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A. D. Ribeiro

Publications and source records attributed to A. D. Ribeiro.

13 recordsLinked to original sources
arXiv2023

Identifying primes from entanglement dynamics

Factorization is the most fundamental way to determine if a number $n$ is prime or composite. Yet, this approach becomes impracticable when considering large values of $n$, a difficulty that is exploited by cryptographic protocols. We propose an alternative method to decide the primality of a natural number, that is based on the analysis of the evolution of the linear entanglement entropy. Specifically, we show that a singular behavior in the amplitudes of the Fourier series of this entropy is associated with prime numbers. We also discuss how this idea could be experimentally implemented and examine possible connections between our results and the zeros of the Riemann zeta function.

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arXiv2020

Wave-particle duality using the Compton effect

Thought experiments based on the double-slit interferometer had a crucial role to develop ideas concerning the wave-particle duality and the Bohr's complementarity principle. Ideally, a slit with a sufficiently low mass recoils due to the passage of the photon. This motion denounces the path taken by the light and suppresses any attempt to observe an interference pattern. In real life, however, available which-way information in such a setup is significantly impaired by the typical magnitudes of photons and slits, making the verification of the effect almost impossible. Here, we extend this discussion by applying similar ideas to the Mach-Zehnder interferometer. That is, we study the consequences of the beam-splitter recoil, during the passage of the photon, over the interference pattern produced by the device. Unlike the double-slit experiment, this recoil can now be encoded in the wavelength of the photon itself, which, in principle, is more easily accessed. Fortuitously, the model used to describe the interaction between the idealized beam-splitter and the photon clearly indicates that an interferometer based on Compton's effect could be build to study wave-particle duality. We follow this hint, finding realistic experimental parameters needed to observe the trade-off between wave and corpuscular behaviors in such a modified interferometer.

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arXiv2015

Wave-particle duality: an information-based approach

Recently, Bohr's complementarity principle was assessed in setups involving delayed choices. These works argued in favor of a reformulation of the aforementioned principle so as to account for situations in which a quantum system would simultaneously behave as wave and particle. Here we defend a framework that, supported by well-known experimental results and consistent with the decoherence paradigm, allows us to interpret complementarity in terms of correlations between the system and an {\em informer}. Our proposal offers formal definition and operational interpretation for the dual behavior in terms of both nonlocal resources and the couple work-information. Most importantly, our results provide a generalized information-based trade-off for the wave-particle duality and a causal interpretation for delayed-choice experiments.

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arXiv2012

Kinematics and dynamics in noninertial quantum frames of reference

From the principle that there is no absolute description of a physical state, we advance the approach according to which one should be able to describe the physics from the perspective of a quantum particle. The kinematics seen from this frame of reference is shown to be rather unconventional. In particular, we discuss several subtleties emerging in the relative formulation of central notions such as vector states, the classical limit, entanglement, uncertainty relations, and the complementary principle. A Hamiltonian formulation is derived as well which correctly encapsulates effects of fictitious forces associated with the accelerated motion of the frame. Our approach shows, therefore, how to formulate nonrelativistic quantum mechanics within noninertial reference frames which can be consistently described by the theory, with no appeal to classical elements.

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arXiv2012

Entanglement dynamics via semiclassical propagators in systems of two spins

We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. Also, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin $j$.

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arXiv2011

Integrability in time-dependent systems with one degree of freedom

The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the time-dependent Hamiltonian we prove the existence of two invariants in involution which are shown to obey the criterion of functional independence. The implication of this result is that chaotic motion cannot exist in these systems. In addition, if the invariant manifold is compact, then the system is Liouville integrable. As an application, we discuss regimes of integrability in models of dynamical tunneling and parametric resonance, and in the dynamics of two-level systems under generic classical fields. A corresponding quantum algebraic structure is shown to exist which satisfies analog conditions of Liouville integrability and reproduces the classical dynamics in an appropriate limit within the Weyl-Wigner formalism. The quantum analog is then conjectured to be integrable as well.

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arXiv2011

Semiclassics around a phase space caustic: an illustration using the Nelson Hamiltonian

The semiclassical formula for the coherent-state propagator is written in terms of complex classical trajectories of an equivalent classical system. Depending on the parameters involved, more than one trajectory may contribute to the calculation. Eventually, however, two contributing trajectories coalesce, characterizing what is called phase space caustic. In this case, the usual semiclassical formula for the propagator diverges, so that a uniform approximation is required to avoid this singularity. In this paper, we present a non-trivial application illustrating this scenario, and showing the accuracy of the uniform formula that we have previously derived.

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arXiv2011

Entanglement dynamics via coherent-state propagators

The dynamical generation of entanglement in closed bipartite systems is investigated in the semiclassical regime. We consider a model of two particles, initially prepared in a product of coherent states, evolving in time according to a generic Hamiltonian, and derive a formula for the linear entropy of the reduced density matrix using the semiclassical propagator in the coherent-state representation. The formula is explicitly written in terms of quantities that define the stability of classical trajectories of the underlying classical system. The formalism is then applied to the problem of two nonlinearly coupled harmonic oscillators and the result is shown to be in remarkable agreement with the exact quantum measure of entanglement in the short-time regime. An important byproduct of our approach is a unified semiclassical formula which contemplates both the coherent-state propagator and its complex conjugate.

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arXiv2008

A conjugate for the Bargmann representation

In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z*, whereas the creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ play the role of multiplication and differentiation with respect to z*, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of $\hat{a}^\dagger$ and $\hat{a}$ are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented.

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arXiv2007

Controlling Phase Space Caustics in the Semiclassical Coherent State Propagator

The semiclassical formula for the quantum propagator in the coherent state representation $<\mathbf{z}'' | e^{-i\hat{H}T/\hbar} | \mathbf{z}'>$ is not free from the problem of caustics. These are singular points along the complex classical trajectories specified by $\mathbf{z}'$, $\mathbf{z}''$ and $T$ where the usual quadratic approximation fails, leading to divergences in the semiclassical formula. In this paper we derive third order approximations for this propagator that remain finite in the vicinity of caustics. We use Maslov's method and the dual representation proposed in Phys. Rev. Lett. {\bf 95}, 050405 (2005) to derive uniform, regular and transitional semiclassical approximations for coherent state propagator in systems with two degrees of freedom.

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arXiv2005

Semiclassical coherent state propagator for systems with spin

We derive the semiclassical limit of the coherent state propagator for systems with two degrees of freedom of which one degree of freedom is canonical and the other a spin. Systems in this category include those involving spin-orbit interactions and the Jaynes-Cummings model in which a single electromagnetic mode interacts with many independent two-level atoms. We construct a path integral representation for the propagator of such systems and derive its semiclassical limit. As special cases we consider separable systems, the limit of very large spins and the case of spin 1/2.

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arXiv2005

Semiclassical Propagation of Wavepackets with Real and Complex Trajectories

We consider a semiclassical approximation for the time evolution of an originally gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed gaussian approximation. The other approximations are non-gaussian and may involve several trajectories determined by mixed initial-final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive gaussian potential, and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-gaussian approximations with real trajectories work well in some cases, whereas the thawed gaussian works only in very simple situations.

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