Optimal quantum driving of a thermal machine Andrea Mari Vasco Cavina Vittorio Giovannetti Alberto Carlini Workshop on Quantum Science and Quantum Technologies – ICTP, Trieste, 12-09-2017
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
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
Master equations Classical Markov process Quantum Markov process Liouvillian matrix Liouvillian superoperator
Equilibrium states is a fixed point of the map = equilibrium state corresponds to an eigenvector of with eigenvalue zero There is at least one equilibrium state (trace preserving condition) is unique the master equation is usually called “ mixing ” or “ relaxing ” If (assuming convergence from every initial state) Mixing process
Slowly driven master equations Time dependent master equation: If is relaxing for every : unique instantaneous equilibrium state Slow driving regime [external driving time-scale] [characteristic time-scale of the system] Quasi-static limit
Slowly driven master equations Time dependent master equation: If is relaxing for every : unique instantaneous equilibrium state Slow driving regime [external driving time-scale] [characteristic time-scale of the system] Finite driving time
Perturbation theory of slowly driven quantum systems Time scaling time-length of the process “shape” of the process Perturbation series ansatz: might not converge! Solution: Projector on the traceless subspace
Example: slowly driven two-level system modulation (sinusoidal in this case) Exact solution 1 st order approx. 2 nd order approx. 0 th order (quasi-static limit )
Finite-time thermodynamics Thermal master equations: Quasi-static evolution Finite-time corrections Reversible Irreversible corrections thermodynamics
First order irreversible corrections 1 st law 2 nd law Important property : is invariant for a time reversed protocol
Finite-time Carnot cycle Time reversed Isothermal expansion isothermal compression at temperature at temperature Adiabatic compression Adiabatic expansion
Efficiency at maximum power Initial conditions are lost and Limit of many cycles also the quantum state becomes periodic, 1 st order perturbation theory Power Carnot efficiency Efficiency Max Power Efficiency at max Power Schmiedl, Seifert. EPL 81.2 20003 (2007) We know how to compute Esposito et al., PRL 105, 150603 (2010) finite-time heat corrections
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) Universal scaling for all protocols Spectral density exponent
Efficiency at maximum power Efficiency at maximum power Thermal bath spectral density Curzon, Ahlborn, AJP 43, 22 (1975) Flat bath Chambadal, L.c..n. , 4 1-58 (1957) Ohmic bath Schmiedl, Seifert. EPL 81.2 20003 (2007) Infinitely Esposito et al., PRL 105, 150603 (2010) super-Ohmic bath Schmiedl, Seifert. EPL 81.2 20003 (2007) Benenti, et al. ArXiv:1608.05595 (2016) Infinitely sub-Ohmic bath
Efficiency at maximum power Curzon-Ahlborn upper bound Schmiedl-Seifert Carnot lower bound Only within 1 st order perturbation theory Only for sufficiently smooth cycles
Efficiency at maximum power Exact simulation based on a single qubit in flat or Ohmic thermal baths: 0.0020 0.0015 0.0010 0.0005 upper bound Curzon-Ahlborn 0.0000 0.6 0.7 0.8 0.9 1.0 Schmiedl-Seifert Carnot lower bound
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
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 )
Optimal control of a thermal machine Hamiltonian driving Dissipative control Heat released by the system: Optimal control problem minimize Work done by the system: with respect to all control strategies for fixed:
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)
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 for all 3. the pseudo Hamiltonian is constant
Thermodynamic link between and maximum power Does 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: YES Check if Determine ? 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 .
Optimal cycle for a d-level quantum heat engine Upper bound on the total dissipation rate: The optimal control for and turns out to be of “ bang-bang ” type: (strong coupling only with the cold bath) 2 alternatives: (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
Example: full solution for a 2-level system control on the energy level Gibbs thermalizing dissipators Quantum state (diagonal): Pontryagin's costate: Pseudo Hamiltonian: (constant of motion) Pseudo Hamilton equations: (master equation) (costate equation)
Optimal solutions for a 2-level system Optimal trajectories in the plane Cold isotherm Hot isotherm
Optimal solutions for a 2-level system is also a continuous cycle Carnot cycle at fixed Populations for adiabats completely determines the Carnot cycle.
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
Maximum power cycle for a 2-level system Optimal state Optimal energy levels Optimal control
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
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).
Recommend
More recommend
