In this thesis, we investigate two quantum models.The first model is a Kerr-nonlinear parametric oscillator (KPO) coupled to a bath of harmonic oscillators.We assume that the detuning is zero, then the Hamiltonian of the KPO has two degenerate ground states.We also assume that the parametric pump is much stronger than the Kerr-nonlinearity.In this case, the approximate forms of the first two excited states are known.Using these four states, we derive the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-type Markovian master equation for a KPO from the microscopic Hamiltonian.We compare our GKSL equation with the GKSL equation in the literature.We show that a KPO under our GKSL equation is more robust to excitation errors than that under the GKSL equation in the literature.In particular, we show that in the low-temperature limit of the bath, the state of a KPO under our GKSL equation is mostly confined to the subspace spanned by KPO's two degenerate ground states.The second model consists of coupled harmonic oscillators in a star configuration, where the central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath).We define and investigate the nonequilibrium thermodynamic entropy of the total system.In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature.This enables us to define timedependent thermodynamic entropy for each harmonic oscillator, and thereby define the total nonequilibrium thermodynamic entropy as their summation.We analytically confirm that our total thermodynamic entropy satisfies the third law of thermodynamics.Our numerical solutions show that, though the finite-time dynamics of the system is well approximated by the GKSL equation, the total thermodynamic entropy production rate can be negative, while the total thermodynamic entropy satisfies the second law of thermodynamics.i I am really indebted to Prof. Naomichi Hatano for teaching me how to conduct research: boldly tackling new frontiers; persistently addressing a problem; believing in myself even if I am confronted with difficulty and for giving me a lot of keen comments and valuable advice.
In this thesis, we investigate two quantum models.The first model is a Kerr-nonlinear parametric oscillator (KPO) coupled to a bath of harmonic oscillators.We assume that the detuning is zero, then the Hamiltonian of the KPO has two degenerate ground states.We also assume that the parametric pump is much stronger than the Kerr-nonlinearity.In this case, the approximate forms of the first two excited states are known.Using these four states, we derive the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-type Markovian master equation for a KPO from the microscopic Hamiltonian.We compare our GKSL equation with the GKSL equation in the literature.We show that a KPO under our GKSL equation is more robust to excitation errors than that under the GKSL equation in the literature.In particular, we show that in the low-temperature limit of the bath, the state of a KPO under our GKSL equation is mostly confined to the subspace spanned by KPO's two degenerate ground states.The second model consists of coupled harmonic oscillators in a star configuration, where the central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath).We define and investigate the nonequilibrium thermodynamic entropy of the total system.In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature.This enables us to define timedependent thermodynamic entropy for each harmonic oscillator, and thereby define the total nonequilibrium thermodynamic entropy as their summation.We analytically confirm that our total thermodynamic entropy satisfies the third law of thermodynamics.Our numerical solutions show that, though the finite-time dynamics of the system is well approximated by the GKSL equation, the total thermodynamic entropy production rate can be negative, while the total thermodynamic entropy satisfies the second law of thermodynamics.i I am really indebted to Prof. Naomichi Hatano for teaching me how to conduct research: boldly tackling new frontiers; persistently addressing a problem; believing in myself even if I am confronted with difficulty and for giving me a lot of keen comments and valuable advice.
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermo-dynamic entropy for each harmonic oscillator and total non-equilibrium thermodynamic entropy as the summation of them. We analytically confirm that the total thermodynamic entropy satisfies the third law of thermodynamics. Our numerical solutions show that, even when the dynamics of the system is well approximated by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-type Markovian master equation, the total thermodynamic entropy production rate can be negative, while the total thermodynamic entropy satisfies the second law of thermodynamics. This result is a counterexample to the common belief that the total entropy production rate is non-negative when the system is under the GKSL-type Markovian dynamics.
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy for each harmonic oscillator and total nonequilibrium thermodynamic entropy as the summation of them. We analytically confirm that the total thermodynamic entropy satisfies the third law of thermodynamics. Our numerical solutions show that, even when the dynamics of the system is well approximated by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-type Markovian master equation, the total thermodynamic entropy production rate can be negative, while the total thermodynamic entropy satisfies the second law of thermodynamics. This result is a counterexample to the common belief that the total entropy production rate is non-negative when the system is under the GKSL-type Markovian dynamics.
We present a systematic thermodynamic analysis of nonsecular GKLS master equations. When some energy transitions in the system are comparable with the dissipative rate, the secular approximation is not valid, and the positivity of the dynamics must be ensured via other approximations. The dynamics then presents features reminiscent from strong coupling to the environment: e.g. the equilibrium state with an environment at thermal equilibrium is not a Gibbs state of the local system Hamiltonian and the dynamics couples energy coherences and populations. Nonetheless, we build a unified, consistent thermodynamic framework for those dynamics. We focus on two approximations beyond the full secular approximation (the local and the more accurate geometric-arithmetic approximations), and then employ systematic perturbation theory on microscopic expressions for the first and second law of thermodynamics which relate irreversibility to the increase of system-environment correlations. We show that, in spite of the weak system-bath coupling, the system-bath interaction energy participates to the energy balance, and is directly linked to the Lamb-shift Hamiltonian. We compare this microscopic entropy production with the definition based on the contractivity of the reduced system dynamics (Spohn inequality). We show that, unlike for secular master equations, the two entropy production rates differ because of the presence of non-vanishing stationary coherences in the energy eigenbasis. However, in the case of a single thermal bath, the difference is purely transient, and no work can be cyclically extracted from the steady-state despite its non-Gibbs form. Finally, we illustrate our results with simple examples, clarifying and completing the thermodynamic picture of Markovian dynamics in the quantum regime.
In the following, we study the dissipative time-evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the unitary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) approach to open quantum systems, one can perform the rotating wave approximation with respect to the interacting, or non-interacting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different time-scales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behaviour of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical inter-oscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger inter-oscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of inter-oscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
In the following, we study the dissipative time evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the uni- tary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad approach to open quantum systems, one can perform the rotating-wave approximation with respect to the interacting or noninteracting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different timescales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behavior of fidelity between them versus interoscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical interoscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger interoscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of interoscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.