# Catalytic advantage in Otto-like two-stroke quantum engines

Marcin Łobejko  Institute for Theoretical Physics and Astrophysics, Faculty of
Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk,
Poland  International Centre for Theory of Quantum Technologies, University of
Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland  Tanmoy Biswas  International
Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza
63, 80-308 Gdansk, Poland  Theoretical Division (T4), Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA.  Paweł Mazurek  Institute of
Informatics, Faculty of Mathematics, Physics and Informatics, University of
Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland  International Centre for
Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308
Gdansk, Poland  Michał Horodecki  International Centre for Theory of Quantum
Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland

###### Abstract

We demonstrate how to incorporate a catalyst to enhance the performance of a
heat engine. Specifically, we analyze efficiency in one of the simplest
engines models, which operates in only two strokes and comprises of a pair of
two-level systems, potentially assisted by a $d$-dimensional catalyst. When no
catalysis is present, the efficiency of the machine is given by the Otto
efficiency. Introducing the catalyst allows for constructing a protocol which
overcomes this bound, while new efficiency can be expressed in a simple form
as a generalization of Otto’s formula:
$1-\frac{1}{d}\frac{\omega_{c}}{\omega_{h}}$. The catalyst also provides a
bigger operational range of parameters in which the machine works as an
engine. Although an increase in engine efficiency is mostly accompanied by a
decrease in work production (approaching zero as the system approaches Carnot
efficiency), it can lead to a more favorable trade-off between work and
efficiency. The provided example introduces new possibilities for enhancing
performance of thermal machines through finite-dimensional ancillary systems.

††preprint: APS/123-QED

With inspiration coming from the field of chemistry, a catalyst has been
introduced in quantum information theory as an auxiliary system that expands
the range of possible state transformations while remaining unchanged
throughout the protocol [1, 2]. Catalysis was subsequently also applied to
quantum thermodynamics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In
particular, catalysis was used to enhance performance of cooling [15, 16]. In
the context of thermal machines working in a cyclic fashion, it emerges as the
most apparent generalized thermodynamic resource.

In this Letter, we want to characterize the catalyst’s ability to enhance the
performance of quantum heat engines. We show that the presence of a catalyst
not only boosts the efficiency, but also can drive a machine that was not
working as an engine to the engine regime. Moreover, the catalyst can also
boost engine’s power (i.e., work per cycle) or lead to a more favorable power
vs. efficiency trade-off. These results are obtained within a class of two-
stroke engines closely related to the Otto engine.

The conventional Otto cycle is one of the most well-known paradigms for
constructing heat engines at the microscopic scale and has been thoroughly
investigated both theoretically [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32] and experimentally [33, 34, 35, 36, 37, 38, 29]. Its
simple operational mode is based on interaction with the environment
alternating with energy level transformations, collectively forming the four-
stroke thermodynamic cycle. The central question in the Otto engine studies is
how to enhance its performance. Most of the research primarily focuses on
dynamical optimization, aiming to achieve a better balance between power and
efficiency by considering different system-environment couplings [39, 40, 36,
41]. Yet, there has been limited attention given to fundamentally improving
efficiency alone, solely through a less dissipative heat-to-work conversion.
In this case, there seems to be no novel ideas explored beyond the obvious
approaches like the usage of the non-thermal baths [42, 43, 44, 45, 46].

We consider an operational simplification of the Otto cycle, which reduces it
from four to just two strokes [47, 48, 49, 50, 51, 52, 53, 54]. The
fundamental concept of the two-stroke engine involves breaking down the cycle
into two distinct stages: (i) work extraction through an isoentropic process
and (ii) heat exchange via thermalization with heat baths. This division
significantly streamlines the engine’s operation, as after the work-stroke,
the cycle can be promptly completed by bringing the working body into contact
with the respective environments. The primary trade-off for this
simplification is an increase in the engine’s dimension. Specifically, the
simplified, two-stroke engine requires simultaneous operation on the pair of
two-level systems (TLSs), while only a single TLS is required for the four-
stroke Otto engine. Here, we focus on the two stroke-model representing an
analogue of the Otto cycle. However, the concept of two-stroke engines extends
beyond this, and detailed analysis, including optimality proofs, will be
presented for this general model in the accompanying paper [55],

Within this paradigm, we provide a systematic methodology how to include a
$d$-dimensional catalyst into an operation of the two-stroke engine that leads
to the generalized “$d$ \- Otto” efficiency, given by the formula:

$\displaystyle\eta_{d}=1-\frac{1}{d}\frac{\omega_{c}}{\omega_{h}},$ (1)

with a positive work production for

$\displaystyle
d\in\left(\frac{\omega_{c}}{\omega_{h}},\frac{\beta_{c}\omega_{c}}{\beta_{h}\omega_{h}}\right).$
(2)

We identify regions of parameters (temperatures $\beta_{c}$ and $\beta_{h}$
and frequencies $\omega_{c}$ and $\omega_{h}$) where presence of the catalyst
is a neccessary condition for the machine to work as an engine. In other
regimes, he catalyst brings higher efficiency and work production.

Although we concentrate here on engines, the idea can be easily generalized
and applied for refrigerators and heat pumps as well. Generally, it opens a
new range of possibilities for increasing the performance of thermal machines.
Notably, in contrast to most of the other studies, in our case, the catalyst
is included explicitly as the finite-dimensional system.

_Two-stroke engine._ We commence by introducing a two-stroke engine, which is
conceptually depicted in Figure 1. The engine operates in two fundamental
steps: the work-stroke and the heat-stroke. In the initial stage, two thermal
systems denoted as $\tau_{h}$ and $\tau_{c}$ (coming from hot and cold bath,
respectively) are brought into contact with a third system labeled as
$\rho_{s}$, functioning as a catalyst. The work-stroke, in general, involves
the implementation of the cyclic isoentropic process, which serves the purpose
of reducing the overall energy of the composite system. Here, we concentrate
solely on the unitary processes, such that together with cylicity condition,
the work-stroke corresponds to the ergotropy extraction process [56, 57, 58,
59, 60]. From a physical standpoint, due to unitarity and energy conservation,
we interpret this process as work extraction carried out by an external agent.
In the final stage, i.e., the heat-stroke, the hot and cold systems are once
again thermally equilibrated, which also uncorrelates the catalyst and
restores it to its initial state. With this construction the thermodynamic
cycle is completed, and the engine is ready to repeat the whole process again.

A mathematical description of the proposed engine is based on the following
assumptions: (i) initial state of the engine is defined as the product state:
$\varrho=\tau_{h}\otimes\tau_{c}\otimes\rho_{s}$, where
$\tau_{h,c}\propto\exp(-\beta_{h,c}H_{h,c})$ are thermal states with
corresponding (inverse) temperatures $\beta_{h}<\beta_{c}$ and Hamiltonians
$H_{h,c}$; (ii) a single cycle of the engine is described by a unitary process
$U$, such that $\varrho\to U\varrho U^{\dagger}$; (iii) the final marginal
state of the catalyst is equal to the initial one: $\Tr_{hc}[U\varrho
U^{\dagger}]=\rho_{s}.$ Then, thermodynamics of the engine is introduced based
on the following exchanged heat definitions:

$\displaystyle Q_{h,c}$ $\displaystyle=$
$\displaystyle\Tr[H_{h,c}(\varrho-U\varrho U^{\dagger})],$ (3)

which is precisely the amount of energy that needs to be provided by external
baths in order to thermalize the systems. In accordance with the First Law,
the work that is provided by external agent is equal to $W=Q_{c}+Q_{h}$, such
that efficiency of the engine is given by:

$\displaystyle\eta=\frac{W}{Q_{h}}=1+\frac{Q_{c}}{Q_{h}}.$ (4)

One may prove that in this framework the Second Law is satisfied, such that
the Clausius inequality holds: $\beta_{h}Q_{h}+\beta_{c}Q_{c}\leq 0$.

Figure 1: Two-stroke engine assisted by a catalyst. The operational principle
of a two-stroke engine is to combine thermal resources, such as hot and cold
thermal two-level systems, and use an assisted catalyst to reduce the total
energy of the system by performing work $W$ on an external agent. The first
work-stroke is performed on an isolated system and is considered a unitary
process. After that, the used fuels are removed and thermalized again,
triggering corresponding hot $Q_{h}$ and cold $Q_{c}$ heat flows. The
catalyst, after removing the fuel, returns to its initial state, allowing the
entire process to be repeated. As a main result, we demonstrate that a
$d$-dimensional catalyst can enhance the efficiency of the process to the
generalized Otto’s formula: $\eta=1-\frac{1}{d}\frac{\omega_{c}}{\omega_{h}}.$

In this Letter, we are interested in one of the simplest self-contained
engines that is consisted of the hot $\tau_{h}$ and cold $\tau_{c}$ two-level
systems (TLSs), and the $d$-dimensional catalyst $\rho_{s}$. The initial
thermal states of TLSs are
$\tau_{k}=\mathcal{Z}_{k}^{-1}(\outerproduct{0}{0}_{k}+e^{-\beta_{k}\omega_{k}}\outerproduct{1}{1}_{k})$
with Hamiltonians $H_{k}=\omega_{k}\outerproduct{1}{1}_{k}$, where
$\mathcal{Z}_{k}=1+e^{-\beta_{k}\omega_{k}}$ (for $k=h,c$). The catalyst is
described by the density matrix
$\rho_{s}=\sum_{i}p_{i}\outerproduct{i}{i}_{s}$ with an arbitrary Hamiltonian.
Moreover, we concentrate solely on unitary operations $U$ given by the set of
transpositions of the energy levels (the so-called swaps). For example: the
swap $\ket{ijk}\leftrightarrow\ket{i^{\prime}j^{\prime}k^{\prime}}$,
corresponds to the unitary action
$U\ket{ijk}=\ket{i^{\prime}j^{\prime}k^{\prime}}$ and
$U\ket{i^{\prime}j^{\prime}k^{\prime}}=\ket{ijk}$.

Figure 2: Graphical representation of disjoint swaps that lead to $d$ \- Otto
efficiency. The plot represents all the energy levels of the composite system,
where $\ket{ijk}\equiv\ket{i}_{h}\ket{j}_{c}\ket{k}_{s}$. Each column
represent a $k$-th state of the catalyst, such that cyclicity condition
reflects the conservation of the sum of probabilities within the column. For
disjoint swaps this boils down to the equal flow of the probability $\delta p$
between the connected columns. Connected levels (either via red or blue arrow)
are swapped within the work extraction stage. The red and blue arrows
represents the energy changes of hot and cold TLS, respectively, where the up-
arrow correspond to increase and down-arrow to decrease in energy. The process
is designed in such a way that if $\delta p>0$, the hot TLS is deexcited in
each transposition and the cold TLS is excited in one of the transposition.
For $d$ transpositions, we get the following heat flows: $Q_{h}=d\delta
Q_{h}=d\omega_{h}\delta p$ and $Q_{c}=-\delta Q_{c}=-\omega_{c}\delta p$,
which implies $\eta=1+Q_{c}/Q_{h}=1-\omega_{c}/(d\omega_{h})$.

_Otto efficiency (without a catalyst)._ Let us start with a protocol with two
TLSs (without the assisted catalyst), which leads to the Otto efficiency. In
this case, to extract positive work $W>0$ via the transposition of the energy
levels, there must be an inversion of the population in the composite state of
$\tau_{h}\otimes\tau_{c}$. In the following, we represent the initial
(diagonal) density matrix by the vector of probabilities
$\vec{p}_{0}=(\mathcal{Z}_{h}\mathcal{Z}_{c})^{-1}(1,e^{-\beta_{h}\omega_{h}},e^{-\beta_{c}\omega_{c}},e^{-\beta_{h}\omega_{h}-\beta_{c}\omega_{c}})$
(with corresponding energies $0,\omega_{h},\omega_{c}$ and
$\omega_{h}+\omega_{c}$). From here we see that the ground state $\ket{00}$ is
the most occupied state and the excited state $\ket{11}$ is the least. Thus,
the only possibility to extract work from the system via swap is to transpose
$\ket{01}\leftrightarrow\ket{10}$ providing that
$e^{-\beta_{h}\omega_{h}}>e^{-\beta_{c}\omega_{c}}$ and
$\omega_{h}>\omega_{c}$ (i.e., the inversion of population is present). In
this case, we achieve the Otto efficiency:

$\displaystyle\eta_{1}=1-\frac{\omega_{c}}{\omega_{h}}$ (5)

with the work production:

$\displaystyle
W_{1}=\frac{(\omega_{h}-\omega_{c})(e^{-\beta_{h}\omega_{h}}-e^{-\beta_{c}\omega_{c}})}{(1+e^{-\beta_{h}\omega_{h}})(1+e^{-\beta_{c}\omega_{c}})}.$
(6)

One may also prove that this is the optimal efficiency in the whole set of
unitary operations [55].

_2 - Otto efficiency (two-dimensional catalyst)._ Let us now reveal how the
efficiency of the process may be increased via the assisted catalyst, which
essentially boils down to increasing the hot heat $Q_{h}$ while keeping the
cold one $Q_{c}$ constant.

From now on, we concentrate solely on disjoint swaps. Then, one may consider
an _internal_ swap $\ket{ijk}\leftrightarrow\ket{i^{\prime}j^{\prime}k}$, that
does not affect the state of the catalyst, or the _external_ swap
$\ket{ijk}\leftrightarrow\ket{i^{\prime}j^{\prime}k^{\prime}}$, with the
change in the catalyst state. Clearly, the protocol based solely on internal
swaps satisfies the cyclicity at the prize of trivializing the problem. Thus,
we consider at least one external swap of the type
$\ket{ijk}\leftrightarrow\ket{i^{\prime}j^{\prime}k^{\prime}}$. Now, let us
label the initial set of occupation probabilities by
$\vec{p}_{l}=(p_{1}^{(l)},p_{2}^{(l)},p_{3}^{(l)},p_{4}^{(l)})$, for $l=1,2$
representing two different states of the catalyst, such that $\vec{p}_{1}=p\
\vec{p}_{0}$ and $\vec{p}_{2}=(1-p)\ \vec{p}_{0}$, where $p\in[0,1]$ describes
the state of the catalyst. The final state of the system (after the
permutation) is labeled by
$\vec{s}_{l}=(s_{1}^{(l)},s_{2}^{(l)},s_{3}^{(l)},s_{4}^{(l)})$, such that the
cyclicity constraint is given by $\sum_{i}p_{i}^{(l)}=\sum_{i}s_{i}^{(l)}$.

Then, let us consider a protocol with only one external swap. We label it by
$i\leftrightarrow j$, when the $i$th state from the first block is swapped
with the $j$th state from the second one. E.g. a swap $1\leftrightarrow 3$
results with a set of occupation probabilities:
$\vec{s}_{1}=(p_{3}^{(2)},p_{2}^{(1)},p_{3}^{(1)},p_{4}^{(1)})$ and
$\vec{s}_{2}=(p_{1}^{(2)},p_{2}^{(2)},p_{1}^{(1)},p_{4}^{(2)})$. In general,
for the $i\leftrightarrow j$ swap, one can show that the heat flow is given
by:

$\displaystyle
Q_{h,c}=(\varepsilon_{i}^{h,c}-\varepsilon_{j}^{h,c})(p_{i}^{(1)}-p_{j}^{(2)}),$
(7)

where $\varepsilon_{i}^{h,c}$ are the corresponding energies of the hot or
cold TLS associated with the $i$th state. However, one can easily show that to
satisfy the cyclicity condition, the equality $p_{i}^{(1)}=p_{j}^{(2)}$ has to
be obeyed, which consequently results in no heat and work flow at all. The
same reasoning is true for the process with three external swaps since the
blocks for $k=1$ and $k=2$ are equivalent, and the three-swap process is
equivalent to the one-swap process.

As a conclusion, we see that the only advantage that we can get from the
presence of the catalyst is for a process with two external swaps. Let us then
consider a two disjoint swaps: $i\leftrightarrow j$ and $n\leftrightarrow m$.
From this we get a formula for the heat flow:

$\displaystyle
Q_{k}=(\varepsilon_{i}^{k}-\varepsilon_{j}^{k})(p_{i}^{(1)}-p_{j}^{(2)})+(\varepsilon_{n}^{k}-\varepsilon_{m}^{k})(p_{n}^{(1)}-p_{m}^{(2)}),$
(8)

and the cyclicity condition translates to the relation:

$\displaystyle p_{i}^{(1)}+p_{n}^{(1)}=p_{j}^{(2)}+p_{m}^{(2)},$ (9)

such that $p_{i}^{(1)}-p_{j}^{(2)}=-(p_{n}^{(1)}-p_{m}^{(2)})\equiv\delta p$.
Finally, we have

$\displaystyle
Q_{k}=(\varepsilon_{i}^{k}+\varepsilon_{m}^{k}-\varepsilon_{j}^{k}-\varepsilon_{n}^{k})\delta
p.$ (10)

Then, to maximize the efficiency one has to increase the hot heat flow and
decrease the cold one. This can be achieved by putting $i=m=3$ and $j=2,n=1$.
In accordance, we get

$\displaystyle Q_{h}=2\omega_{h}\delta p,\quad Q_{c}=-\omega_{c}\delta p,$
(11)

which results in the so-called “$2$ \- Otto” efficiency:

$\displaystyle\eta_{2}=1-\frac{\omega_{c}}{2\omega_{h}}.$ (12)

To calculate the work production $W=Q_{h}+Q_{c}$, one needs to solve the
equation (9) for $\delta p$, which results in:

$\displaystyle
W_{2}=\frac{(2\omega_{h}-\omega_{c})(e^{-2\beta_{h}\omega_{h}}-e^{-\beta_{c}\omega_{c}})}{(1+e^{-\beta_{h}\omega_{h}})(1+e^{-\beta_{c}\omega_{c}})f_{2}},$
(13)

where $f_{2}=1+e^{-\beta_{c}\omega_{c}}+2e^{-\beta_{h}\omega_{h}}$.

Figure 3: Regimes of the engine mode. According to inequality (24) the regime
of the engine mode (i.e., with $W>0$) is provided solely by the frequency
ratio $\omega_{c}/\omega_{h}$ and temperature ratio $\beta_{c}/\beta_{h}$. The
graph presents regimes for different dimensions of the catalyst $d$.

_$d$ \- Otto efficiency ($d$-dimensional catalyst)._ The generalization to
$d$-dimensional catalyst with disjoint transpositions is straightforward. To
simplify the notation, we parameterize swaps in the following way. An $i$th
swap $u_{i}\leftrightarrow d_{i}$ is a transposition of the energy levels
$\ket{u_{i}}$ (“up”) and $\ket{d_{i}}$ (“down”), such that the corresponding
energies satisfy: $\varepsilon_{u_{i}}>\varepsilon_{d_{i}}$. In accordance,
the heat flows can be written as:

$\displaystyle Q_{k}=\sum_{i}\omega_{i}^{k}(p_{u_{i}}-p_{d_{i}}),$ (14)

where $\omega_{i}^{k}=\bra{u_{i}}H_{k}\ket{u_{i}}-\bra{d_{i}}H_{k}\ket{d_{i}}$
and $p_{u_{i}}=\bra{u_{i}}\varrho\ket{u_{i}}$,
$p_{d_{i}}=\bra{d_{i}}\varrho\ket{d_{i}}$, while the cyclicity condition is
given by:

$\displaystyle\sum_{i}(\delta_{l,u_{i}}-\delta_{l,d_{i}})(p_{u_{i}}-p_{d_{i}})=0$
(15)

for each $l=1,2,\dots,d$. Let us note here that within this generalized
description, the trivial case $d=1$ refers to no catalyst present. In analogy
to the two-dimensional case, the strategy is to design the swaps in such a way
that: $\omega_{i}^{h}=\omega_{h}$ (for all $i$), $\omega_{j}^{c}=-\omega_{c}$
(for some $j$) and $\omega_{i}^{c}=0$ (for all $i\neq j$). According to this
idea, one can easily generalize a two-swap case to the $d$-swap protocol
putting:

$\displaystyle\ket{u_{i}}=\ket{10i}\quad\text{for}\quad i=1,2,\dots,d$ (16)
$\displaystyle\ket{d_{i}}=\ket{00i}\quad\text{for}\quad i=1,2,\dots,d-1$ (17)
$\displaystyle\ket{d_{d}}=\ket{01d}.$ (18)

Moreover, with this choice the cyclicity reduces to the condition:

$\displaystyle\delta p_{i}=\delta p_{j}\equiv\delta p,$ (19)

for all $i$ and $j$, where $\delta p_{i}=p_{u_{i}}-p_{d_{i}}$.

We arrive with the heat currents

$\displaystyle Q_{h}=d\omega_{h}\delta p,\quad Q_{c}=-\omega_{c}\delta p,$
(20)

which leads to the generalized efficiency and work production formulas for
$d$-dimensional case:

$\displaystyle\eta_{d}=1-\frac{1}{d}\frac{\omega_{c}}{\omega_{h}},$ (21)
$\displaystyle
W_{d}=\frac{(d\omega_{h}-\omega_{c})(e^{-d\beta_{h}\omega_{h}}-e^{-\beta_{c}\omega_{c}})}{(1+e^{-\beta_{h}\omega_{h}})(1+e^{-\beta_{c}\omega_{c}})f_{d}}.$
(22)

Above,

$f_{d}=\frac{(e^{-d\beta_{h}\omega_{h}}-1)(e^{-\beta_{c}\omega_{c}}-1)}{(1-e^{-\beta_{h}\omega_{h}})^{2}}\\\
+\frac{d(e^{-d\beta_{h}\omega_{h}}-e^{-\beta_{c}\omega_{c}})(e^{-\beta_{h}\omega_{h}}-1)}{(1-e^{-\beta_{h}\omega_{h}})^{2}}.$
(23)

Figure 4: Efficiency vs. work production trade-off. For a fixed ratio of
temperatures $\beta_{c}/\beta_{h}$, ratio of frequencies
$\omega_{c}/\omega_{h}$ and hot temperature $\beta_{h}\omega_{h}$, the trade-
off curve is constructed via changing the dimension of the catalyst $d$
(points on the curve). According to formula (21), the efficiency increases
with growing dimension $d$; however, the growth in work production is also
observed from non-catalytic ($d=1$) to catalytic engine $(d=2)$ in the top
panel. The range of possible dimensions is given by Eq. (25).

Regime of operation. According to Eq. (21), the catalyst provides an
enhancement in engine efficiency $\eta_{d}$ provided the machine works in the
engine’s mode, i.e., whenever the provided work (22) is positive $W_{d}>0$.
The condition boils down to the following inequality:

$\displaystyle\frac{\beta_{h}}{\beta_{c}}<\frac{\omega_{c}}{d\omega_{h}}<1,$
(24)

that proves that the engine’s efficiency is always smaller than Carnot, i.e.,
$\eta_{d}<\eta_{c}=1-\beta_{h}/\beta_{c}$. Equivalently, for fixed bath
temperatures we provide the range of all possible dimensions of the catalyst:

$\displaystyle
d\in\left(\frac{\omega_{c}}{\omega_{h}},\frac{\beta_{c}\omega_{c}}{\beta_{h}\omega_{h}}\right).$
(25)

(A larger range of parameters for which the engine with $d$-dimensional
catalyst is operating was obtained via a more general protocol given in [55]).
Fig. 3 presents the range of operation of the engine assisted by the
$d$-dimensional catalyst, given by the ratio of frequencies
$\omega_{c}/\omega_{h}$ and temperatures $\beta_{c}/\beta_{h}$, with
extensions due to catalyst visible. Nevertheless, for every $d>1$, the range
for $d$ does not cover the entire range for $d-1$, so that there exist a
regime where only standard two-stroke Otto engine works (i.e., with $d=1$),
and catalytic approach is ineffective in improving its efficiency or work
production.

Efficiency vs. work production. Finally, one may ask what is the trade-off
between efficiency and work production (per cycle) for engines assisted by the
catalyst, and how the Carnot limit is approached. In Fig. 4 and 5, we provide
an analysis in two different ranges of parameters. In the former, the trade-
off is explored via changing the dimension of the catalyst $d$, whereas in the
latter via changing the ratio of frequencies $\omega_{c}/\omega_{h}$. In Fig.
4 we observe an increase in work production when going from the non-catalytic
($d=1$) to the catalytic engine (e.g., $d=2$), before the work production goes
to zero at Carnot’s efficiency point. On the other hand, Fig. 5 shows that,
depending on the temperature regime, maximizion of work production may single
out settings without a catalyst. Nevertheless, in the limit of high
efficiencies, the catalytic approach can still prevail.

Figure 5: Efficiency vs. work production trade-off. For a fixed ratio of
temperatures $\beta_{c}/\beta_{h}$, hot temperature $\beta_{h}\omega_{h}$ and
dimension of the catalyst $d$, the trade-off curve is constructed via changing
the ratio of frequencies $\omega_{c}/\omega_{h}$. In the top figure for high
temperature $\beta_{h}\omega_{h}=0.01$, the trade-off is favorable for
catalytic engines, whereas for lower temperature $\beta_{h}\omega_{h}=0.3$
(bottom panel), a transition to the opposite regime is observed.

_Discussion._ The letter outlines a microscopic heat engine composed of a pair
of two-level systems that completes the thermodynamic cycle in just two
strokes. This so-called two-stroke Otto engine operates at the optimal
efficiency given by the Otto’s formula [55]. Our main contribution lies in
formulating protocols that exceeds that efficiency by incorporating a catalyst
into its construction. Notably, we have demonstrated that even a catalyst as
basic as a two-level system is sufficient to surpass the Otto efficiency. We
have extended this protocol to include a $d$-level system as a catalyst,
revealing that the efficiency can be elevated to the $d$-Otto efficiency (21).
Furthermore, we have analytically determined the amount of work generated by
such an engine and analyzed the trade-off curve between work and efficiency.
The work-efficiency trade-off curve demonstrates that a catalyst enables us to
simultaneously exceed the optimal work production and efficiency, which is a
particularly intriguing result.

Our findings pave the way for future research. Using the formalism described
in this letter, it is relatively straightforward to enhance the performance of
other thermal machines, such as coolers or heat pumps. Addressing these
questions would not only facilitate the development of more efficient
microscopic thermal machines but also provide insights into the deseign of
energy-efficient quantum algorithms for computation and information processing
tasks (see e.g. [61]). Another avenue of research emerges from the question:
Can construction principles of a two-stroke thermal machine be translated to
an autonomous thermal machine that produces work and exchanges heat
continuously and simultaneously [62, 63, 64]? Establishing such a
correspondence would enable the replication of the efficiency of two-stroke
thermal machines in a control-free manner, which is more experimentally
feasible. For instance, experiments could be designed to implement two-stroke
thermal machines using superconducting qubits or within the framework of
cavity quantum electrodynamics [61, 65]. Furthermore, establishing the
connection between stroke-based discrete thermal machines and autonomous
thermal machines would provide estimates for the time required to complete a
thermodynamic cycle.

_Acknowledgement:_ TB acknowledges Pharnam Bakhshinezhad for insightful
discussions and comments during a visit in quantum information and
thermodynamics group at TU, Vienna. The part of the work done at Los Alamos
National Laboratory (LANL) was carried out under the auspices of the US DOE
and NNSA under contract No. DEAC52-06NA25396. TB also acknowledges support by
the DOE Office of Science, Office of Advanced Scientific Computing Research,
Accelerated Research for Quantum Computing program, Fundamental Algorithmic
Research for Quantum Computing (FAR-QC) project. MŁ, PM and TB acknowledge
support from the Foundation for Polish Science through IRAP project co-
financed by EU within the Smart Growth Operational Programme (contract
no.2018/MAB/5). MH acknkowledges the support by the Polish National Science
Centre grant OPUS-21 (No: 2021/41/B/ST2/03207). MH is also partially supported
by the QuantERA II Programme (No 2021/03/Y/ST2/00178, acronym Ex-TRaQT) that
has received funding from the European Union’s Horizon 2020.

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11institutetext: LASIGE, Faculdade de Ciências da Universidade de Lisboa
11email<EMAIL_ADDRESS>
# Benchmark datasets for biomedical knowledge graphs with negative statements

Rita T. Sousa  Sara Silva  Catia Pesquita

###### Abstract

Knowledge graphs represent facts about real-world entities. Most of these
facts are defined as positive statements. The negative statements are scarce
but highly relevant under the open-world assumption. Furthermore, they have
been demonstrated to improve the performance of several applications, namely
in the biomedical domain. However, no benchmark dataset supports the
evaluation of the methods that consider these negative statements.

We present a collection of datasets for three relation prediction tasks -
protein-protein interaction prediction, gene-disease association prediction
and disease prediction - that aim at circumventing the difficulties in
building benchmarks for knowledge graphs with negative statements. These
datasets include data from two successful biomedical ontologies, Gene Ontology
and Human Phenotype Ontology, enriched with negative statements.

We also generate knowledge graph embeddings for each dataset with two popular
path-based methods and evaluate the performance in each task. The results show
that the negative statements can improve the performance of knowledge graph
embeddings.

###### Keywords:

Biomedical Knowledge Graphs Biomedical Ontologies Gene Ontology Human
Phenotype Ontology Negative Statements Protein-Protein Interaction Prediction
Gene-Disease Association Prediction Disease Prediction

##  1 Introduction

Knowledge Graphs (KGs) have been used to represent knowledge about real-world
entities and their relationships. Most KGs use ontologies as a backbone to
describe entities through ontology-based annotation, which associates an
entity with a class. These annotations are commonly represented as positive
statements establishing that an ontology class describes an entity. For
example, in the biomedical domain, positive statements express that a protein
$P1$ performs $intracellular\\_galactose\\_homeostasis$ as defined in the Gene
Ontology (GO) [6]. Negative statements are extremely rare but can be used to
declare that a given protein $P2$ does not perform $glucose\\_homeostasis$
(Figure 1).

Figure 1: A GO KG subgraph with positive and negative statements describing
two proteins.

The lack of negative statements is a significant issue because KGs operate
under the open-world assumption. Therefore, this lack of information can lead
to confusion regarding whether the absence of a positive statement is due to a
lack of knowledge or the actual absence of the relationship. Moreover, the
importance of negative statements to produce more accurate representations of
entities in a KG [8, 4] and improving performance in different applications
[3, 15] is increasingly recognized in the biomedical domain.

While there have been attempts to enhance current KGs with interesting
negative statements, to the best of our knowledge, no benchmark datasets have
been established to evaluate learning tasks over those KGs. With this in mind,
we enrich existing biomedical KGs with negative statements and propose a
collection of datasets for different biomedical tasks of relation prediction.
The biomedical domain was selected because biomedical KGs are usually back-
boned by biomedical ontologies that can express negation. Additionally,
negative statements have been considered relevant for different biomedical
applications [7]. Our datasets are grouped according to the task: protein-
protein interaction (PPI) prediction, gene-disease association (GDA)
prediction and disease prediction. Regarding the KGs, we enrich two successful
biomedical ontologies: GO which covers distinct semantic aspects of gene
products’ function, and Human Phenotype Ontology (HP) which describes the
universe of concepts related to phenotypic abnormalities found in human
hereditary diseases.

##  2 Related Work

Several approaches to enriching existing KGs with interesting negative
statements have been proposed. Arnaout et al. [1] proposed a method to enrich
Wikidata by including interesting negative statements, which led to
improvements in tasks involving entity summarization and decision-making.

In the biomedical domain, several approaches tackle the lack of negative
statements in biomedical ontologies, such as GO. The number of functions that
a protein does not have is larger than the number of functions it has.
Therefore, the number of negative statements describing proteins in the GO
should be several orders of magnitude greater than the number of positive
statements. Youngs et al. [17] designed two algorithms to predict negative
statements for GO and populate the NoGo database, one based on empirical
conditional probability and the other on topic modeling applied to genes and
annotation. Fu et al. [3] introduced NegGOA, a new method to enrich the GO
with relevant negative statements indicating that a protein does not perform a
given function. This method exploits the GO by using hierarchical semantic
similarity between GO terms. The enriched GO was used for protein function
prediction. Later, Vesztrocy et al. [15] presented a benchmark based on a
balanced test set of positive and negative statements. The negative statements
are generated from expert-curated annotations of protein families on
phylogenetic trees. The results of this work demonstrated that negative
statements improve protein function prediction. Regarding the HP, although the
importance of negative statements in gene-phenotype prediction is recognized,
the enrichment with negative statements has yet to be investigated [8].

##  3 Building the Datasets

We present a collection of datasets that work over two enriched KGs for three
relation prediction tasks: PPI prediction, GDA prediction, and disease
prediction. Each benchmark dataset comprises several pairs of biomedical
entities (or instances) that can be of the same type (protein-protein) or
distinct types (gene-disease and disease-patient) with the respective label (1
for the positive pairs and zero for the negative pairs). Tables 1 and 2 show
the KGs’ and datasets’ statistics for each task. Since for GDA prediction and
disease prediction, the target relation happens between two types of instances
(genes and diseases for GDA prediction and diseases and patients for disease
prediction), the instance numbers in Table 2 appear separately. Moreover, in
the case of PPI prediction, we exclusively employ the GO KG that has been
subjected to a negative statement enrichment approach. However, when it comes
to GDA prediction and disease prediction, we rely on the HP KG, which lacks a
negative statement enrichment approach, resulting in a significant imbalance
between the number of positive and negative statements.

Table 1: Statistics for each ontology regarding classes, nodes, edges. | GO | HP
---|---|---
Classes | 50918 | 17060
Literals and blank nodes | 532373 | 442246
Edges | 1425102 | 1082859
Table 2: Statistics for each task’s dataset regarding the number of instances, pairs, positive and negative statements. | PPI prediction |  | GDA prediction |  | Disease Prediction
---|---|---|---|---|---
Instances | 440 |  | 174 + 107 |  | 1033 + 660
Positive Pairs | 1024 |  | 107 |  | 660
Negative Pairs | 1024 |  | 107 |  | 681120
Positive statements | 7364 |  | 14828 |  | 38130
Negative statements | 8579 |  | 9191 |  | 179

To build these datasets, we adopt three main steps. The first one consists of
enriching the KGs. The KG is constructed using the owlready2
package111https://owlready2.readthedocs.io/en/v0.37/, which parses the
ontology file in OWL format and processes the annotation file. The annotation
file contains positive and negative statements used to describe entities. We
use the guidelines established by the
W3C222https://www.w3.org/TR/owl2-mapping-to-rdf/ to define the negative
statements as negative object property
assertions333https://www.w3.org/TR/owl2-syntax/#Negative_Object_Property_Assertions.
To do so, we use metamodeling and represent each ontology class as a class and
an individual. This situation translates into using the same IRI. Then, we use
a negative object property assertion to state that the individual representing
a biomedical entity is not connected by the object property expression to the
individual representing an ontology class, as depicted in Figure 2. The second
step consists of extracting pairs of entities from bioinformatic databases.
The third step involves selecting the pairs containing KG entities that are
well described with positive and negative statements.

Figure 2: Example of how the negative statements are defined in the OWL file.

The following subsections describe in more detail the KGs as the
characteristics of each task.

###  3.1 Biomedical Knowledge Graphs

Two KGs back-boned by biomedical ontologies are used: the GO KG and the HP KG.
Table 1 shows the statistics for each ontology.

The GO is used to describe gene products (proteins or genes) according to the
molecular functions they perform, the biological processes they are involved
in, and the cellular components where they act. The GO KG is built by
integrating three sources: the GO444The GO was downloaded on September 2021.
It is available at
http://release.geneontology.org/2021-09-01/ontology/index.html itself, the GO
Annotation data555The GO positive annotations were downloaded on January 2021.
It is available at
http://release.geneontology.org/2021-01-01/annotations/index.html. [5], and
negative GO associations produced in [15]666The negative annotations were
downloaded from https://lab.dessimoz.org/20_not.

A GO annotation links a specific gene product with a particular GO class. The
majority of GO annotation data corresponds to positive statements. However,
the GO annotation has the qualifier ‘NOT’ for a few cases, meaning that a gene
product has been proven not to carry out a specific function. The annotations
that possess this qualifier were added as negative statements. In addition to
these negative statements, the GO KG was also enriched with negative
statements derived from expert-curated annotations of protein families on
phylogenetic trees. The idea is that, if no evidence exists to suggest
otherwise, gene function is maintained over time through evolution. Therefore,
after expert curators have annotated ancestral states in gene phylogenies with
GO classes, they check if the annotations are propagated down the phylogeny.
When there is evidence that the function is absent in a specific sub-tree, a
negative statement is added to that protein. These enriched negative
statements were filtered so there were no contradictions with the GO
annotation data.

HP characterizes phenotypic abnormalities discovered in human hereditary
diseases according to five semantic aspects: phenotypic abnormalities, mode of
inheritance, clinical course, clinical modifier and frequency. HP annotations
can link diseases, patients or genes to HP classes via positive and negative
statements. The construction of HP KG777The HP was downloaded on October 2022,
while the HP annotations were downloaded on November 2021. A link to these
versions is no longer available. is similar to that of the GO KG. A negative
annotation from HP that includes ’NOT’ indicates that a disease does not cause
that phenotype, so they are included as negative statements.

###  3.2 Protein-Protein Interaction Prediction Dataset

Predicting PPIs is a fundamental task in molecular biology for understanding
biological systems. Given the high cost of experimentally determining PPI,
many computational approaches for PPI prediction based on available functional
information described by the GO [6] have been proposed to find protein pairs
likely to interact and thus provide a selection of good candidates for
experimental analysis. Therefore, the GO KG is used to describe the proteins
of the dataset.

The positive examples are extracted from the STRING [13] database. Our
selection of protein pairs was based on the following criteria: (i)
interactions between proteins had to be curated or experimentally determined
rather than computationally determined; (ii) interactions needed to have a
confidence score above 0.950 to ensure high confidence; (iii) each protein
must have at least one positive statement for a GO class and one negative
statement for another GO class. The negative examples are generated by random
negative sampling over the set of proteins of the positive examples.

###  3.3 Gene-Disease Association Prediction Dataset

Knowing which genes are associated with a specific disease is crucial to
understanding the disease mechanisms and recognising potential biomarkers or
therapeutic targets. However, once again, validating these associations in the
wet lab is expensive and time-consuming. This has prompted the evolution of
computational methods to identify the most promising associations to be
further validated.

The two KGs are used for the GDA prediction task dataset. GO KG describes the
genes, and HP KG describes the diseases. The target relations to predict are
extracted from DisGeNET [11]. Adapting the approach described in [10], we
considered the following criteria to select gene-disease pairs: (i) each gene
must have at least one positive statement for a GO class and one negative
statement for another GO class; (ii) each disease must have at least one
positive statement for an HP class and one negative statement for an HP class.
We sampled random negative examples of the same genes and diseases to create a
balanced dataset.

###  3.4 Disease Prediction Datasets

Since human diseases are a complex phenomenon, disease prediction is an
essential but still complicated task that must be executed accurately and
efficiently. Therefore, using computational methods to help physicians
prioritize diseases is highly advantageous.

The dataset to predict if a synthetic patient has been diagnosed with a
specific disease is generated by adapting the methodology proposed in [9].
Thirty-three mendelian diseases for which they knew the penetrance of each
phenotype are selected. Penetrance indicates the likelihood that a patient
suffering from a specific disease will exhibit a particular phenotype. For
each of these 33 diseases, 20 synthetic patients diagnosed with that disease
are created. The patients’ positive annotation is determined by the disease’s
penetrance and the patient’s gender. The gender is defined randomly with an
equal likelihood for both genders. For example, the ’Aarskog-Scott syndrome’
is annotated with the phenotype ’Ptosis’ with a penetrance of 0.5061, meaning
that approximately half of the synthetic patients diagnosed with that disease
will have a positive statement for this phenotype. The negation of phenotypes
does not have a penetrance associated, so synthetic patients inherit the
negative phenotypes related to the disease. For example, since the disease
’Aarskog-Scott syndrome’ is annotated with ’NOT Decreased Fertility’, each
patient will have a negative statement for this phenotype. Furthermore, 1000
diseases were randomly chosen to add complexity to the task. These diseases
are annotated with positive and negative statements.

Random annotations can also be added to patients to emulate a more realistic
situation where a patient is associated with phenotypes unrelated to the
patient’s disease. In addition to the disease prediction dataset, we present
three versions with random annotations. The number of random annotations is
defined by a percentage Noi (Noi=[0, 0.1,0.2,0.4]) concerning a given
patient’s total number of annotations. For example, if Noi=0.5, half of the
full annotations of a given patient are added. Table 3 shows the number of
positive and negative statements for each noise version.

Table 3: Statistics for each noise version regarding the number of positive and negative statements. Noise | Positive Statements | Negative Statements
---|---|---
0.1 | 40592 | 216
0.2 | 37195 | 214
0.4 | 38242 | 192

##  4 Validation of the Datasets

KG embedding methods [14] have been successfully employed in several
biomedical applications [14]. Since these methods map KGs into low-dimensional
spaces, they have emerged as a popular way to generate features for machine
learning tasks. Therefore, we use two KG embedding methods to evaluate our
datasets - RDF2Vec [12] and OWL2Vec* [2]. RDF2Vec is a path-based method that
generates random walks in the KG that constitutes the corpus of word sequences
given as input to a neural language model. OWL2Vec* was designed to learn
ontology embeddings and it also employs direct walks on the graph to learn
graph structure. These embedding methods generate representations of the
biomedical entities that are combined using the binary Hadamard operator to
represent the pair.

The pair representations are then fed into a Random Forest algorithm for
training using Monte Carlo cross-validation (MCCV) [16]. MCCV is a variation
of traditional $k$-fold cross-validation in which the data is divided into
training and testing sets (with $\beta$ being the proportion of the dataset to
include in the test split) $M$ times. Our experiments use MCCV with $M=30$ and
$\beta=0.3$ for PPI and GDA prediction. Given the large number of pairs for
disease prediction, we use MCCV with $M=5$ and $\beta=0.3$.

Each embedding method is run with two different KGs, one with only positive
statements and the other with both positive and negative statements. Table 4
reports each task’s median of recall, precision and weighted average
F-measure.

Table 4: Median precision, recall and weighted average F-measure (Pr, Re, and F1) for PPI, GDA, and disease prediction using only positive statements (Pos) or positive and negative statements (Pos+Neg). Method | Statements | PPI |  | GDA |  | Disease
---|---|---|---|---|---|---
Pr | Re | F1 |  | Pr | Re | F1 |  | Pr | Re | F1
RDF2Vec | Pos | 0.831 | 0.826 | 0.828 |  | 0.623 | 0.625 | 0.615 |  | 0.994 | 0.742 | 0.850
Pos+Neg | 0.847 | 0.844 | 0.845 |  | 0.654 | 0.600 | 0.645 |  | 1.000 | 0.771 | 0.870
OWL2Vec* | Pos | 0.833 | 0.806 | 0.823 |  | 0.652 | 0.656 | 0.646 |  | 0.975 | 0.584 | 0.730
Pos+Neg | 0.860 | 0.812 | 0.840 |  | 0.625 | 0.661 | 0.630 |  | 0.980 | 0.563 | 0.713

Figure 3 compares the impact of using only positive statements versus both
positive and negative statements on our datasets. The bars represent the
difference in performance for precision, recall and weighted average
F-measure, with upward bars indicating improved performance with both positive
and negative statements and downward bars indicating decreased performance.

(a) PPI prediction

(b) GDA prediction

(c) Disease prediction

Figure 3: Barplots showing the differences in precision, recall, weighted
average F-measure (Pr, Re, and F1) between using positive and negative
statements or only positive statements.

The experiments show that the added information given by negative statements
generally improves the performance of RDF2Vec. However, for OWL2Vec*, the
performance only improves for PPI prediction.

##  5 Using the Benchmark

All datasets are available on Zenodo888https://doi.org/10.5281/zenodo.7709195
under a CC BY 4.0 license. For each dataset, we provide access to two types of
files: (1) one TSV file containing pairs of entities and information about
whether a relationship exists between them or not; (2) OWL files containing
the KG used to describe the biomedical entities that appear in the TSV file.
Together, these files can be used to perform relation prediction tasks since
the TSV file provides the specific entities and relations that need to be
predicted, while the OWL file provides the necessary background knowledge for
generating the features.

##  6 Conclusions

Benchmark datasets are essential for evaluating and comparing the performance
of different approaches that work over KGs. This paper presents a collection
of datasets for three relation prediction tasks in the biomedical domain: PPI
prediction, GDA prediction, and disease prediction. The biomedical domain is
chosen since it is already demonstrated that the inadequacy of approaches to
take into consideration negative statements is a limitation for several
biomedical applications. However, although the datasets are domain-specific,
they can be used to evaluate approaches outside the biomedical domain.

The datasets are validated using two popular KG embedding methods to generate
features that are then given as input for a classifier. The results highlight
the importance of incorporating negative statements into KGs to create more
accurate representations of KG entities.

#### Acknowledgements

C. P., S. S., R. T. S. are funded by the FCT through LASIGE Research Unit
(ref. UIDB/00408/2020 and ref. UIDP/00408/2020), and the FCT PhD grant (ref.
SFRH/BD/145377/2019). It was also partially supported by the KATY project,
which has received funding from the European Union’s Horizon 2020 research and
innovation programme under grant agreement No 101017453, and by HfPT: Health
from Portugal under the Portuguese Plano de Recuperação e Resiliência. The
authors thank Lina Aveiro for the preliminary results of this work.

## References

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  * [2] Chen, J., Hu, P., Jimenez-Ruiz, E., Holter, O.M., Antonyrajah, D., Horrocks, I.: OWL2Vec*: Embedding of OWL ontologies. Machine Learning pp. 1–33 (2021)
  * [3] Fu, G., Wang, J., Yang, B., Yu, G.: NegGOA: negative GO annotations selection using ontology structure. Bioinformatics 32(19), 2996–3004 (06 2016)
  * [4] Gaudet, P., Dessimoz, C.: Gene Ontology: pitfalls, biases, and remedies. In: The Gene Ontology Handbook, pp. 189–205. Humana Press, New York, NY (2017)
  * [5] GO Consortium: The Gene Ontology resource: enriching a GOld mine. Nucleic Acids Research 49(D1), D325–D334 (2021)
  * [6] GO Consortium: The Gene Ontology Resource: 20 years and still GOing strong. Nucleic Acids Research 47(D1), D330–D338 (11 2018)
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  * [8] Liu, L., Zhu, S.: Computational methods for prediction of human protein-phenotype associations: A review. Phenomics 1(4), 171–185 (2021)
  * [9] Masino, A.J., Dechene, E.T., Dulik, M.C., Wilkens, A., Spinner, N.B., Krantz, I.D., Pennington, J.W., Robinson, P.N., White, P.S.: Clinical phenotype-based gene prioritization: an initial study using semantic similarity and the human phenotype ontology. BMC bioinformatics 15(1), 1–11 (2014)
  * [10] Nunes, S., Sousa, R.T., Pesquita, C.: Predicting gene-disease associations with knowledge graph embeddings over multiple ontologies. In: ISMB Annual Meeting \- Bio-Ontologies (2021)
  * [11] Piñero, J., Ramírez-Anguita, J.M., Saüch-Pitarch, J., Ronzano, F., Centeno, E., Sanz, F., Furlong, L.I.: The DisGeNET knowledge platform for disease genomics: 2019 update. Nucleic Acids Research 48(D1), D845–D855 (11 2019)
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  * [13] Szklarczyk, D., Gable, A.L., Nastou, K.C., Lyon, D., Kirsch, R., Pyysalo, S., Doncheva, N.T., Legeay, M., Fang, T., Bork, P., Jensen, L.J., von Mering, C.: The STRING database in 2021: customizable protein–protein networks, and functional characterization of user-uploaded gene/measurement sets. Nucleic Acids Research 49(D1), D605–D612 (11 2020)
  * [14] Wang, Q., Mao, Z., Wang, B., Guo, L.: Knowledge graph embedding: A survey of approaches and applications. IEEE Transactions on Knowledge and Data Engineering 29(12), 2724–2743 (2017)
  * [15] Warwick Vesztrocy, A., Dessimoz, C.: Benchmarking Gene Ontology function predictions using negative annotations. Bioinformatics 36(Supplement_1), i210–i218 (07 2020)
  * [16] Xu, Q.S., Liang, Y.Z.: Monte Carlo cross validation. Chemometrics and Intelligent Laboratory Systems 56(1), 1–11 (2001)
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# First experience with He conditioning of an SRF photoinjector

I. Petrushina<EMAIL_ADDRESS>Department of Physics and Astronomy, Stony
Brook University, Stony Brook, NY  Y. Jing  Collider-Accelerator Department,
Brookhaven National Laboratory, Upton, NY  V.N. Litvinenko  Department of
Physics and Astronomy, Stony Brook University, Stony Brook, NY  Collider-
Accelerator Department, Brookhaven National Laboratory, Upton, NY  J. Ma  F.
Severino  G. Narayan  T. Hayes  J. C. Brutus  L. Smart  K. Decker  Collider-
Accelerator Department, Brookhaven National Laboratory, Upton, NY  S.
Belomestnykh  Fermi National Accelerator Laboratory, Batavia, IL

###### Abstract

The recent achievements in the performance of superconducting RF (SRF)
photoinjectors have opened a new era in the development of the reliable high-
brightness CW electron sources. While the SRF guns become one of the most
promising technologies, the compatibility of SRF environment with the complex
photocathodes remains on the forefront of the modern accelerator science. The
SRF cavities operate at cryogenic temperatures providing the ultra-high vacuum
(UHV) environment highly beneficial for the photocathode performance. However,
the necessity to keep the photocathodes at room temperature while being
surrounded by the cavity walls that are kept at cryogenic temperatures creates
an additional complexity for the SRF gun design and operation. The complex and
volatile chemical compounds used for photocathodes have a high chance of
contaminating the surfaces of an SRF cavity. When deposited, such compounds
could create centers for cold electron emission which degrade performance of
the SRF guns. Such a circumstance would require development of the _in-situ_
processing techniques for restoring the SRF cavity performance. This paper
presents the results of the successful implementation and application of the
He conditioning method for cavity restoration using the existing SRF
photoinjector at Brookhaven National Laboratory (BNL). The method has proven
to be extremely effective and resulted in a dramatic improvement of the BNL
gun performance.

††preprint: APS/123-QED

##  I Introduction

The future development of modern X-ray Free Electron Lasers (XFELs),
$\gamma$-ray sources, fixed target, and collider experiments for Nuclear and
High Energy Physics (NP&HEP), and a variety of other scientific, industrial,
and medical applications, rely heavily on the ability to produce low emittance
and high current CW electron beams [1]. The development of reliable electron
sources has been a major enabling technology for achieving such beams. The
recent progress in the field of SRF photoinjectors have demonstrated a
successful performance confirming that, indeed, the SRF environment and high
quantum efficiency (high-QE) photocathodes are compatible and can produce
high-brightness beams with low transverse emittances [2]. Nevertheless, the
nature of the photoinjector operation requires systematic cathode exchange,
which puts the cleanliness of the photoinjector cavity at risk. A reliable
_in-situ_ cavity restoration procedures can be of great importance when it
comes to the cavity performance and maintenance.

A successful performance of an SRF photoinjector is a delicate balance of
maintaining the good quality of operation for both the photocathodes and the
cavity itself. The introduction of a photocathode into an SRF environment
comes with a great risk of the cavity contamination by the particulates
produced during the process of the cathode insertion/removal, and deposition
of volatile or highly emissive compounds from the cathode (Cs, Na, K, etc.)
onto the walls of the cavity during operation. Such contaminants create
emission centers on the cavity surface and result in a significant cavity
performance degradation caused by excessive production of _dark current_ and
enhanced strength of _multipacting_ barriers.

_Dark current_ is an undesirable emission of electrons from the regions of
high electric fields on the cavity surface caused by a particulate
contamination. An excessive dark current results in an increased generation of
X-rays, higher consumption of liquid He, and, possibly, quenching of the
cavity. By aggravating the dark current, the presence of a photocathode within
the cavity body increases the risk of the reduced operational accelerating
voltage of the photoinjector, and subsequent degradation of the beam quality.

The resonant build-up of the secondary electrons within the cavity body—
_multipacting_ (MP)—strongly depends on the Secondary Emission Yield (SEY) of
the cavity wall material and is directly impacted by the presence of external
contaminants. MP prevents the cavity from reaching operational voltage, causes
local overheating of the cavity surface, and may result in the cavity
quenching.

Both phenomena described above demonstrate an example of the damage to the
cavity coming from the introduction of the photocathode into an SRF
environment. On the other hand, if a cavity has a presence of strong MP
barriers, the occurring electron bombardment can significantly damage the
cathode surface and dramatically decrease the cathode lifetime. To address the
issue of the inevitable contamination of the cavity walls, the reliable in-
situ cavity restoration techniques must be developed. A variety of surface
treatment techniques applicable for SRF cavities have been developed
throughout the years, but their applications to the restoration of the SRF
photoinjectors are still under investigation.

In this paper we present our experience with handling of the 113 MHz SRF
photoinjector with a warm $\textrm{CsK}_{2}\textrm{Sb}$ cathode throughout
several years of operation. We report the complications encountered throughout
these years and the solutions that were found to overcome these challenges.
First, we will provide a brief overview of a few major SRF cavity restoration
techniques available to date in application to the SRF accelerating
structures. We will then proceed by providing a description of the 113 MHz SRF
gun cavity at Brookhaven National Laboratory (BNL) and its incredible
performance in Section III. Section IV describes our experience in application
of the He conditioning technique for restoration of the BNL SRF gun
performance and reports on the progress achieved using this treatment.

##  II Cavity restoration techniques

With the variety of restoration techniques applicable for the SRF cavities,
there are a few that currently stand out when it comes to the prevention of
the field emission: High Power Processing (HPP), He processing and plasma
treatment [3, 4]. These methods are generally implemented as _in-situ_
restoration techniques and have demonstrated significant improvement in the
cavity performance.

CW RF processing or Cold Conditioning (CC) technique is fairly straightforward
and involves operation of the cavity in a CW mode for an extended period of
time while monitoring the RF losses and X-ray radiation levels from the
cavity. An RF conditioning in a sense is quite similar to a DC breakdown when
an applied surface electric field causes an explosive event resulting in
vaporization of a surface emitter. The event of an emitter explosion is
accompanied by a rapid drop in the field emission current, RF losses,
radiation activity, and results in an immediate improvement of the cavity
performance that reaches higher operational voltage levels.

High Power Processing (HPP) is a natural second step in the cavity
conditioning after the CC method has reached its limits. HPP involves pulsed
operation of the cavity which is beneficial for the emitter vaporization due
to the ability to achieve higher surface electric fields at a reduced power
dissipation and liquid helium consumption compared to the CC.

Helium Conditioning (HeC) is usually performed at the operational cryogenic
temperatures of the cavity by filling the cavity volume with pure He gas at
pressures ranging from 10-6 to 10-5 Torr. Subsequent application of short high
voltage RF pulses creates electric field at the surface of the dark current
emitters above the critical level. When such high electric fields are applied
to the impurity, the emitted burst of electrons ionizes the surrounding He
gas, and He ions bombard the source and efficiently destroy such emitters.

Plasma Treatment (PT) of the SRF cavities is a relatively new method, and is
actively pursued by the Fermi National Accelerator Laboratory (FNAL) team in
application to the 1.3 GHz LCLS-II SRF cavities. PT is performed at room
temperature and involves filling the cavity with a mixture of noble gases
(such as Ne or Ar) and oxygen at pressures of 70-200 mTorr followed up with an
ignition and sustenance of an RF discharge. A low percentage of oxygen is
added to react with the hydrocarbons adsorbed on the cavity surface. The
volatile byproducts are pumped out of the cavity and monitored with a Residual
Gas Analyzer (RGA). Ne and O2 are mixed before reaching the cavity and their
ratio is controlled using an RGA. The _in-situ_ PT technique was developed
mostly to mitigate hydrocarbon related field emission [3]. Starting from the
SNS experience [5, 6] and using the new ignition method, PT has been applied
to multiple 1.3 GHz cavities at FNAL and extended to using HOM (Higher Order
Modes) couplers to ignite the glow discharge with only a few watts of RF power
required [4, 7]. In multi-cell cavities, the plasma can be ignited one cell at
a time using a superposition of HOMs.

When the appropriate restoration methods are applied, the performance of an
SRF cavity can be significantly improved resulting in a higher achieved
electric fields, lower consumption of liquid He, and lower radiation levels.
The investigation of application of such methods to SRF photoinjectors is
crucial for the advancement in the development of novel SRF electron sources.

##  III 113 MHz SRF gun

Figure 1: Schematic of the CeC SRF gun assembly indicating the major system
components.

The 113 MHz SRF gun was built for the Coherent electron Cooling Proof of
Principle (CeC PoP) [8] experiment at BNL, and is a VHF-type quarter-wave
resonator (QWR) made of bulk Niobium. The gun operates at a very low frequency
which provides the beneficial condition for photoelectrons to be generated at
the peak of the accelerating field (10-20 MV/m at the photocathode surface).
While the detailed design of the cavity body and its auxiliaries can be found
in [9, 10, 11], here we will limit the description only to a general system
overview with the focus on the specifics of the cathode delivery system as
it’s critical for the purpose of this paper. The full assembly of the QWR
cavity in the cryostat with its surroundings is shown in Fig. 1. The cavity is
maintained at 4 K and uses the liquid He (LiHe) from the refrigeration systems
of the Relativistic Heavy Ion Collider (RHIC). The green 532 nm laser light is
transported to the cathode surface through the laser cross and the hollow
fundamental power coupler (FPC) that serves two purposes—delivers power from
the 4 kW solid-state amplifier to the cavity and acts as a frequency tuner.
The room temperature hollow Cu-coated stainless steel stalk resides inside the
4 K SRF gun. The stalk has an impedance transformer and functions as a half-
wave choke that is shorted outside the cryostat. The cathode is delivered
inside the cavity through the stalk with the use of the magnetic cathode
manipulator system. The cathode delivery system has three UHV arms used to
transport the cathode pucks from a storage unit called “garage” into the
cavity. The long manipulator arm delivers the cathode to the end of the stalk,
where it is grounded to the stalk by the RF spring fingers (the details of the
cathode and cathode manipulator end effector are illustrated in Fig. 2).
Controlling the depth of the stalk insertion with respect to the cavity nose
allows for the control of the primary beam focusing near the cathode surface,
which significantly improves the beam quality at the exit of the gun.

The $\textrm{CsK}_{2}\textrm{Sb}$ photocathode material is deposited onto the
polished surface of Mo puck using a dedicated cathode deposition system. The
garage, equipped with mobile UHV ion and sublimation pumps, can be attached to
the system via a load-lock, and can hold up to three photocathodes at a time.
When it is detached, the garage is transported to RHIC tunnel, where it is
connected to the gun’s load-lock. After a brief bake-out, the cathodes can be
transferred into the gun. Overall, the cathode exchange between the gun and
the garage takes about 30 minutes.

The cavity was manufactured by Niowave, but has never achieved its design
voltage of 2 MV due to the limitations during the production process. The
routine CeC PoP operation requires the gun to deliver 1.25 MeV electron beams
with a headroom of 200 kV, which was easily achievable. Overall, the cavity
has shown an impeccable performance by sustaining CW operation during the CeC
experimental runs (continuous use of the gun for 5-6 months each year). The
early stages of operation were challenged by the presence of persistent low-
level MP barriers, but the issue was methodically evaluated and successfully
resolved resulting in complete elimination of MP as an obstacle for the cavity
operation. The detailed description of our MP studies, and the dedicated
start-up procedure developed to avoid the MP issue can be found in [12].

The cavity performance as a photoinjector has been rather impressive: the gun
demonstrated the ability to deliver high-brightness electron bunches with low
transverse emittances (e.g. 0.15 mm-mrad slice emittance for 100 pC, 400 ps
bunches), and has also generated record high charges per bunch (up to 20 nC)
[2]. The cavity employs high quantum efficiency (QE) room temperature CsK2Sb
photocathodes, and has demonstrated the ability to maintain the cathode’s QE
of 3-4% for several months of continuous operation [13].

Figure 2: Detailed schematic of the cathode puck and cathode manipulator end
effector assembly. Figure 3: Detailed view of the SRF gun cryostat with the
critical vacuum components. The cathode insertion is on the right side of the
schematic, and the FPC is on the left side. Here ip = ion pump; tc = thermal
conductivity gauge, reads pressure from atmosphere to 1e-3 Torr; cc = cold
cathode gauge, reads 1e-3 to 1e-11 Torr; sv = vacuum valve.  Figure 4:
Schematic of the clean He gas delivery system for the conditioning of the CeC
SRF cavities. Here ip = ion pump; tc = thermal conductivity gauge; cc = cold
cathode gauge; sv = vacuum valve; NEG = non-evaporable getter pump; UHP =
ultra high purity grade.

##  IV He Conditioning in application to the BNL SRF gun

During the multiple years of the gun operation, we have learned that
development of _in-situ_ methods of cavity performance restoration is crucial
for the routine photoinjector maintenance. There are always multiple sources
of possible SRF gun cavity contamination, including particulates generated by
inserting and removing cathodes or those migrating from other parts of the
vacuum system, which can result either in reduced performance of the SRF gun
or even in loss of valuable piece of equipment. One of the successful methods
that we have implemented for improving performance of our SRF gun is He
conditioning (HeC). The system for HeC of the CeC SRF accelerator, designed
and built by the BNL vacuum group, is shown in Figs. 3 and 4. Below we will
discuss two cases when the HeC procedure was used in application to the BNL
SRF gun and demonstrated a dramatic improvement in the cavity performance.

###  IV.1 He Conditioning: Case 1

At the beginning of the CeC experimental run 2021, the gun showed signs of
significant performance degradation: rapid decay of the cathode quantum
efficiency, appearance of new low-level multipacting barriers, increased
radiation and LiHe consumption compared to the routine operation levels. These
observations were then followed by the series of dramatic changes in the beam
dynamics within the cavity. The measured axis of the beam propagation has
shown a significant deviation from the straight path: the beam axis was tilted
by 22 mrad horizontally indicating the corresponding tilt of the cathode
surface (see [14] for the detailed description of the gun axis measurement
procedure). This fact was then confirmed by the direct observation of the
asymmetry in the cathode placement within the cavity nose (see Fig. 5). The
reported behavior indicated a potential damage to the cathode delivery system
and inevitable presence of the cavity surface contamination. Upon a detailed
inspection, the damage to the cathode end effector was found to be the issue.
When the puck was removed a close-up inspection found evidence that the
manipulator had been damaged due to either poor or inattentive handling of the
end effector or horizontal manipulator. As can be seen from Fig. 6, at least
one of the RF spring fingers clearly showed a burn mark, which is the evidence
that particulates were generated that may have escaped through the gap between
the puck and the end effector registration surface.

(a)

(b)

Figure 5: (a) Cathode installed at the end of the cathode stalk. An asymmetric
circular pattern can be seen here. (b) After the cathode was extracted from
the gun, this image shows a gap that had opened up between the puck and the
end effector.

(a)

(b)

Figure 6: (a) Spring finger element. One of the fingers clearly shows a burn
mark. (b) Shows the plastic deformation of the spring finger being “pulled
forward”.

After the unsuccessful attempts to bring the cavity to the regular operational
state by using CC and HPP, it was decided to proceed with He conditioning. The
He conditioning of the CeC gun cavity typically adheres to the following
procedure. First, before the He is introduced into the cavity, with the He
leak valve and all the surrounding valves of the CeC system closed and the
cathode puck retracted, the gun voltage must be brought up well above
multipacting levels, usually at a baseline of 700-750 kV. The protection
mechanism for monitoring the cavity voltage is then enabled to prevent the
cavity from falling into multipacting for a prolonged period of time. The
implemented protocol is registering the cavity voltage, and once the level is
detected to be below the multipacting threshold (usually below 400 kV), the
cavity is immediately turned off. The pulsing with 10 Hz repetition and pulse
width of 1-2 ms is then introduced on top of the baseline similar to one shown
in Fig. 7b. Once the pulsing event is set, the He leak valve is opened to
introduce He into the cavity at a desired pressure level, typically in the
range of 2-5$\times 10^{-6}$ Torr. The conditioning procedure consists of a
gradual increase of the peak cavity voltage that can be achieved through
either increase of the pulse length of the pulsing event or increase of the
baseline level. Since the increase of the baseline level requires higher LiHe
consumption, the conditioning is usually performed by the adjustment of the
pulse length of the pulsing event. One needs to gradually increase the pulse
width in the steps of $\sim$0.5 ms, stay at each level for an extended period
of time (10-15 mins) and simultaneously monitor the gun voltage behavior,
radiation levels, mass flow of liquid He, pressure in the gun cavity, and
gauges of the ion pumps providing the pressure levels of the He used for
conditioning. During the HeC, He and the loose debris are being constantly
pumped out of the cavity by a turbo pump. The overall process of HeC is rather
delicate and sudden increases of the applied power could result in
interruption by the cavity machine protection system (MPS). If the gun trips,
the protection mechanism turns the cavity off, one must close the He leak
valve and wait for the vacuum to recover to $\sim 1\times 10^{-7}$ Torr before
turning the system back on.

Figure 7: The highest achieved CW (a) and pulsed (b) cavity voltage after the
first case of He conditioning. Figure 8: Gun voltage, pressure in the cavity
and radiation levels during HeC.

After a couple of days of He conditioning, the cavity showed significant
performance improvement: the gun reached the all-time highest voltage
demonstrating 1.5 MV CW and 1.7 MV pulsed. The radiation level at the CeC
operational level of 1.25 MV CW has also improved dramatically and reduced to
$\sim$380 mR/hr with $\sim$2 g/sec LiHe consumption. Figure 7 shows the final
test of the cavity operation in CW and pulsed modes after the completion of
the He conditioning. The CW voltage limit is defined by the maximum available
LiHe consumption of 8 g/sec. A quench-like limit occurs at 1.7 MV in pulsed
mode as indicated by the steep drop of the cavity voltage in Fig. 7b. The
mechanism of capturing the cavity voltage when it falls below the MP level
during the HeC pulsing procedure was implemented after this event, hence the
appearance of the small spike in Fig. 7b is an indication that the cavity
wasn’t turned off soon enough after the cavity tripped.

###  IV.2 He Conditioning: Case 2

In February 2022, during a mundane cathode exchange, it was found that the
cathode puck wasn’t fully engaged with the end effector of the cathode
manipulator once the old cathode was extracted. While attempting to insert a
replacement cathode, an obstacle was encountered inside the cathode stalk that
prevented the team from inserting the cathode, unless a significant adjustment
of the horizontal angle of the insertion arm was applied. The cathode was
installed, but the cavity demonstrated an immediate performance degradation:
although the MP barriers were surpassed without an issue, the cavity would not
be able to exceed 600 kV, showing significant amount of dark current in the
cathode area and several spots of visibly glowing emitters. An attempt to
clean the cavity by using traditional CW or high-power processing (HPP) was
successful at first, but eventually resulted in a cavity trip followed by
significantly aggravated MP levels, that were impossible to overcome. The
cathode was extracted, and after several attempts of HPP, it was decided to
switch to HeC. Within the course of a few days, we performed HeC, and studied
in detail how the amount of the delivered He affects the cleaning process. The
evolution of the cavity voltage and respective cavity pressure and radiation
levels are shown in Fig. 8. The same procedure as in the previous case was
applied, but the He pressure levels were adjusted in the range between
2$\times 10^{-6}$ to 6$\times 10^{-6}$ Torr based on the cavity behavior.
Overall, HeC procedure was highly successful and resulted not only in the
complete cavity recovery, but also in an all-time low X-ray radiation from the
cavity ($\sim$250 mR/hr, compared to $\sim$380 mR/hr before the incident and
$\sim$750 mR/hr right after the incident) at the operational voltage of 1.25
MV.

##  V Conclusions

The development of cavity restoration techniques in application to the SRF
photoinjectors is highly valuable for the future development and maintenance
of the SRF electron sources. On the example of the BNL SRF photogun we have
proven that He conditioning technique is highly successful in restoration of
the cavity performance and allows to effectively remove contamination, reduce
LiHe consumption and radiation levels, resulting in the improved perfromance
of the gun cavity.

###### Acknowledgements.

Work is supported by Brookhaven Science Associates, LLC under Contract No.
DEAC0298CH10886 with the U.S. Department of Energy, and the Department of
Energy office of Nuclear Physics grant DE-FOA-0002670.

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