma_{j}$
can lead to asymptotic breaking of reciprocity. To that end we assume that for
the first part we have $\gamma_{0,2}=0$, $\gamma_{1}=\gamma$ and for the
second part we have $\gamma_{0,1}=0$, $\gamma_{2}=\gamma$ in Eq.(14). I.e., we
have only two-photon loss in the first part and only the NCL in the second
part of our device. Two-photon loss does not affect states in the subspace of
single and zero photon states. Asymptotically, two-photon loss drives an
initial state with the number of photon much larger than unity toward a mixed
single-photon state with one-half photons on average [46, 23]. The NCL
described by $L_{2}$ also does not affect states in the subspace of single and
zero photon states. But asymptotically it drives an initial state with the
number of photon much larger than unity toward the single-photon state [24,
37]. So, for the initial coherent state with the average number of photons
much larger than unity, the bipartite system will produce asymptotically the
state with $1/2$ photon on average for the $1\rightarrow 2$ propagation, and
the single-photon state for propagation in the opposite direction.

Fig. 3(b) shows that reciprocity is broken for such a system also for a finite
time even in the regime when the systems’ state is far from the stationary
state. For the simulation we have taken that the system is also subject to the
common single-photon loss, and that the two-photon loss is present also in the
second part of the device (which corresponds to the realistic three-waveguide
scheme [25, 45]). So, for the first part we have taken
$\gamma_{0}=\gamma_{1}=\gamma$, $\gamma_{2}=0$; for the second part we have
taken $\gamma_{j}=\gamma$, $\forall j$. The initial state is taken to be the
coherent one with the amplitude $20$. One can see from Fig. 3(b) that the
average number of photons at outputs is different, and the photon number
distributions of the output states (shown in the inset in (b)) are also quite
different. Both output states are sub-Poissonian, but for the $1\rightarrow 2$
propagation the state is considerable more photon-number squeezed, with the
Mandel parameter $Q\approx-0.7$. Whereas in the opposite direction,
$Q\approx-0.5$.

The single-mode device analyzed above can be realized with the waveguide
arrangement schematically shown in Fig. 3(c). The device consists of three
parts: the part 1 and part 2 realize different kinds of nonlinear loss by the
dissipative coupling scheme shown in Fig. 1(a). The symmetric coupling between
the side self-Kerr nonlinear waveguides and the lossy central one allows to
realize the two-photon loss of the collective mode $A_{1}\propto a_{1}-b_{1}$,
where $a_{1}$, $b_{1}$ are the modes guided in corresponding waveguides. The
mode $A_{1}$ would be a stationary mode of the part 1 in absence of
nonlinearity and uncorrelated single-photon loss. In the part 2 the coupling
is asymmetric, so, NCL arises described by the Lindblad operator $L_{2}$ in
addition to the two-photon loss for the collective mode $A_{2}\propto\delta
a_{2}-b_{2}$, the parameter $\delta$ describing the coupling asymmetry [24,
37]. Notice that for initial states with large number of photons, on the
initial stage of dynamics (i.e., when the average number of photons still
remains large) NCL dominates over two-photon loss making its influence on the
photon number dynamics negligible even though the loss rates are close in
value [25]). The third part connecting parts 1 and 2 in Fig. 3(a) provides for
the effectively single-mode operation of the device transforming the
collective mode $A_{1}$ into $A_{2}$ and vice versa. Initially one should also
excite only $A_{1}$ for the propagation $1\rightarrow 2$, and $A_{2}$ from the
propagation in the opposite direction.

Notice that for demonstrating non-reciprocity one might simplify the set-up of
Fig. 3(c) getting rid of the central unitary coupled part, i.e., leaving only
two conjoined nonlinear dissipative couplers as in Fig. 1(d) . But the single-
modality of the operation would be lost, and both linear (considered in
Section 3) and nonlinear effects of the reciprocity breaking would be
combined.

##  VI Breaking reciprocity by dephasing

We have already shown with examples of both linear and nonlinear dissipative
dynamics that dissipative systems possessing several stationary states can
break reciprocity. Now let us consider another example of quite simple and
widespread (actually, even virtually ubiquitous) kind of dissipative dynamics
with several (and possibly even infinite number of) stationary states that can
also break reciprocity. Also, this reciprocity violation cannot be captured by
a simple classical criterion implementing the amplitude scattering matrix (7).

This kind of dissipation is dephasing [47]. The essence of it is as follows.
The dephasing with respect to some orthogonal basis $\\{|\psi_{j}\rangle\\}$
(i.e, $\langle\psi_{k}|\psi_{j}\rangle=\delta_{kj}$) means that asymptotically
any state is transformed to the state with only the diagonal components in
this basis

$\rho=\sum\limits_{\forall
j,k}c_{jk}|\psi_{j}\rangle\langle\psi_{k}|\rightarrow\bar{\rho}=\sum\limits_{\forall
j}c_{jj}|\psi_{j}\rangle\langle\psi_{j}|,$ (15)

where the coefficients $c_{jk}$ describe the representation of the state
density matrix in the basis $\\{|\psi_{j}\rangle\\}$. The transformation (15)
is realized by the Kraus operators $D_{j}=|\psi_{j}\rangle\langle\psi_{j}|$,
and any state commuting with all these Kraus operators is stationary.

It is easy to see that a bipartite system with each part performing dephasing
with respect to different basis breaks reciprocity. Indeed, let us take that
the $j$-th part dephases the input state with respect to the basis
$\\{|\psi_{k}^{(j)}\rangle\\}$, and at least for some $j$ and $k$ one has
$\langle\psi_{k}^{(1)}|\psi_{j}^{(2)}\rangle\neq 0$. Than the corresponding
Kraus operators $D^{(1)_{k}}$ and $D^{(2)_{j}}$ do not commute and the
dynamics is not reciprocal.

Physically, realization of a single mode dephasing in different bases is not
simple and involves state transformation corresponding to transition to the
different basis. However, reciprocity breaking can be realized conjoining
unitary and feasible dephasing parts. Dephasing in the energy basis is quite
simply realized by random fluctuation of frequency, or, equivalently, by
inducing random fluctuation of modal phase [47]. Let us show how it is
possible to break reciprocity in propagation of correlations by a bipartite
system consisting of a simple 50/50 beam-splitter and the energy dephaser.

We take a two-mode system as shown in Fig. 1(c). The part 1 is a beam-splitter
performing the following standard transformation

$a_{out}=\frac{1}{\sqrt{2}}(a_{in}+ib_{in}),\quad
b_{out}=\frac{1}{\sqrt{2}}(ia_{in}+b_{in}).$ (16)

The part 2 is the dephaser in the energy basis. Let us consider the following
pure input state

$\rho_{in}=|\psi\rangle\langle\psi|,\quad|\psi\rangle=x|1_{a}\rangle|0_{b}\rangle+\sqrt{1-x^{2}}|0_{a}\rangle|1_{b}\rangle,$
(17)

where the vector $|n_{x}\rangle$ is the Fock state of the mode $x$ with
$n_{x}$ photons, and a real $x$ satisfies $|x|\leq 1$.

Now as it was discussed in Section 2, let us consider an observable $P=\lambda
a^{\dagger}b+\lambda^{*}b^{\dagger}a$, where $\lambda$ is a scalar parameter.
It is easy to see that the value of this observable for the initial state (17)
is

$p=\mathrm{Tr}\\{P\rho_{in}\\}=(\lambda+\lambda^{*})x\sqrt{1-x^{2}}.$

For the propagation $1\rightarrow 2$ we have from Eqs.(16,17)
$\overrightarrow{p}=0$. For the propagation $2\rightarrow 1$ we have

$\overleftarrow{p}=\frac{i}{2}(\lambda-\lambda^{*})(1-2x^{2}).$

So, according to the definition given by Eqs.(4,5), for
$\lambda\neq\lambda^{*}$ and $|x|\neq 1/\sqrt{2}$ our device is indeed
breaking reciprocity. In our case, modal correlations are non-reciprocally
transmitted through our bipartite device. Notice that the classical definition
(2) cannot work for the case. For the initial state (17) modal amplitudes are
zero.

Figure 4: (a) A scheme of bipartite three-mode dissipatively coupled rotator.
We consider only the propagation $1\rightarrow 2$. Panels (b-d) shows dynamics
of modal amplitudes as given by Eqs. (18,19) with decay rates
$\gamma_{x,j}=\gamma$, $\forall x,j$, and asymmetry parameters
$y_{1}=-z_{2}=-0.9$, $y_{2}=-z_{1}=0.1$. The border between the parts
corresponds to $\gamma t=5$. In the panels (b-d) solid, dashed and dash-dotted
lines correspond to the modal amplitudes of the modes in the waveguides
$x=A,B,C$. The panels (b,c,d) correspond to the initial coherent excitation
with the unit amplitude of the modes of the waveguides A,B,C with vacuum of
other two modes.

##  VII Asymmetric dissipatively coupled circulator

Finally, let us demonstrate how purely dissipative coupling might serve for
building an asymmetric optical ”circulator”, i.e. a device that asymmetrically
transfers modal energy between the modes in dependence on the initial
excitation. We consider a bipartite structure with each part having three
physical modes A, B, C, where dissipative coupling is only between the modes
A,B and B,C (which makes the structure feasible to fabricate using planar
arrangement of strongly coupled low-loss waveguides A, B, C and high-loss
waveguides D as shown in Fig. 4(a); such a structure is a simple
generalization of ones discussed in the Appendix).

Each part of our bipartite set-up is described by the following master
equation ($j=1,2$)

$\displaystyle\frac{d}{dt}\rho=\sum\limits_{x=AB,BC}\gamma_{x,j}\Bigl{(}2L_{x,j}\rho
L_{x,j}^{\dagger}-\rho
L_{x,j}^{\dagger}L_{x,j}-L_{x,j}^{\dagger}L_{x,j}\rho\Bigr{)},$ (18)

where $\gamma_{xj}$ are corresponding loss rates and the Lindblad operators
are given by

$L_{AB,j}=a+y_{j}b,\quad L_{BC,j}=b+z_{j}c,$ (19)

where $y_{j}$, $z_{j}$ are the asymmetry parameters. We consider only the
propagation from the part 1 to the part 2.

To uncover specifics of the non-reciprocal circulation between modes A, B and
C performed by the device shown in Fig. 4(a), let us consider first the
asymptotic regime when the single-photon uncorrelated loss is absent and the
lengths of both parts are sufficiently large for transition to the stationary
state. From Eqs. (18) one gets the following relation between the input and
output values of the modal amplitudes

$\begin{bmatrix}\langle a_{out}\rangle\\\ \langle b_{out}\rangle\\\ \langle
c_{out}\rangle\end{bmatrix}=\mathbf{S}_{j}\begin{bmatrix}\langle
a_{in}\rangle\\\ \langle b_{in}\rangle\\\ \langle
c_{in}\rangle\end{bmatrix},\quad\mathbf{S}_{j}=\frac{1}{F_{j}}\begin{bmatrix}(y_{j}z_{j})^{2}&-y_{j}z_{j}^{2}&-y_{j}z_{j}\\\
-y_{j}z_{j}^{2}&z_{j}^{2}&-z_{j}\\\ -y_{j}z_{j}&-z_{j}&1\end{bmatrix}$ (20)

where $F_{j}=z_{j}^{2}(1+y_{j}^{2})+1$. From Eq.(20) one has

$\left|\frac{[\mathbf{S}_{1}\mathbf{S}_{2}]_{12}}{[\mathbf{S}_{1}\mathbf{S}_{2}]_{21}}\right|=\left|\frac{y_{1}}{y_{2}}\right|,\quad\left|\frac{[\mathbf{S}_{1}\mathbf{S}_{2}]_{23}}{[\mathbf{S}_{1}\mathbf{S}_{2}]_{32}}\right|=\left|\frac{z_{1}}{z_{2}}\right|,\quad\left|\frac{[\mathbf{S}_{1}\mathbf{S}_{2}]_{13}}{[\mathbf{S}_{1}\mathbf{S}_{2}]_{31}}\right|=\left|\frac{y_{1}z_{1}}{y_{2}z_{2}}\right|.$
(21)

So, whereas it is hardly possibly to realize an ideal circulator (which
provides for the unidirectional transfer, i.e. $A\rightarrow B$, $B\rightarrow
C$ and $C\rightarrow A$) with the device Fig. 4(a), nevertheless, energy
transfer between the waveguides is strongly asymmetric.

In Fig. 4 (b-d) we show an example of asymmetric optical circulation for the
bipartite structure as given by Eqs. (18,19) with the all equal decay rates
$\gamma_{x,j}=\gamma$, $\forall x,j$, and mirrored asymmetry parameters in
different parts, $y_{1}=-z_{2}=-0.9$, $y_{2}=-z_{1}=-0.1$. In all panels Fig.
4 (b-d) solid, dashed and dash-dotted lines correspond to the modal amplitudes
of the modes a, b, c. The panels (b,c,d) correspond to the initial coherent
excitation with the unit amplitude of the modes a,b,c with vacuum of other two
modes. The transformation of the modal amplitudes at the end of the part 2 as
shown in Fig. 4 (b-d) is described by the following scattering matrix

$\mathbf{S}_{1}\mathbf{S}_{2}=\begin{bmatrix}0.0075&0.0070&-0.0529\\\
0.0692&0.0669&-0.5306\\\ -0.0768&-0.0743&0.5897\end{bmatrix}.$ (22)

The matrix (22) shows of about 20 dB asymmetry in power transfer between the
waveguides A and B, and B and C. Notice that due to strong coupling asymmetry
even for the interaction time much exceeding the inverse decay rate
$\gamma^{-1}$, the dynamics shown in Fig. 4 (b-d) is still not close to the
asymptotic regime.

Dynamics shown in Fig. 4 (b-d) demonstrate another curious feature: namely,
that the asymmetry in state distribution does not always require and is not
always connected to breaking of reciprocity (for the discussion, see, for
example, Ref. [4]) Each part of the structure depicted in Fig. 4(a) does break
reciprocity by itself. However, for example, the part 1 by itself provides for
quite asymmetric state propagation from the waveguide A to the waveguide B,
and from the waveguide C to the waveguide B. Excitation of the waveguide A
creates strong excitation of the mode B, whereas excitation the mode C leaves
the mode B non-exited. It is interesting to notice that an asymmetric but
reciprocal state distributor was suggested in Ref.[34] for a dissipatively
coupled chain of waveguides. There excitation was shown to spread either to
the left waveguide chain or to the right one in dependence on the excitation
of the control waveguide.

##  VIII Conclusions

Here we have shown that dissipatively coupled systems can break reciprocity
even without simultaneous conjoining unitary and dissipative coupling. Even
for the linear systems, dynamics corresponding to different dissipative
coupling is not commutative. This can give rise to practical linear optical
diode devices providing for unidirectional propagation of certain classes of
states. We outlines the way to fabricate such devices using laser-written
waveguides in glass or generic integration platform on the basis of such
standard materials as InP (in the Appendix we outline practical ways to create
such planar integrated structures).

We show also that designed nonlinear loss enables one to create a single-mode
optical insulator. We demonstrate the way to do that with nonlinear coherent
loss, and show it is possible to achieve a single-mode non-asymptotic breaking
of reciprocity with a practically realizable system of waveguides with self-
Kerr nonlinearity. Such system can realize both two-photon loss and nonlinear
coherent loss able asymptotically to produce a single-photon state from the
initial coherent state. We also show that dephasing is able to induce non-
reciprocity, and demonstrate non-reciprocity of correlation transfer with a
simple example of bipartite system including a beam-splitter and a dephaser.
We demonstrate an example of non-reciprocal optical circulator is a three-mode
system with purely dissipative coupling.

Figure 5: Possible implementations of the dissipative coupler. a – active
unbiased waveguide, b – passive waveguide in isolation cross-section with
metal, c – active waveguides with applied bias. In each cross-section, two
side waveguides and a middle waveguide with extra loss are shown.

## Acknowledgments

The authors (D.M. and I.P.) gratefully acknowledges support from the EU
project PhoG 820365 and the NATO project NATO SPS - G5860; D.P. would like to
acknowledge the Dutch Stimulus OPZuid programme through the Open Innovation
Photonic ICs project (PROJ-00315). The authors thank Prof. K. Williams for
useful criticism and fruitful discussion.

## Appendix: Possibilities for practical implementation

In this section we propose several options for practical realization of
dissipative coupling which can be used to construct the schemes discussed
before (i.e., constitute components of the bipartite set-ups discussed there).
The suggested solutions use a mature COBRA / Smart Photonics InP integration
platform [35] which features a large set of standard building blocks, and thus
can be used to integrate the dissipative couplers in more complex circuits.

The suggested dissipative coupler consists of three parallel single-mode
waveguides A, B, and C, where C corresponds to the common loss reservoir as
shown in Fig. 1(a). The waveguide C is placed between A and B, and has
excessive loss $\Gamma$ as compared to them. The light coupling occurs in the
waveguide pairs A–C, and C–B.

Platform description. In the selected integration platform [35], there are two
types of waveguides: active and passive, see Fig. 5. The passive refers to a
lattice-matched InGaAsP material which is transparent in the C band (1530–1565
nm). It is used as a waveguiding layer with InP as upper and lower cladding
layers with a lower refractive index. The active is a quantum well-based
structure (QW), which uses quaternary InGaAsP material with a composition
chosen to have the emission wavelength and bandgap around 1550 nm. This leads
to presence of absorption at this wavelength and provides means to create
controlled amount of excess loss in waveguide C.

We have considered three possible ways of achieving excess loss in the middle
waveguide, shown in Fig. 5.

Unbiased active sections. This option is shown in Fig. 5a. The middle
waveguide is an active waveguide with a high absorption at 1550 nm if unbiased
[48]. This is the most natural solution considering the absorption spectra of
the active and passive waveguides. However, taking into account the required
distance between the waveguides (below 3 $\mathrm{\mu m}$), the reliable
integration of such narrow active stripes by passive regrowth is not possible
with the current fabrication flow.

Passive waveguides with metal. This option is shown in Fig. 5b. The middle
waveguide is fabricated in the isolation cross-section, in which the strongly
conductive part of the top p-type cladding layer is removed. If we add the
metal on top of this waveguide, it will overlap with the optical mode and
cause extra losses. This option does not require any electrical signals to be
applied to the coupler. However, there is no possibility to tune the excess
loss on the middle waveguide.

Active waveguides with applied bias. This option is shown in Fig. 5c. All
three waveguides are fabricated in the active cross-section, and the two side
waveguides have metal contacts connected on top of them. By applying reverse
bias voltage or pump current to these metal pads, one can control the
absorption in the QW layer, and thus tune the amount of excess loss in the
middle waveguide. This scheme allows some tuneability, but requires active
driving electronic to be permanently connected. And similar to the first
option, small distance between the waveguides means that the opening between
the metal stripes will be 3 to 4 $\mathrm{\mu m}$, which is on the limit of
the fabrication technology.

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