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arXiv:0905.1900 · New Journal of Physics

Blind spots between quantum states

Eduardo Zambrano, Alfredo M. Ozorio de Almeida

Published Nov 24, 2009• 5 citations
arXivHTMLPDF

Abstract

The overlap of a large quantum state with its image, under tiny translations, oscillates swiftly. We here show that complete orthogonality occurs generically at isolated points. Decoherence, in the Markovian approximation, lifts the correlation minima from zero much more quickly than the Wigner function is smoothed into a positive phase space distribution. In the case of a superposition of coherent states, blind spots depend strongly on positions and amplitudes of the components, but they are only weakly affected by relative phases and the various degrees and directions of squeezing. The blind spots for coherent state triplets are special in that they lie close to a hexagonal lattice: further superpositions of translated triplets, specified by nodes of one of the sublattices, are quasi-orthogonal to the original triplet and to any state likewise constructed on the other sublattice.

DOI
10.1088/1367-2630/11/11/113044

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The overlap of a large quantum state with its image, under tiny translations, oscillates swiftly. We here show that complete orthogonality occurs generically at isolated points. Decoherence, in the Markovian approximation, lifts the correlation minima from zero much more quickly than the Wigner function is smoothed into a positive phase space distribution. In the case of a superposition of coherent states, blind spots depend strongly on positions and amplitudes of the components, but they are only weakly affected by relative phases and the various degrees and directions of squeezing. The blind spots for coherent state triplets are special in that they lie close to a hexagonal lattice: further superpositions of translated triplets, specified by nodes of one of the sublattices, are quasi-orthogonal to the original triplet and to any state likewise constructed on the other sublattice.
It is shown that a particular combination of so-called even coherent state dubbed the orthogonal-even coherent state, while exhibiting no ``ordinary'' or second-order squeezing, can furnish near-optimal simultaneous-quadrature fourth-order squeezing [where the latter term is defined in the sense of Hong and Mandel, Phys. Rev. Lett. 54, 323 (1985)]. The Wigner function of the particular quantum-mechanical superposition of states is calculated and various plots compared to the Wigner function of a simple statistical mixture of the same states. Rather uninteresting behavior is found in the latter case, which illustrates that the higher-order squeezing effect is due to nonclassical, quantum-mechanical ``interference in phase space.''
Characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the Fourier transform of the Wigner function: a quantum characteristic function , or the chord function. However, knowledge of a finite number of moments does not allow for accurate determination of the chord function. This provides the overlap of the state with all its possible rigid translations (or displacements). We here present a semiclassical approximation of the chord function for large Bohr-quantized states, which is accurate right up to a caustic, beyond which the chord function becomes evanescent. It is verified to pick out blind spots, which are displacements for zero overlaps. These occur even for translations within a Planck area of the origin. We derive a simple approximation for the closest blind spots, depending on the Schrödinger covariance matrix, which is verified for Bohr-quantized states.
The probability distribution for finding a quantum state of the light field with a particular phase can be described by a number of theoretical methods. One of the most frequently used in quantum optics is the Pegg–Barnett phase distribution which identifies the phase distribution as the (necessarily positive) modulus square of the overlap of the quantum state with the relevant phase state. Other phase distributions have been proposed which employ the phase-sensitivity of various quantum quasi-probabilities such as the Wigner function. In this paper we discuss the connections between these approaches and illustrate their similarities and important differences for a number of pertinent quantum states of light. For some of these states, quantum interference can generate significant regions of phase space for which the Wigner quasi-probability is negative, and as we show, this may render the quasi-probability phase distribution negative. We discuss the general significance of this phenomenon.
Quantum information and quantum communication are both strongly based on concepts of quantum superposition and entanglement. Entanglement allows distinct bodies, that share a common origin or that have interacted in the past, to continue to be described by the same wave function until evolution is coherent. So, there is an equivalence between coherence and entanglement. In this paper, we show the relation between quantum coherence and quantum interference and the negative parts of the Wigner quasi-distribution, using the Wigner signed-particle formulation. A simple physical problem consisting of electrons in a nanowire interacting with the potential of a repulsive dopant placed in the center of it creates a quasi two-slit electron system that separates the wave function into two entangled branches. The analysis of the Wigner quasi-distribution of this problem establishes that its negative part is principally concentrated in the region after the dopant between the two entangled branches, maintaining the coherence between them. Moreover, quantum interference is shown in this region both in the positive and in the negative part of the Wigner function and is produced by the superposition of Wigner functions evaluated at points of the momentum space that are symmetric with respect to the initial momentum of the injected electrons.
The cover page shows snapshots of a decaying Schrödinger cat Wigner function. This is a quasi-probability distribution for the amplitude of a microwave field stored in an ultra-high quality superconducting cavity. A single circular Rydberg atom interacting with a coherent state produces a quantum superposition of two fields with different phases. The two positive peaks in the Wigner function (front) are centered on the classical amplitudes of these components. The interference pattern in-between indicates the quantum nature of the superposition. As time proceeds (from front to rear), the interference pattern vanishes, revealing the rapid decoherence of a mesoscopic quantum state superposition.
The cover page shows snapshots of a decaying Schrödinger cat Wigner function. This is a quasi-probability distribution for the amplitude of a microwave field stored in an ultra-high quality superconducting cavity. A single circular Rydberg atom interacting with a coherent state produces a quantum superposition of two fields with different phases. The two positive peaks in the Wigner function (front) are centered on the classical amplitudes of these components. The interference pattern in-between indicates the quantum nature of the superposition. As time proceeds (from front to rear), the interference pattern vanishes, revealing the rapid decoherence of a mesoscopic quantum state superposition.