Photon filters in a microwave cavity

Abstract

In an earlier paper we concluded that time-dependent parameters in the atom-mode interaction can be utilized to modify the quantum field in a cavity. When an atom shoots through the cavity field, it is expected to experience a trigonometric time dependence of its coupling constant. We investigate the possibilities this offers to modify the field. As a point of comparison we use the solvable Rosen-Zener model, which has parameter dependences roughly similar to the ones expected in a real cavity. We do confirm that by repeatedly sending atoms through the cavity, we can obtain filters on the photon states. Highly non-classical states can be obtained. We find that the Rosen-Zener model is more sensitive to the detuning than the case of a trigonometric coupling.     

Determining the state of a single cavity mode from photon statistics

Abstract

We present various schemes for measuring the quantum state of a single mode of the electromagnetic field. These involve measuring the photon statistics for the mode before and after an interaction with either one or two two-level atoms. The photon statistics conditioned on the final state of the atoms, for two choices of the initial set of atomic states, along with the initial photon statistics, may be used to calculate the complete quantum state in a simple manner. Alternatively, when one atom is used, two unconditioned sets of photon statistics, each after interaction with a single atom in different initial states, along with the initial photon statistics may be used to calculate the initial state in a simple manner. When the cavity is allowed to interact with just one atom, only pure cavity states which do not contain zeros in the photon distribution may be reconstructed. When two atoms are used we may reconstruct pure states which do not contain adjacent zeros in the photon distribution. Coherent states and number states are among those that may be measured with one-atom interaction, and squeezed states and ?Schrödinger cats‘ are among those that may be measured with a two-atom interaction.     

Quasi mode theory of macroscopic canonical quantization in quantum optics and cavity quantum electrodynamics

Abstract

A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.     

Counter-intuitive vacuum-stimulated raman scattering

Abstract

Vacuum-stimulated Raman scattering in strongly coupled atomcavity systems allows one to generate free-running single photon pulses on demand. Most properties of the emitted photons are well defined, provided spontaneous emission processes do not contribute. Therefore, electronic excitation of the atom must not occur, which is assured for a system adiabatically following a dark state during the photon-generation process. We experimentally investigate the conditions that must be met for adiabatic following in a time-of-flight driven system, with atoms passing through a cavity and a pump beam oriented transverse to the cavity axis. From our results, we infer the optimal intensity and relative pump-beam position with respect to the cavity axis.     

Comparisons of spectra determined using detector atoms and spatial correlation functions

Abstract

We show how two level atoms can be used to determine the local time dependent spectrum. The method is used in a one-dimensional cavity. The spectrum obtained is compared with the mode spectrum determined using spatially filtered second-order correlation functions. The spectra obtained using two level atoms give identical results with the mode spectrum. One benefit of the method is that only one time averages are needed. It is closely related to a realistic spectrum measurement.     

Coherence phenomena in coupled pptical resonators

Abstract

We predict a variety of photonic coherence phenomena in passive and active coupled ring resonators. Specifically, the effective dispersive and absorptive steady-state response of coupled resonators is derived, and used to determine the conditions for coupled-resonator-induced transparency and absorption, lasing without gain, and cooperative cavity emission. These effects rely on coherent photon trapping, in direct analogy with coherent population trapping phenomena in atomic systems. We also demonstrate that the coupled-mode equations are formally identical to the two-level atom Schrödinger equation in the rotating-wave approximation, and use this result for the analysis of coupled-resonator photon dynamics. Notably, because these effects are predicted directly from coupled-mode theory, they are not unique to atoms, but rather are fundamental to systems of coherently coupled resonators.     

Interaction between dual cavity modes in a planar photonic microcavity

Abstract

We theoretically study the interaction between dual cavity modes in a planar photonic microcavity structure in the optical communication wavelength range. The merging and splitting of cavity mode is analysed with realistic microcavity structures. The merging of dual cavity resonance into a single cavity resonance is achieved by changing the number of layers between the two cavities. The splitting of single cavity resonance into dual cavity resonance is obtained with an increase in the reflectivity of mirrors in the front and rear side of the microcavity structure. The threshold condition for the merging and splitting of cavity mode is established in terms of structural parameters. The physical origin of the merging of dual cavity modes into a single cavity resonance is discussed in terms of the electric field intensity distribution in the microcavity structure. The microcavity structure with dual cavity modes is useful for the generation of entangled photon pairs, for achieving the strong-coupling regime between exciton and photon and for high-resolution multi-wavelength filters in optical communication.     

On the possibility of preparing decoherence-free states in a cavity

The effect of decoherence in a quantum system can be viewed as a consequence of the interaction with the environment. As has been pointed out first by Dicke, in a system of N two-level atoms where each of the atoms is individually dipole coupled to the environment, there are collective subradiant states that have no dipole coupling to photon modes, and therefore they are expected to decay more slowly. We have recently proposed a scheme which is intended to create such states in a detuned cavity. We shall examine here the conditions under which our scheme can be used and compare them with the experimental possibilities. The analysis shows that our proposal can be implemented with present-day techniques achieved in atom—cavity interaction experiments.     

The standard model in cavity quantum electrodynamics. II. Coupling constants and atom field interaction

Abstract

Specific forms of the travelling and trapped vector mode functions for a three-dimensional Fabry-Perot cavity are developed from the general results of the preceding paper, with parameters describing the output cavity mirror chosen for a typical high Q cavity case. Cavity and external quasi-mode functions associated with the quasi-mode theory of macroscopic canonical quantization are then obtained via an idealized choice of output mirror parameters. The coupling constants describing photon exchange processes between the single cavity quasi-mode associated with each Fabry-Perot resonance and various external quasi-modes are calculated, and their slow dependence on the external quasi-mode frequency shows that the conditions for irreversible Markovian damping of the cavity quasi-mode are satisfied. For radiative atoms placed in the cavity the coupling constants for energy exchange processes with sideways travelling external quasi-modes also vary slowely, so that Markovian spontaneous emission damping occurs for the radiative atoms. However, their coupling with the isolated cavity quasi-modes is associated with reversible photon exchanges as represented via one photon Rabi frequencies. The standard model in cavity quantum electrodynamics, in which the basic processes are described by a cavity damping rate, a radiative atom spontaneous decay rate and an atom-cavity mode coupling constant has now been justified in terms of the quasi-mode theory of macroscopic canonical quantization.     

Interaction of a System of Initially Unexcited Two-level Atoms with a Weak Cavity Field

Abstract

Using a perturbation method, constructed in terms of SU(2) group representations, the interaction of N initially unexcited two-level atoms and a weak single-mode cavity field is studied. The field is assumed to be initially either in a Fock state with a number of photons equal to n or in a coherent state. In the case of the photon-number state with n  3, the pure phenomenon of collective collapses and revivals manifests itself. For the initially coherent field the phenomenon of collapses and revivals arising from the photon number distribution mechanism is additionally modulated by this collective mechanism. The problem of the interaction of excited atoms with an initially coherent field has already been solved numerically by Barnett and Knight. For n=1 2 and 3 the approximate solution is compared with the exact solutions also given in this paper and the limit of applicability of our approach is established.     

Higher-generation Schrödinger cat states in cavity QED

Abstract

Higher-generation Schrödinger cat states of the quantized electromagnetic field can be produced in a high-Q cavity, starting from a coherent state, through the passage of prepared Rydberg atoms interacting dispersively across it. These states are natural generalizations of the even and odd coherent states, the N ^th-generations corresponding to specific superpositions of 2 ^N states on a circle in phase space with well defined parity, and present very peculiar properties. Their photon statistics interchange between super- and sub-Poissonian behaviours and the nature of the photon bunching oscillates as the field intensity in the cavity is varied. For higher-generation even states, the minimum value of the Mandel factor almost reaches ?1.0 and the state represents the Fock state |2 ^N ). Squeezing properties and the Wigner function of these higher-generation Schrödinger cat states are also considered.     

Cavity QED analog of spin

Abstract

Using the even and odd coherent states, we show that a single mode cavity field, prepared in a coherent state by a classical source and manipulated by both dispersive and resonant interactions with atoms, is analogous to a spin one-half particle interacting with Stern–Gerlach magnets where the parity of the field is the analog of spin. Because the number of photons in the cavity may be large, the system we describe can exhibit quantum effects on at least a mesoscopic scale. We show that entangled two and three cavity systems can be generated. The three cavity case can be used to demonstrate the contradiction between local realistic theories and quantum mechanics in the manner proposed by Greenberger, Horne and Zeilinger in 1989 [13].     

Fock states generation in a kicked cavity with a nonlinear medium

Abstract

We discuss a model of a cavity filled with a passive nonlinear ?Kerr‘ medium and periodically kicked by a series of ultra-short laser pulses. The nonlinear medium is described by the (2q ? 1)th nonlinearity X ^(2q?1). We find analytical formulas describing the field states inside the cavity. We show that such a system can produce, depending on the order of the nonlinearity, superpositions of several Fock states with the small photon numbers (0,1; 0,1,2; etc). In particular, the one-photon state can be approached during the evolution of the system with X ⁽³⁾ nonlinearity provided the cavity losses are negligible. The purity of states generated in this process, however, can be seriously degraded by the cavity damping. We perform numerical calculations to validate our analytical results.     

A Nanophotonic Structure Containing Living Photosynthetic Bacteria

Photosynthetic organisms rely on a series of self‐assembled nanostructures with tuned electronic energy levels in order to transport energy from where it is collected by photon absorption, to reaction centers where the energy is used to drive chemical reactions. In the photosynthetic bacteria Chlorobaculum tepidum, a member of the green sulfur bacteria family, light is absorbed by large antenna complexes called chlorosomes to create an exciton. The exciton is transferred to a protein baseplate attached to the chlorosome, before migrating through the Fenna–Matthews–Olson complex to the reaction center. Here, it is shown that by placing living Chlorobaculum tepidum bacteria within a photonic microcavity, the strong exciton–photon coupling regime between a confined cavity mode and exciton states of the chlorosome can be accessed, whereby a coherent exchange of energy between the bacteria and cavity mode results in the formation of polariton states. The polaritons have energy distinct from that of the exciton which can be tuned by modifying the energy of the optical modes of the microcavity. It is believed that this is the first demonstration of the modification of energy levels within living biological systems using a photonic structure.     

An open-systems approach to calculating time-dependent spectra

Abstract

A new method to calculate the spectrum using cascaded open systems and a master equation is presented. The method uses two-state analyser atoms which are coupled to the system of interest, whose spectrum of radiation is read from the excitation of these analyser atoms. The ordinary definitions of a spectrum uses two-time averages and Fourier transforms. The present method uses only one-time averages. The method can be used to calculate time-dependent as well as stationary spectra.     

A Non-markov Model of Cavity Radiation

Abstract

A detailed analysis is provided of a non-Markov model of cavity population in which each emission gives rise to a random dead-time during which no further emissions can take place. Differential equations are derived for the generating functions governing the cavity population size. The moment structure of the population size is examined and explicit expressions for the first two moments are provided. The photon population statistics are shown to be antibunched for a wide choice of the population parameters. The detection process is analysed and an explicit expression is provided for the Hanbury Brown-Twiss type of correlation. It is further shown that an appropriate choice of the population parameters can produce a thermal stream with Lorentzian profile.     

Eavesdropping on practical quantum cryptography

Abstract

Practical implementations of quantum cryptography use attenuated laser pulses as the signal source rather than single photons. The channels used to transmit are also lossy. Here we give a simple derivation of two beamsplitting attacks on quantum cryptographic systems using laser pulses, either coherent or mixed states with any mean photon number. We also give a simple derivation of a photon-number splitting attack, the most advanced, both in terms of performance and technology required. We find bounds on the maximum disturbance for a given mean photon number and observed channel transmission efficiency for which a secret key can be distilled. We start by reviewing two incoherent attacks that can be used on single photon quantum cryptographic systems. These results are then adapted to systems that use laser pulses and lossy channels.     

Dissipation and Amplification of Jaynes-Cummings Superposition States

Abstract

There have recently been several proposals for generation of optical superposition states in the resonant atom-field interaction and more practically in microwave cavities. In the present paper we study the influence of the vacuum reservoir on properties of the near-superposition state of the cavity field which is described by the Jaynes-Cummings model at one-half of the revival time. Instead of introducing the cavity loss from the first instance of the atom-field interaction we consider the cavity loss only after the near-superposition state is produced and after the atom leaves the cavity. We solve the corresponding master equation with the initial condition being the Jaynes-Cummings field at one-half of the revival time. We find that under the influence of the vacuum reservoir the photon number distribution of the superposition state we study exhibits certain asymmetry around the mean photon induced by the decay process. We show that an analogous effect can be seen when the Jaynes-Cummings superposition state is amplified. For a basic test of our approach we study the dissipation and amplification of Fock states.     

Can the kicked top be realized?

Abstract

Collections of identical two-level atoms can give rise to (quantum) chaotic behaviour if non-resonantly coupled to a resonator mode and periodically driven. Observation of such chaos would require a new generation of experiments on microwave superradiance, or optical variants thereof which would exploit the strong coupling characteristic of very small cavities. Similarly, collections of identical three-level atoms non-resonantly coupled to two cavity modes could provide ‘SU(3) laboratories’, capable of realizing the semiclassical and classical limits of SU(3) dynamics, both integrable and chaotic. Some of the more interesting modes of behaviour of such systems are discussed.     

Adding and subtracting energy quanta of the harmonic oscillator

Abstract

We propose a scheme to add/subtract excitations to/from an arbitrary quantum state or the harmonic oscillator. The method displaces the photon-number distribution and leaves its shape unchanged for a wide range of displacements. Mathematically this is realized by the action of phase operators of the Susskind-Glogower type onto the initial quantum state. Consequently, the shape of the phase distribution is preserved unless the number statistics are modified due to displacing it by subtraction onto the vacuum state. Starting with an initially coherent state one may realize pure quantum states displaying either amplitude or phase squeezing. The implementation of the method is based on interactions of the Jaynes-Cummings type, in the case of subtracting quanta one additionally needs to perform measurements on the electronic quantum state of the atoms. Our approach could be used for adding and subtracting both photons on a cavity field and motional quanta of a trapped ion.