Collective heat capacity for quantum thermometry and quantum engine enhancements

On the one hand, the philosophy behind this paper is to investigate a particular type of collective effect and analyse how beneficial it can be for certain thermodynamic tasks. Then, depending on the extent of the benefit, one can decide to start thinking of how to actually realise such collectively-enhanced devices. In this perspective, we briefly discuss in the following some possibilities for experimental realisations of collective coupling between spins and bath. The aim of this paper is to suggest that the benefits are worth the 'experimental battle'.

Ideally, we would think of adapting the aforementioned designs to include a spin ensemble collectively coupled to the bath. This is certainly possible in [1] since therein the baths are emulated by an external magnetic field, offering the possibility of addressing collectively an ensemble of spins. While it might be possible to upgrade the other designs [2–5] to collective bath coupling, it is less obvious than in [1].

Beyond that, we stress that there are several known platforms realising collective coupling between a spin ensemble and electromagnetic modes [31–34] or even phonons [15, 35] and mechanical oscillators [36]. Note that spins commonly used in realisations of collective coupling are from atoms of strontium [33, 34, 37] and rubidium [36]. Using such platforms, one can imagine the implementation of collective coupling between the spin ensemble and a bath or a sample of interest. Indeed, if the intermediary system—the electromagnetic or phononic mode—is coupled to the bath (or sample), the effective dynamics of the spin ensemble can be a collective dissipation. A required condition for that is having a coupling between the intermediary system and the bath larger than the coupling between the spin ensemble and the intermediary system. This is shown explicitly in [31, 33] where the intermediary system is a cavity mode coupled to the external electromagnetic field playing the role of the thermal bath. This can be extended directly to other platforms since the core mechanism is the same, namely a spin ensemble interacting collectively with a bosonic mode which is itself interacting with a thermal bath.

The conclusion of this section is that collective coupling of a spin ensemble with a thermal bath is tangible in several platforms, and even readily realisable in the experimental design used in [32–34].

The heat capacity C of a system in a thermal state at temperature T determines how much energy must be absorbed (or released) to increase (or decrease) the system's temperature by an amount δT. It is naturally given by $C=\frac{\partial {E}^{\text{th}}\left(\beta \right)}{\partial T}=-{k}_{\mathrm{B}}{\beta }^{2}\frac{\partial {E}^{\text{th}}\left(\beta \right)}{\partial \beta }$, where E^th(β) = Tr Hρ^th(β) is the energy of the system in the thermal state ρ^th(β) := Z⁻¹(β)e^−βH at inverse temperature β := 1/k_BT, k_B is the Boltzmann constant, Z(β) = Tr e^−βH is the partition function and H is the free Hamiltonian of the system. As one could expect, the heat capacity plays a central role in thermometry and thermal machines. We will come back on this aspect in sections 3 and 4. For now we focus on the heat capacity of an ensemble of spins interacting collectively with a thermal bath. The main idea is that since the steady state energy of a spin ensemble interacting collectively with a thermal bath [20, 38] is different from the thermal equilibrium energy—reached when all spins interact independently with the bath, the collective and independent heat capacities should also be different.

More precisely, we consider an ensemble of n identical spins of dimension s and free Hamiltonian H = ℏωJ_z, where ${J}_{z}{:=}{\sum }_{k=1}^{n}{j}_{z,k}$ is the sum of the z-component of the local angular momentum operators associated to each spin k (similar notations for the x and y components). The collective coupling with a thermal bath corresponds to a coupling Hamiltonian of the form V = gJ_xO_B, where O_B is an unspecified bath observable, and g represents the coupling strength. Under the usual Born, Markov, and secular approximations [39, 40], the dynamics of the collective dissipation is of the form [20]

$$\begin{equation}\frac{\mathrm{d}\rho }{\mathrm{d}t}={\Gamma}\left(\omega \right)\left({J}^{-}\rho {J}^{+}-{J}^{+}{J}^{-}\rho \right)+{\Gamma}\left(-\omega \right)\left({J}^{+}\rho {J}^{-}-{J}^{-}{J}^{+}\rho \right)+\mathrm{h}.\mathrm{c}.,\end{equation} \tag{ 1 }$$

where J^± := J_x ± iJ_y are the collective jump operators of the spin ensemble, ${\Gamma}\left(\omega \right)={\hslash }^{2}{g}^{2}{\int }_{0}^{\infty }{\mathrm{e}}^{i\omega u}\enspace \mathrm{Tr}\enspace {\rho }_{\mathrm{B}}{O}_{\mathrm{B}}\left(u\right){O}_{\mathrm{B}}\enspace \mathrm{d}u$ is the 'half Fourier transform' of the bath correlation function, ρ_B is the density operator of the thermal bath at temperature T, and O_B(u) denotes the interaction picture of O_B. The steady state of the above collective dissipation (1) can be expressed in a relative simple way using the collective basis |J,m⟩_i [41] made of the common eigenvectors of J_z and ${\mathcal{J}}^{2}{:=}{J}_{x}^{2}+{J}_{y}^{2}+{J}_{z}^{2}$,

$$\begin{align}\hfill {\mathcal{J}}^{2}{\vert J,m\rangle }_{i}& =\hslash J\left(J+1\right){\vert J,m\rangle }_{i}\hfill \\ \hfill {J}_{z}{\vert J,m\rangle }_{i}& =\hslash m{\vert J,m\rangle }_{i},\hfill \end{align} \tag{ 2 }$$

with −J ⩽ m ⩽ J and J ∈ [J₀; ns], where J₀ = 0 if s ⩾ 1 and J₀ = 1/2 if s = 1/2 and n odd. The index i belongs to the interval [1; l_J], where l_J denotes the multiplicity of the eigenspaces associated to the eigenvalue J of the operator ${\mathcal{J}}^{2}$. With these notations, the steady state takes the form [20]

$$\begin{equation}{\rho }^{\text{ss}}\left(\beta \right)=\sum _{J={J}_{0}}^{ns}\sum _{i=1}^{{l}_{J}}{p}_{J,i}{\rho }_{J,i}^{\text{th}}\left(\beta \right)\end{equation} \tag{ 3 }$$

where ${p}_{J,i}{:=}{\sum }_{m=-J}^{{J}_{i}}\langle J,m\vert {\rho }_{0}{\vert J,m\rangle }_{i}$ is the weight of the initial state of the spin ensemble ρ₀ in the eigenspace of total spin J and

$$\begin{equation}{\rho }_{J,i}^{\text{th}}\left(\beta \right){:=}{Z}_{J}{\left(\beta \right)}^{-1}\sum _{m=-J}^{J}{\mathrm{e}}^{-m\hslash \omega \beta }{\vert J,m\rangle }_{i}\langle J,m\vert ,\end{equation} \tag{ 4 }$$

with ${Z}_{J}\left(\beta \right){:=}{\sum }_{m=-J}^{J}{\mathrm{e}}^{-m\hslash \omega \beta }$. Note that if the initial state contains some coherences of the type ${\;}_{i}\langle J,m{\vert {\rho }_{0}\vert J,m\rangle }_{{i}^{\prime }}$, with i ≠ i', it is not proven that the corresponding steady state has exactly the form (3) (see [20]). A short note is provided in appendix C regarding the stability of the collective steady state under the emergence of small spin–spin interactions or small disorder and inhomogeneities altering the local energy levels of each spin. Additionally, the fact that the steady state retains some dependence on the initial state is a consequence of a dynamics with multiple steady states. The steady state actually reached is determined by the initial conditions [20, 38, 42]. This should not be seen as a contradiction with the Markovian dynamics. In particular, one can verify that in the present dynamics there is not backflow of information as it typically occurs in non-Markovian dynamics.

The energy of the spin ensemble when it reaches the steady state (3) is

$$\begin{align}\hfill {E}^{\text{ss}}\left(\beta \right)& {:=}\hslash \omega \enspace \mathrm{Tr}\enspace {J}_{z}{\rho }^{\text{ss}}\left(\beta \right)=\sum _{J={J}_{0}}^{ns}\sum _{i=1}^{{l}_{J}}{p}_{J,i}\enspace {\mathrm{e}}_{J}\left(\beta \right),\hfill \end{align} \tag{ 5 }$$

with ${\mathrm{e}}_{J}\left(\beta \right){:=}\hslash \omega \enspace \mathrm{Tr}\enspace {J}_{z}{\rho }_{J,i}^{\text{th}}\left(\beta \right)=\hslash \omega {\sum }_{m=-J}^{J}m\frac{{\mathrm{e}}^{-m\hslash \omega \beta }}{{Z}_{J}\left(\beta \right)}$. Then, quite naturally, we can define the collective heat capacity as the derivative with respect to the bath temperature of the steady state energy reached via collective dissipation, namely

$$\begin{align}\hfill {C}^{\text{col}}\left(\beta \right)& {:=}-{k}_{\mathrm{B}}{\beta }^{2}\frac{\partial {E}^{\text{ss}}\left(\beta \right)}{\partial \beta }=\sum _{J={J}_{0}}^{ns}\sum _{i=1}^{{l}_{J}}{p}_{J,i}{C}_{J}\left(\beta \right),\hfill \end{align} \tag{ 6 }$$

with

$$\begin{align}\hfill {C}_{J}\left(\beta \right)& {:=}-{k}_{\mathrm{B}}{\beta }^{2}\frac{\partial {\mathrm{e}}_{J}\left(\beta \right)}{\partial \beta }\hfill \\ \hfill & ={k}_{\mathrm{B}}{b}^{2}\left[{\left(\frac{1/2}{\mathrm{sinh}\enspace b/2}\right)}^{2}-{\left(\frac{J+1/2}{\mathrm{sinh}\left(J+1/2\right)b}\right)}^{2}\right],\hfill \end{align} \tag{ 7 }$$

where b := ℏωβ.

One can verify that C_J(β) > C_J'(β) for J > J' and for all β, even for negative effective bath temperature—relevant in some specific situations like in presence of several thermal baths [43, 44] or spin baths [45, 46]. This implies that the largest collective heat capacity is obtained with initial state such that p_J=ns = 1. Such states span a subspace sometimes called the Dick or symmetrical subspace. In particular, some experimentally simple states like thermal states at inverse temperature |β₀| ≫ 1/ℏω belong to such subspace. As expected, the applications to thermometry and thermal engines seek the largest heat capacity. Therefore, now we know that the largest advantage obtained from collective interactions is achieved for initial state belonging to the symmetrical subspace. In the following we compare the best case scenario, ${C}_{+}^{\text{col}}\left(\beta \right){:=}{C}_{J=ns}\left(\beta \right)$, to the independent heat capacity.

Comparison with independent heat capacity. The independent heat capacity is the derivative with respect to the bath temperature of the thermal equilibrium energy E^th(β)—the energy reached when each spin interacts independently with the bath,

$$\begin{align}\hfill {C}^{\text{ind}}\left(\beta \right)& {:=}-{k}_{\mathrm{B}}{\beta }^{2}\frac{\partial {E}^{\text{th}}\left(\beta \right)}{\partial \beta }=n{C}_{J=s}\left(\beta \right),\hfill \end{align} \tag{ 8 }$$

where

$$\begin{equation}{E}^{\text{th}}\left(\beta \right){:=}\hslash \omega \enspace \mathrm{Tr}\enspace {J}_{z}{\rho }^{\text{th}}\left(\beta \right)=n\enspace {\mathrm{e}}_{J=s}\left(\beta \right),\end{equation} \tag{ 9 }$$

${\rho }^{\text{th}}\left(\beta \right)=Z{\left(\beta \right)}^{-1}{\mathrm{e}}^{-\hslash \omega \beta {J}_{z}}$, and $Z\left(\beta \right)=\mathrm{Tr}\enspace {\mathrm{e}}^{-\hslash \omega \beta {J}_{z}}$.Then, we are left to compare ${C}_{+}^{\text{col}}\left(\beta \right)={C}_{J=ns}\left(\beta \right)$ and C^ind(β) = nC_J=s(β). The expansion of the expression (7) at ℏω|β| ≫ 1 gives respectively ${C}_{+}^{\text{col}}\left(\beta \right)\underset{\hslash \omega \vert \beta \vert \gg 1}{\sim }{\left(\hslash \omega \beta \right)}^{2}{\mathrm{e}}^{-\hslash \omega \vert \beta \vert }$, and ${C}^{\text{ind}}\left(\beta \right)\underset{\hslash \omega \vert \beta \vert \gg 1}{\sim }n{\left(\hslash \omega \beta \right)}^{2}{\mathrm{e}}^{-\hslash \omega \vert \beta \vert }$. In particular,

$$\begin{equation}{C}_{+}^{\text{col}}\left(\beta \right)/{C}^{\text{ind}}\left(\beta \right)\underset{\hslash \omega \vert \beta \vert \gg 1}{\sim }{n}^{-1}.\end{equation} \tag{ 10 }$$

By contrast, for ℏω|β| ≪ 1, we obtain

$$\begin{equation}{C}_{+}^{\text{col}}\left(\beta \right)=\frac{1}{3}ns\left(ns+1\right){\left(\hslash \omega \beta \right)}^{2}+\mathcal{O}\left[{\left(ns\hslash \omega \beta \right)}^{4}\right],\end{equation} \tag{ 11 }$$

and

$$\begin{equation}{C}^{\text{ind}}\left(\beta \right)=\frac{n}{3}s\left(s+1\right){\left(\hslash \omega \beta \right)}^{2}+\mathcal{O}\left[n{\left(s\hslash \omega \beta \right)}^{4}\right],\end{equation} \tag{ 12 }$$

implying ${C}_{+}^{\text{col}}\left(\beta \right)/{C}^{\text{ind}}\left(\beta \right)\underset{\hslash \omega \vert \beta \vert \ll 1}{=}\frac{ns+1}{s+1}+\mathcal{O}\left[n{\left(n\hslash \omega \beta \right)}^{2}\right]$. For the intermediary regime between these asymptotic limits, the behaviour of ${C}_{+}^{\text{col}}\left(\beta \right)$, C^ind(β), and ${C}_{+}^{\text{col}}\left(\beta \right)/{C}^{\text{ind}}\left(\beta \right)$ is represented in figures 1 and 2 as functions of k_BT/ℏω for several value of n and s.

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Importantly, the critical value T_cr of the bath temperature such that ${C}_{+}^{\text{col}}\left(1/{k}_{\mathrm{B}}{T}_{\text{cr}}\right)={C}^{\text{ind}}\left(1/{k}_{\mathrm{B}}{T}_{\text{cr}}\right)$ is well approximated by the function

$$\begin{equation}\frac{{k}_{\mathrm{B}}{T}_{\text{cr}}\left(n,s\right)}{\hslash \omega }\simeq {\left(\frac{4ns\left(s+1\right)+1}{12}\right)}^{1/2}.\end{equation} \tag{ 13 }$$

A comparison with numerical solutions gives very good agreement, see figures 1(b) and 2(b). The above expression (13) was obtained expanding the expressions of C^ind(β) and ${C}_{+}^{\text{col}}\left(\beta \right)$ using the assumptions that ℏω/k_BT_cr ≪ 1 and (ns + 1/2)ℏω/k_BT_cr ≫ 1 for growing n. Such assumptions are based on observations of the numerical solutions.

We consider the situation where the sample we want to estimate the temperature is much larger than the probe—our spin ensemble. Under weak coupling, one can consider that the spin ensemble reach a steady state without indeed affecting the sample. In other words, the sample plays the role of a thermal bath. This is the general framework considered by quantum thermometry in thermal equilibrium [11]. Additionally to this framework, and accordingly to the above study of heat capacity, we consider two different situations. In what we call the independent-dissipation situation, each spin of the ensemble interacts independently with the sample so that the spin ensemble eventually reaches a thermal state at the sample's temperature T. In the other situation, the collective-dissipation situation, the spin interact collectively with the sample and reaches the steady state ρ^ss(β) of equation (3).

The maximal information extractable from the spin ensemble about the sample's temperature is usually quantified by the quantum Fisher information $\mathcal{F}\left(T\right)$ [55–57]. Quite intuitively, the maximal precision of the estimation of the sample's temperature can be related to this maximal extractable information $\mathcal{F}\left(T\right)$. This is indeed established by the Cramer–Rao bound [58]. Therefore, $\mathcal{F}\left(T\right)$ directly informs about the maximal achievable precision of the estimation of the sample's temperature. As mentioned in the introduction, it was shown in [11, 21–23] that $\mathcal{F}\left(T\right)=C\left(T\right)/{k}_{\mathrm{B}}{T}^{2}={{\Delta}}^{2}\langle H\rangle /{T}^{2}$, where C(T) is the heat capacity of the probe and Δ²⟨H⟩ is its energy variance. However, such property is valid when the probe is in a thermal state, and in principle not valid for non-thermal states. Therefore, before applying the results of the last section regarding collective heat capacity we have to show that the maximum precision for the temperature estimation using non-thermal states of the form ρ^ss(β) is indeed given by the collective heat capacity. This is shown in appendix A where we establish in particular that the quantum Fisher information for any state of the form ρ^ss(β) is

$$\begin{equation}{\mathcal{F}}^{\text{col}}\left(T\right)={C}^{\text{col}}\left(T\right)/{k}_{\mathrm{B}}{T}^{2}.\end{equation} \tag{ 14 }$$

Then, it follows from the analysis of the last section that the largest enhancements from collective coupling are obtained for spin ensembles initially in a thermal state at very large inverse temperature |β₀| ≪ 1/ℏω, or more generally for initial states belonging to the symmetrical subspace. We denote by ${\mathcal{F}}_{+}^{\text{col}}\left(T\right)$ the corresponding quantum Fisher information. This is to be compared with the quantum Fisher information ${\mathcal{F}}^{\text{ind}}\left(T\right)$ obtained from independent dissipation of each spin, which is equal to ${\mathcal{F}}^{\text{ind}}\left(T\right)={C}^{\text{ind}}\left(T\right)/{k}_{\mathrm{B}}{T}^{2}$ since the steady state in this case is a thermal state. Then, the ratio ${\mathcal{F}}_{+}^{\text{col}}\left(T\right)/{\mathcal{F}}^{\text{ind}}\left(T\right)$ is equal to ${C}_{+}^{\text{col}}\left(\beta \right)/{C}^{\text{ind}}\left(\beta \right)$ which is represented for some values of n and s in figures 1(b) and 2(b).

In terms of the relative precision ΔT/T, where ΔT represents the standard deviation⁴ of the estimated value of T, we have

$$\begin{equation}\frac{{\Delta}T}{T}{\geqslant}{\nu }^{-1/2}{\left(\frac{{\Delta}T}{T}\right)}_{\mathrm{min}}{:=}\frac{1}{\sqrt{\nu {C}_{+}^{\text{col}}\left(T\right)/{k}_{\mathrm{B}}}}\end{equation} \tag{ 15 }$$

where ν is the number of measurements used to realise one estimation of T. The above inequality can be saturated for instance when choosing the maximum likelihood estimator. Importantly, for large sample's temperature k_BT ≫ ℏω one obtains from (11) an Heisenberg scaling [11]: ΔT ∼ 1/n. Figure 3(a) presents the plots of the minimal relative standard deviation ${\left(\frac{{\Delta}T}{T}\right)}_{\mathrm{min}}$ for collective and independent coupling with the sample, denoted respectively by ${D}_{+}^{\text{col}}$ and D^ind, in function of the sample's temperature. Figure 3(b) provides the ratio ${D}_{+}^{\text{col}}/{D}^{\text{ind}}$ of the minimal relative variances. We have the following asymptotic scaling for large sample's temperature ${D}_{+}^{\text{col}}/{D}^{\text{ind}}\underset{{k}_{\mathrm{B}}T\gg \hslash \omega }{\sim }\sqrt{\left(s+1\right)/\left(ns+1\right)}$.

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In particular, the temperature of the sample can be estimated with a higher precision thanks to collective interaction as long as T > T_cr(n, s). To have an estimate of what would be the critical temperature we can use experimental data from [3]. The energy splitting of the cesium quasispin is about ℏω ∼ 2.4 × 10⁻³⁰J. It means that for an ensemble of n = 2 cesium atoms, collective interaction become advantageous for sample's temperature of the order or superior to T_cr(n = 2, s = 7/2) ≃ 5.5 nK (the cesium quasispin is of dimension s = 7/2), and increasing the precision by a factor up to 1.8. For larger ensembles of for instance n = 10, the critical temperature is T_cr(n = 10, s = 7/2) ≃ 12 nK and collective interactions can increasing the precision by a factor 8. For the NV-centre spins used in [4, 5], the energy splitting is of the order of ℏω ≃ 1.9 × 10⁻²⁴J, implying that collective interactions become advantageous from T_cr(n = 10, s = 1/2) ≃ 0.22 K for an ensemble of n = 10 NV center spins, with precision increased by a factor up to 4.

Finally, there are two interesting remarks to be made around the proof of equation (14). First, while the maximum precision is given by the collective heat capacity C^col(T), it is not equal to the variance of the energy unlike thermal states. In fact, the collective heat capacity is in general strictly smaller than the energy variance Δ²⟨H⟩. Secondly, the best measurement is not the energy measurement but a non-local measurements corresponding to projections onto the collective states ${\left\{{\vert J,m\rangle }_{i}\right\}}_{{J}_{0}{\leqslant}J{\leqslant}ns,-J{\leqslant}m{\leqslant}J,1{\leqslant}i{\leqslant}{l}_{J}}$. Since such measurements sounds experimentally unrealistic, we should consider only the precision yielded by local energy measurements. Fortunately, in the particular situation where the initial state of the spin ensemble belongs to the symmetrical subspace, local energy measurements turn out to be optimal. The conclusion of appendix A is that when considering the best case scenario—the spin ensemble initially in the symmetrical subspace—the maximal precision of the temperature estimation can be reached by usual local energy measurements and is given by the collective heat capacity.

4.1. Work per cycle near Carnot efficiency

In this section we consider a quantum engine operating according to the quantum Otto cycle [59, 60]. The working medium is an ensemble of n spins of dimension s of Hamiltonian H(λ_t) = λ_tℏωJ_z, where λ_t is the compression factor which varies continuously between λ_c and λ_h during the isentropic strokes. The two isochoric strokes realised alternatively in contact with hot and cold baths bring the spin ensemble to the usual thermal equilibrium state ${\rho }^{\text{th}}\left({T}_{x},{\lambda }_{x}\right)={Z}^{-1}\left({T}_{x},{\lambda }_{x}\right){\mathrm{e}}^{-H\left({\lambda }_{x}\right)/{k}_{\mathrm{B}}{T}_{x}}$, with x = c, h and $Z\left({T}_{x},{\lambda }_{x}\right)=\mathrm{Tr}\enspace {\mathrm{e}}^{-H\left({\lambda }_{x}\right)/{k}_{\mathrm{B}}{T}_{x}}$ if each spin interacts independently with the bath. However, if the spins interact collectively with the successive bath, the isochoric strokes result in the steady state ${\rho }^{\text{ss}}\left({T}_{x},{\lambda }_{x}\right)={\sum }_{J={J}_{0}}^{ns}{\sum }_{i=1}^{{l}_{J}}{p}_{J,i}{\rho }_{J,i}^{\text{th}}\left({T}_{x},{\lambda }_{x}\right)$, where

$$\begin{equation}{\rho }_{J,i}^{\text{th}}\left({T}_{x},{\lambda }_{x}\right){:=}\sum _{m=-J}^{J}\frac{{\mathrm{e}}^{-m{\lambda }_{x}\hslash \omega /{k}_{\mathrm{B}}{T}_{x}}}{{Z}_{J}\left({T}_{x},{\lambda }_{x}\right)}{\vert J,m\rangle }_{i}\langle J,m\vert ,\end{equation} \tag{ 16 }$$

with ${Z}_{J}\left({T}_{x},{\lambda }_{x}\right){:=}{\sum }_{m=-J}^{J}\enspace {\mathrm{e}}^{-m{\lambda }_{x}\hslash \omega /{k}_{\mathrm{B}}{T}_{x}}$. Importantly, the weight ${p}_{J,i}={\sum }_{m=-J}^{{J}_{i}}\langle J,m\vert {\rho }_{0}{\vert J,m\rangle }_{i}$ in each eigenspace of total spin J is constant throughout the cycles [20] and is determined by ρ₀, the state of the spin ensemble before the engine is switched on.

Then, for an Otto cycle operating close to the Carnot efficiency, the extracted work per cycle is [24] (see also a brief derivation in appendix D)

$$\begin{equation}W={\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{C\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right)\end{equation} \tag{ 17 }$$

where, ${\Delta}\eta ={\eta }_{\mathrm{c}}-\eta =\frac{{\lambda }_{\mathrm{c}}}{{\lambda }_{\mathrm{h}}}-\frac{{\beta }_{\mathrm{h}}}{{\beta }_{\mathrm{c}}}$ is the difference between the Carnot efficiency ${\eta }_{\mathrm{c}}{:=}1-\frac{{\beta }_{\mathrm{h}}}{{\beta }_{\mathrm{c}}}$ and the actual efficiency, θ_x := λ_xβ_x and C(θ_h) denotes generically the collective or independent heat capacity depending whether the spins interact collectively or independently with the baths.

The central question is what are the parameters yielding the largest output work and which of the independent or collective spin machine gives the largest work per cycle? Considering the best case scenario for the collective spin machine, meaning that ρ₀ belongs to the symmetrical subspace, we have to compare ${W}_{+}^{\text{col}}{:=}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{{C}_{J=ns}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right)$ with ${W}^{\text{ind}}{:=}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)n\frac{{C}_{J=s}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right)$. We are looking for the parameters maximising the output work at constant efficiency. Then, as expected, we find that the larger Δβ := β_c − β_h, the larger the output work. Considering now Δβ fixed, we are left with two parameters, λ_h and θ_h. Fixing firstly λ_h (one can verify that θ_h can be changed while keeping Δη, Δβ and λ_h fixed), the best choice is taking β_h to zero. This is because C_J(θ)/θ² is monotonic decreasing (even though C_J(θ) is not monotonic, see for instance figure 1). Since the maximum of $\frac{{C}_{J}\left(\theta \right)}{{\theta }^{2}}$ is $\frac{{\hslash }^{2}{\omega }^{2}}{12}\left[{\left(2J+1\right)}^{2}-1\right]$, we obtain for all λ_h, Δη (to the first order), and Δβ,

$$\begin{equation}{W}^{\text{ind}}{\leqslant}{W}_{\mathrm{max}}^{\text{ind}}{:=}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\beta }_{\mathrm{c}}\frac{{\hslash }^{2}{\omega }^{2}}{12}n\left[{\left(2s+1\right)}^{2}-1\right],\end{equation} \tag{ 18 }$$

and

$$\begin{equation}{W}_{+}^{\text{col}}{\leqslant}{W}_{\mathrm{max}}^{\text{col}}{:=}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\beta }_{\mathrm{c}}\frac{{\hslash }^{2}{\omega }^{2}}{12}\left[{\left(2ns+1\right)}^{2}-1\right].\end{equation} \tag{ 19 }$$

Note that both maximal values are reached for ℏωβ_h going to 0 and that ${W}_{\mathrm{max}}^{\text{col}}=\frac{sn+1}{s+1}{W}_{\mathrm{max}}^{\text{ind}}$. We recover the asymptotic result of [20]. For intermediary value of ℏωβ_h, the plots of ${w}_{+}^{\text{col}}{:=}{W}_{+}^{\text{col}}/\left({\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\Delta}\beta \right)={C}_{+}^{\text{col}}\left({\theta }_{\mathrm{h}}\right)/{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}$ and ${w}^{\text{ind}}{:=}{W}^{\text{ind}}/\left({\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\Delta}\beta \right)={C}^{\text{ind}}\left({\theta }_{\mathrm{h}}\right)/{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}$ are given in figure 4 for ensembles of n = 2 to n = 100 spins.

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Observation about reaching asymptotically the Carnot bound at finite power. In [24, 25] the authors show that one could in principle reach asymptotically the Carnot bound while having a non-zero output power. One possibility is for instance to use phase transitions where the heat capacity scales up super-linearly in the number of atoms/subsystems in the working medium. Then, naturally, one could think that the collective effects on heat capacity shown here could be also useful for such purposes. This is in fact not the case as we explain in the following.

For finite value of ℏωβ_h and for arbitrary n, the amplification is upper bounded by ${W}^{\text{col}}{\leqslant}{\left(\hslash \omega \right)}^{2}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right){\left(\frac{1/2}{\mathrm{sinh}\enspace \hslash \omega {\theta }_{\mathrm{h}}/2}\right)}^{2}$. In other words, if we consider β_h ≠ 0 (as in realistic conditions) and fixed, increasing the size of the working medium will increase the output power only until it reaches the above saturation limit. Therefore, it cannot be helpful to reach asymptotically the Carnot bound at finite power (where the idea is to take advantage of a super-linear scaling in the power per cycle to increase slowly the efficiency).

The above observation emphasises a drawback. At finite T_h, there is always a critical number n_cr of spins such that for spin ensembles larger than n_cr the independent-spin engine performs better than the collective-spin one. One can estimate the critical spin number from (13) and obtains ${n}_{\text{cr}}\left({T}_{\mathrm{h}},{\lambda }_{\mathrm{h}},s\right)\simeq \frac{3{\left({T}_{\mathrm{h}}/\hslash \omega {\lambda }_{\mathrm{h}}\right)}^{2}-1/4}{s\left(s+1\right)}$. In the same spirit, for fixed bath temperatures, there is always a critical value λ_h,cr of the compression factor above which the collective-spin engine become less performant than the independent-spin engine. From (13) we have ${\lambda }_{\mathrm{h},\mathrm{c}\mathrm{r}}\left({T}_{\mathrm{h}},n,s\right)\simeq {T}_{\mathrm{h}}\sqrt{12}/\left[\hslash \omega \sqrt{4ns\left(s+1\right)+1}\right]$. On the other hand, the range of compression factors is always experimentally limited so that for finite n—since n is also experimentally limited—the collective spin engine performs always better that the independent one as soon as T_h/λ_h ⩾ T_cr(n, s).

4.2. What about the output power?

The length time of each cycle is mainly determined by the thermalisation time—the isentropic strokes can be made in principle on a timescale much smaller than the thermalisation time as long as one considers a driving such that [H(t), H(t')] = 0 for all t, t', which we assumed here. The thermalisation time can be estimated from the dynamics and can be very different between collective and independent bath coupling. This phenomenon was indeed exploited in [16]. In appendix B we show that for spin ensembles initially in a thermal state at inverse temperature β₀ such that ℏω|β₀| ≫ 1 the timescale to reach the steady state through collective interactions is at least n times shorter than the equilibration timescale for independent dissipation. Although stemming from collectively-enhanced dissipation rates as superradiance does, this accelerated equilibration is a different phenomenon. In particular, it happens for any state in the symmetrical subspace (J = ns) whereas superradiant states are limited to values of m close to zero. Thanks to this accelerated equilibration, the timescale τ_col of the cycles of the collective spin engine can set to be n times shorter than τ_ind, the timescale of the cycles of the independent spin engine. Then, with τ_col = τ_ind/n the output power of the Otto machines are given by

$$\begin{equation}{\mathcal{P}}^{\text{ind}}=\frac{1}{{\tau }_{\text{ind}}}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{n{C}_{s}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right)\end{equation} \tag{ 20 }$$

and

$$\begin{equation}{\mathcal{P}}_{+}^{\text{col}}=\frac{n}{{\tau }_{\text{ind}}}{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{{C}_{ns}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right).\end{equation} \tag{ 21 }$$

In particular we have

$$\begin{equation}\frac{{\mathcal{P}}_{+}^{\text{col}}}{{\mathcal{P}}^{\text{ind}}}\enspace \enspace \underset{{T}_{\mathrm{h}}\gg {\lambda }_{\mathrm{h}}{T}_{\text{cr}}\left(n,s\right)}{\sim }\enspace n\frac{ns+1}{s+1}.\end{equation} \tag{ 22 }$$

Note that due to the same issue of saturation commented above, collective effects still cannot be used to reach asymptotically the Carnot efficiency at finite output power. However, we have now that at fixed T_h, when the size of the working medium is much larger than the critical size n_cr(T_h, λ_h, s), the output powers of the two machines become equivalent (instead of ${W}^{\text{th}}/{W}_{+}^{\text{col}}\sim n$ for the extracted work per cycle). Therefore, in terms of output power, the collective Otto engine performs always better than or equal to the independent Otto engine. To illustrate this important point, the plots of ${p}_{+}^{\text{col}}{:=}{\mathcal{P}}_{+}^{\text{col}}{\tau }_{\text{ind}}/\left({\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\Delta}\beta \right)$, ${p}^{\text{ind}}{:=}{\mathcal{P}}^{\text{ind}}{\tau }_{\text{ind}}/\left({\Delta}\eta {\lambda }_{\mathrm{h}}^{2}{\Delta}\beta \right)$ and ${\mathcal{P}}_{+}^{\text{col}}/{\mathcal{P}}^{\text{ind}}$ are shown in figure 5 for several size of the spin ensemble.

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Finally, it is worth mentioning that the (n + 2)/3-fold power enhancement reported in continuous machines using collective coupling with spins 1/2 [18] do not coincide with the above n(ns + 1)/(s + 1)-fold enhancement. However, it coincides with the first enhancement mentioned above for the extracted work per cycle. This emphasises that there are two different sources of enhancement in our model. One coming from alterations of the steady state properties of the spin ensemble due to collective coupling and bath-induced coherences [20, 38, 47], which appears in continuous engines [18, 19] in the form of a kind of 'steady-state' superradiance [48]. The other source of enhancement comes from the fast equilibration mentioned above, and is not present in continuous machines since they operate and remain in steady state. Note that this manifest non-equivalence between cyclic and continuous engines is not in contradiction with [49] since the setup considered here and in [18, 19] do not belong to the small action regime used in [49] to derive the equivalence between cyclic and continuous engines.

We show that collective coupling between spins and a thermal bath leads to a collective heat capacity which takes very different values when compared to the independent heat capacity—when all spin interact independently with the thermal bath. Beyond being a curiosity by itself, we show two applications. The first one is related to quantum thermometry. We show that using a probe made of spins collectively coupled to the sample can bring large precision enhancements at high temperature, which can be of interest in some technological or bio-medicinal applications. In terms of spin number n contained in the ensemble, the precision scales as ΔT ∼ 1/n, corresponding to the famous Heisenberg scaling. Interestingly, such Heisenberg scaling is achieved with a probe initially prepared in a pure classical state (namely thermal states at inverse temperature satisfying ℏω|β₀| ≫ 1) whereas it usually requires initial states with genuine quantum characteristics like squeezing [50, 51] or entanglement [52–54]. However, the price to pay is a potentially complex experimental setup in order to realise collective interactions between the spins and the sample, as commented in section 1. We also determine the critical temperature T_cr below which collective couplings stop being beneficial.

In a second part of the paper, we apply the results on collective heat capacity to Otto engines using an ensemble of n spins of dimension s as working medium and operating close to the Carnot efficiency. Our results show that the output power of a collective spin engine (when the spins interact collectively with the baths) is always larger than or equal to the output power of an independent spin engine (when the spins interact independently with the baths). The enhancement is two-fold: a steady-state effect relying on bath-induced coherences bringing a (ns + 1)/(s + 1)-fold enhancement and a fast equilibration bringing an additional n-fold enhancement. Combining both, the largest enhancements happen at high hot bath temperatures, reaching asymptotically levels of n(ns + 1)/(s + 1)-fold enhancements.

One can wonder if similar enhancements can happen in low-dissipative Carnot engines since the crucial role of heat capacity in such engines was recently pointed out [25]. While one can show that the role of the heat capacity remains prominent for an ensemble of spins interacting independently with the thermal baths, it is not obvious that it can be extended to spins interacting collective with the baths. Indeed, the dynamics of Tr ρJ_z does not follow a simple exponential decay so that a more complex treatment is needed [25], involving numerical methods. Such complexity emphasises that although it is quite intuitive to see the heat capacity playing a central role in thermal machines, for certain designs it is a highly non-trivial conclusion. Still, it would be interesting to try to overcome these obstacles to investigate collective couplings in low-dissipative Carnot engines as new phenomena might emerge.

Finally, one could say that the performances of the best case scenario, when the initial state belongs to the symmetrical subspace, can be reproduced by a larger single spin of dimension ns. Furthermore, one could add that since the dissipative dynamics of a single spin does not involve generation of coherences or quantum correlations, there is nothing genuinely quantum in the performances of the spin ensemble collectively coupled to the sample or thermal bath. While it is true that the best-case scenario performances of the spin ensemble reproduce the ones of a single spin of dimension ns, the comparison is a bit unfair because the systems are not the same. Classical and quantum performances should be compared with the same systems and the same resources. Beyond that, in any experimental implementations the size of the number of spins are limited. If one wants to increase the performances beyond this classical limitation, one can use collective bath couplings. Overall, our results advocate for increasing efforts towards experimental realisations of collective couplings.

CLL wants to thank Mohammad Mehboudi for interesting and enthusiastic discussions related to the thermometry part of the paper. This work is based upon research supported by the South African Research Chair Initiative, Grant No. UID 64812 of the Department of Science and Technology of the Republic of South Africa and National Research Foundation of the Republic of South Africa.

In this section we show that the maximal precision related to the temperature estimation when using collective interaction with the sample are determined by the collective heat capacity C^col(β). In order to show that we have to compute the quantum Fisher information associated with the collective steady state, ρ^ss(β). The first issue is that there is no general explicit expression of the quantum Fisher information for mixed states.

Before continuing we must introduce some concepts of quantum metrology. The quantum Fisher information is defined as the maximum over all possible measurements—described by a POVM—of the Fisher information [61]. The Fisher information represents the amount of information about the parameter of interest contained in the output statistics of a given measurement. For instance, let us consider a measurement described by the operators {E(m)}_m forming a positive-operator-valued measure (POVM). The probability distribution of the result m is given by

$$\begin{equation}p\left(m\vert T\right){:=}\mathrm{Tr}\enspace {\rho }^{\text{ss}}\left(T\right)E\left(m\right).\end{equation} \tag{ A.1 }$$

The information about the temperature T that one can extract from the measurement outputs statistics is the Fisher information

$$\begin{equation}{\mathcal{F}}_{E\left(m\right)}\left(T\right)=\sum _{m}\frac{1}{p\left(m\vert T\right)}{\left(\frac{\partial p\left(m\vert T\right)}{\partial T}\right)}^{2}.\end{equation} \tag{ A.2 }$$

The quantum Fisher information can be formally defined as ${\mathcal{F}}^{\text{col}}\left(T\right){:=}{\mathrm{M}\mathrm{a}\mathrm{x}}_{\left\{E\left(m\right)\right\}}\enspace {\mathcal{F}}_{E\left(m\right)}\left(T\right)$. However, the direct maximisation over all possible measurements is usually not tractable. One alternative involves purifications in larger Hilbert spaces [50, 62]. An other alternative is to use the 'symmetric logarithmic derivative' operator of ρ^ss(β), implicitly defined by [55–57]

$$\begin{equation}\frac{\partial }{\partial T}{\rho }^{\text{ss}}\left(\beta \right)=\frac{1}{2}{L}_{T}{\rho }^{\text{ss}}\left(\beta \right)+\frac{1}{2}{\rho }^{\text{ss}}\left(\beta \right){L}_{T},\end{equation} \tag{ A.3 }$$

which has the interesting property of providing the quantum Fisher information through the relation

$$\begin{equation}{\mathcal{F}}^{\text{col}}\left(T\right)=\mathrm{Tr}\enspace {\rho }^{\text{ss}}\left(\beta \right){L}_{T}^{2}.\end{equation} \tag{ A.4 }$$

For thermal states one can easily verify that ${L}_{T}=\frac{{\left(\hslash \omega \right)}^{2}}{{\left({k}_{\mathrm{B}}T\right)}^{2}}\left({J}_{z}-\langle {J}_{z}\rangle \right)$, which leads immediately that the maximal precision is determined by the variance of the energy or equivalently by the heat capacity—for thermal states. Beyond thermal states, it is in general very hard to find one symmetric logarithmic derivative operator, and unfortunately this includes the case of collective steady states. Still, one can show that

$$\begin{equation}\frac{\partial }{\partial T}{\rho }^{\text{ss}}\left(\beta \right)=\sum _{J={J}_{0}}^{ns}\sum _{i=1}^{{l}_{J}}\frac{{p}_{J,i}}{2}\left({L}_{T,J}{\rho }_{J,i}^{\text{th}}\left(\beta \right)+{\rho }_{J,i}^{\text{th}}\left(\beta \right){L}_{T,J}\right),\end{equation} \tag{ A.5 }$$

with ${L}_{T,J}={k}_{\mathrm{B}}^{-1}{T}^{-2}\left[\hslash \omega {J}_{z}-{\mathrm{e}}_{J}\left(\beta \right)\right]$, which is not of the form (A.3) but instead a sum of symmetric logarithmic derivative operators acting on each eigenspace J. Thus, the relation (A.4) does not hold automatically. Even though, using twice the Cauchy–Schwarz inequality one can show

$$\begin{align}\hfill {\mathcal{F}}^{\text{col}}\left(T\right)& {\leqslant}\sum _{J={J}_{0}}^{ns}\sum _{i=1}^{{l}_{J}}{p}_{J,i}\enspace \mathrm{Tr}\enspace {\rho }_{J,i}^{\text{th}}\left(\beta \right){L}_{\beta ,J}^{2}\hfill \\ \hfill & ={T}^{-2}\left[{\left(\hslash \omega \beta \right)}^{2}{\langle {J}_{z}^{2}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}-{\beta }^{2}\enspace \overline{{\mathrm{e}}_{J}^{2}\left(\beta \right)}\right]\hfill \\ \hfill & {\leqslant}{T}^{-2}{\left(\hslash \omega \beta \right)}^{2}{{\Delta}}^{2}{J}_{z}\hfill \end{align} \tag{ A.6 }$$

where $\overline{{\mathrm{e}}_{J}^{2}\left(\beta \right)}{:=}{\sum }_{J={J}_{0}}^{ns}{\sum }_{i=1}^{{l}_{J}}{p}_{J,i}\enspace {\mathrm{e}}_{J}^{2}\left(\beta \right)$ and ${{\Delta}}^{2}{J}_{z}{:=}\left[{\langle {J}_{z}^{2}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}-{\langle {J}_{z}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}^{2}\right]$.

Now that we have an upper bound, the next step is to show that there exists one measurement {E(x)}_x such that the associated Fisher information ${\mathcal{F}}_{\left\{E\left(x\right)\right\}}\left(T\right)$ saturates the upper bound. Naturally, one can think of energy measurements since it is the best measurement for thermal states [11]. The energy measurements is described by the following POVM ${{\Pi}}_{m}{:=}{\sum }_{J=\vert m\vert }^{ns}{\sum }_{i=1}^{{l}_{J}}{\vert J,m\rangle }_{i}\langle J,m\vert$, with m ∈ [−ns; ns] denoting the eigenvalues of the Hamiltonian H_S = ℏωJ_z of the spin ensemble. One can show that ${\mathcal{F}}_{{{\Pi}}_{m}}\left(t\right)$ does not reach the upper bound (A.6).

There is indeed one measurement which can extract more information about T than the energy measurement. This is the measurement described by Π_J,m,i := |J,m⟩_i⟨J, m|, which corresponds to the projection onto the collective states |J,m⟩_i. One can show easily that

$$\begin{equation}{\mathcal{F}}_{{{\Pi}}_{J,m,i}}={T}^{-2}\left[{\left(\hslash \omega \beta \right)}^{2}{\langle {J}_{z}^{2}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}-{\beta }^{2}\overline{{\mathrm{e}}_{J}^{2}\left(\beta \right)}\right].\end{equation} \tag{ A.7 }$$

This allows us to conclude that the upper bound (A.6) is indeed an equality,

$$\begin{equation}{\mathcal{F}}^{\text{col}}\left(T\right)={T}^{-2}\left[{\left(\hslash \omega \beta \right)}^{2}{\langle {J}_{z}^{2}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}-{\beta }^{2}\overline{{\mathrm{e}}_{J}^{2}\left(\beta \right)}\right].\end{equation} \tag{ A.8 }$$

Finally, one can also show that the collective heat capacity C^col(β), defined in (6), can alternatively be expressed as ${C}^{\text{col}}\left(\beta \right)={k}_{\mathrm{B}}\left[{\left(\hslash \omega \beta \right)}^{2}{\langle {J}_{z}^{2}\rangle }_{{\rho }^{\text{ss}}\left(\beta \right)}-{\beta }^{2}\overline{{\mathrm{e}}_{J}^{2}\left(\beta \right)}\right]$, so that

$$\begin{equation}{\mathcal{F}}^{\text{col}}\left(T\right)={k}_{\mathrm{B}}{\beta }^{2}{C}^{\text{col}}\left(\beta \right),\end{equation} \tag{ A.9 }$$

as announced in the main text.

One remark is in order. The optimal measurement yielding an information equal to the quantum Fisher information is ${\left\{{{\Pi}}_{J,m,i}\right\}}_{J,m,i}$, which is a non-local measurements. Therefore, it is not really realistic to consider that it is actually possible to experimentally saturates the quantum Fisher information and the estimate the temperature's sample at the corresponding precision. However, in the best case scenario where the spin ensemble initially belongs to the symmetrical subspace, like in particular for thermal states at extreme inverse temperature |β₀| ≫ 1/ℏω, the energy measurement ${\left\{{{\Pi}}_{n}\right\}}_{m}$ indeed yields an information equal to the quantum Fisher information. Then, the precision announced in the main text equation (15) is achievable experimentally, at least from the point of view of the measurements.

In this section we show that, for initial state belonging to the symmetrical subspace, collective interaction with the bath yields a dissipation timescale n times shorter than independent dissipation. We start from the dynamics of the collective dissipation provided in equation (1). Assuming that the ensemble is initially in a thermal state, it is initially diagonal and will remain diagonal in the collective basis {|J,m⟩_i}, J ∈ [J₀; ns], m ∈ [−J; J], i ∈ [1; l_J]. Therefore, the dynamics is given by the populations p_J,m,i := _i⟨J, m|ρ|J,m⟩_i only. Using the relation [41]

$$\begin{equation}{J}^{{\pm}}{\vert J,m\rangle }_{i}=\hslash \sqrt{\left(J\mp m\right)\left(J{\pm}m+1\right)}{\vert J,m{\pm}1\rangle }_{i},\end{equation} \tag{ B.1 }$$

one obtains

$$\begin{align}\hfill {\dot {p}}_{J,m,i}& =G\left(\omega \right)\left[\left(J-m\right)\left(J+m+1\right){p}_{J,m+1,i}-\left(J+m\right)\left(J-m+1\right){p}_{J,m,i}\right]\hfill \\ \hfill & \quad +G\left(-\omega \right)\left[\left(J+m\right)\left(J-m+1\right){p}_{J,m-1,i}-\left(J-m\right)\left(J+m+1\right){p}_{J,m,i}\right],\hfill \end{align} \tag{ B.2 }$$

where G(ω) := Γ(ω) + Γ*(ω), and Γ(ω) is the 'half Fourier transform' of the bath correlation function introduced in (1). By contrast, the dynamics of the independent dissipation follows

$$\begin{equation}\frac{\mathrm{d}\rho }{\mathrm{d}t}={\Gamma}\left(\omega \right)\sum _{i}^{n}\left({j}_{i}^{-}\rho {j}_{i}^{+}-{j}_{i}^{+}{j}_{i}^{-}\rho \right)+{\Gamma}\left(-\omega \right)\sum _{i}^{n}\left({j}_{i}^{+}\rho {j}_{i}^{-}-{j}_{i}^{-}{j}_{i}^{+}\rho \right)+\mathrm{h}.\mathrm{c}..\end{equation} \tag{ B.3 }$$

Again, since the ensemble is assumed to be initially in a thermal state, all coherences are and remain null (this would not be true for collective dissipation, reason why the collective basis was used there). Then, the independent dissipation is described by the populations only,

$$\begin{align}\hfill {\dot {p}}_{{m}_{1},\dots ,{m}_{n}}& {:=}\langle {m}_{1},\dots ,{m}_{n}\vert \dot {\rho }\vert {m}_{1},\dots ,{m}_{n}\rangle \hfill \\ \hfill & =G\left(\omega \right)\sum _{i=1}^{n}\left[\left(s-{m}_{i}\right)\left(s+{m}_{i}+1\right){p}_{{m}_{1},\dots ,{m}_{i}+1,\dots ,{m}_{n}}-\left(s+{m}_{i}\right)\left(s-{m}_{i}+1\right){p}_{{m}_{1},\dots ,{m}_{i},\dots ,{m}_{n}}\right]\hfill \\ \hfill & \quad +G\left(-\omega \right)\sum _{i=1}^{n}\left[\left(s+{m}_{i}\right)\left(s-{m}_{i}+1\right){p}_{{m}_{1},\dots ,{m}_{i}-1,\dots ,{m}_{n}}-\left(s-{m}_{i}\right)\left(s+{m}_{i}+1\right){p}_{{m}_{1},\dots ,{m}_{i},\dots ,{m}_{n}}\right].\hfill \end{align} \tag{ B.4 }$$

One can see that the non-zero coefficients appearing in (B.2), which determines the rate of each transition and consequently the timescale of the dissipation, range from 2G(±ω)J to G(±ω)J(J + 1) for m ∈ [−J, J]. By contrast, the rates of transition in (B.4) range from 2G(±ω)s to G(±ω)s(s + 1). One recovers in particular that the equilibration timescale (for independent dissipation) is of the order $G{\left(\omega \right)}^{-1}\sim {\left({g}^{2}{\tau }_{\mathrm{c}}\right)}^{-1}$, where τ_c is the bath correlation time. Since J can take value from 0 or 1/2 to ns, the timescale to reach the steady state is in general of the same order or even larger for collective dissipation than for independent dissipation. However, for a thermal state at initial inverse temperature ℏω|β₀| ≫ 1 (or more generally belonging to the symmetrical subspace), all components of J < ns are null implying that all transition rates involved in the collective dissipation are at least n times larger than the transition rates of the independent dissipation. Then, for initial states such that ℏω|β₀| ≫ 1 the collective dissipation happens on a timescale at least n times faster than the independent dissipation. Note that this interesting result shares some similarities with superradiance [48]. Indeed, both phenomena rely on the collective damping rates appearing in (B.2), larger than the independent damping rates of (B.4) for states confined to the symmetrical subspace (J = ns). However, superradiance is an increase of the rate of energy change whereas this 'super-equilibration' is purely about the dynamics of the spin ensemble's state itself. Note also that superradiance appears for states |J = ns,m⟩_i with m close the zero while the accelerated equilibration occurs for any states in the symmetrical subspace. As a consequence of this accelerated equilibration, the timescale τ_col of one cycle of the collective Otto machine can be reduced by a factor n compared to the timescale τ_ind of the independent Otto machine.

For very small imperfections like tiny spin-spin interactions or inhomogeneities and disorder altering the local energy levels of each spin, both tending to break down the spin exchange symmetry (or equivalently, the spin indistinguishability), it was shown in [20] that the steady state (3) was still reached as long as δ ≪ g²τ_c, where δ stands for the order of magnitude of the energy involved in the imperfections, g is the coupling strength with the bath, and τ_c is the bath correlation time. Furthermore, in the applications to thermometry and engines we are mostly interested in initial states such that p_J=ns ≃ 1 for which the equilibration time is of the order of (see appendix B). Therefore, for such initial states, the condition on the magnitude of the imperfections is relaxed to δ ≪ ng²τ_c.

In this section we detail briefly the derivation of the expression of the output work per cycle. During the isochoric stroke in contact with the hot bath, the spin ensemble is brought to the state ρ₁ := ρ^ss(T_h, λ_h). The next stroke is isentropic, preserving the state of the spin ensemble while realising the relaxation λ_h → λ_c. Then, follows the second isochoric stroke, taking the spin ensemble to ρ₂ := ρ^ss(T_c, λ_c). The last isentropic stroke is a compression λ_c → λ_h, closing the cycle. Note that, as mentioned in the main text, the initial weights p_J,i are preserved throughout the cycles and have a crucial impact on the properties and performances of the engine. The work W^col extracted per cycle by the engine is the sum of the work realised during the two isentropic strokes,

$$\begin{align}\hfill {W}^{\text{col}}& =\mathrm{Tr}\enspace {\rho }_{1}\left[H\left({\lambda }_{\mathrm{c}}\right)-H\left({\lambda }_{\mathrm{h}}\right)\right]+\mathrm{Tr}\enspace {\rho }_{2}\left[H\left({\lambda }_{\mathrm{h}}\right)-H\left({\lambda }_{\mathrm{c}}\right)\right]\hfill \\ \hfill & =\hslash \omega \left({\lambda }_{\mathrm{c}}-{\lambda }_{\mathrm{h}}\right)\mathrm{Tr}\enspace {J}_{z}\left[{\rho }^{\text{ss}}\left({T}_{\mathrm{h}},{\lambda }_{\mathrm{h}}\right)-{\rho }^{\text{ss}}\left({T}_{\mathrm{c}},{\lambda }_{\mathrm{c}}\right)\right]\hfill \\ \hfill & =\left({\lambda }_{\mathrm{c}}-{\lambda }_{\mathrm{h}}\right)\left[{E}^{\text{ss}}\left({\theta }_{\mathrm{h}}\right)-{E}^{\text{ss}}\left({\theta }_{\mathrm{c}}\right)\right],\hfill \end{align} \tag{ D.1 }$$

where θ_x := λ_xβ_x, for x = h, c. Note that the only way of having work extraction (W^col < 0) is with compression factors satisfying the condition $1{< }\frac{{\lambda }_{\mathrm{h}}}{{\lambda }_{\mathrm{c}}}{< }\frac{{\beta }_{\mathrm{c}}}{{\beta }_{\mathrm{h}}}$. The work extraction efficiency is defined as $\eta {:=}-\frac{{W}^{\text{col}}}{{Q}_{\mathrm{h}}}$, where Q_h = Tr H(λ_h)[ρ^ss(T_h, λ_h) − ρ^ss(T_c, λ_c)] is the heat transferred from the hot bath to the spin ensemble. One recovers the usual expression for the efficiency, $\eta =1-\frac{{\lambda }_{\mathrm{c}}}{{\lambda }_{\mathrm{h}}}$, and the difference with the Carnot efficiency ${\eta }_{\mathrm{c}}=1-\frac{{\beta }_{\mathrm{h}}}{{\beta }_{\mathrm{c}}}$ is ${\Delta}\eta =\frac{{\lambda }_{\mathrm{c}}}{{\lambda }_{\mathrm{h}}}-\frac{{\beta }_{\mathrm{h}}}{{\beta }_{\mathrm{c}}}$. One can rewrite the output work in terms of Δη to obtain

$$\begin{equation}{W}^{\text{col}}={\lambda }_{\mathrm{h}}\left({\Delta}\eta +\frac{{\beta }_{\mathrm{h}}}{{\beta }_{\mathrm{c}}}-1\right)\left[{E}^{\text{ss}}\left({\theta }_{\mathrm{h}}\right)-{E}^{\text{ss}}\left({\theta }_{\mathrm{h}}+{\lambda }_{\mathrm{h}}{\beta }_{\mathrm{c}}{\Delta}\eta \right)\right].\end{equation} \tag{ D.2 }$$

Taking the limit of near Carnot efficiency, Δη → 0, the output work takes the form

$$\begin{equation}{W}^{\text{col}}=-{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{{C}^{\text{col}}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right).\end{equation} \tag{ D.3 }$$

One can repeat the same reasoning with the alternative situation where each spin interacts independently with the baths. One obtains the expression found in [24]

$$\begin{equation}{W}^{\text{ind}}=-{\Delta}\eta {\lambda }_{\mathrm{h}}^{2}\left({\beta }_{\mathrm{c}}-{\beta }_{\mathrm{h}}\right)\frac{{C}^{\text{ind}}\left({\theta }_{\mathrm{h}}\right)}{{k}_{\mathrm{B}}{\theta }_{\mathrm{h}}^{2}}+\mathcal{O}\left({\Delta}{\eta }^{2}\right).\end{equation} \tag{ D.4 }$$

These are the expressions used in the main text. Note that in the main text we consider implicitly the absolute value of the extracted work so that the front minus sign is dropped in the above expressions.