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Annual Meeting Optical Society of America

Single-beam squeezing in a Doppler broadened medium

Seng‐Tiong Ho, N. C. Wong, Jeffrey H. Shapiro, Prem Kumar

Published Jan 1, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1988.tul1

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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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Annual Meeting Optical Society of America•Jan 1, 1986•0 citations

Single-beam squeezed-state generation in sodium vapor and its self-focusing limitations

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Observation of phase-sensitive noise in sodium vapor

Mari W. Maeda, Prem Kumar, Jeffrey H. Shapiro

Squeezed states of light exhibit phase-sensitive quantum noise measurable via optical homodyne detection. They have been predicted to be generated via degenerate four-wave mixing, resonance fluorescence, and nonlinear propagation.1 We report observing phase-sensitive noise via optical homodyning on a cw dye laser beam which has propagated through sodium vapor near the D lines. A p-polarized dye laser beam of ~5-W/cm2 intensity was passed through sodium vapor near its D lines. Homodyne measurements were made on both the s- and the p-polarized components of the transmitted beam using the dual-detector subtraction scheme.2 A spectrum analyzer monitored the noise in a 6–15-MHz band. In the absence of sodium, the noise in this band was vacuum-state quantum noise limited and was insensitive to the relative local-oscillator beam/transmitted laser beam phase θ. In the presence of sodium, the noise was sensitive to θ, with dependence of the form As,p + Bs,p cos2(θ − ϕs,p), where ϕs = 0 for s polarization and ϕp = π/2 for p polarization. For one set of data we measured As = 2 dB, Bs = 3 dB, Ap = 3 dB, and Bp = 2.5 dB above the noise levels in the absence of sodium. The s-poiarization data were collected with the local-oscillator power PLO far in excess of the s component of the transmitted pump power PTP; for p polarization, PLO ~ PTP prevailed. The measurement uncertainty in the spectrum analyzer data is ±0.5 dB. Possible physical mechanisms to explain these results are discussed.

Annual Meeting Optical Society of America•Jan 1, 1985•0 citations

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

Roy S. Bondurant, Prem Kumar, Jeffrey H. Shapiro +1 more

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.

Physical Review A•Jul 1, 1984•130 citations
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.
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.
Squeezed states of light exhibit phase-sensitive quantum noise which falls below the semiclassical shot noise limit. We report results from experiments in sodium vapor to observe squeezed light. In a single-beam experiment, a dye laser beam transmitted through sodium vapor was homodyned. Phase-sensitive noise on the radio frequency off­set modes was observed at or above the semiclassical limit.1 In a two-beam correlation experiment, the transmitted probe and the reflected con­jugate beams in forward four-wave mixing were direct detected. Although this configuration does not generate squeezed light, the subtracted photo­current exhibits subshot noise fluctuations.2 Phase-insensitive noise above the semiclassical limit was observed with a strong positive correlation between the probe and the conjugate photo­currents. In a squeezed state experiment, a probe-conjugate combination mode of forward four-wave mixing was homodyned.3 With the input beam blocked, the noise showed nonclassical phase-sensitivity with a minimum of 0.2 dB below the vacuum state limit. When only the probe or the conjugate beam was homodyned, the noise remained phase-insensitive near the vacuum state level.
We describe our efforts to observe squeezing, and related nonclassical effects, in three forward four-wave mixing experiments using sodium vapor as a nonlinear medium. In a single-beam experiment with homodyne detection, we found optically phase-sensitive noise, which was radio-frequency phase insensitive, in a cw dye-laser beam that had propagated through sodium vapor. The minima of the phase-sensitive noise were always at or above the coherent-state value. In our forward four-wave mixing experiment with probe–conjugate-beam direct detection and correlation, we found that the sodium-vapor interaction produced positive noise correlation between the probe and conjugate beams. The correlation, however, did not exceed the excess noises on these beams, so that a nonclassical behavior was not demonstrated. Finally, our forward four-wave mixing experiment with probe–conjugate-beam combination and homodyne detection did demonstrate the generation of squeezed-state light in our setup. Optically phase-sensitive noise with a minimum falling 4% below the coherent-state level was observed.
Squeezed states of light exhibit phase-sensitive quantum noise measurable via optical homodyne detection. They have been predicted to be generated via degenerate four-wave mixing, resonance fluorescence, and nonlinear propagation.1 We report observing phase-sensitive noise via optical homodyning on a cw dye laser beam which has propagated through sodium vapor near the D lines. A p-polarized dye laser beam of ~5-W/cm2 intensity was passed through sodium vapor near its D lines. Homodyne measurements were made on both the s- and the p-polarized components of the transmitted beam using the dual-detector subtraction scheme.2 A spectrum analyzer monitored the noise in a 6–15-MHz band. In the absence of sodium, the noise in this band was vacuum-state quantum noise limited and was insensitive to the relative local-oscillator beam/transmitted laser beam phase θ. In the presence of sodium, the noise was sensitive to θ, with dependence of the form As,p + Bs,p cos2(θ − ϕs,p), where ϕs = 0 for s polarization and ϕp = π/2 for p polarization. For one set of data we measured As = 2 dB, Bs = 3 dB, Ap = 3 dB, and Bp = 2.5 dB above the noise levels in the absence of sodium. The s-poiarization data were collected with the local-oscillator power PLO far in excess of the s component of the transmitted pump power PTP; for p polarization, PLO ~ PTP prevailed. The measurement uncertainty in the spectrum analyzer data is ±0.5 dB. Possible physical mechanisms to explain these results are discussed.
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.