# State-independent quantum tomography of a single-photon state by photon-
number-resolving measurements

Rajveer Nehra<EMAIL_ADDRESS>Department of Physics, University of
Virginia, 382 McCormick Rd, Charlottesville, VA 22904-4714, USA  Aye Win  Now
at: Department of Physics, University of Oklahoma, 440 W. Brooks St., Norman,
OK 73019, USA  Department of Physics, University of Virginia, 382 McCormick
Rd, Charlottesville, VA 22904-4714, USA  Miller Eaton  Department of Physics,
University of Virginia, 382 McCormick Rd, Charlottesville, VA 22904-4714, USA
Niranjan Sridhar  Now at: Google, Inc., 1600 Amphitheatre Parkway, Mountain
View, CA, USA  Department of Physics, University of Virginia, 382 McCormick
Rd, Charlottesville, VA 22904-4714, USA  Reihaneh Shahrokhshahi  Now at:
Xanadu, 372 Richmond Street West, Toronto, Ontario M5V1X6, Canada  Department
of Physics, University of Virginia, 382 McCormick Rd, Charlottesville, VA
22904-4714, USA  Thomas Gerrits  National Institute of Standards and
Technology, 325 Broadway, Boulder, CO 80303, USA  Adriana Lita  National
Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303, USA
Sae Woo Nam  National Institute of Standards and Technology, 325 Broadway,
Boulder, CO 80303, USA  Olivier Pfister  Department of Physics, University of
Virginia, 382 McCormick Rd, Charlottesville, VA 22904-4714, USA

###### Abstract

The Wigner quasiprobability distribution of a narrowband single-photon state
was reconstructed by quantum state tomography using photon-number-resolving
measurements with transition-edge sensors (TES) at system efficiency 58(2)%.
This method makes no assumptions on the nature of the measured state, save for
the limitation on photon flux imposed by the TES. Negativity of the Wigner
function was observed in the raw data without any inference or correction for
decoherence.

##  I Introduction

Single and multiphoton sources prepared in Fock states are of fundamental
importance: not only do they enable experiments that epitomize the wave-
particle “duality” of quantum mechanics, they also can only be described by
quantum theory due to the non-positivity of their Wigner quasi-probability
distribution. Leonhardt1997 ; Walls1994

Eugene Wigner originally defined the continuous phase-space quasiprobability
distribution function to study quantum corrections to classical statistical
systems Wigner1932 . For a quantum state of density operator $\hat{\rho}$, the
Wigner function is given by

$\displaystyle W(q,p)$
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{\infty}e^{2ipy}\left\langle\,q-y\,\right|\hat{\rho}\left|\,q+y\,\right\rangle
dy,$ (1)

where $q$ and $p$ are the respective eigenvalues of the position and momentum
operators or, in our case, of the amplitude-quadrature,
$Q=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$, and phase-quadrature,
$P=i(\hat{a}^{\dagger}-\hat{a})/\sqrt{2}$, of the quantized electromagnetic
field, $\hat{a}$ and $\hat{a}^{{\dagger}}$ being the photon annihilation and
creation operators, respectively. The Wigner function is uniquely defined and
contains all the information about the quantum system. It is normalized over
phase space and its marginal distributions correspond to the probability
density distributions of the quadratures

$\displaystyle\int_{-\infty}^{\infty}dp\,W(p,q)$ $\displaystyle=|\psi(q)|^{2}$
(2) $\displaystyle\int_{-\infty}^{\infty}dq\,W(p,q)$
$\displaystyle=|\psi(p)|^{2}.$ (3)

However, unlike classical distributions, the quantum Wigner distribution
$W(q,p)$ can’t always be interpreted as a joint probability distribution
because it can be nonpositive (hence non-Gaussian for pure states Hudson ),
e.g. for Fock states with $n>0$. This acquires major significance in the
context of quantum information and quantum computing over continuous variables
(CVQC) Lloyd1999 ; Bartlett2002  as it is well known that all-Gaussian- (gates
and states) CV quantum information suffers from no-go theorems for Bell
inequality violation Bell1987 , entanglement distillation Eisert2002 , and
quantum error correction Niset2009 . However, none of these no-go theorems
apply to CVQC when including non-Gaussian states or gates Gottesman2001 ;
Menicucci2006 ; Menicucci2014ft . Non-Gaussian resources are therefore
essential to CVQC and can be implemented, for example, by Fock-state
generation or detection Ghose2007 . It is therefore important to be able to
characterize Fock states fully and efficiently, possibly in real time. One
standard method of state tomography is Wigner function reconstruction. Quantum
state tomography in phase space Smithey1993  can be performed by
reconstructing the Wigner function from the measurement statistics of the
generalized quadrature ${Q}\cos\phi+{P}\sin\phi$, measured by balanced
homodyne detection (BHD) where phase $\phi$ is the tomographic angle. This was
first done for heralded single photon states in 2001 Lvovsky2001  and recently
improved Morin2012 .

An issue with BHD-based tomography is that the reconstruction process is
computationally intensive, using the inverse Radon transform, or maximum
likelihood algorithms Lvovsky2009 . A more direct approach to reconstruct the
Wigner function was proposed by Wallentowitz and Vogel Wallentowitz1996  and
by Banaszek and Wodkiewicz Banaszek1996 . It is based on the following
expression of the Wigner function Royer1977

$W_{{\hat{\rho}}}(\alpha)=\frac{1}{\pi}Tr[{\hat{\rho}}\hat{D}(\alpha)(-1)^{\hat{N}}\hat{D}^{\dagger}(\alpha)],$
(4)

where $\alpha=(q+ip)/\sqrt{2}$,
$\hat{D}(\alpha)=\exp(\alpha\hat{a}^{{\dagger}}-\alpha^{*}\hat{a})$ is the
displacement operator, and $\hat{N}=\hat{a}^{{\dagger}}\hat{a}$ is the number
operator. Equation (4) reveals that the Wigner function at a particular phase
space point $\alpha$ is the expectation value of the displaced parity operator
$\hat{D}(-1)^{\hat{N}}\hat{D}^{{\dagger}}$ over $\hat{\rho}$ or, equivalently,
the expectation value of the parity operator $(-1)^{{N}}$ over the displaced
density operator $\hat{D}^{{\dagger}}\hat{\rho}\hat{D}$. This provides a
direct measurement method given that one has access to photon-number-resolving
(PNR) measurements. In particular, the value of the Wigner function at the
origin is the expectation value of the photon number parity operator

$W(0)=\frac{1}{\pi}\sum_{n}^{\infty}(-1)^{n}\rho_{nn}.$ (5)

Hence, the PNR detection statistics of a quantum system of density operator
$\hat{\rho}$ yield a direct determination of the Wigner function at the
origin. In order to recover the Wigner function at all points, one can simply
displace $\hat{\rho}$ by a raster scan of complex number $\alpha$. This can be
done by interference at a highly unbalanced beamsplitter Paris1996  of
transmission to refelection coefficient ratio $t/r\ll 1$, as depicted in
Fig.1.

Figure 1: Implementation of a displacement by a beamsplitter. The initial
coherent state amplitude is $\beta$ with $|\beta|\gg 1$, so that we can have
$t\ll 1$ in order to preserve the purity of the quantum signal $\hat{\rho}$,
while still retaining a large enough value of $|\alpha|=t|\beta|$, as needed
for the raster scan of the Wigner function in phase space.

This technique is commonplace in quantum optics and was used, for example, to
implement Bob’s CV unitary in the first unconditional quantum teleportation
experiment Furusawa1998 . In all rigor, the resulting Wigner function is a
more general one, the $s$-ordered Wigner function, $W(s\alpha;s)$, which tends
toward $W(\alpha)$ when $s=-t/r\to 0$ Cahill1969 . This method was implemented
for quantum state tomography of phonon Fock states of a vibrating ion
Leibfried1996 , as well as microwave photon states in cavity QED Haroche2002 ;
Vlastakis607 . For quantum states of light, it has been experimentally
realized for the positive Wigner functions of vacuum and coherent states, as
well as phase-diffused coherent-state mixtures, initially detecting no more
than one photon Banaszek1999a  and subsequently detecting several photons
Bondani2009 ; Sridhar2014a . The nonpositive Wigner function of a single-
photon state was reconstructed using PNR measurements by time-multiplexing
non-PNR, low efficiency avalanche photodiodes, albeit with the use of a priori
knowledge of the input state in order to deconvolve the effect of losses
Laiho2010 .

Our work is the first demonstration of state-independent photon-counting
quantum state tomography of a nonpositive Wigner function. The only assumption
made here is that the initial quantum state consists of low photon numbers to
avoid the saturation limit of the detector, which is less than five photons
per microseconds for the superconducting Transition Edge Sensor (TES) used in
our experiment. Since no other prior knowledge is assumed about the state to
be measured, this technique is equally applicable to any arbitrary quantum
state with low photon flux. We directly observe negativity of the Wigner
function with no correction for detector inefficiency.

##  II Experimental setup and methods

###  II.1 Cavity-enhanced narrowband heralded single-photon source

####  II.1.1 General model

Our single photon source is based on type-II spontaneous parametric
downconversion (SPDC) in a periodically poled $\rm KTiOPO_{4}$ (PPKTP)
crystal. A pump photon at $\omega_{\scriptscriptstyle p}$ is downconverted
into a cross-polarized signal-idler photon pair at $\omega_{s,i}$, such that
$\omega_{p}=\omega_{s}+\omega_{i}$, and the presence of the signal photon is
heralded by detecting the idler photon Lvovsky2001 . All tomographic
measurements were therefore conditioned to the detection of an idler photon.
The SPDC Hamiltonian is given by Grice1997 ; Christ2012

$H\propto i\hbar\chi^{(2)}\int d^{3}\vec{r}\
{E}^{(-)}_{p}(\vec{r},t){E}^{(+)}_{i}(\vec{r},t){E}^{(+)}_{s}(\vec{r},t)+H.c.$
(6)

where $\chi^{(2)}$ is the crystal’s nonlinearity and the fields in the
Heisenberg picture take the form,

$\displaystyle E^{(-)}_{j=p,s,i}(\vec{r},t)$
$\displaystyle=E^{(+)}(\vec{r},t)^{{\dagger}}$ $\displaystyle=\int
d\omega_{\scriptscriptstyle j}\ A(\vec{r},\omega_{\scriptscriptstyle
j})\,\hat{a}_{j}\,e^{i[k_{j}(\omega_{\scriptscriptstyle
j})r-\omega_{\scriptscriptstyle j}t]},$ (7)

where $A(\vec{r},\omega_{\scriptscriptstyle j})$ is an approximately slowly
varying amplitude and $\hat{a}_{j}$ is the annihilation operator for the mode
of frequency $\omega_{\scriptscriptstyle j}$. Solving for the state under the
evolution of the Hamiltonian in Eq. (6) for low parametric gain regime and a
non-depleted classical pump yields the output quantum state

$|\psi\rangle=\int d^{3}\vec{k}_{s,i}\,d\omega_{\scriptscriptstyle
s,i}\,\phi(\vec{k}_{s},\omega_{\scriptscriptstyle
s},\vec{k_{i}},\omega_{\scriptscriptstyle
i})\,\hat{a}^{\dagger}_{s}\hat{a}^{\dagger}_{i}\left|\,0\,\right\rangle_{s}\left|\,0\,\right\rangle_{i}$
(8)

where $\phi(\vec{k}_{s},\omega_{\scriptscriptstyle
s},\vec{k_{i}},\omega_{\scriptscriptstyle i})$ determines the spectral and
spatial properties of the SPDC, depending on the pump field and the non-linear
crystal (phase matching bandwidth around
$\vec{k}_{p}=\vec{k}_{s}+\vec{k}_{i}$). We can see from Eq. (8) that the
signal and idler photon pairs are emitted in a multitude of spatial and
spectral modes. Therefore, any measurement on a particular idler mode will
collapse the quantum state given by Eq. (8) to a mixture of signal-mode
states. As a result, the heralded signal state will not be a pure quantum
state, which limits its applications in quantum information processing
Gisin2007 ; Clauser1978 . This is because a nonzero vector phase-mismatch can
lead to a detected, heralding idler photon with a “twin” signal photon
completely out of alignment and therefore undetectable, even in the absence of
losses, which greatly diminishes the experimentally accessible quantum
correlations. One therefore needs to emit photon pairs in the well defined
spatial and spectral modes which are optimally coupled to the detectors. This
involves spectral and spatial filtering and has been widely studied both
theoretically and experimentally Ou1999 ; JeronimoMoreno2010 ; Virally2010 ;
Scholz2009 ; Hockel2011 ; Bao2008 . Our spectral and spatial filtering steps
are discussed in the next section.

####  II.1.2 Spectral and spatial filtering

Spectral and spatial filtering was achieved by using optical resonators: both
actively, by placing the nonlinear crystal in a resonant cavity — thereby
building an optical parametric oscillator (OPO) — and passively, by using a
filtering cavity (FC) and an interference filter (IF) after the OPO. The OPO
was used in the well-below-threshold optical parametric amplifier (OPA)
regime. The OPO cavity enhanced the SPDC at doubly resonant (signal and idler)
frequencies by a factor of the square of the cavity finesse Kuklewicz2002 .
However, this enhancement was still masked by the “sea” of nonresonant SPDC
photons until we filtered the idler with a short FC, which selected only one
OPO mode, and with an IF, which selected only one of the FC modes. After
filtering, we are allowed to consider the simpler OPA Hamiltonian

$H=i\hbar\kappa\,\hat{a}_{s}^{\dagger}\hat{a}_{i}^{\dagger}+H.c.,$ (9)

where $\kappa$ is the product of the pump amplitude and $\chi^{(2)}$. This
yields the well-known two-mode squeezed state

$|\psi\rangle=(1-\zeta^{2})^{\frac{1}{2}}\sum_{n=0}^{\infty}\zeta^{n}|n\rangle_{s}|n\rangle_{i},$
(10)

where $\zeta=\tanh(\kappa t)$. In the weak pump regime, both $\kappa t$ and
$\zeta\ll 1$, and Eq. (10) can be approximated by

$|\psi\rangle\simeq|0\rangle_{s}|0\rangle_{i}+\zeta|1\rangle_{s}|1\rangle_{i}.+\mathcal{O}(\zeta^{2})$
(11)

A detection of a single photon in the idler mode thus projects the signal mode
into a single-photon state.

Note that, since the heralding process consists in postselection of the idler
channel, filtering losses in this channel are unimportant. Indeed, if the pump
power is kept low enough that practically no pairs from different modes ever
overlap in time, one can then reasonably claim that the detected, the heralded
signal photon will be the twin of the filtered, heralding idler photon, as per
Eq. (11).

It is important to also note that the situation will change if one seeks to
herald a multi-photon state by using PNR detection for heralding, as per Eq.
(10). In that case, losses in the heralding channel cannot be tolerated as
they will lead to errors.

A significant contribution to photon loss is any mode mismatch between the OPO
and the FC, which must also be locked on resonance simultaneously, as detailed
in the next section. By careful modematching of a seed OPO beam to the FC, we
were able to achieve $83\%$ transmission of the OPO mode through the FC.

###  II.2 Experimental setup for quantum tomography

####  II.2.1 Setup description

The experiment, depicted in Fig.2, built upon our previous demonstration of
coherent-state tomography Sridhar2014a  with the addition of the heralded
single-photon source. The OPO was pumped by a stable frequency-doubled 532 nm
Nd:YAG nonplanar ring oscillator laser (1 kHz FWHM). A type-II (YZY) quasi-
phasematched PPKTP crystal, of period 450 $\mu$m, was used in the doubly
resonant OPO. The two-mirror OPO cavity, as mentioned above, was one-ended,
with a finesse of F $\simeq$ 300, an FSR of 1.5 GHz and a FWHM of 5 MHz. One
mirror’s inside facet was 99.995% reflective for the signal and idler fields
near 1064 nm and 98% transmissive for the pump field at 532 nm (the outside
mirror facet was uncoated); the other mirror’s inside facet was 98% reflective
at 1064 nm and 99.95% reflective at 532 nm (its outside facet was
antireflection coated at 1064 nm). The cavity was near-concentric with a
super-Invar structure, the mirrors’ radius of curvature being 5 cm and the
mirrors $\simeq$10 cm apart. The FC was made of two 5 cm-curvature, 99%
reflective mirrors placed $\simeq$0.5 mm apart.

Figure 2: Experimental setup. The red dotted lines denote the locking beam
paths for the on/off Pound-Drever-Hall (PDH) servo loops of the OPO and the
FC. The displacement operation is contained in the black dash-dotted box at
the top. BP: Brewster prism. DM: Dichroic Mirror. EOM: Electro-optic
modulator. FI : Faraday Isolator. FR: Faraday Rotator. HWP: Half-wave plate.
IF : Interference Filter. LO: Local Oscillator input to the displacement
field. PBS: Polarizing Beamsplitter. PD: Photodiode. POL: Polarizer. PZT:
Piezoelectric transducer.

The OPO mode was aligned and mode-matched to all parts of the experiment (FC,
TES fibers) by using a seed beam which was injected into the OPO through its
highly (99.995%) reflecting mirror and exited through its output coupler. The
seed beam was carefully mode-matched to the OPO so as to be a pure $\rm
TEM_{00}$ mode before being sent to the rest of the setup. It was also used
for interference visibility optimization with the displacement field.

####  II.2.2 Stabilization procedure

Both the OPO and the FC cavities were Pound-Drever-Hall (PDH) locked
Drever1983  to a reference laser beam provided by the undoubled output of the
pump laser. This was achieved by way of an “on/off” locking system, effected
by a system of computer-controlled diaphragm shutters. In the “on” locking
phase, the input to the single-photon sensitive PNR detectors was closed and
the reference laser was unblocked and sent into both the OPO and the FC
(dotted lines in Fig.2) whose PDH lock loops were closed for a few seconds.
Because of its super-Invar structure, the OPO drift was low and the PDH loops
could then be open, in the “off” phase, with their correction signals held
constant. The shutter of the reference laser was closed and the paths between
the OPO and the PNR detectors were open for data acquisition, for as long as 3
seconds, see Fig.3.

Figure 3: On/off cycles of alternated active locking and data acquisition.
Data collection begins for a period of 800ms while the auxiliary locking beam
is blocked, followed by a period of 200ms where the auxiliary locking (broken
red line in the experiment schematic) is enabled for active locking and the
signal channel is blocked. This process occurs cyclically during data
collection to prevent excess photon flux from damaging the TES while ensuring
a stable OPO cavity mode.

This procedure allowed us to lock the OPO to its doubly resonant, frequency
degenerate mode at $\omega_{\scriptscriptstyle s}=\omega_{\scriptscriptstyle
i}=\omega_{\scriptscriptstyle p}/2$. This was essential as the displacement
field, also provided by the undoubled output of the pump laser, had to be at
the same frequency as the OPO’s quantum signal beam and phase-coherent with
it. Note that finding this frequency degenerate, doubly resonant OPO mode is
nontrivial since the double resonance condition

$\displaystyle\omega_{\scriptscriptstyle s}$
$\displaystyle=\omega_{\scriptscriptstyle i}$ (12)
$\displaystyle\Leftrightarrow\frac{m_{s}}{L+n_{s}(T)\ell}$
$\displaystyle=\frac{m_{i}}{L+n_{i}(T)\ell}$ (13)

features two different indices of refraction $n_{s}\neq n_{i}$ ($L$ is the
cavity length in air only, $\ell$ the crystal length and
$m_{s,i}\in\mathbb{N}$ are the mode numbers). It is, however, possible to
temperature-tune the OPO crystal to achieve stable frequency degeneracy
Feng2003 ; Feng2004 ; Feng2004b . This required temperature control of the
PPKTP crystal to the level a few millidegrees, around $27.810^{\circ}$ C,
using a commercial temperature controller.

####  II.2.3 PNR detection

Our PNR detection system is comprised of two transistion edge sensors (TES),
consisting of tungsten chips in a cryostat, coupled through standard telecom
fiber. A detailed description of the TES system can be found in Refs. 46; 29.
The TESs are cooled using an adiabatic demagnetization fridge at a stable
temperature of 100 mK, at the bottom edge of the steep superconducting
transition slope (resistance versus temperature). When one photon is detected,
its energy is absorbed by the tungsten chip, yielding a sharp increase in its
resistance which is detected by a SQUID over a rise time on the order of 100
ns. The heat is then dissipated through a weak thermal link, over a time on
the order of 1 $\mu$s. During this time, the TES is still active (as opposed
to, say, of nanowire detectors or avalanche photodiodes). Due to the
finiteness of its superconducting transition slope, the TES can resolve up to
5 photons. The absolute maximum photon flux sustainable by the TES without the
tungsten driven into the normal conductive regime is therefore 5
photons/$\mu$s in the continuous-wave regime, i.e., a power of 1 pW. The OPO’s
average power was kept at 100 fW by setting the pump power to 200 $\mu$W (the
OPO threshold was 200 mW). We observed that the background counts were
negligible when the TES signal was suppressed by rotating the pump’s linear
polarization by 90∘, thereby completely phase-mismatching the nonlinear
interaction in PPKTP.

####  II.2.4 Displacement calibration

The displacement operator was implemented by interfering the OPO signal mode
with a phase- and amplitude-shifted coherent-state displacement field at a
highly unbalanced beamsplitter with a reflectivity $r^{2}=0.97$. The
interference visibility between the seed OPO beam and the displacement field
was 90%. The amplitude shift $|\alpha|$ was effected by a homemade,
temperature-stabilized electro-optic modulator consisting in an X-cut, 20 mm-
long rubidium titanyl arsenate ($\rm RbTiOAsO_{4}$) crystal; the phase shift
$\arg(\alpha)$ was effected by a piezoelectric transducer- (PZT) actuated
mirror. Both the EOM and the PZT mirror were driven by homemade, low-noise,
high voltage drivers, fed by computer-controlled lock-in amplifiers.

The amplitude displacement was varied in 20 steps from $|\alpha|=0$ to
$|\alpha_{\text{max}}|=0.796(7)$, fixed by the TES’ photon flux limit of 5
photons/$\mu$s. The phase displacement was varied in 10 steps from 0 to
$2\pi$. The amplitude steps $|\alpha|=\sqrt{\eta}|\beta|$, where $\eta$ is the
overall detection efficiency, were directly calibrated by comparing the TES
photon statistics to that of a Poisson distribution

$P(n)=e^{-|\alpha|^{2}}\frac{|\alpha|^{2n}}{n!},$ (14)

with the OPO beam blocked. This allowed us to determine the displacement
amplitude

$|\alpha|=\left[\frac{2P(2)}{P(1)}\right]^{\frac{1}{2}}.$ (15)

Note that this method requires the presence of 2-photon detection events,
i.e., $|\alpha_{\text{min}}|\simeq 0.15$ for the very first displacement
amplitude, besides the zero displacement for which we blocked the displacement
beam. Photon number statistics were averaged over 2 seconds to ensure an
average calibration accuracy

$\triangle|\alpha|=3\times 10^{-3}$ (16)

of the displacement amplitude. However, the error on the maximum displacement
was somewhat larger

$\triangle|\alpha_{\text{max}}|=7\times 10^{-3},$ (17)

due to the photon pileups occurring at higher flux which make the continuous-
wave TES signals harder to analyze. We observed the long-term power stability
of the laser to be on the order of 1% over an hour. The laser’s short-term
intensity noise was much lower as ensured by a built-in “noise eater”
intensity servo. Moreover, the temperature stability of the EOM was on the
order of 1 mK. Because of all this, we consider the error $\triangle|\alpha|$
on the displacement calibration to be valid over the course of our data
acquisition time of several minutes.

The phase steps were calibrated by scanning the interference fringe between
the OPO seed beam and the displacement field, which provided a set of 10
voltage values for the PZT mirror. Experimental data runs were conducted by
scanning the amplitude at fixed phases, with the phase PZT voltage being
refreshed at every amplitude EOM voltage step. For each point of the quantum
phase space, a continuous stream of data was acquired at 5 MS/s, digitized
using an PCI board, and stored for subsequent photon statistical analysis. A
detailed discussion of our data analysis of continuous-wave photon counting
can be found in our previous paper on coherent state tomography using PNR
measurements Sridhar2014a .

##  III Experimental results

###  III.1 Heralding ratio

The heralding ratio determines the quality of the single-photon source. It is
the probability of seeing one photon in the OPO signal (heralded) beam with no
displacement field, provided one photon was detected in the filtered idler
(heralding) beam. The pump power was kept low enough so as to suppress two-
photon events in the OPO signal in the absence of a displacement field.
Results are displayed in Table 1.

$\begin{array}[]{c|c|c|c}\hline\cr&N_{s}&N_{i}&N_{c}\\\ \hline\cr\text{Single-
photon events}&54320\pm 90&1556\pm 30&903\pm 17\\\ \hline\cr\end{array}$ Table
1: Experimentally measured number of single-photon counts on both channels.
$N_{s}$: number of photons in the (heralded) signal channel, $N_{i}$: number
of photons on the (heralding) idler channel, $N_{c}$: number of coincident
counts.

We can see that the heralded channel has a lot more counts than the heralding
channel, as expected since the latter is filtered by the FC and the IF. The
heralding efficiency was

$\displaystyle\eta_{h}$ $\displaystyle=\frac{N_{c}}{N_{i}}$ (18)
$\displaystyle=0.58\pm 0.02$ (19)

and can also be considered the overall detection efficiency of the heralded
channel, i.e., of the quantum signal.

###  III.2 Photon probability distributions versus displacement amplitude

Figure 4 displays the measured photon number distributions for a heralded
single-photon input when $|\alpha|$ = 0 (left) and 0.25 (right). For no
displacement, the histogram reflects the exact same measurement as in Table 1
and Eq. (18), and the result yields a compatible value of $0.58(2)$. The two-
photon counts are essentially absent, which results in a very low second-order
coherence $g^{2}(0)=0.07(5)$. For $|\alpha|$ = 0.25, the two-photon peak grows
from the presence of the displacement field. In both cases, the observations
agree with the theoretical distribution, calculated with $\eta$= 0.58.

Figure 4: Measured photon-number distributions, left: $\alpha=0$ and right :
$|\alpha|=0.25$. Error bars (1$\sigma$) are calculated from the statistics of
the measurements.

As expected, the single-photon component decreased while the vacuum and higher
photon components increased. It can be thought of as follows: If we displace a
pure single-photon state, then we obtain

$\displaystyle D(\alpha)|1\rangle$
$\displaystyle=D(\alpha)a^{{\dagger}}D^{{\dagger}}(\alpha)|\alpha\rangle$
$\displaystyle=a^{{\dagger}}|\alpha\rangle-\alpha^{*}|\alpha\rangle.$

Clearly, the first term has no vacuum component, as does the initial state
$|1\rangle$, but the second term does have a vacuum component. Therefore, the
displacement of a single-photon state increases its vacuum component
probability amplitude, somewhat unintuitively. In the case where our initial
state is a mixture of vacuum and single-photon, then it can be seen that for
low enough displacement amplitudes, the vacuum component still increases from
its previous value. As the displacement becomes large, the vacuum component
will eventually decrease.

###  III.3 Model Wigner function and loss analysis

Before we turn to the tomography results, we outline the Wigner function model
that accounts for the aforementioned nonideal system detection efficiency.
There are several sources of losses in our experiment: photon absorption and
general scattering out of the mode due to mismatch. As mentioned above, losses
in the heralding channel can be factored out in the generation of a heralded
single-photon state provided that the OPO output never contains more than one
photon per mode during the detection window, which was the case in this work.

It is also important to note that the TES fiber is single-mode at telecom
wavelengths but not at our operating wavelength of 1064 nm. Hence we need to
address the possibility of multimode coupling into the TES fiber. A simple
reasoning shows that this is not a matter of concern if there are no losses in
the fiber. Indeed, the coupling of the input field into each of the different,
orthogonal propagation modes of the fiber can be accurately described by as
many beamsplitting operations into distinguishable outputs. While each of
these beamsplitting operations does bring in vacuum fluctuations, all
beamsplitter outputs are still detected and the final TES detection is simply
that of the total photon number of all the fiber modes. In the absence of
losses, the multimode fiber is a passive optical element which conserves the
total photon number and the final total photon number measurement must
therefore give the same exact result as the initial one, before the quantum
light is coupled into the fiber. An argument could be made that fiber losses
could be mode dependent, with higher-order modes being more likely to leak out
of the fiber; we assume that this is negligible in our case because the
operating wavelength was close enough to the specified single-mode wavelength
that the mode order should not be that high.

We measured the coupling efficiency, $\eta_{\text{OFC}}$, into the TES fiber
on the optical table by cleaving the fiber to insert a power meter and re-
fusing it to the TES thereafter. To minimize the coupling to higher modes, we
optimized our fiber coupling to as high as 90% with the seed beam (discussed
in the “spectral and spatial filtering” section above) and we also measured
the intensity variations of about one percent at the output of the fiber. This
ensures that most of the fiber coupling was to the fundamental mode of the
fiber. However, we didn’t measure the overall fiber transmission into the TES
cryostat. This was bundled with the TES quantum efficiency in
$\eta_{\text{TES}}$, which was inferred from all other measured efficiencies,
as summarized in Table 2.

$\begin{array}[]{cccc|c}\hline\cr\eta_{\text{TES}}&\eta_{\text{OT}}&\eta_{\text{BS}}&\eta_{\text{OFC}}&\eta\\\
\hline\cr 0.71(3)&0.93(1)&0.97(1)&0.90(2)&0.58(2)\\\ \hline\cr\end{array}$
Table 2: $\eta_{\text{TES}}$: TES quantum efficiency (including fiber
transmissivity); $\eta_{\text{OT}}$ : optical transmission of single-photon
signal field from the OPO to the displacement operation; $\eta_{\text{OFC}}$:
optical fiber coupling. The overall efficiency
$\eta=\eta_{\text{TES}}\times\eta_{\text{OT}}\times\eta_{\text{BS}}\times\eta_{\text{OFC}}$.

We modeled losses by considering a fictitious beamsplitter of transmissivity
$\eta$ and reflectivity $(1-\eta)$, placed between the displacement and a
detector of unity efficiency as shown in Fig.5. The input state of this system
is

Figure 5: Loss model. The beamsplitter transmission and reflection
coefficients are $\sqrt{\eta}$ and $\sqrt{1-\eta}$ respectively.

$\hat{\rho}_{\text{in}}=|1\rangle_{a}\,{}_{a}\\!\langle
1|\otimes|0\rangle_{b}\,{}_{b}\\!\langle 0|.$ (20)

After applying the displacement $\hat{D}_{a}(\beta)$ and beamsplitter
$\hat{U}_{ab}$ operators we obtain the reduced, detected density operator by
tracing out the vacuum mode

$\displaystyle\hat{\rho}_{\text{out}}$
$\displaystyle=\mathrm{Tr}_{b}\left[\hat{U}_{ab}\hat{D}(\beta)|1\rangle_{a}\,{}_{a}\\!\langle
1|\otimes|0\rangle_{b}\,{}_{b}\\!\langle
0|\hat{D}^{\dagger}(\beta)\hat{U}_{ab}^{\dagger}\right]$ (21)
$\displaystyle=\hat{D}(\sqrt{\eta}\beta)\left[\eta|1\rangle_{a}\,{}_{a}\\!\langle
1|+(1-\eta)|0\rangle_{a}\,{}_{a}\\!\langle
0|\right]\hat{D}(\sqrt{\eta}\beta)^{\dagger}.$ (22)

From Eq. (22) we can see that displacement by $\beta$ followed by losses
$\eta$ is essentially the same as introducing losses first by mixing the pure
single-photon state with vacuum, and then applying a displacement by the
reduced amount $\sqrt{\eta}\beta$. Due to the linearity of the Wigner
function, Eq. (22) shows that the experimentally reconstructed Wigner function
will in fact be

$W(p,q)=\eta W_{|1\rangle\langle 1|}(p,q)+(1-\eta)W_{|0\rangle\langle
0|}(p,q).$ (23)

As expected, losses (1-$\eta$) add a Gaussian vacuum function to the original
nonpositive Wigner function of the single-photon state. In particular, the
undisplaced photon-number distribution will yield the overall transmissivity
of the whole experiment $\eta$, as in Fig.4, left.

###  III.4 Quantum tomography of a single-photon state

We now turn to the state tomography results. Figure 6 shows the reconstructed
Wigner function along with a fit with Wigner function Eq. (23), of free
parameter $\eta$.

Figure 6: Top, reconstructed Wigner function. Black points: reconstructed
values from raw data. Solid surface: least-square Wigner-function fit, Eq.
(23). Bottom, fit residuals.

Figure 7: Phase-averaged Wigner function. The Wigner function fit yielded
$\eta=0.57(3)$, which is consistent with the heralding efficiency $0.58(2)$.
Error bars are discussed in the text.

The Wigner function is plotted for experimentally measured values of
$|\alpha|$ where phase space coordinates are
$(q,p)=(\sqrt{2}|\alpha|\cos\phi,\sqrt{2}|\alpha|\sin\phi)$, where $\phi$ is
the tomographic angle. We can clearly see the negativity around the origin of
the phase space,

$W(0,0)=-0.035\pm 0.005.$ (24)

Errors in the displacement amplitudes were considered to be negligible due to
the long-term amplitude stability of the laser producing the displacement
field and the high-accuracy of the calibration as mentioned in the
“displacement calibration” section. The Wigner function error bars (1$\sigma$)
at zero-displacement were obtained from the statistics of multiple data sets
with the displacement field blocked. At non-zero displacement, in order to
speed up the measurement process and minimize experimental drifts, we decided
to use the statistics of the measurement results at 10 different phases for
the same displacement amplitude. This procedure yields a conservative estimate
of the Wigner function error bars (1$\sigma$), in the particular case of a
single-photon Fock state, because it assumes that the measured Wigner function
has the required cylindrical symmetry about the origin of phase space. The
results are plotted on Fig.7. Note that the fact that Wigner function isn’t
significantly altered by this averaging — in fact, both the 2D fit in Fig.6
and the 1D fit in Fig.7 yield $\eta=0.57(3)$ — speaks to the high quality of
the phase-space rotational symmetry of our data. One can notice that the fit
residuals are reasonably small around the origin of phase space but grow
larger in the outskirts of the function, near our maximum displacement values.
These correspond to larger detected photon numbers on the TES, for which
photon pileups during the TES’ cooling time make data analysis more arduous
Sridhar2014a .

##  IV Conclusion

We have demonstrated state-independent photon-counting quantum state
tomography with PNR measurements using a superconducting TES system and
evidenced clear negativity in the single-photon Fock Wigner function with no
correction for photon loss. This work has been limited by two factors: when
working with continuous-wave detection, photon fluxes become overwhelming to
the TES when $|\alpha|\to 1$. Moreover, photon pileups, in particular during
the TES cooling time, greatly complicate data analysis Sridhar2014a . In the
future, we will multiplex several TES channels in order to to access larger
displacement amplitudes, i.e., larger regions of phase space. This will also
reduce the photon pileup effect. Finally, owing to the intrinsic simplicity of
photon-counting quantum tomography, we believe it is possible to herald and
visualize Fock state Wigner functions in real time for quantum information
applications.

## Funding Information

This work was supported by NSF grants PHY-1521083 and PHY-1708023.

## Acknowledgments

The authors thank Rafael Alexander, Carlos Andreas Gonzalez Arciniegas, Xu Yi,
Avi Pe’er, Chun-Hung Chang, Jacob Higgins, Chaitali Joshi, and Xuan Zhu for
helpful discussions. We would also like to thank Scott Glancy and Arik Avagyan
for valuable comments and feedback during the revision process.

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# In-vehicle Sensing and Data Analysis for Older Drivers with Mild Cognitive
Impairment
*Corresponding Author ††thanks: This research was supported by the U.S. National Institute of Health (NIH) (R01 AG068472 02S1) and National Science Foundation (NSF) (OAC 1948066).

1st Sonia Moshfeghi  College of Engg and Computer Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>2nd Muhammad Tanveer Jan  College of Engg and Computer
Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>3rd Joshua Conniff  Charles E. Schmidt College of Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>4th Seyedeh Gol Ara Ghoreishi  College of Engg and Computer
Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>5th* Jinwoo Jang  College of Engg and Computer Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>6th Borko Furht  College of Engg and Computer Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>7th Kwangsoo Yang  College of Engg and Computer Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>8th Monica Rosselli  Charles E. Schmidt College of Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>9th David Newman  Christine E. Lynn College of Nursing
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>10th Ruth Tappen  Christine E. Lynn College of
Nursing
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>11st Dana Smith  College of Engg and Computer Science
Florida Atlantic University
Boca Raton, USA
<EMAIL_ADDRESS>
###### Abstract

Driving is a complex daily activity indicating age and disease-related
cognitive declines. Therefore, deficits in driving performance compared with
ones without mild cognitive impairment (MCI) can reflect changes in cognitive
functioning. There is increasing evidence that unobtrusive monitoring of older
adults’ driving performance in a daily-life setting may allow us to detect
subtle early changes in cognition. The objectives of this paper include
designing low-cost in-vehicle sensing hardware capable of obtaining high-
precision positioning and telematics data, identifying important indicators
for early changes in cognition, and detecting early-warning signs of cognitive
impairment in a truly normal, day-to-day driving condition with machine
learning approaches. Our statistical analysis comparing drivers with MCI to
those without reveals that those with MCI exhibit smoother and safer driving
patterns. This suggests that drivers with MCI are cognizant of their condition
and tend to avoid erratic driving behaviors. Furthermore, our Random Forest
models identified the number of night trips, number of trips, and education as
the most influential factors in our data evaluation.

###### Index Terms:

in-vehicle sensing, telematics data, machine learning, cognitive impairment

††publicationid: pubid: pt ©2023 IEEE. Personal use of this material is
permitted. Permission from IEEE must be obtained for all other uses, in any
current or future media, including preprinting/republishing this material for
advertising or promotional purposes, creating new collective works, for resale
or redistribution to servers or lists, or reuse of any copyrighted component
of this work in other works.

##  I Introduction

The World Health Organisation (WHO) and National Institute of Health (NIH)
reports reveal that the overall number of people aged 65 and above quickly
expands due to improved health care and higher life expectancy. The
predictions show that by 2050, the percentage of the world’s population aged
65 and up will have increased from 10 percent in 2022 to 16 percent. About 19
percent of the people in North America were aged 65 or older in 2022, making
it one of the regions with the highest proportion of older residents [1, 2].
Mild cognitive impairment (MCI) is an anomalous decline in cognitive function
that exceeds the predicted average decline. The clinical concerns persist in
differentiating MCI from otherwise healthy conditions in older people [3]. A
person with MCI has a decline in cognitive abilities like memory, language,
and reasoning skills, having difficulty remembering recent events or
conversations, finding the right words, planning, and organizing tasks.
Although it is a condition that occurs between healthy brain aging and
dementia, they can still perform their usual daily activities. Some cases of
MCI may progress into dementia over time. MCI can be caused by various
factors, including age-related changes and certain medical conditions, such as
Alzheimer’s disease.

According to the Veterans Administration’s work group on Driving Safety for
Veterans with Dementia, older adults with moderate to severe dementia should
limit their driving due to safety concerns [4]. Several studies describe how
emotions in daily life can affect human behavior and facial expression [5].
Driving is a complex diurnal life activity and can indicate age- and disease-
related cognitive declines [6, 7, 8]. Therefore, driving performance deficits
compared with those without mild cognitive impairment (MCI) can reflect
changes in cognitive functioning [9]. Research indicates that older people
with mild to severe dementia have limited physical and mental abilities [3].
Determining age- and cognitive-related ‘fitness-to-drive’ remains unspecified
and controversial since driving licensee varies by state. Current driver
evaluation programs can test only a small number of drivers with cognitive
concerns, missing many who need to know if they can continue to drive safely.
Importantly, no concrete scientific approaches have been investigated to
directly understand the current or former status of cognitively impaired older
drivers in daily life [10].

Advances in wireless technology, local computing, and data science can
contribute to unobtrusively monitoring daily activities to assess real-world
cognitive and functional changes [11, 12, 13]. Sensing methods and monitoring
tools were developed and applied to real-time data to assess patients’ well-
being [14]. The potential of using in-vehicle sensing to identify cognitive
impairment has recently been studied [15]. In-vehicle sensing is a novel and
unobtrusive idea for measuring driver behavior and detecting cognitive
deterioration. In-vehicle sensors and their applications for monitoring driver
behavior have advanced significantly. Some vehicular and behavioral strategies
provide viable and inexpensive solutions for diagnosing early dementia in
older drivers based on driving characteristics. [16, 17]. Older drivers show
age-related changes in driving behavior, such as traveling at a lower speed
and self-regulating driving [18]. The effects of aging on driving safety
prompted the development of a comprehensive program to examine individual and
medical information and to conduct behavioral analysis for operational and
GPS-based data [19]. Machine Learning methods have roles in validating the
interaction between the brain function and sensors [20, 21]. As a supervised
machine learning method, Random Forests demonstrated higher accuracy in
identifying and reporting anomalies [22]. Real-life driving data with machine-
learning approaches for demographic and behavioral factors show that age,
number of trips, race/ethnicity, time, and the number of hard brakes have been
the most accurate predictors of MCI and dementia in older drivers [23]. Some
machine learning models could predict the most critical diagnostic factors,
such as Alzheimer’s disease genetics and individual and behavioral patterns
for older drivers [24]. However, high-quality sensing data and novel
scientific techniques that can distinguish cognitive-decline-related changes
from age-related ones to understand the impact of cognitive impairment on
driving behavior is missing [25]. Therefore, the authors have developed new
programmable in-vehicle sensing units with unobtrusive vision and telematics
sensors to track and record the subtle driver behavior of older drivers. The
novelty of the telematics units used in this study is that they build on
microcomputers and enable programmable and expendable hardware and software
capabilities. The in-vehicle sensors installed 150+ older drivers’ vehicles
since 2022 and tracked changes in older drivers’ cognitive functioning in a
daily-life driving setting. Our programmable in-vehicle sensing units comprise
a set of unobtrusive vision and telematics sensors. The recorded changes in
driver behavior are compared to results from a battery of cognitive tests
(executive function, visuospatial, memory, and language) selected for
sensitivity to early subtle changes in cognition and ability to predict driver
risk. The collected data for the group of participants under the study was
analyzed and visualized using statistical [26] and machine learning methods.

Figure 1: In-vehicle Sensing System

Figure 2: Telematics Unit (TMU)

##  II Design of In-vehicle Sensing

The authors developed a novel in-vehicle sensing system, which includes
telematics and vision sensing units, to collect driver-behavior-related
datasets unobtrusively (see Fig. 1). The in-vehicle sensing system is
installed on participants’ vehicles for three years to collect longitudinal
data streams to capture changes in driving behavior over time. The five-year
study was conducted at Florida Atlantic University and the University of
Central Florida (UCF) for older drivers with or without MCI and at least 65
with a valid driver’s license. The study involved 236 participants in the
third year to evaluate drivers’ cognitive and behavioral status. All
participants completed the initial evaluation form and provided written
approval. Individual sessions are subsequently organized every three months to
examine any changes in drivers’ health and vehicle conditions and to complete
related clinical tests, including the Montreal Cognitive Assessment (MoCA),
Loewenstein-Acevedo Scales of Semantic Interference and Learning (LASSI-L).

TABLE I: DBIs Definition and Statistical Overview

— Driver Indexes —
---
Index | Description | Statistical Remarks
|  | Min | Max | Mean | StDev
MCI |  | The diagnosis for MCI
---
(1: MCI; 0: non-MCI)
0 | 1 | - | -
Age | The participants with 65 years old and above | 65 | 89 | 75.67 | 6.04
Gender | Male; Female | - | - | - | -
Race |  | 1:Black; 2:White; 3:Asian;
---
4:Native American;
5:Multiracial; 6:Others
- | - | - | -
Ethnicity |  | 1:African-American;
---
2:European-American;
3:Hispanic-American;
4:Afro-Caribbean;
5:Asian-American;
6:Others
- | - | - | -
Education |  | Grade school; High school;
---
Some college; Technical school;
Associate degree; Bachelor’s degree;
Post-graduate studies; Master’s degree;
Post-master’s degree; Doctoral degree
- | - | - | -
Retired | No; Yes | - | - | - | -
BMI |  | Body Mass Index
---
(Obese: greater than 30;
non-Obese: lesser than 30)
- | - | - | -
— Driving Indexes —
Total Trip |  | Unit: No.
---
The total number of trips
5 | 693 | 266.57 | 174.81
Night Trip |  | Unit: No.
---
The number of trips at night
0 | 58 | 5.8 | 11.56
Peak Trip |  | Unit: No.
---
The number of trips at peak hours
(7:00 to 9:00 and 16:00 to 18:00)
0 | 596 | 222.97 | 156.09
Duration |  | Unit: s
---
The duration of each trip per second
0 | 38658.26 | 1280.95 | 2029.02
Distance |  | Unit: km
---
The distance of each trip per kilometer
0 | 412.35 | 10.59942 | 20.55902
Speed |  | Unit: km/h
---
The speed in kilometers per hour
0 | 158 | 26.74934 | 10.99516
RPM | The revolution per minute | 0 | 5698.25 | 1118.675 | 194.8706
nH-Acceleration |  | Unit: No.
---
Number of harsh-acceleration
(The number of events with an acceleration
greater than 3.943 $m/s^{2}$ on the X axis)
1 | 233 | 3.005261 | 6.084068
nH-Braking |  | Unit: No.
---
Number of hard-Braking
(The number of events with an acceleration
greater than -3.943 $m/s^{2}$ on the X axis)
1 | 123 | 3.614141 | 9.438562
nH-Turn |  | Unit: No.
---
Number of hard-turns
(The number of events with an acceleration
greater than -3.943 $m/s^{2}$ on the Y axis)
1 | 51 | 2.506608 | 2.671595
Urban Trip |  | Unit: No.
---
The number of trips below 32 kilometers
- | 7742 | - | -
Suburb Trip |  | Unit: No.
---
The number of trips above 32 kilometers
- | 38 | - | -

###  II-A Programmable Telematics Units

This study leverages TMUs developed by AutoPi, which build on Raspberry Pi 4
Model B. Fig. 2 shows an overview of the Raspberry-based TMU units. These
programmable and open-sourced TMUs support the robust system expandability of
hardware and software, allowing the development of customized TMUs to adjust
and modify the sensing parameters (e.g., data collection algorithms and
sampling frequencies). Each TMU comprises 1) a GPS sensor, 2) an inertial
measurement unit (IMU), 3) an OBD connector, 4) a 4G/LTE cellular modem, 5) an
SD card, and 6) a USB flash drive. Tri-axial gyroscopes are added to the TMUs
to capture angular velocity. It continuously monitors the voltage levels from
the port. When the voltage goes down to 12V, it automatically goes into sleep
mode for five minutes. After waiting for another 5 minutes, the device goes
into hibernate mode. The power usage of the AutoPi in its sleep and
hibernation mode is as follows: around 30mA and 10mA.

In-vehicle data can be collected from the Controller Area Network (CAN bus)
[27, 28] and augmented sensor hardware. Within a group of vehicles, CAN allows
the engine control unit (ECU), transmission control unit, brake system control
unit, steering system control unit, and others to exchange information quickly
and efficiently, providing access to vehicle status data such as sensor
readings, control signals, and diagnostic information. One of the most common
ways of obtaining CAN bus data is using On-Board Diagnostics (OBD) connectors.
Available OBD data includes engine RPM, vehicle speed, and fuel system status.
In addition to the OBD data, additional data can be collected from augmented
sensor hardware, such as GPS and inertial measurement unit (IMU) modules.
Furthermore, various unobtrusive sensing units can be installed on a vehicle
to understand the state of drivers [29].

##  III Driver Behavior Indexes

This study used 7794 data points with 19 indexes over two years (see Table I).
The indexes included 19 independent variables as inputs denoted by the matrix
of $X\in\mathbb{R}^{m\times n}$, given by Eq. 1 to predict the presence or
absence of MCI as the dependent or output variable. Considering the values 0
for the absence of MCI (non-MCI) and 1 for the presence of MCI as
$y_{i}\in\\{0,1\\}$ for $i=1,...,m$, the vector of $Y\in\mathbb{R}^{m}$ has
been obtained. $m$ represents the number of samples, and $n$ is the number of
independent variables. Therefore, the values of $m$ and $n$ are 7794 and 19,
respectively.

$X=\begin{bmatrix}x_{11}&\cdots&x_{1n}\\\ \vdots&\ddots&\vdots\\\
x_{m1}&\cdots&x_{mn}\end{bmatrix}{and}~{}~{}~{}Y=\begin{bmatrix}y_{1}\\\
\vdots\\\ y_{m}\end{bmatrix}$ (1)

Some indexes refer to drivers’ characteristics and health circumstances, such
as age, gender, race, ethnicity, education, and body mass index (BMI). Both
males and females were required to be at least 65 years old. We examined their
race/ethnicity to see how their physical traits, nationality, religion,
linguistics, or culture influence their driving patterns. To define
race/ethnicity, we evaluated six categories for each. The education and
employment status of the participants can reveal information about their
social, psychological, and economic situation. We divided them into ten groups
based on their education level and whether they are employed or retired. Body
mass index (BMI) is a unit for calculating body fat by the height and weight
of individuals. We classified participants into obese and non-obese groups. We
defined a set of indicators for driving features. We studied the total number
of trips, the number of trips between 21:00 and 5:00 as night trips, and the
number of trips from 7:00 to 9:00 and 16:00 to 18:00 as peak hours trips.
Driving characteristics are kinematic factors such as trip duration and
distance, speed, rpm, and acceleration. The number of harsh acceleration, hard
braking, and hard turns was determined. When the acceleration on the X-axis
reaches 3.943 m/s2, it is called harsh acceleration; when the acceleration on
the X-axis surpasses -3.943 m/s2, it is deemed hard braking. A hard turn is
defined as an acceleration greater than -3.943 m/s2. We evaluated the types of
trips following their mileage. Urban trips are defined as distances between 0
and 32 km, whereas suburban trips have a distance greater than 32 km [30].

##  IV Data Analysis

The in-vehicle data are further analyzed to characterize driver behavior
indexes (DBIs) that can reflect physical and cognitive functions over time.

###  IV-A Data Prepossessing

We combined the quantile normalization method to normalize data distribution
and quantile-based flooring and capping to treat outliers, with flooring for
lesser values (10th percentile) and capping for higher values (90th
percentile). The skewness value dropped for all features after applying this
method (e.g., nH-Acceleration decreased from 15.33 to 1.56.).

###  IV-B Machine Learning Technique

We used a supervised machine learning algorithm named Random Forests (RF) for
classification purposes. This ensemble learning method integrates multiple
machine learning models to improve their performance. Numerous Random Forest
algorithm decision trees have been built on a different random subset of
training data and features. Each training tree is to predict the target
variable, and the final prediction is based on the mode of all the trees’
predictions. The random forest training algorithm extends the generic bagging
technique to tree learners. Bagging repeatedly ($b$ times) has a random
selection with the substitute of the training set and fits trees to these
samples, given a training set $X=x_{1},...,x_{i}$ with responses
$Y=y_{1},...,y_{i}$ for $B=1,...,b$, where $i$ is the number of training
samples and $b$ shows the number of trees. We developed Random forest
classification models with multiple decision trees of (1) only age; (2) the
number of trips (for total, peak hours, and night hours); (3) driver
variables; (4) driving variables; (5) age with driving variables (6) All the
variables to predict MCI status. These six groups show the contribution of
each variable in predicting MCI status and the model’s performance.

Figure 3: Trip Distribution based on Time of Day

(a) (b)

Figure 4: Quarterly Evaluation Reports ((a)-non-MCI; (b)-MCI)

##  V Results

###  V-A Statististical reports

We analyzed the time of day for a subset of the participants. We divided the
time into four categories: 5:00 to 11:59 as the morning; 12:00 to 16:59 as the
afternoon; 17:00 to 20:59 as the evening; and 21:00 to 4:59 as the night. Fig.
3 represents the number of trips in each category using a pie chart. According
to the results, half of the trips occur between 12:00 and 16:59, meaning
participants prefer to drive more in the afternoon and less at night.
Considering 34.9 percent for morning trips, older drivers prefer driving
during daylight. Fig. 4 depicts the three-month aggregated data for
participants with and without MCI, which presents changes in their driving
patterns during quarterly visits.

###  V-B Random Forests Model

Table II reports the performance metrics and confusion matrix for the
developed model in six groups. The first model represents the contribution of
age as the only variable for evaluation. The obtained accuracy needs to show
better performance for this model. The second model considered only the number
of trips for three categories: the total number of trips, the number of trips
at night, and the number of trips for peak hours. This model offers a better
performance in comparison to the first model. Model 4 is related to only
drivers and their demographic characteristics. This model evaluates age,
gender, ethnicity, education, race, body mass index (BMI), and employment
status. The performance result is approximately similar to the first model.
Models 4, 5, and 6 recruit driving variables with three different preferences
involving driver variables with orders of no driver variables, only age, and
all driver variables. All these three models show a similar performance as a
result.

TABLE II: Evaluation of Random Forest Model Performance Metrics

Model | Input | Accuracy | AUC | Precision | Recall |  | F1
---
Score
| Confusion
---
Matrix
|  |  |  |  |  |  | Predicted | Observed
|  |  |  |  |  |  |  | 0 | 1
1 | Only age | 0.79 | 0.87 | 1.00 | 0.70 | 0.82 | 0 | 1242 | 533
|  |  |  | 0.60 | 1.00 | 0.75 | 1 | 0 | 797
2 |  | Number of trips
---
(total, peak, night)
0.86 | 0.91 | 0.83 | 1.00 | 0.91 | 0 | 1775 | 0
|  |  |  | 1.00 | 0.54 | 0.70 | 1 | 366 | 431
3 | Driver Variables | 0.82 | 0.87 | 0.79 | 1.00 | 0.88 | 0 | 1774 | 1
|  |  |  | 1.00 | 0.41 | 0.58 | 1 | 468 | 329
4 | Driving Variables | 0.78 | 0.93 | 0.75 | 1.00 | 0.86 | 0 | 1775 | 0
|  |  |  | 1.00 | 0.28 | 0.43 | 1 | 576 | 221
5 | Age with driving variables | 0.86 | 0.94 | 0.83 | 1.00 | 0.91 | 0 | 1774 | 1
|  |  |  | 1.00 | 0.54 | 0.70 | 1 | 366 | 431
6 | All the variables | 0.86 | 0.95 | 0.83 | 1.00 | 0.91 | 0 | 1775 | 0
|  |  |  | 1.00 | 0.54 | 0.70 | 1 | 366 | 431

Figure 5: Importance of Features

##  VI Conclusion

This research addresses the pressing problem of identifying and assessing mild
cognitive impairment (MCI) among older drivers using in-vehicle sensing
technology and machine learning methods. This study aims to develop a system
capable of recognizing early signs of cognitive decline and providing
appropriate interventions to ensure the safety and well-being of older
drivers. Integrating sensors such as cameras, accelerometers, and the
telematics unit into vehicles enables the collection of various data points
related to driving behavior, vehicle dynamics, and driver interactions. These
data points serve as valuable indicators, helping to reveal subtle alterations
associated with MCI. The potential benefits of these systems are undeniable.
Early identification of MCI can enable timely intervention and support for
older drivers, improving safety while prolonging independence and mobility. By
harnessing in-vehicle sensing technology and machine learning techniques, this
study opens the way to developing intelligent systems capable of responding in
real-time to diverse driving scenarios and detecting and assessing MCI.

This system can classify and predict MCI in real time by training machine
learning models on diverse datasets. These models may provide timely
interventions, such as alerts or adaptive driving assistance, to support older
drivers with MCI and mitigate potential road risks. Machine learning
techniques offer several advantages in this respect, allowing for the creation
of models that can recognize anomalous patterns or conditions indicative of
MCI, adapt, and improve over time as more data is collected and analyzed. In
addition, quickly processing large amounts of data enables real-time
monitoring and decision-making that allows prompt responses to changing
cognitive states. Depending on the training data’s quality, there can be
various challenges and considerations related to the accuracy and reliability
of machine learning models. Collecting diverse and comprehensive data sets
accurately representing MCI-related driving behaviors and patterns is
protracted. Ensuring the privacy and security of the data collection process
is also essential. Measures taken must anonymize and protect sensitive
personal data while permitting practical analysis.

Overall, this study aims to assist in designing advanced assisting systems for
supporting older drivers while driving. The findings provide insight into
novel solutions that improve road safety and quality of life for older drivers
with mild cognitive impairments.

## Acknowledgment

We want to express our gratitude to the U.S. National Institute of Health
(NIH) (R01 AG068472 02S1) and the National Science Foundation (NSF) (OAC
1948066) for their excellent assistance and financing.

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