Phase transitions and quantum measurements

In a quantum measurement, a coupling $g$ between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling $g$ behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of $N$ interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off-diagonal elements of S+A, rapid due to the large size of $N$. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states. If $g$ is too weak, the measurement fails due to a ``Buridan's ass'' effect. The probability distribution for the magnetization then develops not one but two narrow peaks at the ferromagnetic values. During its evolution it goes through wide shapes extending between these values.Comment: 12 pages, 2 figure

Lectures on dynamical models for quantum measurements

In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component $\hat s_z$ by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S+A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetisation. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called "collapse of the wave function" or the "reduction of the state" appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.Comment: 43 pages, 5 figures. Lectures given in the "Advanced School on Quantum Foundations and Open Quantum Systems", Joao Pessoa, Brazil, summer 2012. To appear in the proceedings and in IJMP

The Quantum Measurement Process: Lessons from an Exactly Solvable Model

The measurement of a spin-\half is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schr\"odinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at the pointer alone. Apparent non-unitary behavior of the measurement process is explained by the arisal of small many particle correlations, that characterize relaxation.Comment: 13 pages, discussion of pre-measurement added. World Scientific style. To appear in proceedings "Beyond The Quantum", Th.M. Nieuwenhuizen et al, eds, (World Scientific, 2007

Mean-field theory of quantum brownian motion

We investigate a mean-field approach to a quantum brownian particle interacting with a quantum thermal bath at temperature $T$, and subjected to a non-linear potential. An exact, partially classical description of quantum brownian motion is proposed, which uses negative probabilities in its intermediate steps. It is shown that properties of the quantum particle can be mapped to those of two classical brownian particles in a common potential, where one of them interacts with the quantum bath, whereas another one interacts with a classical bath at zero temperature. Due to damping the system allows a unique and non-singular classical limit at $\hbar \to 0$. For high $T$ the stationary state becomes explicitly classical. The low-temperature case is studied through an effective Fokker-Planck equation. Non-trivial purely quantum correlation effects between the two particles are found.Comment: 13 pages, 0 figures, revte

The quantum measurement process: an exactly solvable model

An exactly solvable model for a quantum measurement is discussed, that integrates quantum measurements with classical measurements. The z-component of a spin-1/2 test spin is measured with an apparatus, that itself consists of magnet of N spin-1/2 particles, coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state, according to the sign of s_z of the test spin. The quantum measurement goes in two steps. On a timescale 1/\sqrt{N} the collapse takes place due to a unitary evolution of test spin and apparatus spins; on a larger but still short timescale this collapse is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is basically a classical process, that can already be understood from classical statistical mechanics.Comment: 4 pages, presented at the conference "Anomalies and Strange Behavior in Physics: Challenging the conventional", Napels, April, 2003. v2: Elaboration on the statistical interpretation of Q

Thomson's formulation of the second law: an exact theorem and limits of its validity

Thomson's formulation of the second law - no work can be extracted from a system coupled to a bath through a cyclic process - is believed to be a fundamental principle of nature. For the equilibrium situation a simple proof is presented, valid for macroscopic sources of work. Thomson's formulation gets limited when the source of work is mesoscopic, i.e. when its number of degrees of freedom is large but finite. Here work-extraction from a single equilibrium thermal bath is possible when its temperature is large enough. This result is illustrated by means of exactly solvable models. Finally we consider the Clausius principle: heat goes from high to low temperature. A theorem and some simple consequences for this statement are pointed out.Comment: 6 pages Latex, uses aip-proceedings style files. Proceedings `Quantum Limits to the Second Law', San Diego, July 200

Does the quark-gluon plasma contain stable hadronic bubbles?

We calculate the thermodynamic potential of bubbles of hadrons embedded in quark-gluon plasma, and of droplets of quark-gluon plasma embedded in hadron phase. This is a generalization of our previous results to the case of non-zero chemical potentials. As in the zero chemical potential case, we find that a quark-gluon plasma in thermodynamic equilibrium may contain stable bubbles of hadrons of radius $R \simeq 1$ fm. The calculations are performed within the MIT Bag model, using an improved multiple reflection expansion. The results are of relevance for neutron star phenomenology and for ultrarelativistic heavy ion collisions.Comment: 12 pages including 8 figures. To appear in Phys. Rev.

Quantum thermodynamics: thermodynamics at the nanoscale

A short introduction on quantum thermodynamics is given and three new topics are discussed: 1) Maximal work extraction from a finite quantum system. The thermodynamic prediction fails and a new, general result is derived, the ``ergotropy''. 2) In work extraction from two-temperature setups, the presence of correlations can push the effective efficiency beyond the Carnot bound. 3) In the presence of level crossing, non-slow changes may be more optimal than slow ones.Comment: 5 pages. Talk given at Physics of Quantum Electronics (PQE2004), Snowbird, by Th.M. Nieuwenhuize

Small object limit of Casimir effect and the sign of the Casimir force

We show a simple way of deriving the Casimir Polder interaction, present some general arguments on the finiteness and sign of mutual Casimir interactions and finally we derive a simple expression for Casimir radiation from small accelerated objects.Comment: 13 pages, late