Abstract
Quantum heat engines are subjected to quantum fluctuations related to their discrete energy spectra. Such fluctuations question the reliable operation of thermal machines in the quantum regime. Here, we realize an endoreversible quantum Otto cycle in the large quasispin states of Cesium impurities immersed in an ultracold Rubidium bath. Endoreversible machines are internally reversible and irreversible losses only occur via thermal contact. We employ quantum control to regulate the direction of heat transfer that occurs via inelastic spinexchange collisions. We further use fullcounting statistics of individual atoms to monitor quantized heat exchange between engine and bath at the level of single quanta, and additionally evaluate average and variance of the power output. We optimize the performance as well as the stability of the quantum heat engine, achieving high efficiency, large power output and small power output fluctuations.
Introduction
Most engines used in modern society are heat engines. Such machines generate motion by converting thermal energy into mechanical work^{1}. Two central figures of merit of heat engines are efficiency, defined as the ratio of workoutput and heat input, and power characterizing the workoutput rate. Heat engines should ideally have high efficiency, large power output, and be stable, i.e., exhibit small power fluctuations. However, real thermal machines operate far from reversible conditions and their performance is thus reduced by irreversible losses^{2,3}. At the same time, microscopic motors are exposed to thermal fluctuations and, at low enough temperatures, to additional quantum fluctuations, which are associated with random transitions between discrete energy levels. Both fluctuation mechanisms contribute to their instability^{4,5}. An important issue is hence to design and optimize small heat engines in order to maximize both their performance and their stability^{6}.
Nanoscopic heat engines have been implemented recently using a single trapped ion^{7} and a spin coupled to the singleion motion^{8,9}. Indications for quantum effects have been reported in a spin engine consisting of nitrogenvacancy centers interacting with a light field^{10}, and quantum heat engine operation has been shown in nuclear magnetic resonance^{11,12} and singleion^{9} systems. These thermal machines are based on harmonic oscillators or twolevel systems, and the baths mediating heat exchange are simulated by interaction with either laser fields^{7,8,9,10} or radiofrequency pulses^{11,12}.
We here experimentally realize a quantum Otto cycle using a large quasispin system in individual Cesium (Cs) atoms immersed in a quantum heat bath made of ultracold Rubidium (Rb) atoms. Expansion and compression steps are implemented by varying an external magnetic field, changing the energylevel spacing of the engine and performing work^{13}. Heat exchange between system and bath occurs via inelastic endoenergetic and exoenergetic spinexchange collisions^{14}. The increased number of internal engine states, compared to simple twolevel systems, allows for highenergy turnover per cycle, while their finite number naturally limits power fluctuations due to saturation, in contrast to the unbounded spectrum of harmonic oscillators. We employ quantum control of the coherent spinexchange process^{15} to control the direction of heat transfer between system and bath at the level of individual quanta of heat^{14}, independently of the kinetic thermal state of the bath. The precise control of the spin states of both engine and bath effectively suppresses internal irreversible losses in individual collisions, and thus makes the quantum heat engine endoreversible. Endoreversible machines operate internally without dissipation, while (external) irreversible losses only occur via the contact with the bath^{2,3}. They hence outperform fully irreversible engines and have played for this reason a central role in finitetime thermodynamics for forty years^{2,3}. We note that quantum systems generally exhibit internal friction when their Hamiltonian does not commute at different times^{16,17}. We additionally characterize the discrete quantum heat transfer at the level of individual quanta using fullcounting statistics^{18,19} and monitor the population dynamics of the engine from singleatom and timeresolved measurements of the engine’s quasispin distribution along the cycle. We employ this system and techniques to evaluate and optimize the performance as well as the stability of the quantum heat engine, achieving high efficiency, large power output and small power output fluctuations.
Results
Components of the neutralatom machine
We experimentally immerse up to ten lasercooled Cs atoms in the \({F}_{\text{Cs}}=3,{m}_{F,\text{Cs}}=3\rangle\) state into an ultracold Rb gas of up to 10^{4} atoms in the state \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=1\rangle\), both species confined in a common optical dipole trap (Fig. 1a) (Methods A). Here F and m_{F} denote the total atomic angular momentum and its projection onto the quantization axis, respectively. The quantization axis is given by an external magnetic field of B_{1} = 346.5 ± 0.2 mG or B_{2} = 31.6 ± 0.1 mG. The Cs atoms quickly thermalize to the kinetic temperature of T = 950 ± 50 nK of the gas. We operate the quantum heat engine in the spinstate manifold of the seven Cs hyperfine ground states \({F}_{\text{Cs}}=3,{m}_{F,\text{Cs}}\rangle\), m_{F,Cs} ∈ [+3, +2, …, −3], which define its quasispin. These states are energetically equally spaced with Zeeman energy \({E}_{n}^{\,\text{Cs}\,}=n\lambda B\), with \(\lambda = {g}_{F}^{\,\text{Cs}} {\mu }_{\text{B}}\), where \({g}_{F}^{\,\text{Cs}\,}=1/4\) is the Cs Landé factor, μ_{B} Bohr’s magneton and n = 3 − m_{F,Cs}^{20}, with the zeropoint of energy set to the lowestenergy state \({m}_{F,\text{Cs}}=3\rangle\).
Heat between the quantum engine and the bath is exchanged at the microscopic level via inelastic spinexchange collisions (Fig. 1c). Each collision changes the value of the quasispin of the Cs engine by Δm_{Cs} = ∓1ℏ, leading to an energy change of ΔE^{Cs} = ±λB for each Cs atom, and Δm_{Rb} = ±1ℏ for one Rb atom corresponding to the energy change ΔE^{Rb} = ∓κB, with \(\kappa = {g}_{F}^{\,\text{Rb}} {\mu }_{\text{B}}\), where \({g}_{F}^{\,\text{Rb}\,}=1/2\) is the Rb Landé factor^{14}. The spin population thus directly reflects the energy exchange between engine and reservoir at the level of single energy quanta. The direction of the heat transfer is determined by the spin polarization of the Rb bath and by angular momentum conservation during individual collisions^{14}. The spin polarization of the Rb atoms distinguishes a highenergy bath for m_{Rb} = −1 from a lowenergy bath for m_{Rb} = +1. Control over the internal Rb state accordingly permits to either increase or decrease the energy of the quasispin of the engine. Heat exchange automatically stops after six spinexchange collisions, because then the highest/lowest energy state has been reached. One collision transfers the colliding Rb atom to the \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=0\rangle\) state, which forms the exhaust of the engine. Owing to the massive imbalance between the Rb and Cs atom numbers (N_{Rb}/N_{Cs} > 1000), the probability of a second collision with the same Rb atom is indeed vanishingly small. Memory effects are furthermore negligible, making the bath Markovian.
Implementation of the quantum Otto cycle
The quantum Otto cycle consists of four parts: one compression and one expansion step, during which work is performed, and a heating and a cooling stage, during which heat is exchanged^{13}. The corresponding experimental sequence is shown in Fig. 1b. The Cs machine is first driven by up to six spinexchange collisions into energetically higher states (at magnetic field B_{1}), absorbing average heat 〈Q_{H}〉 in time τ_{H} = t_{B}. Mean work 〈W_{BC}〉 is then performed by adiabatically decreasing the magnetic field to B_{2} in τ = t_{C} − t_{B} = 10 ms. This time is much longer than the inverse energy splitting ΔE of the quasispin states, making the process adiabatic. It is, however, fast enough to avoid unwanted spinexchange collisions, implying that no heat is transferred. The engine is subsequently brought into contact with the lowenergy bath by flipping the spins of the Rb bath using microwave (MW) sweeps. The Cs engine is accordingly driven by up to six spinexchange collisions into energetically lower states, releasing heat 〈Q_{C}〉 in time τ_{C} = t_{D} − t_{C}. Work 〈W_{DA}〉 is further performed by adiabatically increasing the magnetic field back to B_{1} in τ = t_{A} − t_{D} = 10 ms. The Rb spins are finally flipped to their initial state with other microwave sweeps, restoring the highenergy bath.
While each single collision is coherent and thus amenable to quantum control^{15}, coupling of the engine to the large number of bath modes in elastic collisions destroys the coherence between the engine’s quasispin levels. Heat is thus associated with changes of occupation probabilities, 〈Q〉 = ∑_{n}E_{n}Δp_{n}, whereas work corresponds to changes of energy levels, 〈W〉 = ∑_{n}p_{n}ΔE_{n}^{13}. In our system, we concretely have \(\langle {Q}_{\text{H}}\rangle ={\sum }_{n}n\left({p}_{n}^{\,\text{B}}{p}_{n}^{\text{A}\,}\right)\lambda {B}_{1}\) for heating and \(\langle {Q}_{\text{C}}\rangle ={\sum }_{n}n\left({p}_{n}^{\,\text{D}}{p}_{n}^{\text{C}\,}\right)\lambda {B}_{2}\) for cooling. On the other hand, the respective work contributions for expansion and compression are given by \(\langle {W}_{\text{BC}}\rangle ={\sum }_{n}n{p}_{n}^{\,\text{B}\,}\lambda ({B}_{2}{B}_{1})\) and \(\langle {W}_{\text{DA}}\rangle ={\sum }_{n}n{p}_{n}^{\,\text{D}\,}\lambda ({B}_{1}{B}_{2})\). In order to evaluate these average quantities, we determine the magnetic fields B_{1} and B_{2} with the help of Rb microwave spectroscopy (Methods). We further detect the Zeeman populations \({p}_{n}^{i}\) of single Cs atoms at arbitrary times by position resolved fluorescence measurements combined with Zeemanstateselective operations (Fig. 1a inset)^{21}. From each individual measurement, we can determine quantized spin transitions for each single atom. This allows us to monitor the resulting quantized heat exchange between engine and environment with a resolution of single quanta at each time. From a series of such measurements, we can further construct the average evolution of the quasispin populations (Fig. 2a, b): the progressive transfer from low (high) energy states to high (low) energy states during heating (cooling) as a function of time is clearly seen (green dots). From the measured heat counting statistics, we compute average (blue and red dots) and variance of heat exchange. We will use these quantities to examine the power output of the quantum machine and its fluctuations.
Performance of the quantum heat engine
We first characterize the performance of the quantum Otto engine by evaluating its efficiency given by^{13},
where 〈Q_{H}〉 − ∣〈Q_{C}〉∣ is the total work produced by the machine, 〈Q_{L}〉 the energy dissipated during the total heat exchange in one cycle, and 〈Q_{H}〉 + 〈Q_{L}〉 the heat emitted by the highenergy bath (Fig. 1d). Indeed, due to the different atomic Landé factors for Rb \(({g}_{F}^{\,\text{Rb}\,}=1/4)\) and Cs \(({g}_{F}^{\,\text{Cs}\,}=1/2)\), only half (\(\gamma ={g}_{F}^{\,\text{Cs}}/{g}_{F}^{\text{Rb}\,}=1/2\)) of the energy change of a bath atom is effectively exchanged with the heat engine during an inelastic spinexchange collision^{21}. As a result, the heat emitted (absorbed) by the bath differs from the energy portions absorbed 〈Q_{H}〉 (emitted 〈Q_{C}〉) by the machine. We macroscopically account for the remaining lost energy, which is irreversibly transferred to the kinetic energy of Rb during an average of ten elastic collisions, by a heat leak^{22} equal to \(\langle {Q}_{\text{L}}\rangle ={\sum }_{n}n\left({p}_{n}^{B}{p}_{n}^{A}\right)\kappa (1\gamma )({B}_{1}{B}_{2})\) with γ = λ/κ the ratio of the Landé factors (Methods). We then obtain,
Its maximum value η_{max}, reached in the absence of irreversible losses (γ = 1), is determined by the ratio of the two magnetic fields. We evaluate the efficiency (2) using experimental data for different cycle durations, τ_{cycle} = τ_{H} + τ_{C} + 2τ, by varying the heating and cooling times and evaluating the average heats 〈Q_{H}〉 and 〈Q_{C}〉 (Fig. 3a). We find a constant value, i.e., independent of the number of spinexchange collisions, of η = 0.478 ± 0.002. We emphasize that the internal efficiency of the quantum Otto engine, η_{int} = 1 − ∣〈Q_{C}〉∣/〈Q_{H}〉 = 0.917 ± 0.009 (Methods C) is close to the maximal value η_{max} = 0.908. We may, therefore, conclude that irreversible losses mainly occur during heat transfer, while the engine itself runs reversibly. The quantum heat engine is hence endoreversible. We further note that, since heat losses are determined by the value of the Landé factors, they can in principle be reduced by choosing different atomic species.
Second, we consider the average power of the quantum heat engine which reads,
We use the heat counting statistics to track its time evolution in Fig. 3b. We observe that the power (blue dots) increases with the number of inelastic collisions and reaches a maximum, \({\left\langle P\right\rangle }_{\max }/{k}_{\text{B}}=30\) nK/ms, for a cycle time of 960 ms. The corresponding number of inelastic collisions responsible for the heat exchange is almost 12 collisions total (6 spinexchange collisions for the heating process and 6 for the cooling). This maximum nearly coincides with full population inversion between these two processes (\({m}_{F,\text{Cs}}=3\rangle \leftrightarrow {m}_{F,\text{Cs}}=3\rangle\)), in analogy to that of a laser. Good agreement with a theoretical model (red solid line) is observed (Methods). From a collisional perspective, the energy transfer with the atomic bath is optimal in the sense that it exchanges the maximum energy of six quanta, which can be stored in the machine, in exactly six spinexchange collisions as a consequence of the precise control of the spin states of machine and bath. The value of \({\left\langle P\right\rangle }_{\max }\) may be further optimized by enhancing the magnetic field difference, as well as the collision rate and the collision crosssection by controlling the temperature or density of the Rb gas.
We finally investigate the stability of the quantum Otto engine by analyzing the relative power fluctuations via the Fano factor, which quantifies the deviation from a Poisson distribution^{23},
where \({\sigma }_{P}^{2}\) is the variance of the power, which we determine from the measured quasispin distributions (Methods). Figure 3c displays the Fano factor as a function of the cycle time, with the absolute fluctuations σ_{P} shown in the inset. We find superPoissonian fluctuations (F_{P} > 1) for short cycle times, indicating that the quantum engine is unstable in this regime, with large relative power fluctuations. However, with increasing cycle time, the power increases faster than its variance, leading to a decrease in relative fluctuations. The transition to a Poissonian statistics (F_{P} = 1) (red dashed line), with strongly reduced power fluctuations and significantly increased stability, is located approximately at maximum power. This behavior follows from the finite Hilbert space of the Cs machine and the saturation effect due to the existence of an upper energy level. Importantly, the latter effect causes even the absolute value of the power fluctuations to decrease after on average six collisions (Fig. 3c inset). Power fluctuations could, in principle, also become subPoissonian (F_{P} < 1), but this regime is not seen experimentally due to experimental imperfections.
Discussion
In conclusion, we have realized an endoreversible quantum Otto cycle using single Cs atoms interacting with a Rb bath. The key asset of this machine is the exquisite control over both the fewlevel engine and the atomic reservoir. This unique feature allows us not only to regulate and monitor the heat exchange between system and environment at the singlequantum level, but also to operate the quantum engine in a regime of high efficiency, large power output and small power output fluctuations. The produced work could in principle be extracted by, e.g., coupling the magnetic moment of another microscopic particle to the magnetic moment of the engine. In a magnetic field gradient, motion of the coupled microscopic particle will directly reveal the work performed. Our system provides a versatile experimental platform to elucidate fundamental new effects generated by quantum reservoir engineering, such as nonequilibrium atomic baths^{24,25} and squeezed baths^{26,27}, as well nonMarkovian heat reservoirs by reducing the size of the Rb cloud^{28,29}.
Methods
Experimental procedures
We start our experimental sequence by preparing an ultracold Rb gas in the magnetic field insensitive state \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=0\rangle\) and, at a distance of ≈200 μm, a small sample of lasercooled Cs atoms. The Cs atoms are further cooled and optically pumped into the \({F}_{\text{Cs}}=3,{m}_{F,\text{Cs}}=3\rangle\) hyperfine ground state by employing degenerate Raman sidebandcooling^{30}. A speciesselective optical lattice^{31} transports the Cs atoms into the Rb cloud. MW radiation prepares the bath atoms in the state \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=1\rangle\). The starting point of the heat engine cycle is defined by switching off the optical lattice potential. After a predefined time t_{i}, the CsRb interaction is stopped by freezing the positions of the Cs atoms using the optical lattice, and pushing the Rb cloud out of the trap with a resonant laser pulse. Stateselective fluorescence imaging of the Cs atoms completes the procedure^{32}.
The highenergy and lowenergy baths are interchanged by transferring the Rb atoms from \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=1\rangle\) to \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=+1\rangle\) and vice versa using two successive Landau–Zener sweeps. The transfer takes ~4.4 ms, which is fast enough to avoid spinexchange interactions during the state change of the bath. The two magnetic fields B_{1} and B_{2} defining the quantization axis for the engine operation, are measured using Rb microwave spectroscopy on the \({F}_{\text{Rb}}=1,{m}_{F,\text{Rb}}=0\rangle \to {F}_{\text{Rb}}=2,{m}_{F,\text{Rb}}=+1\rangle\) transition. The population of the Rb atoms in state \({F}_{\text{Rb}}=2,{m}_{F,\text{Rb}}=+1\rangle\) is detected by standard absorption imaging, using a timeofflight measurement (Fig. 4). We fit the measured data with a standard model to extract the transition frequency, which translates into a magnetic field value using the BreitRabi formula^{33}. We find typical errors of the order of 0.1 mG.
The magnetic field changes extracting work of the engine have to be adiabatic, i.e., preserving the populations p_{n}. The adiabaticity condition writes \(\dot{{\omega }_{\text{lar}}}/{\omega }_{\text{lar}\,}^{2}\ll 1\), where \({\omega }_{\text{lar}}= {g}_{F}^{\text{Rb}} {\mu }_{\text{B}}B/\hslash\) is the Larmor frequency. It can, therefore, be expressed as
Experimentally, we linearly vary the magnetic field from B_{1} = 346.5 ± 0.2 mG to B_{2} = 31.6 ± 0.1 mG in a time scale of 10 ms, yielding values of A(B_{1}) = 0.2 × 10^{−3} and A(B_{2}) = 14 × 10^{−3}, thus fulfilling the adiabatic condition at any time during the variation of the magnetic field. Moreover, the time scale of the magnetic field variation is faster than the time scale associated with the spin exchange collisions (see number of collisions over time in Fig. 3). Hence, the populations p_{n} are constant during the isentropic processes (B → C and D → A).
Microscopic model and number of collisions
The quantum heat exchange between engine and bath is based on the understanding of individual spinexchange collisions. In general, the spincollision rate \({{{\Gamma }}}^{{m}_{F}\to {m}_{F}\pm 1}\) is different both for every initial state m_{F} and for the direction, i.e., Δm_{F} = ±1. The individual rates are well known from coupledchannel calculations of the molecular interaction potential between Rb and Cs^{14}. These rates allow us to describe the evolution with a rate model^{21} that captures the spin dynamics and yields excellent agreement with the experimental data. From these rates, we also compute the mean number of spin collisions N_{spin} within a cycle duration t = t_{D} in two steps. First, we calculate the timeaveraged collision rate as the sum of timeaveraged collision rates during heating (exothermal spin collisions) and cooling (endothermal spin collisions) as
Second, we integrate these rates during the heating and cooling to obtain the number of collisions within cycle time t as
In order to close the cycle, the inital and final Cs states before and after a cycle have to be the equal, leading to the condition N_{A→B} = N_{C→D}.
Efficiency of the endoreversibe machine
We calculate the efficiency by distinguishing two different forms of heat exchange. First, we consider the respective mean energies given (〈Q_{1}〉) and taken (〈Q_{2}〉) by the baths, where 〈Q_{1}〉 − ∣〈Q_{2}〉∣ is the energy turnover of the reservoirs per cycle. Second, we consider the average energies absorbed (〈Q_{H}〉) and rejected (〈Q_{C}〉) from the engine, where 〈Q_{H}〉 − ∣〈Q_{C}〉∣ is the energy turnover of the machine. Both quantities differ because of the different atomic Landé factors of Cs and Rb. The difference \(\langle {Q}_{\text{L}}\rangle =\left(\langle {Q}_{1}\rangle  \langle {Q}_{2}\rangle  \right)\left(\langle {Q}_{\text{H}}\rangle  \langle {Q}_{\text{C}}\rangle  \right)\) is dissipated via elastic collisions and irreversibly lost to the kinetic energy of Rb. We macroscopically model it as a heat leak from the highenergy reservoir. Using the population distribution of the quasispin levels at the cycle points in Fig. 2 of the main text, the individual heats can be calculated, leading to
Owing to preservation of populations during adiabatic strokes, we can further use \({p}_{n}^{\,\text{D}}={p}_{n}^{\text{A}\,}\) and \({p}_{n}^{\,\text{B}}={p}_{n}^{\text{C}\,}\), yielding the expression for the dissipated heat
The efficiency is calculated as the work, ∣〈W〉∣ = 〈Q_{H}〉 − ∣〈Q_{C}〉∣, produced by the engine, divided by the energy provided by the highenergy bath, 〈Q_{H}〉 + 〈Q_{L}〉. Using \({p}_{n}^{\,\text{D}}={p}_{n}^{\text{A}\,}\), \({p}_{n}^{\,\text{B}}={p}_{n}^{\text{C}\,}\) and γ = λ/κ, we find
The internal efficiency of the engine is computed as the ratio of the produced work ∣〈W〉∣ and the heat absorbed by the machine 〈Q_{H}〉:
It corresponds to the efficiency without a leak (γ = 1).
Fluctuations of the quantum machine
To extract the fluctuations of the engine, Eq. (4), we calculate the mean power, Eq. (3), via 〈P〉 = ∣〈W〉∣/τ_{cycle}. The cycle time τ_{cycle} = t_{D} is experimentally controlled, and we assume that it is a fixed parameter not adding further fluctuations to the poweroutput fluctuations. Therefore, we can restrict the calculation to the fluctuations σ_{W} of work 〈W〉 as \({\sigma }_{W}^{2}=\langle {W}^{2}\rangle {\langle W\rangle }^{2}\). The work is given by the difference of energy absorbed by and rejected from the engine ∣〈W〉∣ = 〈Q_{H}〉 − ∣〈Q_{C}〉∣, and hence
The averages and variances of heat absorbed or rejected depend on the energy differences at the different points during the cycle, for example, 〈Q_{H}〉 = E(t_{B}, B_{1}) − E_{0}(t_{0}, B_{1}). Here, \(E({t}_{i},{B}_{j})={\sum }_{n}{p}_{n}^{i}({t}_{i})\ n\lambda {B}_{j}\) can be computed from the measured populations \(\{{p}_{n}^{i}\}\) of level n at point i = A, B, C, D during the cycle and the magnetic field B_{j}(j = 1, 2), together with mean energy and variance. Then, the fluctuations \({\sigma }_{Q}^{2}\) of heat 〈Q〉 exchanged when changing the engine’s probability distribution from point i to point f at a magnetic field B_{j} reads
where, using the notation of Fig. 1b, for 〈Q_{H}〉i = 0, f = B, and B_{j} = B_{1}, and for 〈Q_{C}〉i = C, f = D, and B_{j} = B_{2}. Inserting these expressions into Eq. (12) allows us to compute the work fluctuations for every cycle time τ_{cycle} = t_{A} and thereby the variance of the output power fluctuations \({\sigma }_{P}^{2}\).
Data availability
The data that support the plots and findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank E. Tiemann for providing us with the scattering crosssections underlying our numerical model, and T. Busch and J. Anglin for helpful comments on the manuscript. This work was funded by Deutsche Forschungsgemeinschaft via Sonderforschungsbereich (SFB) SFB/TRR185 (Project No. 277625399) and Forschergruppe FOR 2724.
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Q.B. and J.N. contributed equally to this work. A.W. and Q.B. conceived the experiment and supervised the project. Q.B., J.N., S.B., and D.A. performed the experimental measurements. J.N., S.B., and Q.B. contributed to the microscopic numerical model. E.L. provided the theoretical thermodynamic explanation. All authors contributed to analysis and interpretation of the data, and writing of the manuscript.
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Bouton, Q., Nettersheim, J., Burgardt, S. et al. A quantum heat engine driven by atomic collisions. Nat Commun 12, 2063 (2021). https://doi.org/10.1038/s4146702122222z
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DOI: https://doi.org/10.1038/s4146702122222z
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