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arXiv:quant-ph/9806098 · Physical Review A

Quantum state engineering via unitary transformations

A. Vidiella-Barranco, José Antonio Roversi

Published Oct 1, 1998• 19 citations
arXivHTMLPDF

Abstract

We construct a Hamiltonian for the generation of arbitrary pure states of the quantized electromagnetic field. The proposition is based upon the fact that a unitary transformation for the generation of number states has already been found. The general unitary transformation here obtained would allow the use of nonlinear interactions for the production of pure states. We discuss the applicability of this method by giving examples of generation of simple superposition states. We also compare our Hamiltonian with the one resulting from the interaction of trapped ions with two laser fields.

DOI
10.1103/physreva.58.3349

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We construct a Hamiltonian for the generation of arbitrary pure states of the quantized electromagnetic field. The proposition is based upon the fact that a unitary transformation for the generation of number states has already been found. The general unitary transformation here obtained would allow the use of nonlinear interactions for the production of pure states. We discuss the applicability of this method by giving examples of generation of simple superposition states. We also compare our Hamiltonian with the one resulting from the interaction of trapped ions with two laser fields.
We study a coherent superposition $t\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}+r{\mathrm{a\ifmmode \hat{}\else \^{}\fi{}}}^{\ifmmode\dagger\else\textdagger\fi{}}$ of field annihilation and creation operators acting on continuous variable systems and propose its application for quantum state engineering. Specifically, it is investigated how the superposed operation transforms a classical state to a nonclassical one, together with emerging nonclassical effects. We also propose an experimental scheme to implement this elementary coherent operation and discuss its usefulness to produce an arbitrary superposition of number states involving up to two photons.
It is well-known that any two pure quantum states (in the same Hilbert space) can be mapped to any other using unitary transformations. However, previous approaches to this problem required two explicit bases for the Hilbert space, one each for the initial and target states, and thus their complexity necessarily scales with the dimension of the Hilbert space. In this Letter, we show how to utilize novel algebraic methods to construct a closed-form exponential unitary transformation which achieves this in general, using only a single unitary generator. This construction is independent of any bases and agnostic to the dimension of the Hilbert space. We highlight the usefulness of this tool for studying relationships between systems of pure states in quantum information theory, as well in elementary analyses of quantum circuits and unitary operators.
Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. The algorithm itself is not unitary because it involves a conditional measurement. However, it does yield a constant probability of the conditional measurement which does not depend on the input state of the bosonic system. We consider a physical implementation of unitary transformations acting on one-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.
It has been recently shown [1] that, under suitable conditions, the interaction between a three-level atom and the quantized field of a bimodal microcavity may be effectively reduced to the two-photon coupling between a single two-level atom and the electromagnetic field in the resonator. The corresponding Hamiltonian model is characterized by intensity-dependent Stark shifts due to the presence of an intermediate off-resonance atomic level. Assume the cavity field prepared at in an arbitrary state and let an initially excited atom cross the cavity with an appropriately selected velocity. It is possible to show that, exploiting a single conditional measurement of the atomic state immediately after the atom leaves the resonator, the bimodal field is projected into a new state characterized by the same photon number distribution present at More in detail, the experimental scheme we propose makes it possible manipulating only the phases of the initial probability amplitudes of finding a well defined population in the cavity. We prove that the state into which the cavity collapses after a successful atomic measurement is unitarily related to the initial field state and the explicit form of the accomplishing operator is constructed. We wish to remark that, beside its inherent theoretical interest, this simple experimental scheme might also be of relevance in the socalled field of quantum communication [2], where implementing unitary modifications of a quantum state signal represents an issue of topical and central importance. More in general, our proposal provides a systematic procedure for controlling selected properties of a quantum state. We in fact demonstrate that our experimental scheme may be reliably exploited to generate orthogonal states to a prefixed one or to control the squeezing direction.
Since their introduction by Glauber in 1963 [1], coherent states have been deeply studied [2] due to the fact that, in some sense, the statistical properties related to the first momenta of their Fock distribution exhibit features interpretable in the realm of classical probability theory. Of course the quantum nature of a coherent state, and ultimately the granularity of the energy of a quantum harmonic oscillator, becomes more and more evident when higher momenta of the Fock distribution are considered. It is of relevance that coherent states of both a mechanical quantum oscillator or a quantized mode of an electromagnetic resonator may be currently proposed in accordance with well-defined protocols. Even more interesting is the fact that they provide the effective possibility of realizing interesting experiments able to bring to the light peculiarities of quantum mechanics. In particular, the very famous gedanken Schrodinger Cat State (SCS) experiment [3] consisting in entangling a microscopic quantum object (an atom) with a macroscopic one (a cat) has been realized. No unfortunate feline has been tortured! Trapped ions have been indeed exploited. In fact, such systems provide the possibility of testing fundamental quantum physics aspects giving at experimentalist’s disposal compound quantum systems consisting of a few-level atomic part and a tri-dimensional harmonic oscillator describing the center of mass motion of an ion confined into a Paul trap. Suitably acting upon the system with Raman lasers it is possible to engineer the Schrodinger Cat-like state [4, 5]