Dark periods in a single radiating system

Ured (t; 0) = e?iHred t
(1)
where Hred is a nonhermitian operator. Therefore Ured(t; 0) is not unitary and the norm is not preserved. This can be understood as follows. In the usual statistical interpretation of quantum mechanics a state vector or density matrix describes a whole ensemble of systems and in general not a single system. Our single atom, however, can be thought of as a member of such a (hypthetical) quantum mechanical ensemble E . Now, in this ensemble description Ured (t; 0) describes only the time development of a subensemble, E0, consisting of those systems which have not emitted a photon until time t. The size of the subensemble E0 decreases in time since whenever an atom emits a photon it leaves the subensemble E0. The relative size of the subensemble E0 with respect to E is given by
jump method, which is treated in full generality in 11], is brie y reviewed in Section II. In Section III we discuss the well-known double optical resonance scheme where two widely separated lower levels are coupled by two lasers to a common upper level. We show that dark and light periods exist and that the dark resonance which is known to occur for the two photon resonance can be easily understood through the photon statistics and dark periods of a single atom. In fact, when approaching the detuning parameter of a dark resonance the dark periods of a single atom become longer and longer while the light periods do not. The overall intensity thus decreases. We also show an interesting bunching behaviour of the resonance uorescence near the dark resonance. As opposed to a two-level system where at low intensities the photons tend to come one by one, in the double optical resonance scheme the photons tend to come in pairs or triples. In Section IV we investigate the quantum-beat system irradiated by a single laser where two close lying lower levels couple to a common upper level. We show that for parallel transition dipole moments and almost degenerate lower levels light and dark periods occur. The length of both the light and dark periods is proportional to the square of the inverse of the lower level separation. Finally in Section V we propose a realistic scheme which is unitary equivalent to the quantum-beat system with parallel transition dipole moments. Because of the unitary equivalence the photon statistics of both systems are the same.
Dark periods in a single radiating system
Gerhard C. Hegerfeldt and Martin B. Plenio
Institut fur Theoretische Physik, Universitat Gottingen Bunsenstr. 9, D-37073 Gottingen, Germany
Abstract
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I. Introduction
The interest in single radiating atoms experienced a great stimulus through the development of ion traps making it possible to store, and perform experiments with, a single ion. An interesting example for the new phenomena that became observable are light and dark periods in the resonance uorescence of a single trapped ion. This e ect was predicted by Dehmelt 1] for a three-level V system with two upper levels, one of them metastable, that couple to a common ground state. The ion scatters photons on the strong transition. This uorescence signal is interrupted for relatively long times when the system is excited to the metastable level. The uorescence signal vanishes until the system decays back to the ground state. Since this procedure repeats itself the uorescence is turned on and o at random times. The photon statistics of this scheme have been observed experimentally 2] and calculated quantum mechanically 3]. Later, light and dark periods were predicted for di erent schemes under coherent as well as continuous incoherent pumping 4]. In this paper we investigate the photon statistics of a di erent class of single systems. These are the so-called systems where two lower levels are strongly coupled to a common upper level. For systems with large level separation ensemble properties such as the intensity of the resonance uorescence and the intensity correlation function g(2) have already been investigated both theoretically as well as experimentally 5, 6, 7], but the photon statistics for a single ion has not been investigated completely. We use the the quantum jump method 8] (which is equivalent to the Monte Carlo wave function approach 9] and to the use of quantum trajectories 10]). The quantum 1
II. Description of the quantum jump method
To calculate the photon statistics of a single laser-driven atom we apply the method of quantum jumps. For the complete determination of the photon statistics of a single atom or ion we need to know two quantities, the density matrix immediately after a photon emission (or rather, detection) and the time development of the system under the condition that no photon has been emitted. This time development, in a suitable interaction picture, is given by an operator of the form
We use the quantum jump method to study the photon statistics of a single laserdriven atom in the con guration where both lower levels are strongly coupled to the common upper level. Under certain conditions we show that, for almost degenerate lower levels, light and dark periods occur which are similar to those of the well-known Dehmelt V system. Analytic results for their mean lengths and other statistical properties are given. For large separation of the lower levels we prove an interesting bunching property for the photons in the resonance uorescence near the dark resonance. We propose a realistic system for which these e ects may be observed. PACS numbers: 42.50, 32.90 +a