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Open Systems & Information Dynamics

Counting Photons in the Λ-Experiment

Anita Magdalena Dąbrowska

Published Dec 1, 2002• 2 citations
Publisher

Abstract

Dehmelt's Λ-experiment for a three-level atom with simultaneously driven strong and weak transition is studied within quantum stochastic calculus approach. The statistics of the emitted photons is found by the method of generating functional of the corresponding two-dimensional output counting process. In particular, the average waiting times for a count are calculated.

DOI
10.1023/a:1021810402366

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Dehmelt's Λ-experiment for a three-level atom with simultaneously driven strong and weak transition is studied within quantum stochastic calculus approach. The statistics of the emitted photons is found by the method of generating functional of the corresponding two-dimensional output counting process. In particular, the average waiting times for a count are calculated.
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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.
The starting point is a given semigroup of completely positive maps on the 2×2 matrices. This semigroup describes the irreversible evolution of a decaying two-level atom. By using the integral-sum kernel approach to quantum stochastic calculus, the two-level atom is coupled to an environment, which in this case will be interpreted as the electromagnetic field. The irreversible time evolution of the two-level atom then stems from the reversible time evolution of the atom and the field together. Mathematically speaking, a Markov dilation of the semigroup has been constructed. The next step is to drive the atom by a laser and to count the photons emitted into the field by the decaying two-level atom. For every possible sequence of photon counts, a map is constructed that gives the time evolution of the two-level atom implied by that sequence. The family of maps obtained in this way forms a so-called Davies process. In his book, Davies describes the structure of these processes, which brings us into the field of quantum trajectories. Within the model presented in this paper, the jump operators are calculated and the resulting counting process is briefly described.
The excitation of atomic and molecular systems by propagating light in a two-photon state within the Wigner-Weisskopf approximation has been described using stochastic tools. The problem of a stochastic evolution of the quantum system, depending on the results of the measurement of the output field, was formulated and solved making use of the model of repeated interactions and measurement. We defined the discrete in-time interaction between the quantum system and its environment being the electromagnetic field approximated by a chain or chains of harmonic oscillators. We determined analytical formulae for quantum trajectories associated with one-dimensional and two-dimensional counting processes, corresponding respectively to unidirectional or bidirectional input field prepared in the two-photon states. We derived the formulae for the exclusive probability densities of photon counts that allow us to completely characterize the photon statistics of the output field. Finally, we showed how to apply the quantum trajectories to obtain the formula for the probability of the two-photon absorption for a three-level atom in a ladder configuration. The paper also includes a discussion on the optimal two-photon state that maximizes the two-photon absorption probability.