Focus issue on Quantum Optics in the International Year of Light

The quantum physics of light is a most fascinating field. Here I present a very personal viewpoint, focusing on my own path to quantum entanglement and then on to applications. I have been fascinated by quantum physics ever since I heard about it for the first time in school. The theory struck me immediately for two reasons: (1) its immense mathematical beauty, and (2) the unparalleled precision to which its predictions have been verified again and again. Particularly fascinating for me were the predictions of quantum mechanics for individual particles, individual quantum systems. Surprisingly, the experimental realization of many of these fundamental phenomena has led to novel ideas for applications. Starting from my early experiments with neutrons, I later became interested in quantum entanglement, initially focusing on multi-particle entanglement like GHZ states. This work opened the experimental possibility to do quantum teleportation and quantum hyper-dense coding. The latter became the first entanglement-based quantum experiment breaking a classical limitation. One of the most fascinating phenomena is entanglement swapping, the teleportation of an entangled state. This phenomenon is fundamentally interesting because it can entangle two pairs of particles which do not share any common past. Surprisingly, it also became an important ingredient in a number of applications, including quantum repeaters which will connect future quantum computers with each other. Another application is entanglement-based quantum cryptography where I present some recent long-distance experiments. Entanglement swapping has also been applied in very recent so-called loophole-free tests of Bell's theorem. Within the physics community such loophole-free experiments are perceived as providing nearly definitive proof that local realism is untenable. While, out of principle, local realism can never be excluded entirely, the 2015 achievements narrow down the remaining possibilities for local realistic explanations of the quantum phenomenon of entanglement in a significant way. These experiments may go down in the history books of science. Future experiments will address particularly the freedom-of-choice loophole using cosmic sources of randomness. Such experiments confirm that unconditionally secure quantum cryptography is possible, since quantum cryptography based on Bell's theorem can provide unconditional security. The fact that the experiments were loophole-free proves that an eavesdropper cannot avoid detection in an experiment that correctly follows the protocol. I finally discuss some recent experiments with single- and entangled-photon states in higher dimensions. Such experiments realized quantum entanglement between two photons, each with quantum numbers beyond 10 000 and also simultaneous entanglement of two photons where each carries more than 100 dimensions. Thus they offer the possibility of quantum communication with more than one bit or qubit per photon. The paper concludes discussing Einstein's contributions and viewpoints of quantum mechanics. Even if some of his positions are not supported by recent experiments, he has to be given credit for the fact that his analysis of fundamental issues gave rise to developments which led to a new information technology. Finally, I reflect on some of the lessons learned by the fact that nature cannot be local, that objective randomness exists and about the emergence of a classical world. It is suggestive that information plays a fundamental role also in the foundations of quantum physics.

Dark states were first observed in optical pumping experiments and interpreted as a quenching of fluorescence due to a destructive interference between two absorption amplitudes connecting two ground state sublevels g₁ and g₂ to an excited sublevel e. The atom can absorb light from g₁ and jump to e. Similarly, it can absorb light from g₂ and jump to e. But, if it is in a certain linear superposition of g₁ and g₂, ${c}_{1}{g}_{1}+{c}_{2}{g}_{2}$, the two absorption amplitudes from g₁ to e and from g₂ to e interfere destructively and cancel out. The fluorescence stops. The state ${c}_{1}{g}_{1}+{c}_{2}{g}_{2}$ from which light cannot be absorbed is called a dark state. This paper will start with a description of the first experiments demonstrating the existence of dark states and of the first theoretical interpretations that have been proposed. Dark states appeared to play an essential role in several new physical effects, such as electromagnetically induced transparency, slow light, stimulated Raman adiabatic passage (STIRAP), which will be briefly described. A special emphasis will be given to the applications of dark states in the field of ultracold atoms and molecules. It will be first shown how the use of velocity dependent dark states allows atoms to be cooled below the so-called recoil limit, corresponding to a velocity dispersion of the cooled atoms smaller than the recoil momentum communicated to the atom by the absorption of a single photon. It turned out that a quantitative interpretation of this subrecoil cooling may be given in terms of anomalous random walks and Lévy statistics. Finally, recent applications of dark states to the production of ultracold molecules will be described: production of ultracold polar molecules by the combination of Feshbach resonances and STIRAP; dark states in the photo-association of ultracold atoms allowing a precise determination of the scattering length describing their collisions.

We have developed a simple Newtonian dynamics for the motion of rays as particles that are governed by Snell's law. In Newton's original formulation, the particle moves faster in a relatively higher index of refraction medium. We show that it is the constant mass assumption that leads to this conclusion. We derive an explicit expression for the mass as a function of position and show that the formulation leads to the conclusion that indeed the particle moves slower in a relatively higher index of refraction medium. Our approach leads to a simple Newtonian particle picture where the equations of motion may be simply written. We obtain explicit expressions for the velocity, acceleration, and forces which allow one to write the equations of motion. We also formulate the dynamics in terms of the Lagrangian and Hamiltonian formulations, taking variable mass into account. The solutions to the dynamics are such that the particle always follows Snell's law in a variable index of refraction medium. Exactly solvable analytic examples are given. We also we show that the SOFAR channel phenomenon, where a wave is trapped between two regions, is easily explained in the particle picture.

We use the P-distribution to show that the familiar values 1, 2 and 3 of the normalized second order correlation function at equal times ${g}^{(2)}(0)$ corresponding to a coherent state, a thermal state and a highly squeezed vacuum are a consequence of the number of dimensions these states take up in quantum phase space. Whereas the thermal state exhibits rotational symmetry and thus extends over two dimensions, the squeezed vacuum factorizes into two independent one-dimensional phase space variables, and in the limit of large squeezing is therefore a one-dimensional object. The coherent state is a point in the phase space of the P-distribution and thus has zero dimensions. The fact that for photon number states the P-distribution is even narrower than that of the zero-dimensional coherent state suggests the notion of 'negative' dimensions.

The characterization of the polarization properties of a quantum state requires the knowledge of the joint probability distribution of the Stokes variables. This amounts to assessing all the moments of these variables, which are aptly encoded in a multipole expansion of the density matrix. The cumulative distribution of these multipoles encapsulates in a handy manner the polarization content of the state. We work out the extremal states for that distribution, finding that SU(2) coherent states are maximal to any order, so they are the most polarized allowed by quantum theory. The converse case of pure states minimizing that distribution, which can be seen as the most quantum ones, is investigated for a diverse range of number of photons. Exploiting the Majorana representation, the problem appears to be closely related to distributing a number of points uniformly over the surface of the Poincaré sphere.

I revisit Niels Bohr's famous 1932 'Light and Life' lecture, confronting it with current knowledge. Topics covered include: life origin and evolution, quantum mechanics and life, brain and mind, consciousness and free will, and light as a tool for biology, with special emphasis on optical tweezers and their contributions to biophysics. Specialized knowledge of biology is not assumed.

In the last decade quantum cascade lasers (QCLs) have become the most widely used source of mid-infrared radiation, finding large scale applications because of their wide tunability and overall high performance. However far-infrared (terahertz) QCLs have lagged behind in terms of performance and impact due to the inability so far of achieving room temperature operation. Here we review recent research that has led to a new class of QCL light sources that has overcome these limitations leading to room temperature operation in the terahertz spectral range, with nearly 2 mW of optical power and significant tunability, opening up also this region of the spectrum to a wide range of applications.

In this article we argue that thermal reservoirs (baths) are potentially useful resources in processes involving atoms interacting with quantized electromagnetic fields and their applications to quantum technologies. One may try to suppress the bath effects by means of dynamical control, but such control does not always yield the desired results. We wish instead to take advantage of bath effects, that do not obliterate 'quantumness' in the system-bath compound. To this end, three possible approaches have been pursued by us. (i) Control of a quantum system faster than the correlation time of the bath to which it couples: such control allows us to reveal quasi-reversible/coherent dynamical phenomena of quantum open systems, manifest by the quantum Zeno or anti-Zeno effects (QZE or AZE, respectively). Dynamical control methods based on the QZE are aimed not only at protecting the quantumness of the system, but also diagnosing the bath spectra or transferring quantum information via noisy media. By contrast, AZE-based control is useful for fast cooling of thermalized quantum systems. (ii) Engineering the coupling of quantum systems to selected bath modes: this approach, based on field–atom coupling control in cavities, waveguides and photonic band structures, allows one to drastically enhance the strength and range of atom–atom coupling through the mediation of the selected bath modes. More dramatically, it allows us to achieve bath-induced entanglement that may appear paradoxical if one takes the conventional view that coupling to baths destroys quantumness. (iii) Engineering baths with appropriate non-flat spectra: this approach is a prerequisite for the construction of the simplest and most efficient quantum heat machines (engines and refrigerators). We may thus conclude that often thermal baths are 'more friends than foes' in quantum technologies.

The Hamiltonian optics notion of phase-space modes is shown to be central to understanding self-focusing, multiple filamentation, and the λ² scaling of the self-focusing threshold with the radiation wavelength λ.

Correlations in photodetection signals from quantum light sources are conventionally calculated by application of the source master equation and the quantum regression theorem. In this article we show how the conditioned dynamics, associated with the quantum theory of measurements, allows calculations and offers interpretations of the behavior of the same quantities. Our theory is illustrated for photon counting and field-amplitude measurements, and we show, in particular, how transient correlations between field-amplitude measurements and later photon counting events can be accounted for by a recently developed theory of past quantum states of a monitored quantum system.

We review and interpret a modern approach to laser theory, steady-state ab initio laser theory (SALT), which treats lasing and amplification in a unified manner as a non-unitary scattering problem described by a nonlinear scattering matrix. Within the semiclassical version of the theory the laser line has zero width as the lasing mode corresponds to the existence of an eigenvector of the S-matrix with diverging eigenvalue due to the occurrence of a pole of the scattering matrix on the real axis. In this approach the system is infinite from the outset and no distinction is made between cavity modes and modes of the Universe; lasing modes exist both in the cavity and in the external region as solutions satisfying Sommerfeld radiation boundary conditions. We discuss how such solutions can be obtained by a limiting procedure in a finite box with damping according to the limiting absorption principle. When the electromagnetic and matter fields are treated as operators, quantum fluctuations enter the relevant correlation functions and a finite linewidth is obtained, via a generalization of SALT to include noise (N-SALT). N-SALT leads to an analytic formula for the linewidth that is more general than all previous corrected versions of the Schawlow–Townes formula, and can be evaluated simply from knowledge of the semiclassical SALT modes. We derive a simpler version of this formula which emphasizes that the noise is dominated by the fluctuations in the polarization of the gain medium and is controlled by the rate of spontaneous emission.

Collective modes manifest themselves in a variety of different physical systems ranging from superconductors to superfluid ³He. The collective modes are generated via the Higgs–Anderson mechanism that is based on the symmetry breaking double well potential. Recently, collective modes were explored in superconducting NbN and InO in the presence of a strong terahertz laser field. In both cases a single collective mode that oscillates with twice the frequency of the superconducting energy gap Δ was discovered. Superfluid ³He is the host for a whole variety of collective modes. In particular, in the superfluid ³He B-phase, two massive collective modes were found with masses $\sqrt{8/5}{\rm{\Delta }}$ and $\sqrt{12/5}{\rm{\Delta }}$. We show that for both cases of the superconducting films and for the superfluid ³He B-phase, the collective modes satisfy the Nambu identity that relates the masses of different collective modes to the energy gap parameter Δ.

Stark splitting and quantum interference effects in the absorption spectrum of a probe field in coherently driven closed upper coupled three level atomic schemes (${\rm{\Lambda }}$ and upper-cascade), which are created by a strong driving field are studied using the dressed state representation. In the dressed representation, the absorption due to the Stark splitting can be separated from the absorption due to the interference effects. We explicitly show the presence of both destructive and constructive interference in upper coupled three level atomic schemes. The interference in these atomic schemes is due to the coupling of the two dressed states by the same vacuum modes and the probe. We show that the dephasing rates can change the nature of the interference from constructive to destructive and vice versa, resulting in increased or decreased resonant absorption of the probe field.

We present a simple setup that exploits the interference of entangled photon pairs. 'Signal' photons are sent through a Mach–Zehnder-like interferometer, while 'idlers' are detected in a variable polarization state. Two-photon interference (in coincidence detection) is observed with very high contrast and for significant time delays between signal and idler detection events. This is explained by quantum erasure of the polarization tag and a delayed choice protocol involving a non-local virtual polarizer. The phase of the two-photon fringes is scanned by varying the path length in the signal beam or by rotating a birefringent crystal in the idler beam. We exploit this to characterize one beam splitter of the signal photon interferometer (reflection and transmission amplitudes including losses), using only information about coincidences and control parameters in the idler path. This is possible because our bi-photon state saturates the Greenberger–Yelin–Englert inequality between contrast and predictability.

Single photon emission from a collection of resonantly excited two-level atoms is an expanding field. Recent work has shown single photon superradiance from an extended ensemble yields enhanced directional spontaneous emission. This paper presents an operator which commutes with the observables ${{\bf{R}}}^{2},{R}_{z}$ and breaks their degeneracy for the single photon states. Its eigenvectors are a unimodular basis for the single photon states. A simple scheme is given for writing out these states directly without iterative construction and without requiring recourse to Gram–Schmidt orthogonalization. A relatively simple scheme is proposed for experimental realization. In the final part of the paper the mathematical method is extended to generate cooperative states with smaller cooperativity number R.

We study the feasibility of measuring vacuum birefringence by probing the focus of a high-intensity optical laser with an x-ray free electron laser (XFEL). This amounts to performing a new type of QED precision experiment, employing only laser pulses, hence space- and time-dependent fields. To set the stage, we briefly review the status of QED precision tests and then focus on the example of vacuum birefringence. Adopting a realistic laser beam model in terms of pulsed Gaussian beams we calculate the induced phase shift and translate it into an experimental signal, counting the number of photons with flipped polarization. We carefully design a detailed experiment at the European XFEL operating in self-seeded mode, supplemented by a petawatt class optical laser via the HIBEF project. Assuming all components to represent the current state of the art, in particular the x-ray polarizers, realistic estimates of signal-to-noise ratios plus ensuing acquisition times are provided. This is accompanied by a statistical analysis of the impact of poor laser focus overlap either due to timing and pointing jitter as well as limited alignment accuracy. A number of parasitic effects are analyzed together with appropriate countermeasures. We conclude that vacuum birefringence can indeed be measured upon combining an XFEL with a high-power optical laser if depolarization effects in the x-ray lenses can be controlled.

Quantum optics did not, and could not, flourish without the laser. The present paper is not about the principles of laser construction, still less a history of how the laser was invented. Rather, it addresses the question: what are the fundamental features that distinguish laser light from thermal light? The obvious answer, 'laser light is coherent', is, I argue, so vague that it must be put aside at the start, albeit to revisit later. A more specific, quantum theoretic, version, 'laser light is in a coherent state', is simply wrong in this context: both laser light and thermal light can equally well be described by coherent states, with amplitudes that vary stochastically in space. Instead, my answer to the titular question is that four principles are needed: high directionality, monochromaticity, high brightness, and stable intensity. Combining the first three of these principles suffices to show, in a quantitative way—involving, indeed, very large dimensionless quantities (up to $\sim {10}^{51}$)—that a laser must be constructed very differently from a light bulb. This quantitative analysis is quite simple, and is easily relatable to 'coherence', yet is not to be found in any textbooks on quantum optics to my knowledge. The fourth principle is the most subtle and, perhaps surprisingly, is the only one related to coherent states in the quantum optics sense: it implies that the description in terms of coherent states is the only simple description of a laser beam. Interestingly, this leads to the (not, as it turns out, entirely new) prediction that narrowly filtered laser beams are indistinguishable from similarly filtered thermal beams. I hope that other educators find this material useful; it may contain surprises even for researchers who have been in the field longer than I have.

Absorption is usually expected to be detrimental to quantum coherence effects. However, there have been few studies into the situation for complex absorption spectra. We consider the resonance fluorescence of excitons in a semiconductor quantum well. The creation of excitons requires absorption of the incoming pump-laser light. Thus, the absorption spectrum of the medium acts as a spectral filter for the emitted light. Surprisingly, absorption can even improve quantum effects, as is demonstrated for the squeezing of the resonance fluorescence of the quantum-well system. This effect can be explained by an improved phase matching due to absorption.

Device-independent quantum key distribution (DIQKD) generates a secret key among two parties in a provably secure way without making assumptions about the internal working of the devices used in the protocol. The main challenge for a DIQKD physical implementation is that the data observed among the two parties must violate a Bell inequality without fair-sampling, since otherwise the observed correlations can be faked with classical resources and security can no longer be guaranteed. In spite of the advances recently made to achieve higher detection efficiencies in Bell experiments, DIQKD remains experimentally difficult at long distances due to the exponential increase of loss in the channel separating the two parties. Here we describe and analyze plausible solutions to overcome the crucial problem of channel loss in the frame of DIQKD physical implementations.

Absorption of light at various wavelengths (i.e. absorption spectroscopy) is a powerful tool for identifying the presence of chemical compounds or specific substances in a sample. Cavity ring down spectroscopy (CRDS) is a well-known technique for very high sensitivity absorption spectroscopy. Another technique, integrating cavity spectroscopy has the additional unique feature of providing accurate absorption data even in the presence of severe scattering. This paper describes a combination of these two techniques that has led to an extremely powerful and useful new technology—integrating CRDS.

This article provides a brief review of how various spectroscopies have been used to investitage many-body quantum phenomena in the context of ultracold Fermi gases. In particular, work done with RF spectroscopy, Bragg spectroscopy and lattice modulation spectroscopy is considered. The theoretical basis of these spectroscopies, namely linear response theory in the many-body quantum physics context is briefly presented. Experiments related to the BCS–BEC crossover, imbalanced Fermi gases, polarons, possible pseudogap and Fermi liquid behaviour and measuring the contact are discussed. Remaining open problems and goals in the field are sketched from the perspective how spectroscopies could contribute.

In quantum mechanics the position and momentum operators are related to each other via the Fourier transform. In the same way, here we show that the so-called Pegg–Barnett phase operator can be obtained by the application of the discrete Fourier transform to the number operators defined in a finite-dimensional Hilbert space. Furthermore, we show that the structure of the London–Susskind–Glogower phase operator, whose natural logarithm gives rise to the Pegg–Barnett phase operator, is contained in the Hamiltonian of circular waveguide arrays. Our results may find applications in the development of new finite-dimensional photonic systems with interesting phase-dependent properties.

Squeezed light generation has come of age. Significant advances on squeezed light generation have been made over the last 30 years—from the initial, conceptual experiment in 1985 till today's top-tuned, application-oriented setups. Here we review the main experimental platforms for generating quadrature squeezed light that have been investigated in the last 30 years.

The past few decades have seen dramatic progress in our ability to manipulate and coherently control matter-waves. Although the duality between particles and waves has been well tested since de Broglie introduced the matter-wave analog of the optical wavelength in 1924, manipulating atoms with a level of coherence that enables one to use these properties for precision measurements has only become possible with our ability to produce atomic samples exhibiting temperatures of only a few millionths of a degree above absolute zero. Since the initial experiments a few decades ago, the field of atom optics has developed in many ways, with both fundamental and applied significance. The exquisite control of matter waves offers the prospect of a new generation of force sensors exhibiting unprecedented sensitivity and accuracy, for applications from navigation and geophysics to tests of general relativity. Thanks to the latest developments in this field, the first commercial products using this quantum technology are now available. In the future, our ability to create large coherent ensembles of atoms will allow us an even more precise control of the matter-wave and the ability to create highly entangled states for non-classical atom interferometry.

Quantum optics and classical optics are linked in ways that are becoming apparent as a result of numerous recent detailed examinations of the relationships that elementary notions of optics have with each other. These elementary notions include interference, polarization, coherence, complementarity and entanglement. All of them are present in both quantum and classical optics. They have historic origins, and at least partly for this reason not all of them have quantitative definitions that are universally accepted. This makes further investigation into their engagement in optics very desirable. We pay particular attention to effects that arise from the mere co-existence of separately identifiable and readily available vector spaces. Exploitation of these vector-space relationships are shown to have unfamiliar theoretical implications and new options for observation. It is our goal to bring emerging quantum–classical links into wider view and to indicate directions in which forthcoming and future work will promote discussion and lead to unified understanding.

The waveguide quantum electrodynamics (QED) system may have important applications in quantum device and quantum information technology. In this article we review the methods being proposed to calculate photon transport in a one-dimensional (1D) waveguide coupled to quantum emitters. We first introduce the Bethe ansatz approach and the input–output formalism to calculate the stationary results of a single photon transport. Then we present a dynamical time-dependent theory to calculate the real-time evolution of the waveguide QED system. In the longtime limit, both the stationary theory and the dynamical calculation give the same results. Finally, we also briefly discuss the calculations of the multiphoton transport problems.

Modern quantum optics encompasses a wide field of phenomena that are either related to the discrete quantum nature of light, the quantum wave nature of matter or light–matter interactions. We here discuss new perspectives for quantum optics with biological nanoparticles. We focus in particular on the prospects of matter-wave interferometry with amino acids, nucleotides, polypeptides or DNA strands. We motivate the challenge of preparing these objects in a 'biomimetic' environment and argue that hydrated molecular beam sources are promising tools for quantum-assisted metrology. The method exploits the high sensitivity of matter-wave interference fringes to dephasing and shifts in the presence of external perturbations to access and determine molecular properties.

Classical electromagnetism allows the rapidity of light field oscillations to be inferred from measurement of the speed and wavelength of light. Quantum mechanics connects the rapidity of electronic motion with the energy spacing of the occupied quantum states, accessible by light absorption and emission. According to these indirect measurements, both dynamics, the oscillation of light waves as well as electron wavepackets, evolve within attoseconds. With the birth of attosecond metrology at the dawn of the new millennium, light waving and atomic-scale electronic motion, being mutually the cause of each other, became directly measurable. These elementary motions constitute the primary steps of any change in the physical, chemical, and biological properties of materials and living organisms. The capability of observing them is therefore relevant for the development of new materials and technologies, as well as understanding biological function and malfunction. Here, I look back at milestones along the rocky path to the emergence of this capability, with some details about those my group had the chance to make some contributions to. This is an attempt to show—from a personal perspective—how revolution in science or technology now relies on progress at a multitude of fronts, which—in turn—depend on the collaboration of researchers from disparate fields just as on their perseverance.

Since the beginning of quantum physics, the relation between the properties of the microscopic quantum and the macroscopic classical world has been an important source for the development of the theory, and has led to new insights on the role of the environment in the transition from quantum to classical physics. Decoherence affects both coherence and entanglement of open systems. Quantum optics and cavity quantum electrodynamics have allowed detailed investigations of this phenomenon, within the framework of microwaves and light waves. In this paper, I present a personal account of theoretical and experimental developments that have led to the probing of the subtle frontier between quantum and classical phenomena.

Recent theoretical and experimental studies have given raise to new aspects in quantum measurements and error-disturbance uncertainty relations. After a brief review of these issues, we present an experimental test of the error-disturbance uncertainty relations in photon polarization measurements. Using a generalized, strength-variable measurement of a single photon polarization state, we experimentally evaluate the error and disturbance in the measurement process and demonstrate the validity of recently proposed uncertainty relations.

Artificially engineered light–matter systems constitute a novel, versatile architecture for the quantum simulation of driven, dissipative phase transitions and non-equilibrium quantum many-body systems. Here, we discuss recent experimental as well as theoretical works on the simulation of geometrical frustration in interacting photonic systems out of equilibrium. In particular, we review two recent discoveries at the interface of quantum optics and condensed matter physics: (i) the experimental achievement of Bosonic condensation into a flat energy band and (ii) the theoretical prediction of crystalline phases of light in a frustrated qubit-cavity array. We show that this new line of research leads to novel and unique tools for the experimental investigation of frustrated systems and holds the potential to create new phases of light and matter with interesting spatial structure.

As a contribution to the international year of light, we give a brief history of quantum optics in phase-space, with new directions including quantum simulations of multipartite Bell violations, opto-mechanics, ultra-cold atomic systems, matter-wave Bell violations, coherent transport and quantum fluctuations in the early Universe. We mostly focus on exact methods using the positive-P representation, and semiclassical truncated Wigner approximations.

We study a tripartite system of coupled spins, where a first set of one or two spins is our central system which is coupled to another set considered, the near environment, in turn coupled to the third set, the far environment. The dynamics considered are those of a generalized kicked spin chain in the regime of quantum chaotic dynamics. This allows us to test recent results that suggest that the presence of a far environment, coupled to the near environment, slows decoherence of the central system. After an extensive numerical study, we confirm previous results for extreme values and special cases. In particular, under a wide variety of circumstances an increasing coupling between near and far environment, slows decoherence, as measured by purity, and protects internal entanglement.

The free electron laser (FEL) has lived up to its promise as given in (Madey 1971 J. Appl. Phys. 42 1906) to wit: 'As shall be seen, finite gain is available ...from the far-infrared through the visible region ...with the further possibility of partially coherent radiation sources in the x-ray region'. In the present paper we review the history of the FEL drawing liberally (and where possible literally) from the original sources. Coauthors, Madey, Scully and Sprangle were involved in the early days of the subject and give a first hand account of the subject with an eye to the future.

Coherent multidimensional optical spectroscopy is broadly applied across the electromagnetic spectrum ranging from NMR to UV. These techniques reveal the properties of matter through the correlation plots of signal fields generated in response to sequences of short pulses with variable delays. Here we discuss a new class of multidimensional techniques obtained by the time-and-frequency-resolved photon coincidence counting measurements of N photons, which constitute a $2N$ dimensional spectrum. A compact description of these signals is developed based on time-ordered superoperators rather than the normally ordered ordinary operators used in Glauber's photon counting formalism. The independent control of the time and frequency gate parameters reveals fine details of matter dynamics not available otherwise. These signal are illustrated for application to an anharmonic oscillator model with fluctuating energy and anharmonicity.

We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive-P representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model increases substantially as a result of reducing excess thermal noise and squeezing the incident vacuum fluctuation on the out-coupling port.

We introduce a toy model for a diatomic molecule which is based on coupling electronic and nuclear spins to a rigid rotor. Despite its simplicity, the model can be used scientifically to analyze and understand complex molecular hyperfine spectra. In addition, the model has educational value as a number of fundamental symmetries and conservation laws of the molecule can be studied. Because of its simple structure, the model can be readily implemented as a computer program with comparatively short computing times on the order of a few seconds.

With the wealth of super-resolution techniques available in the literature it is useful to provide a succinct review of the general concepts involved in the different schemes. In this paper we group super-resolution schemes into several broad categories to simplify comparison, and to elucidate the factors limiting their respective resolutions.

In this paper we provide a simple, straightforward example of a specific situation in which weak-value amplification (WVA) clearly outperforms conventional measurement in determining the angular orientation of an optical component. We also offer a perspective reconciling the views of some theorists, who claim WVA to be inherently sub-optimal for parameter estimation, with the perspective of the many experimentalists and theorists who have used the procedure to successfully access otherwise elusive phenomena.

The interaction of three and two-level atomic systems with two electromagnetic fields, a weak near resonant high-frequency and a strong off-resonant low-frequency ones, is analyzed by applying the method of multiple-scale perturbation theory. We consider two examples: (1) a three-level Λ system interacting with a weak XUV field at the high-energy transition and a strong IR field at the adjacent low-energy transition. The IR field will periodically Stark modulate the energy of the most excited state, which will result in generation of sidebands of the XUV field at the frequencies multiple of the IR field modulation frequency; (2) a two-level atom interacting with a weak near-resonant high-frequency and an off-resonant low-frequency field, which is related to the quantum amplification by superradiant emission of radiation problem.