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Scaling-up quantum heat engines efficiently via shortcuts to adiabaticity

M. Beau, J. Jaramillo, A. del Campo

arXiv:1603.06019v3quant-phmath-phmath.MPphysics.atom-ph

TL;DR

Finite-time single-particle quantum heat engines face an efficiency–power tradeoff because nonadiabatic excitations cause friction. This paper combines a many-particle working medium with shortcuts to adiabaticity. It reports zero-friction operation at maximum efficiency with tunable output power.

  • Problem

    Finite-time operation of a single-particle quantum heat engine can increase output power while nonadiabatic friction lowers cycle efficiency.

    Source: §I · p. 1

  • Method

    The paper combines a many-particle working medium with shortcuts to adiabaticity and analyzes a quantum Otto cycle for scale-invariant dynamics.

    Source: §I · p. 1, §IV · p. 12

  • Results

    The many-particle cycle can be engineered with shortcuts to adiabaticity to reach zero friction and the adiabatic Otto efficiency upper bound.

    Source: B. Quantum Otto Cycle and Fundamental… · p. 3, §IV · p. 12, §V · p. 12

  • Takeaways & Limitations

    The proposed engine has enhanced output power from its many-particle medium and reduced cycle time while operating at maximum efficiency.

    Source: §I · p. 2, §V · p. 12

Abstract

from arXiv · show

The finite-time operation of a quantum heat engine that uses a single particle as a working medium generally increases the output power at the expense of inducing friction that lowers the cycle efficiency. We propose to scale up a quantum heat engine utilizing a many-particle working medium in combination with the use of shortcuts to adiabaticity to boost the nonadiabatic performance by eliminating quantum friction and reducing the cycle time. To this end, we first analyze the finite-time thermodynamics of a quantum Otto cycle implemented with a quantum fluid confined in a time-dependent harmonic trap. We show that nonadiabatic effects can be controlled and tailored to match the adiabatic performance using a variety of shortcuts to adiabaticity. As a result, the nonadiabatic dynamics of the scaled-up many-particle quantum heat engine exhibits no friction and the cycle can be run at maximum efficiency with a tunable output power. We demonstrate our results with a working medium consisting of particles with inverse-square pairwise interactions, that includes noninteracting and hard-core bosons as limiting cases.

I. INTRODUCTION

Finite-time single-particle engines trade efficiency for power because nonadiabatic excitations create quantum friction. The paper proposes combining a many-particle working medium with shortcuts to adiabaticity, and studies scale-invariant quantum fluids in a time-dependent harmonic trap.

I. INTRODUCTION

  • Finite-time single-particle quantum heat engines gain power at the cost of friction that lowers efficiency, while their maximum efficiency occurs at vanishing power in the adiabatic limit.Scaling up is proposed as a way to address this efficiency–power tradeoff.Source: §I · p. 1
  • The paper combines a many-particle working medium with shortcuts to adiabaticity to seek zero-friction operation at maximum efficiency and enhanced output power.The stated power enhancement comes from the many-particle medium and reduced cycle time.Source: §I · pp. 1–2

A. Trapped Quantum Fluids as Working Media

  • The working medium is a scale-invariant many-body system in an isotropic time-dependent harmonic trap, with pair interactions homogeneous of degree −2.The family includes noninteracting gases, hard-core fluids, inverse-square interaction models, and two-dimensional Bose gases with s-wave contact interactions.Source: A. Trapped Quantum Fluids as Working… · p. 2
  • The harmonic Calogero–Sutherland model illustrates the family: λ = 0 gives trapped ideal bosons, while λ = 1 gives the Tonks–Girardeau regime with thermodynamics identical to trapped spin-polarized fermions.For other λ values, the model describes particles with fractional exclusion statistics.Source: A. Trapped Quantum Fluids as Working… · p. 3
  • For the Calogero–Sutherland model, interactions affect thermodynamics through a renormalization of the ground-state energy.The passage states that the interaction effect is completely absorbed in that energy shift.Source: A. Trapped Quantum Fluids as Working… · p. 3

B. Quantum Otto Cycle and Fundamental Limits

A quantum Otto cycle alternates unitary frequency-changing strokes with thermalizing strokes. Finite-time excitations lower efficiency relative to the adiabatic Otto bound, but the bound is shared by single- and many-particle cycles.

B. Quantum Otto Cycle and Fundamental Limits

  • The Otto cycle alternates isentropic compression, hot isochore, isentropic expansion, and cold isochore, with work on the isentropes and heat transfer on the isochores.The isentropic strokes change trap frequency at constant temperature; the isochores vary temperature at fixed frequency.Source: B. Quantum Otto Cycle and Fundamental… · p. 3
  • In the adiabatic limit, the Otto efficiency is an upper bound shared by single- and many-particle quantum Otto cycles.The cycle diagram identifies thermal states at A and C and generally nonequilibrium states at B and D.Source: B. Quantum Otto Cycle and Fundamental… · p. 3, Figure 1 · p. 4
  • Finite-time driving introduces nonadiabatic excitations and quantum friction, lowering efficiency; maximum efficiency occurs in the slow-driving limit, where output power vanishes.The cited discussion describes this as the finite-time thermodynamics tradeoff in the many-particle setting as well.Source: B. Quantum Otto Cycle and Fundamental… · p. 4

C. Optimization of Power and Efficiency

The section formulates finite-time power optimization for a fixed cycle time by varying the trap-frequency ratio. Its optimal-work condition is presented for a high-temperature hot bath, with a factor that depends on the stroke protocol.

C. Optimization of Power and Efficiency

  • For a fixed cycle time, the method maximizes output power by optimizing the frequency ratio that determines total output work.The total output work is the negative sum of work over the compression and expansion strokes.Source: §1 · p. 5
  • For a high-temperature hot bath, the optimal work condition gives the frequency ratio as a solution involving a protocol-dependent factor N_F.N_F equals 1 for an adiabatic protocol and x^2 for sudden-quench driving.Source: §1 · p. 5
  • The paper notes that sudden-quench optimal-power efficiency can be enhanced for many particles when σ_c(h) > 1 while remaining much smaller than N.The stated regime is relatively low temperature and is reported as prior work [17].Source: §1 · p. 5

2. Optimizing Adiabatic Output Power

The section evaluates adiabatic output-power optimization across thermal regimes. At high temperatures, the work scales with particle number and efficiency follows the Curzon–Ahlborn result; quantum corrections can reduce performance outside that regime.

2. Optimizing Adiabatic Output Power

  • When σ_c(h) ≪ 1, many-particle effects are negligible and adiabatic efficiency is the Curzon–Ahlborn result, 1−√a.The passage associates this result with a classical heat engine operated in the adiabatic limit.Source: §2 · p. 6
  • For 1 ≤ σ_c ≪ N, the reported efficiency is below the Curzon–Ahlborn value, while the very-low-temperature quantum limit also has lower performance than the classical limit.The intermediate-regime result assumes σ_h ≪ 1 and applies for large N with ℏω_1β_c ≪ 1; the quantum-limit passage describes σ_c ≫ N.Source: §2 · p. 6
  • At high temperatures for both reservoirs, adiabatic work per cycle scales by N relative to the single-particle engine, and efficiency is the Curzon–Ahlborn value.This is identified as the classical regime in which the scaled-up engine's adiabatic efficiency and work output are maximized.Source: §2 · p. 6

III. SUPERADIABATIC MANY-PARTICLE QUANTUM HEAT ENGINES

The section examines finite-time optimization of many-particle quantum heat engines using shortcuts to adiabaticity. In the accidental-STA approach, selected stroke durations recover adiabatic work and efficiency while shortening the cycle.

A. First Approach: Finite-Time Optimization and Accidental STA

  • The approach varies finite stroke duration at fixed bath temperatures and seeks protocols whose nonadiabatic dynamics become adiabatic at selected durations.Those instances are termed accidental STAs and depend on trap frequencies ω1 and ω2.Source: A. First Approach: Finite-Time… · p. 7
  • The accidental protocol interpolates between sudden-quench and adiabatic limits as its duration changes.For γ≤1 the nonadiabatic factor decreases from its sudden-quench value but remains above unity; for γ>1 it oscillates above the adiabatic value.Source: A. First Approach: Finite-Time… · p. 7
  • At accidental-STA contact times, Q*=1 and work and efficiency reproduce slow-driving values; choosing τ1 gives zero friction, maximum efficiency, and reduced cycle time.The contact times are determined by the nonadiabatic factor, and τ1 is the minimum accidental-STA duration.Source: A. First Approach: Finite-Time… · pp. 7–8
  • For λ=0, and typically λ<0.1, optimal-power efficiency can be enhanced drastically when τ≤1/ω1.For τ>1/ω1, nonadiabatic efficiency at optimal power approaches the adiabatic value.Source: A. First Approach: Finite-Time… · p. 8
  • For τ′1<τ<τ1, nonadiabatic optimal-power efficiency can exceed the adiabatic value, explained by a smaller optimal frequency ratio, while remaining below the same-ratio Otto limit.At τ′1 the efficiency matches the adiabatic value even though Q*>1.Source: A. First Approach: Finite-Time… · p. 8

B. Second Approach: Adiabatic Optimization and STA

This approach first optimizes the engine parameters assuming adiabatic dynamics, then replaces the compression and expansion strokes with shortcuts to adiabaticity.

B. Second Approach: Adiabatic Optimization and STA

  • For fixed bath temperatures, the frequency ratio is chosen to maximize adiabatic output power, then STA protocols implement that performance in finite, pre-scheduled stroke times.The section considers three techniques for engineering the required shortcuts.Source: B. Second Approach: Adiabatic… · p. 9

1. Reverse Engineering of the Scaling Dynamics

Reverse engineering the scaling dynamics provides a way to construct shortcuts to adiabaticity by choosing a scaling factor that satisfies endpoint conditions.

1. Reverse Engineering of the Scaling Dynamics

  • Reverse engineering the dynamics constructs STA protocols by imposing boundary conditions on the scaling factor b(t).The associated trap frequency must satisfy the Ermakov equation and its endpoint values.Source: §1 · pp. 9–10
  • For the proposed scaling-factor form, τ must exceed a threshold τc to ensure ω(t)² remains nonnegative throughout the stroke.Numerical analysis finds a unique τc such that the condition holds for all τ>τc.Source: §1 · p. 10
  • The construction gives shortcut times τn=2nπt0; for fixed τ and ω1(2), with ω(t)^2>0 on [0,τ], one protocol has a shorter minimum STA time than accidental STA.Numerically, τ1≤1/ω1 for all values of x.Source: §1 · p. 10

2. Counterdiabatic Driving

Counterdiabatic driving (CD) makes the adiabatic approximation an exact solution without slow driving; local counterdiabatic driving (LCD) maps the auxiliary control to a local potential. Both can match adiabatic performance at the stroke endpoints, with LCD requiring boundary conditions on the trap-frequency modulation.

2. Counterdiabatic Driving

  • CD makes the adiabatic approximation an exact solution of the time-dependent Schrödinger equation without slow driving.For scaling dynamics, the auxiliary term can be computed in closed form and takes the form of a one-body counterdiabatic term.Source: §2 · p. 10
  • Under CD, the mean energy follows its adiabatic value throughout the evolution and the nonadiabatic factor is unity.This holds for an arbitrary protocol ω(t).Source: §2 · p. 11

3. Local Counterdiabatic Driving

  • LCD reduces the auxiliary counterdiabatic term to a local potential for scale-invariant many-body systems.The working-medium Hamiltonian considered belongs to this family.Source: §3 · p. 11
  • LCD returns the nonadiabatic factor to unity at both protocol endpoints when dotω(0) = dotω(τ) = 0.Transient excitations are generated during the protocol and canceled by its completion; arbitrarily small τ is possible if modulation of Ω(t) can be implemented.Source: §3 · p. 11

IV. DISCUSSION

The discussion describes two routes to maximum-efficiency, optimal-power operation of a many-particle Otto engine: finite-time power optimization that can yield accidental shortcuts, or direct zero-friction optimization with designed shortcuts. The conclusion proposes combining a many-particle working medium and shortcuts to reach maximum efficiency with enhanced output power in finite time.

IV. DISCUSSION

  • Both optimization routes lead to maximum efficiency and optimal output power: finite-time power optimization can yield accidental STA, while imposing zero friction allows designed STA.The designed options include reverse engineering, counterdiabatic driving, and local counterdiabatic driving.Source: §IV · p. 12

V. CONCLUSIONS

  • The usual route to maximum efficiency is slow driving, which comes at the expense of vanishing output power.The paper proposes finite-cycle-time operation that matches adiabatic performance using shortcuts to adiabaticity.Source: §V · p. 12
  • The proposed many-particle engine operates with zero friction at the adiabatic Otto upper-bound efficiency shared by single- and many-particle engines, with enhanced output power.The enhancement is attributed to the many-particle working medium and reduced cycle time; the proposed implementation uses an interacting trapped Bose or Fermi gas with suitable harmonic-trap modulation.Source: §V · p. 12

APPENDIX A. NONADIABATICITY OF THE ACCIDENTAL PROTOCOL

The appendix evaluates nonadiabaticity for the accidental STA protocol during Otto-cycle compression and checks that its scaling factor satisfies STA consistency conditions. It derives the scaling factor and identifies times at which its value and derivatives meet the stated conditions.

APPENDIX A. NONADIABATICITY OF THE ACCIDENTAL PROTOCOL

  • The nonadiabatic factor is symmetric between the compression A→B and expansion C→D steps of the Otto cycle.The appendix computes these quantities for compression and checks the scaling factor's consistency for both steps.Source: Appendix A · p. 12

Appendix A.1. Derivation of the Scaling Factor b(t)

  • The appendix derives the scaling factor b(t) for the accidental protocol using a formula attributed to [17].The displayed formula is not legible in the supplied passage.Source: Appendix A.1 · p. 12
  • At times τ_n, b(τ_n)=1 and its first and second derivatives vanish, with τ_n set by Equation (32) as the STA time.The passage gives n=1,2,3,... as positive integers.Source: Appendix A.1 · p. 13

Appendix A.2. Derivation of the Nonadiabatic Factor Q∗

The appendix calculates the nonadiabatic factor Q∗ using the Husimi formula. It uses the squared derivative of a classical fundamental solution and Equation (A2) to obtain Formula (31).

Appendix A.2. Derivation of the Nonadiabatic Factor Q∗

  • The Husimi formula is used to compute the nonadiabatic factor Q∗.The passage attributes the formula to.Source: Appendix A.2 · p. 13
  • The calculation uses the square of the first derivative of the classical fundamental solution.The passage states that this quantity is read from the classical equation's fundamental solution.Source: Appendix A.2 · p. 13
  • Applying Equation (A2) leads to Formula (31) for Q∗.The intervening expressions are not legible in the supplied passages.Source: Appendix A.2 · p. 14
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