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Generation of Glauber Coherent State Superpositions via Unitary Transformations

ANTONINO MESSINA, Benedetto Militello, Anna Napoli

Published Jan 1, 2004• 2 citations

Abstract

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]

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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]

In 1935, Erwin Schrödinger put forward his famous gedanken experiment in which a microscopic degree of freedom - the emission state of a radioactive atom - and a macroscopic, classical state - the life or death of a cat - are entangled [1]. The resulting state is dubbed a Schrodinger-cat state. According to quantum theory, a measurement of the atom in a superposition basis projects the cat into a macroscopic superposition of being alive and dead. Here, we use coherent laser fields with different optical phases as a macroscopic model system and generate entangled Schrodinger-cat states in a deterministic way [3]. We reflect the coherent pulses from an optical cavity that contains a single strongly-coupled atom. This entangles the spin of the atom with the optical phase of the light field [2]. By manipulating and detecting the atomic state, we create a plethora of flying optical coherent-state superpositions. We reconstruct the corresponding Wigner functions (see Fig. 1) to demonstrate the tunability and non-classicality of the produced light state.

A "Schrodinger cat"-like state of matter was generated at the single atom level. A trapped 9Be+ ion was laser-cooled to the zero-point energy and then prepared in a superposition of spatially separated coherent harmonic oscillator states. This state was created by application of a sequence of laser pulses, which entangles internal (electronic) and external (motional) states of the ion. The Schrodinger cat superposition was verified by detection of the quantum mechanical interference between the localized wave packets. This mesoscopic system may provide insight into the fuzzy boundary between the classical and quantum worlds by allowing controlled studies of quantum measurement and quantum decoherence.

We laser-cool single beryllium ions in a Paul trap to the ground (n equals 0) quantum harmonic oscillator state with greater than 90% probability. From this starting point, we can put the atom into various quantum states of motion by application of optical and rf electric fields. Some of these states resemble classical states (the coherent states), while others are intrinsically quantum, such as number states or squeezed states. We have created entangled position and spin superposition states (Schrodinger cat states), where the atom's spatial wavefunction is split into two widely separated wave packets. We have developed methods to reconstruct the density matrices and Wigner functions of arbitrary motional quantum states. These methods should make it possible to study decoherence of quantum superposition states and the transition from quantum to classical behavior. Calculations of the decoherence of superpositions of coherent states are presented.

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.

High-harmonic generation is one of the most fundamental processes in strong laser-field physics that has led to countless achievements in atomic physics and beyond. However, a rigorous quantum electrodynamical picture of the process has never been reported. Here, we prove rigorously and demonstrate experimentally that the quantum state of the driving laser field, as well as that of harmonics, is coherent. Projecting this state on its part corresponding to harmonic generation, it becomes a superposition of a state, amplitude-shifted due to the quantum nature of light, and the initial state of the laser. This superposition interpolates between a Schrodinger kitten, and a genuine Schrodinger cat state. This work opens new paths for ground-breaking investigations in strong laser-field physics and quantum technology. We dedicate the work to the memory of Roy J. Glauber, the inventor of coherent states.

Strong laser-atom interactions can produce highly non-classical states of light by using the process of high-harmonic generation in atoms. When the high-harmonic generation is present, the quantum state of the fundamental mode following the interaction is known as the Schrödinger cat state, which is a superposition of the laser’s initial coherent state and the coherent state with a smaller amplitude that results from its interaction with atoms. Here, we demonstrate that new light states with significantly different Wigner function distributions can be produced by combining two separate Schrödinger cat states. Through the engineering of Schrödinger cat states’ parameters, we are able to produce Wigner functions that exhibit high sensitivity in relation to the system parameter. Our research paves the way for the creation of non-classical light by superposing Schrödinger cat states with application in such as quantum sensing.