# Eigenstate plateau transition and equilibration in 1D quantum lattice models

Wei-Han Li<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex
Systems, 01187 Dresden, Germany  Abbas Ali Saberi<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, University of Tehran, P. O. Box
14395-547, Tehran, Iran  Max Planck Institute for the Physics of Complex
Systems, 01187 Dresden, Germany

###### Abstract

We report on a remarkable spectral phenomenon in a generic type of quantum
lattice gas model. As the interaction strength increases, eigenstates
spontaneously reorganize and lead to plateaus of the interaction energy, with
gaps opening akin to continuous phase transitions. Perturbation theory
identifies a hidden structure underlying eigenstates within each plateau,
resulting in a statistical shift in the wavefunction amplitudes described by
extreme value theory. The structured eigenstates manifest themselves naturally
in far-from-equilibrium dynamics proceeding through multiple universal stages.
Our findings reveal a profound connection between emergent properties in high-
energy states and out-of-equilibrium dynamics, providing insights into the
impact of interactions across the entire energy spectrum. The results are
directly relevant to experiments probing equilibration in quantum spin and
lattice gases.

Introduction.— The study of quantum many-body systems has mainly focused on
their equilibrium properties and low-energy characteristics, particularly the
ground states and associated quantum phase transitions. However, exploring the
exotic out-of-equilibrium phenomena and in the vast high-energy realms delves
into fundamental questions that are still unanswered. Recent years have seen a
burgeoning interest in exploring the equilibration dynamics of closed quantum
systems Eisert2015 ; Gogolin2016 , particularly in low dimensions, propelled
by the advances in ultracold atomic experiments Langen2015 ; Gross2017 ;
Bloch_a . This progress has enabled the realization of archetypal model
Hamiltonians Greiner2002 ; Stoferle2004 ; Spielman2007 ; Esslinger2010  and
the ability to study dynamical equilibration phenomena such as mass expansion
Scheider2012 ; Trotzky2012 ; Ronzheimer2013 ; Xia2015 ; Scherg2018  and spin
transport Hild2014 ; Jepsen2020  in highly controllable settings. A key
frontier has been the exploration of long-range interacting quantum gases
Defenu2023 , where strong inter-site interactions can be engineered in optical
lattices, giving rise to effective long-range spin models Paz2013 ;
Patscheider2020 ; Scholl2021  and extended Hubbard models (EHMs) Baier2016 ;
Su2023 ; Sanchez2021 . The delicate interplay between these tunable
interactions and kinetic energy in such systems can cause novel quantum phases
Torre2006 ; Deng2011 ; Lahaye2009 ; Dutta2015  and rich non-equilibrium
dynamical behavior Barbiero2015 ; WHL1 ; WHL2 ; WHL3 ; Korbmacher2023 .

For various one-dimensional (1D) EHMs and spin models, previous studies have
revealed that inter-site interactions strong enough to drive ground-state
phase transitions can induce intriguing changes in dynamical properties. These
include a qualitative slowdown of quench dynamics Sanchez2021 , distinct
features in the self-return probability Misguich2016 , and transitions in far-
from-equilibrium transport behavior Steinigeweg2017 ; Ljubotina2017 . In this
letter, we consider a generic 1D EHM, which can be mapped to the XXZ model
(spin-$\frac{1}{2}$ Heisenberg chain). Our main focus lies in exploring the
emergent properties of the many-body eigenstates arising from the tunable
inter-site interactions and identifying their signatures in the out-of-
equilibrium dynamics.

We uncover a hidden structure across the spectrum as inter-site interactions
increase, where eigenstates exhibit plateaus of interaction energies. The gaps
between these plateaus serve as order parameters in continuous phase
transitions, remaining stable upon extrapolation to infinite system size and
displaying power-law scaling. Using perturbation theory, we identify a
structure shared among eigenstates within each plateau, our arguments also
predict statistical shifts in the distribution of eigenstates described by
extreme value theory. Although probing the structure of an eigenstate with
specific energy may be impossible in a thermodynamic system, the structures
across the full spectrum impact far-from-equilibrium dynamics that involve
numerous eigenstates. We study the time evolution of Fock states, revealing
multiple equilibration stages that universally apply across different
interaction forms in current experiments. Our findings offer deeper insights
into previous theoretical studies Misguich2016 ; Steinigeweg2017 ;
Ljubotina2017  and a recent experimental observation Sanchez2021 .

The model.— We consider the EHM described by the Hamiltonian

$H=\sum_{j}-t\left(\hat{a}^{\dagger}_{j+1}\hat{a}_{j}+\hat{a}^{\dagger}_{j}\hat{a}_{j+1}\right)+V\hat{n}_{j+1}\hat{n}_{j},$
(1)

where $\hat{a}^{\dagger}_{j}$ ($\hat{a}_{j}$) is the creation (annihilation)
operator at lattice site $j$ and
$\hat{n}_{j}=\hat{a}^{\dagger}_{j}\hat{a}_{j}$ is the on-site occupation with
the hard-core constraint $(\hat{a}^{\dagger}_{j})^{2}=0$. We assume periodic
boundary conditions, and half filling, $N/L=\frac{1}{2}$, for $N$ particles
moving in $L$ sites.

Equation (1) is a generic model capturing 1D quantum lattice gases that
emphasize nearest-neighbor (NN) interactions. The interplay between the
hopping amplitude $t$ and the interaction strength $|V|$ is the core factor
influencing the eigenstates. A well-known example is the quantum phase
transition Nozieres2004  when the EHM is mapped onto the XXZ model
note_mapping . For $|V|/t>2$, the ground state turns from paramagnetic to
ferromagnetic or antiferromagnetic depending on the sign of $V$. Nonetheless,
an increase in $|V|$ does not merely impact the ground state; in principle,
the influence extends to all excited states in the spectrum.

[width=]fig_1.pdf

Figure 1: Emergence of plateaus. The scaled interaction energy
$\varepsilon_{V}$ (blue symbols) for the eigenstates labeled by
$\alpha=1,\cdots,\Omega$ sorted in ascending order of total energy
$\varepsilon$. Where $\Omega$ is the dimension of the Hilbert space for
$N/L=8/16$. (a) The continuum of $\varepsilon_{V}$ for $V/t=1$. (b) The
discrete plateaus of $\varepsilon_{V}$ for $V/t=9$, while $\varepsilon$ (gray
solid line) remains continuous. High mobility indicated by the high modulus of
the kinetic energy $|\varepsilon_{K}|$ (red symbols) appears at the two sides
of each plateau. Vertical dotted lines are the boundaries of plateaus for
$t=0$.

Observables and parameters. Our study focuses on the response of excited
states to $V$, the control parameter. Hence we take $\partial H/\partial
V=\hat{l}$ as a response parameter of interest, where

$\hat{l}=\sum_{j}\hat{n}_{j+1}\hat{n}_{j},$ (2)

is the number of NN links. The ferromagnetism (antiferromagnetism) mentioned
above thus corresponds to the extreme expectation value
$\langle\hat{l}\rangle\to N-1$ ($\langle\hat{l}\rangle\to 0$). Additionally,
states with the same $\langle\hat{l}\rangle$ may have different numbers of
solitary particles or holes, e.g., $\bullet\bullet\bullet\circ\bullet\circ$
and $\bullet\bullet\circ\circ\bullet\bullet$. These particles and holes are
given as

$\hat{s}=\sum_{j}(\hat{n}_{j+1}-1)\hat{n}_{j}(\hat{n}_{j-1}-1)+\hat{n}_{j+1}(\hat{n}_{j}-1)\hat{n}_{j-1},$
(3)

where the first and second term denotes the solitary particles and holes,
respectively.

A solitary particle in Eq. (3) can move freely when no links are altered, as
in the case of $\circ\bullet\circ\circ\leftrightarrow\circ\circ\bullet\circ$.
Similar motion occurs for a hole when one link is broken and another is
created, as in
$\bullet\bullet\circ\bullet\leftrightarrow\bullet\circ\bullet\bullet$. Through
these motions, a solitary particle can turn into a solitary hole and vice
versa, for example,
$\bullet\bullet\bullet\circ\circ\bullet\circ\leftrightarrow\bullet\bullet\bullet\circ\bullet\circ\circ\leftrightarrow\bullet\bullet\circ\bullet\bullet\circ$.
Therefore, double counting between the solitary particles and holes must be
avoided, and as shown in the Supplementary Information (SI), their total
number is constrained. From now on, we will refer to the solitary particles
and holes together as singlons.

The motions of singlons conserve $\hat{l}$ and $\hat{s}$. From the
perturbation point of view, large $|V|$ renders an eigenstate governed by a
set of Fock bases that are mutually connected through the motion of singlons.
We consider these bases to share a common structure, as each of them can be
”mapped” to all the others through singlon motions and vice versa
note_symmetry . These common structures are identified by Eq. (2) and (3);
they are non-local parameters that classify the eigenstates. We will take Eq.
(2) as our main indicator.

Plateau transitions.— Let us first examine the energetic response of the
eigenstates $|\varepsilon\rangle$ to increasing $|V|$. We obtain
$|\varepsilon\rangle$ by exact diagonalization and measure the scaled energies
$\varepsilon_{K}=E_{K}/V=-(t/V)\langle\varepsilon|\sum_{j}\hat{a}_{j+1}^{\dagger}\hat{a}_{j}+h.c.|\varepsilon\rangle$
and $\varepsilon_{V}=E_{V}/V=\langle\varepsilon|\hat{l}|\varepsilon\rangle$,
where $E_{K}$ and $E_{V}$ are the kinetic energy and interaction energy,
respectively. The total energy is thus
$\varepsilon=\varepsilon_{K}+\varepsilon_{V}$. Without loss of generality, we
always choose $V>0$ note_signV . The first observation is a distinct
collective behavior of eigenstates as $V$ increases, where
$|\varepsilon\rangle$, sorted in order of $\varepsilon$, spontaneously
reorganize themselves such that interaction energies $\varepsilon_{V}$
aggregate from a continuum (Fig. 1a) to a configuration of discrete plateaus
(Fig. 1b). Each plateau is associated with an integer in $0,1,\cdots,N-1$
which approximates $\varepsilon_{V}$, meaning that the eigenstates
constructing a plateau are spanned mostly by Fock bases with the same number
of NN links. Common structures that classify eigenstates are identified by the
coexistent splitting of $\langle\varepsilon|\hat{s}|\varepsilon\rangle$ (SI).

[width=]fig_2.pdf

Figure 2: Extrapolations of plateau gaps. Gaps $\Delta_{i}$ versus $V/t$ for
various system sizes ($L=10,12,\cdots,18$, along the arrow direction). The red
solid dots represent the extrapolation to $L\to\infty$, resulting in (a)
$V_{c}/t\sim 4.12$ for $\Delta_{2}$, and (b) $V_{c}/t\sim 3.06$ for
$\Delta_{N-2}$. Insets: $\Delta_{i}$ versus $(V-V_{c})/t$ on logarithmic
scales, confirming the power law relationship (dashed lines) with exponents
$\beta_{2}\sim 0.74(2)$ and $\beta_{N-2}\sim 0.36(1)$.

An immediate question is how the gaps in $\varepsilon_{V}$ open as $V$
increases. To address finite-size effects, we estimate the $i^{\textrm{th}}$
gap $\Delta_{i}$ (where $i=1$ to $N-1$, counting from left to right in Fig.
1b) between two adjacent plateaus by extrapolating $L\to\infty$. Figure 2
shows examples of $\Delta_{2}$ and $\Delta_{N-2}$ as functions of $V/t$ (SI).
Remarkably, the gaps behave similarly to order parameters in equilibrium
critical phenomena that present continuous phase transitions: with a critical
interaction strength $V_{c}$, the gap $\Delta_{i}$ is zero in the subcritical
regime ($V<V_{c}$) and continuously increases to a nonzero value in the
supercritical regime ($V>V_{c}$). Drawing an analogy with phase transitions,
we examine the existence of universal power laws
$\Delta_{i}\propto(V-V_{c})^{\beta_{i}}$ as $(V-V_{c})\to 0^{+}$, where
$\beta_{i}$ is the critical exponent for the $i^{\textrm{th}}$ gap. The insets
in Fig. 2 show a great agreement with the power-law behavior, where fits to
logarithmic scales (dashed lines) measure the values of $\beta_{2}$ and
$\beta_{N-2}$ with impressive precision. Estimations reveal different values
for each gap, with $\beta_{i}$ ranging from $\beta_{1}\approx 1$ to
$\beta_{N-1}\approx 0$ (SI), indicating a distinct universal pattern
associated with the common structures during the formation of plateaus.

Theoretical framework.— To analyze the eigenstates in the presence of common
structures and plateaus, we consider strong interactions and employ a
perturbative treatment of $(t/V)$. This approach renders the zeroth-order
Hamiltonian as $\hat{l}$ augmented by the singlon motions conserving
$\hat{l}$, where the zeroth-order interaction energy, denoted as
$l_{0}=0,1,\cdots,N-1$, dominates the plateaus. As detailed in the SI, higher-
order processes systematically restore $l_{0}$, outweighing its detrimental
effects in the thermodynamic limit. Consequently, $\varepsilon_{V}$ aggregates
around $l_{0}$ and forms plateaus. Existing plateaus directly lead to the
high-mobility states on either side. Since the interaction energies within a
plateau closely approximate $l_{0}$, the states on either side with the
highest or lowest total energy exhibit extreme kinetic energies. As a result,
peaks of $|\varepsilon_{K}|$, indicative of high mobility, appear on both
sides of a given plateau, as illustrated in Fig. 1b.

[width=0.9]fig_3.pdf

Figure 3: Emergent extreme value statistics. The weight $W(\varepsilon)$ is
obtained for all eigenstates $|\varepsilon\rangle$ with various $L$, the
statistics are performed in the middle of the spectrum. (a) and (b) depict the
probability distribution $P(W)$ for $V/t=0.8$ and $5.9$ respectively, with the
mean value $\overline{W}$ and the standard deviation $\sigma_{W}$. Solid lines
show the best fit to the data, where the Gumbel distribution is
$P(x)=\frac{1}{\sigma_{W}}\exp[-(x+e^{-x})]$ with
$x=(W-\overline{W})/\sigma_{W}$ and $\sigma_{W}=0.038(2)$. The dashed line in
(b) is a normal-distribution fit for comparison; the displacement between the
peaks of the two fittings is $\sim 0.58(3)\sigma_{W}$. (c) $\overline{W}$
versus $V/t$, indicating a decrease $\propto(t/V)^{2}$ for $V/t\gtrsim 3.5$,
data for different $L$ collapse after a slight vertical shift.

Extreme value statistics. The quantum nature of the model allows a
wavefunction to seep from its zeroth-order space, leading to a nonequivalence
between $\varepsilon_{V}$ and $l_{0}$, thereby widening the plateaus (Fig.
1b). Treating this as an indicator of the emergence of plateaus, we compute
the Treating this as an indicator of plateau emergence, we compute the leakage
$W(\varepsilon)=\sum_{f^{\prime}}|\langle f^{\prime}|\varepsilon\rangle|^{2}$
with all Fock bases $|f^{\prime}\rangle$ satisfying $\langle
f^{\prime}|\hat{l}|f^{\prime}\rangle\neq l_{0}$, and analyze the probability
distribution $P(W)$ to fully describe all eigenstates. Figs. 3a and 3b
illustrate that $P(W)$ transitions from a symmetric form to a distorted one as
$V$ surpasses $t$ and reaches a value where the interaction energy dominates
the entire spectrum
($\varepsilon_{V}^{max}-\varepsilon_{V}^{min}>\varepsilon_{K}^{max}-\varepsilon_{K}^{min}$).

Our argument suggests that $P(W)$ undergoes a transition from a normal
distribution to a form described by extreme value statistics. For the
probability distribution of an eigenstate, $P_{\varepsilon}(f)=|\langle
f|\varepsilon\rangle|^{2}$, the leakage weight
$W(\varepsilon)=\sum_{f^{\prime}}P_{\varepsilon}(f^{\prime})$ simulates the
mean of the samples at $f^{\prime}$, where $\langle
f^{\prime}|\hat{l}|f^{\prime}\rangle=l^{\prime}_{0}\neq l_{0}$. For a weaker
interaction, eigenstates naturally spread across all bases, thereby
$P_{\varepsilon}$ remains qualitatively the same, and the
$P_{\varepsilon}(f^{\prime})$ act as random samples returning arbitrary values
in $P_{\varepsilon}$. According to the central limit theorem, the distribution
of the mean of random samples, i.e., $P(W)$, converges to a normal
distribution as shown by Fig. 3a. However, for a large interaction causing
wavefunctions to concentrate in the zeroth-order space, the leakage decreases.
$P_{\varepsilon}(f^{\prime})$ no longer presents random samples in
$P_{\varepsilon}$; instead, it goes to vanishingly small values as the
interaction increases. The statistics of $P_{\varepsilon}(f^{\prime})$ in such
a scenario naturally lead to the statistics of minimums, described by the
generalized extreme value theory Majumdar2020 . Assuming that decreased
leakages cause $P_{\varepsilon}(f^{\prime})$ to decay exponentially as
$l^{\prime}_{0}$ moves away from $l_{0}$, then $P(W)$ is predicted to be
governed by the Gumbel distribution. Fig. 3b demonstrates good consistency
between this distribution and our dataset.

Moreover, $W(\varepsilon)$ can be estimated as
$(\frac{t}{V})^{2}|\langle\varepsilon_{1}|\varepsilon_{1}\rangle|^{2}$, the
decrease in leakage is then expected to be proportional to $(V/t)^{-2}$. Fig.
3c illustrates the mean value
$\overline{W}=\frac{1}{\Omega}\sum_{\varepsilon}W(\varepsilon)$ versus $V/t$.
The decrease is observed around $V/t\sim 3.5$, indicating where
$\varepsilon_{V}$ begins to aggregate into plateaus and common structures
start to emerge in eigenstates. This observation aligns with the analytical
estimation in the SI. Comparing with the $V_{c}/t$ in Fig. 2, it becomes
evident that along with the emergence of common structures, most
$\varepsilon_{V}$ aggregate around $l_{0}$ even before gaps open. The latter
stringently requires the two states with exactly the highest and lowest
$\varepsilon_{V}$ in two adjacent plateaus.

[width=]fig_4_2.pdf

Figure 4: Equilibration of Fock states. (a) $\eta(\tau)$ for $L=18$, two
stages of dynamics are observed before and after $\tau t\sim 1$. The inset
shows the universal form $A(\tau)$ with $\epsilon\sim 0.05$ (see the main
text), which governs the dynamics in the presence of eigenstate common
structures. (b) Similar to (a) but with dipole-dipole interactions
$V_{i,j}=V/|i-j|^{3}$ and $\epsilon\sim 0.25$, the universal function
$A(\tau)$ is almost unchanged.

Universal dynamics.— The diminishing leakage of wavefunctions from their
zeroth-order spaces implies a reduction in the traversed area within Hilbert
space during time evolution. As detailed in the SI, the evolution of Fock
states conserve $\hat{l}$ to the first order of $(t/V)$ while allowing
$\hat{s}$ to change. The dynamics are therefore grounded by the initial number
of NN links to a great degree.

To align with experiments, we investigate this impact by examining the
equilibration of Fock states $|f_{0}\rangle$ with the $j_{0}^{\textrm{th}}$
site initially occupied. We compute $\eta(\tau)=[\langle
f_{0}(\tau)|\hat{n}_{j_{0}}|f_{0}(\tau)\rangle-N/L]/(1-N/L)$, where $\eta=1$
indicates that the initially observed particle is localized at $j_{0}$, while
$\eta=0$ implies that it has dissipated into the background. Assuming that the
initial preparation Bakr2009 ; Kuhr2016 ; Gross2021  does not favor particular
configurations, we average $\eta(\tau)$ over all $|f_{0}\rangle$ for a
statistical description. The dynamics depicted in Fig. 4a unveil two stages.
For $\tau t\lesssim 1$, free-moving singlons dominate the dynamics, yet the
final equilibration at $\eta=0$ is not achieved, contrasting with the behavior
of free-moving particles with $V=0$. This suggests that singlons alone do not
furnish sufficient motion to connect $|f_{0}\rangle$ equally to
$j_{0}$-occupied and $j_{0}$-empty states. Nonetheless, equilibration
persists. For $\tau t\gtrsim 1$, the spread of singlons saturates, uncovering
the remaining higher-order processes. These processes compel equilibration to
progress until $\eta=0$.

The slight variations associated with different values of $V/t$ reflect the
interaction dependence of the higher-order processes. The approximation to the
first order allows us to express $\eta(\tau)\sim A(\tau)-B(\tau)\cdot t/V$,
with $A(\tau)$ and $B(\tau)$ being $V$-independent functions representing
dynamics concerning NN links (SI). Assuming $B(\tau)\sim A(\tau)+\epsilon$ for
simplicity, the dynamics can be described by a universal form
$A(\tau)\sim[\eta(\tau)+\frac{t}{V}\epsilon]/(1-\frac{t}{V})$ (Fig. 4a).

Conclusions.— We have studied a generic 1D quantum lattice model and uncovered
a profound relationship between out-of-equilibrium dynamics and emergent
properties in eigenstates across the spectrum. Specifically, a strong yet
finite inter-site interaction induced inherent structures within the
eigenstates, leading to the reorganization of wavefunctions across the Fock
bases and the opening of gaps in interaction energy akin to continuous phase
transitions. These structured eigenstates exhibit statistical patterns
described by extreme value theory and compress the area of time evolution,
wherein the equilibration of Fock states unfolds through multiple stages with
a universal nature.

Our findings, particularly regarding the dynamical aspects, demonstrate
significant consistency across a broad range of realistic interactions (e.g.,
Fig. 4b), suggesting that our observations are not consequences of the non-
thermal feature originating from the NN interacting form quantum_integrability
. This offers a comprehensive understanding of recent observations Sanchez2021
, wherein a qualitative slowdown of quench expansion is confirmed with
interaction strengths close to those associated with eigenstate structures.

The EHM under examination is isomorphic to the XXZ model, where the structured
eigenstates provide deeper insight into the transformation of the self-return
probability Misguich2016  observed previously. Earlier studies have revealed
far-from-equilibrium spin transport in the XXZ model, elucidated by near-
equilibrium theory Steinigeweg2017 ; Ljubotina2017 . However, this behavior
relies on initial conditions embodying a certain equilibrium background. In
contrast, initial Fock states circumvent such limitations, focusing on the
essence of dynamics far from equilibrium. The reduced evolving area of Fock
states presents weakened ergodicity due to interactions, a phenomenon that
should be distinguished from non-thermal states arising from kinetic
constraints QMBS ; localization .

Acknowledgements.— The authors express their gratitude for the insightful
discussions with Roderich Moessner and appreciate his constructive comments
throughout the completion of this project.

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  * (43) V. Khemani, M. Hermele, and R. Nandkishore, Phys. Rev. B 101, 174204 (2020).
  * (44) Eq. (1) is also called the $t$-$V$ model, it describes a hard-core Bose lattice gas, which can be mapped into the XXZ model by Holstein–Primakoff transformation, so, half-filling is associated with zero magnetization. In 1D, the XXZ model can be again mapped into a spinless Fermi lattice gas by Jordan-Wigner transformation.
  * (45) Besides singlons’ connection, a common structure is also shared between translating or reflecting counterparts of Fock bases due to the symmetry of Eq. (1). But this does not affect our concern in this work.
  * (46) For the XXZ model note_mapping , consider $\mathcal{U}=\exp\left({i\pi\sum_{j}j\hat{S}_{j}^{z}}\right)$, a rotation by $\pi$ about $z$-axis for every second spin, where $\hat{S}^{z}=\frac{1}{2}\hat{\sigma}^{z}$. The unitary transform $\mathcal{U}^{\dagger}H(V)\mathcal{U}=-H(-V)$ gives the spectrum relationship $E(-V)=-E(V)$. So $E(-V)$ and $E(V)$ are exact reflections of each other to $E=0$, note that the ground state for $-V$ corresponds to the highest excited state for $V$. Hence, to investigate $E(V)$ with a varying $|V|$, taking $V>0$ will be enough.


# Near-video frame rate quantum sensing using Hong-Ou-Mandel interferometry

Sandeep Singh<EMAIL_ADDRESS>Photonic Sciences Lab., Physical Research
Laboratory, Navrangpura, Ahmedabad 380009, Gujarat, India  Indian Institute of
Technology-Gandhinagar, Ahmedabad 382424, Gujarat, India  Vimlesh Kumar
Photonic Sciences Lab., Physical Research Laboratory, Navrangpura, Ahmedabad
380009, Gujarat, India  Varun Sharma  Photonic Sciences Lab., Physical
Research Laboratory, Navrangpura, Ahmedabad 380009, Gujarat, India  Daniele
Faccio  School of Physics and Astronomy, University of Glasgow, Glasgow G12
8QQ, UK  G. K. Samanta  Photonic Sciences Lab., Physical Research Laboratory,
Navrangpura, Ahmedabad 380009, Gujarat, India

###### Abstract

Hong-Ou-Mandel (HOM) interference, the bunching of two indistinguishable
photons on a balanced beam-splitter, has emerged as a promising tool for
quantum sensing. There is a need for wide spectral-bandwidth photon pairs (for
high-resolution sensing) with high brightness (for fast sensing). Here we show
the generation of photon-pairs with flexible spectral-bandwidth even using
single-frequency, continuous-wave diode laser enabling high-precision, real-
time sensing. Using 1-mm-long periodically-poled KTP crystal, we produced
degenerate, photon-pairs with spectral-bandwidth of 163.42$\pm$1.68 nm
resulting in a HOM-dip width of 4.01$\pm$0.04 $\upmu$m to measure a
displacement of 60 nm, and sufficiently high brightness to enable the
measurement of vibrations with amplitude of $205\pm 0.75$ nm and frequency of
8 Hz. Fisher-information and maximum likelihood estimation enables optical
delay measurements as small as 4.97 nm with precision (Cramér-Rao bound) and
accuracy of 0.89 and 0.54 nm, respectively, therefore showing HOM sensing
capability for real-time, precision-augmented, in-field quantum sensing
applications.

††preprint: APS/123-QED

##  I Introduction

Quantum sensing and metrology is a rapidly growing field due to its
outstanding features in measuring physical parameters with great precision and
accuracy, outperforming the pre-existing technologies based on the principles
of classical physics [1]. In recent years, quantum sensing has enabled the
measurement of several key physical quantities, including the measurement of
the electrical field [2], magnetic field [3], vacuum [4], temperature [5], and
pressure [6] with unprecedented precision and accuracy. Among various quantum
sensors as presented in Ref. [7], Hong-Ou-Mandel (HOM) interferometry, since
its discovery [8], has found a large number of applications within quantum
optics for a variety of advantages, including easy development and
implementation, sensitivity only to photon group delay and not to phase shifts
[9].
HOM interference, purely a non-classical phenomenon, is observed for two
photons that are identical in all degrees of freedom, including spin,
frequency, and spatial mode, when simultaneously entering on a lossless, 50:50
beam splitter through different input ports bunch together into one of the
output ports. The coincidence probability between the two output ports shows a
characteristic low or zero coincidence, known as the HOM interference dip,
directly related to the level of indistinguishability or the degree of purity
of the photons [10]. As a result, the change in coincidence counts can be
treated as a pointer to estimate the time delay between photon properties,
forming the basis of the HOM interferometer-based quantum sensors to sense any
physical process influencing the photon delay. As such, efforts have been made
to use the HOM-based quantum sensors to characterize the single photon sources
[11], precise measurement of time delays between two paths [8, 12, 13],
characterization of ultrafast processes [14], measurement of frequency shifts
[15, 16] and the spatial shift [17]. On the other hand, the use of Fisher
Information (FI), a measure of the mutual information between the interfering
photons, has enabled the maximum precision to access a lower limit decided by
the Cramér-Rao bound [18, 19]. In fact, the use of peak FI of the HOM
interferometer, the point of maximum sensitivity, has enabled five orders of
magnitude enhancement in the resolution of time delay measurements [12, 13,
20]. A typical FI-based measurement approach involves tuning the HOM
interferometer for the peak FI point and performing iterative measurement to
estimate any optical delay between the two photons. However, the number of
iterations/measurements and hence the total time needed to reach the targeted
precision decreases with the increase of FI value. On the other hand, the FI
value can be increased by optimizing the parameters affecting the visibility
and the width of the HOM interference dip. Despite the experimental
demonstration of nanometer path length (few-attosecond timing) precision [12,
21] and a recent theoretical study on the tailoring of the spectral properties
[22, 23] of photon pairs to achieve precision beyond the value reported in the
laboratory condition, the width of the HOM interference dip governed by the
bandwidth of the pair photons remained the limiting factor for optimal
measurement precision [12, 13, 24, 25] in HOM interferometer based dynamic
(real-time) or fast sensing applications. As such, ultra-broadband photon
sources have long been hailed as a vital prerequisite for ultra-precise HOM
interferometry. On the other hand, the dynamic sensing process requires the
single-photon source to have high brightness. On the other hand, a new
strategy has been reported very recently where entangled photons have been
used to produce the HOM beating effect at very different frequencies (visible
and infrared) [21]. As the frequency of the beating is proportional to the
frequency detuning of the entangled photons [21, 26] the HOM beating, in
conjunction with FI analysis, enabled the measurement of low-frequency (0.5 -
1 Hz) vibration with a precision of $\sim$2.3 nm using low (18000) number of
photons pairs [21]
Nonlinear spontaneous parametric down-conversion (SPDC) [27, 28] process-based
sources, where the annihilation of a high energy pump photon produces two
daughter photons simultaneously, have evolved as a workhorse for quantum
optics experiments, including HOM based sensors. However, the development of
high brightness single-photon source requires a nonlinear crystal with high
figure-of-merit (FOM) [29], long crystal length, and high intensity of the
pump beam. As such, periodically poled nonlinear crystals can be used due to
their high intrinsic FOM. However, the increase in the crystal length reduces
the spectral bandwidth of the single-photon source. On the other hand, the
increase of input pump intensity and spectral bandwidth of the generated
photons using ultrafast laser [30] has its own limitations in terms of
expensive and bulky system architecture for any practical on-field uses.
Therefore, it is imperative to explore a continuous-wave (CW) pump-based
bright single-photon source with high brightness and spectral bandwidth for
ultra-precise HOM interferometry.
Recently, we reported a CW pumped high brightness, degenerate single-photon
source at 810 nm using periodically-poled potassium titanyl phosphate (PPKTP)
crystal in non-collinear, type-0, phase-matched geometry [31, 32]. Using the
same phase-matching geometry, here, we report on the experimental
demonstration showing the dependence of spectral bandwidth of degenerate SPDC
photons and subsequent change in the HOM interference dip width on the length
of the PPKTP crystal. Pumping the PPKTP crystal of length 1 mm, using a
single-frequency, CW diode laser at 405.4 nm we observed the spectral width of
the SPDC photons to be as high as 163.42 $\pm$ 1.68 nm producing a HOM
interference dip having a full-width-at-half-maximum (FWHM) as small as 4.01
$\pm$ 0.04 $\upmu$m. Using such small HOM interference dip width, we have
dynamically measured the threshold vibration amplitude corresponding to the
optical delay (twice the vibration amplitude of the piezo mirror) introduced
between the photons as small as 205$\pm$0.75 nm at a frequency as high as 8
Hz. On the other hand, the reduction of crystal length from 30 mm to 1 mm
shows a 17 times enhancement in the magnitude of the peak FI value and
achieves a targeted precision (say $\sim$5 nm or $\sim$16.7 attosecond optical
delay) for a number of iterations/measurements (or the total experiment time)
as low as 3300 (19 minutes).

## Theoretical Background

The spectral brightness of SPDC photons generated from a nonlinear crystal of
length, $L$ can be expressed as follows,

$I_{SPDC}\sim\mbox{sinc}^{2}\left(\Delta kL/2\right)$ (1)

where $\Delta k$ is the momentum mismatch among the interacting photons (pump
and down-converted pair-photons) given as follows

$\Delta k=k_{p}-k_{s}-k_{i}-\frac{2\pi m}{\Lambda}$ (2)

Here, $k_{p,s,i}$ = $\frac{2\pi n_{p,s,i}}{\lambda_{p,s,i}}$ is the wavevector
and $n_{p,s,i}$ is the refractive index of the nonlinear crystal for the pump,
$\lambda_{p}$, signal, $\lambda_{s}$, and idler, $\lambda_{i}$, wavelengths.
$\Lambda$ is the grating period of the periodically-poled nonlinear crystal to
satisfy the quasi-phase-matching (QPM) condition. The odd integer number, $m$,
is the QPM order. For maximum efficiency, we consider $m$ = 1. The momentum
mismatch between degenerate photons around the central angular frequency,
$\omega_{p}/2$, can be represented as [33, 34]

$\Delta k=\Delta
k|_{\omega=\omega_{p}/2}+M|_{\omega=\omega_{p}/2}{\Delta\omega}+\frac{1}{2}K|_{\omega=\omega_{p}/2}{\Delta\omega}^{2}+....$
(3)

Assuming the QPM is achieved at degenerate SPDC photons with angular
frequencies of pump, signal, and idler as $\omega_{p}$, $\omega_{s}$, and
$\omega_{i}$, respectively, for the grating period, $\Lambda$, the first term
of the right-hand of Eq. (3) can be made zero. In this situation, owing to the
energy conservation, $\omega_{p}$ = $\omega_{s}$ \+ $\omega_{i}$, the spectral
bandwidth of the down-converted photons for the fixed pump frequency defined
as $\Delta\omega$ = $\Delta\omega_{s}$ = -$\Delta\omega_{i}$
=$\omega-\omega_{p}/2$, is decided by the second term, M = $[\partial
k_{s}/\partial\omega-\partial k_{i}/\partial\omega]_{\omega=\omega_{p/2}}$,
known as group-velocity mismatch. However, for degenerate
$(\omega_{s}=\omega_{i}=\omega_{p}/2)$, type-0 SPDC process, the group-
velocity mismatch term vanishes or has negligible value. Under this condition,
the spectral acceptance bandwidth of the SPDC photons is essentially
determined by the much smaller third term of Eq. (3), commonly known as group-
velocity dispersion with a mathematical form
$K=\partial^{2}k/\partial\omega^{2}|_{\omega=\omega_{p/2}}$. Under this
condition, the spectral bandwidth of the SPDC photons can be written as,

$\Delta\omega\propto\frac{1}{\sqrt{KL}}$ (4)

Thus, one can utilize the length of the nonlinear crystal as the control
parameter in the SPDC process to generate paired photons of broad spectral
bandwidth near the degeneracy. On the other hand, it is well known that the
optical delay range or the width, $\sigma$, of the HOM interference is
proportional to the coherence length of the photon wave packets [9, 35] or the
ensemble dephasing time of SPDC photons [36] bunching on the beam splitter.
Although one can find the width, $\sigma$, of the HOM interference dip from
the Fourier transform of the spectral density function of the paired photons
[37, 38], we used the coherence length, $L_{c}$, of the photon wave packets
and Eq. (4) to find the dependence of HOM interference dip width on the
crystal length as,

$\sigma\hskip 5.0pt\sim\hskip 5.0ptL_{c}\hskip 5.0pt\propto\hskip 5.0pt2\pi
c\sqrt{KL}$ (5)

where c is the speed of light in vacuum. It is evident from Eq. (5) that the
width, $\sigma$, of the HOM interference dip can be controlled by simply
varying the crystal length even in presence of the single-frequency, CW diode
laser as the pump.

## Experiment

The schematic of the experimental configuration is shown in Fig. 1. The
details of the experiment can be found in the Methods section. A single-
frequency, CW diode laser providing 20 mW of output power at 405.4 nm is used
to create downconverted photons in the periodically poled KTiOPO4 (PPKTP)
crystals of 1 $\times$ 2 mm2 aperture but of five different interaction
lengths, $L$ = 1, 2, 10, 20, and 30 mm. All the crystals have a single grating
period of $\Lambda$ = 3.425 $\upmu$m corresponding to the degenerate, type-0
($e$→$e+e$) phase-matched parametric down-conversion (PDC) of 405 nm at 810
nm.

Figure 1: Schematic of the experimental setup. C1-4: fiber coupler, SMF1-3:
single-mode fiber, HWP1-2: half-wave plate, PBS1-2: polarizing beam splitter
cube, L1-2: lenses, NC: PPKTP crystal of grating period = 3.425 $\upmu$m, NF:
notch filter, PM: prism mirror, M1-4: dielectric mirrors, QWP: quarter wave
plate, BS: 50:50 beam splitter cube, F1,2: high-pass filter, SPCM1-2: single
photon counting module, TDC: time-to-digital converter.

The noncollinear down-converted photon pairs having annular ring spatial
distribution with signal and idler photons on diametrically opposite points
are guided in two different paths and finally made to interact on the two
separate input ports of a balanced beam splitter (BS). The combination of the
piezo-electric actuator (PEA) (NF15AP25), placed on a motorized linear
translation stage (MTS25-Z8), is used to provide both fine and coarse optical
delay. The photons from the output ports of the BS, after being extracted by
long pass filters (F1, F2), are coupled into the single mode fibers, SMF2 and
SMF3, through the fiber couplers, C3, and C4, respectively. The collected
photons are detected using the single-photon counting modules, SPCM1-2
(AQRH-14-FC, Excelitas), and counted using the time-to-digital converter
(TDC). All the data is recorded at a typical pump power of 0.25 mW with a
coincidence window of 1.6 nanoseconds.

## Results and Discussions

### Displacement sensing using HOM interferometer

First, we have characterized the HOM interference for different crystal
lengths and calibrated various system parameters as presented in the Materials
and Methods section. Knowing the dependence of the sensitivity on the crystal
length, we have studied the performance of the HOM interferometer to measure
the static displacement of the mirror, introducing an optical delay between
the photons. Using the 1 mm long PPKTP crystal resulting in HOM interference
dip width of 4.01 $\pm$ 0.04 $\upmu$m, we have moved the mirror using the
motorized stage to a displacement of $\sim$ 1.25 $\upmu$m away from the zero-
optical delay position. This translation resulted in a positive optical delay
of $\sim$ 2.5 $\upmu$m corresponding to the region having maximum sensitivity
of the HOM region of the sensor. We moved the mirror in a step of 60 nm
(resulting optical delay of 200 attoseconds) and measured the coincidence
counts. The results are shown in Fig. 2. It is evident from Fig. 2(a), the
displacement of the mirror by 60 nm results in the change of the coincidence
counts of more than 50 per 20 ms of integration time. Such a large change in
the coincidence count, much higher than the dark count and accidental counts,
confirms the possibility of measurement of static displacement as low as 60
nm.

Figure 2: a. Variation in the coincidence counts (CC) with nanometer
displacement of the optical delay mirror, M2. b. Temporal stability of
coincidence counts at different positions of the delay mirror, M2.

Such a change in the coincidence count is easily detectable, confirming the
displacement measurement as small as 60 nm. Further, to confirm the
reliability of the measurement, we measured the temporal variation of the
coincidence counts over 10 minutes for each mirror position. It is evident
from Fig. 2(b), that the temporal variation (with standard deviation $\sim$
7.4$\%$) of the coincidence counts due to various parameters, including the
laser intensity fluctuation, air current, and local temperature instability in
the laboratory, is much smaller than the change in the coincidence counts due
to the static displacement of the mirror. The possibility of smaller static
displacement can be possible by suitably reducing the fluctuation of the
coincidence counts in the experiment.

### Dynamic vibration sensing using HOM interferometer

Having the calibration data (see Materials and Methods) for the piezo for
static displacement, we have studied the performance of the HOM interferometer
for a dynamic signal. In doing so, we have used the HOM interference dip width
(FWHM) of 4.01 $\upmu$m and adjusted the initial positive delay by a value of
$\sim$3 $\upmu$m. Driving the PEA with a periodic voltage signal in a
triangular wavefront of controllable frequency and amplitude, we have recorded
the temporal variation of the coincidence counts of the HOM with the results
shown in Fig. 3. As evident from Fig. 3(a), the vibrations of the PEA with a
periodic triangular signal of period 1 Hz and the peak-to-peak voltage of 1 V
corresponding to a peak-to-peak optical delay of $\sim$1.0 $\upmu$m, as
estimated from the calibration data (see Materials and Methods), produces a
peak-to-peak variation in the coincidence counts of $\sim$400 over the mean
value of 960 at a period of 1 Hz, the same as the driving frequency. Further,
we changed the peak-to-peak vibration amplitude of PEA from the calibration
data and measured the peak-to-peak optical delay from the coincidence counts
with the results shown in Fig. 3(b). We have repeated this exercise twice and
termed the results as set-1 (red data points) and set-2 (blue data points). As
evident from Fig. 3(b), the peak-to-peak optical delay (which is twice the
displacement of the PEA) measured using the change in the coincidence counts
of HOM exactly follows the set delay at a slope of 0.835 $\pm$ 0.004. Ideally,
the slope should have a value of 1 to maintain one-to-one correspondence of
the set and measured values. However, the discrepancy between the experimental
slope with respect to the ideal value can be attributed to the error in the
coupling of the applied voltage to the PEA and its hysteresis [39, 40]. While
increasing the PEA vibration amplitude through the applied voltage signal, we
realized that the threshold (to replicate the shape of the input waveform
exactly) peak-to-peak vibration amplitude, which can be measured dynamically
using this technique, is found to be $205\pm 0.75$ nm. Subsequently, the
average maximum attainable resolution between two consecutive vibration
amplitudes is found to be $\sim$80 nm. However, one can access smaller peak-
to-peak vibration amplitude ($<$50 nm) (without replicating the shape of the
input wavefront) from the peak-to-peak variation of the coincidence counts.

Figure 3: Vibration measurement using HOM interferometer sensor. a. Variation
of coincidence counts for periodic triangular voltage signal of peak-to-peak
amplitude 1 V at 1 Hz applied to the PZT stage. b. Variation of PEA
displacement, and c. driving frequency measured from the coincidence counts
with respect to the set values. d. Fast Fourier transformation of the time-
varying coincidence counts measuring the driving frequency of the applied
voltage signal.

As the HOM interferometer deals with single photons, it is essential to
integrate the detected coincidence events for a substantial time to get a
tangible number of coincidence counts while making any measurement. Again the
required integration time is highly influenced by the brightness of the
single-photon source. In the current experiment, we used high brightness
single photon source based on PPKTP crystal [31, 32]. As a result, we can keep
the integration time as low as a few tens of milliseconds. Here, we have also
tested the HOM interferometer to measure the vibration frequency of the PEA.
Keeping the experimental parameters the same as previous, we have changed the
frequency of the triangular voltage signal to the PEA at the constant peak-to-
peak voltage of 1 V corresponding to the optical delay of $\sim$1 $\upmu$m.
While we adjusted the set frequency from the function generator, we recorded
the temporal variation of the coincidence counts for each driving frequency
and performed the fast Fourier transform (FFT) to experimentally measure the
driving frequency. As evident from Fig. 3(c), the measured frequency using the
coincidence data exactly matches the driving frequency. The linear fit (line)
to the experimental data (dots) shows a slope of 1, confirming the reliable
measurement of the frequency of the driving field. However, we have restricted
our measurement to 8 Hz as the FFT signal amplitude, despite having the peak
at the driving frequency, as shown in Fig. 3(d), decreases with the increase
of the driving frequency. Such a decrease can be understood as follows. In FFT
analysis, the accuracy depends on the number of acquired samples from the
experimental data. For example, the higher resolution in estimating the
driving frequency using FFT analysis requires many data points. Again, the
number of samples/data points can be increased by increasing the sampling
rate. However, in the present case, the sampling rate is restricted to 50 Hz
due to the requirement of a high exposure/integration time of 20 milliseconds
limited by the generation rate of the SPDC source and the processing speed of
the TDC. Although we have 50 data points to perform FFT analysis for a driving
frequency of 1 Hz, the available data points vary inverse to the driving
frequency as $M=50/f$, where $M$ is the number of samples available for FFT
analysis and $f$ is the driving frequency. As a result, it becomes difficult
to retrace the waveform of the input signal at an increased frequency which
leads to the gradual decrease in the FFT signal amplitude (decrease in signal-
to-noise, SNR, ratio) and the subsequent appearance of its second harmonic
peak. Further, an increase in the driving frequency requires a decrease in
data integration time (to achieve a higher sampling rate) through the increase
of the photon generation rate of the source and the use of TDC with low data
acquisition latency. However, in the best-case scenario, one can, in practice,
use HOM interferometer-based sensor to measure dynamic vibration signals of
unknown frequency restricted to a few tens of hertz only.
We further study the performance of the HOM interferometer-based quantum
sensor to measure any arbitrary vibration signal. Considering the initial
position on the high-sensitivity region of the HOM interference dip as the
zero displacement point, we drove the PEA with an external voltage signal to
the PEA in the triangular waveform of varying amplitude (peak-to-peak voltage
range of 0.7 to 2 V) and frequency (1 Hz to 8 Hz). The results are shown in
Fig. 4. As evident from Fig. 4(a), the displacement of the PEA measured from
the calibration curve (see Materials and Methods) shows a temporal variation
in both amplitude and frequency, confirming the arbitrary vibration. However,
as evident from Fig. 4(b), the variation of the coincidence counts recorded
for an integration time of 20 ms exactly follows the applied signal. Such
observation confirms the possibility of measurement of any arbitrary
vibration, for example, low amplitude and frequency seismic S- and P-waves,
using a HOM interferometer-based quantum sensor.

Figure 4: a. Temporal variation of PEA displacement and b. corresponding
variation in the coincidence counts of the HOM interferometer for the external
vibration.

The current study establishes the potential of the HOM interferometer as a
quantum sensor for real-time detection of time-varying signals in the
frequency ranging from 1 Hz to 8 Hz and any physical process producing optical
delays as low as $0.41\pm 0.06$ $\upmu$m with a precision of $\sim$0.19
$\upmu$m. While the frequency range can further be increased to tens of hertz
by reducing the exposure time of the measurement, the precision in the real-
time measurement of lower optical delay is fundamentally limited by the
statistical and systematic error, especially at lower variation in coincidence
counts [41]. As such, one needs to explore the statistical concept of Fisher
information to attain a precision that has a lower limit governed by the
Cramér-Rao bound [12, 20, 13].

## Precision augmented sensing

The ultimate limit on the precision in measurement is decided by the Cramér-
Rao bound [18, 19] defined as,

$Var(\tilde{x})\hskip 5.0pt\geq\hskip 5.0pt\frac{1}{NF(x)}$ (6)

where $\tilde{x}$ represents an unbiased estimator to estimate the physical
parameter and var($\tilde{x}$) is the variance in the measurement. $F(x)$
represents the Fisher information (FI), a metric describing the amount of
information available from a probability distribution of an unknown parameter,
for estimating the information of the parameter $x$ in a single measurement,
and N is the total number of measurements performed in the experiment. In the
case of HOM interference, the mathematical form of FI is primarily determined
by the temporal mode profile of the signal and idler photons [12]. As the
down-converted photons generated through the type-0 SPDC process have a
Gaussian spectral profile, the FI can be modeled as,

$F=\frac{4s^{2}\alpha^{2}(\gamma-1)^{2}(1+\gamma)}{(e^{s^{2}}-\alpha)(\alpha-\alpha\gamma+e^{s^{2}}(1+3\gamma))\sigma^{2}}$
(7)

here, $\alpha$, $\gamma$, and $\sigma$ represent the HOM interference
visibility, the rate of loss in photon detection, and the FWHM width of the
HOM curve, respectively. $s=\frac{x}{\sigma}$ is the parameter controlling the
distinguishability of the photon arriving at the beam splitter. As evident
from Eq. (7), keeping all parameters constant, one can increase the magnitude
of FI, $F$ by employing the HOM interference of smaller dip width ($\sigma$).
Again, it is evident from Eq. 5 that for a given laser parameter, the dip
width of the HOM interference varies with the square root of the crystal
length. Since the crystal length, a physical parameter controllable in the
experiment, using Eq. (5) in Eq. (7), we can derive a general expression for
FI in terms of crystal length, L, as,

$F=\frac{4x^{2}\alpha^{2}(\gamma-1)^{2}(1+\gamma)}{A^{4}(e^{\frac{x^{2}}{A^{2}L}}-\alpha)(\alpha-\alpha\gamma+e^{\frac{x^{2}}{A^{2}L}}(1+3\gamma))L^{2}}$
(8)

Here, A is a constant determined by the crystal properties. It is evident from
Eq. (6) that for a fixed number of measurements/iterations, $N$, the precision
can be enhanced, or for a fixed precision, the number of
measurements/iterations, $N$, can be reduced with the increase of the
magnitude of $F$ while saturating the Cramér-Rao bound. Therefore, Eq. (8)
sets the foundation for our further study of precision augmented quantum
sensing.

Figure 5: a. Variation of Fisher information as a function of optical delay,
$x$, between the signal and idler photons for different crystal lengths. b.
Dependence of the peak FI value with the length of the nonlinear crystal.
Solid line is the theoretical fit (Eq. 8) to the experimental data.

To get a better perspective on the dependence of FI on the crystal length, we
have plotted Eq. (8) as the function of optical delay, $x$ while keeping the
experimentally measured values of $\alpha$ and $\gamma$ as constant. We have
used crystal lengths, $L$ = 1, 2, 10, 20, and 30 mm, as available in our lab.
The results are shown in Fig. 5(a). As evident from Fig. 5(a), the FI has the
usual double-peak Gaussian profiles for HOM visibility $\alpha<1$. It has been
previously observed [12] that the HOM visibility, $\alpha$ influences the
magnitude and the separation between two peaks of FI, respectively. However,
in the current experiment, we observed that for a fixed value of HOM
interferometer visibility, the shape and peak value of FI depend on the value
of HOM interference dip width, $\sigma$. As evident from Fig. 5(a), the
decrease in the width of the HOM interference dip due to the decrease of
crystal length (see Eq. (5)) not only reduces the separation of the two peaks
of FI but also show a substantial increase in their respective peak values. To
put things in perspective, using the experimental parameters in Eq. (8), we
estimated the peak values of FI for crystal lengths 1, 2, 10, 20, and 30 mm.
The results are shown in Fig. 5(b).
It is evident from Fig. 5(b) that the peak value of FI (black dots) decreases
from 35.06 $\pm$ 1.65) x $10^{-4}$ $\upmu$m-2 to (2.07 $\pm$ 0.19) x $10^{-4}$
$\upmu$m-2 for the increase of the crystal length from 1 mm to 30 mm due to
the corresponding increase in the FWHM width of the HOM interference dip from
4 $\upmu$m to 21.2 $\upmu$m (see Fig. 9). Such observation clearly shows the
enhancement in peak value of FI (here 17$\times$) by simply reducing the
nonlinear crystal length (here 1/30) even using a single-frequency diode
laser. It is also interesting to note that the current peak value of FI shows
a $\sim$24$\times$ enhancement as compared to the FI value obtained using an
ultrafast laser [12], with the further possibility of enhancement by using an
ultrafast laser generating SPDC photons in a thin nonlinear crystal. Such an
increase in the peak value of FI can be useful in faster saturation of the
Cramér-Rao bound as defined in Eq. (6) at the lower number of
measurements/iterations to achieving a predefined precision in the measurement
of optical delay.

### FI based measurement

The precise estimation of a physical parameter such as optical delay, $x$, can
be obtained through the maximum likelihood estimator [12]. In the current
study, the estimator of the maximum likelihood function for measuring the
optical delay can be defined as,

$\tilde{x}=\pm\sigma\sqrt{ln\left(\frac{(N_{1}+N_{2})}{N_{1}-N_{2}(\frac{1+3\gamma}{1-\gamma})}\right)}$
(9)

Here, $N_{1}$ and $N_{2}$ represent singles and coincidence counts,
respectively. The optical delay introduced between the signal and idler
photons can be estimated using Eq. (9) with the proper knowledge of $N_{1}$
and $N_{2}$. To perform the FI-based measurements on the small optical delay
between the paired photons, we first calibrated the 1 mm long PPKTP crystal-
based HOM interferometer to estimate the parameters, $\alpha$, $\gamma$, and
$\sigma$. We performed 16 scans of the HOM dip to ascertain the precise values
of these defining parameters of the HOM interferometer. We tried to resolve
between two selected points on the HOM interference curve having the positive
optical delays of, say, $x_{1}$ and $x_{2}$ with a separation of $\sim$5.51 nm
apart. The optical delay corresponding to the maximum peak value of FI lies
between points $x_{1}$ and $x_{2}$. We set the voltage to the PEA according to
the calibration data shown in Fig. 11 to make the back-and-forth movement
between the positions $x_{1}$ and $x_{2}$ and recorded the singles count,
$N_{1}$, and coincidence counts, $N_{2}$ at both points. Although the exposure
time for this iterative measurement was set to 50 ms, the electronic response
and delay of the Piezo controller and TDC restricted the effective data
acquisition frequency to $\sim$6 Hz. Using all the experimental parameters in
Eq. (9), we have estimated the values of $x_{1}$ and $x_{2}$ with the results
shown in Fig. 6.

Figure 6: Variation of a. cumulative estimates for the PZT positions, $x_{1}$
and $x_{2}$, and b. the estimated optical delay between signal and idler
photons as a function of the number of measurements.

As evident from Fig. 6(a), the cumulative estimates for $x_{1}$ (black line)
and $x_{2}$ (red line) with the increase in the number of
iterations/measurements show a drift due to the drift in the piezo position
over the measurement time. However, the separation between the estimated
values of $x_{1}$ and $x_{2}$, as shown by the inset image of Fig. 6(a),
remains almost constant. We have estimated the separation between the set
points as a function of the number of iterations with the results shown in
Fig. 6(b). As evident from Fig. 6(b), we observe a large uncertainty in
estimating the separation between the set positions for the initial
measurements, which settles quickly toward the set value with the increase in
the number of measurements/iterations. For the set separation value of $\delta
x$ $\sim$ 5.51$\pm$0.75 nm estimated from the difference in the applied
voltage to the PEA, we measured the optical delay to be 4.97 $\pm$ 0.89 nm
even for the experimental measurements/iterations as low as 3300 (as shown in
the inset of Fig. 6(b)). The achievable precision for the complete range of
the measurements can be calculated as $\sqrt{Var(\delta\tilde{x})/N}$, where N
is the total number of independent measurements performed for the cumulative
estimation.
The minute drift in the cumulative estimated data (see from Fig. 6(a)), due to
the drift in the piezo position over the large experiment time (as high as
$\sim$ 139 minutes), can easily be mitigated by adjusting the frequency of
back-and-forth movement of $\sim$ 6 Hz; a rate much faster than the rate of
drift in the position of PEA. Despite such drift, it is to be noted that the
PEA position remains well inside the region ($\sim$ 200 nm), having maximum FI
value, adding to the reliability of the acquired data. Although we have
performed about 25000 independent measurements over an experiment time of
$\sim$139 minutes, one can easily notice from Fig. 6(b) that the true value
has been achieved through the saturation of Cramér-Rao bound for the
experimental measurements/iterations as low as 3380 corresponding to the
experiment time as low as $\sim$19 minutes. This was possible due to the
increase of the peak FI value through manipulating the spectral bandwidth of
the SPDC photons. Further reduction in the experiment time to reach such high
sensitivity can be possible by further enhancing the FI using the ultrafast
pump laser generating SPDC photons in thin crystals.
We repeated these measurements over a wide range of optical delays using PPKTP
crystals of lengths 1, 2, 10, 20, and 30 mm. The estimated optical delays
calculated from the cumulative estimates are shown in Fig. 7. As evident from
Fig. 7, the experimental values of the cumulative estimates (solid dots) of
the optical delay in the range of $\sim$3 nm to $\sim$80 nm measured using
different crystal lengths exactly follow the set true value of $\delta x$
(black line) derived from the peak-to-peak amplitude of the voltage signal
applied for the back-and-forth motion of the PEA. It is worth mentioning that
we have incorporated the piezo hysteresis as shown in Fig. 3 (b) in all the
measurement data. The close agreement of the experimental values of the
cumulative estimate with the set true values of $\delta x$, for all crystal
lengths confirms the reliability and robustness of the current experimental
scheme.

Figure 7: Variation of optical delay measured using the FI analysis with the
set optical delays for different crystal lengths. Solid line represents the
true values.

We further experimentally verify the dependence of the value of FI and the
minimum number of required iterations/measurements while achieving a fixed
precision. It is evident from Eq. (6) and Eq. (8) that for a given precision,
the increase in the value of peak FI reduces the required number of
iterations/measurements. For experimental verification, we set, for example,
the precision of the optical delay between the pair photons to be $\sim$5 nm
and observed the minimum number of measurements/iterations (N) required to
achieve this precision for all available PPKTP crystals. The results are shown
in Fig. 8. For better understanding, we

Figure 8: Variation of peak FI value and the corresponding number of
iterations/measurement required to achieve a fixed precision as a function of
crystal length.

have also shown the variation of the peak value of FI with crystal length. As
evident from Fig. 8, to achieve a set precision, the number of minimum
iterations/measurements (red dots) increases from 3380 to 17500 for the
increase of crystal length from 1 mm to 30 mm due to the subsequent decrease
of peak FI value (black dots) from $\sim$35 x $10\textsuperscript{-4}$ to
$\sim$2 x $10\textsuperscript{-4}$ $\upmu$m-2. It is interesting to note that
the inverse of the product of the number of minimum iterations/measurements
(N) and the square root of the peak value of FI forming the precision [13] is
exactly matching with set precision and independent to the increase of crystal
length from 1 mm to 30 mm. The black line is the theoretical fit to the
experimental data as reproduced from Fig. 5. From this experiment, it is
evident that a further increase of the peak value of FI through the increase
of spectral bandwidth of the SPDC photon using of ultrafast pump can lead to
the realization of FI-based ultra-sensitive measurements in real-time
applications.

## Conclusions

In conclusion, we have experimentally demonstrated easy control in the
spectral bandwidth of the pair-photons through proper selection of the length
of the non-linear crystal. Using a 1 mm long PPKTP crystal, we have generated
paired photons with spectral width as high as 163.42$\pm$1.68 nm even in the
presence of a single-frequency, CW, diode laser as the pump. The use of photon
pairs with such a high spectral bandwidth in a HOM interferometer results in a
narrow-width HOM interference dip enabling sensing of static displacement as
low as 60 nm and threshold vibration amplitude as low as $\sim$205 nm with a
resolution of $\sim$80 nm at a frequency measurement up to 8 Hz. Further, we
experimentally observed the dependence of FI on the spectral bandwidth of the
pair photons and hence the length of the nonlinear crystal. We observed a 17
times enhancement in the FI value while reducing the crystal length from 30 mm
to 1 mm. Such an increase in the peak FI value (35.06 $\pm$ 1.65) x
$10\textsuperscript{-4}$ $\upmu$m-2, which is nearly 24 times higher than the
previous study [12], saturates the Cramér-Rao bound to achieve any arbitrary
precision (say $\sim$5 nm) in a lower number of iterations ($\sim$ 3300),
$\sim$11 times lower than the previous reports. Multiplying the number of
coincidence counts to the number of iterations, one can find the total number
of photons used in the experiment to achieve such high precision. In our case,
this provides 2 $\times$ $10^{6}$ number of photon pairs to achieve the
desired precision. This is somewhat higher than the 2 $\times$ $10^{4}$
photons used e.g. in the experiments by Kwait et.al.[21]. However, our total
measurement time is still lower thanks to the high photon flux rates,
therefore allowing us to achieve overall faster sampling rates and sensitivity
to higher vibration frequencies. The accessibility of high precision in lower
iterations or time establishes the potential of HOM-based sensors for real-
time, precision-augmented, in-field quantum sensing applications. Unlike the
use of bulky and expensive ultrafast pump lasers of broad spectral bandwidth
to enhance the spectral bandwidth of the down-converted photons, the
generation of broadband photons using a single-frequency diode laser in small
crystal length is beneficial for any practical applications.

## Materials and Methods

The schematic of the experimental configuration is shown in Fig. 1. A single-
frequency, CW diode laser providing 20 mW of output power at 405.4 nm is used
as the pump laser in the experiment. The spatial mode filtering unit comprised
of a pair of fiber couplers (C1 and C2) and a single-mode fiber (SMF1) is used
to transform the laser output in a Gaussian (TEM00 mode) beam profile. The
laser power in the experiment is controlled using a combination of $\lambda$/2
plate (HWP1) and a polarizing beam splitter cube (PBS1). The second
$\lambda$/2 plate (HWP2) is used to adjust the direction of linear
polarization of the pump laser with respect to the orientation of the crystal
poling direction to ensure optimum phase matching conditions.The pump laser is
focused at the center of the nonlinear crystal (NC) using the plano-convex
lens, L1, of focal length, $f1$ = 75 mm to a beam waist radius of $w_{o}$ = 58
$\upmu$m. Periodically poled KTiOPO4 (PPKTP) crystals of 1 $\times$ 2 mm2
aperture but of five different interaction lengths, $L$ = 1, 2, 10, 20, and 30
mm, are used as the nonlinear crystal (NC) in the experiment. The crystals
have a single grating period of $\Lambda$ = 3.425 $\upmu$m corresponding to
the degenerate, type-0 ($e$→$e+e$) phase-matched parametric down-conversion
(PDC) of 405 nm at 810 nm. To ensure quasi-phase-matching, the PPKTP crystals
are housed in an oven whose temperature can be varied from room temperature to
200oC with temperature stability of $\pm$0.1oC. The down-converted photon
pairs, after separation from the pump photons using a notch filter (NF), are
collimated using the plano-convex lens, L2 of focal length, $f2$ = 100 mm. Due
to the non-collinear phase-matching, the correlated pair (commonly identified
as “signal” and “idler”) of down-converted photons have annular ring spatial
distribution with signal and idler photons on diametrically opposite points.
Using a prism mirror (PM), the signal and idler photons are separated and
guided in two different paths and finally made to interact on the two separate
input ports of a balanced beam splitter (BS). Since we need to vary temporal
delay between the photons at the BS, and the translation of the optical
component disturbs the alignment, we have devised the alignment preserving
delay path consisting of a mirror, M1, $\lambda/2$ plate (HWP3), PBS2,
$\lambda/4$ plate (QWP), mirror, M2 on the delay stage, and finally, the beam
splitter, BS. The $\lambda/2$ plate rotates the photon’s polarization state
from vertical to horizontal to ensure complete transmission through PBS2. The
$\lambda/4$ having the fast axis at +45o to its polarization axis transforms
the photon polarization (horizontal) into circular (left circular). However,
the handedness of the photon polarization gets reversed (right circular) on
reflection from the mirror, M2, at normal incidence. On the return pass
through the $\lambda/4$ plate, the right circular polarized photons convert
into vertical polarized photons and are subsequently reflected from the PBS2
to one of the input ports of the BS. Due to the normal incident of the photons
on the moving mirror, M2, the varying delay of the photons does not require
tweaking of the experiment. The combination of the piezo-electric actuator
(PEA) (NF15AP25), placed on a motorized linear translation stage (MTS25-Z8),
is used to provide both fine and coarse optical delay. The range (resolution)
of the PEA and motorized linear stage as specified by the product catalog is
25 $\upmu$m (0.75 nm), and 25 mm (29 nm), respectively. Throughout the report,
the optical delay of the photon is twice the displacement of the mirror, M2,
if otherwise mentioned. On the other hand, the photons of the other arm are
guided using mirrors, M3 and M4, to the second input port of BS such that the
optical path length of both arms are the same. The photons from the output
ports of the BS, after being extracted by long pass filters (F1, F2), are
coupled into the single mode fibers, SMF2 and SMF3, through the fiber
couplers, C3, and C4, respectively. The collected photons are detected using
the single-photon counting modules, SPCM1-2 (AQRH-14-FC, Excelitas), and
counted using the time-to-digital converter (TDC). All optical components used
in the experiment are selected for optimum performance at both the pump and
SPDC wavelengths. All the data is recorded at a typical pump power of 0.25 mW
with a coincidence window of 1.6 nanoseconds.

### Characterization of HOM interference

First, we verified the dependence of HOM characteristics on the length of the
nonlinear crystal. Keeping the input pump power to the crystal and the
temporal coincidence window of the TDC constant at 0.25 mW and 1.6 ns,
respectively, we have recorded the variation of coincidence counts as a
function of the relative optical delay between the photons. As evident from
Fig. 9(a), the variation of coincidence counts with the relative optical path
delay has the characteristic HOM interference dip profile, approximated by an
inverted Gaussian function (owing to the temporal Gaussian profile of signal
and idler). However, it is interesting to note that the HOM visibilities
calculated using the formula given in [12] remain constant in the range of 86
- 94 % despite appreciable variation in the span of the HOM interference dip
profile.

Figure 9: a. Variation of HOM interference dip profiles with the length of the
non-linear crystals. b. Dependence of spectral bandwidth (black dots) and the
FWHM width of HOM interference profile (red dots) of SPDC photons on the
length of the non-linear crystal. The solid black line is theoretical fit to
the experimental data.

To gain further insights, we measured the FWHM width of the HOM interference
dip and estimated the corresponding spectral bandwidth of the photon pairs
using Eq. (4) and (5). The results are shown in Fig. 9(b). It is evident from
Fig. 9(b) that the HOM interference dip has FWHM width (red dots) of 4.01
$\pm$ 0.04, 6.16 $\pm$ 0.05, 13.33 $\pm$ 0.40, 17.32 $\pm$ 0.08 and 21.19
$\pm$ 0.13 $\upmu$m corresponding to the estimated FWHM spectral bandwidth
(black dots) of SPDC photons of 163 $\pm$ 1.68, 106 $\pm$ 0.93, 49 $\pm$ 1.44,
38 $\pm$ 0.20 and 30 $\pm$ 0.18 nm for the crystal lengths of 1, 2, 5, 10, 20,
and 30 mm are in close agreement with the Eq. 5 (solid black lines). Using the
available pump power, we experimentally measured the spectral bandwidth of the
SPDC photons using the spectrometer (HR-4000, Ocean Optics) to be $\sim$30 nm,
same as our previous report [31] and $\sim$37 nm for 30 mm and 20 mm long
crystals, respectively. The relatively low parametric gain has restricted the
experimental measurement of the SPDC spectrum for smaller crystal lengths.
However, it is important to note that the experimentally measured spectral
bandwidth matches the estimated spectral bandwidth as shown in Fig. 9(b).
Interestingly, the decrease of crystal length from 30 mm to 1 mm results in a
decrease (increase) of HOM interference dip width (spectral width of SPDC
photons) by more than five times. As a result, one can see the sharp variation
(see black and red dots of Fig. 9(a)) in the coincidence counts for the small
changes in the optical delay away from the zero-point optical delay for HOM
interference with smaller dip width. Such amplification in the rate of change
of coincidence counts with optical delay, i.e., the increase of sensitivity of
the HOM interferometer with the decrease in crystal length, can be useful for
designing HOM interferometer-based sensor to measure any physical process
influencing the optical delay between the photons.
To gain further perspective, we have selected half of the HOM curve of Fig.
9(a) showing coincidence variation from minimum to maximum with positive
optical delay for crystal lengths, $L$ = 1, 2, 10, 20 and 30 mm. The results
are shown in Fig. 10. Since the maximum coincidence counts per second at a
fixed pump power of the HOM curve directly depend on the pair generation rate
and hence the interaction length of the non-linear crystal used, for
comparative study of the slopes of HOM curves for different crystal lengths,
we have normalized the coincidence counts with its maximum. Using the linear
fit (solid lines) to the experimental data (points), we observe

Figure 10: Crystal length dependent variation in the slope of coincidence
counts of HOM interferometer. Solid lines are linear fit to the experimental
results (dots).

the slope (see Fig. 10), of the HOM curve to decrease with the crystal length
from the maximum of 0.215 /$\upmu$m for $L$ = 1 mm to 0.051 /$\upmu$m for $L$
= 30 mm. In terms of the coincidence counts, the slopes of the HOM curve for 1
mm and 30 mm crystals are found to be 450/$\upmu$m and 260/$\upmu$m,
respectively, at a pump power of 0.25 mw and exposure time at 20 ms. As
expected, due to the high slope or sensitivity of the HOM interferometer, the
range is low for $L$ = 1 mm than for $L$ = 30 mm. However, it is evident that
in order to have access to the high sensitivity of the HOM interferometer for
the measurement of any physical quantity, it is advisable to use a shorter
nonlinear crystal length. However, one has to be careful with the overall
coincidence counts arising from the lower parametric gain due to the small
crystal length. On the other hand, in the present study, we have used a CW
diode laser to implement the system in outdoor applications. However, using an
ultrafast pump laser in combination with the thin crystal for lab-based
experiments, one can further enhance the sensitivity of the HOM interferometer
due to the increase of overall spectral bandwidth (decrease of HOM
interference dip width) of the down-converted photons.

### Calibration of piezo actuator

Figure 11: Linear displacement of the PEA stage as a function of applied
voltage.

After successful characterization and static displacement measurement, we used
a piezo-electric actuator (PEA) (NF15AP25), having a travel range of 25
$\upmu$m for the applied voltage 0 - 75 V to study the performance of the HOM
interferometer for dynamic optical delay or mirror vibration sensing
applications. However, due to the unavailability of closed-loop feedback for
the PEA, we had to calibrate the PEA to estimate the displacement with the
applied voltage. In doing so, we have attached the mirror M2 (see Fig. 1) to
PEA placed over the motorized linear translation stage (MTS25-Z8). Using a
crystal of length 20 mm corresponding to the HOM curve represented by the
green dots in Fig. 9, we have adjusted the initial position of the linear
stage corresponding to a positive delay of $\sim$10 $\upmu$m so that the
coincidence count is 50% of maximum value. We moved the linear stage by
$\sim$0.5 $\upmu$m corresponding to the optical delay of $\sim$1 $\upmu$m
towards the zero-delay point resulting in a change in the coincidence counts.
Such change in the coincidence counts is reset by increasing the applied
voltage to the PEA. Under this condition, the applied voltage to the PEA makes
a displacement corresponding to the displacement of the linear stage in the
opposite direction. We have repeated this exercise in steps of $\sim$0.5
$\upmu$m up to the allowed PEA voltage of 75 V with the results shown in Fig.
11. It is evident from Fig. 11 that the displacement (blue dots) of the PEA is
linear to the applied voltage producing a maximum displacement of 35 $\upmu$m,
more than the manufacturer’s specification (25 $\upmu$m) for the maximum
allowed voltage of 75 V. Using the linear fit (red line) to the experimental
data (black dots) we find the displacement of the PEA to be 0.48 $\pm$ 0.02
$\upmu$m per volt. As expected, the calibration of PEA using the HOM
interferometer based on a 1 mm long crystal (black dots) coincides with the
results obtained using the 20 mm long crystal (blue dots).

Acknowledgments We acknowledge the support of Ms. Shreya Mishra, Physical
Research Laboratory, Ahmedabad, for the useful discussion on statistics.

Conflict of Interest: All authors declare no financial/commercial conflicts of
interest.

Data Availability Statement: The data that support the findings of this study
are available from the corresponding author upon reasonable request.

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