We report the first direct experimental characterization of continuous variable quantum Stokes parameters. We generate a continuous wave light beam with more than 3 dB of simultaneous squeezing in three of the four Stokes parameters. The polarization squeezed beam is produced by mixing two quadrature squeezed beams on a polarizing beam splitter. Depending on the squeezed quadrature of these two beams the quantum uncertainty volume on the Poincaré sphere became a `cigar' or `pancake'-like ellipsoid.
The transverse spatial attributes of an optical beam can be decomposed into the position, momentum and orbital angular momentum observables. The position and momentum of a beam is directly related to the quadrature amplitudes, whilst the orbital angular momentum is related to the polarization and spin variables. In this paper, we study the quantum properties of these spatial variables, using a representation in the Stokes-operator basis. We propose a spatial detection scheme to measure all three spatial variables and consequently, propose a scheme for the generation of spatial Stokes operator …
We study polarization squeezing of a light beam initially in the coherent state using the nonlinear interaction hamiltonian $ H=k\big(\hat a_{x}^{\dagger2}+{\hat a_{x}}^2\big)$. For the degree of polarization squeezing, we use a definition written in the final form by the authors and also used in earlier papers. We find that the polarization squeezing can be very high and the degree of polarization squeezing can be less than unity by a very small amount. We achieve the polarization squeezing even for very low beam intensity under some conditions involving phase angles in the polarization modes…
We report on a novel and efficient source of polarization squeezing using a single pass through an optical fiber. Simply passing this Kerr squeezed beam through a carefully aligned lambda/2 waveplate and splitting it on a polarization beam splitter, we find polarization squeezing of up to 5.1 +/- 0.3 dB. The experimental setup allows for the direct measurement of the squeezing angle.
Polarization squeezing is shown to occur in non-degenerate parametric amplification of coherent light and the degree of squeezing at interaction time $T$ can be as large as $1-e^{-2T}$. This gives $86.4\%$ polarization squeezing for $T=1$ and $98.2\%$ for $T=2$. One simple case when this occurs is on taking initially plane polarized light having equal amplitudes in the two modes that finally has equal intensities of two circular polarizations. This suggest the experimental settings of parameters to achieve this extent of polarization squeezing in coherent light.
The dynamic feature of monochromatic bi-modal chaotic optical field, enriched with orthogonally-polarized basis-modes propagating collinearly and undergoing Degenerate Parametric Amplification, is investigated to demonstrate Squeezing in Hidden Optical-Polarization states. The Variance (Noise) of Hidden Optical-Polarization Parameters showing squeezing therein is numerically studied by identifying a Squeezing - Function. This squeezing in HOPS is seen to depend critically on interaction time and an Onset-time responsible for the same is demarcated.
The impact of the operator-valued commutator on nonclassical properties of continuous polarization variables is discussed. The definition of polarization squeezing is clarified to exclude those squeezed states which do not contain any new physics beyond quadrature squeezing. We present a consistent derivation of the general nonseparability criterion for the continuous variables with the operator-valued commutator, and apply it to the polarization variables.
Photonic Crystal Fibers can be tailored to increase the effective Kerr nonlinearity, while producing smaller amounts of excess noise compared to standard silicon fibers. Using these features of Photonic Crystal Fibers we create polarization squeezed states with increased purity compared to standard fiber squeezing experiments. Explicit we produce squeezed states in counter propagating pulses along the same fiber axis to achieve near identical dispersion properties. This enables the production of polarization squeezing through interference in a polarization type Sagnac interferometer. We observ…
A 2.8-dB polarization squeezing of the Stokes operator S2 for the rubidium D1 line (795 nm) is achieved, with the lowest squeezing band at an audio frequency of 2.6 kHz. Two methods are applied for improving the squeezing towards low frequencies: an orthogonal-polarized locking beam that precludes residual unwanted interference and quantum noise locking method that stabilizes the relative phase between the squeezed vacuum and the local oscillator. The squeezing level is limited by absorption-induced losses at short wavelengths, here is 397.5 nm. The generated polarization squeezed light can be…
We show that it is possible to generate continuous-wave fields and pulses of polarization squeezed light by sending classical, linearly polarized laser light twice through an atomic sample which causes an optical Faraday rotation of the field polarization. We characterize the performance of the process, and we show that an appreciable degree of squeezing can be obtained under realistic physical assumptions.
Within the new description of the polarization structure of quantum light (given in Part I) some types of generalized coherent states related to the polarization SU(2) group are examined. With their help we give a quasiclassical description of polarization properties of light fields and discuss the concept of squeezing and uncertainty relations for multimode light in the polarization quantum optics. As a consequence, a new classification of polarization states of quantum light is obtained. We also derive geometric phases acquired by different quantum light beams transmitted through "polarizati…
We report on the generation of polarization squeezing of intense, short light pulses using an asymmetric fiber Sagnac interferometer. The Kerr nonlinearity of the fiber is exploited to produce independent amplitude squeezed pulses. The polarization squeezing properties of spatially overlapped amplitude squeezed and coherent states are discussed. The experimental results for a single amplitude squeezed beam are compared to the case of two phase-matched, spatially overlapped amplitude squeezed pulses. For the latter, noise variances of -3.4dB below shot noise in the S0 and the S1 and of -2.8dB i…
We study the interaction of a nearly resonant linearly polarized laser beam with a cloud of cold cesium atoms in a high finesse optical cavity. We show theoretically and experimentally that the cross-Kerr effect due to the saturation of the optical transition produces quadrature squeezing on both the mean field and the orthogonally polarized vacuum mode. An interpretation of this vacuum squeezing as polarization squeezing is given and a method for measuring quantum Stokes parameters for weak beams via a local oscillator is developed.
We present a theoretical study of an ensemble of X-like 4-level atoms placed in an optical cavity driven by a linearly polarized field. We show that the self-rotation (SR) process leads to polarization switching (PS). Below the PS threshold, both the mean field mode and the orthogonal vacuum mode are squeezed. We provide a simple analysis of the phenomena responsible for the squeezing and trace the origin of vacuum squeezing not to SR, but to crossed Kerr effect induced by the mean field. Last, we show that this vacuum squeezing can be interpreted as polarization squeezing.
We further elaborate the theory of quantum fluctuations in vertical-cavity surface-emitting lasers (VCSELs), developed in Ref. \cite{Hermier02}. In particular, we introduce the quantum Stokes parameters to describe the quantum self- and cross-correlations between two polarization components of the electromagnetic field generated by this type of lasers. We calculate analytically the fluctuation spectra of these parameters and discuss experiments in which they can be measured. We demonstrate that in certain situations VCSELs can exhibit polarization squeezing over some range of spectral frequenc…
We discuss the possibility of generating noncritical quadrature squeezing by spontaneous polarization symmetry breaking. We consider first type-II frequency-degenerate optical parametric oscillators, but discard them for a number of reasons. Then we propose a four-wave mixing cavity in which the polarization of the output mode is always linear but has an arbitrary orientation. We show that in such a cavity complete noise suppression in a quadrature of the output field occurs, irrespective of the parameter values.
We report on nonlinear squeezing effects of polarization states of light by harnessing the intrinsic correlations from a polarization-entangled light source and click-counting measurements. Nonlinear Stokes operators are obtained from harnessing the click-counting theory in combination with angular-momentum-type algebras. To quantify quantum effects, theoretical bounds are derived for second- and higher-order moments of nonlinear Stokes operators. The experimental validation of our concept is rendered possible by developing an efficient source, using a spectrally decorrelated type-II phase-mat…
We observed squeezed vacuum light at 795 nm in 87Rb vapor via resonant polarization self-rotation, and report noise sidebands suppression of ~1 dB below shot noise level spanning from acoustic (30 kHz) to MHz frequencies. This is the first demonstration of sub-MHz quadrature vacuum squeezing in atomic systems. The spectral range of observed squeezing matches well typical bandwidths of electromagnetically induced transparency (EIT) resonances, making this simple technique for generation of optical fields with non-classical statistics at atomic transitions wavelengths attractive for EIT-based qu…
We theoretically study quadrature and polarization squeezing in dispersive optical bistability through a vectorial Kerr cavity model describing a nonlinear cavity filled with an isotropic chi(3) medium in which self-phase and cross-phase modulation, as well as four--wave mixing, occur. We derive expressions for the quantum fluctuations of the output field quadratures as a function of which we express the spectrum of fluctuations of the output field Stokes parameters. We pay particular attention to study how the bifurcations affecting the non-null linearly polarized output mode squeezes the ort…
The Stokes-parameter operators and the associated Poincare sphere, which describe the quantum-optical polarization properties of light, are defined and their basic properties are reviewed. The general features of the Stokes operators are illustrated by evaluation of their means and variances for a range of simple polarization states. Some of the examples show polarization squeezing, in which the variances of one or more Stokes parameters are smaller than the coherent-state value. The main object of the paper is the application of these concepts to bright squeezed light. It is shown that a ligh…
We study polarization squeezing of a pure photon number state which is obviously polarized but the mere change in the basis of polarization leads to simultaneous polarization squeezing in all the components of Stokes operator vector except those falling along or perpendicular to the direction of polarization state, is observed. We use the most general definition of polarization squeezing and discuss the experimental feasibility of the result. We also observe that a squeezing operation like non-degenerate parametric amplification of the state does not reveal simultaneous squeezing in all Stokes…
We apply spin-squeezing techniques to identify and quantify highly multi-partite photonic entanglement in polarization-squeezed light. We consider a practical single-mode scenario, and find that Wineland-criterion polarization squeezing implies entanglement of a macroscopic fraction of the total photons. A Glauber-theory computation of the observable N-photon density matrix, with N up to 100, finds that N-partite entanglement is observable despite losses and without post-selection. We estimate that existing detectors could observe $\sim1000$-partite entanglement from a few dB of polarization s…
In analogy with the spin-squeezing inequality of Wang and Sanders [Physical Review A 68, 012101 (2003)], we find inequalities describing macroscopic polarization correlations that are obeyed by all classical fields, and whose violation implies entanglement of the photons that make up the optical beam. We consider a realistic and exactly-solvable experimental scenario employing polarization- squeezed light from an optical parametric oscillator (OPO) and find that any two photons separated by less than the OPO coherence time are polarization entangled. The polarization entanglement is robust aga…
The traversal of an elliptically polarized optical field through a thermal vapour cell can give rise to a rotation of its polarization axis. This process, known as polarization self-rotation (PSR), has been suggested as a mechanism for producing squeezed light at atomic transition wavelengths. In this paper, we show results of the characterization of PSR in isotopically enhanced Rubidium-87 cells, performed in two independent laboratories. We observed that, contrary to earlier work, the presence of atomic noise in the thermal vapour overwhelms the observation of squeezing. We present a theory …
We report the experimental demonstration of directly produced polarization squeezing at 1064 nm from a type I optical parametric amplifier (OPA) based on a periodically poled KTP crystal (PPKTP). The orthogonal polarization modes of the polarization squeezed state are both defined by the OPA cavity mode, and the birefringence induced by the PPKTP crystal is compensated for by a second, but inactive, PPKTP crystal. Stokes parameter squeezing of 3.6 dB and anti squeezing of 9.4 dB is observed.
A monochromatic double-mode coherent light endowed with orthogonally polarized photons propagating collinearly is studied in Degenerate Parametric Amplification. Generation of Hidden Optical- Polarized States is shown by non-zero values of Index of Hidden Optical-Polarization. Squeezing in HOPS is demonstrated by recognizing a Squeezing function. The Non-Classical feature of HOPS is observed by 'degree of Hidden Optical-Polarization' which attains non-classical value 'greater than unity'. The dynamical nature of Generation, Squeezing and Non-Classicality are numerically presented.
We report new experiments on polarization squeezing using ultrashort photonic pulses in a single pass of a birefringent fiber. We measure what is to our knowledge a record squeezing of -6.8 +/- 0.3 dB in optical fibers which when corrected for linear losses is -10.4 +/- 0.8 dB. The measured polarization squeezing as a function of optical pulse energy, which spans a wide range from 3.5-178.8 pJ, shows a very good agreement with the quantum simulations and for the first time we see the experimental proof that Raman effects limit and reduce squeezing at high pulse energy.
The notion of spin squeezing has been discussed in this paper using the density matrix formalism. Extending the definition of squeezing for pure states given by Kitagawa and Ueda in an appropriate manner and employing the spherical tensor representation, we show that mixed spin states which are non-oriented and possess vector polarization indeed exhibit squeezing. We construct a mixed state of a spin 1 system using two spin 1/2 states and study its squeezing behaviour as a function of the individual polarizations of the two spinors.
We develop a theory of optomechanical cooling with a squeezed input light field. We show that Stokes heating transitions can be \emph{fully} suppressed when the driving field is squeezed below the vacuum noise level at an appropriately selected squeezing phase and for a finite amount of squeezing. The quantum backaction limit to laser cooling can be therefore moved down to zero and the resulting final temperature is then solely determined by the ratio between the thermal phonon number and the optomechanical cooperativity parameter, independently of the actual values of the cavity linewidth and…
We have studied relative-intensity fluctuations for a variable set of orthogonal elliptic polarization components of a linearly polarized laser beam traversing a resonant $^{87}$Rb vapor cell. Significant polarization squeezing at the threshold level (-3dB) required for the implementation of several continuous variables quantum protocols was observed. The extreme simplicity of the setup, based on standard polarization components, makes it particularly convenient for quantum information applications.
Squeezed light is one of the resources of photonic quantum technology. Among the various nonlinear interactions capable of generating squeezing, the optical Kerr effect is particularly easy-to-use. A popular venue is to generate polarization squeezing, which is a special self-referencing variant of two-mode squeezing. To date, polarization squeezing generation setups have been very sensitive to fluctuations of external factors and have required careful tuning. In this work, we report on a development of a new all-fibre setup for polarization squeezing generation. The setup consists of passive …
We present experimental and theoretical analysis of quantum fluctuation in a vacuum field in the presence of orthogonal linearly polarized pump field propagating through a Rb vapor cell. Previously reported theoretical and experimental studies provided somewhat contradictory conclusions regarding the possibility to observe the "squeezed vacuum" -- the reduction of vacuum fluctuations below standard quantum limit -- in this system. Here, using the D1 transitions of Rb in a cell without buffer as as an example, we demonstrate that vacuum squeezing is corrupted by incoherent processes (such as sp…
We report on the generation of non separable beams produced via the interaction of a linearly polarized beam with a cloud of cold cesium atoms placed in an optical cavity. We convert the squeezing of the two linear polarization modes into quadrature entanglement and show how to find out the best entanglement generated in a two-mode system using the inseparability criterion for continuous variable [Duan et al., Phys. Rev. Lett. 84, 2722 (2000)]. We verify this method experimentally with a direct measurement of the inseparability using two homodyne detections. We then map this entanglement into …
We investigate the spatial overlap of nonclassical ultrashort light pulses produced by self-phase modulation effect in electronic Kerr media and its relevance in the formation of polarization-squeezed states of light. The light polarization is treated in terms of four quantum Stokes parameters whose spectra of quantum fluctuations are investigated. We show that the frequency at which the suppression of quantum fluctuations of Stokes parameters is the greatest can be controlled by adjusting the linear phase difference between pulses. By varying the intensity of one pulse one can suppress effect…
Based upon standard angular momentum theory, we develop a framework to investigate polarization squeezing and multipartite entanglement of a quantum light field. Both mean polarization and variances of the Stokes parameters are obtained analytically, with which we study recent observation of triphoton states [L. K. Shalm {\it et al}, Nature \textbf{457}, 67 (2009)]. Our results show that the appearance of maximally entangled NOON states accompanies with a flip of mean polarization and can be well understood in terms of quantum Fisher information.
We report the experimental demonstration of polarization squeezed beam at 795 nm by combining a quadrature squeezed beam with a coherent beam. The quadrature squeezed beam is generated by a degenerate optical parametric amplifier based on a PPKTP crystal. Stokes parameter squeezing of -3.8 dB and anti squeezing of +5.0 dB is observed. This polarization squeezed beam resonant to rubidium D1 line has potential applications in quantum information networks and precise measurement beyond the shot noise limit.
We investigate quantum correlations in continuous wave polarization squeezed laser light generated from one and two optical parametric amplifiers, respectively. A general expression of how Stokes operator variances decompose into two mode quadrature operator variances is given. Stokes parameter variance spectra for four different polarization squeezed states have been measured and compared with a coherent state. Our measurement results are visualized by three-dimensional Stokes operator noise volumes mapped on the quantum Poincare sphere. We quantitatively compare the channel capacity of the d…
We report the generation of optical squeezed vacuum states by means of polarization self-rotation in rubidium vapor following a proposal by Matsko et al. [Phys. Rev. A 66, 043815 (2002)]. The experimental setup, involving in essence just a diode laser and a heated rubidium gas cell, is simple and easily scalable. A squeezing of 0.85+-0.05 dB was achieved.
We investigate polarization squeezing in squeezed coherent states with varying coherent amplitudes. In contrast to the traditional characterization based on the full Stokes parameters, we experimentally determine the Stokes vector of each excitation manifold separately. Only for states with a fixed photon number do the methods coincide; when the photon number is indefinite, our approach gives a richer and deeper description. By capitalising on the properties of the Husimi Q function, we map this notion onto the Poincare space, providing a full account of the measured squeezing.
We propose an operational degree of polarization in terms of the variance of the projected Stokes vector minimized over all the directions of the Poincaré sphere. We examine the properties of this degree and show that some problems associated with the standard definition are avoided. The new degree of polarization is experimentally determined using two examples: a bright squeezed state and a quadrature squeezed vacuum.
We present a scheme for generating and manipulating three-mode squeezed states with genuine tripartite entanglement by injecting single-mode squeezed light into an array of coupled optical waveguides. We explore the possibility to selectively generate single-mode squeezing or multimode squeezing at the output of an elliptical waveguides array, determined solely by the input light polarization. We study the effect of losses in the waveguides array and show that quantum correlations and squeezing are preserved for realistic parameters. Our results show that arrays of optical waveguides are suita…
We report new experiments that test quantum dynamical predictions of polarization squeezing for ultrashort photonic pulses in a birefringent fibre, including all relevant dissipative effects. This exponentially complex many-body problem is solved by means of a stochastic phase-space method. The squeezing is calculated and compared to experimental data, resulting in excellent quantitative agreement. From the simulations, we identify the physical limits to quantum noise reduction in optical fibres. The research represents a significant experimental test of first-principles time-domain quantum dy…
Stokes parameters are a standard tool in quantum optics. They involve averaged intensities at exits of polarizers. If the overall measured intensity fluctuates, as e.g. for states with undefined photon numbers, the instances of its increased value contribute more to the parameters. One can introduce normalized quantum Stokes operators. Operationally, for a given single run of the experiment, their values are differences of measured intensities (or photon numbers) at the two exits of a polarizer divided by their sum. Effects of intensity fluctuations are removed. Switching to normalized Stokes …
A formalism for two-photon Stokes parameters is introduced to describe the polarization entanglement of photon pairs. This leads to the definition of a degree of two-photon polarization, which describes the extent to which the two photons act as a pair and not as two independent photons. This pair-wise polarization is complementary to the degree of polarization of the individual photons. The approach provided here has a number of advantages over the density matrix formalism: it allows the one- and two-photon features of the state to be separated and offers a visualization of the mixedness of t…
A quantum model of the Raman process with the independent Stokes and anti-Stokes nonlinear interactions is developed to study nonclassical correlations between the photons in the Stokes and anti-Stokes fields. The role of the laser pump amplitude, the ratio of the Stokes and anti-Stokes coupling constants and the population and losses of the vibrational mode in forming the correlations is elucidated. The $ g^{(2)} $ intensity cross-correlation function, noise-reduction-factor, two-mode principal squeezing variance, logarithmic negativity, non-classicality depth, steering parameter and the Bell…
We observe the polarization squeezing in the mixture of a two mode squeezed vacuum and a simple coherent light through a linear polarization beam splitter. Squeezed vacuum not being squeezed in polarization, generates polarization squeezed light when superposed with coherent light. All the three Stokes parameters of the light produced on the output port of polarization beam splitter are found to be squeezed and squeezing factor also depends upon the parameters of coherent light.
We obtain a compact analytical solution for the nonlinear equation for the nuclear spin dynamics in the central spin box model in the limit of many nuclear spins. The total nuclear spin component along the external magnetic field is conserved and the two perpendicular components precess or oscillate depending on the electron spin polarization, with the frequency, determined by the nuclear spin polarization. As applications of our solution, we calculate the nuclear spin noise spectrum and describe the effects of nuclear spin squeezing and many body entanglement in the absence of a system excita…
We have numerically solved the Heisenberg-Langevin equations describing the propagation of quantized fields through an optically thick sample of atoms. Two orthogonal polarization components are considered for the field and the complete Zeeman sublevel structure of the atomic transition is taken into account. Quantum fluctuations of atomic operators are included through appropriate Langevin forces. We have considered an incident field in a linearly polarized coherent state (driving field) and vacuum in the perpendicular polarization and calculated the noise spectra of the amplitude and phase q…
The polarization properties of macroscopic Bell states are characterized using three-dimensional quantum polarization tomography. This method utilizes three-dimensional inverse Radon transform to reconstruct the polarization quasiprobability distribution function of a state from the probability distributions measured for various Stokes observables. The reconstructed 3D distributions obtained for the macroscopic Bell states are compared with those obtained for a coherent state with the same mean photon number. The results demonstrate squeezing in one or more Stokes observables.
Squeezed light, which is easily degraded by loss, could benefit from generation directly in optical fiber. Furthermore, highly nonlinear fiber could offer more efficient generation with lower pump power and shorter fiber lengths than standard single-mode fiber. We investigate non-polarization-maintaining highly nonlinear fiber (HNLF) for squeezed-light generation by characterizing possible sources of excess noise, including its zero-dispersion wavelength (ZDW) variation and polarization noise. We find significant ZDW variation and excess polarization noise. We believe the polarization noise is…
We report the demonstration of a magnetometer with noise-floor reduction below the shot-noise level. This magnetometer, based on a nonlinear magneto-optical rotation effect, is enhanced by the injection of a squeezed vacuum state into its input. The noise spectrum shows squeezed noise reduction of about 2 dB spanning from close to 100 Hz to several megahertz. We also report on the observation of two different regimes of operation of such a magnetometer: one in which the detection noise is limited by the quantum noise of the light probe only, and one in which we see additional noise originating…
We demonstrate a diode-laser-pumped system for generation of quadrature squeezing and polarization squeezing. Due to their excess phase noise, diode lasers are challenging to use in phase-sensitive quantum optics experiments such as quadrature squeezing. The system we present overcomes the phase noise of the diode laser through a combination of active stabilization and appropriate delays in the local oscillator beam. The generated light is resonant to the rubidium D1 transition at 795nm and thus can be readily used for quantum memory experiments.
We report the experimental transformation of quadrature entanglement between two optical beams into continuous variable polarization entanglement. We extend the inseparability criterion proposed by Duan, et al. [Duan00] to polarization states and use it to quantify the entanglement between the three Stokes operators of the beams. We propose an extension to this scheme utilizing two quadrature entangled pairs for which all three Stokes operators between a pair of beams are entangled.
We present experimental study of polarization quantum noise of laser radiation passed through optically think vapor of Rb87. We observe a step-like noise spectrum. We discuss various factor which may result in such noise spectrum and prevent observation of squeezing of quantum fluctuations predicted in Matsko et al. PRA 63, 043814 (2001).
