We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.
We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.
Within the framework of a two-dimensional microscopic, purely quantum mechanical model, we analyse the dynamics of single-photon wave packets interacting with optical elements (beam splitters, mirrors), modelled as systems of two-level atoms. That is, we utilize a two-dimensional cavity to simulate the quantum behaviour of simple optical components and networks made thereof. The field is quantized using the canonical procedure, and only the basis states with one unit of excitation are included. This, however, covers linear optical phenomena. The field is taken to interact with localized atoms through a dipole interaction. Using different configurations of atoms, and choosing their frequencies to be resonant or off-resonance, we can model mirrors, beam splitters, focusing devices and multicomponent systems. Thus we can model arbitrary linear networks of optical components. We show the time evolution of a photon wave packet in an interferometer as an example. As the state of the field is known at each instant, spectral properties and spatial coherence can immediately be obtained from the simulations. We also know the states of the two-level atoms constituting the components, which allows us to consider their quantum behaviour. Here the decay of an excited atom into the vacuum state of the electromagnetic field in the two-dimensional cavity is studied.
Spontaneous emission competes with stimulated emission in many interactions of light with matter. In the usual analyses which describe the interaction of an atom with a coherent optical field, the spontaneous emission characteristics, e.g., probability and spectral distribution, are not determined. The spontaneous emission from an atom which interacts with a coherent light wave is considered. The competition between coherent photons and spontaneous photons is treated in detail for a system consisting of a stationary atom, an open cavity and spatial fields. In the model chosen, a multilevel atom which spontaneously decays by interacting with spatial fields has two nondegenerate states coupled by an interaction with a single mode of the cavity. The Laplace-transformed Schrodinger equation is solved for specified initial conditions of the system. It is found that the interaction with the coherent field modifies the spectral distribution of spontaneous radiation from the atom. For spontaneous transitions involving an atomic state which interacts with the coherent field, the spectral distributions can no longer be described by Lorentzian functions. The new distributions exhibit a broadening and splitting for strong interactions between the atom and the coherent field. It is shown that the qualitative features of these distributions can be predicted from the energy-level diagram of the atom-cavity system. The net probability of the system gaining a coherent or cavity photon is calculated by integrating over the emitted spontaneous frequencies. The equivalence of this approach to the method of computing probabilities by integrating over time is demonstrated by using Parseval's theorem.
Introduction Ever since 1916/17 when Einstein argued [45] [46] that spontaneous emission must occur if matter and radiation are to achieve thermal equilibrium, physicists have believed that spontaneous emission is an inherent quantum mechanical property of atoms and that excited atoms inevitably radiate. This view, however, overlooks the fact that spontaneous emission is a consequence of the coupling of quantized energy states of atoms with the quantized radiation field, and is a manifestation of quantum noise or of emission ‘stimulated’ by ‘vacuum fluctuations’. An infinity of vacuum states is available to the photon radiated by an excited atom placed in free space, leading to the effective irreversibility of such emissions. If these vacuum states are modified, as for example by placing an excited atom between closely spaced mirrors or in a small cavity (essentially the Casimir effect), spontaneous emission can be greatly inhibited or enhanced or even made reversible. Recent advances in atomic and optical techniques have made it possible to control and manipulate spontaneous emission. A whole new branch of quantum optics called ‘cavity QED’ has developed since 1987 utilizing these dramatic changes in spontaneous emission rates in cavities to construct new kinds of microscopic masers or micromasers that operate with a single atom and a few photons or with photons emitted in pairs in a two-photon transition.
In our preceding treatments of the interaction of radiation with matter, we assumed that the electric field is classical. In most cases, this assumption is validated by experiment. There are several instances, however, for which a classical field fails to give experimentally observed results, whereas a quantized field succeeds. This is true, for example, of spontaneous emission, a phenom enon which was described phenomenologically in our earlier work (see Chaps. 2 and 7). Another case involves the 2 and 2p \ states in the hydrogen atom, which are predicted to have the same energy not only by the simple theory of Chap. 1, but also by the more sophisticated Dirac theory. In both cases, a classical electric field was assumed. Experimentally, the two levels differ by approximately the Lamb shift, 1057 MHz. A fully quantized treatment of the field and atom system gives impressive agreement with the experimentally observed shift. Derivation of the fluctuations in intensity of a laser near threshold requires the quantum theory o f radiation. In addition there are other problems for which the concept of a “particle,” the photon, is either necessary or convenient. Despite a popular impression to the contrary, however the explanation of the photoelectric effect is not one of these (see Prob. 2-10).
We introduce a simple model for electromagnetically induced transparency in which all fields are treated quantum mechanically. We study a system of three separated atoms at fixed positions in a one-dimensional multimode optical cavity. The first atom serves as the source for a single spontaneously emitted photon; the photon scatters from a three-level Lambda-configuration atom which interacts with a single-mode field coupling two of the atomic levels; the third atom serves as a detector of the total transmitted field. We find an analytical solution for the quantum dynamics. From the quantum amplitude describing the excitation of the detector atom we extract information that provides exact single-photon analogs to wave delays predicted by semi-classical theories. We also find complementary information in the expectation value of the electric field intensity operator.
We investigate (non-relativistic) atomic systems interacting with quantum electromagnetic field (QEF). The resulting model describes spontaneous emission of light from a two-level atom surrounded by various initial states of the QEF. We assume that the quantum field interacts with the atom via the standard, minimal-coupling Hamiltonian, with the A2 term neglected. We also assume that there will appear at most single excitations (photons). By conducting the analysis on a general level we allow for an arbitrary initial state of the QEF (which can be for instance: the vacuum, the ground state in a cavity, or the squeezed state). We derive a Volterra-type equation which governs the time evolution of the amplitude of the excited state. The two-point function of the initial state of the QEF, integrated with a combination of atomic wavefunctions, forms the kernel of this equation.