We present the optical transfer functions for third-order nonlinear cavities that involve an optical carrier frequency and its modulation sideband fields. Our approach is based on linearized transformations and provides a convenient tool to calculate squeezed light sources as well as complex interferometer topologies, containing subsystems that involve intensity dependent phase shifts, i.e., optical Kerr media. As the result we present the noise spectral density of a Michelson interferometer with Kerr nonlinear arm cavities and resonant sideband extraction and find that quantum noise can be squeezed by arbitrary amounts even outside the cavity linewidth. Such a system might apply for future gravitational wave detectors or simply for a continuous wave source of squeezed states.
We present the optical transfer functions for third-order nonlinear cavities that involve an optical carrier frequency and its modulation sideband fields. Our approach is based on linearized transformations and provides a convenient tool to calculate squeezed light sources as well as complex interferometer topologies, containing subsystems that involve intensity dependent phase shifts, i.e., optical Kerr media. As the result we present the noise spectral density of a Michelson interferometer with Kerr nonlinear arm cavities and resonant sideband extraction and find that quantum noise can be squeezed by arbitrary amounts even outside the cavity linewidth. Such a system might apply for future gravitational wave detectors or simply for a continuous wave source of squeezed states.
Quantum noise is one of the limiting factors in laser-interferometric gravitational-wave (GW) detectors. The application of squeezed states in these interferometers allows the reduction of quantum noise in one quadrature. Due to opto-mechanical coupling in a GW detector the squeezed quadrature needs to be rotated within the spectrum to achieve a broadband noise reduction. So far, the implementation of additional filter cavities is considered that allow for the optimal, frequency-dependent rotation of the squeezed quadrature. However, these cavities need to have low loss, a length in the order of \unit[100]{m} and must be situated in the vacuum system, making them cost-intensive. In 2017, Ma and coworkers proposed a scheme for the broadband quantum-noise reduction without the need of additional filter cavities. It was shown by Brown et al. that a similar scheme can be used to broadband-enhance interferometers with a detuned signal-recycling cavity. Here, we performed a proof-of-principle experiment of the proposal on a table-top-scale. Squeezed states were produced detuned to the carrier field of a \unit[2.5]{m}-linear cavity and read out in a bichromatic homodyne detection. The frequencies of the lower and upper local oscillator were at entangled sidebands of the squeezed field. Depending on the relations between the involved frequencies, we can address both variants of the proposal. We show, that the frequency-dependences of the resulting noise spectra fit to a theoretical model we derived from the theory used by Ma et al. With this work we set the path towards an implementation of these schemes in a GW-detector prototype, where the compatibility of the approach with a low-frequency suspended interferometer can be tested. Moreover, we used the same setup to show nonclassical interferometer enhancement at low frequencies by high-frequency squeezed states. Here, a heterodyne readout scheme was implemented to avoid limiting noises at low frequencies. The application of squeezed states centered around the local oscillator frequency yielded an improvement in signal-to-noise ratio of $\unit[3.4]{dB}\pm\unit[0.3]{dB}$. Additionally, I designed, built and characterized a compact source of squeezed vacuum-states at \unit[1064]{nm} with a footprint of just $\unit[0.8]{m^2}$. I show measurements of squeezed states from this source with a reduction of quantum noise of $\unit[10.7]{dB}\pm\unit[0.2]{dB}$ below the vacuum noise and present a noise reduction in the frequency range from \unit[70]{kHz} to \unit[65]{MHz}.
In optical fibers, nonlinear dispersion due to the same third-order susceptibility as that responsible for parametric mixing significantly alters the squeezing behavior in a traveling wave geometry, giving the squeezed quadrature an intensity dependent phase and reducing the overall squeezing. Phase noise is not squeezed at all. In the case of a fiber ring resonator, the nonlinear susceptibility will also lead to dispersive optical bistability and the squeezing will be strongly dependent on pump and sideband detuning. Yurke1 has analyzed squeezing due to backward four-wave mixing in a cavity, but the pump wave did not circulate in the cavity, and bistability was not dealt with. We have developed a theory of the resonant fiber ring interferometer, where the pump wave is near resonance with an interferometer mode and the circulating power is thereby enhanced. For perfect pump resonance and for frequencies corresponding to sideband resonance, the overall enhancement of the squeezing in the output mode compared to the traveling wave case is proportional to the square of the resonator finesse. The best squeezing in the output mode is always found for this condition of sideband resonance. The phase of the minimum noise quadrature is quite sensitive to both pump and sideband detuning. This feature may provide a means for avoiding excess thermal phase noise due to guided acoustic wave Brillouin scattering in the fiber itself. At the critical points for dispersive optical bistability in such a resonator and for sideband resonance (i.e., noise frequencies which are multiples of the resonator free spectral range), perfect squeezing of the output light is predicted. This result was also obtained previously for the case of degenerate mixing in a resonant cavity by Collett and Walls.2 Our current theory does not account for the reduction in squeezing due to the loss present in such a resonator.2 Since the loss is also enhanced by a factor of the resonator finesse, this is of some concern and will limit the maximum useful finesse for quantum noise squeezing. The major source of loss is the single-mode directional coupler, which for currently available technology is of the order of a few percent.
One of the most promising applications for squeezed light is interferometry beyond the shot-noise limit (SNL). To date, only squeezed light generated by parametric down-conversion has been used to improve interferometer sensitivity. Remarkable results have been achieved with this approach, including applications to large-scale gravitational wave detectors. On the other hand, there exists a potentially more robust way to generate squeezed light, that is, using the optical Kerr effect. It occurs almost for free in optical fibers and requires no phase matching. However, no interferometer sensitivity enhancement has been demonstrated so far using this method. One of the reasons for that is that the uncertainty distribution of a Kerr-squeezed state in phase space is tilted with respect to the amplitude or phase quadratures. Additional obstacles include Raman and guided acoustic wave Brillouin scattering in fibers.
Kerr microresonators are integrated nonlinear devices widely used to generate classical optical frequency combs and nonclassical states of light. In particular, they provide a promising platform for squeezed light sources, which are important elements of emerging quantum sensing, communication, and computing technologies. Designing such sources requires numerical tools that efficiently incorporate realistic device physics and noise processes. Here, we show that established semi-classical frameworks based on the Lugiato-Lefever equation and the Ikeda map, instrumental in the development of classical light sources, can also predict the quadrature-noise properties of squeezed light when vacuum fluctuations are properly included. We consider three canonical configurations producing two-mode quadrature squeezing, single-mode quadrature squeezing, and intensity difference squeezing. In each case, the semi-classical simulations produce quadrature variances and noise spectra in close agreement with the quantum theoretical predictions. These frameworks therefore provide practical tools for designing squeezed light sources, with the capability to incorporate additional noise mechanisms, nonlinear effects, and complex resonator arrangements. More fundamentally, our results show that key experimental signatures of squeezing can be modeled as phase-sensitive amplification of properly normalized vacuum fluctuations within stochastic semi-classical field equations. This observation helps distinguish squeezing properties accessible through semi-classical field evolution from those intrinsically requiring a full quantum description.
A nonlinear interferometer uses nonlinear elements as beam splitters to split and to recombine optical waves for interference. As a result, the interference fringe size has a nonlinear dependence on the intensity of the field for phase sensing and leads to an enhanced phase signal. In this paper, a practical scheme of nonlinear interferometry for precision phase measurement is analyzed with parametric amplifiers as the nonlinear beam splitters. It is found that the signal due to phase shift is enhanced by a factor of the amplification gain as compared to a linear interferometer with the same phase-sensing light intensity while the quantum noise is kept at the vacuum level, thus, effectively increasing the signal-to-noise ratio (SNR) beyond the standard quantum limit. Furthermore, the scheme is not as sensitive to the detection loss as the linear scheme with a squeezed state for noise reduction. However, losses inside the interferometer limit the enhancement factor in SNR. We apply the concept to a Michelson interferometer but with parametric amplifiers involved for gravitational-wave detection. We find that effective power is increased by the gain of the amplifiers without actually increasing the cycling power inside the interferometer. Furthermore, the full benefits with squeezed input and variational output or the combination of a quantum nondemolition interferometer for sensitivity beyond the standard quantum limit apply here with even better results. Such a nonlinear interferometer will find wide applications in precision measurements.
We theoretically analyze the noise transformation of a high-power continuous-wave light field that is reflected off a critical Kerr nonlinear cavity (KNLC). Our investigations are based on a rigorous treatment in the time domain. Thereby, realistic conditions of a specific experimental environment including optical intracavity loss and strong classical driving noise can be modeled for any KNLC. We show that, even in the presence of optical loss and driving noise, considerable squeezing levels can be achieved. We find that the achievable squeezing levels are not limited by the driving noise but solely by the amount of optical loss. Amplitude-quadrature squeezing of the reflected mean field is obtained if the KNLC's operating point is chosen properly. Consistently, a KNLC can provide a passive purely optical reduction of laser-power noise as experimentally demonstrated in Khalaidovski et al. [Phys. Rev. A 80, 053801 (2009)]. We apply our model to this experiment and find good agreement with measured noise spectra.