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Physical Review A

Possibility of quantum jumps in a three-level system

Juha Javanainen

Published Mar 1, 1986• 116 citations
Publisher

Abstract

We calculate from the quantum regression theorem the entire photon-counting statistics for the fluorescence from a laser-driven two-level system weakly coupled to a metastable third level, and compare the result with the statistics of a dichotomic Markov process. The comparison shows, that at least in the limit of an infinite intensity of the laser in a single experiment the sequence of photon counts should display randomly alternating periods with no counts and a nonzero counting rate as if the atom executed ``quantum jumps'' to and from the metastable state.

DOI
10.1103/physreva.33.2121

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According to the usual interpretation of quantum mechanics, measurement of an observable yields its eigenvalue. Extrapolation of classical reasoning then suggests that at every instant of time the quantum system actually is in one of the eigenstates of the observable, and as time goes on it executes random jumps between the states. In this spirit, Cook and Kimble1 have recently elaborated Dehmelt’s suggestion2 that fluorescence from a laser-driven two-level system weakly coupled to a third state may turn on and off when the atom jumps between the active two-level system and the inactive third state.

•Jun 9, 1986•0 citations
We calculate from the quantum regression theorem the entire photon-counting statistics for the fluorescence from a laser-driven two-level system weakly coupled to a metastable third level, and compare the result with the statistics of a dichotomic Markov process. The comparison shows, that at least in the limit of an infinite intensity of the laser in a single experiment the sequence of photon counts should display randomly alternating periods with no counts and a nonzero counting rate as if the atom executed ``quantum jumps'' to and from the metastable state.
We compare the statistics of photon counts of the light scattered by a laser-driven two-level system weakly coupled to a third metastable state with the statistics of a dichotomic Markov process. The fluorescence is found to turn abrubtly off and on, as if the atom executed quantum jumps to and from the metastable state. Moreover, the jumping nature of the fluorescence over long time scales survives even though the atom cannot be treated with rate theory and the picture of instantaneous quantum jumps is not strictly justified.
A single-atom optical clock proposed over ten years ago utilizes an amplification scheme which only recently was recognized as a novel problem in quantum statistics. The atom, a three-state system, has two coupled transitions that are driven continuously by two external fields, one being an allowed transition (1\ensuremath{\leftrightarrows}3) and the other a forbidden transition (2\ensuremath{\leftrightarrows}3) where \ensuremath{\Vert}3〉 is the lowest state. It has been argued that the weak transition, which is difficult to detect, could be monitored by the presence or absence of spontaneous emission of the strong transition. Thus, when the atom is shelved in the metastable state \ensuremath{\Vert}2〉, the strong transition is extinguished, but when the atom executes a single quantum jump (2\ensuremath{\rightarrow}3), it triggers a succession of perhaps a million quantum jumps (macroscopic quantum jumps) in the strong transition, an amplification that can be detected easily. This intuitive argument for alternating bright and dark intervals assumes, however, that the atom is always in an eigenstate. Should the atom be in a superposition state, because of coherent excitation, one could imagine that the weak transition would merely reduce the intensity of the strong transition slightly. This issue is resolved, in favor of the first intuitive argument, by calculating the photon-counting statistics, the probability W(n,T) of observing precisely n photon-counting events in a collection T in a quantum-mechanically consistent way. The results cannot be described by classical statistics. A compact analytic form is obtained for W(n,T) by considering the entire hierarchy of correlation functions where the emission interval peaks sharply about a particular n with a Poisson distribution and can be comparable in length to the darkness interval (n=0) while the other values of n display vanishingly small probabilities.
One of the experiments made possible by the trapping and cooling of a single atom is to observe the quantum jumps to a metastable state in the atom by monitoring the resonance fluorescence of a strong transition in which at least one of the states is coupled to the metastable state. When the atomic electron moves to the metastable state, the fluorescence from the strongly driven transition disappears. When the electron drops back into the strongly driven transition, the fluorescence abruptly returns. Statistical analysis of these random interruptions to the fluorescence signal implies photon antibunching in the emission of the weak photon.
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 QQQ 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.
A quantum-statistical theory is developed for a single-atom, three-state quantum system where a strong (1-3) and a weak (2-3) transition are driven resonantly by an incoherent field. From a second-order correlation function of the fluorescence intensity, it is concluded that a quantum jump can occur, in the sense that the single atom can undergo large fluctuations between a state of full emission in the strong transition to a state of no emission when the electron is shelved in the metastable state (2).
According to the usual interpretation of quantum mechanics, measurement of an observable yields its eigenvalue. Extrapolation of classical reasoning then suggests that at every instant of time the quantum system actually is in one of the eigenstates of the observable, and as time goes on it executes random jumps between the states. In this spirit, Cook and Kimble1 have recently elaborated Dehmelt’s suggestion2 that fluorescence from a laser-driven two-level system weakly coupled to a third state may turn on and off when the atom jumps between the active two-level system and the inactive third state.