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Cited by 272Advanced Thermodynamics and Statistical MechanicsAdvanced Thermodynamic Systems and EnginesQuantum Mechanics and Applications

Markovian master equations for quantum thermal machines: Local versus global approach

Open access
Hofer, Patrick P, Perarnau-Llobet, Martí, Miranda, L David M, Haack, Géraldine, Silva, Ralph, Brask, Jonatan Bohr, Brunner, Nicolas·2017·New Journal of Physics
DOI 10.1088/1367-2630/aa964f
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Abstract

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.

AI snapshot

Written from the paper’s full text.

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1088/1367-2630/aa964f

Cites

61 of 62 references verified0 citation errors · 1 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 25 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The second term in the local-master-equation heat current decays exponentially in time.

    “From equations (18), we can infer that the second term in the heat current decays exponentially in time.”
    Quote in the paperNot a substantial claim
  • Findingreasoning

    The local-master-equation efficiency fulfills the Carnot bound.

    “We note that this efficiency fulfills equation (45).”
    Quote in the paperSubstantial claim
  • Findingown data

    The presence of an external field ensures that the local master equation does not result in any violations of the laws of thermodynamics.

    “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.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [58] source

      Not checked in the free run.

    • cites [59] Levy A, 2014, Europhys. Lett. source

      Not checked in the free run.

    • cites [63] González J O, 2017

      Not checked in the free run.

  • Findingown data

    For intermediate inter-system coupling, both the local and the global approaches agree with the exact solution.

    “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.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    For the local master equation, the efficiency is one minus the ratio of the cold to hot oscillator frequencies.

    “resulting in the efficiency h = - W W ( ) 1 . 49 c h”
    Quote in the paperSubstantial claim
  • May combine two claims. 21
  • The quote only partly states this claim. 2
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingown data

    For weak inter-system coupling, the local master equation agrees with the exact solution for thermodynamical observables and the quantum state of the machine.

    “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.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    For weak inter-system coupling in non-equilibrium situations, the global master equation fails.

    “whereas the global approach fails for non-equilibrium situations.”
    • Also stated in: § 1. Introduction · plain
      “On the contrary, the global approach fails in this regime.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the two-qubit entangler, the global approach fails in the weak coupling regime, while for intermediate coupling the local and global approaches agree well.

    “In particular, the global approach still fails in the weak coupling regime, while for intermediate coupling, the two approaches agree well.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    In the heat engine regime, all models fulfill J_c + J_h = P (first law) and η < η_C (second law).

    “Note that all models fulfill + = J J P c h (first law), and h h < C (second law).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the thermal bias regime, the local approach reliably reproduces the steady state for small γ while the global approach works well for large γ.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the thermal bias regime, the occupation numbers obtained from the local approach differ quite a bit from exact numerics for large interactions, but the heat current is still captured very well by the local approach.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    There is good agreement in power and efficiency between the local approach, the global approach, and exact numerics, except when the machine is operated close to the Carnot point.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing). May combine two claims.

    Findingreasoning

    For ω_c ≈ Ω_a, the numerical results do not closely match the analytics because energetically possible transitions are not captured by the bath.

    “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.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In equilibrium, the global approach yields an accurate description of the steady state while the local approach gets progressively worse as the inter-system interaction γ increases.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    When γ becomes comparable to Ω_c, the local approach starts to deviate from exact numerics and the global approach becomes preferable.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    In the heat engine regime, the local approach shows very good agreement with numerics while the global approach breaks down when κ ≪ γ.

    “While the local approach agrees extremely well with exact numerics, the global approach fails for small g.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The Born-Markov equation does not guarantee positive evolution, and further approximations are usually made to obtain a GKSL form ensuring completely positive dynamics.

    “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.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [1] Breuer H-P, 2002, The Theory of Open Quantum Systems

      Not checked in the free run.

    • cites [3] source

      Not checked in the free run.

    • cites [4] source

      Not checked in the free run.

    • The authors state this as an assumption: The baths are in thermal states.

  • Classified as probably not a substantial claim.

    Study design and conductreasoning

    The exact numerical solution serves as a benchmark for evaluating the performance of both the local and global master equations.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    When the sub-systems do not interact, the local master equation yields correct results while the global approach fails, producing a finite energy current even without interaction.

    “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.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: If the sub-systems do not interact, each sub-system thermalizes to its respective bath.

  • May combine two claims.

    Interpretationreasoning

    In the limit of vanishing inter-system coupling, the global master equation does not recover the local master equation and is expected to break down because the secular approximation is no longer justified.

    “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.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The baths are in thermal states.
    • The authors state this as an assumption: The renormalization of the Hamiltonian is neglected.

  • May combine two claims.

    Premisereasoning

    In the global master equation, the bath couples to global states dressed by the external field, so the dissipative terms include the external field and the definitions of heat and work from the local approach are no longer valid.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given

    For the global approach, the first law is enforced, whereas in the local approach it follows from the expressions for heat currents and power.

    “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.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The first law of thermodynamics is enforced in the global approach.

  • May combine two claims.

    Interpretationnone given

    When the hot and cold oscillator frequencies are equal, there is no external power and heat always flows from the hot bath to the cold bath as dictated by the second law of thermodynamics.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    As long as the efficiency is below the Carnot efficiency, the machine is a heat engine; at the Carnot efficiency it is reversible; above the Carnot efficiency it acts as a refrigerator.

    “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].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [44] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data

    The choice ω_c = 3 is large enough to capture the different energy transitions and not too large so that the effect of renormalization can be neglected.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The two-qubit machine can generate entanglement between the two qubits in the steady state, as shown in [51].

    “This machine can generate entanglement between the two qubits in the steady state, as shown in [51].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [51] Brask J B, 2015, New J. Phys.

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The efficiency can only reach the Carnot value if one of the channels carries no energy or if both channels can simultaneously reach the Carnot point, which is the case if equations (17) hold.

    “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).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The local approach is not more phenomenological than the global approach; the approximation leading to the local master equation is completely analogous to the Markov approximation and has a well-defined regime of validity.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation

    This result is consistent with [15], which discusses the absence of any currents as expressed through system observables in the global approach.

    “This is consistent with the results of [15], which discusses the absence of any currents as expressed through system observables in the global approach.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Both the local and global master equations for the two-qubit machine can be written in GKSL form with constant rates and lead to Markovian (semigroup) evolution.

    “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”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [3] source

      Not checked in the free run.

    • cites [4] source

      Not checked in the free run.

25 of 25

  • 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.”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • In the dressed picture, local baths on each buffer spin become global baths acting collectively on the dressed buffer spins.

    “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]

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. Breuer H-P, 2002, The Theory of Open Quantum Systems
  2. 10.1016/bs.aamop.2015.07.002q3 · DOI
  3. Brask J B, 2015, New J. Phys., 17
  4. 10.1007/s10051-002-8904-5 · DOI
  5. Carmichael H, 1991, An Open Systems Approach to Quantum Optics

Cited by

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Related

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)
ORCID
Venue
New Journal of Physics
Publisher
IOP Publishing
Published
2017-12-20
Volume
19
Issue
12
Pages
123037
ISSN
1367-2630 (ISSN-L)
DOI
10.1088/1367-2630/aa964f
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
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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