# The Herbarium 2021 Half–Earth Challenge Dataset

Riccardo de Lutio
ETH Zürich
<EMAIL_ADDRESS>Damon Little
New York Botanical Garden  Barbara Ambrose
New York Botanical Garden  Serge Belongie
Cornell Tech & Google Research

###### Abstract

Herbarium sheets present a unique view of the world’s botanical history,
evolution, and diversity. This makes them an all–important data source for
botanical research. With the increased digitisation of herbaria worldwide and
the advances in the fine–grained classification domain that can facilitate
automatic identification of herbarium specimens, there are a lot of
opportunities for supporting research in this field. However, existing
datasets are either too small, or not diverse enough, in terms of represented
taxa, geographic distribution or host institutions. Furthermore, aggregating
multiple datasets is difficult as taxa exist under a multitude of different
names and the taxonomy requires alignment to a common reference. We present
the Herbarium Half–Earth dataset, the largest and most diverse dataset of
herbarium specimens to date for automatic taxon recognition.

##  1 Introduction

Figure 1: Distribution of training images per taxon. The Herbarium Half–Earth
dataset is highly imbalanced. Featured taxa are from top to bottom: Ericameria
nauseosa (Pall. ex Pursh) G.L. Nesom & G.I. Baird (Asteraceae), Bidens
sulphurea (Cav.) Sch. Bip. (Asteraceae) and Solanum rixosum A.R. Bean
(Solanaceae).

Natural history collections, such as herbarium specimens, contain a plethora
of information from phenotype to genotype. Each specimen is a snapshot in time
and all together provide a history of plants on Earth since the first
herbarium collections were made nearly 500 years ago [18]. Therefore,
herbarium specimens are integral for understanding biodiversity and providing
data to ameliorate the impacts of habitat loss and climate change [1, 5, 13].

Citizen science initiatives such as iNaturalist [10] and Pl@ntNet [12], have
popularised species recognition as a challenging real–world classification
task, with large imbalanced fine–grained datasets. Similarly using computer
vision methods for the automatic classification of herbarium specimens is a
well studied topic, [2, 3, 7, 14, 15, 16, 20, 21, 22, 23, 24, 26]. Many of
these works focus on morphological trait recognition [3, 15, 16, 17, 20, 21,
26], while others focus on species recognition from leaves only [21, 23, 24].

However existing datasets designed for computer vision approaches currently
present some limitations. They are either small, targeted at specific taxa,
only representative of a certain geographic region or coming from a single
institution (see Tab. 1). With the Herbarium Half–Earth dataset, we aim to
address all these limitations and present the largest and most diverse dataset
of herbarium specimens for automatic taxon recognition to date.

Dataset | # Images | # Taxa | # Institutions | Geo. Range
---|---|---|---|---
Dillen et al. [7] | 1’900 | 1’580 | 9 | All Continents
Lorieul et al. [15] | 163’233 | 7’782 | 1 | Americas
Herbarium 255 [2] | 11’071 | 255 | 1 | Costa Rica
Herbarium 1K [2] | 253’733 | 1’204 | 1 | France
Herbarium 2019 [19] | 46’000 | 680 | 1 | Americas
Herbarium 2020 | 1’170’000 | 32’000 | 1 | Americas
Herbarium 2021 | 2’500’000 | 64’500 | 5 | Americas, Oceania and Pacific

Table 1: Summary of existing herbarium sheet datasets. Note that the Herbarium
2019 dataset focuses on the flowering plant family Melastomataceae, while the
other datasets present a wider taxonomic diversity.

##  2 The Herbarium Half–Earth Dataset

The Herbarium Half–Earth dataset
111https://github.com/visipedia/herbarium_comp includes more than 2.5M images
of vascular plant specimens representing nearly 64,500 taxa from the Americas
and Oceania.

The most exact labels are, in many cases, intraspecific (subspecies,
varieties, forms, etc.) or nothospecies (hybrids) neither of which can be
characterized as “species”, thus we use the terms “taxon” and “taxa” as
generic descriptors of taxonomic labels. In addition to labels for
species–level and below, we also include labels at higher levels in the
taxonomic hierarchy: family and order. This allows for experimentation with
methods that address label hierarchy and label similarity. These labels may
also be supplemented by more fine–grained estimates of difference among taxa
available from other sources [11].

The images are provided by the New York Botanical Garden (NY), Bishop Museum
(BPBM), Naturalis Biodiversity Center (NL), Queensland Herbarium (BRI), and
Auckland War Memorial Museum (AK).

This dataset has a long tail; there are a minimum of three images per taxon
(Fig. 1). However, some taxa can be represented by more than 100 images. This
dataset only includes images of vascular plant—the group of plants that
includes lycophytes, ferns, gymnosperms, and flowering plants (Fig. 2). The
extinct forms of lycophytes are the major component of coal deposits, ferns
are indicators of ecosystem health, gymnosperms provide major habitats for
animals, and flowering plants provide almost all of our crops, vegetables, and
fruits.

Figure 2: Example of images in the Herbarium Half–Earth dataset.

###  2.1 Dataset Challenges

The Herbarium 2021 Half–Earth dataset is challenging due to multiple reasons.
First, of course, due to its large imbalance (Fig. 1), the imbalance factor
for the dataset is 1,654.5. Second, the variation within species (Figs. 3) is
high. Herbarium specimens can capture plants at different growth–stages (e.g.,
juvenile versus adult) or with different sets of plant parts (e.g., leaves and
flowers versus leaves and fruit; see Fig. 3). In addition, the techniques used
to press, dry, and mount specimens vary among collectors and collecting
expeditions—these differences can change the appearance of specimens
dramatically (e.g., collecting in alcohol often causes leaves to turn black).
Arbitrary aesthetic decisions made while processing specimens can result in
specimens that differ dramatically in appearance even though they are simply
different parts of the same individual plant (Fig. 4). In a herbarium
collection, every attempt to conserve dried specimens is made, but in practice
older specimens become more fragile and suffer damage as they age leading to
some specimens being less complete and more damaged than others. Third, the
visual similarity among species can be high (Fig. 5). Finally, the diagnostic
morphological features that botanists use to identify species are often very
small and thus require a model that is able to handle high–resolution images
and can focus on specific details [4, 22].

Figure 3: Example of visually different images corresponding to the same
species: Abarema brachystachya (DC.) Barneby & J.W.Grimes (Fabaceae). The
observed differences are primarily due to different reproductive stages: early
flowering, late flowering, and fruit. Figure 4: Different specimens of Arbutus
xalapensis Kunth (Ericaceae) made from the same individual plant at the same
time by the same collector using the same pressing, drying, and mounting
protocol. Figure 5: Example of visually similar images from different Alyssum
species (Brassicaceae): A. alyssoides (L.) L., A. desertorum Stapf, A. simplex
Rudolphi, A. szovitsianum Fisch. & C.A. Mey.

###  2.2 Data Preprocessing

In this section we give an overview of how we preprocessed the dataset. Figure
6 presents some example of herbarium sheets before and after the preprocessing
steps.

####  2.2.1 Label Alignment

Herbarium specimens of the same taxon may have been labeled in various ways
due to differences in the interpretation of taxon circumscriptions,
nomenclatural changes, and/or errors. For example, over time Pilosella
piloselloides (Vill.) Soják (Asteraceae) has been known by at least 526
different names [8]. To ameliorate this situation as much as possible, we have
standardized the image labels to the Leipzig Catalogue of Vascular Plants
(LCVP v1.0.2) [8]. Labels in our dataset have an LCVP status of either
“accepted” or “unresolved”.

The data exported from the institutional databases were first processed to
find labels that exactly matched LCVP. For labels that did not precisely
match, we then searched for long unambiguous partial matches to LCVP: the
label was shortened by removing the rightmost word and then we searched for a
match that produced only one LCVP output taxon; if no match was found, this
was repeated until the label contained only two words. Labels that still did
not unambiguously match LCVP, were matched using tre-agrep [25] allowing an
increasing amount of mismatch (10–30% of label length; all weights were set to
1). Matches returned by tre-agrep were manually reviewed (8,430 labels passed
manual review). Images with labels that could not be coerced into matching
LCVP were excluded from the dataset (c. 73 thousand images).

####  2.2.2 Image Blurring

Herbarium specimens always have a hand–written or printed label on the sheet
(usually lower right–hand corner), which includes information about the name
of the taxon, the geographic location where it was collected, the date of
collection, and the person or team of people who collected it. In addition,
annotation labels are often added to the specimen to correct or update
information on the original label—these are sequentially added in the empty
space above the original label. Specimens often also have institutional labels
or stamps indicating the herbarium in which the specimen is archived and a
barcode label corresponding to an institutional database entry. Specimens may
also include field tags with identification numbers attached directly to the
plant. Images usually include color and measurement scales as well as
institutional logos. All of these labels can of course, help identify the
specimen, thus we blurred this information in the dataset in order to force
models to learn about the plants themselves rather than the label text.

To detect these labels we used a pretrained EAST text detection model [27].
This model outputs bounding boxes around the detected text. We merged the
bounding boxes that overlapped by a sufficient margin, and filtered out those
that were too small. The resulting regions were then heavily blurred. We first
applied a mean blur, then a single Gaussian blur with added noise, and finally
used a smooth alpha map to blend into the original (Fig. 6). Finally, we
excluded images from the dataset where more than 25% of the image was blurred,
as we found those to be, in most cases, wrong predictions from our text
detection model. We deliberately chose to tune the text detection model to
have a high specificity, in order to avoid unnecessarily blurring parts of the
plant. Even though, this means that there are images where part of the labels
are missed by our blurring algorithm.

Figure 6: Example of images before and after preprocessing.

####  2.2.3 Image Resizing

Herbarium sheets are digitized as very high–resolution images to preserve as
much of the detail as possible. A common image size is around $6000\times
4000$. This is very large even for networks that are designed to work with
higher resolutions. We have resized all images in the dataset to a larger
dimension of 1000 (while preserving the aspect ratio), in order to make the
overall size of the dataset more accessible.

####  2.2.4 Dataset Split

Our dataset contains images from 64,500 taxa at the species–level or below
with 2,257,759 in the training set and 243,020 in the test set. The data has
been split to obtain an approximately even number of images across taxa in the
test set. In fact, we capped the maximum number of images per taxon to 10 for
the test set. For taxa that have few images we did a 80%/20% split for
training/test. Each category has a minimum of three images: at least one in
the test set and two in the training set.

##  3 The Herbarium 2021 Half-Earth Challenge

The Herbarium 2021 Half-Earth Challenge is a competition hosted on Kaggle as
part of $8^{th}$ workshop for Fine–Grained Visual Categorization at CVPR 2021.
This is the third iteration of the Herbarium Challenge, in this section we
give a brief description of the previous challenges.

##### The Herbarium 2019 Challenge

[19, 14] focuses on the flowering plant family Melastomataceae. It contains
46,469 digitally imaged herbarium specimens representing 683 species. The
Melastomataceae is a large family with 166 recognized genera and 5,892 species
[8]. The overlap with the iNaturalist 2018 challenge dataset [10] is only 2
out of the 683 species in the Herbarium 2019 dataset.

##### The Herbarium 2020 Challenge

dataset contains over 1M images representing over 32,000 plant species. This
challenge focuses on vascular land plants of the Americas.

##  4 Baseline and Evaluation Metric

As a simple baseline we have trained a ResNet-50 [9] for 10 epochs. We split
the training set in a stratified manner to create a hold-out validation set,
thus the baseline was trained on 80% of the full training set. We used a
balanced sampling strategy, so as to mitigate the impact of the imbalance on
the classifier. We resized the images to $256\times 256$ and used some
standard data augmentations (small rotations, horizontal flips, color-jitter
and finally center-crop to $224\times 224$). We initialised the model with
weights pretrained on ImageNet [6]. Finally we trained the model using the
standard cross-entropy loss, a batch size of $32$, a stochastic gradient
descent with a learning rate of $1\cdot 10^{-3}$ which is further reduced when
a plateau is reached and a momentum factor of $0.9$.

The evaluation metric for the Herbarium 2021 Half-Earth Challenge is the
$F_{1}$ score:

$F_{1}=2\frac{\mathrm{Pre}\cdot\mathrm{Rec}}{\mathrm{Pre}+\mathrm{Rec}}~{},$
(1)

where $\mathrm{Pre}$ denotes the precision and $\mathrm{Rec}$ the recall. Our
baseline achieves an $F_{1}$ score of $0.46$ on the private test set of the
competition. For comparison, the first place solution of the competition
achieved an $F_{1}$ score of $0.76$ on the private test set 222Results still
unverified at the moment of writing..

##  5 Conclusion

We presented the Herbarium Half-Earth dataset to enable the development of
better automatic taxon recognition models. The development of models to
automatically identify specimens will reduce the bottleneck of species
identification and has the potential to advance biodiversity research at an
unprecedented rate.

In the future, we would like to expand the dataset to include specimens
collected world-wide. There are more than 35 million digitized specimens in
electronic databases representing more than 80% of the known vascular plant
diversity.

##### Acknowledgments

We would like to thank our colleagues at the New York Botanical Garden (Damon
Little, Kimberly Watson, Barbara Ambrose) and Kiat Chuan Tan from Google for
their generous support in making this challenge possible. We are grateful for
the participation of our collaborators at Auckland War Memorial Museum
(Dhahara Ranatunga, Yumiko Baba), Bishop Museum (Melissa Tulig), Queensland
Herbarium at Brisbane Botanic Gardens (Gillian Brown, Gordon Guymer, Andrew
Franks), Naturalis Biodiversity Center (Jan Wieringa) and for contributing
images to this dataset.

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# The Multimode Character of Quantum States Released from a Superconducting
Cavity

Maryam Khanahmadi<EMAIL_ADDRESS>Department of Microtechnology and
Nanoscience, Chalmers University of Technology, 412 96 Gothenburg, Sweden
Mads Middelhede Lund  Center for Complex Quantum Systems, Department of
Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark  Klaus Mølmer
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100
Copenhagen,Denmark  Göran Johansson<EMAIL_ADDRESS>Department
of Microtechnology and Nanoscience, Chalmers University of Technology, 412 96
Gothenburg, Sweden

###### Abstract

Quantum state transfer by propagating wave packets of electromagnetic
radiation requires tunable couplings between the sending and receiving quantum
systems and the propagation channel or waveguide. The highest fidelity of
state transfer in experimental demonstrations so far has been in
superconducting circuits. Here, the tunability always comes together with
nonlinear interactions, arising from the same Josephson junctions that enable
the tunability. The resulting non-linear dynamics correlates the photon number
and spatio-temporal degrees of freedom and leads to a multi-mode output state,
for any multi-photon state. In this work, we study as a generic example the
release of complex quantum states from a superconducting resonator, employing
a flux tunable coupler to engineer and control the release process. We
quantify the multi-mode character of the output state and discuss how to
optimize the fidelity of a quantum state transfer process with this in mind.

##  I Introduction

The exchange of quantum states between distant locations is an important
ingredient in secure communication networks and in scalable architectures for
quantum computing kimble2008quantum ; wehner2018quantum . Quantum bits encoded
in the higher dimensional oscillator modes of superconducting cavities have
been demonstrated to withstand photon losses and permit elementary error
correction vlastakis2013deterministically ; leghtas2015confining ;
ofek2016extending ; campagne2020quantum ; puri2020bias ; ni2023beating . It
would be desirable to use such multiphoton quantum states also for quantum
communication purposes grimsmo2020quantum ; PhysRevA.106.042614 .

While a linear mapping between a single oscillator mode and the continuum of
propagating field modes, in principle, transfers the quantum state of the
former to a traveling single-mode pulse, the temporal control of the release
process is not trivial. In superconducting circuits, tunable couplers based on
Josephson junctions are employed to control the evolution and release process
in different architectures, such as fixed-frequency transmons, flux tunable
transmons, or tunable transmission line resonators houck2007generating ;
pierre2014storage ; PhysRevX.4.041010 ; PhysRevA.93.063823 ;
PhysRevApplied.8.054015 ; pfaff2017controlled ; axline2018demand ;
cozzolino2019high ; burkhart2021error ; yang2023deterministic . While the non-
linearity of the Josephson junction enables tunable coupling, it also adds
effective self-Kerr and cross-Kerr terms to the oscillator Hamiltonian. These
non-linear terms may entangle the spatio-temporal release with the photon
number contents of the pulse, and thus the emission becomes multi-mode in
character and it may not function properly in a quantum network.

In this article, we present a general analysis that takes the multi-mode
character of the emission process fully into account. We employ a master
equation approach that readily incorporates both the coherent coupling to the
output field and decay and decoherence channels, and we use the quantum
regression theorem to assess the mode decomposition of the emitted radiation.

For the more quantitative discussion, we consider superconducting circuits.
With low loss rates and strong coupling, these are promising platforms to
efficiently prepare and emit quantum states into propagating modes
krantz2019quantum ; blais2021circuit . Different studies and experiments have
been done with a low number of photons yin2013catch ; pechal2014microwave . In
this paper, we theoretically analyze an experimentally relevant
superconducting circuit architecture for which we can control the out-coupling
strength and compute the accompanying non-linear couplings. The propagation
transfer and the recapture of the field by downstream circuit components can
then be analyzed by the method presented in kiilerich2019input .

The article is structured as follows: In Sec. II, we provide the formalism
determining the characteristics of the output field of the quantum system. In
Sec. III, we describe the superconducting emitter and tunable out-coupler in
detail, and in Sec. IV we provide numerical results and study the quantitative
effects of the nonlinearity on the multi-mode character of different bosonic
quantum states released from the circuit. Finally, we summarize the paper in
Sec. VI.

##  II Multimode Theory

Figure 1: Time-dependent spectra of the output field from a non-linear (a) and
a linear (b) resonator. Panel (a) shows the radiation from a non-linear
resonator with constant frequency $\omega_{0}$, outcoupling rate
$1/\kappa=1\,\mu s$, and Kerr coefficient $\chi/2\pi=.47$ MHz prepared in an
initial Fock state $\ket{n}=\ket{5}$. The insert in panel (a) shows that the
emitted radiation occupies several eigenmodes. In panels (b) we assume a
vanishing Kerr coefficient $\chi=0$ and a time-dependent oscillator frequency
$\omega(t)=\omega_{0}+$max$(0,2\chi(n\exp(-\kappa t)-1))$. As shown in the
inset of panel (b), the output field is a single-mode Fock state with $n=5$.

In this section, we consider a single nonlinear resonator as a toy model for a
non-linear emitter with the Hamiltonian

$\displaystyle H_{S}(t)=\omega(t)a^{\dagger}a+\chi(t)a^{\dagger 2}a^{2},$ (1)

having time-dependent frequency $\omega(t)$ and self-Kerr coefficient
$\chi(t)$, using angular frequency units of energy ($\hbar=1$). If we assume
the emitter is coupled to a waveguide with constant strength $\sqrt{\kappa}$,
the evolution of the reduced density matrix $\varrho(t)$ of the resonator is
described by the Lindblad master equation with a single Lindblad operator
$\sqrt{\kappa}$, describing the dissipation to the waveguide,

$\displaystyle\frac{\partial\varrho(t)}{\partial
t}=-i[H_{S}(t),\varrho(t)]+\kappa\Bigg{(}a\varrho(t)a^{\dagger}-\frac{1}{2}[a^{\dagger}a,\varrho(t)]\Bigg{)}.$
(2)

To find the mode decomposition of the emitted field one can utilize its first-
order autocorrelation function

$\displaystyle\mathcal{G}^{(1)}(t_{1},t_{2})=\kappa\langle
a^{\dagger}(t_{2})a(t_{1})\rangle=\sum_{i}n_{i}v^{*}_{i}(t_{1})v_{i}(t_{2}),$
(3)

which can be expressed in terms of the orthonormal output modes $v_{i}(t)$,
with the mean photon number $n_{i}$. The autocorrelation function
$\mathcal{G}^{(1)}(t_{1},t_{2})$ can be calculated using the quantum
regression theorem breuer2002theory ; gardiner2004quantum

$\displaystyle\mathcal{G}^{(1)}(t_{1},t_{2})=\kappa\mathrm{Tr}\Bigg{[}a^{\dagger}\mathcal{L}(t_{2},t_{1})[a\mathcal{L}(t_{1},0)\varrho_{s}(0)]\Bigg{]},$
(4)

where $\mathcal{L}(t^{\prime},t)$ represents the linear time evolution map of
the master equation (2) from time $t$ to $t^{\prime}$. The time-dependent
spectrum related to the autocorrelation function in Eq. (4) is found by the
Fourier transform

$\displaystyle\mathcal{I}(\omega,t)=\int_{-\infty}^{+\infty}\mathrm{d}s\,\,\,\,\mathcal{G}^{(1)}\bigg{(}t+\frac{s}{2},t-\frac{s}{2}\bigg{)}\,\,e^{-i\omega
s},$ (5)

which provides information about the time-dependent frequency content of the
output field.

Output field spectra for a nonlinear resonator and a linear resonator are
shown in Fig. 1, where in both cases the resonator is initialized in the Fock
state $\ket{\psi}=\ket{5}$. For the parameters in the figure caption, the
time-dependent spectrums show the emission of a wide range of frequencies for
both linear and non-linear resonators. The spectrum in Fig. 1 (a), shows a
visible gap between the frequency pertaining to each emitted photon,
proportional to the amount of the Kerr nonlinearity $2\chi$. The insert shows
that multiple output field modes are populated in the field emitted by the
nonlinear resonator.

For comparison, in Fig. 1(b), we assume a linear resonator, $\chi=0$, with a
time-dependent frequency $\omega(t)$ chosen to give a similar frequency range
of the emitted field as in (a). The insert in panel (b) shows that the output
field in this case only occupies a single (chirped) mode. In the next section,
we model a realistic quantum emitter implemented in superconducting circuits
and we assess how the non-linear elements used for tuning the emission process
affect the mode character and purity of the emitted quantum state.

##  III Physical Model

Several approaches have been used to map a stationary resonator mode to a
propagating pulse mode. To optimally control the release of a quantum state,
we consider the storage system (cavity), initiated in the desired quantum
state $|\psi\rangle$, described by the creation and annihilation operators
$a^{\dagger},a$ and frequency $\omega_{a}$. The storage cavity is dispersively
coupled to a flux-tunable transmon tinkham2004introduction ;
PhysRevA.76.042319 ; blais2021circuit ; rasmussen2021superconducting
described by the operators $c^{\dagger},c$ and frequency $\omega_{c}$, through
the coupling strength $g_{ac}\ll(\omega_{c}-\omega_{a})$. In addition to the
storage cavity, the coupler is also dispersively coupled to a leakage cavity,
described by frequency $\omega_{b}$ and operators $b^{\dagger},b$, with the
coupling $g_{bc}\ll(\omega_{b}-\omega_{c})$; see Fig. 2.

Figure 2: Schematic of the emitter. The non-linear coupler interacts with both
the storage and the leakage cavity. By driving the coupler at the frequency
$\omega^{a}_{01}-\omega^{b}_{01}$, it implements a frequency conversion and
resonant transfer of quanta between the storage and leakage cavity, as
governed by a beam splitter interaction $H_{bs}(t)$ between the two cavities
at frequencies $\omega^{a}_{01}$ and $\omega^{b}_{01}$. Due to the
nonlinearity of the coupler, the beam splitter interaction is accompanied by
self-Kerr and cross-Kerr non-linear interactions; see Eq. (III), which leads
to an output field populating several mode´s $v_{1}(t),v_{2}(t),v_{3}(t),...$
.

The entire system is described by the effective Hamiltonian

$\displaystyle H_{S}(t)=$
$\displaystyle\omega_{a}a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{c}c^{\dagger}c$
$\displaystyle+2E_{J}\bigg{[}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})-\cos(\frac{\varphi_{\mathrm{dc}}}{2})\bigg{]}\frac{\varphi^{2}_{c}}{2}$
$\displaystyle-2E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})\frac{\varphi^{4}_{c}}{4!},$
(6)

where $\varphi_{c}$ and $E_{J}$ correspond to the reduced flux operator of the
transmon and the energy of the junction, respectively. The AC flux drive on
the coupler is
$\varphi_{\mathrm{ext}}(t)=\varphi_{\mathrm{dc}}+F(t)\sin(\omega_{d}t)$ which
is described by the time-dependent amplitude
$F(t)\equiv\delta\tanh{(t/t_{0})}\ll 1$ and frequency $\omega_{d}$, where
$\delta$ and $t_{0}$ correspond to the amplitude and rate of turning on the
drive, respectively. For more details on the derivation of the Hamiltonian
(III) see Appendix A.

In the dispersive regime, the reduced flux of the coupler is found as the
superposition of all dressed modes nigg2012black ; marcos2013superconducting

$\displaystyle\varphi_{c}=$ $\displaystyle\frac{2\pi}{\phi_{0}}\phi_{c},$
$\displaystyle\phi_{c}=$
$\displaystyle\frac{\lambda_{a}(a+a^{\dagger})+\lambda_{b}(b+b^{\dagger})+\lambda_{c}(c+c^{\dagger})}{\sqrt{2}}\equiv\frac{A+A^{\dagger}}{\sqrt{2}},$
(7)

where $\phi_{c}$ is the flux of the coupler and the coefficients
$\lambda_{a,b,c}$ are described in Eq. (A.2). Using the Taylor expansion
(A.1), we insert Eq. (III) in the Hamiltonian (III), thus the second line of
Eq. (III) is obtained as

$\displaystyle H^{l}(t)=\frac{-2E_{J}\pi^{2}}{\phi_{0}^{2}}\Bigg{[}$
$\displaystyle\frac{\sin(\frac{\varphi_{\mathrm{dc}}}{2})F(t)\sin(\omega_{d}t)}{2}$
$\displaystyle+\frac{\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{8}F(t)^{2}\sin(\omega_{d}t)^{2}\Bigg{]}(2A^{\dagger}A+\bar{\lambda}),$
(8)

where $\bar{\lambda}=[A,A^{\dagger}]$. The third line of Eq. (III) (fourth
order of the flux operator), provides the nonlinear interactions as follows

$\displaystyle H_{nl}\approx$
$\displaystyle\frac{-\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})}{3\phi_{0}^{4}}\Bigg{[}12\bar{\lambda}A^{\dagger}A+6A^{\dagger}A^{\dagger}AA\Bigg{]},$
(9)

where we just keep the terms conserving the energy.

We assume that the coupler is initiated in the ground state and mediates the
resonant frequency conversion without itself being excited. This permits the
elimination of its quantum degrees of freedom at all times. If we consider
$\omega_{d}=\omega_{b}-\omega_{a}$ and utilize the rotating wave approximation
and transform to the rotating frame interaction picture with respect to
$\omega_{a}a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{c}c^{\dagger}c$, the
Hamiltonian is obtained as

$\displaystyle H_{S}(t)=$ $\displaystyle
S_{a}(t)a^{\dagger}a+S_{b}(t)b^{\dagger}b+\chi_{a}a^{2\dagger}a^{2}+\chi_{b}b^{2\dagger}b^{2}$
$\displaystyle+\chi_{ab}a^{\dagger}ab^{\dagger}b-ig_{\mathrm{swap}}(t)(b^{\dagger}a-a^{\dagger}b),$
(10)

where the parameters are as follows

$\displaystyle\chi_{a(b)}=$
$\displaystyle\frac{-2\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}}\lambda_{a(b)}^{4},$
$\displaystyle\chi_{ab}=$
$\displaystyle\frac{-8\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}}\lambda_{a}^{2}\lambda_{b}^{2},$
$\displaystyle g_{\mathrm{swap}}(t)=$
$\displaystyle(\frac{\pi^{2}}{\phi_{0}^{2}}-\frac{\bar{\lambda}\pi^{4}}{\phi_{0}^{4}})E_{J}\sin(\frac{\varphi_{dc}}{2})\lambda_{a}\lambda_{b}F(t),$
$\displaystyle S_{a(b)}(t)=$
$\displaystyle(\frac{\bar{\lambda}\pi^{4}}{\phi_{0}^{4}}-\frac{\pi^{2}}{\phi_{0}^{2}})\frac{E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})\lambda_{a(b)}^{2}}{4}F(t)^{2},$
(11)

and $S_{a(b)}(t)$ are the Stark shifts induced by the flux drive. The dressed
mode coefficients $\lambda_{a(b)}$ in Eq. (III,III) are proportional to the
ratio $\lambda_{a(b)}\propto g_{a(b),c}/\Delta_{a(b),c}$, where $g_{a(b),c}$
and $\Delta_{a(b),c}$ correspond to the coupling strength and detuning between
the coupler and the storage (leakage) cavity. The coupler is coupled to the
storage (leakage) cavities through the coupling capacitance $C_{ac}(C_{bc})$
where by changing the capacitance strengths, different values of the
conversion rates $g_{\mathrm{swap}}$ and also nonlinear terms
$\chi_{ab},\chi_{a(b)}$ will be obtained; see Fig. 3. As shown in Fig. 3, the
reduction (increase) of the non-linearities
$\chi_{a(b)}\propto\lambda_{a(b)}^{4}$ yields a similar effect on the swap
rate $g_{\mathrm{swap}}(t)\propto\lambda_{a}\lambda_{b}$, as mentioned
$\lambda_{a(b)}\propto g_{a(b),c}\propto C_{a(b)c}$. It is worth noting that
while a stronger amplitude of the drive $F(t)$, accelerates the transfer and
release process, in this regime the Hamiltonian of the system acquires higher-
order nonlinear interactions.

The leakage cavity decays to the waveguide with a constant decay rate
$\kappa$. The analysis of the emitted radiation is equivalent to the one
presented for the toy model in Sec. II, with the master equation,

$\displaystyle\frac{\partial\varrho(t)}{\partial
t}=-i[H_{S}(t),\varrho(t)]+\kappa\Bigg{(}b\varrho(t)b^{\dagger}-\frac{1}{2}[b^{\dagger}b,\varrho(t)]\Bigg{)},$
(12)

where $H_{S}(t)$ is given in Eq. (III), and the Lindblad operator
$\sqrt{\kappa}b$ describes the dissipation of the leakage cavity to the
waveguide. As for the toy model in Eq. (4), the mode decomposition of the
first-order correlation function of the field operator $b$, determines the
most populated orthonormal output modes.

Figure 3:  The change of the nonlinear parameters $\chi_{a},\chi_{ab}$, swap
rate $g$, detuning from resonant interaction $\chi_{ab}-2\chi_{a}$, and
Purcell rate $4g^{2}/\kappa$ as functions of the value of the coupling
capacitance $C_{ac}$, where $1/\kappa=0.2\,\mu s$. The vertical black dashed
line corresponds to the parameters used to simulate the lower panels in Fig. 4
and Fig. 5. Figure 4: The characteristics of the output field for different
initial quantum states such as a FS $\ket{n}$, a TCCS (13), and a FCCS (14).
The upper panels show (a) the relative occupation of the most populated mode,
(c) the state fidelity of the state occupying that mode with respect to the
optimal target state, see Eq. (17), and (e) the value of the cat state
amplitude $\tilde{\alpha}$ that has the maximum fidelity with the state
occupying the most populated mode.The results are shown for the initial FS
$\ket{n=3}$ and FCCS and TCCS with amplitude parameter $\alpha=\sqrt{3}$. They
are calculated as a function of the non-linear coupling strengths $\chi_{ab}$
and $\chi_{a}$ indicated along the lower and upper axes, which in turn
correspond to a varying capacitance between the storage cavity and the coupler
in the physical system (see Fig. 3). The self-Kerr $\chi_{b}$ is not specified
as the leakage cavity is mostly occupied by the vacuum and the occasional one-
photon state. In the lower panels, we assume non-linear interaction parameters
$\chi_{ab}/2\pi=$-0.11\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$,
$\chi_{a}/2\pi=$-0.017\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, and
$\chi_{b}/2\pi=$-0.04\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, and we show
the same quantities as in the upper panels but for different values of the
initial mean photon number $\langle a^{\dagger}a\rangle(t=0)$; see Sec. V for
details.

As we saw in Sec. II, non-linear terms in the Hamiltonian cause the output
field to populate several temporal field modes of the waveguide. Since the
self Kerr and cross Kerr terms
$(\chi_{a}a^{2\dagger}a^{2},\chi_{b}b^{2\dagger}b^{2}$ and
$\chi_{ab}a^{\dagger}b^{\dagger}ab)$ are inevitable consequences of the
tunable coupling in our system, the output field will indeed populate many
temporal modes, as will be seen in the next section.

In the limit where $\kappa$ is much larger than the other couplings in the
master equation, it is possible to adiabatically eliminate the leakage cavity
and obtain an effective Markovian master equation for the storage cavity mode
with a Purcell damping rate $\simeq 4g_{\mathrm{swap}}^{2}/\kappa$; green dots
in Fig 3. While we do not rely on the quantitative validity of this effective
treatment in our numerical studies, we will refer to the value of the Purcell
rate in the analysis of the results.

Here, we also note that due to the presence of other dissipation channels, the
release process cannot be made arbitrarily slow. Hence, in the experiment,
there will be a trade-off between the loss to many less populated modes in the
rapid-release regime and to other dissipation channels in the slow-release
regime.

##  IV Release of Fock states and cat states

As illustrated in Sec. II, a linear resonator emits any initial quantum state
into a single spatiotemporal mode determined by the time-dependent outcoupling
strength. However, the temporal shape of the output field from a nonlinear
emitter is correlated with the photon number contents. As discussed in the
preceding section, the strength of the coherent swap coupling between the
cavities, the effective Purcell decay rate of the storage cavity, and the non-
linearity are correlated and vary with the physical parameters of the circuit.
In this section, we characterize the output field from the resonator using the
variation of the parameters corresponding to the coupling capacitance strength
between the storage cavity and the coupler which are shown in 3.

In Fig. 4, we investigate the output field of the emitter for different
quantum states such as a Fock state (FS) $\ket{\psi}=\ket{n}$, a two-component
cat state (TCCS)

$\displaystyle\ket{\psi}\propto\ket{\alpha}+\ket{-\alpha}\propto\sum_{n=0}^{\infty}\frac{\alpha^{2n}}{\sqrt{(2n)!}}\ket{2n},$
(13)

which is a promising candidate for correcting dephasing errors
PhysRevA.59.2631 ; PhysRevLett.100.030503  , and a four-component cat state
(FCCS)

$\displaystyle\ket{\psi}\propto\ket{\alpha}+\ket{-\alpha}+\ket{i\alpha}+\ket{-i\alpha}\propto\sum_{n=0}^{\infty}\frac{\alpha^{4n}}{\sqrt{(4n)!}}\ket{4n},$
(14)

which can be used for quantum storage and communication in the presence of
photon loss PhysRevLett.111.120501 ; mirrahimi2014dynamically .

Fig. 4 (a) shows the relative population of the most populated mode
$n_{1}/n_{out}$ as a function of the Kerr nonlinearities. The ratio
$n_{1}/n_{\mathrm{out}}$ reveals the multimode character of the output field
released from a three-photon FS and TCCS and FCCS, composed of coherent states
with amplitude $\alpha^{2}=3$. As one expects, the output field becomes more
multimode with higher values of nonlinearity and with higher photon numbers.
We observe that the FS and the FCCS yield a higher single-mode content than
the TCCS with the same photon number. This difference arises partly because
the FS and FCCS populate a single, dominant Fock state component, while the
TCCS populates both $\ket{n=2}$ and $\ket{n=4}$ components, which are dephased
by the Kerr effect and couple to different frequency components of the output
field. Panel 4 (b) shows $n_{1}/n_{\mathrm{out}}$ as a function of the initial
mean photon number of the storage cavity for the same states using fixed
nonlinear parameters, corresponding to the dashed vertical line in Fig. 3. As
expected, with a higher number of photons, the resonator non-linearity leads
to an output field occupying more modes.

In Fig.4, panel (a), the output field is almost single mode,
$n_{1}/n_{\mathrm{out}}\simeq 1$, until $\chi_{ab}/2\pi$ exceeds the value
$-0.23$ MHz or equivalently the value $C_{ac}\simeq 4.5$fF in Fig 3. This can
be ascribed to the release of photons being meditated by the process,
$\ket{n,0}_{a,b}\xrightarrow{\text{swap}}\ket{n-1,1}_{a,b}\xrightarrow[]{\kappa}\ket{n-1,0}_{a,b}$.
The energy difference between the states $\ket{n,0}_{a,b},\ket{n-1,1}_{a,b}$
is $\Delta E_{n}=(n-1)(2\chi_{a}-\chi_{ab})$ where a small additional shift is
omitted, see Eq. (III). If $\chi_{ab}\simeq 2\chi_{a}$, the transfer is
resonant and faster which results in a more single-mode character. According
to Fig. 3, the energy difference has a smooth behavior until $C_{ac}\simeq
4.5$fF and hereafter, the transition states rapidly become non-resonant and
the slow release in combination with the nonlinearity causes the increasing
multimode character, witnessed by the reduction of the population
$n_{1}/n_{\mathrm{out}}$ and similar reductions in the other quantifiers of
the output field.

##  V Characterizing the most populated mode

Figure 5: Wigner function of the initial FCCS (a) and the state of the most
populated mode of emission (b), the initial TCCS (c) and the corresponding
highest fidelity single mode output state (d). The amplitude of the initial
cat states is $|\alpha^{2}|=2$ and the nonlinear interactions are
$\chi_{ab}/2\pi=-$0.11\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,\chi_{a}/2\pi=-$0.017\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$,
and $\chi_{b}/2\pi=$-0.04\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$; the same
as panels 4 (b,d,f) . The amplitude of the single mode output cat state for
panel (b) is $|\tilde{\alpha}|^{2}=1.96$ with a corresponding fidelity
$\mathcal{F}_{\mathrm{4cat}}=.97$. Panel (d) shows the emitted single mode
TCCS obtained with the optimal drive rate $t_{0}$ for the flux drive
$F(t)=\delta\tanh(t/t_{0})$. This state has a lower amplitude
$|\tilde{\alpha}|^{2}=1.66$ and the fidelity
$\mathcal{F}_{\mathrm{2cat}}=.85$; See Sec. V.1 for more explanation.

In this section, we address the quantum state contents of the most occupied
mode $v_{1}$. This is done by the formalism introduced in kiilerich2019input ,
which, for the theoretical calculation assumes a downstream ideal linear
cavity with mode operators $d,d^{\dagger}$, coupled to the system by the
interaction Hamiltonian

$\displaystyle
H_{b,v_{1}}(t)=\frac{i\sqrt{\kappa}}{2}[g_{v_{1}}(t)d^{\dagger}b-g^{*}_{v_{1}}(t)b^{\dagger}d].$
(15)

The time-dependent coupling between the cavity and waveguide

$\displaystyle
g_{v}(t)=-\frac{v_{1}^{*}(t)}{\sqrt{\int_{0}^{t}\mathrm{d}t^{\prime}|v_{1}(t^{\prime})|^{2}}},$
(16)

ensures that the cavity captures the contents of the temporal mode $v_{1}(t)$.
The dynamics of the cascaded system is described by the Lindblad master
equation with total Hamiltonian $H_{T}=H_{S}(t)+H_{b,v_{1}}(t)$ and a single
Lindblad operator describes dissipation to the waveguide,
$L_{0}(t)=\sqrt{\kappa}b+g_{v_{1}}^{*}(t)d$, representing the interference
between the emitted field of the leakage cavity and the ideal downstream
cavity. This form of the master equation ensures the cascaded nature of the
propagation of the fields, carmichael1993quantum ; gardiner1993driving .

In the three oscillator description (a,b,d), the ideal state transformation is
$\ket{\psi,0,0}_{a,b,d}\rightarrow\ket{0,0,\psi}_{a,b,d}$. While the $a$ and
$b$ oscillators may, indeed, be emptied with certainty, cavity $d$, will, in
general, be occupied by a mixed state $\varrho_{d}$, as it is correlated with
other modes of the output field.

Because the output field is not a single mode, the number of photons in the
most populated output mode $v_{1}$ is less than the initial number of photons
in the storage cavity. As shown in Figs. 5 (b,d), the output state in that
mode may still be a cat-like state $\ket{\tilde{\psi}}$, however, with a
modified amplitude $\tilde{\alpha}$. We thus vary the parameter
$\tilde{\alpha}$ in order to maximize the fidelity

$\displaystyle\mathcal{F}=\bra{\tilde{\psi}}\varrho_{\mathrm{d}}(T)\ket{\tilde{\psi}},$
(17)

over TCCS and FCCS cat states $\ket{\tilde{\psi}}$ where $T$ is a time well
after the emission of the temporal mode $v_{1}(t)$. For the Fock state
fidelities, we calculate the population of the same Fock state
$\ket{\tilde{\psi}}=\ket{\psi}$ as the initial state of the storage cavity.

Panel 4 (c) shows the fidelity $\mathcal{F}$ of the quantum state occupying
the most populated mode $v_{1}$ as a function of the Kerr nonlinearities. For
all three initial states, the fidelity is higher in the low Kerr regime,
however, the fidelity of TCCS is affected more than the other quantum states
which we ascribe to the TCCS occupying more Fock components. In panel 4 (c),
in addition to the rapid reduction of the fidelity, a revival of the fidelity
of the TCCS is observed in the higher nonlinear regime. This can be explained
by considering the different phases $\theta_{2}$ and $\theta_{4}$ induced by
the self Kerr and cross-Kerr coefficients $\chi_{a},\chi_{ab}$ on the Fock
components $\ket{2}$ and $\ket{4}$, respectively. For a certain amount of the
nonlinearities, the phases can be related by $\theta_{4}\approx\theta_{2}+2\pi
k$, recovering fidelity of the TCCS; see Fig. 8 in appendix C for
visualizations of the Wigner functions corresponding to Fig. 4 (c).

Panels 4 (e) and (f) show the size of the cat states $\tilde{\alpha}$ of the
mode $v_{1}$ as a function of the nonlinearities and the initial photon
number, respectively. In Fig. 4 (e), for both cat states, the amplitude
$\tilde{\alpha}$ decreases with increasing the nonlinearity as the less number
of photons populates the most populated mode; see panel 4 (a). Panel 4 (d)
shows the fidelity of the FS, TCCS, and FCCS as the function of the initial
mean photon number where the higher photon numbers, as expected, yield more
reduction in the fidelity.

In the low photon number regime, $n<2$, the loss of photons to other modes is
noticeable, Fig. 4 (b), but is not dominating over the reduction of the
fidelity coming from the Kerr rotation. In the recent experiment
axline2018demand , transferring TCCS and FCCS between a sender and a receiver,
the dynamics is determined by a similar Hamiltonian as Eq. (III). In the
supplementary material of axline2018demand , the fidelity reduction due to an
effective Kerr rotation combined with photon loss is discussed. Our formalism
gives a very similar picture, where we can quantify the cavity dephasing due
to the population of multiple output modes. We find a one percent reduction of
the population of the most populated output mode, which agrees with the
analysis in axline2018demand .

###  V.1 Catching the optimal cat states

As discussed in the previous section, the Kerr rotations affect the FCCS less,
as only two Fock components $\ket{0},\ket{4}$ are essentially populated and
the effect can to a large extent be described by an effective phase acquired
by $\ket{4}$. One realization of the initial and the released FCCS is shown in
Fig. 5, panels (a) and (b), respectively. The initial amplitude is
$|\alpha^{2}|=2$ and the emitted single-mode state has the highest fidelity
with a FCCS with amplitude $|\tilde{\alpha}^{2}|=1.96$; the same as panel 4
(f) and with the fidelity of $\mathcal{F}_{\mathrm{4cat}}=.97$ which is shown
in 4 (d).

In contrast to FCCS, the TCCS has three main Fock components $\ket{0},\ket{2}$
and $\ket{4}$, thus involving two different effective phases, which in general
can not be modeled by a single rotation of the Wigner function. To improve the
fidelity for a specific set of nonlinear parameters, we optimize the flux
drive on the coupler to minimize the effect of the nonlinear rotations and
find an approximate TCCS with relation $\theta_{4}\approx\theta_{2}+2\pi k$
and amplitude $\tilde{\alpha}<\alpha$. Panel (c) shows an initial TCCS with
$|\alpha|^{2}=2$, and panel (d) shows the state released in the most occupied
mode using the optimal drive rate $t_{0}=7.3$\text{\,}\mu\mathrm{s}$$, which
is almost 5 times slower than the one employed for Fig. 4 and Fig. 5 (a,b).
The amplitude of the reabsorbed TCCS is $|\tilde{\alpha}|^{2}=1.66$ with
fidelity $\mathcal{F}_{\mathrm{2cat}}=.85$ up to the arbitrary linear rotation
illustrated in Fig. 5 (d).

##  VI Summary

We studied the characteristics of the quantum state released from a realistic
nonlinear emitter. To optimally release the desired quantum states into a
propagating mode, we utilized a flux-tunable coupler to transfer the quantum
state from the storage cavity to the waveguide. We have shown that due to the
nonlinear interactions in the emitter, the output field obtains a multimode
character, where the shape of the modes and their photon population become
correlated. We investigated the output field for Fock states and two- and
four-component cat states. We also studied the adjustment of the flux drive to
emit an optimal cat state into the most populated mode. Our results showed
that in the low photon number regime, the fidelity reduction due to the
nonlinear interactions is clearly noticeable (on the percent level), but it is
not the dominant contribution to the experimentally observed reduction in
fidelity, seen in recent experiments pfaff2017controlled ; axline2018demand .

Our calculation and simulation results illustrate a trade-off between the
speed of emission and the effective nonlinearities of the emitter. This may
suggest that using more elaborate couplers with more tunability, e.g a SNAIL-
based coupler frattini20173 , may improve the fidelity of the beam-splitter
gate $\propto H_{bs}(t)$ chapman2022high . However, according to our
formalism, in aiming for high-fidelity state transfer of multiphoton states,
the multimode aspects of the transmitted field are unavoidable and need to be
taken into account.

Lastly, it is worth to note that, to actually catch a single mode using a
linear receiver, it is enough to use the time-inversed drive compared to the
one used in a linear transmitter cirac1997quantum . A realistic non-linear
emitter, however, generates a multi-mode output field, and we would need a
non-linear receiver to optimally reabsorb the output field. The concept of
time reversal can be used as a guiding principle, but how to find such a
receiver in practice is still an open question.

###### Acknowledgements.

M. Khanahmadi and G. Johansson acknowledge Simone Gasparinetti for useful
comments and the support from Knut and Alice Wallenberg Foundation through the
Wallenberg Center for Quantum Technology (WACQT). M. M. Lund and K. Mølmer
acknowledge support from Carlsberg Foundation through the “Semper Ardens”
Research Project QCooL.

##  Appendix A Lagrangian of the quantum circuit of the Emitter

Figure 6:  Quantum circuit of the emitter. The coupler is derived to properly
transfer a quantum state from storage to the leakage cavity. Both cavities are
shown by lumped elements which capacitively are coupled to a symmetry flux
tunable transmon.

The quantum circuit of Fig. 2 is shown in detail in Fig. 6. We consider the
two cavities as lumped elements including capacitances $C_{a(b)}$ in parallel
with the inductances $L_{a(b)}$. The coupler is described by the total
capacitance $C_{c}$, junction energy $E_{j}$, and the AC flux drive
$\varphi_{\mathrm{ext}}(t)$. The coupler capacitively is coupled to the
storage and leakage cavity through the capacitance $C_{ac},C_{bc}$,
respectively. The capacitance $C_{bL}$ is the coupling between the leakage
cavity and the transmission line which gives rise to a minor frequency shift
of the leakage cavity. The red dots show the flux of each system with which
the Lagrangian and the Hamiltonian of the system can be evaluated.

To properly drive the effective Hamiltonian we first evaluate the lagrangian
of the quantum circuit $\mathcal{L}=\mathcal{T}-\mathcal{V}$ where
$\mathcal{T}$ and $\mathcal{V}$ correspond to the kinetic and potential
energy, respectively.

According to Fig. 6, the kinetic energy is evaluated as

$\displaystyle\mathcal{T}=\frac{1}{2}\Bigg{[}$ $\displaystyle
C_{a}\dot{\phi}_{a}^{2}+C_{ac}\dot{\phi}_{C_{ac}}^{2}+C_{c}\dot{\phi}_{c}^{2}$
$\displaystyle+C_{bc}\dot{\phi}_{C_{bc}}^{2}+C_{b}\dot{\phi}_{b}^{2}+C_{bL}\dot{\phi}_{C_{bL}}^{2}\Bigg{]}$
(18)

and the potential is

$\displaystyle\mathcal{V}=\frac{1}{2}\Bigg{[}\frac{\phi_{a}^{2}}{L_{a}}+\frac{\phi_{b}^{2}}{L_{b}}\Bigg{]}-2E_{J}\cos(\frac{\pi\phi_{\mathrm{ext}}(t)}{\phi_{0}})\cos(\frac{2\pi\phi_{c}}{\phi_{0}}),$
(19)

where the detail of the derivation of the coupler potential is provided in the
appendix B. Using Kirchoff’s voltage laws

$\displaystyle\dot{\phi}_{C_{ac}}=\dot{\phi}_{c}-\dot{\phi}_{a}\,,\,\,\,\,\dot{\phi}_{C_{bc}}=\dot{\phi}_{b}-\dot{\phi}_{c}\,,\,\,\,\,\dot{\phi}_{C_{bL}}=-\dot{\phi}_{b},$
(20)

and the conjugate relation

$\displaystyle
Q_{i}=\frac{\partial\mathcal{L}}{\partial\dot{\phi_{i}}},i=a,c,b$ (21)

the kinetic energy in the charge basis $[Q_{a},Q_{c},Q_{b}]$ is obtained as

$\displaystyle\mathcal{T}=\frac{1}{2}\vec{Q}\mathcal{C}\vec{Q}^{T}$ (22)

where

$\displaystyle\mathcal{C}\approx\begin{bmatrix}\frac{1}{\mathbf{C_{a}}}&\,\,\,\,\frac{C_{ac}}{\mathbf{C_{a}}\mathbf{C_{c}}}&0\\\
\\\
\frac{C_{ac}}{\mathbf{C_{a}}\mathbf{C_{c}}}&\frac{1}{\mathbf{C_{c}}}&\frac{C_{bc}}{\mathbf{C_{b}}\mathbf{C_{c}}}\\\
\\\
0&\frac{C_{bc}}{\mathbf{C_{b}}\mathbf{C_{c}}}&\frac{1}{\mathbf{C_{b}}}\end{bmatrix}.$
(23)

In the capacity matrix,
$\mathbf{C_{a}}=C_{a}+C_{ac},\mathbf{C_{c}}=C_{c}+C_{bc}+C_{ac},\mathbf{C_{b}}=C_{b}+C_{bc}+C_{bL}$.
According to the capacitance matrix, the coupling strength between the coupler
and the storage ( leakage ) cavity depends on
$g_{ac}\propto\frac{C_{ac}}{\mathbf{C_{a}}\mathbf{C_{c}}},g_{bc}\propto\frac{C_{bc}}{\mathbf{C_{b}}\mathbf{C_{c}}}$,
respectively, where we assume the coupling is weak $C_{ac},C_{bc}\ll
C_{a},C_{b},C_{c}$.

###  A.1 Linear and nonlinear potential of the coupler

From Eq. (19), the potential energy of the coupler includes the contribution
of the nonlinear flux operators and the external drive as

$\displaystyle U_{\mathrm{coupler}}=$
$\displaystyle-2E_{J}\cos(\frac{\pi\phi_{\mathrm{ext}}}{\phi_{0}})\cos(\frac{2\pi\phi_{c}}{\phi_{0}})$
$\displaystyle=$
$\displaystyle-2E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})\cos(\varphi_{c}),$
(24)

where the reduced flux is considered as
$\varphi_{c}=\frac{2\pi\phi_{c}}{\phi_{0}}$ and
$\varphi_{\mathrm{ext}}(t)=\varphi_{\mathrm{dc}}+F(t)\sin(\omega_{d}t).$ Using
the Taylor expansion

$\displaystyle\cos(\varphi_{c})=$ $\displaystyle
1-\frac{\varphi_{c}^{2}}{2}+\frac{\varphi_{c}^{4}}{4!}+\mathcal{O}(\varphi_{c}^{6}),$
(25)

the potential energy of the transmon can be decomposed into a linear
$U_{C}^{l}$ and nonlinear $U_{C}^{nl}$ terms as (the constant terms are
dropped)

$\displaystyle U_{\mathrm{coupler}}=$ $\displaystyle\,U_{C}^{l}+U_{C}^{nl},$
$\displaystyle U_{C}^{l}=$
$\displaystyle-2E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})[1-\frac{\varphi_{c}^{2}}{2}],$
$\displaystyle U_{C}^{nl}=$
$\displaystyle-2E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})[\frac{\varphi_{c}^{4}}{4!}].$
(26)

By expanding the flux drive term

$\displaystyle\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})=$
$\displaystyle\cos(\frac{\varphi_{\mathrm{dc}}}{2})-\frac{\sin(\frac{\varphi_{\mathrm{dc}}}{2})F(t)\sin(\omega_{d}t)}{2}$
$\displaystyle-\frac{\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{8}F(t)^{2}\sin(\omega_{d}t)^{2}+\mathcal{O}(F(t)^{3}),$
(27)

the quadratic part of the potential, second line of (A.1), can be decomposed
into time-independent and time-dependent terms as

$\displaystyle U_{C}^{l}=\,$ $\displaystyle
E_{J}\Bigg{[}\cos(\frac{\varphi_{\mathrm{dc}}}{2})\Bigg{]}\varphi_{c}^{2}$
$\displaystyle-
E_{J}\Bigg{[}\frac{\sin(\frac{\varphi_{\mathrm{dc}}}{2})F(t)\sin(\omega_{d}t)}{2}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{8}F(t)^{2}\sin(\omega_{d}t)^{2}\Bigg{]}\varphi_{c}^{2}.$
(28)

Consequently one can easily find the time-independent potential matrix
$\mathcal{V^{\prime}}$ in basis $\phi=(\phi_{a},\phi_{c},\phi_{b})$

$\displaystyle\mathcal{V^{\prime}}=\begin{bmatrix}\frac{1}{L_{A}}&0&0\\\ \\\
0&\frac{8\pi^{2}E_{J}(\cos(\frac{\varphi_{\mathrm{dc}}}{2}))}{\phi_{0}^{2}}&0\\\
\\\ 0&0&\frac{1}{L_{B}}\end{bmatrix},$ (29)

which makes the potential energy
$\mathcal{V}=\frac{1}{2}\vec{\phi}\mathcal{V^{\prime}}\vec{\phi}^{T}$. Using
the linear potential $\mathcal{V}$ and the Kinetic energy $\mathcal{T}$, the
dressed mode of the circuit can be evaluated. In the dressed mode, the time-
dependent and nonlinear terms of the potential of the coupler provide the
value of the effective nonlinear interaction on the cavities and the optimal
swap operator between the cavities, which are discussed in the next section.

###  A.2 Dressed modes and the effective Hamiltonian

To calculate the dressed mode of the circuit Fig. 6, one can define the linear
and time-independent Hamiltonian as

$\displaystyle
H^{l}=\frac{1}{2}\vec{Q}\mathcal{C}\vec{Q}^{T}+\frac{1}{2}\vec{\phi}\mathcal{V^{\prime}}\vec{\phi}^{T}$
(30)

where $\mathcal{C},\mathcal{V^{\prime}}$ are evaluated in Eqs. (23,29),
respectively. The equation of motion for the Heisenberg operator is given by

$\displaystyle\partial^{2}_{t}\vec{Q}^{T}$
$\displaystyle=-\mathcal{V^{\prime}C}\vec{Q}^{T}$
$\displaystyle\partial^{2}_{t}\vec{\phi}^{T}$
$\displaystyle=-\mathcal{CV^{\prime}}\vec{\phi}^{T},$ (31)

and can be solved by making the ansatz

$\displaystyle\vec{Q}^{T}$
$\displaystyle=i\sum_{n}\sqrt{\frac{\hbar\omega_{n}}{2}}\frac{1}{\sqrt{\mathcal{C}}}\vec{\zeta}^{T}_{n}(a_{n}^{\dagger}e^{i\omega_{n}t}-a_{n}e^{-i\omega_{n}t})$
$\displaystyle\vec{\phi}^{T}$
$\displaystyle=\sum_{n}\sqrt{\frac{\hbar}{2\omega_{n}}}\sqrt{\mathcal{C}}\vec{\zeta}^{T}_{n}(a_{n}^{\dagger}e^{i\omega_{n}t}+a_{n}e^{-i\omega_{n}t}).$
(32)

The frequency of the modes $\omega_{n}$ and its corresponding orthogonal mode-
functions $\vec{\zeta}_{n}$ follow from the eigenvalue equation

$\displaystyle(\sqrt{\mathcal{C}}\mathcal{V^{\prime}}\sqrt{\mathcal{C}}-\omega_{n}^{2}\mathpzc{I})\vec{\zeta}^{T}_{n}=0,$
(33)

where yields the uncoupled Hamiltonian

$\displaystyle H^{l}/h=\sum_{n=1}^{3}\omega_{n}a_{n}^{\dagger}a_{n}.$ (34)

The phase operator of the coupler $\phi_{c}$, in the new basis $a_{i}$, can be
expressed as

$\displaystyle\phi_{c}=$
$\displaystyle\sum_{n=1}^{3}\frac{\lambda_{n}}{\sqrt{2}}(\hat{a}^{\dagger}_{n}+\hat{a}_{n})=\sum_{n=1}^{3}\frac{A_{n}+A^{\dagger}_{n}}{\sqrt{2}}$
$\displaystyle\longrightarrow\lambda_{n}=\sqrt{\frac{\hbar}{\omega_{n}}}\vec{e}_{c}\sqrt{\mathcal{C}}\vec{\zeta}_{n}^{T},$
(35)

where indices $i=1,2,3$ correspond to $a,c,b$, respectively and we introduce
new parameters representation

$\displaystyle\lambda_{a}$
$\displaystyle\equiv\lambda_{1}\,\,,\,\,\lambda_{c}\equiv\lambda_{2}\,\,,\,\,\lambda_{b}\equiv\lambda_{3}$
$\displaystyle a$ $\displaystyle=a_{1},c=a_{2},b=a_{3},$
$\displaystyle\omega_{a}$
$\displaystyle=\omega_{1},\omega_{c}=\omega_{2},\omega_{b}=\omega_{3}.$ (36)

Considering a compact form of the flux of the tunable coupler nigg2012black ;
marcos2013superconducting

$\displaystyle\phi_{c}=\frac{1}{\sqrt{2}}(A_{a}+A_{a}^{\dagger}+A_{b}+A_{b}^{\dagger}+A_{c}+A_{c}^{\dagger})=\frac{A+A^{\dagger}}{\sqrt{2}}$
(37)

and defining
$\bar{\lambda}=[A,A^{\dagger}]=|\lambda_{a}|^{2}+|\lambda_{b}|^{2}+|\lambda_{c}|^{2}$,
the time-dependent parts of Eq. (28) is obtained as

$\displaystyle H^{l}(t)=\frac{-2E_{J}\pi^{2}}{\phi_{0}^{2}}\Bigg{[}$
$\displaystyle\frac{\sin(\frac{\varphi_{\mathrm{dc}}}{2})F(t)\sin(\omega_{d}t)}{2}$
$\displaystyle+\frac{\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{8}F(t)^{2}\sin(\omega_{d}t)^{2}\Bigg{]}(2A^{\dagger}A+\bar{\lambda}),$
(38)

where provides the swap operator and the stark shifts. In addition, the fourth
order of the flux operator, $U_{C}^{nl}$ in Eq. (A.1), provides selfKerr,
crossKerr interaction, and also Stark shifts which in the dressed mode of Eq.
(37) is obtained

$\displaystyle U_{C}^{nl}=H_{nl}\approx$
$\displaystyle\frac{-\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})}{3\phi_{0}^{4}}(A+A^{\dagger})^{4}$
$\displaystyle H_{nl}\approx$
$\displaystyle\frac{-\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{ext}}(t)}{2})}{3\phi_{0}^{4}}\Bigg{[}12\bar{\lambda}A^{\dagger}A+6A^{\dagger}A^{\dagger}AA\Bigg{]}.$
(39)

If we consider the drive frequency $\omega_{d}=\omega_{b}-\omega_{a}$, the
total Hamiltonian $H=$ 34+A.2+39, in the rotating frame
$\omega_{a}a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{c}c^{\dagger}c$ is
obtained

$\displaystyle H=$ $\displaystyle-
ig_{\mathrm{swap}}(t)[b^{\dagger}a-a^{\dagger}b]$
$\displaystyle+g_{Stark}(t)[\lambda_{a}^{2}a^{\dagger}a+\lambda_{b}^{2}b^{\dagger}b+\lambda_{c}^{2}c^{\dagger}c]$
$\displaystyle+g_{\mathrm{selfkerr}}[\lambda_{a}^{4}a^{2\dagger}a^{2}+\lambda_{b}^{4}b^{2\dagger}b^{2}+\lambda_{c}^{4}c^{2\dagger}c^{2}]$
$\displaystyle+g_{\mathrm{crosskerr}}[\lambda_{a}^{2}\lambda_{b}^{2}b^{\dagger}ba^{\dagger}a+\lambda_{a}^{2}\lambda_{c}^{2}c^{\dagger}ca^{\dagger}a+\lambda_{c}^{2}\lambda_{b}^{2}b^{\dagger}bc^{\dagger}c],$
(40)

where the coefficients are evaluated as

$\displaystyle g_{\mathrm{swap}}(t)$
$\displaystyle=(\frac{\pi^{2}}{\phi_{0}^{2}}-\frac{\bar{\lambda}\pi^{4}}{\phi_{0}^{4}})E_{J}\lambda_{a}\lambda_{b}\sin(\frac{\varphi_{\mathrm{dc}}}{2})F(t),$
(41) $\displaystyle g_{\mathrm{Stark}}(t)$
$\displaystyle=(\frac{\bar{\lambda}\pi^{4}}{\phi_{0}^{4}}-\frac{\pi^{2}}{\phi_{0}^{2}})\frac{E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{4}F(t)^{2},$
(42) $\displaystyle g_{\mathrm{selfkerr}}$
$\displaystyle=\frac{-2\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}},$
(43) $\displaystyle g_{\mathrm{crosskerr}}$
$\displaystyle=\frac{-8\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}}.$
(44)

It is worth noting that because the coupler is initialized in the ground
state, the Hamiltonian (A.2), decouples the coupler from both cavities which
means the terms $(c^{\dagger}ca^{\dagger}a,c^{\dagger}cb^{\dagger}b,c^{\dagger
2}c^{2})$ vanish at all times. Comparing Eqs. (III,A.2-44), the following
parameters are introduced

$\displaystyle\chi_{a(b)}=$
$\displaystyle\frac{-2\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}}\lambda_{a(b)}^{4},$
$\displaystyle\chi_{ab}=$
$\displaystyle\frac{-8\pi^{4}E_{J}\cos(\frac{\varphi_{\mathrm{dc}}}{2})}{\phi_{0}^{4}}\lambda_{a}^{2}\lambda_{b}^{2},$
$\displaystyle S_{a(b)}(t)=$ $\displaystyle
g_{\mathrm{Stark}}(t)\lambda_{a(b)}^{2}.$ (45)

##  Appendix B  Calculation of the potential of the coupler

Figure 7: Schematic of a flux-tunable coupler. Two Josephson junction
$E_{J}^{1},E_{J}^{1}$ are in parallel with two parasitic (small) capacitance
$C_{j,1},C_{j,2}$ and shunted by a large capacitance $C_{t}$
rasmussen2021superconducting . The flux drive in the small loop is
$\varphi_{\mathrm{ext}}(t)$ and the bigger loop is described by the flux
operator $\phi_{C}$.

In general, we can consider each junction coupled in parallel with parasitic
capacitance ($C_{j,1},C_{j,2}$), and the total system is shunted by a large
capacitance $C_{t}$; see Fig. 8. The Kinetic and potential energy of this
simple circuit is

$\displaystyle\mathcal{T}$
$\displaystyle=\frac{1}{2}\Bigg{[}C_{t}\dot{\phi}_{C}^{2}+C_{j,1}\dot{\phi}_{{j,1}}^{2}+C_{j,2}\dot{\phi}_{{j,2}}^{2}\Bigg{]}$
$\displaystyle\mathcal{V}$
$\displaystyle=-E_{j,1}\cos(\frac{2\pi\phi_{j,1}}{\phi_{0}})-E_{j,2}\cos(\frac{2\pi\phi_{j,2}}{\phi_{0}})$
(46)

According to the number of nodes in the circuit, it can be described by one
degree of freedom which we consider $\phi_{C}$. Using the Kirchhoff voltage
low, the small loop of junctions obeys the following equation

$\displaystyle\phi_{j,1}-\phi_{j,2}=\phi_{\mathrm{ext}}.$ (47)

In the following, we introduce the independent parameter $\phi_{C}$ as the
function of the two fluxes across the junctions

$\displaystyle m_{1}\phi_{j,1}+m_{2}\phi_{j,2}=\phi_{C}$ (48)

Using Eq. (47,48) the fluxes are obtained as

$\displaystyle\phi_{j,1}$
$\displaystyle=\frac{\phi_{C}+m_{2}\phi_{\mathrm{ext}}}{m_{1}+m_{2}}$
$\displaystyle\phi_{j,2}$
$\displaystyle=\frac{\phi_{C}-m_{1}\phi_{\mathrm{ext}}}{m_{1}+m_{2}},$ (49)

where subsequently, the kinetic part of Eq. (B) can be rewritten as

$\displaystyle\mathcal{T}=\frac{1}{2}\Bigg{[}$
$\displaystyle[\frac{C_{j,1}+C_{j,2}}{(m_{1}+m_{2})^{2}}+C_{t}]\dot{\phi}_{C}^{2}$
$\displaystyle+\frac{2(m_{2}C_{j,1}-m_{1}C_{j,2})}{(m_{1}+m_{2})^{2}}\dot{\phi}_{C}\dot{\phi}_{\mathrm{ext}}+\mathcal{O}(\dot{\phi}_{\mathrm{ext}})^{2}\Bigg{]}.$
(50)

We do not take the second order of the flux fluctuation into account, as we
assume the flux drive is slow in time. To remove the effect of the flux drive
fluctuation $\dot{\phi}_{\mathrm{ext}}$, one can utilize the following two
conditions

$\displaystyle m_{1}+m_{2}$ $\displaystyle=1$ $\displaystyle
m_{2}C_{j,1}-m_{1}C_{j,2}$ $\displaystyle=0,$ (51)

where leads to

$\displaystyle m_{1}$ $\displaystyle=\frac{C_{j,1}}{C_{j,1}+C_{j,2}}$
$\displaystyle m_{2}$ $\displaystyle=\frac{C_{j,2}}{C_{j,1}+C_{j,2}}.$

Hence $\mathcal{T},\mathcal{V}$ are obtained as

$\displaystyle\mathcal{T}$
$\displaystyle=\frac{C_{j,1}+C_{j,2}+C_{t}}{2}\dot{\phi}_{C}^{2}$
$\displaystyle\mathcal{V}$
$\displaystyle=-E_{J,1}\cos(\frac{2\pi[\phi_{C}+m_{2}\phi_{\mathrm{ext}}]}{\phi_{0}})-E_{J,2}\cos(\frac{2\pi[\phi_{C}-m_{1}\phi_{\mathrm{ext}}]}{\phi_{0}}).$
(52)

If we consider symmetry junction with $E_{J,1}=E_{J,2}=E_{J}$ and
$C_{j,1}=C_{j,2}$, consequently $m_{1}=m_{2}=1/2$ and the potential energy can
be simplified as

$\displaystyle\mathcal{V}=-2E_{j}\cos(\frac{2\pi\phi_{C}}{\phi_{0}})\cos(\frac{\pi\phi_{\mathrm{ext}}}{\phi_{0}}),$
(53)

which is equivalent to Eq. (19). Introducing $C_{c}\equiv
C_{j,1}+C_{j,2}+C_{t}$, the same kinetic energy as in Equation (A) is
obtained.

##  Appendix C Wigner Function of Fig. 4 (c)

Figure 8: Different realizations of the Wigner functions of the single output
mode TCCS in Fig. 4 (c).

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