The investigation on quantum radar requires accurate computation of the state vectors of the single-photon processes of the two-level system in free space. However, the traditional Weisskopf-Wigner (W-W) theory fails to deal with those processes other than spontaneous emission. To solve this problem, we provide a new method based on the renormalization theory. We evaluate the renormalized time-ordered Green functions associated with the single-photon processes, and relate them to the corresponding state vectors. It is found that the ultraviolet divergences generated by the Lamb shift and higher-order interactions can be systematically subtracted in the state vectors. The discussions on spontaneous emission and single-photon absorption are then presented to illustrate the proposed method. For spontaneous emission, we obtain the same results of the W-W theory. For single-photon absorption where W-W theory fails, we find that the two-level electric dipole first gets excited rapidly and then decays exponentially, and that the efficiency of the single-photon absorption declines as the bandwidth of the incident photon becomes narrow. The proposed method can improve the investigation on quantum radar.
The investigation on quantum radar requires accurate computation of the state vectors of the single-photon processes of the two-level system in free space. However, the traditional Weisskopf-Wigner (W-W) theory fails to deal with those processes other than spontaneous emission. To solve this problem, we provide a new method based on the renormalization theory. We evaluate the renormalized time-ordered Green functions associated with the single-photon processes, and relate them to the corresponding state vectors. It is found that the ultraviolet divergences generated by the Lamb shift and higher-order interactions can be systematically subtracted in the state vectors. The discussions on spontaneous emission and single-photon absorption are then presented to illustrate the proposed method. For spontaneous emission, we obtain the same results of the W-W theory. For single-photon absorption where W-W theory fails, we find that the two-level electric dipole first gets excited rapidly and then decays exponentially, and that the efficiency of the single-photon absorption declines as the bandwidth of the incident photon becomes narrow. The proposed method can improve the investigation on quantum radar.
Models of the spontaneous emission and absorption of photons coupled to the electronic states of quantum dots, molecules, N-V (single nitrogen vacancy) centers in diamond, that can be modeled as artificial few level atoms, are important to the development of quantum computers and quantum networks. A quantum source modeled after an effective few level system is strongly dependent on the type and coupling strength the allowed transitions. These selection rules are subject to the Wigner-Eckert theorem which specifies the possible transitions during the spontaneous creation of a photonic state and its subsequent emission. The model presented in this dissertation describes the spatio-temporal evolution of photonic states by means of a Dirac-like equation for the photonic wave function within the region of interaction of a quantum source. As part of this aim, we describe the possibility to shift from traditional electrodynamics and quantum electrodynamics, in terms of electric and magnetic fields, to one in terms of a photonic wave function and its operators. The mapping between these will also be presented herein. It is further shown that the results of this model can be experimentally verified. The suggested method of verification relies on the direct comparison of the calculated density matrix or Wigner function, associated with the quantum state of a photon, to ones that are experimentally reconstructed through optical homodyne tomography techniques. In this non-perturbative model we describe the spontaneous creation of photonic state in a non-Markovian limit which does not implement the Weisskopf-Wigner approximation. We further show that this limit is important for the description of how a single photonic mode is created from the possibly infinite set of photonic frequencies νk that can be excited in a dielectric-cavity from the vacuum state. We use discretized central-difference approximations to the space and time partial derivatives, similar to finite-difference time domain models, to compute these results. The results presented herein show that near field effects need considered when describing adjacent quantum sources that are separated by distances that are small with respect to the wavelength of iii their spontaneously created photonic states. Additionally, within the future scope of this model, we seek results in the Purcell and Rabi regimes to describe enhanced spontaneous emission events from these few-level systems, as embedded in dielectric cavities. A final goal of this dissertation is to create novel computational and theoretical models that describe single and multiple photon states via single photon creation and annihilation operators.
In this paper we analyze the dynamics of single-excitation states, which model the scattering of a single photon from multiple two level atoms. For short times and weak atom-field couplings we show that the atomic amplitudes are given by a sum of decaying exponentials, where the decay rates and Lamb shifts are given by the poles of a certain analytic function. This result is a refinement of the "pole approximation" appearing in the standard Wigner-Weisskopf analysis of spontaneous emission. On the other hand, at large times, the atomic field decays like O(1/t3) with a known constant expressed in terms of the coupling parameter and the resonant frequency of the atoms. Moreover, we show that for stronger coupling, the solutions also feature a collection of oscillatory exponentials which dominate the behavior at long times. Finally, we extend the analysis to the continuum limit in which atoms are distributed according to a given density.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
The velocity-dependent spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong-coupling regime and the deflection of a beam of two-level atoms in a classical standing wave inside a cavity are discussed using a renormalization group approach. In this way we are able to renormalize the leading-order solutions for both problems through calculations of the corrections at first order. In fact, the first-order terms are not bounded for large times and no sense can be attached to this higher-order correction unless small times are considered. These are like the divergences of quantum field theory. To make them harmless, the condition for the Raman-Nath regime is recovered. The renormalization group methods permit one to eliminate those divergences generating a renormalized leading-order wave function without any condition of applicability. For the spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong regime, using a Hamiltonian without losses, it is shown that the unperturbed levels are shifted by a term proportional to the zeroth-order Bessel function with an argument yielded by the ratio of the Rabi frequency and the Doppler-shifted frequency of the mode of the cavity. When the detuning is zero, the correction to the leading-order wave function is not present and known results are recovered. For the beam of two-level atoms in a classical standing wave, when the detuning is much larger than the Rabi frequency, it is shown that the renormalization group equation, which gives the correction for the renormalized leading-order wave function, is a time-dependent Schr\"odinger equation for a free particle that induces a spreading of the initial Gaussian wave packet.
Being motivated by the controversial results based on two dispersion models and Weisskopf–Wigner approximation (WWA), we introduce, for the first time to our knowledge, the position-dependent photon-atom interaction into the Green function method of the evolution operator and develop a universal theoretical approach to study spontaneous emission of atoms in photonic crystals (PCs). A position-sensitive generalized Lorentzian formalism (non-Lorentzian shape) for the decay of an excited atom in PCs is derived, and an exact numerical method for calculating the local coupling strength, proportional to the photonic local density of state (LDOS), is presented. For weak interaction PCs with pseudo gaps, the generalized Lorentzian formalism may be reduced to the famous Lorentzian spectrum. In this case, we introduce a lifetime distribution function for an assembly of atoms and find that it depends strongly on the atomic configuration in space, which clarifies successfully the tremendous discrepancy between different experiments. For the PCs with large full gaps, we find that the atomic position can fundamentally change the decay behavior of an excited atom: in strong interaction positions, the atomic decay is non-classical or exhibits an envelope-damped Rabi oscillation, while in weak interaction positions the WWA is valid. Recently, we also predicted giant Lamb shifts for hydrogen atoms in PCs, and revealed that in inhomogeneous electromagnetic environment, the dominant contribution to the Lamb shift comes from real photon emission, while the contribution from emission and reabsorption of virtual photon is negligible, in vast contrast with the case of free space where the virtual photon processes play a key role.
Motivated by the significant controversy between the two dispersion models and Weisskopf-Wigner approximation (WWA), for the first time to our knowledge, we introduce the position-dependent photon-atom interaction into the Green function method of the evolution operator and develop a universal theoretical treatment on spontaneous emission of atoms in photonic crystals (PCs). A position-sensitive generalized Lorentzian formulism (non-Lorentzian shape) for the decay of an excited atom in PCs is derived, an exact numerical method for calculating the local coupling strength, proportional to the photonic local density of state (LDOS), is presented. For weak interaction PCs with pseudo gaps, the generalized Lorentzian formulism may be reduced to the famous Lorentzian spectrum. In this case, we introduced a lifetime distribution function for an assembly of atoms and found that the lifetime distribution strongly depend on the spread configuration of these atoms in space, which clarifies successfully the tremendous discrepancy between different experiments. For the PCs with large full gaps, we found that the atomic position can fundamentally change the decay behavior of an excited atom: in strong interaction positions, the atomic decay is non-classical or exhibits an envelope-damped Rabi oscillation, while in weak interaction positions the WWA is valid. Recently, we also predicted giant Lamb shifts for hydrogen atoms in PCs, and revealed that in inhomogeneous electromagnetic environment, the dominant contribution to the Lamb shift comes from real photon emission, while the contribution from emission and reabsorption of virtual photon is negligible, in vast contrast with the case of free space where the virtual photon processes play a key role.