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Physical Review A

Degenerate four-wave mixing as a possible source of squeezed-state light

Roy S. Bondurant, Prem Kumar, Jeffrey H. Shapiro, Mari W. Maeda

Published Jul 1, 1984• 130 citations
Publisher

Abstract

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 ${g}_{2}$ 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 ${g}_{2}$ 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.

DOI
10.1103/physreva.30.343

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Quantum theory of nondegenerate multiwave mixing III Application to single-beam squeezed-state generation

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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.

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In this study, a semiclassical version of the degenerate four-wave mixing (DFWM) solution without slowly varying amplitude (SVA) approximation is investigated. The quadrature fluctuations of the output modes exhibit an oscillatory behavior with respect to that of the corresponding SVA modes. Furthermore, the variation of the two quadrature fluctuations of the combined mode, plotted against the normalized coupling constant κL, oscillates. Beyond a certain value of κL, the two quadratures oscillate in opposite directions, which implies that the squeezing effect of DFWM without the SVA approximation oscillates around that of the SVA. In addition, the squeezing effect in DFWM without SVA approximation depends on the refractive index of the medium and increases sharply upon the increase of the latter. This study also shows that the super-Poissonian behavior of the conjugate mode is preserved but is oscillatory. PACS No.: 42.65.Hw

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Single-beam squeezing in a Doppler broadened medium

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We have generated squeezed states of light with a single pump beam propagating through Doppler broadened sodium vapor via forward four-wave mixing (FWM) with spatially degenerate pump and probe modes. The two probe-mode inputs are vacuum states at nondegenerate frequencies, which are transformed to squeezed vacuum by the FWM interaction. Compared with previous FWM configurations,1 this single-pass single-pump-beam configuration ensures maximal overlap of the four interacting beams, thus maximizing the squeezing nonlinearity and improving the overall efficiency. The transmitted pump has the wrong phase to act as the local oscillator for detecting the squeezed spectrum.1 Therefore, a cavity is used to filter out most of the transmitted pump intensity, and the squeezed vacuum is detected with another local oscillator using the standard dual-detector scheme. The residual pump beam provides a reference to monitor the phase between the pump and squeezed vacuum. The observed 0.8-dB reduction in the shot noise corresponds to an inferred 50% squeezing of the light at the exit of the vapor medium, which is in reasonable agreement with the theory given by Ho et al. 2 A linearly polarized pump is found to give less squeezing than a circularly polarized pump, possibly due to optical pumping.

Annual Meeting Optical Society of America•Jan 1, 1988•0 citations
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.
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.
Degenerate four-wave mixing has been suggested as a possible generation scheme for squeezed-state light. A recent analysis of the quantum effects of probe-conjugate loss in backward degenerate four-wave mixing has shown that such loss puts an absolute limit on the squeezing that can be obtained via this generation scheme. In this Rapid Communication we show that it is the counter-propagating beam geometry of backward degenerate four-wave mixing that makes it ill suited for squeezed-state generation. On the other hand, the nominally copropagating beam geometry of forward degenerate four-wave mixing is shown to alleviate the absolute probe-conjugate loss limit on squeezing.
We study phase-conjugate degenerate four-wave mixing (DFWM) in an optically pumped, two-level atomic system (the 3s2S1∕23s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}3s2S1∕2​, F=2F=2F=2, ${m}_{F}=2\ensuremath{\rightarrow}3p\phantom{\rule{0.2em}{0ex}}^{2}P_{3∕2}$, F=3F=3F=3, mF=3{m}_{F}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium) using phase-modulated laser beams. In this investigation, we study the detailed line shape of the four-wave mixing signal as a function of the temporal delay between the probe field and the pump fields at different intensities of the cw pump waves and modulation frequencies. We show that phase modulation of the pump and probe beams leads to strong amplitude modulation of the phase-conjugate beam and report excellent agreement between our experimental observations of the DFWM spectra and the results of numerical calculations. Our numerical results suggest an enhancement of the ``local'' DFWM signal when the sidebands of the phase-modulated laser are resonant with dressed-state transitions, but that this resonance is less obvious in the total signal generated. We also show that the complex line shapes depend sensitively upon the delay of the probe beam, with the peak DFWM signal occurring when the probe beam is advanced compared to the pump beam.
Degenerate four-wave mixing (DFWM) in optical fibres can be used to obtain gain and to generate squeezed states, by utilizing a balanced Mach-Zehnder fibre interferometer (MZFI) with two non-linear arms, which behaves as a degenerate parametric amplifier. In a classical derivation for the optical gain; pump depletion and propagation loss are considered. Maximum achievable gain is calculated in the presence of loss. A numerical study compares the signal gain under different input pump-to-signal field ratios and its limit (undepleted pump), which is the assumption for the derivation of squeezing. Pump depletion and fibre loss hinder the validity of using large non-linear phase shifts to generate squeezed states. A semiclassical derivation provides the correct expression for the degree of squeezing in the absence of loss and pump depletion. Quantum-mechanical solutions provide the same result. The squeezing limitation due to loss is studied by a lumped-parameter model.
In this study, a semiclassical version of the degenerate four-wave mixing (DFWM) solution without slowly varying amplitude (SVA) approximation is investigated. The quadrature fluctuations of the output modes exhibit an oscillatory behavior with respect to that of the corresponding SVA modes. Furthermore, the variation of the two quadrature fluctuations of the combined mode, plotted against the normalized coupling constant κL, oscillates. Beyond a certain value of κL, the two quadratures oscillate in opposite directions, which implies that the squeezing effect of DFWM without the SVA approximation oscillates around that of the SVA. In addition, the squeezing effect in DFWM without SVA approximation depends on the refractive index of the medium and increases sharply upon the increase of the latter. This study also shows that the super-Poissonian behavior of the conjugate mode is preserved but is oscillatory. PACS No.: 42.65.Hw
We have generated squeezed states of light with a single pump beam propagating through Doppler broadened sodium vapor via forward four-wave mixing (FWM) with spatially degenerate pump and probe modes. The two probe-mode inputs are vacuum states at nondegenerate frequencies, which are transformed to squeezed vacuum by the FWM interaction. Compared with previous FWM configurations,1 this single-pass single-pump-beam configuration ensures maximal overlap of the four interacting beams, thus maximizing the squeezing nonlinearity and improving the overall efficiency. The transmitted pump has the wrong phase to act as the local oscillator for detecting the squeezed spectrum.1 Therefore, a cavity is used to filter out most of the transmitted pump intensity, and the squeezed vacuum is detected with another local oscillator using the standard dual-detector scheme. The residual pump beam provides a reference to monitor the phase between the pump and squeezed vacuum. The observed 0.8-dB reduction in the shot noise corresponds to an inferred 50% squeezing of the light at the exit of the vapor medium, which is in reasonable agreement with the theory given by Ho et al. 2 A linearly polarized pump is found to give less squeezing than a circularly polarized pump, possibly due to optical pumping.