Shortcut-to-adiabaticity quantum Otto refrigerator

We investigate the performance of a quantum Otto refrigerator operating in finite time and exploiting local counterdiabatic techniques. We evaluate its coefficient of performance and cooling power when the working medium consists a quantum harmonic oscillator with a time-dependent frequency. We find that the quantum refrigerator outperforms its conventional counterpart, except for very short cycle times, even when the driving cost of the local counterdiabatic driving is included. We moreover derive upper bounds on the performance of the thermal machine based on quantum speed limits and show that they are tighter than the second law of thermodynamics.

Heat engines and refrigerators are two prime examples of thermal machines. While heat engines produce work by transferring heat from a hot to a cold reservoir, refrigerators consume work to extract heat from a cold to a hot reservoir [1,2]. Refrigerators thus appear as heat engines functioning in reverse. According to the second law of thermodynamics, the coefficient of performance (COP) of any refrigerator, defined as the ratio of heat input and work input, is limited by the Carnot expression, C = T 1 /(T 2 − T 1 ), where T 1,2 denote the respective temperatures of the cold and the hot reservoirs [1,2]. However, this maximum coefficient of performance is only attainable in the limit of infinitely long refrigerator cycles where the cooling power vanishes. At the same time, any refrigerator that runs in finite time necessarily dissipates irreversible entropy, which reduces its coefficient of performance. An important issue is to optimize the finite-time performance of thermal machines [3]. A central result of the theory of finite-time thermodynamics is the coefficient of performance at maximum figure of merit, * = 1/ 1 − T 1 /T 2 − 1, which is the counterpart of the Curzon-Ahlborn efficiency for heat engines [4][5][6][7].
Techniques based on shortcuts to adiabaticity (STA) [8,9] have recently been suggested as promising candidates to approach such a desired regime of performance-optimized finite-time quantum thermal machines. The implementation of STA methods on an evolving system mimics its adiabatic dynamics in a finite time [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Among the STA techniques put forward so far is the local counterdiabatic (LCD) driving technique, which cancels the possible nonadiabatic transitions induced by the dynamics of a given system by introducing an auxiliary local control potential [14]. It offers a wide range of applicability and has been experimentally successfully realized in state-of-art ion trap setups [18,19]. Such STA strategies hold the potential to enhance the performance of both classical and quantum heat engines [24][25][26][27][28][29][30]. However, these studies have so far focussed on the unattainability of the absolute zero temperature (according to the third law of thermodynamics) in a quantum refrigerator context. Therefore, a full fledged application of STA techniques to enhance the overall performance of a quantum refrigerator is still missing. Moreover, the implementation of STA protocols is not without an energetic cost, which is induced by the additional control potentials. In this regard, only recently the cost of performing STA drivings has been properly taken into account in the performance analysis of quantum heat engines [29][30][31][32][33][34][35].
In this paper, we study the STA quantum Otto refrigerator taking into account the cost of the driving in the performance analysis. We explicitly evaluate the coefficient of performance and cooling power of such refrigerator. We find that its performance can exceed its conventional counterpart even when the cost of the STA driving is included, except for very short cycle times. We further use the concept of quantum speed limits for driven unitary dynamics [36] to derive generic upper bounds on both the coefficient of performance and the cooling rate of the superadiabatic refrigerator.
The remainder of this paper is organized as follows. In Sec. I we introduce the quantum Otto refrigerator and illustrate the formalism and notation in use in the rest of the article. Section II is dedicated to the analysis of the quantum refrigerator under local counterdiabatic STA driving with Sec. III discussing its performance. Section IV is further dedicated to the establishment of upper bounds on such a performance of the refrigerator as set by the use of the quantum speed limit valid for the dynamics that we explore here. Finally, in Sec. V we draw our conclusions and discuss the possibility for further developments opened by our assessment.

I. QUANTUM OTTO REFRIGERATOR
The quantum Otto cycle is a paradigm for thermodynamic quantum devices [6,7,26,[37][38][39][40][41][42][43][44][45][46][47]. The cycle consists of two isentropic and two isochoric processes. At the end of a cycle, work is consumed by the refrigerator to pump heat from a cold to a hot reservoir. In this paper we make the choice of a working medium embodied by a quantum harmonic oscillator with controllable timedependent frequency ω t (see Fig. 1) and corresponding Hamiltonian Here m is the mass of the oscillator while x (p) is its position (momentum) operator. The device is alternately coupled to two heat baths at inverse Concretely, the Otto cycle consists of the following four steps as shown in Fig. 1: 1. Isentropic compression, corresponding to the transformation A(ω 1 , β 1 ) → B(ω 2 , β 1 ) in Fig. 1. The frequency is varied during time τ 1 , while the system is isolated from the baths. The corresponding evolution is unitary and the von Neumann entropy of the oscillator is constant.
2. Hot isochore, associated with the transformation B(ω 2 , β 1 ) → C(ω 2 , β 2 ) in Fig. 1. In this process, the oscillator is weakly coupled to the reservoir at inverse temperature β 2 at fixed frequency and for a time τ 2 . Notice that no request is made for thermalization of the oscillator.
3. Isentropic expansion, described by the transformation C(ω 2 , β 2 ) → D(ω 1 , β 2 ) in Fig. 1. The frequency of the working medium is unitarily changed back to its initial value during time τ 3 . No change of entropy occurs during this stroke.

4.
Cold isochore, at constant frequency, illustrated by the D(ω 1 , β 2 ) → A(ω 1 , β 1 ) process in Fig. 1. This transformation is obtained by weakly coupling the oscillator to the reservoir at inverse temperature β 1 > β 2 and letting the relaxation to the initial thermal state A(ω 1 , β 1 ) occur within a (in general short) time τ 4 .
The total cycle time is τ cycle = 4 j=1 τ j . In the rest of our analysis, we assume, as commonly done [7,39,42,45], that the time needed to accomplish the isochoric transformations is negligible with respect to the compression or expansion times, so that the total cycle time can be approximated to τ cycle τ 1 + τ 3 = 2τ for equal stroke duration. This assumption does not affect the generality of our results.
During the first and third strokes (compression and expansion), the quantum oscillator is isolated and only work is performed by changing the frequency in time. The mean work of the unitary dynamics can be evaluated by using the exact solution of the Schrödinger equation for the parametric oscillator for any given frequency modulation [48,49]. The mean work under scrutiny is thus given by [7] We have introduced the dimensionless quantities Q *

1,3
that, by depending on the speed of the frequency driving [50], embodies a parameter of adiabaticity of the dynamics. In general, we have Q * 1,3 ≥ 1, with the equality being satisfied for a quasi-static frequency modulation. Its expression is not crucial for the present analysis and can be found in Refs. [48,49], to which we refer for more details.
During the thermalization steps (isochoric processes), heat is exchanged with the reservoirs. Such contributions can be quantified by calculating the corresponding variation of energy of the oscillator, which gives us (3) In order to operate as a refrigerator, the system should absorb heat from the cold reservoir, so that Q 4 ≥ 0, and release it into the hot reservoir, which entails Q 2 ≤ 0. According to Eq. (3), the condition for cooling is thus that The coefficient of performance of the quantum Otto refrigerator is given by the ratio of the heat removed from the cold reservoir to the total amount of work performed per cycle, = Q 4 /( W 1 + W 3 ). It explicitly reads [7] = where we have defined x j = β j ω j /2 (j = 1, 2) and the function c(x 1,2 ) = coth(x 1,2 ). For slow (adiabatic) driving processes, Q * 1,2 = 1, the coefficient of performance of the engine becomes [7] which is positive provided that ω 2 > ω 1 . An upper bound to the coefficient of performance in Eq. (4) follows from the second law of thermodynamics which states that the total entropy production of a cyclic thermal device is always positive [1,51]. Employing the quantum relative entropy of two density operators, S(ρ 1 ||ρ 2 ) = tr {ρ 1 ln ρ 1 − ρ 1 ln ρ 2 } > 0, the total entropy production for one complete cycle can be written as where we have used the fact that the quantum relative entropy during the isentropic processes AB and CD is null, as the von Neumann entropy is constant. Moreover, the quantum relative entropy of the isochoric processes BC and DA corresponds to the entropy production associated with the heating and cooling steps. From Eq. (4), the total entropy production is then [41,46] Equality to zero is reached for the Carnot cycle scenario for which β 2 /β 1 = ω 1 /ω 2 . Based on the first law of thermodynamics, we have in addition Combining Eqs. (6) and (8), we obtain the following upper bound on the refrigerator performance The above equation shows that the coefficient of performance of the quantum refrigerator is always bounded by the Carnot coefficient of performance.

II. DRIVING A QUANTUM REFRIGERATOR WITH SHORTCUTS TO ADIABATICITY
Let us now consider the situation when the compression and expansion strokes of the Otto refrigerator cycle is sped up by addition of a counterdiabatic driving control field H CD STA (t) to the original harmonic oscillator Hamiltonian H O (t). Scope of this term is to suppress the non-adiabatic transitions induced by the finite-time evolution of the oscillator and, as a consequence, quench the entropy production all the way down to the value taken in the adiabatic manifold of the initial system Hamiltonian. The resulting effective Hamiltonian reads [10,11] where |n ≡ |n(t) denotes the n th eigenstate of the original Hamiltonian H O (t). For a harmonic working medium, the counterdiabatic term H CD STA (t) is [9,12] H CD Although this additional control removes the requirement of slow driving, the (non-local) counterdiabatic potential -which induces squeezing of the oscillator -makes its experimental application/implementation a challenging task. As a result, in order to circumvent this difficulty, it is natural to construct a unitarily-equivalent Hamiltonian with a local potential. This is achieved by applying the operator U x = exp imω t x 2 /4 ω , which cancels the squeezing term and gives the new effective local counterdiabatic (LCD) Hamiltonian [14] H where the modified time-dependent frequency is Ω 2 (t) = ω 2 t −3ω 2 t /4ω 2 t +ω t /2ω t . By requesting that the initial and final state of the working medium ensuing from H LCD (t) equal that from the original Hamiltonian H O (t), one gets the boundary conditions where ω i,f = ω 1,2 correspond to the initial and final frequency of the compression/expansion strokes. A suitable ansatz is [14,15] with s = t/τ . In order to ensure that the trap is not inverted, one must also guarantee that Ω(t) 2 > 0 is always fulfilled [26]. The mean value of the local counterdiabatic Hamiltonian H LCD (t) may be calculated explicitly for an initial thermal state and reads [29] H where we have introduced the LCD parameter The expectation value of the control field H LCD STA (t) follows therefore as value is non-null. We also remark that the local counterdiabatic control has been implemented in various experimental platforms [18,20], specifically in Paul traps [20] which are a potential candidate for building quantum thermal devices [52]. Figure 2 shows the rate of entropy production ∆S tot /τ cycle as a function of the time τ for adiabatic and nonadiabatic driving. We see that for short cycle time, the entropy production of nonadiabatic transition processes dramatically increases (blue dashed), thus leading to lower performance of the thermal machine. On the other hand, the application of STA methods is effective in suppressing such over-shooting of irreversible entropy (red dotted) to the adiabatic value (black large dashed).

III. PERFORMANCE OF A SUPERADIABATIC QUANTUM REFRIGERATOR
We now study three important quantities characterizing the performance of a refrigerator, namely its coefficient of performance ε, cooling rate J c STA and figure of merit χ. Taking into account the energetic cost of the STA driving, we define the coefficient of performance of the superadiabatic quantum Otto refrigerator as the ratio of the heat removed from the cold reservoir to the total amount of energy added per cycle where H i STA τ = (1/τ ) energetic cost of the STA protocol is ignored (which corresponds to setting H i STA τ = 0 in Eq. (18)), the coefficient of performance reduces to the adiabatic expression AD given by Eq. (5) [7,38,39,44]. Figure 3 shows the coefficient of performance of the superadiabatic quantum refrigerator STA (red dotted) as a function of time τ , together with the adiabatic AD (black large dashed) and nonadiabatic NA = Q 4 /( W 1 + W 3 ) (blue small dashed) counterparts. We observe that the superadiabatic driving significantly enhances the performance of the quantum Otto refrigerator, NA ≤ STA ≤ AD , for all driving times larger than τ ≈ 2.0, even though the energetic cost of the STA is explicitly included. We additionally note that the superadiabatic coefficient of performance STA is remarkably close to the adiabatic value AD for τ ≥ 25, indicating that the energetic STA cost is relatively small for larger times. Yet, the nonadiabatic coefficient of performance NA is already greatly reduced compared to the adiabatic value in this regime. The STA techniques thus appear here to be highly effective in suppressing nonadiabatic transitions at a little cost.
On the other hand, the cooling power of the superadiabatic refrigerator is given by the ratio of heat flowing from the cold reservoir into the system to the cycle time An infinitely long cycle time, which would allow to achieve the maximum coefficient of performance, would thus also gives zero cooling power. In this regard, the main advantage of the STA approach is to realize the same amount of heat output as in the adiabatic case, but in a shorter cycle time. Hence, the STA strategy ensures that J c STA (red dotted) is always greater than the nonadiabatic cooling rate J c NA = Q 4 NA /τ cycle (blue dashed) for fast cycles, as shown in Fig. 4(a). However, there still exists a trade-off between cooling power and coefficient of performance of STA refrigerator for fast cycles.
Following Feldmann and Kosloff [53], such trade-off can be illustrated as in Fig. 4(b), where the dependence of 1/ on the inverse cooling power 1/J c is illustrated for both the STA driving and the nonadiabatic protocol. The former simultaneously enhances both the coefficient of performance and the cooling power, thereby clearly demonstrating the benefits of the STA quantum Otto refrigerator over the conventional ones.
We finally consider the figure of merit χ = Q 4 /τ cycle = J c defined as the product of the coefficient of performance and the cooling power of the refrigerator [5,7,41,42]. The corresponding expression for a heat engine, χ engine = η Q 2 /τ cycle = − W /τ cycle , is equal to its power output Q 2 being in this case the heat absorbed from the hot reservoir. In optimization problems, the maximum figure of merit (and not the maximum cooling power) condition for refrigerators is in direct correspondence to the maximum power criterion for heat engines [5,7,41,42]. Figure 4(c) presents the corresponding values as a function of τ for the case of adiabatic, non-adiabatic and STA strategies. In analogy to the cooling power (19), a clear hierarchy emerges as χ NA ≤ χ STA ≤ χ AD with the equalities holding in the long-time limit. Compared to the nonadiabatic case, the STA approach increases the area under the curve, which determines the overall performance of the device.

IV. PERFORMANCE BOUNDS BY QUANTUM SPEED LIMIT
The maximum performance of a classical thermal machine (refrigerator/engine) is limited by the second law of thermodynamics [2]. However, quantum mechanics imposes restrictions on the time of evolution of quantum processes. Understanding such restrictions is important for the successful implementation of the STA technique [29]. We next derive general upper bounds for both the STA-based coefficient of performance and cool-ing rate of the quantum Otto refrigerator using the concept of quantum speed limits, which can be regarded as an extension of the Heisenberg energy-time uncertainty relation [36,[54][55][56][57].
For unitary driven dynamics, a Margolus-Levitin-type bound [57] on the evolution time given by [58] τ is appropriate. Here L(ρ i , ρ f ) = arccos F (ρ f , ρ i ) denotes the Bures angle between initial and final density operators of the system, with F (ρ f , ρ i ) the fidelity between the two, and H STA τ the time-averaged superadiabatic energy. Eq. (20) becomes a proper bound for the compression and expansion phases, when the engine dynamics is dominated by the STA driving for small τ . From Eqs. (18) and (20), an upper bound on the STAbased coefficient of performance of the quantum Otto refrigerator is obtained as where τ i QSL (i = 1, 3) are the respective speed-limit bounds in Eq. (20) for the compression/expansion phases. In addition, an upper bound for the figure of merit χ follows as The above upper bounds are displayed in Figs. 3, 4(a), 4(b) and 4(c) (green solid). We observe that the quantum bound on the coefficient of performance (Figs. 3 and 4(b)) is much tighter than the adiabatic bound (black large dashed) imposed by the second law of thermodynamics (discussed in Sec. I). They are hence more useful for applications. We emphasize that these results are general and do not depend on the choice of the engine cycle or on the STA driving protocol.

V. CONCLUSIONS
We have studied the performance of a quantum Otto refrigerator with a working medium consisting of a timedependent harmonic oscillator, exploiting STA mechanisms. We have explicitly analyzed the coefficient of performance, the cooling power, as well as the related figure of merit, for the case of local counterdiabatic driving. We have found that the STA quantum refrigerator outperforms its conventional nonadiabatic counterpart, except for short cycle durations, by strongly minimizing the nonequilibrium entropy production, even when the energetic cost of the STA driving is included. We have further derived generic upper bounds on the coefficient of performance of the Otto refrigerator by using the concept of quantum speed limits. Such bounds are tighter than those based on the second law of thermodynamics and therefore more useful. The possibility to achieve simultaneous enhancements of coefficient of performance and cooling power should be of advantage for the future design of micro-and nano-devices operating in the quantum regime.