Current Trends in Quantum Optics

Subhashish Banerjee1, ∗ Arun Jayannavar2, † 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 |ψi = c |Hi + c |V i . in his celebrated work on what is now known 1 2 as Maxwell equations. Basically, he showed Here, |ψi, 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 |c1| + |c2| = 1. The states |Hi and |V i can 2 and the Rayleigh-Jeans catastrophe associated exist simultaneously with probabilities |c1| and 2 with black-body radiation. The former lead to |c2| , 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 Hˆ 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 † [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 {Mm} 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 |ψii 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 † † is hψi|MmMm|ψii. Here, hψi|Mm is the hermitian photon antibunching. Photon bunching is the conjugate of the vector Mm|ψii. 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 |ψii together in time than further apart. In contrast |ψii → |ψf i = q . (1) † to this, in antibunching, the probability of pho- hψi|MmMm|ψii ton pairs detection further apart is greater than The measurement operators satisfy the com- that for pairs close together. Both the phenom- P † 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. † 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 |0i = (1 0)T or |1i = (0 1)T to the states |1i or an ideal system to realize some of the thought ex- |0i, 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 |ψi and |φi. A simple choice those related to quantum computation. How- for the combined state of the combined system ever, one needs to keep in mind that the quan- would |ψi⊗|φi where ⊗ denotes the tensor prod- tum mechanical coherences are subject to decay uct. However, one can think√ of states of the due to interactions with the surroundings [7]. form (|ψi ⊗ |φi ± |φi ⊗ |ψi)/ 2. By no means This leads to losses such as decoherence, i.e., loss can one write this state as a product of two ar- of coherence and dissipation which implies loss bitrary states. Such states are called entangled of energy. Hence, it is imperative that one has states, a well known example of which would be a good understanding of these processes. Ex- the Bell states [10]. A simple illustration gen- perimental progress in this direction is discussed erating a photon entangled state is shown in next. Fig. 1. For two optical modes aˆ and ˆb, suffi- cient condition for entanglement is given by the Hillery-Zubairy criteria: (i) haˆ†aˆˆb†ˆbi < |haˆˆb†i|2 A. Experimental studies on decoherence models (ii) haˆ†aˆihˆb†ˆbi < |haˆˆbi|2. Another important non- classical correlation exhibited by light is the sub- In a series of beautiful Ion Trap experiments, Poissonian statistics, see Fig. 2. The coherent the Wineland group [8] induced decoherence, the state of light is the closest classical description loss of coherence due to interaction with the in which the probability of finding n photons has n¯n −n¯ surroundings, and decay by coupling the atom Poissonian distribution: Pn = n! e . Here n¯ is to engineered reservoirs in which the coupling the average photon number and is equal to the to, and the state of the environment were con- variance, which in this case is ∆n2 =n ¯. Fluc- trolled. Here, the basic tool used was the coher- tuations are expected to increase the departure 4

FIG. 1. A single photon prepared in vertical linear polarization arrives from left, as illustrated by the yellow electric field amplitude. This photon carries zero orbital angular momentum, as illustrated by the yellow flat phase fronts. The single photon passes through the metasurface comprising dielectric nano-antennae (purple), and exits as a quantum entangled state, depicted as a superposition of the red and blue electric field amplitudes and with the corresponding vortex phase fronts opposite to one another [11].

III. Quantum State Engineering

Quantum state engineering (QSE) comprises of three processes: (1) preparation, (2) detection and (3) reconstruction of quantum states. In the recent past, QSE has appeared as the epithet for various proposals and experiments preparing in- teresting states of the quantized electromagnetic field and atomic systems. Their motivation is the potential application of nonclassical states to FIG. 2. Probability distribution tasks such as, teleportation, quantum computa- tion and communication, quantum cryptography and quantum lithography, among others. These harness the power of quantum correlations which occupy a central position in the quest for under- standing and harvesting the utility of quantum mechanics. A field state having holes in its pho- from the mean value and consequently one would ton number distribution (PND) corresponds to have what is called super Poissonian light for a nonclassical state. A challenge in QSE is to which ∆n2 > n¯, an example is the thermal light. make such holes by controlling their position and However, it is found that light under certain cir- depths in the PND. A class of states in quan- cumstance shows sub-Poissonian behavior, i.e., tum optics, of relevance to the present context, ∆n2 < n¯. This phenomenon has no classical are the intermediate states such as the binomial analog and is a quantum effect. Mandel param- states (BS), formed by an interpolation of the 2 eter defined by QM = (∆n − n¯)/n¯ measures the number state, characterized by the number of departure from the Poissonian statistics. There- photons in the state, with the coherent state, as fore, we have QM < 0, QM = 0 and QM > 0 for discussed above. The concept of hole burning sub-Poissonian, Poissonian and super-Poissonian has been extended to the atomic domain as well. light, respectively. Here the generation of spin squeezing proceeds 5

via hole burning of selected Dicke states, coun- ity and permeability. Metamaterials are artifi- terparts of the photonic number states, out of an cially fabricated materials, usually comprising of atomic coherent state, counterparts of the usual nanoscale structures designed to respond to light coherent states, prepared for a collection of N in different ways. They have been heavily stud- two−level atoms or ions. The atoms or ions of ied in last two decades and have been used to the atomic coherent state are not entangled, but demonstrate a wealth of fascinating phenomena the removal of one or more Dicke states gener- ranging from negative refractive index to super ates entanglement, and spin squeezing occurs for resolution imaging. In recent times, metamateri- some ranges of the relevant parameters. Spin als have emerged as a new platform in quantum squeezing in a collection of two-level atoms or optics and have been used to carry out a number ions is of importance for precision spectroscopy. of important experiments using single photons. Sophisticated macro-scale structures have been QSE can be studied using linear optics; in the used to create photon entanglement [12]. Recent from of photon added or subtracted using beam advancements in on-chip quantum photonic cir- splitters (BSs), as illustrated in Fig 3. An iso- cuits have lead to the development of integrated morphism can be set up between BS operations entangled photon sources. The metasurfaces and angular momentum relations. Squeezed made of high refractive index dielectric are resis- states are examples of nonclassical multiphoton tive to plasmonic decoherence and loss. Silicon states of light and can be generated from co- based metasurfaces with nearly 100% efficiency herent states. Of special emphasis is the en- makes them candidates for quantum optics and gineering of nonclassical states of photons and quantum information applications. A simple il- atoms, such as, Fock states, macroscopic super- lustration of a dielectric metasurface generating position states, multiphoton generalized coher- entanglement between the spin and the orbital ent and squeezed states, that are relevant for angular momentum of photons is shown in Fig. applications of forefront aspects of modern quan- 1. tum mechanics. In recent times, advances have Plasmonics is a rapidly developing field at the been made by combining linear and nonlinear boundary of physical optics and condensed mat- optical devices allowing realization of multipho- ter physics. It studies phenomena induced by ton entangled states of the electromagnetic field, and associated with surface plasmons: elemen- either in discrete or in continuous variables, that tary polar excitations bound to surfaces and in- are relevant for applications to efficient quan- terfaces of good nanostructured metals. tum information processing. Multiphoton quan- Electronic systems governed by the rules of tum states are carriers of information, and the quantum mechanics are hard to understand prin- manipulation of these by the methods of quan- cipally due to the strong Coloumb interactions tum optics, has brought about a strong interface between electrons. This makes many interesting between the fields of quantum information and systems difficult to comprehend via simple mod- quantum optics. els as the many-body interactions and effects cannot be ignored nor properly described in a simplistic manner. The Maxwell equations that IV. Metamaterials and Plasmonics govern electromagnetism and the Schrodinger equation of quantum mechanics reduce to the Metamaterials are structured composite mate- same Helmholtz equations under certain circum- rials with periodic subwavelength sized unit cells. stances. This can be used to great effect to use The subwavelength nature of the unit cells does light as a test-bed for quantum effects as the not permit the wave to resolve individual unit photon-photon interactions can be very small cells and permits the description of the compos- and can be accurately controlled via non-linear ite material as an effective medium described by interactions of a medium. Electromagnetics has effective medium properties such as permittiv- greatly benefitted in past two decades from de- 6

(a) (b)

FIG. 3. Beam splitter model for (a) photon addition in which a photon is added to the state ρ2 and (b) photon subtraction ends up taking one photon from the state ρ2, as a result of beam splitter interaction. velopments in metamaterials and plasmonics. chanics. The dissipative environment is, how- Metamaterials having resonant unit cells have ever, an issue, but the systems offer a neat plat- highly dispersive effective medium parameters form to explore the role of decoherence for quan- whereby novel properties and phenomena not tum systems as well. usually found in nature become accessible. The The unique properties offered by metamateri- most famous examples are negative refractive als and plasmonic structured surfaces can offer index, perfect lenses with sub-diffraction image new testing grounds for quantum theories. For resolution, perfect absorbers of light and media example, surface plasmons on the interfaces of with extremely large anisotropy resulting in even metals and nonlinear materials can offer deep hyperbolic dispersion for light. Similarly, plas- insights into Anderson localization of interacting monics that concerns with the study and ma- particles in two dimensions. Similarly, nonlinear nipulation of surface electromagnetic waves on metamaterials and nonlinear surface plasmonic the surfaces of metals (electronic plasmas) and structures can enable controlled experiments of metamaterials, gives rise to enormous possibil- quantum tunnelling in time retarded systems by ities on structured surfaces. With great con- studying evanescent electromagnetic waves. The trol that becomes possible on the dispersion of easily modified photonic density modes here offer the two-dimensional surface plasmons as well as unique possibilities. The extreme control on the their coupling to radiative modes, plasmonics of- generation of and detection of light (microwaves fers immense promise for miniaturization of op- in particular) make these a good platform for im- tical devices that are projected to supplant elec- plementation of many thought experiments that tronic devices due to the enormous bandwidths have hitherto been inaccessible in electronic sys- at optical frequencies. tems. It is known that surface plasmon waves pre- In recent times, different types of metamateri- serve the entanglement of twin photons, which als have being studied. Thus, for example, one is understood by the coherent nature of the sur- can have nanowires that use quantum dots as face plasmon waves inspite of the coupling to dis- unit cells or artificial atoms arranged as periodic sipative modes in the plasmonic medium. Plas- nanostructures. This material has a negative in- monic systems are becoming more important for dex of refraction and effective magnetism and is quantum information purposes. Particularly as simple to build. The radiated wavelength of in- the interaction volumes become smaller and sub- terest is much larger than the constituent diam- wavelength, the interaction of the surface plas- eter. Photonic Band Gap materials, also known mons with emitting molecules or quantum dots as photonic crystals, are materials which have becomes increasingly governed by quantum me- a band gap due to a periodicity in the mate- 7

rials dielectric properties. A photonic bandgap where it is possible to control the quantum state can be demonstrated with this structure, along of mechanical oscillators by their coupling to the with tunability and control as a quantum sys- light field. Recent advances in this area include tem. The band gap in photonic crystals rep- the realization of quantum-coherent coupling of resents the forbidden energy range where pho- a mechanical oscillator with an optical cavity, tons can not be transmitted through the ma- where the coupling rate exceeds both the cav- terial. Quantum metamaterial can also be re- ity and mechanical motion decoherence rate and alized using superconducting devices both with laser cooling of a nanomechanical oscillator to and without Josephson junctions and are being its ground state. Furthermore, new experimen- actively investigated. Recently a superconduct- tal works open up enormous possibilities in the ing quantum metamaterial prototype based on design of hybrid quantum systems whose elemen- flux qubits was realized [13]. tary building blocks are physically implemented by systems of different nature. Cavity optomechanical systems can provide a V. Optomechanical Systems natural platform to induce an interaction be- tween mechanical resonators because there is Recent experiments in cavity quantum elec- an intrinsic coupling mechanism between optical trodynamics (cQED) have explored the interac- and mechanical degrees of freedom. There has tion of light with atoms as well as semiconductor been a lot of interest in the creation of quantum nanostructures inside a cavity. Cavity quantum correlations in macroscopic mechanical systems, elctrodynamics is the study of interaction be- achieved by means of optomechanical models. A tween light confined in a cavity and atoms, where typical optomechanical system driven by a laser the ambient conditions are such that the quan- is shown in Fig. 4. tum nature of light becomes predominant. This On the one hand, there is the highly sensitive field could be traced to the Purcell effect [14], optical detection of small forces, displacements, which is connected to the process of spontaneous masses, and accelerations. On the other hand, emission, a purely quantum effect by which the cavity quantum optomechanics promises to ma- system transitions from an excited energy state nipulate and detect mechanical motion in the to a lower energy state, emitting, in the process, quantum regime using light, creating nonclassi- a quantized amount of energy in the form of a cal states of light and mechanical motion. These photon. The Purcell effect is the process of en- tools form the basis for applications in quan- hancement of the systems spontaneous emission tum information processing, where optomechan- rate by its ambient environment. It is dependent ical devices could serve as coherent light-matter on the quality Q factor of the cavity, which is a interfaces, for example, to interconvert informa- measure of the ratio of the energy stored to the tion stored in solid-state qubits into flying pho- energy dissipated. A good quality cavity would tonic qubits. At the same time, it offers a route have a high Q factor. towards fundamental tests of quantum mechan- Light carries momentum which gives rise to ics in a hitherto unaccessible parameter regime radiation-pressure forces. Recent works have of size and mass. been able to study coupled cavity photons to solid-state mechanical systems containing a large number of atoms [15]. In these systems, there VI. PT Symmetry is an optical cavity, with a movable mirror in one end or a micro-mechanical membrane with The concept of Parity-Time (PT) symmetry mechanical effects caused by light through radi- has played a crucial role in extending quantum ation pressure. Hence, cavity quantum optome- mechanics to the non-Hermitian domain. A non- chanics has emerged as a very interesting area Hermitian Hamiltonian can also have real eigen- for revealing quantum features at the mesoscale, values if it possess PT symmetry. To be precise, 8

FIG. 4. A typical cavity optomechanical system driven by a laser. The left mirror is fixed while the right mirror is attached to spring providing the mechanical mode in the system. the Hamiltonian Hˆ with Hˆ 6= Hˆ †, where the phases with preserved or weakly broken PT sym- symbol † stands for hermitian conjugation, can metry appear. Such interactions could in prin- have a real spectrum of eigenvalues if (PˆTˆ)Hˆ = ciple also appear in other contexts of quantum Hˆ (PˆTˆ), where Pˆ and Tˆ are the Parity and Time- optics, such as the prototypical case of an atom reversal operators, respectively. They have the interacting with the mode of a light field in an following actions on the position (xˆ) and momen- open system. tum (pˆ): PˆxˆPˆ = −xˆ, PˆpˆPˆ = −pˆ; TˆpˆTˆ = −pˆ, In [19], a system was realized whose dynamics TˆxˆTˆ =x ˆ and TˆˆiTˆ = −ˆi. Therefore, for a general is governed by a PT Hamiltonian. Many op- Hamiltonian Hˆ =p ˆ2/2m + Vˆ (ˆx) to be PT sym- tomechanical properties have been investigated, metric, the potential term must satisfy the con- such as the cavity optomechanical properties un- dition Vˆ (ˆx) = Vˆ ∗(−xˆ). Although the idea of PT derlying the phonon lasing action, PT symmet- symmetry was introduced in quantum mechan- ric chaos, cooling of mechanical oscillator, cavity ics in [16], the importance of the phenomenon assisted metrology, optomechanically-induced- has been realized recently [17]. Equivalance of transparency (OMIT) and optomechanically in- a quantum system possesing PT symmetry and duced absorption (OIA). The possibility of the to a quantum system having Hermitian Hamil- spontaneous generation of photons in PT sym- tonian was shown in [18]. metric systems is illustrated in.

A new direction, in the application of PT sym- VII. Quasiprobability distributions and metry, could be the study of non-Hermitian ef- Tomography fects in small-scale devices as well as in atomic and molecular systems, where quantum pro- A. Quasiprobability distributions cesses are known to play a significant role. These include, for instance, driven atomic condensates A very useful concept in the analysis of the dy- in cavities, artificial atoms or hybrid quantum namics of classical systems is the notion of phase systems in cavity quantum electrodynamics as space. A straightforward extension of this to the well as coupled optomechanical resonators with realm of quantum mechanics is however foiled gain and loss, where effects such as phonon las- due to the uncertainty principle. Despite this, it ing near exceptional points could be explored. is possible to construct quasiprobability distribu- Initial theoretical studies in this direction show tions (QDs) for quantum mechanical systems in that the presence of quantum noise leads to sig- analogy with their classical counterparts. These nificantly different physics as compared to that QDs are very useful in that they provide a quan- expected from semiclassical approaches. Novel tum classical correspondence and facilitate the 9 calculation of quantum mechanical averages in from experimentally measured values is of prime close analogy to classical phase space averages. interest for both quantum computation and com- Nevertheless, the QDs are not probability distri- munication. The tomogram is one such candi- butions as they can take negative values as well, date as it is experimentally measurable and is a feature that could be used for the identification obtained as a probability distribution. Further, of quantumness in a system. the quantum state tomography has its applica- The first such QD was developed by Wigner re- tions in quantum cryptography. sulting in the epithet Wigner function (W ) [20]. How to reconstruct a quantum state from ex- Another, very well known, QD is the P function perimentally measured values is of prime inter- whose development was a precursor to the evo- est for both quantum computation and commu- lution of the field of quantum optics. This was nication. Specifically, in [22] it is strongly es- originally developed from the possibility of ex- tablished that tomography and spectroscopy can pressing any state of the radiation field in terms be interpreted as dual forms of quantum com- of a diagonal sum over coherent states [3, 4]. putation. From the experimental perspective, The P function can become singular for quan- a quantum state always interacts with its sur- tum states, a feature that promoted the devel- roundings. Hence, it is important to consider the opment of other QDs such as the Q function as evolution of the tomogram after taking into ac- well as further highlighted the use of the W func- count the interaction of the quantum state with tion which does not have this feature. its environment. A nonclassical state can be used to perform tasks that are classically impossible. This fact motivated many studies on nonclassical states, VIII. Topological photonics for example, studies on squeezed, antibunched and entangled states. The interest in nonclassi- Topological phases in condensed matter, that cal states has increased with the advent of quan- usually arise out of Berry phase effect during adi- tum information processing where several appli- abatic evolution of states, has taken the centre- cations of nonclassical states have been reported. stage of research in the condensed matter com- The fields of quantum optics and information munity for quite some time now. Interesting have matured to the point where intense experi- findings such as topological insulators, where mental investigations are being made. Both from conducting edge/surface states appear in an oth- the fundamental perspective as well as from the erwise bulk-insulating system or Weyl semimet- viewpoint of practical realizations, it is impera- als, where topologically robust Weyl charges, tive to study the evolution of the system of inter- analogous to magnetic monopoles of the Berry est taking into account the effect of its ambient curvature of Bloch bands, appear in pairs within environment. This is achieved systematically by its bulk, are currently being investigated with using the formalism of Open Quantum Systems intense vigour. [7]. Over the last decade, there has been a very ex- citing new development in quantum optics with roots in condensed matter physics, in particular, B. Tomography topological insulators and quantum Hall effects. This is the field of topological photonics [23, 24], There is no general prescription for direct ex- where externally provided photons in photonic perimental measurement of the quasidistribution crystals can induce surprising topological effects. functions, such as Wigner function [21]. In gen- As topological insulators are rare among solid- eral, to detect the nonclassicality in a system state materials, suitably designed electromag- the Wigner function is obtained either by photon netic media (metamaterials) can demonstrate counting or from experimentally measured tomo- a photonic analogue of a topological insulator. grams [21]. Reconstruction of a quantum state They provide topologically non-trivial photonic 10 states, similar to those that have been identi- photonics include the possibility of applications fied for condensed-matter topological insulators. to quantum information processing and topolog- The interfaces of these metacrystals support he- ical quantum computing. Another application lical edge states, robust against disorders. could be topological lasers, the study of laser os- Topology is the study of geometrical conserved cillation in topological systems. quantities and its use, in the context of optics, We have attempted to give a brief overview of allows the creation of new states of light with the recent developments in the field of quantum interesting properties. Thus, for example, one optics. Efforts have been made to connect the could have robust unidirectional waveguides al- modern developments with the roots of the sub- lowing light to propagate around defects without ject. The subject has seen an enormous progress back-reflection. This also provides opportunities in various directions and is believed to provide to realize and exploit topological effects in new the testbed for exploring some of the fundamen- ways. The practical implications of topological tal problems of physics.

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