Quantum coherence, many-body correlations, and non-thermal effects for autonomous thermal machines

One of the principal objectives of quantum thermodynamics is to explore quantum effects and their potential beneficial role in thermodynamic tasks like work extraction or refrigeration. So far, even though several papers have already shown that quantum effect could indeed bring quantum advantages, a global and deeper understanding is still lacking. Here, we extend previous models of autonomous machines to include quantum batteries made of arbitrary systems of discrete spectrum. We establish their actual efficiency, which allows us to derive an efficiency upper bound, called maximal achievable efficiency, shown to be always achievable, in contrast with previous upper bounds based only on the Second Law. Such maximal achievable efficiency can be expressed simply in term of the apparent temperature of the quantum battery. This important result appears to be a powerful tool to understand how quantum features like coherence but also many-body correlations and non-thermal population distribution can be harnessed to increase the efficiency of thermal machines.


A. Expression ofQ SR/j
The heat flow from the bath j = C, H is defined in the main text asQ SR/j := Tr SR L j ρ I SR H SR . The expression announced in the main text is obtained by inserting the expression of the dissipative operator L j in the above definition. One recurrent approximation (sometimes called the adiabatic approximation), is to consider that S is in a thermal state at temperature T C denoted by ρ eq S due to the resonant and continuous contact with the cold bath C. Corrective terms to this approximation are of order g/|ν| and are time-dependent. However, assuming that g λ C , S remains approximately in a thermal state at all times, ensuring therefore that these corrective terms remains very small. As a consequence, when dealing with a term of second order in g/|ν| we approximate ρ I S by ρ eq S . By contrast, when dealing with terms of lower order, such approximation is not valid since we want a final result taking into account up to second order in g/ν. We mention the following identity used throughout this Section, Starting withQ +Tr{Λ j,t (ρ I SR )H SR }, we provide in the following the contribution from each of the three terms present in the above expression of the heat flow.

Contribution from the term (3)
The commutator [ Using the expression of A(ω) given in Method we find [A † (ω), and, Adding together the three contributions the commutator [A † (ω), H SR ] is reduced to Substituting in (3) we obtain, The second line of the above expression of the term (3) can be simplified as follows where [X, Y ] + := XY +Y X denotes the anti-commutator of X and Y . Note that for the bath H the above contribution is trivially null (since G H (ω) = 0). For the bath C, since the above term is already issued from second-order processes we can approximate ρ I S by ρ eq S (so that [N S , ρ I SR ] = O(g/|ν|)) and use (1), and we get so that the term (3) is reduced to for the bath C, and (3) = 0 for the bath H.
Using the expression of A(ω) indicated in Methods, we derive the following expression for A † (ω)A(ω), Finally the term (3) is reduced to using the identity (1) in the last line (since the terms are already of second order in g/ν).

Contribution from the term (4)
The commutator [A † (ω + ν), H SR ] gives so that 3. Contribution from the term (5) Using the expression of the dissipative map Λ j,t given in Methods, we have, We are again dealing with terms from second-order so that we can use the approximation ρ I S = ρ eq S + O(g/|ν|) and the relations [ We now derive a useful identity where we again used ρ I S = ρ eq S + O(g/|ν|). From the above identity (21) one can simplify the term (5) to with the notation G j (ω) := ∂ ω G j (ω). We recall that for the bath H the contribution of the term (5) is trivially null (G H (ω) = 0 for all ω ∈ E S ). For the bath C the expression of (5) can be simplified further. Using (1) one can show in a similar way Furthermore, the equality holds for thermal baths. Combining the two last equalities (23) and (24) we finally obtain, From the expression of C(ω) (in Methods) and using the identity (1) one can rewrite (5) in the form which also gives the right expression for H, namely (5) = 0.

Final expression forQ SR/j
Combining terms (3), (4), and (5) we obtain finally, valid for j = H, C, Since we neglect contributions of order higher than g 2 /|ν| 2 one can once again approximate ρ I S by ρ eq S (or ρ eq SR by ρ eq S ρ I R ), yielding, and, However, the first line of (27) is of order 0 in g/|ν| so that we cannot approximate

B. Expression ofQ S/j
From the above calculations we can easily derive the following expression forQ S/j , the heat flow entering S only, The unique difference from the expression ofQ SR/j comes from the second line which carries an energy ω instead of ω + ν.
In Section I we obtained expressions of the different heat flows up to order g 2 /ν 2 . Accordingly, the expectation value A † S (ω)A S (ω) ρ I S therein has to be evaluated to the same order g 2 /ν 2 . To do so we take the time derivative of A † S (ω)A S (ω) ρ I S and keep only terms up to second order in g/|ν|. The obtained differential equation can be easily solved when S is a harmonic oscillator or a two-level system. In the remainder of the Supplemental Material we focus on such situations and denote by ω 0 the transition frequency of S, implying E S = {−ω 0 , ω 0 }. To simplify further the model and the results we assume that C is resonant only with S, and that H is resonant with only one transition energy of SR, denoted by ω 0 + ν 0 with ν 0 ≥ 0. Then, G H (ω 0 + ν) = 0 for all ν = ν 0 . This corresponds to the model considered in the main text. Starting with we inject expression from Methods ofρ I SR in the above one and after similar calculations as in the last section I we obtain To continue we need to compute the commutators appearing in the differential equation. We do so in the following Sections when S is a harmonic oscillator (Section II A) and a two-level system (Section II B).

A. Harmonic oscillators
We assume here that S is a harmonic oscillator of frequency ω 0 and a, a † are the annihilation and creation operators. Harmonic oscillators usually couple to baths through quadratures operators. We therefore choose P S of the following form, where c is a complex number. As a consequence, A S (ω 0 ) = c * a and A S (−ω 0 ) = ca † . Equation (32) is then simplified to where and In the following we set c = 1 since the phase of c has no observable influence (and the amplitude of c can be included in G(ω)).
A quick analysis of the dynamics of A R (ν)A † R (ν) ρ I R reveals that its time derivative is of order 2 in g/|ν| (the contribution of L 0 clearly gives zero whereas the contribution of gL 1 turns out to be of order g 2 /ν 2 when t τ 1 := G −1 C (ω 0 ). One obtains from this observation that λ varies very slowly, at a rate of order g 4 /ν 4 , so that it can safely be taken as constant. The same conclusion can be drawn for r. Then we conclude that a good approximation of a † a ρ I S is a † a ρ I S = e −λt a † a ρ S (0) + 1 − e −λt λ r = t τ1 r λ .

Equation (40) implies in particular thatĖ
. This is the usual condition of steady state for continuous thermal machines [1][2][3][4] mentioned in the main text and valid for times t much bigger than τ es = [G C (ω 0 )] −1 .

B. Two-level systems
We assume is this section that S is a two-level system of transition frequency ω 0 . Such system usually couples to baths through the operators σ + and σ − (the Pauli matrices) so that a natural choice for P S is where c is a complex number. As a consequence A S (ω 0 ) = c * σ − , A S (−ω 0 ) = cσ + , A S (ω 0 )A † S (ω 0 ) ρ I S = |c| 2 ρ gg , and A † S (ω 0 )A S (ω 0 ) ρ I S = |c| 2 ρ ee , where ρ gg := g|ρ I S |g and ρ ee := e|ρ I S |e , being |g and |e the ground and excited state of S respectively. From (32) we obtain the following dynamicṡ with As in the previous Section the analysis of the dynamics of A † R (ν)A R (ν) ρ I R reveals that its time derivative is of order g 2 /ν 2 implying thatṘ ± = O(g 4 /ν 4 ), justifying that we can safely take R ± as constant. The dynamics of ρ ee = 1−ρ gg is then where R := R + + R − . Substituting in (27), (30), (51) and using (1) one obtains for the heat flows similar expressions as in the previous Section II A, Same identities as for harmonic oscillators (but the expression of the heat flows differs slightly, G C (ω 0 ) + G C (−ω 0 ) in the denominator instead of G C (ω 0 ) − G C (−ω 0 )). As in the previous Section, Eq. (47) implies in particular thaṫ . This is the usual condition of steady state for continuous thermal machines [1][2][3][4] mentioned in the main text and valid for times t much bigger than τ es = [G C (ω 0 )] −1 .

III. EXPRESSION OFĖR
The internal energy of R is defined as (see also main text) E R := H R ρ SR . In term of ρ I SR , the density matrix of SR in the interaction picture with respect to H SR , the internal energy of R can be re-written as with H I R (t) := e itH SR H R e −itH SR = H R . Then, the time derivative of E R is a sum of two contributions: Using Eq. (23) of Methods 'Why dispersive coupling?' (with H I R (t) instead of H SR ) one can show straightforwardly that Trρ I SR (t)H I R (t) = 0. This is because in the interaction picture with respect to H SR , R does not interact with S and therefore "does not see" the baths. Then, only the second term contribute toṘ R . Up to second order in g/|ν| one can show that The term of second order in g/ν is rapidly oscillating with ν = ν , resulting in a contribution of higher order (after time-graining), and terms of ν = −ν sum up to zero since ν∈E R [A R (ν), A † R (ν)] = 0. The first term contains the rapidly oscillating phase e −iνt so that the expression of N S A R (ν) ρ I SR has to be derived retaining only terms oscillating at the frequency ν, becoming non-oscillating terms after multiplying by the phase e −iνt . This can be done (for S harmonic oscillator or two-level system) using Eq. (21) of Methods "Why dispersive coupling?" (since Eq. (36) of Methods "Expression of the baths dissipative operators" was obtained by neglecting fast oscillating terms). We finally findĖ Comparing with Eqs. (39) and (46) when S is a harmonic oscillator and a two-level system, respectively, one obtains the following fundamental relation, valid for t τ es ,

IV. REFRIGERATION CONDITIONS AND EFFICIENCY
In Section II we saw that the heat flowsQ SR/j takes almost the same expression for both two-level systems and harmonic oscillators. Thanks to that the following considerations are valid for both systems. From Section II we havė The ± at the denominator corresponds to the possibility of S being a harmonic oscillator or a two-level system. The refrigeration condition corresponds toQ SR/C ≥ 0, which implies from (52)Ė R ≤ 0, meaning that R supplies energy to the refrigerator. The refrigeration condition is We defined the apparent temperature of R as From (1) (adapted to R) one can show that T R = T R when R is in a thermal state at temperature T R . More properties of the apparent temperature are mentioned in the main text. The refrigeration condition can be rewritten as which is the result announced and discussed in the main text.
The efficiency η is defined as the ratio of the energy extracted from C,Q SR/C , by the energy invested by R, −Ė R , . From the expressions (52) we have and the upper bound is a direct consequence of the above refrigeration condition. This result is also announced and discussed in the main text.
Note that we defined the efficiency in terms of rates of the energy flows. Traditionally it is defined in terms of time-integrated flows of energy, i.e. finite difference of energy (between a time t and the initial time). One can show that if we ignore the small time interval of order τ 1 = G −1 C (ω 0 ) before S reaches the steady state, the efficiency η int :=

V. ENERGY EXTRACTION CONDITIONS AND EFFICIENCY
We derive in this Section the conditions and the efficiency for the reverse operation of refrigeration: the storage in R of energy extracted from the baths. Such storage operation is obtained by reversing the sign of all heat flows with respect to the refrigeration regime. The storage condition isĖ R ≥ 0 which from (52) impliesQ SR/H ≥ 0 anḋ Q SR/C ≤ 0. From Section II we havė which imposes for the extraction condition, just the opposite of the refrigeration condition. The efficiency η e is defined as the energy stored in R, accounted bẏ E R , divided by the cost in thermal energy from the hot bath, accounted byQ SR/H , η e :=Ė Ṙ Q SR/H . From (52) we have that η e = ν0 ω0+ν0 + O(g 3 /ν 3 0 ). The storage extraction condition provides the upper bound, assuming T R ≥ T C (otherwise one can heat up R trivially by thermal contact with C or H). This is the expression mentioned and briefly discussed in the main text.

VI. APPARENT TEMPERATURE OF NON-DEGENERATED SYSTEMS
A. Apparent temperature of squeezed states In this Section we consider that R is a harmonic oscillator with the usual annihilation operator A R (ν 0 ) = a, creation operator A † R (ν 0 ) = a † , and the free Hamiltonian H R = ν 0 a † a, it is straight forward to show that T R := ν 0 [ln (1 + ν 0 /E R )] −1 , which depends only on the average internal energy E R . Some consequences of such property are detailed in the main text. As a special case, we consider a squeezed thermal state of squeezing factor r and thermal excitation corresponding to a temperature T R . The mean energy of such squeezed state is [5] E R /ν 0 = sinh 2 r + (sinh 2 r + cosh 2 r) e ν0/T R − 1 −1 . (61) The expression of the apparent temperature can be rewritten as which is the analogue of the expression used in [6,7] for the effective temperature characterising the upper bound efficiency in presence of squeezed baths.
B. Apparent temperature of a non-degenerated finite-level system Let's consider now a N-equidistant-level system. The ladder operator takes the form where |n is the eigenstate of the level n and c n,n+1 := n|A R |n + 1 . The products of the upwards and downwards ladder operators give From the definition of the apparent temperature [8] provided in the main text, we obtain where ρ n := n|ρ I R |n is the population of the level n. For N ≥ 3 it appears that changing the value of the some ρ n can alter T R but not necessarily the internal energy E R , and reciprocally. It is more apparent if one assumes that the transition amplitudes n|A R |n + 1 are independent of n implying the following simple expression for the apparent temperature, Now we ask the question, given a fixed average energy E R , what is the maximal achievable value of T R ? A simple way to derive the solution is by answering the reverse question: we look for the smallest E R compatible with a fixed apparent temperature T R .
We first assume that T R is positive. This fixed the value of ρ N in term of ρ 1 , ρ N = 1 − e ν0/T R (1 − ρ 1 ), with ρ 1 restricted to the range of values 1 − e ν0/T R ≤ ρ 1 ≤ 1 in order to ρ N be in the interval [0; 1]. In case the population ρ 1 and ρ N do not sum up to 1, ρ 1 + ρ N < 1, some other levels have to be populated. The choice which leads certainly to the lowest mean energy is populating the lowest available level, the level 2 (assuming N ≥ 3). Then, we choose ρ 2 = 1 − ρ 1 − ρ N , with the allowed range of values for ρ 1 : It follows that the double constraint ρ N , ρ 2 ∈ [0; 1] implies 1 − e −ν0/T R ≤ ρ 1 ≤ e ν 0 /T R e ν 0 /T R +1 . The mean energy of such a state is E R = ν 0 ρ 2 + ν 0 (N − 1)ρ N , choosing a ground state energy equal to zero. Substituting ρ 2 and ρ N by their expression in term of ρ 1 we have For N ≥ 3, the minimal energy is E R,min (T R ) = ν 0 e −ν0/T R , achieved for the minimal allowed value ρ 1 = 1 − e −ν0/T R . Reversing the relation we obtained the maximal apparent temperature for a given energy E R ≤ ν 0 , For E R ≥ ν 0 , the maximal apparent temperature is negative. Its expression can be derived in the same way as done above but assuming T R negative.
Finally, a thermal state at temperature T R has an internal energy equal to Then, thermal state of infinite temperature has an energy equal to ν 0 (N − 1)/2. It follows that for N ≥ 4, thermal states of positive temperatures can be manipulated with no energy change into non-thermal states of negative T R . Substituting E th R into (70) one can find numerically the maximum of T R,max /T R .For instance, for N = 3, we find T R,max /T R → 1.5 (achieved for T R going to infinity). For N ≥ 4 the ratio T R,max /T R is indeed unbounded since a thermal energy equal to ν 0 (corresponding to a finite positive temperature T R ) is enough to reach an infinite apparent temperature.