# Asymmetric Transmission Through a Classical Analogue of the Aharonov-Bohm
Ring

Lei Chen  Maryland Quantum Materials Center, Department of Physics, University
of Maryland, College Park, Maryland 20742, USA  Department of Electrical and
Computer Engineering, University of Maryland, College Park, Maryland 20742,
USA  Isabella L. Giovannelli  Maryland Quantum Materials Center, Department of
Physics, University of Maryland, College Park, Maryland 20742, USA  Nadav
Shaibe  Maryland Quantum Materials Center, Department of Physics, University
of Maryland, College Park, Maryland 20742, USA  Steven M. Anlage
Corresponding author<EMAIL_ADDRESS>Maryland Quantum Materials Center,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA  Department of Electrical and Computer Engineering, University of
Maryland, College Park, Maryland 20742, USA

###### Abstract

It has been predicted that new physics and technology are enabled for quantum
systems that suffer from partial decoherence, in the intermediate range
between coherent quantum evolution and incoherent classical physics. We
explore the asymmetric transmission through a classical analogue of the
Aharonov-Bohm (AB) mesoscopic ring that supports a 3:1 asymmetry in
transmission times, augmented with lossy features that act preferentially on
the longer-lingering waves. Such a device is realized as a linear microwave
graph utilizing a gyrator to create the 3:1 transmission time delay asymmetry,
along with both homogeneous and localized losses, to produce an imbalance in
wave transmission through the device. We demonstrate asymmetric transmission
through the microwave-ring graph as a function of loss in both simulation and
experiment, and in both the frequency- and time-domain. The microwave ring-
graph results are compared to a numerical simulation representative of a class
of recent models proposing dephasing-induced transport asymmetry in few-
channel quantum systems, and parallels are noted.

##  I Introduction

There is a deep disparity between the quantum and classical perspectives on
physical laws. Our experience with the classical world is dominated by
equations of motion that include loss, dissipation, and friction, explicitly
breaking time-reversal invariance, and describing phenomena involving large
numbers of interacting particles, creating a perception of the “arrow of time”
[1, 2]. At the microscopic scale we know that the laws of quantum mechanics
are time-reversal invariant, and isolated systems are described by unitary
time evolution of the quantum state. This poses the question: how do the laws
of quantum mechanics segue into those of classical mechanics as one admits
stronger interactions with other degrees of freedom? Such questions have been
addressed in the fields of fundamental quantum mechanics [3, 4, 5, 6, 7, 8],
quantum chaos [9], quantum statistical mechanics [1] and thermodynamics [10],
and mesoscopic physics [11, 12], among others.

A related question is whether or not there is new physics to explore in the
regime between pure quantum evolution, and classical physics? Can systems
described by a mixture of quantum and classical properties show qualitatively
new phenomena that are not anticipated by the properties of systems in either
limit? More specifically, can a finite degree of quantum ‘dephasing’ be
harnessed to perform a new task that is not possible in the fully quantum or
fully classical limits? The answers to these questions are relevant to a wide
variety of phenomena and applications. For example, coupling a quantum system
to additional degrees of freedom can be used for quantum state
preparation,[13] as a resource for universal quantum computation,[14] to
simulate open quantum systems,[15, 16, 17] to enable environment-assisted
quantum transport (ENAQT),[18, 19, 20, 21, 22] to accomplish fast reset of
qubits,[23] and for advanced quantum sensing [24]. However, it is interesting
to see if other qualitatively new phenomena can be discovered and exploited in
this regime.

There has been long-sustained interest in preparing materials and devices that
explicitly break time-reversal invariance (TRI) to create non-reciprocal
transport for electrons between equivalent contacts [11, 12], for example.
Many proposals to create non-reciprocal transport are inspired by Maxwell
demons,[25, 26] thermally-driven ratchets,[27] non-centro-symmetric
materials,[28] and efforts to violate detailed balance in thermal radiation
[29, 30]. Systems with nonlinearity have been utilized to demonstrate
asymmetric transport [31, 32]. One approach to selectively transferring
electrons is to create wavefunction interferometers that preferentially pass
matter waves moving in one direction but not the other, and such devices
require breaking of time-reversal invariance [33, 34]. In particular, the
Aharonov-Bohm matter-wave interferometer, augmented with decoherence to induce
non-reciprocity,[35] has been proposed as one possible setting for this type
of device.

The Aharonov–Bohm (AB) effect is a quantum mechanical phenomenon in which an
electrically charged particle of charge $q$ is affected by an electromagnetic
potential (through the vector potential $\vec{A}$) in the absence of a
magnetic or electric field at the location of the particle [36]. This effect
gives rise to a type of “action at a distance” in which a particle can be
affected by electromagnetic fields even if it does not experience them
directly. The traditional setting for the AB effect is a finite region of
space containing non-zero magnetic flux $\Phi$ surrounded by a field-free
region. A beam of charged particles is sent through the field-free region in a
ring geometry that fully incorporates the region of finite flux within the
ring. The charges show interferometric properties arising from a quantum phase
shift $e^{\frac{iq}{\hbar}\int\vec{A}\cdot d\vec{\ell}}$ upon traversing each
branch of the ring. Upon recombination, parts of the wave packet that pass
through different branches of the ring create interference that depends on
$\Phi/\Phi_{0}$, where $\Phi_{0}=h/q$ is the flux quantum for the charges, and
$h$ is Planck’s constant.

In Refs. [33, 37, 34, 38], the authors consider an Aharonov-Bohm (AB) ring in
a mesoscopic conducting sample that suffers from a certain degree of
“dephasing” in electron transport. Specifically, they consider a single-
channel, 2-terminal AB ring enclosing a DC magnetic flux (as described above).
Fig. 1(a) shows the schematic of the proposed mesoscopic Aharonov–Bohm ring.
This device has two key features that are proposed to bring about new
behavior. First, this device has non-reciprocal transmission times as a result
of the direction dependent phase shift that the electron picks up as it
traverses the ring. An electron wavepacket propagating through the device from
left to right, say, will split into two parts at the left combiner. Along each
of these paths, the respective wave will accumulate a phase. In order for the
waves to constructively recombine at the other end, they must be in phase with
each other, hence the net phase difference accumulated must be $2\pi
n,n\in\mathbb{Z}$. The phase difference accumulated as the electron traverses
the ring is given by,[34] $\Delta\phi=k\Delta l\pm 2\pi\frac{\Phi}{\Phi_{0}}$
where $\Delta l$ is the difference in length between the two arms and $k$ is
the wavenumber of the electron. In [34], the author considers the case where
$k\Delta l=\frac{\pi}{2}$ and $2\pi\frac{\Phi}{\Phi_{0}}=\frac{\pi}{2}$,
leading to $\Delta\phi_{1\rightarrow 2}=0$ and $\Delta\phi_{2\rightarrow
1}=\pi$. Thus, electrons traveling from port 1 to port 2 would suffer no
relative phase shift and immediately exit the ring. However, electrons
traveling from port 2 to port 1 will suffer destructive interference, due to
the $\pi$ phase difference, causing the wave packet to reflect back around the
ring. Hence an electron starting at port 2 will need to traverse the ring 3
times in order to accumulate a net phase difference of $2\pi
n,n\in\mathbb{Z}$, allowing it to coherently recombine and exit the ring. This
consequently leads to a non-reciprocal 3 to 1 ratio of transmission times, as
further discussed in [38].

The second key feature arises when a finite degree of quantum dephasing is
added in the form of a defect or “deep trap” in the semiconducting ring [37,
12, 35], denoted as the red dot in Fig. 1(a). This defect works as a
“dephasing site” by taking a passing electron wavepacket out of the conduction
band with some probability. When an electron is captured, the state is
projected onto an eigenstate of the trap and it loses all memory of its prior
state, including its phase and the direction that it was traveling. The
electron is eventually released back into the conduction band with equal
probability of traveling to the left or right. A more detailed description of
this process is given in [12]. It is argued in [12, 38] that this dephasing
site, combined with the engineered 3 to 1 transmission time ratio, leads to a
net asymmetry in transmission probability. The argument is that since
electrons traveling from port 2 to port 1 need to make three times as many
passes as electrons traveling from port 1 to port 2, they then have three
times as many chances of interacting with the dephasing site. Hence electrons
traveling in that direction have a lower chance of being transmitted.
Supporting simulation results are discussed in [12] where they show that not
only is this asymmetry in transmission probability present, but it has a non-
monotonic dependence on the modelled dephasing rate.

In addition to the explicit proposals made in Refs. [33, 37, 38], a few other
works have hinted at this asymmetric transmission probability effect. The work
of Entin-Wohlman, et al. essentially considers a quantum ring subjected to an
external magnetic field and suffering partial dephasing in electron transport
around the graph. They calculate that this results in a loss of detailed
balance in equilibrium, and the creation of a net current between two points
in the ring [39]. A cold atom version of a dissipative AB-ring in momentum
space, roughly similar to the quantum models cited above [37, 12, 35], has
demonstrated nonreciprocal quantum transport [40]. This approach utilizes a
synthetic magnetic flux and laser-induced loss of atoms in a Bose-Einstein
condensate to break inversion and time-reversal symmetries, and demonstrates
directional atom flow.

Figure 1: Schematic layout of an Aharonov–Bohm ring and a corresponding
classical microwave graph analogue. (a) shows the device layout of a two-port
Aharonov–Bohm ring made of clean metal. The penetrating magnetic flux $\Phi$
creates non-reciprocal transmission times for electron wave packets transiting
the ring. The red dot marks an inelastic scattering center. (b) shows the
classical analogue microwave graph realization. A microwave gyrator (see Supp.
Mat. Fig. 9) creates the $\pi$ phase difference for waves travelling in
opposite directions on the lower bond. The variable attenuators of strength
$\Gamma_{A}/2$ act as a localized loss center. Panel (a) after Ref. [38].

In this paper, our objective is to examine the properties of a classical
analogue of the mesoscopic AB-ring proposed in [33, 37, 38]. In particular, we
are motivated by their proposition that quantum wave interference, combined
with finite dephasing can lead to asymmetric transmission. Our goal is to see
if parallel behavior can be observed in a purely classical setting.
Specifically, we want to understand if classical wave interference effects,
combined with dissipation, give rise to asymmetric wave propagation.

The outline of the paper is as follows. We begin by discussing the background
for the experiment, where we provide a brief overview of the parasitic channel
phenomenology that provides an organizing principle for understanding the
effects of loss on our experimental results. We then present simulations of
the microwave-ring graph and demonstrate the properties required to achieve
asymmetric transmission. A number of complications are added to the
simulations to represent the experimental situation, and the key properties
are shown to survive. Next we present the experimental realization of the
microwave ring-graph, and discuss measurements of the scattering matrix and
time delays in both the frequency and time domains, and find good consistency
with the simulations. Finally we present results on the transmission asymmetry
through the graph in experiment, and compare to expectations based on
simulations.

##  II Background

There is a long history of using microwave circuits and resonant microwave
billiards to emulate the wave interference properties of quantum systems [41,
42]. Quantum billiards are simulated using microwave resonators, where the
analogy between the Schrödinger equation for the wavefunction $\psi$ and the
electromagnetic Helmholtz equation for one component of electric field
$\vec{E}$ is particularly clear in two-dimensions [43, 44, 45, 46, 47, 48, 49,
50, 51, 52]. For example, it was also shown that there is an analogy between
the Poynting vector for energy flow in the quasi-two-dimensional
electromagnetic cavity and the probability current in the corresponding
quantum system [53]. Similarly, the analogy between solutions to the
Schrodinger equation on multi-connected molecules [54] and the wave equation
on networks of microwave graphs has been noted [55].

In this paper we will be adopting the Heidelberg picture of wave scattering
[56, 57, 58, 59, 60, 61, 62, 63, 64] to describe scattering in non-Hermitian
systems. Localized loss is modeled by a matrix $\Gamma_{A}$ that evaluates the
overlap of the $M$ resonant eigenmodes of the closed system at the location of
loss, represented by a set of $L$ absorption channels. The $\Gamma_{A}$ matrix
is added to the Hamiltonian of the closed system to create an effective
Hamiltonian that is non-Hermitian. In addition, one can model uniform
attenuation $\eta$ by evaluating the energy (or equivalently frequency of the
microwaves) with an imaginary offset $E\rightarrow E+i\eta$. We note that this
is a simple phenomenological approach to describing loss that is quite
generic, and therefore applies to all sorts of classical wave scattering
systems (electromagnetic, acoustic, mechanical, etc.). However, it is not a
microscopic theory, and in the case of quantum systems suffering from
decoherence, it altogether fails to capture the quantum degrees of freedom,
and their associate quantum interactions with the scattering system. As such,
the Heidelberg picture has no microscopic justification in describing the
scattering properties of quantum systems suffering from decoherence.

In Ref. [65] it was experimentally demonstrated that the statistical
properties of the scattering matrix of a microwave cavity with variable
uniform loss were described by a simple dephasing-lead model borrowed from a
treatment that utilized the parasitic channel model of dephasing in mesoscopic
transport [66, 67, 68]. The uniform attenuation of the quasi two-dimensional
microwave cavity (parameterized as the dimensionless quantity
$\alpha=k^{2}A/4\pi Q$, where $k=\omega/c$, $\omega$ is angular frequency, $A$
is the area of the billiard, and $Q$ is the typical quality factor of the
resonant modes) was found to be directly proportional to the dimensionless
phenomenological dephasing rate $\gamma$ that governs the statistics of the
scattering (S) matrix and conductance in the corresponding mesoscopic billiard
[67, 68], as $\gamma=4\pi\alpha$ over a wide range of microwave loss. (It can
be shown that $\alpha$ is related to the uniform attenuation as
$\alpha=\frac{\eta}{2\Delta}$, where $\Delta$ is the mean spacing between
modes of the closed system [65, 69].) This demonstrated that the statistical
fluctuations of the wave interference properties of both the classical
microwave experiment, and the model quantum system, have the same dependence
on a single parameter. We refer to this single-parameter dependence of the
scattering matrix statistical properties as the “parasitic channel
phenomenology” (PCP). We shall explore the PCP further in this paper through a
study of wave interference effects in a classical analogue of the AB-ring
graph.

Here we consider quasi-one-dimensional microwave network analogues of quantum
graphs that emulate corresponding few-channel mesoscopic conducting quantum
systems. Quantum graphs are one-dimensional metric graphs with complex
topology that support excitations described by the Schrodinger operator [70,
71] that can be used to describe quantum transport through a variety of
structures, including molecules [54, 72], quantum wires [73, 74], disordered
two-dimensional quantum dots [75, 76, 74], etc. In general, microwave graphs
consist of vertices connected by bonds, and we consider the propagation of
microwave signals on the bonds and the resulting interference that occurs when
multiple bonds meet at a vertex. Microwave graphs have been used to
investigate many aspects of wave scattering theory, including statistics of
the Wigner reaction matrix (analogous to electromagnetic impedance) [55],
topological edge invariants [77], deviations from the predictions of random
matrix theory [78], etc. The PCP has also be used to quantify the effects of
dissipation on the scattering matrix statistics of microwave graphs [79, 80].

We introduce a linear two-terminal, single-channel microwave graph to emulate
the wave interference properties of a mesoscopic Aharonov–Bohm ring, and study
its scattering properties in both the frequency domain and the time domain. In
particular, our objective is to demonstrate asymmetric transmission by
utilizing a specific combination of wave interference and attenuation. The
microwave graph in Fig. 1(b) satisfies all of the conditions for displaying
asymmetric transmission, including broken time-reversal invariance, a sub-
unitary scattering matrix, and a gyrator which produces a broadband constant
$\pi-$phase shift for waves going in one direction [37]. We note that the
gyrator effectively performs the function of a Faraday rotator, as originally
utilized by Rayleigh to create non-reciprocal transmission of light [12, 81,
82]. We demonstrate that the real part of forward and backward transmission
time delays have a 3:1 ratio in a broad range of frequency, which establishes
the first condition for asymmetric transmission. We also introduce a localized
loss $\Gamma_{A}$ (exploiting the PCP to phenomenologically treat dissipation
[65]) in the ring graph, and adjust the left/right transmission coefficients
by varying the attenuation $\Gamma_{A}/2$ of two objects, in a balanced
manner. We demonstrate the dependence of asymmetric transmission on the loss
rate through the attenuation variation.

##  III Simulations

We first simulate the non-reciprocal transmission effect of the microwave
ring-graph in CST Microwave Studio. In particular we use a circuit modeling
package to create a faithful model of the microwave graph (see Appendix A and
Fig. S1). We implement the complete circuit shown in Fig. 1(b) with equal
electrical lengths (i.e. propagation phase shifts) for the upper and lower
bonds.

###  III.1 Frequency Domain Simulations

The complex transmission time delays (see Appendix C) for both directions
($S_{21}$ and $S_{12}$) can be calculated from the complex scattering matrix
data as $\tau_{T}^{12,21}=-i\frac{d}{df}\log(S_{12,21})$ [83, 84, 85, 34, 86,
87, 88]. Figure 2(a) shows the simulation comparison between the real part of
the transmission time delay for the two directions, demonstrating the required
3:1 ratio, modulo small oscillations associated with the shape resonances of
the microwave graph [88]. Fig. 2(b) shows the transmission magnitudes
$|S_{12}|$ and $|S_{21}|$ through the microwave graph under several different
attenuation settings. When the lumped attenuation $\Gamma_{A}/2$ is 0 Nepers
and there are no uniform losses, i.e. no loss in the entire system and a
unitary S-matrix, the two transmission paths have identical transmission
magnitude as a function of frequency. As the lumped attenuation increases,
significant differences begin to show up between the two transmission
amplitudes.

Figure 2: Simulated properties of the microwave AB-ring graph with uniform
attenuation $\eta=0$. (a) Real part of transmission time delays of the
Aharonov-Bohm microwave ring graph simulation for the case of localized
attenuation $\Gamma_{A}/2=0.18$ Np. Blue curve shows $\text{Re}[\tau_{21}]$
and red curve shows $\text{Re}[\tau_{12}]$, and the dashed lines show the mean
values, demonstrating a 3:1 ratio. (b) Linear transmission magnitudes in both
directions for the case of $\Gamma_{A}/2=0$ Np attenuation and
$\Gamma_{A}/2=0.18$ Np local attenuation. Blue corresponds to $|S_{21}|$ while
red corresponds to $|S_{12}|$. With no attenuation the transmission magnitudes
are identical. The scale bar gives the expected periodicity frequency scale
($\Delta=0.5$ GHz) for the shape resonances of a simple ring graph. (c) The
asymmetric transmission $P_{21}-P_{12}$ as a function of frequency for the
case of $\Gamma_{A}/2=0.18$ Np local attenuation.

The asymmetric transmission behavior of the ring graph at each frequency can
be quantified as the transmission probability difference:
$P_{21}-P_{12}=|S_{21}|^{2}-|S_{12}|^{2}$. Note that $P_{21}-P_{12}$ is
bounded between 0 and 1, and is frequency dependent in general. If there is no
asymmetric transmission then $P_{21}-P_{12}=0$, while a non-zero value implies
some degree of asymmetric transmission. Fig. 2(c) shows $P_{21}-P_{12}$ as a
function of frequency for the $\Gamma_{A}/2=0.18$ Np attenuation case,
revealing a frequency-averaged value (dashed green line) of
$\braket{P_{21}-P_{12}}=0.28$.

Figure 3: Asymmetric transmission through microwave AB-ring graph and a model
mesoscopic electron device.(a) Summary of the frequency domain results. In the
upper right is a schematic of how these measurements work. For frequency
domain measurements, a wave with a single frequency is sent through the
device, this is done multiple times over a frequency range. For all curves,
the quantity $P_{21}-P_{12}$ is averaged over the frequency range $8.25-8.75$
GHz, which is denoted as $\braket{P_{21}-P_{12}}$. The solid line is from
simulation where only the localized attenuation from the attenuators is
considered. The dashed line is also from the same simulation but with finite
uniform attenuation added. The discrete diamond points are from experimental
data. (b) Summary of the time domain results expressed in terms of
$\sigma_{V}=(|V_{21}|^{2}-|V_{12}|^{2})/|V_{in}|^{2}$. For these time domain
measurements, a 1 ns wide Gaussian pulse with a carrier frequency of 8.5 GHz
is sent into the device. The legend has the same meaning as in (a). (c) Noise
source input experimental results. This data was taken in the frequency domain
with a microwave noise source input as a function of total attenuation in the
graph. It is plotted as frequency-averaged transmitted power difference,
normalized by the measured incident power,
$(\bar{P}_{21}-\bar{P}_{12})/\bar{P}_{in}$. (d) Tight-binding model of
triangle-shaped mesoscopic device with a dephasing center, as a function of
the average number of inelastic scattering events per electron passage through
the structure. From Refs. [12, 35]. Data provided courtesy of Dr. Jochen
Mannhart, Max Planck Institute for Solid State Research, Stuttgart, Germany.

To explore the asymmetric transmission further, Fig. 3(a) shows
$P_{21}-P_{12}$ as a function of total lumped attenuation $\Gamma_{A}$ for the
model graph. Due to the periodic wiggles arising from the shape resonances of
the ring graph, we perform an average of $P_{21}-P_{12}$ over a range of
frequencies corresponding to one period of the shape resonances $\Delta=0.5$
GHz, and designate it as $\braket{P_{21}-P_{12}}$. The frequency averaged
transmission asymmetry $\braket{P_{21}-P_{12}}$ shows a non-monotonic bell
shaped behavior as the lumped attenuation increases.

We have also simulated the more realistic case where both uniform attenuation
and lumped variable attenuation are present in the AB-ring microwave graph. We
set the parameters of the simulated coaxial cables to approximately match
those used in the experiments discussed below, leading to a frequency
dependent uniform attenuation described by Eq. S1. The resulting transmission
asymmetry $\braket{P_{21}-P_{12}}$ vs. lumped attenuation is also shown in
Fig. 3(a) as a dashed line. The main difference is that uniform attenuation
effectively establishes a finite amount of asymmetric transmission even with
zero lumped attenuation.

###  III.2 Time Domain Simulations

Time-domain simulations have been performed with the model microwave ring-
graph shown in Fig. S1 (see Appendix A and Supp. Mat. section “Simulations”).
Gaussian wave packets with 1 ns width and a chosen carrier frequency are
injected into the graph from each port. The output pulses show the expected
3:1 asymmetry in time delay upon going through the graph, as shown in Fig. S2.
To quantify the transmission probability difference in the time-domain, we
form the quantity $\sigma_{V}=(|V_{21}|^{2}-|V_{12}|^{2})/|V_{in}|^{2}$, where
$V_{21}$ and $V_{12}$ are the voltage amplitudes of the transmitted pulses in
the simulations and $V_{in}$ is the amplitude of the incident pulse. Fig. 3(b)
shows $\sigma_{V}$ as a function of the lumped attenuation (solid line). As
the lumped attenuation increases the transmitted wavepackets first show an
increasingly asymmetric transmission probability. However, beyond about
$\Gamma_{A}=1/2$ Np the wavepackets show reduced transmission asymmetry. The
addition of uniform loss serves to shift the $\sigma_{V}$ curve (dashed line),
just as with the frequency domain simulations in Fig. 3(a).

These simulation results of the microwave ring-graph in both the frequency-
and time-domains establish the basic properties of asymmetric transmission.

##  IV Experiment

The microwave graph consists of a coaxial cable structure that supports a
single mode of propagation for microwave frequencies below the cutoff of
higher-order modes, which is well beyond our operating frequency range [55].
An Aharonov–Bohm ring uses magnetic flux to produce non-reciprocal
transmission times in a simple electron interferometer. In order to mimic that
effect, we use a microwave gyrator [89, 90, 91] to create a uni-directional
$\pi$ phase shift in a microwave ring network. As illustrated in Figs. 9 and
S3 this is achieved by means of two microwave circulators that are configured
with short and open circuits on their third ports to create the requisite
$\pi$ relative phase shift. We also demonstrate in Appendix B that this
relative phase difference is approximately achieved over a broad frequency
range (7-12.4 GHz) with virtually no difference in attenuation of the signals
in the two directions.

Next, we construct the Aharonov–Bohm analogue ring microwave graph (see Fig.
4) using the gyrator design. The key is to add an upper branch to the ring
that has the same electrical length as the gyrator, so that waves travelling
from left to right in both branches would go through the same electrical
length. Equating the electrical lengths of the branches also has the effect of
eliminating the Feshbach modes, leaving only the shape resonances of the ring
graph (see Refs. [88] and [92]). We measured the electrical length of the
gyrator and a series of single coaxial cables, and selected a 12-inch-long
coaxial cable as the upper branch. We then perform some fine-tuning on both
phase trimmers to achieve an electrical length on the lower bond as close as
possible to that of the 12-inch cable. The 12-inch coaxial cable has an
electrical length of $0.4386$ m, and we manage to adjust the electrical length
of the gyrator to be $0.4385$ m while maintaining a near $\pi$ phase
difference for left/right transmission. The resulting microwave analogue of
the AB graph with $\pi$ flux-induced phase shift is shown in Fig. 4(a) (no
lumped loss) and Fig. 4(b) (including lumped loss $\Gamma_{A}$).

Figure 4: (a) Photograph of the Aharonov–Bohm microwave ring graph constructed
with the gyrator from Fig. 9. The upper branch is a single coaxial cable
matching the electrical length of the lower branch. This is the experimental
realization of the schematic shown in Fig. 1(b), without the attenuators. (b)
Microwave graph analogue Aharonov-Bohm ring for the two-attenuator case, with
one attenuator of strength $\Gamma_{A}/2$ on each bond.

The losses in the microwave graph arise from two sources. There exists roughly
uniform attenuation of the microwave signals associated with the insertion
loss of the coaxial cables, circulators, phase trimmers, microwave tees,
adapters, and terminations. The attenuation $\eta$ of the coaxial cables used
here, which dominates the uniform attenuation in the circuit, has been
characterized in Appendix B of Ref. [88]. The second source of loss is
associated with the variable lumped (localized absorbing channels) attenuators
$\Gamma_{A}$, shown schematically in Fig. 1(b). In the experiments both types
of loss in the microwave graph will be considered. Note that in all
comparisons of microwave data and mesoscopic theory we treat the independent
variables frequency $f$ and energy $E$ as interchangeable.

###  IV.1 Frequency Domain Measurements

We measure the scattering ($S$) matrix of the 2-port microwave ring-graph as a
function of frequency. A Keysight N5242B network analyzer (PNA-X) is
calibrated with a Keysight N4691D Electronic Calibration kit over the 7 to
12.4 GHz frequency range with a frequency step size of 168750 Hz. An Agilent
Technologies N5242A vector network analyzer (VNA) is also used over the same
frequency range; it is calibrated with an Agilent N4691-60001 Electronic
Calibration kit with the same frequency step size. The device under test (DUT)
is attached at the calibration planes of the network analyzer and the $2\times
2$ $S$-matrix is measured as a function of frequency for various settings of
the lumped attenuators in the microwave ring-graph.

To measure the response of the microwave ring-graph to incoherent input
signals, a Microwave Semiconductor Corp (MSC) MC65242, 1.0 to 18 GHz noise
source producing output excess noise ratio (ENR) of 31.5 dB, is used to create
broadband noise in the microwave domain. The noise signal is amplified by two
Mini-Circuits amplifiers (ZX60-183A-S+) each with a bandwidth of 6 to 18 GHz.
The two amplifiers are separated by one Narda-MITEQ (94S46) attenuator with a
bandwidth of 4 to 18 GHz. The amplified noise source is connected to one port
of the microwave ring-graph while the other port of the graph is connected to
a network analyzer (Keysight N5242B). The network analyzer is used in
receiving mode, with a resolution bandwidth of 10 kHz and 200 point averaging,
to measure the transmitted power in a frequency resolved manner. The
experiment is repeated with the ports on the microwave ring-graph reversed.

###  IV.2 Time Domain Measurements

The broadband nature of the 3:1 transmission time ratio of our AB-ring graph
analogue allows us to utilize wide-bandwidth pulses to investigate the
asymmetric transmission through the device. We perform pulse propagation
measurements through the microwave ring-graph in both directions,
systematically varying the center frequency of the pulse, and measuring the
time delays and amplitudes of the transmitted pulses. A Tektronix model
AWG70001B Arbitrary Waveform Generator (AWG) is attached to the input port of
the DUT, and the other port is attached to a Keysight/Infiniium model UXR0104A
real-time oscilloscope. The input pulse created by the AWG is designed to be a
1-ns long Gaussian modulated pulse of center frequency $f_{c}$, with $f_{c}$
ranging in value from 7 to 12.4 GHz. The pulse is measured by connecting the
output of the AWG directly to the oscilloscope to establish the amplitude and
timing of the incident pulse. The measurement is then repeated with the
microwave ring-graph present in the other orientation.

We note that prior measurements of pulse transport in microwave graphs has
focused on the delay distribution and identifying orbits due to short closed
loops in the graph [93].

###  IV.3 Frequency-domain Experimental Results

We measure the $2\times 2$ scattering ($S$)-matrix of the Aharonov-Bohm ring
microwave graph shown in Fig. 4(a) (with approximately uniform attenuation
only) from 7 to 12.4 GHz, and plot the results in Fig. 5. The periodic wiggles
in the plot come from the shape resonances of the graph, which have been
thoroughly studied in Refs. [88] and [92], and discussed for the simulations.
The periodicity of the shape resonances depends on the total circumferential
length of the graph $\Sigma=0.8771$ m, which corresponds to a repetition
frequency of $\Delta=\frac{c}{\Sigma}=0.342$ GHz. Note that waves propagating
anti-clockwise around the microwave ring satisfy the ordinary shape resonance
conditions, $f_{n}=\frac{c}{\Sigma}n$, with $n=1,2,3,...$. However, waves
travelling clockwise around the microwave ring suffer an additional $\pi$
phase shift upon passage through the gyrator, resulting in a different
resonance condition, $f_{m}=\frac{c}{\Sigma}(m-\frac{1}{2})$, with
$m=1,2,3,...$, creating a series of modes that begin at $\Delta/2=0.171$ GHz,
and then alternate with the anti-clockwise modes as a function of frequency,
consistent with the periodicity shown in Fig. 5. This creates two distinct
sets of S-matrix poles and zeros for the modes of the microwave ring-graph
(see Supp. Mat. section “Poles and Zeros of the S-Matrix of the Microwave
Ring-Graph”).

Figure 5: Measured $S$-matrix transmission data for the microwave ring-graph
in Fig. 4(a) with approximately uniform attenuation, as a function of
frequency. The scale bar gives the expected periodicity frequency scale
($\Delta=0.342$ GHz) for the shape resonances of a simple ring graph. Note
that $|S_{21}|$ is larger than the $|S_{12}|$ consistently across the full
frequency range. Inset (a) shows the real part of the transmission time delays
of the graph. We see that $\text{Re}[\tau_{T}^{12}]\approx
3\cdot\text{Re}[\tau_{T}^{21}]$ over the full frequency range. The dashed
lines represent the average of the time delays. Inset (b) shows transmission
probability asymmetry of the graph measured in both the frequency domain
($P_{21}-P_{12}$, green line) and time domain ($\sigma_{V}$, red diamonds) as
a function of frequency.

###  IV.4 Transmission Time Delays

The measured results for the real part of transmission time delay in both
directions, $\text{Re}[\tau_{T}^{21}]$, $\text{Re}[\tau_{T}^{12}]$, are shown
in Fig. 5(a). The mean values are clearly different, while the regular
variations of the transmission time delay plot come from the shape resonances
of the ring graph, although here we are more interested in the ratio between
the two transmission time delays. The mean values are
$\braket{\text{Re}[\tau_{T}^{21}]}=$ 1.49 ns and
$\braket{\text{Re}[\tau_{T}^{12}]}=$ 4.47 ns, with a ratio of 2.99. This
demonstrates the average 3:1 ratio for the two transmission time delays in the
Aharonov–Bohm ring microwave graph, based on frequency-domain data.

###  IV.5 Asymmetric Transmission

Figure 5 shows a comparison between forward and reverse transmission
amplitudes $|S_{21}|$ and $|S_{12}|$ in the microwave ring-graph, where it can
be seen that $|S_{21}|>|S_{12}|$ for all frequencies in the bandwidth of the
device. The asymmetric transmission probability $P_{21}-P_{12}$ is plotted as
a function of frequency in Fig. 5(b) (green line), demonstrating the broadband
nature of the effect.

In addition to measuring the microwave ring-graph with coherent sources in the
frequency domain and time domain, we are curious to see how it behaves when
subjected to a broadband noise source. It is interesting to see whether or not
the asymmetric transmission properties are also exhibited when the AB-ring
graph is excited by incoherent noise. Figure 6 shows the results of such an
experiment, in which a microwave noise source is used to illuminate one port
of the graph. In this case the transmitted signal is measured in a frequency-
resolved manner by a network analyzer operating in receiver mode, as discussed
in Section IV.1. The experiment demonstrates broadband asymmetric transmission
over the bandwidth of the device. The bandwidth-averaged asymmetric
transmission probability was measured as a function of total attenuation in
the microwave ring-graph (Fig. 4(b)), and the results are shown in Fig. 3(c).
This result shows that the non-monotonic dependence of transmission asymmetry
exists even for a completely incoherent source of microwaves.

Figure 6: Measurements of asymmetric noise power transmission ($NP_{12}$ and
$NP_{21}$ in units of $dBm$) as a function of frequency through the AB-
microwave graph having only uniform attenuation, when subjected to a broadband
noise source. Averaging ($\overline{NP}_{12}$, $\overline{NP}_{21}$) is
performed by sectioning the data in groups of 50 frequency points and
replacing each of these groups with their respective mean value. Inset shows
the asymmetric transmission power ratio $\frac{NP_{21}}{NP_{12}}$ in linear
scale over the bandwidth of the graph, showing a mean value of $1.35$.
Averaging for $\overline{(\frac{NP_{21}}{NP_{12}})}$ is performed in the same
way as described above for $\overline{NP}_{12}$ and $\overline{NP}_{21}$.

###  IV.6 Time-domain Experiments

The results presented so far have been obtained from microwave scattering
matrix measurements performed entirely in the frequency domain. Electrons in
mesoscopic systems are modeled as quantum wavepackets in the time domain.
Electron wave-packets do not have a single energy, as implied by our time-
delay results obtain from S-matrix data, presented above. After demonstrating
a 3:1 ratio between the two transmission time delays from the frequency-domain
data, the question arises whether or not we can demonstrate the asymmetric
delay directly in the time domain.

Figure 7 shows representative time-domain results of transmitted pulses
emerging at each port when sending in the pulse from the other port. Here, an
arbitrary waveform generator (AWG) generates the gaussian-modulated pulse of
fixed carrier frequency $f_{c}$ used in the time-domain measurements. The
pulse is a 1-ns wide Gaussian amplitude modulation of an $f_{c}=8.5$ GHz
carrier signal. Such a pulse includes approximately $1$ GHz bandwidth, which
is well within the operating bandwidth of our gyrator-based microwave ring-
graph. The pulse will have the effect of averaging the non-reciprocal
properties of the graph over a finite bandwidth. Here DUT is the device under
test (the microwave ring-graph), and the oscilloscope records the incident and
transmitted signals. We measured the pulse transmission in both directions,
and plot them on the same time axis. Shown for reference (yellow pulse in
Figure 7) is the incident pulse on the DUT, illustrating the relatively modest
effects of pulse dispersion in the experiment. Figure 7 demonstrates directly
a 3:1 ratio of the transmission time delays for pulses propagating through the
AB graph in opposite directions. From the plot, the transmitted pulse from
port 2 to port 1, $V_{12}$, has smaller amplitude compared to the transmitted
pulse from port 1 to port 2, $V_{21}$.

Figure 7: Time-domain measurements for the microwave ring-graph (the DUT here)
shown in Fig. 4(a). Insets show the schematic of the time-domain setup for the
$1\rightarrow 2$ and $2\rightarrow 1$ measurements. The yellow pulse is a
measurement of the incident pulse (a 1-ns wide Gaussian amplitude modulation
of a $f_{c}=8.5$-GHz carrier signal) from the arbitrary waveform generator
(AWG) to the oscilloscope and establishes the baseline pulse amplitude. The
blue pulse shows the transmitted pulse from port 1 to port 2, while the orange
pulse shows the transmitted pulse from port 2 to port 1. Note that for the
port 2 to port 1 measurement the DUT orientation is reversed.

The red diamonds in Fig. 5(b) show the pulse transmission asymmetry parameter
$\sigma_{V}=(|V_{21}|^{2}-|V_{12}|^{2})/|V_{in}|^{2}$ vs. pulse center
frequency from the measured time-domain results in Fig. S4. The results are in
good agreement with the transmission probability asymmetry from frequency
domain measurements of the same graph (green line). Figure 3(b) shows a plot
of $\sigma_{V}$ at a pulse center frequency of 8.5 GHz vs. the lumped
attenuation on each bond $\Gamma_{A}/2$, as they are varied together (red
diamonds). Adjusting both attenuators to the same attenuation setting
preserves identical electrical path lengths on the two bonds of the microwave
graph (Fig. 4(b)). We see once again the non-monotonic dependence of
transmission asymmetry on variable lumped attenuation, and good agreement
between frequency domain $\braket{P_{21}-P_{12}}$ and time domain $\sigma_{V}$
measurements.

##  V Discussion

We note some interesting parallels between the measurements on the classical
microwave graph and earlier simulations of asymmetric transport in a model
mesoscopic device with dephasing centers. The mesoscopic calculation in Fig.
3(d) shows a non-monotonic dependence of asymmetric left/right transmission
probability $P_{21}-P_{12}$ as a function of the average number of inelastic
scattering events per passage based on a phenomenological model of the quantum
mesoscopic system, illustrated in the inset [35, 12]. Plotting the lumped
attenuation $\Gamma_{A}$ in units of Nepers (a $1/e$ decay of amplitude) is
roughly analogous to the number of inelastic scattering events per passage of
the wavepacket through the device. A plot of asymmetric transmission vs.
$\Gamma_{A}/2$ from microwave data has essentially the same dependence as
simulated transmission probability asymmetry plotted vs. the average number of
inelastic scattering events per passage, as shown in Fig. 8 (see also Fig. S6
in Supp. Mat. section “Detailed microwave AB-ring graph results”). The
similarities are illustrated in Fig. 8 for (a) frequency-domain, (b) time-
domain, and (c) broadband noise transmission. In the unitary evolution (zero
attenuation) case, there is symmetric transmission due to the purely coherent
properties of the system. At large de-coherence (attenuation) rates, the
scattering events are so frequent that they destroy any bias for the electrons
(microwaves) to follow a particular direction through the device, sometimes
explained as a manifestation of the quantum Zeno effect [94]. Only in the
intermediate de-coherence (attenuation) case does the combination of coherent
transport, along with a finite degree of modeled de-coherence (attenuation)
acting asymmetrically, result in a net transmission asymmetry through the
device.

From the perspective of the PCP, this unique state is not observed in either
the purely coherent or strongly attenuated limits, but is a unique feature of
wave systems augmented by a finite number of parasitic channels. Similar
results on asymmetric transport of a Bose-Einstein condensate through an AB-
ring analogue in cold atoms, as a function of loss rate (analogous the the
attenuation used here), shows the same qualitative dependence as those in
Figs. 3 and 8 [40]. It has also been proposed that transport of excitations
across dissipative quantum networks can be enhanced by local dephasing noise,
and the dependence of transport probability on dephasing rate is non-monotonic
[18, 19]. Experiments on electronic excitation transport through a network of
coupled trapped ions shows evidence of environment-assisted quantum transport
(ENAQT) effects that are also a non-monotonic function of dephasing rate [21,
22]. There are also proposals for dephasing assisted transport in coupled
quantum dot systems [95, 96].

There have also been classical wave transport experiments that demonstrate
similar behavior to that shown in Figs. 3 and 8. A classical optics experiment
involving noisy resonators on a fiber network demonstrated a peak in the
dependence of the transport efficiency as a function of the amount of
‘dephasing noise’ in the network, realizing the phenomenon of ‘noise-assisted
transport’ [97]. Another classical optics experiment utilizes variable input
optical bandwidth to enhance transport [98].

Figure 8: Comparison of AB-graph microwave analogue asymmetric transmission
($\Delta P$ on left axis) and simulated mesoscopic device transmission
probability asymmetry ($\Delta P$ on right axis), as a function of attenuation
$\Gamma_{A}/2$ in Nepers, and average number of inelastic scattering events
per electron passage, on a common log scale. Microwave measurements and
simulation results are obtained in the (a) frequency-domain, (b) time-domain,
and (c) frequency-resolved noise transmission. Lines and symbols have the same
meaning as those in the corresponding parts of Fig. 3. The mesoscopic
simulation results $\Delta P$ (in yellow) are the same in all panels [35, 12].

It is remarkable that the measured microwave transmission asymmetry and the
phenomenological mesoscopic dephasing treatment of asymmetric electron
transport [12, 35] have the same dependence on the loss/dephasing parameter
over such a wide dynamic range. One possible reason for this is that the
dephasing model effectively results in the same non-Hermitian treatment of the
system Hamiltonian, and consequent modification of the scattering matrix, as
observed for classical waves. This conclusion is consistent with all three
observations discussed here, namely the experimental data on the classical
wave analogue of the AB-graph, the simulated results for an asymmetric
mesoscopic scatterer of Refs. [35, 12], and the dissipative cold-atom
experiment modeling an AB-graph in momentum space [40]. In fact, the non-
monotonic dependence of transmission probability on ‘dephasing’ or measurement
rates seems to be a generic property of many treatments of quantum transport,
such as environment-assisted quantum transport, [18, 19, 21] and monitored
quantum devices [99, 100]. This leaves open the question of whether a more
detailed or microscopic treatment of dephasing in quantum systems might arrive
at different conclusions with regards to asymmetric transmission.

The microwave ring-graph elegantly captures the fundamental wave interference,
non-reciprocal and dissipative properties that give rise to asymmetric
transmission. Of course, there are a number of limitations of the analogy
between electromagnetic (EM) and mesoscopic quantum systems. In the EM
analogue there are no Fermi-Dirac statistics, a linear (as opposed to
quadratic) dispersion relation, and no electron-electron interactions. The
physics of entanglement and superposition states are also absent in the
classical case. In the EM case the microwave photons are dissipated, whereas
in the mesoscopic case the particle number is conserved. Finally, the wave
packet time evolution is different in detail, owing to the first-order nature
of the time dependent Schrodinger equation, vs. the second order nature of the
EM wave equation. In addition the PCP approach to accounting for absorption in
classical wave systems (or dephasing in quantum systems) is entirely
phenomenological and generic in nature, detached from microscopic theory. For
example, it fails to capture multi-particle quantum interactions that lead to
dephasing.

Next we comment on several properties of the microwave graph. It is well
established that non-reciprocal wave transport can be created by a simple
microwave device known as an isolator. One might ask: how does the microwave
analogue of the AB-ring graph differ from an isolator? An isolator can be
fabricated from a three-terminal circulator, like those utilized in our design
for the gyrator (see Fig. 9). The second port of the circulator is connected
to a matched load that absorbs energy injected into port 1 and circulated to
port 2. Waves entering port 3 are directed to port 1, creating a two-port
device that passes waves in one direction while absorbing those going in the
opposite direction. We note that photons absorbed and then re-emitted from the
absorptive load will be forced to circulate to port 3, due to the design of
the device. In contrast the microwave AB-ring graph has a crucial difference.
Photons absorbed and re-emitted by the localized attenuators $\Gamma_{A}/2$
are able to exit the device through either port, with equal probability. This
feature is utilized in the phenomenological mesoscopic analogue model to
create the asymmetric flow of electrons in that case [12, 35].

A second important feature of the microwave graph concerns its linearity. It
is well established that non-reciprocal transport can be achieved by
exploiting nonlinearities. A common example is the diode, which has a
nonlinear current-voltage characteristics at large voltage amplitudes, but a
linear response at low amplitude [34]. In Fig. S5 we demonstrate that the
asymmetric transmission of the microwave AB-graph is present over a broad
range of amplitude excitation. It’s ability to create asymmetric transmission
is not based on nonlinear processes or features.

##  VI Conclusions

In this paper, we present a two-port microwave network structure that
demonstrates a 3:1 ratio of the transmission time delays for waves travelling
in opposite directions. The 3:1 ratio of the time delay has been illustrated
in both frequency-domain and time-domain experiments. Using both simulations
and experiments, we also demonstrate the asymmetric transmission through the
microwave ring in both the frequency domain and time domain. The degree of
asymmetric transmission is shown to be a non-monotonic function of localized
attenuation in the ring. The experimental results on asymmetric transmission
exist for both coherent and incoherent signal sources. We find that a
particular phenomenological model of decoherence in mesoscopic quantum systems
produces a very similar dependence of transmission asymmetry on dephasing
rate.

Acknowledgements We acknowledge insightful discussions with Drs. Jochen
Mannhart and Chris Jarzynski, and assistance from Jingnan Cai. This work was
supported by ONR under grant N000142312507, NSF/ECCS/RINGS under grant
2148318, DARPA/WARDEN under grant HR00112120021, ONR/DURIP FY21 under grant
N000142112924, and ONR/DURIP FY22 under grant N000142212263.

##  Appendix A CST Simulations

Figure S1 shows the simulation schematic of the microwave ring-graph as a CST
circuit model. Each port leads to a 3-way tee junction that acts as a beam-
splitter and combiner for the incident waves from either direction, but having
finite reflection for waves arriving on all three transmission lines. The
upper branch is a finite length of uniform coaxial cable transmission line,
while the lower branch contains a model gyrator. The upper and lower branches
are chosen to have the same electrical length to eliminate Feshbach resonances
of the ring [92, 88] and both branches contain a lumped-loss variable
attenuator. The model closely approximates the experimental realization of the
AB-analogue microwave graph. The model allows introduction of lumped loss
$\Gamma_{A}$ (variable attenuator) and uniform loss $\eta$ (in all the coaxial
cables) to the graph.

The shape modes of the ring graph involve standing wave patterns around the
circumference of the ring and having a large overlap with propagating modes on
the two leads [101, 92, 88]. The resulting poles and zeros of the scattering
matrix are located relatively far from the real frequency axis [92, 88]. The
Feshbach modes are an orthogonal set of modes that have minimal overlap with
the extended modes on the leads, and create poles and zeros near the real
frequency axis,[92, 88] giving rise to narrow spectral features.

##  Appendix B Aharonov-Bohm-Analogue Microwave Graph

Figure 9 shows the schematic design and photograph of the microwave gyrator
setup. A wave travelling from port 1 to port 2 will be directed upwards and
reflect from the open circuit on the left circulator, while the wave
travelling in the opposite direction from port 2 will be directed downwards
and reflect from the short circuit on the right circulator. The open/short
circuits are designed to provide a $\pi$ phase difference for waves travelling
in opposite directions, but due to the unequal finite electrical length of the
components inside the circuits, the phase difference is not exactly $\pi$.
Thus, we add a pair of carefully adjusted phase trimmers in the design to
compensate for the difference in electrical lengths. (This basic idea of
creating 0 and $\pi$ phase shifts for left and right-going waves was inspired
by the graph presented in [102])

Figure 9: The schematic design and photograph of a microwave gyrator offering
non-reciprocal $\pi$ phase difference between the left-going waves and right-
going waves traveling between ports 1 and 2. Two nominally identical
circulators are used to guide the wave propagation directions, and a pair of
open/short circuits are used to create the $\pi$ phase difference for waves
travelling in different directions. Between the open/short circuits and the
circulators, there are a pair of phase trimmers for calibrating the phase
shift on the two vertical branches of the circuit.

The measured phase difference between $S_{12}$ and $S_{21}$ between ports 1
and 2 in Fig. 9 as a function of frequency is shown in Fig. S3. The
circulators used in the experiment have a working frequency range of $7-12.4$
GHz, and we measure the $S$-matrix of the gyrator in that frequency range.
This broad frequency range enables the time-domain measurements with wave
packets. After doing some fine-tuning of the two phase trimmers, we are able
to get the phase difference from the gyrator to be close to $\pi$ across the
frequency band. We also verify that the insertion loss of the gyrator is
symmetric (Fig. S3(b)). There are some small wiggles in these plots, which are
due to the imperfection of the circulators and resulting standing waves inside
the circuit. Note that the circulators contain microwave ferrites that are
biased by a dc magnetic field in a fixed direction, dictating the circulation
direction. The operating bandwidth of the circulators is what limits the
overall bandwidth of the entire microwave ring-graph. Further characterization
of the gyrator is given in the Supp. Mat. section “Gyrator Properties.”

Figure 4(b) shows a photograph of the assembled microwave AB-ring graph with
two identical attenuators present. Further characterization of this circuit is
given in the Supp. Mat. section “Detailed microwave ring-graph results.”

##  Appendix C Complex Time Delay

We have introduced a generalized version of time delay applicable to non-
Hermitian (in this case sub-unitary) scattering systems, such as the ring
graph with either uniform or lumped attenuation, or both. The Wigner-Smith
time delay for an $M$-port scattering system described by the $M\times M$
scattering matrix $S$ measured as a function of frequency $f$ is defined as
[84] $\tau_{W}\equiv\frac{-i}{M}\frac{d}{df}\log\det S(f)$. This definition is
a straightforward generalization of the Wigner-Smith time delay, previously
considered only in the context of unitary quantum systems, to include
variations in the magnitude of the scattering matrix with frequency. The
complex Wigner-Smith time delay as a function of frequency can be expressed in
terms of the pole and zero locations of the scattering matrix $S$ [84, 69]
(see Supp. Mat. section “Poles and Zeros of the S-Matrix of the Microwave
Ring-Graph”). We also utilize the complex reflection and transmission time
delays to identify the zeros of the reflection and transmission sub-matrices,
respectively. In this paper we evaluate the asymmetric complex transmission
time delays of the $M=2$ port microwave ring-graph as [88]
$\tau_{T}^{12}\equiv-i\frac{d}{df}\log(S_{12}(f))$ and
$\tau_{T}^{21}\equiv-i\frac{d}{df}\log(S_{21}(f))$. We have previously
utilized all of the complex time delays to characterize the shape and Feshbach
modes of the ordinary (as opposed to AB-like) ring graph in terms of the zeros
and poles of the scattering matrix and its transmission and reflection sub-
matrices [88].

SUPPLEMENTARY MATERIAL

Asymmetric Transmission Through a Classical Analogue of the Aharonov-Bohm Ring

Lei Chen, Isabella L. Giovannelli, Nadav Shaibe, and Steven M. Anlage

Here we provide additional details for the experiments and simulations
described in the text of the paper. Also included are measurements of the
poles and zeros of the scattering matrix of the AB-ring graph, and simulation
and experimental results for an AB-ring graph with an attenuator on just one
of the bonds. Finally, we make some brief comments about dephasing models.

Simulations. Figure S1 shows the schematic of the CST model of the microwave
AB-ring graph used for the simulations presented in the article. All the
transmission line blocks (boxes with yellow and black stripes) represent
homogeneous coaxial cables with inner diameter $a=0.091$ cm, outer diameter
$b=0.298$ cm, and relative dielectric constant $\epsilon_{r}=2.01$. The
coaxial cables have two sources of uniform loss, i.e. the dielectric loss
tangent of the medium $\tan\delta=0.00028$, and the resistivity of the metals
in the cable $\rho=4.4\crossproduct 10^{-3}$ $\Omega$m. These values reflect
the parameters of the physical coaxial cables used in the experiment. The
frequency dependent uniform attenuation is given by (Appendix B of [88]):

$\eta=\frac{1}{2}\left[2\pi f\tan\delta+\sqrt{\frac{2\pi
f\rho}{2\mu_{0}}}\frac{1}{\sqrt{\epsilon_{r}}}\frac{1}{\ln{(b/a)}}\left(\frac{1}{a}+\frac{1}{b}\right)\right].$
(S1)

The attenuator blocks (boxes with resistor symbol) act as lumped loss elements
which have adjustable attenuation.

Fig. S1: Simulation schematic of the two-port microwave ring-graph in CST. A
variable attenuator is included in both branches of the ring graph. The
transmission line blocks represent the coaxial cables used in the experiment.
The coaxial cables have two sources of uniform loss, i.e. the dielectric loss
tangent of the medium $\tan\delta$, and the resistivity of the metals in the
cable conductors, $\rho$. The open and short circuit blocks have a finite
length of 0.013 cm with the same relative dielectric constant of 2.01. The two
circulator blocks (denoted TS) are imported from TOUCHSTONE files as ideal
circulator $S$-matrices that are constant over the range of $0-20$ GHz. The
attenuators (boxes with resistor symbols) add a source of lumped loss. The
gyrator produces an asymmetric $\pi$ phase shift, giving the $\pi$ phase shift
to the right-to-left travelling waves. Note that the blue lines are zero-
length connections between components while the grey squares are ideal 3-way
tee junctions with finite reflection.

For the simulation, we used a total graph length of $\Sigma=0.6$ m, which
corresponds to a repetition frequency of $\Delta=\frac{c}{\Sigma}=0.5$ GHz.
The reason for this choice is that each branch has a total length of $0.3$ m,
so the transmission time through either branch is approximately $1$ ns. We can
then most clearly display, as in Figure S2, the 3:1 ratio of transmission
times as in one direction it will be $1$ ns and in the other it will be $3$ ns
due to the gyrator.

Both frequency domain and time domain simulations were done with this model.
In the frequency domain, we investigated the range of $0-20$ GHz, but only
show a small section of the full range in Fig. 2 since the behavior of
$|S_{12}|$ and $|S_{21}|$ is largely repetitive in frequency. In the case of
finite uniform attenuation $\eta$, both $|S_{12}|$ and $|S_{21}|$ decrease
overall with increasing frequency as the total loss increases, and $|S_{12}|$
decreases faster as the waves travelling from port 2 to port 1 spend more time
in the graph and therefore experience more loss.

For the time domain simulations, signals of varying duration and center
frequency were put in the model as incident pulses. Figure S2 shows the
propagation of a $1$ ns pulse with center frequency $9.7$ GHz through the
microwave ring-graph in both directions. The incident pulse, portrayed in
black, occurs at a normalized time of $0$ ns, and the transmission signal from
port 1 to port 2 in blue occurs $1$ ns later, having the same temporal length
and smaller amplitude. The transmission signal from port 2 to port 1, however,
occurs at $3$ ns and has an even smaller amplitude. There is both uniform
attenuation and a total of $0.35$ Np lumped attenuation in the graph in this
case.

Fig. S2: Simulation of pulse propagation through the microwave ring-graph
(Fig. S1) in two different directions, conducted with both uniform attenuation
and $0.35$ Np total lumped attenuation. Incident signal with center frequency
$9.7$ GHz is displayed in black while the two transmitted pulses are displayed
in blue for the signal traveling from port 1 to 2 (labeled $V_{21}$) and red
for port 2 to 1 (labeled $V_{12}$). The 3:1 time-delay asymmetry is clearly
evident, along with a significant difference in pulse amplitudes.

Gyrator Properties. The gyrator schematic and laboratory realization are shown
in Fig. 9. The measured transmission phase difference for right/left
propagation through the experimentally realized gyrator is shown in Fig. S3.
Inset (a) of that figure shows a closeup of the phase difference between 8.2
and 8.5 GHz, where the device shows nearly ideal behavior. Inset (b) shows the
insertion loss for waves propagating in both directions, showing that they are
nearly identical over the operating range of the gyrator.

Figure 4(a) shows a photograph of the assembled AB-ring microwave graph in the
case where no lumped attenuators are present. Figure 4(b) shows a photograph
of the assembled microwave AB-ring graph with two identical attenuators
present.

Fig. S3: Measured phase difference between $S_{12}$ and $S_{21}$ for the
gyrator shown in Fig. 9. The red line shows a reference phase difference of
$\pi$. Inset (a) shows a zoom-in view of the plot. Inset (b) shows the
comparison between $|S_{12}|$ and $|S_{21}|$ of the gyrator, demonstrating
nearly equal insertion loss in both directions.

Detailed microwave ring-graph results.

Figure S4 shows a plot of the two measured transmission times and pulse
amplitudes, as a function of center frequency of the 1-ns wide gaussian pulse,
over the bandwidth of the AB-ring microwave device with uniform attenuation
only (i.e. no lumped attenuation). These are the results discussed in section
IV.6 and Fig. 7 of the main text. The left axis (green) demonstrates a
consistent 3:1 ratio of the transmission time delays for pulses with different
center frequencies propagating through the microwave AB graph in opposite
directions. The right axis (blue) shows that the transmitted pulse from port 2
to port 1 always has smaller amplitude compared to the transmitted pulse from
port 1 to port 2. As the pulse center frequency increases, both amplitudes
decrease because the uniform attenuation $\eta$ of the microwave graph
increases at higher frequency [88], but the asymmetric amplitudes are
maintained. (A similar dependence of transmission amplitude vs. frequency is
seen in the noise power transmission data in Fig. 6.) The resulting pulse
transmission asymmetry $\sigma_{V}$ vs. center frequency for this time-domain
data is shown in Fig. 5(b).

Fig. S4: The two experimentally measured pulse transmission times and
amplitudes as a function of center frequency $f_{c}$ of the 1-ns wide gaussian
pulse, taken over the bandwidth of the microwave AB-graph device (7-12.4 GHz).
The left axis (green) shows the two transmission time delays, while the right
axis (blue) shows the two amplitudes of the transmitted pulses. The diamond
points represent the measured data from port 1 to port 2, while the plus-sign
points represents the measured data from port 2 to port 1.

Figure S5 shows the measured asymmetric transmission of the microwave AB-ring
with no lumped attenuation as a function of excitation power over a large
dynamic range, from -10 dBm to +10 dBm. The result illustrates that the
microwave graph creates the asymmetric transmission in the linear-response
regime.

Fig. S5: Transmission probability difference $P_{21}-P_{12}$ through the
microwave ring graph at 8 GHz versus input microwave signal power. This is a
frequency domain measurement averaged over a 100 kHz bandwidth centered at 8
GHz. The graph has no lumped attenuation. The result demonstrates the
linearity of the asymmetric transmission probability of the device. Fig. S6:
Asymmetric transmission $P_{21}-P_{12}$ through the microwave AB-ring graph
and a model mesoscopic device with similar asymmetric time delay. The green
circles and squares (with y-axis on the left) correspond to asymmetric
transmission $P_{21}-P_{12}$ of an Aharonov–Bohm microwave ring graph
simulation with zero (green circles and solid line) and finite uniform
attenuation $\eta$ (green squares and dashed line), respectively, averaged
over a period of the shape resonances, $8.25-8.75$ GHz, (denoted as
$\braket{P_{21}-P_{12}}$) as a function of the lumped attenuation
$\frac{\Gamma_{A}}{2}$ on each bond. The black symbols (with y-axis on the
right) correspond to the asymmetric transmission probability in a tight-
binding model of a triangle-shaped mesoscopic device containing an extended
model dephasing center, as a function of the average number of inelastic
scattering events per electron passage through the device, from Refs. [12,
35]. Data provided courtesy of Dr. Jochen Mannhart, Max Planck Institute for
Solid State Research, Stuttgart, Germany. Note the difference in y-axis
scales. The x-axis is the same for both quantities.

Figure S6 shows a plot of the simulated asymmetric transmission
$P_{21}-P_{12}$ for both the simulation of the Aharonov–Bohm microwave ring
graph and the tight-binding model of an asymmetric triangle-shaped mesoscopic
electron device containing an extended dephasing center, from Refs. [12, 35].
Here the asymmetric transmission is presented as functions of total lumped
attenuation in the microwave graph (in green) and average number of inelastic
scattering events per electron passage through the modeled mesoscopic device
(in black), respectively. Note that both independent degrees of freedom are
plotted on a logarithmic scale, as opposed to the linear scale used in Fig. 3.
Also note that both dimensionless rate quantities are plotted on the same
scale. The green circles and solid line are from a simulation with no uniform
attenuation ($\eta=0$) while the squares and dashed line have a finite,
frequency dependent uniform attenuation $\eta$. Remarkably, the plot of the
frequency averaged asymmetric transmission $\braket{P_{21}-P_{12}}$ agrees in
detail over many decades of attenuation/dephasing with the results of the
mesoscopic, tight binding model calculations, although the vertical axes
differ by a factor of 10. The agreement for the degree of asymmetric
transmission vs. attenuation/dephasing suggests a close connection between the
two models for wave transport [65].

Poles and Zeros of the $S$-Matrix of the Microwave Ring-Graph. The AB-ring
graph resonance properties in the frequency domain can be understood from the
locations of the poles and zeros of the scattering matrix in the complex
frequency plane. It is known that uniform attenuation preserves the complex
conjugate relationship of the $S$-matrix zeros and poles for each mode, but
adds an overall offset $f\rightarrow f+i\eta$ in the definition of the
$S$-matrix [84, 88]. The lumped loss $\Gamma_{A}$ will act on the poles and
zeros differently, allowing the real and imaginary parts of the poles and
zeros to vary independently [84]. In particular, the zeros of the $S$-matrix
can move across the real frequency axis and go into the lower-half of the
complex frequency plane [103].

Fig. S7: Locations of the zeros (diamonds) and poles (squares) of the
$S$-matrix in the complex frequency plane from the AB-ring graph simulation in
the case of no uniform attenuation ($\eta=0$) and variable lumped attenuation
$\Gamma_{A}/2$. The zeros and poles are complex conjugates of each other when
the lumped loss $\Gamma_{A}=0$. As the total attenuation on the two bonds
increases, represented by the color shift from blue to green (color bar on the
right), the imaginary components of both the zeros and poles decrease while
the real components remain unchanged. The zeros cross the real frequency axis
at $\Gamma_{A}/2\approx 0.52$ Np. A representative frequency range including 7
modes is shown.

Figure S7 shows the motion of the zeros and poles (represented by diamonds and
squares, respectively) as the lumped loss $\Gamma_{A}/2$ is increased from $0$
to $0.86$ Nepers in the simulation of the AB-like microwave ring graph with
$\eta=0$. The zero and pole locations are obtained by fitting the Wigner-Smith
time delay $\tau_{W}$ as a function of frequency. The complex Wigner-Smith
time delay is defined as $\tau_{W}\equiv\frac{-i}{M}\frac{d}{df}\log\det
S(f)$, where $M$ is the number of ports, and is calculated from the measured
(or simulated) scattering matrix in the frequency domain. These frequency
dependent functions are simultaneously fit to these expressions [84, 88]:

$\text{Re}\>\tau_{W}=\frac{1}{M}\sum\limits_{n=1}^{N}[\frac{\text{Im}[z_{n}]-\eta}{(f-\text{Re}[z_{n}])^{2}+(\text{Im}[z_{n}]-\eta)^{2}}+\\\
\frac{\Gamma_{n}+\eta}{(f-f_{n})^{2}+(\Gamma_{n}+\eta)^{2}}]$ (S2)
$\text{Im}\>\tau_{W}=-\frac{1}{M}\sum\limits_{n=1}^{N}[\frac{f-\text{Re}[z_{n}]}{(f-\text{Re}[z_{n}])^{2}+(\text{Im}[z_{n}]-\eta)^{2}}-\\\
\frac{f-f_{n}}{(f-f_{n})^{2}+(\Gamma_{n}+\eta)^{2}}]$ (S3)

where the complex frequency locations of the poles
($\varepsilon_{n}=f_{n}-i\Gamma_{n}$, with the convention that $\Gamma_{n}>0$
in passive lossy systems) and zeros
($z_{n}=\text{R}e[z_{n}]+i\text{Im}[z_{n}]$) of each mode $n$ are used as
fitting parameters. As before, $\eta$ is the uniform attenuation. For the
simulations used to generate $S$-matrix data that is analyzed in Fig. S7, the
increments in lumped attenuation are uniform, but the changes in the imaginary
components of the zeros and poles are not, with the changes becoming more
pronounced as the zeros approach the real axis. Interestingly, the zeros cross
the real axis at around $\Gamma_{A}/2=0.52$ Np, while the peak in asymmetric
transmission $\braket{P_{21}-P_{12}}$ occurs at a little under
$\Gamma_{A}/2=0.3$ Np, suggesting the two events may be related. Note that the
fits to complex $\tau_{W}$ are dominated by the zero locations when they are
in the vicinity of the real frequency axis, hence the pole locations are not
determined for $\Gamma_{A}/2>0.45$ Np.

Single-attenuator microwave AB-ring graph. A version of the AB-ring graph was
also constructed with a single lumped attenuator on the lower bond, the
schematic of which is shown in Figure S8. This version of the graph is meant
to be a closer analogue of the schematic form shown in Fig. 1(a). A photograph
of the experimentally realized graph is shown in Fig. S11(b). The un-balanced
nature of the attenuation in the two bonds leads to more complicated behavior
of the scattering properties as the attenuation value is increased.

Fig. S8: Simulation schematic of the microwave ring-graph in CST. This differs
from Fig. S1 as a variable attenuator $\Gamma_{A}$ is added to only the branch
with the gyrator. That is the only change between the two schematics.

Figure S9(a) shows the simulated real part of the complex transmission time
delay through the un-balanced version of the AB-ring graph as a function of
frequency. The 3:1 asymmetry in transmission time delays is clearly seen,
along with variations associated with the strong shape resonances. Figure
S9(b) shows the simulated forward and reverse transmission amplitudes as a
function of frequency for the case of lumped attenuation $\Gamma_{A}=0.35$ Np,
with zero uniform attenuation. The case of no loss is not included here as it
is identical to having two attenuators set to $0$ Np, shown in Figure 2,
resulting in the graph having a unitary scattering matrix and symmetric
transmission. In the $0.35$ Np lumped attenuation case the transmission is
asymmetric for all frequencies except discrete values that repeat at intervals
of the periodicity $\Delta=c/\Sigma=0.5$ GHz. Figure S9(c) shows the
difference in simulated transmission probabilities through the graph as a
function of frequency in the unbalanced $0.35$ Np case, while the $0$ Np case
has $\langle P_{21}-P_{12}\rangle=0$ at all frequencies.

Figure S10 is the time domain analogue of Figure S9(a). Similar to Figure S2,
there is both uniform attenuation and $0.35$ Np lumped attenuation in the
simulation, but now all the lumped attenuation is located on one branch. The
unbalanced attenuation creates a favored path for the waves to travel along,
with a larger amplitude propagating through the branch without the lumped
attenuation. When the waves travelling on the un-attenuated upper bond and
attenuated lower bond later meet at a tee junction, they do not have the same
amplitude. So despite the gyrator adding a $\pi$ phase shift to one side in
one direction, the waves don’t perfectly interfere and reflect, and a small
amount is transmitted. This is the cause of the small blue signal seen at $3$
ns and red signal at $1$ ns in Fig. S10. The reflected incident voltage at $2$
ns is due to the tee junctions having some intrinsic finite reflection, and
now due to the unbalanced nature of the graph a small amount of the incident
signal will return to escape through its input port, having made a complete
return trip through the graph.

Fig. S9: (a) Real part of transmission time delays of the Aharonov-Bohm
microwave ring graph simulation for the case of $0.35$ Np attenuation on only
the bottom branch and uniform attenuation throughout (see the schematic in
Fig. S8). Blue curve shows $\text{Re}[\tau_{21}]$ and red curve shows
$\text{Re}[\tau_{12}]$, and the dashed lines show the mean values,
demonstrating a 3:1 ratio. (b) Linear transmission magnitudes in both
directions for the case of $0.35$ Np unbalanced attenuation. Blue corresponds
to $|S_{21}|$ while red corresponds to $|S_{12}|$. With no attenuation the
transmission magnitudes are identical. The scale bar gives the expected
periodicity frequency scale ($\Delta=0.5$ GHz) for the shape resonances of a
ring graph. (c) The asymmetric transmission probability $P_{21}-P_{12}$ for
the case of $0.35$ Np unbalanced attenuation. The $0$ Np case was not
displayed in panels (b) and (c) as it is the same as that shown in Figure 2.
Fig. S10: CST simulation of pulse propagation through the AB-ring graph in two
different directions for the case of attenuation on only one bond (Fig. S8).
The incident signal with center frequency $8.5$ GHz is displayed in black
while the two transmitted pulses are displayed in blue for the signal
traveling from port 1 to 2 and red for port 2 to 1. The 3:1 time-delay
asymmetry is clearly evident, along with a significant difference in pulse
amplitudes. The unbalanced attenuation in the ring causes the waves
propagating along the two bonds to not have equal magnitudes, hence they do
not perfectly interfere at the combiners, causing some degree of reciprocal
transmission and a non-zero reflection at the input port. Compare with Fig.
S2.

Figure S11 shows the experimental results for the asymmetric transmission
through a microwave AB-ring graph having un-balanced attenuation $\Gamma_{A}$
in the bonds, as shown schematically in Fig. S8. The experiment includes
finite uniform attenuation $\eta$ as well. Figure S11(a) shows a schematic of
the AB-ring graph with one single attenuator. Figure S11(b) shows the
experimental realization of Figure S11(a). Figure S11(c) repeats the
asymmetric transmission results obtained from simulations of the quantum
system shown in the inset, as presented in the main text. Figure S11(d) shows
the frequency domain asymmetric transmission results $\braket{P_{21}-P_{12}}$
for both experiment (green diamonds) and simulation (green solid and dashed
lines). Figure S11(e) shows the time domain results $\sigma_{V}$ for both
experiment (red diamonds) and simulation (red solid and dashed lines).

The un-balanced AB-ring-like microwave graph retains all of the asymmetric
properties of the fully balanced version, but these features are obscured to
some extent by the design of the graph. Hence even a literal interpretation of
the graph shown in Fig. 1(a) still shows the same asymmetric transmission
properties of the balanced version of the graph.

Fig. S11: Asymmetric transmission through an AB-analogue microwave ring graph
in the single attenuator case, shown schematically in Fig. S8. (a) Shows a
schematic of the AB-ring graph with only one attenuator on one bond. (b) Shows
the experimental realization of the schematic in (a). (c) Asymmetric
transmission probability in a tight-binding model of triangle-shaped
mesoscopic device with a model dephasing center, as a function of the average
number of inelastic scattering events per electron passage through the
structure. From Refs. [12, 35]. Data provided courtesy of Dr. Jochen Mannhart,
Max Planck Institute for Solid State Research, Stuttgart, Germany. (d) Summary
of the frequency domain asymmetric transmission results. For all curves, the
quantity $P_{21}-P_{12}$ is averaged over the frequency range $8.25-8.75$ GHz,
which is denoted as $\braket{P_{21}-P_{12}}$. The solid line is from
simulation where only the localized attenuation from the single attenuator is
considered. The dashed line is also from simulation but for the more realistic
case where finite uniform attenuation throughout the device is taken into
account. The discrete diamond points are from experimental data. (e) Summary
of time domain results. For these time domain measurements, a 1 ns wide
Gaussian pulse with a center frequency of 8.5 GHz is sent into the device. The
solid line is from simulation where only the localized attenuation from the
single attenuator is considered. The dashed line is also from simulation but
for the more realistic case where finite uniform attenuation throughout the
device is taken into account. The discrete diamond points are from
experimental data.

Comments about dephasing models. Although the work in this paper is strictly
classical, it is of interest to discuss models of quantum dephasing, since
they originally motivated our work. The authors of Refs.[35, 12] contend that
it is advantageous to employ a finite amount of decoherence, implemented
through inelastic scattering, to create asymmetric transmission in a model
mesoscopic device, suggesting that working in the regime between purely
quantum and purely classical physics may be optimal for establishing a net
transmission bias that spontaneously pumps charge from one equivalent
reservoir to another. Their model AB ring graph with partial dephasing
effectively treats excitations moving in one direction more as waves, while
the excitations travelling in the other direction are treated more as
particles [104, 12]. It is an open question whether this approach to creating
asymmetric transmission can be implemented in a mesoscopic metal or molecule.
Proposals have been put forward [33, 38, 8, 35, 12] that broken time-reversal
invariance at the microscopic level can allow one to bypass constraints
imposed by the Onsager relations, or restrictions against spontaneous currents
[105, 33].

Other possible experimental settings in the intermediate range between
coherent quantum and purely classical behavior (in addition to Ref. [40])
include the following. A mesoscopic AB-ring made of a clean metal, or ring-
like molecule, with engineered dephasing centers have been proposed to show
asymmetric transmission in equilibrium [106, 12]. Another possibility is an
AB-ring defined by interfaces supporting topological edge modes, as proposed
for layers of buckled silicene [107]. Including a defect that creates inter-
valley scattering of the electron wavepackets could generate bi-directional
asymmetry of transport through the ring. Finally, it has been proposed that a
class of monitored quantum devices can show a non-reciprocal current between
reservoirs even in the absence of an applied bias [100].

The results in this paper suggest that simple treatments of dephasing may be
effectively equivalent to the PCP approach to describing the effects of loss
in classical wave scattering systems. Although the present work does not
address issues in thermodynamics, we note that improper treatment of quantum
dephasing have been shown to result in erroneous predictions for the
properties of the system, including violations of the second law of
thermodynamics [108, 109]. Alternative models of quantum dephasing have also
been developed, and may be better suited for understanding the properties of
mesoscopic systems. One example of such an approach is the class of
collisional models which consider interactions between the quantum system and
ancillary environmental qubits, through a series of discrete interactions
(collisions)[15, 17]. Many-body treatments of dephasing have also been pursued
[110]. In general, it would appear that a global quantum treatment of the
system and surroundings together is necessary to properly address issues of
dephasing [109].

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