arXiv subjects

Pier A. Mello

Publications and source records attributed to Pier A. Mello.

At least 19 recordsLinked to original sources

arXiv2023

ELECTRON TRANSPORT AND ELECTRON DENSITY INSIDE ONE-DIMENSIONAL DISORDERED CONDUCTORS: An Analysis of the Electronic-Levels Contribution

We consider the problem of electron transport along a one-dimensional disordered multiple-scattering conductor, and study the electron density for all the electronic levels. A model is proposed for the reduced density matrix of the system placed between two reservoirs at different chemical potentials, and the statistical-mechanical expectation value of the electron density is evaluated. An ensemble average is computed over disordered configurations. We compare its predictions with computer simulations. We find that the contribution of low-lying levels is very different from that of the high-lying ones studied in the past. Going down in energy, the wave function penetrates ever less inside the sample. For high-lying levels, this is interpreted in terms of localization from disorder. For low-lying levels, this interpretation gradually gives way to an understanding in terms of the increasing reflection produced by each scatterer, which is seen by the electron as a higher and higher -- and hence impenetrable -- potential barrier. Indeed, the local-density-of-states, LDOS, is gradually depleted in the interior of the system, since the wave function is ever smaller inside. The problem studied here is also of interest in electromagnetic, thermal, and acoustic transport in disordered systems.

cond-mat.dis-nn↗

arXiv2020

The Uncertainty Principle Revisited

We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {\em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $δa$ around $a^{(n)}$. The width of this statistical distribution as function of $δa$ constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.

quant-ph↗

arXiv2020

Electron transport and electron density inside quasi-one-dimensional disordered conductors

We consider the problem of electron transport across a quasi-one-dimensional disordered multiply-scattering medium, and study the statistical properties of the electron density inside the system. In the physical setup that we contemplate, electrons of a given energy feed the disordered conductor from one end. The physical quantity that is mainly considered is the logarithm of the electron density, $\ln {\cal W}(x)$, since its statistical properties exhibit a self-averaging behavior. We also describe a {\em gedanken} experiment, as a possible setup to measure the electron density. We study analytically and through computer simulations the ballistic, diffusive and localized regimes. We generally find a good agreement between the two approaches. The extension of the techniques that were used in the past to find information outside the sample is done in terms of the scattering properties of the two segments that form the entire conductor on each side of the observation point. The problem is of interest in various other branches of physics, as electrodynamics and elasticity.

cond-mat.dis-nn↗

arXiv2017

Single-Parameter Scaling and Maximum Entropy inside Disordered One-Dimensional Systems: Theory and Experiment

The single-parameter scaling hypothesis relating the average and variance of the logarithm of the conductance is a pillar of the theory of electronic transport. We use a maximum-entropy ansatz to explore the logarithm of the energy density, $\ln {\cal W}(x)$, at a depth $x$ into a random one-dimensional system. Single-parameter scaling would be the special case in which $x=L$ (the system length). We find the result, confirmed in microwave measurements and computer simulations, that the average of $\ln {\cal W}(x)$ is independent of $L$ and equal to $-x/\ell$, with $\ell$ the mean free path. At the beginning of the sample, ${\rm var}[\ln {\cal W}(x)]$ rises linearly with $x$ and is also independent of $L$, with a sublinear increase near the sample output. At $x=L$ we find a correction to the value of ${\rm var}[\ln T]$ predicted by single-parameter scaling.

cond-mat.dis-nn↗

arXiv2017

Partial Transposition in a Finite-Dimensional Hilbert Space: Physical Interpretation, Measurement of Observables and Entanglement

We show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of a "momentum-like" variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an "isotropic state", which is dependent on a parameter $r$, and an operator whose variance becomes negative for the partially transposed state for certain values of $r$; for such $r$, the original states are entangled.

quant-ph↗

arXiv2016

A Family of Weyl-Wigner Transforms for Discrete Variables Defined in a Finite-Dimensional Hilbert Space

We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.

quant-ph↗

arXiv2016

Bipartite Entanglement, Partial Transposition and the Uncertainty Principle for Finite-Dimensional Hilbert Spaces

We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter $r$ of a Bell state $|Φ^{+} \rangle$ and the completely incoherent state, $\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.

quant-ph↗

arXiv2015

Measurement and Ergodicity in Quantum Mechanics

The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to $N$ "probes" --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.

quant-ph↗

arXiv2015

Wave Transport in One-Dimensional Disordered Systems with Finite-Size Scatterers

We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $ł_c$ which may contain an arbitrary number $δ/2π$ of wavelengths, where $δ= k l_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $δ$. For weak scatterers, we find: i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$, ii) a regime, to be called II, for $δ$ in the vicinity of $π$, where the system is almost transparent and less localized, and iii) right in the middle of regime II, for $δ$ very close to $π$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $δ$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.

cond-mat.dis-nn↗

arXiv2014

Non-ideal classical measurements and quantum measurements: a comparative study

Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using a measurement probe with finite resolution. We use the von Neumann interaction model to represent the interaction between system and probe. We find that in reality classical systems are affected by measurement in a similar manner as quantum systems. In particular, we derive classical equivalents of Lüders' rule, the "collapse postulate", and the Lindblad equation.

quant-ph↗

arXiv2013

Wigner function and the successive measurement of position and momentum

Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements involving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space, and then turn to the case of a discrete, finite-dimensional Hilbert space.

quant-ph↗

arXiv2013

The von Neumann Model of Measurement in Quantum Mechanics

We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called "Wigner's formula" emerges in the strong-coupling limit, while "Kirkwood's quasi-probability distribution" is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.

quant-ph↗

arXiv2012

Quantum State Tomography Using Successive Measurements

We describe a quantum state tomography scheme which is applicable to a system described in a Hilbert space of arbitrary finite dimensionality and is constructed from sequences of two measurements. The scheme consists of measuring the various pairs of projectors onto two bases --which have no mutually orthogonal vectors--, the two members of each pair being measured in succession. We show that this scheme implies measuring the joint quasi-probability of any pair of non-degenerate observables having the two bases as their respective eigenbases. The model Hamiltonian underlying the scheme makes use of two meters initially prepared in an arbitrary given quantum state, following the ideas that were introduced by von Neumann in his theory of measurement.

quant-ph↗

arXiv2012

Wave transport in one-dimensional disordered systems with finite-width potential steps

An amazingly simple model of correlated disorder is a one-dimensional chain of n potential steps with a fixed width lc and random heights. A theoretical analysis of the average transmission coefficient and Landauer resistance as functions of n and klc predicts two distinct regimes of behavior, one marked by extreme sensitivity and the other associated with exponential behavior of the resistance. The sensitivity arises in n and klc for klc approximately pi, where the system is nearly transparent. Numerical simulations match the predictions well, and they suggest a strong motivation for experimental study.

cond-mat.dis-nn↗

arXiv2009

Order Symmetry of Weak Measurements

Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.

quant-ph↗

arXiv2009

Quantum Scattering and Transport in Classically Chaotic Cavities: An Overview of Past and New Results

We develop a statistical theory that describes quantum-mechanical scattering of a particle by a cavity when the geometry is such that the classical dynamics is chaotic. This picture is relevant to a variety of physical systems, ranging from atomic nuclei to mesoscopic systems and microwave cavities; the main application to be discussed in this contribution is to electronic transport through mesoscopic ballistic structures or quantum dots. The theory describes the regime in which there are two distinct time scales, associated with a prompt and an equilibrated response, and is cast in terms of the matrix of scattering amplitudes S. We construct the ensemble of S matrices using a maximum-entropy approach which incorporates the requirements of flux conservation, causality and ergodicity, and the system-specific average of S which quantifies the effect of prompt processes. The resulting ensemble, known as Poisson's kernel, is meant to describe those situations in which any other information is irrelevant. The results of this formulation have been compared with the numerical solution of the Schroedinger equation for cavities in which the assumptions of the theory hold. The model has a remarkable predictive power: it describes statistical properties of the quantum conductance of quantum dots, like its average, its fluctuations, and its full distribution in several cases. We also discuss situations that have been found recently, in which the notion of stationarity and ergodicity is not fulfilled, and yet Poisson's kernel gives a good description of the data. At the present moment we are unable to give an explanation of this fact.

cond-mat.mes-hall↗

arXiv2009

Quantum Mechanics of Successive Measurements with Arbitrary Meter Coupling

We study successive measurements of two observables using von Neumann's measurement model. The two-pointer correlation for arbitrary coupling strength allows retrieving the initial system state. We recover Luders rule, the Wigner formula and the Kirkwood-Dirac distribution in the appropriate limits of the coupling strength.

quant-ph↗

arXiv2008

Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions

We study the possibility of giving a classical interpretation to quantum projective measurements for a particle described by a pure Gaussian state whose Wigner function is non-negative. We analyze the case of a projective measurement which gives rise to a proper Wigner function, i.e., taking on, as its values, the eigenvalues of the projector. We find that, despite having this property, this kind of projector produces a state whose Wigner function ceases to be non-negative and hence precludes its interpretation as a classical probability density. We also study the general case in which the projected state has a non-negative Wigner function; but then we find that the Wigner function of the projector is not a proper one. Thus, we conclude that a non-negative Wigner function is inadequate to serve as a hidden variable model for quantum processes in which projective measurements take place.

quant-ph↗