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arXiv:0802.4356

On canonical pairs in 2-transverse-mode DOPOs

Carlos Navarrete–Benlloch, Eugenio Roldán, Germán J. de Valcárcel

Published Feb 29, 2008• 0 citations
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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.

DOI
10.48550/arxiv.0802.4356

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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.
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 TEM10{}_{10}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.
In a series of articles we studied the quantum properties of a degenerate optical parametric oscillator tuned to the first family of transverse modes at the subharmonic. We found that, for a cavity having rotational symmetry with respect to the optical axis, a TEM10{}_{10}10​ mode with an arbitrary orientation in the transverse plane is emitted above threshold. We proved then that quantum noise induces a random rotation of this bright TEM10{}_{10}10​ mode in the transverse plane, while the orthogonal mode, the so-called dark mode, has perfect quadrature squeezing irrespective of the distance to threshold (noncritical squeezing). This result was linked to the spontaneous rotational symmetry breaking which occurs when the bright mode is generated, and here we analyze how the squeezing of the dark mode is degraded when the cavity is not perfectly symmetric. We will show that large levels of squeezing are still attainable for reasonable values of this anisotropy, with the advantage that the orientation of the bright and dark modes is basically fixed, a very attractive situation from the experimental point of view.
This communication studies the phenomenon of quadrature squeezing generated via spontaneous rotational symmetry breaking. A degenerate optical parametric oscillator (DOPO) tuned to the first family of transverse modes at the signal frequency, and having perfectly spherical mirrors. When pumped above threshold with a Gaussian beam and within a classical description, it is easy to show that a TEM10mode with an arbitrary orientation emerges at the subharmonic, hence breaking the rotational symmetry of the system in the transverse plane.
We theoretically describe the quantum properties of a large-Fresnel-number degenerate optical parametric oscillator (DOPO) with spherical mirrors that is pumped by a Gaussian beam and is tuned at the subharmonic frequency to a given transverse-mode family. We first analyze the classical problem and find that only the Laguerre-Gauss modes with lowest orbital angular momentum (OAM) are amplified above threshold. The transverse symmetry of the classically emitted signal field depends on the family index fff. If fff is even the lowest available OAM is zero and emission occurs in a radially symmetric mode; on the contrary, if fff is odd the lowest available OAM is 1 and the emitted signal field has the shape of a hybrid Laguerre-Gauss mode (a linear combination of the two Laguerre-Gauss modes with OAM equal to $\ifmmode\pm\else\textpm\fi{}1$) that breaks the rotational invariance of the system. Next we focus on the squeezing properties of this DOPO model. As for the modes with lowest OAM we demonstrate that their quantum properties (in the linear approximation) are equal to the standard single-mode DOPO (for even fff) or to the recently analyzed DOPO tuned to its first transverse-mode family [C. Navarrete-Benlloch, E. Rold\'an, and G. J. de Valc\'arcel, Phys. Rev. Lett. 100, 203601 (2008)]. Concerning the rest of (classically empty) modes (having larger OAM) we find that combinations of Laguerre-Gauss modes with opposite OAM (hybrid Laguerre-Gauss modes) exhibit quadrature squeezing. This property is independent of the even or odd character of the family index and hence has nothing to do with the symmetry of the classically emitted signal field. Noticeably the amount of squeezing does not depend on the pump level (it is thus noncritical squeezing) and can be arbitrarily large for the lower OAM nonamplified modes.
We study the classical and quantum properties of a degenerate optical parametric oscillator (DOPO) tuned to the first family of transverse modes at the down-converted frequency, under the injection of a resonant TEM10{}_{10}10​ mode. Unlike the usual single-mode DOPO with injected signal, large levels of squeezing are predicted even for relatively large injections. We interpret these results in connection with the spontaneous symmetry breaking predicted for this system in the absence of injection and with the existence of a bifurcation giving rise to the switching on of the (noninjected) TEM01{}_{01}01​ mode.