Current Trends in Quantum Optics Subhashish Banerjee1, ∗ Arun Jayannavar2, y 1Indian Institute of Technology Jodhpur, India. 2Institute of Physics, Bhubaneswar, India. Here we review some of the recent developments in Quantum Optics. After a brief introduction to the historical development of the subject, we discuss some of the modern aspects of quantum optics including atom field interac- tions, quantum state engineering, metamaterials and plasmonics, optomechan- ical systems, PT (Parity-Time) symmetry in quantum optics as well as quasi- probability distributions and quantum state tomography. Further, the recent developments in topological photonics is briefly discussed. The potent role of the subject in the development of our understanding of quantum physics and modern technologies is brought out. PACS numbers: 03.65.Yz,03.67.-a I. Introduction needs to treat it quantum mechanically. This lead to the development of the quantum theory Light is intimately connected to existence of of radiation, which, in turn, was the precursor all forms of life. The systematic study of light is to quantum optics [2]. The photon provides an known as optics and could be traced historically example of a simple quantum state labelled by, to [1]. The notion of light as an electromag- say, its hoizontal and vertical polarizations as netic field was made clear by Maxwell resulting j i = c jHi + c jV i : in his celebrated work on what is now known 1 2 as Maxwell equations. Basically, he showed Here, j i, pronounced as ket psi, denotes the that electromagnetic fields in vacuum propa- state of the system and is a vector and is an gate at the speed of light. The next step in example of a superposition state. Also, c1 and the history of this subject were the questions c2 are in general, complex numbers such that 2 2 raised by the Michealson-Moreley experiment jc1j + jc2j = 1. The states jHi and jV i can 2 and the Rayleigh-Jeans catastrophe associated exist simultaneously with probabilities jc1j and 2 with black-body radiation. The former lead to jc2j , respectively. This is the surprising con- the development of special theory of relativity tent of quantum mechanics that makes it stand and the later to Plancks resolution, which pro- apart from classical physics. This feature can be vided the first seeds for the field of quantum me- used as a resource for achieving things not possi- chanics. ble in the classical realms. Further, it should be The notion of photon, essentially considering it noted that in quantum mechanics, we deal with as a particle, was first realized in Einstein’s work operators characterizing the observables. For ex- arXiv:1902.08576v1 [quant-ph] 21 Feb 2019 on the photo-electric effect. A large number of ample, energy is represented by the operator H^ phenomena, related to optics, could be explained called the Hamiltonian of the system. Exper- by invoking the concept of photons involving a iments typically involve making measurements classical electromagnetic field along with vacuum of the operator on the state vector and what is fluctuations. It was soon realized, in the wake obtained as a result of the measurement is the of experimental developments, that in order to eigenvalue of the operator. A simple example understand the full potential of the photon, one of measurement would be a projection operator that acts on the state of the system to be mea- sured and take it to the final state. A basic tenet ∗ [email protected] of quantum mechanics is that the results of the y [email protected] process of measurement are probabilistic in na- 2 ture, and is known as the Born rule. tribute themselves uniformly, i.e., the variance Quantum measurements are described, in gen- of the photon distribution is equal to its mean. eral, by a collection fMmg of measurement op- In contrast to this, in sub-Poissonian statistics, erators, acting on the state space of the sys- the variance is less than the mean and was, first tem being measured. The index m refers to experimentally demonstrated in [6]. The devi- the measurement outcomes that may occur in ation from Poissonian statistics is quantified by the experiment. If the state of the quantum the Mandel QM parameter, discussed below in system is j ii immediately before the measure- the context of quantum correlations. A closely ment, then the probability that result m occurs related albeit distinct phenomenon is that of y y is h ijMmMmj ii. Here, h ijMm is the hermitian photon antibunching. Photon bunching is the conjugate of the vector Mmj ii. The measure- tendency of photons to distribute themselves in ment leaves the system in the state bunches, such that when light falls on a photo detector, more photon pairs are detected close Mm j ii together in time than further apart. In contrast j ii ! j f i = q : (1) y to this, in antibunching, the probability of pho- h ijMmMmj ii ton pairs detection further apart is greater than The measurement operators satisfy the com- that for pairs close together. Both the phenom- P y ena of sub-Poissonian statistics and photon an- pleteness condition m MmMm = 1, express- ing the fact that probabilities sum up to one. tibunching are purely quantum in nature. y For those operators Mm where Mm = Mm and 2 Mm = Mm, this reduces to the projection oper- ators mentioned above. A major technological development due to Quantum optics provides tools to study foun- quantum optics has been the development of dations of quantum mechanics with precision LASER (Light Amplification by Stimulated and is the cause of a number of developments Emission of Radiation), where coherent states in quantum information and quantum technol- play an essential role. Quantum entanglement, ogy. An important development in quantum which is quintessence of the quantum world, optics came with the formulation of coherent has been experimentally demonstrated with pho- states [3, 4], which basically ask the question tons and has proved to be a potential re- of what are the states of the field that most source in quantum computation and communica- nearly describes a classical electromagnetic field. tion. Spontaneous Parametric Down Conversion This was originally introduced by Schrodinger (SPDC) is a frequently used process in quan- [5]. Classically an electromagnetic field, light, tum optics which is very useful for generating has a well defined amplitude and phase, a picture squeezed states, single and entangled photons. that changes in the quantum mechanical sce- nario. A field in a coherent state is a minimum- uncertainity state with equal uncertainities in the two variables, in this context, often termed Having sketched the roots of quantum op- as the two quadrature components. This was tics, we will, in the remaining part of this ar- followed by the development of squeezed states, ticle, try to provide a flavor for some of the where fluctuations in one quadrature component modern developments in the field. In this con- are reduced below that of the corresponding co- text, the following topics will be discussed: (a) herent state. Atom-Field interactions, (b) Quantum state en- In the quest for characterizing the nonclassi- gineering, (c) Metamaterials and Plasmonics, cal behavior of photons, anti-bunching and sub- (d) Optomechanical systems, (e) PT (Parity- Poissonian statistics were investigated. The co- Time) symmetry in quantum optics, (f) Quasi- herent field is represented by Poissonian pho- probability distributions and quantum state to- ton statistics, in which the photons tend to dis- mography and (g) topological photonics. 3 II. Atom-Field Interactions ent control of atoms’ internal states to determin- istically prepare superposition states and extend A cannonical model of quantum optics is the this control to the external (motional) states Rabi model which deals with the response of an of atoms. The Haroche group [9] used Cavity atom to an applied field, in resonance with the Quantum Electrodynamics (QED), i.e., the con- atom’s natural frequency. Rabi considered the trolled interaction of atom with electromagnetic problem of a spin-half magnetic dipole undergo- field in an appropriate cavity to bring out vari- ing precessions in a magnetic field. He obtained ous aspects of decoherence. Microwave photons the probability of a spin-half atom flipping from trapped in a superconducting cavity constitute j0i = (1 0)T or j1i = (0 1)T to the states j1i or an ideal system to realize some of the thought ex- j0i, respectively, by an applied radio-frequency periments that form the foundations of quantum magnetic field. Here T stands for the transpose physics. The interaction of these trapped pho- of the row matrix. In the context of quantum tons with Rydberg atoms, effectively two-level optics, Rabi oscillations would imply oscillations atoms, crossing the cavity illustrates fundamen- of the atom, considered as a two-level system, tal aspects of measurement theory. between the upper and lower levels under the influence of the electromagnetic field. The Rabi model lead to the development of the well-known B. Quantum Correlations Jaynes Cumming model, which is a generaliza- tion of the semi-classical Rabi model in the sense Here we discuss the generation of various that it invokes the interaction of the atom and facets of quantum correlations, including en- the single mode of the electromagnetic field in tanglement, arising from atom-field interactions. a cavity. In contrast to the semiclassical Rabi The widely studied among quantum correlations model, here the electromagnetic field is quan- is the non separability of the quantum state of tized. a system comprised of two or more subsystems. The Jaynes-Cummings model has provided a In the simplest scenario, consider two systems platform for numerous investigations, including with state vectors j i and jφi.