Extending the scheme developed for a single mode of the electromagnetic field in the preceding paper ``Structure of multiphoton quantum optics. I. Canonical formalism and homodyne squeezed states'', we introduce two-mode nonlinear canonical transformations depending on two heterodyne mixing angles. They are defined in terms of hermitian nonlinear functions that realize heterodyne superpositions of conjugate quadratures of bipartite systems. The canonical transformations diagonalize a class of Hamiltonians describing non degenerate and degenerate multiphoton processes. We determine the coherent states associated to the canonical transformations, which generalize the non degenerate two--photon squeezed states. Such heterodyne multiphoton squeezed are defined as the simultaneous eigenstates of the transformed, coupled annihilation operators. They are generated by nonlinear unitary evolutions acting on two-mode squeezed states. They are non Gaussian, highly non classical, entangled states. For a quadratic nonlinearity the heterodyne multiphoton squeezed states define two--mode cubic phase states. The statistical properties of these states can be widely adjusted by tuning the heterodyne mixing angles, the phases of the nonlinear couplings, as well as the strength of the nonlinearity. For quadratic nonlinearity, we study the higher-order contributions to the susceptibility in nonlinear media and we suggest possible experimental realizations of multiphoton conversion processes generating the cubic-phase heterodyne squeezed states.
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local oscillator angle; for n-mode systems they depend on n heterodyne mixing angles. The canonical formalism realizes nontrivial mixings of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems; and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamilton…
We introduce nonlinear canonical transformations that yield effective Hamiltonians of multiphoton down conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behavior of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.
We propose a displacement-operator approach to some aspects of squeezed states for general multiphoton systems. The explicit displacement-operators of the squeezed vacuum and the coherent states are achieved and expresses as the ordinary exponential form. As a byproduct the coherent states of the $q$-oscillator are obtained by the {\it usual exponential} displacement-operator.
Multiphoton heralding generates large-amplitude squeezed Schrödinger cat states and parity-selective Fock superpositions from squeezed vacuum via an OPA
In terms of the characteristic functions of the quantum states, we present a complete operator description of a lossy photon-subtraction scheme. Feeding a single-mode squeezed vacuum into a variable beam splitter and counting the photons in one of the output channels, a broad class of multiphoton-subtracted squeezed vacuum states (MSSVSs) can be generated in other channel. Here the losses are considered in the beginning and the end channels in the circuit. Indeed, this scheme has been discussed in Ref. [Phys. Rev. A 100, 022341 (2019)]. However, different from the above work, we give all the d…
As part of the effort to make use of squeezed states of light for detection of sub-shot-noise optical signals, we study the balanced heterodyne scheme, for which the corresponding spectral density of the photocurrent fluctuations produced at the output of the detector is calculated as the Fourier transform of their autocorrelation function. Our analysis shows that, for maximal signal-to-noise ratio enhancement by use of squeezed states of light, an optical signal to be measured must be carried in the squeezed quadrature of the carrier field. We discuss how "the additional heterodyne noise" can…
In this paper we discuss the quantum properties for superposition of squeezed displaced number states against multiphoton Jaynes-Cummings model (JCM). In particular, we investigate atomic inversion, photon-number distribution, purity, quadrature squeezing, Mandel $Q$ parameter and Wigner function. We show that the quadrature squeezing for three-photon absorption case can exhibit revivals and collapses typical to those occurring in the atomic inversion for one-photon absorption case. Also we prove that for odd number absorption parameter there is a connection between the evolution of the atomic…
We introduce a linear, canonical transformation of the fundamental single--mode field operators $a$ and $a^{\dagger}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding to the linear transformation a nonlinear function of any of the fundamental quadrature operators $X_{1}$ and $X_{2}$, making the original Bogoliubov transformation quadrature--dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set…
We present a new technique for the detection of two-mode squeezed states of light that allows for a simple characterization of these quantum states. The usual detection scheme, based on heterodyne measurements, requires the use of a local oscillator with a frequency equal to the mean of the frequencies of the two modes of the squeezed field. As a result, unless the two modes are close in frequency, a high-frequency shot-noise-limited detection system is needed. We propose the use of a bichromatic field as the local oscillator in the heterodyne measurements. By the proper selection of the frequ…
In this paper, we extend the results of an earlier paper in which we had demonstrated the limitations of the notion of non-resonant multiphoton ionization, in the exploration of photon statistics effects in non-linear processes. Through the quantitative analysis of specific realistic processes, we provide the connection to conditions of intensity and pulse duration necessary in relevant experiments, including a recent seminal experiment demonstrating the effect of superbunching found in squeezed radiation.
We investigate the quantum non-demolition (QND) measurement of an atomic population based on a heterodyne detection and show that the induced back-action allows to prepare both spin-squeezed and Dicke states. We use a wavevector formalism to describe the stochastic process of the measurement and the associated atomic evolution. Analytical formulas of the atomic distribution momenta are derived in the weak coupling regime both for short and long time behavior, and they are in good agreement with those obtained by a Monte-Carlo simulation. The experimental implementation of the proposed heterody…
We consider a multiphoton Bell-type inequality to study nonlocality in four-mode continuous variable systems, which goes beyond two-photon states and can be applied to mixed as well as states with fluctuating photon number. We apply the inequality to a wide variety of states such as pure and mixed Gaussian states (including squeezed thermal states) and non-Gaussian states. We consider beam splitters as a model for leakage and show that the inequality is able to detect nonclassicality of noisy Gaussian states as well. Finally, we investigate nonlocality in pair-coherent states and entangled coh…
In this work, we consider the estimation of single mode Gaussian states using four different measurement schemes namely: i) homodyne measurement, ii) sequential measurement, iii) Arthurs-Kelly scheme, and iv) heterodyne measurement, with a view to compare their relative performance. To that end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out involved calculations. We show that the optimal performance of the Arthurs-Kelly scheme and the sequential measurement is equal to the heterodyne measur…
Cross-correlation heterodyne detectors exhibit the potential for suppression of the detection quantum noise below shot noise without use of optical squeezing for capturing weak optical signals in low frequency bands. To understand the underlying mechanism, we develop a quantum theory to describe the noise performance of cross-correlation heterodyne detectors. By calculating the cross spectral density (CSD) of the photocurrent fluctuations from a cross-correlation heterodyne detector, we prove that its noise performance can break the shot noise limit and exceed that of a regular heterodyne dete…
Quantum measurements of mechanical systems can produce optical squeezing via ponderomotive forces. Its observation requires high environmental isolation and efficient detection, typically achieved by using optical cavities and cryogenic cooling. Here we realize these conditions by measuring the position of an optically levitated nanoparticle at room temperature and without the overhead of an optical cavity. We use a fast heterodyne detection to reconstruct simultaneously orthogonal optical quadratures, and observe a noise reduction of $9\%\pm0.5\%$ below shot noise. Our experiment offers a nov…
The new process of quantum-injection into an optical parametric amplifier operating in entangled configuration is adopted to amplify into a large dimensionality spin 1/2 Hilbert space the quantum entanglement and superposition properties of the photon-couples generated by parametric down-conversion. The structure of the Wigner function and of the field's correlation functions shows a decoherence-free, multiphoton Schroedinger-cat behaviour of the emitted field which is largely detectable against the squeezed-vacuum noise. Furthermore, owing to its entanglement character, the system is found to…
In this work, we introduce a non-Gaussian quantum state named excited squeezed vacuum state (ESVS), which can be ustilized to describe quantum light field emitted from the multiphoton quantum process occurred in some restricted quantum systems. We investigate its nonclassical properties such as Wigner distribution in phase space, photon number distribution, the second-order autocorrelation and the quadrature fluctuations. By virtue of the Hilbert-Schmidt distance method, we quantify the non-Gaussianity of the ESVS. Due to the similar photon statistics, we examine the fidelity between the ESVS …
We present a review of theoretical and experimental aspects of multiphoton quantum optics. Multiphoton processes occur and are important for many aspects of matter-radiation interactions that include the efficient ionization of atoms and molecules, and, more generally, atomic transition mechanisms; system-environment couplings and dissipative quantum dynamics; laser physics, optical parametric processes, and interferometry. A single review cannot account for all aspects of such an enormously vast subject. Here we choose to concentrate our attention on parametric processes in nonlinear media, w…
We show the equivalence between two different communication schemes that employ a couple of modes of the electromagnetic field. One scheme uses unconventional heterodyne detection, with correlated signal and image-band modes in a twin-beam state from parametric downconversion. The other scheme is realized through a complex-number coding over quadrature-squeezed states of two uncorrelated modes, each detected by ordinary homodyning. This equivalence concerns all the stages of the communication channel: the encoded state, the optimal amplifier for the channel, the master equation modeling the lo…
Asymmetry effects in homodyne and heterodyne measurements: Positive operator-valued measures and asymptotic security of Gaussian-continuous-variable quantum key distribution
ITMO University（伊托夫大学）；Laboratory for Quantum Communications, ITMO University（伊托夫大学量子通信实验室）；Laboratory of Quantum Processes and Measurements, ITMO University（伊托夫大学量子过程与测量实验室）
In this paper we consider a system consisting of a two-level atom in an excited state interacting with two modes of a radiation field prepared initially in $l$-photon coherent states. This system is described by two-mode multiphoton (, i.e., $k_1, k_2$) Jaynes-Cummings model (JCM). For this system we investigate the occurrence of the revival-collapse phenomenon (RCP) in the evolution of the single-mode, two-mode, sum and difference quadrature squeezing. We show that there is a class of states for which all these types of squeezing exhibit RCP similar to that involved in the corresponding atomi…
We investigate the cavity excitation spectrum and the photon number distribution in a cavity QED system driven by a broadband squeezed vacuum. In an empty cavity, we show that only states with even number of photons can be measured under resonant condition since the squeezed vacuum consists of states with even number of photons only. When a single atom is trapped in the cavity, the strong coupling between the atom and cavity results in energy splittings of the system, and there exist two peaks in the cavity excitation spectrum at two-photon transition frequencies. At the central frequency, how…
The quantum theory of electromagnetic radiation predicts characteristic statistical fluctuations for light sources as diverse as sunlight, laser radiation and molecule fluorescence. Indeed, these underlying statistical fluctuations of light are associated with the fundamental physical processes behind their generation. In this contribution, we demonstrate that the manipulation of the quantum electromagnetic fluctuations of a pair of vacuum states leads to a novel family of quantum-correlated multiphoton states with tunable mean photon numbers and degree of correlation. Our technique relies on …
Multi-photon processes are the essence of nonlinear optics. Optical harmonics generation and multi-photon absorption, ionization, polymerization or spectroscopy are widely used in practical applications. Generally, the rate of an n-photon effect scales as the n-th order autocorrelation function of the incident light, which is high for light with strong photon-number fluctuations. Therefore `noisy' light sources are much more efficient for multi-photon effects than coherent sources with the same mean power, pulse duration and repetition rate. Here we generate optical harmonics of order 2-4 from…
A general analysis of squeezing transformations for two mode systems is given based on the four dimensional real symplectic group $Sp(4,\Re)\/$. Within the framework of the unitary metaplectic representation of this group, a distinction between compact photon number conserving and noncompact photon number nonconserving squeezing transformations is made. We exploit the $Sp(4,\Re)-SO(3,2)\/$ local isomorphism and the $U(2)\/$ invariant squeezing criterion to divide the set of all squeezing transformations into a two parameter family of distinct equivalence classes with representative elements ch…
Quantum fluctuations of the electromagnetic vacuum impose an observable quantum limit to the lowest temperatures that can be reached with conventional laser cooling techniques. As laser cooling experiments continue to bring massive mechanical systems to unprecedented temperatures, this quantum limit takes on increasingly greater practical importance in the laboratory. Fortunately, vacuum fluctuations are not immutable, and can be "squeezed" through the generation of entangled photon pairs. Here we propose and experimentally demonstrate that squeezed light can be used to sideband cool the motio…
Full-Period Optical Phase Estimation with Heisenberg Scaling Using Displaced Squeezed States and Gaussian Measurements
Institute for Quantum Science and Technology, University of Calgary（卡尔加里大学量子科学与技术研究所）；Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico（新墨西哥大学物理与天文系量子信息与控制中心）；Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究所）
We present an algorithmic approach to calculate the quantum-noise spectral density of photocurrents generated by optical fields with arbitrary discrete classical spectrum in coherent or squeezed states. The measurement scheme may include an arbitrary number of demodulations of the photocurrent. Thereby, our method is applicable to the general heterodyne detection scheme which is implemented in many experiments. For some of these experiments, e.g. in laser-interferometric gravitational-wave detectors, a reliable prediction of the quantum noise of fields in coherent and squeezed states plays a d…
The quantum statistical fluctuations of the electromagnetic field establish a limit, known as the shot-noise limit, on the sensitivity of optical measurements performed with classical technologies. However, quantum technologies are not constrained by this shot-noise limit. In this regard, the possibility of using every photon produced by quantum sources of light to estimate small physical parameters, beyond the shot-noise limit, constitutes one of the main goals of quantum optics. Here we experimentally demonstrate a scalable protocol for quantum-enhanced optical phase estimation across a broa…
We consider a hypothesis testing problem for displacement parameters of n independent copies of an m-mode squeezed quantum Gaussian state whose mixture parameter is known. Given n>1, we construct a quantum measurement as a test using an observable which is invariant by n-fold tensor product of any m-mode squeezing operator. For a pure state case, we calculate the type II error probability of this test. We compare this test with a Hotelling's T-squared test which is based on heterodyne measurements.
Lossless filtering of a single coherent (Schmidt) mode from spatially multimode radiation is a problem crucial for optics in general and for quantum optics in particular. It becomes especially important in the case of nonclassical light that is fragile to optical losses. An example is bright squeezed vacuum generated via high-gain parametric down conversion or four-wave mixing. Its highly multiphoton and multimode structure offers a huge increase in the information capacity provided that each mode can be addressed separately. However, the nonclassical signature of bright squeezed vacuum, photo…
We consider the effect of loss on quantum-optical communication channels. The channel based on direct detection of number states, which for a lossless transmission line would achieve the ultimate quantum channel capacity, is easily degraded by loss. The same holds true for the channel based on homodyne detection of squeezed states, which also is very fragile to loss. On the contrary, the ``classical'' channel based on heterodyne detection of coherent states is loss-invariant. We optimize the a priori probability for the squeezed-state and the number-state channels, taking the effect of loss in…
We introduce a method for efficient, in situ characterization of linear-optical networks (LONs) in randomized boson-sampling (RBS) experiments. We formulate RBS as a distributed task between two parties, Alice and Bob, who share two-mode squeezed-vacuum states. In this protocol, Alice performs local measurements on her modes, either photon counting or heterodyne. Bob implements and applies to his modes the LON requested by Alice; at the output of the LON, Bob performs photon counting, the results of which he sends to Alice via classical channels. In the ideal situation, when Alice does photon …
We propose a model for computing of a certain set of analytical functions based on estimating the output distribution of multiphoton outcomes in an optical scheme with an initial single-mode squeezed vacuum (SMSV) state and photonic states measuring the number of photons in one of the output modes of the beam splitter (BS) by photon number resolving (PNR) detector. The set of considered analytical functions is polynomial expressions including arbitrary derivatives of certain functions which can take on very large values even on small interval in their argument and small values of the parameter…
Resonance fluorescence arises from the interaction of an optical field with a two-level system and has played a fundamental role in the development of quantum optics and its applications. Despite its conceptual simplicity it entails a wide range of intriguing phenomena, such as the Mollow-triplet emission spectrum and coherent photon emission. One fundamental aspect of resonance fluorescence, reduced quantum fluctuations in the single photon stream from an atom in free space, was predicted more than 30 years ago. However, the requirement to operate in the weak excitation regime, together with …
It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cramér-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use t…
Loophole-free Bell tests are essential if one wishes to perform device-independent quantum key distribution (QKD), since any loophole could be used by a potential adversary to undermine the security of the protocol. Crucial work by Eberhard demonstrated that weakly entangled two-qubit states have a far greater resistance to the detection loophole than maximally entangled states, allowing one to close the loophole with detection efficiency greater than 2/3. Here we demonstrate that this same limit holds for photon-counting CHSH Bell tests which can demonstrate non-locality for higher dimensiona…
