Table of contents

A longstanding goal in chemical physics has been the control of atoms and molecules using coherent light fields. This paper provides a brief overview of the field and discusses experiments that use a programmable pulse shaper to control the quantum state of electronic wavepackets in Rydberg atoms and electronic and nuclear dynamics in molecular liquids. The shape of Rydberg wavepackets was controlled by using tailored ultrafast pulses to excite a beam of caesium atoms. The quantum state of these atoms was measured using holographic techniques borrowed from optics. The experiments with molecular liquids involved the construction of an automated learning machine. A genetic algorithm directed the choice of shaped pulses which interacted with the molecular system inside a learning control loop. Analysis of successful pulse shapes that were found by using the genetic algorithm yield insight into the systems being controlled.

We study the dynamics of atomic waves in a two-dimensional light crystal formed by two crossed standing laser fields. The longitudinal modulation of the crystal with the Doppler frequency significantly influences the transversal spatial modulation of the atomic wave. Near the doppleron resonance the atomic density shows a fractional space period. In this case a normally incident wave gives rise to an almost perfect conversion into the first momentum components and the light crystal acts as a highly efficient beamsplitter. The crossing angle, determining the Doppler frequency, is the easy-to-control parameter of the system.

In this paper, we consider a superposition of a weak coherent field and a two-photon source and examine the amplitude-squared squeezing (ASS) properties of the combined field. We demonstrate that the efficient and lasting ASS effect of the field could be observed due to constructive two-photon quantum interference. The relationship between ASS and photon antibunching effects is discussed. It is found that photon antibunching may accompany weak ASS, but in most cases ASS effects appear accompanying photon bunching.

We study a force acting on a three-level Λ-type atom in an inhomogeneous coherent electromagnetic field. We show that the magnitude of the force as well as incoherent scattering of the field is comparable with that of a two-level atom. Using Λ-type three-level atoms, one is able to realize a new kind of quantum nondemolition measurement of a photon number of modes of an open optical cavity.

Quantum theory of self-phase and cross-phase modulation of ultrashort light pulses in the Kerr medium is developed, taking into account the response time of an electronic nonlinearity. The corresponding algebra of time-dependent Bose operators is elaborated. It is established that the spectral region of the pulse where the quadrature fluctuation level is lower than the shot-noise one depends on the value of the nonlinear phase shift, the intensity of another pulse and the relaxation time of the nonlinearity. It is shown that the frequency of the pulse spectrum at which the suppression of fluctuations is maximum can be controlled by adjusting the other pulse intensity.

In classical three-wave interaction there is a regime where only the phases move but no energy is exchanged. This is strictly valid for classical trajectories with sharp amplitudes and phases. Quantum mechanically, such a phase motion regime can be realized only approximately if all three waves start in coherent states because the states deform due to their initial uncertainties during rotation. In the limit of a strong signal or idler wave, what amounts to frequency conversion, we show that exact rotations of coherent states become possible. This completes our understanding of three-wave interaction. Different initial quantum states are exchanged periodically if the strong wave is treated classically and undepleted. This limit is investigated by numerical diagonalization of the exact Hamiltonian via nonclassical states in the pump and coherent states in the signal and idler. Classical phase motion turns out to be very helpful to understand the strong demands for the converter limit. In the converter, we consider the impurity of the two modes and show that the coherent state in one mode, which enables the no-energy transfer regime, has no influence on the impurity of the other wave and can be substituted by a vacuum state. Thus, the impurity in such a regime behaves exactly as under simple one-photon losses although the energy can be held constant. The different rotations are illustrated by calculating the Wigner functions via classical trajectories. In addition, we also consider the equivalent emission from a prepared atomic system (squeezed atoms) into one mode.

We analyse the entanglement nature of quantum complementarity in individual and simultaneous continuous-variable (CV) quantum nondemolition measurements of coordinate and momentum. The complementarity expressed by the uncertainty relations is compared with an information-theoretical analysis of the quantum correlations. To demonstrate the entanglement nature, new CV quantum erasing experiments are suggested, employing only Gaussian states and feasible quantum optical devices, as parametrical processes, beamsplitters and homodyne detections.

A theoretical investigation is carried out through the interaction of the plane running monochromatic light wave, having an arbitrary intensity, with atoms (molecules) of a rarefied gas in the plane cell (at the normal incidence of the wave). Cases of closed and open resonance transitions from the non-degenerate ground (or metastable) quantum level are considered. Possible sub-Doppler resonances are analysed in the wave absorption, caused by the transient establishment of the optical coherence on the transition, Rabi oscillations between its levels, and optical pumping during the free flights of particles between the walls of the cell. Results of the previous works on given problems are generalized, which were obtained at definite restrictions on the wave intensity and cell length. Moreover, non-trivial sub-Doppler spectral structures, resulting as a consequence of the dependence of the absorption saturation on the transit relaxation of particles, have been established and investigated. Such structures may consist of a number of peaks and dips caused by Rabi oscillations between the transition levels. The results obtained can be used in sub-Doppler spectroscopy and for the stabilization of laser frequencies in thin gas cells.

We present two purification schemes for nonmaximally entangled states. We first show that two parties, Alice and Bob, start with shared less-entangled three-particle states to probabilistically produce a three-particle Greenberger–Horne–Zeilinger state by Bell state measurements and positve operator valued measure (POVM) or a unitary transformation. Then, by a straightforward generalization of the schemes, the purification of a multi-particle entangled state can be realized.

This paper studies Bloch oscillations of atoms in a standing laser wave driven by a constant (for example, gravitational) force. For zero or moderate detuning, oscillatory motion of the atom along the standing wave is accompanied by transitions between internal atomic states, which appears to be in phase with Bloch oscillations.

We study the multimode higher-order nonclassical effects of novel trio coherent states. We show that such states exhibit antibunching to all orders in the single-mode case. However, the two-mode higher-order antibunching may or may not exist depending on the parameters. We also show that in such states squeezing is fully absent in both single-mode and two-mode situations. As for the three-mode case, the so-called sum-squeezing is impossible but another kind of squeezing may arise for the orders K that are a multiple of three. The degree of the lowest allowable K = 3 order squeezing can reach a remarkable amount of 18%. Of interest is the following property: when the order grows, the degree of antibunching increases but that of squeezing decreases.

We analyse the entanglement involved in the common eigenvector |ζ⟩ of the centre-of-mass coordinate and mass-weighted relative momentum of two particles. Its Schmidt decomposition exhibits that the |ζ⟩ state maximally entangles two single-mode coordinate (or momentum) eigenstates, of which one is squeezed. The squeezing parameter is related to the ratio of the masses of two particles. The |ζ⟩ state may describe the entangled state produced by a non-50/50 beamsplitter. Its applications in entanglement swapping and quantum dense coding are discussed. The corresponding entangling operators and swapping operators are derived.

This special issue is composed mainly of extended versions of talks and papers presented at the Seventh International Conference on Squeezed States and Uncertainty Relations held in Boston, Massachusetts, USA, on 4-8 June 2001. The conference was hosted by Boston University.

The first meeting of this kind took place in College Park, Maryland, USA in 1991 as a small workshop. The second and third meetings were held in Moscow, Russia and Baltimore, USA in 1992 and 1993, respectively. The fourth meeting took place in Taiyuan, Shanxi Province, China in 1995 as an international conference sponsored by the International Union of Pure and Applied Physics. The fifth and sixth meetings were held in Balatonfüred, Hungary and Naples, Italy in 1997 and 1999, respectively. The eighth conference will take place in Puebla, Mexico in 2003.

The contributions to this special issue address, in the first instance, the foundations of quantum mechanics and quantum optics, and several modern topics which have been intensively developed during the last decade. Quantum fluctuations, squeezing phenomena and different kinds of nonclassical states of light represent some of the most intensively discussed subjects. In particular, several papers deal with the properties of different kinds of `Schrödinger cat states' modelled in the simplest case by the even and odd coherent states. This paradigm exhibits in the most distinct form such cornerstone properties of quantum mechanics as the superposition principle, which manifests itself in quantum interference and entanglement.

The quantum entanglement phenomenon has been studied intensively for the past few years due to its importance, for example, in quantum computer projects. In spite of the fact that this concept was known from the beginning of quantum mechanics, its properties, for example, the most appropriate entanglement measures, are still areas of discussion. Another related aspect of quantum nature is teleportation, which is also a subject of intensive study. The phenomena of dissipations and decoherence of the initial pure states are also very important, because fast decoherence can destroy all the advantages of quantum processes in quantum computing, teleportation and image processing. Due to these aspects, papers concerning various methods of controlling the decoherence, for example, by means of using different kinds of nonlinearities and deformations, are also presented here.

From the very beginning of quantum mechanics, the uncertainty relations were basic inequalities distinguishing the classical and quantum worlds. Developing a deeper understanding of the well-known uncertainty relations and striving to find new ones (connected with purity parameters of states, entropy, etc.) is an important aspect of modern research. Recently developed tomographic methods of measuring quantum states (in terms of the Wigner function or other phase-space characteristics) have resulted in the formulation of quantum mechanics in which the standard probability density describes the quantum states in an alternative way to the density matrix or wave function. Some papers devoted to tomographic methods of measuring quantum states and studying different sorts of uncertainty relations are presented here.

Group-theoretical methods of studying quantum-mechanical and quantum-optical phenomena, including the group-representation description of light polarization, also attract ongoing interest in the conference series discussions.

We hope that this special issue will help readers to obtain an overview of the current trends in modern quantum optics, both experimental and theoretical.

Young S KimUniversity of Maryland, College Park, USAMargarita A Man'koP.N. Lebedev Physical Institute, Moscow, RussiaAlexander Sergienko Boston University, MA, USA

Guest Editors

The results of quantum analysis of the light generated by consecutive and simultaneous quasi-phase-matched (QPM) wave interactions in periodically poled nonlinear crystals (PPNCs) are presented. In the case of the consecutive interactions of waves with multiple frequencies ω, 2ω and 3ω, the parametric amplification at low-frequency pumping is investigated. Generation of the quadrature-squeezed light at frequencies ω and 3ω in the 2ω frequency pumping field, the photon statistics and correlation as well as the entanglement properties for photons produced at different frequencies are studied. In the case of simultaneous QPM spontaneous parametric down-conversion processes in single PPNC, the main attention is paid to obtaining the polarization-entangled states at the collinear geometry of the wave interaction.

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr ² and 37% if the `superpurity' lim _r→0[Tr (^1+1/r)]^r is chosen as a measure of deviation).

Traditional methods of cryptographic key distribution rest on judgments about an attacker. With the advent of quantum key distribution (QKD) came proofs of security for the mathematical models that define the protocols BB84 and B92; however, applying such proofs to actual transmitting and receiving devices has been questioned.

Proofs of QKD security are propositions about models written in the mathematical language of quantum mechanics, and the issue is the linking of such models to actual devices in an experiment on security. To explore this issue, we adapt Wittgenstein's method of language games to view quantum language in its application to experimental activity involving transmitting and receiving devices.

We sketch concepts with which to think about models in relation to experiments, without assuming the experiments accord with any model; included is a concept of one quantum mechanical model enveloping another. For any model that agrees with given experimental results and implies the security of a key, there is an enveloping model that agrees with the same results while denying that security. As a result there is a gap between equations and the behaviour recorded from devices in an experiment, a gap bridged only by resort to something beyond the reach of logic and measured data, well named by the word guesswork.

While this recognition of guesswork encourages eavesdropping, a related recognition of guesswork in the design of feedback loops can help a transmitter and receiver to reduce their vulnerability to eavesdropping.

We study the possibility of exploiting superpositions of coherent states to encode qubits. A comparison between the use of deformed and undeformed bosonic algebra is made in connection with the amplitude damping errors.

Entanglement, including `quantum entanglement', is a consequence of correlation between objects. When the objects are subunits of pairs which in turn are members of an ensemble described by a wavefunction, a correlation among the subunits induces the mysterious properties of `cat-states'. However, correlation between subsystems can be present in purely non-quantum sources, thereby entailing no unfathomable behaviour. Such entanglement arises whenever the so-called `qubit space' is not afflicted with Heisenberg uncertainty. It turns out that all optical experimental realizations of the Einstein, Podolsky and Rosen (EPR) Gedanken experiment in fact do not suffer Heisenberg uncertainty. Examples will be analysed and non-quantum models for some of these described. The consequences for experiments that were to test EPRs contention in the form of Bell's theorem are drawn: valid tests of EPR's hypothesis have yet to be done.

We describe a novel tool for the quantum characterization of optical devices. The experimental set-up involves a stable reference state that undergoes an unknown quantum transformation and is then revealed by balanced homodyne detection. Through tomographic analysis of the homodyne data we are able to characterize the signal and to estimate parameters of the interaction, such as the loss of an optical component or the gain of an amplifier. We present experimental results for coherent signals, with application to the estimation of losses introduced by simple optical components, and show how these results can be extended to the characterization of more general optical devices.

We investigate a modified model of the Jaynes–Cummings model involving multimode and multiphoton processes. The interaction part of the Hamiltonian consists of a two-level atom interacting with two isotropic fields injected simultaneously within a perfect cavity. The equation governing the system are solved and the relevant wave functions for one-and two-photon processes are obtained. In the two-photon case we invoked the SU(1, 1) Lie algebra to reach this solution. The density matrix elements are calculated and the entropy and the atomic inversion are computed. The influence of the mean photon numbers and the detuning on these quantities are studied. Comparison between the present model and the usual nonlinear model is given. The phase properties are also discussed.

If a broadband squeezed vacuum is treated as a reservoir with respect to a two-level atom, the non-zero phase-dependent reservoir correlation functions characterizing the squeezed vacuum introduce `squeezing terms' to the master equation. These terms are responsible, for example, for the well known narrowing of the spectral lines in the resonance fluorescence spectrum of the atom. For a squeezed vacuum reservoir with finite bandwidth it is possible to derive the master equation that is consistent with the Born-Markov approximation by a two-step procedure which consists in dressing the atom first and next coupling it to the finite-bandwidth reservoir. The master equation is valid whenever the bandwidth of the reservoir is much broader than the atomic linewidth but not necessarily broader than the Rabi frequency. This procedure can be applied not only for the squeezed vacuum reservoir but also for the ordinary vacuum with the structure of the mode density that is not flat, e.g., modelled by a Lorentzian function as in a cavity. The master equation for this case, in the operator form, shows some similarities to the master equation for the squeezed vacuum reservoir due to the presence of the `squeezing-like terms'. The similarities and differences of the two master equations are discussed in this paper.

In this paper we investigate the single-mode properties (Wigner function, photon-number distribution and quadrature squeezing) of a dissipative parametric amplifier when the signal and idler modes are initially prepared in the Schrödinger-cat states. We show that there are two sources producing decoherence in the system, which are spontaneous pump photon decay and interaction with the environment. For the latter case, the system collapses to both statistical mixture states as well as thermal states governed by the interaction parameters.

A 532 nm pumped type-II phase-matched, doubly resonant KTP optical parametric amplifier (OPA) was operated near frequency degeneracy to yield an inferred downconverted photon pair production rate of 1.7×10⁶ s^-1 at a pump power of 100 µW. The OPA output consisted of three components: narrowband doubly resonant mode pairs; narrowband singly resonant mode pairs for which either the signal or idler was resonant with the cavity and broadband nonresonant mode pairs. Under frequency-degenerate operation, the broadband nonresonant mode pairs were polarization triplet states. We observed quantum interference between the orthogonally polarized photons of the triplet states when they were analysed with a polarizer set at 45° relative to the OPA's output polarizations, leading to reduced coincidence counts.

We study a scheme for entangling two-level atoms located close to the surface of a dielectric microsphere. The effect is based on medium-assisted spontaneous decay, rigorously taking into account dispersive and absorptive properties of the microsphere. We show that even in the weak-coupling regime, where the Markov approximation applies, entanglement up to 0.35 ebits between two atoms can be created. However, larger entanglement and violation of Bell's inequality can only be achieved in the strong-coupling regime.

We provide a brief overview of the newly born field of quantum imaging, and discuss some concepts that lie at the root of this field.

We found analytically the exact joint Wigner function (WF) of an atom-field system described by the Jaynes-Cummings model in the dispersive limit for a given initial state. Using the solution of a master equation with a dissipative term calculated by Peixoto de Faria and Nemes (Peixoto de Faria J G and Nemes M C 1999 Phys. Rev. A 59 3918) and a general expression for the joint WF proposed by Czirják and Benedict (Czirják A and Benedict M G 1996 Quantum Semiclass. Opt.8 975), we obtained the exact reduced atom and field WFs and analysed their time evolution. Our results show that, at the maximum entanglement times, the atom tends to a statistical mixture state represented by a spherical polar plot. Also, the atomic WF shows how fast the negative-valued part disappears as a function of dissipation constant and the strength of the field coherent state.

We study the problem of quantum state exchange between two coupled modes of the electromagnetic field modelled by quantum oscillators. Analysing the structure of propagators of the Schrödinger equation with the most general weak bilinear resonance coupling, we find the conditions of the exchange in both `narrow' and `wide' senses.

The combinatorial properties of the Bell polynomials are shown to follow from the algebraic properties of the boson operators in an extremely simple way. This treatment sheds further light on the `hidden' combinatorial significance of the boson operators and of the identities they satisfy, and allows a consistent introduction of q-deformations.

We examine the time-dependent squeezing of fluctuations associated with the atomic coherent state of the SU(2) Lie algebra. Based on the exact solutions for a non-dissipative SU(2) system, regions of squeezing and its symmetric/asymmetric patterns are investigated in the cases of weak and strong coupling and for the resonance condition. For a spontaneously damped two-level atom, transient squeezing occurs in the atomic polarization components with initial atomic coherent state |θ,ϕ⟩, with asymmetric structure with respect to the state-of-zero atomic inversion. The rapidly vanishing squeezed regions for <θ<π are accompanied by entropy reduction.

It is shown that any state of two modes of the electromagnetic field (or any other bipartite bosonic system) can be expressed as a coherent superposition of conjugate coherent-state pairs. This representation has a deep connection with both the two-mode squeezing operator and a maximally entangled basis. A description of continuous variable quantum teleportation is presented, as an example of the application of this method.

Dynamical behaviour of a single harmonic oscillator (HO) and of a single and two cooperative atoms in front of a phase-conjugated mirror is investigated without using the rotating-wave approximation. The mean photon number of the HO shows transient oscillation of frequency (2ω₀) and O(γ/ω₀), the ratio of the free-space decay rate to the oscillation frequency, and the fluorescent spectrum becomes asymmetric due to additional resonant and non-resonant dispersive terms. In the single-two-level-atom case, the mean atomic inversion and the fluorescent intensity show steady oscillation O(γ₀/ω₀), the ratio of the A-coefficient to the atomic transition frequency. The amplitude of this steady oscillation at frequency (2ω₀) is larger in the case of two cooperative atoms.

New generalized squeezed states for the time-dependent harmonic oscillator are found through the theory of invariants. Our method gives a comparatively clearer picture than methods using evolution operators because we can establish a direct connection between the classical and quantum solutions. The additional significance of our method is that it is possible to find new generalized quantum squeezed states from just one particular solution to the classical time-dependent oscillator. Accordingly, more general results for the variance of x (squared uncertainty) are found for the cases of linear sweep of the restoring force and compared to recent results encountered in the literature.

Optical quantum nondemolition measurements are performed using a beamsplitter with a nonclassical meter input and a electro-optic feedforward loop. The nonclassical meter input is provided by a stable 4.5 dB amplitude squeezed source generated by an optical parametric amplifier. We show that the implementation of a feedforward loop gives a substantial improvement in the signal transfer efficiency. With a 92% reflective beamsplitter we measure a transfer coefficient of T_s+m = 1.81. The quantum state preparation ability of our system is evaluated using two-dimensional correlation plots. This technique allows the direct visualization of the quantum correlations between the outputs of the QND system.

The propagation of polarized photons in optical media can be effectively modelled by means of quantum dynamical semigroups. These generalized time evolutions consistently describe phenomena leading to loss of phase coherence and dissipation originating from the interaction with a large, external environment. Highly sensitive experiments in the laboratory can provide stringent bounds on the fundamental energy scale that characterizes these non-standard effects.

Semiclassically, the signal-spontaneous beat noise in an optical amplifier can be reduced by 3 dB via a filtering technique. This creates a paradox since the noise figure's quantum limit of 3 dB could then be reduced to 0 dB, implying noiseless amplification even though noise is still being introduced by the amplifier. We discuss ways in which this paradox might be resolved and present experimental evidence that dispenses (up to the noise floor of our detectors) with one of these. A semiclassical treatment of the effects of a real (rather than ideal) filter is also presented and suggestions are made for improved (multi-mode) quantum models of this noise.