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Physical Review Letters

Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion

Bernard Yurke, David Stoler

Published Jul 7, 1986• 1,283 citations
Publisher

Abstract

It is pointed out here that a coherent state propagating through an amplitude dispersive medium will, under suitable conditions, evolve into a quantum superposition of two coherent states 180\ifmmode^\circ\else\textdegree\fi{} out of phase with each other. The response of a homodyne detector to this superposition of macroscopically distinguishable states is calculated. Signatures which an experimentalist might look for in the homodyne detector's output in order to verify the generation of such states are described.

DOI
10.1103/physrevlett.57.13

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It is pointed out here that a coherent state propagating through an amplitude dispersive medium will, under suitable conditions, evolve into a quantum superposition of two coherent states 180\ifmmode^\circ\else\textdegree\fi{} out of phase with each other. The response of a homodyne detector to this superposition of macroscopically distinguishable states is calculated. Signatures which an experimentalist might look for in the homodyne detector's output in order to verify the generation of such states are described.
In conditions of strong pump depletion, four-wave mixers can generate a quantum mechanical superimposition of two macroscopically distinguishable states of the electromagnetic field. We have shown with exactly solvable models1 and computer simulations that a four-wave mixer can, in conditions of strong pump depletion, generate states of the electromagnetic field which consist of a quantum mechanical superimposition of two macroscopically distinguishable states. In particular, in suitable conditions, the four-wave mixer can generate a superimposition of two coherent states of large amplitude but 180° out of phase with respect to each other. Experimentally, one can distinguish such Schrodinger catlike states from a statistical mixture by performing a measurement in which the interference between the two macroscopically distinguishable parts of the wave function is exhibited. For the cases we have considered, this interference can be exhibited as fringes in the probability distribution for the photocurrent delivered by a homodyne detector.
A four-wave mixer model is presented for which exact analytic results can be obtained in a regime where the usual perturbation procedures no longer apply. It is shown that under suitable conditions the output of such a four-wave mixer consists of a superposition of macroscopically distinguishable states. Such states can be identified with a homodyne detector which is able to observe the interference between the macroscopically distinguishable components. Further, it is shown that the device can act as an even-odd filter which will switch incoming pump light between two output ports depending on whether there is an even or odd number of pump photons.
It is usually assumed that a balanced homodyne detector measures a variable called the quadrature phase amplitude of a signal field. We examine this assumption by obtaining analytic expressions for the probability statistics for the output of such a detector for a single-mode signal with both signal and local oscillator treated quantum mechanically. We investigate the conditions under which the statistics of the quadrature phase are reproduced in the actual output of the detector. We show that the most obvious condition---that the number of quanta in the local oscillator be much larger than the number in the signal---is not sufficient to ensure that a balanced homodyne detector acts like an ideal detector of quadrature phases. Furthermore, we obtain the explicit conditions that are necessary and sufficient to reproduce the interference features in the homodyne statistics of a superposition of coherent states.
Thus far, all proposals for an optical evidence of macroscopic quantum interference have been frustrated by the low detector efficiencies. Two innovations permit such an evidence with currently available equipment. We refer to the generation scheme of entangled states by parametric amplification proposed by S. Song, C. M. Caves, and B. Yurke [Phys. Rev. A 41, 5261 (1990)]. Setting a low gain for the parametric amplifier introduces a drastic cutoff in the generated photon-number probabilities that compensates for the broadening of the detection probability in the first detector. Furthermore, we supplement the final homodyne detector with a degenerate type-I parametric amplifier which brings nearer the two previously distant states of the superposition. The availability of optical superposition states with large separation suggests a practical way of transforming them into entangled coherent states associated with different paths, thus providing a new convenient source for a ``which path'' experiment.