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Physical Review A

Properties of the linearized quantum optical bus

Robert Bruckmeier, Stephan Schiller

Published Jan 1, 1999• 3 citations

Abstract

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.

DOI: 10.1103/physreva.59.750

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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 state is injected into the usually unused input port. Here, we specifically investigate the so-called “dark port case,” when the beam splitter is oriented close to 90° to the incoming laser beam, such that nearly all photons go to one output port of the interferometer, and only a small fraction of photons is seen in the other port (“dark port”). For this case it had been suggested that signal amplification is possible without concurrent noise amplification [R. Barak and Y. Ben-Aryeh, J. Opt. Soc. Am. B 25 , 361 (2008)]. We show that by injection of a squeezed vacuum state into the second input port, counting noise is reduced for large values of the squeezing factor, however the signal is not amplified. Signal strength only depends on the intensity of the laser beam.

The properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.

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.

We consider here the problem of quantum noise in light produced by an optical parametric oscillator operating below threshold inside a singlesided cavity. By employing the correct boundary conditions on quantum fields at the output mirror, a procedure that is valid for arbitrary mirror transmission, we derive exact expressions for quantum noise reduction for arbitrary pump phase and mirror out-coupling. We find that for a given out-coupling, squeezing of quantum noise in the intracavity field increases to a final value of 50%, as the pump intensity is increased and the oscillation threshold is approached. The output field on the other hand exhibits perfect squeezing of noise in one quadrature of its central Fourier component at threshold regardless of the mirror out-coupling. The present approach, which is based on boundary conditions, is quite general and applicable to any matter-field interaction problem inside a cavity. In fact, we have previously employed this approach in the calculation of the intrinsic linewidth of a laser with arbitrary out-coupling.1

A beam splitter is the simplest and clearest example of a coupling device with which quantum limited measurements can be performed. The limitations due to the vacuum fluctuations entering the usually unused port are circumvented by injection of a 3.7 dB squeezed wave into this port. Thereby phase-sensitive measurements that satisfy the quantum nondemolition criteria are realized (signal transfer

{T}_{s}{+T}_{m}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.29

, 32% correlation). The dependences of the signal transfer and correlation on beam splitter transmissivity are measured and found to be in quantitative agreement with a calculation that takes into account various inefficiencies. Continuous stable operation for 36 hours is achieved.