# Bright multiplexed source of indistinguishable single photons
with tunable GHz-bandwidth at room temperature

O. Davidson  R. Finkelstein  E. Poem  O. Firstenberg  Physics of Complex
Systems, Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Narrowband single photons that couple well to atomic ensembles could prove
essential for future quantum networks, but the efficient generation of such
photons remains an outstanding challenge. We realize a spatially-multiplexed
heralded source of single photons that are inherently compatible with the
commonly employed D2 line of rubidium. Our source is based on four-wave mixing
in hot rubidium vapor, requiring no laser cooling or optical cavities, and
generates single photons with high rate and low noise. We use Hong-Ou-Mandel
interference to verify the indistinguishability of the photons generated in
two different (multiplexed) channels. We further demonstrate a five-fold
tunability of the photons’ temporal width. The experimental results are well
reproduced by a theoretical model.

##  I Introduction

Single photons are key ingredients in quantum optics experiments and in
quantum communication protocols, such as quantum repeaters and quantum key
distribution (QKD) Eisman_2011_review_single_photons . A prevalent method for
generating single photons is spontaneous parametric down-conversion (SPDC) in
$\chi^{(2)}$ nonlinear crystals 2019_cavity_SPDC_review . However, the photons
generated in SPDC are inherently broadband, and thus it is difficult to
interface them with atomic ensembles, as required for many quantum
communication and computation protocols. Narrowing the source bandwidth by
placing the nonlinear crystals inside an optical cavity generally comes at the
cost of a reduction in other performance parameters and in technical overhead
Cavity_SPDC_Ou_1999 ; 2007_photon_source_cavity_SPDC_Polzik ;
cavity_SPDC_Riedmatten_2016 ; 2017_photon_source_cavity_SPDC_Chuu ;
2018_photon_source_cavity_SPDC_Chen ; cavity_SPDC_Treutlein_2020 ;
2020_photon_source_cavity_SPDC_Zhang ; 2020_photon_source_cavity_SPDC_Walther
; 2021_photon_source_cavity_SPDC_Guo_arxiv ;
2021_photon_source_cavity_SPDC_Shi .

Another successful route for single-photon generation is utilizing single
quantum emitters. A prime example is semiconductor quantum-dots in micro-
cavities photon_source_QD_Senellart_2016 ; 2016_Pan_QD ; 2021_Tomm_QD_cavity
able to generate photons with a bandwidth of several GHz, which could, in
principle, couple to atomic ensembles. However, only limited success has been
achieved to date in interfacing these photons with atomic ensembles
QD_atoms_interface_Akopian_2011 ; 2018_QD_interfacing_atomic_vapor_Michler .
Moreover, it is difficult to achieve indistinguishable photons from two
distinct quantum dots, making these source’s scaling-up challenging. Another
type of single emitters is single trapped atoms photon_source_atom_Kimble_2004
; photon_source_atom_Rempe_2007  and ions 2007_Monroe_single_ion_photon_source
; photon_source_ion_Blatt_2016 , which generate single photons that are
inherently compatible with atomic systems. However, these systems require a
relatively complex apparatus and their photon collection efficiency and
generation rate are relatively low.

As an alternative to single emitters, correlated photon pairs can be generated
using atomic ensembles photon_soure_DLCZ_cold_Kimble_2004 ;
photon_soure_DLCZ_cold_Kuzmich_2005 ; photon_soure_DLCZ_cold_Vuletic_2006 ;
photon_soure_DLCZ_cold_Granjier_2014 ; photon_soure_DLCZ_cold_Riedmatten_2016
; photon_soure_DLCZ_hot_XianMinJin_2018 ; photon_source_CW_cold_Harris_2005 ;
photon_source_CW_cold_Srivathsan_2013 ; photon_source_CW_cold_Kurtriefer_2014
; photon_source_CW_cold_Zhang_2016 ; photon_source_CW_hot_Guo_2012 ;
2015_source_vapor_w_solid_state_Guo ; photon_source_CW_hot_Du_2016 ;
photon_source_CW_hot_Du_2017 ; photon_source_CW_hot_SebMoon_2016 ;
2017_HOM_interference_two_sources_Seb_Moon ;
2018_source_CW_hot_atoms_Franson_interference_Seb_Moon ;
2019_source_CW_hot_atoms_Sagnac_Seb_Moon ;
2020_Slodicka_biphoton_source_hot_vapor ; 2021_photon_source_hot_vapor_Yu . In
the Duan-Lukin-Cirac-Zoller protocol 2001_DLCZ_paper , the photons generation
is done with a write-read pulse sequence: the first photon heralds the
existence of a spin-wave in the atomic ensemble, which is later converted into
a second photon. Alternatively, photon pairs can be generated in continuous
operation by a four-wave mixing (FWM) process, where detecting one photon
heralds the existence of the other. Such heralded sources were realized with
cold photon_soure_DLCZ_cold_Kimble_2004 ; photon_soure_DLCZ_cold_Kuzmich_2005
; photon_soure_DLCZ_cold_Vuletic_2006 ; photon_soure_DLCZ_cold_Granjier_2014 ;
photon_source_CW_cold_Kurtriefer_2014 ; photon_soure_DLCZ_cold_Riedmatten_2016
; photon_source_CW_cold_Zhang_2016  and hot atomic ensembles
photon_source_CW_hot_Du_2016 ; photon_source_CW_hot_Du_2017 ;
photon_source_CW_hot_SebMoon_2016 . Hot atomic vapor provides a simpler setup
and can operate continuously, although often motional-broadening reduces the
atomic coherence and consequently lowers the signal-to-noise ratio (SNR).

Figure 1: Comparison of heralded single-photon sources in terms of the peak of
the normalized bi-photon cross-correlation function
$[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}$ and the generation rate of heralded
single photons $R$. Getting to the top right corner is desirable. We include
bi-photon sources with a bandwidth on the order of a few GHz or smaller,
capable in principle of interfacing efficiently with atomic ensembles. $R$ is
the actual detection rate in kilo-counts per second (kcps). Sources: Hot vapor
2015_source_vapor_w_solid_state_Guo ; photon_source_CW_hot_Du_2017 ;
photon_source_CW_hot_SebMoon_2016 ; 2019_source_CW_hot_atoms_Sagnac_Seb_Moon ;
2020_Slodicka_biphoton_source_hot_vapor ; 2021_photon_source_hot_vapor_Yu ,
cold atoms photon_source_CW_cold_Harris_2005 ;
photon_source_CW_cold_Srivathsan_2013 , cavity SPDC
2007_photon_source_cavity_SPDC_Polzik ; cavity_SPDC_Riedmatten_2016 ;
2017_photon_source_cavity_SPDC_Chuu ; 2018_photon_source_cavity_SPDC_Chen ;
cavity_SPDC_Treutlein_2020 ; 2020_photon_source_cavity_SPDC_Zhang ;
2020_photon_source_cavity_SPDC_Walther ;
2021_photon_source_cavity_SPDC_Guo_arxiv ; 2021_photon_source_cavity_SPDC_Shi
, FWM in MRR (Microring resonators) 2020_source_FWM_in_MRR_Thompson ;
2020_source_FWM_in_MRR_Thew_arxiv ; 2021_source_FWM_in_MRR_Bajoni_arxiv . See
Supplementary Material for more details.  Figure 2: Energy levels and
experimental setup. (a) The relevant atomic levels of ${}^{85}\text{Rb}$. In
the presence of pump and control lasers, phase-matched signal and idler
photons are spontaneously generated. (b) The experimental setup. The pump and
control lasers counter-propagate inside the vapor cell. Signal and idler
photons are generated in the cone-shaped phase-matched direction and collected
from two spatial modes into single-mode fibers. HWP: half-wave plate, IF:
interference filter (FWHM 3 nm), M: mirror, P: Glan calcite polarizer, PBS:
polarizing beam splitter, SMF: single-mode fiber, SPCM: single-photon counting
module.

Here we study a heralded single-photon source based on FWM in hot
${}^{85}\text{Rb}$ vapor. We employ the nearly Doppler-free configuration of
the $5S_{1/2}-5P_{3/2}-5D_{5/2}$ ladder scheme. This system was first used as
a photon source by Lee et al. photon_source_CW_hot_SebMoon_2016 ;
2017_HOM_interference_two_sources_Seb_Moon ;
2018_source_CW_hot_atoms_Franson_interference_Seb_Moon ;
2019_source_CW_hot_atoms_Sagnac_Seb_Moon . As shown in Fig. 1, we
simultaneously achieve high generation rate and high SNR (quantified by the
bi-photon cross-correlation). Furthermore, we demonstrate the multiplexing
capability of the source by collecting single photons from two spatial
channels and verifying their indistinguishability using Hong-Ou-Mandel (HOM)
interference. We investigate the system performance for different optical
depths of the atomic ensemble and demonstrate the ability to control the
photons’ temporal width. The experimental study is accompanied by a detailed
model reproducing the measurements. This system is adequate for interfacing
with other quantum systems and particularly with Rb atoms. When combined with
a quantum memory FLAME_paper , it can be used to construct large multi-photon
states applicable for secure QKD.

##  II Setup

We use the rubidium level system shown in Fig. 2(a). The pump field, coupling
the $5S_{1/2}\rightarrow 5P_{3/2}$ transition, originates from a 780-nm
distributed Bragg reflector laser. It is blue detuned by $\Delta=1$ GHz from
the transition in order to minimize the absorption to the $5P_{3/2}$ level
while maintaining a large third-order susceptibility $\chi^{(3)}$. The control
field, coupling the $5P_{3/2}\rightarrow 5D_{5/2}$ transition, originates from
an external-cavity diode-laser at 776 nm. It is tuned such that the two-photon
transition is on resonance for maximizing $\chi^{(3)}$. The two lasers are
independently phase-locked to stable reference lasers, which are locked to
ultra-stable cavities. We do not perform dedicated optical pumping, and
therefore all the Zeeman states in the ground level are roughly equally
populated.

Figure 2(b) shows the experimental setup. The vertically-polarized pump and
horizontally-polarized control fields, with beam waist radii of 0.45 mm,
counter-propagate through a 25-mm-long cell of pure ${}^{85}\text{Rb}$ vapor.
The cell is placed inside a three-layer mu-metal magnetic shield mounted on a
pitch-yaw rotation stage, which is used to minimize the scattering of light
off the vapor-cell facets into the signal and idler modes. We vary the optical
depth of the medium in the range $\text{OD}=1-15$ (measured for the
$|5S_{1/2},F=3\rangle\rightarrow|5P_{3/2},F=2,3,4\rangle$ transitions) by
tuning the cell temperature. The signal and idler photons are spontaneously
generated in a FWM process in the phase-matched directions. After polarization
and wavelength filtering [interference filters, full width at half maximum
(FWHM) of 3 nm], the horizontally-polarized signal and vertically-polarized
idler photons are coupled into single-mode fibers (SMF) that act as spatial
filters.

Due to the wavelength mismatch in the ladder scheme and due to carefully
tuning the vapor cell orientation, we do not need to use narrow-band Fabry-
Perot cavities to filter out the pump and control light. This increases the
setup transmission and contributes to the high generation rate we achieve. The
signal and idler photons are collected at $\sim$ 1.4° off-axis from both
sides, thus forming two spatially multiplexed collection channels. This angle
is chosen as a compromise between maximizing the phase-matching condition,
which is practically on axis, and minimizing the noise due to the pump and
control light. The signal and idler photons are then detected by single-photon
counting modules (SPCM; Excelitas NIR-14-FC) with a $\sim$68% quantum
efficiency. The SPCMs are connected to a time tagging device (Swabian Time
Tagger Ultra). The combined detection jitter of the time tagger and the two
SPCMs was measured with a sub-ps laser to be approximately a sech function
with FWHM of 590 ps.

To align the signal and idler output-couplers, we pick off some of the control
(pump) light and use it to seed the idler (signal) mode in the first (second)
collection channel. A stimulated FWM signal is then generated in the phase-
matched direction and coupled into the corresponding SMF. Subsequently, since
the spontaneously generated photons have slightly different modes than the
(collimated) seed beams, we perform the final optimization of the phase-
matching conditions by maximizing the spontaneous bi-photon generation rate
$R$ without the seed light. This procedure increases $R$ by more than 40%.

##  III Results

We begin with measurements of the cross-correlation and generation rate of the
bi-photons. The normalized signal-idler cross-correlation function is defined
as $g_{\text{s-i}}^{(2)}(\tau)=\langle
a^{\dagger}_{\text{i}}(t)a^{\dagger}_{\text{s}}(t+\tau)a_{\text{s}}(t+\tau)a_{\text{i}}(t)\rangle/[\langle
a^{\dagger}_{\text{s}}a_{\text{s}}\rangle\langle
a^{\dagger}_{\text{i}}a_{\text{i}}\rangle]$, where
$\tau=t_{\text{s}}-t_{\text{i}}$ is the time difference between the detection
of the signal and idler photons, and $\langle\cdot\rangle$ indicates averaging
over the time $t$. Figure 3(a) shows $g_{\text{s-i}}^{(2)}(\tau)$ measured at
$\text{OD}=9.3$ (cell temperature $\sim 55$°C) and low pump and control
powers. A strong signal-idler temporal correlation is evident, with a peak
value $[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}=709\pm 10$, higher than
previously-reported values in atomic vapor systems. The peak value
$[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$ is often associated with the SNR of
the source photon_source_CW_hot_Du_2016 , and our source violates the
classical Cauchy-Schwartz inequality
$[[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}]^{2}/[g^{(2)}_{\text{s-s}}(0)g^{(2)}_{\text{i-i}}(0)]\leq
1$ classical_inequlities_1986  by a factor greater than $10^{5}$. When
increasing the pump and control powers,
$[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$ decreases, as shown in Fig. 3(b),
while the bi-photon generation rate $R$ increases, as shown in Fig. 3(c). For
$R$, we consider only detected pairs and subtract the background coincidence
counts. The trade-off between $R$ and $[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$,
i.e., between the rate and the SNR, is summarized in Fig. 3(d); the dashed
line in this figure is a fit to a pure inverse relation ($\propto R^{-1}$).

Figure 3: Bi-photon cross-correlation. (a) The normalized cross-correlation
between the signal and idler photons $g_{\text{s-i}}^{(2)}$ versus
$\tau=t_{\text{s}}-t_{\text{i}}$, where $t_{\text{s}}$ ($t_{\text{i}})$ is the
signal (idler) photon detection time. Here the Rabi frequencies of the pump
and control fields are $\Omega_{\text{p}}=4.6$ MHz (power 0.05 mW) and
$\Omega_{\text{c}}=11.5$ MHz (power 5 mW), respectively. The inset shows a
schematic of the detection scheme, where TT stands for time tagger. (b) The
peak of the normalized cross-correlation function
$[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$ versus $\Omega_{\text{p}}$ and
$\Omega_{\text{c}}$ (values given in MHz). (c) The bi-photon generation rate
$R$ as a function of $\Omega_{\text{p}}$ and $\Omega_{\text{c}}$. (d)
$[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$ versus $R$ in the full parameter range
($\Omega_{\text{p}}=5-15$ MHz and $\Omega_{\text{c}}=10-35$ MHz). In all
plots, circles are measured data, and solid curves are numerical calculations
(see text for details of the calculations). Dashed line in (d) is a fit to a
pure $R^{-1}$ curve. Here $\text{OD}=9.3$. Error bars correspond to 1 standard
deviation (s.d.).

In order to demonstrate that the bi-photon source can be used as a heralded
single photon source, we measure the auto-correlation of signal photons,
conditioned on the detection of an idler photon. In this measurement, we use a
Hanbury Brown and Twiss setup, where the signal photons are sent into a fiber
beam-splitter connected to two SPCMs. For a given temporal coincidence-window
$\Delta\tau$, the conditional auto-correlation is given by
$g_{\text{c}}^{(2)}(0;\Delta\tau)=N_{\text{i-s}_{1}\text{-s}_{2}}N_{\mathrm{i}}/[N_{\text{i-s}_{1}}N_{\text{i-s}_{2}}]$
1986_Grangier_cascade_source . Here $N_{\mathrm{i}}$ is the idler count rate,
$N_{\text{i-s}_{1}}(\Delta\tau)$ and $N_{\text{i-s}_{2}}(\Delta\tau)$ are the
count rates of coincidences of an idler photon and a signal photon in one of
the two SPCMs in a coincidence time window $\Delta\tau$, and
$N_{\text{i-s}_{1}\text{-s}_{2}}(\Delta\tau)$ is the three-photon coincidence
count rate. Figure 4(a) shows that $g_{\text{c}}^{(2)}(0;\Delta\tau)\ll 1$ for
all coincidence windows in the range $0<\Delta\tau\leq 4$ ns, indicating
strong anti-bunching. As $\Delta\tau$ decreases, we measure a smaller fraction
$E_{\text{c}}(\Delta\tau)$ of the single-photon wave-packet
[$E_{\text{c}}(\infty)=100\%$]; this fraction is given in Fig. 4(b). For
$\Delta\tau=2.5$ ns, $95.6\pm 1.4\%$ of the heralded single photon intensity
is measured. Figure 4(c) shows $g_{\text{c}}^{(2)}(0;\Delta\tau)$ as a
function of the bi-photon generation rate $R$ for $\Delta\tau=2.5$ ns. The
minimal value measured is $g_{\text{c}}^{(2)}(0;\Delta\tau)=0.012\pm 0.0003$,
with $R=15$ kcps. Even with $R=400$ kcps,
$g_{\text{c}}^{(2)}(0;\Delta\tau)=0.27$ is still well below the two-photon
threshold of $g_{\text{c}}^{(2)}(0;\Delta\tau)=0.5$, thus validating the
single-photon nature of the source.

Figure 4: The single-photon nature of the heralded source. (a) The auto-
correlation of signal photons $g_{\text{c}}^{(2)}(0;\Delta\tau)$, conditioned
on the detection of an idler photon within a temporal coincidence window
$\Delta\tau$. Here the pump and control Rabi frequencies are
$\Omega_{\text{p}}=6.5$ MHz and $\Omega_{\text{c}}=30$ MHz, respectively. (b)
The overall signal counts $E_{\text{c}}$ within $\Delta\tau$ normalized by the
counts in a wide coincidence window ($\Delta\tau>4$ ns). Less than $5\%$ of
the heralded single photons are trimmed for a choice of $\Delta\tau\geq 2.5$
ns. (c) The conditional auto-correlation function versus the heralded single-
photon generation rate $R$ for $\Delta\tau=2.5$ ns. Dark-blue data points in
(a) and (b) correspond to $\Delta\tau=2.5$ ns used in (c). Error bars are 1
s.d. The solid line is a linear fit to the data. The inset in (c) shows a
schematic of the detection scheme.

The detection of an idler photon heralds the generation of a collective state
comprising a single $5P_{3/2}$ excitation that is shared among all atoms. The
subsequent spontaneous emission of a signal photon from this state into the
phased-matched direction is collectively enhanced
Jen_2012_superradiance_in_cascade_systems . The number of atoms, quantified by
the OD, thus plays an important role in the source performance, which we now
examine. As expected and as shown in Figs. 5(a,b), the heralded photon
generation rate increases with OD, while the duration (temporal width) of the
heralded photons decreases. Taking into account the combined electronic time-
jitter for two detectors (590 ps), we find that the photon’s duration can be
varied five-fold by changing the OD [dashed line in Fig. 5(b)].

Figure 5: Source performance as a function of optical depth. (a) Heralded
single-photon generation rate $R$. (b) Duration of the heralded single-
photons. The dashed line shows the inferred duration after removing the
electronic detection time-jitter, demonstrating tunability of the photon
duration by more than $\times 5$. (c) Signal photon heralding efficiency
$\eta$, reaching an optimum at $\text{OD}\approx 5$. (d) The peak of the
normalized cross-correlation function for a constant $R$, reaching an optimum
at $\text{OD}\approx 7-11$. Circles are measured values, and curves are
numerical calculations. In (a) and (b), the pump and control Rabi frequencies
are $\Omega_{\text{p}}=6.5$ MHz and $\Omega_{\text{c}}=21$ MHz, respectively.
In (c) and (d), $\Omega_{\text{p}}$ and $\Omega_{\text{c}}$ are adjusted to
keep a constant $R=100$ kcps.

We now turn to measure the heralding efficiency, defined as
$\eta=R/N_{\text{i}}$, which quantifies the probability to get a signal photon
given a detection of an idler photon. To be conservative, we consider only
detected events and do not correct for the setup transmission or detectors
inefficiencies. For a fixed generation rate $R=100$ kcps, the heralding
efficiency as a function of OD shows an optimum $\eta\approx 10.5\%$ at
$\text{OD}\approx 5$, see Fig. 5(c). The decrease in $\eta$ at high OD can be
explained by reabsorption of the signal photon in the medium, whereas the
increase at low OD is explained by the enhancement in the collective emission
into the phase-matched direction Jen_2012_superradiance_in_cascade_systems .
Figure 5(d) shows $[g_{\text{s-i}}^{(2)}]_{\mathrm{max}}$ versus OD. Here
also, there is an optimal OD that maximizes the SNR, at $\text{OD}\approx
7-11$. This optimal OD is higher than that found for the heralding efficiency
due to the enhanced emission rate of the photons at higher OD.

Figure 6: Photons indistinguishability. Coincidence counts of the heralded
photons (in the two signal channels) as a function of the time difference
between the heralding photons (idler photons) in a HOM interference setup.
Interference visibility of $V=82.7\pm 4.1\%$ is observed without any
background subtraction, demonstrating high indistinguishability between the
two channels. Error bars are 1 s.d., estimated from Poissonian counting
statistics. The inset shows the shape of the heralded photons in the two
signal channels, exhibiting a temporal likeness of $>99\%$. The temporal
window considered for a coincidences detection (gray shaded area) starts at
the peak of the cross-correlation function because of the asymmetry resulting
from the cascaded emission of the idler and signal photons and considering
that the detection time jitter (590 ps) is large compared to the photons’
duration.

Next, we turn to demonstrate the multiplexing capability of the source by
using two collection channels of idler-photon pairs and using HOM interference
to characterize the indistinguishably of the generated photons. Figure 6 shows
the four-fold coincidence counts of the heralded single photons in a HOM
setup, where the two signal photons generated in the two collection channels
enter a 50:50 fiber-BS from different input ports HOM_1987 . The two BS
outputs are connected to SPCMs. The horizontal axis is the time difference
between the detection of the two idler photons, which heralds the generation
of the two signal photons. The raw HOM visibility is $V=82.7\pm 4.1\%$. The
main factor that reduces $V$ from unity is the large detection time-jitter
compared to the photons duration, which reduces the heralded photons’ purity
by $\sim 20\%$ 2015_Du_photon_purity , thus limiting the HOM visibility to
$V\sim 80\%$ 2008_Mosley_JSA_and_purity . For this reason, the measurement was
done with $\text{OD}=4.7$, for which the photons are relatively long, while
the SNR is still high. The inset of Fig. 6 shows the shape of the heralded
photons generated in the two collection channels. The temporal likeness of the
two photons, defined as $|\int\sqrt{G_{1}(t)}\sqrt{G_{2}(t)}dt|^{2}/[\int
G_{1}(t)dt\int G_{2}(t)dt]$, where $G_{1(2)}$ is the non-normalized cross-
correlation function in channel 1(2), is $>99\%$. Here $\Omega_{\text{p}}=7$
MHz and $\Omega_{\text{c}}=10.5$ MHz, and the generation rates in the two
channels are $R=27$ kcps and $R=20$ kcps.

##  IV Model

We base our model on the state-vector formalism of the FWM interaction
Du_2008_bi_photon_state_vector_model . We provide here a brief description of
the model, while details are given in as Supplementary Material (SM).

Let $\omega_{\text{s}}$, $\omega_{\text{i}}$, $\omega_{\text{p}}$, and
$\omega_{\text{c}}$ be the frequencies of the signal, idler, pump, and control
fields, respectively. We write the wave numbers of these fields as
$k_{\text{s}}=\sqrt{1+\chi_{\text{s}}(\omega_{\text{s}})}\omega_{\text{s}}/c$,
$k_{\text{i}}=\sqrt{1+\chi_{\text{i}}(\omega_{\text{i}})}\omega_{\text{i}}/c$,
$k_{\text{p}}=\omega_{\text{p}}/c$, and $k_{\text{c}}=\omega_{\text{c}}/c$,
where for the signal and idler fields we include the linear susceptibility of
the atomic medium, incorporating the effects of gain, loss, and dispersion.
The bi-photon wavefunction is given by Du_2008_bi_photon_state_vector_model

$\psi(\tau)=\frac{L}{2\pi}\int
d\omega_{\text{s}}\kappa(\omega_{\text{s}})\text{sinc}\Big{(}\frac{\Delta
kL}{2}\Big{)}e^{i(k_{\text{s}}+k_{\text{i}})L/2}e^{-i\omega_{\text{s}}\tau},$
(1)

where $L$ is the length of the atomic medium, and $\Delta
k=(k_{\text{s}}-k_{\text{i}})-(k_{\text{p}}-k_{\text{c}})$ is the phase
mismatch of the FWM process, assuming that the angular deviation of the fields
from the $\hat{z}$ axis is small. The term
$e^{i(k_{\text{s}}+k_{\text{i}})L/2}$ represents the free propagation of the
signal and idler fields, the phase matching condition is manifested in
$\text{sinc}(\Delta kL/2)$, and
$\kappa(\omega_{\text{s}})\propto\chi^{(3)}(\omega_{\text{s}})$ represents the
nonlinear interaction; the full expression of $\kappa(\omega_{\text{s}})$ is
given in the SM.

Previously published calculations of $\chi^{(3)}$ have assumed that the atomic
population remains in the ground state, i.e., that the pump field is weak
Du_2008_bi_photon_state_vector_model ; photon_source_CW_hot_Guo_2012 . Here,
we do not make this assumption. We use the finite population in the excited
states to model the signal and idler detection count rates and, together with
the bi-photon wavefunction [Eq. (1)], we numerically calculate the scaling of
$[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}$ and the heralding efficiency with the
pump and control powers and with the OD. Detailed expressions of $\chi^{(3)}$
and exact expressions for the numerical curves in Figs. 3 and 5 are given in
the SM. No fit parameters are used in these calculations except for fixed
rescaling factors of $g_{\text{s-i}}^{(2)}$, $R$, and $\eta$. The scaling
factors are the same for all presented data and account for photon collection
efficiency and for the conversion ratio of the excited-state population to the
signal and idler detection rates.

##  V Discussion

We now return to Fig. 1, comparing between different heralded single-photon
sources in terms of the generation rate $R$ (actual detection rates, without
any corrections) and $[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}$, which quantifies
the SNR. Here we limit ourselves to sources with bandwidth on the order of a
few GHz or lower, which, in principle, could be interfaced with atomic
ensembles or with high-efficiency solid-state memories. Our source outperforms
all other atomic sources in this metric and is comparable to the best solid-
state sources.

Several factors enable the high generation rate and SNR of our source. First,
the nearly Doppler-free level scheme of the $5S_{1/2}-5P_{3/2}-5D_{5/2}$
excitation maintains the spin-wave coherence for a long time. Second, we
minimize photon loss by not using narrow-band etalons, which are usually
employed to filter out the noise originating from the scattering of the
driving fields. Third, we optimize the powers and detunings of the driving
fields and the OD of the medium. Finally, our pump and control lasers are
independently locked to ultra-stable references, further increasing the spin-
wave coherence of the $5S\rightarrow 5D$ transition.

Residual scattering of the driving fields from the vapor cell’s facets weakly
couples into the collection channels, generating noise that degrades the
source performance. The noise becomes a limitation at a pump power below 50
$\mu$W and at a control power above 40 mW. This noise can be minimized by
small changes in the collection angles at the expense of deteriorating the
phase-matching conditions (in our optimizations, minimizing the noise reduces
$R$ by $\sim 25-30\%$). These limitations can be overcome by using better wall
coatings to reduce the scattering of the driving fields. Moreover, the HOM
interference visibility can be greatly improved by using detectors with a
lower electronic detection time jitter, such as superconducting nanowire
detectors.

Our source, like the majority of photon sources, has a greater than 50% vacuum
component. This property makes the generated heralded photons less suitable
for quantum computation applications
positive_Wigner_make_classical_computation_efficient . In the future, this
limitation may be elevated by using detectors with higher quantum efficiency
and by enclosing the signal and idler modes in cavities. At the source’s
current level of the performance, up to $R\approx 200$ kcps, our heralded
single photons are in a quantum non-Gaussian (QNG) state, as estimated from
our $g_{\text{c}}^{(2)}(0;\Delta\tau)$ measurements using the QNG state
criterion $P_{\text{c}}<P_{\text{s}}^{3}/3$ (for $P_{\text{c}}\ll
P_{\text{s}}$) Quantum_non_Gaussian_criteria_2_w_experiment ;
photon_source_ion_Blatt_2016 . Here, $P_{\text{s}}\approx 0.09$ is the
probability, upon a heralding event, to detect exactly one photon in the
signal mode, and $P_{\text{c}}\approx 2.5\times 10^{-5}-1.6\times 10^{-4}$
(for $R=15-126$ kcps) is the probability to detect two photons. The QNG
criterion, which is stricter than the non-classical condition of
$g_{\text{c}}^{(2)}(0)<1$ Quantum_non_Gaussian_criteria_2_w_experiment , is a
sufficient condition for the security of QKD
Quantum_non_Gaussian_sufficient_for_QKD , thus making our source a potential
resource for QKD applications.

In conclusion, we demonstrate a highly bright, spatially-multiplexed, heralded
single-photon source, emitting indistinguishable photons with a tunable
bandwidth. The generated photons are inherently compatible for interfacing
with rubidium atomic ensembles, and we simultaneously achieve high SNR and
high generation rate. These properties make the source adequate for atom-
photon quantum networks. Furthermore, the generated photons are in a QNG
state, making them suitable for secure QKD.

###### Acknowledgements.

We thank Lee Drori and Ohr Lahad for useful discussions and technical
assistance. We acknowledge financial support by the the Israel Science
Foundation, the US-Israel Binational Science Foundation (BSF) and US National
Science Foundation (NSF), the European Research Council starting investigator
grant QPHOTONICS 678674, the Minerva Foundation with funding from the Federal
German Ministry for Education and Research, and the Laboratory in Memory of
Leon and Blacky Broder.

## References

  * [1] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov.  Invited review article: Single-photon sources and detectors.  Review of Scientific Instruments, 82(7):071101, 2011.
  * [2] Oliver Slattery, Lijun Ma, Kevin Zong, and Xiao Tang.  Background and review of cavity-enhanced spontaneous parametric down-conversion.  J. Res. Natl. Inst. Stan., 124:124019, 2019.
  * [3] Z. Y. Ou and Y. J. Lu.  Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons.  Phys. Rev. Lett., 83:2556–2559, 1999.
  * [4] J. S. Neergaard-Nielsen, B. Melholt Nielsen, H. Takahashi, A. I. Vistnes, and E. S. Polzik.  High purity bright single photon source.  Opt. Express, 15(13):7940–7949, 2007.
  * [5] Daniel Rieländer, Andreas Lenhard, Margherita Mazzera, and Hugues de Riedmatten.  Cavity enhanced telecom heralded single photons for spin-wave solid state quantum memories.  New Journal of Physics, 18(12):123013, 2016.
  * [6] Chih-Hsiang Wu, Tsung-Yao Wu, Yung-Chin Yeh, Po-Hui Liu, Chin-Hsuan Chang, Chiao-Kai Liu, Ting Cheng, and Chih-Sung Chuu.  Bright single photons for light-matter interaction.  Phys. Rev. A, 96:023811, 2017.
  * [7] Pin-Ju Tsai and Ying-Cheng Chen.  Ultrabright, narrow-band photon-pair source for atomic quantum memories.  Quantum Science and Technology, 3(3):034005, 2018.
  * [8] Roberto Mottola, Gianni Buser, Chris Müller, Tim Kroh, Andreas Ahlrichs, Sven Ramelow, Oliver Benson, Philipp Treutlein, and Janik Wolters.  An efficient, tunable, and robust source of narrow-band photon pairs at the 87rb d1 line.  Opt. Express, 28(3):3159–3170, 2020.
  * [9] Jianji Liu, Jiachen Liu, Ping Yu, and Guoquan Zhang.  Sub-megahertz narrow-band photon pairs at 606 nm for solid-state quantum memories.  APL Photonics, 5(6):066105, 2020.
  * [10] Amir Moqanaki, Francesco Massa, and Philip Walther.  Novel single-mode narrow-band photon source of high brightness tuned to cesium d2 line.  APL Photonics, 4(9):090804, 2019.
  * [11] Xiao Liu, Jun Hu, Zong-Feng Li, Xue Li, Pei-Yun Li, Peng-Jun Liang, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo.  Experimental demonstration of functional quantum repeater nodes with absorptive quantum memories.  arXiv:2101.04945, 2021.
  * [12] Yin-Hai Li, Zhi-Yuan Zhou, Shi-Long Liu, Yan Li, Shi-Kai Liu, Chen Yang, Shuang Wang, Zhi han Zhu, Wei Gao, Guang-Can Guo, and Bao-Sen Shi.  Compact sub-ghz bandwidth single-mode time-energy entangled photon source for high-speed quantum networks.  OSA Continuum, 4(2):608–620, 2021.
  * [13] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart.  Near-optimal single-photon sources in the solid state.  Nature Photonics, 10(5):340–345, 2016.
  * [14] Xing Ding, Yu He, Z.-C. Duan, Niels Gregersen, M.-C. Chen, S. Unsleber, S. Maier, Christian Schneider, Martin Kamp, Sven Höfling, Chao-Yang Lu, and Jian-Wei Pan.  On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar.  Phys. Rev. Lett., 116:020401, 2016.
  * [15] Natasha Tomm, Alisa Javadi, Nadia Olympia Antoniadis, Daniel Najer, Matthias Christian Löbl, Alexander Rolf Korsch, Rüdiger Schott, Sascha René Valentin, Andreas Dirk Wieck, Arne Ludwig, and Richard John Warburton.  A bright and fast source of coherent single photons.  Nature Nanotechnology, 2021.
  * [16] N. Akopian, L. Wang, A. Rastelli, O. G. Schmidt, and V. Zwiller.  Hybrid semiconductor-atomic interface: slowing down single photons from a quantum dot.  Nature Photonics, 5(4):230–233, 2011.
  * [17] Hüseyin Vural, Simone L. Portalupi, Julian Maisch, Simon Kern, Jonas H. Weber, Michael Jetter, Jörg Wrachtrup, Robert Löw, Ilja Gerhardt, and Peter Michler.  Two-photon interference in an atom-quantum dot hybrid system.  Optica, 5(4):367–373, 2018.
  * [18] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble.  Deterministic generation of single photons from one atom trapped in a cavity.  Science, 303(5666):1992–1994, 2004.
  * [19] Markus Hijlkema, Bernhard Weber, Holger P. Specht, Simon C. Webster, Axel Kuhn, and Gerhard Rempe.  A single-photon server with just one atom.  Nature Physics, 3(4):253–255, 2007.
  * [20] P. Maunz, D. L. Moehring, S. Olmschenk, K. C. Younge, D. N. Matsukevich, and C. Monroe.  Quantum interference of photon pairs from two remote trapped atomic ions.  Nature Physics, 3(8):538–541, 2007.
  * [21] D B Higginbottom, L Slodička, G Araneda, L Lachman, R Filip, M Hennrich, and R Blatt.  Pure single photons from a trapped atom source.  New Journal of Physics, 18(9):093038, 2016.
  * [22] C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble.  Single-photon generation from stored excitation in an atomic ensemble.  Phys. Rev. Lett., 92:213601, 2004.
  * [23] T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich.  Storage and retrieval of single photons transmitted between remote quantum memories.  Nature, 438(7069):833–836, 2005.
  * [24] James K. Thompson, Jonathan Simon, Huanqian Loh, and Vladan Vuletić.  A high-brightness source of narrowband, identical-photon pairs.  Science, 313(5783):74–77, 2006.
  * [25] Erwan Bimbard, Rajiv Boddeda, Nicolas Vitrant, Andrey Grankin, Valentina Parigi, Jovica Stanojevic, Alexei Ourjoumtsev, and Philippe Grangier.  Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory.  Phys. Rev. Lett., 112:033601, 2014.
  * [26] Pau Farrera, Georg Heinze, Boris Albrecht, Melvyn Ho, Matías Chávez, Colin Teo, Nicolas Sangouard, and Hugues de Riedmatten.  Generation of single photons with highly tunable wave shape from a cold atomic ensemble.  Nature Communications, 7(1):13556, 2016.
  * [27] Jian-Peng Dou, Ai-Lin Yang, Mu-Yan Du, Di Lao, Jun Gao, Lu-Feng Qiao, Hang Li, Xiao-Ling Pang, Zhen Feng, Hao Tang, and Xian-Min Jin.  A broadband dlcz quantum memory in room-temperature atoms.  Communications Physics, 1(1):55, 2018.
  * [28] Vlatko Balić, Danielle A. Braje, Pavel Kolchin, G. Y. Yin, and S. E. Harris.  Generation of paired photons with controllable waveforms.  Phys. Rev. Lett., 94:183601, 2005.
  * [29] Bharath Srivathsan, Gurpreet Kaur Gulati, Brenda Chng, Gleb Maslennikov, Dzmitry Matsukevich, and Christian Kurtsiefer.  Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble.  Phys. Rev. Lett., 111:123602, 2013.
  * [30] Gurpreet Kaur Gulati, Bharath Srivathsan, Brenda Chng, Alessandro Cerè, Dzmitry Matsukevich, and Christian Kurtsiefer.  Generation of an exponentially rising single-photon field from parametric conversion in atoms.  Phys. Rev. A, 90:033819, 2014.
  * [31] Peng Qian, Zhenjie Gu, Rong Cao, Rong Wen, Z. Y. Ou, J. F. Chen, and Weiping Zhang.  Temporal purity and quantum interference of single photons from two independent cold atomic ensembles.  Phys. Rev. Lett., 117:013602, 2016.
  * [32] Dong-Sheng Ding, Zhi-Yuan Zhou, Bao-Sen Shi, Xu-Bo Zou, and Guang-Can Guo.  Generation of non-classical correlated photon pairs via a ladder-type atomic configuration: theory and experiment.  Opt. Express, 20(10):11433–11444, 2012.
  * [33] Dong-Sheng Ding, Wei Zhang, Shuai Shi, Zhi-Yuan Zhou, Yan Li, Bao-Sen Shi, and Guang-Can Guo.  Hybrid-cascaded generation of tripartite telecom photons using an atomic ensemble and a nonlinear waveguide.  Optica, 2(7):642–645, 2015.
  * [34] Chi Shu, Peng Chen, Tsz Kiu Aaron Chow, Lingbang Zhu, Yanhong Xiao, M. M. T. Loy, and Shengwang Du.  Subnatural-linewidth biphotons from a doppler-broadened hot atomic vapour cell.  Nature Communications, 7(1):12783, 2016.
  * [35] Lingbang Zhu, Xianxin Guo, Chi Shu, Heejeong Jeong, and Shengwang Du.  Bright narrowband biphoton generation from a hot rubidium atomic vapor cell.  Appl. Phys. Lett., 110(16):161101, 2017.
  * [36] Yoon-Seok Lee, Sang Min Lee, Heonoh Kim, and Han Seb Moon.  Highly bright photon-pair generation in doppler-broadened ladder-type atomic system.  Opt. Express, 24(24):28083–28091, 2016.
  * [37] Taek Jeong, Yoon-Seok Lee, Jiho Park, Heonoh Kim, and Han Seb Moon.  Quantum interference between autonomous single-photon sources from doppler-broadened atomic ensembles.  Optica, 4(10):1167–1170, 2017.
  * [38] Jiho Park, Taek Jeong, Heonoh Kim, and Han Seb Moon.  Time-energy entangled photon pairs from doppler-broadened atomic ensemble via collective two-photon coherence.  Phys. Rev. Lett., 121:263601, 2018.
  * [39] Jiho Park, Heonoh Kim, and Han Seb Moon.  Polarization-entangled photons from a warm atomic ensemble using a sagnac interferometer.  Phys. Rev. Lett., 122:143601, 2019.
  * [40] J Mika and L Slodička.  High nonclassical correlations of large-bandwidth photon pairs generated in warm atomic vapor.  Jr. Phys. B: At. Mol. Opt. Phys., 53(14):145501, 2020.
  * [41] Chia-Yu Hsu, Yu-Sheng Wang, Jia-Mou Chen, Fu-Chen Huang, Yi-Ting Ke, Emily Kay Huang, Weilun Hung, Kai-Lin Chao, Shih-Si Hsiao, Yi-Hsin Chen, Chih-Sung Chuu, Ying-Cheng Chen, Yong-Fan Chen, and Ite A. Yu.  Generation of sub-mhz and spectrally-bright biphotons from hot atomic vapors with a phase mismatch-free scheme.  Opt. Express, 29(3):4632–4644, 2021.
  * [42] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller.  Long-distance quantum communication with atomic ensembles and linear optics.  Nature, 414:413, 2001.
  * [43] Daniel Llewellyn, Yunhong Ding, Imad I. Faruque, Stefano Paesani, Davide Bacco, Raffaele Santagati, Yan-Jun Qian, Yan Li, Yun-Feng Xiao, Marcus Huber, Mehul Malik, Gary F. Sinclair, Xiaoqi Zhou, Karsten Rottwitt, Jeremy L. O’Brien, John G. Rarity, Qihuang Gong, Leif K. Oxenlowe, Jianwei Wang, and Mark G. Thompson.  Chip-to-chip quantum teleportation and multi-photon entanglement in silicon.  Nature Physics, 16(2):148–153, 2020.
  * [44] Farid Samara, Nicolas Maring, Anthony Martin, Arslan S. Raja, Tobias J. Kippenberg, Hugo Zbinden, and Rob Thew.  Entanglement swapping between independent and asynchronous integrated photon-pair sources.  arXiv:2011.08150, 2020.
  * [45] Francesco Garrisi, Federico Andrea Sabattoli, Nicola Bergamasco, Micol Previde Massara, Federico Pirzio, Francesco Morichetti, Andrea Melloni, Marco Liscidini, Matteo Galli, and Daniele Bajoni.  Characterization of photon pairs generated by a silicon ring resonator under electrical self-pumping.  arXiv:2011.03273, 2020.
  * [46] Ran Finkelstein, Eilon Poem, Ohad Michel, Ohr Lahad, and Ofer Firstenberg.  Fast, noise-free memory for photon synchronization at room temperature.  Science Advances, 4(1), 2018.
  * [47] M. D. Reid and D. F. Walls.  Violations of classical inequalities in quantum optics.  Phys. Rev. A, 34:1260–1276, 1986.
  * [48] P. Grangier, G. Roger, and A. Aspect.  Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences.  EPL (Europhysics Letters), 1(4):173, 1986.
  * [49] H. H. Jen.  Positive-$p$ phase-space-method simulation of superradiant emission from a cascade atomic ensemble.  Phys. Rev. A, 85:013835, 2012.
  * [50] C. K. Hong, Z. Y. Ou, and L. Mandel.  Measurement of subpicosecond time intervals between two photons by interference.  Phys. Rev. Lett., 59:2044–2046, 1987.
  * [51] Shengwang Du.  Quantum-state purity of heralded single photons produced from frequency-anticorrelated biphotons.  Phys. Rev. A, 92:043836, 2015.
  * [52] Peter J. Mosley, Jeff S. Lundeen, Brian J. Smith, Piotr Wasylczyk, Alfred B. U’Ren, Christine Silberhorn, and Ian A. Walmsley.  Heralded generation of ultrafast single photons in pure quantum states.  Phys. Rev. Lett., 100:133601, 2008.
  * [53] Shengwang Du, Jianming Wen, and Morton H. Rubin.  Narrowband biphoton generation near atomic resonance.  J. Opt. Soc. Am. B, 25(12):C98–C108, 2008.
  * [54] A. Mari and J. Eisert.  Positive wigner functions render classical simulation of quantum computation efficient.  Phys. Rev. Lett., 109:230503, 2012.
  * [55] Ivo Straka, Ana Predojević, Tobias Huber, Luká š Lachman, Lorenz Butschek, Martina Miková, Michal Mičuda, Glenn S. Solomon, Gregor Weihs, Miroslav Ježek, and Radim Filip.  Quantum non-gaussian depth of single-photon states.  Phys. Rev. Lett., 113:223603, 2014.
  * [56] Mikołaj Lasota, Radim Filip, and Vladyslav C. Usenko.  Sufficiency of quantum non-gaussianity for discrete-variable quantum key distribution over noisy channels.  Phys. Rev. A, 96:012301, 2017.

Supplementary material

##  S1 Numerical model

The starting point of our modeling is Eq. (1) in the main text, provided by Du
et al. [53], which delineates the bi-photon wavefunction

$\psi(\tau)=\frac{L}{2\pi}\int
d\omega_{\text{s}}\kappa(\omega_{\text{s}})\text{sinc}\Big{(}\frac{\Delta
kL}{2}\Big{)}e^{i(k_{\text{s}}+k_{\text{i}})L/2}e^{-i\omega_{\text{s}}\tau}.$
(S1)

Here $\kappa(\omega_{\text{s}})$ is defined as

$\kappa(\omega_{\text{s}})=-i\frac{2\sqrt{\bar{\omega}_{\text{s}}\bar{\omega}_{\text{i}}}}{c}\chi^{(3)}(\omega_{\text{s}})E_{\text{p}}E_{\text{c}},$
(S2)

where $\bar{\omega}_{\text{s}}$ and $\bar{\omega}_{\text{i}}$ are the central
frequencies of the signal and idler photons, $2E_{\text{p}}$ and
$2E_{\text{c}}$ are the pump and control electric-field amplitudes, and
$\chi^{(3)}(\omega_{\text{s}})$ is the third-order susceptibility of the
atomic medium, which is calculated in Sec. S2 below.

The solid curve $g_{\text{s-i}}^{(2)}(\tau)$ in Fig. 3(a) is calculated by
convolving $|\psi(\tau)|^{2}$ with the measured detectors’ time jitter. The
dashed and solid curves in Fig. 5(b) are the FWHM of $|\psi(\tau)|^{2}$ before
and after the convolution, respectively. The generation rate $R$ is calculated
by $R=\int|\psi(\tau)|^{2}d\tau$. The single detection count rates in the
signal and idler modes are given by
$N_{\text{s}}=2\rho_{22}\Gamma\cdot\text{OD}$ and
$N_{\text{i}}=2\rho_{44}\gamma\cdot\text{OD}$, respectively, where $\rho_{jj}$
is the population in state $|j\rangle$ as denoted in Fig. S1, and $2\Gamma$
($2\gamma$) is the decay rate from state $|2\rangle$ ($|4\rangle$). The peak
of the normalized cross-correlation and the heralding efficiency are
calculated as
$[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}=|\psi(\tau)|^{2}_{\mathrm{max}}/[N_{\text{i}}N_{\text{s}}]$
and $\eta=R/N_{\text{i}}$. No fit parameters are used except for a fixed
rescaling factors of $g^{(2)}_{\text{s-i}}$, $R$, and $\eta$, common to all
data in the paper. All experimental parameters of the system were
independently determined from spectroscopic measurements.

##  S2 Susceptibility calculation

We model our system as an ensemble of four-level atoms shown in Fig. S1. The
Hamiltonian in the rotating-wave approximation, assuming a parametric process
where
$\Delta_{\text{s}}=\Delta_{\text{p}}+\Delta_{\text{c}}-\Delta_{\text{i}}$, is

$\begin{split}&H=\hbar\bigg{[}-\Delta_{\text{p}}|2\rangle\langle
2|-\Delta_{\text{s}}|3\rangle\langle
3|-(\Delta_{\text{p}}+\Delta_{\text{c}})|4\rangle\langle 4|\\\ &\ \
-\Big{(}\Omega_{\text{p}}|2\rangle\langle
1|+\Omega_{\text{p}}^{*}|1\rangle\langle 2|+\Omega_{\text{c}}|4\rangle\langle
2|+\Omega_{\text{c}}^{*}|2\rangle\langle 4|\\\ &\ \
+\Omega_{\text{i}}|4\rangle\langle 3|+\Omega_{\text{i}}^{*}|3\rangle\langle
4|+\Omega_{\text{s}}|3\rangle\langle 1|+\Omega_{\text{s}}^{*}|1\rangle\langle
3|\Big{)}\bigg{]}.\end{split}$ (S3)

Here $\Delta$ and $\Omega$ are the fields’ detunings and Rabi frequencies, and
the subscripts p, c, i, and s stand for pump, control, idler, and signal
fields, respectively.

Figure S1: Schematic level structure and optical fields used for calculating
the susceptibility.

For brevity, we define $\delta=\Delta_{\text{p}}+\Delta_{\text{c}}$ and the
following complex decay rates for atoms with velocity $v$ along the optical
axis

$\displaystyle\Gamma_{21}(v)=\Gamma-i(\Delta_{\text{p}}-k_{\text{p}}v)$ (S4a)
$\displaystyle\Gamma_{31}(v)=\Gamma-i(\Delta_{\text{s}}-k_{\text{s}}v)$ (S4b)
$\displaystyle\Gamma_{42}(v)=(\Gamma+\gamma)-i(\Delta_{\text{c}}+k_{\text{c}}v)$
(S4c)
$\displaystyle\Gamma_{43}(v)=(\Gamma+\gamma)+i(\Delta_{\text{i}}+k_{\text{i}}v)$
(S4d)
$\displaystyle\Gamma_{41}(v)=\gamma-i[\delta-(k_{\text{p}}-k_{\text{c}})v]$
(S4e)
$\displaystyle\Gamma_{32}(v)=2\Gamma-i[\Delta_{\text{c}}-\Delta_{\text{i}}+(k_{\text{c}}-k_{\text{i}})v].$
(S4f)

The Master equation for atoms with velocity $v$ gives at the steady-state

$\displaystyle\dot{\rho}_{21}=0=-\Gamma_{21}(v)\rho_{21}+i\Omega_{\text{p}}(\rho_{11}-\rho_{22})-i\Omega_{\text{s}}\rho_{23}+i\Omega_{\text{c}}^{*}\rho_{41}$
(S5a)
$\displaystyle\dot{\rho}_{31}=0=-\Gamma_{31}(v)\rho_{31}+i\Omega_{\text{s}}(\rho_{11}-\rho_{33})-i\Omega_{\text{p}}\rho_{32}+i\Omega_{\text{i}}^{*}\rho_{41}$
(S5b)
$\displaystyle\dot{\rho}_{42}=0=-\Gamma_{42}(v)\rho_{42}+i\Omega_{\text{c}}(\rho_{22}-\rho_{44})-i\Omega_{\text{p}}^{*}\rho_{41}+i\Omega_{\text{i}}\rho_{32}$
(S5c)
$\displaystyle\dot{\rho}_{43}=0=-\Gamma_{43}(v)\rho_{43}+i\Omega_{\text{i}}(\rho_{33}-\rho_{44})-i\Omega_{\text{s}}^{*}\rho_{41}+i\Omega_{\text{c}}\rho_{23}$
(S5d)
$\displaystyle\dot{\rho}_{41}=0=-\Gamma_{41}(v)\rho_{41}-i\Omega_{\text{p}}\rho_{42}-i\Omega_{\text{s}}\rho_{43}+i\Omega_{\text{c}}\rho_{21}+i\Omega_{\text{i}}\rho_{31}$
(S5e)
$\displaystyle\dot{\rho}_{32}=0=-\Gamma_{32}(v)\rho_{32}-i\Omega_{\text{p}}^{*}\rho_{31}-i\Omega_{\text{c}}\rho_{34}+i\Omega_{\text{i}}^{*}\rho_{42}+i\Omega_{\text{s}}\rho_{12}$
(S5f)
$\displaystyle\dot{\rho}_{11}=0=2\Gamma\rho_{22}+2\Gamma\rho_{33}+i\Omega_{\text{p}}^{*}\rho_{21}-i\Omega_{\text{p}}\rho_{12}+i\Omega_{\text{s}}^{*}\rho_{31}-i\Omega_{\text{s}}\rho_{13}$
(S5g)
$\displaystyle\dot{\rho}_{22}=0=-2\Gamma\rho_{22}+\gamma\rho_{44}+i\Omega_{\text{p}}\rho_{12}-i\Omega_{\text{p}}^{*}\rho_{21}+i\Omega_{\text{c}}^{*}\rho_{42}-i\Omega_{\text{c}}\rho_{24}$
(S5h)
$\displaystyle\dot{\rho_{33}}=0=-2\Gamma\rho_{33}+\gamma\rho_{44}+i\Omega_{\text{s}}\rho_{13}-i\Omega_{\text{s}}^{*}\rho_{31}+i\Omega_{\text{i}}^{*}\rho_{43}-i\Omega_{\text{i}}\rho_{34}$
(S5i)
$\displaystyle\dot{\rho}_{44}=0=-2\gamma\rho_{44}+i\Omega_{\text{c}}\rho_{24}-i\Omega_{\text{c}}^{*}\rho_{42}+i\Omega_{\text{i}}\rho_{34}-i\Omega_{\text{i}}^{*}\rho_{43},$
(S5j)

where it is implicit that $\rho_{ij}\equiv\rho_{ij}(v)$.

The third-order susceptibility that generates the signal photons is given by
the third-order term in the fields generating the coherence $\rho_{31}$. Re-
arranging Eq. (S5b), we find

$\rho_{31}=\frac{1}{\Gamma_{31}(v)}\Big{[}i\Omega_{\text{s}}(\rho_{11}-\rho_{33})-i\Omega_{\text{p}}\rho_{32}+i\Omega_{\text{i}}^{*}\rho_{41}\Big{]}.$
(S6)

The first term in the brackets on the right-hand side corresponds to the
first-order susceptibility, and the second and third terms correspond to the
third-order susceptibility.

The relation $|\Omega_{\text{p}}\rho_{32}|\ll|\Omega_{\text{i}}^{*}\rho_{41}|$
is valid in our experiment for the vast majority of atomic velocities. It is
so, as the majority of the population remains in the ground state, and due to
the nearly Doppler-free configuration of the $5S_{1/2}\rightarrow
5P_{3/2}\rightarrow 5D_{5/2}$ transitions. We have numerically verified this
relation for a wide range of parameters. Therefore, we neglect the second term
in the brackets of the right-hand side of Eq. (S6), integrate over all
velocities, and arrive at the third-order susceptibility

$\chi^{(3)}(\Delta_{\text{s}})=in\frac{\mu_{\text{s}}\mu_{\text{i}}}{\varepsilon_{0}\hbar}\frac{1}{E_{\text{p}}E_{\text{c}}}\int
dvf(v)\frac{i\Omega_{\text{i}}^{*}}{\Gamma_{31}(v)}\rho_{41}(v).$ (S7)

Here $n$ is the atomic density, and $\mu_{13}$ and $\mu_{34}$ are the dipole
moments of the $|1\rangle\rightarrow|3\rangle$ and
$|3\rangle\rightarrow|4\rangle$ transitions, respectively.
$f(v)=\exp\big{(}v^{2}/(2v_{\text{T}}^{2})\big{)}/(\sqrt{2\pi}v_{\text{T}})$
is the one-dimensional Maxwell-Boltzmann distribution, and
$v_{\text{T}}=\sqrt{k_{\text{B}}T/m}$ is the thermal velocity, where $T$ is
the temperature and $m$ is the atomic mass of ${}^{85}\text{Rb}$.

We calculate the spinwave coherence per atomic velocity $\rho_{41}(v)$ by
neglecting the signal and idler vacuum fields. Therefore, Eq. (S5) is reduced
to the Master equation of a three-level ladder system, and we solve it
numerically for each $v$. Doing so, we also get the excited-state populations,
used to model the fluorescence into the signal and idler modes.

From Eq. (S6), we find the signal photon’s linear susceptibility

$\chi^{(1)}_{\text{s}}(\Delta_{\text{s}})=n\frac{|\mu_{13}|^{2}}{\varepsilon_{0}\hbar}\int
dvf(v)\frac{i\big{(}\rho_{11}(v)-\rho_{33}(v)\big{)}}{\Gamma_{21}(v)}.$ (S8)

The idler photon’s linear susceptibility is small and can be neglected.

A note on the weak-pump limit

For completeness, we also solve the steady-state of the Master equation
analytically under the weak-pump approximation. To first order in
$\Omega_{\text{p}}$ and $\Omega_{\text{s}}$, the population remains in the
ground state: $\rho_{11}=1,\ \rho_{22}=\rho_{33}=\rho_{44}=0$. Under this
assumption, Eqs. (S5) for the coherences simplify into two decoupled sets:

$\displaystyle 0=-\Gamma_{42}(v)\rho_{42}+i\Omega_{\text{i}}\rho_{32},$ (S9a)
$\displaystyle 0=-\Gamma_{43}(v)\rho_{43}+i\Omega_{\text{c}}\rho_{23},$ (S9b)
$\displaystyle
0=-\Gamma_{32}(v)\rho_{32}-i\Omega_{\text{c}}\rho_{34}+i\Omega_{\text{i}}^{*}\rho_{42},$
(S9c)

and

$\displaystyle
0=-\Gamma_{21}(v)\rho_{21}+i\Omega_{\text{p}}+i\Omega_{\text{c}}^{*}\rho_{41},$
(S10a) $\displaystyle
0=-\Gamma_{31}(v)\rho_{31}+i\Omega_{\text{s}}+i\Omega_{\text{i}}^{*}\rho_{41},$
(S10b) $\displaystyle
0=-\Gamma_{41}(v)\rho_{41}+i\Omega_{\text{c}}\rho_{21}+i\Omega_{\text{i}}\rho_{31}.$
(S10c)

We need to solve only the second set. Neglecting terms second-order in
$\Omega_{\text{s}},\Omega_{\text{i}}$, we get

$\rho_{41}(v)=-\frac{\Omega_{\text{p}}\Omega_{\text{c}}}{\Gamma_{21}(v)\Gamma_{41}(v)+|\Omega_{\text{c}}|^{2}},$
(S11)

and, using Eq. (S11) in Eq. (S10b), we find

$\rho_{31}(v)=\frac{i\Omega_{\text{s}}}{\Gamma_{31}(v)}-i\frac{\Omega_{\text{c}}\Omega_{\text{p}}\Omega_{\text{i}}^{*}}{\Gamma_{31}(v)\big{[}\Gamma_{21}(v)\Gamma_{41}(v)+|\Omega_{\text{c}}|^{2}\big{]}}.$
(S12)

The first term corresponds to the linear susceptibility of the signal photon
and the second term corresponds to the third-order susceptibility. Hence, the
third-order susceptibility averaged over all velocities in the weak-pump limit
is

$\chi^{(3)}(\Delta_{\text{s}})=\frac{n}{i\varepsilon_{0}\hbar^{3}}\int
dvf(v)\frac{\mu_{31}^{*}\mu_{43}^{*}\mu_{21}\mu_{42}}{\Gamma_{31}(v)\big{[}\Gamma_{21}(v)\Gamma_{41}(v)+|\Omega_{\text{c}}|^{2}\big{]}}.$
(S13)

##  S3 Comparison of single-photon sources

Table S1 shows the data of the heralded single-photon sources used to generate
Fig. 1 of the main text. In cases where
$[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}$ could not be evaluated from the
available data, but the coincidence-to-accidentals ratio (CAR) was given, we
estimate $[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}=2\times\text{CAR}$.

Table S1: Photon source parameters used for compiling Fig. 1 in the main text.
SPDC: Spontanoues parametric down-conversion; FWM: Four-wave mixing; MRR:
Microring resonators.

Reference | source | generation rate [kcps] | max cross-correlation
---|---|---|---
This work | Hot vapor | 55* | 254*
[41] | Hot vapor | 1 | 42
[8] | Cavity SPDC | 6 | 200**
[41] | Hot vapor | 1 | 42
[9] | Cavity SPDC | 0.11 | 11
[10] | Cavity SPDC | 2.5 | 11.25
[7] | Cavity SPDC | 6.1 | 92**
[5] | Cavity SPDC | 0.045 | 60
[36] | Hot vapor | 30 | 84
[45] | FWM in MRR | 26 | 360**
[39] | Hot vapor | 12 | 360
[33] | Hot vapor | 30 | 126
[35] | Hot vapor | 40 | 6
[40] | Hot vapor | 8 | 97
[29] | Cold atoms | 0.031 | 5800
[4] | Cavity SPDC | 8 | 65
[6] | Cavity SPDC | 2.9 | 40
[44] | FWM in MRR | 20 | 600**
[43] | FWM in MRR | 20 | 100**
[28] | Cold atoms | 0.36 | 20
[11] | Cavity SPDC | 2.5 | 50
[12] | Cavity SPDC | 42.5 | 600**

* One representative point from the curve shown in Fig. 1 of the main text.
** Here, $[g^{(2)}_{\text{s-i}}]_{\mathrm{max}}$ is estimated from the
coincidence-to-accidentals ratio (CAR).


# Objective Prediction of Tomorrow’s Affect Using Multi-Modal Physiological
Data and Personal Chronicles: A Study of Monitoring College Student Well-being
in 2020

Salar Jafarlou 1, Jocelyn Lai 2, Zahra Mousavi 2, Sina Labbaf 1,
Ramesh C. Jain 1, Nikil D. Dutt 1, Jessica L. Borelli 3, Amir M. Rahmani 1,4
1University of California Irvine, Department of Computer Science 2University
of California, Irvine, Department of Psychological Science 3University of
California Irvine, Department of Cognitive Sciences 4University of California
Irvine, School of Nursing

###### Abstract

Monitoring and understanding affective states are important aspects of healthy
functioning and treatment of mood-based disorders. Recent advancements of
ubiquitous wearable technologies have increased the reliability of such tools
in detecting and accurately estimating mental states (e.g., mood, stress,
etc.), offering comprehensive and continuous monitoring of individuals over
time. Previous attempts to model an individual’s mental state were limited to
subjective approaches or the inclusion of only a few modalities (i.e., phone,
watch). Thus, the goal of our study was to investigate the capacity to more
accurately predict affect through a fully automatic and objective approach
using multiple commercial devices. Longitudinal physiological data and daily
assessments of emotions were collected from a sample of college students using
smart wearables and phones for over a year. Results showed that our model was
able to predict next-day affect with accuracy comparable to state of the art
methods.

##  I INTRODUCTION

National estimates of mental health suggest that as of 2019, 1 in 5 adults in
the U.S. experience mental illness, with young adults at greater risk than
their older counterparts [1]. An important facet of mental health is affect,
subjective and physiological experiences of positive and negative emotions
[2]. Affective disturbances and dysregulation - more specifically, overly
heightened or dampened affect, prolonged negative affect, or instability in
experienced affect are core facets of many types of psychopathology. Since
these marks often precede the onset of psychopathology, they are considered to
be risk factors for psychopathology [3]. Monitoring and increased
understanding of one’s affect contributes to the regulation of affect [4], and
as such is a key component of many forms of intervention or approaches to
manage affective disturbances [5, 6]. In monitoring one’s affect, patients and
their provider can determine contextual factors relevant to experienced
emotions and moods, and in turn develop a plan through medication or therapy
to facilitate cognition and behavior in such a way that will reduce
disturbances. Traditional approaches rely on the patient self-reporting or
monitoring through diary or logging and require assistance from a therapist or
provider. The prevalence of technology, advances in Internet-of-things (IoT),
Wearable IoT (WIoT), and machine learning offer an easily accessible and less
burdensome approach to objectively monitor and predict affect, which can
result in a better understanding of an individual’s affect [7, 8]. These
advancements have opened new gateways in monitoring an individual’s different
physical and mental health aspects in a continuous and uninterrupted way.
While modeling behavioral mental health markers through smartphones already
existed [9], the research community gradually started to utilize wearables
technologies as additional modalities to assess physiology (e.g., autonomic
nervous systems activity in response to mental health stressors) and develop
more in-depth analyses of behavior (e.g., sleep, physical activity, etc.).
This resulted in more holistic information about an individual’s daily life
[10]. One notable study using a holistic approach is (SNAPSHOT) [11] that
assessed participant daily life for one month using two wristbands and a
smartphone per user. Authors found that some indicators of an individual’s
well-being (e.g., mood, stress and health) could be modeled using a set of
objective behavioral and biological features. In addition to objective
assessments, participants in this study completed self-reported assessments
about their activities and interactions. Subjective data assessments require a
user’s dedicated attention, which may result in increased burden on users and
unsatisfactory experience, and thus be subject to higher rates of missing or
inconsistent data. Furthermore, a majority of daily monitoring studies require
participants to track or complete assessments over the course of a few weeks
to a month [12] which depending on when the study is conducted, may be
inadequate in capturing long-term changes or responses to certain stressors
and events. Taylor and colleagues [6] achieved up to % of accuracy in
predicting next day mood using general models on SNAPSHOT dataset. However,
there may still be varying events over a year or large-scale collective
traumas that may relate to an individual’s experiences of emotion and mood
[13]. For example, the COVID-19 pandemic lockdown within the United States was
initiated in March of 2020 and lasted through the first half of 2021, with
variation in mandates across states. The duration of the pandemic has had
large impacts on livelihood and social norms [7]. During this watershed time,
emerging adults, who often are at the height of forming their identities and
social networks, may have experienced greater distress and adjustment in light
of stay-at-home orders that mandated they isolate themselves in the home
environment. Following individuals over the course of a year may offer a more
comprehensive understanding of the fluctuations of affect in relation to
contextual factors in daily life [9]. Thus, the goal of this study is to
address the aforementioned limitations of previous studies (e.g., time span
and partial reliance on subjective assessment), by both expanding the time
length of data collection to about 12-months during the eventful year of 2020
and also by limiting to exclusively use objective measurements to model users’
next day affect. Using only objective measures by commercial devices to
predict an individual’s affect can enable us to monitor mental health in a
more continuous and convenient way; thus making mental health monitoring
service accessible to more people. We monitored the life of 20 college
students pre- and post COVID-19 lock-down. Our results demonstrate that it is
feasible to predict an individual’s affect in a longer timeframe by only
relying on objectively collected data while achieving up to 23% of relative
improvement in accuracy compared to the state-of-the-art.

##  II Affect Prediction Study

In this section we discuss different aspects of collected data in this study,
i.e., collection setup, some insight about features and labels, and the mood
prediction modeling scheme.

###  II-A Data Collection Setup

To develop our fully objective models, we collected data from a sample of
college students (N=20, 65% female, Mage=19.80, SDage=1.0) as part of a larger
study aimed at assessing personalized approaches to understanding mental
health and well-being among emerging adults [10]. Full-time college students
between the ages of 18-22 who were unmarried, English fluent, and used an
Android device as their primary phone were recruited from a large west-coast
university to participate in this extensive longitudinal study conducted over
a 12-month period. Data were collected throughout 2020, specifically during
the COVID-19 pandemic. This study was approved by the Institutional Review
Board prior to its inception. Participants provided written consent prior to
their participation. College students were monitored over a 12-month period
where they wore non-invasive commercial smart devices that captured
physiology, sleep, and physical activity. Students also downloaded a
smartphone application that automatically detected activities and lifelogs
(e.g., staying still, walking, in-vehicle, etc.), movement, and location.
During this monitoring period, participants completed daily surveys using a
separate smartphone application that prompted them to report their affect
based on a list of 20 discrete emotion words at the end of the day.
Participants’ daily self-reported affect was used to evaluate our predictive
models.

Figure 1: Data collection setup; wearable devices collect targeted information
through applications we developed and send the data to our cloud server by
proxying the smartphone internet connection. Smartphones also are used to
collect participant’s daily affect assessments.

We utilized a number of services offered by ZotCare [14] to collect different
types of data. ZotCare is a dynamic and flexible multi-layer (sensor-
smartphone-cloud) platform built in the University of California, Irvine (UCI)
Institute for Future Health (IFH) which provides a variety of services
necessary to run a human study trial in one place (Figure 1 ). It is designed
to support the entire life-cycle of a remote monitoring-intervention study
from planning to deployment and analysis. It offers i) services to interface
with wearable and portable sensors (both native and third party), ii) a multi-
platform mobile app for two-way communication with users, collecting EMAs and
self-reports, sending recommendations and reminders, and implementing remote
interventions, iii) back-end services for storage, computation, control, and
visualization, and iv) a web-based dashboard for workers, researchers, and
end-users interact with the system. In this study we utilized ZotCare to
collect data from participants. We installed custom services on the
participants’ smartwatches to collect, store, and transfer data to the ZotCare
server. Participants’ Oura ring data was also accessible through ZotCare using
a server-to-server connection to Oura’s server. Subjective mood assessment was
collected using ZotCare’s mobile app customized for this study.

Ring | Watch | Phone
---|---|---
Sleep (mins) | Activity | Metabolic | Calorie | Heart | Distance | Environment | Activity
Awake |  REM |  Stay Active |  Meet Daily Activity Target |  Avg MET(*) |  MET Inactive |  Calorie Active |  Heart Rate |  Distance |  Pressure |  Main Activity
Light |  Deep |  Move Every Hour |  Training Frequency |  Minutes Low Activity |  MET Low |  Calorie Total |  Heart Rate std |  Runsteps |  Pressure min |  Key Activity
Total |  |  Training Volume |  Recovery Time |  Minutes Med Activity |  MET Medium |  Target Calories |  Heart Rate Variability |  Remains |  Pressure max |  Location Change
|  |  Daily Movement |  Inactivity Alerts |  Minutes High Activity |  MET High |  Target Miles |  Heart Rate Variability std |  Walk Steps |  |
TABLE I: List of collected features based on the modality (device) and
objectives. MET metabolic-equivalent minutes. For further information about
ring and watch features please refer to their online documentation [15, 16,
17]

###  II-B Features and Labels

Considering the fully objective and portable affect assessment in this study,
we only used commercial monitoring devices (i.e., Oura ring, Samsung Watch,
and Android Phones) and obtained three data modalities using the mentioned
devices. We present below an overview of collectable features from these
modalities (Table 1 shows the full list of all features).

Figure 2: Distribution of selected features. (a) REM and Light sleep
durations, (b) heart rate and heartrate variability collected by ring. (c)
atmospheric pressure (altitude) and step counts collected by smartwatch. (d)
location change and commuting events detected by Personicle app on the phone.

  * •

Smart Ring: Oura Ring [18] is a smart ring that is capable of extracting
meaningful information about sleep (e.g., length of awake, deep and REM sleep
stages), physiology (e.g., heart rate, heart rate variability), and activity
of users (e.g., daily movement and rest time, etc.). From the viewpoint of
missing data, the convenience of wearing the ring and Oura’s built-in data
management makes this modality more continuous and reliable.

  * •

Smart Watch: The Samsung smartwatch is another wearable device used for
capturing physiological data from participants. Accelerometer (ACC) and
photoplethysmography (PPG) are the main measures captured in the smartwatch
that are indicators of a user’s physical activities (e.g., steps, movements)
and the autonomic nervous system’s activities (e.g., heart rate, heart rate
variability). In contrast to smart rings, smart watches are capable of
recording the signals with a higher frequency and resolution. However, this
results in higher power consumption, the need for more frequent charging, and
therefore, greater likelihood of missing data.

  * •

Smart Phone: As mentioned, we also extracted information regarding type of
activities and location of the participant through Personicle android
monitoring application [17] installed on participants’ phone. Using the
Personicle app, we collected data on major physical (e.g., in vehicle, still,
on bicycle) and behavioral activities (e.g., working, commuting, relaxing)
throughout the day. This modality relies on smartphone sensors and
applications, and mostly on movement and location to detect these activities;
it is possible that due to the lockdown situation and movement limitations of
individuals, most of the in-house activities were not detected. This
information is available through the Android API which was collected by the
phones’ physical sensors and location tracking.

  * •

Affect (Daily Assessment): For the daily assessments of affect, participants
rated on a scale of 0-100 (0 = “Very Slightly; 100 = “Extremely”) how they
felt on a series of 20 different discrete emotion items (e.g., “inspired”,
“enthusiastic”, “nervous”, “upset”) over the course of the day. The selected
items were adapted from the Positive and Negative Affect Schedule [19] , a
frequently used scale to assess emotions. Each emotion was examined separately
but also as a composite, such that Positive Affect (PA) was calculated as an
average of the included 10 PA items (M=45.27, SD=20.22, $\alpha$ = .85) and
Negative Affect (NA) was calculated as an average of the included 10 negative
affect items (M=21.79, SD=12.28, $\alpha$ = .91) .

Finally, using these three modalities, we collected 52 features as shown in
Table 1. Some of the features were captured daily whereas some features were
captured intensively (e.g. 5 minutes sliding average for ring) over the course
of the day. Those captured multiple times a day were weighted-averaged, with
respect to the duration.

###  II-C Distributions

Features representing different biological or behavioral aspects of
participants had very different distributions. Figure 2 (a) and (b) represent
(min/max normalized) distributions of sleep and heart related features
collected by the Oura ring. Sleep features had a more normal distribution as
compared to heart-rate features and in gener‘al, values collected by the ring
had distributions closer to normal. Figure 2 (c) depicts atmospheric pressure
and number of steps detected by the smartwatch. As we can see, the
distribution of the number of steps after min-max normalization is very
concentrated (y axis is logarithmic), which could be an effect of the COVID
lockdown circumstance. Finally, Figure 2 (d) shows distributions of selected
features from the phone through the Personicle application. Reliance of this
modality to GPS and motion sensors made its data largely affected by the
pandemic lockdown situation.

Figure 3 demonstrates the averaged affect distributions for participants with
more than 200 days of valid affect values. As we can see, PA and NA follow a
roughly normal distribution with some possible skews and bumps and diversity
in peak values and locations. In general, NA has narrower distribution with
higher peaks, this makes NA values labels less discriminative compared to PA.
For example, in NA, two days with similar feature values could have small
differences but are labeled differently because they are located on different
sides of the median (refer to section II-E for classification scheme).

Figure 3: Smoothed distribution of averaged positive and negative affect of
selected seven participants. NAs tend to have a narrower distribution compared
to PA.

###  II-D Distributions

Managing a one-year human subject study in everyday settings as well as during
a pandemic raised several challenges. In particular, the increased volume of
missing data poses additional challenges for data modeling. Figure 4 shows the
number of days of data available for each participant and modality. We found
that activities detected by phone data (Personicle) was the modality with the
highest number of missing values. We speculate that with participants spending
most of their time at home (self-quarantining), movements and location
detection suffered (e.g., leaving the phone at one’s desk), resulting in poor
performance in activity detection. Rings and watches generally had the most
available data, confirming the necessity of objective measurements through
wearable devices.

For every participant’s missing feature value, we averaged values of that
feature within two nearest days before and after that date in a 5-day window.
No data imputation was applied on the targeted affect values to avoid
inaccuracy in the prediction labels.

Figure 4: Participants’ number of days with valid data collected from
different devices and EMA sorted by affect values.

###  II-E Predictive Modeling Scheme

For prediction, we selected participants with more than 200 days of affect
data available. This included 7 of the 20 participants. The classification
task is defined as predicting if each affect value is above or below the
median of the whole distribution. This is done after removing the middle 20%
of the values in the whole distribution. We used random forests (RF), support
vector machine (SVM), multilayer perceptron (MLP) and K-nearest neighbor (KNN)
models for training on all features. Hyper parameters of each model were tuned
with respect to the accuracy on the 10% held-out portion of the training
(validation set). In general, RF outperformed the other models due to its
capability in learning large feature space. Details of performances are
available in results and discussion.

During preprocessing detected activities by Personicle were fed to the model
as Boolean variables each representing the weighted average of that feature
during the day. Features were z-normalized with respect to the mean and
variance of the training set. Random 5 fold cross validation was performed to
avoid test set contamination and the accuracies were averaged out.

##  III Results and Discussion

We compared our results to two of the state-of-the-art studies conducted on
the SNAPSHOT dataset,Taylor et al. and Spathis et. al [20]. In order to be
able to compare our models with other related works in which an aggregate
single binary value is used to represent mood (i.e., sad/happy), we calculated
the deviation of the PA and NA values from their median of the entire
distribution for each participant, and selected the one with a larger
deviation as the dominant mood and assigned it as the final (sad/happy) mood
label. Our general model for predicting mood and nervousness outperformed the
state-of-the-art in predicting mood and stress. The Random Forest model was
selected for comparison here. ( detailed performance values of the other
models are shown in Figure 10). Figure 5-(a) compares prediction accuracy of
mood and nervousness affect to Taylor et al [6] showing about 16% and 4%
improvement in accuracy for mood and stress (i.e., nervousness) prediction,
respectively. Figure 5-(b) shows the receiver operating characteristic (ROC)
of the mood prediction model yielding 0.82 area under curve (AUC) that has
8.1% improvement compared to the best values reported by Spathis et. al for
mood prediction.

Figure 5: (a) Accuracy of Random Forest classifier on Nervous affect and
compiled mood scores from all affects compared to the models with best
performance on SNAPSHOT by Taylor et al. (b) ROC and AUC for mood scores
prediction

Figure 6 (a) and (b) also show the highest, lowest, and median of PA and NA
with respect to their AUCs. NA models tends to have a higher ROC but in
general lower accuracy (Please see Figure 10 for detailed results). The main
reason is the comparatively narrower distribution of the NA (Figure 3) that
makes the close values get different labels. This indicates that despite
better probabilities being generated for NA, the chance of producing the wrong
label is higher since labeling thresholds are narrower.

Figure 6: (a) Accuracy of RF classifier on Nervous affect and compiled mood
scores from all affects compared to the models with best performance on
SNAPSHOT by Taylor et al. (b) ROC and AUC for mood scores prediction  Figure
7:  Performance of RF trained on modalities separately and all together for PA
and NA

One of the objectives of this study was to explore capabilities of leveraging
multimodal data for modeling affect. In general, multimodal machine learning
could be used to either increase robustness (i.e., providing redundant
information) or improve prediction performance (i.e., obtaining additional
information from different aspects of the event). In this study, we focused on
the later purpose of multimodal machine learning and therefore, assessment
devices were selected in order to cover different aspects of an individual’s
life from physiological (ring and watch) to behavioral (phone). To test the
power of multimodal assessment, we trained models modalities separately and
all together. Figure 7 compares accuracies of models trained with data from
different devices separately to ones trained with data from all devices for NA
and PA. As shown, leveraging all of the modalities improve accuracy by up to
21.8%.

Another unique aspect of this study was its timing–the study began in early
2020 and covered the period before and during the stay-at-home order in the
state of California. The stay-at-home order in most of the states happened
around March and affected almost everyone’s lives, including college students,
in many different ways. In order to investigate the effects of this event, we
used our mood prediction model to predict participants’ mood during the last
week of every month. Then we computed t-value as a metric of difference of
predicted moods between months before and after lockdown order. Figure 8 shows
absolute t-values of predicted mood for participants across these 1-month
intervals. As we observe, March has the highest t-value (difference from other
months) that coincides with the very early stages of lockdown.

To gain a deeper understanding about different modalities, we conducted
correlational analyses to explore associations of physiological and personal
chronicle features with PA and NA. Figure 9 (a) shows correlation of ring
features to PA and NA. Duration of light and deep sleep along with sleep score
and inactivity level were the main indicators of PA, while NA was mainly
correlated with activity-related features. The findings of this study resonate
with prior work suggesting that sleep may predict PA. PA may be one potential
pathway through which sleep has implications for mental and physical health
outcomes. Specifically, individuals with sleep disturbances report more
blunted positive affect associated with rewarding activities such as decreased
energy and happiness compared to individuals with extended sleep [13].

We also conducted more detailed experiment on different models and trained
different machine learning models on all the specific reported affect features
separately. Figure 10 demonstrates the results for all the experiments. Random
forests (RF) outperformed all the other models in terms of accuracy.

Figure 8:  Performance of RF trained on modalities separately and all together
for PA and NA  Figure 9:  Correlation of features collected by objective
assessment with two target labels Figure 10:  Accuracy of RF, SVM, KNN, MLP
models and random prediction(majority class) calculated in 5 fold cross
validation.

##  IV Conclusion and Future Work

This study was designed to investigate the viability of predicting an
individual’s mood and affects only by leveraging objective measurements
without a need to intervene users for collecting feedback. We conducted this
study over the course of one year which is a wider time span compared to
similar works. The chance of observing young adults during the lockdown
situation of COVID-19 pandemic gives a unique dimension to this study.
Different watch, phone and web applications were developed to objectively
assess participants using commercially available devices. Results showed that
we can predict next-day mood with accuracy of 81% using generic machine
learning algorithms. These models outperformed their alternatives in similar
studies even without an individual’s historical information for bringing
personalization or temporal context. Results show that sleep and heart rate
variability features collected by the smart ring are among the best predictors
of mood and positive affect which suggests further investigation.

During the study COVID-19 lockdown circumstances imposed some challenges on
data collection due to the limited access to the participants and their normal
movement and commute patterns. These limitations made missing data a
challenging part for building predictive models. We observed that generic
machine learning models on our dataset resulted in comparable performance to
counterpart studies even using personalization and temporal context.

In our future studies we plan to investigate novel methods that incorporate
patient-specific information into modeling and explore high-resolution signals
(raw photoplethysmography, accelerometer) collected by wearables.
Additionally, monitoring and predicting affect, and eventually developing and
refining interventions to regulate and manage affect using technology is an
important future avenue. Specifically, the findings of this study are
important in showing that the advancements in objective monitoring and
predicting affect using features such as sleep may provide important
opportunities in a less burdensome approach of monitoring and predicting
important aspects of individuals’ health. Future research may benefit from
designing and refining interventions among emerging adults using smartphones.
For instance, monitoring and prediction of affect could be used to identify
the moments to use interventions in hopes of personalization of interventions
targeting affect and other aspects of mental health.

## References

  * [1] Mental illness.  https://www.nimh.nih.gov/health/statistics/mental-illness.
  * [2] Lisa Feldman Barrett and Eliza Bliss-Moreau.  Affect as a psychological primitive.  Advances in experimental social psychology, 41:167–218, 2009.
  * [3] James J Gross and Hooria Jazaieri.  Emotion, emotion regulation, and psychopathology: An affective science perspective.  Clinical psychological science, 2(4):387–401, 2014.
  * [4] James W Pennebaker.  Writing about emotional experiences as a therapeutic process.  Psychological science, 8(3):162–166, 1997.
  * [5] Mor Nahum, Thomas M Van Vleet, Vikaas S Sohal, Julie J Mirzabekov, Vikram R Rao, Deanna L Wallace, Morgan B Lee, Heather Dawes, Alit Stark-Inbar, Joshua Thomas Jordan, et al.  Immediate mood scaler: tracking symptoms of depression and anxiety using a novel mobile mood scale.  JMIR mHealth and uHealth, 5(4):e6544, 2017.
  * [6] Sara Taylor, Natasha Jaques, Ehimwenma Nosakhare, Akane Sano, and Rosalind Picard.  Personalized multitask learning for predicting tomorrow’s mood, stress, and health.  IEEE Transactions on Affective Computing, 11(2):200–213, 2017.
  * [7] U.S. Census Bureau.  Initial impact of covid-19 on u.s. economy more widespread than on mortality.  https://www.census.gov/library/stories/2021/03/initial-impact-covid-19-on-united-states-economy-more-widespread-than-on-mortality.html, Mar 2021.
  * [8] Amir M Rahmani, Jocelyn Lai, Salar Jafarlou, Asal Yunusova, Alex Rivera, Sina Labbaf, Sirui Hu, Arman Anzanpour, Nikil Dutt, Ramesh Jain, et al.  Personal mental health navigator: Harnessing the power of data, personal models, and health cybernetics to promote psychological well-being.  arXiv preprint arXiv:2012.09131, 2020.
  * [9] Jocelyn Lai, Amir Rahmani, Asal Yunusova, Alexander P Rivera, Sina Labbaf, Sirui Hu, Nikil Dutt, Ramesh Jain, Jessica L Borelli, et al.  Using multimodal assessments to capture personalized contexts of college student well-being in 2020: Case study.  JMIR formative research, 5(5):e26186, 2021.
  * [10] Asal Yunusova, Jocelyn Lai, Alexander P Rivera, Sirui Hu, Sina Labbaf, Amir M Rahmani, Nikil Dutt, Ramesh C Jain, and Jessica L Borelli.  Assessing the mental health of emerging adults through a mental health app: Protocol for a prospective pilot study.  JMIR Research Protocols, 10(3):e25775, 2021.
  * [11] Akane Sano.  Measuring college students sleep, stress and mental health with wearable sensors and mobile phones.  PhD thesis, PhD thesis, MIT, 2015.
  * [12] Timothy J Trull and Ulrich Ebner-Priemer.  Ambulatory assessment.  Annual review of clinical psychology, 9:151–176, 2013.
  * [13] Zahra Mousavi, Jocelyn Lai, Asal Yunusova, Alexander Rivera, Sirui Hu, Sina Labbaf, Salar Jafarlou, Nikil Dutt, Ramesh Jain, Amir Rahmani, et al.  193 sleep variability and affect dynamics among college students during covid-19 pandemic.  Sleep, 44(Supplement_2):A78–A78, 2021.
  * [14] Arman Anzanpour.  Health scitech group: Zotcare.  http://healthscitech.nursing.uci.edu/projects/zotcare/.
  * [15] Api documentation.  https://cloud.ouraring.com/docs/.
  * [16] Galaxy watch - build.  https://developer.samsung.com/galaxy-watch-develop.
  * [17] Laleh Jalali, Da Huo, Hyungik Oh, Mengfan Tang, Siripen Pongpaichet, and Ramesh Jain.  Personicle: personal chronicle of life events.  In Workshop on Personal Data Analytics in the Internet of Things (PDA@ IOT) at the 40th International Conference on Very Large Databases (VLDB), Hangzhou, China, 2014.
  * [18] Oura Ring.  Accurate health information accessible to everyone.  Oura. URL: https://ouraring. com/[accessed 2021-02-02].
  * [19] David Watson, Lee Anna Clark, and Auke Tellegen.  Development and validation of brief measures of positive and negative affect: the panas scales.  Journal of personality and social psychology, 54(6):1063, 1988.
  * [20] Dimitris Spathis, Sandra Servia-Rodriguez, Katayoun Farrahi, Cecilia Mascolo, and Jason Rentfrow.  Passive mobile sensing and psychological traits for large scale mood prediction.  In Proceedings of the 13th EAI International Conference on Pervasive Computing Technologies for Healthcare, pages 272–281, 2019.