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The thermodynamic cost of driving quantum systems by their boundaries
Felipe Barra
TL;DR
Thermodynamic frameworks for quantum engines address passive, weak coupling, while actively controlled boundary couplings raise questions about their thermodynamic cost and appropriate heat and work quantities. The paper derives these quantities from unitary system-environment dynamics and obtains boundary-driven Lindblad evolution in an appropriate limit. Its spin-chain examples show distinct single-bath behavior for XX and XY chains and engine, refrigerator, or heater operation with two baths, with efficiencies bounded by Carnot efficiencies.
Problem
Thermodynamic analogues for quantum engines cover passive weak coupling, but actively controlled boundary interactions require appropriate heat, work, and entropy-production quantities.
Source: Introduction · p. 1Method
The authors start from unitary system-environment dynamics to derive thermodynamic quantities and boundary-driven Lindblad evolution for active local coupling.
Source: Introduction · p. 1, Heat, work and boundary-driven… · p. 4Results
An XX chain with one bath relaxes to equilibrium while an XY chain reaches a driven nonequilibrium steady state; with two baths, the XX chain can act as an engine, refrigerator, or heater with efficiencies bounded by Carnot.
Source: Introduction · p. 1, Spin models · pp. 5–6Takeaways & Limitations
Boundary driving can perform work despite appearing work-free, extending thermodynamic analysis to engines with actively controlled boundary coupling.
Source: Introduction · p. 1The repeated-interaction formulation assumes each incoming thermal copy is initially uncorrelated with the system; correlations with earlier copies do not affect its applicability.
Source: The repeated interaction scheme · p. 4
Abstract
from arXiv · showhide
The laws of thermodynamics put limits to the efficiencies of thermal machines. Analogues of these laws are now established for quantum engines weakly and passively coupled to the environment providing a framework to find improvements to their performance. Systems whose interaction with the environment is actively controlled do not fall in that framework. Here we consider systems actively and locally coupled to the environment, evolving with a so-called boundary-driven Lindblad equation. Starting from a unitary description of the system plus the environment we simultaneously obtain the Lindblad equation and the appropriate expressions for heat, work and entropy-production of the system extending the framework for the analysis of new, and some already proposed, quantum heat engines. We illustrate our findings in spin 1/2 chains and explain why an XX chain coupled in this way to a single heat bath relaxes to thermodynamic-equilibrium while and XY chain does not. Additionally, we show that an XX chain coupled to a left and a right heat baths behaves as a quantum engine, a heater or refrigerator depending on the parameters, with efficiencies bounded by Carnot efficiencies.
Introduction
The paper asks how to account for heat, work, and entropy production when a system is actively coupled to its environment at its boundaries. This differs from passive weak-coupling models, for which established master equations provide thermodynamic quantities and second-law behavior.
Introduction
- Active boundary coupling matters for engineered open-system dynamics, while passive weak coupling is the setting addressed by established quantum-engine thermodynamics.Boundary interactions can be localized and repeatedly switched with a controlled probe state.Source: Introduction · p. 1
open system weakly and passively coupled to the environment:
- For passive weak coupling, the Born-Markov-secular master equation gives a proper description and correct heat-flow and irreversible-entropy balances.The stated framework has been applied to thermodynamic properties and engine efficiencies.Source: Introduction · p. 1, open system weakly and passively… · p. 2
Introduction
- The authors derive thermodynamic quantities for active interactions by applying a unitary-system framework in the boundary-driven Lindblad limit.They illustrate the results in XX and XY spin chains.Source: Introduction · pp. 1–2
- Boundary driving can perform work on the system even when the operation appears work-free, and the paper examines its thermodynamic cost.The analysis applies to spin chains coupled to one or two heat baths.Source: Introduction · p. 1
open system weakly and passively coupled to the environment:
- Boundary-driven Lindblad equations generally do not satisfy local detailed balance, so passive weak-coupling thermodynamic results do not directly cover them.The paper uses a unitary system-plus-environment framework to analyze boundary-driven systems.Source: open system weakly and passively… · p. 2
“universe” under unitary evolution
The unitary formulation treats the system and environment together, defining energy changes through their joint evolution and separating heat from work. It also decomposes system entropy change into entropy flow and production, recovering standard expressions in the weak-coupling limit.
“universe” under unitary evolution
- The joint internal energy includes the system Hamiltonian and interaction, and its change is split into work and heat.Work is defined by the change in total Hamiltonian energy; heat contributions are minus the corresponding reservoir energy changes.Source: “universe” under unitary evolution · p. 3
- System entropy change separates into entropy flow, given by the reservoir heat flows weighted by inverse temperatures, and entropy production.The formulation relies on unitary evolution of the full system and environment.Source: “universe” under unitary evolution · p. 3
- In the weak-coupling limit with a Born-Markov-secular Lindblad equation, entropy production is nonnegative and the work and heat expressions reduce to standard forms.The paper says this recovery is shown using full-counting statistics.Source: “universe” under unitary evolution · p. 3
The repeated interaction scheme
The repeated-interaction model couples the system in turn to fresh thermal copies of each reservoir, yielding a boundary-driven evolution in the appropriate limit. It assigns work to interaction switches and tracks heat and entropy production through the outgoing copies and their correlations with the system.
The repeated interaction scheme
- Each reservoir is modeled as a sequence of identical copies, initially thermal and uncorrelated with the system, that interact with it in succession.The interaction form is repeated while the copy changes at intervals of duration τ.Source: The repeated interaction scheme · p. 3
- Work is performed at each switch between reservoir copies because the interaction term changes, with the work increment given by that interaction change.The derivation uses the assumption that the partial trace of the interaction over each reservoir copy is zero.Source: The repeated interaction scheme · pp. 3–4
- Heat from a reservoir is evaluated over the interval in which its copy interacts with the system, using the copy's state after interaction.The outgoing-copy density matrix is obtained by tracing out the system after the joint evolution.Source: The repeated interaction scheme · p. 4
- Entropy production includes the outgoing copy's relative entropy from its initial thermal state and correlations built with the system; it vanishes only if the copy is restored and those correlations are absent.Both contributions are nonnegative, and the stated conditions apply separately to the interacting copy.Source: The repeated interaction scheme · p. 4
Heat, work and boundary-driven Lindblad equation
Repeated interactions produce a boundary-driven Lindblad equation when the coupling is scaled appropriately as the interaction interval shrinks. In that limit, the derived heat, work, and entropy-production expressions provide the thermodynamic quantities for the resulting system dynamics.
Heat, work and boundary-driven Lindblad equation
- Scaling each interaction as v_r/√τ, with zero thermal average for the interaction, yields Lindblad evolution as τ approaches zero.The resulting equation applied to particular systems provides boundary-driven Lindblad equations.Source: Heat, work and boundary-driven… · p. 4
- The limiting expressions define heat and work rates from the repeated-interaction increments and give the system energy balance.The energy change rate is the sum of heat and work rates and equals the change in system energy from its dissipative evolution.Source: Heat, work and boundary-driven… · p. 4
- The paper expresses entropy production as the system entropy-change rate minus entropy flow and states that its equations provide thermodynamic expressions for boundary-driven Lindblad dynamics.The authors then illustrate the framework in spin-1/2 chains.Source: Heat, work and boundary-driven… · p. 4
Spin models
The spin-chain examples show that boundary driving can produce equilibrium or a driven steady state with one bath, while a two-bath XX chain operates in parameter-dependent machine modes. The reported thermodynamic quantities include power supplied or extracted through the boundary coupling and satisfy the stated entropy-production and efficiency bounds.
XX chain coupled to two baths
- The boundary dissipators thermalize an isolated boundary spin but do not satisfy local detailed balance with respect to the chain's Gibbs state.In particular, the Gibbs state is not stationary under either boundary dissipator.Source: Spin models · p. 5
XX chain coupled to one bath
- With one bath, the uniform-field XX chain reaches equilibrium with vanishing entropy production, heat flows, and power; its state is a product of identical single-spin canonical states matching the reservoir copy.The equilibrium is a generalized Gibbs state associated with a conserved quantity commuting with the chain Hamiltonian.Source: Spin models · p. 5
XY chain coupled to a single bath
- With one bath, the XY chain reaches a driven NESS with constant positive entropy production, where βẆ = diS/dt > 0 and Ẇ = −Q̇.In the NESS, the system entropy is steady and the average system-energy change is zero.Source: Spin models · p. 5
XX chain coupled to two baths
- In the two-bath NESS, heat flows can reverse while entropy production remains nonnegative; Q̇L = −hLjs, Q̇R = hRjs, and Ẇ = (hL − hR)js, so equal fields permit current without power.Entropy production vanishes only when βRhR = βLhL.Source: Spin models · pp. 5–6
- For βL ≤ βR, the XX chain operates as an engine when βL/βR < hR/hL < 1, a refrigerator when hR/hL < βL/βR, and a heater when hR/hL > 1; engine and refrigerator efficiencies are bounded by Carnot.The efficiencies are independent of temperature.Source: Spin models · p. 6
L , ˙QNESS
In the non-driven steady state, the heat-flow relation is accompanied by zero power.
L , ˙QNESS
- The steady state has zero power, ˙W^NESS = 0, and is non-driven.The passage relates the left and right heat flows in the NESS before stating this result.Source: L , ˙QNESS · p. 6
Discussion
The repeated-interaction scheme yields a boundary-driven Lindblad equation and corresponding thermodynamic quantities. Reservoir refreshing can require work, while spin-chain outcomes depend on the model and coupling conditions.
Discussion
- In an appropriate limit, repeated interactions provide a physical description yielding a boundary-driven Lindblad equation.The equation’s operators depend on the system–environment interaction, the bath-copy Hamiltonian, and constant refreshing.Source: Discussion · pp. 6–7
- Reservoir refreshing can involve work by an external agent, and this power can drive the system out of equilibrium even with time-independent system Hamiltonians and Lindblad operators.The power is associated with refreshing the reservoir.Source: Discussion · p. 7
- With one bath, a homogeneous-field XX chain reaches thermal equilibrium, whereas an XY chain reaches a driven NESS with nonzero entropy production, diS/dt = β ˙W > 0.For two baths, the XX chain reaches a non-driven NESS when β_R ≠ β_L; homogeneous-field equilibrium occurs when β_L h_L = β_R h_R, while inhomogeneous fields yield a driven NESS.Source: Discussion · p. 7
- A Lindblad equation alone does not determine heat flows: thermodynamic quantities also depend on the environment and how it couples to the system.For weak, passive coupling, the text instead associates thermodynamic quantities with a global Lindblad equation; the paper extends quantum thermodynamics to systems without local detailed balance.Source: Discussion · p. 7
Methods
The section provides details of calculations mentioned in the main text.
Methods
- The section provides calculation details for results discussed in the main text.No further calculation or result is specified in the supplied passage.Source: Methods · p. 7
Work, heat and boundary-driven Lindblad equation from the repeated interaction scheme
The repeated-interaction derivation takes a small-time limit to obtain the boundary-driven evolution and thermodynamic expressions, organizing contributions by reservoir.
Work, heat and boundary-driven Lindblad equation from the repeated interaction scheme
- The unitary evolution is expanded for small τ under a scaled interaction, with terms retained to the orders needed for the derivation.The heat change has leading order O(τ); the other expansion requires U through O(τ^3/2).Source: Work, heat and boundary-driven… · pp. 7–8
- The derivation takes the limit τ → 0 and n → ∞ with t = nτ after dividing the evolution expression by τ.The passage states that Tr_E[vρ_E] = 0 makes the leading term O(τ).Source: Work, heat and boundary-driven… · p. 7
- The derivation separates contributions by reservoir to obtain the dissipator and the stated thermodynamic expressions.The text identifies the reservoir-specific dissipator as Eq. (9) and connects the resulting expressions to Eqs. (10) and (11).Source: Work, heat and boundary-driven… · pp. 7–8
The two spin XX chain with inhomogeneous magnetic field
For a two-site XX chain with an inhomogeneous magnetic field, the dynamics is described through a closed set of correlation functions. In the steady state, work and heat currents are related to the spin current and the field difference.
The two spin XX chain with inhomogeneous magnetic field
- The chain is a two-site XX model with coupling J and site fields h1 = hL and h2 = hR.Its Lindblad dynamics uses the Hamiltonian specified in the main text.Source: The two spin XX chain with… · p. 8
- The dynamics is fully characterized by ⟨σz1⟩t, ⟨σz2⟩t, X, and Y, which satisfy a closed system of equations.Here X = ⟨σx1σx2 + σy1σy2⟩t and Y = i⟨σy1σx2 − σx1σy2⟩t.Source: The two spin XX chain with… · p. 8
- In the steady state, the total work rate is (hL − hR)js, while the heat rates are Q̇L = −hLjs and Q̇R = hRjs.The spin current is js = iJY; the current reported in the main text is obtained by solving the full system in the nonequilibrium steady state.Source: The two spin XX chain with… · p. 8
Heat from a given reservoir in the weak-coupling limit
The paper defines heat from a reservoir using a unitary description of the system and environment, then relates that definition to the weak-coupling Markovian description. In that limit, the heat flow agrees with the system-energy change generated by the reservoir dissipator.
Heat from a given reservoir in the weak-coupling limit
- Heat from the left reservoir is defined by the change in its energy in the unitary system-plus-environment description.The definition is QL ≡ Tr[HL(ρtot(0) − ρtot(t))].Source: Heat from a given reservoir in the… · p. 8
- Full-counting statistics expresses this heat through a counting-field derivative and, equivalently, through the counting-field-dependent generator acting on the system state.The passage gives QL = i∂λG(λ)|λ=0 = −iTr(t∂λLλρS(t))|λ=0.Source: Heat from a given reservoir in the… · p. 8
- In the weak-coupling Markovian limit, the heat flow equals the system-energy change generated by the reservoir dissipator.The passage identifies −iTr[(∂λDL,λ)ρS(t)]|λ=0 with tr(HSDL(ρS)).Source: Heat from a given reservoir in the… · p. 9