# Season combinatorial intervention predictions with Salt & Peper

Thomas Gaudelet, Alice Del Vecchio, Eli M Carrami, Juliana Cudini &
Chantriolnt-Andreas Kapourani
Relation Therapeutics
London, UK
{thomas, alice, eli, juliana<EMAIL_ADDRESS>
&Caroline Uhler
Laboratory for Information and Decision Systems, MIT
Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard
Boston, USA
<EMAIL_ADDRESS>Lindsay Edwards
Relation Therapeutics
London, UK
<EMAIL_ADDRESS>
###### Abstract

Interventions play a pivotal role in the study of complex biological systems.
In drug discovery, genetic interventions (such as CRISPR base editing) have
become central to both identifying potential therapeutic targets and
understanding a drug’s mechanism of action. With the advancement of CRISPR and
the proliferation of genome-scale analyses such as transcriptomics, a new
challenge is to navigate the vast combinatorial space of concurrent genetic
interventions. Addressing this, our work concentrates on estimating the
effects of pairwise genetic combinations on the cellular transcriptome. We
introduce two novel contributions: Salt, a biologically-inspired baseline that
posits the mostly additive nature of combination effects, and Peper, a deep
learning model that extends Salt’s additive assumption to achieve
unprecedented accuracy. Our comprehensive comparison against existing state-
of-the-art methods, grounded in diverse metrics, and our out-of-distribution
analysis highlight the limitations of current models in realistic settings.
This analysis underscores the necessity for improved modelling techniques and
data acquisition strategies, paving the way for more effective exploration of
genetic intervention effects.

##  1 Introduction

Interventions are an essential tool when striving to decode the functioning of
complex systems and in particular, to identify causal dependencies (as
exemplified by causal learning theory (Eberhardt & Scheines, 2007)). In the
field of biology, researchers routinely use chemical or genetic perturbations
to probe and manipulate cellular systems. Notably, interventions are of
fundamental importance to drug discovery, where the goal is to design an
intervention that can suppress or even reverse disease-related biological
processes in targeted and specific ways. Genetic interventions are of special
interest as they allow more precise manipulation of individual genes, as
opposed to (for example) small molecules that often have multiple cellular
targets and wide-ranging effects that are poorly understood (Hopkins, 2008;
Lin et al., 2017). Recent advancements in CRISPR technologies (that allow
precise editing of genetic sequences) have empowered scientists to unravel the
intricate workings of biological systems (Dixit et al., 2016; Frangieh et al.,
2021; Norman et al., 2019; Replogle et al., 2022). From a drug discovery
standpoint, manipulating specific elements and observing outcomes enables
scientists to pinpoint promising hypotheses for the development of new
therapeutics (Liu et al., 2020; Chavez et al., 2023; Kim & Lee, 2023) or
understand the mode of action of existing ones (Jost & Weissman, 2018).

The application of deep learning to the task of estimating intervention
effects in biological systems is a relatively recent development (Ma et al.,
2018; Lotfollahi et al., 2019; Zhang et al., 2023). With the emergence of
large, single-cell resolution datasets, there has been a rapid increase in
research activity aiming to model gene expression patterns (Lopez et al.,
2018) and how they are impacted by interventions (Roohani et al., 2023;
Lotfollahi et al., 2023). However, as elsewhere, uncertainty remains about the
best task framing and most appropriate metrics. For example, straightforward
performance metrics (such as the RMSE or correlation coefficient between
predicted and actual gene expression) often fail to appropriately weight
signals that are likely to be of importance to biologists. By contrast,
selected researchers have moved to using custom performance metrics (e.g. the
error on only the tails of the predicted distributions) that capture more
relevant detail. As a modality of interest, the transcriptome has a number of
benefits. First, there are established and widely used technologies, such as
scRNA-seq, that accurately measure gene expression levels at single cell
resolution. Second, transcriptome variation captures low-level cellular states
that often proxy higher-level effects, acting as a generic foundation for
predictive models designed for specific phenotypic endpoints (this concept –
that a phenotype can be effectively modelled by a gene expression pattern,
sometimes referred to as a ’gene signature’ – is widely used in drug discovery
and biology). Because of this, we can motivate research and model development
around transcriptomics using the same logic as foundation models in machine
learning: addressing a low-level, generic task that can be built upon for
specific applications.

Progress in CRISPR technologies that measure interventional impacts on a
cell’s transcriptome has opened the door to genome-scale analyses in which
most / many genes can be intervened upon individually (Replogle et al., 2022).
Although conducting a genome-wide perturb-seq (the name used for an experiment
where all genes are intervened upon, and the transcriptomic effect measured at
the single-cell level) screen remains prohibitively expensive at present, it
is anticipated that costs will drop considerably in the coming years. The
primary challenge moving forward is expected to revolve around understanding
the complex effects of _combinatorial interventions_ given the vast number of
potential combinations (approximately ${2\times 10^{4}\choose n}$, where $n$
is the number of simultaneous perturbations). Further, studying combinatorial
interventions becomes critical due to the limitations of single-target
approaches in tackling the complexity of biological systems (Hopkins, 2008).
Combinatorial strategies perturbing multiple genes may help unravel the
organisation of complex networks, overcoming redundancies and disrupting
adaptive responses. This is perhaps best exemplified by the combinatorial
therapeutics leveraged to treat some forms of cancer (Shen et al., 2017; Kim &
Lee, 2023).

In this work, we focus only on pairwise genetic combinations due to existing
constraints in available public datasets. Additionally, as we expect the cost
of genome-scale screens to decrease, we assume that we have already measured
the outcomes of targeting each individual gene. Our key contributions are

  * (1)

a careful comparison across current state-of-the-art methods using different
metrics and grounded with Salt, a biologically-motivated, non-parametric
baseline making the assumption that combination effects are mostly additive,

  * (2)

Peper, a deep learning model built from the additive inductive bias from Salt
and achieving state-of-the-art performances,

  * (3)

an out-of-distribution analysis indicating that current solutions are ill-
suited to address realistic scenarios and calling for better modelling
approaches and data acquisition strategies.

##  2 Related work

##### Context transfer.

The task at hand can be related to the context transfer problem, whereby the
effect of an intervention has been measured in some context, for instance a
cell line, and we aim to predict the effect it would have in a different
context (e.g. a different cell line). This task has been garnering interest
from the deep learning community for some years, particularly for
transcriptomics response prediction (Lotfollahi et al., 2019; Wu et al., 2022;
Lotfollahi et al., 2023). Notably, both context transfer and combinatorial
interventions tasks can be addressed in the same framework, as exemplified by
CPA (Lotfollahi et al., 2023). Conceptually, the subtle difference between the
two tasks arise from the sequential nature of context transfer, the context
precedes the intervention, as opposed to the parallel nature of the task at
hand, i.e. all targets are affected simultaneously.

##### Potential outcomes and counterfactual estimation.

A large majority of approaches (Lotfollahi et al., 2019; Wu et al., 2022;
Lotfollahi et al., 2023) tackling either the context transfer or combinatorial
intervention tasks aim to predict what state an input cell, typically a
control cell, would have been in had it been treated or intervened upon. These
can naturally be viewed under the lens of counterfactual estimation from the
potential outcomes causal framework (Rubin, 2005), which is concerned with
resolving similar statements in order to estimate causal effects.

##### Modelling cellular dynamics.

Generating counterfactual predictions would collapse to a simpler task given a
complete model of cellular dynamics that can be sampled at will. Despite a
long history of this class of models in metabolism (e.g. (Edwards et al.,
2011)), dynamic models of the complexity of transcription are rarer (Erbe et
al., 2023; Bhaskar et al., 2023; Ishikawa et al., 2023); the lack of large-
scale, time-resolved, interventional datasets has limited the development of
methods targeting our task of interest through the modelling of cellular
dynamics.

##  3 Methods

\begin{overpic}[width=357.73405pt]{fig1.png}
\par\par\put(1.5,35.0){\hbox{\pagecolor{white}$\displaystyle Intervention$}}
\put(6.1,32.4){\hbox{\pagecolor{white}$\displaystyle control$}}
\put(6.1,29.9){\hbox{\pagecolor{white}$\displaystyle p$}}
\put(6.1,27.5){\hbox{\pagecolor{white}$\displaystyle q$}}
\put(6.1,24.75){\hbox{\pagecolor{white}$\displaystyle p+q$}}
\par\put(11.0,15.0){$\displaystyle\bm{\mu}_{q}$}
\put(25.0,10.0){$\displaystyle\bm{\mu}_{p}$}
\put(28.0,16.0){$\displaystyle\textsc{Salt}$}
\put(55.75,27.5){$\displaystyle\textsc{Peper}$}
\put(77.5,33.0){$\displaystyle\textsc{Peper}$}
\put(87.0,3.25){$\displaystyle\bm{\mu}_{q}$}
\put(73.5,3.25){$\displaystyle\bm{\mu}_{p}$}
\par\put(73.0,10.0){$\displaystyle f_{\bm{\theta}_{1}}$}
\put(87.0,10.0){$\displaystyle f_{\bm{\theta}_{1}}$}
\put(80.0,27.0){$\displaystyle f_{\bm{\theta}_{2}}$} \end{overpic} Figure 1:
Illustration of Salt and Peper predictions for the combinations of
interventions on genes $p$ and $q$.

###  3.1 Notation

In the following, we define $\mathbf{X}_{p}$ as the matrix representing cells
measured after gene p intervention and $\mathbf{\tilde{X}}^{m}_{p}$ as the
matrix of estimated cells for the same intervention predicted by method $m$,
with both matrices structured such that rows correspond to individual cells
and columns to measured genes. We use the letter $c$ to refer to control cells
which have only been treated with non-targeting guides and are typically used
as controls experimentally. We use lower case $\mathbf{x}$ to denote a single
cell, i.e. a row of a matrix as defined previously. If the intervention
correspond to a pairwise combination, we write $\mathbf{X}_{p+q}$ where $p$
and $q$ indicate the two genes targeted.

We introduce the average intervention effect vector as the average difference
between perturbed cells and control cells, we denote it by
$\bm{\mu}_{p}=\overline{\mathbf{X}}_{p}-\overline{\mathbf{X}}_{c}$, where
$\overline{\mathbf{X}}$ denotes the mean over the columns of $\mathbf{X}$,
also known as pseudo-bulk, and gives the average expression of each gene
across cells.

###  3.2 Existing art

We use two published baselines to ground our work: GEARS (Roohani et al.,
2023), the current state-of-the-art, and CPA (Lotfollahi et al., 2023).

CPA follows from a long lineage of autoencoder architectures applied to
single-cell (sc)RNA-seq data (Lopez et al., 2018; Lotfollahi et al., 2019; Wu
et al., 2022). In CPA, the authors model combinatorial effects as an additive
process in a learnt (potentially nonlinear) latent space.

By contrast, GEARS takes its inspiration from the causal and graph neural
network literature, relying on graph mutilation of biological graph priors and
the message passing paradigm. In practice, GEARS applies a fixed,
intervention-specific translation to an input cell. Formally, given a control
cell input (i.e. a cell treated with non-targeting guides), an intervention
targeting gene p and q jointly, GEARS’s counterfactual prediction can be
written as

$\mathbf{\tilde{x}}^{GEARS}_{p+q}=\mathbf{x}_{c}+g_{\bm{\phi}}(p,q|\mathcal{G}),$

where $g_{\bm{\phi}}$ is a graph neural network architecture with parameters
$\bm{\phi}$ and $\mathcal{G}$ denotes the biological graph priors.

###  3.3 Salt

The aim of the study of combinations is often the identification of
interactions such as synergy or antagonism (Norman et al., 2019; Bertin et
al., 2023). However, these interactions are fairly rare (Martin, 2023) and
notably genetic variations tend to be largely additive (Hill et al., 2008). As
such, we expect that a simple additive heuristic would provide a strong non-
parametric baseline for the task. We refer to it as Salt (Simply Assume Linear
combinations of Transcriptomes) and an illustration can be seen on Figure 1.
Following our notations, we can write Salt as

$\mathbf{\tilde{x}}^{\textsc{Salt}}_{p+q}=\mathbf{x}_{c}+\bm{\mu}_{p}+\bm{\mu}_{q}.$

It is worth noting that similar vector arithmetic baselines in transcriptome
space have been used for the context transfer task (Lotfollahi et al., 2019).

###  3.4 Peper

We spice up Salt by introducing a learnable non-linear correction term. This
is reminiscent of statistical models that add a non-interactive term with
interactive terms to model and study synergistic effects (Norman et al., 2019;
Rønneberg et al., 2021). The resulting method – Perturbation Effect Prediction
by Error Reduction (Peper) – can be written as

$\displaystyle\mathbf{\tilde{x}}^{\textsc{Peper}}_{p+q}=\mathbf{\tilde{x}}^{\textsc{Salt}}_{p+q}+f_{\bm{\theta}}(\bm{\mu}_{p},\bm{\mu}_{q}),$

where $f_{\bm{\theta}}$ is a neural network with parameters $\bm{\theta}$. In
practice, we decompose $f_{\bm{\theta}}$ as follows

$f_{\bm{\theta}}(\bm{\mu}_{p},\bm{\mu}_{q})=f_{\bm{\theta}_{2}}(f_{\bm{\theta}_{1}}(\bm{\mu}_{p})+f_{\bm{\theta}_{1}}(\bm{\mu}_{q})),$

with both $f_{\bm{\theta}_{1,2}}$ corresponding to multi-layer perceptrons.
See Figure 1 for illustration. From the definitions, it is important to remark
that GEARS, Salt and Peper would at best capture the average intervention
effect vector, $\bm{\mu}_{p+q}$.

To train Peper we use the Central Moment Discrepancy (CMD) which is a
distributional loss that compares moments (Zellinger et al., 2016). Because
Peper only models the average effect of interventions, we only use the first
moment to compute the loss. Note that with this loss, we batch the data by
intervention, i.e. each batch contains data from a single intervention.

##  4 Experiments

###  4.1 Evaluation

##### Datasets.

We use two distinct CRISPR interventional datasets to evaluate prediction of
the outcome of genetic combinations for our experiments (Norman et al., 2019;
Wessels et al., 2023). Details can be found in Appendix A.

##### Splits.

For each datasets, we define two in-distribution splits, in-distribution-25
and in-distribution-75, and one out-of-distribution split based on
similarities between intervention effects in transcriptomics space, reasoning
that the task is rendered more challenging if a model is forced to estimate
out-of-distribution transcriptomic responses. We detail the split creation in
Appendix A.

##### Hyperparameters selection.

We tune each model on each split, using the validation sets to identify the
best configurations. For CPA and GEARS, we use the tuning configurations
provided by the authors. For Peper, we provide the hyperparameters and
associated ranges considered in Appendix D. We retrain each model post-tuning
with the identified best configurations using $5$ different seeds.

##### Metrics.

Following recommendations from Ji et al. (2023), we use root mean squared
error (RMSE) as our primary metric to evaluate models’ predictions (see
Appendix B for details). We also include energy distance scores, a
distributional metric, in the Appendix. Following previous work (Roohani et
al., 2023), we calculate metrics in two ways: for all genes (labelled All in
tables) and for the top $20$ genes ranked by absolute effect sizes (labeled
Top 20 ES in tables). These sets of $20$ genes are naturally intervention-
specific. As noted in the Introduction, metrics computed on a smaller,
meaningful set of genes are generally preferred by the community as many lowly
expressed genes are noisy and the effect of genetic interventions tend to be
observed on a small set of genes (Peidli et al., 2024). We include both for
completeness and report metrics computed on the held-out test sets using the 5
models obtained from the different seeds.

###  4.2 Results

Peper achieves state-of-the-art for in-distribution splits, as shown in Tables
1 and 2, indicating that our recipe relying on a strong inductive bias
motivated by biological observations is an effective approach to the problem.
The relatively good performances of the Salt heuristics also speak to the
quality of the inductive bias. In particular, it is notable that Salt beats
both CPA and GEARS in the hardest in-distribution-75 split of the norman-
dataset. Our results demonstrate that the inductive bias provided by Salt’s
prior is more useful to the task at hand than the priors that GEARS relies on.

Table 1: RMSE results on the in-distribution-25 splits. The numbers correspond to average score and standard deviation (in parenthesis) across 5 different seeds. | norman-dataset | wessels-dataset
---|---|---
Model | All | Top 20 ES | All | Top 20 ES
Peper | 0.030 (0.002) | 0.104 (0.006) | 0.034 (0.001) | 0.094 (0.002)
GEARS | 0.029 (0.001) | 0.117 (0.0099) | 0.034 (0.001) | 0.101 (0.004)
CPA | 0.032 (0.002) | 0.119 (0.008) | 0.036 (0.000) | 0.131 (0.007)
Salt | 0.034 | 0.124 | 0.061 | 0.143
Table 2: RMSE results on the in-distribution-75 splits. The numbers correspond
to average score and standard deviation (in parenthesis) across 5 different
seeds.
| norman-dataset | wessels-dataset
---|---|---
Model | All | Top 20 ES | All | Top 20 ES
Peper | 0.030 (0.000) | 0.118 (0.000) | 0.043 (0.001) | 0.112 (0.005)
GEARS | 0.038 (0.001) | 0.145 (0.011) | 0.038 (0.002) | 0.113 (0.005)
CPA | 0.032 (0.000) | 0.139 (0.004) | 0.071 (0.004) | 0.164 (0.018)
Salt | 0.033 | 0.134 | 0.065 | 0.160

We investigate further model performances on the in-distribution-75 split for
the norman-dataset by grouping interventions based on the types of interaction
identified by Norman et al. (2019).The definition for the different
interaction types can be found in Appendix E. Overall, we observe similar
performances across the different groups, with Peper outperforming all other
methods (see Figure 2). However, the potentiation group stands out with
significantly worse performances for all models. Note that potentiation occurs
when one of the two interventions combined has no effect on its own but
enhances the effect of specific interventions when combined. Given this
definition, the results could be explained by the fact that the average
intervention effect vector is uninformative and does not help predicting the
enhancing effect of the intervention. This result exemplifies the limitation
of the inductive bias provided by Salt.

It is worth remarking that the only method that actually models single-cell
variation, CPA, does not benefit from it. One could argue that this is an
artifact of the metric choice as RMSE only compares the first moment of the
predicted and actual cell distributions. However, the gap with all three other
approaches grows even larger when considering the energy distributional
distance as reported in Appendix Tables 4 and 5. This might suggests that the
noise-to-signal ratio in single-cell data is too high to model higher moments
accurately for the task and evaluation we chose. In other terms, architectures
that only model average effects and otherwise conserve the higher moments of
the distribution of control cells, such as Peper and GEARS, may be advantaged
due to the noisiness of the single-cell data.

When out-of-distribution, all models have significant performance drops and
Salt becomes the best performing model for all but one metric (see Table 3).
As Peper simply augments Salt, the performance drop is less striking when
compared to GEARS. However, the results are symptomatic of an overfit to the
seen data and a lack of generalisability beyond the support from the training
set.

Table 3: RMSE results on held-out clusters splits. The numbers correspond to average score and standard deviation (in parenthesis) across 5 different seeds. | norman-dataset | wessels-dataset
---|---|---
Model | All | Top 20 ES | All | Top 20 ES
Peper | 0.032 (0.000) | 0.162 (0.000) |  0.049 (0.003) | 0.189 (0.005)
GEARS | 0.041 (0.001) | 0.274 (0.01) | 0.053 (0.003) | 0.3 (0.068)
CPA | 0.051 (0.024) | 0.330 (0.142) | 0.070 (0.014) | 0.566 (0.096)
Salt | 0.031 | 0.157 | 0.065 | 0.181

Figure 2: Score breakdown of GEARS, CPA, Salt, and Peper (lower is better)
based on intervention subgroups identified by Norman et al. (2019). The
definition for each label can be found in Appendix E. As we can see, there are
some variability in performances based on interaction type, but the
potentiation interaction type stands out particularly.

##  5 Discussion & Conclusion

In this paper, we investigated the estimation of the effect of genetic
interventions targeting pairs of genes in diverse data scenarios. We proposed
to use a biologically-motivated, non-parametric baseline, that we call Salt,
to help put results into perspective. We introduced Peper, a deep learning
architecture leveraging Salt’s prior and demonstrated that it achieves state-
of-the-art performances on our key metric.

However, our out-of-distribution splits uncovered significant weaknesses in
all current modeling approaches when predicting previously unseen regions of
the transcriptional response manifold. Unfortunately, this is precisely the
setting in which these models are likely to be the most useful to biologists -
predicting previously unseen phenotypic responses to combinatorial
perturbations. This indicates a need for further research, accompanied by
refined approaches to generating data. Creating situations where we
consistently find ourselves within an in-distribution scenario for a given
biological question is a challenging endeavour. One promising avenue is
through efficient data acquisition strategies, such as active learning
(Settles, 2012; Bertin et al., 2023; Scherer et al., 2022), which provides a
level of control over data generation and experimental design.

Concerning model development, the incorporation of inductive biases derived
from prior knowledge, typified by Bayesian learning theory (Fortuin, 2022) and
here exemplified by Peper, has proven to be a robust approach in many
applications. However, the observed performance drops in out-of-distribution
splits highlight the limitations with current priors, which may not always be
of high quality or suitable for generalization. The insufficiency of the
inductive bias from Salt becomes clear from the performance drop over
combinations that interact through potentiation. Moreover, the use of priors
derived from measurements taken at a single timepoint has inherent
limitations. These static priors lack insight into any underlying dynamics,
and interactions that could arise were sequential measures available. To
address these issues and enhance performance on this task, we need both better
priors, better data and improved architectures capable of leveraging or
rejecting inappropriate prior knowledge.

These challenges can also be indicative of a broader issue related to the
scarcity of data, notably around combinatorial interventions. We do not yet
know whether gene interactions tend to be conserved across different contexts,
facilitating the use of transfer learning approaches to fill in the gap, or
whether significant novel data would need to be generated for each new
context. In all cases, we expect the situation to improve as the volume of
publicly available data increases to a scale of dataset sufficient to train
large modern architectures. In the interim, the community should employ clever
inductive biases and priors while efficiently acquiring data to address
specific questions.

On a conceptual note, our results suggest that modelling higher moments of
single-cell distributions may be detrimental to overall performances on our
metrics of interest. This should be caveated by the fact that our observation
rests only on a single model, namely CPA. However, it can be further
compounded by the fact that Ji et al. (2023) identified mean-squared error
between pseudo-bulks as the best metric to compare single-cell transcriptomics
distributions in multiple applications. Put together, these two results might
imply limited utility gain from considering higher moments for selected
applications; for these, a more productive focus might be centered on
(pseudo-)bulk data. However, this statement needs to be tested in specific
cases, and it is important to recognise that it might not hold for many
possible downstream applications. Lastly, and perhaps most importantly of all,
the choice of metrics should be carefully thought through such that
improvement on a chosen metric captures meaningful improvements in practice.

#### Acknowledgments

This work would not have been possible without the support of the entire
Relation Therapeutics team, with special mentions for Edith M. Hessel, Marco
P. Licciardello, Mohammad Lotfollahi, Rebecca E. McIntyre, Charles E.S.
Roberts, Jake P. Taylor-King, and John C. Whittaker, for the many insightful
discussions.

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##  Appendix A Data

##### Datasets.

The first dataset was generated with CRISPR activation interventions in K562
cells (Norman et al., 2019), and comprise $124$ combinations derived from a
pool of $106$ individual potential targets. The second dataset utilises CRISPR
Cas13 technology in THP1 cells and encompasses $142$ combinations within a set
of $27$ genes (Wessels et al., 2023). In the following, we refer to each
dataset as norman-dataset and wessels-dataset. We detail our data processing
in Appendix A.

##### Pre-processing.

We follow standard pre-processing steps for both datasets. We first remove
cells that do not pass the usual quality control thresholds in terms of total
number of counts, number of genes expressed, and percentage of mitochondrial
counts. We also remove the subset of cells associated to a target if the
subset is too small. Then, we filter out measured genes if they are not
expressed consistently for at least one of the cell subsets associated to
targets. Finally, we normalise cell counts to a $10^{4}$ total and log-
transform the data.

##### Splits.

To create our splits, we first cluster interventions based on pseudo-bulk
profiles using the Leiden algorithm (Traag et al., 2019). From this we obtain
clusters that group interventions that have relatively similar average
responses in terms of transcriptomics. We use the clustering to define two in-
distribution splits and one out-of-distribution split. For the in-distribution
splits, we define train, validation, and test sets from the set of
combinatorial interventions within each cluster. The two splits are defined
with different amounts of held combinations, we hold-out either 25% of
combinations (i.e. 12.5% in val and test sets) or 75% (i.e. 37.5% in val and
test sets). Henceforth, we will refer to these splits as in-distribution-25
and in-distribution-75. We define out-of-distribution splits by holding out
entire clusters of combinations for validation and test, choosing the farthest
clusters away from the cluster containing control cells. Note that all cells
associated to single interventions are added to the training sets for all
splits. Visualisations based on UMAP of the various splits can be found in the
Appendix (Figure 3 and Figure 4).

##  Appendix B Log-fold change and RMSE metric

As discussed in the main document, we compute the root mean-square error
between predicted and actual log-fold change vectors with respect to the
control distribution. Specifically, we use Limma’s definition of log-fold
change (Smyth, 2005), which defines the log-fold change associated to
perturbation targeting gene $p$ with respect to the control distribution as

$\bm{\delta}_{p}=\frac{\bm{\mu}_{p}}{\textnormal{log}(2)},$

assuming the data is log-transformed in base $10$. This formulation of log-
fold change from Limma is motivated by the assumption that gene expression
follows a normal distribution in log space, rather than in count space. We
then denote for each perturbation $p$ the set of $k$ genes that have the
greatest absolute log-fold change value by $\mathcal{S}_{p,k}$. With these
definitions, we can simply write our metrics over a set of perturbations
$\mathcal{P}$ as

$\displaystyle\textnormal{RMSE}_{\textsc{All}}(\mathcal{P})$
$\displaystyle=\frac{1}{|\mathcal{P}|}{{\sum}}_{p\in\mathcal{P}}\sqrt{\frac{\sum_{i=1}^{N}\left(\bm{\delta}_{p}[i]-\tilde{\bm{\delta}}_{p}[i]\right)^{2}}{N}},$
$\displaystyle\textnormal{RMSE}_{\textsc{Top 20 ES}}(\mathcal{P})$
$\displaystyle=\frac{1}{|\mathcal{P}|}{{\sum}}_{p\in\mathcal{P}}\sqrt{\frac{\sum_{i\in\mathcal{S}_{p,20}}\left(\bm{\delta}_{p}[i]-\tilde{\bm{\delta}}_{p}[i]\right)^{2}}{|\mathcal{S}_{p,20}|}},$

where $N$ is the total number of genes post-QC, $|\cdot|$ denotes the cardinal
of a set, and $\tilde{\delta}$ denotes predicted log-fold changes.

##  Appendix C Additional results.

To get a view of the prediction quality in distributional terms, we compute
the energy distance between predicted and actual distribution of cells on the
test set for both in-distribution splits. The results are reported in Tables 4
and 5. The results are mostly consistent with the RMSE scores, apart from CPA
which performs significantly worse on this metric. As discussed in the main
document, this suggest that modelling single-cell variations is detrimental
when compared with methods which conserve the higher moments from the control
cell distribution.

##  Appendix D Peper hyperparameters tuning configuration.

We list below the name and range of the hyperparameters tuned for Peper on
each split.

  * •

batch size: $\left[256,6136\right]$,

  * •

batch accumulation: $\left[1,100\right]$,

  * •

learning rate: $\left[10^{-5},10^{-3}\right]$,

  * •

weight decay: $\left[0,10^{-4}\right]$,

  * •

number of encoder layers $\left(f_{\bm{\theta}_{1}}\right)$:
$\left[1,4\right]$,

  * •

number of decoder layers $\left(f_{\bm{\theta}_{2}}\right)$:
$\left[1,4\right]$,

  * •

latent dimension: $\left[1000,6500\right]$,

  * •

encoder and decoder hidden dimension: $\left[1000,6500\right]$.

##  Appendix E Glossary.

##### Epistasis

A combination is epistatic if the effect of targeting one of the genes mask
the effect of targeting the other.

##### Neomorphism

A combination is neomorphic if the resulting effect is atypically new or
unexpected given the effects observed from intervening on each gene
individually.

##### Potentiation

Potentiation is a sub-type of synergy in that the effect of the combination is
greater than expected. The defining aspect is that one of the target does not
have any effect on its own and only enhances the effect of targeting the
other.

##### Redundancy

A combination is redundant when the two targets have similar effect when
targeted individually and the combination leads also to a similar output.

##### Suppression

There is suppression when the combined intervention on two genes attenuates
the effect of each gene when individually targeted.

##### Synergy

A combination is synergistic when the effect of targeting both genes together
is greater than expected when compared to the effects of targeting each gene
individually. There exist multiple mathematical definition of synergy, each
making different assumption about what should be expected, e.g. Bliss
independence and Loewe additity (Tang et al., 2015).

Table 4: Energy distance results on in-distribution-25 split. The numbers correspond to average score and standard deviation (in parenthesis) across 5 different seeds. | norman-dataset | wessels-dataset
---|---|---
Model | All | Top 20 ES | All | Top 20 ES
Peper | 0.151 (0.013) | 0.056 (0.003) | 0.151 (0.002) | 0.041 (0.001)
GEARS | 0.144 (0.009) | 0.062 (0.007) | 0.126 (0.003) | 0.041 (0.002)
CPA | 1.883 (0.048) | 0.129 (0.012) | 2.365 (0.059) | 0.181 (0.010)
Salt | 0.145 | 0.065 | 0.292 | 0.059
Table 5: Energy distance results on in-distribution-75 split. The numbers correspond to average score and standard deviation (in parenthesis) across 5 different seeds. | norman-dataset | wessels-dataset
---|---|---
Model | All | Top 20 ES | All | Top 20 ES
Peper | 0.151 (0.001) | 0.067 (0.000) | 0.211 (0.006) | 0.050 (0.003)
GEARS | 0.178 (0.005) | 0.085 (0.007) | 0.152 (0.009) | 0.048 (0.003)
CPA | 1.873 (0.028) | 0.132 (0.005) | 2.467 (0.133) | 0.147 (0.020)
Salt | 0.146 | 0.075 | 0.331 | 0.069

Figure 3: UMAP visualisation of pseudo-bulked data from the norman-dataset,
showing the clustering of interventions, their types, and whether they are in
train, validation, or test sets in the different splits we use for the
analysis. The last panel show the interaction labels associated to
combinations by Norman et al. (2019).

Figure 4: UMAP visualisation of pseudo-bulked data from the wessels-dataset,
showing the clustering of interventions, their types, and whether they are in
train, validation, or test sets in the different splits we use for the
analysis.


# Minimal AC injection into Carbon Nanotubes

Kota Fukuzawa1, Takeo Kato1, Thibaut Jonckheere2, Jérôme Rech2, and Thierry
Martin2 1Institute for Solid State Physics, The University of Tokyo, Kashiwa,
Japan
2Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

###### Abstract

We study theoretically the effect of electronic interactions in 1d systems on
electron injection using periodic Lorentzian pulses, known as Levitons. We
consider specifically a system composed of a metallic single-wall carbon
nanotube, described with the Luttinger liquid formalism, a scanning tunneling
microscope (STM) tip, and metallic leads. Using the out-of-equilibrium Keldysh
Green function formalism, we compute the current and current noise in the
system. We prove that the excess noise vanishes when each Leviton injects an
integer number of electrons from the STM tip into the nanotube. This extends
the concept of minimal injection with Levitons to strongly correlated, uni-
dimensional non-chiral systems. We also study the time-dependent current
profile, and show how it is the result of interferences between pulses non-
trivially reflected at the nanotube/lead interface.

##  I Introduction

Controlled electron injection in an electronic system is an important issue,
both for potential applications and for the fundamental study of the many-body
properties of the systemFève  _et al._ (2007); Hermelin  _et al._ (2011);
McNeil  _et al._ (2011). More than twenty years ago, Levitov and co-workers
have shown that applying a generic time-dependent voltage to inject a charge
creates a fundamental disturbance to the system, akin to the Anderson
catastrophe, with the creation of a divergent number of electron-hole pairs in
the Fermi sea, in addition to the injected chargeLevitov  _et al._ (1996);
Ivanov  _et al._ (1997); Keeling  _et al._ (2006). Importantly, they also
showed that, by applying a specific quantized time-dependent voltage on a
electronic conductor, it is possible to excite a single electron above the
Fermi sea, without creating any perturbation to the system. These peculiar
excitations have been called Levitons, and have been realized experimentally
in 2d electron gasesDubois  _et al._ (2013a); Bocquillon  _et al._ (2014);
Jullien  _et al._ (2014). Such injection of a single charge, without any
spurious excitation of the system, has been called minimal injection, and can
be characterized by studying the excess noise.

Coulomb interactions can have a major impact on the many-body state of
electronic systems. This is particularly true for 1d electronic systems, where
interactions lead to a very specific behavior, which can be described at low
excitation energies by the Luttinger liquid theoryGiamarchi (2003). This
applies for example for conducting carbon nanotubesEgger and Gogolin (1998);
Kane  _et al._ (1997); Yoshioka and Odintsov (1999), semiconductor
nanowiresZaitsev-Zotov  _et al._ (2000), edge states of the fractional quantum
Hall effectWen (1995), etc. Important physical quantities of these systems,
such as the tunneling density of states and the current voltage
characteristics typically obey some power-law behavior, with an exponent which
is explicitly dependent on the interaction parameter. Other remarkable
behaviors of 1d interacting systems are the charge fractionalization thanks to
the existence of collective mode, spin-charge separation, etc. When connected
to standard, non-interacting electrodes, non trivial processes at the
boundaries between the interacting system and the electrodes, similar to
Andreev reflection, do occur, creating a complex behavior for the time-
dependent currentMaslov and Stone (1995); Safi and Schulz (1995); Ponomarenko
(1995); Sandler  _et al._ (1998); Acciai  _et al._ (2019); Hashisaka  _et al._
(2021); Glidic  _et al._ (2023).

Figure 1: Schematic of the system composed of a carbon nanotube, metallic
leads, and a STM tip. (a) Two-lead setup. (b) One-lead setup. (c) A model for
general theoretical description.

It is thus natural to ask whether the concept of Levitons can be extended to
1d interacting electronic systems. This question has already been given a
positive answer in the case of chiral edge states of FQHERech  _et al._
(2017a). However, in a non-chiral system, the behavior at the interface
between the interacting system and a normal lead is much more complex, which
may modify strongly the physical response of the system to an AC charge
injection. To answer this question in the case of a non-chiral system, in this
work we consider explicitly AC electron injection into a carbon nanotube (CNT)
from a STM tip (see Fig. 1), and calculate the excess noise induced by the
external AC driving. While the STM-CNT coupling is treated perturbatively as
it operates in the weak tunneling regime, we treat the Coulomb interactions
and the coupling between the CNT and the leads non-perturbatively using the
bosonization method. We show that, although interactions have a deep impact on
the behavior of the system, it is still possible to inject a single electronic
charge with zero excess noise by using Leviton pulses. Furthermore, we provide
an explicit general proof that the excess noise is zero for Levitons which
inject an integer number of electrons. We also discuss the formulas for the
noise and current, which contains the complex physics of Andreev-like
reflection at the interface with the leads.

The rest of this work is organized as follows. In Sec. II, we briefly
summarize our theoretical model for the CNT, metallic leads, and STM tip. We
also formulate the external voltage driving through the STM-CNT coupling. In
Sec. III, we show the analytical formulas for the tunneling current, noise and
excess noise (details of the derivation of these formulas are given in the two
Appendices). In Sec. IV, we show the numerical results for the excess noise
for different values of the relevant parameters, and discuss its properties.
We also provide an analytic proof that the excess noise is zero for Levitons
with integer charge, thus answering rigorously the initial question motivating
this work. Finally, to clarify the importance of the Andreev-like reflection
at the CNT-lead junction, we discuss the current profile as a function of time
in Sec. V. We summarize our results in Sec. VI.

##  II Description of the system

We consider a metallic CNT connected at both ends to two semi-infinite Fermi
liquid leads (see Fig. 1 (a)) or on one end to one Fermi liquid semi-infinite
lead (see Fig. 1 (b)). A STM tip is placed close to the CNT, and a time-
dependent voltage $V(t)$ is applied between the tip and the CNT to allow the
tunneling of electrons. These two setups can be described generally by an
infinite one-dimensional system with inhomogeneous interaction parameters as
shown in Fig. 1 (c), whose Hamiltonian is given with the bosonization
technique as Crépieux  _et al._ (2003); Lebedev  _et al._ (2005); Guigou  _et
al._ (2007)

$\displaystyle\hat{H}=\sum_{j\delta}\int
dx\left[\frac{v_{j\delta}(x)g_{j\delta}(x)}{2}(\partial_{x}\phi_{j\delta})^{2}+\frac{v_{j\delta}(x)}{2g_{j\delta}(x)}(\partial_{x}\theta_{j\delta})^{2}\right].$
(1)

Here, $\phi_{j\delta}(x)$ and $\theta_{j\delta}(x)$ are non-chiral bosonic
fields which satisfy the commutation relation

$\displaystyle[\phi_{j\delta}(x),\theta_{j^{\prime}\delta^{\prime}}(x^{\prime})]=-(i/2)\delta_{jj^{\prime}}\delta_{\delta\delta^{\prime}}\mathrm{sgn}(x-x^{\prime}),$
(2)

$g_{j\delta}(x)$ is the interaction parameter, $v_{j\delta}(x)=v_{\rm
F}/g_{j\delta}(x)$ is the renormalized velocity, $v_{\rm F}$ is the Fermi
velocity in the absence of Coulomb interaction, $j\in({\rm c},{\rm s})$
specifies charge and spin sectors, and $\delta\in(+,-)$ specifies the
symmetric and anti-symmetric sectors with respect to the branch (the valley)
in the CNT. The origin of the coordinate $x$ is set to the position of the STM
tip. Assuming that the screened Coulomb interaction modifies only the
$(j,\delta)=(c,+)$ sector Egger and Gogolin (1998); Yoshioka and Odintsov
(1999), the interaction parameters are set as $g_{c-}=g_{s+}=g_{s-}=1$ and

$\displaystyle g_{c+}=\left\\{\begin{array}[]{ll}g_{1},&(x<-L_{1}),\\\ g_{\rm
N},&(-L_{1}<x<L_{2}),\\\ g_{2},&(L_{2}<x),\end{array}\right.$ (6)

where $L_{1}$ and $L_{2}$ are lengths of the CNT separated by the STM tip (see
Fig. 1 (c)). We also define the total length of the CNT as $L=L_{1}+L_{2}$.
The two-lead case shown in Fig. 1 (a) can be described by setting
$g_{1}=g_{2}=1$. On the other hand, the one-lead case shown in Fig. 1 (b) can
be described by setting $g_{1}=0$ to express an open boundary at $x=-L_{1}$
while $g_{2}$ ($=1$) is left unchanged. We note that for repulsive Coulomb
interaction $g_{\rm N}$ becomes smaller than unity.

The electron operator in the CNT is given by

$\displaystyle\Psi_{r\alpha\sigma}(x)=\frac{\eta_{r\alpha\sigma}}{\sqrt{2\pi
a}}e^{iq_{\rm F}rx+ik_{\rm F}\alpha x+i\varphi_{r\alpha\sigma}(x)},$ (7)

where $\eta_{r\alpha\sigma}$ is a Klein factor, $a$ is a short-length cutoff,
$k_{\rm F}$ is the Fermi wavenumber, and $q_{\rm F}$ ($\ll k_{\rm F}$) is the
momentum mismatch associated with the two modes. Hereafter, we neglect the
Klein factor as it does not affect the results. The bosonic field
$\varphi_{r\alpha\sigma}$ is described with $\phi_{j\delta}(x)$ and
$\theta_{j\delta}(x)$ as

$\displaystyle\varphi_{r\alpha\sigma}(x)=\frac{\sqrt{\pi}}{2}\sum_{j\delta}h_{\alpha\sigma
j\delta}[\phi_{j\delta}(x)+r\theta_{j\delta}(x)],$ (8)

where $h_{\alpha\sigma c+}=1$, $h_{\alpha\sigma c-}=\alpha$, $h_{\alpha\sigma
s+}=\sigma$, and $h_{\alpha\sigma s-}=\alpha\sigma$. For convenience, the STM
is also modeled as a one-dimensional non-interacting system in a bosonized
form

$\displaystyle c_{\sigma}(t)=\frac{1}{\sqrt{2\pi
a}}e^{i\tilde{\varphi}_{\sigma}(t)}.$ (9)

The electron tunneling between the STM tip and the CNT is described by the
Hamiltonian

$\displaystyle\hat{H}_{T}(t)$
$\displaystyle=\sum_{r\alpha\sigma\epsilon}\epsilon\Gamma_{\epsilon}(t)\Psi_{r\alpha\sigma}^{(-\epsilon)}(0,t)c_{\sigma}^{(\epsilon)}(t),$
(10) $\displaystyle\Gamma_{\epsilon}(t)$
$\displaystyle=\Gamma_{0}\exp\left[\frac{i\epsilon
e}{\hbar}\int_{-\infty}^{t}V(t^{\prime})\,dt^{\prime}\right],$ (11)

where $\Gamma_{0}$ and $\Gamma_{\epsilon}(t)$ are the tunneling amplitudes
without and with effect of the time-dependent voltage $V(t)$ and the
superscript $\epsilon$ leaves either operator unchanged ($\epsilon=+$) or
transforms it into its Hermitian conjugate ($\epsilon=-$). The applied voltage
$V(t)$ is divided into DC and AC parts as $V(t)=V_{\rm dc}+V_{\rm ac}(t)$,
where by definition $V_{\rm ac}(t)$ averages to zero over one drive period
$T$. Here, we consider three types of the voltage pulse:

$\displaystyle{\rm Leviton:\ }V(t)=\frac{V_{\rm
dc}}{\pi}\sum_{k}\frac{\eta}{\eta^{2}+(t/T-k)^{2}},$ (12) $\displaystyle{\rm
Cosine:\ }V(t)=V_{\rm dc}(1-\cos\Omega t),$ (13) $\displaystyle{\rm Square:\
}V(t)=2V_{\rm dc}\sum_{k}{\rm rect}(2t/T-k),$ (14)

where $\Omega$ is the driving frequency, $T=2\pi/\Omega$ is the period, ${\rm
rect}(x)=1$ for $x<1/2$ ($=0$, otherwise) is the rectangular function, and
$\eta=W/T$ ($W$: the half-width at half-maximum of the Lorentzian pulse). We
define the Fourier components of $\Gamma_{\epsilon}(t)$ as

$\displaystyle\Gamma_{-}(t)=\sum_{l=-\infty}^{\infty}p_{l}e^{-i(\omega_{0}+l\Omega)t},$
(15)

and $\Gamma_{+}(t)=(\Gamma_{-}(t))^{*}$, where $\omega_{0}\equiv eV_{\rm
dc}/\hbar$ and

$\displaystyle p_{l}$
$\displaystyle\equiv\int_{-T/2}^{T/2}\frac{dt}{T}e^{il\Omega
t}\exp\left[-\frac{ie}{\hbar}\int_{-\infty}^{t}dt^{\prime}V_{\text{ac}}(t^{\prime})\right].$
(16)

For the pulse shapes given in Eqs. (12)-(14), the Fourier components are given
by

$\displaystyle{\rm Leviton:\
}p_{l}=q\sum_{s=0}^{\infty}\frac{\Gamma(q+l+s)}{\Gamma(q+1-s)}\frac{(-1)^{s}e^{-2\pi\eta(2s+l)}}{(l+s)!s!},$
(17) $\displaystyle{\rm Cosine:\ }p_{l}=J_{l}(-q),$ (18) $\displaystyle{\rm
Square:\
}p_{l}=\frac{2}{\pi}\frac{q}{l^{2}-q^{2}}\sin\left[\frac{\pi}{2}(l-q)\right],$
(19)

where $J_{l}(z)$ is the Bessel function. Here, we introduced a dimensionless
quantity $q$ defined as

$\displaystyle q$ $\displaystyle\equiv\frac{eV_{\rm
dc}}{\hbar\Omega}=\frac{\omega_{0}}{\Omega}.$ (20)

This quantity means that the charge injected per period is $qe$.

##  III Formulation of the excess noise

The current operator is expressed by the bosonic field as Crépieux  _et al._
(2003)

$\displaystyle\hat{I}(x,t)=2ev_{\rm
F}\frac{\partial_{x}\phi_{c+}(x,t)}{\sqrt{\pi}}.$ (21)

In the Keldysh formalism, the average current is written in the form:

$\displaystyle I(x,t)$ $\displaystyle=\frac{1}{2}\sum_{\eta}\left\langle
T_{K}{\hat{I}(x,t^{\eta})e^{i\int_{K}H_{T}(t_{1})dt_{1}}}\right\rangle$
$\displaystyle=\left\langle
T_{K}{\hat{I}(x,t^{-})e^{i\int_{K}H_{T}(t_{1})dt_{1}}}\right\rangle,$ (22)

where $T_{K}$ indicates time-ordering operator along the Keldysh contour,
$t^{\eta}$ indicates time on the forward ($\eta=+$) and backward ($\eta=-$)
contour, and $\int_{K}$ indicates an integral over the Keldysh contour K. In
the second line, we fixed the time on the backward contour because the current
average is independent of $\eta$. The second-order perturbation with respect
to $H_{T}$ gives

$\displaystyle I(x,t)$
$\displaystyle=-\frac{1}{2}\sum_{\eta_{1}\eta_{2}}\eta_{1}\eta_{2}\int
dt_{1}dt_{2}$ $\displaystyle\times\langle
T_{K}\\{\hat{I}(x,t^{-})H_{T}(t_{1}^{\eta_{1}})H_{T}(t_{2}^{\eta_{2}})\\}\rangle.$
(23)

In the Keldysh formalism, the current fluctuations can be written in the form:

$\displaystyle
S(x,t,t^{\prime})=\langle\hat{I}(x,t)\hat{I}(x,t^{\prime})\rangle$
$\displaystyle=\left\langle
T_{K}{\hat{I}(x,t^{-})\hat{I}(x,t^{+})e^{i\int_{K}H_{T}(t_{1})dt_{1}}}\right\rangle.$
(24)

The second-order perturbation gives

$\displaystyle
S(x,t,t^{\prime})=-\frac{1}{4}\sum_{\eta\eta_{1}\eta_{2}}\eta_{1}\eta_{2}\int
dt_{1}dt_{2}$ $\displaystyle\hskip 28.45274pt\times\langle
T_{K}\\{\hat{I}(x,t^{\eta})\hat{I}(x,t^{-\eta})H_{T}(t_{1}^{\eta_{1}})H_{T}(t_{2}^{\eta_{2}})\\}\rangle.$
(25)

The current noise $S(x)$ at the position $x$ is obtained by

$\displaystyle S(x)$ $\displaystyle=\lim_{T\rightarrow
0}\frac{1}{T}\int_{-T/2}^{T/2}dt\int_{-T/2}^{T/2}dt^{\prime}\,S(x,t,t^{\prime})$
$\displaystyle=\int_{-\infty}^{\infty}dt\,S(x,t,0).$ (26)

The excess noise, which is a noise induced by ac driving, is defined as

$\displaystyle S^{\rm ex}(x)=S(x)-e\bar{I}(x),$ (27)

where $\bar{I}(x)$ is the time-averaged current and the second term represents
the Poisson noise due to nonequilibrium currents injected from the STM tip.
Hereafter, we focus on the average current and excess noise in the right lead
($x>L_{2}$).

The results of the second-order perturbation are obtained after a somewhat
lengthy calculation, which goes beyond the calculations found in Ref. Guigou
_et al._ , 2007. As we are interested in the low temperature behavior of the
system, we peform the calculations at zero temperature only. We only show the
final results for $x>L_{2}$ as follows:

$\displaystyle\bar{I}(x)$ $\displaystyle=\frac{8e\Gamma^{2}}{N_{\rm
lead}\pi^{2}a^{2}}\sum_{l=-\infty}^{\infty}\left|p_{l}\right|^{2}\int_{0}^{\infty}\\!\\!\\!dt\;\sin((q+l)\Omega
t)\frac{D(t)}{(1+(v_{F}t/a)^{2})^{\frac{1}{2}}}\sin(F(t)+\mathrm{ArcTan}(v_{F}t/a)),$
(28) $\displaystyle S(x)$ $\displaystyle=\frac{8e^{2}\Gamma^{2}}{N_{\rm
lead}\pi^{2}a^{2}}\sum_{l=-\infty}^{\infty}\left|p_{l}\right|^{2}\int_{0}^{\infty}\\!\\!\\!dt\;\cos((q+l)\Omega
t)\frac{D(t)}{(1+(v_{F}t/a)^{2})^{\frac{1}{2}}}\cos(F(t)+\mathrm{ArcTan}(v_{F}t/a)),$
(29) $\displaystyle S^{\rm ex}(x)$
$\displaystyle=\frac{8e^{2}\Gamma^{2}}{N_{\rm
lead}\pi^{2}a^{2}}\sum_{l=-\infty}^{\infty}\left|p_{l}\right|^{2}\int_{0}^{\infty}\\!\\!\\!dt\,\frac{D(t)}{(1+(v_{F}t/a)^{2})^{\frac{1}{2}}}\cos((q+l)\Omega
t+F(t)+\mathrm{ArcTan}(v_{F}t/a)),$ (30) $\displaystyle F(t)$
$\displaystyle=\nu\mathrm{ArcTan}\left[\frac{v_{F}t}{a}\right]+\sum_{k=1}^{\infty}\frac{(b_{1}b_{2})^{k}}{8}\left(g_{\rm
N}+\frac{1}{g_{\rm
N}}\right)\mathrm{ArcTan}\left[\frac{2av_{F}t}{a^{2}+(2kLg_{\rm
N})^{2}-(v_{F}t)^{2}}\right]$
$\displaystyle+\sum_{k=0}^{\infty}\frac{b_{1}^{k+1}b_{2}^{k}}{16}\left(-g_{\rm
N}+\frac{1}{g_{\rm
N}}\right)\mathrm{ArcTan}\left[\frac{2av_{F}t}{a^{2}+((2kL+2L_{1})g_{\rm
N})^{2}-(v_{F}t)^{2}}\right]$
$\displaystyle\left.+\sum_{k=0}^{\infty}\frac{b_{1}^{k}b_{2}^{k+1}}{16}\left(-g_{\rm
N}+\frac{1}{g_{\rm
N}}\right)\mathrm{ArcTan}\left[\frac{2av_{F}t}{a^{2}+((2kL+2L_{2})g_{\rm
N})^{2}-(v_{F}t)^{2}}\right]\right],$ (31) $\displaystyle D(t)$
$\displaystyle=\left(\frac{a^{2}+(v_{F}t)^{2}}{a^{2}}\right)^{-\nu/2}$
$\displaystyle\times\prod_{k=1}^{\infty}\left[\left(\frac{a^{2}+(2kLg_{\rm
N})^{2}-(v_{F}t)^{2}}{a^{2}+(2kLg_{\rm
N})^{2}}\right)^{2}+\left(\frac{2av_{F}t}{a^{2}+(2kLg_{\rm
N})^{2}}\right)^{2}\right]^{-\frac{(b_{1}b_{2})^{k}}{16}\left(g_{\rm
N}+\frac{1}{g_{\rm N}}\right)}$
$\displaystyle\times\prod_{k=0}^{\infty}\left[\left(\frac{\alpha^{2}+((2kL+2L_{1})g_{\rm
N})^{2}-(v_{F}t)^{2}}{\alpha^{2}+((2kL+2L_{1})g_{\rm
N})^{2}}\right)^{2}+\left(\frac{2\alpha v_{F}t}{a^{2}+((2kL+2L_{1})g_{\rm
N})^{2}}\right)^{2}\right]^{-\frac{b_{1}^{k+1}b_{2}^{k}}{32}\left(-g_{\rm
N}+\frac{1}{g_{\rm N}}\right)}$
$\displaystyle\times\prod_{k=0}^{\infty}\left[\left(\frac{a^{2}+((2kL+2L_{2})g_{\rm
N})^{2}-(v_{F}t)^{2}}{a^{2}+((2kL+2L_{2})g_{\rm
N})^{2}}\right)^{2}+\left(\frac{2av_{F}t}{a^{2}+((2kL+2L_{2})g_{\rm
N})^{2}}\right)^{2}\right]^{-\frac{b_{1}^{k}b_{2}^{k+1}}{32}\left(-g_{\rm
N}+\frac{1}{g_{\rm N}}\right)},$ (32)

where $\nu=(6+g_{\rm N}+g_{\rm N}^{-1})/8$, $N_{\rm lead}$ is the number of
metallic leads and $b_{1,2}=(g_{N}-g_{1,2})/(g_{N}+g_{1,2})$ are the
reflection coefficients. We note that the calculated current and current noise
are independent of $x$ as far as $x>L_{2}$. Details of the derivation are
given in Appendix A.

While the formulas for the $F$ and $D$ factors are somewhat heavy, the
physical meaning of the different terms is quite simple. For each term in the
sum defining $F$, there is a corresponding term in the product defining $D$.
The first term correspond to the contribution for an infinite nanotube, with
direct propagation from the injection point located at the origin to the
measurement point $x$. All the other terms correspond to the propagation after
a given number of reflection on the CNT-leads interface: $b_{1}$ (resp.
$b_{2}$) is the reflection coefficient at the left (resp. right) interface, so
for example a term with the $(b_{1}b_{2})^{k}$ corresponds to $k$ reflection
at both interface, and one can see that this term also contains a factor
$2kLg_{N}$ which stands for $k$ times the length of one round trip along the
CNT. Note that $b_{1}<0$ and $b_{2}<0$ for the two leads case, which means
that the reflections are similar to an Andreev reflection, with an electron-
like excitation being reflected as a hole-like excitation.

##  IV Excess Noise

In this section, we discuss the properties of the excess noise as a function
of $q=eV_{dc}/\hbar\Omega$. We mainly focus on the excess noise when Leviton
pulses are injected. We define the charge velocity in the CNT as $v_{\rm
N}=v_{\rm F}/g_{\rm N}$ and the unit of time and frequency as
$T_{0}=2L_{2}/v_{\rm N}$ (the time needed for an electron to travel from the
tip to the right lead and back) and $\Omega_{0}=2\pi/T_{0}$, respectively.

###  IV.1 Two-lead case

Figure 2: Excess noises as a function of $q$ for the two-lead case. The unit
of the noise is given by $S_{0}=4e^{2}\Gamma^{2}/\pi^{2}a^{2}$. (a) Leviton
pulses for $\Omega=\Omega_{0}\equiv\pi v_{\rm N}/L_{2}$ and
$g_{N}=1.0,0.7,0.5$, and $0.2$. (b) Three types of pulses for
$\Omega=\Omega_{0}$ and $g_{N}=0.2$. (c) Leviton pulses for $g_{N}=0.2$ and
$\Omega/\Omega_{0}=2.5,2.0,1.5,1.0$, and $0.5$. In all the plots, we set
$L_{1}=L/3$, $L_{2}=2L/3$, $\eta=0.1$, and $\omega_{c}=100\Omega_{0}$.

Let us first discuss the two-lead setup shown in Fig. 1 (a). In Fig. 2 (a), we
show the excess noise as a function of $q$ when Lorentzian voltage pulses
(i.e. Levitons) are applied. The interaction parameter of the CNT is set as
$g_{\rm N}=1,0.7,0.5$ and $0.2$ and the other parameters are given in the
caption. The most striking feature is the fact that the excess noise is zero
when $q$ is an integer, independently of the value of the interaction
parameter. This is a well-known property of a Leviton drive for a non-
interacting electronic system.Levitov  _et al._ (1996); Keeling  _et al._
(2006); Dubois  _et al._ (2013b) Fig. 2 (a) shows that this property remains
valid even for strongly interacting non-chiral 1d electronic systems. An
analytical proof of this remarkable property is given in Sec. IV.4. For non-
integer values of $q$, we observe that the excess noise is reduced a little as
the Luttinger parameter $g_{\rm N}$ is reduced from unity.

Fig. 2 (b) shows the pulse-shape dependence of the excess noise, for $g_{\rm
N}=0.2$. The excess noise vanishes only for Lorentzian pulses, while it does
not for the cosine and square pulses. These observations confirm that the
exceptional features of the Levitons compared to all other voltage drives
survive even for this interacting electron system. We note that, contrarily to
the case of quasi-particle tunneling into edge states of the fractional
quantum Hall effect Rech  _et al._ (2017b), the excess noise does not show a
singular behavior near integer values of $q$. This is because we are
considering here injection of electrons (rather than fractional quasi-
particles) from the STM tip.

Fig. 2 (c) shows the excess noise for $g_{\rm N}=0.2$ as a function of $q$ for
$\Omega/\Omega_{0}=0.5,1.0,1.5,2.0$ and 2.5 and $\omega_{c}/\Omega_{0}=100$,
where $\Omega_{0}$ is the unit of frequency. Although the excess noise for a
non-integer value of $q$ is almost proportional to $\Omega$, it grows a little
faster than expected from linear dependence.

###  IV.2 One-lead case

Figure 3:  Excess noises as a function of $q$ for the one-lead case. The unit
of the noise is given by $S_{0}=4e^{2}\Gamma^{2}/\pi^{2}a^{2}$. (a) Leviton
pulses for $\Omega=\Omega_{0}\equiv\pi v_{\rm N}/L_{2}$ and
$g_{N}=1.0,0.7,0.5$, and $0.2$. (b) Three types of pulses for
$\Omega=\Omega_{0}$ and $g_{N}=0.2$. (c) Leviton pulses for $g_{N}=0.2$ and
$\Omega/\Omega_{0}=2.5,2.0,1.5,1.0$, and $0.5$. In all the plots, we set
$L_{1}=L/3$, $L_{2}=2L/3$, $\eta=0.1$, and $\omega_{c}=100\Omega_{0}$.

Next, we discuss the one-lead setup shown in Fig. 1 (b). We show the excess
noise as a function of $q$ in Fig. 3 (a) for $g_{\rm N}=1,0.7,0.5$ and 0.2,
where the other parameters are the same as Fig. 2. We find that the
qualitative features of the excess noise are similar to the two-lead case; for
non-integer $q$ the excess noise is reduced a little as the interaction
parameter $g_{\rm N}$ is reduced, while it vanishes when $q$ is an integer.
This means that the exceptional features of the Levitons are also valid for
the interacting electrons systems for this setup. Fig. 3 (b) and (c) show the
pulse-shape and ac-frequency dependence, respectively. By comparing with Fig.
2 (b) and (c), we find that the qualitative features are common with the two-
lead case. Compared to the two-lead case, the one-lead setup produces only
small quantitative changes in the excess noise for non-integer values of $q$.

###  IV.3 Interference effects

Figure 4: Geometry dependence of the excess noise for (a) the two-lead case
and (b) the one-lead case. We set the parameters as $(L_{1},\Omega)=(L/3,3\pi
v_{\rm N}/2L)$ (purple line), $(L/2,2\pi v_{\rm N}/L)$ (green line),
$(2L/3,3\pi v_{\rm N}/2L)$ (blue line), and $(L/2,\pi v_{\rm N}/L)$ (orange
line). The other parameters are set as $\eta=0.1$, $\omega_{c}/\Omega=100$,
and $g_{\rm N}=0.2$.

As will be discussed in Sec. V, the Leviton pulses reflect at the boundary
between the CNT and the leads in a complex manner and can interfere with
subsequent Leviton pulses injected from the STM tip. To see this interference
effect, we show the excess noise in Fig. 4 for four geometries, i.e., four
sets of the STM-tip position and two boundaries of the CNT. Fig. 4 (a) and (b)
shows the two- and one-lead cases, respectively. Although the geometry of the
STM-tip and CNT boundaries controls the interference between Lorentzian
pulses, the excess noise keeps its main features; it always vanishes when $q$
is an integer. We find that the interference effect causes small changes in
the excess noise for non-integer $q$ in both the two- and one-lead cases. We
conclude that the interference between the pulses only has a minor effect on
the excess noise.

###  IV.4 Proof that excess noise is zero for integer Leviton

As observed in Figs. 2, 3 and 4, the excess noise is always zero for a
periodic drive of Levitons when $q$ is integer, independently of the other
parameters of the system (number of leads, position of the STM tip, frequency
of the drive, etc.). We give here an analytical proof of this property.

The excess noise is defined as $S^{\rm ex}(x)=S(x)-e\bar{I}(x)$ in Eq. (30).
Using $D(t)=D(-t)$ and $F(t)=-F(-t)$, we can recast Eq. (30) as:

$\displaystyle S^{\rm ex}(x)$ $\displaystyle=\frac{4e^{2}\Gamma^{2}}{N_{\rm
lead}\pi^{2}a^{2}}\Re\sum_{l=-\infty}^{\infty}\left|p_{l}\right|^{2}\int_{-\infty}^{\infty}\\!\\!\\!dt\,\frac{D(t)}{(1+(v_{F}t/a)^{2})^{\frac{1}{2}}}$
$\displaystyle\times\mbox{exp}\left[i((q+l)\Omega
t+F(t)+\mathrm{ArcTan}(v_{F}t/a))\right],$ (33)

where $F(t)$ and $D(t)$ are given by Eqs. (31)-(32), and $\Re$ denotes the
real part. $F(t)$ is composed by an infinite sum of terms, and each term has a
corresponding term in the infinite product which composes $D(t)$. A typical
term in $F(t)$ can be written as:

$A\times\mbox{ArcTan}(\alpha(t)/\beta(t))$ (34)

and the corresponding term in $D(t)$ is:

$\left[(\beta(t)/\gamma)^{2}+(\alpha(t)/\gamma)^{2}\right]^{-A/2},$ (35)

with $\alpha(t)=2av_{F}t$, $\beta(t)=a^{2}+(2kLg_{\rm N})^{2}-(v_{F}t)^{2}$
(or a similar expression with $2kL$ replaced by $2kL+2L_{1}$ or
$2k_{L}+2L_{2}$), and $A>0$, $\gamma>0$ are time-independent constants. In the
integral of Eq. (33), this combination of terms appears as:

$\frac{\mbox{exp}(iA\times\mbox{ArcTan}(\alpha(t)/\beta(t)))}{\left[(\beta(t)/\gamma)^{2}+(\alpha(t)/\gamma)^{2}\right]^{A/2}}=\frac{\gamma^{A}}{(\beta(t)-i\alpha(t))^{A}}.$
(36)

From the expressions of $\alpha(t)$ and $\beta(t)$, one can check that the
expression of Eq. (36) has poles in the $t$ plane at:

$t=-\frac{ia}{v_{F}}\pm\frac{2kLg_{\rm N}}{v_{F}}.$ (37)

Importantly, these two poles are in the lower half-plane. The integrand of Eq.
(33) thus contains an infinite product of terms like Eq. (36), which have
poles and branch cuts in the lower half-plane only. This product is multiplied
by $\mbox{exp}(i(q+l)\Omega t)$; this factor allows to perform the integration
by closing the contour in the upper (lower) complex plane for $q+l>0$
($q+l<0$). As a Leviton drive for an integer charge $q$ is characterized
byLevitov  _et al._ (1996); Ivanov  _et al._ (1997); Keeling  _et al._ (2006);
Dubois  _et al._ (2013b); Rech  _et al._ (2017a)

$p_{l}=0,\quad\mbox{ for }\quad l\leq-q,$ (38)

one can see that only terms with $q+l>0$ exist for a Leviton drive with
integer charge $q$, which leads to a zero integral for the excess noise as the
integrand has no poles in the upper complex plane. The excess noise for a
Leviton drive with an integer charge $q$ is thus always zero, independently of
the value of the other parameters, including the interaction parameter $g_{N}$
in the nanotube. We note that suppression of the excess noise has recently
been discussed for a different setup using the non-chiral Luttinger liquid
Acciai  _et al._ (2019).

##  V Current profiles

Figure 5: Time-dependent current profile as a function of position, inside the
CNT (yellow background) and inside the normal leads (light gray background),
for the two-lead case (left panels) and the one-lead case (right panels) in
the case of $L_{1}=L/3$. Each panel shows the current at three different times
as indicated. The unit of the current is given by
$I_{0}=4e\Gamma^{2}/\pi^{2}a^{2}$. The period of the ac voltage is given
$T=4L_{1}/v_{\rm N}=2L_{2}/v_{\rm N}$. Other experimental parameters are taken
as $\eta=0.01$, $\omega_{c}T/2\pi=100$, and $g_{\rm N}=0.2$.

Indeed, as the figures for the excess noise are qualitatively similar to the
ones for a non-interacting system, one may think that the system behaves
overall as a non-interacting one. However, this is not the case, and it will
be clear from the current profiles which are fundamentally different from the
case of a non-interacting system. It is thus quite remarkable that the excess
noise is robust with respect to interactions, and in particular that it goes
to zero for integer value of $q$ for a Leviton drive.

An essential element to understand the transport in the system is the
reflection at each boundary between the nanotube and the leads. These are
characterized by the reflection coefficients
$b_{1,2}=(g_{N}-g_{1,2})/(g_{N}+g_{1,2})$. For the two-lead case, we have
$b_{1}=b_{2}=(g_{N}-1)/(g_{N}+1)<0$, which means that an ‘electron’-like
excitation is converted into a ‘hole’-like excitation by the reflection. This
peculiar reflection, which is inherent to one-dimensional interacting electron
systems, is called Andreev reflection in analogy with that of a
superconductor/normal junctionSafi and Schulz (1995); Crépieux  _et al._
(2003). For the one-lead case, one has $b_{1}=1$; the reflection at the right
boundary is still an Andreev reflection, while at the left boundary it is a
simple reflection. The current profile will be the result of the interference
of the injected pulses and their multiple reflections at the boundaries.

We first consider the two-lead case. To visualize the Andreev reflection and
interference with different Leviton pulses, we show the time-dependent current
profile for $L_{1}=L/3$ and $L_{2}=2L/3$ in Fig. 5 (a)-(c). The expressions
used to compute the time-dependent current profile are detailled in Appendix
B. The period of the ac voltage is taken as $T=4L_{1}/v_{\rm N}=2L_{2}/v_{\rm
N}$, where $v_{\rm N}=v_{F}/g_{\rm N}$ is the charge velocity in the CNT. The
three curves correspond to $t=0,T/9,2T/9$ in (a), to $t=3T/9,4T/9,5T/9$ in
(b), and to $t=6T/9,7T/9,8T/9$ in (c), respectively. Fig. 5 (a) shows that the
applied voltage pulse at $t=0$ creates two pulses propagating in opposite
directions from the STM tip at $x=0$, as indicated by the blue and red arrows.
The heights of these two pulses are different because of the resonance
condition $T=4L_{1}/v_{\rm N}=2L_{2}/v_{\rm N}$, which means that pulses
reflected one or several times at the left and right boundaries can interfere
with new pulses created periodically at the tip position. In Fig. 5 (b), the
pulses are divided into a reflected part (in the CNT) and a transmitted part
(in the leads) at the two boundaries of the CNT. We note that the current
carried by the reflected pulse has the same sign as the incident pulse despite
the reversal of the propagation direction. This is because the electron pulses
are converted into hole pulses at the boundary due to the ‘Andreev’ reflection
characteristic of the Luttinger liquid Crépieux  _et al._ (2003); Guigou  _et
al._ (2007). In Fig. 5 (c), the reflected pulses continue to propagate in the
CNT. Here, because the lengths, $L_{1}$ and $L_{2}$, between the STM tip and
the leads 1 or 2 are different, the timing of arrival of pulses at the origin
is also different for the two reflected pulses. For our choice of parameters,
the pulse reflected at $x=L_{2}$ returns at the origin precisely after the
period $T$ (see the blue arrow). The fact that there is a strong interference
between the pulse created at the STM tip and the pulses reflected on the right
lead, while there is no such interference for the pulse reflected at the left
lead, explains the different current profile in the two metallic leads; the
time-dependent current is alternatively positive and negative in the right
lead, while it is always negative in the left lead.

The time-dependent current profile for the one-lead case is shown in Fig. 5
(d)-(f) for the same geometry. In contrast with the two-lead case, the system
has one open boundary at $x=-L_{1}$, where the reflection is a standard one
which keeps the sign of the charge excitation unchanged. Therefore, the
current carried by the pulse changes its sign after reflection at $x=-L_{1}$
(see Fig. 5 (e)) as in a usual reflection. This change in the reflection
properties affects the sign of the current in the right lead; the current is
always positive there in contrast with the two-lead case (see Fig. 5 (a)-(c)).

Figure 6:  Time-dependent current profile as a function of position, inside
the CNT (yellow background) and inside the normal leads (light gray
background), for the two-lead case (left panels) and the one-lead case (right
panels) in the case of $L_{1}=2L/3$. Each panel shows the current at three
different times as indicated. The unit of the current is given by
$I_{0}=4e\Gamma^{2}/\pi^{2}a^{2}$. The period of the ac voltage is given
$T=2L_{1}/v_{\rm N}=4L_{2}/v_{\rm N}$. Other experimental parameters are taken
as $\eta=0.01$, $\omega_{c}T/2\pi=100$, and $g_{\rm N}=0.2$.

We further show the current profile for a different geometry with $L_{1}=2L/3$
and $L_{2}=L/3$ in Fig. 6. Although the reflection properties at the
boundaries of the CNT are the same as the previous case, the geometry affects
the current profile in the metallic leads due to the modifications of the
interference conditions between pulses. For the two-lead case (Fig. 6
(a)-(c)), the current is now always positive in the right lead, while it can
be both negative and positive in the left one. This result is the inverse of
what was observed with the previous geometry (Fig. 5 (a)-(c)). Also for the
one-lead case (Fig. 6 (d)-(f)), the current can now be negative in the right
lead, in contrast with the previous geometry (Fig. 5 (d)-(f)).

##  VI Summary

We theoretically studied the effect of electron correlations in single
electron injection into a carbon nanotube (CNT) coupled with metallic leads.
We formulated the current and the excess noise induced by an AC voltage
driving by describing the CNT in terms of a Luttinger liquid, and studied how
electron correlations modified (or not) the property of minimal excess noise
for Lorenzian pulse (Leviton) injection. We showed both analytically and
numerically that the excess noise vanishes when each Leviton pulse includes an
integer number of electrons, as observed in non-interacting electron systems.
This indicates that the electron correlations in the CNT do not change the
physics of minimum noise pulses, i.e. Levitons, at all. For a non-integer
electron injection, the excess noise depends on both the geometry of the
system and the ac driving frequency. We also showed that the time-dependent
current profile depends on the geometry and the ac frequency through the
interference condition between injected pulses. We demonstrated that injected
pulses induces Andreev-like (resp. normal) scattering at the junction with a
metallic lead (resp. at the open boundary). This affects the excess noise in
non-integer electron injection. Our findings show the universality of the
minimum noise properties for the Leviton pulses, which holds even in
interacting electron systems. Detailed setup and estimate for experimental
verification of our results are left as a future problem. Our calculations
could also apply to artificially created non-chiral Luttinger liquid systems
in the quantum Hall effect.Brasseur  _et al._ (2017)

###### Acknowledgements.

This French-Japanese collaboration is supported by the CNRS International
Research Project “Excitations in Correlated Electron Systems driven in the
GigaHertz range” (ESEC). This work received support from the French government
under the France 2030 investment plan, as part of the Initiative d’Excellence
d’Aix-Marseille Université - A*MIDEX, through the institutes IPhU
(AMX-19-IET-008) and AMUtech (AMX-19-IET-01X). T. K. acknowledges support from
the Japan Society for the Promotion of Science (JSPS KAKENHI Grant No.
JP20K03831).

##  Appendix A Green functions

In this appendix, we show the main steps of the calculation to obtain the
Green function of the bosonic fields for an inhomogeneous infinite one-
dimensional system, with an interaction parameter being a piecewise constant
function defining three different regions, without any asumption of spatial
symmetry.Guigou  _et al._ (2007)

Let us first consider the imaginary-time Green functions defined by

$\displaystyle G^{XY}_{j\delta}(x,x^{\prime},\tau)$
$\displaystyle=\left\langle
X_{j\delta}(x,\tau)Y_{j\delta}(x^{\prime},0)\right\rangle,$ (39)

for $0<\tau<\hbar\beta$, where $X,Y=\theta$ or $\phi$ and
$X_{j\delta}(x,\tau)=e^{H\tau/\hbar}X_{j\delta}(x)e^{-H\tau/\hbar}$ indicates
the imaginary-time evolution. From the Hamiltonian (1), the Green functions,
$G^{\phi\phi}_{j\delta}(x,x^{\prime},\tau)$ and
$G^{\theta\theta}_{j\delta}(x,x^{\prime},\tau)$, obey the following equations:

$\displaystyle-\left(\frac{g_{j\delta}(x)}{v_{j\delta}(x)}\partial_{\tau}^{2}+\partial_{x}v_{j\delta}(x)g_{j\delta}(x)\partial_{x}\right)G^{\phi\phi}_{j\delta}(x,x^{\prime},\tau)$
$\displaystyle\hskip 128.0374pt=\delta(x-x^{\prime})\delta(\tau),$ (40)
$\displaystyle-\left(\frac{\partial_{\tau}^{2}}{v_{j\delta}(x)g_{j\delta}(x)}+\partial_{x}\frac{v_{j\delta}(x)}{g_{j\delta}(x)}\partial_{x}\right)G^{\theta\theta}_{j\delta}(x,x^{\prime},\tau)$
$\displaystyle\hskip 128.0374pt=\delta(x-x^{\prime})\delta(\tau),$ (41)

where $\delta(x)$ is a delta function. The mixed Green functions,
$G^{\phi\theta}_{j\delta}$ and $G^{\theta\phi}_{j\delta}$, can be obtained
from $G^{\phi\phi}_{j\delta}$ and $G^{\theta\theta}_{j\delta}$ as

$\displaystyle i\partial_{\tau}G_{j\delta}^{\phi\theta}(x,x^{\prime},\tau)$
$\displaystyle=\frac{v(x)}{g(x)}\partial_{x}G_{j\delta}^{\theta\theta}(x,x^{\prime},\tau),$
(42) $\displaystyle
i\partial_{\tau}G_{j\delta}^{\theta\phi}(x,x^{\prime},\tau)$
$\displaystyle=v(x)g(x)\partial_{x}G_{j\delta}^{\phi\phi}(x,x^{\prime},\tau).$
(43)

Using the fact that the interaction parameters $g_{j\delta}(x)$ and the
velocities $v_{j\delta}(x)$ are piecewise constant functions, with three
different domains: $x<-L_{1}$ (left lead), $-L_{1}<x<L_{2}$ (nanotube) and
$x>L_{2}$ (right lead), one can solve for the Green functions
$G^{\phi\phi}_{j\delta}$ and $G^{\theta\theta}_{j\delta}$ in the Matsubara
frequency space. For calculation of the current and current noise, the Green
functions at $x=x^{\prime}=0$ (which is the injection point) and its spatial
derivatives in which $x$ or $x^{\prime}$ is set as zero are required.

Since the full derivation is rather lengthy, we only explain here the
calculation of $G_{c+}^{\phi\phi}(x,x^{\prime},\tau)$. The other types of the
Green functions can be calculated in a similar way. By the Fourier
transformation of Eq. (40), we obtain

$\displaystyle\left(\frac{g_{c+}(x)}{v_{c+}(x)}\omega^{2}-\partial_{x}v_{c+}(x)g_{c+}(x)\partial_{x}\right)G_{c+}^{\phi\phi}(x,x^{\prime},\omega)$
$\displaystyle\hskip 85.35826pt=\delta(x-x^{\prime}),$ (44)

where $\omega$ is a Matsubara frequency, that can be regarded as a real number
at zero temperature, and

$\displaystyle G^{\phi\phi}_{c+}(x,x^{\prime},\omega)=\int
d\tau\,G_{c+}^{\phi\phi}(x,x^{\prime},\tau)e^{i\omega\tau}.$ (45)

It is straightforward to solve this differential equation with respect to $x$.
For example, the Green function in the range of $-L_{1}<x^{\prime}<L_{2}$ is
obtained as

$\displaystyle G^{\phi\phi}_{c+}(x,x^{\prime},\omega)$
$\displaystyle=\left\\{\begin{array}[]{ll}A(x^{\prime})e^{\frac{|\omega|x}{v_{1}}},&(x<-L_{1}),\\\
B(x^{\prime})e^{\frac{|\omega|x}{v_{\rm
N}}}+C(x^{\prime})e^{-\frac{|\omega|x}{v_{\rm N}}},&(-L_{1}<x<x^{\prime}),\\\
D(x^{\prime})e^{\frac{|\omega|x}{v_{\rm
N}}}+E(x^{\prime})e^{-\frac{|\omega|x}{v_{\rm N}}},&(x^{\prime}<x<L_{2}),\\\
F(x^{\prime})e^{-\frac{|\omega|_{n}x}{v_{2}}},&(L_{2}<x),\end{array}\right.$
(50)

where

$\displaystyle A(x^{\prime})$ $\displaystyle=\frac{2g_{\rm N}}{g_{1}+g_{\rm
N}}e^{\frac{|\omega|L_{1}}{v_{1}}-\frac{|\omega|L_{1}}{v_{\rm
N}}}B(x^{\prime}),$ (51) $\displaystyle B(x^{\prime})$
$\displaystyle=\frac{1}{2g_{\rm
N}|\omega|}\frac{e^{-\frac{|\omega|}{v_{N}}x^{\prime}}+\gamma_{2}e^{\frac{|\omega|}{v_{N}}x^{\prime}}}{1-\gamma_{1}\gamma_{2}},$
(52) $\displaystyle C(x^{\prime})$ $\displaystyle=\gamma_{1}B(x^{\prime}),$
(53) $\displaystyle D(x^{\prime})$ $\displaystyle=\gamma_{2}E(x^{\prime}),$
(54) $\displaystyle E(x^{\prime})$ $\displaystyle=\frac{1}{2g_{\rm
N}|\omega|}\frac{e^{\frac{|\omega|}{v_{\rm
N}}x^{\prime}}+\gamma_{1}e^{-\frac{|\omega|}{v_{\rm
N}}x^{\prime}}}{1-\gamma_{1}\gamma_{2}},$ (55) $\displaystyle F(x^{\prime})$
$\displaystyle=\frac{2g_{\rm N}}{g_{2}+g_{\rm
N}}e^{\frac{|\omega|L_{2}}{v_{2}}-\frac{|\omega|L_{2}}{v_{\rm
N}}}E(x^{\prime}).$ (56)

Here, the coefficients, $\gamma_{1}$ and $\gamma_{2}$, are given as

$\displaystyle\gamma_{1}$ $\displaystyle=\frac{g_{\rm N}-g_{1}}{g_{\rm
N}+g_{1}}e^{\frac{-2|\omega|L_{1}}{v_{\rm N}}}\equiv
b_{1}e^{-\frac{2|\omega|L_{1}}{v_{\rm N}}},$ (57) $\displaystyle\gamma_{2}$
$\displaystyle=\frac{g_{\rm N}-g_{2}}{g_{\rm
N}+g_{2}}e^{\frac{-2|\omega|L_{2}}{v_{\rm N}}}\equiv
b_{2}e^{\frac{-2|\omega|L_{2}}{v_{\rm N}}}.$ (58)

By setting $x,x^{\prime}\rightarrow 0$, we obtain

$\displaystyle G_{c+}^{\phi\phi}(0,0,\omega)=\frac{1}{2|\omega|g_{\rm
N}}\frac{(1+\gamma_{1})(1+\gamma_{2})}{1-\gamma_{1}\gamma_{2}}.$ (59)

Here, we further use the expansion

$\displaystyle\frac{1}{1-\gamma_{1}\gamma_{2}}=\sum_{k=1}^{\infty}(b_{1}b_{2})^{k}e^{-\frac{2kL|\omega|}{v_{\rm
N}}},$ (60)

where $k$ can be regarded as the number of round trips of a pulse in the CNT.
The inverse Fourier transformation can easily be performed by using the
formula

$\displaystyle\partial_{\tau}\int_{0}^{\infty}\frac{d\omega}{\omega}e^{\omega(x\pm
i\tau)}=\frac{\mp i}{x\pm i\tau},$ (61)

and by the analytic continuation $\tau=it+a/v_{\rm F}$ the Green function is
calculated as

$\displaystyle\tilde{G}_{c+}^{\phi\phi}(0,0,t)$ $\displaystyle=-\frac{1}{2\pi
g_{\rm N}}\Biggl{\\{}\log(1+iv_{\rm
F}t/a)+\sum_{k=1}^{\infty}(b_{1}b_{2})^{k}I_{+}(2kL)$ $\displaystyle\hskip
28.45274pt+\frac{1}{2}\sum_{k=0}^{\infty}b_{1}(b_{1}b_{2})^{k}I_{+}(2kL+2L_{1})$
$\displaystyle\hskip
28.45274pt+\frac{1}{2}\sum_{k=0}^{\infty}b_{2}(b_{1}b_{2})^{k}I_{+}(2kL+2L_{2})\Biggr{\\}},$
(62)

where

$\displaystyle I_{\pm}(x)$ $\displaystyle=\pm\log\left(1+\frac{iv_{\rm
F}t}{a+ixg_{\rm N}}\right)$ $\displaystyle+\log\left(1+\frac{iv_{\rm
F}t}{a-ixg_{\rm N}}\right).$ (63)

Here, the real-time Green’s function is defined as

$\displaystyle\tilde{G}^{XY}_{j\delta}(x,x^{\prime},t)$
$\displaystyle=\left\langle
X_{j\delta}(x,t)Y_{j\delta}(x^{\prime},0)\right\rangle,$
$\displaystyle-\left\langle X_{j\delta}(x,t)^{2}\right\rangle-\left\langle
Y_{j\delta}(x^{\prime},0)^{2}\right\rangle,$ (64)

where $X,Y=\theta$ or $\phi$ and
$X_{j\delta}(x,t)=e^{iHt/\hbar}X_{j\delta}(x)e^{-iHt/\hbar}$ indicates the
real-time evolution. In a similar way, the other Green functions can be
calculated from Eqs. (41)-(43) as

$\displaystyle\tilde{G}_{c+}^{\theta\theta}(0,0,t)$
$\displaystyle=-\frac{g_{\rm N}}{2\pi}\Bigl{\\{}\log(1+iv_{\rm
F}t/a)+\sum_{k=1}^{\infty}(b_{1}b_{2})^{k}I_{+}(2kL)$ $\displaystyle\hskip
28.45274pt-\frac{1}{2}\sum_{k=0}^{\infty}b_{1}(b_{1}b_{2})^{k}I_{+}(2kL+2L_{1})$
$\displaystyle\hskip
28.45274pt-\frac{1}{2}\sum_{k=0}^{\infty}b_{2}(b_{1}b_{2})^{k}I_{+}(2kL+2L_{2})\Bigr{\\}},$
(65)
$\displaystyle\tilde{G}_{c+}^{\theta\phi}(0,0,t)=-\frac{1}{4\pi}\Bigl{\\{}\sum_{k=0}^{\infty}b_{1}(b_{1}b_{2})^{k}I_{-}(2kL+2L_{1})$
$\displaystyle\hskip
28.45274pt-\sum_{k=0}^{\infty}b_{2}(b_{1}b_{2})^{k}I_{-}(2kL+2L_{2})\Bigr{\\}},$
(66)
$\displaystyle\tilde{G}_{c+}^{\phi\theta}(0,0,t)=\frac{1}{4\pi}\Bigl{\\{}\sum_{k=0}^{\infty}b_{1}(b_{1}b_{2})^{k}I_{-}(2kL+2L_{1})$
$\displaystyle\hskip
28.45274pt-\sum_{k=0}^{\infty}b_{2}(b_{1}b_{2})^{k}I_{-}(2kL+2L_{2})\Bigr{\\}},$
(67)

For other modes, i.e., $(j,\delta)=(c,-),(s,+),(s,-)$, the Green functions are
easily obtained from the above results by setting $g_{1}=g_{2}=g_{\rm N}=1$
(the two-lead case) or $g_{1}=0$ and $g_{2}=g_{\rm N}=1$ (the one-lead case).

Next, we calculate spatial derivatives of the Green functions for the channel
$(j,\delta)=(c,+)$. For simplicity, we drop the subscripts assigning the mode
hereafter. By solving the Fourier transformation of Eqs. (40)-(43), we obtain

$\displaystyle
G^{\phi\phi}(x,0,\omega)=G^{\phi\phi}(0,x,\omega)=\frac{1}{g_{\rm
N}+g_{2}}\frac{1}{|\omega|}G_{+}(x),$ (68) $\displaystyle
G^{\theta\theta}(x,0,\omega)=G^{\theta\theta}(0,x,\omega)=\frac{g_{\rm
N}g_{2}}{g_{\rm N}+g_{2}}\frac{1}{|\omega|}G_{-}(x),$ (69) $\displaystyle
G^{\phi\theta}(x,0,\omega)=\frac{g_{\rm N}}{g_{\rm
N}+g_{2}}\frac{1}{\omega}G_{-}(x),$ (70) $\displaystyle
G^{\phi\theta}(0,x,\omega)=-\frac{g_{2}}{g_{\rm
N}+g_{2}}\frac{1}{\omega}G_{+}(x),$ (71) $\displaystyle
G^{\theta\phi}(x,0,\omega)=\frac{g_{2}}{g_{\rm
N}+g_{2}}\frac{1}{\omega}G_{+}(x),$ (72) $\displaystyle
G^{\theta\phi}(0,x,\omega)=-\frac{g_{\rm N}}{g_{\rm
N}+g_{2}}\frac{1}{\omega}G_{-}(x),$ (73) $\displaystyle
G_{\pm}(x)=\frac{1\pm\gamma_{1}}{1-\gamma_{1}\gamma_{2}}e^{-b|\omega|/v_{\rm
N}-(x-b)|\omega|/v_{2}},$ (74)

for $x>b$. By using the same techniques for calculation of
$\partial_{\tau}G^{XY}(0,0,\tau)$, the spatial derivatives can be calculated.
For example, we obtain

$\displaystyle\partial_{x}G^{\phi\phi}(x,0,t)=\partial_{x}G^{\phi\phi}(0,x,t)$
$\displaystyle=-\frac{1}{2\pi(g_{\rm N}+g_{2})v_{2}}$
$\displaystyle\times\sum_{k=0}^{\infty}(b_{1}b_{2})^{k}\Bigl{[}\frac{1}{\alpha_{k}(x)+t-i\tau_{0}}+\frac{1}{\alpha_{k}(x)-t+i\tau_{0}}$
$\displaystyle\hskip
14.22636pt+\frac{b_{1}}{\beta_{k}(x)+t-i\tau_{0}}+\frac{b_{1}}{\beta_{k}(x)-t+i\tau_{0}}\Bigr{]},$
(75) $\displaystyle\alpha_{k}(x)=\frac{2kL+L_{2}}{v_{\rm
N}}+\frac{x-L_{2}}{v_{2}},$ (76)
$\displaystyle\beta_{k}(x)=\alpha_{k}(x)+\frac{2L_{1}}{v_{\rm N}},$ (77)

after analytic continuation $\tau=it+\tau_{0}$, where $\tau_{0}=a/v_{\rm F}$
is a short-time cutoff.

The Green functions obtained above can be related to the Keldysh Green
functions defined as

$\displaystyle G^{XY}_{K}(x,x^{\prime},t)=\langle
T_{K}\\{X(x,t^{\eta_{1}})Y(x^{\prime},0^{\eta_{2}})\\}\rangle,$ (78)

where $\eta_{1}$, $\eta_{2}$ ($=\pm 1$) represents the forward ($+1$) or
backward ($-1$) contours. The Keldysh Green functions are expressed in a
matrix form as

$\displaystyle
G^{XY}_{K}(x,x^{\prime},t)=\left(\begin{array}[]{cc}G^{XY}_{(++)}(x,x^{\prime},t)&G^{XY}_{(+-)}(x,x^{\prime},t)\\\
G^{XY}_{(-+)}(x,x^{\prime},t)&G^{XY}_{(--)}(x,x^{\prime},t)\end{array}\right)$
(81)
$\displaystyle=\left(\begin{array}[]{cc}G^{XY}(x,x^{\prime},|t|)&G^{XY}(x^{\prime},x,-t)\\\
G^{XY}(x,x^{\prime},t)&G^{XY}(x^{\prime},x,-|t|)\end{array}\right).$ (84)

The current and current noise are calculated combining these expressions for
$G^{XY}_{(\eta_{1}\eta_{2})}(x,x^{\prime},t)$ with the results given in Ref.
Guigou  _et al._ , 2007.

The Keldysh Green functions of the bosonic field describing the STM tip is
calculated as

$\displaystyle g_{\sigma(\eta_{1}\eta_{2})}(t_{1}-t_{2})=\langle
T_{K}\\{\tilde{\varphi}_{\sigma}(t_{1}^{\eta_{1}})\tilde{\varphi}_{\sigma}(t_{2}^{\eta_{2}})\\}\rangle$
$\displaystyle\hskip
14.22636pt=-\log\Bigl{[}1+i(\eta_{1}+\eta_{2})\frac{v_{F}|t_{1}-t_{2}|}{2a}$
$\displaystyle\hskip
42.67912pt-i(\eta_{1}-\eta_{2})\frac{v_{F}(t_{1}-t_{2})}{2a}\Bigr{]}.$ (85)

##  Appendix B Current Profile

The calculation of $I(x,t)$ for arbitrary time and position is a rather long,
tedious, but straightforward extension of that in Refs. Crépieux  _et al._ ,
2003; Guigou  _et al._ , 2007. We only show the final expression as follows:

$\displaystyle I(x,t)$
$\displaystyle=\frac{-8iev_{F}\Gamma^{2}}{\pi^{2}a^{2}}\sum_{l,l^{\prime}}p_{l}p_{l^{\prime}}^{*}e^{i(l-l^{\prime})\Omega
t}$
$\displaystyle\qquad\times\left[\frac{i\theta((l^{\prime}-l)\Omega)}{(g_{\rm
N}+g_{2})v_{2}}\left(\frac{1+b_{1}e^{-i\frac{2(l-l^{\prime})\Omega
L_{1}}{v_{N}}}}{1-b_{1}b_{2}e^{-i\frac{2(l-l^{\prime})\Omega}{v_{N}}L}}e^{-i\frac{(l-l^{\prime})\Omega
L_{2}}{v_{N}}}e^{-i\frac{(l-l^{\prime})\Omega}{v_{2}}(x-L_{2})}-{\rm
H.c.}\right)X_{1,(ll^{\prime})}\right.$
$\displaystyle\qquad\qquad\left.-\frac{i}{(g_{\rm
N}+g_{2})v_{2}}\left(\frac{1+b_{1}e^{-i\frac{2|(l-l^{\prime})\Omega|L_{1}}{v_{N}}}}{1-b_{1}b_{2}e^{-i\frac{2|(l-l^{\prime})\Omega|}{v_{N}}L}}e^{-i\frac{|(l-l^{\prime})\Omega|L_{2}}{v_{N}}}e^{-i\frac{|(l-l^{\prime})\Omega|}{v_{2}}(x-L_{2})}\right)X_{2,(ll^{\prime})}\right],\qquad(x>L_{2}),$
(86) $\displaystyle I(x,t)$
$\displaystyle=\frac{-8iev_{F}\Gamma^{2}}{\pi^{2}a^{2}}\sum_{l,l^{\prime}}p_{l}p_{l^{\prime}}^{*}e^{i(l-l^{\prime})\Omega
t}$
$\displaystyle\qquad\times\left[\frac{i\theta((l^{\prime}-l)\Omega)}{2v_{F}}\left(\frac{1+b_{1}e^{-i\frac{2(l-l^{\prime})\Omega
L_{1}}{v_{N}}}}{1-b_{1}b_{2}e^{-i\frac{2(l-l^{\prime})\Omega}{v_{N}}L}}(e^{-i\frac{(l-l^{\prime})\Omega}{v_{N}}x}-b_{2}e^{i\frac{(l-l^{\prime})\Omega}{v_{N}}(x-2L_{2})})-{\rm
H.c.}\right)X_{1,(ll^{\prime})}\right.$
$\displaystyle\qquad\qquad\left.-\frac{i}{2v_{F}}\left(\frac{1+b_{1}e^{-i\frac{2|(l-l^{\prime})\Omega|L_{1}}{v_{N}}}}{1-b_{1}b_{2}e^{-i\frac{2|(l-l^{\prime})\Omega|}{v_{N}}L}}(e^{-i\frac{|(l-l^{\prime})\Omega|}{v_{N}}x}-b_{2}e^{i\frac{|(l-l^{\prime})\Omega|}{v_{N}}(x-2L_{2})})\right)X_{2,(ll^{\prime})}\right],\qquad(L_{2}\geq
x\geq 0),$ (87) $\displaystyle X_{1,(ll^{\prime})}=\int_{0}^{\infty}d\tau
D(\tau)\sin\left[\left(\omega_{0}+\frac{l+l^{\prime}}{2}\Omega\right)\tau\right]\left(\frac{\sin\left(\frac{l-l^{\prime}}{2}\Omega\tau\right)e^{-iF(\tau)}}{1+iv_{F}\tau/a}-{\rm
Im}\left[\frac{e^{i(l-l^{\prime})\Omega\tau/2}e^{-iF(\tau)}}{1+iv_{F}\tau/a}\right]\right),$
(88) $\displaystyle X_{2,(ll^{\prime})}=\int_{0}^{\infty}d\tau
D(\tau)\sin\left[\left(\omega_{0}+\frac{l+l^{\prime}}{2}\Omega\right)\tau\right]\left(\frac{\sin\left(\frac{l-l^{\prime}}{2}\Omega\tau\right)e^{iF(\tau)}}{1-iv_{F}\tau/a}+{\rm
Im}\left[\frac{e^{-i(l-l^{\prime})\Omega\tau/2}e^{-iF(\tau)}}{1+iv_{F}\tau/a}\right]\right).$
(89)

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