Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic

Transmission estimation at the quantum Cramér-Rao bound with macroscopic quantum light

The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.

Optimal bright multimode quantum squeezing via multi-seeding energy-level cascaded four-wave mixing

Quantum Squeezing is one of the most important quantum resources in quantum optics and quantum information. In particular, multimode quantum squeezing, with ultra-low quantum fluctuations and quantum correlations amongst many optical modes, is essential for realizing multipartite entanglement and quantum precision measurements. In this paper, we propose an all-optically controlled scheme to generate three-mode bright quantum correlated beams from energy-level cascaded four-wave mixing (ELC-FWM). By using a linear modes transform approach, the input-output relation and the covariance matrix of the produced states are obtained. Moreover, single-, double- and triple-seeding conditions are investigated to measure the quantum squeezing properties. We find that various permutations of two- and three-mode quadrature squeezing can be generated and optimized to reach the corresponding limit, via only modulating the ratio of the multiple seeds, without need of any post-operating linear optics, e.g., beam splitters. Such weak seeding light controlled scheme suggests the modulation and the optimization of multimode quantum states might be operated at photons-level, providing a reconfigurable and integrated strategy for complex quantum information processing and quantum metrology.

Multi-Way Noiseless Signal Amplification in a Symmetrical Cascaded Four-Wave Mixing Process

According to the fundamental laws of quantum optics, vacuum noise is inevitably added to the signal when one tries to amplify a signal. However, it has been recently shown that noiseless signal amplification can be realized when a phase-sensitive process is involved. Two phase-sensitive schemes, a correlation injection scheme and a two-beam phase-sensitive amplifier scheme, are both proposed to realize multi-way noiseless signal amplification in a symmetrical cascaded four-wave mixing process. We theoretically study the possibility of the realization of four-way noiseless signal amplification by using these two schemes. The results show that the correlation injection scheme can only realize one-way noiseless signal amplification, but that the two-beam phase-sensitive amplifier scheme can lead to four-way noise figure values below 1. Our results here may find potential applications in quantum information processing, e.g., the realization of quantum information tap and quantum non-demolition measurement, etc.

All-Optical Entanglement Swapping

Entanglement swapping, which is a core component of quantum network and an important platform for testing the foundation of quantum mechanics, can enable the entangling of two independent particles without direct interaction both in discrete variable and continuous variable systems. Conventionally, the realization of entanglement swapping relies on the Bell-state measurement. In particular, for entanglement swapping in continuous variable regime, such Bell-state measurement involves the optic-electro and electro-optic conversion, which limits the applications of the entanglement swapping for constructing broadband quantum network. In this Letter, we propose and demonstrate a measurement-free all-optical entanglement swapping. In our scheme, a high-gain parametric amplifier based on the four-wave mixing process is exploited to realize the function of Bell-state measurement without detection, which avoids the introduction of the optic-electro and electro-optic conversion. Our results provide an all-optical paradigm for implementing entanglement swapping and pave the way to construct a measurement-free all-optical broadband quantum network.

Multidimensional four-wave mixing signals detected by quantum squeezed light

Quantum light and its statistics provide powerful tools for the study of properties of matter that are difficult to retrieve with classical light. Novel spectroscopic and sensing techniques based on quantum light sources can reveal information about complex material systems that is not accessible by varying the frequencies or time delays of classical light pulses. Here, based on a four-wave mixing process, we report an experimental study of the 2D quantum noise spectra of two-beam intensity difference squeezing. External noise erodes the resolution of classical measurements, while quantum signals remain intact. Our results pave the way for exploiting quantum correlations of squeezed light for spectroscopic applications.

Four-wave mixing (FWM) of optical fields has been extensively used in quantum information processing, sensing, and memories. It also forms a basis for nonlinear spectroscopies such as transient grating, stimulated Raman, and photon echo where phase matching is used to select desired components of the third-order response of matter. Here we report an experimental study of the two-dimensional quantum noise intensity difference spectra of a pair of squeezed beams generated by FWM in hot Rb vapor. The measurement reveals details of the χ(3) susceptibility dressed by the strong pump field which induces an AC Stark shift, with higher spectral resolution compared to classical measurements of probe and conjugate beam intensities. We demonstrate how quantum correlations of squeezed light can be utilized as a spectroscopic tool which unlike their classical counterparts are robust to external noise.

Quantum Plasmonic Sensors

The extraordinary sensitivity of plasmonic sensors is well-known in the optics and photonics community. These sensors exploit simultaneously the enhancement and the localization of electromagnetic fields close to the interface between a metal and a dielectric. This enables, for example, the design of integrated biochemical sensors at scales far below the diffraction limit. Despite their practical realization and successful commercialization, the sensitivity and associated precision of plasmonic sensors are starting to reach their fundamental classical limit given by quantum fluctuations of light-known as the shot-noise limit. To improve the sensing performance of these sensors beyond the classical limit, quantum resources are increasingly being employed. This area of research has become known as "quantum plasmonic sensing", and it has experienced substantial activity in recent years for applications in chemical and biological sensing. This review aims to cover both plasmonic and quantum techniques for sensing, and it shows how they have been merged to enhance the performance of plasmonic sensors beyond traditional methods. We discuss the general framework developed for quantum plasmonic sensing in recent years, covering the basic theory behind the advancements made, and describe the important works that made these advancements. We also describe several key works in detail, highlighting their motivation, the working principles behind them, and their future impact. The intention of the review is to set a foundation for a burgeoning field of research that is currently being explored out of intellectual curiosity and for a wide range of practical applications in biochemistry, medicine, and pharmaceutical research.

Nonlinear interferometric surface-plasmon-resonance sensor

A nonlinear interferometer can be constructed by replacing the beam splitter in the Mach-Zehnder interferometer with four-wave mixing (FWM) process. Meanwhile, the conventional surface plasmon resonance (SPR) sensors can be extensively used to infer the information of refractive index of the sample to be measured via either angle demodulation technique or intensity demodulation technique. Combined with a single FWM process, a quantum SPR sensor has been realized, whose noise floor is reduced below standard quantum limit with sensitivity unobtainable with classical SPR sensor. Therefore, in this work we have theoretically proposed a nonlinear interferometric SPR sensor, in which a conventional SPR sensor is placed inside nonlinear interferometer, which is called as I-type nonlinear interferometric SPR sensor. We demonstrate that near resonance angle I-type nonlinear interferometric SPR sensor has the following advantages: its degree of intensity-difference squeezing, estimation precision ratio, and signal-noise-ratio are improved by the factors of 4.6 dB, 2.3 dB, and 4.6 dB respectively than that obtained with a quantum SPR sensor based on a single FWM process. In addition, the theoretical principle of this work can also be expanded to other types of sensing, such as bending, pressure, and temperature sensors based on a nonlinear interferometer.

Characterization of quantum squeezing generated from the phase-sensitive and phase-insensitive amplifiers in the ultra-low average input photon number regime

We give the general expressions of intensity-difference squeezing (IDS) generated from two types of optical parametric amplifiers [i.e. phase-sensitive amplifier (PSA) and phase-insensitive amplifier (PIA)] based on the four-wave mixing process, which clearly shows the IDS transition between the ultra-low average input photon number regime and the ultra-high average input photon number regime. We find that both the IDS of the PSA and the IDS of the PIA get enhanced with the decrease of the average input photon number especially in the ultra-low average input photon number regime. This result is substantially different from the result in the ultra-high average input photon number regime where the IDS does not vary with the average input photon number. Moreover, under the same intensity gain, we find that the optimal IDS of the PSA is better than the IDS of the PIA in the ultra-low average input photon number regime. Our theoretical work predicts the presence of strong quantum correlation in the ultra-low average input photon number regime, which may have potential applications for probing photon-sensitive biological samples.

Observation of the interplay between seeded and self-seeded nondegenerate four-wave mixing in cesium vapor

Nondegenerate four-wave mixing (NFWM) is a practical and effective technique for generating or amplifying light fields at different wavelengths, and could be used to create color correlation and entanglement. Here we experimentally investigate the NFWM process in diamond atomic system via two-photon excitation with two pumps at 852 nm and 921 nm, demonstrating that a seeded NFWM with a third laser at 895 nm and two self-seeded NFWMs due to amplified spontaneous emission (ASE) occur simultaneously. We compare the two kinds of processes and show that the single- and two-photon detunings hold the key role in distinguishing them. As a result, the enhancement of seeded NFWM is obtained by selecting large one- and two-photon detunings, in which case the ASE induced self-seeded NFWM can be largely suppressed. In contrast, the ASE and its induced NFWM are effectively achieved with one- and two-photon resonant excitations allowing for population inversion for efficient ASE.

Tripartite Einstein-Podolsky-Rosen steering with linear and nonlinear beamsplitters in four-wave mixing of Rubidium atoms

Multipartite Einstein-Podolsky-Rosen (EPR) steering is an essential resource for secure one-sided device-independent quantum secret sharing. Here, we analyze the EPR steering properties exhibited in three-mode Gaussian states created by four-wave mixing (FWM) in Rubidium atoms combined with a linear beamsplitter and a nonlinear beamsplitter (second FWM), respectively. By quantifying Gaussian steerability based on a measure determined by the covariance matrix of the produced states, we compare the performance of two schemes to achieve one-way, collective, and genuine tripartite steering, as well as the monogamy constraints for distributing steering among three parties. We show that the scheme with nonlinear beamsplitter is feasible to create stronger bipartite steering and genuine tripartite steering and has more flexibility to manipulate the monogamy relation through the cooperation of the two cascaded FWM processes.

Attenuation-free non-diffracting Bessel beams

We report on a versatile method to compensate the linear attenuation in a medium, independently of its microscopic origin. The method exploits diffraction-limited Bessel beams and tailored on-axis intensity profiles, which are generated using a phase-only spatial light modulator. This technique for compensating one of the most fundamental limiting processes in linear optics is shown to be efficient for a wide range of experimental conditions (modifying the refractive index and the attenuation coefficient). Finally, we explain how this method can be advantageously exploited in applications ranging from bio-imaging light sheet microscopy to quantum memories for future quantum communication networks.

Interference-Induced Quantum Squeezing Enhancement in a Two-beam Phase-Sensitive Amplifier

We experimentally demonstrate a method for realizing quantum squeezing enhancement which is induced by the interference in a two-beam phase-sensitive amplifier (PSA) based on a four-wave mixing process. Compared to the normal phase-insensitive amplifier with an intensity-difference squeezing (IDS) of 8.97±0.24  dB or 8.76±0.26  dB, the IDS of our two-beam PSA is enhanced to 10.13±0.21  dB under the same experimental situation. Furthermore, we study how various parameters influence the quantum squeezing enhancement of the PSA. These results clearly show that the physical mechanism inducing the IDS enhancement of the two-beam PSA is its intrinsic interference nature. Our results may find potential applications in improving the fidelity of quantum information processing and the precision of quantum metrology.

Quantum beam splitter for orbital angular momentum of light: quantum correlation by four-wave mixing operated in a nonamplifying regime

Nondegenerate four-wave mixing (FWM) process based on a double-Λ scheme in hot alkali metal vapor is a versatile tool in quantum state engineering, quantum imaging, and quantum precision measurements. In this Letter, we investigate the generation of quantum correlated twin beams which carry nonzero orbital angular momentums (OAMs) based on the FWM process in hot cesium vapor. The amplified probe beam and the newly generated conjugate beam in the FWM process have the same and opposite topological charge as the seed beam, respectively. We also explore the FWM process operated in a nonamplifying regime where quantum correlated twin beams carrying OAMs can still be generated. In this regime, the FWM process plays the role of quantum beam splitter for the OAM of light; that is, a device that can split a coherent light beam carrying OAM into quantum-correlated twin beams carrying OAMs. More generally, our setup can be used as a quantum beam splitter of images.

Beam Focusing and Reduction of Quantum Uncertainty in Width at the Few-Photon Level via Multi-Spatial-Mode Squeezing

We show for the first time that it is possible to realize laser beam focusing at the few-photon level in the four-wave-mixing process, and at the same time reduce the quantum uncertainty in width. The reduction in quantum uncertainty results directly from the strong suppression of local intensity fluctuations. This surprising effect of simultaneous focusing and reduction of width uncertainty is enabled by multi-spatial-mode (MSM) squeezing, and is not possible via any classical optical approach or single-spatial-mode squeezing. Our results open promising possibilities for quantum-enhanced imaging and metrology; as an example, the limit on the measurement of very small beam displacement can be enhanced within feasible experimental parameters because of beam focusing and the noiseless amplification in the MSM squeezing process.

Improved measurement of two-mode quantum correlations using a phase-sensitive amplifier

We demonstrate the ability of a phase-sensitive amplifier (PSA) to pre-amplify a selected quadrature of one mode of a two-mode squeezed state in order to improve the measurement of two-mode quantum correlations that exist before degradation due to optical and detection losses. We use four-wave mixing (4WM) in ⁸⁵Rb vapor to generate bright beams in a two-mode squeezed state. One of these two modes then passes through a second 4WM interaction in a PSA configuration to noiselessly pre-amplify the desired quadrature of the mode before loss is intentionally introduced. We demonstrate an enhancement in the measured degree of intensity correlation and intensity-difference squeezing between the two modes.

Quantum steering in cascaded four-wave mixing processes

Quantum steering is used to describe the "spooky action-at-a-distance" nonlocality raised in the Einstein-Podolsky-Rosen (EPR) paradox, which is important for understanding entanglement distribution and constructing quantum networks. Here, in this paper, we study an experimentally feasible scheme for generating quantum steering based on cascaded four-wave-mixing (FWM) processes in hot rubidium (Rb) vapor. Quantum steering, including bipartite steering and genuine tripartite steering among the output light fields, is theoretically analyzed. We find the corresponding gain regions in which the bipartite and tripartite steering exist. The results of bipartite steering can be used to establish a hierarchical steering model in which one beam can steer the other two beams in the whole gain region; however, the other two beams cannot steer the first beam simultaneously. Moreover, the other two beams cannot steer with each other in the whole gain region. More importantly, we investigate the gain dependence of the existence of the genuine tripartite steering and we find that the genuine tripartite steering exists in most of the whole gain region in the ideal case. Also we discuss the effect of losses on the genuine tripartite steering. Our results pave the way to experimental demonstration of quantum steering in cascaded FWM process.

Hybrid interferometer with nonlinear four-wave mixing process and linear beam splitter

Optical interferometer has played an important role in optics. Up to now, many kinds of interferometers have been realized and found their applications. In this letter, we experimentally construct an interferometer by using parametric amplifier as a wave splitter and beam splitter as a wave combiner. We make measurements of interference fringes and explore the relationships between the interference visibility of the interferometer and various system parameters, such as the gain of the parametric amplifier, the one-photon detuning and the temperature of the Rb-85 vapor cell.

Nonlinear Sagnac interferometer based on the four-wave mixing process

A new nonlinear Sagnac interferometer (NSI) is proposed by replacing the beam-splitter in the traditional Sagnac interferometer (TSI) with a four-wave mixing process. Such a NSI has better angular velocity sensitivity than the one of the TSI. The standard quantum limit can be beaten and the Heisenberg Limit can even be reached for the ideal case by the NSI. We study the effect of the losses on the angular velocity sensitivity of the NSI and find that the optimal angular velocity, where the best angular velocity sensitivity can be obtained, of the NSI may be dependent on the losses inside the interferometer. Such a NSI has its advantages compared with the TSI and may find its potential applications in quantum metrology.

Characterization of Pairwise Correlations from Multiple Quantum Correlated Beams Generated from Cascaded Four-Wave Mixing Processes

We theoretically characterize the performance of the pairwise correlations (PCs) from multiple quantum correlated beams based on the cascaded four-wave mixing (FWM) processes. The presence of the PCs with quantum corre- lation in these systems can be verified by calculating the degree of intensity difference squeezing for any pair of all the output fields. The quantum correlation characteristics of all the PCs under different cascaded schemes are also discussed in detail and the repulsion effect between PCs in these cascaded FWM processes is theoretically predicted. Our results open the way for the classification and application of quantum states generated from the cascaded FWM processes.

High conversion efficiency in resonant four-wave mixing processes

We propose a new scheme of the resonant four-wave mixing (FWM) for the frequency up or down conversion, which is more efficient than the commonly-used scheme of the non-resonant FWM. In this new scheme, two control fields are spatially varied such that a probe field at the input can be converted to a signal field at the output. The efficiency of probe-to-signal energy conversion can be 90% at medium's optical depth of about 100. Our proposed scheme works for both the continuous-wave and pulse cases, and is flexible in choosing the control field intensity. This work provides a very useful tool in the nonlinear frequency conversion.

Dressed Gain from the Parametrically Amplified Four-Wave Mixing Process in an Atomic Vapor

With a forward cone emitting from the strong pump laser in a thermal rubidium atomic vapor, we investigate the non-degenerate parametrically amplified four-wave mixing (PA-FWM) process with dressing effects in a three-level “double-Λ” configuration both theoretically and experimentally. By seeding a weak probe field into the Stokes or anti-Stokes channel of the FWM, the gain processes are generated in the bright twin beams which are called conjugate and probe beams, respectively. However, the strong dressing effect of the pump beam will dramatically affect the gain factors both in the probe and conjugate channels, and can inevitably impose an influence on the quantum effects such as entangled degree and the quantum noise reduction between the two channels. We systematically investigate the intensity evolution of the dressed gain processes by manipulating the atomic density, the Rabi frequency and the frequency detuning. Such dressing effects are also visually evidenced by the observation of Autler-Townes splitting of the gain peaks. The investigation can contribute to the development of quantum information processing and quantum communications.

Parametric amplification and cascaded-nonlinearity processes in common atomic system

For the first time, we study the parametric amplification process of multi-wave mixing (PA-MWM) signal and cascaded-nonlinearity process (CNP) in sodium vapors both theoretically and experimentally, based on a conventional phase-conjugate MWM and a self-diffraction four-wave mixing (SD-FWM) processes, which are pumped by laser or amplified spontaneous emission (ASE), respectively. For laser pumping case, SD-FWM process serves as a quantum linear amplifier (a CNP) out (inside) of the resonant absorption region. While for ASE case, only the CNP occurs and the output linewidth is much narrower than that of the MWM signal due to the second selected effect of its electromagnetically induced transparency window. In addition, the phase-sensitive amplifying process seeded by two MWM processes is discussed for the first time. Theoretical fittings agree well with the experiment. The investigations have important potential applications in quantum communication.

Compact diode-laser-pumped quantum light source based on four-wave mixing in hot rubidium vapor

Using a nondegenerate four-wave mixing process in hot rubidium vapor, we demonstrate a compact diode-laser-pumped system for the generation of intensity-difference squeezing down to 8 kHz with a maximum squeezing of -7 dB. To the best of our knowledge, this is the first demonstration of kilohertz-level intensity-difference squeezing using a semiconductor laser as the pump source. This scheme is of interest for experiments involving atomic ensembles, quantum communications, and precision measurements. The diode-laser-pumped system would extend the range of possible applications for squeezing due to its low cost, ease of operation, and ease of integration.