# Powering quantum Otto engines only with q-deformation of the working
substance

Fatih Ozaydin<EMAIL_ADDRESS>Institute for International Strategy, Tokyo
International University, 1-13-1 Matoba-kita, Kawagoe, Saitama 350-1197, Japan
CERN, CH-1211 Geneva 23, Switzerland  Özgür E. Müstecaplıoğlu  Department of
Physics, Koç University, Sarıyer, İstanbul, 34450, Türkiye  TÜBİTAK Research
Institute for Fundamental Sciences, 41470 Gebze, Türkiye  Tuğrul Hakioğlu
Energy Institute, İstanbul Technical University, Sarıyer, İstanbul, 34467,
Türkiye  Department of Physics Engineering, İstanbul Technical University,
Sarıyer, İstanbul, 34467, Türkiye  Department of Physics, Northeastern
University, Boston, MA 02115, USA

(August 28, 2024)

###### Abstract

We consider a quantum Otto cycle with a $q$-deformed quantum oscillator
working substance and classical thermal baths. We investigate the influence of
the quantum statistical deformation parameter $q$ on the work and efficiency
of the cycle. In usual quantum Otto cycle, a Hamiltonian parameter is varied
during the quantum adiabatic stages while the quantum statistical character of
the working substance remains fixed. We point out that even if the Hamiltonian
parameters are not changing, work can be harvested by quantum statistical
changes of the working substance. Work extraction from thermal resources using
quantum statistical mutations of the working substance makes a quantum Otto
cycle without any classical analog.

††preprint: APS/123-QED

##  I Introduction

Quantum Heat Engines (QHEs) are devices that can harvest work using a quantum
working substance between hot and cold reservoirs [1, 2]. After the
foundations of the quantum engines have been established, many researchers
have devoted intense theoretical and experimental efforts to find new
breakthroughs [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
Enhancement of work and efficiency of such quantum machines, together with
exploring their fundamental bounds, are among the significant goals of the
emerging field of quantum thermodynamics. For that aim, non-linear, many-body,
fermionic or bosonic working systems have been studied to reveal their
differences and relative advantages [3]. Here, we contribute to these research
endeavors by addressing two questions. First, how the engine performance
depends on quantum statistics in general if we mutate the particle statistics?
Second, can we consider quantum statistics as another control parameter such
that if all the system parameters remain the same, we can harvest work from a
heat bath by only changing the quantum statistics of the working substance?

In the 1970s, the concept of deformed algebras was first initiated as a
generalization of Weyl-Heisenberg algebra [18, 19]. The theory of the
$q$-oscillators was stated previously by Mcfarlane, and Biedenharn [20, 21].
Since then, $q$-deformation has been considered in various research areas
including statistical physics and quantum information [22], nuclear and atomic
physics [23], thermodynamics [24], open quantum systems and optomechanical
systems  [25].

It is pointed out that there is a correspondence between $q$-deformed
Heisenberg algebra and effective non-linear interaction of the cavity mode
[26], and an isomorphism between the $q$-deformed harmonic oscillator and an
anharmonic oscillator model was discovered [27]. Physical realization of the
deformation parameter $q$ has been searched for heavily, among which are the
quantum Yang-Baxter equation [28], deformed Jaynes-Cummings model [29],
quantum phase problem [30, 31, 32], relativistic $q$-oscillator [33, 34],
Morse oscillator [35], and Kepler problem [36]. Deformed algebras have been
explored by subjecting the non-deformed ones to non-linear invertible
transformations [37, 38, 39]. The $q$-deformation parameter was considered for
deriving generalized uncertainty and information relations [40], Tsallis
entropy, and other relative entropy measures [41, 42, 43].

In atomic and nuclear physics, $q$-deformation was considered from theoretical
and experimental perspectives [23, 44, 45, 46, 47, 48, 49]. It was considered
for obtaining generalizations of quantum spin chains with exact valence-bond
ground states [50], and self-localized solitons of $q$-deformed quantum
systems have been recently explored [51]. Deformed algebra is also used in
open quantum systems to show the relationship between the efficiency of QHEs
and the degree of the non-Markovianity cycle processes [24]. One of the recent
works showed that two linearly coupled $q$-deformed cavities could be tuned to
provide enhancement of non-classical phenomena [25].

In one of the earliest works on $q$-deformed quantum information, entanglement
and noise reduction techniques were studied between $q$-deformed harmonic
oscillators [52]. Nonclassical properties of noncommutative states [53, 54],
coherent and cat states in Fock representation [55, 56] and entanglement in
nonlinear quantum systems were analyzed in q-deformed settings [57]. Quantum
states and logic gates were defined for two- and three-level $q$-deformed
systems [58, 22, 59], and $q$-deformed relative entropies were studied in
quantum metrology [60].

Our work presents a unique perspective on the quantumness of heat engines,
which reflects the genuine quantum statistical character of the working system
in harvesting work from classical thermal resources. The usual method to
characterize the quantum nature of a heat engine is to look for quantum-
enhanced performance over its classical analog. Our case is another yet more
direct reflection of the quantumness of a heat engine as the cycle mechanism,
which is based upon harvesting work by changing the quantum statistical
character of its working substance, has no classical analog.

This paper is organized as follows: Sec. II introduces the $q$-deformed
quantum oscillator as the working system of the quantum Otto cycle. Sec. III
presents the necessary tools to construct a $q$-deformed heat engine by
discussing the fundamental thermodynamic quantities such as entropy and
internal energy. Then, equipped with the theoretical tools presented in
previous sections, we present our results in Sec. IV, where we explicitly show
how work harvesting from thermal resources can be achieved by varying the
particle statistics. We compare our results with previous investigations and
discuss the effectiveness of our approach in Sec. V. We conclude in Sec. VI.

##  II Working system: $q$-deformed oscillator

We consider a q-deformed harmonic oscillator as the working substance of the
Otto cycle as illustrated in Figure 1. The usual commutation relation of the
Weyl-Heisenberg algebra of the quantum harmonic oscillator is deformed in the
case a $q$-deformed quantum oscillator according to [24, 61]

$\left[\hat{a},\hat{a}^{\dagger}\right]_{q}=\hat{a}\hat{a}^{\dagger}-q^{-1}\hat{a}^{\dagger}\hat{a}=q^{\hat{N}}$
(1)

where ${\hat{N}}$ is a number operator with eigenstates $|n\rangle$ such that
${\hat{N}}|n\rangle=n|n\rangle$, $a$ and $a^{\dagger}$ are lowering and
raising operators in the spectrum of ${\hat{N}}$ such that
$a^{\dagger}a|n\rangle=[n]|n\rangle$, and $q$ is the deformation parameter.

Figure 1: Entropy-temperature (S-T) diagram of the quantum Otto cycle. In the
isochoric processes, the system is in contact with hot and cold baths, while
during the quantum adiabatic stages, quantum statistical character of the
oscillator changes. Populations, therefore the entropies remain the same
during the quantum adiabatic stages, differentiating the quantum Otto cycle
from its classical counterpart.

In terms of $\hat{a}$ and $\hat{a}^{\dagger}$, the Hamiltonian of the
$q$-deformed quantum oscillator with natural frequency $\omega$ is written as

$\hat{H}=\frac{\hbar\omega}{2}\left(\hat{a}a^{\dagger}+\hat{a}^{\dagger}\hat{a}\right).$
(2)

Using the $q$-number notation

$[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}\,,$ (3)

the eigenenergies of the Hamiltonian (2) can be expressed in the form

$E_{n}=\frac{\hbar\omega}{2}\left(\left[n\right]+\left[n+1\right]\right).$ (4)

These eigenenergies are associated with the deformed Fock number eigenstates

$\left|n\right\rangle=\frac{\left(a^{+}\right)^{n}}{\sqrt{\left[n\right]!}}\left|0\right\rangle,$
(5)

where the $q$-factorial is defined to be $[n]!=[n][n-1]\cdot\cdot\cdot[1]$.

With a real valued $r$, the deformation parameter can be considered to be a
pure phase factor as $q=\text{exp}(ir)$, or to be a real number as
$q=\text{exp}(r)$ [62].

Here we focus on the nonlinear characteristics of the quantum oscillator
associated with $q$-deformation, and the deformation of bosonics system, hence
we consider only real values with $0<q<1$ [62, 61]. The usual quantum harmonic
oscillator relations can be recovered by substituting $q=1$ corresponding to
the non-deformed case.

$E_{n}=\frac{\hbar\omega}{2}\frac{\sinh\left[r\left(n+\frac{1}{2}\right)\right]}{\sinh\left(\frac{r}{2}\right)},$
(6)

where $r=\ln{(q)}$ and $n=0,1,…<\infty$ is the energy eigen index. $E_{n}$ is
plotted in Figure 2 as a function of $q$ for $n=1,2,3,4$.

Figure 2: Energy eigenvalues as a function of the deformation parameter $q$
for energy eigen index $n=1,2,3,4$.

Deformation parameter $q$ gives strong nonlinear character to the quantum
oscillator, reflected in the anharmonicity of the energy spectrum. Without the
deformation ($q=1$), quantum Otto cycle can only be implemented by changing
$\omega$ in the quantum adiabatic stages. When $\omega$ is increased, the
uniform energy gaps ($\hbar\omega$) between the eigenenergies increase, which
can be compared to the case of compressing the volume of the oscillator. In
contrast, if $\omega$ is decreased, the energy gaps shrink as if the volume of
the oscillator is increasing. Accordingly, an effective piston like behavior
can be translated to the quantum oscillator by $\omega$ variation. For the
case of a deformed quantum oscillator ($q\neq 1$), oscillator has non-uniform
energy gaps and their splitting can be further controlled by $q$. By keeping
$\omega$ constant, we can induce a piston-like behavior to a deformed
oscillator by changing its quantum statistics per se. Such a $q$-deformed
oscillator uses $q$ as the quantum adiabatic control (expansion and
compression) parameter and it can exploit the anharmonic energy spectrum (for
more expansion and compression effects) to enhance the work harvesting and
efficiency. Furthermore, alternatively we can still keep $q$ parameter
constant and vary $\omega$ as usual, but determine a critical quantum
statistics (critical $q$) for which the work harvesting and (or) efficiency
would be maximum.

Figure 3:  Population differences $P_{n}(B)-P_{n}(A)$ for $n$ and $q_{A}$ at
$T_{h}=0.5$ with $q_{C}=1$ at $T_{c}=0.1$.

##  III Quantum Otto Cycle

The canonical partition function of the working system with Hamiltonian
$\hat{H}$ in Eq. (2) at temperature $T$ is given by

$\displaystyle
Z=\text{Tr}\left[\text{exp}(-\hat{H}/T)\right]=\sum_{n}\mathrm{e}^{-E_{n}/T},$
(7)

where we take the Boltzmann constant $k_{B}$ as unity. Such a non-deformed
structure of the partition function relies on the assumption of the non-
deformed Gibbsian form of the thermal equilibrium state
$\rho\sim\text{exp}(-\beta\hat{H})$ with $\beta$ = 1/T and the expectation
value $\langle\hat{A}\rangle=\text{Tr}(\rho\hat{A})$; and it is associated
with the assumption that Boltzmann-Gibbs form of the entropy function $S=\log
W$ is preserved [63] where $W$ is the work output.

The quantum version of the classical Otto cycle [64] has been experimentally
realized with quantum working substances [65, 66]. As illustrated in Figure 1,
it consists of two quantum adiabatic and two isochoric heating/cooling stages.
We can describe the cycle in energy-population space for a quantum number $n$.
At point $A$, the system is given with energy levels $E_{n}(A)$ and their
populations $P_{n}(A)$. Under isochoric heating, the system is transformed
($A\rightarrow B$) to a thermal state at point $B$ such that
$E_{n}(B)=E_{n}(A)$ and population is changed to $P_{n}(B)$. Subsequently, the
bath is removed and system is quantum adiabatically transformed ($B\rightarrow
C$) obeying the conditions $P_{n}(C)=P_{n}(B)$ while the energy eigenvalues
are adiabatically changed to $E_{n}(C)$. In the third stage, the system is
isochorically cooled ($C\rightarrow D$) to point $D$ such that
$E_{n}(D)=E_{n}(C)$ while the population is modified to $P_{n}(D)$. Finally,
the system is reset to its starting point $A$, ($D\rightarrow A$), by closing
the cycle with another quantum adiabatic transformation under the condition
$P_{n}(A)=P_{n}(D)$.

Figure 4:  Net variation of energy eigenvalue $E_{n}(A)-E_{n}(C)$ during the
adiabatic transformations for $n$ and $q_{A}$ at $T_{h}=0.5$ with $q_{C}=1$ at
$T_{c}=0.1$.

In classical Otto cycles, the control parameter in the isentropic steps is
typically the physical volume of the system. Here, we consider two types of
quantum Otto cycles: (i) We keep the frequency of the quantum oscillator
$\omega$ constant and vary the deformation parameter $q$ during the quantum
adiabatic steps. (ii) We keep $q$ constant and change $\omega$ in the quantum
adiabatic stages. During these steps, the system is uncoupled from the thermal
baths and the system is quantum adiabatically transformed such that occupation
probabilities $P_{n}$ of the eigenenergies $E_{n}$ do not change. During these
steps, the system is uncoupled from the thermal baths and the system is
quantum adiabatically transformed such that occupation probabilities $P_{n}$
of the eigenenergies $E_{n}$ do not change.

For a classical heat engine, it is sufficient to make the transformation
faster than the rate of heat exchange, instead of physically uncoupling the
system from the environment, to ensure adiabatic condition. In the quantum
case, transformation needs to be slower than characteristic time scale for the
transitions between the energy levels to ensure $P_{n}$ remains the same.
Quantum case is also an isentropic process, while the classical case does not
necessarily satisfy the constant population condition. During the other two
steps, isochoric heating and cooling, thermal baths are coupled to the
oscillator system while the parameters of the Hamiltonian and the quantum
statistical parameter $q$ are kept constant.

Figure 5:  Extractable work for $n$ and $q_{A}$ at $T_{h}=0.5$ with $q_{C}=1$
at $T_{c}=0.1$.

Using the energy eigenvalues $E_{n}$ in Eq. (4) and associated populations

$P_{n}=\exp(-E_{n}/T)/Z,$ (8)

we evaluate the work output ($W$) of the quantum Otto cycle according to the
formula [3]

$W=\sum_{n}\left[E_{n}(A)-E_{n}(C)\right]\left[(P_{n}(B)-P_{n}(A)\right].$ (9)

$P_{n}(A)$ is the population of $n$-th energy level at the beginning of the
isochoric heating. We can use the quantum adiabatic condition
$P_{n}(A)=P_{n}(D)$ to evaluate it using by the thermal distribution as the
system is in thermal equilibrium at point $D$. $P_{n}(B)$ is calculated from a
thermal distribution as well, as it is the level population at the end of
isochoric heating. The factor $E_{n}(A)-{\color[rgb]{0,0,0}E_{n}(C)}$ is the
net variation of the energy eigenvalue during the quantum adiabatic
transformations.

Thermal efficiency of the engine is defined by

$\eta=\frac{W}{Q_{\text{in}}}.$ (10)

Here, the injected heat into the system is given by

$Q_{\text{in}}=\sum_{n}E_{n}(A)\left[P_{n}(B)-P_{n}(A)\right],$ (11)

where again we can use $P_{n}(A)=P_{n}(D)$ to evaluate it using thermal
population distribution at point $D$.

##  IV RESULTS

###  IV.1 Work harvesting by statistical mutation

Figure 6: Work ($W$) by only changing the quantum statistics of the working
substance ($q_{A}$) for the hot isochore ($T_{h}=0.5$) for three choices of
$q_{C}$ for the cold isochore ($T_{c}=0.1$) with fix oscillator frequency
$\omega=1$. Global maxima for $W$ are at $q_{A}=0.379$, $q_{A}=0.369$ and
$q_{A}=0.338$, respectively for the first three choices in descending order of
$q_{C}$.  The inset shows the domain that positive work condition is not
satisfied, i.e., $q_{A}<0.101$, $q_{A}<0.984$, and $q_{A}<0.887$,
respectively, for $q_{C}=1$, $q_{C}=0.8$, and $q_{C}=0.6$.

Let us start changing the deformation parameter in the quantum adiabatic
stages of the cycle. For a q-deformed quantum oscillator, a decrease in $q$
yields an exponential increase in the energy gaps at higher energy levels.
Thus, we take a smaller $q$ value ($q_{A}$) for hot isochore ($T_{h}$) and a
higher $q$ value ($q_{C}$) for cold isochore ($T_{c}$). During the isentropic
stages, $q$ varies between the values used in hot and cold isochores.
Hamiltonian parameter remain fixed at $\omega=1$, and only the $q$-deformation
parameter of the working substance is varied in the cycle.

Population differences $P_{n}(B)-P_{n}(A)$, the factor $E_{n}(A)-E_{n}(C)$,
and their multiplication in Eq. (9) for $n$-th energy level and $q_{A}$ at
$T_{h}=0.5$ are presented in Figures 3, 4 and 5, respectively, for $q_{C}=1$
at $T_{c}=0.1$. Here, $q_{C}=1$ corresponds to non-deformed case showing that
positive work can also be extracted by changing the statistics of the
substance only for the hot isochore. Efficiency is meaningful and well-defined
only when this positive work condition is satisfied. Analytical expression for
the positive work condition could be possible if the energy gaps would change
uniformly with $q$ [3], which is not the case for our deformed oscillator.
Accordingly, we resort the numerical calculations of work for a range of
temperature and $q$ to determine the positive work domains.

Figure 7:  Efficiency of quantum Otto cycle as a function of $q_{A}$ (at
$T_{c}=0.1$) with three choices of $q_{C}$ (at $T_{h}=0.5$). The functions are
plotted not for the entire range of $q_{A}$ but only in the positive work
domain as presented in the inset of Figure 6. As $q_{A}$ approaches its
minimum within the positive work domain, the quantum Otto engine’s efficiency
$\eta$ approaches the Carnot limit $\eta_{\text{Carnot}}=1-T_{c}/T_{h}=0.8$.

Summing the work values over $n$ in Figure 5, the extractable work (W) with
respect to $q_{A}$ is presented in Figure 6 at $T_{h}=0.5$ for three choices
of $q_{C}$ at $T_{c}=0.1$.  Figure 7 presents the efficiency as a function of
$q_{A}$ with the same choices of $q_{C}$ and bath temperatures. Quantum Otto
engine’s efficiency approaches the Carnot limit only by $q$-deformation as
$q_{A}$ gets smaller within the positive work domain which is numerically
found and shown in the inset of Figure 6. Note that it is straightforward to
check numerically for any choice of $q$ and bath temperatures that in the
positive work condition, the Carnot limit is satisfied. For example, setting
the hot isochore $T_{h}=1$ with the same choices as in Figure 6, the positive
work condition is satisfied until $q_{A}\approx 0.05$ at which the quantum
Otto efficiency approaches the Carnot limit $\eta_{\text{Carnot}}=0.9$.

Figure 8: Extractable work as a function of $\omega$ with $q_{A}=0.4$ at
$T_{h}=1$ and $q_{C}$ at $T_{c}=0.1$. For small $\omega$, an optimal choice of
$q_{C}$ yields maximum extractable work.

###  IV.2 Critical quantum statistics for optimal work harvesting

For a given temperature difference between hot and cold reservoirs as a
classical resource, we can point out a critical quantum statistics of a
deformed quantum oscillator to maximize the harvested work or efficiency. Ref.
[67] has shown that a bosonic system has higher engine performance than a
fermionic system and stated that the performance difference occurs due to the
difference in internal energies arising from the Pauli exclusion principle. In
the same spirit, we find out that one can optimize the engine performance for
a given thermal resource in terms of the quantum statistical character of the
working substance. As shown in Figure 6, $q_{A}\approx 0.4$ is the optimal
value for the working substance for the hot isochore.

We now keep $q_{A}=0.4$ for various $q_{C}$ and induce the cycle for different
$\omega$. As shown in Figure 8, for $\omega<0.2$, there is an optimal $q_{C}$
that achieves maximum extractable work.

Figure 9: Extractable work as a function of $\omega_{A}$ with $\omega_{C}=0.5$
with $T_{c}=0.1$ and $T_{h}=0.5$ with the substance deformed at same level in
adiabatic steps ($q_{A}=q_{C}$).

Lastly, for the second type of quantum Otto engine, we keep $q$ constant
(i.e., $q_{A}=q_{C}$) and change $\omega$ in the quantum adiabatic steps,
i.e., we choose different $\omega_{A}$ and $\omega_{C}$. As can be seen in
Figure 9, greater $q$ implies greater extractable work and maximum in the
nondeformed case ($q_{A}=q_{C}=1$). This result shows that the advantage of
$q$-deformation with no classical analogue is due not to a simply deformed
working substance, but rather to utilizing the working substance with
different deformation levels at the adiabatic stages even with the same
$\omega$.

##  V Discussion

Let us consider an anharmonic Hamiltonian

$\hat{H}_{a}=\mu_{1}a^{{\dagger}}a+\mu_{2}a^{{\dagger}}aa^{{\dagger}}a$ (12)

where nonlinearity can be represented by deformation as

$a_{q}=a+{\epsilon\over 4}aa^{{\dagger}}a$ (13)

with $\epsilon=1-q\ll 1$. Then, as shown in [68], Kerr medium or s-wave
scattering in atomic BEC can be simulated through nonlinear interactions of
the deformed oscillator.

It is found in accordance with previous works [69, 70, 71, 72, 73] that the
extractable work and efficiency are not monotonic with respect to the
interaction parameter, but rather exhibit optimum values as presented in
Figure 6.

We remark that we do not include explicitly the work reservoir into the engine
model as it is a common treatment in quantum heat engines. It was shown in
[74] that in a thermodynamic cycle using particles with nonlinear interactions
as the working substance, adjusting the Feshbach resonances [75] can tune the
nonlinear interaction strength to produce work by modifying the volume of the
gas. Similarly in our model, it should be understood that the work of the
engine will be done against the magnetic field used to modify Feshbach
resonances to change the $q$-parameter associated with the nonlinearity of the
working substance.

The effect of non-linearity due to $q$-deformation can be observed more
clearly with greater temperature difference between the cold and the hot bath
as presented in Figures 10 and 11 for work and efficiency, respectively. Note
that due to weak nonlinearity achievable in experiments, we considered a small
$\epsilon$ in Eq. (13). However, considering that strong nonlinearities and
other physical phenomena could also be associated with $q$-deformation, we
present our results for the entire range of $q$-deformation parameter in the
positive work domain satisfying the Carnot limit.

Algebras corresponding to both $q$-deformed fermions and $q$-deformed bosons,
as well as their transformations and unification are widely studied [23, 62,
76, 77, 78]. Considering potential realizations in atomic BEC systems, also in
accordance with the findings of Ref. [67] that bosonic working system achieves
a higher performance than fermionic systems in the non-deformed setting
($q=1$), in this work we focused on $q$-deformed bosonic system as the working
substance. Nevertheless, it would be interesting as future research to study
different performance characteristics of complex deformations, and
$q$-deformed fermionic systems as well, especially following the unified
representation by Lavagno and Swamy [78].

In another work, the effect of $q$-deformation was studied to show a
relationship between the efficiency of QHE and the non-Markovianity in the
engine cycle [24]. It states that the $q$-parameter, which causes non-
equilibrium dynamics, helps to build a relationship between theoretical and
experimental results. Here, we did not examine finite time engine cycles; the
$q$ parameter, in our case, plays a more active and direct role as the engine
cycle’s control parameter.

Figure 10: Work with respect to $q_{A}$ with a greater temperature difference
between hot isochore ($T_{h}=5$) and cold isochore ($T_{c}=0.1$). Figure 11:
Efficiency of quantum Otto cycle as a function of $q_{A}$ at $T_{c}=0.1$ and
three choices of $q_{C}$ at $T_{h}=100$ with $\omega=1$.

##  VI CONCLUSION

Classical heat engines harvest work from a thermal resource by converting a
heat flow between a hot and cold bath to an ordered work. To realize this
task, parameters of their classical working system, for example, the volume of
working gas, are varied in an engine cycle. Quantum heat engines use a quantum
working material, and in addition to external degrees of freedom, Hamiltonian
parameters and internal degrees of freedom can also be utilized for work
extraction. Profound quantum effects, particularly improving engine
performance over its classical counterpart, such as via quantum correlations
[14], are possible with quantum heat engines.

Here, we show that the quantum statistical character of the working substance
can be used as another control parameter of a quantum heat engine.
Specifically, we consider a quantum oscillator with a fixed frequency $\omega$
but deformed quantum statistics characterized by the $q$-deformation parameter
and verify that such a $q$-deformed oscillator can harvest work from a thermal
resource by variation of the $q$ parameter. Alternatively, we can optimize the
harvested work or efficiency for a given thermal resource by choosing a
critical quantum statistics of the deformed oscillator.

While we consider the Otto cycle as a paradigmatic model, we expect that our
fundamental conclusion holds for other engine cycles as well. Variation of
particle statistics can be experimentally challenging relative to the
traditional way of variation of Hamiltonian parameters. However,
$q$-deformation can be envisioned and mapped to nonlinear terms in
Hamiltonians and effective engineered deformed oscillator models ranging from
semiconductor cavity QED [79] to atomic Bose-Einstein condensates [80] could
be explored for physical embodiment of the statistical mutation route of work
extraction.

###### Acknowledgements.

This study was funded by Istanbul Technical University BAP-41181. F.O.
acknowledges the Personal Research Fund of Tokyo International University.

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CMS experiment

Inclusive and differential cross-sections measurements in the single top tW
e-mu channel with CMS

Alejandro Soto Rodríguez for the CMS Collaboration

Instituto de Ciencias y Tecnologías Espaciales de Asturias (ICTEA)
Universidad de Oviedo

> Inclusive and normalised differential cross sections measurements are
> presented for the production of single top quarks in association with a W
> boson, in proton-proton collisions at a centre-of-mass energy of 13 TeV.
> Events containing one muon and one electron in the final state are analysed.
> For the inclusive measurement, a multivariate discriminant, exploiting the
> kinematic properties of the events, is used to separate the signal from the
> dominant t$\overline{\mathrm{t}}$ background. For the differential
> measurements, a fiducial region is defined according to the detector
> acceptance, and the requirement of exactly one b-tagged jet. The resulting
> distributions are unfolded to particle-level and compared with predictions
> at next-to-leading order in perturbative QCD. Within current uncertainties,
> all predictions agree with the data.

> PRESENTED AT
>
>
>
>
> $14^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (videoconference), 13–17 September, 2021

##  1 Introduction

Single top quarks were observed for the first time by the D0 [1] and CDF [2]
Collaborations at the Fermilab Tevatron collider. There are three main
production modes in proton-proton (pp) collisions: the production and decay of
a virtual W boson (s channel), the exchange of a virtual W boson (t channel),
and the associated production of a top quark and a W boson (tW channel).

The tW process interferes with top quark pair production
($\mathrm{t}\overline{\mathrm{t}}$) at NLO in QCD. In this analysis, two
schemes are defined to treat this interference: diagram removal (DR), in which
all doubly resonant diagrams are removed from the matrix element (ME)
calculation, and diagram subtraction (DS), in which a gauge invariant term is
introduced in the ME calculation that locally cancels the doubly resonant
diagrams. The tW production cross section is computed at approximate next-to-
next-to-leading order (NNLO). The corresponding theoretical prediction for the
tW cross section in pp collisions at $\sqrt{s}=13$ TeV, assuming a top quark
mass ($m_{\mathrm{t}}$) of 172.5 GeV [3], is
$\sigma^{\mathrm{SM}}_{\mathrm{tW}}=71.7\pm 1.8(\mathrm{scale})\pm
3.4(\mathrm{PDF})\ \mathrm{pb.}$

This document reports the first measurement from the CMS Collaboration of the
inclusive [4] and differential [5] cross sections of the tW process in pp
collisions at $\sqrt{s}=13$ TeV. The measurement is performed using final
states with one electron and one muon of opposite charge. The analysed data
was recorded by the CMS detector [6] during 2016 and corresponds to
$35.9\mathrm{fb}^{-1}$ of integrated luminosity.

##  2 Event selection

The analysis uses the $\mathrm{e}^{\pm}\mu^{\mp}$ channel of the tW process,
which corresponds to a final state with one electron and one muon of opposite
charge, the two corresponding neutrinos and a jet resulting from the
fragmentation of the bottom quark. The main background contribution arises
from the $\mathrm{t}\overline{\mathrm{t}}$ process, which can give very
similar final states with only one additional b quark. To select signal events
and reduce the background contributions an event selection is performed.

Events are required to pass either a dilepton or single-lepton trigger.
Leptons (electrons or muons) are required to be well isolated and to have
$p_{\mathrm{T}}$ $\ >20$ GeV and $|\eta|<2.4$. In events with more than two
leptons passing the selection, the two with the largest $p_{\mathrm{T}}$ are
selected for further study. Jets are reconstructed using the
anti-$\kappa_{\mathrm{T}}$ clustering algorithm with a distance parameter of
0.4. Jets are required to have $p_{\mathrm{T}}$ $\ >30$ GeV, and $|\eta|<2.4$.
To avoid double counting issues, jets within a cone of $\Delta R=0.4$ with
respect to the selected leptons are not considered. An additional category of
jets called “loose jets” is defined to be jets passing the above selection but
with $p_{\mathrm{T}}$ between 20 and 30 GeV.

Some sources of background, such as Drell-Yan (DY), do not contain jets
originated from a b quark. The identification of these jets results in a
significant reduction in background. Jets are identified as b jets using the
algorithm CSVv2 [7].

Events are classified as belonging to the $\mathrm{e}^{\pm}\mu^{\mp}$ final
state if the two leptons with larger $p_{\mathrm{T}}$ (leading leptons)
passing the above selection criteria are an electron and a muon of opposite
charge. We also require that the leading lepton has $p_{\mathrm{T}}$ $\ >25$
GeV. To reduce the contamination from low mass resonances, the invariant mass
of the dilepton pair must be greater than 20 GeV. Different regions for the
inclusive and differential measurements are defined based on the number of
jets and b-tagged jets. Figure 1 shows this distribution and the data/Monte
Carlo (MC) comparison (left). For the inclusive measurement the regions with
one b-tagged jet (1j1b), two jets and one of them b-tagged (2j1b) and two
b-tagged jets (2j2b) are used, while for the differential measurements, only
the 1j1b region is used with an additional selection in the number of loose
jets, Fig. 1 (right), rejecting events with one or more loose jets.

(a)  (b)

Figure 1: Yields observed in data, compared with those expected from
simulation, as a function of the number of jets and number of b-tagged jets
for events passing the baseline dilepton selection (left) and as a function of
the number of loose jets passing also the 1j1b selection (right). The error
band includes the statistical and all systematic uncertainties, except those
from background normalisation. The bottom of each panel shows the ratios of
data to the sum of the expected yields [4, 5].

##  3 Inclusive measurement

Following the event selection the data sample in the 1j1b region consists
primarily of $\mathrm{t}\overline{\mathrm{t}}$ events with a significant
number of tW events. Given that there is no single observable that
discriminates between $\mathrm{t}\overline{\mathrm{t}}$ and tW, two boosted
decision trees (BDT) are trained, one in the 1j1b region and the other in the
2j1b region, to discriminate between them. The BDT implementation is provided
by the TMVA [8] package. The 2j2b region is used as a
$\mathrm{t}\overline{\mathrm{t}}$ control region to constrain this main source
of background.

The signal is extracted by performing a simultaneous maximum likelihood (ML)
fit to the distributions of the BDT output in the 1j1b and 2j1b regions and
the subleading jet $p_{\mathrm{T}}$ in the 2j2b region. The binning of the BDT
outputs is chosen such that each bin contains approximately the same amount of
$\mathrm{t}\overline{\mathrm{t}}$ events. This ensures that enough background
events populate all the bins, helping to constrain the systematic
uncertainties. Figure 2 shows the postfit distributions of the three variables
that enter in the fit.

The measured inclusive cross section of the tW process that results in the
best fit to the data is $\sigma^{\mathrm{exp}}_{\mathrm{tW}}=63.1\pm
1.8(\mathrm{stat})\pm 6.4(\mathrm{syst})\pm 2.1(\mathrm{lumi})\ \mathrm{pb,}$
consistent with the SM expectations.

(a)  (b)

(c)

Figure 2: Comparison of the BDT output in the 1j1b (upper left) and 2j1b
(upper right) regions and the $p_{\mathrm{T}}$ of the subleading jet in the
2j2b region (lower) distributions after the fit is performed for the observed
data and simulated events [4].

##  4 Differential measurement

For the differential measurement, the signal is extracted by subtracting
background to data. Unfolding techniques are used to take into account
migration effects when extrapolating to the fiducial region at particle level.
The fiducial region is defined by the same selection requirements employed in
the event selection but applied on particle-level objects. The differential
cross sections are measured as function of the leading lepton $p_{\mathrm{T}}$
, jet $p_{\mathrm{T}}$ , $\Delta\phi(\mathrm{e}^{\pm},\mu^{\mp})$,
$p_{\mathrm{z}}(\mathrm{e}^{\pm},\mu^{\mp},j)$,
m$(\mathrm{e}^{\pm},\mu^{\mp},j)$ and
$\mathrm{\textit{m}}_{\mathrm{T}}(\mathrm{e}^{\pm},\mu^{\mp},j,p_{\mathrm{T}}^{\mathrm{miss}})$
(the definition of these variables can be found in [5]) and normalised to the
fiducial cross section.

Figure 3 shows the normalised differential cross sections as a function of the
leading lepton $p_{\mathrm{T}}$ and $\Delta\phi(\mathrm{e}^{\pm},\mu^{\mp})$.
Good agreement, within the uncertainties, with the predictions from POWHEG DR,
POWHEG DS and MADGRAPH5_aMC@NLO DR is observed.

(a)  (b)

Figure 3: Normalised differential tW production cross section as a function of
the $p_{\mathrm{T}}$ of the leading lepton and
$\Delta\phi(\mathrm{e}^{\pm},\mu^{\mp})$ (right). The solid band represents
the total uncertainty. Predictions from POWHEG and MADGRAPH5_aMC@NLO are also
shown. In the bottom panel, the ratio between predictions and data is shown
[5].

##  5 Summary

A measurement of the inclusive and differential cross sections of the tW
process at 13 TeV with the CMS detector is presented. The analysis employs the
$\mathrm{e}^{\pm}\mu^{\mp}$ channel and classifies the events in terms of the
number of jets and b-tagged jets to perform the measurement. The inclusive
cross section is obtained using a maximum likelihood fit to the distribution
of the boosted decision tree discriminants in two categories, and the
subleading jet $p_{\mathrm{T}}$ in a third category. The measured cross
section is found to be $63.1\pm 1.8(\mathrm{stat})\pm 6.4(\mathrm{syst})\pm
2.1(\mathrm{lumi})\ \mathrm{pb,}$ achieving a relative uncertainty of 11% in
agreement with the SM. Finally, the measured differential cross sections are,
in general, consistent with the expectations from the models used to simulate
the tW signal.

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