Synchronization Lower Bounds the Efficiency of Near-Degenerate Thermal Machines

Quantum coherence has been a vital resource [1] in quantum thermodynamics and has been a topic of intense research in the recent past [2, 3, 4, 5]. The presence of coherence typically boosts the performance metrics of thermal machines such as engines [6], refrigerators [7], and batteries [8, 9, 10, 11]. Moreover, coherences, which can be thought of as phases, have also been subject to resource counting studies from an information theoretic perspective [12]. In the case of non-degenerate systems, that possess diagonal steady states in the absence of driving, such coherences are usually generated using a coherent drive. Such systems do not allow coherences to interact and thus a thorough understanding of the performance of quantum thermal machines with interacting coherences is lacking.

Classically, interacting phases are well studied in coupled Kuramoto models where phase pulling can produce a well-known second-order phase transition involving cooperative phase locking called synchronization [13]. Besides this, differences in coupling of the Kuramoto model can lead to competition as well, giving rise to a variety of behavior such as anti-synchronization and chimera, a phenomenon explored even in the quantum regime [14]. Specifically, in quantum systems, such coupled phase oscillator models arise due to a variety of different reasons, one of them being degeneracies. Recently, for non-degenerate quantum systems coupled to diagonal baths, it was highlighted that synchronization measures fully account for all coherences in the system [15] and can help explain the performance of quantum thermal machines.

In contrast to coherently driven quantum systems, incoherently driven near-degenerate levels can generate bath-induced coherences. Since the energetic cost of generating coherences is (nearly) zero for making transitions between (near-)degenerate levels, such systems cause the thermodynamic analysis to be decoupled from synchronization. Moreover, the (near-)degeneracies cause such coherences to interact with each other and induce cooperation or competition between the different phases.

In an accompanying manuscript, we show that synchronization of driven, dissipative exactly-degenerate open quantum systems exhibit a crossover from cooperation to competition. In this manuscript, we highlight that this synchronous behavior aids in the understanding of nanoscale thermal machines. We show that while coherent driving of near-degenerate four-level thermal masers has a tendency to entrain the quantum system to the external drive, coupled coherences exert a simultaneous force that can be either competitive or cooperative. This interplay between cooperation and competition leads to a rich dynamical regime whose implications on the thermal maser are observed in terms of its performance metrics (power, heat, and efficiency) being bounded by the synchronization measure.

In Sec. II below, we begin with a discussion of the near-degenerate four-level thermal maser. Then, we present an analysis of competition and cooperation in four-level thermal masers taking into account near-degeneracy and bath-induced coherence in Sec. III. Then, in Sec. IV and Sec. V we connect the competition and cooperation to relevant thermodynamic quantities of the thermal machine, specifically its power and heat current. In [16], the coherent power output of a non-degenerate three-level maser was shown to be related to the measures of synchronisation and it was noted in [17] that this relationship does not hold for degenerate systems where there is no energetic cost of generating coherences. Furthermore such degeneracies were related to synchronisation blockade recently [18]. We investigate this relationship in detail and connect synchronization to steady state heat current. The latter is then used to derive lower-bound on the efficiency of near-degenerate thermal machines in the mutual coupling dominant regime. Finally, we summarize our main results in Sec. VI.

We begin with the analysis of a four-level thermal maser that operates under a temperature difference to create a population inversion as depicted in Fig. 1. The original Scovil–Schulz-DuBois thermal maser is a minimal setup that comprises of three-levels and is one of the first models of a thermal heat engine [19], which has been recently extended to four levels [20] and beyond [21, 22] to include effects of bath-induced coherence. As a generalization of this, we consider the four-level model consisting of a free Hamiltonian

with $\omega_{3}>\omega_{2}>\omega_{1}>\omega_{0}=0$. The near-degenerate levels ($\omega_{2},\omega_{3}$) are spread over a small interval $\Delta\equiv\omega_{3}-\omega_{2}\ll(\omega_{1}-\omega_{0})$ and $\Delta\ll(\omega_{2}-\omega_{1})$. The system is driven out-of-equilibrium by a hot (temperature $T_{h}$) and a cold bath (temperature $T_{c}$) as well as an external time-periodic drive of strength $\lambda$ and frequency $\Omega\approx(\omega_{2}-\omega_{1})$. The role of the baths is to establish population inversion between the near-degenerate manifold $\{\ket{2},\ket{3}\}$ and the first-excited state $\ket{1}$. Specifically, when population inversion is achieved, the external drive can easily trigger stimulated emission, thereby producing power, i.e. the system acts as an engine. On the other hand, if there is no population inversion, the system absorbs power from the external drive, i.e. the system acts as a refrigerator.

When the baths are weakly coupled to the machine working fluid $H_{0}$, the reduced dynamics of the system is governed by a quantum master equation for the reduced density matrix $\rho$ [20] that reads,

The operator $V(t)$ is a collective drive [22],

that stimulates transitions between all the near-degenerate energy levels and the first-excited state. The cold-bath dissipator $\mathcal{D}_{c}[\rho]$ takes the Gorini-Kassakowski-Sudarshan-Lindblad (GKSL) [23, 24] form

with jump operators $c_{1}=c_{2}^{\dagger}=\ket{0}\bra{1}$ and the decay rates satisfying local detailed balance $\Gamma_{c_{1}}=\gamma_{c}(1+n_{c})$ and $\Gamma_{c_{2}}=\gamma_{c}n_{c}$. Here, $\gamma_{c}$ is the coupling strength squared between the system and the cold bath.

The hot-bath dissipator $\mathcal{D}_{h}[\rho]$ connects the first excited state to the near-degenerate manifold. The near-degeneracy causes a breakdown of the secular approximation [25] and results in a Bloch-Redfield form of the dissipator [26, 27, 28, 29, 20],

The operators $h_{1}^{i}=h_{2}^{i}=\ket{0}\bra{i}$ and pair-wise decay rates $\Gamma_{1}^{ij}=P_{ij}\sqrt{\gamma_{h}^{i}\gamma_{h}^{j}}\,(1+n_{h}^{(j)})$ and $\Gamma_{2}^{ij}=P_{ij}\sqrt{\gamma_{h}^{i}\gamma_{h}^{j}}\,n_{h}^{(j)}$. Here, $\gamma_{h}^{i}$ denotes the squared coupling strength of the bath that induces transitions between the ground state $\ket{0}$ and the $i$th state of the near-degenerate manifold $\ket{i}$ ($i=2,3$). The mean-bosonic hot-bath distribution $n_{h}^{(i)}=(\exp[\omega_{i}/T_{x}]-1)^{-1}$ encodes information about the hot bath temperature $T_{h}$ (similar definition for the cold-bath with $\omega_{i}=\omega_{1}$). In this paper, we will throughout set $\gamma_{h}^{(2)}=\gamma_{h}^{(3)}=\gamma_{h}$ and work with the Redfield form that despite not being positive in all parameter ranges [30, 31] provides accurate results when used appropriately in the weak system-bath coupling regime [32, 33].

The coefficients $P_{ij}$ are elements of the symmetric correlation matrix [34]

where $|p_{ij}|\leq 1$. Above since we are focused on a four-level thermal maser, the matrix $P$ is a $2\times 2$ matrix with its off-diagonal elements $|p|\leq 1$. The elements $p$ originate from the dipole-alignment factor [28, 29] with $p=\mathbf{d_{2}}\cdot\mathbf{d_{3}}/|\mathbf{d_{2}}||\mathbf{d}_{3}|$ such that $\mathbf{d}_{i}=\braket{i}{\mathbf{d}}{i}$ and $\mathbf{d}$ being the dipole operator. Intuitively, $|p|$ is the strength of quantum interference between thermalization pathways $\ket{0}\leftrightarrow\ket{2}$ and $\ket{0}\leftrightarrow\ket{3}$ while its sign encodes whether the interference is constructive or destructive. The parameter $p$ determines the strength of the noise-induced coherence with $p=0$ yielding no coherence due to the hot bath and $|p|=1$ yielding maximum coherence. The above Redfield hot-bath dissipator takes the completely positive Lindblad form ¹¹1By ‘Lindblad’ we refer to the general Lindblad form $d\rho/dt=-i[H,\rho]+\sum_{i,j}K_{ij}(2A_{i}\rho A_{j}^{\dagger}-\{A_{j}^{\dagger}A_{i},\rho\}$ where $A_{i}\equiv h_{i}$ are the jump operators and $K_{ij}$ is the positive Kassakowski matrix. when $\Gamma_{\mu}^{ij}=\Gamma_{\mu}^{ji}~{}\forall~{}i,j=2,3$, which can be satisfied in the exact degenerate limit ($\Delta=0$ wherein $n_{h}^{(i)}=n_{h}$). In this limit, the Kossakowski matrix becomes proportional to the correlation matrix and thus positivity can be ensured if the correlation matrix is positive definite, i.e., for $-1\leq p\leq 1$. When the correlation matrix has a zero eigenvalue ($p=\pm 1$), the dissipator possesses a dark state [36, 37] that can lead to multiple steady states [38, 39].

The above quantum master equation (2) can be transformed to a rotating frame such that any operator $\tilde{O}=\exp[i\tilde{H}t]O\exp[-i\tilde{H}t]$ with

The above transformation eliminates the explicit time dependency from the coherent evolution and leaves the dissipators invariant leading to a quantum master equation in the rotating frame,

with $\tilde{V}=\lambda\sum_{j=2}^{3}\ket{j}\bra{1}+\text{h.c.}$. Throughout this work, we will focus our attention on the four-level thermal maser investigating the effects of near degeneracy $\Delta$ and the dipole-alignment factor $p$ (noise-induced coherence strength) on quantum synchronization and the thermodynamic observables of the thermal maser. In particular, unlike Ref. [22] we will not explore the effect of several degenerate levels (generalized Scovil–Schulz-DuBois thermal maser) and restrict ourselves to the easily tractable and physically intuitive four-level heat machine.

The four-level thermal maser forms the minimal model in which entrainment and mutual coupling coexist. In the accompanying work [22], we show how competition and cooperation manifest in the synchronization measure for degenerate multilevel thermal masers. Depending on the thermodynamic functionality of the maser, the phases compete when the system behaves as an engine and cooperate when it acts as a refrigerator.

The aim of this section is to show that the same phenomena persist even in presence of noise-induced coherence ($p\neq 0$) and near-degeneracy ($\Delta\neq 0$) by analytically examining the minimal model. We derive the formula for phase-space synchronization measure [40] $S_{max}$ for the exactly degenerate ($\Delta=0$) four-level maser in presence of noise-induced coherence. This measure not only captures the strength of steady state coherences, but also its phase-matching condition [22]. The general $SU(D)$ quantum synchronization measure applicable to $D$-level systems is presented in Ref. [22]. Here, we apply the general result for our specific case of $D=4$, to obtain,

where $k=\gamma_{h}(1+n_{h})(1+p)/\lambda$ is the dissipation-to-driving ratio. The superscript of $\tilde{\rho}$ denotes the steady state, whose analytical formula can be derived for $\Delta=0$ and $p\neq 0$ (see Appendix A). The analytical solution for $\tilde{\rho}^{ss}$ can then be used to derive Eq. (9) by following the recipe in the Supplementary Material of [22]. In fact, Eq. (9) has the same form as the result obtained in [22] with the exception that the dissipation-to-driving ratio $k$ now depends on the noise-induced coherence strength $p$.

Equation (9) displays cooperation and competition between entrainment and mutual coupling in different thermodynamic regimes. The mutual-coupling (entrainment) contribution is represented by (non-) degenerate coherences $|\tilde{\rho}_{ij}^{ss}|(|\tilde{\rho}_{1j}^{ss}|)$ for $i,j=2,3$. In the refrigerator regime ($n_{c}<n_{h}$), entrainment and mutual coupling cooperate to increase the overall synchronization of the maser since all steady state coherences contribute positively to $S_{max}$. Meanwhile, in the engine case, we observe competition between coherences for $k<2$. The competition and cooperation is due to different phase configurations preferred by the two mechanisms, i.e., both prefer in-phase synchronization in the refrigerator case while one prefers in-phase and the other prefers out-of-phase in the engine case (see Fig.2). Moreover, in the engine case for $k<2$, synchronization is dominated by mutual coupling contribution $|\tilde{\rho}_{23}^{ss}|>|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho}_{13}^{ss}|$.

The $p$-dependence of $k$ allows us to explore different synchronization regimes by tuning the strength of noise-induced coherence. For example, in absence of noise-induced coherence ($p=0$), the deep mutual coupling dominant regime $k\ll 2$ can only be explored in the strong-driving regime $\lambda\gg 2\gamma_{h}(1+n_{h})$. However, if the driving is strong, one starts to deform the limit cycle and deviate away from the typical synchronization paradigm [13]. Yet, in presence of noise-induced coherence ($p\neq 0$), the deep mutual coupling dominant regime of $k\ll 2$ can be easily explored in presence of total destructive interference ($p\rightarrow-1$).

Figures 2a-b show the quasi-probability phase distribution $S(\varphi_{21},\varphi_{31})$, where $\varphi_{ij}\equiv\phi_{i}-\phi_{j}$ is the relative phase between states $\ket{i}$ and $\ket{j}$. They are computed in the engine regime for $p=0.5$ and $p=-0.99$. Figure 2a shows the regime where entrainment is stronger than mutual coupling so that despite their competition, the phases localize in the rotating frame. On the other hand, Fig. 2b shows the deep mutual-coupling dominant regime where entrainment is effectively lost but the relative phase is still fixed. Note that in both figures, the near-degenerate gap is set to be non-zero ($\Delta\neq 0$).

In this section, we will derive the key thermodynamic observables, specifically steady state power and heat current following the standard recipe for weak-coupling thermodynamics [41]. We follow the standard energy partitioning [42] that defines the internal energy as

The internal energy here is defined as that of the bare Hamiltonian $H_{0}$ and differs from the standard definition that involves the full Hamiltonian [41]. Importantly, such a definition remains the same in all pictures (Schrödinger, Heisenberg, and Interaction) and thereby avoids the inconsistencies when one moves from one picture to another. Using the quantum master equation (8), the change in internal energy can be expressed as,

The energy flux is separated into three terms. The first term in (11) is power and the following terms are heat fluxes from the hot and cold bath respectively [42, 43],

In the engine regime, heat flows from the hot bath to the cold bath and power is produced in the steady state, i.e. $\dot{Q}_{h}^{ss}>0,\;\dot{Q}_{c}^{ss}<0,P^{ss}<0$. Conversely, in the refrigerator regime, heat flows from the cold bath to the hot bath and power is consumed, i.e. $\dot{Q}_{h}^{ss}<0,\;\dot{Q}_{c}^{ss}>0,\;P^{ss}>0$.

The heat current to the cold bath $\dot{Q}_{c}$ depends solely on populations whereas the heat from the hot bath $\dot{Q}_{h}$ depends on populations and coherences. Intuitively, since the hot bath connects the ground state and the near-degenerate manifold, a finite dipole alignment factor $p$ leads to noise-induced coherence causing the hot heat current to be dependent on both populations and coherences. Thus, the incoherent and coherent contribution to the hot bath’s heat current [44, 45] can be expressed as,

Note that the coherent heat current is proportional to $p$ which we set to be zero in the accompanying work [22].

The coherent heat either suppresses or enhances the natural inclination of heat flow. For example, in the exactly degenerate ($\Delta=0$) engine regime, a net heat flow coming from the hot bath to the system creates a population inverted state such that $\dot{Q}_{h}^{inc,ss}>0$. From Eq. (16), we see that the sign of $\dot{Q}_{h}^{coh,ss}$ depends on the sign of $p$ [$\because\text{Re}(\tilde{\rho}^{ss}_{23})<0$, see Eq. (45)]. If noise-induced coherence strength $p>0$ ($p<0$), then the heat flow is enhanced (suppressed). Similarly, in the exactly degenerate ($\Delta=0$) refrigerator regime, we observe enhancement or suppression of heat dumped into the hot bath due to either constructive ($p>0$) or destructive ($p<0$) interference.

In general the synchronization measure ($S_{max}$) and thermodynamic observables ($P$, $\dot{Q}_{c}$, and $\dot{Q}_{h}$) are unrelated. However, in the case of a three-level thermal maser, the steady state power of the maser is bounded by the synchronization measure [16] (power-synchronization bound) connecting two seemingly distinct quantities. In this section, we build the general framework of connecting the mathematically abstract notion of synchronization to physical thermodynamic observables for a four-level thermal maser. Despite several efforts analyzing the thermodynamic properties of the four-level thermal maser [20, 36, 21], there have so far been no connections between the synchronizing ability of such a thermal maser and its thermodynamic observables.

There is intense debate regarding the optimal design of quantum thermal machines. Almost all design paradigms assume the systems are well modeled by Markovian master equations and furthermore pursue coherence from an information-theoretic resource cost point of view or a non-equilibrium cost point of view. While such analyses are definitely sufficient for non-degenerate systems, secular approximation begins to fail to describe near-degenerate systems interacting with a bath. Furthermore, interacting coherences can be well understood under the paradigm of coupled oscillator models, an analysis that has tremendously benefited classical dynamical systems.

In this manuscript, we argue fundamentally that such a coupled oscillator model is indispensable to understanding the subtle dynamics of interacting coherences, which arise due to a variety of situations like those described in this manuscript. We developed the synchronization dynamics for a four-level atom coupled to two baths, generalizing the canonical thermal maser analysis by Scovil and Schulz-DuBois. We show that coherences interact with each other in two distinct qualitative ways. The first of these is a cooperative phase-locking phenomenon understood as entrainment dynamics of coherences being driven by an external field. Such entrainment dynamics and their contribution to the performance of a thermal maser were discussed earlier [16]. In this manuscript, we add to this entrainment dynamics and highlight the competition between coherences that can arise due to the bath coupling in our model. This coexistence of cooperation and competition gives rise to a richer tapestry of dynamics, uncoupling the power output from the thermal maser from the synchronization measure. The bound on power from the synchronization measure of the standard maser is modified to include coherent contributions to heat.

In presence of noise-induced coherence, in addition to the power-synchronization bound, we also find that coherent heat is bounded by synchronization measure showing that synchronization not only influences the useful work [46, 47] but can also bound the wasteful heat. The result of this work can be summarized in the following table namely,

where the check mark indicates that the bound is satisfied whereas the cross means the bound is violated. The lower bound of COP is included in the refrigerator column of E-S bound for compactness. Unlike the standard upper bounds like the Carnot efficiency, our analysis shows that efficiency can be lower bounded in presence of quantum synchronization. In other words, a quantum synchronous working substance can boost the efficiency of the machine. Even though our analysis was limited to a four-level thermal maser, it would be an interesting future avenue to investigate other coupled oscillator models for interacting coherences to inform the future designs of quantum thermal machines.