# Precision Measurements of the PMNS Parameters with T2K Data

Ali Ajmi
on behalf of the T2K Collaboration University of Winnipeg, Canada, R3B2E9

###### Abstract

T2K is a long baseline neutrino experiment which exploits a neutrino and
antineutrino beam at JPARC to perform precision measurements of neutrino
oscillation parameters $\Delta{\rm m}^{2}_{\rm 32}$, $\sin^{2}\theta_{23}$
(besides the CP-violating phase $\delta_{\rm CP}$). The latest results for the
measurement of PMNS parameters in the disappearance mode are presented here,
highlighting the main systematic uncertainties limiting the precision. The
future strategy to improve the precision on the measurement of PMNS parameters
are also discussed.

##  I Introduction

The T2K (Tokai to Kamioka) t2k  is a long baseline neutrino experiment,
located in Japan, sending muon-(anti)neutrino beam, produced from 30 GeV
protons at the Japan Proton Accelerator Research Complex (JPARC) at Tokai to
Kamioka, along the 295 km baseline. The 2.5∘off-axis beam peaks around 0.6
GeV. The near detectors for this experiment are located 280m downstream from
the source at JPARC in Tokai, and the water-cherenkov detector, Super-
Kamikande at Kamioka plays the role of the far detector for studying neutrino
oscillations.

The basics of neutrino oscillations, the T2K experiment and the results from
the appearance mode at T2K are described in Joe . The latest results nature ;
pdun  from the disappearance mode are discussed here. The aim is to measure
the value of the parameters $\Delta{\rm m}^{2}_{32}$ and the $\theta_{23}$
with maximum possible precision. This requires one to ensure minimum
systematic uncertainties in the experiment, which is addressed through a
number of contributing factors, like the flux, the neutrino interaction
models, and the constraints from the near detectors, to name a few.

These factors are discussed in the following sections, along with the new
improvements being worked upon at T2K for more precise oscillation analyses in
the near future.

##  II Data for the presented results

The T2K experiment has been taking data since 2010 and has accumulated a total
of 1.99$\times$1021 protons-on-target (POT) in $\nu_{\mu}$ mode and
1.65$\times$1021 POT in $\bar{\nu}_{\mu}$ mode until 2020. The near detector
used 1.15$\times$1021 POT in neutrino mode and 0.83$\times$1021 POT in
antineutrino mode for the presented analysis results. The details of data used
at the far detector are mentioned in Joe . The oscillation analysis procedure
followed at T2K is explained in Joe ; pdun .

##  III Flux-prediction uncertainties

The neutrino flux for the T2K is simulated with the FLUKA simulation package
flux , and then tuned with external data constraints from the NA61/SHINE
hadron production experiment at CERN. Earlier, a thin graphite target was used
for this purspose, but changing it to the target configuration replica of that
of T2K replica , the systematic uncertainties on the flux reduced from 8% to
5%, as shown in Fig. 1. The current analysis uses this uncertainty.

Further improvements can be achieved by using the NA61/SHINE 2010 data shine ,
which adds kaons and protons yields along with increased statistics, reducing
the uncertainties further to $\sim$4%. Results with this will be presented
soon.

Figure 1: Flux prediction uncertainties vs neutrino energy in the neutrino
mode for T2K, at the far detector site. The arrows highlight the reduction in
the uncertainties from the earlier thin target case (dotted line) to the
replica target case (bold line).

##  IV The Near Detector (ND) Constraints

The detector along the axis of the beam, built out of scintillator-iron layers
(INGRID) monitors the incident neutrino beam and its stability. The off-axis
near detector suite (ND280) comprises of a magnet, two fine-grained detectors
(FGD1: scintillators, FGD2: scintillator-water layers) in between three time
projection chambers (TPCs) to act as the tracker, the Electromagnetic
Calorimeter (scintillator-lead), the Pi-Zero detector (scintillator-water
bags) and the Side-Muon Range Detector (scintillator plates). The magnetic
field of 0.2 T helps in charge identification of the particles.

The dominant interaction type at the energy range for T2K, is the charged-
current quasi-elastic reaction (CCQE). Other charged-current interactions are
also present, like mainly the resonant pion production (CCRES), and the deep
inelastic scattering (CCDIS) channels.

The events in the near detector are classified according to the topology based
on the reconstructed pion multiplicity.

###  IV.1 Neutrino Interaction Models:

One of the largest and most complex systematic uncertainty in the
interpretation of the data for the neutrino oscillation measurement is due to
the modelling of neutrino-nucleus interactions. The T2K collaboration
developed a new version of the model of neutrino-nucleus interactions based on
the Spectral Function approach and with refined tuning of the nucleons removal
energy.

The T2K Collaboration has been improving the NEUT neutrino event generator
iteratively with every analysis. Significant updates have been applied on the
recent NEUT 5.4.0 model neut . A tuned Benhar Spectral Functiforon benhar  to
describe CCQE interactions is being used now instead of the earlier used
Relativistic Fermi Gas Model with Random Phase Approximations. The Shell model
is built largely from the electron-scattering data elecdata , with the nuclear
ground states better defined and the outgoing nucleon kinematics better
predicted. This model is being used for the current results.

Further modifications are underway, one of which is the $|q_{3}|$-dependent
removal energy treatment from comparing NEUT to electron scattering data
(where, q3 is the three-momentum transfer in nuclear models). All these
modifications will make NEUT a more robust neutrino interaction model and
amount to reduced uncertainties from the interaction modelling.

###  IV.2 Events Selections

Events at the ND280 are selected with mainly three types of topologies: the
charged current no-pion (CC0pi) events, the CC1pi (events containing only one
pion track besides the lepton) and the CC-others (events containing multiple
tracks), which are enriched in the CCQE, CCRES, CCDIS channels respectively. A
total of 18 such samples (FGD1, FGD2, both modes and wrong-sign component for
the anti-neutrino mode) are used for this analysis, with larger data
statistics in comparison to earlier results.

Further improvements in the selection processes are being made for future
analyses, by implementing proton/gamma tagging methods kamil . The newer
criteria are based on the energy and charge depositions, the likelihood ratios
and the electro-magnetic shower topologies, all resulting to an increase in
the purity of the samples by 5-10%. Newer samples with multi-ring topologies
at SuperK are also being worked upon lsm .

###  IV.3 Near Detector Fits

An extended binned likelihood fit to the ND sample as a function of muon
kinematics is made to constrain the predicted number of events. The resulting
post-fit ND events distributions match the observed data well, with a prior
model p-value of 74%. The fitting introduces anti-correlation between the flux
and the cross section model parameters.

The flux prediction of the muon neutrinos at the Far-Detector (FD) is made
thereafter, and the ND-fit reduces the uncertainty to a great extent. The
systematic uncertainties at SK are hence 3.0 (4.0)% in $\nu$-mode
($\bar{\nu}$-mode), as can be seen in Fig.2. The uncertainties in the
different samples are listed in Table 1.

Figure 2: Predicted flux at the FD site with the systematic uncertainties shown before and after the ND fits, for the Forward Horn Current (FHC) or the neutrino mode.  Table 1: Systematic Uncertainties at the far detector before and after using flux, cross section model, ND constraints. (d.e. stands for ‘decay electron’)  SuperK Sample | Pre-ND fit error | Post-ND fit error
---|---|---
$\nu_{\mu}$ 1Rμ | 11.1% | 3.0%
$\bar{\nu}_{\mu}$ 1Rμ | 11.3% | 4.0%
$\nu_{e}$ 1Re | 13.0% | 4.7%
$\bar{\nu}_{e}$ 1Re | 12.1% | 4.9%
$\nu_{e}$ 1Re 1 d.e. | 18.7% | 14.3%

##  V Results presented

The oscillation analysis was done using five SK samples for T2K Run 1-10, as
elaborated in Joe . The upper octant is slightly preferred with a probability
of 77.1%. Similar results are also noticed by the Frequentist approach, as
shown in Fig. 3. The ND-fits and the FD-fits are done consecutively in the
Frequentist approach, while the ND and the FD fits are done simultaneously in
the Bayesian approach. The normal ordering of the neutrino mass hierarchy is
preferred at 80.8% probability, and is well in agreement with the Frequentists
confidence intervals. The Bayesian results are listed in Table 2 and 3.

Figure 3: Best fit results from T2K (reactor constraints included) for the $\Delta{\rm m}^{2}_{\rm 32}$ and $\sin^{2}\theta_{23}$.  Table 2: Best fit results from T2K (with and without reactor constraints by the Bayesian method) for the $\Delta{\rm m}^{2}_{\rm 32}$.  | Hierarchy |  Most Probable Value | Range
---|---|---|---
|  |  (in 10-3 eV2) | (in 10-3 eV2)
T2K only | Normal |  2.487 | [2.437, 2.537]
| Inverted |  2.457 | [2.407, 2.507]
T2K + reactor | Normal |  2.485 | [2.436, 2.536]
| Inverted |  2.457 | [2.406, 2.506]
Table 3: Best fit results from T2K (with and without reactor constraints by the Bayesian method) for the $\sin^{2}\theta_{23}$.  | Hierarchy |  Most Probable Value |  Range
---|---|---|---
T2K only | Normal |  0.471 |  [0.452, 0.508] and [0.530, 0.568]
| Inverted |  0.469 |  [0.449, 0.508] and [0.531, 0.565]
| Both |  0.471 |  [0.451, 0.508] and [0.530, 0.567]
T2K + reactor | Normal |  0.559 |  [0.504, 0.583]
| Inverted |  0.560 |  [0.519, 0.585]
| Both |  0.559 |  [0.507, 0.584]

##  VI T2K Upgrade

The T2K beam will be upgraded with increased power of $\sim$750kW in 2022 and
an upgraded version of the ND280 near detector is being assembled to exploit
the increased statistics beamupgrade ; t2kupgrade .

Two of the beam’s magnetic focussing horns are being replaced at JPARC, the
horn power supply is being upgraded to enable faster beam repetition rate, and
efforts are going on to improve the cooling capability of the beam target.

Moreover in 2020, the Super-Kamiokande detector was loaded with 0.01% of
Gadolinium skupgrade , enhancing its neutron tagging capabilities. The T2K
Run-11 data was taken with this configuration, the analysis of which is
currently going on.

The near detector is undergoing a major upgrade. A Super-FGD comprising of
$\sim$2 million scintillator cubes, and High Angle TPCs are being added for
more phase space acceptance, and improved resolution in the particle
kinematics, thus increasing the detection efficiency of protons, neutrons,
lower momentum pions, and also aid in tagging decay electrons.

##  VII Summary

Latest precise measurements of the neutrino oscillation parameters from the
T2K experiment, using 3.8$\times 10^{21}$ POT data have been presented here
for the disappearance mode. A slight preference for Non-maximal mixing is
observed. The data shows a preference for the upper octant for $\theta_{23}$
with a 77.1% probability. The normal ordering of the neutrino mass hierarchy
is also preferred by the observed data with 80.8% probability.

However, several efforts are being made to achieve improved sample selections,
reduced predicted flux uncertainties, and more robust cross section models.
These will result into more precise values of the measured oscillation
parameters which will be presented to the physics community very soon.

## References

  * (1) K. Abe et al. (T2K), Nucl. Inst. Meth. A 659, 106 (2011).
  * (2) Previous talk by Dr. Joseph Walsh in this conference (“Wed21415”).
  * (3) K. Abe et al. (T2K), Nature 580, 339-344 (2020).
  * (4) P. Dunne, Latest neutrino oscillation results from T2K, NEUTRINO 2020, DOI: 10.5281/zenodo.4154355.
  * (5) K. Abe et al. (T2K), Phys. Rev. D 87, 012001 (2013).
  * (6) N. Abgrall et al. (NA61/SHINE), Eur. Phys. J.C 76, 617 (2016).
  * (7) N. Abgrall et al. (NA61/SHINE), Eur. Phys. J.C 79 (2019) 2, 100.
  * (8) Y. Hayato and L. Pickering, The NEUT Neutrino Interaction Simulation, arXiv:2106.15809 [hep-ph].
  * (9) O. Benhar, A. Fabrocini, S. Fantoni, G.A. Miller, V.R. Pandharipande, I. Sick. Phys. Rev. C, 44 (1991), p. 2328.
  * (10) O. Benhar et al., An archive for quasi-elastic electron-nucleus scattering data, arXiv:nucl-ex/0603032v1.
  * (11) K. Skwarczynski, Improving T2K near detector fit by adding proton information, poster at WIN 2021.
  * (12) L. Mohan, Multi-ring charged current $\nu$1$\pi^{+}$ samples at T2K far detector, poster at WIN 2021.
  * (13) K. Abe et al. J-PARC Neutrino Beamline Upgrade Technical Design Report. 8 2019.
  * (14) K. Abe et al. T2K ND280 Upgrade - Technical Design Report. 1 2019
  * (15) K. Abe et al. First gadolinium loading to Super-Kamiokande. Nucl. Instrum. Meth. A, 1027:166248, 2022


# Quantum Reservoir Computing Implementations for Classical and Quantum
Problems

Adam Burgess<EMAIL_ADDRESS>Leverhulme Quantum Biology Doctoral
Training Centre, University of Surrey, Guildford, GU2 7XH, United Kingdom
Advanced Technology Institute, University of Surrey, Guildford, GU2 7XH,
United Kingdom  Department of Physics, University of Surrey, Guildford, GU2
7XH, United Kingdom  Marian Florescu<EMAIL_ADDRESS>Advanced
Technology Institute, University of Surrey, Guildford, GU2 7XH, United Kingdom
Department of Physics, University of Surrey, Guildford, GU2 7XH, United
Kingdom

###### Abstract

In this article we employ a model open quantum system consisting of two-level
atomic systems coupled to Lorentzian photonic cavities, as an instantiation of
a quantum physical reservoir computer. We then deployed the quantum reservoir
computing approach to an archetypal machine learning problem of image
recognition. We contrast the effectiveness of the quantum physical reservoir
computer against a conventional approach using neural network of the similar
architecture with the quantum physical reservoir computer layer removed.
Remarkably, as the data set size is increased the quantum physical reservoir
computer quickly starts out perform the conventional neural network.
Furthermore, quantum physical reservoir computer provides superior
effectiveness against number of training epochs at a set data set size and
outperformed the neural network approach at every epoch number sampled.
Finally, we have deployed the quantum physical reservoir computer approach to
explore the quantum problem associated with the dynamics of open quantum
systems in which an atomic system ensemble interacts with a structured
photonic reservoir associated with a photonic band gap material. Our results
demonstrate that the quantum physical reservoir computer is equally effective
in generating useful representations for quantum problems, even with limited
training data size.

††preprint: APS/123-QED

##  I Introduction

Open quantum many-body problems are notoriously challenging. Most often, the
approaches used to deal with them employ various approximations whose
applicability is typically limited to weak coupling or rapid relaxation of the
environment [4]. Recently, the use of neural network approaches to represent
and analyze quantum many-body states has emerged as a new area of research
with great potential[19]. Previous studies have demonstrated the general
efficacy of neural networks in capturing quantum mechanical aspects [34, 19,
18] and neural networks have been deployed to deal with quantum many-body
problems [7] and in designing quantum experiments[25]. In particular,
recurrent neural networks (RNN) have been shown to be particularly efficient
in solving open quantum systems problems wherein a quantum mechanical system
of interest is coupled to a large environment [1, 5]. This is due to the
internal structures of a RNN and the associated internal temporal dynamic
behaviour [9] . Furthermore, the RNN architectures are ideally suited for
dealing with non-Markovian or memory effects in the quantum dynamics of open
systems as they can locally store relevant information about previous quantum
states and capture their influence on the temporal evolution of the system. A
schematic for this is depicted in Fig. 1(a). Recently, a new paradigm for
information processing was devised based on exploiting a physical system’s
dynamics to perform effective computations. This concept has been termed
‘physical reservoir computing’ [32]. Generally speaking, reservoir computing
represents a scheme for training RNNs that has been around since the turn of
the millennia [29]. Contrary to the typical back-propagation approach to
updating weights inside a neural network, reservoir computing allows for the
data structure to be passed through a large ‘reservoir’ containing many non-
linear components and then passed into a neural network of reduced size.
Weight calibration is then performed on the smaller network placed after the
reservoir, massively reducing the number of computations required for a
typical back-propagation approach. Conceptually, this is extremely important
as many highly successful machine learning projects, such as natural language
processing [17] and systems like AlphaGo [38], require extremely powerful
high-performance computers and distributed systems to perform their
calculations. Devising a scheme that can reduce computational overhead is not
only preferable but necessary[33]. Furthermore, because the reservoir weights
are not updated, the reservoir can be used to multitask feeding the reservoir
outputs into multiple independent neural networks[41]. It is then possible to
extend this reservoir computing concept to a physical reservoir[39]. For a
large physical system with many internal degrees of freedom that are strongly
interacting, we can have a significant degree of non-linearity in the
dynamics[8, 40, 27], satisfying one of the requirements of a reservoir
computer. If the physical reservoir considered is a many-body quantum system,
the subsystem entanglement (the nontrivial correlations encoded in the many-
body wave function) provides access to many internal degrees of freedom. This
makes quantum physical reservoir computers (QPRCs) a desirable prospect. On
the other hand, we need them to interact with their environment to perform
readouts on quantum mechanical systems. As such they are no longer isolated
systems and we need to deploy a open quantum system framework to describe
QPRCs, as depicted in Fig. 1(b). We note that machine learning approaches
utilizing RNNs have already been used to explore the physics of open quantum
systems [1, 5], quantum-classical generative algorithms for image generation
[36] using ion trap quantum computers, and quantum reservoir computing using
arrays of Rydberg atoms [3].

In this paper, we exploit the dynamics of an open quantum system as a generic
quantum physical reservoir to construct a reservoir computing framework. In
Section II, we introduce the physical platform for implementing a QRPC. In
Section III, we utilize the QPRC to study its effectiveness when applied to a
typical pattern recognition problem in machine learning. Finally, in Section
IV we deploy the QPRC approach to study an archetypal open quantum mechanical
problem: a quantum emitter coupled to a structured photonic reservoir.

(a)

(b)

Figure 1: (a) Schematic of a recurrent neural network, showing the input to
the output pathways including GRU layers [9] (a type of RNN layer) .
Information is passed vertically through layers (blue arrows) and horizontally
(orange arrows) between GRU timesteps to embed memory of previous timesteps.
In such an RNN, many internal weights of the GRU layer need to be updated by
back-propagation, making it an effective but computationally expensive
architecture. (b) Schematic of the Quantum Physical Reservoir Computing
network, an input is fed into the open quantum system and the time series of
the quantum state evolution is fed into the dense layer at the end of the
network. In this case, only the dense layer weights are updated.

##  II Quantum Physical Reservoir System

A prerequisite of an effective physical reservoir computer is that the
reservoir behaves predictably and produces the same output for a given input.
A dissipative system, such as an atomic system embedded in a lossy cavity,
will tend towards an equilibrium state if the system’s evolution is well
controlled. Then by reducing the temperature, this equilibrium state will
roughly coincide with the ground state of the atomic systems and the local
reservoir. Laser drives can be used to generate effective initial conditions
from a given input data set, ensuring that the system acts predictably[13, 30,
37]. In order to effectively deploy our QPRC approach, we need to identify a
large open quantum system model that can be sampled effectively. The model
system we are focusing on is many-body atomic systems placed inside of a
single lossy photonic cavity [6]. The dynamics of the composite system is
governed by the Hamiltonian:

$\displaystyle H=$
$\displaystyle\sum_{i=1}^{N}\omega_{0}\sigma_{i}^{+}\sigma_{i}^{-}+\sum_{\lambda}\omega_{\lambda}a^{\dagger}_{\lambda}a_{\lambda}+$
$\displaystyle
i\sum_{i,\lambda}g_{\lambda}(a_{\lambda}\sigma^{+}_{i}-a^{\dagger}_{\lambda}\sigma^{-}_{i})+\sum_{i\neq
j}\Lambda\sigma^{+}_{i}\sigma_{j}^{-}.$ (1)

where $\sigma_{i}^{+}$and $\sigma_{i}^{-}$ are the excitation and de-
excitation operators for the $i^{\textrm{th}}$ atomic system respectively,
$a_{\lambda}$ and $a^{\dagger}_{\lambda}$ are the bosonic field annihilation
and creation operators, $\omega_{0}$ and $\omega_{\lambda}$ are the atomic
transition and the $\lambda$-boson mode frequencies, $g_{\lambda}$ is the
coupling strength of the atomic system and the $\lambda$-boson mode and
$\Lambda$ is the dipole-dipole coupling strength between atoms [35, 43, 22].
This is equivalent to a Dicke model [12] for $N$-atoms with an inter-atom
coupling. We have chosen a highly symmetric atomic system to generate time-
resolved data efficiently. To this extent, we also limit our investigation to
the single-excitation regime. We also assume an initial condition given by

$\phi(0)=c_{0}\psi_{0}+\sum_{i}^{N}c_{i}(0)\psi_{i}+\sum_{\lambda}c_{\lambda}(0)\psi_{\lambda},$
(2)

where $\psi_{i}=\ket{i}_{A}\ket{0}_{B}$ is the state wherein the
$i^{\textrm{th}}$atom is in its excited state and all other atoms and the
photonic cavity are in the ground state.
$\psi_{\lambda}=\ket{0}_{A}\ket{\lambda}_{B}$ represents all atoms in their
ground state, and the cavity system has its $\lambda$ mode excited.
$\psi_{0}=\ket{0}_{A}\ket{0}_{B}$ denotes the ground state of the entire
system.

Then the time evolution of this initial state is given by

$\phi(t)=c_{0}\psi_{0}+\sum_{i}^{N}c_{i}(t)\psi_{i}+\sum_{\lambda}c_{\lambda}(t)\psi_{\lambda}.$
(3)

Using an adiabatic cancellation of the photonic mode variables $c_{\lambda}$
and introducing the memory kernel

$G(t)=\sum_{\lambda}g_{\lambda}^{2}e^{i(\omega_{0}-\omega_{\lambda})t},$ (4)

the equations of motion for the atomic degrees of freedom can be expressed as

$\displaystyle\dot{c}_{i}=$
$\displaystyle-i\Lambda(c_{+}-c_{i})-\int^{t}_{0}G(t-t_{1})c_{+}(t_{1})dt_{1},$
(5)

where $c_{+}=\sum_{i}^{N}c_{i}$. The non-Markovian or memory effects in the
system’s dynamics are associated with this convolution over the memory kernel
at previous times. This is precisely the aspect of our QPRC implementation
that we intend to leverage.

The time evolution of this system is then determined by the distribution of
the coupling strengths given by the spectral density

$J(\omega)=\sum g_{\lambda}^{2}\delta(\omega-\omega_{\lambda}),$ (6)

which is related to the memory kernel by the simple relation

$G(t)=\int d\omega J(\omega)e^{i(\omega_{0}-\omega)t}.$ (7)

For a lossy photonic cavity on-resonance with the atomic transitions, it is
common to model the spectral density by a Lorentzian distribution of the form

$J(\omega)=\frac{\lambda\gamma}{\gamma^{2}+(\omega-\omega_{0})^{2}}.$ (8)

$\lambda$ is the coupling strength of the atoms to the cavity mode, and
$\gamma$ determines the spectral width of the lossy cavity [28, 2, 26].

As a result, the time evolution of the excited state amplitudes for the
$i^{\textrm{th}}$atom is given by

$\displaystyle c_{i}(t)$
$\displaystyle=(c_{i}(0)-\frac{c_{+}(0)}{N})e^{i\Lambda t}$
$\displaystyle+\frac{c_{+}(0)}{N}e^{-\mu^{*}t}\left(\cosh(\Gamma
t)+\frac{\mu}{\Gamma}\sinh(\Gamma t)\right),$ (9)

where we have introduced the following for convenience the following notations

$\displaystyle\Lambda_{FC}$ $\displaystyle=J(N-1)$ $\displaystyle\Gamma$
$\displaystyle=\frac{1}{2}\sqrt{\gamma^{2}-2i\gamma\Lambda_{FC}-\Lambda_{FC}^{2}-4\lambda
N}$ $\displaystyle\mu$
$\displaystyle=\frac{1}{2}\big{(}\gamma-i\Lambda_{FC}\big{)}.$ (10)

In order to utilize this physical system as a QPRC layer, we require a
procedure for inputting data into the QPRC layer and subsequently outputting
it into the neural network. For a stream of input data given by $\mathcal{I}$,
we need to define a bijection $f$ that maps $\mathcal{I}$ to the multi-valued
ensemble of values $c=\\{c_{i}\\}$. Then we require an map $g$ that transfers
$\phi(t)$ to $\mathcal{O}$, the output of the QPRC layer, which is then passed
directly into the conventional neural network layers. The particular map $g$
we utilize in the following is the real and imaginary components of the
excited state amplitudes at various timesteps $\\{t_{j}\\}$,
$g:\phi(t)\rightarrow\\{\Re{c_{i}(t_{j})},\Im{c_{i}(t_{j})}\\}$ akin to
performing non-demolition measurements on the system or quantum state
tomography[10, 16] .

##  III Quantum Physical Reservoir for Image Recognition

(a)

(b)

(c)

(d)

Figure 2: (a) Schematic of the Quantum Physical Reservoir network used to
implement image recognition; pixel data is converted into an initial condition
for the open quantum system, and the time series of the quantum state is fed
into the dense layer at the end of the network. Only dense layer weights are
updated. (b-d) The accuracy, as a fraction of correct predictions, of the QPRC
approach (blue) and the conventional neural network without the QPRC layer
(orange) for (b,c) varying sizes of training data (c) varying numbers of
training epochs.

Having defined the dynamical process of our QPRC framework, we deploy it to
solve an archetypal problem in machine learning, and pattern recognition using
the standard MNIST database[11]. The MNIST database is a collection of
monochrome images of handwritten numbers from 0-9 and the task is to classify
what number each image represents successfully. In order to effectively deploy
our reservoir computing approach, we need to identify an isometry between the
pixel data for each image and an initial condition for the state of our
system. To this end, we use the pixel brightness of the $2n$ and $2n+1$ pixel
and map it directly to the real and imaginary component of $c_{n}(0)$ and then
normalize the wavefunction $\phi(0)$.

Mathematically, for an image with $i^{\textrm{th}}$pixel brightness given by
$P_{i}$ we have the mapping $c_{i}=(P_{i}+iP_{i+1})/\mathcal{N}$, where
$\mathcal{N}$ is a normalisation constant. Then the wavefunction is time-
evolved (see per Eqn. 9) and sampled at $n=50$ equal time steps up to
$n\cdot\omega_{0}^{-1}$. The time series’ real and imaginary components are
then passed into a reduced neural network. The reduced neural network consists
of a single dense layer of dimension $n_{\textrm{nn}}=128$. A pictorial
schematic of this procedure is given in Fig.2(a). The elegance of this
approach is that we need only perform the gradient descent on the small final
dense layer of the network, reducing the most complex step of readjusting
network weights. With this procedure developed, we performed two numerical
experiments. First, how does the QPRC network compare to a neural network
without the QPRC layer for varying sizes of training data? The results of this
experiment are presented in Fig. 2(b) and (c). The QPRC neural network
outperforms the conventional neural network for intermediate and large values
of the training data, a regime in which we approach $>90\%$ accuracy in our
prediction capabilities. Furthermore, we can identify a crossover point at
which the QPRC begins to outperform the standard neural network at around 900
training sets. This suggests that the QPRC approach may outperform
conventional approaches for large data sets. This is a remarkable result
since. With the ever-increasing relevance of big-data and the massively
growing nature of computational costs associated with extremely large neural
networks used in natural language processing systems, it becomes more and more
necessary to be find more efficient ways to handle large data sets.
Furthermore, our results suggest that the QPRC neural network generates a more
effective representation of the data set being studied. To demonstrate this,
we compared the QPRC network performance against a conventional neural network
for varying numbers of training epochs. The results presented in Fig.2(d)
clearly show that the QPRC system outperforms the non-QPRC network at all
values of the training epoch sampled. All numerical experiments shown in Fig.
2 have been repeated five times at each sampling point, and the errors on the
graph are the standard deviations from these five samples. In the experiments,
we used a single dense layer of dimension 128, with a $\tanh$ activation
function; the Adam optimizer was used with a learning rate of 0.001 and 30
training epochs.

The QPRC platform introduced here can further be optimized to increase its
performance and accuracy. Here we could exploit the correlations in the data
set to guide the QPRC system choice. Due to pixel data displaying strong
nearest-neighbour correlations, a nearest neighbour type of interaction in the
quantum system would increase the approach’s efficacy. Such a system may
consist of a 2D lattice of atomic systems embedded in an array of photonic
cavities with controllable cavity-cavity positioning. This architecture, this
2D lattice geometry, would lead to a nearest neighbour interaction between the
atomic systems and thus generate commensurate nearest neighbour correlations
to the pixel data.

##  IV Quantum Physical Reservoir Computing Approach for Open Quantum Systems

(a)

(b)

(c)

(d)

(e)

Figure 3: (a) Schematic of the Quantum Physical Reservoir network used to
predict the dynamics of atoms embedded in a photonic band gap material. (b)
Population dynamics of a single atom coupled to another atom inside of a
photonic crystal and inside of a photonic cavity. The photonic crystal allows
for non-zero steady state populations due to strong coupling to the band edge
mode. (c) The prediction of the QPRC approach (dotted) and the real dynamics
(dashed) of two coupled atoms inside of a photonic band gap material for
randomly sampled initial conditions. (d,e) Heatmap to show the average error
of the QPRC for varying values of both the reservoir and objective system’s
spectral width $\gamma$. The heatmap depicts average $\log_{10}$ of the errors
for QPRCs for two simulated run times (d) $10\omega_{0}^{-1}$ (e)
$100\omega_{0}^{-1}$.

In the previous section, we demonstrated the QPRC approach’s usefulness in
enhancing conventional neural networks’ performance in pattern recognition
tasks. Here, we focus on extending this approach to deal with quantum
mechanical problems. Specifically, we study the efficacy of the QPRC approach
in understanding the dynamics of open quantum systems schematically presented
in Fig. 3(a). It is often impossible to describe the dynamics of the combined
quantum system of interest and environment in open quantum systems. Various
approximations are deployed to model the influence of the environment’s
degrees of freedom on the system of interest evolution. However, these
approximations also have some limitations. A standard approximation in the
field is the Markov approximation; wherein there is a unidirectional
dissipation of information from the system into the environment. This
approximation is only valid when the environment relaxes on timescales much
faster than the systems such that the system barely perturbs the environment,
akin to a weak coupling limit assumption.

This method has been successful in analyzing many systems, however it falls
short in capturing quantum-induced memory effects. For instance, in highly
structure photonic systems, such as photonic crystals, the local density of
states for the electromagnetic field undergo rapid fluctuation quickly for
frequencies close to the band edges of the photonic band gap. When a two-level
atomic system is embedded with its transition energy close to the band edge of
a photonic bandgap, the Markov approximation is rendered invalid due to the
strong fluctuations [5, 15, 20], and extremely non-Markovian dynamics emerge.
As a result of the strong interaction between the atomic system and the
photonic reservoir, the atomic states become highly intertwined with the
properties of the photonic reservoir and the degrees of freedom of the atoms.
This results in temporal oscillations, fractional atomic population decay,
spectral splitting, and sub-natural line-widths of atomic transitions [21].

Furthermore, it has been demonstrated that the coherence lifetime of multi-
qubit systems may be increased by effectively characterising the non-Markovian
noise [42]. This emphasises the necessity of learning more about how many-
bodied systems interact with non-Markovian environments. Understanding these
control processes may help design and create more reliable and performant
quantum technologies, including quantum clocks [24] and quantum networks [23].
Beyond the preservation of entanglement, another challenge faced by quantum
technology is the creation of such entanglement. An intriguing recent
discovery demonstrates that it is viable to use the environment’s non-
Markovianity to create entanglement between atomic systems that were
previously uncorrelated [31, 14].

Here we explore the efficacy of the lossy cavity atomic ensemble QPRC in
capturing the dynamics of a strongly non-Markovian system, in which a two-
level atomic ensemble coupled to a photonic reservoir associated with a
photonic band gap material. We have used the isotropic one-dimensional
photonic crystal band edge model for our photonic crystal. Such systems can be
constructed using a photonic crystal waveguide. This fascinating system
generates strong atom-photon coupling. This is due to a divergence in the
local density of states of the electromagnetic field modes around the band
edge frequency $\omega_{I}$
($\rho(\omega)\propto(\omega-\omega_{I})^{-\frac{1}{2}}$). The Hamiltonian for
this system is as in Eqn.1, leading to the convolution of states given by
Eqn.5. However, due to the divergence in the density of states around the
photonic band edge, the memory kernel takes the form

$\tilde{G}(s)=\beta^{3/2}e^{-i\pi/4}(s-i(\omega-\omega_{I}))^{-\frac{1}{2}},$
(11)

where $\beta$ is the coupling strength. This divergence in the memory kernel
leads to strong non-Markovian effects in the system generating a plethora of
new features such as fractional decay of the atomic systems and photon-atom
bound states [21, 20, 15]. In Fig. 3(c), we present the dynamics of a coupled
pair of atoms inside a Lorentzian cavity and a photonic band gap material. We
note that these two systems undergo very different dynamics. In the single
atom case, the lossy cavity will dissipate all of the energy in the excited
state at the steady state. In contrast, the atom placed inside the photonic
band gap materials displays strongly non-Markovian dynamics. Due to the
splitting of the energetic states by strong coupling to the lower photonic
band edge some of the excitation is maintained even to the steady state giving
rise to the fractional decay. Next, we train the QPRC network to model the
non-Markovian dynamics of the atoms in the photonic band gap system. To
achieve this, we initialised our QPRC system with the same initial conditions
as the photonic band gap system we wanted to model. That is, our input $c(0)$
is the same for both the QPRC and the photonic band gap system. The output of
the QPRC is then fed into a small neural network of 3 dense layers, which are
subsequently updated via standard back-propagation. The results presented in
Fig. 3(b) demonstrate the ability of two-atom QPRC to predict the dynamics of
two atoms bound within the photonic band gap. Even for a relatively small
sample size of 1500 data-sets and despite the distinctive dynamics of the two
systems, the QPRC framework is very accurate in predicting the complex
dynamics of the system. Next, we tested the efficacy of cavities of different
spectral widths against one another; here we have the same setup as in the
previous experiment. To test multiple configurations of the cavity systems, we
varied the spectral width $\gamma$ of both the QPRC system and the target
system we wanted to model. The results shown in Figs.3(d) and (e) demonstrate
that the QPRC system is highly effective at capturing the dynamics of similar
systems, showing average errors of $10^{-7}$ across the data sets.

Furthermore, we extended the run time of the reservoir from
$10\omega_{0}^{-1}$ to $100\omega_{0}^{-1}$ (see Fig. 3(d)), and remarkably we
find that efficacy of the reservoir increases greatly now with average errors
in the range of $10^{-8}$. The QPRC approach is not only simply maintaining a
representation of the input state but is performing useful computation due to
the expansion of the input into a higher dimensional computational space
brought about by its intricate non-Markovian dynamics.

##  V Conclusion

In summary, we have introduced a QPRC system consisting of two-level atomic
systems coupled to Lorentzian photonic cavities as a model open quantum
system. We have applied the QPRC approach to a classic machine learning
challenge of recognizing images from the MNIST database. The efficacy of the
QPRC was compared to a neural network with the same design but without a QPRC
layer. Remarkably, our results revealed that the QPRC is underperforming
against the traditional neural network for relatively small training data sets
but rapidly starts to outperform it for larger data set sizes. This result has
significant implications since modern neural networks are quickly bottlenecked
by the size of the data they can effectively model. Furthermore, we used a
fixed data set size to assess the efficacy versus the number of training
epochs and demonstrated that the QPRC outperformed the traditional neural
network approach at each epoch number examined. Finally, we have demonstrated
the efficiency of the QPRC approach in modelling the dynamics of atomic
systems coupled to cavities of various quality, atomic systems in photonic
band gap materials, and open quantum systems challenges. The QPRC was very
efficient in producing accurate representations of capturing the complexities
of the quantum dynamics of these systems, even with a small amount of training
data. The physical platform introduced here for QPRC paves the way for a new
approach to scalable quantum machine learning, vastly outperforming
conventional recurrent neural network approaches.

###### Acknowledgements.

This work was supported by the Leverhulme Quantum Biology Doctoral Training
Centre at the University of Surrey were funded by a Leverhulme Trust training
centre grant number DS-2017-079, and the EPSRC (United Kingdom) Strategic
Equipment Grant No. EP/L02263X/1 (EP/M008576/1) and EPSRC (United Kingdom)
Grant EP/M027791/1 awards to M.F. We acknowledge helpful discussions with the
members of the Leverhulme Quantum Biology Doctoral Training Centre.

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#  $B_{\rm 1g}$ phonon anomaly driven by Fermi surface instability at
intermediate temperature in YBa2Cu3O7-δ

Dongjin Oh  Dongjoon Song  Younsik Kim  Center for Correlated Electron
Systems, Institute for Basic Science, Seoul, 08826, Korea  Department of
Physics and Astronomy, Seoul National University, Seoul, 08826, Korea  Shigeki
Miyasaka  Setsuko Tajima  Department of Physics, Osaka University, Osaka
560-0043, Japan  Yunkyu Bang  Asia Pacific Center for Theoretical Physics,
Pohang 37673, Korea  Seung Ryong Park<EMAIL_ADDRESS>Department of
Physics, Incheon National University, Incheon 22012, Republic of Korea
Changyoung Kim<EMAIL_ADDRESS>Center for Correlated Electron Systems,
Institute for Basic Science, Seoul, 08826, Korea  Department of Physics and
Astronomy, Seoul National University, Seoul, 08826, Korea

###### Abstract

We performed temperature- and doping-dependent high-resolution Raman
spectroscopy experiments on YBa2Cu3O7-δ to study $B$1g phonons. The
temperature dependence of the real part of the phonon self-energy shows a
distinct kink at $T=T_{\rm B1g}$ above $T$c due to softening, in addition to
the one due to the onset of the superconductivity. $T$B1g is clearly different
from the pseudogap temperature with a maximum in the underdoped region. The
region between $T$B1g and $T$c resembles that of superconducting fluctuation
or charge density wave order. While the true origin of the $B$1g phonon
softening is not known, we can attribute it to a gap on the Fermi surface due
to an electronic order. Our results may reveal the role of the $B$1g phonon
not only in the superconducting state but also in the intertwined orders in
multilayer copper oxide high-$T$c superconductors.

In strongly correlated electron systems, instabilities with complex order
parameters are often accompanied by electronic orders that cause various
broken symmetries and Fermi surface instabilities in the ground state. The
instabilities generically lead to an energy gap at the Fermi surface to lower
the energy of the system. The most representative cases are Cooper pair and
Peierls instabilities which are associated with superconductivity and charge
density wave (CDW), respectively. In the former case, the superconducting
ground state is created through coherent superposition of Cooper pairs and the
superconducting gap is opened in the presence of a weak attractive interaction
between electrons. Similarly in the latter case, coherent electron-hole pairs
create CDW gap Gruner (2018); Li and Comin (2019).

The gap opening driven by an electronic instability affects not only the
electronic but also phononic excitation spectra. For example, it is well known
that the $B$1g oxygen bond-buckling phonon in copper oxide high-$T$c
superconductors shows a drastic softening when the temperature is lowered
below $T$c Altendorf  _et al._ (1993); Limonov  _et al._ (2000); Hewitt  _et
al._ (2004); Bakr  _et al._ (2009). The origin of the phonon anomaly was
understood to be from a phonon self-energy effect caused by the opening of the
superconducting gap (2$\Delta$); the phonon peak in the Raman spectrum
exhibits narrowing due to the reduced scattering rate upon gap opening. In
contrast, phonons with $\omega$$>$2$\Delta$ show broadening since, for such
phonons, scattering channels are available and their scattering rate increases
due to the increased quasiparticle density of states (DOS). In this way, we
can indirectly infer the existence of a superconducting gap. In addition to
the superconductivity, this explanation can be also applied to other
electronic orders such as CDW which gives rise to an energy gap Raichle  _et
al._ (2011).

Among the various phonon modes in copper oxide superconductors, only the $B$1g
phonon shows a drastic renormalization in the Raman spectra as we mentioned
above. This result drew much interest since it may reflect the intimate link
between superconductivity and $B$1g phonon. Indeed, the coupling matrix
between electrons and $B$1g phonon at the Brillouin zone center ($q$=0)
possesses a $d$-wave nature, which is the symmetry of the superconductivity as
well as CDW form factor in cuprates Devereaux  _et al._ (1995); Opel  _et al._
(1999); Fujita  _et al._ (2014); Comin  _et al._ (2015); Hamidian  _et al._
(2016). Such character allows the $B$1g phonon to strongly couple with the
electrons near the antinode where the superconducting gap is the maximum. As a
result, a strong renormalization in the electronic structure appears in the
form of kink or peak-dip-hump feature near the anti-nodal region Devereaux
_et al._ (2004); Cuk  _et al._ (2004). A recent theoretical work pointed out
that the $B$1g phonon may induce charge order in underdoped copper oxide
superconductors Banerjee  _et al._ (2020a, b).

In spite of the predicted close relationship between the $B$1g phonon and
electronic orders, a direct evidence for the relationship has not been
observed. Discovery of such evidence may allow us to unify the understanding
of the novel multiple orders in the phase diagram of high $T$c materials. In
this Letter, we present results of comprehensive doping- and temperature-
dependent high-resolution Raman spectroscopy studies on YBa2Cu3O7-δ. Owing to
the very high-statistics data, our results clearly show that phonon softening
occurs not only in the superconducting state, but also at a temperature higher
than $T$c which has not yet been reported. The softening is robust in the
underdoped region but is not observed for the most overdoped sample. The onset
temperature of the phonon anomaly, $T$B1g, is distinguished from the pseudogap
temperature $T$∗. It is rather close to the CDW temperature $T$CDW Blanco-
Canosa  _et al._ (2014) or superconducting fluctuation temperature $T$pair
Rullier-Albenque  _et al._ (2011); Li  _et al._ (2010); Tallon  _et al._
(2011). Even though our experimental results cannot point to the origin of the
phonon anomaly, it is clear that a Fermi surface instability of an electronic
order causes the softening of the $B$1g phonon. Our results not only highlight
the important role of $B$1g phonon in the intertwined orders but also shed
light on the possibility of their unified understanding in copper oxide
superconductors.

Figure 1: Schematic of the low energy electronic structure of a superconductor
(a) below and (b) above the critical temperature, $T$c. (c) Sketch of the
phonon renormalization due to the phonon self-energy effect. The red curve
represents a phononic Raman spectrum in the normal state. Blue and green
curves correspond to the spectrum in the ordered state for the phonon energy
$\omega<2\Delta$ and $\omega>2\Delta$, respectively. Figure 2: (a)
Temperature-dependent Raman response of optimally doped YBa2Cu3O7-δ (OP94)
single crystal below 110K. The peak at 340 cm-1 corresponds to the $B$1g
phonon and the broad peak around 550 cm-1 is due to pair breaking peak.
Compared are $B$1g phonon spectra measured at (b) 300 and 150 K, (c) 150 and
95 K, and (d) 95 and 65 K. Temperature-dependent frequency (e) and linewidth
(f) extracted from a Fano line shape fitting. The black dashed curve
corresponds to simulated frequency and linewidth of $B$1g phonon from the
phonon anharmonicity model. The blue and red vertical dashed lines represent
the superconducting transition temperature ($T$c) and onset temperature of the
phonon anomaly ($T$B1g), respectively. Figure 3: Temperature dependent $B$1g
phonon frequency from YBa2Cu3O7-δ single crystals with different dopings.
Results for UD62 (a), UD68 (b), UD85 (c), OD91 (d), OD89 (e). The hole doping
concentration was determined using the method in Ref. Liang  _et al._ (2006).
$T$c and ${T}$B1g are indicated by blue and red vertical dashed lines,
respectively. Figure 4: YBa2Cu3O7-δ phase diagram with various phases
determined by various techniques. $T$B1g extracted from our Raman spectroscopy
data is plotted as red circles. Superconducting fluctuation temperature
($T$pair) in YBa2Cu3O7-δ measured by various methods are also plotted:
magnetoconductivity (blue circles) Rullier-Albenque  _et al._ (2011), Nernst
(violet inverted triangles) Cyr-Choinière  _et al._ (2018), torque
magnetometry (orange diamonds) Li  _et al._ (2010) and specific heat (purple
diamonds) Tallon  _et al._ (2011). $T$CDW measured by resonant x-ray
scattering (RXS) Blanco-Canosa  _et al._ (2014) is plotted as yellow circles.
Black shaded line represents the $T$∗ that can be found elsewhere Zhao  _et
al._ (2017); Loret  _et al._ (2019).

We performed Raman spectroscopy with back-scattering geometry using 532 nm
(2.3 eV) diode pumped solid state (DPSS) laser (Cobolt). The incident light
was focused on the surface of the sample by a $\times$40 (N.A. 0.6) objective
lens (Olympus). A spot size of the focused light was approximately 2 $\times$
2 $\mu$m2 and laser power was set below 0.6 mW to avoid laser-induced local
heating. To reject the stray light close to 10 cm-1, BragGrate notch filters
(OptiGrate) were used. The superachromatic waveplate (Thorlabs) was used for
polarization resolved Raman spectroscopy. The polarization of the incident and
scattered light was set to be perpendicular to each other and parallel to the
diagonal direction of the CuO2 plaquette [Fig. 2(a), inset] to probe the
signal from the $B$1g channel. The direction of the Cu$-$O bonding was
determined by the angle-resolved polarized Raman scattering measurement. The
Raman scattered light was collected by the same objective lens and collimated
before entering the focusing lens in front of the spectrometer. The collected
light was dispersed by a Jobin-Yvon Horiba iHR320 spectrometer with 1800
grooves/mm grating and detected by a thermoelectric cooled CCD. YBa2Cu3O7-δ
single crystals were mounted on an optical cryostat (Oxford) for temperature-
dependent measurements. Optimally doped YBa2Cu3O7-δ single crystals were
annealed in a tube furnace with oxygen flow to control the oxygen content. The
superconducting transition temperature was determined from the diamagnetic
signal measured with a magnetic property measurement system (Quantum Design).

We first briefly touch upon how the gap in the electronic phase can affect the
phonon peaks in Raman spectra. Plotted in Fig. 1 is a schematic of the low
energy single particle spectral function across the superconducting gap and
its effect on a phonon peak in Raman spectra. In the superconducting state,
the superconducting gap opens at the Fermi surface while quasiparticle DOS at
$\omega$ = $\pm$$\Delta$ is enhanced. It results in a singularity at $\omega$
= 2$\Delta$ in the real and imaginary parts of the phonon self-energy Zeyher
and Zwicknagl (1990); Nicol  _et al._ (1993). At $T<T_{\rm c}$, for a phonon
with $\omega$ph smaller (larger) than 2$\Delta$, softening (hardening) and
narrowing (broadening) of the phonon is expected, in comparison to the
$T>T_{\rm c}$ case [Fig. 1(c)]. This phenomenon can be intuitively understood
as follows. In the normal state, all phonons can be scattered by coupling to
the electrons near the Fermi level [Fig. 1(b)]. In the superconducting state,
however, phonons with $\omega_{\rm ph}<2\Delta$ cannot be scattered by
electrons due to the lack of scattering channels. On the other hand, the
scattering rate of $\omega_{\rm ph}>2\Delta$ phonons increases due to the
enhanced quasiparticle DOS [Fig. 1(a)].

Fig. 2(a) shows the $B$1g Raman response of optimally doped ($T$c = 94 K)
YBa2Cu3O7-δ. As the temperature decreases below $T$c, a broad electronic Raman
continuum appears near 550 cm-1. It is a signature of superconducting pair
breaking, representing the superconducting gap near the antinodal region in
the Brillouin zone [Fig. 2(a), inset] Devereaux and Hackl (2007); Devereaux
_et al._ (1994). The strongest peak at 340 cm-1 corresponds to the $B$1g
phonon. Consistent with previous results, our results also show hardening in
the normal state [Fig. 2(b)] and strong softening below $T$c [Fig. 2(d)]
Altendorf  _et al._ (1993); Limonov  _et al._ (2000); Hewitt  _et al._ (2004);
Bakr  _et al._ (2009). However, an additional weak phonon softening was
observed for a temperature range just above $T$c [Fig. 2(c)]. To quantify the
phonon softening, we extracted the frequency and linewidth of the $B$1g phonon
by fitting the peak with a Fano line shape Altendorf  _et al._ (1993); Hewitt
_et al._ (2004); Bakr  _et al._ (2009); Driza  _et al._ (2012). Figs. 2(e) and
(f) show the frequency and linewidth of the $B$1g phonon, respectively, as a
function of temperature. The black dashed curves represent the simulated
frequency and linewidth behavior using a phonon anharmonicity model Altendorf
_et al._ (1993); Bakr  _et al._ (2009). Above 150 K, phonon hardening was well
described by the phonon anharmonicity. The frequency of the $B$1g phonon
begins to deviate from the anharmonic behavior around 150 K. The frequency
decreases slowly until the temperature reaches $T$c, then decreases rapidly
below $T$c. In Fig. 2(f), the linewidth of the $B$1g phonon also appears to
deviate from the black dashed curve above $T$c. However, if higher order terms
are added to the phonon anharmonicity model, the linewidth data can be fitted
well from $T$c to 300 K. Hence, we define the onset temperature of the phonon
anomaly, dubbed as $T$B1g, from the temperature versus frequency data.

To verify the doping dependence of $T$B1g, we performed the same experiment
and analysis on samples with various dopings. We find that not only the
optimally doped sample but also underdoped samples (UD62, UD68 and UD85) [Fig.
3(a)-(c), respectively] clearly show the $B$1g phonon anomaly between $T$c and
$T$B1g. On the other hand, the phonon anomaly significantly weakens for
slightly overdoped sample [Fig. 3(d)] and vanishes for the most overdoped
sample [Fig. 3(e)]. This alludes to the fact that the instability that causes
the $B$1g phonon anomaly disappears in the overdoped region.

Plotted in Fig. 4 are the measured $T$B1g (filled red circle) with the onset
temperatures of various phases. The value of $T$B1g is highest near slightly
under doping ($p$=0.144) and decreases as the hole doping concentration
increases or decreases. It is clearly seen that the doping dependence of
$T$B1g is quite different from that of $T$∗ because $T$∗ monotonously
increases with decreasing hole doping concentration. This strongly suggests
that the $B$1g phonon anomaly is not caused by the pseudogap. Instead, $T$B1g
appears to be close to the temperature scales of superconducting fluctuation,
$T$pair. Except for $T$pair measured by Nernst effect Cyr-Choinière  _et al._
(2018), $T$pair measured by magnetoconductivity Rullier-Albenque  _et al._
(2011), torque magnetometry Li  _et al._ (2010) and specific heat Tallon  _et
al._ (2011) are quite close to $T$B1g. In addition, there is also a reasonable
similarity between $T$CDW and $T$B1g Blanco-Canosa  _et al._ (2014).

From the point of view that $T$B1g corresponds to $T$pair, it can be
understood that the phonon softening above $T$c is caused by phonon self-
energy effect due to formation of preformed Cooper pairs. Above $T$c,
preformed Cooper pairs open a gap at the Fermi surface and enhance the
quasiparticle DOS at $\omega$ = $\pm$$\Delta$ which has been clearly observed
in a recent angle-resolved photoemission spectroscopy (ARPES) study on Bi2212
Chen  _et al._ (2019). As a result, a singularity appears in the phonon self-
energy which in turn causes the phonon softening as we mentioned earlier (Fig.
1). On the other hand, we also note that $T$B1g is considerably higher than
$T$pair measured by Nernst effect Cyr-Choinière  _et al._ (2018). The
difference might be due to the uncertainty in the analysis to define $T$pair.
Since the Nernst effect is very sensitive to formation of Cooper pairs, it is
a powerful way to measure $T$pair. Nevertheless, there is a debate on how to
determine the $T$pair since the pseudogap also contributes to the Nenst
signal. In the case of YBa2Cu3O7-δ, superconductivity and pseudogap contribute
to a positive and negative Nernst signal, respectively Cyr-Choinière  _et al._
(2018). The pseudogap contribution interferes with the signal from preformed
Cooper pairs and makes it difficult to precisely define $T$pair. As a result,
the $T$pair measured by the Nernst effect underestimates the actual $T$pair
where Cooper pairs start forming.

Although $T$B1g does not match previously reported $T$CDW Blanco-Canosa  _et
al._ (2014) very well, we cannot entirely rule out the possibility that CDW
order is the origin of $B$1g phonon softening in the intermediate temperature
range since the overall doping dependence of $T$B1g is very similar to that of
$T$CDW. CDW is the most common electronic order which softens phonon through
the Kohn anomaly Gruner (2018); Le Tacon  _et al._ (2014). In this case, a
giant phonon anomaly can occurs at $q$ = $q$CDW. However, $B$1g phonon
softening in our results cannot be reconciled with the Kohn anomaly since
Raman spectroscopy can probe only $q\sim 0$ phonons. It is more natural to
interpret that phonon softening comes from the phonon self-energy effect due
to the Fermi surface instability induced by CDW Raichle  _et al._ (2011).
Based on these considerations, the mismatch between $T$CDW and $T$B1g can be
understood as follows. Although no CDW peak was observed in resonant X-ray
scattering (RXS) studies, inelastic neutron scattering results showed the
softening of $B$1g phonon at $q=q_{\rm CDW}$ even for the most overdoped
YBa2Ca3O7 Raichle  _et al._ (2011); Blanco-Canosa  _et al._ (2014). It
suggests existence of CDW fluctuation even for the doping range where no
static CDW order is observed. If the dynamic CDW affects the phonon self-
energy, $T$B1g may exhibit a different temperature scale than that measured by
RXS. This calls for further studies with various experimental tools in the
doping range where CDW order exists.

If the observed $B$1g phonon softening is indeed due to a CDW order, it is
suggestive of an important role played by the $B$1g phonon in both the high
temperature superconductivity and charge order. This notion is indeed
supported by the result of a recent electronic Raman study in which the energy
scale of 2$\Delta$CDW was found to be the same as that of antinodal
superconducting gap 2$\Delta$AN Loret  _et al._ (2019). This result suggests
the possibility that the $B$1g phonon might play a role (a secondary, if not
the primary role) for the superconductivity and CDW orders. Our result may
lend another evidence for the connection between the superconductivity and
CDW.

Regardless of its origin, it is likely that the $B$1g phonon softening occurs
due to a gap at the Fermi surface. Both Cooper pairing and Peierls
instabilities result in an energy gap at the Fermi surface in their single
particle excitation spectra even though the pairing channels for
superconductivity (electron-electron) and CDW (electron-hole) are different
Gruner (2018). Since these two orders induce the Fermi surface gap, in
particular, share the maximum gap near the antinodal region, both can induce
the renormalization in the phonon self-energy and softening of the $B$1g
phonon.

It is also worth noting that both electronic orders have a $d$-wave nature.
The link between $d$-wave symmetry and $B$1g phonon has been considered for a
long time Devereaux  _et al._ (1995); Newns and Tsuei (2007) and has recently
attracted renewed attention He  _et al._ (2018); Banerjee  _et al._ (2020a);
Hamidian  _et al._ (2016). It is argued that the electron-phonon coupling in
the presence of strong electronic correlation can create electronic orders
such as CDW with a $d$-wave symmetry form factor and $d$-wave
superconductivity. Within this picture, our experimental results may be viewed
as an additional evidence that the $B$1g phonon plays an important role in the
formation of electronic orders with a $d$-wave character.

The remaining question is why $B$1g phonon anomaly is not observed in the
pseudogap phase. A possible explanation may come from the different nature of
the pseudogap. Spectroscopic studies have shown that the pseudogap phase are
not accompanied by a sharply depleted Fermi surface gapping (and thus no
quasiparticle DOS enhancement at $\omega$ = $\pm$$\Delta$), which should
result in a smooth change in the phonon self-energy. Such effect can be
expected in the pair density wave (PDW) scenario, where the spatial modulation
of the $d$-wave pairing should lead to an almost cancelled coherence factor
Agterberg  _et al._ (2020). Further investigations of the pseudogap and
electronic order (CDW or preformed pairs) using complementary experimental
techniques should lead to a comprehensive understanding of the novel phases.

We acknowledge the fruitful discussion with S. Jung and D. Wulferding. This
work was supported by the Institute for Basic Science in Korea (Grant No.
IBS-R009-G2). S. R. P. acknowledges support from the NRF (Grant No.
2020R1A2C1011439).

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