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Markovian master equations for quantum thermal machines: local versus global approach
Open accessAbstract
The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations. Two approaches are mainly followed. On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths. On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine. There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium. Here we compare the local and global approaches against an exact solution for a particular class of thermal machines. We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine. Our results show that the use of a local master equation is generally well justified. In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations. For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable. These results are backed by detailed derivations of the regimes of validity for the respective approaches.
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TL;DR
The paper benchmarks local versus global Markovian master equations for quantum thermal machines against exact numerics for a quadratic heat engine. It finds the local approach is generally well justified for weakly interacting subsystems, while the global approach is preferable for strong inter-system coupling and in equilibrium.
Key findings
- For the harmonic-oscillator heat engine, the local master equation agrees well with exact numerics when the inter-system coupling satisfies \(g \lesssim \Omega_a\), whereas the global approach fails out of equilibrium when \(g \lesssim \kappa_a\), even predicting a finite heat current as \(g \to 0\).
- At intermediate coupling both approaches agree with exact numerics; at strong coupling the global approach becomes preferable. In equilibrium, the global approach outperforms the local approach except in the limit \(g \to 0\).
- The local approach gives the heat-engine efficiency \(\eta = 1 - \Omega_c/\Omega_h\) and, with an external drive, does not violate thermodynamic laws. The global approach requires dressed-state definitions of heat currents; using local-like definitions instead gives \(P=0\) and \(J_h=-J_c\).
- For the two-qubit entangler, the global approach again predicts a constant heat current as \(g \to 0\), while the local model predicts vanishing current; the two models agree well for intermediate coupling. No exact benchmark is available for this case.
Methods
- The main model is a heat engine of two harmonic oscillators with frequencies \(\Omega_c\) and \(\Omega_h\), each coupled to a thermal bath and coupled to each other via an external drive. The full Hamiltonian is quadratic, so exact unitary dynamics of system plus finite baths is simulated at the level of covariance matrices.
- Exact numerics use baths of about 400 oscillators, cutoff \(\omega_c=3\), and evolution time \(t \approx 20\,\tau_{\rm eq}\). These are compared with local and global Markovian master equations for steady-state heat currents, power, efficiency, and fidelity of reduced Gaussian states.
- A finite-dimensional two-qubit entangler is also analyzed by comparing local and global GKSL master equations, without an exact benchmark.
Limitations
- Stated: the exact benchmark is available only for the harmonic-oscillator heat engine; for the qubit entangler, conclusions are conjectural because no exact solution is provided.
- Stated: the Hamiltonian renormalization is neglected, which causes a dip in local-approach fidelity near \(g \approx |\Sigma_h|\) and a small dependence on the bath cutoff in the appendix.
- Stated/evident: the validity regimes are asymptotic and model-specific, derived for bosonic Ohmic baths; the local approach fails at strong coupling, the global approach fails at weak coupling and out of equilibrium, and analogous fermionic-bath conditions are noted as usually not fulfilled at low temperatures.
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![Figure 1 Sketch of the heat engine. The system consists of two harmonic oscillators with different frequencies. Each oscillator couples to a thermal bath modeled by a collection of harmonic oscillators. An external fi eld mediates a weak coupling between the two oscillators. A thermal gradient can result in a heat fl ow from the hot to the cold bath, injecting some of the energy into the external fi eld which can in principle be used to charge a battery. A thermoelectric implementation of this machine in superconducting circuits is proposed in [ 44 ] . In this work, we compare different Markovian master equations for the description of this machine, using exact numerics as a benchmark.](/paper-figures/16e35ad93abe41438fdc6f3c08470a1b/markovian-master-equations-for-quantum-thermal-machines-local-versus-global-figure-1.png)
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24 Own interpretation · 23 Own result · 12 Not checked yet · 12 No basis given · 4 Own method
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“From equations (18), we can infer that the second term in the heat current decays exponentially in time.”
§ Markovian master equations for quantum thermal machines: local versus global approach > B h c c h
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“We note that this efficiency fulfills equation (45).”
§ Markovian master equations for quantum thermal machines: local versus global approach > B h c c h
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“In contrast to [58,59,63], which lack an external field, we find that the presence of such a field ensures that the local master equation does not result in any violations of the laws of thermodynamics.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 3. Heat engine
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“For intermediate coupling, the local and the global approach both agree with the exact solution”
- Also stated in: § 1. Introduction · plain
“Moving to the regime of intermediate coupling, we find that both approaches provide good descriptions of the system.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Abstract
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“resulting in the efficiency h = - W W ( ) 1 . 49 c h”
§ Markovian master equations for quantum thermal machines: local versus global approach > B h c c h
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“As we show in section 7, the local approach gives reliable predictions even for interactions that are several times the broadening.”
§ Markovian master equations for quantum thermal machines: local versus global approach > B
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“In the steady state, we find that the state is a product of thermal states with occupation numbers á ñ = s ˆ † a a n as expected.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 4.2. Global master equation
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“for strong coupling, the global approach is preferable.”
- Also stated in: § 1. Introduction · plain
“For very strong inter-system coupling strengths, the local approach starts to fail while the global approach still yields a faithful description of the system.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Abstract
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“In particular, the numerical value for the efficiency is very close to the universal value h = -W W 1 c h obtained from the local master equation.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7. Results
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“Finally, we focus on the heat engine regime which requires both a thermal bias as well as an external field.Again, the global approach breaks down for small values of g as expected.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7. Results
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“In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations.”
- Also stated in: § 1. Introduction · boosted
“In the weak inter-system coupling regime, we see that the local approach provides an accurate description of the system, capturing both the thermodynamical observables and the quantum state.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Abstract
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“whereas the global approach fails for non-equilibrium situations.”
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“On the contrary, the global approach fails in this regime.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Abstract
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“In particular, the global approach still fails in the weak coupling regime, while for intermediate coupling, the two approaches agree well.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 1. Introduction
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“Note that all models fulfill + = J J P c h (first law), and h h < C (second law).”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.3. Thermal bias and external field
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“As expected, the local approach reliably reproduces the steady state for small values of g while the global approach works well for large values of g.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.2. Thermal bias
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“Note that the occupation numbers obtained from the local approach differ quite a bit from the ones obtained from exact numerics for large interactions.Nevertheless, the heat current is still captured very well by the local approach.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.2. Thermal bias
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“We find good agreement in the power as well as the efficiency between the local approach, the global approach, and the exact numerics.Only when the machine is operated close to the Carnot point (W = W T T h h c c ) do the efficiencies deviate considerably.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.3. Thermal bias and external field
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“For w » W a c , the results do not closely match the analytics, which is expected because energetically possible transitions are not captured by the bath.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Appendix. Details on the simulation
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“As expected, the global approach yields an accurate description of the steady state while the local approach gets progressively worse as the inter-system interaction g increases.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.1. Equilibrium
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“Note that when g becomes comparable to W c , the local approach starts to deviate from the exact numerics and the global approach becomes preferable.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7. Results
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“While the local approach agrees extremely well with exact numerics, the global approach fails for small g.”
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“Since it does not guarantee positive evolution [1], further approximations are usually made to obtain a master equation in Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form [3,4] ensuring completely positive dynamics.”
§ Markovian master equations for quantum thermal machines: local versus global approach > S H t i S
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“This exact numerical solution serves as a benchmark for evaluating the performance of both the local and global master equations.”
- Also stated in: § 5. Exact numerics · plain
“In this section we briefly describe how we obtain exact numerics which are used as a benchmark when comparing the different master equations.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 1. Introduction
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“While the local master equation yields correct results in this case, the global approach fails, resulting in a finite energy current through the system even in the absence of an interaction.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 2. Local versus global
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- Rests on an assumption: If the sub-systems do not interact, each sub-system thermalizes to its respective bath.
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“We note that in the limit g 0, the local master equation is no longer recovered.Because the secular approximation is no longer justified in this limit, we expect the global master equation to break down.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 4.2. Global master equation
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“In the global master equation, the bath couples to global states which are dressed by the external field.Therefore, the dissipative terms include the external field and the definitions introduced in the last subsection are no longer valid.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 6.1.2. Global master equation
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“We note that for the global approach, we enforce the first law while in the local approach, it follows from the expressions for heat currents and power.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 6.1.2. Global master equation
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“Finally, we note that for W = W h c , where there is no external power, heat always flows from the hot bath to the cold bath as dictated by the second law of thermodynamics.”
§ Markovian master equations for quantum thermal machines: local versus global approach > B h c c h
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“As long as h h < C , our machine is thus a heat engine, h h = C denotes the point of reversibility, where all the energy currents vanish, and for h h > C , the machine acts as a refrigerator, using power to induce a heat current from the cold bath to the hot bath [44].”
§ Markovian master equations for quantum thermal machines: local versus global approach > B h c c h
What the sources say
- Not checked yetcites [44] source
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“The choice w = 3 c is hence large enough to capture the different energy transitions, and at the same time not too large so that the effect of the renormalization can be neglected.”
§ Markovian master equations for quantum thermal machines: local versus global approach > Appendix. Details on the simulation
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“This machine can generate entanglement between the two qubits in the steady state, as shown in [51].”
§ Markovian master equations for quantum thermal machines: local versus global approach > 8. Qubit entangler
What the sources say
- Not checked yetcites [51] Brask J B, 2015, New J. Phys.
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“The efficiency can only reach the Carnot value if one of the channels carries no energy (i.e.c = 1 or c = 0) or if both channels can simultaneously reach the Carnot point (which is the case if equations (17) hold).”
§ Markovian master equations for quantum thermal machines: local versus global approach > 6.1.2. Global master equation
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“We note that the local approach is by no means more phenomenological than the global approach.Indeed, the approximation leading to the local master equation is completely analogous to the Markov approximation and has a well defined regime of validity.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 9. Conclusions
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“This is consistent with the results of [15], which discusses the absence of any currents as expressed through system observables in the global approach.”
§ Markovian master equations for quantum thermal machines: local versus global approach > 7.3. Thermal bias and external field
What the sources say
- Not checked yetcites [15] source
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“Both the local and global master equations can be written in GKSL form with constant rates [3,4], and so lead to Markovian (specifically semigroup) evolution”
§ Markovian master equations for quantum thermal machines: local versus global approach > 8. Qubit entangler
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The authors state this as an assumption.
The baths are in thermal states.
“where we assumed the baths to be in thermal states.”
The authors state this as an assumption.
The renormalization of the Hamiltonian is neglected.
“In the following, we will thus neglect this renormalization.”
“As for the local approach, we will neglect the renormalization of the Hamiltonian in the following.”
The paper relies on this without support.
If the sub-systems do not interact, each sub-system thermalizes to its respective bath.
“Clearly, if the sub-systems do not interact, each sub-system should thermalize to its respective bath.”
The authors state this as an assumption.
The dip in fidelity for the local approach arises from neglecting the renormalization of the Hamiltonian.
“The dip in the fidelity for the local approach occurs at » S | | g h and is therefore assumed to arise from neglecting the renormalization of the Hamiltonian; see equation (16).”
The authors state this as an assumption.
The first law of thermodynamics is enforced in the global approach.
“We note that for the global approach, we enforce the first law while in the local approach, it follows from the expressions for heat currents and power.”
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“In the dressed picture, the separate baths acting locally on each buffer spins become global baths [102,103] acting collectively on the dressed buffer spins.”
From Quantum Models of Consciousness from a Quantum Information Science Perspective (2025) · cites this paper as [103]
Citations
Cites
On the generators of quantum dynamical semigroups
Lindblad, G. (1976) · Communications in Mathematical Physics · doi:10.1007/bf01608499 · Cited by 7,548 · cited 2×
“Since it does not guarantee positive evolution [1], further approximations are usually made to obtain a master equation in Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form [3,4] ensuring completely positive dynamics.”
BackgroundCompletely positive dynamical semigroups of N -level systems
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522979 · Cited by 3,998 · cited 2×
“Since it does not guarantee positive evolution [1], further approximations are usually made to obtain a master equation in Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form [3,4] ensuring completely positive dynamics.”
BackgroundKinetic equations from Hamiltonian dynamics: Markovian limits
Spohn, H. (1980) · Reviews of Modern Physics · doi:10.1103/revmodphys.52.569 · Cited by 1,013
The role of quantum information in thermodynamics—a topical review
Goold, J., Huber, M., Riera, A., del Rio, L., & Skrzypczyk, P. (2016) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/49/14/143001 · Cited by 973
Vinjanampathy, S., & Anders, J. (2016) · Contemporary Physics · doi:10.1080/00107514.2016.1201896 · Cited by 940
Davies, E. B. (1974) · Communications in Mathematical Physics · doi:10.1007/bf01608389 · Cited by 934
Rivas, ́., & Huelga, S. F. (2011) · SpringerBriefs in physics · doi:10.1007/978-3-642-23354-8 · Cited by 876
Quantum Thermodynamics: A Dynamical Viewpoint
Kosloff, R. (2013) · Entropy · doi:10.3390/e15062100 · Cited by 857
Weiß, U. (1993) · Series in modern condensed matter physics · doi:10.1142/1476 · Cited by 799
Quantum thermodynamic cycles and quantum heat engines
Quan, H. T., Liu, Y.-X., Sun, C. P., & Nori, F. (2007) · Physical Review E · doi:10.1103/physreve.76.031105 · Cited by 789
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Metadata
- Authors
- Hofer, P. P., Perarnau-Llobet, M., Miranda, L. D. M., Haack, G., Silva, R., Brask, J. B., & Brunner, N.
- Affiliations
- Max Planck Institute of Quantum Optics (DE)
- University of Geneva (CH)
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2017-12-20
- Volume
- 19
- Issue
- 12
- Pages
- 123037
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W2740733769
- Semantic Scholar
- 0ed3d4abdd81b77e7873e95081dde2094a2201d1
- Cited by
- 272
- References
- 61
- Citations by year
- 2026: 19 · 2025: 34 · 2024: 36 · 2023: 22 · 2022: 28 · 2021: 43 · 2020: 25 · 2019: 37 · 2018: 25 · 2017: 3
- FWCI
- 12.75
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (publishedVersion)
- edoc (University of Basel) (acceptedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- Lund University Publications (Lund University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Alexander von Humboldt-Stiftung
- European Cooperation in Science and Technology (Thermodynamics in the quantum regim)
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (200021_169002, Marie-Heim Vögtlin, 164466, QSIT, Starting Grant DIAQ)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Engineering › Mechanical Engineering › Advanced Thermodynamic Systems and Engines
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum thermal machines, Markovian master equation, open quantum systems, heat current, non-equilibrium thermodynamics, weak and strong coupling, exact solution comparison
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