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16th Congress of the International Commission for Optics: Optics as a Key to High Technology

One dimensional representations in quantum optics

József Janszky

Published Jul 26, 1993• 0 citations
Publisher

Abstract

As a generalization of the optical Schroedinger cats discrete and continuous sets of coherent states are considered on a circle and on a straight line in the a-plane. The interference fringes between the coherent states of the Schroedinger cats form the characteristic pattern of the nonclassical states in Wigner function picture. Continuos superpositions of coherent states on these one-dimensional manifolds can serve as an orthonormal basis for representation of any states.

DOI
10.1117/12.2308445

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As a generalization of the optical Schroedinger cats discrete and continuous sets of coherent states are considered on a circle and on a straight line in the a-plane. The interference fringes between the coherent states of the Schroedinger cats form the characteristic pattern of the nonclassical states in Wigner function picture. Continuos superpositions of coherent states on these one-dimensional manifolds can serve as an orthonormal basis for representation of any states.
As a generalization of the optical Schr\"odinger cats, discrete sets of coherent states are considered on a circle in the \ensuremath{\alpha} plane. It is shown that simple superpositions of Schr\"odinger cats exhibit amplitude squeezing, similarly to the case of a superposition of several coherent states along a straight line that shows quadrature squeezing. The interference fringes between the coherent states form the annuli of the Fock states in the Wigner-function picture. It is also shown that a continuous superposition of coherent states on a circle can serve as a basis for the representation of any state.
Approximating a continuous superposition by a discrete one leads to an understanding of the role of quantum interference in generating squeezed coherent states, and in its quantum properties. It was shown that discrete superposition with a large enough number of coherent states with appropriate coefficients can have practically the same features as the squeezed coherent state. Dependent on the parameters W, r and 8, different numbers of coherent states are required for the interference fringe to form a Wigner function resembling that of the squeezed vacuum state and to reduce the deviation below a certain value. For the case of rotating the direction of squeezing, about 20 states were required to lower the deviation to 0.05%, while 640 states would be necessary when a unit displacement along the imaginary axis is included. When only a few coherent states are superimposed, the resulting state changes chaotically. In the case when the approximated squeezed coherent state was displaced along the imaginary axis an interesting Schrodinger cat-like state was found with about 20 coherent states on the real axis. At this point a significant peak occurs in the variances of the quadratures. It turned out that non-classical states can be effectively produced by a discrete superposition of coherent states with the coefficients derived from the one-dimensional representation of the given state.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider ``cat'' states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.