For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spinless electrons in coupled quantum dots. We show that the geometrical pumping is prohibited for the case of non-interacting electrons if we modulate only temperatures and chemical potentials of reservoirs, while the geometrical pumping occurs in the presence of an interaction between electrons.

We study macroscopic entanglement of various pure states of a one-dimensional N -spin system with N ≫ 1. Here, a quantum state is said to be macroscopically entangled if it is a superposition of macroscopically distinct states. To judge whether such superposition is hidden in a general state, we use an essentially unique index p: A pure state is macroscopically entangled if p = 2, whereas it may be entangled but not macroscopically if p < 2. This index is directly related to fundamental stabilities of many-body states. We calculate the index p for various states in which magnons are excited with various densities and wavenumbers. We find macroscopically entangled states (p = 2) as well as states with p = 1. The former states are unstable in the sense that they are unstable against some local measurements. On the other hand, the latter states are stable in the senses that they are stable against any local measurements and that their decoherence rates never exceed O(N ) in any weak classical noises. For comparison, we also calculate the von Neumann entropy S N/2 (N ) of a subsystem composed of N/2 spins as a measure of bipartite entanglement. We find that S N/2 (N ) of some states with p = 1 is of the same order of magnitude as the maximum value N/2. On the other hand, S N/2 (N ) of the macroscopically entangled states with p = 2 is as small as O(log N ) ≪ N/2. Therefore, larger S N/2 (N ) does not mean more instability. We also point out that these results are partly analogous to those for interacting many bosons. Furthermore, the origin of the huge entanglement, as measured either by p or S N/2 (N ), is discussed to be due to spatial propagation of magnons.

For open systems described by the quantum Markovian master equation, we study a possible extension of the Clausius equality to quasistatic operations between nonequilibrium steady states (NESSs). We investigate the excess heat divided by temperature (i.e., excess entropy production) which is transferred into the system during the operations. We derive a geometrical expression for the excess entropy production, which is analogous to the Berry phase in unitary evolution. Our result implies that in general one cannot define a scalar potential whose difference coincides with the excess entropy production in a thermodynamic process, and that a vector potential plays a crucial role in the thermodynamics for NESSs. In the weakly nonequilibrium regime, we show that the geometrical expression reduces to the extended Clausius equality derived by Saito and Tasaki (J. Stat. Phys. {\bf 145}, 1275 (2011)). As an example, we investigate a spinless electron system in quantum dots. We find that one can define a scalar potential when the parameters of only one of the reservoirs are modified in a non-interacting system, but this is no longer the case for an interacting system.Comment: 28 pages, 3 figures. 'Remark on the fluctuation theorem' has been revised in ver. 2. Brief Summary has been added in Sec. 1 in ver.

We propose the second moment of the Husimi distribution as a measure of complexity of quantum states. The inverse of this quantity represents the effective volume in phase space occupied by the Husimi distribution, and has a good correspondence with chaoticity of classical system. Its properties are similar to the classical entropy proposed by Wehrl, but it is much easier to calculate numerically. We calculate this quantity in the quartic oscillator model, and show that it works well as a measure of chaoticity of quantum states.

We define generalized Husimi distributions using generalized coherent states, and show that their moments are good measures of complexity of many-body quantum states. Our construction of the coherent states is based on the single-particle transformation group of the system. Then the coherent states are independent-particle states, and, at the same time, the most localized states in the Husimi representation. Therefore delocalization of the Husimi distribution, which can be measured by the moments, is a sign of many-body correlation (entanglement). Since the delocalization of the Husimi distribution is also related to chaoticity of the dynamics, it suggests a relation between entanglement and chaos. Our definition of the Husimi distribution can be applied not only to the systems of distinguishable particles, but also to those of identical particles, i.e., fermions and bosons. We derive an algebraic formula to evaluate the moments of the Husimi distribution.