`Nonclassical' states in quantum optics: a `squeezed' review of the first 75 years

Gouy phase and quantum interference with cross-Wigner functions for matter-waves

The Gouy phase is essential for accurately describing various wave phenomena, ranging from classical electromagnetic waves to matter waves and quantum optics. In this work, we employ phase-space methods based on the cross-Wigner transformation to analyze spatial and temporal interference in the evolution of matter waves characterized initially by a correlated Gaussian wave packet. First, we consider the cross-Wigner of the initial wave function with its free evolution, and second for the evolution through a double-slit arrangement. Different from the wave function which acquires a global Gouy phase, we find that the cross-Wigner acquires a Gouy phase difference due to different evolution times. The results suggest that temporal like-Gouy phase difference is important for an accurate description of temporal interference. Furthermore, we propose a technique based on the Wigner function to reconstruct the cross-Wigner from the spatial intensity interference term in a double-slit experiment with matter waves.

Geometric phaselike effects of driven transport in presence of reservoir squeezing

In a bare bosonic site coupled to two reservoirs, we explore the statistics of boson exchange in the presence of two simultaneous processes: squeezing the two reservoirs and driving the two reservoirs. The squeezing parameters compete with the geometric phaselike effect or geometricity to alter the nature of the steady-state flux and noise. The even (odd) geometric cumulants and the total minimum entropy are found to be symmetric (antisymmetric) with respect to exchanging the left and right squeezing parameters. Upon increasing the strength of the squeezing parameters, loss of geometricity is observed. Under maximum squeezing, one can recover a standard steady-state fluctuation theorem even in the presence of phase-different driving protocol. A recently proposed modified geometric thermodynamic uncertainty principle is found to be robust.

Light squeezing enhancement by coupling nonlinear optical cavities

In this paper, we explore the squeezing effect generated by two coupled optical cavities. Each cavity contains a second-order nonlinear material and coherently pumped by a laser. Our results show that light intensity is strongly improved due to the presence of the nonlinearities and mainly depends on the detunings between external laser frequencies and cavity modes. More interestingly, the proposed scheme could enhance light squeezing for moderate coupling between cavities : the squeezing generated by one cavity is enhanced by the other one. For resonant interaction, highest squeezing effect is obtained near resonance. When fields are non resonant, squeezing increases near resonance of the considered cavity, but decreases for large detunings relative to the second cavity. Further, when the dissipation rate of the second cavity is smaller than the first, the squeezing could be improved, attaining nearly the perfect squeezing. While the temperature elevation has a negative impact overall on the nonclassical light, squeezing shows an appreciable resistance against thermal baths for appropriate parameter sets.

Generalized sub-Poissonian states of two-beam fields

Two-beam states obtained by partial photon-number-resolving detection in one beam of a multi-mode twin beam are experimentally investigated using an intensified CCD camera. In these states, sub-Poissonian photon-number distributions in one beam are accompanied by sub-shot-noise fluctuations in the photon-number difference of both beams. Multi-mode character of the twin beam implying the beam nearly Poissonian statistics is critical for reaching sub-Poissonian photon-number distributions, which contrasts with the use of a two-mode squeezed vacuum state. Relative intensities of both nonclassical effects as they depend on the generation conditions are investigated both theoretically and experimentally using photon-number distributions of these fields. Fano factor, noise-reduction parameter, local and global nonclassicality depths, degree of photon-number coherence, mutual entropy as a non-Gaussianity quantifier, and negative quasi-distributions of integrated intensities are used to characterize these fields. Spatial photon-pair correlations as means for improving the field properties are employed. These states are appealing for quantum metrology and imaging including the virtual-state entangled-photon spectroscopy.

Enhancing nonclassical properties of quantum states of light using linear optics

In this Letter, we present a simple and versatile scheme for enhancing the nonclassical properties of light states using only linear optics and photodetectors. By combining a coherent state |α〉 and an arbitrary pure state of light |ϕ〉 (excluding coherent states) at two beam splitters, we show that the amplitude α of the coherent state can be tuned to filter out specific Fock components and generate states of light with increased nonclassical features. We provide two examples of input states and demonstrate the effectiveness of our scheme in enhancing the sub-Poissonian statistics or the quadrature squeezing of the output states.

Quantification of Quantum Correlations in Two-Beam Gaussian States Using Photon-Number Measurements

Identification, and subsequent quantification of quantum correlations, is critical for understanding, controlling, and engineering quantum devices and processes. We derive and implement a general method to quantify various forms of quantum correlations using solely the experimental intensity moments up to the fourth order. This is possible as these moments allow for an exact determination of the global and marginal impurities of two-beam Gaussian fields. This leads to the determination of steering, tight lower and upper bounds for the negativity, and the Kullback-Leibler divergence used as a quantifier of state nonseparability. The principal squeezing variances are determined as well using the intensity moments. The approach is demonstrated on the experimental twin beams with increasing intensity and the squeezed super-Gaussian beams composed of photon pairs. Our method is readily applicable to multibeam Gaussian fields to characterize their quantum correlations.

Optimizing the generation of polarization squeezed light in nonlinear optical fibers driven by femtosecond pulses

Bright squeezed light can be generated in optical fibers utilizing the Kerr effect for ultrashort laser pulses. However, pulse propagation in a fiber is subject to nonconservative effects that deteriorate the squeezing. Here, we analyze two-mode polarization squeezing, which is SU(2)-invariant, robust against technical perturbations, and can be generated in a polarization-maintaining fiber. We perform a rigorous numerical optimization of the process and the pulse parameters using our advanced model of quantum pulse evolution in the fiber that includes various nonconservative effects and real fiber data. Numerical results are consistent with experimental results.

Superior Resilience of Non-Gaussian Entanglement against Local Gaussian Noises

Entanglement distribution task encounters a problem of how the initial entangled state should be prepared in order to remain entangled the longest possible time when subjected to local noises. In the realm of continuous-variable states and local Gaussian channels it is tempting to assume that the optimal initial state with the most robust entanglement is Gaussian too; however, this is not the case. Here we prove that specific non-Gaussian two-mode states remain entangled under the effect of deterministic local attenuation or amplification (Gaussian channels with the attenuation factor/power gain κi and the noise parameter μi for modes i=1,2) whenever κ1μ22+κ2μ12<14(κ1+κ2)(1+κ1κ2), which is a strictly larger area of parameters as compared to where Gaussian entanglement is able to tolerate noise. These results shift the “Gaussian world” paradigm in quantum information science (within which solutions to optimization problems involving Gaussian channels are supposed to be attained at Gaussian states).

Quantum Metrology of Noisy Spreading Channels

We provide the optimal measurement strategy for a class of noisy channels that reduce to the identity channel for a specific value of a parameter (spreading channels). We provide an example that is physically relevant: the estimation of the absolute value of the displacement in the presence of phase randomizing noise. Surprisingly, this noise does not affect the effectiveness of the optimal measurement. We show that, for small displacement, a squeezed vacuum probe field is optimal among strategies with same average energy. A squeezer followed by photodetection is the optimal detection strategy that attains the quantum Fisher information, whereas the customarily used homodyne detection becomes useless in the limit of small displacements, due to the same effect that gives Rayleigh's curse in optical superresolution. There is a quantum advantage: a squeezed or a Fock state with N average photons allow to asymptotically estimate the parameter with a sqrt[N] better precision than classical states with same energy.

Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources

Quantum energy coherences represent a thermodynamic resource, which can be exploited to extract energy from a thermal reservoir and deliver that energy as work. We argue that there exists a closely analogous classical thermodynamic resource, namely, energy-shell inhomogeneities in the phase space distribution of a system’s initial state. We compare the amount of work that can be obtained from quantum coherences with the amount that can be obtained from classical inhomogeneities, and find them to be equal in the semiclassical limit. We thus conclude that coherences do not provide a unique thermodynamic advantage of quantum systems over classical systems, in situations where a well-defined semiclassical correspondence exists.

Minimal Products of Coordinate and Momentum Uncertainties of High Orders: Significant and Weak High-Order Squeezing

We consider the problem of minimization of products of mean values of the high powers of operators x and p. From this point of view, we study several two-term superpositions of the Fock states, as well as three popular families of infinite superpositions: squeezed states, even/odd coherent states, and orthogonal even coherent states (or compass states). The new element is the analysis of products of the corresponding (co)variances and the related generalized (Robertson-Schrödinger) intelligent states (RSIS). In particular, we show that both Fock and pure Gaussian homogeneous states are RSIS for the fourth powers (but not for the sixth ones). We show that lower bounds of the high-order uncertainty products can be significantly below the vacuum values. In this connection, the concept of significant and weak high-order squeezing is introduced.

Phase-Space Inequalities Beyond Negativities

We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.

Supersymmetric approach to coherent states for nonlinear oscillator with spatially dependent effective mass

A nonlinear oscillator with variable mass is studied in the approach of Supersymmetric quantum mechanics. Ladder operators in association with the shape invariance condition allowed us to find the coherent states of the system in the sense of Barut and Girardello. The statistical properties of these particular states are studied using the probability of distribution. In addition, the vibrational partition function is calculated. We see that the thermodynamic functions of the system such as mean energy and entropy depend on the nonlinearity parameter, except for the specific heat.

Waves in spatio-spectral and -temporal coherence of evolving ultra-intense twin beams

Waves in the spatio-spectral and -temporal coherence of evolving ultra-intense twin beams are predicted: Twin beams with low intensities attain maximal coherence in the beam center until certain threshold intensity is reached. Then the area of maximal coherence moves with increasing intensity from the beam center towards its edges leaving the beam center with low coherence (the first coherence wave). For even larger intensities, a new coherence maximum is gradually built in the beam center with the increasing intensity and, later, it again moves towards the beam edges forming the second coherence wave. Rotationally-symmetric twin beams are analyzed within a three-dimensional model that couples spectral and spatial degrees of freedom. Relation between the twin-beam coherence and its local density of modes during the nonlinear evolution is discussed.

Squeezed vacuum states of light for gravitational wave detectors

A century after Einstein's formulation of general relativity, the detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) made the first direct detection of gravitational waves. This historic achievement was the culmination of a world-wide effort and decades of instrument research. While sufficient for this monumental discovery, the current generation of gravitational-wave detectors represent the least sensitive devices necessary for the task; improved detectors will be required to fully exploit this new window on the Universe. In this paper, we review the application of squeezed vacuum states of light to gravitational-wave detectors as a way to reduce quantum noise, which currently limits their performance in much of the detection band.

Strong mechanical squeezing in an electromechanical system

The mechanical squeezing can be used to explore quantum behavior in macroscopic system and realize precision measurement. Here we present a potentially practical method for generating strong squeezing of the mechanical oscillator in an electromechanical system. Through the Coulomb interaction between a charged mechanical oscillator and two fixed charged bodies, we engineer a quadratic electromechanical Hamiltonian for the vibration mode of mechanical oscillator. We show that the strong position squeezing would be obtained on the currently available experimental technologies.

Entanglement and nonclassicality in four-mode Gaussian states generated via parametric down-conversion and frequency up-conversion

Multipartite entanglement and nonclassicality of four-mode Gaussian states generated in two simultaneous nonlinear processes involving parametric down-conversion and frequency up-conversion are analyzed assuming the vacuum as the initial state. Suitable conditions for the generation of highly entangled states are found. Transfer of the entanglement from the down-converted modes into the up-converted ones is also suggested. The analysis of the whole set of states reveals that sub-shot-noise intensity correlations between the equally-populated down-converted modes, as well as the equally-populated up-converted modes, uniquely identify entangled states. They represent a powerful entanglement identifier also in other cases with arbitrarily populated modes.

Single-mode squeezing in arbitrary spatial modes

As the generation of squeezed states of light has become a standard technique in laboratories, attention is increasingly directed towards adapting the optical parameters of squeezed beams to the specific requirements of individual applications. It is known that imaging, metrology, and quantum information may benefit from using squeezed light with a tailored transverse spatial mode. However, experiments have so far been limited to generating only a few squeezed spatial modes within a given setup. Here, we present the generation of single-mode squeezing in Laguerre-Gauss and Bessel-Gauss modes, as well as an arbitrary intensity pattern, all from a single setup using a spatial light modulator (SLM). The degree of squeezing obtained is limited mainly by the initial squeezing and diffractive losses introduced by the SLM, while no excess noise from the SLM is detectable at the measured sideband. The experiment illustrates the single-mode concept in quantum optics and demonstrates the viability of current SLMs as flexible tools for the spatial reshaping of squeezed light.

Exciting Polaritons with Quantum Light

We discuss the excitation of polaritons-strongly coupled states of light and matter-by quantum light, instead of the usual laser or thermal excitation. As one illustration of the new horizons thus opened, we introduce "Mollow spectroscopy"-a theoretical concept for a spectroscopic technique that consists of scanning the output of resonance fluorescence onto an optical target-from which weak nonlinearities can be read with high precision even in strongly dissipative environments.

Cross-polarized photon-pair generation and bi-chromatically pumped optical parametric oscillation on a chip

Nonlinear optical processes are one of the most important tools in modern optics with a broad spectrum of applications in, for example, frequency conversion, spectroscopy, signal processing and quantum optics. For practical and ultimately widespread implementation, on-chip devices compatible with electronic integrated circuit technology offer great advantages in terms of low cost, small footprint, high performance and low energy consumption. While many on-chip key components have been realized, to date polarization has not been fully exploited as a degree of freedom for integrated nonlinear devices. In particular, frequency conversion based on orthogonally polarized beams has not yet been demonstrated on chip. Here we show frequency mixing between orthogonal polarization modes in a compact integrated microring resonator and demonstrate a bi-chromatically pumped optical parametric oscillator. Operating the device above and below threshold, we directly generate orthogonally polarized beams, as well as photon pairs, respectively, that can find applications, for example, in optical communication and quantum optics.

Inducing nonclassical lasing via periodic drivings in circuit quantum electrodynamics

We show how a pair of superconducting qubits coupled to a microwave cavity mode can be used to engineer a single-atom laser that emits light into a nonclassical state. Our scheme relies on the dressing of the qubit-field coupling by periodic modulations of the qubit energy. In the dressed basis, the radiative decay of the first qubit becomes an effective incoherent pumping mechanism that injects energy into the system, hence turning dissipation to our advantage. A second, auxiliary qubit is used to shape the decay within the cavity, in such a way that lasing occurs in a squeezed basis of the cavity mode. We characterize the system both by mean-field theory and exact calculations. Our work may find applications in the generation of squeezing and entanglement in circuit QED, as well as in the study of dissipative few- and many-body phase transitions.

First experimental realization of the Dirac oscillator

We present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yield the spectrum of the one-dimensional Dirac oscillator with and without a mass term. The flexibility of the experimental setup allows the implementation of other one-dimensional Dirac-type equations.

Generation of nonclassical optical fields by a beam splitter with third-order nonlinearity

We propose a quantum-mechanical model of a nonlinear beam splitter having third-order nonlinearity. We show that nonclassical features, such as squeezing and sub-Poissonian photon statistics of optical fields, can be generated in the output when we mix coherent light beams via such a nonlinear beam splitter.

Directly observing squeezed phonon states with femtosecond x-ray diffraction

Squeezed states are quantum states of a harmonic oscillator in which the variance of two conjugate variables each oscillate out of phase. Ultrafast optical excitation of crystals can create squeezed phonon states, where the variance of the atomic displacements oscillates due to a sudden change in the interatomic bonding strength. With femtosecond x-ray diffraction we measure squeezing oscillations in bismuth and conclude that they are consistent with a model in which electronic excitation softens all phonon modes by a constant scaling factor.

Observation of squeezed light with 10-dB quantum-noise reduction

Squeezing of light's quantum noise requires temporal rearranging of photons. This again corresponds to creation of quantum correlations between individual photons. Squeezed light is a nonclassical manifestation of light with great potential in high-precision quantum measurements, for example, in the detection of gravitational waves [C. M. Caves, Phys. Rev. D 23, 1693 (1981)10.1103/PhysRevD.23.1693]. Equally promising applications have been proposed in quantum communication [H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978)10.1109/TIT.1978.1055958]. However, after 20 years of intensive research doubts arose whether strong squeezing can ever be realized as required for eminent applications. Here we show experimentally that strong squeezing of light's quantum noise is possible. We reached a benchmark squeezing factor of 10 in power (10 dB). Thorough analysis reveals that even higher squeezing factors will be feasible in our setup.

Communication via the statistics of photon-number squeezed light

A method of communication employing the second order statistics of photon-number squeezed light is demonstrated. The technique encodes the information content in both nonstationary noise processes and in the average optical power, thereby creating two orthogonal channels and increasing the transmission capacity. Communication via the fragile quantum state has potential applications for privatized communication.

Single-shot measurement of quantum optical phase

Although the canonical phase of light, which is defined as the complement of photon number, has been described theoretically by a variety of distinct approaches, there have been no methods proposed for its measurement. Indeed doubts have been expressed about whether or not it is measurable. Here we show how it is possible, at least in principle, to perform a single-shot measurement of the canonical phase using beam splitters, mirrors, phase shifters, and photodetectors.