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.     

Macroscopic canonical quantization in quantum optics: Properties of quasi mode annihilation and creation operators

Abstract

Macroscopic canonical quantization of the EM field and radiative atom systems occurring in quantum optics experiments involving linear classical optics devices can be carried out via expansion of the vector potential either in terms of true mode functions for the optical device or in terms of approximate or quasi mode functions. The relationship between the true mode and quasi mode annihilation, creation operators is determined and shown to involve a Bogolubov transformation. Analytic properties are also examined and it is found that the annihilation, creation operators times the square root of the angular frequency are analytic functions of the variables specifying the modes.     

Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics. II. Application to reflection and refraction at a dielecric interface

Abstract

The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being nonzero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.     

Quasimode theory of quantum optical processes in photonic band gap materials

This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.     

Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics I. Theory

Abstract

The quasi mode theory of macroscopic quantization in quantum optics and cavity QED developed by Dalton, Barnett and Knight is generalized. This generalization allows for cases in which two or more quasi permittivities, along with their associated mode functions, are needed to describe the classical optics device. It brings problems such as reflection and refraction at a dielectric boundary, the linear coupler, and the coupling of two optical cavities within the scope of the theory. For the most part, the results that are obtained here are simple generalizations of those obtained in previous work. However the coupling constants, which are of great importance in applications of the theory, are shown to contain significant additional terms which cannot be ‘guessed’ from the simpler forms. The expressions for the coupling constants suggest that the critical factor in determining the strength of coupling between a pair of quasi modes is their degree of spatial overlap. In an accompanying paper a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary is given as an illustration of the generalized theory. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes.     

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.     

Influence of Kerr-like medium on the dynamics of a two-mode Raman coupled model

Abstract

We study the quantum dynamics of an effective two-level atom interacting with two modes via Raman process inside an ideal cavity in the presence of Kerr non-linearity. The cavity modes interact both with the atom as well as the Kerr-like medium. The unitary transformation method presented here, not only solves the time-dependent problem, but also provides the eigensolutions of the interacting Hamiltonian at the same time. We study the atomic-population dynamics and the dynamics of the photon statistics in the two cavity modes. The influence of the Kerr-like medium on the statistics of the field is explored and it is observed that Kerr medium introduces antibunching in mode 1 and this effect is enhanced by a stronger interaction with the non-linear medium. In the high non-linear coupling regime anticorrelated beam become correlated. Kerr medium also introduces non-classical correlation between the two modes.     

Commutation Rules and Eigenvalues of Spin and Orbital Angular Momentum of Radiation Fields

Abstract

We investigate the separation of the total angular momentum J of the electromagnetic field into a ‘spin’ part S and an ‘orbital’ part L. We show that both ‘spin’ and ‘orbital’ angular momentum are observables. However, the transversality of the radiation field affects the commutation relations for the associated quantum operators. This implies that neither S nor L are angular momentum operators. Moreover their eigenvalues are not discrete. We construct field modes such that each mode excitation (photon) is in a simultaneous eigenstate of S _z and L _z. We consider the interaction of such a photon with an atom and the resulting effect on the internal and external part of the atomic angular momentum.     

Spin-resolved Purcell effect in a quantum dot microcavity system

We demonstrate the spin selective coupling of the exciton state with cavity mode in a single quantum dot (QD)-micropillar cavity system. By tuning an external magnetic field, each spin polarized exciton state can be selectively coupled with the cavity mode due to the Zeeman effect. A significant enhancement of spontaneous emission rate of each spin state is achieved, giving rise to a tunable circular polarization degree from -90% to 93%. A four-level rate equation model is developed, and it agrees well with our experimental data. In addition, the coupling between photon mode and each exciton spin state is also achieved by varying temperature, demonstrating the full manipulation over the spin states in the QD-cavity system. Our results pave the way for the realization of future quantum light sources and the quantum information processing applications.     

Highly directional emission and photon beaming from nanocrystal quantum dots embedded in metallic nanoslit arrays

We demonstrate a directional beaming of photons emitted from nanocrystal quantum dots that are embedded in a subwavelength metallic nanoslit array with a divergence angle of less than 4°. We show that the eigenmodes of the structure result in localized electromagnetic field enhancements at the Bragg cavity resonances, which could be controlled and engineered in both real and momentum space. The photon beaming is achieved using the enhanced resonant coupling of the quantum dots to these Bragg cavity modes, which dominates the emission properties of the quantum dots. We show that the emission probability of a quantum dot into the narrow angular mode is 20 times larger than the emission probability to all other modes. Engineering nanocrystal quantum dots with subwavelength metallic nanostructures is a promising way for a range of new types of active optical devices, where spatial control of the optical properties of nanoemitters is essential, on both the single and many photons level.     

Quantization of coupled modes propagation in integrated optical waveguides

Abstract

A quantum mechanical analysis of the propagation of coupled modes in integrated optical waveguides is given. The modal orthonormalization property on a cross-section of an optical waveguide, the vector structure of the guided optical modes and the reversal-time symmetry are taken into account to derive the quantum momentum operator and Heisenberg's equations giving a quantum-consistent formulation of the coupled mode propagation as a function of forward and backward creation and annihilation operators.     

Quantum dynamics and nonclassical photon statistics of coherently driven Raman transition in bimodal cavity

We study a classically driven two-level system in a harmonic trap and a lossy two-mode cavity, with the first mode being resonant to the driving field and an electronic transition, and the second mode being off-resonant, forming a vibrational-assisted Raman transition. Using an exact numerical method, we compute the steady state as well as the time evolution of the photon statistics. We further investigate the photon correlations of both the cavity modes and identify the laser parameters and coupling strength that give the nonclassical sub-Poissonian property. The work is useful for coherent control of photon statistics and photon correlations in the trapped two-level system.     

On the quantization of the electromagnetic field in conducting media

In this work we investigate the quantization of electromagnetic waves propagating through homogeneous conducting linear media with no charge density. We use Coulomb's gauge to reduce the problem to that of a time-dependent harmonic oscillator, which is described by the Caldirola–Kanai Hamiltonian. Furthermore, we obtain the corresponding exact wave functions with the help of quadratic invariants and of the dynamic invariant method. These wave functions are written in terms of a particular solution of the Milne–Pinney equation. We also construct coherent and squeezed states for the quantized electromagnetic waves and evaluate the quantum fluctuations in coordinates and momentum as well as the uncertainty product for each mode of the electromagnetic field.     

Cavity QED with semiconductor nanocrystals

We report on a strongly coupled cavity quantum electrodynamic (CQED) system consisting of a CdSe nanocrystal coupled to a single photon mode of a polymer microsphere. The strong exciton-photon coupling is manifested by the observation of a cavity mode splitting variant Planck's over 2piOmega(exp) between 30 und 45 microeV and photon lifetime measurements of the coupled exciton-photon state. The single photon mode is isolated by lifting the mode degeneracy in a slightly deformed microsphere cavity and addressing it by high-resolution imaging spectroscopy. This cavity mode is coupled to a localized exciton of an anisotropically shaped CdSe nanocrystal that emits highly polarized light in resonance to the cavity mode and that was placed in the maximum electromagnetic field close to the microsphere surface. The exciton confined in the CdSe nanorod exhibits an optical transition dipole moment much larger than that of atoms, the standard system for CQED experiments, and a low-temperature homogeneous line width much narrower than the high-Q cavity mode width. The observation of strong coupling in a colloidal semiconductor nanocrystal-cavity system opens the way to study fundamental quantum-optics phenomena and to implement quantum information processing concepts that work in the visible spectral range and are based on solid-state nanomaterials.     

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.     

Controlling decoherence of a two-level atom in a lossy cavity

Abstract

By use of external periodic driving sources, we demonstrate the possibility of controlling the coherent as well as the decoherent dynamics of a two-level atom placed in a lossy cavity. The control of the coherent dynamics is elucidated for the phenomenon of coherent destruction of tunnelling (CDT), i.e. the coherent dynamics of a driven two-level atom in a quantum superposition state can be brought practically to a complete standstill. We study this phenomenon for different initial preparations of the two-level atom. We then proceed to investigate the decoherence originating from the interaction of the two-level atom with a lossy cavity mode. The loss mechanism is described in terms of a microscopic model that couples the cavity mode to a bath of harmonic field modes. A suitably tuned external cw-laser field applied to the two-level atom slows down considerably the decoherence of the atom. We demonstrate the suppression of decoherence for two opposite initial preparations of the atomic state: a quantum superposition state as well as the ground state. These findings can be used to decrease the influence of decoherence in qubit manipulation processes.     

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.     

Quantum Dynamics of a Hyper-Raman Coupled Model Interacting with Two Quantized Cavity Fields

The quantum dynamics of a hyper-Raman coupled model interacting with two modes of the quantized cavity field is described. The model consists of a four-level atom in a v configuration where transitions between the ground and excited states occur through the absorption (emission) of two photons from one mode and the emission (absorption) of one photon from the other and where two intermediate states are taken to be far off-resonance and are adiabatically removed. This is a multiphoton extension of the Jaynes-Cummings model and is exactly solvable. We study the atomic inversion and investigate the production of non-classical light exhibiting antibunching, violations of the Cauchy-Schwartz inequality, and squeezing.     

The standard model in cavity quantum electrodynamics. I. General features of mode functions for a Fabry-Perot cavity

Abstract

In the present and the accompanying paper a justification of the standard model of cavity quantum electrodynamics is given in terms of a quasi-mode theory of macroscopic canonical quantization. The coupling of the cavity quasi-mode to external quasi-modes is treated for the representative case of the three-dimensional Fabry-Perot cavity. The general form of the travelling and trapped mode functions for this cavity are derived in this paper and the mode-mode coupling constants are calculated in the accompanying paper. The slow dependence of the coupling constants with the mode frequency difference demonstrates that the conditions for Markovian damping of the cavity quasimode are satisfied. As also discussed in the accompanying paper, the interaction of radiative atoms with cavity quasi-modes is associated with reversible energy exchanges between atom and cavity and represented by Rabi coupling constants. The interaction of radiative atoms located within the cavity with sideways travelling external quasi-modes involves slowly varying coupling constants and is associated with irreversible spontaneous emission dampling. The basic processes represented in the standard cavity quantum electrodynamics model and the associated coupling constant and decay rates thereby follow from the quasi-mode theory.     

Optical properties of a laterally confined semiconductor microcavity containing a single quantum well

The semiconductor microcavity with a thin oxide-aperture layer is fabricated, and linear optical transmission spectrum measured for various aperture diameters. First, the observation of bare cavity modes is demonstrated in this microstructure which is capable of confining light field three-dimensionally. Several transverse modes are observed as transmission peaks, which manifests the lateral field confinement achieved well down to 2 μm aperture diameter. And the transmission spectrum of cavity modes coupled to the excitonic resonance is measured for the same microcavity system containing a single quantum well. The result shows that each transverse mode couples to an exciton independently as it approaches the excitonic resonance frequency, giving rise to an anti-crossing behavior between coupled modes.