Abstract The task of finding optimal protocols that minimize the energetic cost of thermodynamic processes of long yet finite duration τ is a pressing one. We approach this problem here in a rigorous and systematic fashion by means of the adiabatic perturbation theory of closed Hamiltonian quantum systems. Our main finding is a 1/τ 2 scaling of the excess work for large τ in gapped systems. This result is at odds with the asymptotic 1/τ prediction of the geometric approach to optimization, which is predicated on the slow evolution of open systems close to canonical equilibrium. In contrast, our approach does not lead to an obvious geometric interpretation. Furthermore, as the thermodynamic work does not depend on how an isolated quantum system is split into a system of interest and its environment, our results imply the failure of the geometric approach prediction even for open systems. Additionally, we provide alternative optimization procedures, both for slowly-varying processes described by adiabatic perturbation theory and for weakly-varying processes described by linear response theory. Our findings are benchmarked and confirmed through the application to the driven transverse-field Ising chain.
Abstract The task of finding optimal protocols that minimize the energetic cost of thermodynamic processes of long yet finite duration τ is a pressing one. We approach this problem here in a rigorous and systematic fashion by means of the adiabatic perturbation theory of closed Hamiltonian quantum systems. Our main finding is a 1/τ 2 scaling of the excess work for large τ in gapped systems. This result is at odds with the asymptotic 1/τ prediction of the geometric approach to optimization, which is predicated on the slow evolution of open systems close to canonical equilibrium. In contrast, our approach does not lead to an obvious geometric interpretation. Furthermore, as the thermodynamic work does not depend on how an isolated quantum system is split into a system of interest and its environment, our results imply the failure of the geometric approach prediction even for open systems. Additionally, we provide alternative optimization procedures, both for slowly-varying processes described by adiabatic perturbation theory and for weakly-varying processes described by linear response theory. Our findings are benchmarked and confirmed through the application to the driven transverse-field Ising chain.
The task of finding optimal protocols that minimize the energetic cost of thermodynamic processes of long yet finite duration τ is a pressing one. We approach this problem here in a rigorous and systematic fashion by means of the adiabatic perturbation theory of closed Hamiltonian quantum systems. Our main finding is a 1/τ2 scaling of the excess work for large τ in gapped systems. This result is at odds with the 1/τ prediction of the geometric approach to optimization, which is predicated on the slow evolution of open systems close to canonical equilibrium. In contrast, our approach does not lead to an obvious geometric interpretation. Furthermore, as the thermodynamic work does not depend on how an isolated quantum system is split into a system of interest and its environment, our results imply the failure of the geometric approach prediction even for open systems. Additionally, we provide alternative optimization procedures, both for slowly-varying processes described by adiabatic perturbation theory and for weakly-varying processes described by linear response theory. Our findings are benchmarked and confirmed through the application to the driven transverse-field Ising chain.
In this thesis, we study nonequilibrium statistical mechanics from a geometric perspective.In particular, we (i) examine the relation between quantum information geometry and linear response theory, and (ii) analyze nonequilibrium processes in thermodynamic control and derive a geometric expression for work.The first original study in this thesis is to understand quantum information geometry based on linear response theory.We show that the quantum Fisher information can be determined by measuring the linear response functions.For that purpose, we generalize the fluctuationdissipation theorem, and establish the quantitative relation between linear response functions and the generalized covariance, which contains the same amount of information on the quantum state.Based on the generalized fluctuation-dissipation theorem, we can determine the generalized covariance and the quantum Fisher information by measuring linear response functions, such as the dynamical susceptibilities and the complex admittances for all frequencies.We demonstrate that our result is applicable to experimental determination of the skew information, and a validation of skew information-based uncertainty relations.The second original study is on the analysis of work in thermodynamic control.We extend the thermodynamic metric-based expression for work into two directions.One is to obtain a systematic expansion of the average work from a phenomenological argument, and the other is to obtain an expansion of the work distribution for overdamped Langevin systems.First, we derive a systematic expansion of the work in terms of a small parameter ϵ that characterizes how slowly we control the system.The leading-order contribution is given by the thermodynamic metric expression.The next leading-order contributions to the thermodynamic metric contribution can be detected by comparing the excess work in a forward control and a backward control, and are predicted to scale as 1/T 2 as a function of the total control time T .Since the expansion is derived without assuming any specific microscopic dynamics, it is valid as long as the perturbation series expansion is valid.Finally, we examine the work distribution in overdamped Langevin systems.We derive the time evolution equation for the moment generating function of the work, and solve it from the lower-order contributions in ϵ.The O(ϵ) contribution to the generating function reproduces two known facts: the work distribution is Gaussian, and the average work is given by the thermodynamic metric.When we take up to O(ϵ 2 ) contributions into account, the work distribution exhibits nonzero skewness, which means that the fluctuation-dissipation relation is violated with scaling 1/T 2 .Furthermore, from the analytic calculation with numerical supports, we conjecture that the nth cumulant of the work scales as 1/T n-1 for n ≥ 1.
The quantum Ising chain has shortcuts to adiabaticity when operated with weak processes. However, when exactly do the non-equilibrium effects of the Kibble-Zurek mechanism, inherent to the system, appear in the optimal protocols in such a context? I propose here that such contrasting difference occurs due to the manner by which one measures the excitation spent energy of the system. Therefore, in this work, I made a qualitative analysis of a quantum annealing procedure of the time-averaged excess work, where the system acquires as a diverging decorrelation time the heuristic Kibble-Zurek mechanism relaxation time. Four important effects are then observed: the absence of shortcuts to adiabaticity, the pausing effect around the critical point in the optimal protocol when the Kibble-Zurek mechanism holds, the persistence of the time-averaged work to avoid slowly-varying regime even for large switching times, and diverging fluctuations of the time-averaged work. In the end, by comparing the excess and the time-averaged excess works, I conclude that this last one is not useful to measure the excitation spent energy in weak processes, although brings an intuition to what happens in the strong driving case.
The analytical expression for shortcuts to adiabaticity for any switching time and any thermally isolated system performing a finite-time and weakly driven process is presented. It is based on the analytical solution of the optimal protocols of weak processes for open systems. The extension to adiabatic processes was made by employing the concept of waiting time. The shortcuts to adiabaticity are proven by showing that the excess power is null in all instants, indicating that no nonequilibrium excitation occurs along the driving. Two examples are solved to verify the validity of such shortcuts: the typical case of an oscillatory relaxation function and the transverse-field quantum Ising chain. In this last case, it is shown that the nonquenching process outperforms the quenching one by suppressing the nonequilibrium excitations until the critical point is not achieved.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary sufficiently slow protocols using adiabatic perturbation theory, and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those which drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses, or target individual excited states in quantum chaotic systems, the geometric bound for the QSL can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in non-integrable spin chains using optimal control.
We develop a perturbation theory of quantum (and classical) master equations with slowly varying parameters, applicable to systems which are externally controlled on a time scale much longer than their characteristic relaxation time. We apply this technique to the analysis of finite-time isothermal processes in which, differently from quasistatic transformations, the state of the system is not able to continuously relax to the equilibrium ensemble. Our approach allows one to formally evaluate perturbations up to arbitrary order to the work and heat exchange associated with an arbitrary process. Within first order in the perturbation expansion, we identify a general formula for the efficiency at maximum power of a finite-time Carnot engine. We also clarify under which assumptions and in which limit one can recover previous phenomenological results as, for example, the Curzon-Ahlborn efficiency.