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Physical Review A

Quantum jumps in a coherently driven Raman system

G. S. Agarwal, S. V. Lawande, Richard D’Souza

Published Jan 1, 1988• 22 citations
Publisher

Abstract

Early results of Agarwal and Jha [Z. Phys. B 35, 391 (1979)] are used to examine the question of quantum jumps in a Raman (\ensuremath{\Lambda}) system driven by two coherent fields. We discuss both short-time (${\ensuremath{\gamma}}_{1}$t\ensuremath{\lesssim}1) and long-time (${\ensuremath{\gamma}}_{2}$t\ensuremath{\approxeq}1) behavior of the intensity correlations assuming that the Stokes transition is very weak, ${\ensuremath{\gamma}}_{2}$\ensuremath{\ll}${\ensuremath{\gamma}}_{1}$. We calculate the counting statistics of the dark periods using the higher-order intensity correlations. The results differ from those obtained using arguments based on population trapping which predict only a very rare ``jump'' event. The intermittence in fluorescence produced by a coherently driven \ensuremath{\Lambda} system is interpreted in terms of an equivalent two-state Markov process.

DOI
10.1103/physreva.37.444

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Early results of Agarwal and Jha [Z. Phys. B 35, 391 (1979)] are used to examine the question of quantum jumps in a Raman (\ensuremath{\Lambda}) system driven by two coherent fields. We discuss both short-time (\ensuremathγ1{\ensuremath{\gamma}}_{1}\ensuremathγ1​t\ensuremath{\lesssim}1) and long-time (\ensuremathγ2{\ensuremath{\gamma}}_{2}\ensuremathγ2​t\ensuremath{\approxeq}1) behavior of the intensity correlations assuming that the Stokes transition is very weak, \ensuremathγ2{\ensuremath{\gamma}}_{2}\ensuremathγ2​\ensuremath{\ll}\ensuremathγ1{\ensuremath{\gamma}}_{1}\ensuremathγ1​. We calculate the counting statistics of the dark periods using the higher-order intensity correlations. The results differ from those obtained using arguments based on population trapping which predict only a very rare ``jump'' event. The intermittence in fluorescence produced by a coherently driven \ensuremath{\Lambda} system is interpreted in terms of an equivalent two-state Markov process.
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.
In an externally driven multilevel quantum system observation that the NEXT jump has not yet happened affects its future development. In previous work [Phys. Rev. A36, 929 (1987)] it was shown that this class of measurement makes it possible to observe remarkably long dark intervals -- or intermittency -- in the atomic fluorescence of an atom with 3 or more levels. Those calculations were carried out when the driven oscillations or Rabi flopping between the ground state and a strongly fluorescing state were fast compared to its lifetime. In systems with solid state Qubits the accessible parameter space is generally limited to the regime where oscillations are slower than the lifetime. In this paper we evaluate intermittency in atomic transitions, due to measurements with a null result, in this limit. During the dark periods the wave function of the continuously measured multilevel system is coherent.
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.
We present three additional theoretical signatures of quantum jumps displayed by the behavior of the Mandel Q(T) function, transient squeezing, and the time-dependent spectra, for a Raman system resonantly driven by coherent fields. As in our earlier formalism we assume that one or both of the driving fields are intense and invoke the secular approximation to derive a Markovian master equation for the atomic-density operator. This is then used to obtain analytical expressions for Q(T), transient squeezing parameters, and the time-dependent spectra. These analytical results provide further characteristics of the quantum jumps.
In this paper, we formulate a quantum jump approach associated with single fluorescent systems whose radiative decay is defined by a Lindblad-like equation with a memory kernel. We base our results in a generalized Born–Markov approximation, which allows us to get our results from a full microscopic description. In the weak-laser-excitation regime, the photon-emission process can be described through a delay function, which in a natural way provides the basis for unravelling the dynamics in terms of a stochastic density matrix. As in the Markovian case, the stochastic dynamics consists in periods where the evolution is smooth and non-unitary, interrupted at random times where the photon-detection events produce a sudden change in the system state. The phenomenon of non-Poissonian intermittent fluorescence induced by complex environments (Budini 2006 Phys. Rev. A 73 061802) is analysed in the context of this approach.