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Physical Review A

Quantum behavior of a four-wave mixer operated in a nonlinear regime

Bernard Yurke, David Stoler

Published Jun 1, 1987• 67 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.35.4846

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Four-wave mixers have been investigated theoretically as potential sources of squeezed-coherent radiation 1,2 and there are a number of experimental efforts underway to demonstrate the production of squeezed-coherent radiation via such devices.3,4,5 Here the results of some wideband calculations of the output of cavity four-wave mixers is presented. The wideband squeezed-coherent radiation generated by such devices may be detected via a wideband homodyne detector.6 Wideband squeezing will manifest itself in the noise-power spectrum of the homodyne detector's output as regions where the noise is reduced below the level that is observed when no light enters the input port of the homodyne detector. Wideband homodyne detection allows one to look for squeezing at frequencies far from d.c. This gives the experimentalist the freedom to look for squeezing in the frequency range where his detector performs best.

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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.
A model of a four-wave mixer operated in a nonlinear regime, studied by Yurke and Stoler [Phys. Rev. A 35, 4846 (1987)], is reexamined. Yurke and Stoler have shown that this device, under a certain condition, acts as an even-odd filter, switching even-photon-number states from the pump mode to the signal mode. An initial coherent state in the pump is converted into an entanglement of even and odd coherent states with vacuum states in the output signal and pump modes. We point out that under a different condition and with an even-photon-number state initially in the pump and a vacuum in the signal, the device creates a maximally entangled state between the number state and the vacuum. Using the device to replace the first beamsplitter of a Mach-Zehnder interferometer, phase uncertainties at the Heisenberg limit $(\ensuremath{\Delta}\ensuremath{\varphi}=1/n)$ can be obtained. Since number states are difficult to generate, we point out that an even coherent state obtained from the output of one device can be used as input to a second to achieve the phase uncertainties $\ensuremath{\Delta}\ensuremath{\varphi}{=1/n}_{e},$ where ne{n}_{e}ne​ is the average photon number of the even coherent state.
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.
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.
Four-wave mixers have been investigated theoretically as potential sources of squeezed-coherent radiation 1,2 and there are a number of experimental efforts underway to demonstrate the production of squeezed-coherent radiation via such devices.3,4,5 Here the results of some wideband calculations of the output of cavity four-wave mixers is presented. The wideband squeezed-coherent radiation generated by such devices may be detected via a wideband homodyne detector.6 Wideband squeezing will manifest itself in the noise-power spectrum of the homodyne detector's output as regions where the noise is reduced below the level that is observed when no light enters the input port of the homodyne detector. Wideband homodyne detection allows one to look for squeezing at frequencies far from d.c. This gives the experimentalist the freedom to look for squeezing in the frequency range where his detector performs best.