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Physical Review A

Superposition states at finite temperature

G. Huyet, Sonja Franke‐Arnold, Stephen Mark Barnett

Published Mar 15, 2001• 11 citations
Publisher

Abstract

We describe a method to create superposition states from a mixed state of a harmonic oscillator. If the initial state is described by a thermal state, then the resulting superposition state will be a ``hot'' superposition state. Such a state can be distinguished from a statistical mixture by its coherence properties. Here we suggest how to demonstrate the coherence of the superposition state by observing interference fringes when the two parts of the wave packet are overlapped. In the case of the mixed superposition state, a partial overlap may not be sufficient to observe the presence of fringes. We introduce therefore the idea of a coherence length for the wave packet and demonstrate its relevance for interferometry of mixed states. We illustrate our ideas with the example of superposition states for the motion of a single trapped ion.

DOI
10.1103/physreva.63.043812

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We describe a method to create superposition states from a mixed state of a harmonic oscillator. If the initial state is described by a thermal state, then the resulting superposition state will be a ``hot'' superposition state. Such a state can be distinguished from a statistical mixture by its coherence properties. Here we suggest how to demonstrate the coherence of the superposition state by observing interference fringes when the two parts of the wave packet are overlapped. In the case of the mixed superposition state, a partial overlap may not be sufficient to observe the presence of fringes. We introduce therefore the idea of a coherence length for the wave packet and demonstrate its relevance for interferometry of mixed states. We illustrate our ideas with the example of superposition states for the motion of a single trapped ion.
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.
Summary form only given. We introduce the concept and mathematical description of a coherence length and a coherence momentum for matter states, and explain their significance via the example of the motional states of a single cold trapped ion. These parameters describe the coherence properties of matter waves analogously to the way in which optical coherence determines the coherence time and spectral width. They become apparent in the interference pattern of a matter or light wave interferometer, respectively. While the coherence length is a characteristic length by which two parts of a matter wave packet can be separated so that they still interfere, the coherence momentum describes the time interval of this overlap.
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.
Using only elementary quantum-mechanical concepts, the statistical properties of various harmonic oscillator states that are linear superpositions of its energy eigenfunctions are described. These superpositions include coherent states and squeezed (or two-photon coherent) states. The resulting Gaussian, minimum-uncertainty wave packets are shown to oscillate back and forth for both coherent and squeezed states, but with an oscillating ‘‘width’’ for the squeezed states. Also examined are the principles underlying the production of squeezed electromagnetic waves via parametric amplification or four-wave mixing, their measurement by homodyne detection, and the connection between squeezing and non-Poissonian counting statistics.
We study the decoherence of superpositions of displaced quantum states of the form (where | g ⟩ is an arbitrary 'fiducial' state and is the usual displacement operator) within the framework of the standard master equation for a quantum damped or amplified harmonic oscillator interacting with a phase-insensitive (thermal) reservoir. We compare two simple measures of the degree of decoherence: the quantum purity and the height of the central interference peak of the Wigner function. We show that for N > 2 'mesoscopic' components of the superposition, the decoherence process cannot be characterized by a single decoherence time. Therefore, we distinguish the 'initial decoherence time' and 'final decoherence time' and study their dependence on the parameters α k and N . We obtain approximate formulae for an arbitrary state | g ⟩ and explicit exact expressions in the special case of | g ⟩ = | m ⟩, i.e., for (symmetrical) superpositions of displaced Fock states of occupation number m . We show that the superposition with a large number of components N and rich 'internal structure' ( m ∼ |α| 2 ) can be more robust against decoherence than simple superpositions of two coherent states (with m = 0), even if the initial decoherence times coincide. Also, we show how initial pure quantum superpositions are transformed into highly mixed and totally classical superpositions in the case of a phase-insensitive amplifier.
The time evolution of thermal states of a mirror released from a tight harmonic trap is studied. After the release no dissipation is assumed to be present and the mirror is, after a time delay, kicked into a momentum superposition state. The thermal character of the initial state washes out the telltale interference patterns of the superposition but no loss of coherence is found. This investigation resolves a controversy about decoherence--without--dissipation and shows that entrained measurements can be surprisingly insensitive to temperature effects.