Many-body quantum heat engines with shortcuts to adiabaticity

Quantum heat engines are modeled by thermodynamic cycles with quantum-mechanical working media. Since high engine efficiencies require adiabaticity, a major challenge is to yield a nonvanishing power output at finite cycle times. Shortcuts to adiabaticity using counter-diabatic (CD) driving may serve as a means to speed up such, otherwise infinitely long, cycles. We introduce local approximate CD protocols for many-body spin quantum heat engines and show that this method improves the efficiency and power for finite cycle times considerably. The protocol does not require a priori knowledge of the system eigenstates and is thus realistic in experiments.


I. INTRODUCTION
Heat engines are thermodynamic machines that cyclically convert heat into work [1]. Recently, the concept of heat engines has also been successfully applied in the quantum domain and constitutes an important and very active research direction within the emergent field of quantum thermodynamics, both theoretically [2][3][4][5][6][7][8] and experimentally [9][10][11][12][13][14]. The quantum counterparts of, e.g., the Otto cycle, are considered to consist of a quantum system that is cyclically put into contact with two (hot and cold) heat baths and a work reservoir [7].
A major difference between quantum and classical heat engines is the role of the adiabatic condition. Quantummechanically it does not suffice to implement the workexchange strokes (devoid of any dissipative coupling to the environment) in an isentropic fashion to make the heat-to-work conversion as efficient as possible. Much rather, these strokes must be adiabatic in the quantum sense. Quantum mechanically, a process is adiabatic if a system remains in its instantaneous eigenstate under an external change of the Hamiltonian, which requires the latter to be slow [15][16][17][18]. By contrast, fast changes would excite coherences in the system, i.e., the population of nondiagonal elements in its density matrix. Hence, while quantum mechanically an adiabatic process is always isentropic-as the system evolves in a unitary fashion according to the von Neumann equation-the converse, however, is not true.
In this work we propose a four-stroke (two isentropic and two thermal) many-body quantum Otto engine that is sped up by shortcuts to adiabaticity to deliver finite power at finite speed. The isentropic strokes are driven using a recently developed approximate local counterdiabatic (CD) Hamiltonian following Refs. [41] and [46]. The engine's working medium is an Ising spin chain with nearest-neighbor interactions. We analytically derive the expressions for the CD terms and numerically simulate the engine for up to eight spins. These simulations reveal a considerable enhancement of the performance of these STA heat engines compared to their finite-time, and therefore nonadiabatic, analogs with respect to both efficiency and power. We stress that the derivation of the additional approximate counter-diabatic term does not require a priori knowledge of the system eigenstates and can be implemented efficiently. We analyze the energetic balance of the cycle to gain further insight into the operational principles of the sped-up engine. Strikingly, we find that owing to the approximate nature of the STA the heat engine may be undesirably converted into a hybrid engine that is energized not only by heat but also by work stemming from the external control device that implements the CD drive. This paper is organized as follows. In Sec. II we explain the major properties of many-body quantum heat engines, in particular, the quantum Otto cycle, and describe our models for the many-body working medium. In Sec. III we introduce shortcuts to adiabaticity using local counterdiabatic driving and its application to the quantum Otto cycle. We discuss the operational meaning of these STAs in Sec. IV and numerically analyze the performance of  The working medium (spin chain) interacts with two thermal baths at temperatures Tc and T h , respectively. Work to the load is extracted via the time-dependent protocol H0(t) implemented by a work reservoir (the load). In the underlying quantum Otto cycle this protocol must ideally be adiabatic to avoid the occurrence of "quantum friction" that would reduce the engine efficiency. The infinite cycle time required by the adiabaticity condition causes vanishing output power (work per cycle divided by the cycle time). A "shortcut to adiabaticity" is realized if an additional controller implements the right additional counter-diabatic protocol H CD (t) on the working medium. The original (adiabatic) cycle is then significantly sped up such that the engine yields finite power. (b) Adiabatic quantum Otto cycle in the λ-entropy plane. It consists of two unitary (hence isentropic) strokes (1 and 3) with corresponding work W 1 ad and W 3 ad and two thermal strokes (2 and 4) with corresponding heat Qc and Q h , respectively. In the sped-up cycle the adiabatic protocol H0(t) is supported by an additional counter-diabatic drive H CD (t) implemented by an external control device.
the sped-up engine in Sec. V. In Sec. VI we conclude and give an outlook on future research.

A. Quantum Otto cycle
A four-stroke quantum Otto cycle [7] consists of two heat-exchange strokes, wherein the working medium (WM) is alternatingly coupled to two (hot and cold) ther-mal baths at temperatures T h and T c , respectively, and two work-exchange strokes. In the latter, the WM is isolated from the environments and its Hamiltonian H 0 (λ(t)) is externally controlled via the time-dependent working parameter λ(t). To operate as an engine (produce work), the adiabatic Otto cycle is traversed in the following order [see Fig. 1 Hamiltonian H 0 (λ c ), where λ c := λ(t = 0) and Z(λ c ) = Tr[e −βcH0(λc) ] is the partition function. The working parameter λ(t) is adiabatically increased from λ c to λ h := λ(t = τ 1 ) such that the populations of the instantaneous eigenstates of H 0 (λ(t)) remain invariant. At the end of the stroke the WM attains the state ρ B . Hence, the work is performed on the WM.
is imparted by the hot bath.
3. Stroke 3: Adiabatic expansion (C → D). The working parameter decreases adiabatically from λ h to λ c in the stroke time τ 3 and the WM attains the state ρ D . Hence, the work is extracted from the WM.
is transferred to the cold bath.
Note that engine operation (work extraction) corresponds to W 1+3 ad := W 1 ad + W 3 ad < 0. Here we use the expressions "compression" and "expansion" in analogy with the classical Otto engine where W 1 ad > 0 and W 3 ad < 0. Depending on the physical implementation, however, the roles of the two strokes (i.e., the signs of W 1 ad and W 3 ad ) may be interchanged [47].
The efficiency of this adiabatic cycle is the net work performed by the WM on the piston divided by the heat transferred from the hot bath to the WM, i.e., and is limited by the Carnot efficiency, η ≤ 1 − T c /T h . The power of the engine is given by the work done by the WM divided by the total cycle time τ cycle = 4 l=1 τ l . If the Hamiltonian does not commute with itself at all times, [H 0 (t), H 0 (t )] = 0 ∀t, t , the adiabaticity condition requires infinitely long durations of strokes 1 and 3, τ 1 , τ 3 → ∞. Consequently, in the adiabatic limit the power vanishes, which renders the engine practically useless.
The first way to circumvent this issue would be to apply the protocol H 0 (t) in a finite time, thus giving up the strict requirement for adiabatic compression or expansion. The price of these nonadiabatic dynamics is the occurrence of so-called "quantum friction" (excitation of coherences) [48][49][50], which reduces the output work. Being traversed in a finite time, this nonadiabatic engine yields finite power (recall that a negative sign indicates power output), This approach, however, has two caveats: (i) Quantum friction may significantly reduce the engine efficiency (as the work per cycle is reduced), and (ii) for too short cycle times the machine may cease to act as an engine, P na > 0, due to W 1+3 na := W 1 na + W 3 na becoming positive, which corresponds to work consumption rather than work extraction.
A possible solution of this dilemma is to introduce an external contol device that applies an additional counter-diabatic drive H CD (t) to the working medium [see Fig. 1(a)]. The resulting protocol H 0 (t) + H CD (t) is known as a shortcut to adiabaticity (STA) [19, 21-24, 26-32, 34]: It allows performance of the transformation (ρ A , H(λ c )) → (ρ B , H(λ h )) (and similarly for the third stroke) in finite time. This way the work per cycle equals its adiabatic counterpart but the cycle time is finite. Hence, the engine yields finite power while maintaining the ideal adiabatic efficiency.
Exact counter-diabatic protocols can be analytically derived in special cases, e.g., for a harmonic oscillator, a single spin, or two interacting spins [21,26,30,31,38,39,51,52]. By contrast, for a many-body working medium (see Appendix A for the single-body case) we have to rely on an approximate counter-diabatic drive. This, as we see, entails important operational consequences for the heat-engine operation.

B. Many-body quantum working medium
We consider an Ising spin chain with nearest-neighbor interactions and Hamiltonian where N is the total number of spins, h j (t) and b j (t) are the time-dependent strengths of the magnetic fields at site j in the x and z directions, respectively, and J j (t) is the time-dependent strength of the interaction between spins at sites j and j + 1, where we impose periodic boundary conditions, i.e., σ N +1 = σ 1 . Recent progress in controlling many-body quantum systems has made it possible to experimentally realize and study similar many-body Hamiltonians using quasi one-dimensional Ising ferromagnets [53,54] or cold atoms [55].
The abstract working parameters λ c and λ h in the transferred energies, Eq. (1), then correspond to The explicit forms of the magnetic fields h j (t) and b j (t) and the interactions J j (t) are given in Eqs. (B1) in Appendix B 1.
STAs have been successfully applied to QHEs with a single-body working medium [21,26,30,31,38,39]. It is thus a natural question whether they can also be applied to many-body quantum heat engines, where we have to rely on approximate solutions for the counterdiabatic drive. We note that we apply shortcuts only on the originally adiabatic strokes, as these are typically much slower than the thermalization strokes. Techniques such as shortcuts to equilibration for speeding up the dynamics of open quantum systems have recently been proposed [29,39,72,73].

A. Approximate counter-diabatic driving
For finite times, the original protocol H 0 (t) [here, Eq. (5)] induces a nonadiabatic (diabatic) evolution by generating coherences in the working medium ("quantum friction"). To avoid these coherences, shortcuts to adiabaticity are realized by evolving the working medium according to the Hamiltonian rather than only H 0 (t). The additional counter-diabatic Hamiltonian H CD (t) compensates those undesirable nonadiabatic effects [37]. The determination of its exact form requires a priori knowledge of the system eigenstates for all times, which, in the case of complex many-body working media, is impracticable for both numerical computations and experimental implementations (see Appendix C).
With this challenge in mind we resort to a recently proposed variational method for finding the counter-diabatic Hamiltonian [41,74] where A ϑ (t) is the so-called adiabatic gauge potential and ϑ(t) a control function. The goal is to find an approximate expression for the CD protocol H * CD (t) by making a local ansatz A * ϑ (t) that approximates the solution of

B. Otto cycle with counter-diabatic driving
For the original Hamiltonian, Eq. (5), we use the local ansatz to approximate the counter-diabatic Hamiltonian, Eq. (7), and where A * t (t) =θ(t)A * ϑ (t) is the adiabatic gauge potential with respect to time t. This ansatz consists of applying additional magnetic fields in the y direction for each spin. As shown in Appendix D the optimal solution for these fields evaluates to and thus the local shortcut-to-adiabaticity Hamitonian, Eq. (6), adopts the form where the asterisk denotes that the Hamiltonian is inexact.
which assures smoothness at the beginning and end of the strokes (see Appendix B 1 for more information). Since is an inexact, approximate counter-diabatic drive, the resulting states at points B and D in Fig. 1 will not exactly be ρ B and ρ D , respectively, but different states, ρ B and ρ D , with the same entropy but different energy. The reliability of H * CD (t), i.e., how well the target state is reached, can be greatly improved by a variation of the global strength parameter ϑ 0 in Eq. (11) [46].

IV. WORK UNDER STA PROTOCOLS
During the unitary strokes [strokes 1 and 3 in Fig. 1(b)] the dynamics of the working medium is governed by the time-dependent, Hamiltonian H STA (t). Consequently, the energy change of the WM corresponds to the total exchanged work [2,75,76], where j ∈ {1, 3} denotes the corresponding isentropic stroke and H STA (t) = H 0 (t)+H CD (t) [Eq. (6)] is the timedependent shortcut-to-adiabaticity Hamiltonian. If H 0 (t) and the counter-diabatic drive H CD (t) are implemented by two independent work reservoirs, the division of the total work, Eq. (13), into the two components, is operationally interpreted as the individual work components exchanged between the working medium (the spin chain) and the two work reservoirs. Physically, this would correspond to the situation where the additional field in the y direction in Eq. (10) is implemented by a second control unit (the external "controller"), independent of the one that implements the original protocol (the piston or "load") [cf. Fig. 1(a)]. Note, however, that in general W 0 is not the same work as in the original adiabatic Otto cycle since ρ(t) is now determined by H STA (t) rather than H 0 (t). As we see below, this crucially depends on how good the chosen, approximate STA protocol, Eq. (10), reproduces the ideal, exact STA protocol. In order to understand the consequences of the division, Eq. (14), on the operation of the Otto engine we must distinguish between the cases where over a cycle W 1+3 CD := W 1 CD + W 3 CD ≤ 0 (heat engine) and W 1+3 CD > 0 (thermomechanical engine). Irrespective of W 1+3 CD , the machine operates as an engine (produces useful work)

A. Heat-engine regime
If over a cycle W 1+3 CD ≤ 0, the machine works as a genuine heat engine that converts thermal energy into work. This work, however, is performed not only on the work reservoir that implements H 0 (t) but also on the work reservoir that implements H CD (t). Hence, not all the work performed by the engine is available to the load. Hence, the useful power generated by the engine is since W CD is not available to the load. Consequently, the engine efficiency, as experienced by the load, is Note that in this regime (where W 1+3 CD < 0) the finite-time engine is solely energized by the heat Q h > 0 stemming from the hot bath, which characterizes a genuine heat engine. This heat input is converted into the useful mechanical work output W 1+3 0 < 0. Our numerical simulations (Sec. V) show that the engine performing work on the control device is an artifact of inexact counter-diabatic driving, i.e., when the final state generated by H STA (t) in either stroke 1 or stroke 3 differs from the final state generated by the original protocol in the respective stroke: We currently do not have a general analytic proof but in our numerical simulations we could clearly observe that W 1+3 CD = 0 if the counter-diabatic term is exact (see the discussion of the single-body working medium in Appendix A), meaning that then W 1+3 0 equals its counterpart W 1+3 ad in the adiabatic Otto cycle in Sec. II A. This observation clearly demonstrates the importance of striving for a perfect counter-diabatic protocol to speed up the Otto cycle. For a many-body working medium, however, the exact protocol is typically hard to find analytically and, even if it is known, may be extremely challenging to implement in an actual experiment since it will be of a nonlocal nature. Namely, it will not only involve single-body terms as in Eq. (10) but higher-order terms, possibly up to complicated N -body interactions.
On the other hand, this feature enables us to optimize (to some extent) the counter-diabatic drive by trying to minimize |W 1+3 CD | experimentally.

B. Hybrid thermomechanical engine regime
If the counter-diabatic protocol is not exact, we may also encounter situations in which W 1+3 CD > 0 over a cycle, which has striking operational consequences for the engine: Rather than being a genuine heat engine that converts thermal energy into useful work, the machine now acts as a hybrid thermomechanical engine [77][78][79][80] that is powered by thermal energy Q h > 0 and an external battery that provides W 1+3 CD > 0. Nominally, W 1+3 0 may now strongly surpass its adiabatic counterpart but this work does not solely stem from converted thermal energy. Such sped-up engines could be compared to QHEs powered by nonthermal baths, e.g., squeezed-thermal baths, which are hybrid engines and as such are not bounded by the Carnot efficiency [77,78,80]. Naturally, despite its increased output power, speeding up a heat engine to the price of rendering it thermomechanically can be undesirable.
Whereas the power of such a hybrid engine is still given by Eq. (15), its efficiency differs from its heat-engine counterpart, Eq. (16), and reads Note that in this regime (where W 1+3 CD > 0) the finitetime engine is energized by the heat Q h > 0 stemming from the hot bath as well as by the work W 1+3 CD > 0 stemming from the external controller. This characterizes a hybrid thermomechanical engine. The combined heat and work input is converted into useful mechanical work output W 1+3 0 < 0. While the power of this hybrid engine formally appears to be the same as for the heat engine [Eq. (15)], the physical origin of W 1+3 0 strongly differs and its magnitude may strongly surpass its counterpart from the adiabatic Otto engine.
We note that the above considerations only apply to the case where the working medium interacts with two independent work reservoirs. If the counter-diabatic protocol H CD (t) is also implemented by the piston, the controller ceases to be an external resource. The division, Eq. (14), then becomes operationally irrelevant (even unmeasurable) and the machine operates as a genuine heat engine in either case with the total useful work output W 1+3 STA < 0 and the energy input Q h > 0. While in the nominal heatengine regime |W 1+3 STA | > |W 1+3 0 |, in the nominal hybrid regime |W 1+3 STA | < |W 1+3 0 | (we call these regimes "nominal" in the single-work-reservoir setup as they do not have an operational meaning). We note, however, that in most experimental setups the controller and load being two independent work reservoirs is probably the more natural situation.
Finally, despite being detrimental to the engine operation if negative, we stress that the notion of W 1+3 CD being the work exchanged between the working medium and the controller strongly differs from other cost quantifiers discussed in the literature [23,24,26,30,31,38,81,82]: First, these costs pertain to exact protocols (where W 1+3 CD = 0), and, second, they quantify the extra energy that is required to implement H CD (t) for a certain time, e.g., the required intensity of an electric field. These costs of course become larger the shorter the unitary strokes become (as the fields become stronger and stronger). Since these costs are very strongly implementation-dependent we do not discuss them further in this paper, but note that they may significantly reduce the efficiency of sped-up engines [82].

V. NUMERICAL PERFORMANCE ANALYSIS
As the main part of this work, we are interested in the performance of the sped-up many-body quantum Otto cycle. To this end we numerically compare the performance of Otto engines with [H * STA (t); Eq. (10)] and without [H 0 (t); Eq. (5)] the counter-diabatic drive H * CD (t) for a system size of N = 8 spins. Namely, we numerically integrate the von Neumann equations iρ STA (t) = [H * STA (t), ρ STA (t)] and iρ 0 (t) = [H 0 (t), ρ 0 (t)] for each isentropic stroke. After reaching points B and D in Fig. 1(b), with corresponding states ρ B and ρ D , respectively, the latter get thermalized in the two thermalization strokes until they reach the thermal states ρ C and ρ A at points C and A, respectively. The initial strengths of the magnetic fields at point A in Fig. 1(b) of the first isentropic stroke are h j,i = 0.5 and b j,i = 0 for each spin, respectively, with vanishing interactions, J j,i = 0. The final magnetic fields at point B in Fig. 1(b) are h j,f = 0 and b j,f = 1. In order to test the practical applicability of our local counter-diabatic term, Eq. (12) In order to further improve the shortcut-to-adiabaticity Hamiltonian, Eq. (10), we numerically optimize the free control parameter ϑ 0 in Eq. (11): For each instance of J j,f and isentropic duration time τ we determine this optimized parameter via an iterative numerical update until we maximize the success fidelity where ρ [ρ] are the final states at the end of each isentropic stroke with H * STA (t) [H 0 (t)]. We restrict the values of ϑ 0 to be in [0, 1] to keep the strengths of the additional magnetic fields Y j (ϑ 0 , t) at a reasonable level, i.e., not overwhelmingly larger than the other fields in H 0 (t). Since the counter-diabatic protocol for the third stroke is the time-reversed version of H * CD (t) for the first stroke we only need to optimize ϑ 0 for the latter and use the same value for the former. All simulations were implemented with QuTip 4.2 [83]. Figure 2 shows the power P, work components W , efficiency η, and success fidelity F of our many-body quantum Otto engine (i) with shortcuts to adiabaticity [Eq. (10)] and (ii) with the original nonadiabatic protocol [Eq. (5)] for different durations τ = τ 1 = τ 3 of the isentropic strokes. Figure 2(a) reveals that the finite-time cycle under STA always acts as an engine, i.e., it provides useful work (blue area where P < 0). The engine still works in the limit τ 1 , τ 3 → 0, where the cycle time τ cycle is dominated by thermalization. By contrast, the original protocol H 0 (t) becomes nonadiabatic and quantum friction impacts its operation. Indeed, for too fast isentropic strokes (τ 10) the final states ρ B and ρ D become so different from their adiabatic counterparts ρ B and ρ D that the cycle ceases to describe an engine-rather than delivering power it consumes power. Figure 2(b) presents an operational insight into the engine by depicting the work components W 1+3 0 [Eq. (14a)], attributed to useful work extracted by the piston (load), and W 1+3 CD [Eq. (14b)], stemming from the external control device. Since the counter-diabatic Hamiltonian is not exact a finite amount of work is exchanged between the WM and the control. Up to moderate stroke durations of τ 5 a part of the work generated from the heat input is directed into the control and is thus lacking for the piston, i.e., |W 1+3 0 | < |W 1+3 STA |. In this region [green-shaded area in Fig. 2(b)] the cycle operates as a finite-power heat engine (cf. Sec. IV A). By contrast, for 5 τ 30 (yellow-shaded area), the engine is of a hybrid thermomechanical nature where not only work stemming from the converted heat input but also work stemming from the controller is transferred to the piston, i.e., |W 1+3 0 | > |W 1+3 STA | (cf. Sec. IV B). Finally, the third (gray-shaded) area represents the adiabatic limit where the counter-diabatic term H * CD (t) is very small and thus the entire work W 1+3 STA ≈ W 1+3 0 is performed on the piston. Namely, the work done by the external control device naturally converges towards 0. A more detailed discussion of the work components in the individual strokes is given in Appendix B 2.
The efficiency η of the engine [Eq. (16)] for the greenshaded and Eq. (17) for the yellow-shaded area, respectively) is shown in Fig. 2(c). As expected, the efficiency of the nonadiabatic engine [governed by the protocol H 0 (t) for finite stroke duration] strongly decreases with decreasing stroke duration (red line). By contrast, the STA cycle keeps operating as an engine whose efficiency, albeit being lower than the adiabatic one, is still reasonably high and does not decrease further as the cycle time is further reduced. Figure 2(d) depicts the success fidelity F [Eq. (18)], i.e., the overlap between the final states ρ B and ρ B for the first isentropic stroke (point B) and ρ D and ρ D for the second isentropic stroke (point D), respectively. The strong decay of the fidelity F 1 = F (ρ B , ρ B ) after the first isentropic stroke for decreasing isentropic stroke times in the original Otto cycle coincides with the corresponding drops in power and efficiency. It, however, does not decay to 0, as in the quench limit τ → 0 the state barely changes such that F (ρ B , ρ B ) ≈ F (ρ A , ρ B ). By contrast, the fidelity F 3 = F (ρ D , ρ D ) of the third stroke remains close to unity for all τ due to T h being so high that the eigenstate populations are almost uniform. In the adiabatic limit the fidelity approaches unity, as expected (gray-shaded area).
The situation strongly changes in the presence of the counter-diabatic protocol H * CD (t): Except for a small dip in the thermomechanical-engine regime the fidelities remain close to unity for all times and both isentropic strokes. It is remarkable that this also holds in the limit τ → 0 but this limit requires strong magnetic fields for implementing the CD protocol. We further note that the fidelity behaves very similarly for the different instances of the interaction strength. In Appendix B 3 we show that the fidelity using our local method decreases with increasing standard deviation of J j,f . This indicates the need for higher-order counter-diabatic protocols, i.e., the addition of controlled spin interactions rather than only adding local magnetic fields, in such situations.
We expect a positive W 1+3 CD and thus the occurrence of the thermomechanical engine's being an artifact of the inexact counter-diabatic drive of the many-body working medium. In Appendix A we present the above analysis for a single-body quantum Otto cycle where an exact counter-diabatic term can be found (see also Ref. [31]).
Finally, Fig. 3 shows the power P [Eq. (15)] of our manybody quantum Otto cycle as a function of the system size N for both protocols for different isentropic stroke durations τ . Note that the original protocol operates as an engine only for τ 10, whereas the STA protocol also works for shorter stroke durations [cf. Fig. 2(a)]. It is shown that in either case and independent of τ , the power scales linearly with the number of spins. For future work it would be interesting to consider possible cooperative effects [84].

VI. DISCUSSION AND OUTLOOK
In this work we have presented a finite-time many-body quantum heat engine with finite power output. It is composed of four strokes (two isentropic and two thermal) and a spin system as its working medium. In its isentropic strokes, work is either extracted from or performed on the WM. The caveat of highly efficient but adiabatic cycles is their requirement for almost infinitely long cycle times, which results in vanishing power (work divided by cycle time). In the context of QHEs the detrimental effect of nonadiabatic (diabatic) evolution on the engine efficiency has been dubbed "quantum friction" [7,[48][49][50]. Its illustrative explanation is that the excitation of coherences costs an extra amount of energy but these coherences are "dissipated away" in the subsequent thermalization strokes in which the working medium is put into contact with a thermal bath. Hence, while in classical heat engines the non-reversibility of adiabatic strokes due to entropyincreasing friction causes the engine efficiency to drop, quantum-mechanically, the quantum nonadiabatic behavior may create quantum friction devoid of any change in the WM entropy. Mathematically, quantum friction can only occur if the control Hamiltonian does not commute with itself at different times [7].
A possible solution to overcome this problem of zero power output at finite cycle times is so-called shortcuts to adiabaticity (STA), which have been developed in the context of adiabatic quantum computation and have later also been applied in the field of quantum thermodynamics [19-32, 35, 36]. A major obstacle has been to find an easy-to-implement STA method for many-body systems since the additional counter-diabatic term normally requires a priori knowledge of the system eigenstates at all times. In this work we employ local CD driving [41,46] to a many-body QHE where the additional CD drive consists of adding a local magnetic field in the y direction to speed up the engine while minimizing the quantum friction. The latter is further reduced by an iterative variation of the free control parameter ϑ 0 for the magnetic-field strength.
To assess the performance of the many-body QHE we compare (i) the engine with the STA protocol, i.e, H * STA (t), and (ii) the engine without the STA protocol, i.e., H 0 (t). The sped-up QHE with STA shows large improvement in power output, efficiency, and success fidelity for finite times compared to the original, nonadiabatic QHE without STA, in particular, for short isentropic stroke durations where the QHE without STA ceases to work as an engine. However, for such short cycle times the additional magnetic field required by the STA may be much larger than the other fields. Namely, the dynamics of the working medium may then become dominated by the CD protocol H * CD (t). As our additional CD term is not exact, we have to take care of the energetic balance of the external controller that implements H * CD (t). In particular, we have to distinguish cases where the controller receives work from or provides work to the engine, respectively: If the controller receives the work W 1+3 CD , the QHE works as a genuine heat engine where only thermal energy is converted into mechanical work. By contrast, if the controller provides the work W 1+3 CD , the QHE works as a hybrid heat engine where thermal as well as mechanical work is converted into mechanical work. As we aim for a sped-up engine powered by heat rather than by an external battery, we want to avoid the latter case. The additional input work can be seen as an artifact of the inexact counter-diabatic drive and thus shows the importance of striving for an exact CD drive where the useful work is only done on the piston. This exact drive, however, may be very challenging to implement experimentally, as it may involve controlled many-body interactions rather than simply applying local additional magnetic fields on each spin. We note that this trade-off between the exactness of the protocol and its experimental realizability naturally occurs in the manybody case. By contrast, in the single-body case, exact and conceptually simple protocols can often be found. Note further, that these operational costs conceptually differ from the costs of implementing the CD Hamiltonian, for which different quantifiers have been suggested in the literature [23,24,26,30,31,38,81,82]. Incorporating the latter, the efficiency of our sped-up QHE at very short cycle times may decrease considerably. Implementation costs are highly system dependent and may thus be difficult to assess. By contrast, our operational approach is motivated by an engineer that (i) wants to measure the works W 0 and W CD , respectively, and (ii) can distinguish between work output and heat input, which is an intuitive way to define the efficiency of a heat engine.
For future research, we aim at proposing a sped-up many-body quantum Otto engine using an experimentallyfeasible lattice gauge architecture [85] and to apply our local counter-diabatic method to open many-body quantum systems. A further topic of interest is many-body quantum refrigerators, which may be sped up in analogy to the engines in this work. The many-body quantum Otto cycle presented in the text extends the ideas of a shortcut-to-adiabaticity quantum Otto cycle with a single-body working medium which we consider in this section (see also Ref. [31]). To this end we consider the single-spin Landau-Zener (LZ) model with Hamiltonian For the explicit forms of the magnetic fields h x (t) and b z (t) we chose The exact counter-diabatic term for Hamiltonian (A1) reads [31,52]   where the function f CD (t) = 1 2ḣ is shown in Fig. 4. Thus, the shortcut-to-adiabaticity Hamiltonian reads In analogy to Fig. 2 in Sec. V of the text, Fig. 5 shows the power P, work components W , efficiency η and success fidelity F for this single-body quantum Otto engine (i) with shortcuts to adiabaticity [Eq. (A5)] and (ii) with the original nonadiabatic protocol [Eq. (A1)] for different isentropic-stroke durations τ = τ 1 = τ 3 . Figure 5(a) shows that our STA cycle always works as an engine, i.e., it provides useful work (blue-shaded area) also for short cycle times. By contrast, the Otto cycle governed by the original protocol H 0 (t) is hampered by quantum friction and ceases to work as an engine for τ 25. As for the many-body engine in Fig. 2(a), for too short cycle times the final states ρ B and ρ D are so different from the adiabatic states ρ B and ρ D that the cycle consumes rather than delivers power. Figure 5( . Namely, the external controller optimally assists the piston (greenshaded area). The gray-shaded area corresponds to the adiabatic limit.  Finally, Fig. 2(d) depicts the success fidelity F [Eq. (18)] as the overlap between the final states ρ B and ρ B for the first isentropic stroke [point B in Fig. 1(b)] and ρ D and ρ D for the isentropic stroke 3 (point D), respectively. As the counter-diabatic term is exact, the corresponding fidelity is always unity. By contrast, as in Fig. 2(d) the fidelities in the original strokes decrease with decreasing cycle times and converge towards their quench values F (ρ A , ρ B ) and F (ρ C , ρ D ), respectively.

Effect of the different interaction strengths
Our STA protocol, Eq. (10), being of a local nature, the question of how good this local approximation works for the operation of a sped-up quantum Otto engine naturally arises. In Sec. V we have introduced different final interaction strengths J j,f drawn from a Gaussian distribution with zero mean and standard deviation σ = 0.1. As shown by the distribution of power, work, efficiency, and success fidelity in Fig. 2, for this value of σ our ansatz performs reasonably well.
As a variation of Fig. 2(d), we have investigated the impact of larger standard deviations of the final interaction strengths on the success fidelity of the first stroke for a fixed stroke duration of τ 1 = 0.1 (Fig. 7). It is shown that on average the larger σ, the smaller the fidelity. This indicates the need for higher-order counter-diabatic protocols, i.e., the addition of controlled spin interactions, rather than only local magnetic fields, in such situations.

Appendix C: Approximate adiabatic gauge potential
Here we describe in detail the derivation of the adiabatic gauge potential following the method in Ref. [41]. For the sake of readability we mainly omit explicit time dependences throughout this section.
The Hamiltonian H 0 in the rotating frame with respect to a unitary U (ϑ(t)) has the form whereH 0 = U † H 0 U is the diagonalized (stationary) instantaneous Hamiltonian andÃ ϑ the adiabatic gauge potential in the rotating frame with respect to the timedependent variable ϑ describing the dynamics of the system. The HamiltonianH 0 is diagonal and thus all diabatic  transitions occur due to the adiabatic gauge potential in the second term in Eq. (C1).
The form of the Hamiltonian in the moving frame, i.e., Eq. (C1), can be derived by evolving a quantum state |ψ according to the Schrödinger equation i ∂ t |ψ = H 0 (ϑ(t)) |ψ with a time-dependent Hamiltonian H 0 (ϑ(t)) in a rotating frame ψ = U † |ψ . This leads tõ with the adiabatic gauge potential DifferentiatingH 0 (ϑ) = U † (ϑ)H 0 (ϑ)U (ϑ) with respect to the system's dynamical parameter ϑ, we obtain Transforming back to the laboratory frame and using that the gauge potential eliminates the off-diagonal terms of H m , i.e., [∂ ϑH0 ,H 0 ] = 0, we obtain The solution A ϑ of this equation gives the exact counterdiabatic Hamiltonian H CD (t) =θ(t)A ϑ (t) in Eq. (7) (see Ref. [74] for more details). In the instantaneous eigenbasis, it reads [18] H CD (t) = i n (|∂ t n n| − n|∂ t n |n n|) Equation (C6) requires a priori knowledge of all eigenstates at all times during the sweep and is therefore impracticable, especially in a many-body setup. Hence, we strive for an approximate solution A * ϑ for the adiabatic gauge potential that can relatively easily be implemented in experiments. To this end we employ the variational principle method of Ref. [41], namely, that solving Eq. (C5) is equivalent to minimizing the Hilbert-Schmidt norm of the Hermitian operator with respect to the parameters of A * ϑ and ≡ 1. Here, we seek the minimum of the operator distance between the exact G ϑ (A ϑ ) and the approximate G ϑ (A * ϑ ). Minimizing this operator distance is equivalent to minimizing the action associated with the parameters of the approximate adiabatic gauge potential A * ϑ , i.e., δS(A * ϑ ) δA * ϑ = 0, where δ denotes the functional derivative (see Refs. [41] and [74] for more details).