Squeezed states of the electromagnetic field have been generated by nondegenerate four-wave mixing due to Na atoms in an optical cavity. The optical noise in the cavity, comprised of primarily vacuum fluctuations and a small component of spontaneous emission from the pumped Na atoms, is amplified in one quadrature of the optical field and deamplified in the other quadrature. These quadrature components are measured with a balanced homodyne detector. The total noise level in the deamplified quadrature drops below the vacuum noise level.
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... Squeezed states are specially designed non-canonical initial states which have a smaller variance in quadrature than that of a coherent state . These quantum states form an exotic choice of initial states when studying atom-photon interactions in open quantum systems [5,6]. Squeezed states of light have been shown to exhibit non-trivial effects on system observables like quadrature autocorrelations in atom-cavity systems and higher order correlated photon pairs from MgO:LiNbO 3 crystals. ...
... In our case, we numerically evaluate the quantity F A as a function of x L and x R and show the results graphically in Figs. (5) and (6). We see that the inequality holds for the entire range of x L and x R for a wide range of n L and n R values. ...
We explore the effects of quantum mechanical squeezing on the nonequilibrium fluctuations of bosonic transport between two squeezed harmonic reservoirs and a two level system. A standard full counting statistics technique based on a quantum master equation is employed. We derive a nonzero thermodynamic affinity under equal temperature setting of the two squeezed reservoirs. The odd cumulants are shown to be independent of squeezing under symmetric conditions, whereas the even cumulants depend nonlinearly on the squeezing parameters. The odd and even cumulants saturate at two different but unique values which are identified analytically. Further, squeezing always increases the magnitude of the even cumulants in comparison to the unsqueezed case. We also recover a steady state fluctuation theorem with squeezing dependent thermodynamic affinity and demonstrate the robustness of a steadystate thermodynamic uncertainty relationship.
... A prominent example of non-classical states is the set of squeezed states, in which the fluctuation associated with one quadrature component is below the vacuum state 39 . Early theoretical work in the 60s and 80s led to the conclusion that quantum fluctuations can be reduced below the shot noise in many forms of nonlinear optical interactions  . For example, squeezed states are produced in nonlinear processes called degenerate parametric down-conversion, where a "classical" electromagnetic field drives a nonlinear www.nature.com/scientificreports/ ...
At the heart of quantum thermodynamics lies a fundamental question about what is genuine “quantum” in quantum heat engines and how to seek this quantumness, so that thermodynamical tasks could be performed more efficiently compared with classical protocols. Here, using the concept of P-representability, we define a function called classicality, which quantifies the degree of non-classicality of bosonic modes. This function allows us to explore the role of non-classicality in quantum heat engines and design optimal protocols for work extraction. For two specific cycles, a quantum Otto and a generalised one, we show that non-classicality is a fundamental resource for performing thermodynamic tasks more efficiently.
... In the context of multimodal emission due to spatial hole burning, FWM guarantees a well-defined phase relation among the modes [24,25,26]. Interestingly, FWM is also able to induce non-classical intensity correlations among the different modes, already observed in other materials and devices such as optical fibers and microresonators [27,28,29,30,31], but never studied in QCLs, so far. The study of this phenomenon motivates this work, giving a perspective on the possibility of observing such correlations in Mid-Infrared (MIR) QCL-combs. ...
A novel study on harmonic frequency combs emitted by Quantum Cascade Lasers (QCLs) is here presented, demonstrating the presence of intensity correlations between twin modes characterising the emission spectra. These originate from a Four-Wave Mixing (FWM) process driven by the active medium's third-order non-linearity. The study of such correlations is essential for the engineering of a new generation of semiconductor devices with the potential of becoming integrated emitters of light with quantum properties, such as squeezing and entanglement. Starting from experimental results, the limits of state-of-the-art technology are discussed as well as the possible methodologies that could lead to the detection of non-classical phenomena, or alternatively improve the design of QCLs, in the compelling perspective of generating quantum correlations in mid-infrared light.
... The typical kinds of photon squeezing have amplitude-phase and displacement-momentum noise reductions. Slusher et al reported the first experimental realization by applying four-wave mixing technique to investigate light crossing an atom beam , which results in a reduction of light noise below the vacuum counterpart. The photon squeezing has already been successfully investigated in various hybrid quantum platforms, ranging from cQEDs , optomechanics , to electromechanics [64,65]. ...
Nonclassical two-photon statistics and photon squeezing are considered as representative features of the nonclassicality of light. In this work we investigate two-photon correlation function and quadrature photon squeezing in the dissipative mixed quantum Rabi model (QRM), which includes both the one-photon and two-photon qubit–resonator interactions. The quantum dressed master equation combined with squeezed-coherent states is applied to obtain the steady state. Based on the zero-time delay two-photon correlation function, it is found that with the increase of the two-photon qubit–resonator interaction strength the photon antibunching behavior is monotonically suppressed, whereas the photon bunching signature persists. One additional giant photon bunching feature is unraveled at deep-strong two-photon coupling, which mainly stems from efficient successive transition trajectories. The finite-time delay two-photon correlation function asymptotically approaches the unit by raising the delayed time. Moreover, the steady-state quadrature photon squeezing becomes significant at strong two-photon coupling, which may become perfect in the zero temperature limit.
... The generation of squeezed states usually requires a nonlinear optical process due to its nonlinear photon statistics. Squeezed light was first produced using atomic sodium as a nonlinear medium via four-wave mixing in 1985  and was soon followed with experiments employing optical fibers , nonlinear crystals  and semiconductor laser . After that, a variety of schemes and more substantial squeezing (up to 15 dB ) have been predicted theoretically and realized experimentally with the rapidly development of quantum technology. ...
We propose that the squeezed light accompanied by hyperradiance is induced by quantum interference in a linear system consisting of a high quality optical cavity and two coherently driven two-level qubits. When two qubits are placed at the crest and trough of the standing wave in the cavity respectively (i.e., they have the opposite coupling coefficient to the cavity), we show that squeezed light is generated in the hyperradiance regime under the conditions of strong coupling and weak driving. Simultaneously, the Klyshko's criterion alternates up and down at unity when the photon number is even or odd. Moreover, the orthogonal angles of the squeezed light can be controlled by adjusting the frequency detuning pressure between the driving field and the qubits. It can be implemented in a variety of quantum systems, including but not limited to two-level systems such as atoms, quantum dots in single-mode cavities.
A concurrent parametric amplifier consisting of two pump beams is used to investigate the possibility of generating multi-mode correlation and entanglement. The existence of three-mode entanglement is demonstrated by analyzing the violation degree of three-mode entanglement criteria, including the sufficient criterion, i.e., two-condition and optimal single-condition criterion, and necessary and sufficient criterion, i.e., positivity under partial transposition (PPT) criterion. Besides, two-mode entanglement generated from any pair is also studied by using the Duan criterion and PPT criterion. We find that three-mode entanglement and two-mode entanglement of the two pairs are present in the whole parameter region. Our results pave the way for the realization and application of multi-mode correlation and entanglement based on the concurrent parametric amplifiers.
Photonic time crystals – materials whose dielectric permittivity is modulated periodically in time – offer new concepts in light manipulation. We study theoretically the emission of light from a radiation source placed inside a photonic time crystal, and find that radiation corresponding to the momentum band gap is exponentially amplified, whether initiated by a macroscopic source, an atom, or by vacuum fluctuations, drawing the amplification energy from the modulation. The radiation linewidth becomes narrower with time, eventually shrinking to the middle of the bandgap, which enables to propose the concept of non-resonant tunable photonic time-crystal laser. Finally, we find that the spontaneous decay rate of an atom embedded in a photonic time-crystal vanishes at the band edge due to low density of photonic states.
Quantum non-Gaussian states of photons and phonons are conclusive and direct witnesses of higher-than-quadratic nonlinearities in optical and mechanical processes. Moreover, they are proven resources for quantum sensing, communication and error correction with diverse continuous-variable systems. This review introduces theoretical analyses of nonclassical and quantum non-Gaussian states of photons and phonons. It recapitulates approaches used to derive operational criteria for photons tolerant to optical losses, their application in experiments and their nowadays extension to quantum non-Gaussian photon coincidences. It extends to a recent comparison of quantum non-Gaussianity, including robustness to thermal noise, and sensing capability for high-quality phononic Fock states of single trapped cooled ions. The review can stimulate further development in the criteria of quantum non-Gaussian states and experimental effort to prepare and detect such useful features, navigating the community to advanced quantum physics and technology.
In this project, we study the time dynamics of quantum gates proposed in J. Phys. B: At. Mol. Opt. Phys 52, 205502 (2019) in a system of coupled harmonic oscillators. In particular, we focus on the realization of two-qubit gates such as the CNOT gate and quantum phase gate. These gates operate on qubits which could be prepared by truncating the infinite-dimensional energy levels of the harmonic oscillators.
A simple but rigorous analysis of the important sources of noise in homodyne detection is presented. Output noise and signal-to-noise ratios are compared for direct detection, conventional (one-port) homodyning, and two-port homodyning, in which one monitors both output ports of a 50-50 beam splitter. It is shown that two-port homodyning is insensitive to local-oscillator quadrature-phase noise and hence provides (1) a means of detecting reduced quadrature-phase fluctuations (squeezing) that is perhaps more practical than one-port homodyning and (2) an output signal-to-noise ratio that can be a modest to significant improvement over that of one-port homodyning and direct detection.
Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.
Cavity degenerate-parametric-amplifier and cavity four-wave-mixing configurations capable of producing a large amount of squeezing near the threshold of oscillation are discussed. Cavities allow one to reduce the pump power delivered to the gain medium, or to select a gain medium which has less gain per unit length, but may be better suited for squeezed-state production in other respects.
The performance of phase-sensing interferometers employing squeezed states and homodyne detection is analyzed and compared to the performance of systems employing direct detection. Standard differenced direct-detection Michelson and Mach-Zehnder interferometers are shown to be suboptimal in the sense that an observation/measurement-noise coupling occurs, which can degrade performance. Homodyne-detection interferometers in which the phase shift in one arm is the conjugate of that in the other arm do not suffer from the preceding drawback. Overall, however, the performance of differenced direct-detection and homodyne-detection interferometers is similar in single-frequency operation. In particular, both detection schemes reach the standard quantum limit on position-measurement sensitivity in single-frequency interferometric gravity-wave detectors at roughly the same average photon number. This limit arises from back action in the form of radiation pressure fluctuations entering through the energy-phase uncertainty principle. Multifrequency devices can circumvent this uncertainty principle, as illustrated by the conceptual design given for a two-frequency interferometer which can greatly surpass the standard quantum limit on position sensing. This configuration assumes that ideal photodetectors respond to photon flux rather than energy flux.
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.
The properties of a unique set of quantum states of the electromagnetic field are reviewed. These 'squeezed states' have less uncertainty in one quadrature than a coherent state. Proposed schemes for the generation and detection of squeezed states as well as potential applications are discussed.
It is shown that degenerate four-wave mixing generates two-photon coherent states (TCS) of the radiation field for modes that are proper combinations of the output object and image waves. TCS light has novel quantum behavior, which can be probed by homodyne detection, intensity interferometry, or photocount statistics. Numerical estimates indicate that the generation and detection of TCS light via degenerate four-wave mixing and homodyne detection can be accomplished with current technology.
The interferometers now being developed to detect gravitational waves work by measuring the relative positions of widely separated masses. Two fundamental sources of quantum-mechanical noise determine the sensitivity of such an interferometer: (i) fluctuations in number of output photons (photon-counting error) and (ii) fluctuations in radiation pressure on the masses (radiation-pressure error). Because of the low power of available continuous-wave lasers, the sensitivity of currently planned interferometers will be limited by photon-counting error. This paper presents an analysis of the two types of quantum-mechanical noise, and it proposes a new technique: the ''squeezed-state'' technique: that allows one to decrease the photon-counting error while increasing the radiation-pressure error, or vice versa. The key requirement of the squeezed-state technique is that the state of the light entering the interferometer's normally unused input port must be not the vacuum, as in a standard interferometer, but rather a ''squeezed state'': a state whose uncertainties in the two quadrature phases are unequal. Squeezed states can be generated by a variety of nonlinear optical processes, including degenerate parametric amplification.
Homodyne detection has been proposed as a means of detecting squeezed coherent radiation. Here the response of a balanced homodyne detector to wideband squeezed coherent states is presented. In order to carry out the analysis the theory of wideband photodetection is reviewed and in order to determine the ultimate performance limits of photoemissive detectors small terms of order Deltaomega/omega0 that are usually neglected, where omega0 is the optical carrier frequency and Deltaomega is the electronics bandwidth, have been kept. It is shown that the ultimate noise reduction that can be achieved in the noise-power spectrum of a homodyne detector, detecting squeezed coherent radiation, is a factor of 2 worse when photoemissive detectors are used instead of power flux detectors.
Wideband calculations of the response of a homodyne detector to the outputs of various four-wave-mixer configurations are presented. It is shown that the noise-power spectrum of the homodyne detector output can exhibit regions where the noise is greatly reduced below the shot-noise level even at frequencies far from dc. Hence, in the detection of noise squeezing via homodyne detectors, 1/f noise and other low-frequency noise sources may be avoided by observing the homodyne detector’s noise power at frequencies far from dc.