## Tesi etd-06032018-181313 |

Thesis type

Tesi di laurea magistrale

Author

ABIUSO, PAOLO

URN

etd-06032018-181313

Title

Quantum Thermodynamics: control and non-Markovianity

Struttura

FISICA

Corso di studi

FISICA

Commissione

**relatore**Prof. Giovannetti, Vittorio

Parole chiave

- quantum thermodynamics
- quantum information
- Markovianity
- thermal machines

Data inizio appello

25/06/2018;

Consultabilità

secretata d'ufficio

Riassunto analitico

This Master’s Degree thesis is located in the field of Quantum Thermodynamics and its links with Information Theory.

The starting purpose of the work is to study how memory effects (non-Markovian dynamics) can affect the thermodynamic performances of a quantum system. Incidentally we work out an optimal control theory for a class of simple thermal machinesperforming Carnot cycles.

The mathematical treatment is based on tools from Open Quantum System Dynamics (OQS): evolution is described, under the most general physical assumptions, by means of Completely Positive Trace-preserving linear maps, acting on the density matrix of the system under consideration. We stand by the standard approach which mathematically identifies the Markovian character of a quantum process with its CP-divisibility. Under this assumption a standard form can be derived for the Master equation generating the evolution, that is the Gorini-Kossakowski-Sudarshan-Lindblad form.

Based especially on OQS, our study of Quantum Thermodynamics focuses on the analysis of the performance of thermal engines. In this framework thermalisation effects induced by the interaction with a thermal reservoir will be modelled with Master Equations having the Gibbs state as the only stationary state. To simulate sensible restraints on the system, it will be also assumed external control on dynamical parameters such as the coupling constant to the thermal baths, as well as the local Hamiltonian of the system. Once defined the basic thermodynamic quantities and potentials, we review the main quantum thermodynamic cycles, i.e. the Quantum Carnot Cycle and the Quantum Otto Cycle. In the above setting we develop an optimal control theory for simple quantum engines such as qubits in a wide class of dynamical models. For this kind of study it is necessary to depart from the idealizations of quasistatic processes and analyse Finite Time Thermodynamics (FTT), which we study using a perturbative technique for the study of slowly controlled open quantum systems, called Slow-Driving approximation (S-D). The perturbative solution is very useful for it neglects initial conditions and it is relatively simple to compute, hence we employ it to detect the optimal working points of quantum thermal machines. For the paradigmatic case of a quantum Carnot cycle we find that it is possible to pinpoint three main features which determine, in the S-D scheme, the thermodynamic performance:

• the shape of the protocol on the control parameters,

• the speed of the protocol (for fixed shape),

• a model dependent S-D correction amplitude, namely how large the perturbative correction to the steady state solution is.

The S-D technique allows us to perform optimisation on the shape and duration of the control. The last contribution encodes, for a given protocol "how distant" the Slow-Driving corrections are from the quasistatic case. It is a model dependent amplitude which we then exploit as the main figure of merit in the study e.g. of non-Markovian models.

We define a precise setup for the study of non-Markovian influence on a quantum thermodynamic system: we imagine to have an engine S in contact with the thermal baths, allowing some degrees of freedom of the reservoirs (namely, a system C for the cold bath and H for the hot one) to be taken into account explicitly and couple with S by an interaction Hamiltonian. In the simplest model we study H, S and C are qubits which interact via an exchange Hamiltonian between S and C (and S and H ). When coupled to the cold reservoirs, on both S and C the action of the bath is given by GKSL thermalising dissipators, and the controller has the ability to change the energy gap of the local Hamiltonian of S; the situation is specular when coupled to the hot bath. We study this model in the Slow-Driving approximation; in particular we look for the expression of the S-D amplitude as a function of the setup parameters, to study how the rates defining the GKSL operators influence it.

We find that in a vast region of parameters space the introduction of SH and SC coupling can have improving effects on the FTT performance. We also examine the exact solution of an Otto cycle, confirming that the presence of the n-M enhances the power output.

The starting purpose of the work is to study how memory effects (non-Markovian dynamics) can affect the thermodynamic performances of a quantum system. Incidentally we work out an optimal control theory for a class of simple thermal machinesperforming Carnot cycles.

The mathematical treatment is based on tools from Open Quantum System Dynamics (OQS): evolution is described, under the most general physical assumptions, by means of Completely Positive Trace-preserving linear maps, acting on the density matrix of the system under consideration. We stand by the standard approach which mathematically identifies the Markovian character of a quantum process with its CP-divisibility. Under this assumption a standard form can be derived for the Master equation generating the evolution, that is the Gorini-Kossakowski-Sudarshan-Lindblad form.

Based especially on OQS, our study of Quantum Thermodynamics focuses on the analysis of the performance of thermal engines. In this framework thermalisation effects induced by the interaction with a thermal reservoir will be modelled with Master Equations having the Gibbs state as the only stationary state. To simulate sensible restraints on the system, it will be also assumed external control on dynamical parameters such as the coupling constant to the thermal baths, as well as the local Hamiltonian of the system. Once defined the basic thermodynamic quantities and potentials, we review the main quantum thermodynamic cycles, i.e. the Quantum Carnot Cycle and the Quantum Otto Cycle. In the above setting we develop an optimal control theory for simple quantum engines such as qubits in a wide class of dynamical models. For this kind of study it is necessary to depart from the idealizations of quasistatic processes and analyse Finite Time Thermodynamics (FTT), which we study using a perturbative technique for the study of slowly controlled open quantum systems, called Slow-Driving approximation (S-D). The perturbative solution is very useful for it neglects initial conditions and it is relatively simple to compute, hence we employ it to detect the optimal working points of quantum thermal machines. For the paradigmatic case of a quantum Carnot cycle we find that it is possible to pinpoint three main features which determine, in the S-D scheme, the thermodynamic performance:

• the shape of the protocol on the control parameters,

• the speed of the protocol (for fixed shape),

• a model dependent S-D correction amplitude, namely how large the perturbative correction to the steady state solution is.

The S-D technique allows us to perform optimisation on the shape and duration of the control. The last contribution encodes, for a given protocol "how distant" the Slow-Driving corrections are from the quasistatic case. It is a model dependent amplitude which we then exploit as the main figure of merit in the study e.g. of non-Markovian models.

We define a precise setup for the study of non-Markovian influence on a quantum thermodynamic system: we imagine to have an engine S in contact with the thermal baths, allowing some degrees of freedom of the reservoirs (namely, a system C for the cold bath and H for the hot one) to be taken into account explicitly and couple with S by an interaction Hamiltonian. In the simplest model we study H, S and C are qubits which interact via an exchange Hamiltonian between S and C (and S and H ). When coupled to the cold reservoirs, on both S and C the action of the bath is given by GKSL thermalising dissipators, and the controller has the ability to change the energy gap of the local Hamiltonian of S; the situation is specular when coupled to the hot bath. We study this model in the Slow-Driving approximation; in particular we look for the expression of the S-D amplitude as a function of the setup parameters, to study how the rates defining the GKSL operators influence it.

We find that in a vast region of parameters space the introduction of SH and SC coupling can have improving effects on the FTT performance. We also examine the exact solution of an Otto cycle, confirming that the presence of the n-M enhances the power output.

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