Optimal quantum driving of a thermal machine
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1 Optimal quantum driving of a thermal machine Andrea Mari Vasco Cavina Vittorio Giovannetti Alberto Carlini Workshop on Quantum Science and Quantum Technologies ICTP, Trieste,
2 Outline 1. Slow driving of quantum thermal machines (close to thermodynamic equilibrium) - General theory of slowly driven master equations - Efficiency at maximum power for heat engines 2. Optimal driving of quantum thermal machines (strongly out of equilibrium) - Optimality of finite-time Carnot cycles - Full solution for a two-level system heat engine
3 Outline 1. Slow driving of quantum thermal machines (close to thermodynamic equilibrium) - General theory of slowly driven master equations - Efficiency at maximum power for heat engines 2. Optimal driving of quantum thermal machines (strongly out of equilibrium) - Optimality of finite-time Carnot cycles - Full solution for a two-level system heat engine
4 Master equations Classical Markov process Liouvillian matrix Quantum Markov process Liouvillian superoperator
5 Equilibrium states is a fixed point of the map = equilibrium state corresponds to an eigenvector of (trace preserving condition) If with eigenvalue zero There is at least one equilibrium state is unique the master equation is usually called mixing or relaxing (assuming convergence from every initial state) Mixing process
6 Slowly driven master equations Time dependent master equation: If is relaxing for every : unique instantaneous equilibrium state Slow driving regime [external driving time-scale] Quasi-static limit [characteristic time-scale of the system]
7 Slowly driven master equations Time dependent master equation: If is relaxing for every : unique instantaneous equilibrium state Slow driving regime [external driving time-scale] Finite driving time [characteristic time-scale of the system]
8 Perturbation theory of slowly driven quantum systems Time scaling time-length of the process shape of the process Perturbation series ansatz: Solution: Projector on the traceless subspace might not converge!
9 Example: slowly driven two-level system modulation (sinusoidal in this case) Exact solution 1st order approx. 2nd order approx. 0th order (quasi-static limit )
10 Finite-time thermodynamics Thermal master equations: Quasi-static evolution Finite-time corrections Reversible thermodynamics Irreversible corrections
11 First order irreversible corrections 1st law 2nd law Important property: is invariant for a time reversed protocol
12 Finite-time Carnot cycle Isothermal expansion at temperature Time reversed isothermal compression at temperature Adiabatic compression Adiabatic expansion
13 Efficiency at maximum power Limit of many cycles Initial conditions are lost and also the quantum state becomes periodic, 1st order perturbation theory Power Efficiency Carnot efficiency Max Power Efficiency at max Power We know how to compute finite-time heat corrections Schmiedl, Seifert. EPL (2007) Esposito et al., PRL 105, (2010)
14 Efficiency at maximum power If is continuous and differentiable (depends on the particular protocol) Pseudo-time reversal symmetry of the cycle Scaling properties of thermal Liouvillians (derives from macroscopic derivation) Spectral density exponent Universal scaling for all protocols
15 Efficiency at maximum power Thermal bath spectral density Flat bath Efficiency at maximum power Curzon, Ahlborn, AJP 43, 22 (1975) Chambadal, L.c..n., (1957) Ohmic bath Schmiedl, Seifert. EPL (2007) Infinitely super-ohmic bath Infinitely sub-ohmic bath Esposito et al., PRL 105, (2010) Schmiedl, Seifert. EPL (2007) Benenti, et al. ArXiv: (2016)
16 Efficiency at maximum power Curzon-Ahlborn Schmiedl-Seifert upper bound Carnot lower bound Only within 1st order perturbation theory Only for sufficiently smooth cycles
17 Efficiency at maximum power Exact simulation based on a single qubit in flat or Ohmic thermal baths: Curzon-Ahlborn Schmiedl-Seifert lower bound upper bound Carnot
18 Outline 1. Slow driving of quantum thermal machines (close to thermodynamic equilibrium) - General theory of slowly driven master equations - Efficiency at maximum power for heat engines 2. Optimal driving of quantum thermal machines (strongly out of equilibrium) - Optimality of finite-time Carnot cycles - Full solution for a two-level system heat engine
19 General questions What is the optimal driving of a thermal machine? Given a d-level quantum system and two heat baths, what is the maximum power that we can extract? Methods Slow-driving perturbation theory (because we are far from equilibrium) Optimal control theory approach (Pontryagin's minimum principle)
20 Optimal control of a thermal machine Hamiltonian driving Dissipative control Heat released by the system: Optimal control problem Work done by the system: minimize with respect to all control strategies for fixed:
21 Pontryagin's approach (similar to Hamiltonian formalism applied to control theory) Extended functional Lagrange multipliers normalization master equation Pseudo Hamiltonian Analogue of Hamilton equations: Analogue of energy conservation: (constant conserved quantity)
22 Pontryagin's minimum principle Necessary conditions for optimal control strategies minimizing the extended functional are such that: 1. there exists a non-zero costate evolving according to: 2. the pseudo Hamiltonian is minimized by the control function 3. the pseudo Hamiltonian is constant for all
23 Thermodynamic link between Does and maximum power have a physical meaning? Assume that we want to maximize the power of a cyclic engine Its variation w.r.t. is: optimal solutions must satisfy: Optimization procedure: Determine Check if YES? NO Try a larger The optimal driving of a generic quantum heat engine reduces to the optimization of a single degree of freedom within its accessible region.
24 Optimal cycle for a d-level quantum heat engine Upper bound on the total dissipation rate: The optimal control for and 2 alternatives: turns out to be of bang-bang type: (strong coupling only with the cold bath) (strong coupling only with the hot bath) Optimal control for the Hamiltonian turns out to be given by differentiable solutions (isothermal processes) separated by discontinuous jumps (adiabatic quenches). Maximum power quantum heat engines are achieved by a finite-time Carnot cycle Power maximization: take the minimum such that
25 Example: full solution for a 2-level system control on the energy level Gibbs thermalizing dissipators Quantum state (diagonal): Pontryagin's costate: Pseudo Hamiltonian: Pseudo Hamilton equations: (constant of motion) (master equation) (costate equation)
26 Optimal solutions for a 2-level system Optimal trajectories in the Cold isotherm plane Hot isotherm
27 Optimal solutions for a 2-level system is also a continuous cycle Populations for adiabats completely determines the Carnot cycle. Carnot cycle at fixed
28 Optimal solutions for a 2-level system is also a continuous cycle Carnot cycle at fixed Populations for adiabats completely determines the Carnot cycle. The maximum power is achieved for corresponding to an infinitesimal cycle performed around the optimal non-equilibrium state
29 Maximum power cycle for a 2-level system Optimal state Optimal energy levels Optimal control
30 Maximum power cycle for a 2-level system Maximum power Efficiency at maximum power (high power limit) Remark: same efficiency as for a quasi-static Otto cycle
31 Conclusions 1. Slow driving of quantum thermal machines [1] - Perturbation theory of slowly driven master equations - Universal formula for the efficiency at maximum power 2. Optimal driving of quantum thermal machines [2] - Optimal control theory approach (Pontryagin's minimum principle) - Optimal processes are finite-time Carnot cycles - Maximum power = conserved quantity of the control problem: - Full solution for a two-level system heat engine [1] Cavina, AM, Giovannetti, Phys. Rev. Lett. (2017). [2] Cavina, AM, Carlini, Giovannetti, arxiv: (2017).
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