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Physical Review A

Quantum theory of nonlinear loop mirrors

K. J. Blow, R. Loudon, Simon J. D. Phoenix

Published Jun 1, 1992• 21 citations
Publisher

Abstract

We formulate a continuous-mode quantum theory of nonlinear loop mirrors. We exploit the ``conservation of squeezing'' property at a beam splitter to provide a particularly simple description of the output noise spectra from such devices. We also derive a general condition for which one may obtain a squeezed vacuum from one of the output ports of the loop mirror.

DOI
10.1103/physreva.45.8064

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We formulate a continuous-mode quantum theory of nonlinear loop mirrors. We exploit the ``conservation of squeezing'' property at a beam splitter to provide a particularly simple description of the output noise spectra from such devices. We also derive a general condition for which one may obtain a squeezed vacuum from one of the output ports of the loop mirror.
When a coherent electromagnetic wave passes through a beam splitter (BS), it is divided equally into two parts. However, the quantum noise associated with the resulting coherent states, despite being reduced in amplitude by half, remains fundamentally constrained by the quantum noise limit, independent of the intensity. By placing a mirror at the unused input port of the BS, a standing wave is formed in the vicinity of the mirror, which influences the vacuum fluctuations of the coherent state at the BS output. Using semi-classical and quantum mechanical approaches, we calculate the vacuum fluctuations induced by the mirror and demonstrate that the vacuum noise originating from the mirror side periodically reaches zero at the BS output. Leveraging this effect, we show that the vacuum fluctuations of the light split by the BS can be readily reduced below the quantum noise limit. Furthermore, through feedback mechanisms, the vacuum fluctuations of the electromagnetic field at the other output port can also be suppressed below the quantum noise limit. These findings provide a pivotal insight into the manipulation of electromagnetic noise, with broad implications for all experiments involving quantum noise control.
Machida and Yamamoto used negative feedback from a conventional photodetector to stabilize a laser's intensity and thus to obtain sub-shot-noise photon statistics within the feedback loop. Out-of-Ioop light extracted at a beam splitter, however, displayed super-shot-noise statistics. This behavior can be understood both quantum mechanically and semiclassically. Quantum mechanically the culprit is the vacuum field that enters the beam splitter’s second input port. I describe a technique for extracting sub-shot-noise light by replacing the culprit vacuum field with squeezed light generated by a nonlinear device—a squeezer—that is pumped by a portion of the laser light. Moreover, even if the unstabilized laser is shot-noise-limited, the extracted light can have a better intensity SNR; this signal-to-noise improvement requires both feedback and squeezing.
We analyze numerically the generation of photon-number squeezed light using an asymmetric fiber loop mirror. For photon-number squeezed light generation, the maximum squeezing can be obtained using this method. The evolution of the quantum noise for the optical pulse spectra is calculated using the quantized nonlinear Schrödinger equation and linearization approximation. It is shown that the squeezed light is generated by removing the increased quantum noise part of the spectrum and enhancing the decreased quantum noise part of the spectrum, by the interference of two pulses at the beam splitter. Although the mechanism of squeezing is similar to that of a spectral filter, the maximum squeezing is as large as -13 dB due to the interference.
The familiar input-output relations for an optical beam splitter are generalized to allow for linear absorption by the medium forming the mirror. Beam-splitter losses generally affect the noise levels detectable in experiments involving nonclassical light. When employed to investigate two-photon interference effects, a lossy beam splitter can lead to apparent nonlinear absorption, which, in the most extreme case, leads to either both or neither of the photons being absorbed. The degree of second-order coherence of antibunched light can be maintained on transmission through the beam splitter but any amplitude squeezing in the incident light is degraded.
A formalism is developed that allows one to calculate the propagation of quantum noise, technical noise, and signals through a linearized quantum optical bus, i.e., a complex composite multibeam optical quantum system. General expressions for the output squeezing, the correlation coefficient, and quantum correlation coefficient between output observables and the transfer coefficients are derived and discussed in detail. The total correlation is introduced as a measure of the overall correlation of a set of output quadratures. From its geometric interpretation a multidimensional uncertainty relation is derived. Direct quantum-state preparation (QSP) is introduced as a method complementary to the conventional, indirect QSP. The minimal variance achievable by direct QSP is shown to provide superior noise suppression, including quantum noise suppression, than the conditional variance provided by indirect QSP. As an application we calculate and discuss the properties of a balanced bus of beam splitters with squeezed vacua at the usually unused ports. An important experimental effect is imperfect beam mode match, which is shown to lead to dramatic effects. A theory is derived which quantitatively matches measurements. Analytic expressions for the imperfectly modematched squeezed-light beam splitter including detection losses are given. Finally, a direct method is used to calculate the input-output matrix for the multiport subthreshold degenerate optical parametric amplifier for arbitrary parametric and mirror coupling strengths and detuning.
We study the photon counting noise in optical interferometers used for gravitational wave detection. In order to reduce quantum noise, a squeezed vacuum is injected into the usually unused input port. It is investigated under which conditions the gravitational wave signal may be amplified without increasing counting noise concurrently. Such a possibility was suggested as a consequence of the entanglement of the two output ports of a beam splitter. We find that amplification without concurrent increase of noise is not possible for reasonable squeezing parameters. Photon distributions for various beam splitter angles and squeezing parameters are calculated.