# Probing Ultra-Fast Dephasing via Entangled Photon Pairs

Xinghua Liu  Institute for Quantum Science and Engineering, Department of
Physics and Astronomy, Texas A&M University, College Station, Texas
77843-4242, USA<EMAIL_ADDRESS>Tian Li  Institute for Quantum Science and
Engineering, Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843-4242, USA  Department Department of Biological
and Agricultural Engineering, Texas A&M University, College Station, TX 77843,
USA  (Current address) Departments of Chemistry and Physics, The University of
Tennessee, Chattanooga, TN 37403, USA  Jiaxuan Wang  Institute for Quantum
Science and Engineering, Department of Physics and Astronomy, Texas A&M
University, College Station, Texas 77843-4242, USA  Mrunal R. Kamble
Institute for Quantum Science and Engineering, Department of Physics and
Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
Aleksei M. Zheltikov  Institute for Quantum Science and Engineering,
Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843-4242, USA  Girish S. Agarwal  Institute for Quantum Science and
Engineering, Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843-4242, USA  Department Department of Biological
and Agricultural Engineering, Texas A&M University, College Station, TX 77843,
USA

###### Abstract

We demonstrate how the Hong-Ou-Mandel (HOM) interference with polarization-
entangled photons can be used to probe ultrafast dephasing. We can infer the
optical properties like the real and imaginary parts of the complex
susceptibility of the medium from changes in the position and the shape of the
HOM dip. From the shift of the HOM dip, we are able to measure 22 fs dephasing
time using a continuous-wave (CW) laser even with optical loss > 97 %, while
the HOM dip visibility is maintained at 92.3 % (which can be as high as 96.7
%). The experimental observations, which are explained in terms of a rigorous
theoretical model, demonstrate the utility of HOM interference in probing
ultrafast dephasing.

##  1 Introduction

The Hong-Ou-Mandel interferenceHong et al. (1987); Bouchard et al. (2020);
Hochrainer et al. (2022) was a turning point in the field of optical
interference. It opened up several new pathways- it established the fact that
interference between two independent but identical photons is possible,
leading to the remarkable possibility of detecting time delays at the
femtosecond scale and even lowerLyons et al. (2018); Volkovich and Shwartz
(2020). The original HOM has been generalized in many different ways,
including the observation of HOM in frequency domainJoshi et al. (2020);
Hiekkamäki and Fickler (2021); Devaux et al. (2020); Cerf and Jabbour (2020).
Recently, many different applications of HOM interference have appeared Crespi
et al. (2012); Yang et al. (2019). It has now become a standard tool to
establish the indistinguishable nature of photons Santori et al. (2002);
Schofield et al. (2022). The nonidentical photons can also result in HOM dip
but with reduced visibility depending on the nature of photons Ollivier et al.
(2021); Wiegner et al. (2011); Deng et al. (2019); Koong et al. (2022); Su et
al. (2022). The HOM with quantum dot sourcesPolyakov et al. (2011); Ollivier
et al. (2021) and plasmonic systemsDi Martino et al. (2014); Cai et al.
(2014); Gupta and Agarwal (2014) have been studied. Applications of HOM in the
context of quantum microscopy and quantum imaging have been consideredNdagano
et al. (2022); Camphausen et al. (2021); Defienne et al. (2022). A significant
application of HOM is to produce NOON state, which is of particular importance
in quantum metrology Dowling (2008); Wang and Agarwal (2021) and quantum
microscopy Ono et al. (2013). Recently several papers have discussed the
possibility of determining the absorptive and dispersive properties of matter
by using HOM Kalashnikov et al. (2017); Eshun et al. (2021); Dorfman et al.
(2021). The paper by Eshun et al. Eshun et al. (2021) successfully used
organic samples through their HOM dip visibilities were only around 40 % due
to wide spectral ranges of pulsed lasers.

In this paper, we show how HOM interference can be a useful probe of optical
absorption and dispersion at the ultra-fast scale. We use a CW pump to produce
polarization-entangled photons and change the polarization of the signal
photon to study the HOM with identical photons. One of the two photons is sent
to a medium, which is effectively characterized by a frequency-dependent
optical susceptibility. The results show that the frequency-dependence of the
complex susceptibility breaks the indistinguishability of the two single
photons leading to changes in the interference dip. These changes are in the
HOM dip location, visibility, and symmetry. Using a CW laser, the tunable
dephasing time can be measured down to 22 fs or more. Our HOM dip visibility
can be as high as 96.7 % and maintains 92.3 % even for an open system with
optical loss > 97 %. Moreover, we found that both the absorption and
dispersion of a sample can be inferred from the tilt and shift of the HOM dip,
which provides another useful approach for characterizing a sample’s optical
properties at the ultra-fast scale. The experimental observations are fully
explained in terms of a rigorous theoretical model.

##  2 Experiment

Figure 1: (a) Experimental setup comprised of an SPDC and a HOM
interferometer. BBO: $\beta$-Barium bOrate crystal cut for type-II phase
matching for 404 nm, DM: long-pass dichroic mirror, IF: interference filter,
PBS: polarizing beam splitter, BS: 50:50 non-polarizing beam splitter, APD:
avalanche photodiode for single-photon counting. (b) A testing sample is
mimicked by a PBS in conjunction with a Multi-Order (MO) or a Zero-Order (ZO)
half-wave plate (HWP) mounted on a motorized rotational stage. (c) Image of
the SPDC rings captured by an EMCCD camera.

Our experimental setup is shown in Fig 1(a), where an entangled photon pair
generation by spontaneous parametric down-conversion (SPDC) and a HOM
interferometer comprises the bulk of it. A fiber-coupled 404 nm CW laser
(Toptica iBeam Smart) is used to pump a 1 mm-thick $\beta$-Barium bOrate (BBO)
crystal cut for type-II phase-matching. The down-converted photon pairs
(signal and idler) are frequency degenerate
($\omega_{s}=\omega_{i}=\frac{1}{2}\omega_{p}$) and are orthogonally
polarized. The BBO crystal is mounted on a 3-axis rotational stage to satisfy
the co-linear phase-matching condition, where the generated signal and idler
rings are tangent. The phase-matching condition is verified by an Electron-
Multiplying CCD (EMCCD) camera (Andor iXon 897), as shown in Fig. 1(c). A 505
nm long-pass dichroic mirror (DM) and an 810-10 nm interference filter (IF)
are used to block residual pump photons and to select the entangled photon
pairs.

The polarization-entangled photon pairs are then separated by a polarizing
beam splitter (PBS) so that the signal and idler photons can be in different
optical paths. After the PBS, the idler photons transverse an optical delay
stage composed of a retro-reflecting hollow roof prism (Thorlab HR1015-AG,
ultrafast-enhanced silver coating) mounted on a motorized translational stage
(Thorlab PT1-Z8). In this configuration, the relative optical delay between
the photon pairs can be adjusted by increasing or decreasing the path length
of the idler photons while keeping their propagation direction unchanged. The
motorized stage has a resolution of 29 nm, corresponding to a time delay
resolution of 0.2 fs. In the optical path of signal photons, a zero-order (ZO)
half-wave plate (HWP) is used to match their polarization to that of the idler
photons. The signal photons then pass a ‘mock sample’ composed of an HWP
followed by a PBS (Fig. 1.(b)), and are combined with the idler photons on a
50:50 non-polarizing beam splitter (BS) to form a HOM interferometer. The
outputs of the BS are coupled through two single-mode fibers to two single-
photon avalanche photodiodes (Excelitas, SPCM-AQRH-13-FC). Single photon
counting and coincidence countings rates are acquired by a coincidence counter
(IDQ, ID900) within a 4 ns sampling window.

To achieve the photon pair indistinguishability required by the HOM
interferometer, the signal and idler photons must be overlapped perfectly in
space. Due to the SPDC conversion ratio for BBO crystal being in the order of
$10^{-11}$, it’s unsuitable to use the entangled photon pairs themselves to
perform alignment for the interferometer. Therefore, an additional 780 nm CW
external cavity diode laser ( not shown in Fig 1(a))) is used for the
alignment purpose. We again use the EMCCD camera to verify that this ‘guiding’
laser is aligned with the down-converted photons both in the near and far
fields. With the aid of the guiding laser, the perfectness of overlap between
signal and idler photons can be judged by interference fringes having good
visibilities. After this interference alignment, the ECDL is then replaced by
the 404 nm diode laser. Since the coherence length of the ECDL is much longer
than the down-converted photons, additional pre-scanning is required to find
the 0-delay position, where the optical paths of the signal and idler photons
are identical.

In this experiment, two types of HWPs at 808 nm are used in conjunction with
PBS to mimic the transmission property of a sample. One type of HWP is a
Multi-Order (MO) HWP, and the other is a Zero-Order (ZO) HWP. The MO HWP is
made of crystalline quartz, a type of birefringent material that has different
refraction indexes along two orthogonal axes (ordinary axis refraction index
$n_{o}(\omega)$ and extraordinary axis refraction index $n_{e}(\omega)$).

$\Delta\Phi=\frac{2\pi L(n_{o}(\omega)-n_{e}(\omega))}{\lambda},$ (1)

where $\Delta\Phi$ is the phase delay between the ordinary and extraordinary
axes, and L is the length of the wave plate. For an MO HWP at 808 nm, L is
selected to maintain $\Delta\Phi=\pi$ only for photons with a wavelength of
808 nm. Therefore, the MO HWP would introduce different phase delays for
different wavelengths, as shown in Fig. 2(b). On the other hand, the ZO HWP,
which is composed of two MO HWPs, can offer less wavelength dependency for the
phase delay, and is this considered a "perfect" HWP in this experiment.

Figure 2: (a) Experimental (blue cross, scale on the left) and simulated (red
line, normalized, with the scale on the right) coincidence counts without a
sample. (b) Experimental and linear-fitted phase delay $\phi(\omega)$ for the
ZO HWP (triangle, blue with scale on the left) and the MO HWP (blue circle,
with the scale on the left). Transmission data (red circle, with the scale on
the right) and super-Gaussian fit (red dash line, with the scale on the right)
for the band-pass (BP) filter. Data was acquired from the Thorlab
specification Thorlab (a, b).

##  3 Results and Analysis

The typical bandwidth of entangled photon pairs produced by a type-II BBO
crystal is 20 nm Kim and Grice (2005). In our experiment, it’s determined by
the 810-10 nm premium hard-coated band-pass filter (Thorlabs, FBH810-10),
whose transmission curve is shown in Fig. 2(b) (red circles, with the scale on
the right). The band-pass filter also blocks residual pump photons.
Furthermore, the filter applies an additional spectral selection for the
entangled photons to reduce photon pair distinguishability and thus increase
the HOM dip depth Grice et al. (1998). We can fit the transmission function of
the band-pass filter with a super-Gaussian function,
$T_{f}(\omega)=ae^{-(\frac{\omega-b}{c})^{6}}+d$, as shown in Fig. 2(b)(red
dash line, with the scale on the right). When the pump power is $50$ mW, and
the coincidence window is $4$ ns, we are able to acquire $1.84\times 10^{4}$
photon pairs per 10 s. The visibility V of a HOM dip is defined as:

$V=\frac{R_{c}(\infty)-R_{c}(0)}{R_{c}(\infty)+R_{c}(0)},$ (2)

where $R_{c}(\infty)$ is the coincidence counts rate far from the coherence
region, and $R_{c}(0)$ is the coincidence counts rate for the zero delay
between the signal and idler photons (detailed derivation of $R_{c}$ is in the
next section). Our HOM dip’s visibility is measured to be ($96.7\pm 0.1\%$) as
shown in Fig. 2(a).

To achieve a continuously adjustable absorption and dispersion, we put the
MO/ZO HWP on a rotational stage, so that the angle $\theta$ between the fast
axis of the HWP and the polarization of the signal photon can be turned
according to a complex transmission function $T(\omega)$:

$T(\omega)=\text{cos}\frac{\phi(\omega)}{2}-i\text{sin}\frac{\phi(\omega)}{2}\text{cos}2\theta,$
(3)

where $\phi(\omega)$ is the phase delay between the fast and slow axes, as
shown in Fig. 1(b) (detailed derivation of this equation can be found in the
next section). For a ‘perfect’ HWP, the phase delay should be independent of
wavelength, i.e., $\phi(\omega)=\pi$. In this ideal case, the transmission
function reduces to $T(\omega)=-i\text{cos}2\theta$ and
$|T(\omega)|^{2}=\text{cos}^{2}(2\theta)$. Therefore, loss for ’perfect’ HWP
in our system will be

$\displaystyle Loss=1-\text{cos}^{2}(2\theta)$ (4)

The corresponding loss function with respect to $\theta$ for this ‘perfect’
HWP is shown in Fig. 5(a) (red curve).

Figure 3: Experimental and simulation results for optical loss less than
$97\%$
($\theta=0^{\circ},8^{\circ},16^{\circ},24^{\circ},32^{\circ},40^{\circ}$).
(a) Experimental results for the ZO HWP. (b) Experimental results for the MO
HWP. (c) Simulation results for the ZO HWP. (d) Simulation results for the MO
HWP. The simulation results are based on the Theoretical Model, Eqn. 13.

Figures 3 and 4 are the experimental and simulation results for the normalized
HOM dips, respectively. The dips are normalized to the average background
counts in the uncorrelated region, i.e., $R_{c}(\infty)$ in Eq. 2. The data
are collected within two regimes: 1) $0^{\circ}\leq\theta\leq 40^{\circ}$ and
2) $40^{\circ}\leq\theta\leq 45^{\circ}$, which are separated by whether the
optical loss is greater than $97\%$. Figure 3 is for the first regime where
the angle changes every $8^{\circ}$ and acquisition time is 10 s per data
point. The left-hand-side panels are the experimental results (Fig. 3(a)) and
simulation results (Fig. 3(c)) for the ZO HWP, while the right-hand-side
panels are the experimental results (Fig. 3(b)) and simulation results (Fig.
3(d)) for the MO HWP. We can see that in the case of ZO HWP, all HOM dips
overlap pretty well, which implies wavelength-independent loss (no dispersion
dependence) has no effect on the indistinguishability of the photon pairs.
Whereas in the case of MO HWP, the minima of these HOM dips shift to the
$-\tau$ direction as $\theta$ increases.

The experimental and simulation results for the second regime where
$\theta\geq 40^{\circ}$ are shown in Fig. 4. The HOM dips now were taken with
only $1^{\circ}$ increment, i.e.,
$\theta=40^{\circ},41^{\circ},42^{\circ},43^{\circ}$, and acquisition time is
50 s per data point. Same as Fig. 3, the left- and right-hand side panels are
the experimental (Fig. 4(a) and (b))and simulation (Fig. 4(c) and (d)) results
for the ZO and MO HWPs respectively. We can still see that in the case of the
ZO HWP, all HOM dips overlap except that their depths reduce as the angle
(i.e., absorption) increases, which can be attributed to the fact that the
artificial coincidence now takes a larger portion of the total coincidence.
The artificial coincidence $R_{AC}$ is defined as the average coincidence
counts within a time window when both detectors register fully random
(uncorrelated) events. In our case,
$R_{AC}\sim(I_{s}+I_{i})^{2}\sim(I_{s}+\bar{t}I_{i0})^{2}$, where
$I_{s},I_{i}$ are the signal and idler photon numbers, $I_{i0}$ is the photon
number in the idler beam path if there is no sample present, $\bar{t}$ is the
average transmission rate of the sample. Since the coincidence counts from
correlated photons are proportional to the product of signal and idler photon
numbers ($R_{c}\sim I_{s}\times I_{i}\sim\bar{t}I_{s}I_{i0})$, thus the ratio
of an artificial coincidence to real coincidence
($R_{AC}/R_{c}\sim\frac{1+2\bar{t}}{\bar{t}}$) approaches $\infty$ as
$\bar{t}\rightarrow 0$. In the case of the MO HWP, in addition to the
locations of these HOM dips shift to the $-\tau$ direction as in Fig. 3(b)(d),
shapes of these HOM dips are not symmetric anymore, which implies a change of
spectral mode of the signal photons.

Figure 4: Experimental and simulation results for optical loss greater than
$97\%$ ($\theta=40^{\circ},41^{\circ},42^{\circ},43^{\circ}$). (a)
Experimental results for the ZO HWP. (b) Experimental results for the MO half-
wave. (c) Simulation results for the ZO HWP. (d) Simulation results for the MO
HWP. The simulation results are based on the Theoretical Model, Eqn. 13.

To further investigate the difference between the wavelength-independent and
the wavelength-dependent transmission, we plot the average coincidence counts
$R_{c}(\infty)$ change as a function of optical loss introduced by a ’perfect’
HWP in Fig. 5(a) blue curve. We also plot the optical loss as a function of
$\theta$ in Fig. 5(a) red curve.

Figure 5: Experimental and simulation results for (a) Change of average
coincidence counts versus optical loss (blue, with the scale on the left
axis). Optical loss as a function of $\theta$ (red, with the scale on the
right axis). (b) Experimental and simulation results for visibility versus
optical loss (blue, with the scale on the left axis). Shifted position of HOM
dip versus optical loss (red, with the scale on the right axis), the dephasing
time $T_{2}$ can be extracted from this information (Fig. 7 (b)). Optical loss
is calculated based on a ‘perfect’ HWP as discussed in Eqn. 4.

The measured average coincidence counts are normalized by the coincidence
counts in the incoherent regime $R_{c}(\infty)$ where there is no loss. We can
see that the average coincidence counts maintain a linear behavior with
respect to the optical loss, which implies that both ZO and MO HWPs act like a
photon number attenuator as their spectral modes now is an irrelevant point of
concern. However, their spectral modes are modified with large dispersion, and
the photon pair indistinguishability is subsequently reduced.

Moreover, we plot the change in visibility and shift of HOM dip versus optical
loss in Fig. 5(b). It contains the simulated and experimental results for
visibility changes of MO HWP and ZO HWP, respectively (the blue curve, with
the scale on the left axis). Again, for small absorption, visibilities remain
unchanged for both ZO and MO HWPs, whereas for large absorption, visibilities
for wavelength-independent transmission (ZO HWP) degrade significantly. This
observation can still be attributed to the effect of aforementioned artificial
coincidence. Both experimental and theoretical result shows that the
visibility could reach 95.7 % for zero loss and remain 92.7 % at the optical
loss of 97 % for wavelength-independent transmission (ZO HWP). This
observation can still be attributed to the effect of the aforementioned
artificial coincidence. The visibility starts at 93.5 % at zero loss and drops
to 86.1 % with 97 % optical loss for wavelength-dependent transmission (MO
HWP). We can also see that visibilities deteriorate more aggressively for
wavelength-dependent transmission (using MO HWP) due to the combined impact of
both the artificial coincidence and distortion of the HOM dip.

This effect can also be seen in the change of HOM dip position versus optical
loss (Fig. 5(b) red curve, with the scale on the right axis). For wavelength-
independent transmission (using ZO HWP), the center positions of the HOM dips
remain unchanged when optical loss increases. In contrast, for wavelength-
dependent transmission (using MO HWP), the HOM dips hardly shift with small
dispersion ($\delta\tau<5$ fs when the loss is less than $40~{}\%$), yet with
large dispersion, they shift $\sim~{}40$ fs towards $-\tau$ direction. The
shift and distortion of HOM dip caused by wavelength-dependent transmission
(using MO HWP) could be modeled as the effect of a fictitious oscillator with
ultra-fast dephasing (Eqn. 25).

$\displaystyle T(\omega)$ $\displaystyle=$
$\displaystyle\text{exp}({\frac{iBL/c}{\Omega-\omega-i/T_{2}}})$ (5)

where $B$ is the Bouguer coefficient, $L$ is the length of the sample, c is
the speed of light, $\Omega$ is the resonant frequency, and $T_{2}$ is the
dephasing time. The detailed derivation is shown in the next section. From
this model, we can extract the ultra-fast dephasing time for the fictitious
oscillator as shown in Fig. 7(b).

##  4 Theoretical Model

The entangled twin-photon state generated by SPDC can be written as

$\displaystyle\ket{\Phi}=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}d\omega_{s}d\omega_{i}f(\omega_{s},\omega_{i})a^{\dagger}_{s}(\omega_{s})a^{\dagger}_{i}(\omega_{i})\ket{0_{s},0_{i}},$
(6)

where $a^{\dagger}_{s}$ ($a^{\dagger}_{i}$) is the creation operator for the
signal (idler) photon, and $f(\omega_{s},\omega_{i})$ is the joint spectral
amplitude (JSA) of the entangled photons, which depends both on the spectrum
of the pump and the proprieties (length, cutting angle and phase-matching
condition) of the BBO crystal. A polarizing beam splitter separates the twin
photons. In one of the paths, the signal photon travels through the sample of
interest

$\displaystyle
a^{{}^{\prime}}_{s}(\omega)=T(\omega)a_{s}(\omega)+\eta(\omega)a_{v}(\omega),$
(7)

where $T(\omega)$ is the transmission function that carries the sample’s
optical properties to the photon field; $a_{v}(\omega)$ is the vacuum noise
operator, and $\eta(\omega)=\sqrt{1-|T(\omega)|^{2}}$.
$a^{{}^{\prime}}_{s}(\omega)$ and
$a^{{}^{\prime}\dagger}_{s}(\omega^{{}^{\prime}})$ satisfy the commutation
relations
$[a^{{}^{\prime}}_{s}(\omega),a^{{}^{\prime}\dagger}_{s}(\omega^{{}^{\prime}})]=\delta(\omega-\omega^{{}^{\prime}})$.
Note that the vacuum noise term does not contribute to the correlation because
it is normal-ordered. $T(\omega)$ can also be written in terms of sample
length $L$, speed of light $c$, and complex refractive index $n+i\kappa$ of
the sample as

$\displaystyle T(\omega)=e^{i\omega(n(\omega)+i\kappa(\omega))L/c}.$ (8)

Meanwhile, the idler photon in the other path acquires a phase delay $\tau$
introduced by a retro-reflecting mirror mounted on a translational stage

$\displaystyle a^{{}^{\prime}}_{i}(\omega)=a_{i}(\omega)e^{-i\omega\tau}.$ (9)

The photons are then recombined on a 50:50 beam splitter and produce the
outputs modes ($c(\omega)$, $d(\omega)$)

$\displaystyle\begin{pmatrix}c(\omega)\\\
d(\omega)\end{pmatrix}=\frac{1}{\sqrt{(}2)}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}\begin{pmatrix}a^{{}^{\prime}}_{s}(\omega)\\\
a^{{}^{\prime}}_{i}(\omega)\end{pmatrix},$ (10)

$c(\omega)$, $d(\omega)$ are then collected by two SPADs and analyzed by the
coincidence counter. The detector response time $\tau_{d}$ is in the order of
$1\;ns$ while the correlation time of SPDC photons is usually $0.1-1\;ps$
Blauensteiner et al. (2009). Thus, we can assume the detector response time is
much longer than the correlation time of the entangled photons, then the
averaged coincidence rate ($R_{c}(\tau)$) is given by

$\displaystyle
R_{c}(\tau)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}dtdt^{{}^{\prime}}\langle
d^{\dagger}(t)c^{\dagger}(t+t^{{}^{\prime}})c(t+t^{{}^{\prime}})d(t)\rangle$
(11)

Changing the modes to frequency domain $d(t)=\frac{1}{\sqrt{2\pi}}\int
d\omega_{1}d(\omega_{1})e^{-i\omega_{1}t}$ and
$c(t+t^{{}^{\prime}})=\frac{1}{\sqrt{2\pi}}\int
d\omega_{2}c(\omega_{2})e^{-i\omega_{2}(t+t^{{}^{\prime}})}$, we obtain

$\displaystyle R_{c}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}d\omega_{1}d\omega_{2}f^{*}(\omega_{1},\omega_{2})f(\omega_{1},\omega_{2})T^{*}(\omega_{1})T(\omega_{1})$
(12) $\displaystyle-$
$\displaystyle\frac{1}{8\pi^{2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}d\omega_{1}d\omega_{2}f^{*}(\omega_{1},\omega_{2})f(\omega_{2},\omega_{1})T^{*}(\omega_{1})T(\omega_{2})e^{-i(\omega_{1}-\omega_{2})\tau}.$

With super-Gaussian fit for band-pass filter
$T_{f}(\omega)=ae^{-(\frac{\omega-b}{c})^{6}}+d$, Eqn. 12 is modified to

$\displaystyle R_{c}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}d\omega_{1}d\omega_{2}f^{*}(\omega_{1},\omega_{2})f(\omega_{1},\omega_{2})T^{*}(\omega_{1})T(\omega_{1})|T_{f}(\omega_{1})|^{2}|T_{f}(\omega_{2})|^{2}$
(13) $\displaystyle-$
$\displaystyle\frac{1}{8\pi^{2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}d\omega_{1}d\omega_{2}f^{*}(\omega_{1},\omega_{2})f(\omega_{2},\omega_{1})T^{*}(\omega_{1})T(\omega_{2})$
$\displaystyle|T_{f}(\omega_{1})|^{2}|T_{f}(\omega_{2})|^{2}e^{-i(\omega_{1}-\omega_{2})\tau}.$

It is not possible to obtain analytical result now and therefore all
simulations are done numerically using Eq. 13.

In this study, we consider a Gaussian entangled two-photon distribution with
both photons centered at half of the pump frequency $\omega_{p}/2$

$f(\omega_{1},\omega_{2})=\frac{1}{\sqrt{2\pi\sigma_{P}\sigma_{-}}}e^{-(\omega_{1}+\omega_{2}-\omega_{p})^{2}/(16\sigma_{P}^{2})}e^{-(\omega_{1}-\omega_{2})^{2}/(4\sigma_{-}^{2})},$
(14)

where $\omega_{p}$ and $\sigma_{p}$ are the center frequency and bandwidth of
the pump laser. Note that $f(\omega_{1},\omega_{2})$ is normalized by
$\int\int|f(\omega_{1},\omega_{2})|^{2}d\omega_{1}d\omega_{2}=1$. For the
$405\;nm$ CW laser in this experiment, we have $\omega_{p}=3.069\;eV$ and
$\sigma_{p}=0.149\;meV$. $\sigma_{-}$ ($6.8\;meV$) is the bandwidth of photon
pairs, where $\sigma_{-}\sim\frac{1}{T_{e}}$. $T_{e}$ is the entanglement time
of the photon pairs and obtained from the width of the HOM dip (Fig. 2.(a)).
$\omega_{1},\omega_{2}$ are frequencies for signal and idler photon,
respectively. The center frequencies of signal photon and idler photon are
$\omega_{1}=\omega_{2}=\frac{\omega_{p}}{2}=1.535\;eV$.

Figure 6: (a)Transmission spectrum ($|T(\omega)|^{2}$) for MO HWP and PBS. (b)
Transmission spectrum ($|T(\omega)|^{2}$) for ZO HWP and PBS. (c)Phase change
spectrum ($\Delta\Upsilon$) for MO HWP and PBS. (d)Phase change spectrum
($\Delta\Upsilon$) for ZO HWP and PBS.

The test sample in this experiment consists of an HWP on a rotational stage
and a PBS (Fig. 1.(b)). The HWP transforms the orthogonal components of a
light state as Agarwal (2012)

$\left(\begin{array}[]{c}H_{out}\\\
V_{out}\end{array}\right)=J\,^{\prime}\left(\omega\right)\left(\begin{array}[]{c}H_{in}\\\
V_{in}\end{array}\right).$ (15)

The transformation matrix ($J^{{}^{\prime}}(\omega)$) is the rotation
transformation of the Jones matrix for a wave plate, where

$J\,^{\prime}\left(\omega\right)=R^{-1}\left(\theta\right)J\left(\omega\right)R\left(\theta\right).$
(16)

The Jones matrix ($J(\omega)$) for a wave plate can be written as,

$J\left(\omega\right)=\left(\begin{array}[]{cc}e^{i\frac{\phi\left(\omega\right)}{2}}&0\\\
0&e^{-i\frac{\phi\left(\omega\right)}{2}}\end{array}\right),$ (17)

where $\phi(\omega)$ is the phase delay between the wave plate’s fast and slow
polarization axis. The rotation matrix ($R(\theta)$) can be represented as,

$R\left(\theta\right)=\left(\begin{array}[]{cc}\text{cos}\theta&\text{sin}\theta\\\
-\text{sin}\theta&\text{cos}\theta\end{array}\right).$ (18)

where $\theta$ is the angle between the vertical direction and fast axis of
the HWP, as shown in Fig.1(b).
We obtain

$J\,^{\prime}\left(\omega\right)=\left(\begin{array}[]{cc}\text{cos}\frac{\phi\left(\omega\right)}{2}+i\text{sin}\frac{\phi\left(\omega\right)}{2}\text{cos}2\theta&i\text{sin}\frac{\phi\left(\omega\right)}{2}\text{sin}2\theta\\\
i\text{sin}\frac{\phi\left(\omega\right)}{2}\text{sin}2\theta&\text{cos}\frac{\phi\left(\omega\right)}{2}-i\text{sin}\frac{\phi\left(\omega\right)}{2}\text{cos}2\theta\end{array}\right).$
(19)

For an input state $\left|V_{in}\right\rangle$ with only the vertical
polarization and output state $\left|V_{out}\right\rangle$ selected by PBS to
maintain only vertical polarization component,

$\left|V_{out}\right\rangle=[\text{cos}\frac{\phi\left(\omega\right)}{2}-i\text{sin}\frac{\phi\left(\omega\right)}{2}\text{cos}2\theta]\left|V_{in}\right\rangle.$
(20)

Thus, the transmission function can be written as

$T(\omega)=\text{cos}\frac{\phi(\omega)}{2}-i\text{sin}\frac{\phi(\omega)}{2}\text{cos}2\theta:=|T(\omega)|e^{i\Delta\Upsilon(\omega)}.$
(21)

Where
$\Delta\Upsilon(\omega)=\text{tan}^{-1}[-\text{tan}\frac{\phi(\omega)}{2}\text{cos}2\theta]$.
$|T(\omega)|^{2}$ corresponds to transmission spectrum and the phase
$\Delta\Upsilon(\omega)$ corresponds to phase delay. They are shown in Fig.6
as functions of wavelength for $\theta=40^{\circ}$.

In the frequency range of interest, $\phi(\omega)$ can be approximately
treated as a linear function $\phi(\omega)=\alpha\omega+\beta$ as shown in
Fig. 2(b). $\alpha$ is $40.30\;eV^{-1}$ for a MO HWP and $2.18\;eV^{-1}$ for a
ZO HWP. $\beta$ is $-60.23$ for a MO HWP and $-0.21$ for a ZO HWP. Carrying
out the double integral Eqn. (12) (note that the super-Gaussian fit
$T_{f}(\omega)$ is not applied here), we obtain

$\displaystyle R_{c}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\\{1+\frac{\text{cos}2\theta}{2}e^{-\frac{\sigma_{-}^{2}(\alpha^{2}+4\tau^{2})}{8}}[\text{sinh}(\sigma_{-}^{2}\alpha\tau)-\text{cosh}(\sigma_{-}^{2}\alpha\tau)\text{cos}2\theta]$
(22) $\displaystyle-$
$\displaystyle\frac{\text{sin}^{2}2\theta}{2}[e^{-\frac{1}{2}\sigma_{P}^{2}\alpha^{2}}(1-e^{-\frac{1}{8}\sigma_{-}^{2}\alpha^{2}})\text{cos}(\beta+\frac{\alpha\omega_{p}}{2})+1]\\}.$

In this experiment, a frequency filter is placed before the PBS to get rid of
the influence of the pump laser and can be approximated by a super-Gaussian
function, as shown in Fig. 2 (c).

The linear susceptibility for a damped harmonic oscillator with the
eigenfrequency resonant at $\Omega$ can be modeled with a Lorentz
functionStenzel (2022):

$\chi(\omega)=\frac{q^{2}}{m}\frac{1}{-\omega^{2}-\frac{2i\omega}{T_{2}}+\Omega^{2}}$
(23)

Where $\Omega$ is the resonance frequency, $T_{2}$ is the dephasing time, $q$
is the charge $m$ is the mass. For a near-resonant absorption
$\omega\sim\Omega$ ($\Omega$ = 808 nm while $\omega_{s}$ range from 803 nm to
813 nm), we simplified the above equation to

$\chi(\omega)=\frac{B}{\Omega-\omega-i/T_{2}}$ (24)

Where $B=\frac{q^{2}}{2m\Omega}$. From Eqn. (8), the corresponding
transmission function:

$\displaystyle T(\omega)=\text{exp}({\frac{iBL/c}{\Omega-\omega-i/T_{2}}})$
(25)

Comparison of transmission spectrum $|T(\omega)|^{2}$ of system (Eqn. 21)
versus Lorentz fit (Eqn. 25) is shown on Fig. 7(a). From the formula, one can
obtain the dephasing time $T_{2}$ of the sample versus $\theta$. The
calculated dephasing time $T_{2}$ for a MO HWP and PBS is shown in Fig. 7(b),
which shows the capacity of measuring $22\;fs$ dephasing time with the
experimental scheme.

Figure 7: (a) Comparison of transmission spectrum $|T(\omega)|^{2}$ of the
system (Eqn. 21) and Lorentz fit (Eqn. 25) for MO HWP and PBS orientated at
different angles. (b) Dephasing time ($T_{2}$) versus optical loss based on
the Lorentz model for a two-level system. Optical loss is calculated based on
a ‘perfect’ HWP as discussed in Eqn. 4.

##  5 Conclusion

We demonstrate an experimental scheme and present a theoretical framework to
study the effects of tunable optical properties of a sample on photon
indistinguishability in a HOM interferometer. We continuously change a mock
sample’s transmission/dispersion by varying the orientation of a ZO/MO HWP
followed by a PBS. Our results indicate that the structures of a HOM dip (dip
location, visibility, and symmetry) would contain information about a sample’s
optical properties. Specifically, a wavelength-dependent
transmission/dispersion would partially break the indistinguishability and
thus change the dip location, visibility, and symmetry of the HOM dip, as is
also predicted by our theoretical model. Moreover, a 22 fs dephasing time is
obtained using our experimental scheme even with optical loss $>97\%$ (while
HOM dip visibility is maintained at $92.3~{}\%$), which provides a useful
approach for characterizing a sample’s optical properties even in an
enormously lossy environment.

## Funding

This research was supported in part by the Robert A. Welch Foundation (Grant
A-1801-20180324, A-1943-20210327) and the Air Force Office of Scientific
Research (Award No. FA-9550-20-1-0366).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

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