We study the transition of a quantum system S from a pure state to a mixed one, which is induced by the quantum criticality of the surrounding system E coupled to it. To characterize this transition quantitatively, we carefully examine the behavior of the Loschmidt echo (LE) of E modelled as an Ising model in a transverse field, which behaves as a measuring apparatus in quantum measurement. It is found that the quantum critical behavior of E strongly affects its capability of enhancing the decay of LE: near the critical value of the transverse field entailing the happening of quantum phase transition, the off-diagonal elements of the reduced density matrix describing S vanish sharply. Introduction: Nowadays quantum -classical transitions described by a reduction from a pure state to a mixture [1, 2] renew interests in many areas of physics, mainly due to the importance of quantum measurement and decoherence problem in quantum computing. To study this transition, some exactly-solvable models were proposed for a system coupled to the macroscopic [3,4,5] or classical [6,7] surrounding systems. Relevantly, in association with the quantum -classical transition in quantum chaos, the concept of Loschmidt echo (LE) from NMR experiments was introduced to describe the hypersensitivity of the time evolution to the perturbations experienced by the surrounding system [8,9]. In this letter, by a concrete example, we will show how quantum phase transition (QPT) [10] of the surrounding system can also sensitively affect the decay of its own LE, which means a dynamic reduction of its coupled system from pure state to a mixed one. Here, we note that a QPT effect has been explored for the Dicke model at the transition from quasi-integrable to quantum chaotic phases [11].

In order to describe quantum heat engines, here we systematically study isothermal and isochoric processes for quantum thermodynamic cycles. Based on these results the quantum versions of both the Carnot heat engine and the Otto heat engine are defined without ambiguities. We also study the properties of quantum Carnot and Otto heat engines in comparison with their classical counterparts. Relations and mappings between these two quantum heat engines are also investigated by considering their respective quantum thermodynamic processes. In addition, we discuss the role of Maxwell's demon in quantum thermodynamic cycles. We find that there is no violation of the second law, even in the existence of such a demon, when the demon is included correctly as part of the working substance of the heat engine.

The Jarzynski equality relates the free-energy di erence between two equilibrium states to the work done on a system through far-from-equilibrium processes-a milestone that builds on the pioneering work of Clausius and Kelvin. Although experimental tests of the equality have been performed in the classical regime, the quantum Jarzynski equality has not yet been fully verified owing to experimental challenges in measuring work and work distributions in a quantum system. Here, we report an experimental test of the quantum Jarzynski equality with a single 171 Yb + ion trapped in a harmonic potential. We perform projective measurements to obtain phonon distributions of the initial thermal state. We then apply a laser-induced force to the projected energy eigenstate and find transition probabilities to final energy eigenstates after the work is done. By varying the speed with which we apply the force from the equilibrium to the far-from-equilibrium regime, we verify the quantum Jarzynski equality in an isolated system. T here is increasing interest in non-equilibrium dynamics at the microscopic scale, crossing over quantum physics, thermodynamics and information theory as the experimental control and technology at such a scale have been developing rapidly. Most of the principles in non-equilibrium processes are represented in the form of inequalities, as seen in the example of the maximum work principle, W − F ≥ 0, where the average work W is equal to the free-energy difference F only in the case of the equilibrium process. In close-to-equilibrium processes, the fluctuation-dissipation theorem is valid and connects the average dissipated energy W diss ≡ W − F and the fluctuation of the system σ 2 /2k B T . Here σ is the standard deviation of the work distribution, T is the initial temperature of the system in thermal equilibrium and k B is the Boltzmann constant. Beyond the nearequilibrium regime, no exact results were known until Jarzynski found a remarkable equality 1 that relates the free-energy difference to the exponential average of the work done on the system:The Jarzynski equality (1) is satisfied irrespective of the protocols of varying parameters of the system even when the driving is arbitrarily far from equilibrium. The relation enables us to experimentally determine F of a system by repeatedly performing work at any speed. Experimental tests of the classical Jarzynski equality and its relation to the Crooks fluctuation theorem 2 have been successfully performed in various systems 3-12 .In classical systems, work can be obtained by measuring the force and the displacement, and then integrating the force over the displacement during the driving process. In the quantum regime, however, as a result of Heisenberg's uncertainty principle, we cannot determine the position and the momentum simultaneously-thus invalidating the concepts of force and displacement. Instead of measuring these classical observables, it is necessary to carry out projective measurements over the energy eigenstates to determine the work d...

We describe a simple and solvable model of a device that -like the "neat-fingered being" in Maxwell's famous thought experiment -transfers energy from a cold system to a hot system by rectifying thermal fluctuations. In order to accomplish this task, our device requires a memory register to which it can write information: the increase in the Shannon entropy of the memory compensates the decrease in the thermodynamic entropy arising from the flow of heat against a thermal gradient. We construct the nonequilibrium phase diagram for this device, and find that it can alternatively act as an eraser of information. We discuss our model in the context of the second law of thermodynamics.

Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

We extend to finite temperature the fidelity approach to quantum phase transitions (QPTs). This is done by resorting to the notion of mixed-state fidelity that allows one to compare two density matrices corresponding to two different thermal states. By exploiting the same concept we also propose a finite-temperature generalization of the Loschmidt echo. Explicit analytical expressions of these quantities are given for a class of quasi-free fermionic Hamiltonians. A numerical analysis is performed as well showing that the associated QPTs show their signatures in a finite range of temperatures.Comment: 7 pages, 4 figure

For closed quantum systems driven away from equilibrium, work is often defined in terms of projective measurements of initial and final energies. This definition leads to statistical distributions of work that satisfy nonequilibrium work and fluctuation relations. While this two-point measurement definition of quantum work can be justified heuristically by appeal to the first law of thermodynamics, its relationship to the classical definition of work has not been carefully examined. In this paper, we employ semiclassical methods, combined with numerical simulations of a driven quartic oscillator, to study the correspondence between classical and quantal definitions of work in systems with 1 degree of freedom. We find that a semiclassical work distribution, built from classical trajectories that connect the initial and final energies, provides an excellent approximation to the quantum work distribution when the trajectories are assigned suitable phases and are allowed to interfere. Neglecting the interferences between trajectories reduces the distribution to that of the corresponding classical process. Hence, in the semiclassical limit, the quantum work distribution converges to the classical distribution, decorated by a quantum interference pattern. We also derive the form of the quantum work distribution at the boundary between classically allowed and forbidden regions, where this distribution tunnels into the forbidden region. Our results clarify how the correspondence principle applies in the context of quantum and classical work distributions and contribute to the understanding of work and nonequilibrium work relations in the quantum regime.

We study a new quantum heat engine (QHE), which is assisted by a Maxwell's demon. The QHE requires three steps: thermalization, quantum measurement, and quantum feedback controlled by the Maxwell demon. We derive the positive-work condition and operation efficiency of this composite QHE. Using controllable superconducting quantum circuits as an example, we show how to construct our QHE. The essential role of the demon is explicitly demonstrated in this macroscopic QHE. Introduction.-A Maxwell demon is a construct that can distinguish the velocities of individual gas molecules and then separate hot and cold molecules into two domains of a container, so that the two domains will have different temperatures [1]. This result seems to contradict the second law of thermodynamics, because one can put a heat engine between them to extract work. The solution of this puzzle [1] refers to the so-called Landauer's principle [2,3] that essentially links information theory with fundamental physics [4]. Several quantum heat engines (QHEs) assisted by Maxwell's demons have been proposed in Refs. [5,6,7].Here, we propose a new QHE model integrated with a built-in quantum Maxwell's demon performing both: the quantum measurement on the working substance, and the feedback control for the system according to the measurement. We demonstrate the role of Maxwell's demon in a fully quantum manner. The thermodynamic cycle in our setup contains three fundamental stages: (i) a CNOT operation, making a pre-measurement to extract information from the working substance; (ii) the feedbackaction of the demon controlling the working substance to extract work; and (iii) the disentanglement process that thermalizes the working substance and the demon by two separate thermal baths. The demon plays a role in the first two steps.We further illustrate how to implement our QHE using superconducting qubit circuits [8,9]. In our setup, the demon-assisted working substance does work via two CNOT operations, which can be realized by single-qubit operations and an easily realized i-SWAP operation. The CNOT operation performs the basic functions of the quantum demon.Maxwell's demon-assisted thermodynamic cycle in twoqubit system.-Our QHE cycle is similar to a quantum Otto cycle [10] described in Ref. [11] and generalized in Ref. [12]. Here, the QHE, shown in Fig. 1, is a composite system consisting of two qubits: the "working substance" S and the quantum Maxwell's demon D. They