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Physical Review A

Photon-number-state preparation in nondegenerate parametric amplification

Catherine A. Holmes, Gerard J Milburn, Daniel F. Walls

Published Mar 1, 1989• 52 citations
Publisher

Abstract

In nondegenerate parametric amplification of light, one pump photon is destroyed and two photons are simultaneously created, one in each of two modes with different frequencies (the signal and idler). If m signal photons are counted in time t, what is the state of the idler conditioned on this result? The projection postulate cannot be used to answer this question as photon-counting measurements are not ``first-kind'' measurements. We use a theory of continual quantum-counting measurements to show that in general the idler field is left in a classical mixture of number states with at least m quanta. In an appropriate limit this mixture reduces to an exact number state with m quanta.

DOI
10.1103/physreva.39.2493

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In nondegenerate parametric amplification of light, one pump photon is destroyed and two photons are simultaneously created, one in each of two modes with different frequencies (the signal and idler). If m signal photons are counted in time t, what is the state of the idler conditioned on this result? The projection postulate cannot be used to answer this question as photon-counting measurements are not ``first-kind'' measurements. We use a theory of continual quantum-counting measurements to show that in general the idler field is left in a classical mixture of number states with at least m quanta. In an appropriate limit this mixture reduces to an exact number state with m quanta.
Noise spectra of the in-phase and quadrature-phase components of the signal and idler fields of a nondegenerate optical parametric amplifier are calculated. It is shown that, as is suggested by the operator Manley-Rowe relations, the external in-phase (amplitude) fluctuations of the signal and idler fields are correlated in the frequency region lower than the signal and idler cavity bandwidths. If the idler photon flux is continuously measured and the photocurrent fluctuations are fed back to control the pump intensity, the signal output will feature an amplitude noise spectral density reduced to below the standard quantum limit (amplitude squeezing) in the frequency region below the feedback loop and signal and idler cavity bandwidths. An all-optical realization of such a negative-feedback parametric oscillator is presented. It is also shown that a nondegenerate parametric amplifier with idler measurement feedback can add a fixed number of photons to a photon-number state.
Signal and idler waves produced by a nondegenerate parametric amplifier feature strong positive correlation between their photon numbers, or in-phase amplitudes. They feature equally strong negative correlation between their phases, or quadrature-phase amplitudes. These photon twins can produce arbitrarily squeezed states via state reduction by appropriate measurements of an idler wave. When the quadrature amplitude of an idler wave is measured, the signal wave is collapsed to a quadrature-amplitude squeezed state. When the two-quadrature amplitudes of an idler wave are measured simultaneously, the signal wave becomes a coherent state.
If the idler photon flux is continuously measured in a photon-counting detector and the photocurrent fluctuations are fed back to control the pump intensity, the output signal state will be an amplitude-squeezed state (near-number state) at low frequencies and high feedback loop gain. Electromagnetic field in number-phase minimum-uncertainty states (amplitude- squeezed states), and its limit (photon-number states) have potential applications in optical communication, precision measurement, and atomic spectroscopy, and gravitational wave detection. These states have been the focus of considerable attention.
An exact quantum treatment reveals that signal and idler photon number operators are not well-behaved dynamical operators for studying photon statistics in parametric amplification/down-conversion processes. Contrary to expectations, the mean signal-idler photon number difference varies with time, while the corresponding signal-idler photon number cross-correlation function is complex and experiences an interference phenomenon driven by the interaction parameters. The intensity operators and related polarization operators of the polarized signal-idler photon pair in positive and negative helicity states are identified as the appropriate operators specifying a conservation law and the dynamical symmetry group (SU(1, 1)) of the parametric amplification process. The conservation of the mean positive and negative helicity photon intensity difference and the purely real positive-negative helicity intensity cross-correlation function correctly account for the simultaneous production of polarized signal and idler photons in positive and negative helicity states.
We show that a continuous range of nonclassical states of light can be generated using conditional measurements on the idler mode of an optical parametric amplifier. The output state is prepared by introducing a coherent state in the signal mode of the amplifier with a single photon in the idler mode, followed by a conditional measurement of a single photon in the output idler mode. By varying the gain of the amplifier, this approach can produce a coherent state, a photon-added state, a displaced number state, or a continuous range of other nonclassical states with intermediate properties. We note that this approach can generate a photon-added state even though the postselected amplifier does not add any photons to the signal or idler modes. The ability to generate a continuous range of nonclassical states may have practical applications in quantum information processing.