Multi-spin counter-diabatic driving in many-body quantum Otto refrigerators

Quantum refrigerators pump heat from a cold to a hot reservoir. In the few-particle regime, counter-diabatic (CD) driving of, originally adiabatic, work-exchange strokes is a promising candidate to overcome the bottleneck of vanishing cooling power. Here, we present a finite-time many-body quantum refrigerator that yields finite cooling power at high coefficient of performance (CoP), that considerably outperforms its non-adiabatic counterpart. We employ multi-spin CD driving and numerically investigate the scaling behavior of the refrigeration performance with system size. We further prove that optimal refrigeration via the exact CD protocol is a catalytic process.


I. INTRODUCTION
Heat engines and refrigerators are a cornerstone of modern physics and indispensable in today's society [1]. Unravelling their fundamental laws in the few-particle regime has lead to the study of so-called quantum heat engines and refrigerators [2][3][4][5][6][7][8][9]. With the recent progress in controling quantum systems [10][11][12], such quantum heat engines could already be experimentally realized using single-body quantum working media [13][14][15][16][17][18]. Whereas heat engines convert thermal energy into work, their counterparts, namely refrigerators, cool down a cold bath by pumping heat from the cold to the hot reservoir, thereby consuming work [7,8,[19][20][21][22][23][24]. The maximum coefficient of performance (CoP) of refrigerators is limited by the Carnot CoP [25]. For these infinitely long (adiabatic) cycle times, the corresponding cooling power naturally converges to zero. A natural question thus arises whether such quantum refrigerators can be driven in finite time, yet produce a finite cooling power.
So called shortcuts to adiabaticity (STA) [26][27][28][29] are a promising candidate to overcome this fundamental bottleneck, due to minimizing quantum friction [30][31][32] during the work-exchange strokes. These STA methods [33][34][35][36][37][38] including counter-diabatic (CD) driving [38][39][40][41][42][43] where an additional CD Hamiltonian is applied to suppress any transitions between the system's eigenstates during the Hamiltonian's dynamics, have recently been applied in the field of quantum thermodynamics to enhance the performance of quantum heat engines [44][45][46][47][48] and refrigerators [49,50] using single-body quantum working media (WM). For the latter, exact local CD terms can be found analytically [35,37]. In general, identifying the exact counter-diabatic term requires a priori knowledge of the system's eigenstates at all times during the Hamiltonian's dynamics and which is numerically and experimentally impracticable for many-body WM. With this challenge in mind, Sels and Polkovnikov * Andreas.Hartmann@uibk.ac.at † glen.mbeng@uibk.ac.at [41] have developed a variational principle where approximate multi-spin CD terms can be found. Based on this method, a quantum heat engine using a many-body quantum working medium and local 1-spin CD Hamiltonian could be efficiently implemented [51].
In this work, we present a finite-time many-body quantum refrigerator (QR) with finite cooling power using additional approximate multi-spin counter-diabatic (CD) terms. We numerically demonstrate a large enhancement in cooling power and coefficient of performance (CoP) along with improved scaling behaviour of the cooling power with the system size for the sped-up QR compared to its original non-adiabatic counterpart. For the QR with single-body quantum working medium, we find an analytical expression for the CoP. For the many-body WM, we provide an analytical proof that exact CD driving implies a zero work component of the additional external control device. Remarkably, the latter fully assists the piston to run the QR in this case, mimicking the adiabatic quantum cycle, yet in finite time.
This work is structured as follows: In Sec. II, we introduce the quantum Otto refrigerator using a many-body spin system as its quantum working medium and present multi-spin local counter-diabatic driving. In Sec. III, we numerically analyze the performance of the corresponding refrigerators and conclude our results in Sec. IV while giving an outlook on future research.

A. Quantum Otto refrigerator
Quantum Otto refrigerators cyclically pump heat from a cold to a hot reservoir by consuming work. Its corresponding four-stroke quantum Otto cycle [8] consists of two heat-exchange strokes -where the quantum working medium (WM) is alternatingly coupled to two heat baths -and two work-exchange strokes.
The four strokes are (cf. Fig. 1

Adiabatic expansion (C → D):
The Hamiltonian H 0 (λ h ) changes back to H 0 (λ c ) during the stroke duration τ 3 until the WM attains the state ρ D .

Cold isochore (D → A):
The WM is brought in contact with the cold bath at temperature T c and working parameter λ c and cools down during the duration τ 4 to its originally initial thermal state ρ A .
During one cycle, the WM extracts the heat Q ad c : . A positive (negative) sign of the work components corresponds to work performed on (extracted from) the WM. Analogously, a positive (negative) sign of the heat corresponds to heat extracted from (imparted to) the thermal bath.
The cooling power is defined as the pumped heat Q ad c over the total cycle time τ cycle = 4 l=1 τ l , i.e., (2) In the adiabatic limit -where the isentropic strokes with durations τ 1 and τ 3 are infinitely long -the cooling power of these adiabatic quantum refrigerators goes to zero, i.e., lim τ1,τ3→∞ J ad → 0. To overcome this bottleneck, one can implement so called shortcuts-to-adiabaticity (STA) techniques for the two work-exchange strokes.

B. Quantum working medium
As our quantum working medium, we consider an allto-all connected Ising spin chain with Hamiltonian where h j and b j , respectively, are the time-dependent transverse and longitudinal magnetic field strengths at site j and J jk the interaction strength between spins at sites j and k. ϑ(t) is a continuous function that fulfills ϑ(t = 0) = 0 and ϑ(t = τ 1 ) = 1. For the second isentropic stroke the initial and final values of the function ϑ(t) are interchanged. Throughout this work, we parametrize the working parameters λ c and λ h with the magnetic fields and interaction strengths at each point of Fig. 1 and J jk,i are the values of the magnetic fields and interaction strengths at points A and D, and h j,f , b j,f and J jk,f at points B and C, respectively. The explicit forms of the functions h j (t), b j (t) and J jk (t) as well as the sweep function ϑ(t) are given in Appendix A.

C. Multi-spin counter-diabatic driving
The underlying idea of counter-diabatic (CD) driving [38][39][40][41][42][43] is to efficiently drive an adiabatic evolution of a Hamiltonian in finite time by suppressing transitions between its eigenstates. Thus, we always track the instantaneous eigenstates during the whole sweep. Finding the exact CD term requires a priori knowledge of the system's eigenstates for every time during the sweep which is numerically and experimentally challenging.
In order to overcome this bottleneck, an analytical variational principle has been developed recently [41,42] to find approximate CD terms.
In this work, we drive the WM during the isentropic strokes with the total Hamiltonian where H 0 (t), Eq. (3), is the original non-adiabatic and H CD (t) the additional counter-diabatic Hamiltonian that suppresses coherences in the WM that cause quantum friction in finite-time sweeps [30][31][32].
The additional counter-diabatic Hamiltonian reads with A ϑ (t) the exact adiabatic gauge potential (AGP) [41,42,52] andθ(t) the derivative of the sweep function of Eq. (3). We rely on an approximate adiabatic gauge potential A * ϑ that contains p-spin terms (with p ≤ N ) and an odd number of σ y terms (e.g., σ y j for p = 1, σ y j , σ y j σ x k and σ y j σ z k for p = 2, etc.). For p = N , we obtain the solution of the exact adiabatic gauge potential that entails all combinations of N -spin terms (cf. Ref. [53] in the case of quantum criticality). As an example, for p = 1 we apply the ansatz A * ϑ = N j=1 α j σ y j and solve for the optimal solution of each coefficient α j by minimizing the operator distance between the exact and approximate AGP. For more details, see Appendix B and Refs. [41,42,52].
We note, that we apply the counter-diabatic Hamiltonian only in the isentropic strokes as these are normally much longer than the thermalization strokes. However, techniques to speed up the latter have been also developed recently [47,[54][55][56][57].

D. Quantum refrigerator under STA
The introduction of the additional approximate counterdiabatic term H * CD (t) in the Hamiltonian's dynamics during the, originally non-adiabatic, work-exchange strokes requires a careful definition of work, cooling power and coefficient of performance. In Ref. [51], the division dt the total exchanged work for each of the two work-exchange strokes l ∈ {1, 3} with duration τ l has been introduced. The corresponding work components thus read where the work W l 0 stems from the piston and W l CD from the external control device that implements H * CD (t). Remarkably, the work component W l CD is zero if the additional adiabatic gauge potential and thus the CD Hamiltonian H * CD (t) is exact, i.e., A * ϑ (t) = A ϑ (t) and thus H * CD (t) = H CD (t). In Appendix C, we provide a detailed proof of this statement. Hence, the work component W 1+3 CD stemming from the external control device during one cycle can be seen as an artifact of inexact CD driving. Experimentally it is advantageous to have a vanishing contribution from the external control device as we want the external control device to fully assist the piston instead of just draining it to run the quantum refrigerator. In other words, the exact CD drive is catalytic in the sense that it allows for speeding up the cycle without the external control device being altered (charged or discharged) after a cycle.
The cooling power under STA is given by and the coefficient of performance (CoP) reads where W 1+3 STA = W 1 STA + W 3 STA > 0 is the total work performed on (consumed by) the working medium per cycle. Note the difference between the pumped heat Q ad c for the adiabatic [cf. Eqs. (1) and (2)] and Q c for the sped-up cycle [cf. Eqs. (9) and (10)].
For the case of a single-body quantum working medium that is modelled by Eq. (3) with N = 1, i.e., this reduces to the Landau-Zener (LZ) model, the coefficient of performance evaluates to where h x,i is the initial value of the transverse magnetic field in the first isentropic stroke and b z,f the final value of the longitudinal magnetic field strength, respectively. This expression is positive provided that b z,f > h x,i and is limited by the Carnot CoP C = T c /(T h − T c ) [25] (see Appendix D for more details).

III. NUMERICAL RESULTS
In this section, we present the numerical results of the proposed quantum Otto refrigerator. To this end, we used the QuTip 4. . We numerically solved the von Neumann equation for the isentropic strokes and computed the heat Q c and Q h for the two thermalization strokes as the energy difference between points B and C as well as D and A (cf. Fig. 1), respectively. Throughout this numerical performance analysis, the temperatures for the cold and hot bath are set to T c = 0.2 and T h = 0.4, respectively. The values for the working parameters at points A and D as well as B and C in

A. Scaling with system size
Our primary goal is to speed up the quantum refrigerator, i.e., we want to pump as much heat Q c as possible from the cold to the hot reservoir in the shortest amount of time. Thus, we are particularly interested in the cooling power J during one cycle and its scaling behaviour for different system sizes N . enhancement in cooling power decreases the higher p becomes. The major relative enhancement can be made by applying 1-spin or 2-spin CD terms. Including multi-spin terms (e.g. 3 and 4-body in the cases studied) only give a relatively slight improvement compared to 1-spin or 2-spin CD driving. We note, that there is a trade-off between enhanced cooling power with increasing p and implementation complexity for an experiment. In the adiabatic limit (black-dashed line), the cooling power naturally converges to zero.
From a practical point of view, we deem it more favorable if the increased cooling power J STA is due to the piston rather than draining the external control device. Consequently, we are interested in the work component W 1+3 CD , Eq. (8). As shown in Appendix C the latter is zero if the applied CD Hamiltonian is exact. Figure 3 depicts the absolute value of the work component W 1+3 CD stemming from the external control device during one cycle for different system sizes N up to p = 4. We see that W 1+3 CD is zero as long as p > N . Namely, the CD Hamiltonian must comprise all kinds of interactions up to order N , i.e., involving all the spins in the chain. By contrast, for N > p, i.e., more spins N in the WM than order of interaction p in H * CD (t), we see that the absolute value of W 1+3 CD adopts a non-zero value which, according to Appendix C, implies that the Hamiltonian H * CD (t) is not exact anymore. We therefore conclude, that including a CD Hamiltonian with all combinations up to N -body terms leads to an exact expression when the working medium contains N spins which is consistent with Refs. [41,42]. These results encourage the goal to strive for an exact CD drive. For the latter, the external control device fully assists the piston that optimally "compresses" and "expands" the quantum working medium.

B. Dependence on cycle time
We are further interested in the pumped heat Q c per cycle and the corresponding coefficient of performance (CoP) for different cycle durations τ . Figure 4(a) shows the heat Q c extracted from the cold reservoir over one cycle for a system size N = 6. For the quantum Otto cycle with non-adiabatic Hamiltonian H 0 (t), we see that heat Q na c > 0 is pumped from the cold reservoir to the hot bath only for isentropic stroke durations τ = τ 1 = τ 3 28 (yellow-shaded area, refrigerator regime). For shorter isentropic stroke durations, the obtained states ρ B and ρ D at the end of each isentropic stroke (cf. Fig. 1) are so far away from the ideal adiabatic states ρ B and ρ D that the quantum Otto cycle ceases to describe a refrigerator. For large stroke durations, the refrigerator pumps the maximal possible cooling heat (gray-shaded area, adiabatic regime, τ ≥ 250 where Q na c ≈ Q ad c ). By contrast, the sped-up cycle with Hamiltonian H * STA (t) including 1-up to 4-spin CD terms H * CD (t) even pumps heat from the cold reservoir in the quench limit τ 1 , τ 3 → 0 where the cycle duration is dominated by thermalization. For increasing p, the cooling heat Q c increases to its maximally possible value Q ad c for all durations studied. Note, however, that p = 2 appears to be efficient even for the intermediate regime where p = 1 does not yield optimal results. This is in accordance with Fig. 2. Note that for the adiabatic regime, the strength of the additional H * CD (t) converges to zero (asθ(t) ∝ 1/τ , cf. Appendix A) and hence Q c → Q ad c . Figure 4(b) depicts the corresponding coefficient of performance (CoP) , Eq. (10). For the sped-up cycle with H * STA (t), Eq. (4), we see that the higher p-spin terms we use for our CD Hamiltonian H * CD (t), Eq. (5), the larger the CoP becomes. Note that the CoP appears to be more sensitive to the value of p than the pumped heat Q c . Namely, converging to the optimal CoP ad , Eq. (2), generally requires a larger p than for the optimal exctracted heat Q ad c which is due to a non-zero W 1+3 CD .

IV. DISCUSSION AND OUTLOOK
In this work, we have presented a many-body quantum Otto refrigerator that efficiently operates at finite-time. The many-body quantum working medium (WM) is modelled by an all-to-all connected Ising spin chain where the working parameter is parametrized via magnetic fields and interactions. In order to speed up the quantum Otto cycle, we apply an additional approximate counter-diabatic (CD) Hamiltonian. The latter outperforms its non-adiabatic counterpart in pumped heat per cycle. The additional CD Hamiltonian contains p-spin terms with an odd number of σ y terms and p ≤ N . For p = N , we obtain the exact CD Hamiltonian. We provide an analytical proof and give numerical evidence that the work component stemming from the external control device is zero if the CD Hamiltonian is exact. In this case, the external control device fully assists the piston to pump heat from a cold to a hot reservoir. Note that while an exact CD protocol implies zero work contribution from the external control device over a cycle, the converse, however, is not true. It is therefore not possible to find an exact CD Hamiltonian via minimizing this work component.
We numerically demonstrate an enhanced cooling power and coefficient of perfomance (CoP) for the sped-up quantum Otto refrigerator with higher p-spin CD driving compared to its non-adiabatic counterpart. Furthermore, we show that increasing p improves the scaling behaviour in cooling power with system size. For the Otto cycle with single-body quantum WM, i.e., described via the Landau-Zener model, we find an analytical expression for the CoP that only depends on the applied magnetic field strengths. The analytical and numerical performance analysis reveals that quantum Otto refrigerators can be scaled up efficiently by increasing the number of spins in the working medium.
The results presented in this work disclose some important operational features. We note that there is a trade-off between experimental feasibility of the additionally applied CD terms and vanishing work component stemming from the external control device. For the latter, we need to apply non-local multi-spin CD terms (for example through additional laser fields) that are hard to implement in current experiments. On the other hand, applying only local 1-body CD terms may result in enhanced cooling power compared to its non-adiabatic counterpart, yet under the price of extensive use of the external control device rather than the piston.
We note that several cost identifiers for the additional CD Hamiltonian have been introduced [45,46,48,50,[59][60][61][62]. We note, further, that these implementational costs are conceptually different to the operational costs described by the work W 1+3 CD stemming from the external control device [51]. In fact, in the case of exact CD driving we go on the optimal, i.e., adiabatic, path from points A to B and C to D in Fig. 1, respectively. Although the operational cost will then be zero, i.e., W 1+3 CD = 0, there will still be a cost of implementing the additional CD Hamiltonian.
For future research, we intend to study the robustness of the applied CD protocols with respect to external noise and possible cooperative effects [63]. Further, we aim at developing a many-body quantum refrigerator where work and heat exchanges occur simultaneously. N38, the Hauser-Raspe foundation, and the European Union's Horizon 2020 research and innovation program under grant agreement No. 817482. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0068. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA.

Appendix A: Protocols for magnetic fields and interactions
For the many-body quantum working medium in the text, the explicit time dependence of the magnetic field and interaction strengths for the non-adiabatic [H 0 (t), Eq. (3)] and shortcut-to-adiabaticity Hamiltonian is the sweep function and h j, J jk,f = J jk (t = τ l ) the initial and final values for each isentropic stroke l ∈ {1, 3} of duration τ l , respectively. Its derivative with respect to time readṡ and is applied to the CD Hamiltonian, Eq. (5).

Appendix B: Multi-spin CD driving
For the counter-diabatic Hamiltonian H CD (t), Eq. (5) in the text, we apply an approximate adiabatic gauge potential A * ϑ . Here, we follow the variational method introduced in Ref. [41].
We make an ansatz A * ϑ with p-spin Pauli matrices and an odd number of σ y terms for the adiabatic gauge potential (AGP) and calculate the Hermitian operator The goal is to minimize the operator distance D 2 (A * ϑ ) = Tr{[G ϑ (A ϑ ) − G ϑ (A * ϑ )] 2 } between the exact, A ϑ , and approximate AGP, A * ϑ . This minimization is equivalent to minimizing the action with respect to its parameters in front of every Pauli matrix term, i.e., δS(A * ϑ )/δA * ϑ = 0. As an example, for p = 1, i.e., 1-spin CD driving, we apply the ansatz A * ϑ = N j=1 α j σ y j and calculate the Hermitian operator G ϑ (A * ϑ ) as well as the action S(A * ϑ ). By minimizing the latter with respect to the coefficients α j , we find the optimal solution for each spin. For p = 2, we apply the ansatz A * ϑ = N j=1 α j σ y j + k<j β jk (σ y j σ z k + σ z j σ y k ) + γ jk (σ y j σ x k + σ x j σ y k ) and solve the corresponding action with respect to all coefficients α j , β jk and γ jk . With this variational method, we can also include multi-spin terms, potentially up to N -body terms σ z 1 · · · σ y j · · · σ x N . We solve the optimal form of each coefficient numerically. = 0 (C7) which proves the statement above.

Appendix D: Single-body working medium
In analogy to Ref. [48], we derive the coefficient of performance (CoP) of the quantum refrigerator with a single-body working medium.
The CoP, Eq. (10), in the text can be rewritten in terms of the energies E i with i ∈ {A, B, C, D} at each point (cf. Fig. 1) of the cycle. It reads where the energies are due to the uniform scaling of the energy levels in this two-level system after each stroke. Inserting the latter, Eq. (D1) consequently reads which is equivalent to Eq. (11) from the text.
The pumped heat (D4) is positive for the Otto cycle to describe a refrigerator. From the latter and the fact that tanh is a monotonously increasing function, it follows that Inserting this into Eq. (D3), we finally obtain