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Scientific Reports

Generation of single-mode and two-mode quantum squeezed states of light by degenerate four-wave mixing in a plasmonic waveguide

Abbas Ghasempour Ardakani, Hossein Bornak

Published May 28, 2026• 0 citations
Publisher

Abstract

Squeezed states of light represent one of the most fundamental concepts in quantum optics. They play a crucial role in various fields of science, quantum technology, and photonics, with applications in quantum information processing, quantum computation, quantum sensing, and gravitational wave detection. In this paper, we propose a plasmonic waveguide for the generation of squeezed states via four-wave mixing. The designed waveguide consists of two gold layers deposited on a SiO 2 substrate. DDMEBT, an organic material with a high third-order nonlinear refractive index, is placed on the gold layers and fills the gap between them. To determine the effective refractive indices and corresponding mode wavelengths, the waveguide was simulated using the Mode Analysis module of COMSOL Multiphysics. The coupled equations describing degenerate four-wave mixing for the pump, signal, and idler fields were derived. Furthermore, we present the quantum mechanical formulation of four-wave mixing in terms of expectation values and fluctuations, introducing both single-mode and two-mode squeezing factors. The influence of pump power on the squeezing factor is analyzed, revealing that higher pump powers enhance the squeezing level and reduce the required propagation length to reach maximum squeezing. Compared with previous studies based on second-harmonic generation, the required waveguide length to achieve maximum squeezing in our proposed structure is reduced by three order of magnitude. The detrimental effects of optical losses on the squeezing process are also investigated. Finally, the impact of the interlayer gap width on the squeezing factor is examined, demonstrating that increasing the gap between the gold layers leads to a degradation in the squeezing quality.

DOI
10.1038/s41598-026-45164-2

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Squeezed states of light represent one of the most fundamental concepts in quantum optics. They play a crucial role in various fields of science, quantum technology, and photonics, with applications in quantum information processing, quantum computation, quantum sensing, and gravitational wave detection. In this paper, we propose a plasmonic waveguide for the generation of squeezed states via four-wave mixing. The designed waveguide consists of two gold layers deposited on a SiO 2 substrate. DDMEBT, an organic material with a high third-order nonlinear refractive index, is placed on the gold layers and fills the gap between them. To determine the effective refractive indices and corresponding mode wavelengths, the waveguide was simulated using the Mode Analysis module of COMSOL Multiphysics. The coupled equations describing degenerate four-wave mixing for the pump, signal, and idler fields were derived. Furthermore, we present the quantum mechanical formulation of four-wave mixing in terms of expectation values and fluctuations, introducing both single-mode and two-mode squeezing factors. The influence of pump power on the squeezing factor is analyzed, revealing that higher pump powers enhance the squeezing level and reduce the required propagation length to reach maximum squeezing. Compared with previous studies based on second-harmonic generation, the required waveguide length to achieve maximum squeezing in our proposed structure is reduced by three order of magnitude. The detrimental effects of optical losses on the squeezing process are also investigated. Finally, the impact of the interlayer gap width on the squeezing factor is examined, demonstrating that increasing the gap between the gold layers leads to a degradation in the squeezing quality.
In this article, a full numerical method to study the squeezing procedure through second harmonic generation process is proposed. The method includes complex nonlinear coupling coefficient, phase mismatch, and pump depletion. Attention has been also paid to the effects of accumulated noises in this work. The final form of the numerical formula seems to be much simpler than the analytical solutions previously reported. The function of this numerical method shows that it works accurately for different mechanisms of squeezed state generations and does not suffer from instabilities usually encountered even for non-uniform, coarse steps. The proposed method is used to examine the squeezing procedure in an engineered nonlinear plasmonic waveguide. The results show that using the nonlinear plasmonic waveguide, it is possible to generate the squeezed states for the pump and the second harmonic modes with high efficiency in a propagation length as short as 2 mm which is much shorter than the needed length for the traditional nonlinear lithium niobate- based optical waveguides being of the order of 100 mm. This new method of squeezed states generation may find applications in optical communication with a noise level well below the standard quantum limit, in quantum teleportation, and in super sensitive interferometry.
Using a general Hamiltonian treatment, we theoretically study the generation of degenerate quadrature squeezing in a dual-pumped integrated microring resonator coupled to a waveguide. Considering a dual-pump four-wave mixing configuration in an integrated Si3N4{\text{Si}}_{3}{\text{N}}_{4}Si3​N4​ platform, and following the coupled-mode theory approach, we investigate the effects of parasitic quantum nonlinear optical processes on the generation of squeezed light. Considering five resonance modes in this approach allows us to include the most important four-wave mixing processes involved in such a configuration. We theoretically explore the effects of the pump detunings on different nonlinear processes and show that the effects of some of the parasitic processes are effectively neutralized by symmetrically detuning the two pumps. This yields a significant enhancement in the output squeezing quality without physically changing the structure, but suffers from the trade-off of requiring substantially higher pump power for a fixed target level of squeezing.
We apply our quantum theory of nondegenerate multiwave mixing [ Phys. Rev. A37, 2017 ( 1988)] to squeezed-state generation experiments with two-level atoms. Our main interest is to predict the amount of squeezing achievable with a Doppler-broadened two-level medium. We are particularly interested in the single-beam configuration, in which all four interacting beams are spatially degenerate. We analytically solve the coupled-mode quantum Langevin equations for nondegenerate four-wave mixing. The solutions are used to compute the amount of squeezing. In the computation the effects of pump–probe phase mismatch, collisions, Doppler broadening, and Gaussian-intensity variation are comprehensively taken into account for the first time to our knowledge. Simple rules of thumb as to where one can see squeezing in both degenerate- and nondegenerate-frequency cases are derived by examining the limit of a short medium. We then present the case of an infinitely long medium, in which maximum squeezing is achieved when there is no pump–probe phase mismatch. With the inclusion of pump–probe phase mismatch, however, the maximum amount of squeezing is obtained with a finite-length medium instead. This prompts us to investigate in detail the finite-length medium case. Our results show that the effects of Doppler broadening and Gaussian-intensity variation can be largely circumvented by detuning the pump frequency more than three Doppler half-widths from resonance and that good broadband squeezing can be achieved even with a Doppler-broadened medium that has a moderate amount of collision broadening. Under these circumstances it is found that the effect of pump self-focusing or defocusing will be the major factor that limits the amount of achievable squeezing. In particular, the spatially varying nonlinear refractive indices seen by the pump and the probe modes are quite different, which causes the former to become spatially mismatched with the latter in the region in which strong squeezing is otherwise expected.
We theoretically investigate the frequency-nondegenerate and frequency degenerate squeezed lights with a four-wave mixing process (4WM) driven by two pump fields crossing at a small angle. Different from a 4WM process driven by a single pump field, the refractive index of the corresponding probe field, n p, can be converted to a value that is greater than 1 or less than 1 by an angle adjustment. In the new region with n p < 1, the bandwidth of the gain is relatively large due to the slow change in the refractive index with the two-photon detuning. In this region with an exchange of the roles of the pump and probe beams, the frequency degenerate and spatial nondegenerate twin beams can be generated, which has potential application in quantum information and quantum metrology.
The suitability of degenerate four-wave mixing (DFWM) as a source of squeezed-state (two-photon coherent state) light is investigated, both theoretically and experimentally. In previous semiclassical theory, which neglected pump quantization and loss in the nonlinear medium, Yuen and Shapiro [Opt. Lett. 4, 334 (1979)] showed that such states would be generated by 50%-50% combination of the transmitted probe (TP) and the phase-conjugate (PC) output beams of DFWM. Here the effects of quantum fluctuations in the amplitudes and phases of the pump beams are calculated in a traveling-wave finite-interaction-length DFWM configuration. It is shown that in the limit of strong pump beams and weak nonlinear coupling, the classical-pumps assumption of Yuen and Shapiro is valid. With use of this assumption, loss in the nonlinear medium is incorporated into the model by coupling the interacting modes to a set of reservoir modes. It is shown that loss presents an absolute limit on the degree of squeezing that can be achieved with DFWM. Also, for comparison with experimental work, the photoelectron statistics of the TP, PC, and 50%-50% combination modes are calculated for lossy DFWM; significant deviations are found from the predictions for lossless operation. Quantum-noise measurements on the PC and TP beams generated in sodiumvapor DFWM are described. The mixer is pumped and probed by the nearly transform-limited pulses produced by a continuous-wave-oscillator-pulse-amplifier chain dye laser at 589 nm wave-length. The normalized second-factorial moment g2{g}_{2}g2​ of the PC and TP beams is determined from photoelectron-counting measurements. Inability to simultaneously achieve high DFWM reflectivity and acceptably low scattered-plus-fluorescent background, coupled with low photomultiplier quantum efficiency, restrict the experiment to a DFWM regime wherein theory predicts Poisson statistics. Experimental g2{g}_{2}g2​ values corroborate this prediction, indicating that systematic and excess noises have been eliminated. No attempt was made to measure the quantum statistics of the 50%-50% combination mode.
This paper discusses the possibility of using two-mode squeezed light to improve the performance of existing sensor technology with the focus on its miniaturization under realistic losses. Therefore, we analyze a system consisting of a part for two-mode squeezed light generation, a sensor region, and a detection stage. Based on a general four-wave mixing (FWM) Hamiltonian caused by third-order susceptibility, we formulate linearized equations that describe the FWM process below the threshold and are used to analyze the squeezing quality of the generated optical signal and idler modes. For a possible realization, the focus is set on chip-integrated generation using microring resonators. To do so, the impacts of the design and the pump light are considered in the derived equations. These equations are used to analyze the usage of two-mode squeezed light in quantum metrology and the application in a Mach-Zehnder interferometer. Due to the impact of losses in realistic use cases, we show that the main usage is for small and compact devices, which can lead to a quantum improvement of up to a factor of 10 in comparison with using coherent light only. This enables the use of small squeezing-enhanced sensors with a performance comparable to larger classical sensors.