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Journal of Experimental and Theoretical Physics

Discrete photodetection of quantum jumps on the V configuration of atomic levels

George P. Miroshnichenko

Published Aug 1, 2009• 3 citations

Abstract

A theory of the discrete photodetection of quantum jumps on the V configuration of atomic levels has been developed. A three-level source atom is placed in a cavity excited by a resonance fluorescence field. The cavity is tuned to exact resonance with an atomic transition. The cavity mode state is tested by a flux of unexcited (at the entrance) probe atoms passing through the cavity. The energy states of the outgoing probe atoms are detected by ionization chambers, which are assumed ideal. This a posteriori statistical information is indirectly related to the numerical characteristics of a measured quantum system consisting of the source atom and cavity mode. The “tuning” conditions for a discrete photodetector, i.e., the rules for choosing the parameters and durations of the interactions of the cavity mode with the probe and source atoms, intensities of the pump and probe fields that are necessary for observing quantum jumps from the “bright” state to the “dark” one and vice versa, have been determined. A two-state model that describes the dynamics of a quantum jump has been analyzed. The formulas have been obtained for the observable characteristics of quantum jumps: the mean residence time of the quantum system in quasistationary states (durations of the bright and dark periods), probabilities of quantum jumps, mean excitation levels of the quantized cavity mode, etc.

DOI: 10.1134/s1063776109080032

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A theory of the discrete photodetection of quantum jumps on the V configuration of atomic levels has been developed. A three-level source atom is placed in a cavity excited by a resonance fluorescence field. The cavity is tuned to exact resonance with an atomic transition. The cavity mode state is tested by a flux of unexcited (at the entrance) probe atoms passing through the cavity. The energy states of the outgoing probe atoms are detected by ionization chambers, which are assumed ideal. This a posteriori statistical information is indirectly related to the numerical characteristics of a measured quantum system consisting of the source atom and cavity mode. The “tuning” conditions for a discrete photodetector, i.e., the rules for choosing the parameters and durations of the interactions of the cavity mode with the probe and source atoms, intensities of the pump and probe fields that are necessary for observing quantum jumps from the “bright” state to the “dark” one and vice versa, have been determined. A two-state model that describes the dynamics of a quantum jump has been analyzed. The formulas have been obtained for the observable characteristics of quantum jumps: the mean residence time of the quantum system in quasistationary states (durations of the bright and dark periods), probabilities of quantum jumps, mean excitation levels of the quantized cavity mode, etc.

Complete density-matrix equations and a one-photon counting probability are used to examine the photon statistics of fluorescence light of a single three-level atom, weakly coupled to an external cavity of low

Q

value. The internal structure of the atom is supposed to be characterized by a weak and a strong transition. The strong transition is coherently driven by an external laser field, whereas no external field is acting on the weak transition. In free space the system exhibits quantum jumps. By placing the atom within the mirrors of an external cavity, tuned close to the transition frequency of the two upper levels, modifications in the quantum jump behavior of the atom are shown due to a modified spontaneous decay rate between the two states. In particular, it is demonstrated that the effect of the cavity is to modify the repetition rate of quantum jumps into the metastable dark state and the average duration of bright periods of fluorescence, whereas the average duration of dark periods is not changed in first order of the spontaneous decay rates involved. In view of current experiments in our laboratory a quantitative analysis is presented for the special case of a single indium ion located between the mirrors of a high-order symmetric confocal cavity.

We consider a single three-level atom placed inside a resonant cavity and simultaneously probed in a double-resonance ladder configuration by a relatively strong laser beam. It is shown that photocounting statistics of a fluorescence signal induced by the probe, in certain circumstances, exhibits dark periods (i.e., quantum jumps are present), reflecting the statistical properties of the field inside the cavity. Comparisons are made between the number, thermal, coherent, and squeezed states. Specific features of the latter are pointed out.

Recent experiments with single atomic ions isolated in electromagnetic traps have demonstrated abrupt changes in observed fluorescence intensities, which have been interpreted as evidence for quantum jumps as the atom changes energy levels. This presents the seeming paradox of discontinuously changing observables from a system whose dynamical evolution is governed by the continuous optical Bloch equations. In this paper I present an analysis of the dynamics of a model three-level atom interacting with two laser fields in terms of the dressed states of the atom-laser system. These dressed states are eigenstates of a Hamiltonian which includes laser-atom interactions, and thus they simplify the discussion of the time evolution of the combined laser-atom system. This dressed-state description is particularly useful in the regime in which both laser intensities are strong enough to saturate their respective transitions. I calculate the radiative lifetimes of the different species of dressed states and the branching ratios for decay to other dressed states, and demonstrate the consequences of these quantities for the observed fluorescence pattern. The calculated duration of bright and dark periods shows a pronounced dependence on the detunings of the lasers.

For pt.I see Phys. Rev. A, vol.37, p.4303 (1988). The principle of wavefunction collapse by non-detection of photons leads to simple but rigorous expressions for the average dark and bright periods, T D and T B , of intermittent quantum jump fluorescence from a V-configuration three-level atom. Here further consequences of this principle are explored, including: the derivation of some new correlation functions which exhibit the physical characteristics of extended dark and bright periods; the expressions for T D and T B when one transition is driven coherently and the other incoherently; and a general expression for the characteristic time to be sure that a quantum jump has occurred for any type of driving field.

We consider a three-level atomic system driven strongly on one transition and weakly on the other. The excited state on the weak transition is assumed to be metastable. We give an analysis of the fluorescence from the strong transition in terms of the elementary probability density

{p}_{[0}

,t)(

{\mathrm{t}}_{1}

,

{t}_{2}

,...,

{t}_{n}

) which gives us the probability density that exactly n photons are emitted at times

{t}_{1}

,

{t}_{2}

,...,

{t}_{n}

by the atom in the time interval [0,t). We show that

{p}_{[0}

,t) essentially factorizes into products of conditional densities c\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\tau}) that, given a photon is emitted at time zero, the next photon emission occurs at time \ensuremath{\tau}. This enables a simulation of the individual photon emissions to be given which shows directly the existence of prolonged dark windows in the fluorescence corresponding to the shelving of the electron in the metastable state or ``quantum jumps.''