Abstract
We explore means of maximizing the power output of a heat engine based on a periodicallydriven quantum system that is constantly coupled to hot and cold baths. It is shown that the maximal power output of such a heat engine whose “working fluid” is a degenerate Vtype threelevel system is that generated by two independent twolevel systems. Hence, level degeneracy is a thermodynamic resource that may effectively double the power output. The efficiency, however, is not affected. We find that coherence is not an essential asset in such multilevelbased heat engines. The existence of two thermalization pathways sharing a common ground state suffices for power enhancement.
Introduction
The rapport between quantum mechanics and thermodynamics is still an open problem^{1,2}. Its technological and fundamental implications have motivated numerous proposals of heat engines based on quantum systems^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}. Two main issues underlie such proposals: What are the bounds on the performance of quantum heat engines, i.e., their power output and efficiency^{1,2,23,24,25,26} and what thermodynamic properties (or resources) of quantum systems determine these bounds^{27,28,29,30,31}? A pioneering approach addressing these issues^{32,33} has suggested that steadystate coherence^{34,35,36} between the levels of a quantum system is a thermodynamic resource.
Here we wish to elucidate these issues from first principles. To this end we resort to a fully solvable model of a steadystate, continuouscycle, heat engine that is based on a periodicallydriven quantum system (“working fluid”) constantly coupled to hot and cold baths^{15,21}. Consistency with the first and second laws of thermodynamics is enforced in this theory by the construction of appropriate heat currents flowing between the baths via the system^{21,37}.
To account for the possible rôle of coherences we extend this theory, hitherto applied to a twolevel system (TLS) working fluid^{15,21}, to an analogous heat engine based on a Vtype threelevel system as depicted in Fig. 1a. We have chosen a Vsystem for being the simplest working fluid wherein coherences may persist at steady state and possibly affect the engine performance. The performance of an engine based on such a Vsystem is compared to a TLSbased heat engine (cf. Fig. 1b), where steadystate coherence is absent. We show that the power output of the Vsystem may be boosted by up to a factor of 2 compared to its TLS counterpart. This boost is associated with correlations that arise between the possible thermalization channels in the Vsystem that constitute a hitherto unexploited thermodynamic resource. Such correlations exist even in the absence of coherence, because the degenerate excited states exchange populations with each other via their common ground state. However, steadystate coherence does not affect the efficiency, nor does maximal power boost necessarily require coherence, since thermalization correlations may be incoherent.
Qubitbased heat machine revisited
A continuouscycle quantum heat machine based on a single qubit (TLS) as working fluid has been studied in ref. 21. This TLS is simultaneously and permanently coupled to cold and hot heat baths, while its transition energy is periodically modulated by some external field according to the Hamiltonian
This external field plays the rôle of a piston and allows for work extraction or supply. The dipolar coupling of the atom to the cold and hot baths in the rotatingwave approximation reads^{38}
with the transitiondipole moment d and the Pauli operator describing the excitation of the atom and its adjoint describing deexcitation.
As detailed in^{37} the periodicity of the modulation implies that the dynamics of the system’s density matrix in the interaction picture is governed by a linear combination of “subbath” Lindblad operators, i.e., operators associated with the two baths i ∈ {c, h}, evaluated at the harmonic (Floquet) sidebands q = 0, ±1, ±2,… of the modulation frequency Ω. The master equation in the weakcoupling limit then reads
with the Liouvillian superoperators of the (i, q) “subbaths”
Here P(q) is the weight of the qth harmonic (determined by the modulation form)^{37} and the dissipator reads for any system operators a, b. The factors G_{i}(±ω) are the coupling spectra to the ith bath and depend on the bath autocorrelation functions , where denotes the kth component of B_{i}(t) in the interaction picture. These spectra fulfill the KMS condition^{39} , where for a bosonic bath, , γ_{i}(ω) being the frequencydependent transition rate induced by the ith bath and denoting the corresponding number of thermal quanta at inverse temperature β_{i} = 1/k_{B}T_{i}.
The heat currents between the cold and the hot baths and the TLS evaluate to^{21}
and the power (time derivative of the work) according to the first law, reads
Here is the ratio between the excited and the groundstate steadystate populations of the qubit. We here follow the convention that negative power means work extraction (operation as an engine).
This conceptually simple heat machine can be operated “on demand” as a heat engine (the extracted work is manifested by a coherent amplification of the external field) or as a refrigerator, depending on the modulation rate Ω. The machine behaves as an engine if the rate is below some critical value, whereas above this value it acts as a refrigerator^{21}.
A detailed analysis of the heat currents (5) and the power (6) reveals that at the critical rate the switchover from the engine to the refrigeration mode ensures compatibility with the second law—this is precisely the rate at which the engine reaches Carnot efficiency and yields vanishing power. Strikingly, the engine’s efficiency at maximum power can surpass the Curzon–Ahlborn efficiency^{40} under certain conditions on the bath spectra^{21}.
This heat machine operates at the steadystate (limit cycle) of the corresponding dissipative time evolution of the working fluid. Naturally, coherence is absent in the system’s steady state. In order to study the effects of coherences, we now extend this TLSbased model to a degenerate threelevel system.
Steadystate treatment of Vsystem heat machines
We consider a Vtype threelevel system with degenerate excited states and , ground state and transition frequency ω_{0}. To operate a heat machine, we simultaneously connect this system to two (hot and cold) baths, which induce transitions and . The “piston” periodically modulates both excited states^{21}, which results in the same periodic transition frequency ω_{0} + ω(t) as for a TLS (see Eq. 1), with , where Ω denotes the modulation rate. The dipolar system–bath interaction is described by the following generic Hamiltonian [a generalization of the case presented in ref. 39 and in Eq. (2)] in the rotatingwave approximation,
where and are the excitation (deexcitation) Pauli operators for the jth transition, d_{j} is the transition dipole between the excited state and the ground state and B_{i} is the hot (h) or cold (c) bath operator. For simplicity we here restrict the treatment to real dipoles of equal strength, . These transition dipoles need not be parallel (aligned), as discussed below.
Based on the interaction Hamiltonian (7), the Floquetexpanded master equation in the weakcoupling limit has the same form as (3),
but the Liouvillian superoperators for the degenerate Vtype threelevel system, coupled to the (i, q) “subbaths” are now generalizations of (4) (see supplemental material),
Here the dissipators and describe emission and absorption involving separate transitions ( and ) via their common ground state and hence population transfer between and . These processes give rise to population correlations of the two excited states. By contrast, and describe crosscorrelations between the two transitions, allowing for bathinduced quanta exchange between the two excited states and thereby generating coherences between these states. Thus, the effect of the degeneracy is to mix the diagonal and the offdiagonal terms, via the crosscorrelations in Eq. (9). We note that the evolution of this degenerate system is governed by a wellestablished master equation (see supplemental material)^{34,35,39,41,42,43}.
A key parameter in the ensuing analysis is the dipolealignment factor
Analysis
The energy that is continuously exchanged between the threelevel system and the heat baths is related, according to the first law, to the power (the rate of work W extracted by the piston) by^{44}
This expression involves the sum of heat currents from both baths, which can be derived from the dynamical version of the second law^{2}. Their explicit expression for the ith bath (i ∈ {c, h}) is , where the heat current for the qth harmonic “subbath” () in Eq. (9) reads^{2,37}
Here, denotes the local steady state for a single heat bath at temperature T_{i} evaluated at the sideband ω_{0} + qΩ, i.e., . We stress that the global steady state ρ^{ss} (fulfilling ) ensures the correct description of heat transport in this correlated threestate system, avoiding inconsistencies with the second law due to the improper use of local variables, as discussed in^{45}. Since every Liouvillian in the master equation (8) has the same functional dependence (9) on the atomic operators, the correct global solution can be directly obtained from the local one.
We here search for the steadystate solution of the master equation (8) and the resulting expressions for J_{h(c)}. At this point we still do not know the bound for these currents and its dependence on alignment. These heat currents are therefore compared to the corresponding expressions (5)–(6) for a twolevel system (TLS) with the same transitiondipole strength d and modulated transition frequency ω_{0} + ω(t)^{21}.
The master equation (8) can be reduced to an analytically solvable inhomogeneous system of linear differential equations
for the vector of matrix elements
This system of ordinary differential equations (ODEs), where the matrix and the vector b are defined in Eqs. (31) and (32) in the Methods section, describes two very distinct dynamical regimes corresponding to aligned and misaligned transition dipoles, as detailed in what follows.
(i) Let us first consider the very general steadystate regime obtained for misaligned transition dipoles, . Note that this regime also includes the case of orthogonal dipoles (). The threelevel system then thermalizes to the diagonal steady state (without coherences)
with an effective inverse temperature β_{eff} defined by the Boltzmann factor
This effective temperature determines the steadystate populations of the periodically modulated system coupled to both baths. We can control β_{eff} by engineering the modulation Floquet coefficients P(q) that determine the overlap of the sideband peaks (q = ±1, ±2,…) at the frequency harmonics ω_{0} + qΩ with the response spectra G_{i}(ω) of the two baths, as sketched in Fig. 2a.
Upon computing the heat currents (12), we find that J_{h}, J_{c} and the power are modified (relative to their TLS counterparts in Eqs. (5) and (6)^{21}) by the same factor
This means that the power enhancement relative to a TLS heat machine is determined by the ratio of the steadystate groundstate population in the Vsystem to its TLS counterpart. Namely, in this fully thermalized incoherent regime the enhancement factor (18) only depends on the effective temperature (17).
(ii) For fully degenerate excited states we find that the coefficient matrix (31) of the ODE above is singular () for aligned dipole moments ( = 1). The same result holds for antiparallel dipoles, which justifies the restriction of to nonnegative values. This singularity implies that an infinite number of steadystate solutions may exist. Indeed, in this regime the dynamics is constrained by the existence of a dark state , for which
which renders the steadystate solution dependent on the initial conditions (in agreement with the expressions found for a single bath in refs. 34 and 46). The steadystate solution now depends on the overlap of the initial state ρ(0) with the nondark states (i.e., the ground state and the bright state ) of the full Liouvillian in Eqs. (8) and (9). The rôle of these states becomes apparent upon diagonalizing the steadystate solution, which yields the populations
in the basis spanned by . Here
and
denote the bright and dark states, respectively. Whilst the darkstate population cannot change, i.e., it is a constant of motion (consistent with the one obtained in^{35} for a single zerotemperature bath and external driving), the bright and groundstate populations, ρ_{bb} and ρ_{00}, respectively, thermalize. The same results hold for antiparallel dipoles ( = −1) upon interchanging the dark and the bright states.
Proceeding as before in the nonaligned case, we find the power ratio
Hence, the power as well as the heat currents are enhanced in the aligned regime relative to their TLS counterparts by at most a factor of two, just as in the misaligned regime [Eq. (18)]. Yet, contrary to the latter, the ratio (26) does not depend on the bath spectra or the environmental temperatures, but solely on the initial populations of the nondark states. Enhancement in Eq. (26) requires , or, equivalently, , i.e., at least half of the initialstate population has to be nondark. Maximal enhancement occurs when the initial state is amenable to full thermalization, i.e., it is nondark.
For a given initial groundstate population ρ_{00}(0), the states providing the maximum possible power boost are characterized by . These are the states with the maximally allowed modulus of the ρ_{21}(0) coherence (for a fixed groundstate population) and the correct phase. We have plotted the maximum power output under sinusoidal modulation for a TLS, a nonaligned and an aligned Vsystem in Fig. 2b. The spectra are chosen as in ref. 21 such that only G_{c}(ω_{0}) and G_{h}(ω_{0} + Ω) contribute (as sketched in Fig. 2a) and the modulation frequency has been tuned to the value maximizing the power output.
We stress that a nondark initial state does not correspond to a steady state with maximal coherence when rotating Eq. (21) back to the original basis spanned by . In fact, the coherence is maximized for an initial dark state, which does not exchange energy with the baths and gives zero power, see Fig. 2c.
It is natural to ask: How much initial overlap with the dark state is allowed such that the aligned configuration still outperforms its misaligned counterpart? The answer is, for
The value on the r.h.s. is the initial overlap for which the steadystate coherences vanish in the aligned case (see Fig. 2c).
So far we have made the comparison between the heat currents and the power, respectively, obtained for a threelevel system relative to a twolevel system. We now strive for a direct comparison of the enhancement factors (18) and (26) for the misaligned ( < 1) and aligned ( = 1) regimes. Their ratio is determined by the respective steadystate populations in the ground state, which is directly related to the power or heatcurrent ratio via
We consider this ratio in two limiting cases (assuming no initial overlap with the dark state in the aligned case):
(i) As β_{eff} → 0 (high effective temperature) the thermalized state corresponds to equipartition amongst the levels. For parallel dipoles, the thermalized threelevel system behaves as a TLS (formed by the ground and the bright states) with an effective dipolar transition enhanced by the number of thermalization pathways, in this case two. Hence, in steady state half of the population is found in level (if the initial state had no dark component). For misaligned dipoles, by contrast, thermal equilibrium corresponds to the equipartition amongst the three levels , and . Consequently, only a third of the population is found in the ground state. The 3/2 ratio of the respective groundstate populations according to Eq. (28) explains the ratio of the maximal enhancement factors in the aligned and misaligned regimes at high T_{eff}.
(ii) For large β_{eff}, i.e., low T_{eff}, however, Eq. (28) implies that the maximal enhancement for misaligned dipoles coincides with its counterpart for aligned dipoles (the latter is maximized for an initial state perpendicular to the dark state), since only is then appreciably populated in either regime.
Both regimes still retain the maximal enhancement factor of 2, stemming from their double thermalization pathways instead of one for a genuine TLS. We have summarized these results in Fig. 3. A beneficial influence of alignment on power output is only expected for effective temperatures . For optical transitions this corresponds to a few hundred Kelvin, whereas for microwave transitions the benefit of alignment is already expected for a few hundred milliKelvin.
Realization considerations
Vsystems with degenerate upper states are commonly found in atoms free of hyperfine interactions, e.g., mercury (Hg) or hydrogen (H). In particular, the three transitions in such atoms are degenerate but have orthogonal transition dipoles. However, even such misalignment (orthogonality) does not hamper the Vsystem power boost at low T_{eff} (see above). The simultaneous coupling of such systems to hot and cold baths with controlled spectra can realize the misaligned case.
The case of degenerate upper states and parallel transition dipoles (which, as discussed, is beneficial for power enhancement only at high T_{eff}), is obtainable only for transitions between a lower state with angular momentum l and magnetic number m and degenerate upper states with the same m^{46}. In atomic degenerate Vsystems such parallel transition dipoles are forbidden by selection rules. However, dressed states stemming from driven Λsystems may effectively realize such parallel Vsystems^{46} (see Fig. 4). Unfortunately, if we examine this system more closely, we see that it presents several difficulties: (i) The resulting transitions between the excited state doublet
and the ground state, where ϑ is the mixing angle determined by the Rabi frequency Ω_{R} of the splitting field^{46}, occur at rates that scale with and , where γ is the decay rate of the bare excited state . For maximal splitting (ϑ = π/4), γ_{1} = γ_{2} = γ/2. Hence the power boost is canceled by the reduction of the decay rate. (ii) In order to periodically modulate the transition frequency, we need an auxiliary field that induces an ac Stark shift only on the ground state. (iii) The dressed states are nondegenerate, which limits our results to time scales shorter than the inverse level splitting . The latter, however, can be longer than the experimental time scale.
Molecules may be a more promising possibility due to their rich level structure involving rotational and vibrational degrees of freedom, as discussed in (See supplementary information in ref. 47).
Discussion
Regardless of the transitiondipole misalignment or alignment, the maximally enhanced power output of a degenerate Vsystem heat engine is that generated by two independent twolevel systems. The key to enhancement is the system to have degenerate upper levels sharing a common ground state. Hence, level degeneracy is found to be a thermodynamic resource that may effectively boost the power output. Yet, it does not affect the efficiency: Since the same modifying factors [Eqs. (18) and (26)] are obtained for the heat currents and the power, the efficiency
of the degenerate threelevel heat machine is the same as for a twolevel system. Thus, the same universal dependence of the efficiency on the modulation rate found in ref. 21 holds for the present system. In particular, as the heat currents (12) (by construction) fulfill the second and the first laws, they adhere to the Carnot bound^{26}.
As shown in refs. 28 and 29, the efficiency of a continuouscycle heat engine based on a TLS coupled to a quantized harmonicoscillator “piston” is determined by the effective temperature and entropyproduction rate of the piston: This efficiency may surpass the standard Carnot bound over many cycles if the piston is initially prepared in a smallamplitude coherent state. It is therefore possible that the extension of this model to a Vsystem may allow not only for a power boost but also an efficiency higher than the Carnot bound.
At effective temperatures significantly larger than the level spacing, the aligneddipoles regime, where steadystate coherences arise, can outperform all misaligned cases. On the other hand, as discussed here, aligned transition dipoles can only be realized in a fielddressed atom, but such dressing divides the transitiondipole strength of the bare atom between two dressedstate transitions and thereby cancels the power boost of the dressedatom machine compared to its bareatom counterpart.
This limitation of fielddressed atoms prompts the need for an alternative realization of aligned dipoles, free of such limitations, e.g., in molecules. Let us, however, assume that such a scheme can be realized and focus on conditions under which the aligned regime is advantageous in terms of its power boost compared to a TLS. We may not attribute the power enhancement to steadystate (or initial) coherences between the excited states but rather to the ability of the initial state to completely thermalize. We therefore conclude that (initially induced or steadystate) coherences are not an essential asset in the considered threelevelbased heat machine. The existence of two thermalization pathways sharing a common ground state, regardless of whether they are coherent or incoherent, suffices for power enhancement.
Methods
The coefficients of the ODE for x: = (ρ_{21}, ρ_{12}, ρ_{00}, ρ_{22})^{T} read
and
Note that the coherences between the ground and the excited states (ρ_{10} and ρ_{20}) do not appear as they follow a decoupled dynamics (leading to vanishing steadystate values).
Additional Information
How to cite this article: GelbwaserKlimovsky, D. et al. Power enhancement of heat engines via correlated thermalization in a threelevel “working fluid”. Sci. Rep. 5, 14413; doi: 10.1038/srep14413 (2015).
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Acknowledgements
This work has been supported by the ISF, BSF, AERI and CONACYT.
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The results are an equal contribution of D.G.K. and W.N. D.G.K., W.N., P.B. and G.K. contributed to the writing of the article.
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GelbwaserKlimovsky, D., Niedenzu, W., Brumer, P. et al. Power enhancement of heat engines via correlated thermalization in a threelevel “working fluid”. Sci Rep 5, 14413 (2015). https://doi.org/10.1038/srep14413
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