# Plug&Play subcarrier wave quantum key distribution with deep modulation

Oleg I. Bannik1 and E. S. Moiseev1,2  1Kazan Quantum Center, Kazan National
Research Technical University, 18a Chetaeva str.,
Kazan, 420111, Russia
2KQT LLC, 41-022 Galiaskar Kamal str., Kazan, 420021, Russia
<EMAIL_ADDRESS>
###### Abstract

We report a quantum key distribution using subcarrier wave encoding in
Plug&Play configuration. Our detailed study shows how subcarrier encoding
operates in a presence of Rayleigh back-scattering, chromatic dispersion and
other experimental negative factors that occur in Plug&Play configuration. We
propose and experimentally demonstrate a novel regime for the subcarrier wave
encoding, where simultaneous detection of two logical states in a single basis
is combined with robustness and simplicity of the original protocol. This deep
modulation regime shows reliable quantum communication with losses up to 20 dB
and promises a twice increase in secret key generation rate in comparison with
the original subcarrier wave quantum key distribution in one-way or Plug&Play
configurations.

††journal: oe††articletype: Research Article

##  1 Introduction

Nowadays, quantum key distribution (QKD) is one of the most developed quantum
technologies. In QKD, two authenticated parties, commonly referred as Alice
and Bob, create secure keys by exploiting quantum properties of light. Modern
conventional cryptography is based on the assumption of computational
complexity and is vulnerable to an attacker with sufficient computing power,
while QKD combined with a one-time pad encryption technique is able to provide
a way for achieving unconditional confidentiality of information transmission
[1]. Over the last two decades a huge effort has been carried out to build
commercial QKD systems [2, 3].

One of the promising practical QKD encoding methods is subcarrier wave (SCW)
phase encoding [4]. SCW QKD originally is a one-way system and it is notable
for its interferometer-free optical scheme and robustness to environmental
effects on a fiber line [4, 5]. Meanwhile, Plug&Play configuration [6, 7, 8]
is convenient for several applications such as client-server QKD networks [9],
practical SCW twin-field implementations [10] and creating a cheaper client
device. The system in the Plug&Play configuration uses the double-pass scheme:
Bob sends a reference laser beam to Alice, which she subsequently modulates to
encode information and redirects it back to Bob. Double-pass scheme is a way
to compensate for polarization distortions in an optical fiber and to avoid
complex active compensation. Plug&Play configuration allows to combine most of
the bulky and expensive parts like a laser source, an optical filter,
detectors and optical switches on the Bob’s side. Alice’s client device can be
greatly simplified, resulting in reduced deployment cost and improved
scalability of a such quantum client-server network. Thus evaluation of SCW
QKD in Plug&Play configuration is of certain practical interest.

Rayleigh backscattering from the strong reference beam is one of the key
problems in QKD systems with the double-pass optical design. It considerably
increases false counts and hence limits communication distance [11]. Several
methods have been proposed to circumvent this problem such as an introduction
of time separation between signal and Rayleigh backscattered radiation through
the storage line [12], use of a separate fiber line to deliver the reference
radiation [8], or use of several spectral channels and sequential switching
between them [13]. These methods successfully help to reduce negative effect
of backscattered radiation, but at the cost of limiting the communication
speed, or increasing complexity and overall cost of a QKD setup. Considered
SCW type of encoding is less susceptible to negative effects of Rayleigh
backscattering. This is due to the fact that signal sideband photons carrying
encoded information are partially frequency separated from the back-scattered
photons and there is also no need to send powerful optical pulses except for
the carrier providing the required power level at the output of Alice’s
device.

Here we theoretically analyze and experimentally implement SCW QKD in
Plug&Play configuration. First we develop a mathematical model of SCW quantum
key distribution in the Plug&Play configuration. We derive an expression for
secret key generation for the given set of control parameters taking into
account different practical aspects such as losses, Rayleigh backscattering,
chromatic dispersion, and non-ideal optical filters and detectors. Next, we
propose and analyze a new regime of SCW QKD, namely deep modulation, which
allows to increase key generation rate twice. The key improvement is an
ability to distinguish two logical states in a single measurement in the deep-
modulation regime in contrast to one state in the original scheme [5]. Then we
find the optimal control parameters for Plug&Play SCW QKD depending on the
communication distance. Finally, we describe the experimental setup and
perform an experimental SCW QKD with deep modulation. At the end, we discuss
the results and perspective of Plug&Play SCW QKD and the proposed new regime.

##  2 Mathematical description

###  2.1 Basic model

In the Plug&Play configuration, Bob provides an optical reference signal to
Alice via an optical channel with power attenuation coefficient $\eta$. It
consists from a train of coherent-state pulses with duration $T$ each, carrier
frequency $\omega_{0}$ and complex amplitude $\alpha_{0}$. Alice encodes
information by applying a sinusoidal phase-modulation with modulation depth
$\beta$, modulation frequency $\Omega$, and phase $\phi_{A}$ to the received
reference light. An optical state after Alice’s phase modulator is described
by wavefunction

$\displaystyle\ket{\psi}=\bigotimes^{\infty}_{m=-\infty}\ket{\sqrt{\eta}\alpha_{0}J_{m}(\beta)e^{im\phi_{A}}}_{m},$
(1)

where multiplication is carried over $m\in\mathbb{Z}$ indices of sidebands
with frequencies $\omega_{0}+im\Omega$, $J_{m}(\beta)$ is m-th order Bessel
function of the first kind.

Following BB84-family of protocols [14], a single basis set is formed by pairs
of states (1) with $\phi_{A}$ being shifted by $\pi$ with respect to each
other. To implement BB84 protocol at each clock cycle Alice chooses random
value for basis and logical bit. It is done by randomly picking $\phi_{A}$
either from $\\{``0"\rightarrow 0,``1"\rightarrow\pi\\}$ or from
$\\{``0"\rightarrow\pi/2,``1"\rightarrow 3\pi/2\\}$ sets. Here a right arrow
indicates correspondence between logical state marked by braces and phase to
be chosen to the right of the arrow.

Alice sends state (1) to Bob by reflecting it back through the same lossy
fiber. To decode Alice’s bit at each clock cycle, Bob applies phase modulation
to his optical channel with the same modulation depth $\beta$, modulation
frequency $\Omega$ and his own phase $\phi_{B}$. After that, Bob separates
signal on the sidebands from a carrier frequency by sending signal through an
optical frequency filter with extinction ratio $\rho$. The subsequent photon
detection in the sidebands reveals the encoded information, if Bob has sifted
both basis and phases $\phi_{A}=\phi_{B}$ correctly.

Probability of the detector to click is

$\displaystyle
P_{\text{det}}(\phi_{a},\phi_{b})=\left(\eta_{D}\frac{n_{\text{ph}}(\phi_{A},\phi_{B})}{T}+\gamma_{\text{det}}+\eta_{D}\frac{n_{\text{rs}}}{T}\right)\delta
t,$ (2)

where $\eta_{D}$ is a quantum efficiency of a single photon detector, $\delta
t$ is its gating time and $\gamma_{\text{det}}$ is a dark count rate,
$n_{\text{ph}}$ and $n_{\text{rs}}$ are average numbers of Alice’s and
Rayleigh back-scattered photons, that reach Bob’s detector, respectively. To
quantify the number of photons that reaches Bob’s detector $n_{\text{ph}}$, we
consider two situations where Bob’s phase equals to Alice’s one
($\phi_{A}=\phi_{B}$) or differs by $\pi$ ($\phi_{A}=\phi_{B}+\pi$), whereas
we assume that Bob has sifted the basis correctly. Ideally, if Bob’s and
Alice’s phases coincide, the sidebands are populated as if it was a double
modulation, otherwise all photons are demodulated into the carrier.

A finite extinction ratio of the frequency filter leads photons on a carrier
frequency being detected. We model filter as a Bragg grating mirror, that
reflects only a carrier frequency ($m=0$) with power coefficient $r$ and
transmits with power coefficient $\tau$. Meanwhile, photons on the sidebands
($m\neq 0$) are transmitted through the filter with power coefficient
$1-\varrho$ and reflected with power coefficient $\varrho$, where $\varrho$ is
called as filter’s sideband suppression factor. Taking into account this
effect, the average number of Alice’s photons at Bob’s detector is

$\displaystyle
n_{\text{ph}}(\phi_{A},\phi_{B})=|\alpha_{0}|^{2}\eta^{2}\cdot\begin{cases}(1-\varrho)(1-J^{2}_{0}(2\beta))+\tau
J^{2}_{0}(2\beta)\quad&\phi_{A}=\phi_{B}\\\ \tau\quad&\phi_{A}=\phi_{B}+\pi\\\
\end{cases},$ (3)

where we assume negligibly small jitter between Alice’s and Bob’s modulation
oscillators over a repetition period.

The detector catches not only signal photons arriving from Alice, but as well
noisy photons generated by Rayleigh backscattering $n_{\text{rs}}$. Their
amount is mainly determined by an intensity of the reference beam
$\alpha_{0}$, length of the fiber and Rayleigh backscattering coefficient
($\beta_{\text{RS}}\approx-40$ dB for a single-mode fiber) [11]. Portion of
these photons is frequency shifted into modulation sidebands by Bob’s phase
modulation, that reduces the number of detected noise photons by factor of
$1-J^{2}_{0}(\beta)$. The resultant number of photons from Rayleigh
backscattering at Bob’s detector is

$\displaystyle n_{\text{rs}}\approx\left((1-\rho)(1-J^{2}_{0}(\beta))+\tau
J^{2}_{0}(\beta)\right)(1-\eta^{2})\beta_{\text{RS}}|\alpha_{0}|^{2}.$ (4)

Next, we simulate secret key generation rate against beam splitter collective
attack. Secret key rate $K$ is lower limited by Devetak-Winter bound [15, 16]:

$\displaystyle K=\nu P_{B}(1-f(Q)H(Q)-\chi(A:E)),$ (5)

where $\nu=1/T$ is a repetition rate and $P_{B}$ is probability of bit
detection at Bob’s side per pulse. Probability $P_{B}$ can be represented as a
total probability of detecting a photon with correct and wrong bit decoding.
Namely it is a probability that Alice encodes logical bit value x and Bob get
the same bit value x or opposite bit value: $(x+1)\text{mod}~{}2$. For $x=0$
it is $P_{B}=(P_{\text{det}}(0,0)+P_{\text{det}}(0,\pi))/2$, where factor
$1/2$ comes from the 2 bases used in BB84 protocol. The product $f(Q)H(Q)$
defines an amount of information revealed during error correction with $f(Q)$
being an efficiency of error correcting protocol and $H(Q)$ being a binary
Shannon entropy:

$\displaystyle H(Q)=-Q\log_{2}(Q)-(1-Q)\log_{2}(1-Q).$ (6)

Here $Q$ is a quantum bit error probability (QBER), which is a ratio of
detection with false decoding to the probability of detection $P_{B}$:

$\displaystyle
Q=\frac{P_{\text{det}}(0,\pi)}{P_{\text{det}}(0,0)+P_{\text{det}}(0,\pi)}.$
(7)

The term $\chi(A:E)$ is an upper bound for an amount of information that is
accessible to an eavesdropper Eve for a given collective attack. It was shown
[16] that a collective beam splitter attack provides the following information
to Eve:

$\displaystyle\chi(A:E)=H\left(\frac{1}{2}\left(1-e^{-|\alpha_{0}|^{2}\eta(1-\eta)(1-J_{0}(2\beta))}\right)\right).$
(8)

In practice, there are different losses in a QKD system that do not contribute
to Rayleigh backscattering (4) such as insertion losses of the optical
elements, monitoring photodiodes, etc. We include these losses into the model
by dividing overall losses $\eta=\eta_{A}\eta_{B}\eta_{F}$ into single pass
losses in the Alice’s encoder $\eta_{A}$, losses in Bob’s detection path
$\eta_{B}$ and losses in optical fiber $\eta_{F}$. Next we assume, that
Rayleigh backscattering is associated only with fiber-related losses
$\eta_{F}$. It is done by replacing $(1-\eta^{2})$ in Eq. (4) with
$(1-\eta_{F}^{2})\eta_{B}$. The single pass in Bob’s detection path and double
pass in fiber and Alice’s device are attributed by replacing $\eta^{2}$ with
$\eta^{2}_{A}\eta^{2}_{F}\eta_{B}$ in Eq. (3). We assume Alice’s and Bob’s
devices are trusted, i.e. photons that are lost in the devices are not
intercepted by Eve. In this case, the mutual information in Eq. (7) is
modified by replacing $\eta(1-\eta)$ with $\eta^{2}_{A}\eta_{F}(1-\eta_{F})$.
Eventually, we estimate the key rate by first combining Eq. (3) and Eq. (4) in
Eq. (2), that is used to calculate QBER (7). The resulted QBER entropy and
eavesdropper’s mutual information from Eq. (8) is used to calculate the key
rate bound in Eq. (5).

###  2.2 Chromatic dispersion

So far we have ignored an effect of chromatic dispersion on the QKD
performance. It is a valid approximation if dispersion compensator is used on
Alice’s side. Otherwise the dispersion should be included into analysis at
distances longer than 25 km [17].

The chromatic dispersion in optical fibers manifests itself as nonlinear
frequency-dependant phase shift. It is usually expressed as the Taylor
expansion of a wavenumber in the vicinity of central frequency $\omega_{0}$
[18]:

$\displaystyle k(\omega-\omega_{0})=k_{0}+\frac{\partial
k}{\partial\omega}\bigg{|}_{\omega_{0}}\left(\omega-\omega_{0}\right)+\frac{1}{2}\frac{\partial^{2}k}{\partial^{2}\omega}|_{\omega_{0}}\left(\omega-\omega_{0}\right)^{2}+...$
(9)

The values $\frac{\partial k}{\partial\omega}=1/v_{g}$ and
$\frac{\partial^{2}k}{\partial^{2}\omega}=\text{GVD}$ are usually expressed as
inverse group delay ($1/v_{g}$) and group velocity dispersion (GVD)
respectively. Further on we assume value of GVD to be $-2.0407\cdot 10^{-23}$
$\text{s}^{2}/$km as in G.652 single mode fiber.

In SCW QKD, where Alice sends a wavepacket with equidistantly separated
sidebands, the chromatic dispersion modifies the state (1) sent by Alice to
Bob by an additional nonlinear phase modulation acquired over distance $d$:

$\displaystyle\ket{\Psi}\rightarrow\ket{\Psi^{\prime}(\beta,\phi_{A})}=\bigotimes^{\infty}_{m=-\infty}\ket{\eta_{A}\eta_{F}\sqrt{\eta_{B}}\alpha_{0}J_{m}(\beta)e^{im\phi_{A}+im\Omega(1/v_{g}+m\cdot\text{GVD}\cdot\Omega/2)d}}_{m},$
(10)

where distance $d$ is related to the fiber losses as $\eta_{F}=d\cdot
0.2\text{dB/km}$. The effect of inverse group delay is easily compensated by
an adjusting phase difference between Alice’s and Bob’s modulation signals by
$d\Omega/v_{g}$. The contribution of quadratic effect is not that
straightforward, since opposite sideband pairs accumulate destructive phase
difference at different distance [17]. We include an effect of GVD on the
secret key rate generation by replacing simplified value of average number of
photons in Eq. (3) with:

$\displaystyle
n^{\prime}_{\text{ph}}(\phi_{A},\phi_{B})=(1-\rho)\bra{\Psi^{\prime}(\beta,\phi_{A})}U^{\dagger}(\beta,\phi_{B})\left(\sum_{m\neq
0}a^{\dagger}_{m}a_{m}\right)U(\beta,\phi_{B})\ket{\Psi^{\prime}(\beta,\phi_{A})}$
$\displaystyle+\tau\bra{\Psi^{\prime}(\beta,\phi_{A})}U^{\dagger}(\beta,\phi_{B})a^{\dagger}_{0}a_{0}U(\beta,\phi_{B})\ket{\Psi^{\prime}(\beta,\phi_{A})},$
(11)

where $a^{\dagger}_{m}$ ($a_{m}$) is a photon creation (annihilation) operator
at $m$’th sideband, $U(\beta,\phi_{B})$ is beam-splitter-like unitary operator
representing an action of Bob’s phase modulator such that

$\displaystyle
U^{\dagger}(\beta,\phi_{B})a_{k}U(\beta,\phi_{B})=\sum^{\infty}_{l=-\infty}J_{l}(\beta)a_{k+l}e^{il\phi_{B}}.$
(12)

It is worth noting, that in general more sophisticated model of electro-
optical modulator [19, 20] should be used as infinite number of indices $l$
leads to nonphysical sidebands with negative frequencies. However for
practical modulators and moderate values of modulation depths ($\beta<4$)
simple model as in Eq. (12) is a good approximation as we can limit summation
to the first five pairs of sidebands which contain more than 99% of total
energy in all sidebands. By substituting Eq. (11) instead of Eq. (3) in Eq.
(2), we calculate the key rate bound by Eq. (5) similarly as in case without
chromatic dispersion.

###  2.3 Deep modulation

In original SCW QKD systems, a strong laser beam is phase modulated by Alice
to produce weak sidebands with given phases, that encode quantum information
with respect to 0-th order of modulation. The measurement is performed as
single photodetection at sidebands wavelengths only [4]. In the case of using
deep modulation regime where $\beta=\beta_{\text{DM}}\approx 1.2$ such that
$J_{0}(2\beta_{\text{DM}})=0$, detection can be carried out on both carrier
and subcarrier wavelengths. Thus, it is easy to see, that the proposed regime
allows to distinguish two phases $\phi_{A}=\phi_{B}$ and
$\phi_{A}=\phi_{B}+\pi$ during a single measurement.

Let Bob choose phase $\phi_{B}$ at each clock cycle out of two values 0 or
$\pi/2$. If Bob picked the Alice’s basis, all optical energy is transferred
into the sidebands when $\phi_{A}=\phi_{B}$. However, if
$\phi_{A}=\phi_{B}+\pi$ the energy is completely transferred into the carrier.
For small amplitudes $|\alpha_{0}|^{2}J^{2}_{0}(\beta_{\text{DM}})<1$ presence
of a photon at carrier or sideband frequencies can be detected by placing
single photon counting modules at carrier and sidebands channels after the
spectral filter, that separates carrier from the sidebands. Therefore, Bob can
distinguish $\phi_{A}$ in given basis without an explicit match as in the
original SCW encoding. At a proper intensity level, the protocol is capable of
performing QKD with roughly twice faster key generation rate as we show below.

We still use Eq. (5) to bound key generation rate. The only things we
recalculate are QBER and $P_{b}$ Eq. (7), as now we can distinguish two
logical states. We find these values by first introducing two identical
detectors Dc and Dsb, that detect photons after reflection from the filter
(carrier mode) or after transmission trough the filter (sideband’s modes)
respectively. The probability of getting a click for the detector Dsb is
similar to Eq. (2):

$\displaystyle
P_{\text{Dsb}}(\phi_{A},\phi_{B})=\bigg{(}\frac{\eta_{D}\tau}{T}\bra{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}U^{\dagger}(\beta_{\text{DM}},\phi_{B})a^{\dagger}_{0}a_{0}U(\beta_{\text{DM}},\phi_{B})\ket{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}$
$\displaystyle+\frac{\eta_{D}(1-\rho)}{T}\bra{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}U^{\dagger}(\beta_{\text{DM}},\phi_{B})\bigg{(}\sum_{m\neq
0}a^{\dagger}_{m}a_{m}\bigg{)}U(\beta_{\text{DM}},\phi_{B})\ket{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}$
$\displaystyle+\gamma_{\text{Det}}+\frac{\eta_{D}}{T}n_{\text{rs}}\bigg{)}\delta
t,$ (13)

in turn, probability of detecting photon at the carrier frequency is

$\displaystyle
P_{\text{Dc}}(\phi_{A},\phi_{B})=\bigg{(}\frac{\eta_{D}r}{T}\bra{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}U^{\dagger}(\beta_{\text{DM}},\phi_{B})a^{\dagger}_{0}a_{0}U(\beta,\phi_{B})\ket{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}$
$\displaystyle+\frac{\eta_{D}\varrho}{T}\bra{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}U^{\dagger}(\beta_{\text{DM}},\phi_{B})\left(\sum_{m\neq
0}a^{\dagger}_{m}a_{m}\right)U(\beta_{\text{DM}},\phi_{B})\ket{\Psi^{\prime}(\beta_{\text{DM}},\phi_{A})}+\frac{\eta_{D}}{T}n_{\text{rs}}^{\prime}+\gamma_{\text{Det}}\bigg{)}\delta
t.$ (14)

Here the number of noise photons reaching detector Dc due to Rayleigh
backscattering $n_{\text{rs}}^{\prime}$ differs from Eq. (4) by replacing
$(1-\rho)(1-J^{2}_{0}(\beta))$ and $\tau J^{2}_{0}(\beta))$ with
$rJ^{2}_{0}(\beta_{\text{DM}})$ and $\rho(1-J^{2}_{0}(\beta_{\text{DM}}))$,
respectively.

Similarly as in Eq. (7), we calculate QBER as a ratio between probability of
erroneous bit decoding to total probability of bit decoding from Alice to Bob

$\displaystyle Q=\frac{P_{E}}{P_{C}+P_{E}},$ (15)

with probability of erroneous bit decoding is

$\displaystyle
P_{E}=P_{\text{Dsb}}(\phi_{A},\phi_{A}+\pi)(1-P_{\text{Dc}}(\phi_{A},\phi_{A}+\pi))+P_{\text{Dc}}(\phi_{A},\phi_{A})(1-P_{\text{Dsb}}(\phi_{A},\phi_{A}))$
(16)

and probability of correct bit decoding is:

$\displaystyle
P_{C}=P_{\text{Dsb}}(\phi_{A},\phi_{A})(1-P_{\text{Dc}}(\phi_{A},\phi_{A}))+P_{\text{Dc}}(\phi_{A},\phi_{A}+\pi)(1-P_{\text{Dsb}}(\phi_{A},\phi_{A}+\pi)).$
(17)

The derived QBER (15) together with Eq. (8) is used in Eq. (5) to estimate key
generation rate in deep modulation regime. For the beam splitter attack in
deep modulation regime the amount of information accessible to Eve is still
quantified by Eq.(8) as we show in the appendix.

The deep modulation regime can be straightforwardly used with decoy states
method to counter photon number splitting attack [21, 22, 23], if Alice
applies amplitude modulation and optical phase randomization to the pulses at
her site. The asymptotic secret key rate can be estimated by modified
Gottesman, Lo, Lutkenhaus, and Preskill bound [22, 24]:

$\displaystyle
K=\frac{\nu}{2}\left(P_{1}\left(1-H(Q_{1})\right)-P_{\alpha}f(Q)H(Q)\right),$
(18)

where $P_{\alpha}$ is a fraction of detection events per pulse (gain) due to
signal coherent state with given amplitude $\alpha$, $P_{1}$ is a fraction of
single photon detection events, $Q_{1}$ is a QBER for single photon component.
The later two values are estimated using decoy states method as described in
[24]. However in our experiment we were not able to implement decoy states
method due to electronic hardware’s constrains. Thus for a sake of consistency
in differential comparison of original SCW and deep modulation regime further
on we consider key generation rate in both regimes in a presence of the
collective beam-splitter attack.

###  2.4 Optimization

Parameter | Description
---|---
$\eta_{D}=0.1$ | Detector quantum efficiency
$\Omega=2\pi\cdot$ 5 GHz | Modulation frequency
$\gamma_{\text{det}}=50$ Hz | Detector’s dark count rate
$\tau=0.01$ | Filter’s carrier transmittance
$r=0.99$ | Filter’s carrier reflection
$\varrho=-38$ dB | Filter’s sideband suppression factor
$\text{GVD}=-2.0407\cdot 10^{-23}$ $\text{s}^{2}/$km | Group velocity dispersion
$1/v_{g}=4.9\cdot 10^{-6}$ s/km | Inverse group delay
$\delta t=$3.3 ns | Gating time
$T=$10 ns | Pulse repetition period
$\eta_{A}=-6$dB | Alice’s device losses
$\eta_{B}=-6$dB | Bob’s device losses
Table 1: Model parameters

We numerically optimize the key generation rate (5) at given $\eta_{F}$ for
the original and proposed deep modulation regime of SCW QKD in Plug&Play
configuration. In our calculation we use experimentally accessible parameters,
which are summarized in Table 1. We consider Alice and Bob both having
identical losses $\eta_{A}=\eta_{B}=-6$ dB in their devices and assume
moderate efficiency of reconciliation protocol $f(Q)=1.25$. In Fig. 1 (a) we
present a key generation rate for optimal control parameters with and without
dispersion compensation.

As it was expected the deep modulation regime increases key generation rate
twice at low and moderate losses with and without dispersion compensation.
Similarly to one-way SCW QKD communication [17], the chromatic dispersion
decreases communication distance in Plug&Play configuration. Especially
chromatic dispersion effects the deep modulation regime, where more energy is
distributed in high-order sidebands ($m>1$), which experience stronger
dispersion. Eventually at approximately 60 km dispersion completely suppresses
key generation rate, while original protocol could be capable of delivering
few hundreds of secret bit/s.

Figure 1: (a) Key generation rate $K$ vs communication distance $d$ at optimal
parameters with losses in Alice’s and Bob’s devices $\eta_{A}=\eta_{B}=-6$dB
for original protocol with (black dot-dashed) and without (black solid)
dispersion compensation. $K$ in deep modulation regime with (red dot-dashed)
and without (red solid) dispersion compensation. (b) The optimal values of
$|\alpha_{0}|$ (black) and $\beta$ (green) vs single pass power loss
$\eta_{F}$. Dot-dashed and solid lines differentiate cases with and without
dispersion compensations respectively.

In Fig. 1 (b) we present the optimal control parameters for the original
protocol with and without dispersion. Without dispersion compensation, optimal
$\beta$ tends to be around $0.38\pm 0.05$ as it is depicted in Fig. 1(b). This
value of modulation depth is a compromise between reduction of the Rayleigh
backscattering and increase of QBER due to chromatic dispersion. Use of
dispersion compensation removes an increase of QBER with larger $\beta$. In
turn, it allows to increase $\alpha_{0}$ at higher losses without dramatic
increase of QBER by Rayleigh backscattering. At deep modulation, $\alpha_{0}$
is a single control parameter, since $\beta$ is fixed. We find that the
optimal $\alpha_{0}$ does not significantly change over a full range of $\eta$
and lies within $1.5\pm 0.1$ region.

Figure 2:  $K$ for deep modulation regime with $\eta_{A}=-5$ dB and
$\eta_{B}=-7.5$ dB and carrier detection or original protocol with
$\eta_{A}=-5$ dB and $\eta_{B}=-10.5$ dB. The device losses are chosen for our
experimental setup.

Despite that developed model determines SCW in deep modulation with dispersion
compensation as an optimal protocol at moderate distances, there are numbers
of practical issues interfering with this conclusion. The typical solutions
for compensating dispersion at Alice’s side are Bragg gratings or dispersion
compensating fiber. The latter option is cumbersome with significant losses.
The Bragg gratings could be designed with little extra losses, but with major
overhead in the final price of the setup. The extra detector would also
increase final price in comparison with single detector configuration. In
turn, detecting single photon at the sidebands may require additional optical
filter to protect the detector from stray light. Extra filter would introduce
additional losses ( $\sim-3$ dB for typical in-fiber design) into Bob’s
device, that would decrease key rate proportionally. There is no such problem
for SCW QKD in deep modulation regime with only a single detector monitoring
carrier, where spectral filter in situ removes stray light. We compare the key
generation rate of SCW QKD in deep modulation with single detector and
original protocol with extra device loss of 3 dB, that simulates additional
filter. We find no difference between these two protocols in terms of key rate
generation as present in Fig. 2. Next we experimentally implement Plug&Play
SCW QKD in deep modulation with single detector.

##  3 Experiment

Figure 3:  Principle scheme of SCW Plug&Play QKD system, that consists from
laser diode (LD), optical isolator (OI), amplitude modulator (AM), optical
attenuator (ATN), polarization beam splitter (PBS), single mode fiber line
(SMF), phase modulators (PM), Faraday mirror (FM), optical circulator (OC),
spectral filter (SF), modulation phase synchronization unit (SYNC), single
photon counting module (SPC). Here red arrow indicates the direction of
reference light emitted by Bob, blue arrows indicate the direction of Alice’s
photons with encoded information, black arrow within PM indicates a direction
of optical signal for which phase modulation is possible.

The proposed scheme uses the same basic components of one-way SCW QKD system
[5], whereas we convert it into Plug&Play configuration as depicted in Fig. 3.
Here the laser source is a continuous-wave fiber Bragg grating stabilized
laser diode with linewidth < 1 MHz operating at a wavelength tunable within
C-band. The wavelength is adjusted to match the spectral filter reflection
band. The laser diode is protected by an optical isolator, while variable
optical attenuator is used to set carrier power to the necessary level taking
into account the losses in the fiber-optic line. $\text{LiNb}\text{O}_{3}$
Mach-Zehnder amplitude modulator is used to generate 3.3 ns pulses with a
repetition rate of 100 MHz to concentrate the signal photons inside the
detection gate clocked at 100 MHz. Prepared reference radiation is routed
through the H-port of a three-port fiber polarization beam splitter (PBS) to
its common port, which is connected to Alice via 25-km single mode fiber link
with optical losses of 5 dB.

Alice uses special composite phase-modulation unit to make communication
polarization insensitive. It consists out of two polarization sensitive
$\text{LiNb}\text{O}_{3}$ phase modulators $\text{PM}_{1}$ and $\text{PM}_{2}$
with the same principal axis but opposite direction of microwave tracks.
During the first pass only a single polarization component is phase modulated
by $\text{PM}_{1}$, where direction of microwave and optical signals
coincides. After reflection from the Faraday-mirror, the polarization is
flipped and orthogonal polarization component is modulated by $\text{PM}_{2}$.

Each PM is driven by 5 GHz sine-wave signal generator. The relative phase
between Alice’s and Bob’s drivers is dynamically adjusted by synchronization
circuit (SYNC). SYNC consists of the phaselock loop unit that locks the phase
of 5 GHz generator to 100 MHz logical clock. Small form-factor pluggable
optical modules are used to distribute the clock from Bob to Alice via
separate fiber link. Similar phase lock unit at Alice’s side adjusts the phase
of her 5 GHz signal generator. At each 100 MHz clock cycle Alice’s and Bob’s
control electronics randomly choose phases states of their 5 GHz microwave
signals from a set of four shifted by $\pi/2$ to encode information and
measurement basis respectively. Alice’s and Bob’s modulation bases are
synchronised automatically within each key generation cycle.

Since the signal is returned to Bob over the same fiber path, polarization
distortion is compensated and thus, linearly polarized signal is routed to the
V-port of PBS. Next, after the signal package is phase modulated again at
$\text{PM}_{3}$, the carrier is separated from sidebands on 5 GHz Fiber Bragg
Grating spectral filter and directed to the single photon counter (SPC) by an
optical circulator. In the experiment, we applied registration only at carrier
wavelength due to the limitations of our current hardware, which supports only
one SPC. Photon counting is performed by an avalanche photodiode-based SPC (ID
Quantique ID210) with detection efficiency $\eta_{D}=0.1$, detection window
$\delta t\approx 3.3$ ns, dead time $10$ $\mu$s resulting in dark count rate
of $\gamma_{\text{det}}=50$ Hz. The data was collected in blocks of about 2500
bits and errors were corrected by LDPC-codes [25].

Figure 4:  Normalized histograms of QBER (a) and secret key rate (b) acquired
during 24 hours of continuous communications.

Single pass losses from Alice’s Faraday mirror to Bob’s SPC is 17.5 dB. They
consists from 5 dB losses at Alice’s phase modulators (2.5 dB each), 5 dB and
7.5 dB losses in 25 km fiber spool and in Bob’s detection path respectively.
The latter losses are composed from 3 dB, 2.5 dB and 2 dB insertions losses of
optical circulator, $\text{PM}_{3}$ and spectral filter respectively.

We performed a 24-hour quantum key distribution in Plug&Play configuration
with deep modulation and single photon detection on the carrier wavelength.
Alice and Bob were connected by 25 km optical fiber with loss of 5 dB. No
chromatic dispersion compensation were employed. We achieved a mean secret key
rate of approximately 480 bits/s with average QBER of 3.18%. Both values show
Gaussian statistics with respective root mean square of 102 bits/s and 0.32%
as presented in Fig. 4(a) and Fig. 4(b). The performed test showed robustness
of the communications as temperature in the test area was varying by 10
${}^{\text{o}}$C over 24 hours.

##  4 Conclusion and Discussion

In summary, we proposed and performed first experimental subcarrier wave
quantum key distribution in Plug&Play configuration. We conducted thoughtful
analysis of the main factors affecting on QKD performance like non-ideal
devices, Rayleigh backscattering and chromatic dispersion. The obtained
experimental results indicate that SCW QKD in Plug&Play configuration with
deep modulation can be successfully applied at moderate distances for
intracity networks. Moreover the proposed deep modulation regime can increase
communication speed 2 times in SCW QKD.

It worth noting, that few points should be addressed in any future
implementations of the proposed SCW QKD. It is well known that Plug&Play QKD
systems are particularly susceptible to Trojan horse attacks on Alice’s
modulator [26]. To address this issue the signal, before getting to Faraday
mirror, can be sampled by 50-50 non-polarizing beam splitter and measured by
2-channel single photon detection module (omitted in experimental scheme for
implementation simplicity). The count and coincidence rates have to be
monitored to ensure single photon level and exclude Trojan Horse attack. For
deep modulation regime an absence of the strong reference [27, 28, 5] would
require a use of decoy state method against photon-number-splitting attack
[22, 23]. It can be achieved by an extra amplitude modulation of the reference
beam at Alice’s site. The minimization of the losses in Alice’s and Bob’s
devices is an important future task, that could be solved in on-chip fashion
[29, 30].

## Appendix: Holevo bound for deep modulation regime

Here we consider collective beam splitter attack in deep modulation regime. As
in analysis for original protocol [16] we assume, that Eve intercepts all
photons lost in the optical channel between Alice and Bob and records them
into her quantum memory unit. For sake of simplicity we assume that Eve
“emulates” losses by inserting beam splitter with power reflectively
coefficient $1-\eta_{F}$ at the beginning of the optical channel. Eve records
the reflected photons into her memory, while she sends the transmitted photon
to Bob via lossless optical channel. For a given value of Alice’s phase $\phi$
the wavefunction of Eve memory unit is

$\displaystyle\ket{\psi_{E}(\phi)}=\bigotimes^{\infty}_{m=-\infty}\ket{\eta_{A}\sqrt{(1-\eta_{F})}\alpha_{0}J_{m}(\beta_{\text{DM}})e^{im\phi}}.$
(19)

However $\phi$ is yet unknown to Eve, thus before the measurement Eve stores
even mixture of the state (19) with phases $\phi\in\\{0,\pi/2,\pi,3\pi/2\\}$
in her memory unit. Once Alice and Bob finish basis sifting and announce the
basis Eve has to distinguish two $\pi$-phase shifted states in a fully mixed
state composed out of these two states. Next we consider states with phases 0
and $\pi$ as similar analysis can be applied to set of $\pi/2$ and $3\pi/2$,
which are converted to 0 and $\pi$ by a proper unitary rotation [16]:

$\displaystyle\rho=\frac{1}{2}\ket{\psi_{E}(0)}\bra{\psi_{E}(0)}+\frac{1}{2}\ket{\psi_{E}(\pi)}\bra{\psi_{E}(\pi)}.$
(20)

The information that is accessible to Eve from density matrix $\rho$ being a
mixture of two pure states is value of von Neumann entropy of $\rho$ [31]. In
turn, the von Neumann entropy can be calculated as a Shannon entropy of
eigenvalues of operator $\rho$. The eigenvalues of $\rho$ are

$\displaystyle\lambda_{1,2}=\frac{1}{2}\left(1\pm|I(0,\pi)|\right),$ (21)

where $I(\phi_{1},\phi_{2})$ is an overlap between two states:

$\displaystyle I(\phi_{1},\phi_{2})=\bra{\psi_{E}(0)}\psi_{E}(\pi)\rangle.$
(22)

It is calculated using an expression for scalar product of two coherent
states:

$\displaystyle\langle\alpha|\beta\rangle=\exp\left(-\frac{1}{2}\left(|\alpha|^{2}+|\beta|^{2}\right)+\alpha^{*}\beta\right),$
(23)

that leads to the following expression for the overlap

$\displaystyle
I(\phi_{1},\phi_{2})=\exp\left(-|\alpha_{0}|^{2}\eta^{2}_{A}(1-\eta_{F})\sum^{\infty}_{m=-\infty}\left(|J_{m}(\beta_{\text{DM}})|^{2}(1-e^{im(\phi_{1}-\phi_{2})})\right)\right),$
(24)

for $\phi_{1}=0,\phi_{2}=\pi$ we obtain

$\displaystyle
I(0,\pi)=\exp\left(-|\alpha_{0}|^{2}\eta^{2}_{A}(1-\eta_{F})\left(1-J_{0}(2\beta_{\text{DM}})\right)\right)=\exp\left(-|\alpha_{0}|^{2}\eta^{2}_{A}(1-\eta_{F})\right),$
(25)

where we used the properties of Bessel function of first kind:

$\displaystyle\sum^{\infty}_{m=0}J^{2}_{2m+1}(\beta_{\text{DM}})=\frac{1-J_{0}(2\beta_{\text{DM}})}{2}=\frac{1}{2}.$
(26)

Thus the Holevo bound is

$\displaystyle\chi(A:E)=H\left(\frac{1}{2}\left(1-e^{-|\alpha_{0}|^{2}\eta^{2}_{A}(1-\eta_{F})}\right)\right).$
(27)

Acknowledgments Authors thank S.A. Moiseev and A. Tashchilina for fruitful
discussions. We are grateful to K. S. Melnik for technical assistance. Authors
appreciate support within framework project # 00075-02-2020-051/1 from
02.03.2020. Disclosure The authors declare no conflicts of interest. Data
availability Data underlying the results presented in this paper may be
obtained from the authors upon reasonable request.

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#  Independent Rydberg Atom Sensing using a Dual-Ladder Scheme

Samuel Berweger  Alexandra B. Artusio-Glimpse  Nikunjkumar Prajapati  Andrew
P. Rotunno  National Institute of Standards and Technology, Boulder, CO 80305,
USA  Noah Schlossberger  National Institute of Standards and Technology,
Boulder, CO 80305, USA  Department of Physics, University of Colorado,
Boulder, CO, 80309, USA  Dangka Shylla  National Institute of Standards and
Technology, Boulder, CO 80305, USA  Department of Physics, University of
Colorado, Boulder, CO, 80309, USA  Kaitlin R. Moore  SRI International,
Princeton, NJ, 08540, USA  Matthew T. Simons  Christopher L. Holloway
National Institute of Standards and Technology, Boulder, CO 80305, USA
<EMAIL_ADDRESS>
###### Abstract

Rydberg atom-based electric field sensing can provide all-optical readout of
radio frequency fields in a dielectric environment. However, because a single
set of optical fields is typically used to prepare the Rydberg state and read
out its response to RF fields, it is challenging to perform simultaneous and
independent measurements of the RF field(s). Here we show that using two
independent schemes to prepare and read out the same Rydberg state can be used
to perform independent measurements in general, which we demonstrate
specifically by resolving the the RF polarization. We expect this work will be
useful for fiber-coupled sensor heads where spatial multiplexing is
challenging, as well as for complex multi-level sensing schemes.

Rydberg atom-based electric field sensing is an emerging technology that uses
the interaction between optical and radio frequency (RF) electromagnetic
fields with atomic vapors to enable all-optical detection of RF fields in a
dielectric environment [1, sedlacek2012, 2]. In such a measurement, two or
more optical beams are overlapped inside a vapor cell containing alkali metal
(typically rubidium or cesium) to prepare a Rydberg state and subsequently
read out its RF response. As such, the sensing volume is effectively the
spatial extent of the overlapped beams within the vapor cell. Because the
optical fields are used to prepare and read out specific quantum mechanical
states, simultaneous yet independent measurements beyond frequency
multiplexing [3] are not readily possible within the sensing volume. One
possible approach for enabling entirely independent measurements within a
single sensing volume is to generate Rydberg states by several excitation
different paths (“ladders”) with no shared quantum coherence.

In this work we achieve two independent ladders using acousto-optic modulators
(AOM) to generate beams that couple to the same Rydberg state using different
hyperfine levels of the intermediate excited state. We spatially overlap these
beams in a cross-polarized configuration within a vapor cell in order to
perform simultaneous polarization-sensitive measurements. We clearly show that
the ladders respond independently to incident fields with polarizations
aligned along or perpendicular to the polarization of the optical fields of
each ladder. In contrast, equivalent measurements performed with both optical
fields using the same ladder no longer retain the capability to provide
independent sensing. We expect that this approach will prove useful for sensor
head designs using optical fibers as well as loop schemes and other wave-
mixing processes.

Figure 1: Experimental Schematic.

(a) Illustration of the rubidium dual-ladder scheme used, with ladders A and B
designed to use the F = 4 and F = 2 hyperfine states of the 5P3/2 intermediate
state, respectively. (b) Schematic of the experimental setup, which uses a
combination of acousto-optic modulators (AOMs, not shown) and polarizing beam
displacers (PBD) to generate the two ladders in a cross-polarized
configuration at the same spatial position in the cell along with their
respective reference beams. Shown in (c) are the simultaneously acquired EIT
spectra for both ladders.

Our approach uses rubidium-85 (85Rb) atoms and is based on the two-photon
scheme using electromagnetically induced transparency (EIT) that is commonly
used for Rydberg atom electric field sensing [4, 5, 1, 2]. A linearly
polarized ground state excitation (probe laser, 780 nm) is used to excite the
85Rb ground state (5S1/2) to the first excited state (5P3/2) along the D2
transition, while a second co-polarized counter-propagating field (coupling
laser) then excites the atom from the 5P3/2 state to the 60D5/2 Rydberg state
(480 nm). As schematically illustrated in Fig. 1(a) the two independent
ladders, denoted A and B, are designed to utilize two hyperfine levels of the
5P3/2 state. As illustrated in Fig. 1(b), AOMs are used to generate two beams
from each laser that are shifted by $\delta$ $\approx$ 180 – 200 MHz to
produce two ladders that independently couple to different states. These beams
are then combined using a polarizing beam displacer (PBD) to overlap the two
ladders in a cross-polarized fashion at the same location in the cell while
also producing two spatially separated reference beams. After traversing the
cell, the two beams are separated again using a PBD and independently detected
using balanced detectors.

We begin by examining the spectral characteristics of our dual-ladder scheme
and their interactions. Shown in Fig. 1(c) are the EIT spectra simultaneously
acquired for both ladders as the coupler laser frequency is swept across the
60D5/2 Rydberg state. The frequency axis is zeroed based on the position of
the 60D5/2 F = 4 peak [6] of ladder A, with the weaker 60D3/2 seen 53 MHz
lower in frequency. The main peak seen for ladder B is the F = 3 peak at
$\Delta_{C}$ = 125 MHz that arises from the doppler-shifts enabled by the
large velocity distribution of our room-temperature vapor. Both the coupler
and probe beams are detuned by $\Delta f_{0}$ $\approx$ 83 MHz from the F = 3
state, and the frequency shift is thus given by the ratio of the two
wavelengths $\Delta f$ = $\frac{\lambda_{p}}{\lambda_{C}}\cdot\Delta f_{0}$.
With the AOM set to 200 MHz, the F = 2 peak is expected in ladder B near the
zero-frequency position, though it is weak and and is not readily discernible
from any residual cross-talk from the primary peak of ladder A.

Due to the spatial overlap between the two ladders, additional peaks emerge
due to the interaction between individual fields of the two ladders and their
counterparts in the other ladder, i.e., the probe from ladder B will see the
coupler field from ladder A and vice versa. These peaks are indicated by an
asterisk and can be identified by selectively blocking the individual coupling
laser fields, or by modulating the AOM in the coupler beam path, which results
in out-of-phase oscillation between the peaks corresponding to the “pure”
ladders peaks and the peaks arising from cross-talk between the ladders. These
peaks correspond to the F = 4 probe transition of ladder A seeing the coupler
from ladder B, and the F = 3 probe transition of ladder B seeing the coupler
from ladder A. These peaks are shifted by 200 MHz, as expected due to the
frequency offset between the two coupler fields. Because of the strength of
the signal, we focus our efforts on the F = 3 peak of ladder B while noting
that AOMs with an appropriate frequency shift (120 MHz) could be used to
spectrally overlap the main peaks of the two ladders.

Figure 2: Polarization-Resolved Detection. (a) Schematic of the polarization
discrimination with dual-ladders. Two horn antennas are positioned
orthogonally such that each polarization is parallel with one of the optical
field polarizations of either ladder A or B. (b-c) False color plots showing
the measured EIT spectra as the RF field strength was increased for ladders A
and B as indicated with only Horn A on. (d-e) Same EIT spectra but with only
Horn B on. (f-g) EIT spectra with only Horn B on, but only one ladder is used.
Here (f) and (g) are the two readouts from the two different polarizations,
but with no shift in the frequency (delta=0) between the two probes and
couplers.

In order to explore the utility of a dual ladder scheme for the independent
sensing of two fields we examine two orthogonal RF polarizations. As
schematically shown in Fig 2(a), we position two horn antennas in an
orthogonal configuration to generate RF fields at a frequency of 9.71 GHz to
drive the transition from 60D5/2 to 61P3/2. The polarization of these RF
fields is parallel with the optical fields of either ladder A or ladder B.
Shown in Fig. 2 are the simultaneously acquired false-color plots of the
evolution of the ladder A (F = 4) and ladder B (F = 3) EIT peaks as a function
of the applied RF field strength applied to ladder A (b and c, respectively)
and ladder B (d and e, respectively). These plots show distinctly different
behavior for the two ladders. For each applied RF field we see the
characteristic Autler-Townes (AT) splitting expected for the corresponding
ladder with all optical and RF fields co-polarized [7]. However, the other
ladder shows weak peaks with the same AT splitting but a significantly
stronger residual main EIT peak that is unaffected by the RF field, which is
also as expected when the optical and RF fields are orthogonally polarized
[7]. This clearly shows that the two ladders respond to the RF fields
independently and can be used for simultaneous sensing.

Lastly, we contrast our dual-ladder approach with a single-ladder approach. In
this case, we operate the system identically to the dual-ladder configuration,
but the beams from both ladders are sourced from the same AOM output ($\delta$
= 0). Shown in Fig. 2(f,g) are the simultaneously acquired false-color plots
from both single-ladder beams with the RF field applied along the ladder B
polarization. Here we clearly see that the responses are identical, showing
that a single-ladder approach can no longer be decomposed into orthogonal
polarization for independent measurements.

Our simple demonstration on polarization sensing reveals how a simultaneous
and independent dual-ladder approach can be used to reduce a measurement that
required active polarization rotation [7] into a dual-channel measurement.
This may open up the path to improved transmission bandwith or independent
sensing using polarization multiplexing. However, this approach can also be
used for simultaneous sensing in general with, e.g, different RF transitions
and is readily compatible with loop schemes [8] implemented on each ladder.
More broadly, the use of independent hyperfine states can be used to
independently link the ground state for six wave mixing and other complex loop
schemes [9, 10]. Although two ladders could be generated using separate
lasers, the inherent simplicity of using AOM-generated sidebands limits the
experimental complexity and cost.

The overall EIT peak height of the two ladders depends sensitively on the
relative probe laser field strengths. Although for weak probe fields the EIT
peak height for either ladder increases as the respective probe power is
increased, at higher powers increasing the probe power for one ladder will
increase the corresponding signal while reducing that of the other ladder.
This is the result of the spatially overlapped beams that see the same atomic
vapor and thus are competing for ground state atoms. This is the direct
consequence of our desired effect, where the two ladders enable independent
measurements by interrogating two separate atomic populations, which is not
possible using a single ladder. Since the combined sensitivity of the two
ladders is limited by the ground state population, this could in principle be
avoided by spatially separating the two fields.

The ability for independent sensing in overlapping beams holds several
advantages. First, spatial multiplexing presents significant challenges in
fiber-coupled cells , and the capability for simultaneous yet independent
measurements using a single fiber significantly simplifies technical
challenges. Secondly, because of the influence of dielectric atom vapor cells
on the spatial RF field distribution [11, 12], independent sampling of a
single field should be performed at the same spatial location to enable direct
comparison. It should be emphasized that our choice of orthogonal
polarizations for both ladders is one of convenience rather than necessity.
The orthogonal polarizations can be used to readily spatially separate and
detect the two ladders, though separate detection could also be achieved based
on the frequency separation of the two ladders using, e.g., optical heterodyne
detection.

In this work we have implemented a dual-ladder scheme to independently
generate Rydberg atoms within the sensing volume for RF field detection. We
reveal that this approach can readily be used for simultaneous and independent
sensing, which we demonstrate by polarization-resolved detection. This
approach is generally applicable and could find applications in fiber-based
sensor heads as well more complex multi-state sensing schemes.

Acknowledgements: This research was developed with funding from the Defense
Advanced Research Projects Agency (DARPA). The views, opinions and/or findings
expressed are those of the author and should not be interpreted as
representing the official views or policies of the Department of Defense or
the U.S. Government. A contribution of the US government, not subject to
copyright in the United States.

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