Noncritical quadrature squeezing in two-transverse-mode optical parametric oscillators

Abstract

In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down-converted frequency. Recently we found [C. Navarrete-Benlloch et al., Phys. Rev. Lett. 100, 203601 (2008)] that above threshold a TEM 10 mode following a random rotation in the transverse plane emerges in this system (we denote it as the bright mode), breaking thus its rotational invariance. Then, owing to the mode orientation being undetermined, we showed that the phase quadrature of the transverse mode orthogonal to this one (denoted as the dark mode) is perfectly squeezed at any pump level and without an increase in the fluctuations on its amplitude quadrature (which seems to contradict the uncertainty principle). In this article we go further in the study of this system and analyze some important features not considered previously. First we show that the apparent violation of the uncertainty principle is just that-"apparent"-as the conjugate pair of the squeezed quadrature is not another quadrature but the orientation of the bright mode (which is completely undetermined in the long term). We also study a homodyne scheme in which the local oscillator is not perfectly matched to the dark mode, as this could be impossible in real experiments due to the random rotation of the mode, showing that even in this case large levels of noise reduction can be obtained (also including the experimentally unavoidable phase fluctuations). Finally, we show that neither the adiabatic elimination of the pump variables nor the linearization of the quantum equations are responsible for the remarkable properties of the dark mode (which we prove analytically and through numerical simulations, respectively), which were simplifying assumptions used in Navarrete-Benlloch et al. [Phys. Rev. Lett. 100, 203601 (2008)]. These studies show that the production of noncritically squeezed light through spontaneous rotational symmetry breaking is a robust phenomenon.

Noncritical generation of nonclassical frequency combs via spontaneous rotational symmetry breaking

“…In particular, such device will emit a (classical) frequency comb with the transverse profile of a TEM 10 mode (bright mode), and a perfectly squeezed mode with the same spectral profile but an orthogonal TEM 01 spatial profile (dark mode). As shown in previous works [31,32], such perfect squeezing must be facilitated by the complete quantum indeterminacy of the bright and dark modes' orientation θ. Unfortunately, within the familiar picture of quantum Langevin equations in which operators evolve, keeping track of the quantum dynamics of such a phase is highly non trivial (since the corresponding operator is quite intricate [51,52]), and a consistent linearization procedure including the phase dynamics has to be performed within a stochastic representation.…”

“…Recently, however, it has been predicted in the context of a degenerate OPO pumped by a monochromatic Gaussian beam, that when the cavity is tuned to the first transverse mode family at the subharmonic frequency [28][29][30], the signal field displays a level of squeezing above threshold which equals that at threshold [14,31,32], i.e., the squeezing production in this case is noncritical [17,[31][32][33][34][35][36]. The physics behind such a remarkable result lies in the spontaneous rotational symmetry breaking around the cavity axis brought about by the (above-threshold) classical field, which necessarily has the shape of a TEM 10 mode of arbitrary orientation because of orbital angular momentum (OAM) conservation (see Fig.…”

“…Hence, the modes' orientation diffuses with time, but at a slower rate the further we are from threshold (as |ρ| 2 gets larger). In previous works we proved that a fixed local oscillator which does not follow the rotation of the dark mode is therefore still capable of measuring high levels of squeezing [32], and, furthermore, either the injection of a weak seed with the spatiotemporal shape of the bright mode or a small anisotropy in the system are able to fix the orientation of the latter without degrading too much the squeezing of the dark mode [49,50]. All these properties extend naturally to our proposed SPOPO configuration, which will then allow access to this elusive and rich regime of operation where spontaneous symmetry breaking dominates the quantum properties of the output light.…”

“…Quantum properties of the emitted field.-In previous works [14,31,32] we have studied how quantum noise affects the phase θ undefined at the classical level, proving that it diffuses linearly with time, driven by quantum noise. The analysis requires a rather technical procedure based the positive P representation [48], and hence we present it in the supplemental material.…”

“…1). It follows that the TEM 01 mode orthogonal to this field has perfect quadrature squeezing at any operating point above threshold [14,31,32]. The main problem with OPOs under monochromatic pump is that, in practice, it is not possible to ensure that phase-matching is maximized for the degenerate process [37][38][39], what results in the oscil-lation a pair of non-degenerate modes above threshold.…”