Coherences and the thermodynamic uncertainty relation: Insights from quantum absorption refrigerators

QTMs can be modeled as open quantum systems consisting of a central system ($s$) coupled to multiple (counted by $v$) bosonic heat baths ($b$). The total system-bath Hamiltonian reads (setting $\hbar=1$ and $k_{B}=1$ hereafter)

Here and in what follows, we imply the tensor product with the identity so that $\hat{H}_{s}$ will be used instead of $\hat{H}_{s}\otimes\mathbb{I}_{b}$ with $\mathbb{I}_{b}$ an identity matrix in the bath subspace and so on. The working substance $\hat{H}_{s}$ constitutes a few-level quantum system. The bosonic heat baths are assumed to be harmonic,

with $\hat{b}_{kv}^{\dagger}$ ($\hat{b}_{kv}$) creating (annihilating) a harmonic mode $k$ with frequency $\omega_{kv}$ in $v$ bath. Here, we assume that each bath is described by a thermal equilibrium state characterized by a temperature $T_{v}$. $\hat{S}_{v}$ and $\hat{B}_{v}$ are system and reservoir operators that form the coupling between the system and $v$ bath, respectively. We consider bilinear system-bath interactions and take $\hat{B}_{v}$ to be the displacement operators,

with $\lambda_{kv}$ characterizing the coupling strength between the system and the $v$ reservoir.

To study the thermodynamics of the generic QTMs defined above, we combine the full counting statistic formalism Levitov and Lesovik (1993); Levitov et al. (1996) with the Redfield master equation Redfield (1957, 1965). This formalism was recently described in details in Ref. Friedman et al. (2018). To do so, we assign each bath a counting field $\chi_{v}$. The moment generating function is defined with the two-time measurement protocol as Esposito et al. (2009)

Here, operators are written in the Heisenberg picture. $\hat{\rho}(0)$ denotes the initial factorized density matrix of the system (s) and bath (b), $\hat{\rho}(0)=\hat{\rho}_{s}(0)\otimes\hat{\rho}_{b}(0)$; $\hat{\rho}_{b}(0)=\prod_{v}\exp(-\beta_{v}\hat{H}_{v})/Z_{v}$ with $\beta_{v}=T_{v}^{-1}$ and $Z_{v}$ the inverse temperature and partition function for the $v$ bath, respectively. After some simple manipulations, we arrive at Friedman et al. (2018)

where the counting-field dressed total density matrix reads

with $\hat{H}^{-\chi}\equiv e^{-i\sum_{v}\chi_{v}\hat{H}_{b}^{v}/2}\hat{H}e^{i\sum_{v}\chi_{v}\hat{H}_{b}^{v}/2}$ denoting the counting-field dressed total Hamiltonian. Notice that we have $[\hat{\rho}^{\chi}]^{\dagger}=\hat{\rho}^{-\chi}$. Equation (6) can be written as a differential equation, generalizing the Liouville Equation for the density matrix (in the Schrödinger picture),

The moment generating function is obtained by solving this equation of motion, then tracing $\hat{\rho}^{\chi}$.

Using the explicit form given by Eq. (1), we get

where $\hat{B}_{v}^{-\chi_{v}}=e^{-i\chi_{v}\hat{H}_{b}^{v}/2}\hat{B}_{v}e^{i\chi_{v}\hat{H}_{b}^{v}/2}=\sum_{k}\lambda_{kv}(e^{-i\chi_{v}\omega_{kv}/2}\hat{b}_{kv}^{\dagger}+\mathrm{H.c.})$ with ‘H.c.’ denoting Hermitian conjugate hereafter. Proceeding from Eq. (7) in the interaction representation, treating the counting-field dressed system-bath coupling as a perturbation, and then transforming back to the Schrödinger picture, the reduced density matrix dynamics $\rho_{s}^{\chi}(t)\equiv{\rm Tr}_{b}[\rho^{\chi}(t)]$ can be described by the following $\chi$-RME in the energy basis $\{|n\rangle\}$ of $\hat{H}_{s}$ (detailed derivation can be found in, e.g., Ref. Friedman et al. (2018)):

Here, $\rho_{s,nm}^{\chi}(t)\equiv\langle n|\rho_{s}^{\chi}(t)|m\rangle$, $\Delta_{ij}=E_{i}-E_{j}$ are energy gaps with $E_{i}$ eigenenergies of the subsystem in the global (eigenenergy) basis. The superscript ‘$\ast$’ denotes complex conjugate. The standard Redfield equation is recovered when counting parameters are taken to zero, $\mathcal{R}_{nm,lk}^{v}(\omega)=\left.\mathcal{R}_{nm,lk}^{\chi_{v}}(\omega)\right|_{\chi_{v}=0}$. The transition coefficients satisfy

with $\Omega_{v}(\tau)\equiv\langle\hat{B}_{v}(\tau)\hat{B}_{v}(0)\rangle_{b}$ denoting the bath correlation function evaluated using the state $\hat{\rho}_{b}(0)$. Its explicit form reads

with $\gamma_{v}(\omega)=2\pi\sum_{k}\lambda_{kv}^{2}\delta(\omega-\omega_{kv})$ and $n_{B}^{v}(\omega)$ the spectral density and Bose-Einstein distribution of $v$ bath, respectively. Without loss of generality, here we consider an Ohmic function $\gamma_{v}(\omega)=\alpha_{v}\omega e^{-\omega/\omega_{c}}$; $\alpha_{v}$ is a dimensionless system-bath coupling strength. We assume all baths have the same cutoff frequency $\omega_{c}$, which defines the largest energy scale in the problem.

With Eq. (II.2), we can evaluate the above transition coefficients as $\mathcal{R}_{nm,lk}^{\chi_{v}}(\omega)=S_{v}^{nm}S_{v}^{lk}\Gamma_{\chi_{v}}(\omega)$ with

Here, ‘$|A|$’ takes the absolute value of $A$. We neglected the Lamb shifts, which is justified in the weak system-bath coupling limit. To assess the role of nonzero system coherences, we contrast the full $\chi$-RME given by Eq. (9) against its secular counterpart in which coherences between energy eigenstates vanish Friedman et al. (2018),

Hereafter, we refer to Eq. (15) as the secular $\chi$-RME with the understanding that it represents the incoherent limit of the full $\chi$-RME. Below, we use “secular limit” and “incoherent limit” interchangeably.

To obtain the current and its higher-order cumulants, we recast the $\chi$-RME in the Liouville space as

where $|\rho_{s}^{\chi}\rangle\rangle\equiv(\rho_{s,11}^{\chi},\rho_{s,12}^{\chi},\cdots,\rho_{s,nm}^{\chi},\cdots,\rho_{s,NN}^{\chi})^{T}$ denotes an $N^{2}\times 1$ vector with $N$ the dimension of the system Hilbert space. $\mathbb{L}_{\chi}$ is an $N^{2}\times N^{2}$ matrix representing the $\chi$-dependent Liouvillian superoperator (noting that $\mathbb{L}_{\chi}$ reduces to an $N\times N$ matrix in the secular limit). In the steady state limit, the cumulant generating function (CGF) $G(\chi)\leavevmode\nobreak\ =\leavevmode\nobreak\ \lim_{t\to\infty}\ln\mathcal{Z}(\chi,t)/t$ is given by Esposito et al. (2009)

where $\mathcal{E}_{0}(\chi)$ is the ground-state energy (or the eigenvalue of the smallest real part) of the superoperator $\mathbb{L}_{\chi}$. In scenarios with multiple bath, $\chi$ should be understood as a collection of counting fields, namely, $\chi=\{\chi_{v}\}$.

We note that $\mathcal{E}_{0}(0)=0$, that is, without counting the smallest eigenstate is zero, corresponding to the steady state solution. The CGF supplies all cumulants, specifically the steady state heat current $\langle J_{v}\rangle$ out of the $v$th reservoir and its fluctuation $\langle\langle J_{v}^{2}\rangle\rangle\equiv\langle J_{v}^{2}\rangle-\langle J_{v}\rangle^{2}$,

Here, $\mathcal{E}_{0}(-\chi)$ is the ground-state energy of $\mathbb{L}_{-\chi}$. In numerical simulations presented below, we adopt a real value of $\chi_{v}=0.001-0.005$ to calculate $\langle J_{v}\rangle$ and $\langle\langle J_{v}^{2}\rangle\rangle$ according to the second lines of the above definitions. We verified that the final results of $\langle J_{v}\rangle$ and $\langle\langle J_{v}^{2}\rangle\rangle$ were independent of the value of $\chi_{v}$ we set.

With the ability to calculate currents and fluctuations, we further study the validity of the thermodynamic uncertainty relation (TUR) Barato and Seifert (2015); Gingrich et al. (2016)

with $\langle\sigma\rangle=-\sum_{v}\langle J_{v}\rangle\beta_{v}$ the total entropy production rate. Here we adopt the convention that $\langle J_{v}\rangle>0$ when flowing into the system. In what follows, we refer to the quantity ‘$\frac{\langle\langle J_{v}^{2}\rangle\rangle}{\langle J_{v}\rangle^{2}}\langle\sigma\rangle$’ as the TUR ratio. Note that below we find that our models for QAR always satisfy the original TUR (19). As such, we obviously satisfy the generalized—and less tight bounds as discussed in Refs. Guarnieri et al. (2019); Horowitz and Gingrich (2019).

In sum, in our procedure we construct the Liouvillians $\mathbb{L}_{\chi}$ for both the full Redfield and the secular Redfield equations and find their smallest eigenvalues, the CGF. We obtain the steady state heat current from bath $v$ and the current noise by numerically calculating the first and second derivatives of the CGF, respectively, taken with respect to the counting parameter of bath $v$.

Numerous studies in past years had examined the regime of validity and accuracy of second-order Markovian quantum master equations, specifically comparing the Redfield master equation to the local Lindblad master equation (LLME), which is performed in the site basis, and the eigenbasis Lindblad master equation (ELME) Wichterich et al. (2007); Levy and Kosloff (2014); Purkayastha et al. (2016); Hofer et al. (2017); González et al. (2018); Chiara et al. (2018); Cattaneo et al. (2019), which is performed in the global basis. The Redfield equation reduces to the ELME after making the secular approximation. The LLME is derived in the site basis, and it is known to miss proper thermalization further showing incorrect transport properties Levy and Kosloff (2014). Comparing the predictions of the RME to exact results (when available), it has been generally concluded that the RME is superior over both the ELME and the LLME Wichterich et al. (2007); Purkayastha et al. (2016); González et al. (2018).

On the other hand, the Redfield dissipator does not necessarily satisfy the condition of complete positivity (unlike the Lindblad dissipator). How does this deficiency impact thermodynamical properties? To the best of our knowledge, there are no reports on the impact of deviations from complete positivity on steady state behavior. However, for a driven system it was shown in Ref. Argentieri et al. (2014) that the departure from complete positivity lead to the violation of the second law of thermodynamics, with a negative entropy production at intermediate times.

For a system coupled to three baths ($h,w,c$) with no internal leaks, the cumulant generating function satisfies the steady state exchange fluctuation symmetry (SSFS) Esposito et al. (2009); Campisi et al. (2011); Segal (2018); Andrieux et al. (2009); Liu et al. (2018),

A fundamental question that is still not settled is whether the full $\chi$-RME satisfies this relation, thus is consistent with nonequilibrium thermodynamics. The secular limit of the $\chi$-RME, that is the eigenbasis Lindblad master equation, satisfies the SSFS for QAR Segal (2018). On the other hand, if the fluctuations symmetry is not satisfied in the full $\chi$-RME, it is important to find out what is the impact of this deviation on the accuracy of calculated cumulants of transport. In other words, whether this deviation influences the small $\chi$ behavior. This point is critical to our analysis since we are using the $\chi$-RME to calculate the current and its fluctuations under the influence of quantum coherences.

To address this issue we point out the following: (i) We are only interested in autonomous (non-driven) systems, and in their steady state properties. We are not aware of examples showing that the RME is thermodynamically improper in this case (unlike the transient regime). (ii) Across all parameters regimes that we had analyzed here, we found that levels population were positive (physical). (iii) The V-shaped model [see Eq. (22)] captures the behavior of coherences in the 4-level QAR, as discussed in Ref. Kilgour and Segal (2018). Below, we test the fluctuation symmetry in this (two-bath) model and show that it is obeyed with a small numerical error. While it is intriguing to prove that the $\chi$-RME satisfies the SSFS, our numerical simulations provide a strong support for this assertion, backing up our calculations of the current and its noise with Eq. (II.2). (v) We tested the entropy production rate in our simulations, and found that for results presented below it was always positive.

In fact, in all the tests that we carried out in this study, which were steady state calculations, we observed that the RME lead to positive level population, positive entropy production rate, and the validity of the SSFT. Our simulations provide a strong numerical support for the thermodynamical consistency of the RME in steady state, even when coherences play a critical role, calling for analytic investigations.

For simplicity, we now consider a model involving just two heat baths ($v=c,\leavevmode\nobreak\ h$). In steady state, the input heat at the terminal $v=h$ is equal to the heat output at $v=c$. As such, it is sufficient to look at the fluctuation symmetry of heat exchange with a single counting parameter. The exchange fluctuation symmetry states the following relation for the cumulant generating function Esposito et al. (2009); Campisi et al. (2011),

with $\Delta\beta=1/T_{h}-1/T_{c}$.

We simulate the V-shaped system coupled to two heat baths as an example, given that its behavior closely resembles that of Model I below [see Fig. 4 (a)] for the QAR Kilgour and Segal (2018). The system Hamiltonian involves a ground ($g$) state $|g\rangle$ and two excited ($e$) states $\{|e_{1}\rangle,|e_{2}\rangle\}$, which are coherently coupled,

Here $\epsilon_{g(e)}$ are the ground (excited) state energy. $\mu$ is the coupling strength between the two excited states and its magnitude can be made arbitrarily large. The transitions between ground and excited states are driven by the cold ($c$) and hot ($h$) heat baths with the following system operators associated with $\hat{H}_{sb}$:

First, we verify Eq. (21) in the secular limit using Eq. (15) for which analytic treatments exist (see, for instance, Ref. Segal (2018)). In simulations, we determine the cumulant generating function by calculating numerically the eigenvalue with the smallest real part, Eq. (17). The dependence of $G(\chi_{c})$ and $G(\Delta\beta-\chi_{c})$ on the counting field $\chi_{c}$ and inter-state coupling strength $\mu$ is depicted in Fig. 1. By comparing the upper and lower panels of Fig. 1, it is evident that our simulations preserve the fluctuation symmetry Eq. (21) in the secular limit. Deviations between $G(\chi_{c})$ and $G(\Delta\beta-\chi_{c})$ were 11 orders of magnitude smaller than their value, for both real and imaginary parts (a quantitative analysis is included in Fig. 3). Since we know that the SSFS is obeyed in the secular limit, we reason that these deviations arise from numerical errors from the various stages involved in the procedure, such as matrix inversion. Particularly, we find that the real (imaginary) part of the cumulant generating function is an even (odd) function of the counting field $\chi_{c}$, hence Eq. (II.2) allows us to get real values for the current and its noise.

Next, we turn to simulations with the full $\chi$-RME Eq. (9). The comparison between $G(\chi_{c})$ and $G(i\Delta\beta-\chi_{c})$ is shown in Fig. 2. We observe that the fluctuation symmetry Eq. (21) is preserved by the full $\chi$-RME, thereby suggesting the thermodynamic consistency of the method for evaluating the currents and fluctuations from the cumulant generating function according to Eq. (II.2). Interestingly, we find that the magnitude of the cumulant generating function obtained using the full Redfield dynamics is suppressed in the weak coupling regime of $\mu$ compared with the secular limit, as illustrated in Fig. 1. We note that Ref. Kilgour and Segal (2018) demonstrated current suppression in the same model arising due to the presence of finite system coherence for weak $\mu$. Our results further imply that the so-observed suppression occurs at the level of cumulant generating function and persists for finite $\chi_{c}$.

We interrogate the SSFS in a quantitative way in Fig. 3 by looking at the deviation of the ratio $G(\chi_{c})/G(i\Delta\beta-\chi_{c})$ from unity, separately for the real and imaginary parts. The errors in the secular $\chi$-RME are certainly numerical since the SSFS is obeyed in this case. Turning to the full $\chi$-RME, we note higher deviations from unity for both real and imaginary parts compared to the secular case. However, deviations from perfect symmetry do not depend on the intersite coupling $\mu$ or the counting parameter $\chi$. Particularly for $\mu$, coherence effects show up in this model for $\log\mu\lesssim-3.5$ Kilgour and Segal (2018). The fact that the agreement between $G(\chi_{c})$ and $G(i\Delta\beta-\chi_{c})$ does not depend on $\mu$ suggests that the error is accumulated by numerical operations rather than reflecting a fundamental violation.

Note that the secular $\chi$-RME calculation on the V-shaped model is performed by studying the eigenvalue of a $3\times 3$ matrix. In contrast, the full $\chi$-RME calculation is performed by diagonalizing a $9\times 9$ matrix. As such, the computational effort, and thus error accumulation in these two cases is quite different.

To the best of our knowledge, simulations in Figs. 2-3 are the first strong numerical indication of the validity of the SSFS in the full Redfield formalism for multi-level systems, and for arbitrarily large $\chi$. We highlight that in fact we found that all eigenvalues of $\mathbb{L}_{\chi}$ obey the SSFS symmetry. Finally, predictions from the $\chi$-RME are obviously not exact given the perturbation approximation involved. In validating the SSFS for the $\chi$-RME we point out that though inaccurate, calculations of high order cumulants are physical (thermodynamically consistent).

In the local site basis, the working medium of Model I (see Fig. 4 (a)) is described by a Hamiltonian

The system includes a ground (g) state $|g\rangle$, an excited (e) state $|e\rangle$, and two degenerate intermediate (I) levels ($|\mathrm{I}_{1}\rangle,\leavevmode\nobreak\ |\mathrm{I}_{2}\rangle$) connected by a coherent hoping rate $\mu$. Note that the levels are degenerate in the local basis, and non-degenerate in the global basis. We set the reference energy at $\epsilon_{g}=0$. The system’s operators involved in the system-bath interaction $\hat{H}_{sb}$ have the forms

After ordering the labels of eigenstates of $\hat{H}_{s}$ such that $\hat{H}_{s}=\sum_{i=1}^{4}E_{i}|i\rangle\langle i|$ with $E_{i}<E_{i+1}$, we rewrite the above system operators in the energy basis

In Fig. 5 we display simulation results for the current $\langle J_{c}\rangle$ (when referring to as cooling power we implicitly imply $\langle J_{c}\rangle>0$) and its noise $\langle\langle J_{c}^{2}\rangle\rangle$. From panels 5 (c) and (d) or (h) it is evident that finite system coherences, as quantified by the real or imaginary part of the steady state density matrix element $\rho_{s,23}$ is responsible for the suppression of the cooling power in the weak $\mu$ regime relative to the secular limit Kilgour and Segal (2018). Note that the deep blue background in panels (a) and (b) marks the no-cooling region, with $\langle J_{c}\rangle<0$. In other words, for presentation purposes we present the no-cooling region as zero cooling currents. The cooling region is marked in Fig. 6, where it appears when $\epsilon_{I}\lesssim 0.332$. Note that for the secular limit, results in all figures here and below were independent of $\mu$ for the small $\mu$ considered.

In Fig. 5 (e)-(f), we further show the current fluctuations. We find from panel 5 (e) and (f) that fluctuations become pronounced in the non-cooling region. Interestingly, the system coherence suppresses the current fluctuation but in the non-cooling region, as can be seen from the comparison between Fig. 5 (e) and (f). Noting that the signs of system coherence in the cooling and non-cooling regions are opposite [see Fig. 5 (d) and (h)], it is then clear that in Model I negative coherence in the cooling region induces power suppression, while positive coherence in the non-cooling region is responsible for the suppression of current fluctuations. Nevertheless, as we are interested in the cooling region, such a fluctuation suppression is of no practical use. Intriguingly, the marginal regions between finite coherences in Fig. 5 (d) and (h) marks the boundary of cooling and non-cooling regions, namely, the transition from cooling to non-cooling regions is associated with a sign change of system coherence.

We now look at the relative noise, $\langle\langle J_{c}^{2}\rangle\rangle/\langle J_{c}\rangle^{2}$ and the TUR ratio $\langle\langle J_{c}^{2}\rangle\rangle\langle\sigma\rangle/\langle J_{c}\rangle^{2}$, see Fig. 6. In panel 6 (a) we observe that the relative noise obtained from the full $\chi$-QME is greatly enhanced due to the presence of coherences in both the cooling and non-cooling regions, compared to the secular limit. Ideally, the relative noise tends to infinity at the exact boundary of cooling and non-cooling regions as $\langle J_{c}\rangle=0$, however, we can only see finite peak structures centered around the boundary from panel 6 (a) since we utilize discretized values for $\epsilon_{I}$ in simulations and may not be able to reach the exact boundary.

The TUR ratio is plotted in panel 6 (b). We observe that: (i) The TUR ratio is always above the classical bound set by the TUR Eq. (19). (ii) The TUR ratio saturates to the bound at the point when the entropy production is exactly zero and the refrigerator crosses into the no-cooling region. (iii) Coherences slightly reduce the TUR ratio, relative to the secular limit, yet the behavior is non-monotonic. Nevertheless, coherences only mildly reduce the TUR ratio from the secular limit. The calculation of the TUR ratio at the cooling-no-cooling boundary region is nontrivial numerically. This is due to the nontrivial cancellation between entropy production and the relative noise taking place when approaching the boundary. As such, very close to the boundary region the TUR ratio was not evaluated.

Back to the fundamental question as to whether the non-secular Redfield equation is suitable (thermodynamically consistent) for studying current noise. In Fig. 7 we verify that the entropy production rate is always positive with our parameters. In fact, we have not observed negative entropy production rates in any of our simulations. As such, we hypothesize that the Redfield equation is thermodynamically consistent in the steady state limit with $\langle\sigma\rangle>0$, beyond our case study. This result does not exclude the possibility of observing fundamental deficiencies with the Redfield equation in the transient regime.

Concluding this Section: The TUR, Eq. (19), is satisfied by Model I in the presence of system coherences. Notably, the TUR ratio of Model I is almost independent of the inter-site coupling strength $\mu$, and it almost coincides with that of the secular limit, thereby implying that the coherence-induced enhancement of the relative noise is compensated by the coherence-induced reduction of entropy production rate, which is proportional to the heat currents.

We turn to Model II for a QAR, as illustrated in Fig. 4 (b). Previously, Refs. Holubec and Novotný (2018, 2019) demonstrated that this model can have a coherence-induced enhancement of the cooling power; below we confirm this scenario and further show in the Appendix that this model also permits an adverse effect of system coherence on the cooling power, when varying the system-bath coupling strengths.

The working medium of Model II is described by a four level Hamiltonian in the local basis

The system involves a ground (g) state $|g\rangle$, an intermediate (I) state $|\mathrm{I}\rangle$ and two degenerate excited (e) states ($|e_{1}\rangle,\leavevmode\nobreak\ |e_{2}\rangle$) that are connected by a coherent hoping rate, $\mu$. We set the reference energy at $\epsilon_{g}=0$. After ordering the labels of eigenstates of $\hat{H}_{s}$ such that $\hat{H}_{s}=\sum_{i=1}^{4}E_{i}|i\rangle\langle i|$ with $E_{i}<E_{i+1}$, we get the following system operators, involved in the system-bath interaction $\hat{H}_{sb}$, in the energy basis Holubec and Novotný (2018)

The real-valued parameters $\alpha_{vk}$ are dimensionless; they are taken from the spectral density function $\gamma_{v}(\omega)$ in Eq. (14); recall that we model the spectral density function as $\gamma_{v}=\alpha_{v}\omega e^{-\omega/\omega_{c}}$. Here for convenience we absorbed $\alpha_{vk}$ into the definitions of $\hat{S}_{h,w}$ so as to allow scenarios where different transitions induced by the same bath are enhanced by different coupling strengths.

Fig. 8 depicts an example where system coherences boost the cooling power, as predicted in Refs. Holubec and Novotný (2018, 2019). This is highlighted in panel 8 (c), where we show the current difference $\langle J_{c}\rangle_{\mathrm{FR}}-\langle J_{c}\rangle_{\mathrm{S}}$ in both the cooling and no-cooling regions. As can be seen, the cooling power $\langle J_{c}\rangle_{\mathrm{FR}}$ in the cooling region is slightly enhanced compared with $\langle J_{c}\rangle_{\mathrm{S}}$ when $\mu$ is relatively small. We attribute this cooling power enhancement to the finite positive system coherence in that region as indicated in Fig. 8 (d) and (h). However, panel 8 (g) shows that this cooling power enhancement comes at the price of an enhanced power fluctuation. Similarly to Model I, here we also find that the marginal region between nonzero coherences (Fig. 8 (d) and (h)) marks the boundary between the cooling and the no-cooling regions.

While in Fig. 8 system coherences enhance the cooling current, in the Appendix we show the opposite effect within the same model, but using a different value for $\alpha_{h1}$. This coherence-induced suppression of the cooling power (similarly to Model I) becomes evident by combining the information of Fig. A1 (c) and (d) or (h).

Even though system coherences can either suppress or enhance the cooling power in Model II (depending on the system-bath coupling strengths), in Fig. 9 (a) we again observe that the relative noise obtained from the full $\chi$-RME is larger than that obtained from the secular $\chi$-RME. Furthermore, the relative noise approaches the secular limit near the boundary between the cooling and no-cooling regions. Nevertheless, from Fig. 9 (b) (together with Fig. A2 (b) in the Appendix) we find that the TUR holds as well in Model II. Interestingly, the TUR ratio is enhanced by coherences relative to the secular limit: In Fig. 9 (b), we see that the TUR ratio in the secular case is very close to the bound within the whole range of $\epsilon_{I}$, while it is factor of 3 greater in the coherent case.

Overall, we found that in model II the effect of coherences on the cooling current was very mild; we did not perform detailed simulations to identify region of more substantial cooling effect as this was not the objective of this work. Our main conclusion, which holds for all cases examined here is that while coherences may boost or suppress the cooling current, their effect on fluctuations is adverse, thus validating the standard TUR.