Application of the displaced oscillator basis in quantum optics

Abstract

Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonicoscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, "two-level" atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the 6m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, e A /2mc, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency. An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small "ripple" on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as "interference" between two displaced oscillator-field states that represents the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.PACS number(s): 42.50.Lc, 03.65.Ge

Dynamics of a two-level system strongly coupled to a high-frequency quantum oscillator

“…Graham and Höhnerbach refer to this regime as the "quasidegenerate limit" and give the same lowest-order expressions as we derive. [12][13][14] Schweber utilized the Bargmann Hilbertspace representation, 15 and Crisp solved recurrence relations; 16 both of these authors found higher-order corrections beyond what we present. We take yet a different approach.…”

“…This structure was noted in Ref. 16 and interpreted as an interference between states displaced in opposite directions. In the limit / 0 → ϱ the distance between the wells becomes infinite and the overlap ͗N − ͉ N + ͘ → 0.…”

“…This point is discussed in Ref. 16, and a higher-order formula is derived which is valid even near the critical points. Some caution must therefore be used in making predictions for Fock-state initial conditions from our lowest-order formula.…”

“…We take yet a different approach. First, by neglecting the self-energy of the two-level system, we derive the basis in which the rest of the calculation will be performed, called the "displaced oscillator" 16 basis. In this basis the Hamiltonian may be truncated to a block-diagonal form and the blocks solved individually.…”

On the Canonical Formulation of Electrodynamics and Wave Mechanics

“…Crisp himself has contributed some of the finest work in semiclassical theory. He computed the radiation reaction associated with a rotating charge distribution [50], the atomic radiative level shifts resulting from the solution of the semiclassical nonlinear integro-differential equations [51], the interaction of an atomic system with a single mode of the quantized electromagnetic field [52,53], and the extension of the semiclassical theory to include relativistic effects [54].…”