Littérature scientifique sur le sujet « Squeezing state »

Sum and difference squeezing of the field amplitude are higher-order squeezing effects. These effects are studied in the Raman process under the short-time approximation based on a fully quantum mechanical approach. It is shown that for uncorrelated modes, the normal squeezing in the sum and difference-frequency field depends on the sum and difference squeezing of input field modes respectively, which can generate normal squeezing in the sum and difference-frequency field mode. All the possibilities for obtaining sum and difference squeezing in two modes and its dependence on squeezing of individual field modes are investigated. We have also shown that if the high-frequency mode is in a coherent state and the low-frequency mode is squeezed, the field state will be difference squeezed if the amplitude of the high-frequency mode is large enough; otherwise the state may or may not be difference squeezed. If both modes are squeezed, then the state may or may not be difference squeezed. These higher-order squeezing effects are useful in the production of squeezing in the Raman process.

Under the short-time approximation, Giri and Gupta1 investigated the sum and difference squeezing of the field amplitude in the Raman process. The relations between the normal squeezing and the sum and difference squeezing of the field amplitude are explicitly exhibited. They argued that the sum and difference squeezing of the field amplitude are essentially the squeezing in higher-order sense. Upon verification of the results of Giri and Gupta, we observe that the entire calculations and the conclusions thereof are misleading and baseless. To substantiate our claim, we give the correct expressions for the useful variances. Interestingly, there is no possibility of getting normal squeezing, sum squeezing and difference squeezing of the input state prepared in coherent state.

We consider the problem of an atomic three-level system in interaction with a radiation field. The time evolution of the system, in atomic ladder and Λ configurations, is solved exactly assuming a coherent-state as the initial atomic state. We calculate the atomic spin-squeezing, the atomic entropy-squeezing, and their variances. We show that the spin-squeezing and the entropy-squeezing exhibit similar time dependence.

By virtue of the technique of integration within an ordered product of operators (Fan et al., Ann. Phys.321 (2006) 480) we construct a kind of three-mode entangled squeezed state in the Fock space, which exhibits stronger squeezing in one quadrature than that of the usual two-mode squeezed vacuum state.

Dipole squeezing of the atom in the presence of another atom is investigated for the one-photon and two-photon transition mechanism with the initial atoms in the two-atom squeezed state and the field in a vacuum, a coherent, or a squeezed state. For a vacuum input, the degree of squeezing is shown to depend on the photon multiplicity 'm' and the superposition angles of the atoms, θ1 and θ2. One of the quadratures of the atomic polarization is found to exhibit permanent squeezing only for some nonzero values of θ1, as well as θ2. The effect of θ2 on the dipole squeezing, however, is found to be negligible for the initial field in a coherent or a squeezed state. A comparison with the dipole squeezing of a single atom is also presented.

By extending the usual two-mode squeezing operator S2 = exp[iλ(Q1P2 + Q2P1)] to the three-mode squeezing operator S3 = exp{iλ[Q1(P2 + P3) + Q2(P1 + P3) + Q3(P1 + P2)]}, we obtain the corresponding three-mode squeezed coherent state. The higher order properties of this state, such as higher order squeezing and higher order sub-Possonian photon statistics, are investigated. It is found that the new squeezed state not only can be squeezed to all even orders but also exhibits squeezing enhancement compared with the usual cases. In addition, we examine the violation of the Bell inequality for the three-mode squeezed states by using the formalism of Wigner representation.

In this paper, we construct and investigate the nonclassical properties of the squeezing and rotating coherent state (SRCS) via a generalized squeezed operator by introducing a novel parameter r. In a particular case r = 0, SRCS reduces to the normal squeezed coherent state (SCS). The influence of r is studied in terms of squeezing and quantum statistical properties. Our results show that in the SCS not only is squeezing enhanced but also is rotated by the parameter r. Our discussion about the fidelity between SRCS and SCS shows that the fidelity values decrease with increasing parameter r. The experimental scheme for producing SRCS is also given via a self-Kerr medium and non-degenerate parametric amplifier.

The processing of information based on the generation of common quantum optical states (e.g. coherent states) and the measurement of the quadrature components of the light field (e.g. homodyne detection) is often referred to as continuous-variable quantum information processing. It is a very fertile field of investigation, at a crossroads between quantum optics and information theory, with notable successes such as unconditional continuous-variable quantum teleportation or Gaussian quantum key distribution. In quantum optics, the states of the light field are conveniently characterized using a phase-space representation (e.g. Wigner function), and the common optical components effect simple affine transformations in phase space (e.g. rotations). In quantum information theory, one often needs to determine entropic characteristics of quantum states and operations, since the von Neuman entropy is the quantity at the heart of entanglement measures or channel capacities. Computing entropies of quantum optical states requires instead turning to a state-space representation of the light field, which formally is the Fock space of a bosonic mode.

This interplay between phase-space and state-space representations does not represent a particular problem as long as Gaussian states (e.g. coherent, squeezed, or thermal states) and Gaussian operations (e.g. beam splitters or squeezers) are concerned. Indeed, Gaussian states are fully characterized by the first- and second-order moments of mode operators, while Gaussian operations are defined via their actions on these moments. The so-called symplectic formalism can be used to treat all Gaussian transformations on Gaussian states, including mixed states of an arbitrary number of modes, and the entropies of Gaussian states are directly linked to their symplectic eigenvalues.

This thesis is concerned with the Gaussian transformations applied onto arbitrary states of light, in which case the symplectic formalism is unapplicable and this phase-to-state space interplay becomes highly non trivial. A first motivation to consider arbitrary (non-Gaussian) states of light results from various Gaussian no-go theorems in continuous-variable quantum information theory. For instance, universal quantum computing, quantum entanglement concentration, or quantum error correction are known to be impossible when restricted to the Gaussian realm. A second motivation comes from the fact that several fundamental quantities, such as the entanglement of formation of a Gaussian state or the communication capacity of a Gaussian channel, rely on an optimization over all states, including non-Gaussian states even though the considered state or channel is Gaussian. This thesis is therefore devoted to developing new tools in order to compute state-space properties (e.g. entropies) of transformations defined in phase-space or conversely to computing phase-space properties (e.g. mean-field amplitudes) of transformations defined in state space. Remarkably, even some basic questions such as the entanglement generation of optical squeezers or beam splitters were unsolved, which gave us a nice work-bench to investigate this interplay.

In the first part of this thesis (Chapter 3), we considered a recently discovered Gaussian probabilistic transformation called the noiseless optical amplifier. More specifically, this is a process enabling the amplification of a quantum state without introducing noise. As it has long been known, when amplifing a quantum signal, the arising of noise is inevitable due to the unitary evolution that governs quantum mechanics. It was recently realized, however, that one can drop the unitarity of the amplification procedure and trade it for a noiseless, albeit probabilistic (heralded) transformation. The fact that the transformation is probabilistic is mathematically reflected in the fact that it is non trace-preserving. This quantum device has gained much interest during the last years because it can be used to compensate losses in a quantum channel, for entanglement distillation, probabilistic quantum cloning, or quantum error correction. Several experimental demonstrations of this device have already been carried out. Our contribution to this topic has been to derive the action of this device on squeezed states and to prove that it acts quite surprisingly as a universal (phase-insensitive) optical squeezer, conserving the signal-to-noise ratio just as a phase-sensitive optical amplifier but for all quadratures at the same time. This also brought into surface a paradoxical effect, namely that such a device could seemingly lead to instantaneous signaling by circumventing the quantum no-cloning theorem. This paradox was discussed and resolved in our work.

In a second step, the action of the noiseless optical amplifier and it dual operation (i.e. heralded noiseless attenuator) on non-Gaussian states has been examined. We have observed that the mean-field amplitude may decrease in the process of noiseless amplification (or may increase in the process of noiseless attenuation), a very counterintuitive effect that Gaussian states cannot exhibit. This work illustrates the above-mentioned phase-to-state space interplay since these devices are defined as simple filtering operations in state space but inferring their action on phase-space quantities such as the mean-field amplitude is not straightforward. It also illustrates the difficulty of dealing with non-Gaussian states in Gaussian transformations (these noiseless devices are probabilistic but Gaussian). Furthermore, we have exhibited an experimental proposal that could be used to test this counterintuitive feature. The proposed set-up is feasible with current technology and robust against usual inefficiencies that occur in optical experiment.

Noiseless amplification and attenuation represent new important tools, which may offer interesting perspectives in quantum optical communications. Therefore, further understanding of these transformations is both of fundamental interest and important for the development and analysis of protocols exploiting these tools. Our work provides a better understanding of these transformations and reveals that the intuition based on ordinary (deterministic phase-insensitive) amplifiers and losses is not always applicable to the noiseless amplifiers and attenuators.

In the last part of this thesis, we have considered the entropic characterization of some of the most fundamental Gaussian transformations in quantum optics, namely a beam splitter and two-mode squeezer. A beam splitter effects a simple rotation in phase space, while a two-mode squeezer produces a Bogoliubov transformation. Thus, there is a well-known phase-space characterization in terms of symplectic transformations, but the difficulty originates from that one must return to state space in order to access quantum entropies or entanglement. This is again a hard problem, linked to the above-mentioned interplay in the reverse direction this time. As soon as non-Gaussian states are concerned, there is no way of calculating the entropy produced by such Gaussian transformations. We have investigated two novel tools in order to treat non-Gaussian states under Gaussian transformations, namely majorization theory and the replica method.

In Chapter 4, we have started by analyzing the entanglement generated by a beam splitter that is fed with a photon-number state, and have shown that the entanglement monotones can be neatly combined with majorization theory in this context. Majorization theory provides a preorder relation between bipartite pure quantum states, and gives a necessary and sufficient condition for the existence of a deterministic LOCC (local operations and classical communication) transformation from one state to another. We have shown that the state resulting from n photons impinging on a beam splitter majorizes the corresponding state with any larger photon number n’ > n, implying that the entanglement monotonically grows with n, as expected. In contrast, we have proven that such a seemingly simple optical component may have a rather surprising behavior when it comes to majorization theory: it does not necessarily lead to states that obey a majorization relation if one varies the transmittance (moving towards a balanced beam splitter). These results are significant for entanglement manipulation, giving rise in particular to a catalysis effect.

Moving forward, in Chapter 5, we took the step of introducing the replica method in quantum optics, with the goal of achieving an entropic characterization of general Gaussian operations on a bosonic quantum field. The replica method, a tool borrowed from statistical physics, can also be used to calculate the von Neumann entropy and is the last line of defense when the usual definition is not practical, which is often the case in quantum optics since the definition involves calculating the eigenvalues of some (infinite-dimensional) density matrix. With this method, the entropy produced by a two-mode squeezer (or parametric optical amplifier) with non-trivial input states has been studied. As an application, we have determined the entropy generated by amplifying a binary superposition of the vacuum and an arbitrary Fock state, which yields a surprisingly simple, yet unknown analytical expression. Finally, we have turned to the replica method in the context of field theory, and have examined the behavior of a bosonic field with finite temperature when the temperature decreases. To this end, information theoretical tools were used, such as the geometric entropy and the mutual information, and interesting connection between phase transitions and informational quantities were found. More specifically, dividing the field in two spatial regions and calculating the mutual information between these two regions, it turns out that the mutual information is non-differentiable exactly at the critical temperature for the formation of the Bose-Einstein condensate.

The replica method provides a new angle of attack to access quantum entropies in fundamental Gaussian bosonic transformations, that is quadratic interactions between bosonic mode operators such as Bogoliubov transformations. The difficulty of accessing entropies produced when transforming non-Gaussian states is also linked to several currently unproven entropic conjectures on Gaussian optimality in the context of bosonic channels. Notably, determining the capacity of a multiple-access or broadcast Gaussian bosonic channel is pending on being able to access entropies. We anticipate that the replica method may become an invaluable tool in order to reach a complete entropic characterization of Gaussian bosonic transformations, or perhaps even solve some of these pending conjectures on Gaussian bosonic channels.

Doctorat en Sciences de l'ingénieur
info:eu-repo/semantics/nonPublished

Les capteurs atomiques sont un outil de référence pour les mesures de précision du temps, des champs électriques et magnétiques et des forces d'inertie. Cependant, en absence d’une corrélation quantique entre atomes, le bruit de projection quantique constitue une limite fondamentale pour ces capteurs, appelée la limite quantique standard (SQL). Les meilleures horloges actuelles ont déjà atteint cette limite. Cependant, elle peut être surmonté en utilisant l’intrication quantique, dans un état comprimé de spin notamment. Ce dernier peut être crée par mesure quantique non-destructive (QND), en particulier dans le cadre de l’électrodynamique quantique en cavité (cQED). Dans cette thèse, je présente la deuxième génération de l'horloge à atomes piégés sur puce TACC, dans laquelle nous combinons une horloge atomique compacte avec une plateforme cQED miniature pour tester les protocoles de métrologie quantique à un niveau de précision métrologique. Dans une mesure Ramsey standard, nous mesurons une stabilité de 6E-13 à 1 s. Nous démontrons la compression de spin par mesure QND, atteignant 8(1) dB pour 1.7E4 atomes, limitée actuellement par la décohérence due au bruit technique. Les collisions entre atomes froids jouent un rôle important à ce niveau de précision, donnant lieu à une riche dynamique de spin. Nous constatons que l’interaction entre mesures par la cavité et dynamique collisionnelle de spin se manifeste dans un effet d'amplification du signal de la cavité. Un modèle simple est proposé et confirmé par des mesures préliminaires. De nouvelles expériences sont proposés pour éclairer davantage la physique à N corps surprenante dans ce système d'atomes froids
Atomic sensors are among the best devices for precision measurements of time, electric and magnetic fields, and inertial forces. However, all atomic sensors that utilise uncorrelated particles are ultimately limited by quantum projection noise, as is already the case for state-of-the-art atomic clocks. This so-called standard quantum limit (SQL) can be overcome by employing entanglement, a prime example being the spin-squeezed states. Spin squeezing can be produced in a quantum non-demolition (QND) measurement of the collective spin, particularly with cavity quantum electrodynamical (cQED) interactions. In this thesis, I present the second-generation trapped-atom clock on a chip (TACC) experiment, where we combine a metrology-grade compact clock with a miniature cQED platform to test quantum metrology protocols at a metrologically-relevant precision level. In a standard Ramsey spectroscopy, the stability of the apparatus is confirmed by a fractional frequency Allan deviation of 6E-13 at 1 s. We demonstrate spin squeezing by QND measurement, reaching 8(1) dB for 1.7E4 atoms, currently limited by decoherence due to technical noise. Cold collisions between atoms play an important role at this level of precision, leading to rich spin dynamics. Here we find that the interplay between cavity measurements and collisional spin dynamics manifests itself in a quantum amplification effect of the cavity measurement. A simple model is proposed, and is confirmed by initial measurements. New experiments in this direction may shed light on the surprising many-body physics in this sytem of interacting cold atoms

Neste trabalho exploramos propriedades físicas dos cristais líquidos liotrópicos na sua fase lamelar e dentro desse utilizamos um sistema de spins quadrupolares para a criação e manipulação de estados coerentes de spin nuclear com técnicas de RMN. Os spins nucleares utilizados eram provenientes do núcleo de césio-133, com spin 7/2, presentes em uma molécula de pentadecafluoroctanoato de césio com estrutura líquido-cristalina. Sobre esse núcleo, aplicamos um novo conceito de pulsos fortemente modulados suaves para gerar os estados pseudo-puros correspondentes aos estados coerentes de spin nuclear. Com esses estados pudemos realizar experimentos de compressão de estado coerente, um conceito quântico muito importante quando vinculado ao conceito de emaranhamento. Outro estudo foi a observação de dinâmica clássica e efeitos de bifurcação nesse sistema quântico. Em ambas aplicações se destaca o controle dos spins nucleares no desenvolvimento dos protocolos tanto na implementação do conceito de estado coerente em sistemas de spin nuclear, quanto nas leituras dos estados quânticos via tomografia de estado quântico.
In this work we use a quadrupolar spin system inside a lyotropic liquid crystal in the lamellar phase and explore its physical properties to create and manipulate nuclear spin coherent states with NMR techniques. The nuclear spins come from the cesium-133 nucleus, spin 7/2, contained in the cesium-pentadecafluoroctanoate with liquid crystalline structure. On this nucleus, we apply a new concept of smooth strongly modulating pulses to create the pseudo-pure states corresponding to nuclear spin coherent states. With these coherent states we were able to perform coherent state squeezing, an important concept closely related to entanglement. In another study we observed the classical dynamics and bifurcation on this quantum system. Both applications highlight the quantum control of the nuclear spins in developing the protocols for the creation of nuclear spin coherent states as well as for performing the readout using the quantum state tomography procedure.

In the last few years cavity optomechanics experiments have achieved the significant progress of the preparation and observation of macroscopic me- chanical oscillators in non-classical states. An indicator of the oscillator non-classical proprieties is important also for applications to quantum tech- nologies. In this work, we compare two procedures minimizing the necessity of system calibrations. As first result we compare the homodyne spectra with the measurement of the motional sideband asymmetry in heterodyne spectra. Moreover, we describe and discuss a method to control the probe detuning, that is a crucial parameter for the accuracy of the latter, intrin- sically superior procedure. From it we can use the sidebands asymmetry as indicator of the quantumness of the mechanical oscillator, which is originated by the non-commutativity between the oscillator ladder operators. Starting from it the sidebands assume a peculiar shape when a parametric modula- tion is applied on a oscillator embedded in an optical cavity. A parametric effect is originated by a suitable combination of optical fields. The asymme- try shape is related to the modified system dynamics, while the asymmetric features reveal and quantify the quantum component of the squeezed oscilla- tor motion. The results show that it is possible to use the spectral shape of motional sidebands as a signature of a quantum mechanical squeezed state, without the necessity of absolute calibrations, in particular in the regime where residual fluctuations in the squeezed quadrature are reduced below the zero-point level.

Over the last three decades, atom interferometry has been developed rapidly and has become an important tool in quantum metrology. It has been widely applied both in the test of fundamental physics and in the precise measurement of gravity and gravity gradients. Atom interferometers based on the alkaline-earth (like) atoms such as strontium (Sr) and ytterbium (Yb) have attracted increasing attention due to the existence of narrow intercombination transitions and ultra-narrow clock transitions. The bosonic 88Sr is a good candidate for transportable and space-borne atom interferometers due to the immunity to stray magnetic fields in its electronic ground state, long coherence time and low collision rate. It can therefore be used in space projects for precision measurement of gravity and gravity gradients. However, there is a fundamental limit to the precision in a phase shift measurement with atom interferometers, which is set by the number of atoms involved. This limit is known as the standard quantum limit. It is possible to surpass this limit by introducing correlations in the atomic ensembles thus reducing the phase uncertainty at the expense of an increase in the population uncertainty. In this case the spin-squeezed states are generated and can be used to improve the phase resolution of atom interferometers. In this thesis, a method to generate spin squeezed states in 88Sr momentum states for atom interferometry is considered and the necessary technology that allows its implementation will be presented. Spin squeezing is achieved by resolving the Doppler effect due to momentum state superposition via cavity- enhanced nondestructive measurement. An optical ring cavity is designed and constructed for quantum nondestructive measurements. However, one major obstacle that blocks the way to spin squeezing via cavity-enhanced measurement arises from cavity length fluctuations, which can totally mask the atomic signal if no appropriate scheme is adopted. Therefore, a method to cancel the cavity length fluctuations in measuring the atom-induced phase shift is proposed and close to 30 dB reduction of the cavity noise down to the noise floor has been demonstrated. We further apply the demonstrated noise-reduced measurement scheme in the simulated squeezing experiment, where we mimic the atom-induced cavity phase shift by varying the frequency of one of the two circulating beams. The noise cancellation scheme demonstrates an improvement of a factor of 40 in phase sensitivity with a phase resolution of 0.7 mrad. With this improvement we estimate that the cavity noise will no longer play an important role in a real spin squeezing measurement.

Marketization creates four dilemmas that lead to change in governance. (1) Cost versus quality: squeezing prices siphons resources out of services that could be used to employ qualified workers on the front line, and the prescription of services that price-based competition requires drains the capacity of providers to innovate. (2) Payment by results versus equal access to services: while payment by results is consistent with the ethos of market governance it bears the risk of “creaming and parking.” (3) User choice versus user compulsion: NPM principles of consumerism are difficult to reconcile with the principles of compulsion built into the work-first welfare state. (4) Openness/transparency and transaction costs: openness, transparency, and equal treatment require costly administrative capacity. Insourcing provides one solution by taking services out of the market, but it is not a panacea.

This text reviews fundametals and incorporates key themes of quantum physics. One theme contrasts boson condensation and fermion exclusivity. Bose–Einstein condensation is basic to superconductivity, superfluidity and gaseous BEC. Fermion exclusivity leads to compact stars and to atomic structure, and thence to the band structure of metals and semiconductors with applications in material science, modern optics and electronics. A second theme is that a wavefunction at a point, and in particular its phase is unique (ignoring a global phase change). If there are symmetries, conservation laws follow and quantum states which are eigenfunctions of the conserved quantities. By contrast with no particular symmetry topological effects occur such as the Bohm–Aharonov effect: also stable vortex formation in superfluids, superconductors and BEC, all these having quantized circulation of some sort. The quantum Hall effect and quantum spin Hall effect are ab initio topological. A third theme is entanglement: a feature that distinguishes the quantum world from the classical world. This property led Einstein, Podolsky and Rosen to the view that quantum mechanics is an incomplete physical theory. Bell proposed the way that any underlying local hidden variable theory could be, and was experimentally rejected. Powerful tools in quantum optics, including near-term secure communications, rely on entanglement. It was exploited in the the measurement of CP violation in the decay of beauty mesons. A fourth theme is the limitations on measurement precision set by quantum mechanics. These can be circumvented by quantum non-demolition techniques and by squeezing phase space so that the uncertainty is moved to a variable conjugate to that being measured. The boundaries of precision are explored in the measurement of g-2 for the electron, and in the detection of gravitational waves by LIGO; the latter achievement has opened a new window on the Universe. The fifth and last theme is quantum field theory. This is based on local conservation of charges. It reaches its most impressive form in the quantum gauge theories of the strong, electromagnetic and weak interactions, culminating in the discovery of the Higgs. Where particle physics has particles condensed matter has a galaxy of pseudoparticles that exist only in matter and are always in some sense special to particular states of matter. Emergent phenomena in matter are successfully modelled and analysed using quasiparticles and quantum theory. Lessons learned in that way on spontaneous symmetry breaking in superconductivity were the key to constructing a consistent quantum gauge theory of electroweak processes in particle physics.

Physical abuse is the state of harming the child's body by the people around him. Scars of unknown cause, traces resulting from burning, spots due to impact on any part of the body, traces caused by a bite by another person, bruises and stains caused by squeezing with any tool, burns and scars caused by smoking cigarettes on the victim reflect physical abuse. Problems such as rumination disorders, problems in socialization, lack of self-respect, depression, fear of all situations, withdrawal behaviors, incompatibility, colic, problems in mental perception processes, and decreased academic achievement are observed in these children.

This chapter questions whether the neurological and molecular levels are the most appropriate domains to guide the actions of the State. The reductionism of such thinking creates a form of scientific “tunnel vision” dangerously constraining the direction of future inquiry. The Chapter explores the consequences of the prevailing moral and scientific settlements, demonstrating how these have shifted preferred policy responses towards those that are individualised and increasingly medicalized. A preoccupation with prevention, early intervention and the privileging of certain forms of evidence (that furnished by clinical trials, biological evidence) are squeezing out conversations about different, and potentially more desirable and sustainable, actions to make people’s lives better.

We study a single mode of the radiation field subjected to ideal k-photon parametric amplification (photons created or destroyed k at a time). In an attempt to generalize ordinary squeezing (k = 2), Fisher et al.1 considered the case k > 2. They concluded that there is something seriously wrong with the evolution operator after discovering that for k > 2 its matrix elements in the number-state basis have divergent Taylor series expansions in time. We show that this divergence is due to a branch cut along the negative time axis, and we obtain useful information by treating these Taylor series as asymptotic expansions. We investigate the states generated by a k-photon paramp by calculating the evolution of the quantum O-function. We watch the vacuum evolve into a state whose O-function has k-fold symmetry. The classical behavior which corresponds to a k-photon parametric interaction is described by phase-space trajectories near an unstable fixed point. We compare the classical and quantum behavior and find that the quantum corrections smear out classical phase-space features which are smaller than allowed by the uncertainty principle. Finally, we consider the evolution of an initial coherent state which has large amplitude, and we find that a three-photon paramp generates squeezing at a rate proportional to the amplitude of the initial coherent state.

A broadband or two-mode (nondegenerate) squeezed state is the natural two-mode analog of a single-mode (degenerate) squeezed state. The squeezing is a result of correlations between photons in the two modes. Two-mode squeezed states are produced by the same kind of physical process and the same kinds of physical device that produce single-mode squeezed states (e.g., a parametric amplifier), simply by moving away from degeneracy. They are to be contrasted with another kind of two-mode state, one produced by separately squeezing two single modes. The latter are produced by a different kind of physical process and different physical devices from those that produce two-mode squeezed states. States that are products of two single-mode squeezed states do not exhibit squeezing, for the photons in the two modes are not correlated. There is a formal sense, however, in which these different kinds of two-mode states are (unitarily) equivalent. This formal equivalence tells one that, to achieve the desired squeezing by separately squeezing two modes, one would have to use a frequency-converting device before and after squeezing the two modes separately. This process produces the required correlations, hence the squeezing, but it Is clearly not the natural way to obtain a broadband squeezed state.

Noise suppression in an interferometer, employing a nonlinear Mach-Zehnder squeezer, was proposed[1] and experimentally confirmed[2,3]. The noise characteristics of this squeezer were derived for a coherent state input[4]. It may be advantageous to put the squeezer directly into a modelocked laser resonator where higher power pulses may be utilized. In this case the light acquires a large phase uncertainty. Thus, it is of interest to analyze squeezing with an input state of large phase uncertainty. One approach is to expand the input state in terms of coherent states[5]. Another approach is to use number states, which have total phase uncertainty and also present some interesting quantum features.

Normal squeezing is described in terms of variables which are linear in the field creation and annihilation operators. Amplitude-squared squeezing, on the other hand, is defined in terms of variables which are quadratic in these operators. As a result processes, such as second-harmonic generation, which couple one mode amplitude to the square of another convert this kind of squeezing into normal squeezing. In particular, the variables one considers are the hermitian part and one over i times the antihermitian part of the square of the annihilation operator. The variances of these observables obey an uncertainty relation. In order to better understand the properties of amplitude-squared squeezing it is useful to find the minimum uncertainty states corresponding to this uncertainty relation. These states are described by two parameters; the first,, gives the expectation of the square of the annihilation operator and the other λ, describes the amount of amplitude-squared squeezing. AH of these states have a mean amplitude of zero. These states for which =0 have unusual noise properties in that their Q functions have a four-fold rotational symmetry. We also find that a squeezed vacuum state is an amplitude-squared minimum uncertainty state.

Four-wave mixers have been investigated theoretically as potential sources of squeezed-coherent radiation 1,2 and there are a number of experimental efforts underway to demonstrate the production of squeezed-coherent radiation via such devices.3,4,5 Here the results of some wideband calculations of the output of cavity four-wave mixers is presented. The wideband squeezed-coherent radiation generated by such devices may be detected via a wideband homodyne detector.6 Wideband squeezing will manifest itself in the noise-power spectrum of the homodyne detector's output as regions where the noise is reduced below the level that is observed when no light enters the input port of the homodyne detector. Wideband homodyne detection allows one to look for squeezing at frequencies far from d.c. This gives the experimentalist the freedom to look for squeezing in the frequency range where his detector performs best.