On canonical pairs in 2-transverse-mode DOPOs

TL;DR

This paper investigates the quantum properties of a 2-transverse-mode DOPO, revealing how quantum noise induces mode rotation and clarifying the canonical nature of quadratures related to squeezing and uncertainty principles.

Contribution

It proves that the quadratures of the rotating orthogonal mode do not form a canonical pair, explaining the observed quantum noise and squeezing phenomena.

Findings

Quantum noise causes random rotation of the emitted mode.

Squeezing is observed in quadratures aligned with the local oscillator.

The quadratures of the orthogonal mode are not canonical pairs, resolving the paradox.

Abstract

In ref. [1] we analyzed the properties of a Degenerate Optical Parametric Oscillator (DOPO) tuned to the first transverse mode family at the signal frequency. Above threshold, a Hermite-Gauss mode with an arbitrary orientation in the transverse plane is emitted, and thus the rotational invariance of the system is broken. When quantum effects were taken into account, it was found on the one hand, that quantum noise is able to induce a random rotation on this classically emitted mode. On the other hand, the analysis of a balanced homodyne detection in which the local oscillator (LO) was orthogonal to the excited mode at any time, showed that squeezing in the quadrature selected by the LO was found for every phase of this one, squeezing being perfect for a pi/2 phase. This last fact revealed an apparent paradox: If all quadratures are below shot noise level, the uncertainty principle seems to be violated. In [1] we stated that the explanation behind this paradox is that the quadratures of the rotating orthogonal mode do not form a canonical pair, and the extra noise is transferred to the diffusing orientation. Thes notes are devoted to prove this claim.