#### 3.1. The Equivalence of Heat Machines in the Non-Markovian Regime

The construction of various heat machine types in the same physical system was studied in [

1], and it is based on operator splitting techniques. In particular, the Strang splitting [

80,

81,

82] for two non commuting operators

A and

B is

${e}^{(A+B)dt}={e}^{\frac{1}{2}Adt}{e}^{Bdt}{e}^{\frac{1}{2}Adt}+O\left(d{t}^{3}\right)$. Starting with the simultaneous machine operator where all terminals are connected simultaneously:

where

${U}_{0}={e}^{-i({\mathcal{H}}_{e}+{\mathcal{H}}_{c}+{\mathcal{H}}_{h}+{\mathcal{H}}_{w}){\tau}_{cyc}}$, the single-particle coherence evolution operator can be singled out from the total evolution operator since

$[{\mathcal{H}}_{e}+{\mathcal{H}}_{c}+{\mathcal{H}}_{h}+{\mathcal{H}}_{w},{H}_{int}]=0$. All of the population change is generated by

${U}_{cyc}^{simul}$ In fact,

${U}_{cyc}^{simul}$ is the evolution operator in the interaction picture. Energy observables like heat look the same in the interaction picture (

${U}_{0}^{\u2020}{H}_{c,h,w}{U}_{0}={H}_{c,h,w}$). In practice, all states should be evolved with

${U}_{cyc}^{simul}$ only. The bare Hamiltonians

${H}_{k}$ are used only for calculating the energy observables. Thus, the single-particle coherences associated with interaction-free time evolution

${U}_{0}$ do not affect the population dynamics and observables like energy that are diagonal in the energy basis. The fact that

${U}_{0}$ commutes with

${U}_{cyc}^{simul}$ means that outcome of the operation does not depend on the time the operation is carried out (time invariance).

This type of single-particle coherences should be distinguished from inter-particle coherences. Since the energy gaps in the machine and the terminal are matched, the inter-particle coherences are between degenerate states. For example, the states

$\left|{0}_{c}{1}_{e}\right.\u232a$ and

$\left|{1}_{c}{0}_{e}\right.\u232a$ are degenerate, and so are the pairs

$\{\left|{0}_{h}{3}_{e}\right.\u232a,\left|{1}_{h}{0}_{e}\right.\u232a\}$ and

$\{\left|{0}_{w}{3}_{e}\right.\u232a,\left|{1}_{w}{2}_{e}\right.\u232a\}$. The crossed lines in

Figure 1b show the pairs of two-particle degenerate states. These inter-particle coherences are essential for the dynamics. Their complete suppression leads to a Zeno effect that halts all the dynamics in the engine. The inter-particle coherences are generated and modified by the interaction terms and hence cannot be separated from the rest of the evolution like the single-particle coherences. Note that changes in inter-particle coherence translate to population changes in the subspaces of individual particles.

When starting in a product state where the inter-particle coherences are zero, the energy transfer (population changes) is of order

$d{t}^{2}$ while the coherence generation is of order

$dt$. This is due to the fact that unitary transformation converts population to coherences and coherence to population (see Figure 8 in [

1]). In thermodynamic resource theory, phases are often dismissed as non-essential, but we stress that this is true only for the single-particle coherences.

To study the relations between the simultaneous engine and the two-stoke engine, we apply the Strang decomposition which yields the following product form

where

s is the “engine norm action”

$s=({\u2225{\mathcal{H}}_{ec}\u2225}_{sp}+{\u2225{\mathcal{H}}_{eh}\u2225}_{sp}+{\u2225{\mathcal{H}}_{ew}\u2225}_{sp}){\tau}_{cyc}$ and

${\u2225\xb7\u2225}_{sp}$ is the spectral norm of the operator [

1]. When this number is small compared to ℏ,

${U}_{cyc}^{II-stroke}\to {U}_{cyc}^{simul}$. Note that the first term and the third term in

${U}_{cyc}^{II-stroke}$ are two parts of the same stroke. The operator splits this way since the Strang splitting can only create symmetric units cells. A similar splitting can be done for the four-stroke engine exactly as shown in [

1]. One immediate conclusion follows from the equivalence of the one cycle evolution operators: if different machine types

start in the same initial condition, their state will coincide when monitored stroboscopically at

${t}_{n}=n{\tau}_{cyc}$. While at

${t}_{n}=n{\tau}_{cyc}$ the states of different machine types will differ by

$O\left[\left(\frac{s}{\hslash}\right){}^{3}\right]$ at the most, at other times they will differ by

$O\left[\frac{s}{\hslash}\right]$. This expresses the fact that the machine types are never identical at all times. They differ in the strongest order possible

$O\left[\frac{s}{\hslash}\right]$, unless complete cycles are considered. Since the one cycle evolution operators are equivalent, it follows from Equation (2) that

for the same initial engine state:

where

${Q}^{simul}$ refers to the heat transferred in time of

${\tau}_{cyc}$ in the particle machine. The ≅ stands for equality up to correction

$\u2225{H}_{c\left(h\right)}\u2225O\left[\left(\frac{s}{\hslash}\right){}^{4}\right]$. Note that the cubic term does not appear in Equation (6). Due to lack of initial coherence, the

$O\left[\left(\frac{s}{\hslash}\right){}^{3}\right]$ correction contributes only to the inter-particle coherence generation but not to population changes. Hence, the population changes differ only in order

$O\left[\left(\frac{s}{\hslash}\right){}^{4}\right]$. In transients, the system energy changes from cycle to cycle so, in general, it is not correct to use energy conservation to deduce work equality from heat equality. Nevertheless, work equality follows from Equations (2) and (5) when using

${H}_{w}$ instead of

${H}_{c\left(h\right)}$.

This establishes the equivalence of heat (and work) even very far away from steady state operation or thermal equilibrium

provided all engines start with the same state. This behavior is very similar to the Markovian equivalence principle [

1], but there is one major difference. Since each cycle starts in a product state, the leading order in heat and work is

$O\left[\left(\frac{s}{\hslash}\right){}^{2}\right]$ and not

$O\left[\frac{s}{\hslash}\right]$ as in the Markovian case. The linear term in the work originates from the single-particle coherence generated by the classical driving field. Without this coherence, the power scales as

$({Q}_{h}^{cyc}+{Q}_{c}^{cyc})/{\tau}_{cyc}\propto {\tau}_{cyc}$. Thus, as shown in

Figure 2a, for small action, the power grows linearly with the cycle time. On the other hand, as explained earlier, the correction to the power is only of order

$O\left[\left(\frac{s}{\hslash}\right){}^{4}\right]/{\tau}_{cyc}=O\left[\left(\frac{s}{\hslash}\right){}^{3}\right]$ since there is no cubic correction to the work.

Let us consider now the

steady state operation. Despite Equation (6), it is not immediate that the heat will be the same for different machine types in steady state. In Equation (6), the initial density matrix is the same for all machine types. However, different types may have slightly different initial states, which may affect the total heat. To study equivalence in steady state operation, we first need to define what steady state means when the bath and batteries are included in the analysis. The whole system is in a continuous transient: the hot bath gets colder, the cold bath gets hotter, and the batteries are charged. Nonetheless, the reduced state of the engine relaxes to a limit cycle

${\overline{\rho}}_{e}\left(t\right)={\overline{\rho}}_{e}(t+\tau )$ or explicitly

${\overline{\rho}}_{e}=t{r}_{\ne e}\left[{U}_{cyc}({\rho}_{c}\otimes {\rho}_{h}\otimes {\rho}_{w}\otimes {\overline{\rho}}_{e}){U}_{cyc}\right]$. To see the relation between steady states of different machines, we choose the steady state of one machine, for example

${\overline{\rho}}_{e}^{simul}$, and apply the two-stroke evolution operator to it:

The cubic order is absent because, if there are no initial inter-particle coherences, then the cubic term only generates inter-particle coherences. Hence, the reduced state of the system is not modified by the cubic correction of the two-stroke evolution operator. From Equation (7), we conclude that the steady states are equal for both machines up to quartic corrections in the engine action. From Equation (6), it follows that heat and work in steady state are also equal in all machine types, up to quartic correction.

Figure 2a shows the power in steady state for the three main types of machines as well as for a higher order six-stroke machine that will be discussed in the last section. Let the power of the machine (work per cycle divided by cycle time) be denoted by

P. In

Figure 2b, we plot the normalized power

$P/{P}^{simul}$ where it is easier to see that the correction in the power of one machine

with respect to the other is quadratic. This graph shows that the equivalence of non-Markovian machines is actually similar to that of Markovian machines. The difference is that the reference simultaneous power is constant (in action) in the Markovian case and linearly growing (small action) in the present case.

Figure 2b shows that the equivalence is a phenomenon that takes place in a regime and not only at the (ill defined) point

${\tau}_{cyc}=0$. The same holds for the Markovian case.

At this point, we wish to discuss the quantumness of the equivalence principle in the current setup. In [

1], it was suggested to use dephasing in the energy basis to see if the machine is stochastic or quantum. If a dephasing stroke is carried out before the unitary stroke and the result is not affected, then the machine operates as a stochastic machine. In the four-stroke engine and in the two-stroke engine described in Equation (5), the battery is accessed twice during the cycle. The first interaction with the battery creates some inter-particle coherence between the battery and the engine. As a result, the next interaction with the battery (the second work stroke) starts with nonzero inter-particle coherence. Thus, adding dephasing after the first work stroke will affect the power gained in the next work stroke. This is shown by the red and blue dashed curves in

Figure 2a. The power of the simultaneous engine is zero if we continuously dephase the system (Zeno effect). We conclude that, although there is no coherence that carries over from one cycle to the next, as in the Markovian case, coherence is still needed for the equivalence principle to hold. This time, the coherence is an inter-particle coherence between degenerate states.

So far, we ignored the nature of the energy transferred to the battery, i.e., if it is heat or work. If it is pure work, the device is an engine, whilst if it is heat, the device functions as an absorption machine (only heat bath terminals). However, the equivalence principle is indifferent to this distinction. If the action is small, two-stroke, four-stroke, and simultaneous machines will perform the same. In the next section, however, we study the conditions under which the entropy of the batteries is not increased and the device performs as a proper engine.

#### 3.3. Higher Order Splittings

The regime of equivalence studied in

Section 3.1 and in [

1] is determined by the use of the Strang decomposition for the evolution operator. Although higher order decompositions do exist, they involve coefficients with alternating signs [

85]. In the Markovian case, this is not physical since a bath that generates evolution of the form

$exp(-\mathcal{L}t)$ is not physical (where

$exp(+\mathcal{L}t)$ is the standard Lindblad evolution). In the present paper, instead of non unitary evolution of the reduced state of the engine, we consider the global evolution operator of all the components. The global evolution is unitary and its generators, the interaction Hamiltonian, are all Hermitian. Hence, there is no problem to have for example

$exp(+i{H}_{ec}t)$ instead of

$exp(-i{H}_{ec}t)$. It simply means an interaction term with opposite sign. This facilitates the use of higher order decompositions in order to make machines with more strokes and a larger regime of equivalence.

In [

86], Yoshida introduced a very elegant method to construct higher order decompositions. Let

${U}_{{s}^{2}}\left(t\right)={U}^{simul}\left(t\right)+O\left[\left(\frac{s}{\hslash}\right){}^{4}\right]$ stand for an evolution operator that has a correction of order

${s}^{3}$ with respect to

${U}^{simul}\left(t\right)$. It can be, for example, a four-stroke or two-stroke engine. As shown in [

86], a fourth order evolution operator

${U}_{{s}^{4}}\left(t\right)$ can be constructed from

${U}_{{s}^{2}}\left(t\right)$ in the following way:

where

${U}_{{s}^{4}}\left(t\right)={U}^{simul}\left(t\right)+O\left[\left(\frac{s}{\hslash}\right){}^{5}\right]$. By applying the same arguments as before, when the cycle starts with fresh uncorrelated bath and battery particles, the correction to the work and heat are

$O\left[\left(\frac{s}{\hslash}\right){}^{6}\right]$. The Yoshida method is powerful since it can be repeated, with different

${x}_{0},{x}_{1}$ coefficients, to gain operators that are even closer to the simultaneous machine. Physically, Equation (10) can be interpreted as a regular

${U}_{{s}^{2}}\left(t\right)$ machine where the stroke durations alternate every cycle.

Figure 2b shows the ratio of the power of various engines with respect to the simultaneous engine. While in the Strang four-stroke and two-stroke machine, the

power deviation from the simultaneous machine is second order in the action, the power of the Yoshida engine of order four deviates from the simultaneous machine only in the fourth order in the action.

Two-stroke and four-stroke engines naturally emerge from practical considerations. Two-stroke engines emerge when it is easier to thermalize simultaneously the hot and cold manifolds. Four-stroke engines emerge when it is easier to thermalize one manifold at a time. In contrast, the Yoshida decomposition Equation (10) does not split the simultaneous engine into more basic or simpler operations compared to the two-stroke and four-stroke machines. Thus, the practical motivation for actually constructing Yoshida-like higher order machines is not obvious at all at this point. Nevertheless, our main point in this context is that higher order machines are forbidden in Markovian dynamics and are allowed in the non-Markovian machines studied here.