Abstract
A quantum heat engine (QHE) based on the interaction driving of a manyparticle working medium is introduced. The cycle alternates isochoric heating and cooling strokes with both interactiondriven processes that are simultaneously isochoric and isentropic. When the working substance is confined in a tight waveguide, the efficiency of the cycle becomes universal at low temperatures and governed by the ratio of velocities of a Luttinger liquid. We demonstrate the performance of the engine with an interacting Bose gas as a working medium and show that the average work per particle is maximum at criticality. We further discuss a work outcoupling mechanism based on the dependence of the interaction strength on the external spin degrees of freedom.
Introduction
Universality plays a crucial role in thermodynamics, as emphasized by the description of engines, that transform heat and other resources into work. The working substance can often be described as an ideal gas and the control of interparticle interactions or the equation of state is generally beyond reach. As a result, the role of the working substance and the presence of manyparticle effects are secondary in paradigmatic cycles (Carnot, Otto, etc.).^{1}
In the quantum domain, this state of affairs should be revisited as suggested by recent works on manyparticle quantum thermodynamics. The quantum statistics of the working substance can substantially affect the performance of a quantum engine.^{2} The need to consider a manyparticle thermodynamic cycle arises naturally in an effort to scaleup thermodynamic devices^{3,4,5,6} and has prompted the identification of optimal confining potentials^{7} as well as the design of superadiabatic protocols,^{8,9,10,11} first proposed in a singleparticle setting.^{12,13,14} The realization of superadiabatic strokes with ultracold atoms using a Fermi gas as a working substance has been reported.^{10,11,15} Intrinsically manyparticle effects in the working substance, such as the divergence of energy fluctuations near a secondorder phase transition^{16} and manybody localization,^{17} can prove advantageous. Manyparticle thermodynamics can lead to a quantumenhanced behavior, whereby a single manyparticle cycle can outperform an ensemble of independent singleparticle machines.^{8,18}
Quantum technologies have also uncovered avenues to design thermodynamic cycles. Traditionally, interactions between particles are generally considered to be “fixed by Nature” in condensed matter. However, a variety of techniques makes possible to modify interparticle interactions in different quantum platforms. A paradigmatic example is the use of Feschbach and confinementinduced resonances in ultracold atoms.^{19} The engineering of interparticle interactions is similarly at reach in digital quantum simulation in trapped ions, Rydberg atoms, and superconducting qubits.^{20}
We introduce a thermodynamic cycle that exploits the manyparticle nature of the working substance and thus have no singleparticle counterpart. It consists of four isochoric strokes, alternating heating and cooling with a modulation of the interparticle interactions. Using Luttinger liquid theory, the efficiency of the cycle is shown to be universal in the lowtemperature regime of a onedimensional (1D) working substance, i.e. solely depending on the sound velocities at the states A and B (Fig. 1). The universal efficiency of the cycle is also shown to characterize the regime of strong interactions in the working substance, in which the spectrum is approximately scale invariant. When an interacting Bose gas is used as such, the average work output per particle is maximum at criticality.
Results
Interactiondriven thermodynamic cycle
Consider a quantum heat engine (QHE) with a working substance consisting of a lowdimensional ultracold gas tightly confined in a waveguide.^{25,26} Ultracold gases have previously been explored in quantum cycles where work is done via expansion and compression processes, both in the noninteracting^{7,27} and interacting regimes.^{8,28,29,30,31,32} We propose the implementation of a quantum cycle consisting of four isochoric strokes, in which heating and cooling strokes are alternated with isentropic interactiondriven processes. In the latter, work is done onto and by the working substance by increasing and decreasing the interatomic interaction strength, respectively. This work can be transferred to other degrees of freedom as we shall discuss below. The working substance consists of \(N\) particles with interparticle interactions parameterized by the interaction strength \(c\). Both the particle number \(N\) and the system size \(L\) are kept constant throughout the cycle and any equilibrium point is parameterized by a point \((c,T)\) indexed by the temperature \(T\) and the interaction strength \(c\). Specifically the interactiondriven quantum cycle, shown in Fig. 1 for the 1D LiebLinger gas,^{21,22} involves the following strokes:
(1) Interaction rampup isentrope (A \(\to\) B): The working substance is initially in the thermal state \(A\) parameterized by \(({c}_{{\rm{A}}},{T}_{{\rm{A}}})\) and decoupled from any heat reservoir. Under unitary evolution, the interaction strength is enhanced to the value \({c}_{{\rm{B}}}\) and the final state is nonthermal. (2) Hot isochore (B \(\to\) C): Keeping \({c}_{{\rm{B}}}\) constant, the working substance is put in contact with the hot reservoir at temperature \({T}_{{\rm{C}}}\) and reaches the equilibrium state \(({c}_{{\rm{B}}},{T}_{{\rm{C}}})\). (3) Interaction rampdown isentrope (C \(\to\) D): The working substance is decoupled from the hot reservoir and performs work adiabatically while the interaction strength decreases from \({c}_{{\rm{B}}}\) to \({c}_{{\rm{A}}}\), reaching a nonthermal state. (4) Cold isochore (D \(\to\) A): The working substance is put in contact with the cold reservoir with temperature \({T}_{{\rm{A}}}\) keeping the interaction strength constant until it reaches the thermal state \(({c}_{{\rm{A}}},{T}_{{\rm{A}}})\). The work extracted from the heat engine is given by \(W={W}_{3}{W}_{1}={Q}_{2}{Q}_{4}\) while the efficiency of the heat engine reads
Interacting Bose gas as a working substance
Consider as a working substance an ultracold interacting Bose gas tightly confined in a waveguide,^{33,34} as realized in the laboratory.^{35,36,37} The effective Hamiltonian for \(N\) particles is that of the LiebLiniger model:^{21,22}
where \({x}_{j\ell }={x}_{j}{x}_{\ell }\), with \(2m=\hslash =1\). The spectral properties of the Hamiltonian of Eq. 2 can be found using coordinate Bethe ansatz.^{21,22} We consider a boxlike trap^{38,39} where any energy eigenvalue can be written as \(E={\sum }_{i}{k}_{i}^{2}\) in terms of the ordered quasimomenta \(0<{k}_{1}<{k}_{2}<\cdots <{k}_{N}\). The latter are the (Bethe) roots \(\{{k}_{i}\}\) of the coupled algebraic equations
determined by the sequence of quantum numbers \(\{{I}_{i}\}\) with \(i=1,\ 2,\ \cdots \ ,\ N\). As a function of \(c\) and \(T\), the 1D Bose gas exhibits a rich phase diagram. We first consider the regime of strong interactions and use a Taylor series expansion in \(1/c\). For a given set of quantum numbers \({{\mathcal{I}}}_{n}=\{{I}_{i}^{(n)}\}\), the corresponding energy eigenvalue takes the form
where the interactiondependent factor \({\lambda }_{c}\) reads
The spectrum of a strongly interacting Bose gas is thus characterized by eigenvalues with scaleinvariant behavior, i.e., \({\epsilon }_{n}(c)/{\epsilon }_{n}(c^{\prime} )={\lambda }_{c}/\lambda_{c} ^{\prime}\). This scale invariance in the spectrum allows us to assign an effective temperature along the isentropes; see Supplementary Information Section I. In this regime, the work output is thus set by \(W={Q}_{2}{Q}_{4}=[1({\lambda }_{{c}_{{\rm{A}}}}/{\lambda }_{{c}_{{\rm{B}}}})]{Q}_{2}\) and the efficiency
becomes independent of temperature of the heat reservoirs, as we show in Supplementary Information. Being the spectrum scaleinvariant, the efficiency can be expressed in terms of the scaling factor.^{40} This feature is as well shared by the wellknown harmonic Otto cycle.^{41} This is a consequence of the fact that both the cycles involve the driving of an isolated quantum system with a scaleinvariant energy spectrum. However, the interactiondriven cycle generally involves a change of density of states which can be regarded as the change of the generalized exclusion statistics in an ideal gas;^{42} also see Supplementary Information Sections II and III. Moreover, the scale invariant character of the energy spectrum in the strongly interacting regime is also present for multicomponent Bose and Fermi gases, preserving the universal efficiency given by Eq. 6.
For weaker interactions, we resort to a numericallyexact solution of Eqs. 3 for finite particle number \(N\). We enumerate all the possible sets \({{\mathcal{I}}}_{n}\) of quantum numbers for lowenergy states and solve Eqs. 3 numerically for given \(c\). With the resulting quasimomenta \(\{{k}_{n,1},{k}_{n,2},\cdots \ \}\), and the corresponding energy eigenvalues \({\epsilon }_{n}={\sum }_{i}{k}_{n,i}^{2}\), the probability that the Bose gas at temperature \(T\) is found with energy \({\epsilon }_{n}\) is set by the Boltzmann weights \({p}_{n}={e}^{{\epsilon }_{n}/T}/{\sum }_{m}{e}^{{\epsilon }_{m}/T}\) (with \({k}_{{\rm{B}}}=1\)). The equilibrium energy of the states \(A\) and \(C\) is set by thermal averages of the form \(\langle E\rangle ={\sum }_{n}{p}_{n}{\epsilon }_{n}\) that in turn yield the expressions for \({Q}_{2}\) and \({Q}_{4}\). Here, a proper cutoff of the possible sets \(\{{{\mathcal{I}}}_{n}\}\) can be determined by \({p}_{n}/{p}_{{\rm{G}}}\ll 1\), where \({p}_{{\rm{G}}}\) is the probability for the working substance to be in the ground state. The numerical results for the efficiency \(\eta\) and output work \(W\) are shown in Fig. 2 as a function of the interaction strength. For fixed values of \({T}_{{\rm{C}}}\) and \({T}_{{\rm{A}}}\), the maximum work is studied as a function of \({c}_{{\rm{A}}}\) while keeping \({c}_{{\rm{B}}}\) constant (Fig. 2). The efficiency is well reproduced by Eq. 6 at strong coupling, that captures the monotonic decay with increasing interaction strength. The efficiency is found to be essentially independent of the temperature in the strong interaction regime, whereas the work output is governed by the temperature and interaction strength.
Universal efficiency at low temperature
In the thermodynamic limit (where \(N\) and \(L\to \infty\) with \(n=N/L\) being kept constant), the equilibrium state of the 1D Bose gas is determined by the YangYang thermodynamics^{23,24,42,43,44} (see “Methods”). At low energy, excitations are phonons and the behavior is described by the TomonagaLuttinger liquid (TLL) theory,^{45,46,47,48,49} in which the free energy density reads
where \({{\mathcal{E}}}_{0}\) is the energy density of the ground state, \({v}_{{\rm{s}}}\) is the sound velocity which depends on particle density \(n\) and interaction \(c\). The entropy density \(s\) can be obtained as the derivative of free energy, \(s=\partial {\mathcal{F}}/\partial T=\pi T/3{v}_{{\rm{s}}}\). The expression for the heat absorbed and released are respectively given by energy differences
where \({s}_{i}=\pi {T}_{i}/3{v}_{s}^{i}\) and \({v}_{{\rm{s}}}^{i}\) with \({v}_{{\rm{s}}}^{{\rm{B}}}={v}_{{\rm{s}}}^{{\rm{C}}}\) and \({v}_{{\rm{s}}}^{{\rm{A}}}={v}_{{\rm{s}}}^{{\rm{D}}}\) denote the entropy density and the sound velocity of the state \(i\), respectively. Using the fact that the strokes (1) and (3) are isentropes, it follows that
where \({T}_{{\rm{B}}}\) and \({T}_{{\rm{D}}}\) are the temperatures at states B and D, respectively. As a result, the efficiency and work output are given by
where \(\kappa ={T}_{{\rm{A}}}/{T}_{{\rm{C}}}\). Since \({T}_{{\rm{A}}}\;<\;{T}_{{\rm{B}}}\;<\;{T}_{{\rm{C}}}\), we have \(0\;<\;\kappa \;<\;\xi \;<\;1\). This efficiency (11) is universal regardless the microscopic physics of the working substance with a lowenergy TLL description. The work output for fixed \({T}_{{\rm{A}}}\) and \({T}_{{\rm{C}}}\) is maximized at \(\xi ={\xi }_{{\rm{c}}}\) as shown in Supplementary Information Section IV,
As the TLL theory describes the universal lowenergy behavior of 1D manybody systems, Eqs. 11–13 provide a universal description of the efficiency and work of QHE with a 1D interacting working substance at low temperatures, which are applicable to any cycle in which work and heat exchange occurs in different strokes. In particular, in the strongly interacting regime, the sound velocity of 1D Bose gases is given by \({v}_{{\rm{s}}}\simeq 2\pi n(14n{c}^{1}+12{n}^{2}{c}^{2})\).^{24} In this regime, the result 11 thus reduces to Eq. 6. On the other hand, in the weak interaction regime, \({v}_{{\rm{s}}}\simeq 2n\sqrt{\frac{c}{n}\frac{1}{2\pi }{\left(\frac{c}{n}\right)}^{\frac{3}{2}}}\),^{24} and thus the efficiency reaches the asymptotic value
indicating an enhancement of the performance with respect to the strongly interacting case.
Quantum critical region
We next focus on the performance of an interactiondriven QHE across the phase diagram of the 1D Bose gas and characterized the role of quantum criticality.^{50} The 1D Bose gas displays a rich critical behavior as a function of the temperature \(T\) and chemical potential \(\mu\) (Fig. 3). In the region of \(\mu /T\ll 0\), i.e. the mean distance between atoms is much larger than the thermal wave length \({\lambda }_{{\rm{th}}}=\sqrt{2\pi {\hslash }^{2}/(m{k}_{{\rm{B}}}T)}\), and the system behaves as a classical gas (CG). A quantum critical region (QC) emerges between two critical temperatures (see the white dashed lines in Fig. 3c) fanning out from the critical point \({\mu }_{{\rm{c}}}=0\). In this region, the energy gap closes universally as \(\Delta \sim  \mu {\mu }_{{\rm{c}}}{ }^{z\nu }\), in terms of the correlation length and dynamic critical exponents \(\nu =1/2\) and \(z=2\). The TLL region characterized by a phononic spectrum is found with \(\mu >0\) and \(T\) below the right critical temperature.
For fixed cycle parameters (\({c}_{{\rm{A}}}\), \({c}_{{\rm{B}}}\), \({T}_{{\rm{A}}}\), and \({T}_{{\rm{C}}}\)), we study the performance of the engine across the QC region by changing \(n\). We numerically calculate the efficiency \(\eta\) and average work \(W/N\) by using the TBA equation 16, and show that near the QC region \(W/N\) has a maximum value; see Fig. 3. We set \({c}_{{\rm{A}}}=1\), \({c}_{{\rm{B}}}=3\), \({T}_{{\rm{A}}}=1\), and \({T}_{{\rm{C}}}=5\) for the heat engine and let the density \(n\) increase from \(0.1\) to \(23\). The red, green, and blue dashed lines in Fig. 3 correspond to the densities \(n\simeq 0.3\), \(1.4\), and \(3.0\), respectively. In order to understand the maximum of \(W/N\), we also plot these three engine cycles (A \(\to\) B \(\to\) C \(\to\) D \(\to\) A) in the phase diagram of specific heat in the \(T\mu\) plane; see Fig. 3 (c). The performance of the cycle is optimal at the critical region (green dashed line). Specifically, when the stroke A \(\to\) B is near the TLL boundary and the stroke C \(\to\) D is in the critical region, the average work output per particle is maximized. In this scenario, the two adiabatic processes (A \(\to\) B and C \(\to\) D) pass through zones in which the change in the energy dispersion is maximized, as shown in Fig. 3c. Qualitatively, the process A \(\to\) B passes though the region where the specific heat is maximum, whereas the C \(\to\) D evolves through the region with lowest specific heat. In the adiabatic process, the particle number \(N\) and entropy \(S\) are all fixed. Figure 3d shows the work for the different cycles, where the work for each adiabatic process is given by \(W={\int }_{{c}_{{\rm{initial}}}}^{{c}_{{\rm{final}}}}\left(\partial E/\partial c\right)dc\). Even if the process is not rigorously adiabatic,^{51} this maximum work output still holds, as further discussed in Supplementary Information Section IV.
Discussion
The modulation of the interaction strength \(c\) is associated with the performance of work, that has recently been investigated for different working substances.^{52,53,54} An experimentallyrealizable work outcoupling mechanism can be engineered when the coupling strength depends on the configuration of other degrees of freedom. This is analogous to the standard Carnot or Otto cycles where the working substance is confined, e.g., in a boxlike trap.^{41} The latter is endowed with a dynamical degree of freedom that is assumed to be slow (massive) so that in the spirit of the BornOppenheimer approximation it can be replaced by a parameter. Similarly, the modulation of the coupling strength in an interactiondriven cycle can be associated with the coupling to external degrees of freedom.
As an instance, consider the choice of an interacting SU(2) 1D spinor Fermi gas in a tight waveguide as a working substance. The Hamiltonian can be effectively mapped to the LiebLiniger model (see ref. ^{55} and Supplementary Information Section V), with a coupling strength \(c=c(\langle {\hat{{\bf{S}}}}_{j}\cdot {\hat{{\bf{S}}}}_{\ell }\rangle )\) that depends on the spinor degrees of freedom,^{55,56} an approximation corroborated by the exact solution in a broad range of parameters.^{57} Another example relies on the use of confinementinduced resonances^{19,58} that govern the scattering properties of a quasi1D working substance in a waveguide. The transverse frequency \({\omega }_{\perp }\) of the latter directly determines the interaction strength \(c\), i.e., \(c=c({\omega }_{\perp })\), and the role of \({\omega }_{\perp }\) parallels that of the trap parameters in the conventional Carnot and Otto cycles. The possibility of extracting work from quench dynamics was recently studied.^{59,60}
Finally, we note that an interactiondriven cycle can also be used to describe QHEs in which the interactiondriven strokes are substituted by processes involving the transmutation of the particle quantum exchange statistics, e.g., a change of the statistical parameter of the working substance. The Hamiltonian of Eq. 2 can be used to describe 1D anyons with pairwise contact interactions with coupling strength \(\tilde{c}\) and statistical parameter \(\theta\) characterizing the exchange statistics, smoothly interpolating between bosons and fermions.^{61,62} The spectral properties of this LiebLiniger anyons can be mapped to a bosonic LiebLiniger model given by Eq. 2 with coupling strength \(c=\tilde{c}/\cos (\theta /2)\).^{61,62} The modulation of \(c\) can be achieved by the control of the particle statistics, tuning \(\theta\) as proposed in ref. ^{63}
In summary, we have proposed an experimental realization of an interactiondriven QHE that has no singleparticle counterpart: it alternates heating and cooling strokes with processes that are both isochoric and isentropic and in which work is done onto or by the working substance by changing in the interatomic interaction strength. This cycle can be realized with a Bose gas in a tight waveguide as a working substance. Using Luttinger liquid theory, the engine efficiency has been shown to be universal in the low temperature limit, and set by the ratio of the sound velocities in the interactiondriven strokes. The optimal work can be achieved by changing the ratio of the sound velocity, e.g., by tuning the interaction strength. An analysis of the engine performance across the phase diagram of the Bose gas indicates that quantum criticality maximizes the efficiency of the cycle.
Our proposal can be extended to Carnotlike interactiondriven cycles in which work and heat are simultaneously exchanged in each stroke. Exploiting effects beyond adiabatic limit may lead to a quantumenhanced performance.^{8,18} The use of nonthermal reservoirs^{64,65,66} and quantum measurements^{67,68} constitutes another interesting prospect. Our results identify confined Bose gases as an ideal platform for the engineering of scalable manyparticle quantum thermodynamic devices.
Methods
In the YangYang thermodynamics of the 1D Bose gas, the pressure is given by
where the “dressed energy” \(\varepsilon (k)\) is determined by thermodynamic Bethe Anstatz (TBA) equation
The particle density \(n\) and the entropy density \(s\) can be derived from the thermodynamic relations
in terms of which the internal energy density reads \({\mathcal{E}}=p+\mu n+Ts\). Both interactiondriven strokes are considered to be adiabatic. As a result, the heat absorbed during the hot isochore stroke (B \(\to\) C) is given by \({Q}_{2}=L[{\mathcal{E}}({c}_{{\rm{B}}},{T}_{{\rm{C}}}){\mathcal{E}}({c}_{{\rm{B}}},{T}_{{\rm{B}}})]\), while the heat released during the cold isochore (D \(\to\) A) equals \({Q}_{4}=L[{\mathcal{E}}({c}_{{\rm{A}}},{T}_{{\rm{D}}}){\mathcal{E}}({c}_{{\rm{A}}},{T}_{{\rm{A}}})]\). Here \({T}_{{\rm{B}}}\) and \({T}_{{\rm{D}}}\) can be determined in terms of the isentropic strokes, in which \(s({c}_{{\rm{A}}},{T}_{{\rm{A}}})=s({c}_{{\rm{B}}},{T}_{{\rm{B}}})\) and \(s({c}_{{\rm{B}}},{T}_{{\rm{C}}})=s({c}_{{\rm{A}}},{T}_{{\rm{D}}})\), where \({T}_{{\rm{A}}}\) and \({T}_{{\rm{C}}}\) are the temperature of the cold and hot reservoir, respectively. The efficiency and work can then be obtained by numerically solving the TBA equation. This article was previously published as a preprint.^{69}
Data availability
All data generated during this study are included in the supplementary information files.
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Acknowledgements
We thank F. J. GómezRuiz and Zhenyu Xu for comments on the manuscript. Funding support from the John Templeton Foundation, the National Key R&D Program of China No. 2017YFA0304500 and the key NSFC grant No. 11534014 and NSFC grant No. 11874393, is further acknowledged. G.W. was supported by the Zhejiang Provincial Natural Science Foundation Key Project (Grant No. LZ19A050001), by NSF of China (Grant No. 11674283), by the Fundamental Research Funds for the Central Universities (2017QNA3005, 2018QNA3004), by the Zhejiang University 100 Plan, and by the Thousand Young Talents Program of China. Y.Y.C. was supported by the China Postdoctoral Science Foundation Grant (No. 2019M650145).
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D.C. conceived the project. Y.Y.C. and X.W.G. led its theoretical development. Y.Y.C. and Y.C.Y. performed the numerical analysis. All authors contributing to completing the development of the project, analyzing the results, and writing the manuscript.
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Correspondence to XiWen Guan or Adolfo del Campo.
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Chen, Y., Watanabe, G., Yu, Y. et al. An interactiondriven manyparticle quantum heat engine and its universal behavior. npj Quantum Inf 5, 88 (2019). https://doi.org/10.1038/s4153401902045
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