A Hybrid Anyon-Otto thermal machine

TL;DR

This paper introduces a quantum thermal machine based on the 1D anyon Hubbard model, showing how interactions and anyonic statistics can enhance work extraction at low temperatures, with potential experimental realization using ultracold atoms.

Contribution

It proposes a hybrid anyon-Otto cycle leveraging anyonic statistics and interactions, demonstrating enhanced thermodynamic performance over traditional particles.

Findings

Maximum work at pseudo-fermionic limit without interactions.

Work peaks at intermediate angles with weak interactions.

Experimental protocol outlined for ultracold atom implementation.

Abstract

We propose a four-stroke quantum thermal machine based on the 1D anyon Hubbard model, which is capable of extracting the excess energy arising from anyon exclusion statistics at low temperature into finite work. Defining a hybrid anyon-Otto (HAO) cycle, we find that the low-temperature work, in the absence of any interactions, is maximized in the pseudo-fermionic limit, where the anyons most closely resemble free fermions. However, when weak interactions are introduced, the work output is no longer maximized at the bosonic or pseudo-fermionic extremes but instead peaks at intermediate statistical angles. This clearly demonstrates that interactions and anyonic statistics conspire non-trivially to enhance performance, with interacting anyons offering greater quantum thermodynamic advantage than either bosons or pseudo-fermions, in this regime. Furthermore, we also outline an experimental protocol to realize the HAO cycle using ultracold atoms in an optical lattice.

Equations

$$H=-J\sum _{j=1}^L\left(b_j^{†}{e}^{-i{n}_j\theta }{b}_{j+1}+h.c.\right)+\frac{U}{2}\sum _{j=1}^L{n}_j\left(n_j-1\right),$$

$$H=-J\sum _{j=1}^L\left(a_j^{†}{a}_{j+1}+h.c.\right)+\frac{U}{2}\sum _{j=1}^L{n}_j\left(n_j-1\right),$$

$$W_{otto}=\int \mathrm{Tr}\left[\rho \frac{\partial H}{\partial t}\right]dt,$$

$$Q_{otto}=\int \mathrm{Tr}\left[\frac{\partial \rho }{\partial t}H\right]dt,$$

$$W_{12}=-[\mathrm{Tr}\left({\rho }_{2}H({\theta }_1,{\lambda }_2)\right)-\mathrm{Tr}\left({\rho }_{1}H({\theta }_1,{\lambda }_1)\right)].$$

$$Q_B=[\mathrm{Tr}\left({\rho }_{3}H({\theta }_2,{\lambda }_2)\right)-\mathrm{Tr}\left({\rho }_{2}H({\theta }_1,{\lambda }_2)\right)].$$

$$W_{21}=-[\mathrm{Tr}\left({\rho }_{4}H({\theta }_2,{\lambda }_1)\right)-\mathrm{Tr}\left({\rho }_{3}H({\theta }_2,{\lambda }_2)\right)].$$

$$Q_A=[\mathrm{Tr}\left({\rho }_{1}H({\theta }_1,{\lambda }_1)\right)-\mathrm{Tr}\left({\rho }_{4}H({\theta }_2,{\lambda }_1)\right)].$$

$${\Delta }_{B(A)}=\mathrm{Tr}\left[{\rho }_{2(1)}\right(H({\theta }_{2(1)},{\lambda }_{2(1)})-H({\theta }_{1(2)},{\lambda }_{2(1)})\left)\right].$$

Full text

I Introduction

In recent times, a plethora of models of quantum thermal machines (QTMs) have been proposed and analyzed to understand the emergence of thermodynamic principles from quantum dynamics [kosloff14, souravReview21]. The existence, albeit mostly theoretical, of such a large number of models can be attributed to two reasons. Firstly, the nondeterministic and destructive nature of generic quantum measurements makes it difficult to uniquely and unambiguously define the quantum equivalents of classical work and heat. Secondly, genuine quantum phenomena have been shown to alter the performance of QTMs in a myriad of ways, depending on how they are incorporated into the model, for example, in the form of coherence in the system [hardal15, camati19, mohan25], engineered heat baths [scully03, lutz14, manzano16, klaers17, Niedenzu18, ray23, bera24, sen25], many-body effects [campisi16critical, fazio16, hartmann20, revathy20, piccitto22, solfanelli23, arezzo24, mukherjee24, brollo25], etc. In addition to theoretical investigations, progress has also been made in the experimental realization of QTMs [von19, gluza21, Koch2023, Aamir2025, uusnäkki25experiment].

The simplest and most commonly studied models of QTMs are those based on the four-stroke quantum Otto cycle [souravReview21]. The standard Otto cycle consists of two * work strokes* and two thermalization strokes. In the former, a system is made to evolve unitarily through a quench in some parameter of the Hamiltonian, with the energy change of the system associated with work. The work strokes are interceded by the thermalization strokes in which the system evolves dissipatively in contact with a heat bath, the energy change being associated with the heat transferred. At the end of the full cycle, the system returns to its initial state with a net conversion of quantum heat into quantum work or vice versa. The direction of heat flow and work output determines the operating mode of the cycle, with the thermodynamically allowed (respecting the second law) modes being the engine, refrigerator, accelerator, and heater. The engine (refrigerator) mode is characterized by a net heat transfer from the hotter (colder) to the colder (hotter) bath, with a net work output (input). In accelerator mode, input work is used to boost the heat transfer from the hotter to the colder bath, while in heater mode, input work is fully converted into heat and dumped into each of the baths.

The Otto cycle operating in the engine mode, by definition, relies on thermal energy for work output; the latter thus vanishes in the limit of zero temperature of the heat baths. However, quantum statistical properties depend crucially on the nature of particles at low temperatures, and this has recently been exploited to design and experimentally realize the Pauli engine [Koch2023]. In this variant of the regular Otto engine, the thermalization strokes are replaced by the so-called Pauli strokes, in which the system undergoes a bosonization or fermionization process. Specifically, rather than thermalizing with a heat bath, the statistical nature of the system is altered by tuning it from a molecular Bose condensate to a Fermi gas or vice versa. By virtue of the Pauli exclusion principle, the system has a higher energy ground state in the fermionic state as compared to the bosonic state. The Pauli engine operates by extracting this difference in ground state energy, defined as the Pauli energy, into useful work. Importantly, the Pauli engine has no classical analogue and is purely driven by quantum statistical phenomena. To this end, we note that the relation between the performance of QTMs and the statistical properties of the system has been explored in a number of other works, especially with respect to fermionic and bosonic statistics [myers20, watanabe20, souza22, myers22, ghosh23, Boubakour2023, Menon2025].

In parallel, it has also been shown in recent times that exotic quantum particles such as anyons [Kundu99, anyonRev08, sternRev08, Iqbal24, Zhang25, Himanshu25, Ghosh25, Yutushui25], including impurity excitations in fractional quantum Hall liquids which exhibit fractional angular momentum and effective anyonic statistics [grass20, baldelli21], and paraparticles [hazzard25, Mostaan25], satisfy statistical properties different from bosons or fermions. The performance of QTMs based on systems satisfying such non-trivial statistics has largely remained unexplored, except recent models based on few-body anyonic systems [Myers2021, manianyonotto24, manianyoncarnot25, Dunlop25]. In this work, we establish how anyonic statistics leads to non-trivial performance enhancement in a many-body QTM which, importantly, can be realised using current experimental platforms. Specifically, we define and analyze a hybrid Anyon-Otto (HAO) cycle (see Fig.1), based on the anyon Hubbard model (AHM) [keilmann11, Longhi12, anyonhubbard15, Tang15, ArcilaForero16, Zhang17, Zhang20, Bartolomei20, Bonkhoff21, Bonkhoff23, greinerExpt24, Dhar2025, Wang25, Bonkhoff2025, Theel25], in which the statistical properties of the particles can be continuously tuned from boson-like to pseudo-fermion-like (satisfy anti-commutation relations only when particles are on different sites). We contrast the operation of the HAO cycle with the Otto cycle in the same temperature and parameter regime except that the statistical properties of the system remain unchanged throughout the Otto cycle. We find that the HAO cycle can produce a finite work output even in the limit of vanishing temperatures of the baths, due to the presence of an anyon energy, defined analogously to the Pauli energy. In the absence of explicit interactions between the anyons, we demonstrate that the HAO cycle can operate in an inverse accelerator mode for finite but small temperatures. This mode, characterized by a heat transfer from colder to hotter bath and a net work output, exhibits an apparent violation of the second law which is reconciled when the work production in the anyonization stroke is appropriately accounted for. In presence of weak interactions between the anyons, we observe that anyonic statistics boosts the performance of the QTM at low temperatures, manifested in the form of higher work output as compared to bosonic or pseudo-fermionic statistics. Finally, we propose how the HAO cycle can be realized experimentally through a feasible extension of the experimental protocol used for the recent realization of the AHM [keilmann11, greinerExpt24, Dhar2025], thus rendering the possibility of direct experimental verification of our results.

II Results

Anyon Hubbard Model

The 1D AHM is realized by introducing a synthetic gauge field in the hopping term of the Bose Hubbard model [anyonhubbard15],

[TABLE]

where $L$ is the total number of sites, $b_{j}$ are the bosonic annihilation operators, $n_{j}=b_{j}^{\dagger}b_{j}$ is the number operator, $J$ is the tunneling amplitude, $U$ is the interaction strength, and $\theta$ is a phase parameter. Defining the annihilation operators $a_{j}=e^{-i\theta\sum_{1\leq l\leq j-1}n_{j}}b_{j}$, the equivalent anyon model is obtained,

[TABLE]

where $n_{j}=b_{j}^{\dagger}b_{j}=a_{j}^{\dagger}a_{j}$. The operators $a_{j}$ satisfy the anyonic commutation relations, $a_{j}a_{k}^{\dagger}-e^{i\theta{\rm sgn}(j-k)}a_{k}^{\dagger}a_{j}=\delta_{jk}$ and $a_{j}a_{k}-e^{i\theta{\rm sgn}(j-k)}a_{k}a_{j}=0$. The phase parameter $\theta$ thus provides a direct control over the statistical properties of the particles. It is straightforward to see that the bosonic statistics are recovered in the limit $\theta\to 0$. In contrast, $\theta\to\pi$ leads to pseudo-fermionic statistics as the anyon operators anti-commute on different lattice sites but commute on the same site.

Hybrid Anyon-Otto cycle

The HAO cycle that we propose here is a four-stroke cycle that inherits operational characteristics from both the Pauli and Otto cycles. Thus, a hallmark of this cycle is its ability to function both as an anyon engine (defined analogously to the Pauli engine) at low temperature and as the standard Otto engine at high temperature. Before defining the HAO cycle, we recall that in the Otto cycle, work and heat are commonly associated with the energy exchange in strokes undergoing exclusively unitary or dissipative dynamics, respectively,

[TABLE]

where $\rho$ is the density operator representing the state of the system. In other words, a unitary evolution with a time-dependent Hamiltonian only contributes to the work, while a dissipative evolution, such as thermalization with a heat bath, only results in heat transfer in the Otto cycle.

We now define the HAO cycle as follows. At the start of the cycle, we assume that the system with Hamiltonian $H(\theta_{1},\lambda_{1})$ is in equilibrium with a thermal bath with temperature $T_{A}$ (inverse temperature $\beta_{A}$), and $\lambda$ representing either of the parameters $J$ or $U$ appearing in the Hamiltonian. The initial state is therefore a thermal Gibbs state $\rho_{1}\equiv e^{-\beta_{A}H(\theta_{1},\lambda_{1})}/{\rm Tr}\left[{e^{-\beta_{A}H(\theta_{1},\lambda_{1})}}\right]$. The HAO cycle then consists of the following sequential strokes (see Fig. 1):

Unitary expansion: ($\lambda_{1}\to\lambda_{2}$, isolated) - The Hamiltonian parameter is ramped from $\lambda_{1}$ to $\lambda_{2}$ over a time $\tau$. The system evolves unitarily to a state $\rho_{2}=U_{12}^{\dagger}(\tau)\rho_{1}U_{12}(\tau)$ during this time interval. The change in the energy expectation value is thus associated with work performed,

[TABLE] 2. 2.

Thermalization B: ($\theta_{1}\to\theta_{2}$, contact with heat bath) - This stroke consists of two sub-strokes – the phase parameter is tuned from $\theta_{1}$ to $\theta_{2}$ followed by thermalization with a heat bath of temperature $T_{B}$. Note that the two sub-strokes can be carried out simultaneously, provided that the phase parameter is changed over a time interval much shorter than the thermalization time-scale. The steady state reached is thus given by, $\rho_{3}=e^{-\beta_{B}H(\theta_{2},\lambda_{2})}/{\rm Tr}\left[{e^{-\beta_{B}H(\theta_{2},\lambda_{2})}}\right]$. The change in the energy expectation value in this stroke has two contributions, one from the energy dissipated to the heat bath and the other from the work done in changing the phase parameter. However, for reasons that we shall clarify below, we associate the total energy change in this stroke with heat transfer,

[TABLE] 3. 3.

Unitary compression: ($\lambda_{2}\to\lambda_{1}$, isolated) - As in the case of the unitary expansion stroke, the system evolves unitarily to $\rho_{4}=U_{21}^{\dagger}(\tau)\rho_{3}U_{21}(\tau)$ and the change in the energy expectation value corresponds to work performed $W_{21}$,

[TABLE] 4. 4.

Thermalization A: ($\theta_{2}\to\theta_{1}$, contact with heat bath) - In the final stroke, the phase parameter is restored to its initial value $\theta_{1}$ and the subsequent thermalization with heat bath at temperature $T_{A}$ results in the system returning back to its initial state $\rho_{1}$. Thus,

[TABLE]

The total work $W=W_{12}+W_{21}$, $Q_{A}$ and $Q_{B}$, satisfies energy conservation (first law) $W=Q_{A}+Q_{B}$. The difference of the HAO cycle defined above with the standard quantum Otto cycle lies in the two modified thermalization strokes. In the latter case, the statistical properties of the quantum model remain unchanged, and thus the heat change is solely associated with the energy gained or lost to the heat baths. In contrast, thermalization in the HAO cycle is accompanied by a explicit change of statistical properties through the parameter $\theta$. It is important to note that ramping the phase parameter modifies the Hamiltonian itself and thus technically amounts to performing a certain amount of work. However, in the anyon equivalent of the model defined in Eq. (2), $\theta$ implicitly controls the statistical properties and hence the equilibrium configuration of the system. Since the purpose of the HAO engine is to convert both thermal energy and anyon energy into useful work, we associate the total energy change in the modified thermalization strokes only with heat energy.

The non-interacting limit $U=0$

We first consider the operation of the HAO cycle in the non-interacting limit $U=0$ with $\lambda\equiv J$. In this limit, the Hamiltonians at different instants of time during the work strokes commute with each other, implying adiabatic evolution and thus rendering the cycle operation independent of the work stroke duration $\tau$. We also assume that the total number $N$ of particles remain conserved throughout the cycle. In Fig. 2-2, we examine the possible modes of operation for the standard Otto cycle ($\theta_{1}=\theta_{2}=0$), choosing without loss of generality $J_{1}=2.0$, $J_{2}=1.0$, $L=8$ and $N=4$. Depending on the sign of $W$, $Q_{A}$ and $Q_{B}$, and the relative magnitude of $T_{A}$ and $T_{B}$, three modes of cycle operation can be identified - namely the engine mode (E) with $Q_{A},W>0$, $Q_{B}<0$; the refrigerator mode (R) with $Q_{A},W<0$, $Q_{B}>0$ for $T_{A}>T_{B}$, and the accelerator mode (A) with $Q_{A},W>0$, $Q_{B}<0$ for $T_{A}<T_{B}$. The E-R transition can be identified by the line $W=Q_{A}=Q_{B}=0$. For $T_{A},T_{B}\to 0$, the work output and the heat vanish identically.

Having established the modes of operation in the Otto cycle, we now consider the situation $\theta_{1}=\pi$, $\theta_{2}=0$, that is, the statistical properties of the system are altered between bosonic and pseudo-fermionic during the thermalization strokes. From Fig. 2-2, it is evident that in the large temperature limit of the baths, $T_{A},T_{B}\gg 1$, the HAO cycle is identical to the Otto cycle owing to the fact that the anyonic statistics can be well-approximated by Boltzmann statistics in this limit. However, at low temperatures, $T_{A},T_{B}\to 0$, the excess energy resulting from the development of anyonic exclusions begins to manifest itself in the form of the non-zero work output and heat exchanges. This leads to the emergence of an inverse accelerator (IA) mode - heat flows from the colder bath to hotter bath with a net work extraction, $Q_{A},W>0$, $Q_{B}<0$ for $T_{A}<T_{B}$. As we shall show later, the IA mode does not violate the second law of thermodynamics since the anyon energy gained by the system during the anyonization process involves an additional work cost, which when included in the definition of $W$ restores the second law.

The finite work output in the limit of vanishing temperature manifests itself not only in the pseudo-fermionic limit $\theta_{1}=\pi$, but for any $\theta_{1}\neq 0$. This is demonstrated in Fig. 3, where we observe that the average work output per particle $W/N$ increases monotonically with increasing $\theta_{1}$ in the inverse accelerator mode ($T_{A}=0.2$, $T_{B}=0.4$), provided $T_{A}\to 0$. Similarly, the work input decreases with increasing $\theta_{1}$ in the accelerator mode ($T_{A}=0.2$, $T_{B}=0.8$) in the same limit of $T_{A}$. Note that it is sufficient for only $T_{A}$ to be small as the anyonization process occurs in contact with the heat bath of temperature $T_{A}$.

Weak interaction $U\ll J$

We now consider the situation in which work is performed by ramping the interaction parameter $U$ during the work strokes, holding $J$ to a constant value. In this case, the Hamiltonian does not commute with itself at different times during the work strokes, which leads to non-adiabatic excitations for finite stroke duration $\tau$. However, for simplicity, we shall assume that $\tau\to\infty$ as the results discussed in the following do not change qualitatively in the presence of non-adiabatic excitations. In order to ensure that the anyon energy is not affected by the interaction, we choose $U_{1}=0$. The work output $W$ at low temperature, $T_{A}=T_{B}=0.1$, as a function of the statistical parameter $\theta_{1}$ is shown in Fig. 4 for different interaction strengths $U_{2}$. In the weakly interacting limit $U\ll J$, we observe that $W$ is maximized for $\theta_{1}=\theta^{*}$ such that $0<\theta^{*}<\pi$, for $N\geq L/2$. However, as the interaction strength is increased, the maxima shifts towards the pseudo-fermionic limit $\theta^{*}=\pi$.

To understand the above results, we first note that the dependence of $W$ on $\theta_{1}$ arises only from the unitary expansion stroke, that is, when the interaction strength is increased from $U_{1}=0$ to $U_{2}$ at a fixed $\theta_{1}$. In Fig. 4, we see that the change in the ground state energy $\Delta E_{G}$ during this stroke is also minimized at intermediate values of $\theta_{1}$ between [math] and $\pi$ for $U\ll J$ and $N\geq L/2$. The maximization of $W$ (recall that it is defined with a negative sign) observed in Fig. 4 is a direct consequence of this non-trivial behavior of the ground state energy. The latter can be understood as follows. For $U\ll J$, $\Delta E_{G}$ can be approximated by the first order perturbative correction to the ground state energy of the non-interacting part of the Hamiltonian, considering the interaction as a perturbation of magnitude $U$. This can be seen from Fig. 4 where the dashed lines correspond to $E_{per}=\braket{\psi_{0}|H_{int}|\psi_{0}}$, with $H_{int}=\frac{U}{2}\sum_{j}n_{j}\left(n_{j}-1\right)$ and $\ket{\psi_{0}}$ is the ground state of the system with $U=0$. Since $\sum_{j}n_{j}=N$, the relatively smaller magnitude of $E_{per}$ can therefore be attributed to higher probability of double occupancy or greater per site for $\theta=0$ or $\theta=\pi$ and lesser probability for the same at other intermediate values.

The higher probabilities of double or greater occupancy in the bosonic and pseudo-fermionic limit can be understood by reverting back to the bosonic Hamiltonian defined in Eq. (1). Since the hopping amplitude acquires a density dependent phase for any $\theta\neq 0$, the system minimizes energy by preferring lesser contributions from configurations in which a given site is doubly or more occupied. For less than half-filling $N<L/2$, this is easily achieved as there are sufficient number of empty sites to satisfy lesser or equal to unit occupancy per site. Thus, the probability for higher occupancy decreases monotonically with increasing value of $\theta$. For $N\geq L/2$, the situation is not straightforward and the system is forced to choose configurations with greater than unit occupancy for some sites. However, as $\theta\to\pi$, the density dependence phase reduces to unity even for higher occupancy, and thus the probability of more than unit occupancy increases again. To support the above argument, we plot in Fig. 5 the probability $P(n_{j})$ that the $j^{th}$ site is occupied by $n_{j}$ particles for a couple of nearby sites $j=3,4$ in the system with $L=8$, $J=1.0$ and $U=0$. Although $P(n_{j}=0,1)>P(n_{j}=2,3,4)$ for both the sites, $E_{per}$ has no contributions from the zero or unit occupancies as the interaction energy vanishes identically for $\braket{\psi_{0}|n_{j}|\psi_{0}}=0,1$. It can be seen from Fig. 5 that the $P(n_{j}=2)$ decreases initially as $\theta$ is increased from zero, before increasing again on approaching $\theta=\pi$. Importantly, the increase in the value of $P(n_{4}=2)$ is accompanied by an increase in the value of $P(n_{3}=0)$ and a decrease in the value of $P(n_{3}=1)$, suggesting that double occupancy is preferred at the expense of unit occupancy in the limit $\theta\to\pi$.

Reconciliation with the second law

In the previous sections, we have argued that the apparent violation of the second law of thermodynamics in the IA mode can be reconciled if the work associated with the buildup of anyon energy is accounted for. To show this explicitly, we modify the definition of work in Eq. (4) and Eq. (6) to incorporate the energy change associated with the anyonization and bosonization processes. Likewise, we subtract the same from the heat exchanges to ensure consistency with the first law. Furthermore, in this case, we assume that the anyonization/bosonization process is carried out much faster than the time-scale of thermalization with the baths. This ensures that energy change during the anyonization/bosonization process is only due to work performed by explicit change of the parameter $\theta$ in the Hamiltonian. Thus, the redefined work and heat are, $\overline{W}_{12(21)}={W}_{12(21)}-\Delta_{B(A)}$, $\overline{W}=\overline{W}_{12}+\overline{W}_{21}$, $\overline{Q}_{A(B)}=Q_{A(B)}-\Delta_{A(B)}$, where

[TABLE]

Since the HAO cycle operates like a regular Otto cycle in terms of $\overline{W}$ and $\overline{Q}$, it now also becomes meaningful to define the efficiency $\eta=\overline{W}/\overline{Q}_{A(B)}$ for $T_{A(B)}>T_{B(A)}$, provided the cycle is operating in the engine mode. Examining the non-interacting situation with $U=0$, $J_{1}=2.0$ and $J_{2}=1.0$, it is evident from Figs. 6-6 that the inverse accelerator mode does not emerge even for $\theta_{1}\neq 0$. However, the effect of changing the statistical properties with $\theta_{1}$ still manifests itself at low temperatures. In particular, for $\theta_{1}\geq 2.2$, the cycle is able to operate as an accelerator for certain range of small but finite temperatures for $T_{A}>T_{B}$. Likewise, for $T_{A}<T_{B}$, the cycle switches from accelerator to engine mode $\theta_{1}\geq 2.2$ at small temperatures. Finally, we also note the emergence of heater mode in the vicinity of the temperatures at which the direction of heat flow switches for the two baths.

It is fascinating to note that, the maximum possible efficiency of the cycle when operating in engine mode is significantly enhanced for $\theta_{1}\geq 2.2$. In fact, the maximum efficiency is found to be from the engine mode at $T_{A}<T_{B}$, which emerges for $\theta_{1}\geq 2.2$ (bottom left in Figs. 6 and 6). This is illustrated in Fig. 6, where we plot the maximum efficiency across all temperatures as a function of $\theta_{1}$. In the same figure, we also plot $\mathrm{sgn}(T_{A}^{max}-T_{B}^{max})$ (dashed lines), where $T_{A(B)}^{max}$ are the temperatures of the bath for which the maximum efficiency is obtained for a fixed $\theta_{1}$. For $N=4$, we can clearly see that $\eta_{max}$ increases drastically for $\theta_{1}>2.5$. This coincides with $T_{A}^{max}$ becoming smaller than $T_{B}^{max}$, signifying that the higher efficiency is extracted from the newly emerged engine regime at small temperature.

Experimental realization

The 1D AHM has already been realized experimentally using ultracold ${}^{87}\textrm{Rb}$ atoms in an optical lattice [greinerExpt24], where the tunneling amplitude is engineered to depend on the particle density in accordance with Eq. (1). In this section, we propose how the HAO thermal machine can be realized using the same experimental setup. The main challenges associated with implementing the HAO cycle on the optical lattice platform of [greinerExpt24] are the following - (i) realization of a thermal bath with which the system interacts during the thermalization strokes, (ii) measurement of the energy change of the system at the end of each stroke, and (iii) requirement of higher particle density ($N>2$) to observe the superior performance of the cycle for $0<\theta<\pi$.

To overcome the first challenge, we propose to introduce a second set of atoms in the lattice, which acts as a bath for the atoms already present in the system. Specifically, the additional atoms that form the bath will be prepared in the superfluid state ($\theta_{\rm bath}\to 0$) to facilitate a controlled tunability of the effective temperature of the bath. Similar realization of controlled bath environments has been proposed and experimentally implemented in a number of other works [Seetharam_2015, Rubio_Abadal_2019, L_onard_2023, Schnell_2024], and allows for the controlled engineering of dissipation. In our setup, the bath will be located along the length $L$ of the system, where it is prepared in the superfluid state using the Floquet scheme described in [Cardarelli_2016]. The size of the bath will be much larger than the system such that the system can couple to continuum modes. Independent control of the statistical parameter $\theta$ will be necessary for the bath and the system, as $\theta_{\rm bath}\to 0$ while $\theta_{\rm system}$ changes throughout the HAO cycle, and can be realized by implementing the Floquet scheme using a digital micromirror device (DMD) rather than directly modulating the intensity of the lattice. Using the DMD to modulate the intensity allows for spatial control of the statistical parameter, as the DMD can be programmed such that one section projects modulating light to realize the system while another realizes the bath. The system will be separated from the bath by an optically projected wall, implemented either by the same or another DMD, where the height of the wall is determined by the intensity of the light. By changing the intensity of the light, and therefore the height of the wall, the system can be coupled or isolated from the bath during the thermalization or work strokes, respectively.

Concerning the second challenge, we shall resort to time-of-flight (TOF) measurements to find the energy $E(\lambda,\theta)$ at the end of each stroke. We illustrate the measurement protocol only for the non-interacting case ($U=0$) here, as the same protocol will also work for weak interactions. In the absence of interactions, we recall that the work strokes consist of adiabatically tuning the parameter $J$ as the system evolves unitarily, while the heat strokes correspond to a dissipative evolution of the system in contact with thermal baths along with a simultaneous tuning of the phase parameter $\theta$. The measurement protocol will therefore be as follows. Let us assume that the system is initially prepared in the state $\rho_{3}$, i.e., in thermal equilibrium with the bath with temperature $T_{B}$ and $\theta=0$. We first perform a TOF measurement to determine the energy $E(J_{2},0)$ of the system in this state. The system is confined by optically projected walls along both $x$ and $y$, with the walls along $y$ defining the length $L$, and to perform TOF, the walls along $y$ are quenched to allow the system to expand within a 1D tube. Fluorescence images of the atoms are taken at successively longer times after expansion to track the expansion profile of the system, from which a velocity and therefore the energy may be extracted. Since the TOF measurement destroys the initial state, we reset the system to $\rho_{3}$ before starting the HAO cycle. The compression work stroke is performed by decoupling the bath and adiabatically tuning the hopping amplitude $J_{2}\to J_{1}$ over a time-interval $\tau_{W}$, during which the system evolves to $\rho_{4}$. At $t=\tau_{W}$, the TOF measurement is performed again to find the energy $E(J_{1},0)$. The work $W_{21}$ is therefore obtained as $W_{21}=\langle E(J_{1},0)-E(J_{2},0)\rangle$.

In the next step, we once again prepare the system in state $\rho_{3}$, evolve unitarily to $\rho_{2}$, followed by coupling the system to bath with temperature $T_{A}$. The phase parameter is then tuned to $\theta=\theta_{1}$ while in contact with the bath, and we ensure that the system thermalizes with the bath for sufficiently long time after tuning of the phase parameter is completed, before performing the TOF measurement to find $E(J_{1},\theta_{1})$. The heat exchanged is thus calculated as $Q_{A}=\langle E(J_{1},\theta_{1})-E(J_{2},0)\rangle-W_{21}$. Similarly, to calculate $W_{12}$ and $Q_{B}$, we shall reset the system to $\rho_{3}$ in each case, and then perform TOF measurements at the end of the unitary expansion and thermalization B strokes, respectively, to find $W_{12}=\langle E(J_{2},\theta_{1})-E(J_{2},0)\rangle-W_{21}-Q_{A}$ and $Q_{B}=\langle E(J_{2},0)^{\prime}-E(J_{2},0)\rangle-W_{21}-Q_{A}-W_{12}$, where $E(J_{2},0)^{\prime}$ is the measured energy of the system after completing the cycle. We note that $E(J_{2},0)^{\prime}\neq E(J_{2},0)$ if the thermalization is not perfect in the heat exchange strokes, which will indicate an incomplete cycle.

Finally, regarding the third challenge, achieving a higher particle density ($N>2$) is possible as long as the filling $n=N/L\lesssim 0.36$ [Str_ter_2016]; otherwise, the Floquet implementation does not faithfully realize the AHM. When exploring behavior at higher filling, it may be possible that the essential physics is preserved despite differences between the driven system and the AHM. Other implementations that do not involve shaking may be considered to realize systems at higher filling [Fr_lian_2022].

III Methods

All the numerical simulations were performed in the bosonic representation of the AHM Hamiltonian (Eq. (1)) using the QuSpin package [quspin1, quspin2]. The density dependent phase of the hopping terms cannot be directly implemented using the standard Quspin routines for constructing Hamiltonians. Hence, we explicitly encoded the hopping terms as sparse matrices and added the same to the onsite interaction terms constructed directly using QuSpin routines [code_ahm_hamiltonian]. The full Hamiltonian was then diagonalized within Quspin for generation of required data.

IV Discussion

In summary, we have proposed a four-stroke hybrid anyon Otto cycle based on the 1D anyon Hubbard model which, like its Pauli counterpart, relies on exclusion statistics of anyons to derive work at low temperatures. In the absence of an explicit interaction among the anyons, we see a monotonous increase in the low temperature work output as the statistical parameter is increased from the bosonic limit to the pseudo-fermionic limit. The presence of a finite anyon energy at low temperature leads to the emergence of an inverse accelerator mode of the cycle, which is prohibited by the second law in a regular Otto cycle. When finite but weak interactions are introduced, and at half or higher filling, the low-temperature work output no longer peaks at the bosonic or pseudo-fermionic limits. Instead, it reaches a maximum at an intermediate value of the statistical parameter, thus demonstrating that in the interacting regime, anyonic statistics can be harnessed to achieve greater work extraction than is possible with either bosonic or fermionic statistics alone. We reiterate that the inverse accelerator mode does not violate the second law in the HAO cycle; they arise from treating the anyonization stroke - which can be interpreted as a work stroke - as a “heat” source, as clarified in the final discussion. By properly incorporating the anyonic contribution and redefining the work, we show that at low temperatures the accelerator mode is replaced by an engine mode. This engine mode exhibits maximum efficiency in the anyonic limit at high $\theta$, highlighting the crucial role of anyonic statistics in enhancing cycle performance. Finally, we outline an experimental protocol to verify our results using a similar experimental platform that was used to realize the AHM.

Authors contribution

SB and UB conceived the project. MB led the numerical calculations, with additional contributions from SB. SB and UB performed the main analysis and interpretation of the results, with further contributions from ML, TG, and RC. AP and MAG-M also contributed to a part of the analysis and interpretation. JK developed the experimental proposal. All authors contributed to preparation of the manuscript.

Acknowledgements.

ICFO-QOT group acknowledges support from: European Research Council AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project funded by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR"; (PRTR-C17.I1), FPI); QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European Union’s Horizon 2020 program under Grant Agreement No 101017733; Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund FEDER and CERCA program; Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, European Commission, European Climate, Infrastructure and Environment Executive Agency (CINEA), or any other granting authority. Neither the European Union nor any granting authority can be held responsible for them (HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-OPEN OPTOlogic, Grant No 899794, QU-ATTO, 101168628), EU Horizon Europe Program (This project has received funding from the European Union’s Horizon Europe research and innovation program under grant agreement No 101080086 NeQSTGrant Agreement 101080086 — NeQST); ICFO Internal “QuantumGaudi” project. R.W.C acknowledges support from the Polish National Science Centre (NCN) under the Maestro Grant No. DEC- 2019/34/A/ST2/00081. U.B. is also grateful for the financial support of the IBM Quantum Researcher Program. T.G. acknowledges the financial support received from the IKUR Strategy under the collaboration agreement between the Ikerbasque Foundation and DIPC on behalf of the Department of Education of the Basque Government, as well as funding by the Department of Education of the Basque Government through the project PIBA_2023_1_0021 (TENINT), and by the Agencia Estatal de Investigación (AEI) through Proyectos de Generación de Conocimiento PID2022-142308NA-I00 (EXQUSMI). A.P. and M.L.B. acknowledge for support from the project PID2023-152724NA-I00, with funding from MCIU/AEI/10.13039/501100011033 and FSE+, the Severo Ochoa Grant CEX2023-001292-S, Generalitat Valenciana grant CIPROM/2022/66, the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call - QUANTUM SPAIN project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda, and by the CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-QTEP+). Also funding from Horizon Europe EU projects MSCA-SE CaLIGOLA, Project ID: 101086123, and MSCA-DN CaLiForNIA, Project ID: 101119552. M.A G-M acknowledges support from the Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call—Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan—NextGenerationEU within the framework of the Digital Spain 2026 Agenda: also from Projects of MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and by Generalitat Valenciana, with reference 20220883 (PerovsQuTe) and COMCUANTICA/007 (QuanTwin).

Data Availability

All data generated or analysed during this study are included in this article.

Code Availability

The code for constructing the AHM Hamiltonian can be accessed via the link https://doi.org/10.5281/zenodo.17873350.

Competing Interests

All authors declare no financial or non-financial competing interests.