# A Generative Approach for Financial Causality Extraction

Tapas Nayak<EMAIL_ADDRESS>IIT KharagpurIndia ,  Soumya Sharma
<EMAIL_ADDRESS>IIT KharagpurIndia ,  Yash Butala
<EMAIL_ADDRESS>IIT KharagpurIndia ,  Koustuv Dasgupta
<EMAIL_ADDRESS>Goldman SachsIndia ,  Pawan Goyal
<EMAIL_ADDRESS>IIT KharagpurIndia and  Niloy Ganguly
<EMAIL_ADDRESS>IIT Kharagpur, IndiaLeibniz University, Germany

(2022)

###### Abstract.

Causality represents the foremost relation between events in financial
documents such as financial news articles, financial reports. Each financial
causality contains a cause span and an effect span. Previous works proposed
sequence labeling approaches to solve this task. But sequence labeling models
find it difficult to extract multiple causalities and overlapping causalities
from the text segments. In this paper, we explore a generative approach for
causality extraction using the encoder-decoder framework and pointer networks.
We use a causality dataset from the financial domain, FinCausal, for our
experiments and our proposed framework achieves very competitive performance
on this dataset.

financial information extraction, financial causality extraction, generative
models, pointer networks

††journalyear: 2022††copyright: acmcopyright††conference: Companion
Proceedings of the Web Conference 2022; April 25–29, 2022; Virtual Event,
Lyon, France††booktitle: Companion Proceedings of the Web Conference 2022 (WWW
’22 Companion), April 25–29, 2022, Virtual Event, Lyon, France††price:
15.00††doi: 10.1145/3487553.3524633††isbn: 978-1-4503-9130-6/22/04††ccs: Deep
learning Generative models††ccs: Natural language processing Information
extraction††ccs: Event extraction causality extraction

##  1\. Introduction

Causality extraction from financial text is an important task for the analysis
of financial documents such as financial news articles, financial reports.
Previously, sequence labeling models (Li et al., 2021; Kao et al., 2020;
Becquin, 2020) were proposed to solve this task and they assign ‘BIO’ tags for
cause and effect span to each token in the text. Although these models perform
quite well in this task, they are not modeled to handle the challenges of
multiple and overlapping causalities present in the financial text. In this
paper, we explore a generative approach using an encoder-decoder framework for
causality extraction. We incorporate pointer networks into our decoding
framework for structured prediction of the cause and effects spans in the
text. The encoder-decoder approach extracts the causalities in a sequence thus
the challenges of variable-length causality extraction and overlapping
causalities extraction are solved. The pointer network-based decoding
identifies the cause and effect spans using the start and end positional index
in the text. So the cause and effect spans of different lengths are modeled
uniformly in this approach. We use FinCausal (Mariko et al., 2020), a dataset
containing text segments from financial news articles, for our experiments.
Our proposed model achieves very competitive performance in this dataset. We
release our code and data for future research at
https://github.com/nayakt/CEPN.

##  2\. The CEPN Framework

We present CEPN, a Causality Extraction framework using Pointer Network-based
encoder-decoder model. To formally define this task, given a text segment
$S=\\{w_{1},w_{2},...,w_{n}\\}$ with $n$ tokens, the goal is to extract a set
of causalities $T=\\{y_{1},y_{2},....,y_{m-1}\\}$ where
$y_{t}=(c_{s}^{t},c_{e}^{t},e_{s}^{t},e_{e}^{t})$ is the $t^{th}$ causality,
$m-1$ is the number of causalities. $c_{s}^{t}$, $c_{e}^{t}$, $e_{s}^{t}$, and
$e_{e}^{t}$ represent the positions of the start and end tokens of the cause
and effect span of the $t^{th}$ causality in the text segment $S$,
respectively. In Fig 1, we give an overview of our proposed model which is
inspired from the models proposed in similar structure prediction tasks such
as joint entity-relation extraction (Nayak and Ng, 2020) and aspect-sentiment
triplet extraction (Mukherjee et al., 2021).

We use a pre-trained BERT model (Devlin et al., 2019) to encode the source
text. We concatenate the part-of-speech tag embeddings of the tokens with the
BERT vectors to obtain the encoder hidden states
$\mathbf{h}_{i}^{E}\in\mathbb{R}^{d_{h}}$. We add a special token ‘[unused0]’
at the front of the text and use its positional index ‘$0$’ to stop the
decoding process. We use BERT_Base_Cased of dimension $768$ and
BERT_Large_Cased of dimension $1,024$ model for encoding, and we refer them as
CEPN_Base and CEPN_Large. We set the dimension of the part-of-speech tag
embeddings at $32$, and initialize them randomly. The hidden dimension $d_{h}$
of encoder for CEPN_Base and CEPN_Large model is $768+32=800$ and
$1,024+32=1,056$, respectively.

We consider the causalities as a sequence $T=\\{y_{1},y_{2},....,y_{m-1}\\}$.
We use two special tuples to model this task as a sequence generation problem.
We start the sequence generation process with a special tuple $y_{0}$, and we
mark the end of this generation process with another special tuple,
$y_{m}=(0,-1,-1,-1)$. Here, only $c_{s}^{t}=0$ (corresponds to [unused0]
token) is enough to mark the end of the sequence generation process, and we
ignore the other indexes of value $-1$. In the decoder, we use an LSTM cell
for generating the causality sequence. We set the hidden dimension of this
LSTM cell at $d_{h}$ that is the same as the hidden dimension of the encoder
vectors. We pass the encoder context representation,
$\mathbf{e}_{t}\in\mathbb{R}^{d_{h}}$, and the average of the vectors of
already generated causalities, $\mathbf{y}_{avg}$ to this LSTM cell to
generate the next hidden state, $\mathbf{h}_{t}^{D}\in\mathbb{R}^{d_{h}}$. We
use two pointer networks for identifying the cause and effect span of the
causalities. Each pointer network consists of a BiLSTM layer with hidden
dimension $d_{p}$ and two feed-forward layers (FFN) of dimension $d_{p}\times
1$ with Softmax activation. We set $d_{p}=d_{h}$ as the hidden dimension of
the BiLSTM layer in the pointer networks. The output of this FFN/Softmax layer
is a scalar score corresponding to each token in the source text. These scalar
scores represent the probability distribution of the source tokens being the
start or end token of the cause/effect span. We can extract either of the
cause or effect spans first, then use the vector representation of that span
to extract the other span of a causality. In one version, we extract cause
span first, we refer to them using ‘CF’ such as CEPN_Base_CF. We experiment
with another version where we extract the effect span first, and we refer to
them using ‘EF’ such as CEPN_Base_EF.

Figure 1. The architecture of CEPN framework.

##  3\. Experiments

We use the FinCausal 2020111http://wp.lancs.ac.uk/cfie/fincausal2020/ and
FinCausal 2021222http://wp.lancs.ac.uk/cfie/fincausal2021/ datasets for our
experiments. The text segments in these two datasets contain a maximum of
three sentences and the cause and effect span of causality can appear in
different sentences in the segment. The number of average tokens in the text
segments is $\sim$ 40 and in the cause and effect span is $\sim$ 17-18. As the
gold-label causalities of the blind test split are not publicly available, we
do a 5-fold cross-validation on the training datasets for experiments. We
compare our model against the previous SOTA models such as UPB (Ionescu et
al., 2020), DOMINO (Chakravarthy et al., 2020), PAMNet (Szántó and Berend,
2020), GBe (Becquin, 2020), NTUNLPL (Kao et al., 2020). We report token-level
F1 score and exact-match F1 score for comparison.

We include the results on FinCausal 2020 and FinCausal 2021 datasets in Table
1. We see that our CF and EF versions of the model perform almost equally on
both of these datasets. Compare to GBe_Large model, our CEPN_Large_EF model
achieves 1.4% higher token-level F1 score and 3.3% higher exact match-based F1
score on the FinCausal 2020 dataset. On the FinCausal 2021 dataset, compare to
GBe_Large model, our CEPN_Large_CF model achieves 0.9% higher token-level F1
score and 1.4% higher exact match-based F1 score in this dataset. We also
perform a statistical significance test (two-tailed and paired) between
CEPN_Large_CF model and GBe_Large model on the FinCausal 2020 and 2021
datasets and find that our model is statistically significant with $p<0.001$
(achieves a 1.1% higher mean token-level F1 score on both datasets).

| FinCausal 2020 | FinCausal 2021
---|---|---
Model | Token_F1 | EM_F1 | Token_F1 | EM_F1
UPB_Base (Ionescu et al., 2020) | 0.689 | - | - | -
DOMINO_Base (Chakravarthy et al., 2020) | 0.796 | - | - | -
PAMNet_Large* (Szántó and Berend, 2020) | 0.767 | 0.485 | 0.776 | 0.533
GBe_Large* (Becquin, 2020) | 0.860 | 0.710 | 0.883 | 0.761
NTUNLPL_Base (Kao et al., 2020) | 0.869 | - | - | -
CEPN_Base_CF | 0.866 | 0.733 | 0.885 | 0.769
CEPN_Base_EF | 0.857 | 0.722 | 0.883 | 0.761
CEPN_Large_CF | 0.870 | 0.739 | 0.892 | 0.775
CEPN_Large_EF | 0.874 | 0.743 | 0.891 | 0.767
Table 1. Performance comparison of CEPN model on the FinCausal datasets
against the previous SOTA models. * marked baseline scores are reproduced by
us. The remaining baseline scores are taken from their papers. ‘Base’ and
‘Large’ refer to the use of BERT_Base and BERT_Large models for source
encoding. We report the average of the 5-folds in a cross-validation setting.

##  4\. Conclusion

In this paper, we explore a generative approach for causality extraction in
financial text using the encoder-decoder framework and pointer networks. Our
model is designed to handle different challenges in this task such as
extracting unknown variable-length causalities, identifying text with no
causality, and extracting overlapping causalities from the text in an end-to-
end architecture. Experimental results on the FinCausal datasets show the
effectiveness of our proposed framework on financial causality extraction.

###### Acknowledgements.

This research was partially supported by Goldman Sachs under the research
grant FTHS (FinTalk: Research towards creating a platform for highlight
generation and summarization of financial documents while taking into account
user feedback).

## References

  * (1)
  * Becquin (2020) Guillaume Becquin. 2020\.  GBe at FinCausal 2020, Task 2: Span-based Causality Extraction for Financial Documents. In _Proceedings of the 1st Joint Workshop on Financial Narrative Processing and MultiLing Financial Summarisation_.
  * Chakravarthy et al. (2020) Sharanya Chakravarthy, Tushar Kanakagiri, Karthik Radhakrishnan, and Anjana Umapathy. 2020.  Domino at FinCausal 2020, Task 1 and 2: Causal Extraction System. In _Proceedings of the 1st Joint Workshop on Financial Narrative Processing and MultiLing Financial Summarisation_.  https://aclanthology.org/2020.fnp-1.15
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In _NAACL_.
  * Ionescu et al. (2020) Marius Ionescu, Andrei-Marius Avram, George-Andrei Dima, Dumitru-Clementin Cercel, and Mihai Dascalu. 2020.  UPB at FinCausal-2020, Tasks 1 & 2: Causality Analysis in Financial Documents using Pretrained Language Models. In _Proceedings of the 1st Joint Workshop on Financial Narrative Processing and MultiLing Financial Summarisation_.  https://aclanthology.org/2020.fnp-1.8
  * Kao et al. (2020) Pei-Wei Kao, Chung-Chi Chen, Hen-Hsen Huang, and Hsin-Hsi Chen. 2020.  NTUNLPL at FinCausal 2020, Task 2: Improving Causality Detection Using Viterbi Decoder. In _Proceedings of the 1st Joint Workshop on Financial Narrative Processing and MultiLing Financial Summarisation_.
  * Li et al. (2021) Zhaoning Li, Qi Li, Xiaotian Zou, and Jiangtao Ren. 2021\.  Causality Extraction based on Self-Attentive BiLSTM-CRF with Transferred Embeddings.  _Neurocomputing_ (2021).
  * Mariko et al. (2020) Dominique Mariko, Hanna Abi Akl, Estelle Labidurie, Stephane Durfort, Hugues De Mazancourt, and Mahmoud El-Haj. 2020.  Financial Document Causality Detection Shared Task (FinCausal 2020).  _arXiv preprint arXiv:2012.02505_ (2020).
  * Mukherjee et al. (2021) Rajdeep Mukherjee, Tapas Nayak, Yash Butala, Sourangshu Bhattacharya, and Pawan Goyal. 2021\.  PASTE: A Tagging-Free Decoding Framework Using Pointer Networks for Aspect Sentiment Triplet Extraction. In _EMNLP_.
  * Nayak and Ng (2020) Tapas Nayak and Hwee Tou Ng. 2020.  Effective modeling of encoder-decoder architecture for joint entity and relation extraction. In _AAAI_.
  * Szántó and Berend (2020) Zsolt Szántó and Gábor Berend. 2020.  ProsperAMnet at FinCausal 2020, Task 1 & 2: Modeling causality in financial texts using multi-headed transformers. In _Proceedings of the 1st Joint Workshop on Financial Narrative Processing and MultiLing Financial Summarisation_.


# Photonic interface between subcarrier wave and dual rail encodings

K. S. Melnik  Kazan Quantum Center, Kazan National Research Technical
University, 18a Chetaeva str., Kazan, 420111, Russia  E. S. Moiseev  Kazan
Quantum Center, Kazan National Research Technical University, 18a Chetaeva
str., Kazan, 420111, Russia<EMAIL_ADDRESS>
###### Abstract

Quantum key distribution with multimode subcarrier wave encoding is propitious
for being robust against environmental disturbance. For application in long-
distance quantum communications, this encoding has to be compatible with
entanglement-assisted quantum repeaters that are commonly designed to work
with dual rail encodings. We propose and demonstrate an interface between
subcarrier wave and dual rail encodings with a fidelity of $0.95\pm 0.03$
using a linear optical circuit. The developed scheme opens the way for future
heterogeneous quantum networks that combine subcarrier wave quantum key
distribution with trusted and non-trusted nodes.

##  I Introduction

The communication distance of point-to-point quantum key distribution (QKD) is
fundamentally limited by losses in the optical channel to hundreds of
kilometres Pirandola  _et al._ (2017). Current long-distance QKD systems use a
chain of connected trusted nodes that translate the generated key from one end
of the chain to another Elliott (2002); Sasaki  _et al._ (2011); Dynes  _et
al._ (2019); Chen  _et al._ (2021). Different from point-to-point QKD
protocols, the twin-field QKD improves the secure key rate scaling with
distance. The recent result demonstrates long distance QKD and may provide a
solution to reduce the number of trusted relays Xu  _et al._ (2020); Chen  _et
al._ (2020). Alternatively, the development of the quantum internet pursuits
the creation of entanglement-enriched quantum networks, such as quantum
repeater, that could be used for QKD without a need of the trusted nodes
Briegel  _et al._ (1998); Guha  _et al._ (2015). For a cohesive operation of
the existing QKD lines with the future quantum networks, a mutual encoding of
quantum information is necessary. Conventional discrete variable quantum
repeater provides photonic entanglement in the form of the Bell pairs in dual
rail encoding (e.g., polarization) Sangouard  _et al._ (2011); Yu  _et al._
(2020).

The creation of interfaces between different quantum informational encodings
is a significant milestone for the development of a quantum internet Wehner
_et al._ (2018). It would allow for the connection of local quantum networks
with different encodings and their combination into a global heterogeneous
network with tolerable losses at the interconnections. Recently, several
experiments were performed on conversion between dual and single rail qubits
Drahi  _et al._ (2021), teleportation-based conversion between single rail and
wave-like qubits Ulanov  _et al._ (2017), between dual rail and wave-like
qubits Sychev  _et al._ (2018) and on the interface between hybrid and dual
rail entanglement Guccione  _et al._ (2020) to name a few.

The subcarrier wave (SCW) encoding is a promising for point-to-point Mérolla
_et al._ (1999); Gleim  _et al._ (2016), Plug&Play Bannik and Moiseev (2021),
continuous variables Mel’nik  _et al._ (2018); Samsonov  _et al._ (2020) and
twin-field QKD Chistiakov  _et al._ (2019) due to its robustness to
environmental effects acting on a fiber line, interferometer-free optical
scheme and its capacity for spectral multiplexing Mora  _et al._ (2012). In
SCW-QKD the information is encoded in a value of the phase of the applied
sinusoidal optical phase modulation, that leads to specific multi-mode state.
However, this multi-mode state complicates a creation entanglement
distribution protocol for this encoding that limits potential long-distance
SCW quantum communications. One way to circumvent this issue would be a
conversion of an information from SCW encoding into the one with available
quantum repeater protocol, e.g. dual rail encoding.

Conceptually, the interface between SCW and dual rail encondings may be used
as it is sketched in Fig.1a. Alice and Bob generate a non-local polarization
entangled Bell state $|\,\Psi_{AB}\,\rangle$ with the help of a quantum
repeater, while Alice and Charlie are connected with an optical link. Charlie,
who is located in a trusted node of the SCW-QKD network, sends his SCW state
$|\,\Psi_{\text{SCW}}\,\rangle$ to Alice, where the state is routed to a
polarization Bell state measurement device via the interface. The interface
maps the useful information from $|\,\Psi_{\text{SCW}}\,\rangle$ into a
polarization qubit’s degree of freedom $|\,\Psi_{\text{POL}}\,\rangle$. In
turn, the polarization qubit is interfered with Alice’s part of the shared
state. The proper detection at the Bell state measurement device together with
local polarization rotation at Bob’s part teleports Charlie’s information to
Bob Bouwmeester  _et al._ (1997). However, a scheme for an appropriate
interface has not been addressed until now.

In this work, we propose and demonstrate a proof of principle experiment on
interfacing the SCW with dual rail encodings. First, we describe the scheme of
the interface and its simple theoretical model. Next, we portray the
experimental setup and present its results. Finally, we discuss the
perspectives and further possible improvements of the interface.

Figure 1: a) The possible application of the interface (INT) for teleporting
information from Charlie’s SCW state $|\,\Psi_{\text{SCW}}\,\rangle$ to Bob
with a use of Bell state measurement device (BSM) and entangled state between
Alice and Bob $|\,\Psi_{AB}\,\rangle$ being generated via quantum repeater
chain. b) The principle scheme of the interface that consist of phase
modulator (PM), spectral filter (SF), controlling avalanche photodetector
(APD), non-polarizing beam splitter 50:50 (BS), polarization beam splitter
(PBS), optical circulator (OC). The $\\{-1,0,1\\}$ indices in the circles
indicate the presence of the sidebands with their phases. c) The experimental
setup with laser diode (LD), optical isolator (OI), optical attenuator (ATN),
single photon counting module (SPC), Faraday rotator (FR), fiber output
coupler (FOC), quarter wave plate ($\lambda/4$), half wave plate
($\lambda/2$). Red solid and Blue dashed arrows indicate the carrier and
sidebands direction, respectively. Double arrow and circled dot represent
horizontal and vertical polarizations, respectively.

##  II Principle scheme

In SCW, the information is encoded by applying a sinusoidal phase modulation
with modulation depth $\beta$, modulation frequency $\Omega$, and phase
$\phi_{in}$ to a coherent state with complex amplitude $\alpha_{0}$ and
carrier frequency $\omega_{0}$. An optical state after a phase modulator is
described by wavefunction Miroshnichenko  _et al._ (2017):

$\displaystyle|\,\Psi_{\text{SCW}}\,\rangle=\bigotimes^{S}_{m=-S}|\,\alpha_{0}J_{m}(\beta)e^{im\phi_{in}}\,\rangle_{m},$
(1)

where multiplication is carried over 2S+1 sidebands with $m\in\mathbb{Z}$
being indices of sidebands with frequencies $\omega_{0}+m\Omega$,
$J_{m}(\beta)$ is m-th order Bessel function of the first kind. The value of
the phase $\phi_{in}$ encodes useful information. For example, in typical SCW-
QKD with BB-84-like protocol, Alice sends to Bob logical bits in two non-
orthogonal bases with a single basis formed by pairs of states (1) with
$\phi_{in}$ being shifted by $\pi$ with respect to each other, such as
${0,\pi}$ and ${\pi/2,3\pi/2}$. In turn, Bob measured Alice’s state in given
basis by phase modulating it with the same $\beta$ and $\Omega$, but his own
subcarrier phase and performing photon detection on sidebands ($m\neq 0$) that
effectively acts as a projection on a given state Miroshnichenko  _et al._
(2017). For interfacing the state (1) with some dual encoding, the phase
$\phi_{in}$ has to be mapped into a superposition of two modes.

The principle scheme of the interface is depicted in Fig. 1b. The input state
$|\,\Psi_{\text{SCW}}\,\rangle$ with horizontal polarization enters spectral
filter that reflects the carrier component ($m=0$) with power coefficient $r$
and transmits it with power coefficient $1-r$. Meanwhile the sidebands ($m\neq
0$) are transmitted through the filter with power coefficient $1-\varrho$ and
reflected with power coefficient $\varrho$, where $\varrho$ is called as
filter’s sideband suppression factor. The reflected carrier is routed by an
optical circulator to a phase modulator, that modulates the carrier with the
same modulation depth $\beta$, modulation frequency $\Omega$, but a fixed
phase $\phi_{\text{LO}}$. After the phase modulator another spectral filter
with the same $r$ and $\varrho$ separates the carrier frequency component from
the generated sidebands.

The sidebands from the original state and the ones generated locally are
overlapped on the 50:50 beam splitter. The wavefunction for two spatial modes
at the output of the beam splitter labelled as ’$+$’ and ’$-$’ is

$\displaystyle|\,\Psi\,\rangle=|\,\psi_{+}\,\rangle\otimes|\,\psi_{-}\,\rangle,$
(2)

with

$\displaystyle|\,\psi_{\pm}\,\rangle=|\,\alpha_{0}\frac{J_{0}(\beta)((1-r)\pm
J_{0}(\beta)r(1-r))}{\sqrt{2}}\,\rangle_{0}\cdot$
$\displaystyle\bigotimes_{m\neq
0}|\,\alpha_{0}J_{m}(\beta)(1-\varrho)\frac{e^{im\phi_{in}}\pm
e^{im\phi_{\text{LO}}}rJ_{0}(\beta)}{\sqrt{2}}\,\rangle_{m}.$ (3)

For good enough filters ($r\approx 1$, $\varrho\approx 0$) and small
modulation depth ($J_{0}(\beta)\approx 1$), the state in Eq. (3) represents
the dual rail encoding with two spatial modes for each individual sideband
with $m\neq 0$, e.g., the states with a phase difference of 0 or $\pi$ with
respect to $\phi_{LO}$ will follow different channels of the beam splitter.
Both modes are combined on the polarization beam splitter, with one mode’s
polarization being rotated by $90^{o}$ with the half-waveplate and the mode’s
phase being shifted by $e^{i\phi}$ that is defined by the optical path
difference. The resulted state for the given first order sidebands ($m=\pm 1$)
up to a first order in photon number is:

$\displaystyle|\,\Psi_{\pm 1}\,\rangle\approx|\,0_{H}0_{V}\,\rangle_{\pm
1}+\sqrt{2}\alpha_{0}J_{\pm 1}(\beta)e^{\pm
i(\phi_{in}+\phi_{\text{LO}})/2}\cdot$
$\displaystyle\bigg{(}\cos\frac{\phi_{in}-\phi_{\text{LO}}}{2}|\,1_{H}0_{V}\,\rangle_{\pm
1}\pm\sin\frac{\phi_{in}-\phi_{\text{LO}}}{2}|\,0_{H}1_{V}\,\rangle_{\pm
1}\bigg{)},$ (4)

where we set the phase shift of V mode to be $e^{i\phi}=-i$. Once
$\phi_{\text{LO}}$ is fixed to 0, the phase information $\phi_{in}$ is mapped
on the probability amplitude of a photon being in a given polarization mode,
as in dual rail encoding.

Figure 2:  a) Dependence of the normalized count rate for $|\,H\,\rangle$ (red
triangles) and $|\,V\,\rangle$ (blue dots) states at different values of phase
difference between two microwave generators. The data is obtained with
modulation depth $\beta=0.15$. The dashed and solid lines represent
theoretical fits. b) Dependence of interference visibility on modulation
depth. The solid curve is a theoretical fit.

Figure 3:  Real and imaginary parts of the reconstructed density matrices for
polarization states at different values of the phase difference in
$\\{|\,H\,\rangle,|\,V\,\rangle\\}$ basis.

##  III Experiment

The experimental scheme is depicted in Fig. 1. The fiber-bragg grating
stabilised laser diode (TeraXion PureSpectrum LM) protected by an optical
isolator together with an optical attenuator are used to generate continuous
wave at 1550 nm. The laser is guided through an optical circulator into a
4-port fiber 50:50 beam splitter (BS). Two SCW states are generated using the
same carrier in a Sagnac-type interferometer, where clockwise and counter-
clockwise travelling waves are modulated by two separate
$\text{LiNb}\text{O}_{3}$ phase modulators, PM1 an PM2, respectively. The
phase modulators are driven by two phase-locked microwave generators with an
identical frequency of $\Omega=2\pi\cdot 4.8$ GHz but different phases,
$\phi_{\text{in}}$ and $\phi_{\text{LO}}$, respectively. The implemented
design helps to circumvent a problem with the need for two identical spectral
filters, which is necessary for the principle scheme in Fig. 1b.

The generated optical sidebands are interfered on 50:50 BS and routed to ports
1 and 2 according to their phase difference. The carrier component, meanwhile,
is routed back to the incoming port 1 with isolation greater than 30 dB due to
constructive interference between clockwise and counter-clockwise waves in the
Sagnac loop. The polarization of the sidebands travelling from port 2 is
rotated by 90 degrees with the fiber Faraday rotator. The radiation from port
1 is guided by the optical circulator to a fiber polarization beam splitter
(PBS), where this radiation is combined with sidebands from port 2 having
orthogonal polarization. The polarization insensitive fiber Bragg grating
spectral filter (TeraXion OF) with r=0.99 and $\rho=10^{-4}$ separates the
prepared polarization state of the sidebands from the carrier. The light is
sent to free space via a fiber output coupler for further analysis. The
polarization of the sidebands is characterized with a conventional
polarization analyzer consisting of quarter and half waveplates together with
PBS, where both channels are coupled into single photon counting modules (ID
Quantique ID210) with quantum efficiency of $\epsilon=$10% and dark count rate
of $\gamma\sim$100 Hz. The detectors are gated at 100 MHz with a gate duration
of $\delta t=3.3$ ns. For this time window, 1.0 pW of incident carrier power
corresponds to a value of $\alpha_{0}\sim 0.15$ that results in a maximal
count rate of $\sim 10^{4}$ cps at single sideband. Multi-channel frequency
counter is used to monitor the count rates from both single photon counting
modules.

For estimating the conversion rate, we define the efficiency as the ratio
between the number of photons in sidebands with $m\neq 0$ before and after the
interface. The power efficiency of the conversion is determined by the highest
optical losses out of the losses in the two channels of the interferometer.
The identical losses are necessary for the indistinguishability of coherent
states interfering on a 50:50 beam splitter. The loss budget is composed of
insertion losses in the phase modulator (-3.1 dB), optical circulator (-0.7
dB), non-polarizing beam splitter 50:50 (-0.5 dB), Faraday rotator (-0.6 dB)
and polarization beam splitter (-0.2 dB). The losses of -4.5 dB in both
channel correspond to the efficiency of 35%.

##  IV Results and Discussion

First, we scan $\phi_{\text{in}}-\phi_{\text{LO}}$ between two SCW states by
manually changing phase of a single microwave generator. The normalized count
rate of horizontally and vertically polarized photons at different value of
the phase difference are presented in Fig. 2a, where each point is accumulated
during $T=10$ s. The interference with visibility of $\sim 95\%$ clearly
indicates the transfer of the phase information from SCW state into
polarization degree of freedom according to (4).

We measure visibility at different values of modulation depth $\beta$ with all
sidebands being detected simultaneously. The results are depicted in Fig 2b.
At a small value ($\beta<0.1$) of modulation depth, the number of detected
photons drops below the dark count rate that leads to low visibility. At a
large value of $\beta$ ($\beta>1$) the significant portion of energy is
distributed at higher order sidebands ($m\geq 2$) that interfere as
$\sim\cos(\frac{m(\phi_{in}-\phi_{\text{LO}})}{2})$ and reduce overall
visibility when all sidebands are detected simultaneously by a single
detector. At the same time, the visibility of $92\pm 3\%$ is achievable within
the range of $0.1\leq\beta\leq 0.7$. It is worth noting that the measured
decrease in visibility at low $\beta$ is not fundamental but rather
implementation specific and can be circumvented, if a detector with a lower
dark count rate and a spectral filter with a larger extinction ratio $r/(1-r)$
are used. In principle, the negative effect of the higher order ($|\,m\,|>1$)
sidebands can be mitigated by separating the first order sideband with an
additional spectral filter. However, the additional filter would introduce
extra losses and proportionally reduce the conversion efficiency.

We model the number of clicks with given polarization as:

$\displaystyle n_{H(V)}\approx\left(\gamma+\frac{\epsilon}{\delta
t}\langle\,\psi_{+(-)}\,|n|\,\psi_{+(-)}\,\rangle\right)\cdot T,$ (5)

where $n=\sum_{m\neq 0}\hat{a}^{\dagger}_{m}\hat{a}_{m}$ is number operator
with $\hat{a}_{m}$ ($\hat{a}^{\dagger}_{m}$) being annihilation (creation)
operator for m’th spectral mode, $\epsilon$ is the quantum efficiency of the
detector, $\delta t$ is gate duration time, $T$ is sampling time, $\gamma$ is
dark count rate. The experimentally acquired data of normalized clicks and
visibility is fitted using (5).

The polarization states on the opposite sidebands with m=1 and m=-1 have a
$\pi$ phase shift with respect to their polarization states, which results in
a mixed polarization for $\phi_{in}-\phi_{\text{LO}}\in\\{\pi/2,3\pi/2\\}$ if
both sidebands are measured simultaneously. To characterise the conversion, we
perform the polarization tomography for only a single sideband ($m=+1$) with
$\beta=0.15$ by adjusting the laser’s frequency with respect to the spectral
filter’s reflection band such that only one sideband is transmitted. For each
value of phase difference ($0,\pi,\pi/2,3\pi/2$) we measure polarization in
different polarization basis. The acquired data is used to reconstruct the
polarization density matrix by the maximum likelihood method Řeháček  _et al._
(2001). The reconstructed density matrices are presented in Fig. 3 with
corresponding fidelities summarised in Table 1.

The small imaginary part of non-diagonal elements of the density matrix we
attribute to a jitter in optical path difference between paths for
$|\,H\,\rangle$ (port 1 to PBS) and $|\,V\,\rangle$ (port 2 to PBS) photons.
The jitter introduces extra complex phase between $|\,H\,\rangle$ and
$|\,V\,\rangle$ that leads to unwanted ellipticity of the polarization. We
passively stabilized the optical scheme by placing it in thermally and
acoustically isolating foam and reducing the fiber connection length. However,
some sort of active stabilization may still be required for achieving fidelity
$\geq 99$%. It must be noted, that typical teleporation fidelity is around
$89\pm 2\%$ with current technologies Valivarthi  _et al._ (2020). Hence, the
interface with fidelity $\geq 99\%$ should not limit the teleportation in
practical applications.

$\phi_{in}-\phi_{LO}$ | Polarization state | Fidelity
---|---|---
$0$ | $|\,H\,\rangle$ |  $96\pm 3$ %
$\pi$ | $|\,V\,\rangle$ |  $96\pm 3$ %
$\pi/2$ | $(|\,H\,\rangle+|\,V\,\rangle)/\sqrt{2}$ |  $95\pm 3$%
$3\pi/2$ | $(|\,H\,\rangle-|\,V\,\rangle)/\sqrt{2}$ |  $95\pm 3$%
Table 1: The correspondence between the phase and polarization with the
achieved fidelities.

The proposed interface may be implemented in integrated quantum optical
fashion Sibson  _et al._ (2017). This approach would minimize the phase
fluctuations, device size and hence increase its stability. Moreover, this
technology would allow to build two identical spectral filters for effective
conversion of the external SCW state as in the proposed scheme. The lithium
niobate on insulator platform seems a promising candidate for implementing the
proposed interface. The principle components of the interface such as compact
phase modulators Wang  _et al._ (2018), high finesse microring resonator Zhang
_et al._ (2017), polarizing and non-polarizing beam splitter Boes  _et al._
(2018) were demonstrated.

In this case, this scheme can be used in SCW-QKD for simultaneous detection of
two states in a single basis. If single photon detectors are placed at the
output of each channel of a 50:50 beam splitter, the states with phases
$\phi_{\text{LO}}$ and $\phi_{\text{LO}}+\pi$ are distinguishable by a click
of the corresponding detector. In the appendix, we compare the theoretical
asymptotical key rate of the QKD with the interface used as two-orthogonal
state discriminator and the conventional approach Miroshnichenko  _et al._
(2018) with a single state detection. Simulation shows that use of the
proposed scheme in point-to-point SCW encoded QKD may increase the key rate by
about twice compared to the conventional approach.

It is worth mentioning that the potential eavesdropper may try to corrupt the
security of the communication between Charlie and Alice by photon number
splitting or unambiguous state discrimination attacks on multiphoton
components Horoshko  _et al._ (2019). For this reason, the carrier after the
phase modulator should be monitored by a photodetector (APD in Fig. 1 b) for
detecting the attacks with the strong reference method Koashi (2004). This
countermeasure imposes a condition of trustworthiness only on Alice’s node in
the quantum repeater but not the whole quantum repeater chain, which is a
reasonable cost for interconnecting the heterogeneous network. Alternatively,
one can implement additional phase states for detecting the attacks Gaidash
_et al._ (2022). At the same time, the number resolved detectors in a Bell-
state measuring device may mitigate the multiphoton issues in quantum
teleportation that occur with a practical source Krovi  _et al._ (2016).

In summary, we presented the way for conversion of phase information from SCW
into the dual rail encoding. The experimentally realized in-fiber conversion
from SCW into polarization encoding shows fidelities above 92%. If the first-
order sideband is additionally filtered from the higher-order sidebands, the
main limitations for further increasing visibility are the extinction ratio of
the spectral filter and interference stability. The proposed interface opens
one more way for connecting existing quantum networks with different physical
layers and increasing QKD distance and rate. The reverse conversion from dual
rail into SCW is also of particular interest for long-range transmission.
However, this topic may be a part of a separate study since it requires
mapping from discrete to continuous variables, which usually demands non-
classical resources and heralded preparation as in Ulanov  _et al._ (2017);
Sychev  _et al._ (2018).

###### Acknowledgements.

Authors appreciate support within framework project # 00075-02-2020-051/1 from
02.03.2020. Authors profound thank S.A. Moiseev for fruitful discussions. We
are grateful to M.M. Minnegaliev and M.A. Smirnov for lending us optomechanics
and fiber-optics.

##  Appendix A Key rate comparison

Figure 4:  a) The traditional one-way SCW encoded QKD. b) The use of the
proposed interface in one-way SCW encoded QKD for increasing key generation
rate. The principle schemes consist from phase modulator (PM), spectral filter
(SF), beam blocker (BB), non-polarizing beam splitter 50:50 (BS), optical
circulator (OC). Red solid and Blue dashed arrows indicate the carrier and
sidebands direction, respectively.

Figure 5:  a) Simulated key generation rate $K$ of point-to-point SCW encoded
QKD with the proposed interface used as state discrimination (black solid) and
conventional detection system (red dashed). b) Optimal values of amplitude
$|\,\alpha_{0}\,|$ (black bottom) and modulation depth $\beta$ (green upper)
for two-state discrimination (solid) and single discrimination (dashed). c)
Simulated key generation rate $K$ with the same parameters and labeling as in
a) but with $\gamma=1$ Hz. d) Optimal values of amplitude and modulation depth
with labeling as in b) for $\gamma=1$ Hz.

Next we present the theoretical comparison of secret key generation rate with
BB84 between traditional one-way SCW encoded QKD setup and SCW encoded QKD
with the proposed interface being used as two logical states discriminator.

###  A.1 Traditional scheme

The scheme of the original one-way SCW encoded QKD is depicted in Fig.4a. Here
Alice sends trains of pulses to Bob via an optical channel with losses $\eta$.
Each of the pulses is in SCW state, with subcarrier phase $\phi_{A}$ being
randomly set by Alice out of two sets that correspond to different bases. The
Alice’s state upon arrival to Bob is

$\displaystyle|\,\Psi_{A}\,\rangle=\bigotimes^{S}_{m=-S}|\,\sqrt{\eta}\alpha_{0}J_{m}(\beta)e^{im\phi_{A}}\,\rangle_{m},$
(6)

where all parameters are described in section “The principle scheme of the
interface" of the main text.

At his side, Bob applies phase modulation to the received state with the same
modulation depth $\beta$ but a randomly chosen value for the subcarrier phase
$\phi_{B}$. The modulated state is transmitted through the spectral filter
that separates the carrier ($m=0$) from the sidebands ($m\neq 0$). The
sidebands are transmitted with power coefficient $1-\rho$, while the carrier
is reflected with power coefficient $r$ and is transmitted with power
coefficient $1-r$. After the filter, a non-discriminating single photon
counter with quantum efficiency $\epsilon$ and a dark count rate $\gamma$
monitors the presence of a photon. At low intensity, the probability of the
detector clicking is

$\displaystyle
P_{\text{det}}(\phi_{a},\phi_{b})=\left(\epsilon\frac{n_{\text{ph}}(\phi_{A},\phi_{B})}{T}+\gamma\right)\delta
t,$ (7)

where the $n_{\text{ph}}(\phi_{A},\phi_{B})$ is a number of photons after the
spectral filter:

$\displaystyle
n_{\text{ph}}(\phi_{A},\phi_{B})=|\alpha_{0}|^{2}\eta\cdot\begin{cases}(1-\varrho)(1-J^{2}_{0}(2\beta))+(1-r)J^{2}_{0}(2\beta)\quad&\phi_{A}=\phi_{B},\\\
(1-r)\quad&\phi_{A}=\phi_{B}+\pi,\\\ \end{cases}$ (8)

where we assume negligibly small jitter between Alice’s and Bob’s modulation
oscillators over a repetition period and the linear regime of the detector.

Asymptotic secret key generation rate against beam splitter collective attack
$K$ is lower limited by Devetak-Winter bound Devetak and Winter (2005);
Miroshnichenko  _et al._ (2018):

$\displaystyle K=\nu P_{B}(1-f(Q)H(Q)-\chi(A:E)),$ (9)

where $\nu=1/T$ is a repetition rate and $P_{B}$ is probability of bit
detection at Bob’s side per pulse. Probability $P_{B}$ can be represented as a
total probability of detecting a photon with correct and wrong bit decoding.
Namely, it is a probability of the event that Alice encodes logical bit value
x and Bob get the same bit value x or opposite bit value:
$(x+1)\text{mod}~{}2$. For $x=0$ it is
$P_{B}=(P_{\text{det}}(0,0)+P_{\text{det}}(0,\pi))/2$, where factor $1/2$
comes from the 2 bases used in BB84 protocol. The product $f(Q)H(Q)$ defines
an amount of information revealed during error correction with $f(Q)$ being an
efficiency of error correcting protocol and $H(Q)$ being a binary Shannon
entropy:

$\displaystyle H(Q)=-Q\log_{2}(Q)-(1-Q)\log_{2}(1-Q).$ (10)

Here $Q$ is a quantum bit error probability (QBER), which is a ratio of
detection with false decoding to the probability of detection $P_{B}$:

$\displaystyle
Q=\frac{P_{\text{det}}(0,\pi)}{P_{\text{det}}(0,0)+P_{\text{det}}(0,\pi)}.$
(11)

The term $\chi(A:E)$ is Holevo bound for an amount of information that is
accessible to an eavesdropper for a given collective attack. We assume Eve
intercepts all photons lost in the optical channel between Alice and Bob and
records them into her quantum memory unit. For sake of simplicity, we assume
that Eve “emulates” losses by inserting a beam splitter with a power
reflectivity coefficient $1-\eta$ at the beginning of the optical channel. Eve
records the reflected photons into her memory, while she sends the transmitted
photon to Bob via a lossless optical channel. It was shown Miroshnichenko  _et
al._ (2018) that a collective beam splitter attack provides Eve with the
following information:

$\displaystyle\chi(A:E)=H\left(\frac{1}{2}\left(1-e^{-|\alpha_{0}|^{2}(1-\eta)(1-J_{0}(2\beta))}\right)\right).$
(12)

By combining (12) together with (11) and (10) in (9) we simulate the secret
key, assuming moderate efficiency of the error-correction protocol $f(Q)=1.25$
Assche (2006).

###  A.2 Scheme with two state discriminator

We consider the same one-way SCW-QKD as in previous case but with Bob being
using the interface for measuring both state in a given bases during a single
measurement. The sketch of the setup is presented in Fig. 4b, where Alice’s
state is sent on the interface which output is subjected to single photon
detection. As it is covered in the main text, the spectral filter separates
the carrier frequency from the sidebands of the Alice’s state. The carrier is
phase modulated with subcarrier phase $\phi_{B}=\phi_{\text{LO}}$. The second
spectral filter separates the generated sidebands from the carrier. The
generated sidebands are overlapped at the 50:50 beamsplitter, whose output
ports in modes “$+$" and “$-$" are monitored by a single photon counter with
dark count rate $\gamma$. The probabilities of the detectors in mode “$+$" or
“$-$" getting a click are

$\displaystyle
P_{\pm}(\phi_{A},\phi_{B})=\left(\epsilon\frac{n_{\pm}(\phi_{A},\phi_{B})}{T}+\gamma\right)\delta
t,$ (13)

where

$\displaystyle n_{\pm}(\phi_{A},\phi_{B})$ $\displaystyle=\sum_{m\neq
0}|\,\sqrt{\eta}\alpha_{0}J_{m}(\beta)(1-\varrho)\frac{e^{im\phi_{A}}\pm
e^{im\phi_{B}}rJ_{0}(\beta)}{\sqrt{2}}\,|^{2}$
$\displaystyle+|\,\sqrt{\eta}\alpha_{0}\frac{J_{0}(\beta)((1-r)\pm
J_{0}(\beta)r(1-r))}{\sqrt{2}}\,|^{2}.$ (14)

In the traditional scheme Bob discriminate particular state out of given basis
by a particular choice of a subcarrier phase $\phi_{B}$ that corresponds to
the target state. In contrast, in the proposed scheme Bob can discriminate two
state with subcarrier phase different by $\pi$ in a single measurement. Bob
chooses the particular basis by setting the phase of local oscillator
$\phi_{\text{LO}}$ of either 0 or $\pi/2$.

QBER is calculated as a ratio between the probability of an erroneous bit
decoding and the total probability of bit decoding from Alice to Bob

$\displaystyle Q=\frac{P_{E}}{P_{C}+P_{E}},$ (15)

where probability of erroneous bit decoding is

$\displaystyle P_{E}$
$\displaystyle=P_{+}(\phi_{A},\phi_{A}+\pi)(1-P_{-}(\phi_{A},\phi_{A}+\pi))$
$\displaystyle+P_{-}(\phi_{A},\phi_{A})(1-P_{+}(\phi_{A},\phi_{A}))$ (16)

and probability of correct bit decoding is:

$\displaystyle P_{C}$
$\displaystyle=P_{+}(\phi_{A},\phi_{A})(1-P_{-}(\phi_{A},\phi_{A}))$
$\displaystyle+P_{-}(\phi_{A},\phi_{A}+\pi)(1-P_{+}(\phi_{A},\phi_{A}+\pi)).$
(17)

The probability of Bob detecting a bit erroneously or correctly is modified to
$P_{B}=(P_{E}+P_{C})/2$ accordingly. Since Alice’s state has not changed, we
may use the same Holevo bound for the eavesdropper as in the traditional case.
Hence, the Eq.(15) together with Eq.(12) are used in Eq. (9) to estimate the
key generation rate cod .

Parameter | Description
---|---
$\epsilon=0.1$ | Detector quantum efficiency
$\gamma=50$ Hz | Detector’s dark count rate
$r=0.99$ | Filter’s carrier reflection
$\rho=-40$ dB | Filter’s sideband suppression factor
$\delta t=$3.3 ns | Gating time
$T=$10 ns | Pulse repetition period
Table 2: Model parameters

The key generation rates for the traditional and the proposed scheme are
presented in Fig. 5a with the parameters being summarized in Table 2. For each
value of $\eta$ the control parameters such as $\alpha_{0}$ and $\beta$ are
numerically optimized. The optimized values are depicted in Fig. 5b. The
proposed scheme demonstrates a key rate about twice as large as the
conventional scheme at the given set of parameters. However, the higher-order
sidebands ($|\,m\,|\gg 1$) in the proposed scheme introduce an error
proportional to $\alpha_{0}\sum_{|\,m\,|\geq 2}|\,J_{m}(\beta)\,|^{2}$ that is
absent in the conventional scheme. It becomes especially clear when using a
detector with lower dark count rates. The key rate for a model with the same
parameters but $\gamma=1$ Hz is depicted in Fig. 5c. At large values of losses
(>50 dB), the conventional protocol apparently takes over due to the mentioned
contribution of the higher-order sidebands. To reduce this effect, the optimal
$\beta$ for the proposed scheme is decreased at larger losses, as shown in
Fig. 5d. One way to circumvent this issue is to detect only one sideband,
which may also help mitigate the chromatic dispersion Kiselev  _et al._
(2020). It is worth noting, that we compare two QKD against the collective
beam-splitter attack for a purpose of differential comparison only and use the
same parameters for both case. For a complete security analysis more advanced
attacks Sajeed  _et al._ (2021) should be incorporated.

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# Unusual Diffusivity in Strongly Disordered Quantum Lattices: Random Dimer
Model

Ilia Tutunnikov  Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA  ITAMP, Center for Astrophysics | Harvard & Smithsonian, Cambridge, Massachusetts 02138, USA  Jianshu Cao<EMAIL_ADDRESS>Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

###### Abstract

Recent advances in transport properties measurements of disordered materials
and lattice simulations, using superconducting qubits, have rekindled interest
in Anderson localization, motivating our study of highly disordered quantum
lattices. Initially, our statistical analysis of localized eigenstates reveals
a distinct transition between weak and strong disorder regimes, suggesting a
random distribution of dimers in highly disordered systems. Subsequently, the
random dimer model predicts an oscillating diffusivity that decays as
$t^{-1/2}$, is inversely proportional to the disorder strength, and maintains
a constant frequency with an initial phase shift of $\pi/4$. The first peak
exhibits a universal scaling of $\sigma^{-1}$ both in peak time and amplitude.
Finally, we find that stochastic noise suppresses these oscillations and
induces hopping between localized eigenstates, resulting in constant diffusion
over long times. Our predictions challenge the conventional understanding of
incoherent hopping under strong disorder. This offers new insights to optimize
disordered systems for optoelectrical and quantum information technologies.

Nature abounds in various forms of disorder, evident in both natural and
synthetic materials. In his seminal work (Anderson, 1958), P. W. Anderson
introduced the concept of quantum (Anderson) localization of a single electron
wave function in a disordered lattice. Experimentally, observing localization
remains challenging due to interparticle interactions and lattice fluctuations
(Abanin  _et al._ , 2019). Despite these challenges, localization has been
demonstrated experimentally in various systems ranging from Bose-Einstein
condensates (Roati  _et al._ , 2008; Billy  _et al._ , 2008) to disordered
quantum circuits (Karamlou  _et al._ , 2022). In a broader context,
localization is now recognized as a universal wave phenomenon that can be
realized with microwaves (Chabanov  _et al._ , 2000), ultrasound waves (Hu
_et al._ , 2008), and light (Segev  _et al._ , 2013), and it plays a pivotal
role in the optical and transport properties of molecular solids (Davydov,
1971; Silbey, 1976; May and Kühn, 2011; Nitzan, 2006). Theoretically, our
understanding has advanced significantly through the development of scaling
theory (Abrahams  _et al._ , 1979) and quantum phase transition theory (Evers
and Mirlin, 2008), among others (Abrahams, 2010). Crucially, in one-
dimensional (1D) systems, any amount of disorder can localize all eigenstates,
whereas in 3D, localization depends critically on disorder strength.

While numerous studies have focused on weak disorder (Yamada  _et al._ , 1992;
Skipetrov and van Tiggelen, 2004; Cui  _et al._ , 2023) and diffusive
(Aslangul  _et al._ , 1989; Cao and Silbey, 2009; Hoyer  _et al._ , 2010; Moix
_et al._ , 2012; Chuang  _et al._ , 2016; Cao  _et al._ , 2020) regimes, the
transient diffusivity in highly-disordered 1D quantum lattices remains poorly
understood, which defines the focus of this Letter. Our study reveals the
nature of the oscillating diffusivity, which differs markedly from diffusivity
in ordered or weakly disordered lattices. Furthermore, thermal fluctuations
suppress the oscillations in diffusivity and induce hopping between localized
eigenstates, which leads to constant diffusion in the long-time limit. These
diverse phenomena are unified within the proposed ‘random dimer model’, which
challenges the conventional picture of incoherent hopping in the strong
disorder regime and aids in interpreting recent measurements in quantum dot
superlattices (Blach  _et al._ , 2022, 2023) and superconducting circuits
(Karamlou  _et al._ , 2022).

_Stationary distribution in a closed system._ –We consider an isolated 1D
quantum lattice consisting of $N$ sites described by the Anderson Hamiltonian
(Anderson, 1958; Thouless, 1974)

$\displaystyle\hat{H}$
$\displaystyle=\sum_{n}\epsilon_{n}\ket{n}\bra{n}+J\sum_{n}(\ket{n}\bra{n+1}+\ket{n+1}\bra{n}),$
(1)

where the site energies $\epsilon_{n}$ are Gaussian random variables with zero
mean and standard deviation $\sigma$, and $J$ is the coupling constant. In
this Letter, we set $\hbar=1$, energy is measured in units of $|J|$, and time
is measured in units of $1/|J|$. Additionally, the lattice is equally spaced
with lattice constant $a$, and distances are measured in units of $a$.

Figure 1: (a) Average stationary population distributions.
$P_{\epsilon}(n,t_{m})$ is added for reference, and is obtained by direct
solution of the time-dependent Schrödinger equation and averaging over
disorder realizations; $t_{m}$ is sufficiently large time. Here, $N=201$,
$\sigma=|J|$. (b) Standard deviation of the average stationary distribution as
a function of disorder strength.

When the 1D system is initiated with a single populated site $n=n_{0}$ at
$t=0$, a finite size stationary state is eventually reached for any disorder
strength, $\sigma>0$ (Abrahams  _et al._ , 1979; Domínguez-Adame and Malyshev,
2004). The population distribution can be characterized in two approximately
equivalent ways. First, we carry-out time averaging

$\displaystyle\overline{P}(n-n_{0};n_{0})$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}P(n-n_{0},t;n_{0})\,dt$
$\displaystyle=\sum_{k=1}^{N}|\braket{n_{0}}{v_{k}}|^{2}|\braket{n-n_{0}}{v_{k}}|^{2},$
(2)

where $\ket{v_{k}}$ are the eigenstates of $\hat{H}$. Then, we consider two
average distirbutions: (i) $\overline{P}(n;n_{0}=0)$ averaged over the random
site energies, $\overline{P}_{\epsilon}$, and (ii)
$\overline{P}(n-n_{0};n_{0})$ averaged over the initial position $n_{0}$,
$\overline{P}_{n_{0}}(n)$. $\overline{P}_{\epsilon}$ and
$\overline{P}_{n_{0}}$ are approximately equal [see Fig. 1(a)], because moving
the initial site $n_{0}$ is equivalent to generating a new random sequence of
site energies. In the thermodynamic limit $N\rightarrow\infty$, the two
distributions become equal, consistent with the concept of self-averaging.

There are several metrics for characterization of the stationary state width
(Moix  _et al._ , 2012; Moro  _et al._ , 2018). Here, we use the standard
deviation (with $\overline{P}_{\epsilon}$, for convenience)

$W(\sigma)=\sqrt{\sum_{n}n^{2}\overline{P}_{\epsilon}},$ (3)

Figure 1(b) suggests a qualitative transition between weak and strong disorder
regimes. For relatively weak disorder ($\sigma<|J|$), $W$ scales approximately
as $\sigma^{-2}$ (localization length scaling in 1D quantum lattice (Moix  _et
al._ , 2013)) . For strong disorder ($\sigma>10|J|$), the distributions is
highly localized ($W<1$), and $W$ scales approximately as $\sigma^{-1/2}$. See
Supplemental Material (SM) (Sup, ) for further qualitative discussion of the
scaling.

A more detailed picture is revealed when considering the standard deviations
of eigenstates $\ket{v_{k}}$, defined as

$w_{k}^{2}=\sum_{n}n^{2}|\\!\braket{n}{v_{k}}\\!|^{2}-\left[\sum_{n}n|\\!\braket{n}{v_{k}}\\!|^{2}\right]^{2}.$
(4)

Figure 2(a) shows that for $\sigma=20|J|$ the distribution of $w_{k}$ has a
major peak at $w_{k}\approx 0.05$ and a minor peak at $w_{k}\approx 0.5$, and
it rapidly decays for $w_{k}>0.5$. The major peak at $w_{k}=0.05$ corresponds
to states consisting of practically a single site, i.e., monomers. Figure 2(c)
shows ten eigenstates with $w_{k}\approx 0.5$ that mostly belong to dimers
denoted by curly brackets. Evolution of the initial state can be described by
representing the initial state as a superposition of lattice eigenstates. Due
to the demonstrated effective partitioning of the lattice into monomers and
dimers, the wave packet size is limited to a few sites.

Figure 2: (a, b) Distributions of eigenstate standard deviations, $w_{k}$ [see
Eq. (4)] for several values of disorder strength, $\sigma$. (c) Ten
eigenstates with $w_{k}\approx 0.5$ in a lattice of $N=51$ sites at
$\sigma=20|J|$. (d) Five eigenstates in a lattice of $N=51$ sites at
$\sigma=0.5|J|$.

As $\sigma$ decreases, the fraction of monomers diminishes. The tail of the
distribution becomes thicker for $\sigma<10|J|$, and the dimer picture breaks
down. Figure 2(b) shows the distributions for lower disorder strength. In this
regime, the distributions have a single peak which shifts to the right with
decreasing $\sigma$. Figure 2(d) shows five eigenstates at $\sigma=0.5|J|$
spanning multiple sites.

_Transient dynamics in the strong-disorder regime._ – To analyze the transient
wave packet spreading, we focus on the time-dependent rate of wave packet
expansion, i.e., the diffusivity

$2D(t)=\frac{d\braket{n^{2}}}{dt}-\frac{d\braket{n}^{2}}{dt},$ (5)

where $\braket{n^{k}}=\sum_{n}n^{k}P(n,t)$. In an ordered system, a localized
initial state expands ballistically, i.e., $\braket{n^{2}}=2J^{2}t^{2}$, and
$D(t)=2J^{2}t$. The ballistic expansion results from the overlap between the
initial state with all the extended eigenstates of the lattice (Bloch states).
In contrast, in a disordered 1D system, all the eigenstates have finite extent
(Anderson, 1958; Thouless, 1974; Domínguez-Adame and Malyshev, 2004) resulting
in transient expansion and eventual localization.

As suggested earlier, for sufficiently strong energy disorder, the chain is
effectively partitioned into monomers and dimers. Thus, we can approximate the
diffusivity in the chain by the diffusivity of a random dimer (Mazza and
Grigolini, 1998). This model traces its root to the standard tunneling model,
which was originally proposed to predict the thermal and transport properties
of amorphous solids (Phillips, 1972; Anderson  _et al._ , 1972; Heuer and
Silbey, 1993). Most significant is the application of the model to explain the
unusual spectral diffusion measured by early single molecule experiments (Geva
_et al._ , 1996; Moerner and Orrit, 1999). Evidently, the dimers in our highly
disordered Anderson model give rise to the random distribution of the
incoherent two-level systems (TLS) in the standard tunneling model.

Let the two sites be labeled by $0$ and $1$, and the initial state
$P(n,t=0)=\delta_{n0}$, the population on site $n=1$ at time $t$ reads

$P_{1}(t)=\frac{2J^{2}}{4J^{2}+\epsilon_{01}^{2}}\left[1-\cos\left(t\sqrt{4J^{2}+\epsilon_{01}^{2}}\right)\right],$
(6)

where $\epsilon_{01}=\epsilon_{0}-\epsilon_{1}$. Note that the monomers are
correctly accounted for in this model – when the energy difference between the
two sites is much higher than $|J|$, the bond is effectively broken. The
disorder-averaged diffusivity reads

$\displaystyle D(t)$
$\displaystyle=\frac{1}{2\sigma\sqrt{\pi}}\int_{-\infty}^{\infty}\dot{P}_{1}(t)\exp\left(-\frac{\epsilon_{01}^{2}}{4\sigma^{2}}\right)\,d\epsilon_{01},$
(7)

where we used the fact that $\epsilon_{01}$ is a normal random variable with
variance $2\sigma^{2}$. Asymptotically (for sufficiently large $t$ and
$\sigma\gg|J|$, see SM (Sup, ) for details),

$D_{A}(t)\sim\frac{|J|^{2/3}}{\sigma}\frac{\sin(2|J|t+\pi/4)}{\sqrt{t}}.$ (8)

Figure 3(a) compares the diffusivity obtained numerically by solving the time-
dependent Schrödinger equation for a long lattice vs the asymptotic formula in
Eq. (8). For $t>1/|J|$, the diffusivity oscillates at frequency $2|J|$; the
amplitude is proportional to $(\sigma\sqrt{t})^{-1}$; and the first peak
introduces a phase shift of $\pi/4$. This behaviour differs markedly from the
behaviour at low-moderate disorder, where the diffusivity has a single peak,
and decays steadily to zero after that (Blach  _et al._ , 2023; Cao Group,
2025). The discrepancy between the numerically exact diffusivity and Eq. (8)
grows with decreasing $\sigma$, due to the increasing contributions from
larger clusters (trimers, etc., see SM for an example (Sup, )). Within the
random dimer model, $W\propto\sqrt{D_{A}}\propto\sigma^{-1/2}$ consistent with
the scaling in Fig. 1 (also see SM (Sup, )).

Figure 3: (a) Diffusivity of a localized initial state in a lattice of $N=21$
sites vs the asymptotic diffusivity of a disordered dimer, $D_{A}(t)$ [see Eq.
(8)]. (b) Diffusivity of Gaussian initial state of width $w$ vs $D_{A}(t)$.
Superimposed high frequency noise diminishes with increasing number of site
energies realizations used in averaging. Here, $\sigma=20|J|$.

Close to $t=0$, the diffusivity grows linearly, and $\dot{D}=2J^{2}$ as in an
ordered lattice. After the first peak, the diffusivity continues approximately
according to Eq. (8). In the turnover region, the diffusivity is approximately
given by (see SM for details (Sup, ))

$\displaystyle D(t)$
$\displaystyle\approx\frac{\sqrt{\pi}J^{2}}{\sigma}(1-J^{2}t^{2})\text{erf}(\sigma
t).$ (9)

Figure 4(a) compares $D(t)$ near the turnover in a chain and a dimer. For
$\sigma/|J|\geq 15$, the approximate expression captures the peak height
relatively well. The turnover time, $t_{p}$ (a point where $\dot{D}=0$) is
approximately given by (see SM for details (Sup, ))

$t_{p}\approx\frac{1}{\sigma\sqrt{2}}\sqrt{\ln\left(\frac{2\sigma^{4}}{\pi
J^{4}}\right)-\ln\left[\ln\left(\frac{2\sigma^{4}}{\pi J^{4}}\right)\right]}.$
(10)

With increasing $\sigma$, $t_{p}$ scales slower than $\sigma^{-1}$. Figures
4(b) and 4(c) compare the numerically exact $t_{p}(\sigma)$ and $D(t_{p})$ in
a dimer and a chain vs the analytical approximations. While the dimer model
captures the decay of $t_{p}(\sigma)$ correctly, the values are slightly
underestimated, since the chain still contains some fraction of larger
clusters (trimers, etc.).

Figure 4: (a) Diffusivity near turnover, numerical results in a chain (solid)
vs dimer model (dashed). (b, c) Numerically exact results in dimer (solid) and
chain (triangles) vs. Eqs. (9) and (10) (squares).

Preparing an initial state with a single excited site is feasible in special
systems, e.g., in highly controllable quantum superconducting circuits
(Karamlou  _et al._ , 2022). In contrast, optical excitation results in
initial spatially extended states (Blach  _et al._ , 2022, 2023). Generally,
in presence of either disorder or noise, the transient diffusivity of
spatially extended initial states qualitatively differs from the diffusivity
of a localized initial state (Cui  _et al._ , 2023; Tutunnikov  _et al._ ,
2023). In the strong disorder regime, however, the picture simplifies
considerably. Figure 3(b) shows the time-dependent diffusivity of Gaussian
initial states of various initial widths, $w$. Diffusivity is nearly
independent of $w$, which stems from the effective partitioning of the lattice
into monomers and dimers due to disorder. The extended initial state overlaps
with multiple localized eigenstates, and the disorder averaging results in a
diffusivity close to the diffusivity of a random dimer, $D_{A}(t)$ in Eq. (8).

_Diffusivity in an open system._ – Next, we consider the diffusivity in an
open quantum system coupled to a noisy environment. Long-time diffusion in
strongly disordered lattices is an intriguing problem, where traditional
random walk models, such as the Forst or Marcus rate, fail to capture
transient quantum coherence features, like slow non-exponential relaxation,
and quantum beatings. Here, the environment influence is accounted for by
introducing stochastic site energy fluctuations within the framework of the
Haken-Strobl-Reineker (HSR) model (Kenkre and Reineker, 1982). In the HSR
model the reduced density matrix obeys the Lindblad master equation

$\displaystyle\dot{\rho}$
$\displaystyle=-i[H,\rho]-\frac{\Gamma}{2}\sum_{n}[V_{n},[V_{n},\rho]],$ (11)

where $(V_{n})_{j,k}=\delta_{j,n}\delta_{k,n}$, and $\Gamma$ is the dephasing
rate. We can rewrite Eq. (11) as $\dot{\rho}=-i[H,\rho]-\Gamma\rho_{H},$where
$\rho_{H}$ is a ‘hollow density matrix’, with elements
$(\rho_{H})_{m,n}=(1-\delta_{m,n})\rho_{m,n}.$ The latter form shows that the
$\Gamma$ term damps the off-diagonal elements of the density matrix
(coherences). The diffusivity of a localized initial state in the ordered HSR
model is given by (Moix  _et al._ , 2013) $D(t)=(2J^{2}/\Gamma)[1-\exp(-\Gamma
t)].$ On the short time scale ($\Gamma t\ll 1$), when the dephasing effect is
negligible, one recovers the ballistic behavior, i.e., $D(t)\propto t$. In
contrast, on the long time scale, the coherence is lost and
$D(t\rightarrow\infty)=2J^{2}/\Gamma$.

Figure 5(a) compares the diffusivities in isolated and HSR lattices in the
regime of strong disorder. In contrast to the closed system, the diffusivity
oscillations are further damped, and it slowly tends to a non-zero steady-
state value. Asymptotically, the diffusivity of a random HSR dimer is a sum of
two terms (see SM for details (Sup, ))

$\displaystyle D_{A}^{HSR}(t)$
$\displaystyle=D_{1}^{HSR}(t)+2\mathrm{Re}[D_{2}^{HSR}(t)],$ (12a)
$\displaystyle D_{1}^{HSR}(t)$
$\displaystyle\sim\frac{|J|}{2\sigma}\sqrt{\frac{\Gamma}{t}}-\frac{J^{2}\Gamma}{\sigma^{2}},$
(12b) $\displaystyle 2\mathrm{Re}[D_{2}^{HSR}(t)]$
$\displaystyle\sim\frac{\sqrt{2}J^{2}e^{-\Gamma
t/2}}{\sigma\sqrt{t}(\Gamma^{2}+4J^{2})^{1/4}}$
$\displaystyle\times\sin\left[2t|J|\\!-\\!\frac{1}{2}\arg(\Gamma\\!-\\!2i|J|)\right],$
(12c)

where $D_{1}^{HSR}(t)$ is a slowly decaying baseline, and $D_{2}^{HSR}(t)$
describes the oscillations. Figure 5(b) compares the exact diffusivity of a
HSR dimer with the asymptotic formulas. For $\Gamma=0$, the baseline vanishes,
while $D_{2}^{HSR}(t)$ turns into Eq. (8). The differences in the oscillating
term in Eq. (12c) and the diffusivity of the isolated dimer in Eq. (8) are:
(i) in the HSR model, the oscillations are damped by the exponential factor,
$\exp(-\Gamma t/2)$, (ii) the amplitude depends on gamma,
$\propto\sigma^{-1}(\Gamma^{2}+4J^{2})^{-1/4}$, and (iii) there is an
additional $\Gamma$-dependent phase shift $\arg(\Gamma-2i|J|)/2$. Note that in
a dimer, $D(t\rightarrow\infty)\rightarrow 0$, thus, the constant term in
$D_{1}^{HSR}(t)$ is a consequence of the approximations made (see SM for
details (Sup, )).

On the short time scale, the height of the turnover point is practically
insensitive to the dephasing rate (for $0\leq\Gamma<|J|$), see Figs. 5(a) and
5(b). In contrast, the first minimum at $t=t_{m}\approx 5\pi/(8|J|)\approx
2/|J|$ becomes shallower with increasing $\Gamma$, as there is enough time for
the noise effect to appear at this point. The position of the first minimum
remains approximately fixed (for $0\leq\Gamma<|J|$) allowing to approximate
$D_{A}^{HSR}(t=t_{m};\Gamma)\approx D_{A}^{HSR}(t=2/|J|;\Gamma)$. In 5(c), we
show the $\Gamma$-dependence of the first minimum depth. In the considered
examples, the dimer model slightly overestimates the minimum depth.

Figure 5: (a) Diffusivities in isolated and HSR lattices ($N=21$). (b)
Diffusivity in disordered HSR lattice compared to $D_{A}^{HSR}$ in Eq. (12a).
(c) First minimum of $D(t)$ as a function of $\Gamma$. Here, $\sigma=20|J|$.

While Eqs. (12a)-(12c) are good approximations for a dimer within the HSR
model, they fail to capture the long-time behavior in a chain [see Fig. 5(a)].
Noise destroys the localization ultimately resulting in classical-like
diffusion (Moix  _et al._ , 2013; Chuang  _et al._ , 2016), while in the dimer
model the diffusivity tends to zero. The mechanism behind the transition from
the oscillatory diffusivity to classical-like diffusion is the noise-induces
hopping between eigenstates. The mechanism can be understood by considering
the equations of motion for the density matrix expressed in the lattice’s
eigenbasis (see SM (Sup, ))

$\displaystyle\dot{\tilde{\rho}}_{i,j}$
$\displaystyle=-i(\omega_{i,j}-i\Gamma)\tilde{\rho}_{i,j}+\Gamma\sum_{k,l}\kappa_{k,l}^{(i,j)}\tilde{\rho}_{k,l},$
(13a) $\displaystyle\kappa_{k,l}^{(i,j)}$
$\displaystyle=\sum_{m}U_{i,m}U_{m,j}^{\dagger}U_{m,k}^{\dagger}U_{l,m},$
(13b)

where $\tilde{\rho}$ is the reduced density matrix expressed in the
eigenbasis, $\omega_{i,j}$ is the difference of $i$-th and $j$-th
eigeneneregies, and $U$ is an orthogonal matrix with the real eigenvectors
arranged in rows. These equations shows that, in contrast to the closed
system, the populations of different eigenstates are coupled, allowing the
particle to spread beyond the initial region.

Equations (13a) and (13b) can be further simplified by applying the often used
secular approximation (Nitzan, 2006) that captures the long-time dynamics. In
this approximation, the eigenstates’ populations are decoupled from the
coherences, and obey (see SM for details (Sup, ))

$\displaystyle\dot{\tilde{\rho}}_{i,i}$
$\displaystyle=-\Gamma\tilde{\rho}_{i,i}+\Gamma\sum_{j}\kappa_{j,j}^{(i,i)}\tilde{\rho}_{j,j},$
(14)

where the coupling constants, $\kappa_{j,j}^{(i,i)}$ are determined by the
overlap of eigenstates’ probability densities,
$\kappa_{j,j}^{(i,i)}=\sum_{m}|U_{i,m}|^{2}|U_{j,m}|^{2}$. The distributions
of eigenstate widths in Fig. 2(a) show that in the strong disorder regime all
the eigestates are highly localized and correspond mainly to dimers and
monomers. Thus, Eq. (14) shows that in an open system the excitation hops from
the initially populated eigestates to adjacent spatially overlapping
eigenstates, and the hopping rate between eigenstates $i$ and $j$ is
$\Gamma\kappa_{j,j}^{(i,i)}$. The dashed line (HSR appr.) in Fig. 5(a) shows
the diffusivity obtained using Eq. (14), which is in excellent agreement with
the full HSR model on the long time scale.

_Conclusions._ –To summarize, this Letter systematically explored the
transient dynamics in a highly disordered 1D quantum lattice. The scaling of
the localization width with disorder allows us to identify a transition
between the weak and strong disorder regimes. In the strong disorder regime,
the lattice can be approximately partitioned into monomers and dimers, thus
establishing the random dimer model. The resulting diffusivity exhibits
unusual oscillations, with amplitude decaying as $t^{-1/2}$ and inversely
proportional to the disorder strength. The first peak introduces a phase shift
of $\pi/4$, with the peak time and amplitude inversely proportional to the
disorder strength. In an open system, noise suppresses the oscillations on
short time scales, relaxes localization on intermediate scales, and leads to
constant diffusion for asymptotically large times. The mechanism underlying
the phenomenon is noise-induced hopping between lattice eigenstates.

The scaling predictions in this Letter not only help interpret recent
measurements but can also be quantitively tested on several platforms.
Recently, Anderson localization in 1D and 2D quantum lattices was emulated by
a fully controllable array of superconducting qubits (Karamlou  _et al._ ,
2022). The experimental results confirm the oscillatory behavior of the mean
squared displacement in the strong disorder regime and suggest the possibility
of experimentally verifying the quantitative predictions reported in this
Letter. Recent microscopy measurements of exciton dynamics in disordered
solids (Akselrod  _et al._ , 2014; Delor  _et al._ , 2020; Blach  _et al._ ,
2022, 2023) also reveal oscillatory diffusivity, demonstrating the striking
differences between weak and strong disorder regimes. In follow-up papers, the
current study will be unified with a quantitative analysis of weak-
intermediate disorder (Cao Group, 2025) and is being extended to cavity
polaritons (Engelhardt and Cao, 2023; Md Qutubuddin et. al., 2024). Together,
this and subsequent studies will provide insights into optimizing transport
properties for optoelectrical and quantum information applications.

###### Acknowledgements.

The work is supported by the NSF (Grants No. CHE1800301 and No. CHE2324300),
and the MIT Sloan fund, and is partly motivated by the joint experiment-theory
study in (Blach  _et al._ , 2023).

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