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.     

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.     

Field quantization,photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.     

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.     

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.     

Analytical model for low finesse, external cavity, fiber Fabry-Perot interferometers including multiple reflections and angular misalignment

We present an analytical model for single mode, multiply reflected, external cavity, optical fiber Fabry-Perot interferometers in the low finesse regime using simple geometry and the Gaussian beam approximation. The multiple reflection model predicts attenuation of the peak-to-peak interference as the fiber to mirror distance approaches zero, as well as fringe asymmetry in the presence of nonabsorbing mirrors. A series of experiments are conducted in which a series of fiber Fabry-Perot cavities are constructed using uncoated, single mode glass fibers, and mirrors of varying reflectivity. The cavity length is swept, and the predictions of the model are found to be in good agreement with the experimental interferograms.     

Measurements of the phase shift on reflection for low-order infrared Fabry-Perot interferometer dielectric stack mirrors

A simple technique based on a Fizeau interferometer to measure the absolute phase shift on reflection for a Fabry-Perot interferometer dielectric stack mirror is described. Excellent agreement between the measured and predicted phase shift on reflection was found. Also described are the salient features of low-order Fabry-Perot interferometers and the demonstration of a near ideal low-order (1-10) Fabry-Perot interferometer through minimizing the phase dispersion on reflection of the dielectric stack. This near ideal performance of a low-order Fabry-Perot interferometer should enable several applications such as compact spectral imagers for solid and gas detection. The large free spectral range of such systems combined with an active control system will also allow simple interactive tuning of wavelength agile laser sources such as CO(2) lasers, external cavity diode lasers, and optical parametric oscillators.     

Photon coincidence spectroscopy for two-atom cavity quantum electrodynamics

We show that photon coincidence spectroscopy can provide an unambiguous signature of two atoms simultaneously interacting with a quantized cavity field mode. We also show that the single-atom Jaynes—Cummings model can be probed effectively via photon coincidence spectroscopy, even with deleterious contributions to the signal from two-atom events. In addition, we have explicitly solved the eigenvectors and eigenvalues of two two-level atoms coupled to a quantized cavity mode for differing coupling strengths.     

Optomechanical effect on the Dicke quantum phase transition and quasi-particle damping in a Bose–Einstein condensate: a new tool to measure weak force

Abstract

We make a semi-classical steady state analysis of the influence of mirror motion on the quantum phase transition for an optomechanical Dicke model in the thermodynamic limit. An additional external mechanical pump is shown to modify the critical value of atom–photon coupling needed to observe the quantum phase transition. We further show how to choose the mechanical pump frequency and cavity–laser detuning to produce extremely cold condensates. The present system can be used as a quantum device to measure weak forces.     

Meeting Report

Abstract

The cooperative dynamics of a microlaser consisting of two threelevel atoms interacting with a pump field and two quantized cavity modes forming a radiative cascade are studied. Adiabatic elimination of one mode leads to a strong dynamical entanglement between the internal states of the atoms which allows us to study the effects of a cavity-mediated dipole-dipole interaction. We show that the coherent dynamics of the two-atom system will preferentially couple symmetrical linear combinations of internal states. If this coupling dominates the dynamics, the two-atom system will behave like a single atom with correspondingly larger dipole moment, that is a superradiant two-atom system. Even very small spontaneous decay causes transitions from symmetrical to antisymmetrical states and conversely. The hopping between two subsets of the state space can give rise to intriguing phenomena such as bistability of the laser mode intensity. By a randomization of the two coupling phases we recover the standard independent-atom laser theory.     

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.     

Light transport in chiral and magnetochiral random media

We present a microscopic approach to study electromagnetic wave propagation in media with broken mirror symmetry. We introduce and calculate the transport mean free path l*C associated with the residual polarization of diffuse light in chiral systems. In chiral media subject to an external magnetic field B, all symmetry requirements exist to create a macroscopic "super" light current in the direction of B that persists even in the absence of a spatial photon density gradient. However, we show that such a current is identically zero in our model. We finally show the existence of a linear magnetotransmission in magnetochiral media.     

Cooling neutral particles in multimode cavities without spontaneous emission

Abstract

We discuss a scheme to cool, trap and manipulate an ensemble of polarizable particles moving in the field of a multimode optical cavity via the correlated dynamics of the field and the particle motion. Using a large detuning between the atoms and the field, spontaneous emission plays a negligible role in the dynamics and the cooling scheme only requires a sufficiently large optical dipole moment. Increasing the particle number slows down the cooling process but it can be accelerated using an increasing number of field modes with higher pump amplitudes. For the special case of a two mode ring cavity and assuming small deviations of the particle positions from the potential minima, the frequencies and damping rates of the vibrational excitation modes can be explicitly calculated. We find a rapid damping of the centre-of-mass motion and relatively slow damping rates for the relative particle oscillations. These predictions agree quite well with a quantum treatment of the atomic motion as used for the excitations of a non-interacting Bose gas (at T = 0) inside the cavity field. Due to the position-dependent mode coupling, the cooling process in a multimode configuration in general happens much faster than for a standing wave geometry. These analytical results are confirmed by N-particle simulations of the semiclassical equations and show even enhanced damping due to the anharmonicity of the full potential.     

Aligning a reflection cavity by Anderson's method

The sounding rocket principle of equivalence measurement uses a set of four laser gauges operating in Fabry-Perot cavities to determine the relative acceleration of two test masses that are chemically different. One end of each cavity is a flat mirror on a test mass. Because the test masses are unconstrained and thus expected to rotate slightly during measurement, and because the distance measurements are made at the sub-picometer level, it is essential to have real-time alignment of the beam entering the cavity. However, the cavity must be used in reflection and space is limited. We show that Anderson's alignment method can be used in reflection, but that it requires that the Fabry-Perot cavity have mirrors with significantly unequal reflectivities.     

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.     

Optical Lindblad operator in non-equilibrium thermo field dynamics

The Lindblad operator of the Markovian quantum master equation for an oscillator system is diagonalized by means of linear transformations of annihilation and creation operators in non-equilibrium thermo field dynamics. When the linear attenuation and amplification processes are considered, the left and right eigenstates of the Lindblad operator are constructed in terms of the transformed operators. In the linear attenuation process, the results are compared with those obtained by the damping base of the Lindblad superoperator. The base describing the time-evolution of the linear amplification process is derived and compared with the damping base. The method for diagonalizing the Lindblad operator is useful for calculating average values of physical quantities in the quantum Markovian process.     

Pulsed measurement of high-reflectivity mirror phase retardances

The linear phase retardances of high-reflectivity mirrors are measured by use of the progressive modification of the polarization of a light pulse propagating in a passive Fabry-Perot cavity. This method permits determination of the linear phase retardance of a mirror with a precision of better than 0.05° and a sensitivity of better than 0.1° for any angle of incidence and any wavelength in the high-reflectivity domain of the mirror. This method also leads to a precise determination of the null-phase-retardance wavelength of the mirror.     

Exponential localization of moving end mirror in optomechanics

We discuss the dynamics of moving end mirror of an optomechanical system that consists of a Fabry-Perot cavity loaded with dilute condensate and driven by a single-mode optical field. It is shown that quantum mechanical phenomenon of dynamical localization occurs both in position and momentum space for moving end mirror in the system. The parametric dependencies of dynamical localization are discussed. We also provide a set of parameters which makes this phenomenon experimentally feasible.     

Effect of decoherence on the radiative and squeezing properties in a coherently driven trapped two-level atom

Analysis of the effects of decoherence on the radiative and squeezing properties of a coherently driven two-level atom trapped in a resonant cavity applying the corresponding master equation is presented. The atomic dynamics as well as the squeezing and statistical properties of the emitted radiation are investigated. It is found that the atom stays in the lower energy level more often at steady state irrespective of the strength of the coherent radiation and thermal fluctuations entering the cavity. Moreover, a strong external coherent radiation results in the splitting of the line of the emission spectrum, whereas the decoherence broadens the width and significantly decreases the height. It is also found that the emitted radiation exhibits photon anti-bunching, super-Poissonian photon statistics and squeezing, despite the presence of the decoherence which is expected to destroy the quantum features.     

Properties of far-field fluorescence from an ensemble of interacting Sr atoms

We systematically investigate the far-zone fluorescence from a number of laser-driven two-level atoms with the cooperative effects stemming from the virtual exchange of photons between identical atoms. Due to the large interaction-induced energy level shifts, i.e. the collective Lamb shifts, both atomic excitation and photon emission do not maximize at the resonant light-atom coupling. In addition, the long-range dissipative interactions among atoms lead to the disproportion between the atomic excitation and the photon emission. Moreover, we discuss the temporal properties of the far-field fluorescence light from the atomic ensemble. The first-order correlation function indicates a damped light-atom dynamics combined with a strong background of the elastic photon scattering. The second-order correlation function illustrates that in the weak-excitation limit the scattered light exhibits the nonclassical photon antibunching behavior, while the bunched photons arrive at the photodetector in the saturated-excitation limit.