Out-of-equilibrium operation of a quantum heat engine: The cost of thermal coupling control
Michael Wiedmann Jürgen T. Stockburger Joachim Ankerhold








Institute for Complex Quantum Systems and IQST, University of Ulm, 89069 Ulm, Germany





Real quantum heat engines lack the separation of time and length scales that is characteristic for classical engines. They must be understood as open quantum systems in non-equilibrium with time-controlled coupling to thermal reservoirs as integral part. Here, we present a systematic approach to describe a broad class of engines and protocols beyond conventional weak coupling treatments
starting from a microscopic modeling. For the four stroke Otto engine the full dynamical range down to low temperatures is explored and the crucial role of the work associated with the coupling/de-coupling to/from reservoirs in the energy balance is revealed. Quantum correlations turn out to be instrumental to enhance the efficiency which opens new ways for optimal control techniques. 


Out-of-equilibrium operation of a quantum heat engine:
 The cost of thermal coupling control
    Joachim Ankerhold
    December 30, 2023
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Introduction–Macroscopic thermodynamics was developed for very practical reasons, namely, to understand and describe the fundamental limits of converting heat into useful work. In ideal heat engines, components are always in perfect thermal contact or perfectly insulated, resulting in reversible operation. The work medium of real macroscopic engines is typically between these limits, but internally equilibrated, providing finite power at a reduced efficiency. Any reduction of engine size to microscopic dimensions calls even this assumption into doubt.


At atomic scales and low temperatures, quantum mechanics takes over, and concepts of classical thermodynamics may need to be modified <cit.>. This is not only of pure theoretical interest but has immediate consequences in the context of recent progress in fabricating and controlling thermal quantum devices <cit.>. While the first heat engines implemented with trapped ions <cit.> or solid state circuits <cit.> still operated in the classical regime, more recent experiments entered the quantum domain <cit.>. In the extreme limit, the work medium may even consist of only a single quantum object <cit.>.

Theoretically, one is thus faced with the fundamental challenge that a separation of time and length scales on which conventional descriptions of thermal engines is based, may no longer apply. This has crucial consequences: First, the engine's operation must be understood as a specific mode of the cyclic dynamics of an open quantum system with the coupling/de-coupling processes to/from thermal reservoirs being integral parts of the time evolution; second, thermal coupling strength and thermal times ħ/k_ B T at low temperatures T may match characteristic scales of the work medium.  The latter requires a non-perturbative treatment beyond standard weak-coupling approaches <cit.> to include medium-reservoir quantum correlations and non-Markovian effects <cit.>. The former implies that  coupling work W_I may turn into an essential ingredient in the energy balance whereas typically only exchanged heat Q and work for compression/expansion of the medium (driving work) W_d is adressed, see Fig. <ref>. Roughly speaking, while in a classical engine the cylinder is much bigger than the valve so that |W_d|≫ |W_I|, for a quantum device this separation of scales may fail and |W_d|∼ |W_I|. This is particularly true for scenarios to approach the quantum speed limit in cyclic operation <cit.>. Can a quantum engine under these conditions be operated at all? 




To adress this question, in this Letter we push forward a non-perturbative treatment and apply it to a finite-time generalization of the Otto cycle (Fig. <ref>). It is based on an exact mapping of the Feynman-Vernon path integral formulation <cit.> onto a Stochastic Liouville-von Neumann equation <cit.> which has been successfully applied before <cit.>. Here, we extend it to accommodate time-controlled thermal contact between medium and reservoirs and thus to arrive at a systematic treatment of quantum heat engines at low temperatures, stronger coupling and driving. Work media with either a single harmonic or anharmonic degree of freedom are discussed to make contact with current experiments. We demonstrate the decisive role of the coupling work W_I which inevitably must enter the energy balance. Its dependence on quantum correlations opens ways for optimal control <cit.>. 



Modeling–A quantum thermodynamic device with cyclic operation involving external work and two thermal reservoirs is described by the generic Hamiltonian

    H(t)=H_m(t)+  H_c + H_I, c(t) +  H_h + H_I, h(t) ,

where H_m, H_c/h denote the Hamiltonians of the work medium and the cold/hot reservoirs, respectively, with interactions H_I, c/h. Not only the working medium is subject to external control, but also the couplings—this is required
in a full dynamical description of the compound according to specific engine protocols. We consider a particle in a one-dimensional potential, H_m(t) = p^2/(2 m) + V(q,t), also motivated by recent ion-trap experiments <cit.>. Reservoirs are characterized not only by their temperatures T_c<T_h, but also by a coupling-weighted spectral density <cit.>. Assuming the free fluctuations of each reservoir to be Gaussian, these can be modeled in a standard way <cit.> as a large collection of independent effective bosonic modes with coupling terms of the form
H_I, c/h(t)=-λ_c/h(t) q∑_k c_k, c/h (b_k, c/h^†+b_k,c/h)+1/2 q^2 λ_c/h^2 (t) μ_c/h. The coefficients μ_c/h are conventionally chosen such that only the dynamical impact of the medium-reservoir coupling matters <cit.>. In a quasi-continuum limit the reservoirs become  infinite in size; thermal initial conditions are therefore sufficient to ascertain their roles as heat baths.

The dynamics of this setting will be explored over sufficiently long times such that a regime of periodic operation is reached, without limitations on the ranges of temperature, driving frequency, and system-reservoir coupling strength. The nature of the quantum states encountered, either as quantum heat engine (QH) or refrigerator (QR), is not known a priori. 

In order to tackle this formidable task, we start from the Feynman-Vernon path integral formulation <cit.>. It provides a formally exact expression for the reduced density operator ρ_m(t)= Tr_R{ρ_ tot(t)} of the working medium. The quantum correlation functions L_c/h(t-t')=⟨ X_c/h(t) X_c/h(t')⟩ with X_c/h = ∑_k c_k, c/h (b_k, c/h^†+b_k,c/h) are memory kernels of a non-local action functional, representing the influence of the reservoir dynamics on the distinguished system as a retarded self-interaction. This formulation can be exactly mapped onto a Stochastic Liouville-von Neumann equation (SLN) 
<cit.>, an approach which remains consistent in the regimes of strong coupling, fast driving, and low temperatures <cit.>, where master equations become speculative or inaccurate.



Here we extend an SLN-type method for ohmic dissipation <cit.>
to time-dependent control of the system-reservoir couplings, including stochastic representations of key reservoir observables <cit.>. The resulting dynamics is given by

    ρ̇_ξ(t)=-i/ħ[H_m(t),ρ_ξ]+ℒ_h[ρ_ξ]+ℒ_c[ρ_ξ]

which contains, in addition to terms known from the master equation of Caldeira and Leggett <cit.>, further terms related to the control of system-reservoir couplings and to finite-memory quantum noise ξ_ c/h,

    ℒ_α   =     - mγ_α/2ħ^2λ_α^2(t)[q, i{p,ρ_ξ}+ 2 k_ B T_α [q,ρ_ξ]]
          - i/ħλ_α(t){m γ_α/2ħλ̇_α(t)[q^2,ρ_ξ]- ξ_α(t) [q,ρ_ξ]}  .

Averaging over samples of the operator-valued process ρ_ξ(t) yields the physical reduced density ρ_m(t)=𝔼[ρ_ξ(t)]. The independent noise sources ξ_α(t) are related to the reservoir correlation functions through ⟨ξ_α(t)ξ_α(t')⟩ =  L_α(t-t')-2mγ_α/ħβ_αδ(t-t'). 


The time local Eqs. (<ref>) and (<ref>) thus provide a non-perturbative,  non-Markovian simulation platform  for quantum engines with working media consisting of single or few continuous or discrete (spin) degrees of freedom; different protocols can be applied with unambiguous identification of per-cycle energy transfers to work or heat reservoirs. Next, we will apply it to a four stroke Otto cycle.

Engine cycle–For this purpose, steering of both the time-dependent potential V(q, t) and time-dependent couplings λ_c/h(t) in an alternate mode is implemented, see Fig. <ref>. For simplicity, ohmic reservoirs with equal damping rate γ are assumed. A single oscillator degree of freedom represents the working medium as a particle moving in

    V(q,t)=1/2 m ω^2(t) q^2 + 1/4 m κ q^4

with a parametric-type of driving ω(t) and anharmonicity parameter κ≥ 0. We consider ω(t) varying around a center frequency ω_0  between ω_0±Δω/2, (Δω>0), within the time τ_d during the isentropic strokes of expansion (B → C) and compression (D → A); it is kept constant along the hot and cold isochores (A → B and C → D) (cf. Fig. <ref>). The isochore strokes are divided into an initial phase raising the coupling parameter λ_c/h from zero to one with duration τ_I, a relaxation phase of duration τ_R, and a final phase with λ_c/h→ 0, also of duration τ_I.

The cycle period is thus T = 4τ_I + 2τ_d + 2τ_R, as indicated in Fig. <ref>. The total simulation time covers a sufficiently large number of cycles to approach a periodic steady state (PSS)  with ρ_m(t)=ρ_m(t+T). Conventionally, one neglects what happens during τ_I;  one assumes that modulating the thermal interaction has no effect on the energy balance (see also Ref. <cit.>). In the quantum regime, such effects may, on the contrary, play a crucial role as will be revealed in the sequel. 



 



Periodic steady state–In Fig. <ref>(a-c) results are shown for a purely harmonic system, for which analytical results have been derived in limiting cases <cit.>. We use it as a starting point to refer to the situation in ion trap experiments <cit.> and to identify in (d) the role of anharmonicities. After an interval of transient dynamics (a), the elements of the covariance matrix  settle into a time-periodic pattern with damped oscillations near frequencies ω_0±Δω/2; the  time to reach a PSS typically exceeds a single period. The presence of qp-correlations manifests broken time-reversibility which implies that a description in terms of stationary distributions with effective temperatures is not possible.  Indeed, the PSS substantially deviates from a mere sequence of equilibrium states as also illustrated by the von Neumann entropy S_vN <cit.>.
Further insight is gained by taking the oscillator at its mean frequency ω_0 as a reference and employ the corresponding Fock state basis to monitor populations p_n(t)=⟨ n|ρ(t)|n⟩ and coherences ρ_nm(t)=⟨ n|ρ(t)|m⟩, n≠ m (b, c). Population from the ground and the first excited state is transferred to (from) higher lying ones during contact with the hot (cold) bath. In parallel, off-diagonal elements ρ_nm(t) are maintained. These are dominated by ρ_02(t) contributions according to the parametric-type of driving during the isentropic strokes. While this Fock state picture has to be taken with some care for dissipative systems, it clearly indicates the presence of coherences associated with qp-correlations in the medium <cit.>. The impact of anharmonicities for stiffer potentials in (<ref>) is depicted in (d). In comparison to the harmonic case, dynamical features display smoother traces with enhanced (reduced) variations in ⟨ p^2⟩ (⟨ q^2⟩). Non-equidistant energy level spacings may in turn influence the efficiency (see below).


Work and heat–The key thermodynamic quantities of a QH are work and heat per cycle. Note that even though we operate the model with a medium far from equilibrium, these quantities have a sound and unique definition in the context of fully Hamiltonian dynamics involving reservoirs of infinite size. An assignment of separate contributions of each stroke to heat and work is not needed in this context. Moreover, any such assignment in a system with finite coupling would raise difficult conceptual questions due to system-reservoir correlations <cit.>.


[id=JS]In the context of full system-reservoir dynamics, heat per steady-state cycle is uniquely defined as the energy change of the reservoirDuring the isochores heat is exchanged between medium and reservoirs with the heat 

    Q_c/h = - ∫_0^T dt  Tr{H_c/h ρ̇_ tot(t)} [id=JS].

[id=JS]released from a reservoir per cycle.
Within the SLN the integrand consists of separate terms associated with reservoir noise and medium back action <cit.>.

[id=JS]Similarly, workWork is [id=JS]obtainedmeasured as injected power, i.e., 

    W=∫_0^T dt  Tr{∂ H(t)/∂ tρ_ tot(t)}  ,

[id=JS]where separate driving and coupling work contributions W_d and W_I can be discerned from ∂_t H = ∂_t H_m + ∂_t H_I,c + ∂_t H_I,h.

    W_d=∫_0^T dt  Tr{∂ H_m(t)/∂ tρ_ tot(t)}  ,

while work associated with the coupling/de-coupling processes between medium and respective baths W_I, c/h (coupling work) during the isochores is obtained in the same way from H_I, c/h. Potential ambiguities in thermodynamic quantities arising from finite coupling energies ⟨ H_I, c/h⟩ <cit.> do not arise in a complete PSS cycle.[id=JS]While ambiguities in the definition of the internal energies of system and reservoirs can arise from finite terms ⟨ H_I, c/h⟩ <cit.>, these terms do not enter heat and work of a complete cycle. Careful analysis indicates <cit.> that the coupling work W_I is completely dissipated [This has been previously shown for a Markovian classical heat engine model <cit.>]. 

Figure <ref> (a) displays the strong coupling dependence of the net work W_d+W_I. It turns from negative (net work output) to positive, thus highlighting the coupling work W_I as an essential contribution in the work balance. The SLN approach allows to reveal its two parts, i.e. W_I=W_I, cl+W_I, qm, where the first one, determined by ⟨ q^2⟩, also exists at high temperatures while the second one is a genuine quantum part depending on qp-correlations. One can show that W_I, cl>0 dominates  while, with increasing compression rate Δω/ω_0, W_I, qm contributes substantially with a sign depending on the phase of the qp-correlations[id=JS] relative to the timing of the coupling control. By choosing τ_d,τ_I as in Fig. <ref>, one achieves W_I, qm<0, thus counteracting W_I, cl (b). In turn, control of qp-quantum correlations opens ways to tune the impact of W_I on the energy balance <cit.>. Heat Q_h, see (c),  follows a non-monotonous behavior with γ, also a genuine quantum effect that cannot be captured by standard weak coupling approaches. Its decrease beyond a maximum can be traced back to enhanced momentum fluctuations due to damping.



We are now in a position to discuss the ratio

    η = - W_d+W_I/Q_ h

which describes the efficiency of a QH if W_d+W_I<0[id=JS] since Q_h>0. 



In regimes where η is nominally negative, the system is not a QH, but merely a dissipator. The theory of the adiabatic Otto cycle and its extension using an adiabaticity parameter predicts some regimes of pure dissipation [See Eq. (6) of <cit.>.], however, without recognizing the coupling work W_I as an essential ingredient. As seen above, its detrimental impact can be soothed by quantum correlations, see (d). 

The combined dependence on γ and thermalization adiabaticity parameter ω_0τ_I yields a phase diagram pointing out QH phase (η>0) and a dissipator phase (η≤ 0) over a broad range of thermal couplings up to γ/ω_0∼ 1, Fig. <ref>. A QH is only realized if τ_I exceeds a certain threshold which grows with increasing medium-reservoir coupling. To make this more quantitative, progress is achieved for small compression ratios to estimate W_d and W_I and derive <cit.> from W_d+W_I<0 the relation

    ω_0τ_I > 2γ/Δω1+R/1-R  .

[id=JS]with R =(⟨ q^2⟩_A+⟨ q^2⟩_D)/(⟨ q^2⟩_C+⟨ q^2⟩_B)Here, 0≤ R≤ 1 is determined by the variances in position at the cycle times A–D in and where 0≤ R ≤ 1. Since W_d∼ω̇(t) τ_d∼Δω and W_I∼λ̇^2(t)τ_I∼ 1/τ_I, for short cycle times W_I always dominates.
Qualitatively, for the parameters in Fig. <ref>, very weak coupling γτ_R,γτ_I≪ 1 leads to (1-R)/γ≈ const. and thus ω_0τ_I≈ const.; for larger coupling with more efficient heat exchange, the γ dependence of R is less relevant <cit.> so that ω_0τ_I∼γ/Δω as in Fig. <ref>. 





As expected, values obtained for η are always below the Curzon-Ahlborn and the Carnot efficiencies <cit.>, but yet, even beyond weak coupling, they do exceed η∼ 0.2. The coupling work appears as an essential ingredient also to predict for the power output the optimal cycle time and correct peak height, Fig. <ref>(a). If it is ignored, misleading data are obtained. Beyond the harmonic case, i.e. for stiffer anharmonic potentials, dynamical features discussed in Fig. <ref>(d)  reduce the efficiency, Fig. <ref>b. They have a similar impact as  enhanced thermal couplings, both having the tendency to suppress (increase) fluctuations in position (momentum). 





In conclusion, by simulating non-perturbatively and within a systematic formulation the dynamics of quantum thermal machines with single degrees of freedom as work medium, we have obtained a complete characterization of their properties. The medium-reservoir boundary appears as an internal feature of the model so that full control over the medium as well as its thermal contact to reservoirs is possible. The example of the four stroke Otto engine demonstrates the decisive role of the coupling work that must be considered as an integral part of the total energy balance. Its overall impact is detrimental to the efficiency of quantum heat engines, however, can be reduced by quantum correlations if they are properly controlled. A simple example 
is shown in Fig. <ref>, where a slight change in the expansion stroke of the Otto engine's protocol modifies the phase of qp-correlations such that the opposite happens: coupling work is further enhanced and the efficiency further suppressed. This sensitivity of quantum heat engines to changes in the driving protocol can be exploited by optimal control techniques in future devices. The presented approach provides the required tools to follow theoretically these activities.

We thank R. Kosloff, E. Lutz, M. Möttönen, and J. Pekola for fruitful discussions. Financial support from the Land BW through the LGFG program (M.W.), the IQST and the German Science Foundation (DFG) through AN336/12-1 (J.A.) are gratefully acknowledged.




