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The quantum harmonic Otto cycle

Ronnie Kosloff, Yair Rezek

arXiv:1612.03582v3quant-ph

TL;DR

The review examines the quantum harmonic oscillator linked to the Otto cycle, whose quantum reciprocating engines resemble macroscopic counterparts. It describes Hamiltonian evolution on adiabatic segments and dissipative driving toward thermal states, while highlighting time constraints and quantum coherence as performance considerations.

  • Problem

    Reaching the target state raises the question of the shortest achievable time, while compression ratios require significant time.

  • Method

    The review studies the quantum harmonic oscillator linked to the Otto cycle and describes its dynamics through Hamiltonian adiabatic evolution and dissipative thermal driving.

  • Results

    Quantum reciprocating engines exhibit strong resemblance to macroscopic counterparts, with endoreversible-engine efficiency converging at high temperature to the Novikov–Curzon–Ahlborn prediction.

  • Takeaways & Limitations

    Coherence transferred to the system can be used to increase the engine’s performance under squeezed thermal-bath conditions.

Abstract

from arXiv · show

The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables to study finite time cycles limiting time on the adiabtic and the thermalization times. Explicit finite time solutions are found which are frictionless, meaning that no coherence is generated also known as shortcuts to adiabaticity. The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators.

I. INTRODUCTION

The quantum Otto cycle applies thermodynamic reasoning to a single quantum working medium, especially a harmonic oscillator that remains analytically tractable and experimentally realizable. The review develops its open-system description, cycle strokes, and connections to macroscopic engine behavior.

  • Finite-time operation: Finite-time thermodynamics focuses on the tradeoff between maximum efficiency and maximum power in engines operating under irreversible conditions.Quantum engines incorporate dynamics into thermodynamics, making finite-time operation and irreversible costs central considerations.
  • Motivation: The quantum Otto cycle is a primary reciprocating quantum-engine example because it is easier to analyze than other cycles.Earlier reciprocating-engine studies used qubits as working media, while Otto cycles provide a central framework for comparison with macroscopic engines.
  • Comparison with macroscopic engines: Quantum reciprocating engines can closely resemble macroscopic counterparts, including convergence of endoreversible efficiency at maximum power to the Novikov–Curzon–Ahlborn result at high temperature.The deviations can remain small at low temperature even when the heat-transport law differs; an identified quantum feature is the discrete energy structure.
  • Motivation: A harmonic oscillator provides a sufficiently complex yet analytically tractable working medium for studying generic quantum thermal-device phenomena.The model has also motivated experimental studies and can describe a single atom in a harmonic trap.
  • Theoretical framework: The framework connects quantum mechanics with thermodynamics through weak system–bath coupling, allowing work and heat to be defined in the quantum regime.The analysis is based on quantum open-systems theory; strong coupling remains a regime where the system–bath partition is unclear and the thermodynamic connection is not established.
  • Quantum Otto cycle: The review formulates each Otto-cycle stroke as a completely positive propagator acting on the working-medium state.The four strokes are hot isochore, expansion adiabat, cold isochore, and compression adiabat; their ordered product forms the cycle propagator.

A. Quantum Dynamics of the Working Medium

The quantum Otto cycle uses a harmonic oscillator as its working medium, with trap-frequency control driving unitary adiabats and Lindblad dynamics modeling thermalization on isochores. A closed set of thermodynamical observables reduces the dynamical description while preserving the system state.

  • Working medium: A harmonic oscillator in a controlled trap provides an analytically tractable working medium for the quantum Otto cycle.The trap parameter k(t) controls the potential and energy scale.
  • Adiabatic dynamics: Adiabatic evolution is unitary and generated by the time-dependent Hamiltonian, whose noncommutation at different times produces quantum friction.The corresponding formal solution defines a propagator for the stroke.
  • Thermalization: Isochore dynamics is modeled as an open quantum system with a static Hamiltonian and a Lindblad dissipator driving the oscillator toward thermal equilibrium.Finite hot- and cold-isochore durations generally prevent complete equilibration.
  • Cycle structure: The four-stroke cycle separates power production or consumption on adiabats from heat transfer on isochores.The frequency changes between ωh and ωc during expansion and compression, while baths act during thermalization strokes.
  • Thermodynamical observables: Canonical invariance allows the state to remain describable through a closed set of thermodynamical observables, including H, L, C, and the identity.For the harmonic oscillator, the relevant operators remain closed under both the unitary and dissipative dynamics.
  • Thermodynamical observables: The observable-space formulation reduces the propagator representation from N^2 dimensions to M dimensions, where M is the size of the closed operator set.The state can be reconstructed from the expectations of these observables under the maximum entropy principle.

Entropy Balance

The entropy analysis distinguishes state entropy from energy-measurement entropy for nonequilibrium oscillator states. This distinction supports an internal temperature and a work measure associated with coherence.

  • Entropy definitions: The von Neumann entropy is the minimum entropy required to completely specify the quantum state.It depends only on the state and satisfies SVN ≤ SŌ for measurement-based entropies.
  • Entropy definitions: Energy entropy generally exceeds von Neumann entropy and becomes equal to it when the state is diagonal in the energy representation.Thermal equilibrium is an example where the two entropies coincide.
  • Coherence: The relative entropy between a state and its energy-diagonal representation equals the difference SE−SVN, providing a measure of coherence-related information.The diagonal state preserves the energy-level populations of the original state.
  • Internal temperature: The oscillator’s energy entropy defines an inverse internal temperature through the equilibrium temperature–energy relation.This internal temperature is then used to define the work required to generate coherence.
  • Coherence: For a squeezed pure state, the von Neumann entropy is zero while the energy entropy remains nonnegative.This illustrates that energy entropy can capture information absent from the state entropy.

IV. THE DYNAMICS OF THE QUANTUM OTTO CYCLE

The quantum Otto cycle is analyzed as four dynamical strokes whose finite durations determine heat transport, power, and refrigeration or engine performance. Isochore propagators act on the observables H, L, C, and I.

  • Cycle performance: Quantum-engine performance involves a tradeoff between efficiency and power, with every cycle segment contributing to the final outcome.The analysis optimizes individual segments before considering global performance.
  • Isochores: The isochores extract and reject heat from thermal reservoirs while the working medium approaches thermal equilibrium.Finite isochore durations mean equilibration is generally incomplete.
  • Isochores: Isochore dynamics generates a propagator on the vector space of observables H, L, C, and I.The propagator includes exponential relaxation and oscillatory terms determined by the heat conductance and oscillator frequency.
  • Isochores: The isochore propagator does not generate coherence from the energy observable H, because the coherence observables L and C are not coupled to H.The dynamics is therefore dominated by approach toward thermal equilibrium.

B. The Dynamics on the Adiabats and Quantum Friction

Finite-time adiabatic control mixes energy and coherence, producing quantum friction, but specially designed protocols can eliminate this mixing. The shortest frictionless times depend on control and energy constraints.

  • Quantum friction: Nonadiabatic dynamics couples energy and coherence through the dimensionless parameter µ = ω̇/ω^2.When µ → 0, energy decouples from coherence and the cycle can be described by energy-level populations.
  • Quantum friction: Generating coherence from an energy-diagonal state consumes power, and the associated friction power vanishes under adiabatic conditions.For µ → 0, ⟨L⟩ = 0 and Pf = 0.
  • Quantum friction: The energy cost of quantum friction scales as µ^2, so slow operation suppresses the cost but requires large cycle times and yields low power.This establishes the finite-time tradeoff between friction reduction and power.
  • Frictionless protocols: Periodic protocols can generate and then consume coherence so that the propagator becomes diagonal, eliminating energy–coherence mixing at the stroke boundary.These frictionless solutions occur when cos(Ωθ(τa)) = 1.
  • Frictionless protocols: Shortcuts to adiabaticity achieve frictionless finite-time performance by controlling ω(t) so the state is diagonal in energy when it reaches an isochore.Invariant-based and optimal-control approaches are used to construct such protocols.
  • Time optimization: Under reasonable control constraints, the minimum frictionless time scales as O(1/(√ωc√ωh)).Other constraints can shorten the time, but their energetic cost can diverge.

C. The Influence of Noise on the Adiabats

The review models amplitude and phase noise on frictionless adiabats and shows that imperfect control generates additional friction, with distinct effects for each noise type.

  • Noise modeling: Control-frequency fluctuations are modeled as Markovian Gaussian white noise and represented by a dissipative Lindblad term.The noise arises from fluctuations in the oscillator frequency ω(t).
  • Amplitude noise: Amplitude noise is analyzed by factorizing the hot-to-cold propagator into an adiabatic propagator and an interaction-picture noise propagator.A closed-form solution is obtained in the frictionless limit µ →0 using a Magnus expansion.
  • Amplitude noise: The shortest frictionless protocol minimizes amplitude-noise effects, although some friction-like behavior remains.For large protocol-period count l or µ →0, the friction contribution δf diverges and nulls the adiabatic solution even for small γa.
  • Phase noise: Phase noise models timing errors as dephasing on the adiabats and preserves the frictionless condition at first order in µ.The second-order Magnus term introduces a noise correction.

D. The Sudden Limit

The sudden limit arises when adiabat durations vanish, producing a sudden-quench propagator that mixes energy and coherence. Its non-Hermitian generator has an exceptional point at |µ| = 2.

  • Sudden limit: Vanishing adiabat time τa ≪1/ωc produces the sudden propagator, also termed sudden quench dynamics.The sudden propagator is used in frictionless bang-bang solutions.
  • Sudden limit: When the compression ratio differs from one, the propagator mixes energy and Ĥ-related observables, generating coherence.This mixing is part of the sudden propagator’s role in bang-bang protocols.
  • Exceptional point: At the exceptional point, the dynamics changes from oscillatory to exponential.The same threshold corresponds to the transition from an underdamped to an overdamped oscillator after time rescaling.
  • Cycle performance: Exceptional points in the total cycle propagator are expected to signal a drastic change in cycle performance.Depending on parameters, the complete cycle operates as an engine, refrigerator, or dissipative cycle.

A. Limit Cycle

The cycle propagator is a completely positive map whose fixed point defines the limit cycle. Finite-time friction, frictionless protocols, and thermalization times determine whether the device operates as an engine, refrigerator, or dissipative cycle.

  • A. Limit Cycle: A limit cycle is reached when the working medium’s internal variables attain a periodic steady state with no accumulated energy or entropy.The resulting balance is between external driving and dissipation.
  • A. Limit Cycle: When the cycle time is shortened, friction accumulates heat in the working medium and increases the temperature gaps and dissipation.Overdriving can dissipate power to both the hot and cold baths.
  • A. Limit Cycle: The cycle is a product of completely positive evolution maps, and a unique invariant state causes initial states to approach the limit cycle monotonically.The eigenvalue 1 identifies the fixed point, while the other eigenvalues determine the approach rate.
  • A. Limit Cycle: The non-compact harmonic oscillator challenges compact-map limit-cycle results, and studies identify parameter regimes where a limit cycle is not obtained.The model assumes an unbounded Hamiltonian and Lindblad operator, while the resulting map’s non-compact character remains unresolved.
  • B. Frictionless Conditions: For adiabatic and frictionless cycles, performance is determined by switching-point energies, with no coherence generated in the adiabatic limit.Infinite-time adiabatic operation maximizes work but has zero power because the cycle time is infinite.
  • B. Frictionless Conditions: Frictionless solutions achieve Otto efficiency in finite time, while optimizing power requires finite allocations across the cycle segments.Thermalization on the isochores transports heat and produces entropy.
  • B. Frictionless Conditions: Additional energy generated during frictionless driving can cancel friction and function as a catalyst, but imperfect controls introduce extra work and entropy production.The work in the limit cycle is Wq = ℏ(ωc −ωh)(N B −N D).

3. The Engine in the Sudden Limit

The sudden limit compresses all cycle strokes toward zero duration while retaining finite engine power through coherence. Its performance is constrained by friction, thermalization assumptions, and the transition to dissipative operation.

  • Friction and performance: The engine’s efficiency is lower when friction constrains it than under endo-reversible operation, which itself remains below ideal Carnot efficiency.The comparison is stated for the maximum-work efficiencies derived from the corresponding limits.
  • Friction and performance: Frictional work increases with the temperature ratio and can balance all useful work at the frictionless-optimal compression ratio, beyond which the engine becomes a dissipator.In that regime, entropy is generated at both baths.
  • Sudden-cycle dynamics: The complete sudden limit assigns short times to all strokes, including isochores, producing vanishing cycle times and approaching a continuously operating engine.The isochore times are assumed much shorter than both the oscillator period and the inverse heat-conductance scale.
  • Coherence: The sudden cycle exploits coherence present at all four corners, unlike the frictionless engine, where coherence is canceled during the protocol.The limit-cycle vector contains both energy and coherence-related observables.
  • Friction and performance: Even at zero power, entropy production can remain positive because heat leaks from the hot bath to the cold bath.The cited example is the compression ratio C = 1.
  • Sudden-cycle dynamics: Finite power persists as both thermalization times approach zero, but additional dephasing removes useful power and makes the cycle a dissipator.The sudden engine therefore relies on coherence surviving the short isochore strokes.

4. Work Fluctuation in the Engine Cycle

Work fluctuations arise from energy fluctuations at the cycle corners and increase with coherence. For complete thermalization, the variance is smallest at the Carnot compression ratio.

  • Work fluctuations: Work fluctuations are calculated from energy fluctuations at the four corners of the cycle.For generalized Gibbs states, energy variance is related to the internal temperature by Var(E) = (kBT_int)^2.
  • Work fluctuations: For complete thermalization, work variance is smallest at the Carnot compression ratio C = T_h/T_c.This condition applies when the oscillator reaches the bath temperature.
  • Work fluctuations: Generating coherence increases the energy variance and therefore affects fluctuations in the engine’s work.The review identifies this increase as a direct energetic cost of coherence.

5. Quantum Fuels: Squeezed Thermal Bath

A squeezed thermal bath supplies the oscillator with energy and coherence beyond ordinary equilibrium thermalization. Modified frictionless protocols can convert the transferred coherence into additional extractable work.

  • Energy and coherence resources: Coherence can reduce the fuel’s von Neumann entropy, allowing higher efficiency without violating the second law.A squeezed thermal bath is presented as a fuel supplying both heat and coherence, so work can be extracted from one heat bath within thermodynamic laws.
  • Squeezed-bath model: Squeezing modifies the bath Hamiltonian, system–bath interaction, and thermalization master equation through a squeezing operator and parameter.The heat conductance remains defined by the transition-rate difference, with rates satisfying detailed balance.
  • Squeezed-bath model: The squeezed-bath invariant state contains quantum coherence or correlations in addition to the equilibrium oscillator state.Thermalization also generates mutual system–bath correlations.
  • Energy and coherence resources: The squeezed bath delivers extra energy, with its oscillator energy exceeding or equaling the equilibrium value at the same nominal hot-bath temperature.The review interprets this as an effectively higher temperature for the working fluid.
  • Energy and coherence resources: Transferred coherence can be converted into more work by an adiabatic protocol that ends with lower energy and a state diagonal in energy.The protocol is modified from the frictionless case to exploit initial coherence rather than merely cancel generated coherence.
  • Refrigerator protocols: For frictionless refrigerators, optimal-control and superadiabatic protocols scale the expansion time as τ_hc ∝ 1/√(ω_cω_h), outperforming a linear frequency ramp.The refrigerator analysis frames performance as a tradeoff between cooling power and efficiency.

2. The Sudden Refrigerator

The sudden refrigerator can retain finite cooling power as cycle time vanishes by exploiting coherence, but its cooling power vanishes toward absolute zero, consistent with dynamical third-law restrictions.

  • Sudden refrigerator: Despite excitation from short adiabats, coherence can support finite cooling power in the vanishing-cycle-time limit.This is the refrigerator analogue of the sudden engine’s coherence-assisted operation.
  • Sudden refrigerator: The sudden-limit cooling rate becomes zero below a sufficiently low cold-bath temperature, so the formula cannot be used in that regime.The passage identifies this as a low-temperature scope boundary for the sudden-limit expression.
  • Third-law behavior: Cooling power vanishes as T_c approaches zero, expressing a dynamical version of the third law.The result is linked to the Nernst heat law and the unattainability principle.
  • Third-law behavior: The third-law constraint requires a positive cooling-power scaling exponent α > 0 so that cold-bath entropy production vanishes at absolute zero.The second-law analysis alone permits a weaker condition, while the first formulation of the third law strengthens it.
  • Third-law behavior: The dynamical unattainability principle forbids cooling to absolute zero in finite time and constrains system–bath interaction and cold-bath properties.The associated cooling-rate scaling depends on the cold bath’s heat-capacity exponent.
  • Protocol limits: Optimizing the cold frequency gives ℏω_c = k_BT_c, while frictionless schedules yield positive scaling exponents and reproduce the dynamical Nernst heat law.The reported exponents depend on the chosen frictionless scheduling protocol.
  • Protocol limits: If the adiabat is shorter than the minimum shortcut time, the oscillator cannot reach arbitrarily low final energies and approaches the sudden-adiabat limit.In the modeled refrigerator, this can impose a nonzero minimum reachable cold temperature.

VI. OVERVIEW

The review uses the analytically tractable quantum harmonic Otto cycle to connect quantum thermodynamics with macroscopic heat-engine concepts. It develops propagator-based dynamics and examines coherence, finite-time operation, friction, power–efficiency tradeoffs, and low-temperature refrigeration.

  • Overview: The harmonic oscillator Otto cycle provides an analytically tractable template connecting microscopic quantum devices with macroscopic heat engines.The model has motivated theoretical and experimental work, including extensions to interacting particles, many modes, and many-body dynamics.
  • Overview: The review describes cycle dynamics by composing completely positive stroke propagators in a Heisenberg operator-algebra framework.The state dynamics can be represented using three thermodynamically relevant variables, H, L, and C, associated with energy and coherence.
  • Finite-time dynamics: Frictionless cycles occur when coherence vanishes at stroke boundaries, requiring specially scheduled frequency protocols that can achieve finite-time maximum efficiency.The minimum-time scheduling is related to quantum speed limits, while faster control requires unreasonable stored-energy constraints.
  • Limitations: Noise prevents perfect friction avoidance, and beyond a minimum adiabat time friction cannot be eliminated.The review treats complete frictionlessness as an idealization and notes that control errors can also undermine extreme low-temperature cooling.
  • Finite-time thermodynamics: The model exposes a fundamental efficiency–power tradeoff: finite power requires out-of-equilibrium operation and dissipation, while maximum power coincides with maximum entropy production.Entropy production is linearly related to power in the reviewed case, and the Carnot-efficiency limit has zero power.
  • Sudden cycles: The transition from frictionless to sudden adiabats occurs at an exceptional point of the non-Hermitian adiabatic propagator.Sudden cycles require coherence for power production; without coherence, the model acts as a dissipator generating entropy in both baths.
  • Quantum refrigeration: At extremely low cold-side temperatures, cooling is limited by adiabatic expansion or vanishing heat transport, with optimal frictionless expansion times scaling as the inverse square root of the cold temperature.The resulting entropy-production rate vanishes as the cold temperature approaches zero, illustrating a dynamical version of the Nernst heat law.
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